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Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide range of topics by publishing timely reviews which are more extensive than just literature surveys but normally less than a full monograph. Each Report deals with one speciﬁc subject. These reviews are specialist in nature but contain enough introductory material to make the main points intelligible to a non-specialist. The reader will not only be able to distinguish important developments and trends but will also ﬁnd a suﬃcient number of references to the original literature. Submission In principle, papers are written and submitted on the invitation of one of the Editors, although the Editors would be glad to receive suggestions. Proposals for review articles (approximately 500–1000 words) should be sent by the authors to one of the Editors listed below. The Editor will evaluate proposals on the basis of timeliness and relevance and inform the authors as soon as possible. All submitted papers are subject to a refereeing process. Editors J.V. ALLABY (Experimental high-energy physics), EP Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail: [email protected] D.D. AWSCHALOM (Experimental condensed matter physics), Department of Physics, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] J.A. BAGGER (High-energy physics), Department of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore MD 21218, USA. E-mail: [email protected] C.W.J. BEENAKKER (Mesoscopic physics), Instituut–Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: [email protected] E. BREZIN (Statistical physics and ﬁeld theory), Laboratoire de Physique The´orique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex, France. E-mail: [email protected] G.E. BROWN (Nuclear physics), Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11974, USA. E-mail: [email protected] D.K. CAMPBELL (Non-linear dynamics), Dean, College of Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] G. COMSA (Surfaces and thin ﬁlms), Institut fur . Physikalische und Theoretische Chemie, Universit.at Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] J. EICHLER (Atomic and molecular physics), Hahn-Meitner-Institut Berlin, Abteilung Theoretische Physik, Glienicker Strasse 100, 14109 Berlin, Germany. E-mail: [email protected] M.P. KAMIONKOWSKI (Astrophysics), Theoretical Astrophysics 130-33, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA. E-mail: [email protected]

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Instructions to authors

M.L. KLEIN (Soft condensed matter physics), Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. E-mail: [email protected] A.A. MARADUDIN (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] D.L. MILLS (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] R. PETRONZIO (High-energy physics), Dipartimento di Fisica, Universita" di Roma – Tor Vergata, Via della Ricerca Scientiﬁca, 1, I-00133 Rome, Italy. E-mail: [email protected] S. PEYERIMHOFF (Molecular physics), Institute of Physical and Theoretical Chemistry, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] I. PROCACCIA (Statistical mechanics), Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] E. SACKMANN (Biological physics), Physik-Department E22 (Biophysics Lab.), Technische Universit.at Munchen, . D-85747 Garching, Germany. E-mail: [email protected] A. SCHWIMMER (High-energy physics), Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] R.N. SUDAN (Plasma physics), Laboratory of Plasma Studies, Cornell University, 369 Upson Hall, Ithaca, NY 14853-7501, USA. E-mail: [email protected] W. WEISE (Physics of hadrons and nuclei), Institut fur . Theoretische Physik, Physik Department, Technische Universit.at Munchen, . James Franck Strae, D-85748 Garching, Germany. E-mail: [email protected] Manuscript style guidelines Papers should be written in correct English. Authors with insuﬃcient command of the English language should seek linguistic advice. Manuscripts should be typed on one side of the paper, with double line spacing and a wide margin. The character size should be suﬃciently large that all subscripts and superscripts in mathematical expressions are clearly legible. Please note that manuscripts should be accompanied by separate sheets containing: the title, authors’ names and addresses, abstract, PACS codes and keywords, a table of contents, and a list of ﬁgure captions and tables. – Address: The name, complete postal address, e-mail address, telephone and fax number of the corresponding author should be indicated on the manuscript. – Abstract: A short informative abstract not exceeding approximately 150 words is required. – PACS codes/keywords: Please supply one or more PACS-1999 classiﬁcation codes and up to 4 keywords of your own choice for indexing purposes. PACS is available online from our homepage (http://www.elsevier.com/locate/physrep). References. The list of references may be organized according to the number system or the nameyear (Harvard) system. Number system: [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53 (1974) 249–315. [2] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Instructions to authors

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[3] B. Ziegler, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York, 1986) p. 293. A reference should not contain more than one article. Harvard system:

Ablowitz, M.J., D.J. Kaup, A.C. Newell and H. Segur, 1974. The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53, 249–315. Abramowitz, M. and I. Stegun, 1965, Handbook of Mathematical Functions (Dover, New York). Ziegler, B., 1986, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York) p. 293. Ranking of references. The references in Physics Reports are ranked: crucial references are indicated by three asterisks, very important ones with two, and important references with one. Please indicate in your ﬁnal version the ranking of the references with the asterisk system. Please use the asterisks sparingly: certainly not more than 15% of all references should be placed in either of the three categories. Formulas. Formulas should be typed or unambiguously written. Special care should be taken of those symbols which might cause confusion. Unusual symbols should be identiﬁed in the margin the ﬁrst time they occur.

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from the version accepted by the Editor of the journal are permissible without the prior and explicit approval by the Editor. Such changes should be clearly indicated on an accompanying printout of the ﬁle.

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Physics Reports 363 (2002) 1 – 84 www.elsevier.com/locate/physrep

Ordering and growth of Langmuir–Blodgett &lms: X-ray scattering studies J.K. Basu, M.K. Sanyal ∗ Surface Physics Division, Saha Institute of Nuclear Physics, 1=AF, Bidhannagar, Calcutta 700 064, India Received June 2001; editor: A:A: Maradudin Contents 1. Introduction 2. Langmuir–Blodgett deposition technique 2.1. Preparation of Langmuir monolayer 2.2. Langmuir monolayer and LB &lm deposition 3. Grazing incidence X-ray re6ectivity (GIXR) technique 3.1. Experimental set-up 3.2. Types of scans 3.3. Instrumental e:ects 4. Scattering from surfaces and interfaces 4.1. Basic formalism 4.2. Re6ectivity from multiple interfaces 5. Scattering in Born approximation 5.1. Scattering from single surface 5.2. Scattering from multiple interfaces 6. Application and limitations of GIXR technique 6.1. Extraction of small density variations in thin &lms: a scheme based on distorted wave Born approximation 6.2. Other schemes 6.3. Extraction of compositional pro&les of near-ideal multilayer thin &lms: a scheme based on Born approximation

3 5 5 8 12 13 14 15 17 17 20 21 22 28 30 31 35

6.4. Data analysis 7. Growth and morphology of surfaces and interfaces 7.1. Some examples 7.2. Growth model of LB &lms 7.3. X-ray scattering studies of LB &lms 8. Formation of nanostructures 8.1. Nanoparticle formation in LB &lms 8.2. GIXR studies of CdS nanostructures formed in LB &lms 9. Melting of LB &lms 9.1. Earlier studies 9.2. X-ray re6ectivity studies of LB melting 9.3 AFM studies 10. Conclusion Acknowledgements Appendix A Appendix B Appendix C C.1. Capillary wave theory C.2. X-ray scattering from capillary waves References

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∗

Corresponding author. Tel.: +91-33-337-5346; fax: +91-33-337-4637. E-mail address: [email protected] (M.K. Sanyal). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 8 3 - 7

41 43 44 45 47 52 54 55 58 58 60 67 71 72 72 73 75 75 78 80

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Abstract The interplay of ordering, con&nement and growth in ultrathin &lms gives rise to various interesting phenomena not observed in bulk materials. The nature of ordering and interfacial morphology present in these &lms, in turn, depends on their growth mechanism. Well-ordered metal–organic &lms, deposited using an enigmatic Langmuir–Blodgett (LB) technique, are not only ideal systems for understanding the interplay between growth and structure of ultrathin &lms but also for studying chemical reactions and phase transitions in con&ned geometries. Studies on these LB &lms also enhance our understanding of the fundamental interactions of amphiphilic molecules important for biological systems. Advent of grazing incidence X-ray scattering techniques has enabled us to probe the interfacial structure of these multilayer &lms at very high resolution and as a result has improved our knowledge about the mechanism of growth processes and about physical=chemical properties of ultrathin &lms. In this review we will focus our attention on recent results obtained using these X-ray scattering techniques to understand the mechanism of growth leading to formation of remarkably well-ordered LB &lms after giving a brief outline of these scattering techniques. In addition, we also review recent results on growth and structure of nanoparticles formed by suitable chemical processes within the ordered matrix of LB &lms. Finally, we will discuss the work done on melting of LB &lms and its implications in our understanding of melting process in lower dimensions. In all these studies, especially those on as-deposited LB &lms results of atomic force microscopy measurements have provided important c 2002 Elsevier Science B.V. All rights reserved. complementary morphological information. PACS: 68.18.+p; 61.10.−i; 81.15.Aa; 68.60.Dv; 61.46.+w KEY: Langmuir–Blodgett &lms; X-ray di:raction and scattering; Theory and models of &lm growth; Thermal stability; Thermal e:ects; Clusters; nanoparticles and nanocrystalline materials

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1. Introduction Over the course of the past century, condensed matter physics (CMP) has had a spectacular evolution and has become by far the largest sub&eld of physics at the turn of the century [1]. Surfaces are omnipresent in all condensed matter systems and interactions between such systems necessarily takes place through the surface=interface. Thus developments in our understanding of condensed matter systems have also led to improvements in our understanding of surface science, in general and surface physics, in particular. In turn, the tremendous growth in our understanding of the physics of surfaces and materials over the last two decades has largely been instrumental not only in the evolution of our understanding of the existing traditional areas in CMP but also expanding the frontiers of CMP to such hitherto unchartered areas like soft matter physics [2–5] and physics of mesoscopic and nanoscopic systems [6 –9]. As we learned more about surfaces, it became quite obvious that surface is not a mere inert slice of bulk materials. It may have its own structure, electronic con&gurations and dynamics. The advent of various growth techniques [10,11] has enabled us to deposit ultrathin &lms with control at atomic=molecular level. Using these sophisticated growth techniques, one can thus create arti&cial surfaces—interfaces—between dissimilar materials, to form layered structures, which may not otherwise occur in nature. These &lms are thus ideal two-dimensional (2D) systems to study the surface=interface properties and the e:ect of con&nement and reduced dimensionality on the fundamental processes in nature. Rapid development of surface characterisation techniques over the last few decades is intimately related to the growth of semiconductor industry and to the never-ending quest for smart miniaturised devices, where number of atoms at the surfaces and interfaces becomes comparable to those present in the bulk. Electrons and ion beams were quite obvious choices for studying surfaces of solid materials, primarily due to the lower penetration depth of these radiations as compared to X-rays and neutrons. Although surface characterisation of materials using X-rays and neutrons were demonstrated before 50 years, most of the text books on surface science emphasised the use of techniques based on electrons and ion beams, for characterisation of surfaces. The role of X-rays and neutrons in non-destructive characterisation of surfaces and interfaces of materials [12] has become important for several reasons including availability of intense synchrotron X-ray and neutron sources, improvement in our theoretical understanding of scattering processes from surfaces and interfaces and realisation of the importance of interfacial structure and roughness in determining the physical and chemical properties of materials. These techniques have already been used to study various surfaces and interfaces of condensed matter systems such as liquids [12–15], solid thin &lms [12,16 –19], liquid crystals [20 –22], microemulsions [23], polymers [12,16 –19,24], biological membranes [25], Langmuir–Blodgett (LB) &lms [26 –31] and magnetic &lms [12,32]. LB &lms have been widely studied [33–36] as model systems to enhance our understanding regarding the e:ect of con&nement and reduced dimensionality on various physical and chemical processes. This ability stems from the molecular level control which is achievable during growth of these &lms. This control also allows designing of systems with tailor-made properties useful for various technological and biological applications. Deposition of multilayer LB &lms on solid substrates requires the formation of a monolayer &lm at the air–water interface (or liquid–gas interface, in general). This monolayer &lm is termed as Langmuir monolayer

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[33–37]. The earliest scienti&c work on Langmuir monolayer &lms apparently dates back to 1774 [38] when Benjamin Franklin studied the e:ect of oil &lms on water. He spread an oil &lm on Clapham pond and from the number of molecules transferred and the area of N which is of the orthe &lm, it could be estimated that the &lm thickness was around 25 A der of molecular dimensions. However, the signi&cance of this experiment was not realised at that time and hence not much scienti&c work was done till around 1890 when Agnes Pockels described a method for manipulating oil &lms on water [39]. Further details regarding the other signi&cant contributions and the evolution of this &eld can be obtained from the book by Gaines [33]. Since we intend to review the work done on the fundamental aspects of growth, ordering and melting of LB &lms, we will restrict our discussions here to &lms of simple fatty acid molecules only. X-rays interact with a medium by inducing individual electrons to form re-radiating dipoles. The electromagnetic waves generated by these re-radiating dipoles, having amplitude proportional to the classical electron radius r0 , in turn produces the scattered beam carrying information about the scattering density pro&le of the medium [40 – 42]. X-ray specular re6ectivity measurements from the surface of a material provides information about the one-dimensional (1D) electron density pro&le perpendicular to the surface [43– 45]. Similar measurements can also be carried out with neutron beams. X-ray and neutron re6ectivity measurements provide complementary information [24,46]. The advantage of X-ray re6ectivity over neutron re6ectivity lies in the availability of higher beam 6ux, especially with synchrotron sources, enabling data to be taken within shorter time with better counting statistics and also over a wider range in momentum space implying better spatial resolution. It is also possible to induce arti&cial contrast, between elements lying adjacent in the periodic table, by performing anomalous scattering experiments [47] around X-ray absorption edges of speci&c elements present in the specimen. With neutron re6ectivity, this ability to create enhanced contrast comes naturally by using isotopic substitution. But the main advantage of using neutron re6ectivity is its sensitivity to magnetic interaction, by virtue of the spin 12 of neutrons. Although X-ray magnetic scattering can be performed, the amplitude is much weaker [48] compared to neutron magnetic scattering and hence enables magnetic depth pro&ling of samples quite conveniently. Neutrons also have greater penetration depth and lower absorption coeOcients and hence can provide greater depth information as compared to X-rays. It should be mentioned here that although this review is restricted to X-ray scattering studies, most of the discussions and formalisms presented here are equally valid for neutron scattering as well [49,50]. The extraction of signi&cant and physically meaningful information from experimental re6ectivity data often proves to be very diOcult. Most often the main diOculty, apart from the phase problem, lies in the sensitivity of X-rays to low density contrast present in many systems. Various schemes have been put forward with a view to overcoming this limitation in X-ray re6ectivity technique which has enhanced the strength of this technique and led to its increased application [51,52]. We shall brie6y review some of these data analysis schemes. We will also discuss the relevant results of atomic force microscopy (AFM) studies which provide complementary information about the surface morphology of LB &lms. Scanning probe microscopy, in general, and AFM, in particular, provide real space information about surfaces ranging from molecular resolution over few tens of Angstroms in surface crystallography to surface morphological information over several microns. A large body of work exists where the

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complementary techniques of high resolution AFM [53] and grazing incidence X-ray di:raction [25] have been used to obtain molecular resolution information on LB &lms. Both these &elds have been well reviewed and in this report we will restrict our discussions to grazing incidence X-ray re6ectivity and di:use scattering studies on LB &lms. X-ray re6ectivity and di:use scattering studies had shown that the interfacial correlation of LB &lms can vary from self-aOne fractal [27–29,31], observed in diverse physical systems [54 –58], to long range logarithmic [28], characteristic of capillary waves on liquid surfaces [14,15]. It is expected that systematic studies of interfacial morphology of LB &lms can provide us clue to the growth mechanism of these &lms, especially because a lot of theoretical and simulation studies have been performed to link the evolution of the interfacial morphology with the possible growth mechanism of thin &lms [54,55]. Understanding the formation of nanometer sized particles that exhibit &nite size e:ect in band structure is of seminal importance [59 – 62]. Although there are several methods to prepare these nanostructures, it has been shown recently that multilayer LB &lms provide an ideal medium to form these con&ned structures with a reasonable amount of control over the size of these particles. Speci&cally we will discuss the formation of semiconductor nanoparticles which are also very promising materials for future technology due to the possibility of preparing tailor-made materials with size dependent tuning of various physical properties essential for di:erent applications [59 – 66]. In comparison to the investigations on structure and growth of LB &lms not much work has been done to understand the melting process of these &lms. Study of the melting of multilayer LB &lms, having few monolayers, should be quite interesting since these quasi-2D systems are expected to follow the predictions for melting of 2D solids [67,68]. The exact nature of the phase transitions in LB &lms and its dependence on various parameters like &lm thickness and interfacial morphology, etc. is still not very well understood. Here we review some recent work [69 –76] done in understanding the complex mechanism of melting in LB &lms. 2. Langmuir–Blodgett deposition technique 2.1. Preparation of Langmuir monolayer In 1917, Irving Langmuir developed the experimental and theoretical concepts which underlie the present day understanding of the behaviour of molecules in insoluble monolayers on water [77]. The &rst study on a deposition of multilayers of long chain carboxylic acid onto a solid substrate was carried out by Katherine Blodgett [78] and this started the &eld of LB &lm deposition. The molecules which can form a Langmuir monolayer are called amphiphiles. One part (hydrophilic) of these molecules are soluble in water while the other part (hydrophobic) is insoluble. On water, such molecules take a preferential orientation with the hydrophilic part (also called the head) immersed in water while the hydrophobic part (also called the tail) stays away from water as is shown schematically in Fig. 1. The most common class of molecules which has this property are the long chain fatty acids like stearic acid and its higher homologues. The amphiphilicity of a molecule depends on the balance between the hydrophilicity of the

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Fig. 1. Schematic of typical amphiphilic molecule: stearic acid on water. The region below the water surface is the head or hydrophilic part while the hydrocarbon backbone is the tail or hydrophobic part.

head group and the hydrophobicity of the tail, the latter depending on the chain length. These molecules are also commonly called surfactants. The most important property of the Langmuir monolayer is its surface tension. In studies of these monolayer &lms, one measures the di:erence in surface tensions between a clean or pure liquid (denoted as the subphase) surface and that covered by the &lm. Langmuir &rst pointed out the analogy between the surface tension di:erential and the force or pressure exerted by the &lm [77]. The surface pressure, , is de&ned as, = 0 − ;

(1)

where 0 is the surface tension of the pure liquid and the surface tension of the &lm covered surface. The Langmuir monolayer of any material is generally formed by spreading this material on water. Although some materials spread spontaneously on water most of the materials of interest need a spreading solvent. It is desirable that the solvent should be capable of dispersing the molecules of the &lm-forming material at the air–water interface and then evaporate completely so that the &lm is not contaminated. Usually, the spreading solvent chosen is volatile and insoluble in water. A volatile solvent which is intended to evaporate must do so within a reasonably short time; extremely volatile solvents, however, present diOculties because evaporation prevents accurate determination of solution concentrations. Solvents which have boiling point ◦ in the range 40 –80 C are generally most suitable for experiments under ordinary conditions. High water solubility can also lead to serious problems since some of the &lm-forming material

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may be carried into the subphase and may be precipitated rather than appearing on the surface. If a drop of liquid is placed on a solid or another liquid, the extent to which it will spread out or wet the surface is governed by Young’s equation LV cos = SV − SL ;

(2)

where the ’s are the interfacial tensions at the various interfaces between the solid (S), liquid (L) and gas (G) phases and is the contact angle between the solid and liquid surface at the solid–liquid–gas interface. For complete wetting, there is no &nite contact angle, i.e. the liquid spreads completely on the solid or another liquid. A quantity called the spreading coe5cient can be de&ned as [79] Sb=a = a − b − ab ;

(3)

where Sb=a is the spreading coeOcient for liquid b on liquid a, a and b are the respective surface tensions, and ab is the interfacial tension between the two liquids. If the value of Sb=a is positive, spreading will occur, while if its negative, liquid b will dewet and hence rest as a lens on liquid a. The application of the spreading solution to the surface is accomplished by allowing one drop at a time to fall from a micro-pipette, containing a measured quantity of the solution, held slightly above the subphase. Some time is allowed for the spreading solvent to evaporate completely before the monolayer properties are studied. There are several properties of the monolayer like surface pressure, surface potential, surface viscosity, etc. in terms of which the monolayer can be characterised [33–36]. Here we will discuss the most commonly characterised property of a Langmuir monolayer—surface pressure and its variation with surface area and temperature. Fig. 2 shows the schematic of a typical pressure–area isotherm for a Langmuir monolayer. In reality, the isotherms are much more complicated and the monolayer passes through several intermediate phases [37].

Fig. 2. Schematic of a typical surface pressure versus area isotherm. The di:erent phases are indicated. Actual monolayers have much more complicated phase diagrams with several intermediate phases and undergoing both &rst and second order phase transitions [37].

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Fig. 3. Schematic of a single bath trough: (a) bath, usually made of Te6on; (b) moving barrier allowing control of monolayer pressure; (c) motor for moving the barrier; (d) surface pressure (actually surface tension) sensor and controller; (e) balance for measurement of surface pressure; (f) motor for controlling the motion of substrate; (g) solid substrate.

2.2. Langmuir monolayer and LB 6lm deposition The instrument which is used to prepare the Langmuir monolayer and subsequently deposit the LB &lms is called a trough. The trough has undergone several modi&cations since the &rst one developed by Agnes Pockels. The modern day troughs are fully computerised with state-of-the-electronics involved in the design. Fig. 3 shows the schematic of a single bath trough. The essential elements of a trough are (a) the bath, usually made of a hydrophobic material like Te6on (PTFE), (b) the mobile barriers for controlling the area and hence pressure of the monolayer, (c) a balance that measures the surface pressure and (d) a dipper for dipping the substrate in the monolayer. We have used a KSV 5000 alternating trough for depositing our LB &lms. Fig. 4(a) shows the front view of this trough while Fig. 4(b) shows a closer view of the dipper arms. While in a conventional single bath trough there is only one dipping arm for dipping the substrate, in the alternate trough there are two dipper arms. The advantage of this trough as compared to the ordinary trough with a single bath is that it can be used to deposit two di:erent types of materials in alternate cycles to form an ABAB::::: type &lm. This system has two baths each having a maximum e:ective area of 120 × 240 mm2 with independent sets of two barriers which are hydrophilic to ensure that the &lm material does not go under the barriers during compression. The maximum and minimum speeds of the barriers, which are controlled by a micro step driven stepping motor, are 0:5 and 85 mm= min, respectively. In addition, each bath also contains a Wilhelmy plate made of sandblasted platinum. There is an elevator to position both these plates inside the subphase simultaneously, the usual practice being to submerge one-third of the plates inside the subphase. The upper arm is used to mount the substrate initially and for bringing it upto the subphase while the lower arm is used to transport the

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Fig. 4. KSV 5000 alternating trough. (a) Front view (b) Closer view of the two dipper arms used to transfer substrate from one bath to another.

substrate inside the subphase and between the two baths. The upper arm never goes below the subphase while the lower arm always remains within the subphase. The dipping speed of the substrate can be varied from 0.5 to 85 mm= min. The dipper arms are controlled by independent stepper motors. For our work, we have used water as the subphase. The water used is ultrapure water (Millipore) having very high resistivity of 18 MS cm. The temperature of the water can ◦ be regulated in the range 5–60 C by circulating water from an external temperature controlled water bath inside the circulator &tted beneath the trough. To deposit an LB &lm the precursor monolayer is compressed to a pressure such that the monolayer is in the solid region of the –A isotherm (Fig. 2). During deposition the monolayer pressure is held constant by compensating for the material transferred onto the substrate in terms of the reduction of the area occupied

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by the monolayer. It is well-known that monolayers of salts of the fatty acids are much easily transferred onto a solid support than the acids. Hence, most LB &lms are prepared from salts of the fatty acids [33,80]. There are two di:erent ways in which LB &lms of salts of long chain fatty acids, the prototypical amphiphilic molecules, may be deposited. The most common method is to spread the monolayer of the corresponding acid on a controlled subphase, of the suitable inorganic salt, kept at de&nite pH (usually alkaline) [33]. In the other method, the salt is actually prepared and spread on water at normal pH (around 5.65). Although chemically, these two methods are expected to give identical results, the isotherms [81] are found to be di:erent for these methods. The substrates used to deposit these &lms also have to be treated specially for good transfer (they are made hydrophilic in most depositions). Usually, the transfer is quanti&ed by a para-

Fig. 5. Schematic of an LB &lm deposited on a hydrophilic substrate. The interfaces have been labelled as shown. The dashed line shown alongside is the electron density pro&le (EDP) for ideal &lm structure while solid line is for disordered structure indicated by missing molecules in the top three bilayers. Roughness and waviness of the substrate moves the heads in the tail region and increases the electron densities in these regions, as we will see later.

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11

meter called transfer ratio which is de&ned as the ratio of the area swept by the moving barriers during transfer of a single monolayer onto the substrate to the area of &lm deposition on the substrate. Ideally, this should be close to unity. But in reality, this quantity di:ers from the ideal value due to several reasons like partial transfer or loss of molecules to the subphase. Fig. 5 indicates, schematically, the e:ect of partial transfer of molecules during each monolayer transfer on the structure of a typical multilayer LB &lm. Substrates are usually made hydrophilic in which case the &rst stroke of the dipper arm should be from below water for eOcient transfer. If the substrate is made hydrophobic, then the &rst stroke should start from above water. In addition to simple fatty acids, several other classes of organic compounds like phospholipids, alcohols, esters, acetates as well as polymers can form stable monolayers although not all can be used to transfer good quality LB &lms. Several good books [33–36] and reviews [53] discuss the details of the transfer of such LB &lms. Here, we will concentrate on the most widely studied class of LB &lms—those of simple fatty acids and their metal salts—since deposition process for even these simple molecules have remained an enigma. Conventionally, LB deposition is visualised as a process in which a solid substrate gets deposited with a monomolecular layer during each up=down stroke through the Langmuir monolayer of the amphiphilic molecules to be transferred. It is generally believed that molecules deposited during each upstroke have their hydrophilic headgroups oriented towards the substrate while those deposited during each downstroke are deposited with the hydrophobic tails oriented towards the substrate. Although this model may be valid for fatty acid LB &lms, recent results for fatty acid salt LB &lms, especially those for multivalent metal salts, seems to indicate that this simple model may not be valid for such systems. Nevertheless, this conventional type of LB deposition generally leads to Y type centrosymmetric multilayer &lms. A simple deposition parameter, the transfer ratio (TR), is traditionally used to quantify the quality of &lm deposition. It is de&ned as the ratio of the decrease in Langmuir monolayer surface area on the trough for every pass of the substrate through the monolayer to the total surface area of the substrate. Conventionally, a TR close to unity is treated as a signature of good quality deposition. In certain situations, it is possible that this TR is unity for every upstroke and zero for all downstrokes and vice versa. In such cases, one obtains Z and X type &lms, respectively. It may be mentioned here that the above discussion is based on a very simpli&ed model of LB &lm deposition whereby the structure on the Langmuir monolayer is assumed to be just replicated during transfer of molecules to the substrate. In reality, it has been found that signi&cant molecular re-arrangements takes place both during and after transfer especially during the transfer of the &rst layer on the substrate as might be expected due to the di:erence in water–molecule and substrate–molecule interactions. Using a specially arranged set-up of 6uorescence microscopy, it was shown [82] that pH dependent liquid-to-solid phase transitions can occur locally on the 1D water–substrate–monolayer meniscus during a typical LB transfer, at surface pressures substantially below the pressures at which such phase transitions occur on the Langmuir monolayer. Using [83] a combination of AFM, transmission IR and contact angle goniometry it was shown that LB &lms of pentadecanoic and hexadecanoic acid deposited from a Langmuir monolayer in the liquid-expanded (LE) phase condensed into densely packed islands after transfer onto mica substrates similar in morphology to that observed for &lms deposited from monolayer in the

12

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liquid-condensed (LC) phase. This transition was also found to be dependent on the temperature of deposition. These results also suggested the existence of a surface tension gradient driven interfacial 6ow during the condensation process. Molecular re-arrangements can also take place after deposition. Several studies indicating this have been made earlier [53]. X-ray di:raction studies [84] on several odd-layered &lms of lead arachidate indicated that instead of the expected head–head con&guration expected from the simple layer-by-layer transfer, the molecular con&guration in these &lms was similar to that found in bulk crystals of lead arachidate with the two hydrophobic tails lying on two opposite sides of the lead ion in all the layers except the &rst layer. This con&guration is only possible if the salt molecules 6ip to such a con&guration from the head–head con&guration expected in these &lms. Extensive studies have been made on the dependence of the &lm quality on various parameters like temperature, pH, deposition speed, etc., Details of these studies can be found in the review by Schwartz [53] and also in Ulman’s book [35]. Here, we will just discuss two recent studies which provide useful insight to the formation of defects in LB &lms. Recently, a series of investigations [85 –87] was made on the linked in6uence of pH and type of cations present in the subphase to the monolayer integrity and the morphology of the transferred &lms. They found that both the stability and integrity of a monolayer of behenic acid is greatly improved if Cd 2+ and Mn2+ are added to the subphase at pH 5.75 and 6, respectively, and under these conditions the transfer is very eOcient (as evident from TR) and good quality multilayer LB &lms are obtained as evident from Normarski microscopy studies. They also noted that if the pH is increased to higher values, the monolayer integrity and quality of LB &lms deteriorates. Recently, the e:ect of variation of pH on the domain structure of lead arachidate LB &lms was investigated [88] by means of AFM and X-ray specular and di:use scattering. It was concluded that both the lateral as well as the vertical domain sizes in the multilayer LB &lms decreased with pH increasing from 4.7 to 7.0. Both the above results and earlier studies seem to indicate the existence of a pH window, depending on the acid and the counterion present in the subphase, within which good quality LB &lms can be deposited. 3. Grazing incidence X-ray re"ectivity (GIXR) technique In the last two decades, we have observed tremendous growth in the family of grazing incidence X-ray and neutron scattering techniques not only in terms of the availability of synchrotron and neutron sources with higher brilliance but also development of new techniques based on these sources and with it the increased applicability of these techniques to hitherto uncharted territories of practically every discipline of science. Whether it is a laboratory tube (sealed or rotating anode) source or a synchrotron source, there are two modes in which X-ray scattering experiments can be performed—the energy dispersive mode and the angle dispersive mode. In the energy dispersive mode, the white beam emerging from the source is directly used for scattering and the scattering cross-section as a function of energy (or wavelength) is recorded. Such experiments are usually performed with a &xed experimental geometry and are very useful for studying kinetics, in general [89]. In the angle disper-

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13

Fig. 6. Possible motor movements of goniometer. The ; 2 and motors are interfaced to a computer while the other movements have to be performed manually. Several combinations of movements of the di:erent motors (; 2 and ) are possible. Fig. 7. Schematic of sample stage for in situ X-ray studies at elevated temperatures. S1, S2, S3 are collimating slits of &xed dimensions, M is a silicon monochromator (1 1 1), D is the NaI scintillation detector coupled to a PMT.

sive set-up, the incoming white beam has to be monochromatised before it is made incident on the sample and the experimental geometry has to be varied to obtain a given spectrum [45,90]. 3.1. Experimental set-up In a typical laboratory angle dispersive set-up, like ours [91], the X-ray beam emerging from the surface of the anode is the characteristic copper emission superimposed on the bremsstrahlung radiation. This beam on emerging from the chamber through the beryllium window is collimated by a slit before it is monochromatised. The monochromator, which is a Si(1 1 1) crystal, is aligned to select the Cu K line. For our experiments, a set of crossed vertical and horizontal slits is placed at the end of the collimator before the beam is incident on the sample. The horizontal slit has an aperture of 5 mm in the vertical direction while the vertical slit has an aperture of 100 m in the horizontal direction (in the scattering plane; refer Fig. 6). The slits are placed in slots and the aperture size can be varied by using a di:erent slit. For our work, the 100 m slit is placed in such a way so as to cut o: the Cu K2 line and the K1 line with N is selected. The goniometer has several stepper motors for angular as wavelength of 1:540562 A well as translational movements. The various possible motor movements are indicated in Fig. 6. Di:erent types of sample mounting stages can be attached to the goniometer. Schematic of such a stage which can be used for performing in situ high temperature GIXR studies under controlled atmosphere is shown in Fig. 7. Details of this cell has been published earlier [91]. We have used this set-up for the investigation of melting of LB &lms. A NaI scintillation

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detector coupled with a photomultiplier tube (PMT) and preampli&er is used in our set-up. The detector set-up is mounted on the 2 arm of the goniometer. In order to reduce background counts due to air scattering, evacuated collimator pipes are placed between the monochromator and the sample stage and between the sample stage and the detector. Moreover, we used a tight energy window in the single channel analyser before the counter. For our set-up, the background level is around 0:05 counts= s and the direct beam count is around 2 × 106 counts= s which allows re6ectivity measurements spanning over eight orders of magnitude. 3.2. Types of scans A typical experiment starts with the alignment of the goniometer. This involves determining the position of the axis of the goniometer and ensuring that all motors execute respective motions about this axis. For this, a wedge-shaped metallic object is mounted on the sample mounting stage and the motors are given movements and the shift in the position of a spot (made intentionally) on the centre of the wedge is detected using a telescope. Once the common axis of the motors of the goniometer is determined, the next step is to align the axis in the direction of the beam and the detector such that the edge of the wedge and hence, the axis passes through the middle of the beam (the half-cut position). Once this alignment is completed the wedge is removed and the sample is placed in position to be aligned. The alignment of the sample involves two steps. First, the sample is aligned in the direct beam by setting the detector angle to zero and translating the sample holder to halve the intensity of the direct beam (similar to the half-cut alignment of the wedge). Then the parallelism of the sample face with direct beam is ensured by rotating the sample about its axis (a relative or rocking scan). One such typical scan is shown in Fig. 8. In this case, the sample is correctly aligned. If the sample is not correctly aligned, then the intensity will increase above that at the central position of the scan as the sample is rocked. There are three di:erent scan types used normally in GIXR studies—specular re6ectivity scan, transverse di:use scan and longitudinal o:-specular di:use scan. In specular scans, the scattered intensities are measured as a function of scattering angle by keeping equal incident and scattered angle

8000

Intensity (Arb. Units)

7000 6000 5000 4000 3000 2000 1000 0 -4

-2

0

2

4

θ (mrad)

Fig. 8. Half-cut alignment. This is a rocking scan performed about = 0 to check whether the sample surface is parallel to the incident beam and whether the beam falls symmetrically with respect to the sample. Any distortion from this ideal shape of the pro&le can be used as an indication of possible misalignment of sample.

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15

(i = f ) of X-rays to the sample surface normal (this is specular condition). In this scan, the total wavevector is along the qz direction and varies with the incident angle. A transverse di:use scan is performed about a specular point by scanning motor with &xed 2 motor position. In a typical experiment, several such scans are usually performed about di:erent specular points. In these scans, transverse component of wavevector qx changes continuously and keeps normal component of wavevector qz almost unchanged at small angular positions of the motors. The longitudinal di:use data are taken along the specular direction by o:setting the motor suitably so that this scan in qz is just outside the specular pro&le in qx direction. It can be noted here that collection of specular data is same as longitudinal one except for the fact that for specular data collection qx o:set is kept as zero. All these measurements require a very high accuracy in incident angle and precision movements of sample and detector. In a typical measurement, ◦ in our laboratory, an elementary displacement of 0:001 is used with stepping motors. 3.3. Instrumental e:ects 3.3.1. Corrections to data In the analysis of all the above types of experimental data, it is very important to take into consideration the instrumental e:ects. An important instrumental e:ect which has to be corrected for especially specular re6ectivity measurements is the footprint of the beam on the sample. Fig. 9 shows this e:ect for a beam of thickness, T , and footprint, F, on the sample at an angle of incidence i on the sample of dimension L along the direction of the incident beam. The X-ray beam incident on the sample has the shape of a ribbon with the width determined by the aperture of the incident slit perpendicular to the scattering plane while the thickness is determined by the aperture of the slit in the scattering plane (refer Fig. 7). The footprint, F,

Fig. 9. Incident beam footprint e:ect at very grazing angles on the sample. All specular re6ectivity data should be corrected for this e:ect which results in loss of scattered beams away from the specular direction if the beam footprint is larger than the sample dimension along the incident beam direction.

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of the beam of thickness, T , is given by F = T= sin i :

(4)

The thickness of the incident beam depends both on the intrinsic beam pro&le and the size of the slit aperture. Usually, the intrinsic beam pro&le is Gaussian so that the beam pro&le coming out of the rectangular slit is also Gaussian with the thickness T = 2, where is the full-width at half-maximum (FWHM) of the Gaussian. For specular re6ectivity measurements, it is essential to choose the aperture of the incident slit properly such that F ¡ L, for i = c , c being the critical angle for total re6ection. In this situation, one can simply compensate the loss of re6ectivity below c , expected to be unity otherwise, by multiplying the experimental data with F=L. One should be careful in correcting experimental data above c because the simple correction formula of Eq. (4) does not include beam divergence and non-uniformity of X-ray wavefront. The footprint e:ect also has to be taken into account in di:use scattering measurements especially near Yoneda wings in transverse di:use scans. 3.3.2. Resolution function The resolution function is very crucial in all GIXR measurements and the calculated pro&les have to convoluted with the relevant resolution functions before it can be compared with the measured data [92,93]. With reference to Fig. 7, the wavevector transfer q can be resolved into two components in the x–z system of coordinates as qx = k0 (cos f − cos i ) ;

(5)

qz = k0 (sin i + sin f ) ;

(6)

where k0 = 2=. In the usual geometry, the aperture of the detector slits perpendicular to the scattering plane is kept wide open so that the scattered intensity in the qy direction is integrated out in GIXR measurements. Di:erentiating the above equations with respect to i and f , we get d qx = k0 (sin i d i − sin f d f ) − d k0 (cos i − cos f ) ;

(7)

d qz = k0 (cos i d i + cos f d f ) + d k0 (sin i + sin f ) ;

(8)

where d k0 = −k0 d =. With the assumption that d i and d f are randomly distributed, the resolution widths in qx and qz are given by [92,93] Tqx2 = k02 (sin2 i Ti2 + sin2 f Tf2 ) + Tk02 (cos i − cos f )2 ;

(9)

Tqz2 = k02 (cos2 i Ti2 + cos2 f T f2 ) + Tk02 (sin i + sinf )2 :

(10)

In the specular condition (i = f = ) and neglecting the e:ect of the wavelength dispersion as compared to the angular spread, the resolution widths are given by Tqx = k0 sin [(Ti2 + Tf2 )]1=2 ;

(11)

Tqz = k0 cos [Ti2 + Tf2 )]1=2 :

(12)

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

17

14000

Intensity (Arb Units)

12000

400µm

10000 8000 6000 4000 100µm

2000 0 -2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

θ (mrad)

Fig. 10. Direct beam or detector scans. The width of the incident slit is 100 m while the detector slit width is 100 and 400 m, respectively. This pro&le primarily decides the instrumental resolution.

De&ning [Ti2 + Tf2 ]1=2 as the half-width at half-maximum (HWHM), d , of the direct beam we have, Tqx = qz d =2 ;

(13)

TqZ = k0 d cos :

(14)

Typical direct beam scans (detector scan) with 100 m incident and detector slits and with 100 m incident and 400 m detector slits is shown in Fig. 10. In our experiments, the specular scans were performed with the latter con&guration of slits while the former con&guration has been used for di:use scattering measurements. In the next section, we present the theoretical formalism for X-ray scattering from rough surfaces. 4. Scattering from surfaces and interfaces 4.1. Basic formalism It is possible to arrive at the basic expression for X-ray specular re6ectivity for a smooth surface by solving the Maxwell wave equations for scattering of electromagnetic waves at the surface under the appropriate boundary conditions for the continuity of the electric &eld and its derivative [40,41,52]. The wave equation for the electric &eld, (r), can be written in the form of the Helmholtz equation as ∇2 (r) − k 2 (r) = −%(r) (r) ;

(15)

where k is the wavevector in air and %(r) is the scattering length density of the medium. For neutrons, it is possible to write down a similar wave equation based on the corresponding SchrUodinger equation for the interaction of a neutron beam in a medium represented by the interaction potential V (r) [94 –97], 2˝2 V (r) = %(r) ; (16) m

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in terms of its wave function, (r). Here k = (2mE= ˝2 ). This potential is usually referred to as the Fermi pseudo-potential. This establishes the basis for treating X-ray and neutron re6ectivity on the same footing [49,50] although the basic interaction of X-rays and neutrons with a material medium is vastly di:erent. The refractive index n, of a homogeneous medium, with multiple species, can be written as [49,97]

n = 1 − $ + i% : For neutrons $ can be written as, 'i bi =Ai $ = (2 =2)NA i

2

(17)

= ( =2)

Ni bi

i

2

= ( =2)

%i :

(18)

i

In the above expressions, bi , Ni and 'i are the scattering length, nuclear number density and mass density, respectively, of the ith element with atomic weight Ai , the wavelength of radiation used and NA is the Avogadro number. The imaginary component can be written as, % = (2 =2)NA 'i bi =Ai : (19) i

b

Here, = a =2 and a is the neutron absorption cross-section. The above expression for refractive index is valid for a non-magnetic medium without absorption, which turns out to be negligible for most materials. If the medium is magnetic with magnetisation parallel to the surface, then the above expression (Eq. 18) gets modi&ed to [44,52] $ = (2 =2) NA 'i bi =Ai − (m=2˝2 ))n · B ; (20) i

where )n and B are the neutron magnetic moment and the magnetic induction of the medium, respectively. For X-rays $ is given by $ = (r0 2 =2)NA 'i (Zi + fi )=Ai 2

= (r0 =2)

i

Ni

i

= (2 =2)

%i :

(21)

i

% is related to the imaginary anomalous dispersion factor f (which is related to the X-ray absorption coeOcient) and can be written as % = (r0 2 =2)NA 'i fi =Ai : (22) i

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19

In the above expressions, r0 is the classical electron radius given by e2 =mc2 (which is equal to 2:82 × 10−13 cm), Zi and Ni are the atomic number and electron number density of the ith element and f is the real anomalous dispersion factor. It may be noted that although we have used separate terms to denote the nuclear and electron number densities, we have used the same symbol to represent them to underline the essential unifying feature of this quantity for both X-rays and neutrons. We have used di:erent nomenclature only to indicate that neutron scattering is sensitive to number of nuclei of a given species while X-ray scattering is sensitive to the number of electrons of a given species in a medium. In the literature, it is common to &nd the nuclear number density as being referred to just the number density while the electron number density as being referred to just electron density, which in some cases can be misleading. Henceforth our discussions will be based on X-rays although they are equally valid for neutrons with the possible exception of the di:erences arising as a result of the di:erence in the expressions for refractive index. We will also use the symbol ' to denote the electron number density or electron density as it will be henceforth called. In vacuum, the z-component of the wavevector (normal to the surface or x–y plane) is given by Kz = qz =2 = (sin i + sin f )= ;

(23)

where i is the grazing angle of incidence (refer Fig. 7). In the specular condition, where the angle of incidence i is equal to the angle of re6ection f , this is the same as the total wavevector K. At an interface separating two media with refractive indices n1 and n2 , the direction of the refracted beam can be obtained from the Snell law as n1 cos i = n2 cos i :

(24)

For the substrate with n1 = 1 and n2 = n0 , we can obtain a real angle of incidence for which angle of refraction i becomes zero provided n2 ¡ n1 (which is the case for any material having &nite electron density) and the corresponding grazing incident angle, known as critical angle can be written as cos c = n0 :

(25)

In the small angle approximation, the critical angle can be written as c = (2$)1=2 = [2 NA 'r0 =(A)]1=2 ;

(26)

where $ is given by Eq. (21). Using the de&nition of refractive index n0 = K2z =Kz , we can get the expression (which is in general complex) for the z-component of wavevector in the medium, K2z = [K 2 − KC2 ]1=2 ;

(27)

where KC is the critical wavevector given by KC = qc =2 = 2 sin C = :

(28)

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Again using small angle approximation, for a medium with n0 ¡ 1, we can calculate KC and rewrite Eq. (27) as K2z = [Kz2 − 4NA 'r0 =A]1=2 :

(29)

It is interesting to note that under small angle approximation, KC is independent of wavelength in X-ray energy range and depends only on electron density of the material. The re6ectance, r12 , for the substrate can then be written as, Kz − K2z qz − q2z r12 = = : (30) Kz + K2z qz + q2z ∗ can then be written in two equivalent forms as The specular re6ectivity R = r12 r12 sin i − n0 sin i 2 ; R = sin i + n0 sin i K − K 2 − K 2 2 z z C : R = Kz + Kz2 − KC2

(31) (32)

The above expressions are known as the Fresnel law of re6ectivity. 4.2. Re<ectivity from multiple interfaces In the case of a thin &lm of &nite thickness d, we have to solve the wave equations at two interfaces, namely &lm–vacuum=air (at z = 0) and substrate–&lm (at z = d). It is interesting to note that the continuity condition at z = d will generate an extra factor, which in turn will give us the re6ectance at the &lm–substrate interface as K2z − K3z r23 = exp(−2iK2z d) : (33) K2z + K3z With the help of simple algebra and noting the fact that r21 = −r12 we can write the re6ectance from this thin &lm–substrate system as r12 + r23 r0 = : (34) 1 + r12 r23 We can easily extend the above calculation to the case of re6ectivity for a system having M such thin layers (strati&ed homogeneous media), having smooth interfaces. We denote the thickness of each layer by dn . A set of simultaneous equations similar to Eq. (34) can be solved and one can arrive at a recursive formula [98] given by rn; n+1 + Fn−1; n rn−1; n = exp(−2iK(n−1)z dn−1 ) ; (35) 1 + rn; n+1 Fn−1; n where Fn−1; n =

K(n−1)z − Knz : K(n−1)z + Knz

(36)

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

21

To obtain the re6ectivity of this system, one solves this recursive relation given by Eq. (35) from the bottom layer with the knowledge that rn; n+1 = 0 since the thickness of this medium (normally the substrate) can be taken as in&nite. It is to be noted that d0 is also in&nite. It should be mentioned here that the re6ectivity calculation of strati&ed homogeneous media having smooth boundary interfaces given by Eq. (35) is not only used for multilayer thin &lms, but also widely utilized to approximate realistic scattering density pro&le. This is done by subdividing the continuous scattering density of a &lm into a series of discrete layers [19,99]. So far, we have dealt only with smooth surfaces and interfaces. At this stage, we can introduce the concept of roughness. It should be mentioned here that roughness is di:erent from the waviness of a surface. In fact, this di:erence becomes apparent when the variation of scattering density is plotted as a function of depth (z-axis). In the length scale probed by the X-ray (few Angstroms), the density pro&le for rough surface will be a slowly varying function but the pro&le for surface having waviness will be a sharp one. Roughness for solid &lms can vary from a few tens of Angstroms for very rough &lm to a few Angstroms for &lms grown using MBE technique. Intrinsic roughness for liquid &lms are high and can be correlated to the molecular diameter of the liquid [15]. It also plays an important role in the fabrication of strained multilayers. It is now accepted that X-ray scattering technique will play an important role in characterisation of interfacial roughness even during the growth process. It is known that re6ectivity of a rough surface is smaller than the re6ectivity of a smooth surface and this deviation increases with qz . One can calculate the e:ect of roughness by approximating the scattering density pro&le with a series of discrete layers as discussed above and using iterative scheme of Eq. (35). Actually, the &rst derivative of the scattering pro&le can be described, for most of the cases, as a Gaussian function and as a result by using Born approximation [43], one can write the re6ectance of a rough surface as F 2 rn−1; n = rn−1; n exp(−0:5q(n−1)z qnz n ) :

(37)

The parameter n is the measure of roughness between (n−1)th and nth interfaces, and roughness acts like a Debye–Waller Factor. Eq. (37) also explains the observation that re6ectance of rough F surface rn−1; n deviates more from the re6ectance for smooth surface rn−1; n as q(n−1)z and qnz increases with qz . For a surface separating two media 1 and 2, Eq. (37) can be simpli&ed, as [100], to obtain the re6ectivity as 2 Rrough = RF exp(−qz2 12 );

(38)

where RF is the Fresnel re6ectivity given in Eqs. (31) and (32) and 12 represents the roughness between interface separating media 1 and 2. In general, the roughness of the &lm surface brings down the re6ectivity curve faster; on the other hand, the roughness of the interface of the &lm and the substrate reduces the amplitudes of multilayer peaks. 5. Scattering in Born approximation In the previous section, we discussed the basic formalism for X-ray and neutron re6ectivity from smooth surfaces and interfaces. Although we did discuss rough surfaces, the e:ect of roughness was introduced only by using the Born approximation. Moreover, our discussion

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was only restricted to specular re6ectivity even for the rough surfaces. A large body of work exists on the scattering of electromagnetic radiation (mostly light or radio waves) from rough surfaces [101–105]. In this regime, the interaction of the electromagnetic waves with matter is quite strong and the solution of the relevant wave equations turns out to be quite diOcult involving matching of boundary conditions over random rough surfaces and several simplifying assumptions have to be invoked for their solution. X-rays and neutrons, on the other hand, interact quite weakly with matter and hence the solution of the wave equations can be performed under Born approximation. For a plane wave incident on a medium with scattering length density ', the solution of the wave equation (15) can be written in general as [106] (r) = eiki ·r +

eikr

r

f(kf ; ki ) ;

where the scattering amplitude, f(kf ; ki ), is given by f(ki ; kf ) = dr %(r ) (r )eikf ·r

(39)

(40)

with kf ; ki being the incident and scattered wavevectors, respectively. The above expression for scattering amplitude includes multiple scattering e:ects. If the interaction between the incident plane wave and the medium is suOciently weak, then the &rst Born approximation, can be applied. In this approximation, the scattered wave function 1(r ) can be assumed to be the same as the incident plane wave and hence the scattering amplitude takes a simpler form f(kf ; ki ) = dr %(r )eq·r ; (41) where q = kf − ki . The quantity that is measured experimentally is not the scattering amplitude but the di:erential scattering cross-section de&ned as d = |f(kf ; ki )|2 : d2

(42)

5.1. Scattering from single surface Let us consider a schematic of a typical rough surface as shown in Fig. 11. Applying Eqs. (41) and (42) to the above surface, one can write z(x;y) 2 ∞ d 2 2 −qz z −qx x −qy y dz e d x dy e e (43) = ' r0 : d2 −∞ −∞ Here, % = 'r0 with ' being the electron density in the case of X-rays and and r0 is the scattering length (or classical radius) of electrons. For neutrons, the corresponding quantities are the number density and the neutron scattering length b, respectively.

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

23

Fig. 11. A typical rough surface. For an isotropic rough surface the height correlation functions depends on r, irrespective of the location of the points (x; y) and (x ; y ).

The above equation can be further simpli&ed to [43] ∞ '2 r02 ∞ d d x dy d x d y exp − qz [z(x; y) − z(x ; y )] = 2 d2 qz −∞ −∞ ×exp − [qx (x − x ) + qy (y − y )] :

(44)

To proceed further, we need to formulate a geometrical description of a rough surface. It has been found that description of surface geometry can be done conveniently using the concept of fractals [107]. Referring to Fig. 11, let us assume that the quantity U = z(x; y) − z(x ; y ) is a Gaussian random variable [108] whose probability distribution depends on the relative coordinates (X; Y ) = (x − x ; y − y ). Using this variable, one can de&ne a quantity called the height di:erence correlation function g(X; Y ) as, g(X; Y ) = [z(x ; y ) − z(x; y)]2 ;

(45)

where the average denotes an ensemble average over all possible con&gurations of the surface. Here, an implicit assumption has been made about the stationarity and ergodicity of the Gaussian random variable U [108]. There are several advantages of using this formulation for description of surface morphology. It can be shown that for random variables, U , U 2 = U 2 + 2 , where = U − U 2 since for the rough surface the heights are de&ned with respect to a reference surface U = 0. Hence, in this case U 2 = 2 . In the context of description of surface morphology, 2 is de&ned as the rms roughness of the surface. For a commonly observed type of rough surface, the rms roughness scales as a self-aOne fractal. Since statistically g(X; Y ) is equivalent to 2 , for isotropic self-aOne rough surfaces g(X; Y ) can be written as (46) g(X; Y ) = g(r) = Ar 2 (0 ¡ ¡ 1) ; where r= (X 2 + Y 2 ). This is of course an ideal description of a surface since for most surfaces in nature g(r) scales as a self-aOne fractal only within a &nite length scale, usually limited by

24

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

system size. Using the de&nition of g(X; Y ) = g(r) in Eq. (45) we can write g(r) = z(0)2 + z(r)2 − 2z(0)z(r) = 22 − 2C(r) :

(47)

Here z(0)2 = z(r)2 = 2 . This is valid only under the assumption of stationarity of U . C(r) = z(0)z(r) is de&ned as the height–height correlation of the surface. The height–height correlation can have di:erent forms but the form that is used most commonly is [43] C(r) = 2 exp(−r=6)2 :

(48)

Here, 6 is the cuto: length for height–height correlation of the surface. Using the above de&nitions, g(r) for isotropic rough surfaces with cuto: can be written as g(r) = 22 [1 − exp(−r=6)2 ] :

(49)

Having developed the formalism for the statistical description of rough surfaces, we can now write down the expression for the structure factor and hence intensity of the scattered beam from a rough surface, under the Born approximation, using the variables de&ned above. To calculate the scattering cross-section, a con&gurational average has to be performed over the di:erent possible con&gurations of the random rough surface [43]. Thus d = d 2 becomes '2 r 2 d = 20 d2 qz

∞

−∞

d x dy

∞

−∞

d x d y exp−qz [z(x; y) − z(x ; y )]

×exp(−[qx (x − x ) + qy (y − y )]) :

(50)

For Gaussian random variables exp − qz [z(x; y) − z(x ; y )] = exp(− 12 qz2 [z(x; y) − z(x ; y )]2 ) :

(51)

Hence, with the appropriate change of variables, we can write '2 r 2 d = 20 A d2 qz

∞

−∞

2

d X d Y e−qz g(X;Y )=2 e−(qx X +qy Y ) :

(52)

Here, A is the area of the surface illuminated by the incident beam. The actual intensity detected in an experiment is obtained by convolution of the appropriate instrumental resolution function with the calculated scattering cross-section [92,93]. We next show how it is possible to arrive at the expression for specular re6ectivity for a single smooth surface under the Born approximation from Eq. (52). For a smooth surface g(r) = 0.

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

25

Hence, putting g(r) = 0 in Eq. (52), we get '2 r 2 d = 42 2 0 A$(qx )$(qy ) ; d2 qz where 1 $(qx ) = 2

∞

−∞

d X e−iqx X :

(53)

(54)

Re6ectivity is de&ned as R = Idet =Iinc , where the incident beam intensity, Iinc , is given by incident beam 6ux × area of the beam normal to the surface which is A sin i . Hence, we can write re6ectivity R as 1 d R= d2 A sin i d2 42 2 2 1 = 2 ' r0 $(qx )$(qy ) d 2 : (55) qz sin i Using transformation of coordinates, d 2 can be written as d2 =

d qx d qy : k02 sin i

(56)

Using Eq. (56) in Eq. (55), re6ectivity R can be written as R=

162 '2 r02 : qz4

(57)

The above expression is the Fresnel re6ectivity of a smooth surface under the Born approximation and can be obtained in the asymptotic limit of the exact Fresnel re6ectivity shown earlier in Eq. (32). For any other non-zero form of g(r), the total scattering consists of both the specular and the o:-specular or di:use components. We will now discuss the scattering for the two general forms of g(r)—with and without cuto: in correlation. For isotropic self-aOne rough surfaces without cuto:, the scattering cannot be separated into specular and di:use components, hence total intensity has to be calculated [43]. The expression for scattering cross-section in Eq. (52) can then be simpli&ed using a change of variables from rectilinear to polar coordinates. Using the substitutions X = r cos and Y = r sin we can write d X d Y = r d r d . Eq. (52) can then be written as ∞ '2 r 2 d 2 d X d Y e−qz g(X;Y )=2 e−(qx X +qy Y ) = 20 A d2 qz −∞ ∞ 2 '2 r02 2 = 2 A d exp[ − (qx r cos + qy r sin )] d r r e−(1=2)qz g(r) : (58) qz 0 0

26

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

Using the de&nition [109] 1 2 x sin e d ; J0 (x) = 2 0 the above equation can be written as ∞ 2'2 r02 d 2 A d r r e−(1=2)qz g(r) J0 (qr r) ; = 2 d2 qz 0 where qr = qx2 + qy2 . The total scattering intensity can then be written as 2'2 r02 ∞ 2 I = I0 2 d r r e−(1=2)qz g(r) J0 (qr r) ⊗ R(qx ; qy ; qz ) : qz sin i 0

(59)

(60)

(61)

Here R(qx ; qy ; qz ) is the instrumental resolution function. In most experimental situations, the resolution is set much coarser in one direction (usually qy ) and hence one component of scattering is e:ectively integrated out. In such cases, Eq. (61) can be written as I (qx ; qz ) = I0

R(qz )qz × F(qx ; qz ) × G(qz ) + D0 ; 2k0 sin i

(62)

where R(qz ) is the specular re6ectivity in the Born approximation and D0 is the detector background due to both the detector dark counts as well as ambient scattering. The functions G(qz ) and F(qx ; qz ) depend on the nature of the correlation function, g(r). For a perfectly smooth surface G(qz ) is unity and F(qx ; qz ) can be de&ned as F0 (qx ; qz ) = $qxc ⊗ R(qx − qxc ) :

(63)

For a system with self-aOne fractal height correlation (without cuto:) with exponent = 0:5; G(qz ) and F(qx ; qz ) can be de&ned as [29] GS (qz ) = qz3 Iconv (qx ; qz )qx =0

(64)

FS (qx ; qz ) = Iconv (qx ; qz )=Iconv (qx ; qz )qx =0 ;

(65)

where Iconv (qx ; qz ) is a resolution convoluted Lorentzian A ⊗ R(qx − qxc ) : Iconv (qx ; qz ) = 2 qxc + [ A2 qz2 ]2

(66)

Here qxc is a dummy integration variable over which the convolution is performed with respect to the Gaussian resolution function, R, centred at qx . It may be noted here that in addition to self-aOne correlation, another type of long range correlation which is commonly observed is the logarithmic correlation characteristic of thermally generated spectrum of capillary waves on liquid surfaces [13–15]. Recently, this type of correlation has also been observed in organic thin &lms [28,29,110 –112]. The scattering from interfaces with this type of correlation also cannot

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

27

be clearly separated into specular and di:use components and has been discussed in detail in Appendix C. In case of a system with logarithmic height correlation with cuto:, the above functions can be de&ned as [29] 1 1−; 2 2 GL (qz ) = exp(−qz e: ) √ : (67) 2 1 − ; 1 −qx2 <2 FL (qx ; qz ) =1F1 ; (68) ; ; 2 2 42 where 1 F1 is the Kummer function, ; = Bqz2 =2 and e: is the e:ective interface roughness [15,28,29] as explained in Appendix C. In most cases [43], however, di:use scattering, which arises due to correlation of heights as a function of lateral displacement, is weak compared to the specular component and it is possible to separate out the two. The correlation function in such cases takes the form as given in Eq. (48) and hence using this form of g(r) Eq. (52) can be written as ∞ '2 r02 d 2 −qz2 2 d X d Y eqz C(X;Y ) e−i(qx X +qy Y ) : (69) = 2 Ae d2 qz −∞ The above equation can be split into two parts—specular and di:use [43]. The specular component yields the specular re6ectivity following Eq. (57) as R=

162 '2 r02 −qz2 2 e : qz4

The di:use intensity, Id , can be written as '2 r02 −qz2 2 2 d X d Y (eqz C(X;Y ) − 1) e−(qx X +qy Y ) ⊗ R(qx ; qy ; qz ) : e Id = I0 2 qz sin i

(70)

(71)

In most experiments, the out-of-plane (qy ) resolution is kept coarse to e:ectively integrate out this component of scattering. In that situation, the observed intensity can be written as '2 r02 −qz2 2 qz2 C(X;Y ) −qx X d X d Y (e d qy eqy Y ⊗ R(qx ; qz ) I d = I0 2 e − 1) e qz sin i '2 r02 −qz2 2 2 d X (eqz C(X;0) − 1) e−qx X ⊗ R(qx ; qz ) : e (72) = I0 2 qz sin i In Fig. 12, Id =I0 is plotted, as a function of qx for di:erent values of the parameters (; 6 and ) de&ning the self-aOne correlation functions. It is instructive to note the strong di:use intensity around qx = 0 (the specular point) for the case, where 6 and=or is high and is low. Clearly, in such cases, the assumption of separability of specular and di:use components is not valid. For a more detailed discussion on the relation between the scattering cross-section

28

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84 -3

10

σ = 2.0 Å σ = 5.0 Å

-4

10

50

ξ = 500 Å

40

α = 0.2

30

σ = 2.0 Å

20

σ = 5.0 Å

ξ = 500Å α = 0.2 10

-5

10

0 60

σ = 5.0 Å

α = 0.8

2

g(x)/c(x) (Å )

σ = 5.0 Å α = 0.8 -3

Id/I0 (Arb. Units)

10

ξ = 500 Å

40

ξ = 5000 Å

20

-4

10

ξ = 500Å ξ = 5000Å

0

-5

10

ξ = 5000 Å

σ = 5.0 Å

α = 0.8

40

α = 0.2

-3

10

σ = 5.0 Å ξ = 5000 Å

30

20 -4

10

α = 0.2

10

α = 0.8 0 -5 -0.0010

10

-0.0005

0.0000 -1

qx (Å )

0.0005

0.0010

0

2000

4000

6000

8000 10000

X (Å)

Fig. 12. Correlation function (C(x) (Eq. (48)) and g(x) (Eq. (49))) and the corresponding di:use re6ectivity (Id =I0 ) as a function of qx for di:erent values of the parameters ; 6 and . The di:use scattering pro&les have been generated for a &xed qz value.

and the height–height correlation for di:erent types of surfaces, the paper of Sinha [43] and the review by Dietrich and Haase [113] should be referred. 5.2. Scattering from multiple interfaces In multilayer &lms, apart from in-plane correlation length discussed in previous section, another correlation length, 6z , in the growth direction comes into play. Generally, an interface in a multilayer &lm tends to follow the morphology of the interface just below it and if the total thickness of the &lm d ¡ 6z , all the interfaces become statistically similar and the interfaces are called conformal [114,115]. All the expressions for scattered intensity derived for single surface can be extended for multiple interfaces [114,115] with necessary modi&cations in the formalism. The situation becomes more complicated for systems with non-conformal interfaces with d ¿ 6z and a formalism for calculation of scattering cross-section from this type of multilayer systems, based on DWBA, has been developed [16–18,114,115]. Here, we will discuss the cases for systems with conformal interfaces which has been found to be valid for most of the LB systems.

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

29

Even for systems with partly conformal interfaces, these expressions can be a very good &rst approximation and often the essential physical information about the system one is looking for can be obtained using these expressions. For systems where di:use and specular components of scattering cannot be clearly separated, Eq. (61) gets modi&ed, for multiple conformal interfaces, to I = I0

2r02 T'i T'j eqz (zi −zj ) qz2 sin i ij

∞

0

2

d r r e−(1=2)qz gij (r) J0 (qr r) ⊗ R(qx ; qy ; qz ) :

(73)

Of course, Eq. (73) can be further simpli&ed to the generalised form, Eq. (62), with the only modi&cation that in this case R(qz ) is the specular re6ectivity for multiple interfaces. Similarly, for systems where specular and di:use scattering components can be clearly separated, the scattered intensity for multiple interfaces can be written as [115]

Id = I0

ij T'i T'j

2

2

2

e−qz (zi −zj ) e−(1=2)qz (i +j )

qz2 sin i

2

d X (eqz Cij (X;0) − 1) e−qx X ⊗ R(qx ; qz ) :

(74)

So far, we have discussed the general formalism to obtain intensity of scattered X-rays for di:erent types of rough surfaces characterised by di:erent height di:erence correlation functions, g(r). It was shown that scattering from a surface can, generally, be separated into two components, specular and di:use. The specular component arises due to scattering from the average surface while the di:use or o:-specular component comes from scattering by height 6uctuations about this average surface. The specular component is directional while the di:use component is distributed around the specular direction. The type of interfacial height–height correlation decides the nature of the distribution of di:usely scattered photons around the specular direction and also the ratio of the number of photons scattered in the specular and di:use directions. In general, for a given incident angle, an increase in the height 6uctuations or roughness leads to an increase in the number of photons scattered into the di:use channels at the expense of the photons scattered in the specular channel keeping the sum total of the scattered photons same. In addition to increase in roughness, there may be decrease in number of specularly scattered photons due to the variation in electron density gradients normal to the surface or interfaces of the system from which scattering is being studied. Such variations do not contribute to the diffuse scattering channels although a reduction in the number of photons scattered in the specular channel takes place. In such situations, it is often convenient to calculate the di:use integrated re6ectivity, i.e. total re6ectivity after integration over di:use intensity pro&le (intensity vs. qx ) around each point along the specular ridge. By integrating over the di:use scattering pro&le, the e:ect of lateral height 6uctuations and hence roughness, in the specular re6ectivity, is taken care of and the resulting re6ectivity pro&le depends only on the vertical structure of a system. One clear advantage of this ansatz is the possibility of separating the contribution of roughness and interdi:usion (chemical gradient) in the re6ectivity [91]. We illustrate this scheme for the case of systems where di:use and specular scattering cannot be clearly separated and total intensity has to be calculated since for such systems it is diOcult to clearly separate interdi:usion

30

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

and roughness. The di:use integrated re6ectivity, ID , can be de&ned in general form, Eq. (62), as ID = d qx I (qx ; qz ) R(qz )qz = I0 × G(qz ) × 2k0 sin i

d qx F(qx ; qz )

R(qz )qz × G(qz ) × FD (qz ) : (75) 2k0 sin i Here, FD (qz ) = d qx F(qx ; qz ). In specular re6ectivity measurements, the width of the detector slits (in our case, 400 m) are kept much wider than the incident slit (100 m here) to intercept the full specular beam, including divergence, and also to e:ectively integrate over the di:use component (qx ) of scattering. Thus the intensity, I (0; qz ), at the specular position (qx = 0) can be written, from Eq. (62), as, = I0

I (0; qz ) = I0 R(qz ) × FW (qz ) × G(qz ) + D0 ; where FW (qz ) =

+w

−w

d qx F(qx ; qz ) :

(76) (77)

Here, w is the half-width of the detector aperture used for specular re6ectivity measurements. The quantity that is actually measured experimentally is I (0; qz ). The di:use integrated re6ectivity, ID (qz ), can then be related to the experimentally measured re6ectivity as ID (qz ) FD = RS (qz ) : (78) = I (0; qz ) FW For systems where specular and di:use components of scattering can be clearly separated a similar procedure can be applied by analysing the total intensity rather than separating the di:use and specular. But this scheme is not essential since alternative schemes exist whereby it is possible to separate interdi:usion and roughness in such systems from scattering measurements [116]. The details of the analysis procedures for specular and di:use re6ectivity, in general, and for the di:use integrated re6ectivity analysis scheme, in particular, will be discussed in the following section. 6. Application and limitations of GIXR technique In specular X-ray re6ectivity intensity of re6ected X-ray beam is measured as a function of grazing incidence angle with the grazing incoming and outgoing angles equal (specular condition). However, as we measure intensity of re6ected X-ray beam, not its amplitude and phase, the determination of the electron density pro&le (EDP) from the re6ectivity pro&le becomes non-trivial. Normally, one assumes an a priori distribution of EDP across the thin &lm and calculates specular re6ectivity pro&le by approximating this EDP by a series of slabs having constant electron densities using a recursive technique starting at the substrate–&lm interface

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

31

[98]. The calculated re6ectivity pro&le is then &tted to the experimental data by varying some of the parameters such as the electron density of each slab, thickness of the slabs and roughness of interfaces [18,99]. This conventionally used technique works well for systems in which actual EDP is close to the a priori assumption of EDP with which the non-linear &tting process is started and only a few parameters are involved in &tting. Due to the recursive non-linear relationship between real space parameters, e.g. thickness and electron density of each slab and re6ectivity pro&le, determination of parameters by &tting becomes problematic when initial guess of real space parameters are far away from the actual solution. 6.1. Extraction of small density variations in thin 6lms: a scheme based on distorted wave born approximation The method presented here can extract, most of the time, unknown EDP from the specular X-ray re6ectivity data without any a priori assumption. Moreover, this method can detect small variations in EDP in the resultant re6ectivity pro&le. The sensitivity of the method to density variations makes the specular X-ray re6ectivity technique more powerful. The method is based on distorted wave born approximation formalism [43,47,117] and requires one re6ectivity data taken either with laboratory X-ray sources or with a synchrotron source. Nevertheless, it should be mentioned here that, unlike the anomalous X-ray re6ectivity method [47], this scheme involves a &tting process and, as a result, can sometimes produce unphysical EDP. To avoid these unphysical solutions, we calculated all re6ectivity pro&les, &nally, with recursive technique [98] using the obtained EDP and compared with the data. We demonstrate this method with the help of simulated data and measured X-ray re6ectivity data of polystyrene (PS) &lm deposited on silicon single crystal. In this scheme, we treat the EDP of a thin &lm as '0 + T'(z), where '0 is the average electron density over the total thickness of the &lm and T'(z) is the variation of electron density as a function of the &lm depth z (the &lm surface is taken as z = 0, with z positive into the substrate) [47,117]. We are interested in &nding T'(z). In our method, we have considered the &lm to be composed of a number of thin slices or boxes of equal thickness d with 'i (given by '0 +T'i ) as the electron density of the ith box. We can then write T'(qfz ) as ∞ T'(qfz ) = T'(z) exp[iqfz ] d z −∞

= T'1

0

d

exp[iqfz ] d z + · · · T'N

Nd

(N −1)d

exp[iqfz ] d z :

(79)

Then, by collecting terms after integration, we get    j=N i  T'(qfz ) = (T'j − T'j−1 ) exp(iqfz (j − 1)d) + T'1 − T'N exp(iqfz Nd) ; qfz j=2

(80) where N is the total number of boxes used to represent the total &lm of thickness D = Nd. In the above expression, qfz [ = 2(kz2 − kc2 )1=2 ] is the z-component of momentum transfer vector in the &lm, kc being the critical wavevector for the average &lm of electron density '0 .

32

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

We have used the above expression of T'(qfz ) to calculate the specular re6ectivity, given in the DWBA [47,117] by (refer Appendix B) 2 4r 0 2 2 ? R(qz ) = ir0 (qz ) + [a (qz )T'(qfz ) + b (qz )T' (qfz )] ; (81) qz where r0 (qz ) is the specular re6ectance coeOcient of the &lm with average electron density (AED) '0 and has already been de&ned in Eq. (34). a(qz ) and b(qz ) are the coeOcients for the transmitted and re6ected amplitudes for the average &lm having electron density, '0 and are given by 1 + r12 a(qz ) = (82) 1 + r12 r23 and b(qz ) = a(qz )r23 :

(83)

The quantities rij=1; 2; 3 have been appropriately de&ned in Eqs. (30) and (33). By selecting appropriate number of slices and '0 of the &lm, we &t Eq. (81), after convolution with a Gaussian resolution function, with data keeping T'i as the &t parameters. We have veri&ed this analysis scheme by generating re6ectivity curves for various types of &lms with known EDP using Parratt formalism [98] and then &tting these curves with the expression given in Eq. (81) to get back the EDP ('0 + T'i ) of these &lms. The present method works well provided total &lm thickness is known and the box size used in the method is smaller than any features in the &lm. These two informations are normally known from the growth process but total &lm thickness, in any case, can be determined independently, most of the time, from the total 6lm oscillation or Kiessig fringes in the data. We also observed, in certain simple cases, that if the &lm thickness is overestimated the &tted pro&le generates substrate density pro&le at the proper place. To demonstrate this simulation study, we consider here &lm having two layers with thicknesses N and AED of 1.174 and 0:926 electrons= A N 3 (e = A N 3 ), respectively. The electron density 40 and 60 A 3 N . This system represents an Al0:5 Ga0:5 As and AlAs of the substrate was taken to be 1:368 e= A bilayer &lm on GaAs substrate. For &tting the re6ectivity curve generated using Parratt technique, N each and AED ('0 ) equal to 1:0254 e= A N 3 were chosen for the ten slices of thickness 10 A N above 100 A &lm. No a priori distribution was assumed in EDP and the starting estimate of all T'i were taken as zero. In Fig. 13(c), we show the obtained EDP of the &lm from the present scheme. This EDP matches well with the actual pro&le but we notice a small 6uctuation (6 0:4% around the actual value). It is interesting to note from Figs. 13(a) and (b) that the re6ectance amplitude and phases, plotted as a function of qz , generated from both these EDPs using Parratt formalism are identical. This result indicates the error limit in the EDP obtained using this technique. In order to estimate the minimum detectable contrast in EDP by X-ray re6ectivity, we have generated re6ectivity data using a structure resembling an LB &lm. The length of the head N with AED of 0:55 and 0:5 e= A N and the tail 24 A N 3 , respectively. The group was taken to be 6 A re6ectivity &t for three and half bilayers of this LB &lm is shown in Fig. 14(b) and the EDP is shown in the inset. For comparison, we have included the re6ectivity curve of a single uniform

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

33

0

10

0.55

Electron density

-1

10

-2

10

-3

10

0.54 0.53 0.52 0.51 0.50

Reflectivity

0.49 0

-4

10

50

100

150

200

z (Å)

-5

10

-6

10

-7

10

a

-8

10

-9

10

b

-10

10

0.0

0.1

0.2

0.3

0.4

0.5

qz (Å-1)

Fig. 13. Re6ectance (a) amplitude and (b) phase of a (c) model bilayer system (Al0:5 Ga0:5 As=AlAs) is shown. Line in (c) shows the actual system and ◦ represents the boxes obtained by analysis following the DWBA based scheme presented. In (a) [(b)] ◦ represents re6ectance amplitude [phase] of actual system and line represents those of the obtained pro&le. Fig. 14. EDP of a model LB &lm is shown by line in the inset; ◦ in inset represents the boxes obtained by the analysis scheme presented. X-ray re6ectivity curves of (a) uniform thin &lm having electron density equal to the average of EDP in inset and (b) thin &lm having the EDP in inset are shown. The re6ectivity pro&les have been shifted vertically for clarity. ◦ represents re6ectivity of model systems and line represents re6ectivity calculated from the obtained EDP.

&lm with the same thickness as the LB &lm and with electron density equal to the average of this EDP. It is seen that though the re6ectivity curves are almost same yet the EDP of the &lm comes out correctly from the present scheme. It is diOcult to quantify this minimum detection limit exactly because it is dependent on the EDP under study. Nevertheless, we &nd in this N 3 (i.e. 4% higher than particular problem that presence of head group having AED of 0:52 e= A tail region) can be detected using the present analysis scheme. It became obvious after few such simulation studies that this scheme can detect small variations in EDP of thin &lms from the specular re6ectivity data. We used this technique to perform analysis of specular re6ectivity data of PS &lm (mol: wt: = 186 kDa) deposited using spin coating technique. The re6ectivity data were collected in our laboratory with step size of (in ) ◦ of 0:003 .

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J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84 0 .4 0

10 10

R e fle c tiv ity

10 10 10 10 10 10 10

2

E le c tro n d e n s ity

10

1

0

-1

0 .3 8 0 .3 6 0 .3 4 0 .3 2 0 .3 0 0 .2 8 0 .2 6 0

200

400

600

800

z (Å )

-2

-3

-4

-5

-6

-7

0 .0 0

0 .0 5

0 .1 0

0 .1 5

0 .2 0

0 .2 5

0 .3 0

0 .35

-1

q z (Å )

Fig. 15. Measured X-ray re6ectivity curve of polystyrene &lm is shown (in ◦). Lower curve (in line) is calculated assuming an uniform thin &lm having constant electron density (shown by line in the inset). Upper curve (in line) is calculated using EDP shown by ◦ in the inset, obtained from the DWBA based analysis scheme.

In Fig. 15 we present re6ectivity data of this polymer &lm. The oscillations due to its total N −1 and we could collect data up to seven thickness continues even up to a qz value of 0:3 A orders of magnitude. This data was &rst analysed using Parratt formalism assuming a uniform N 3 . The &lm thickness was found to be 756:4 A; N the polymer–air and &lm of AED 0:34 e= A N polymer–silicon interface roughnesses were found to be 3:2 and 3:56 A, respectively. We notice that the calculated curve (lower curve in solid line in Fig. 15) deviates from the experimental data. To understand the reason for this deviation, we have used the new scheme with 36 boxes N each to model the polymer &lm. The obtained EDP is shown in the inset of Fig. 15. of 21 A We have used this EDP to calculate the re6ectivity using Parratt formalism which is shown along with experimental data in the upper curve of Fig. 15. These calculated data are now in N −1 good agreement with the measured data; even the modulation envelope around qz = 0:15 A is correctly represented. It is interesting to note that we obtain small peaks (6 –7% above the baseline) in the EDP of the PS &lm. The position of these peaks (obtained by Gaussian &tting) N respectively. It can be noted here that the average separation of are 165:2, 354:6 and 553:2 A, N is close to twice the radius of gyration (Rg ) of PS (122 A) N [117,118]. these peaks (∼ 194 A) Polymers have a tendency to form nearly spherical coils [3] inside a solvent liquid, which is

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used to deposit thin polymer &lms by spin coating. It has been shown that polystyrene in coiled state exhibit Lennard–Jones (L–J) type potential [119]. From the above observations, we assume that the PS &lm is formed mainly by coiled polymer balls, which obey L–J type potential. As N compared to the size of the the roughness of the two con&ning interfaces is small (∼ 3 A) N these balls have formed layers. As a result, the EDP is showing polymer balls (Rg = 122 A), small peaks near the centre of the balls as a function of depth. It has been demonstrated [120] recently by using this analysis scheme that, due to con&nement, layering of simple liquid atoms also occur as a function of depth from the surface provided spread in the surface (roughness) is small as compared to the size of the species forming layers. Layering is also observed on the surface of several liquid metals [121–124]. We also note that near silicon–polymer interface, there as a reduction of electron density. In earlier X-ray scattering studies [15,125], it was shown that although thick polymer &lms behave like liquid &lms, polymer &lms undergo dewetting transition and microdroplets are formed, when the thickness of the &lm becomes less than that of the radius of gyration of the polymer. If we assume that part of the polymer chain is grafted to the substrate [126] and the rest is in coiled con&guration, we can argue that near the silicon–polymer interface, there will be polymer mushroom formation [127] having lower electron density. 6.2. Other schemes The scheme presented above, although model independent, does not guarantee a unique solution since phase information still cannot be extracted from the re6ectivity data. Several schemes have been put forward to circumvent the phase problem, as was referred earlier, and here we will discuss two of these schemes. The &rst scheme is called the reference layer method developed over the years and applied to neutron re6ectivity measurements [128,129]. The scheme is useful for extraction of the complex re6ectance and phase of the scattering amplitude from an unknown system by adding to the system a ferromagnetic layer with known scattering length density. Then by using an external magnetic &eld three independent neutron re6ectivity measurements, can be made with known scattering length densities (of the ferromagnetic layer). From the three re6ectivity measurements, it is possible to extract the complex re6ectance for the unknown system exactly. The three independent measurements can be realised by magnetising the ferromagnetic layer in the direction parallel to the layer and using polarised neutrons parallel and perpendicular to the direction of magnetisation and by demagnetising the ferromagnetic layer by aligning the magnetic moments perpendicular to the layer thickness so that the e:ective magnetic potential in that layer is zero. Realising the third measurement has been found to be diOcult to execute experimentally and the original schemes have been further simpli&ed so that it is now possible to get the phase information even from two measurements only provided the physical solution, out of the two possible solutions obtained from the two independent measurements, can be identi&ed [130]. In addition, it has been shown that this scheme is valid even in the multiple scattering regime at low wavevectors, where Born approximation breaks down. But the major drawback of this scheme is to ensure that the insertion of the known reference layer does not a:ect the properties of the unknown system which is being investigated.

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The second scheme has its origins in the isomorphic replacement method used commonly in crystallography to circumvent the phase problem. This method is based on Eq. (81) and requires two independent X-ray re6ectivity measurements at energies close to and away from the absorption edge of the substrate on which a &lm has been deposited [47]. The method uses the fact that the re6ectivity of a thin &lm on a substrate involves an interference between the re6ectivity of the &lm and that of the substrate and hence contains phase information. By changing the amplitude and phase of the re6ection from the substrate in a known manner, by tuning the X-ray energy across an absorption edge of the substrate, one can modify r0 (qz ) in Eq. (81). Then by solving Eq. (81) for the two re6ectivities obtained from two independent measurements, one can obtain the quantity which remains unchanged in these two measurements—T'(qfz )—the real and imaginary parts of the Fourier transform of the density pro&le in the direction normal to the &lm. Fourier inversion then yields the unique density pro&le. The re6ectance coeOcient of the substrate and its variation across the edge are measured in a subsidiary experiment to obtain r0 (qz ) for these two energies. In order to obtain the substrate scattering parameters the re6ectivity pro&les of the bare substrate, made at two di:erent wavelengths, are &tted using the standard Parratt recursive scheme. As is obvious from the discussion, these experiments have to be necessarily performed at a synchrotron source, which in some sense is a disadvantage. Moreover, it may not always be possible to obtain a suitable absorption edge for a given substrate. Nevertheless, this technique is very useful and does not have the ambiguity and complicacies arising out of introduction of an additional layer into the systems being investigated. It may be mentioned here that this scheme can also be used in neutron re6ectivity measurements where the bare substrate re6ectivity may be changed by spin polarisation techniques. Recently, a useful comparative study of these di:erent schemes has been made [51]. An effort has been made to demonstrate the experimental feasibility of another scheme, which has been coined as phaseless inversion scheme, based on the possibility of extracting the scattering potential of a given system from re6ectivity measurements using only the amplitude information [131]. The phase information is not explicitly extracted from any other independent measurements but is assumed to be implicitly present in the amplitude information through the logarithmic dispersion relation. This scheme is again valid only in the Born approximation and also does not guarantee a unique solution. In the next section, we present another scheme based on Born approximation. The advantage of this scheme [116] is that it requires only a single measurement and does not involve any &tting. The scheme is useful for extracting small variations about a known scattering density which often prove to be quite useful for various applications. 6.3. Extraction of compositional pro6les of near-ideal multilayer thin 6lms: a scheme based on Born approximation Advent of excellent growth techniques [10,11,132] has enabled us to deposit multilayered thin &lms especially of epitaxial semiconductor systems almost according to the designed parameters, viz. thicknesses with atomic monolayer precision and uniform composition over each layer. Nevertheless, there still exist undesirable deviations in compositional and structural pro&le of the deposited &lm from the designed parameters and a non-destructive characterisation is highly desirable to improve the growth of these materials. A scheme [116] has been presented here to

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extract the actual compositional and structural pro&le in the growth direction of these multilayer thin &lms from X-ray scattering measurements. In the present scheme, the actual EDP is &rst calculated, with the desired pro&le as the initial guess, from di:use-subtracted true specular re6ectivity data. Information of correlated roughness [133,115] obtained from o:-specular di:use measurement is then used to extract the compositional depth pro&le. The merit of this ansatz is demonstrated using computer simulated and measured re6ectivity pro&les of quantum well N InP cap layer) structure N quantum well (with ∼ 500 A structures. Epitaxial InGaAs (∼ 100 A) grown on InP substrate by low pressure metal organic vapour phase epitaxy (MOVPE) was used here. We had presented earlier an expression for specular re6ectivity in the Born approximation (Eq. (57)). Here, we present an alternative expression for re6ectivity in Born approximation for a thin &lm [26]: 2 ∞ 1 R(qfz ) = RF (qfz ) d z ' (z) exp(iqfz z) : (84) '∞ −∞ Here, '∞ and RF (qz ) are the electron density and the Fresnel re6ectivity of the substrate, respectively; ' (z) = d '(z)= d z is the derivative pro&le. If we assume a model derivative pro&le, 'm (z), which is quite close to the actual derivative pro&le, 'e (z), that represents the experimentally observed re6ectivity data, Re (qfz ), then by taking simple ratio and by making a further assumption that these two derivative pro&les are so close that the phase factor generated in the Fourier transform (given by Eq. (84)) for both observed re6ectivity and model re6ectivity pro&le, Rm (qfz ), are identical, one can write: Re (qfz ) −1 'e (z) = F F['m (z)] : (85) Rm (qfz ) In the above expression, F and F−1 are forward and inverse Fourier transform pair. In practice, the above mentioned phase factors are not equal and an iterative procedure using Eq. (85) is required to obtain the 'e (z) that represents Re (qfz ), through intermediate derivative pro&les generated in each iteration. In analysing the re6ectivity data, an iterative procedure is started using Eq. (85), where we initially used model pro&le 'm (z) to generate the pro&le Rm (qfz ). The deviation of Rm (qfz ) z from Re (qfz ) was used in Eq. (85) to generate modi&ed pro&le, 'e (z) [ = 0 d z'e (z)]. Model re6ectivity pro&le, Rm (qfz ), for the next iteration is now calculated by setting 'm (z) equal to 'e (z) of the previous iteration with the assumption that obtained derivative pro&le 'e (z) is non-zero only in the interval (0; T ), where T is the total thickness of the &lm. One can use either Parratt formalism [98] or Eq. (85) for calculating Rm (qz ) and we found by simulation that in absence of absorption same result can be obtained by both the methods. Here, we have N −1 ) are quite high for these used Parratt formalism because the absorption coeOcients (∼ 10−6 A quantum well systems. The iteration scheme suggested here can be considered as a box-re&nement technique [134,135] and convergence of such technique has been rationalised by Crowther [136]. The obtained EDP within box (0; T ) from this iterative scheme can only be said to be consistent with the re6ectivity and may not be an unique solution. However, recently some work [131,137,138] has been done

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to resolve remaining ambiguities of the solution by using logarithmic dispersion relations and zeros of the re6ectance. We shall not use these techniques here because the starting guess EDP is assumed to be very close to the solution and the iterative ansatz of Eq. (85) presumably converges to a solution whose phase is, in some sense, closest to the phase of the starting re6ectance. The di:use scattering intensity as a function of wavevector Q = (qx ; qy ; qz ) can be written in Born approximation, after simplifying Eq. (71), as [43,115,116,133] qz I (qx ; qz ) = I0 R(qz ) d x[exp(qz2 C(x)) − 1] exp(−iqx x) : (86) 2k sin i In the above expression, we assumed [115,116] that all the interfaces are conformal and resolution out of the scattering plane is relaxed in such a way that integration over qy has been performed during data collection. The height–height correlation function for conformal self-aOne rough interfaces, de&ned earlier (Eq. (48)), was used here. Before analysing the experimental re6ectivity data of quantum well, we simulated similar re6ectivity data from known EDP with Parratt formalism and analysed these data using the present scheme and obtained back the density pro&les. These pro&les were then compared with the original EDP. These analyses reveal that the iterative procedure of the present scheme converges quite well even when initial guess is quite far away from the solution. This is particularly interesting here because features in re6ectivity data are not very prominent in these epitaxial systems with low electron density contrast at the interfaces. For all the simulation work N −1 which is the same as the experimental range here and hence (qz )max was taken as 0:28 A N slices. Although the re6ectivity is required in each iteration 'e (z) was obtained with 11:30 A for evaluation of Eq. (85), we also calculated the complex re6ectance to monitor the approach of phase factor towards the correct value, known in simulation studies. One such analysis with simulated re6ectivity pro&le (curve A, Fig. 16(a)) is shown along with the initial guess EDP and &nal EDP (Fig. 16(b)), having a slope of electron density in the quantum well. It is interesting to note that we could obtain back the EDP using the above scheme almost exactly, including the rounding of the edges due to &nite roughness, given by an error function. The presence of 6uctuations of high spatial frequency are due to the &nite cuto: in the wave number range, as has also been observed in other inversion schemes [44,52]. The best &t of the experimental re6ectivity data (curve B) of the quantum well system using above scheme is shown in Fig. 16(a) along with the corresponding EDP (Fig. 16(b)). The same pro&le was obtained with various initial guess pro&les and it was con&rmed that presence of oscillatory electron density near the surface is essential to represent the hump in the re6ectivity N −1 . data near qz = 0:175 A For clarity, we have normalised all these re6ectivity data with respect to the Fresnel re6ectivity and presented in Fig. 17(b). In this &gure, we have also shown (Fig. 17(a)) the real and imaginary parts of re6ectance, for &nal and initial guess EDP, to indicate the nature of phase as a function of wavevector. It is instructive to note that the correct phase evolves iteratively; the phase for the Fresnel re6ectivity of the substrate is also shown for comparison. Analysis of the transverse di:use data at three di:erent qz values using Eq. (86) (refer N 10 000 A N and 0.45, respectively. The same set Fig. 18) yields values of , 6 and as 5:5 A, of parameters were used to calculate self-consistently the longitudinal di:use scattering pro&le (Fig. 16(a), curve C). This pro&le follows the specular re6ectivity pro&le (curve B) closely,

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Fig. 16. (a) Specular re6ectivity for model system (×10) (A), quantum well sample (B) and longitudinal o:-specular re6ectivity for quantum well sample (C). Parratt simulated and experimental data are shown by ◦ and &tted curves by solid line. Fitted curve for model system has been shifted further (×2) as it exactly matches with the simulated data. (b) Electron N 3 ) as a function of depth, z, for various systems are shown. The actual and obtained density pro&le (EDP) (', in e= A N 3 for clarity) and &tted EDP for the experimental system are shown by solid line, EDP for the model (shifted by 0:02 e= A &lled circles and open circles, respectively. The initial EDP used for both simulated and experimental data is shown by N respecdotted line. The thickness of the cap layer and quantum well for the sample was found to be 528 and 112 A, tively.

indicating conformality. It is known that EDP obtained from X-ray specular re6ectivity study is actually convolution of compositional and interfacial roughness pro&les. As the roughness here is conformal, one can obtain compositional pro&le by deconvoluting the ' (z) with the N corresponding to the interfacial roughness. This deconvolution can Gaussian having = 5:5 A, be performed in Fourier space by utilising the fact that the Fourier transform of the convolution of two functions is the product of the Fourier transforms of the functions. Gaussian like (with variance of t ) derivative pro&le of EDP is shown (Fig. 17(c)) along with the roughness N obtained from di:use scattering analysis, at both the interfaces of the Gaussian ( = 5:5 A), N for quantum well. The values of t , found by &tting Gaussian functions, were 12 and 9 A the quantum well interfaces with the cap layer and substrate, respectively. This indicates that

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J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84 1 .2 x 1 0 -3

I m ( r ) ( Å- 2 )

(a) 8 .0 x 1 0 -4

q

2 z

4 .0 x 1 0 -4

10-1

0 .0

- 4 .0 x 1 0 -4 - 5 .0 x 1 0 -4

0 .0

q

5 .0 x 1 0 -4 2 z

1 .0 x 1 0 -3

10-2

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q = 0.085 Å-1

R e ( r ) ( Å -2 )

z

1 .0 x 1 0 0

(b)

10-3

Intensity

8 .0 x 1 0 -1

F

R /R

6 .0 x 1 0 -1 4 .0 x 1 0 -1 2 .0 x 1 0 -1

10-4 q = 0.103 Å-1 z

10-5

0 .0 0 .0 0

0 .0 5

0 .1 0

q

z

0 .1 5

0 .2 0

10-6

0 .0 0 4

dρ/ d z

0 .2 5

( Å -1 )

q = 0.127 Å-1

(c)

z

10-7

0 .0 0 2 0 .0 0 0 -0 .0 0 2

10-8 -0.002

-0 .0 0 4 500

550

600

z (Å)

650

700

-0.001

0.000

0.001

0.002

q (Å-1) x

Fig. 17. (a) Phase variation as a function of qz is shown by plotting the real and imaginary part of re6ectance (r). The re6ectance corresponding to Fresnel, initial and &nal pro&le are shown by dashed, dotted and solid lines, respectively. (b) Specular re6ectivity normalised by Fresnel re6ectivity as a function of qz for experimental data (◦), initial pro&le N (- - - -) corresponding (- - - -) and &nal pro&le (—). (c) Derivative of &nal EDP (—) and Gaussian function with = 5:5 A to the conformal RMS interfacial roughness in the quantum well region. Fig. 18. Transverse di:use scattering intensity (◦) as a function of qx for three di:erent values of qz for quantum well sample along with &t (—).

the interfacial pro&le is dominated by interdi:usion and substrate–quantum well interface is sharper than the quantum well–cap layer interface. It is also interesting to note that we obtained high in-plane correlation length for this epitaxial system having predominant interdi:usion, as observed earlier [139,140]. Deconvoluted compositional pro&le is not presented here because the obtained pro&le is not much di:erent from the EDP. It should be mentioned here that the procedure presented here for specular data analysis can be used with suitable modi&cation for several other systems as well. This has been demonstrated recently for analysis of X-ray specular re6ectivity data for thin liquid hexane &lms on silicon wafers [141] using a similar scheme.

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6.4. Data analysis The Born approximation formalism developed above is strictly valid in the regime of weak scattering. For the case of GIXR, it breaks down near the region of total external re6ection where multiple scattering becomes strong. The DWBA is a slight improvement on the Born approximation in that it partially takes into account the multiple scattering e:ects near the critical angle in terms of refraction of the incident beam but is still not valid below this angle. In our analysis of specular re6ectivity data, we have replaced the calculated re6ectivity in the Born approximation by the corresponding re6ectivity in the more exact recursive method [98]. This substitution is intuitive and no rigorous proof of the validity of this substitution exists. As discussed above, for systems in which the di:use scattering is weak compared to the specular re6ectivity separate analysis of these two scattering components can be performed. Even when di:use scattering is weak, one has to be careful in actual interpretation of the extracted parameters from analysis of experimental data. The essential steps in analysis of specular re6ectivity data are (a) Correction of raw data for footprint e:ect and normalisation by maximum counts to obtain the re6ectivity data. Fig. 19 shows the e:ect of footprint correction on the raw data.

Fig. 19. Footprint correction and raw data treatment procedure. The specular re6ectivity pro&le is obtained from raw intensity pro&le after footprint correction and normalisation. The three transverse di:use scattering pro&les are normalised to the re6ectivity at the respective qz values along the specular ridge. Inset shows the region where footprint correction (- - - -) is e:ective on raw data.

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(b) Preliminary analysis of transverse and longitudinal di:use data to understand the nature of di:use scattering. If di:use scattering appears to be strong, especially around the specular position, then the specular re6ectivity has to be analysed as total intensity or re6ectivity using Eq. (61) or (73). On the other hand, if di:use scattering is weak, then a di:use background has to be subtracted from the specular data, i.e. the di:use scattering at the specular position has to be subtracted from the experimental specular data to obtain true specular re6ectivity and di:use scattering is analysed using Eq. (72) or (74). An idea about the di:use scattering background can be obtained from the nature of the transverse scan. For instance, if the scan typically looks like the 6at pro&le in Fig. 12(b), then with a fair deal of accuracy it can be assumed that the di:use intensity at the specular position is the same as that slightly away from it. By performing a longitudinal di:use scan at a suitable qx o:set, it is possible to obtain the variation of di:use intensity with qz —this pro&le can then be subtracted from the specular re6ectivity data. This can be assumed to be true even for the pro&les in Fig. 12(a) also although with lesser accuracy but this formalism is certainly not valid for the pro&les in Fig. 12(c). (c) Calculation of specular re6ectivity using the Parratt recursive scheme [98] or using schemes described in Section 6.3. (d) The next step is to 6t the calculated pro&le to the corrected specular data using a Levenberg–Marquardt non-linear least-squares &tting method [142]. It should be mentioned that the &tting procedure does not guarantee the uniqueness of the solution—this is an open question in the inverse scattering problem [143–146] as discussed in the previous sections. In case of analysing di:use scattering data, the data have to be &rst normalised with respect to the specular re6ectivity pro&le (refer Fig. 19). Then depending on the nature of the transverse scan line pro&le either a total intensity analysis or simply di:use intensity calculation is matched to the data. In the former case, the integral in Eq. (61) or (73) has to be evaluated numerically for self-aOne correlation, except for = 0:5 and 1.0 for which analytic solutions exist. In the latter, the integral in Eq. (72) or (74) has to be evaluated numerically for all values of . The above discussion is valid for separate analysis of specular (or total) and di:use re6ectivity. For analysis of di:use integrated re6ectivity, the following steps have to be taken: (1) Analyse the transverse di:use scattering pro&les to assess whether the measured specular re6ectivity data is already e:ectively integrated over such di:use pro&les during specular measurement, the specular measurements being assumed to be performed with detector slits of much wider aperture in the scattering plane as compared to the di:use measurements. If the specular measurements are automatically di:use integrated, then standard analysis procedure may be followed, as explained earlier. If not, then: (2) The quantities FD and FW have to be evaluated either analytically, where possible, or numerically. These quantities have to be calculated from the measured transverse di:use scattering pro&les which have to be taken at several points along the specular ridge. The ratio, RS (qz ), so obtained for each value of qz has to be interpolated over the range of the measured specular re6ectivity pro&le, I (0; qz )=I0 vs. qz . (3) The measured re6ectivity pro&le has to be scaled with the interpolated RS (qz ) to obtain the di:use integrated re6ectivity pro&le, ID vs. qz . (4) Finally, re6ectivity calculated using standard schemes, as explained earlier, is &tted to the di:use integrated re6ectivity data.

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7. Growth and morphology of surfaces and interfaces In nature, one comes across surfaces and interfaces of diverse physical systems with di:erent types of morphology. In general, the morphology is found to depend on the length scale of observation and one can &nd similar morphology in apparently disparate systems, at di:erent length scales. The mathematical formalism for the description of surface morphology has been developed [43] over the past few years using the concept of fractals. The symmetry inherent in the underlying similarity of the morphology of di:erent physical systems has led to the classi&cation of surface morphology into several universality classes based on their underlying symmetry [54,55]. So far we have talked about statics, that is surfaces after they are formed. But surfaces can also be studied during deposition of materials as in thin &lms or due to erosion or etching. In either case, it is important to study the dynamics of evolution of surface morphology. From the technological point of view, the ever-increasing demand for new compact devices has led to an explosion in research on thin &lms. In most cases, the physical properties of these &lms deviate from their expected behaviour due to the roughness of these &lms. The roughness in these &lms is inherent due to the non-equilibrium, stochastic, nature of the growth processes. It is evident that a quantitative understanding of the e:ect of growth mechanism on the surface morphology is essential to understand and control the formation of surfaces. The growth of surface morphology is in6uenced by a large number of factors and, often, it is diOcult to distinguish the e:ect of all these factors on the growth process. Nevertheless, in most cases, it is possible to construct a model for surface growth which contains the basic physical aspects of the growth process. There are essentially two theoretical approaches to the problem of understanding the formation of surfaces. One method is to perform a computer simulation of the dynamics of the surface growth based on a discrete model for the evolution of surface morphology. The predictions of the model are the growth and roughness exponents, of the system, % and , respectively. These exponents are a measure of interfacial width of the system and can be de&ned in terms of the following scaling relations as ∼ r as t → ∞, where r is the measured length scale, and ∼ t % as t → 0, with t being the measured time. These can then be compared to the exponents obtained from relevant experiments to verify the accuracy of the growth models. The other convenient method for studying the dynamics and morphology of surfaces is in terms of the continuum equations of growth [54,55]. These are stochastic di:erential equations which describe the asymptotic behaviour of surfaces unlike the case of computer simulations where the model encompasses microscopic details. In principle, it should be possible to construct the growth equation for a given system if the corresponding discrete model is known for the system. Alternatively, it is possible to arrive at the growth equations using the symmetry properties of the system—the symmetries of the system must be re6ected in the growth equation for the system. In case of simple systems, the equations can usually be solved exactly. But in many cases, the equations cannot be solved exactly in which case one can either use scaling concepts to extract the exponents of the system or use Renormalisation group technique for a rigorous solution. The continuum equations have a general form, 9h =F +N ; (87) 9t

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where F is called the deterministic part, N the stochastic part or noise, h the height above a mean reference surface. The explicit form of F is determined by the symmetries of the relaxation mechanism and conservation laws obeyed by the growth process. The growth equations can be classi&ed into various universality classes according to the following properties of the system: (a) conservation law obeyed by the deterministic part, (b) whether the system is linear or non-linear and (c) the nature of noise in the system. 7.1. Some examples In the following, we will discuss two commonly used growth equations—the Edwards– Wilkinson (EW) equation [147] and the MBE growth equation [54] and also a growth equation which has recently been found to be valid for slow electrodeposition process [148] and which has some resemblance to the equation we have used to model the growth of LB &lms. All equations are similar in that they are (a) linear, (b) dynamics is conservative and (c) the noise is uncorrelated and non-conservative. Conservative dynamics implies that particle number is conserved in the system during the relaxation process after deposition of particles. The uncorrelated noise term usually arises from the randomness of the deposition process and is characterised by the property N (X; t) = 0, where the average implies several realisations of the noise both in space and time. This noise term is also non-conservative in the sense that it may not be zero for a particular realisation implying that particle number may not be conserved. There could also be a random but conservative noise, Nd , which is characterised by the property d X Nd (X; t) = 0 at every moment. The integral is made over the entire surface and the result implies conservation of particle number. Due to the di:erence in the deterministic term, decided by the nature of the respective growth processes they represent, the exponents obtained from the above two equations are di:erent. The EW equation is the simplest linear equation representing growth by random deposition and relaxation, driven only by the tendency for surface height minimisation. The EW equation has the form 9h = E∇ 2 h + N : 9t

(88)

Here, E is a coeOcient describing surface relaxation or the adsorption=desorption process governed by surface tension. It may be noted here that higher order linear terms may be included in the above equation but they are not relevant in deciding the exponents of the growth process which are essentially dependent on the asymptotic properties of the system. These terms may however become relevant in determining the short length scale morphology of the system. The exponents and % for EW equation are given by, = (2 − d)=2 and % = (2 − d)=4, respectively, where d is the dimensionality of the system. Thus in 1D = 0:5 and % = 0:25 while in 2D they are both 0. The physical mechanism of relaxation which represents MBE growth very well is surface di:usion of the deposited particles. One way of understanding the origin of this di:usion process is the tendency to minimise the surface chemical potential in MBE growth. This has to be contrasted with the tendency of minimisation of gravitational potential in the case of the growth

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45

process governed by the EW equation. The simplest equation of MBE growth, therefore, has the form 9h (89) = −F∇4 h + N : 9t It is possible that a conservative di:usive noise term could be present in addition to the nonconservative noise but this equation has been found to represent the MBE growth process quite well in the &rst approximation. With this background and introduction, we next discuss the growth mechanism of LB &lms. In the absence of non-linear terms determining the scaling process, the most general linear equation for interface evolution can be written as [54], 9h (90) = E∇2 h − F∇4 h + N : 9t Although Eqs. (88) and (89) have been found to be applicable for di:erent growth processes, the above equation was only recently shown to be valid for electrodeposition of copper &lms [148]. The interesting aspect of this equation, due to the presence of two competing deterministic terms, is the prediction of a crossover of exponents depending on length scale. Thus, a length scale 6(∼ F=E) can be de&ned below which the exponents of the growth process, satisfying Eq. (90), and %, are decided predominantly by di:usion (second term in Eq. (90)). Above this length scale, the exponents of the growth process are predicted to be decided predominantly by the adsorption=desorption process mediated by surface tension. The crossover behaviour was actually veri&ed from AFM measurements for the electrodeposited copper &lms [148]. With this background, we now discuss in detail the growth process and the possible equation governing this process for LB &lms. 7.2. Growth model of LB 6lms In Section 7.1, we had discussed the general aspects of the growth and morphology of LB &lms. But so far not much work has been done to obtain a quantitative understanding of the growth of LB &lms similar to what has been done for MBE growth or sputter deposition process. The LB deposition process is shown schematically in Fig. 20. As indicated earlier, the process involves repeated up-down movements of a vertical substrate from water to air [(a) → (b) → (c)] and vice versa [(c) → (b) → (a)] through this monolayer (refer Fig. 20(A)) at a &xed rate (3 mm= min here). A drying time is given between up and down cycle (10 min in our case). Molecules are transferred to the substrate at the air–water–substrate contact line, a 1D interface, and considerable molecular re-arrangement can take place during this transfer (position (b) in Fig. 20(A)). These transferred molecules form a 2D layer [53,82,149] on the substrate with previously transferred molecules, and this 2D layer settles through a desorption process during drying time and during the time when the &lm is immersed in water (position (c) and (a), respectively, in Fig. 20(A)). To explain the growth process of LB &lms, we have applied a general linear stochastic equation for the interface evolution of solid &lms, to the growth of these &lms: 9h (91) = E∇2 h − F∇4 h + Nd + N : 9t

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J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

Fig. 20. Schematic view of the deposition mechanism of multilayer LB &lms. The solid substrate on which the amphiphilic molecules from the spread monolayer &lm of these molecules on water gets transferred to form the multilayer structure as the substrate is moved through the monolayer, as indicated in steps (a), (b) and (c) in A. The direction of movement of the substrate is indicated by the ↑. At each instant during the motion of the substrate a 1D interface is formed between the substrate and the monolayer covered water surface when the amphiphilic molecules are transferred. After each cycle of immersion and emmersion of a portion of the substrate in the monolayer covered water a 2D layer of molecules is formed on it. After emmersion from the water the substrate is left in air for drainage of entrained water for a suitable period of time (called the drying time).

Here Nd is conservative di:usive noise arising due to surface di:usion while N is nonconservative deposition noise and E and F are coeOcients related to surface tension mediated adsorption=desorption and di:usion, respectively, as explained earlier. If Eq. (91) does represent the LB growth process correctly, then a crossover length scale, as found earlier for electrodeposition process satisfying Eq. (90); should also exist below which the growth of these &lms will be mainly dominated by di:usion process. In our growth model, we assumed further that this di:usion process is occurring predominantly at the 1D substrate–water interface (refer Fig. 20) and in this process the dominating noise is Nd . As a result, the second and third terms

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47

in Eq. (91) are the dominant terms in deciding the scaling behaviour of the system and , % can be written as (2 − d)=2 and (2 − d)=8 [54], respectively, with dimension d = 1 here. On the other hand, during 2D adsorption=desorption process N dominates because of loss of molecules so that in Eq. (91) the &rst and fourth terms dominate in deciding scaling behaviour of the system and , % becomes (2 − d)=2 and (2 − d)=4, respectively (here d = 2) [54]. Hence, a crossover ( = 1=2 → 0; % = 1=8 → 0) to a slow logarithmic growth of the EW [147] type should occur above 6, for any growth process which can be described by Eq. (91). Although on the basis of measured saturated roughness exponent , other 1D transfer processes [54] having same cannot be ruled out here, the basic nature of our growth model remains valid. This model has given us a uni&ed height di:erence correlation function, g(r), for the LB &lms consistent with the measured and 6. The correlation function, g(r), (above a lower cuto: of about molecular diameter) for conformal interfaces can be written as [29] g(r) = [h(0) − h(r)]2 Gr = 22 + B E + ln [1 − exp(−(r=6)2 )] : (92) 2 2 (refer Appendix C), is the Euler constant and G is a lower cuto: Here, 2 = 02 + int E −5 N −1 here) for logarithmic correlation, B = kB T= and is the interfawavevector (∼ 10 A cial tension [15,28] (refer Appendix C). X-ray scattering probes this correlation function and depending on the relative magnitude of the X-ray coherence length, < (≈ 10 m here), and correlation length, 6 one of the terms in this correlation function will dominate. If 6< the &rst term in Eq. (92) is dominant and hence the interface exhibits logarithmic correlation as was observed earlier [28]. In another extreme, for 6<, the second term in Eq. (92) dominates and g(r) is found to scale in a self-aOne manner without cuto: 6, as g(r)=Ar 2 . In the intermediate region (6 ¡ <) g(r) takes the standard form (Eq. (49)) and the value of 6 can be extracted from X-ray scattering measurements [27]. 7.3. X-ray scattering studies of LB 6lms In Section 7.2, we reviewed some results on the morphology of LB &lms. Here we will mainly discuss the X-ray scattering studies of the morphology of LB &lms. X-ray scattering studies have revealed that LB &lms have remarkably well ordered vertical structure independent of the condition of deposition. Not much attention was paid to the in-plane structure of these &lms. Using specular re6ectivity and di:use scattering, it was shown [31] that the interfaces in LB &lms are self-aOne fractal in nature. They investigated a system consisting of eight bilayers of cadmium tricosanoate and found that the vertical correlation was very good, indicating a presence of interfaces with a high degree of conformality. Hence, they could analyse their transverse di:use scattering data with only a single roughness exponent which was found to be 0.5. In a later work [27], it was shown that indeed the interfaces in LB &lms were self-aOne but the roughness exponents could vary. They performed specular re6ectivity and di:use scattering as well as grazing incidence di:raction on nine monolayer and 11 monolayer LB &lms of cadmium arachidate (CdA) deposited on two silicon substrates with widely di:erent roughnesses. The di:use scattering data were analysed using the scheme of Holy and Baumbach [114]. Inspite

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J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

of this di:erence in roughness, the two &lms were found to have nearly identical structures, with very large vertical correlation length. They concluded that this behaviour is only possible if the &rst deposited layer e:ectively damps the propagation of substrate roughness and subsequent layers replicate the &rst layer morphology. To verify the validity of our proposed growth model, discussed earlier in Section 7.2, we have performed X-ray scattering, with a rotating anode Cu K 1 source as described earlier, and AFM studies on LB &lms of CdA deposited on di:erent types of silicon and quartz substrates. The LB &lms were prepared using standard process described earlier. The LB &lms that we have studied were found to have either self-aOne interfacial correlation without cuto: or long range logarithmic correlation. As was mentioned earlier, for both these types of correlation function, separation of the scattered intensity into specular and di:use components is not possible and hence only total intensity can be calculated. In analysing the X-ray specular and di:use re6ectivity data of these LB &lms, we have used Eq. (62) along with Eqs. (64) – (66) for the &lms with self-aOne height–height correlation (with roughness exponent = 0:5) and Eq. (62) along with Eqs. (67) – (68) for the &lms with logarithmic correlation characteristic of capillary waves. In Figs. 20(b) and (c) the specular re6ectivity data and &t for the nine monolayer (ML) LB &lms on silicon and quartz, respectively, are shown. Also shown in the inset of the respective panels are the EDP obtained from the &t. The respective longitudinal di:use data for the two types of &lms, shown in the same &gure, follows the specular data closely con&rming that interfaces are conformal. For both the &lms, the total thickness and average bilayer spacing, N respectively, indicating stacking of untilted molecules in these came out to be 247:5 and 55 A, deposited &lms. Although the essential features of the specular data, for both the &lms, are obtained using the simple model (10 slices) the re6ectivity pro&les obtained using the detailed N (50 slices), matches with the data almost exactly. model EDP, with slices of thickness 5 A Thus, the average vertical structure seems to be nearly same for both types of &lms. Although the average vertical structure in these &lms are nearly identical the in-plane morphology is quite di:erent as is evident from Fig. 21 which shows the transverse di:use pro&les for &lms on silicon and quartz substrates. It may be noted here that similar di:erence in transverse di:use pro&les, and hence in in-plane morphology of &lms, was also observed for &lms deposited on two di:erent types of silicon substrates, di:ering in their miscut angles, also and these data have been presented later. For the &lms on silicon, the experimental di:use pro&les were best &t by Lorentzians, as given in Eq. (66), the only &t parameter being A which comes N The &lms on quartz substrate were best &t by Kummer function, as out to be 0:02 ± 0:002 A. N 2 , e: = 2:5 ± 0:1 A, N respectively. given in Eq. (68), the only &t parameters being B = 2:1 ± 0:1 A The di:erence in the scaling of the widths and the specular to di:use intensity ratio for the two pro&les FS and FL , as a function of qz , is clearly evident in inset of Fig. 21. We also investigated the structure of 1, 2, 3, 5, 7 ML &lms of CdA deposited under the same conditions as the earlier 9 ML &lms. Fig. 24(a) shows the specular re6ectivity data for the 1, 2, 3, 5, 7 ML &lms while Fig. 24(b) shows a few of the transverse di:use data for the same &lms. It is evident that although the characteristic Bragg peaks become sharper with increasing layer number indicating evolution of ordered bilayer structure along the direction of growth, the respective line shapes of the transverse di:use pro&les for the &lms on di:erent substrates are identical for all the &lms, with di:erent layer numbers, on the same substrate. This indicates

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49

100 S

10-3 10-2

10-4

L

R

q = 0.228 Å-1 z

10-4 10-6

10-4

10-2

10-5 10-6 100 S

10-3 10-4

10-2

q = 0.342 Å-1

R

z

10-4 10-6

L

10-4

10-2

Reflectivity

10-5 10-6 100

10-4 q = 0.456 Å-1 10-5

S

10-2

R

L

z

10-4 10-6

10-4

10-2

10-6 10-7 100 S

10-4

L

10-2

R

q = 0.569 Å-1 10-5

z

10-4 10-6

10-4

10-2

10-6 -0.004

-0.002

0.000

q

x

0.002

0.004

(Å-1)

Fig. 21. Transverse di:use scattering data at four LB multilayer Bragg peak positions for &lm on silicon and quartz. The qz values in each panel indicate the respective positions of these peaks along the specular ridge (qx = 0) around which these data were collected. The data are shown in symbols while the calculated pro&les are shown in solid lines. In the inset, the functions FS (qx ; qz ) and FL (qx ; qz ) are plotted against qx in log–log scale about the same qz positions as indicated in the respective panels alongside. Here L and S implies FS (qx ; qz ) and FL (qx ; qz ), respectively. Also shown is the transverse resolution function R used in our analysis, at the same qz positions.

that the in-plane morphology of the &lms are decided during deposition of the &rst monolayer itself and this is conformally replicated in subsequent layers. 7.3.1. AFM studies of LB 6lms As mentioned earlier AFM can provide local structural information of surfaces which cannot be obtained from X-ray scattering measurements alone. AFM has proved to be quite useful in obtaining the surface morphology of LB &lms and hence to provide crucial quantitative information regarding the growth of these &lms [29]. AFM (Autoprobe CP, Park Scienti&c) images of LB &lms, presented here, were collected in constant force contact mode using silicon nitride tip in ambient condition with a 100 m scanner. Images of the &lms having two di:erent

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J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

Fig. 22. AFM images of scan size 0:8 m × 0:8 m for the LB &lms on (a) silicon (b) quartz. Typical line pro&les drawn through each image are shown at the bottom of the &gure.

types of morphology are shown in Fig. 22. The rms roughness () obtained from the average of several scans of a particular scan size are plotted as a function of scan length for both the &lms in Fig. 23. It can be readily observed that the roughness of the &lm on silicon increases considerably with the scan size up to a certain cuto: length scale ∼ 15 m and after that it N the molecular length. This corresponds to the crossover in roughness gets saturated to ∼ 25 A exponent from 0.5 to ∼ 0:0, corresponding to slow logarithmic variation as predicted by our model. This cuto: length scale is the same as in-plane correlation length, 6, which is larger than the coherence length of our X-ray beam and hence cannot be detected by X-ray measurements. Below 6, the variation of of the &lm on silicon follow ∼ r with exponent ∼ 0:5, as obtained from linear &t (refer to Fig. 23) and con&rms our X-ray results. The variation of 2 for the &lm on quartz, also shown in Fig. 23, is slow and follows a logarithmic relation as observed in X-ray data. This case corresponds to the other extreme where the length scale of crossover between the two regimes with di:erent roughness exponents

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51

100

80 10

40

σ 2 (Å2)

σ (Å )

60

20

0

1 103

104

105

106

r (Å)

Fig. 23. Variation of roughness () with scan length for the LB &lms. log vs. log r for the &lm on silicon (◦). Solid line is the linear &t to the data below 6 ∼ 15 m. 2 vs. log r for the &lm on quartz ( ). Dashed line is the linear &t to the data.

is smaller than the limit of sensitivity of our AFM measurements and hence we observe only the asymptotic slow logarithmic variation. From the slope of the linear &t obtained in the semi-log plot of Fig. 23, one can calculate the surface tension, . The value of comes out to be ∼ 7 mN= m, which is quite small compared to the value of ∼ 63 mN= m estimated from the &t to the X-ray data. This di:erence arises probably due to the di:erent surface tensions of the amphiphilic tails at the top surface, as measured by AFM, and that of the metal ions in buried interfaces, which contribute predominantly to X-ray scattering. It is also to be noted from the images that both the &lms contain lot of defects of di:erent size and height. The typical size and height of the defects can be observed by drawing line pro&les (refer to Fig. 22). N wide and ∼ 55 A N deep) craters, could be seen in the &lm deposited on Large sized (∼ 700 A quartz. This may be a signature of 2D desorption process. On the other hand, strong 6uctuations around mean surface was observed for the &lm deposited on silicon. This may be a signature of domains but without well de&ned size. As a result, like previous attempts [150] even with better resolution we could not detect signature of monodispersed domains, which would have led to reduction of width of transverse pro&les as a function of qz . It may be mentioned here that although logarithmic correlation was obtained for LB &lms on quartz, recently we also found this type of correlation for &lms deposited on silicon substrates also. The di:erence between the two types of silicon substrates, giving rise to self-aOne and logarithmic correlation in LB &lms, is in the respective miscut angles. AFM was also used to obtain quantitative information about the pH dependent morphology of lead arachidate LB &lms in conjunction with X-ray re6ectivity and di:use scattering [88]. The height–height correlation function of the surface of the 11 ML lead arachidate LB &lms

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Counts/Sec (Normalised)

Reflectivity

10

10 8 10 6 10 4 10 2 10 0 10 -2 10 -4 10 -6 10 -8 10 -10 10 9 10 10 10 10 10 10

2 ML 1 ML Sub 5 ML 3 ML 7 ML

3ML 7ML

7

5

3

1

-1

-3

10 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

θ (rad) Fig. 24. Upper panel: Specular re6ectivity data for 1, 2, 3, 5, 7 ML CdA LB &lms on silicon. The re6ectivity for the substrate is also shown. Lower panel: Specular re6ectivity for the 2 and 7 ML LB &lms along with the respective transverse di:use data.

were evaluated and the correlation length and roughness exponents extracted directly. These values were compared with similar results obtained from independent X-ray di:use scattering measurements on the same samples. Although the scaling of the lateral correlation lengths as a function of pH, as obtained from AFM measurements, were similar to that obtained from X-ray measurements, the values of the correlation lengths were in general smaller than that obtained from X-ray measurements. It was pointed that this could be due to the fact that in general AFM and X-ray scattering methods are sensitive to two di:erent parts of the multilayer LB &lm. While AFM is only sensitive to the overall surface morphology of the LB &lm and hence to the tail region, X-ray scattering is sensitive mainly to the interfacial metal region. Since the surface of the &lms contain the tail region and is likely to be less ordered compared to the metal sheets, AFM measurements is likely to lead to lower correlation lengths as compared to the X-ray measurements.

8. Formation of nanostructures The &eld of nanomaterials and nanotechnology has emerged very strongly over the last decade. A very good introduction at a general level about the enormous possibilities that this

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53

&eld presents can be found in the classic talk, There’s Plenty of Room at the Bottom, of Richard Feynman delivered at California Institute of Technology way back in December 1959 [151]. Recently, several books and reviews have been written covering di:erent aspects of this interdisciplinary &eld [61,62,152]. The great interest in these materials essentially stems from the ability to tune the fundamental physical properties of solids ranging from phase transitions to electrical conductivity by varying the size of these nanoparticles. The length scale of interest in these synthetic nanostructures is typically from a few nm to a few hundred nm. In this range, it is possible to obtain wide variations of physical properties of a particular material without changing its chemical composition. Of the di:erent types of nanomaterials, semiconductor nanostructures are the most widely studied systems because of the tremendous potential for future device applications. These nanostructures can be classi&ed according to the extent of con&nement as quantum wells (con&nement in one dimension), wires (con&nement in two dimensions) and dots or clusters (con&nement in three dimensions). Here we will mainly discuss about semiconductor clusters in general and in particular their formation in LB &lms. There are two major e:ects which governs the size variation of physical properties of the nanoclusters—the nature of the surface of these clusters and the intrinsic properties of the interior. Many methods have been developed for the synthesis of these materials [152,153]. In most cases, the clusters prepared by these methods have poorly de&ned external surfaces and a relatively broad size distribution. Synthesis of monodisperse clusters with well de&ned surfaces remains a major goal in this &eld. Recently, the growth of semiconducting nanoparticles in LB &lms has attracted great attention [63– 66,154] primarily to exploit the highly ordered structure as a matrix to obtain better control over the size distribution, geometry and stabilization of the particles. The nanoparticles are formed by exposing preformed LB &lms, usually of the metal salt of long chain carboxylic acids, to reducing gases like hydrogen sulphide or hydrogen selenide. The most commonly formed particles are CdS=CdSe or PbS=PbSe. Several techniques have been used to characterise the nanostructures formed in the LB &lms. Even though a number of techniques have been used, descriptions of the mechanisms of particle growth in the &lm, or the processes which control the size of the particles formed in the &lm are not well developed. It has been predicted that controlled exposure of the reducing gas leads to formation of metal sulphides at the metal sites. This has been con&rmed by mass uptake measurements using quartz crystal microbalance [65] and XPS measurements [154]. But there is some controversy regarding the shape, size and geometry of the nanostructures formed estimated mainly from UV–Vis absorption and XRD data. The onset of absorption of light by semiconductor materials is accompanied by creation of a bound electron–hole pair called an exciton [59,60]. In bulk systems, these excitons are manifested in the form of excitonic levels slightly below the band gap. In nanostructures where the dimensions become comparable to the size of the excitonic wave functions, strong quantum con&nement e:ects come into play leading to shift in the e:ective band gaps of these materials. This can be detected from the shift in the absorption onset as compared to the bulk, in typical absorption spectra. Theoretical calculations have been performed to provide an estimate of the expected shift in band gap as a function of the dimension of con&nement in several nanostructures and hence absorption spectroscopy [28,63– 66] has become a common tool to infer about nanostructure formation and size estimation. In addition, XRD [155] also provides information about the nature of the nanostructures formed.

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8.1. Nanoparticle formation in LB 6lms Nanoparticles of CdS were prepared [64] by exposing LB &lms of cadmium arachidate, of di:erent thicknesses, to H2 S gas. A combination of transmission electron microscopy (TEM) and energy dispersive spectroscopy (EDS) con&rmed the formation of CdS nanoparticles. The absorption onset of the nanoparticles was blue-shifted with respect to that of bulk CdS by N which would span a bilayer of 0:4 eV indicating formation of particles with dimension of 50 A arachidic acid. The change in thickness of the &lms of 3, 5 and 7 layers after exposure to H2 S was estimated from ellipsometric measurements and the possibility of formation of platelets or discs of CdS instead of large spherical particles was indicated. The formation of PbS particles were studied [155] by exposure of H2 S gas to LB &lms of lead stearate using X-ray di:raction, IR and UV–Vis spectroscopy. The X-ray di:raction results indicated that the bilayer separation of the &lms after exposure was nearly the same as that of the unexposed lead stearate &lm. The IR measurements also suggested that the PbS particles formed remained con&ned with the cage of the carboxylic ion group of adjacent layer. From the UV–Vis absorption spectrum of the exposed &lms, it was suggested that PbS particles existed either in the form of 2D sheets or lines. The onset of the absorption edge was blue-shifted by 1:4 eV with respect to the bulk. Recent FTIR and X-ray di:raction measurements [156] by the same group on lead salts of amphiphilic oligomers (with variable carboxylic group to hydrocarbon chain ratio as compared to the 1:1 ratio in simple fatty acids) also indicated that PbS nanoparticles were formed without disturbing the layered structure of the unexposed &lms. The UV–Vis measurements not only con&rmed, as earlier, that the PbS nanoparticles were either in the form of 2D domains or lines but also showed that their size and hence the extent of blue-shift increases with the increase in the ratio between carboxylic group and hydrocarbon. On the other hand, in another study [154] of CdS formation in CdA LB &lms, it was observed a blue-shift in UV–Vis measurements which indicated typical particle sizes of 2–3 nm in diameter. From complimentary TEM measurements they found the particle size distribution to be polydisperse. The most common particle size observed in TEM was found to be close to that calculated from optical measurements. A combination of X-ray di:raction and UV–Vis absorption spectroscopy was used to study the formation of CdS in CdA LB &lms [63]. From the extent of the blue-shift in the absorption N in radius. The X-ray spectra, they estimated the size of the nanoparticles formed to be 30 A di:raction data of the exposed &lms seem to indicate the presence of two types of layered structures with di:erent bilayer thickness—one the same as that for the unexposed &lm while the other slightly smaller than this. They interpreted their results in terms of formation of spherical CdS nanoparticles within the domains having tilted arachidate molecules while leaving the layered structure in the remaining part of the &lm unaltered. We have studied [28] the formation of CdS nanostructures in LB &lms of CdA, each 9 ML, ◦ deposited on quartz substrates at a surface pressure and temperature of 30 mN= m and 10 C, respectively. The &lms were exposed to H2 S gas for duration of 10, 30 and 60 min, respectively. The absorption edge for the H2 S exposed &lms was found to be at around 455 ± 5 nm, as compared to that at 520 nm for bulk CdS, indicating the formation of nanocrystalline CdS in these &lms. From existing theoretical calculations, the size of the nanostructures formed could

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84 10 10 10 10

R e fle c tiv ity

10 10 10 10 10 10 10 10 10 10 10 10

55

7 6

ASP

5 4

10 M in

3

30 M in

2

60 M in

1 0

-1 -2 -3 -4 -5 -6 -7 -8

0 .0

0 .1

0 .2

0 .3

0 .4

0 .5

0 .6

0.7

-1

q z (Å )

Fig. 25. Specular re6ectivity data for 9 ML CdA LB &lm and &lms exposed to H2 S gas for 10, 30 and 60 mins. Systematic decay of the fourth Bragg peak (indicated by the arrow) with increased time of exposure to H2 S is clearly evident.

N The absorption spectra does not, however, indicate presence be estimated to be around 60 A. of any excitonic peaks indicating that the particles formed may be polydisperse. Nevertheless, we performed specular re6ectivity and di:use scattering measurements on two of these &lms, namely the as-prepared (ASP) and 60-min-H2 S exposed &lm (60M), to understand the role of LB interfaces in the formation of CdS nanostructures. Apart from developing basic understanding of these con&ned nanostructured materials, there is a motivation of preparing optical devices from these studies on CdS nanoparticles. It is possible to recover CdS nanostructure by washing away the organic part [65]. It is also possible to form tips of scanning tunneling microscopes from these LB-grown CdS structure [66] for investigating local physical properties of thin &lms and multilayer. However, it should be mentioned here that this &eld is just beginning to unfold and the applications mentioned here are only to indicate some of the many upcoming ideas. 8.2. GIXR studies of CdS nanostructures formed in LB 6lms X-ray specular re6ectivity and di:use scattering measurements were performed on these &lms in order to obtain a more detailed information about the nanostructures formed in the LB &lms. Fig. 25 shows the specular re6ectivity data for the as-prepared &lm as well as for the 10-, 30and 60-min-H2 S exposed &lms. A systematic decay of the fourth Bragg peak is clearly evident. No signi&cant change could be detected on further exposure and hence it can be assumed that the complete conversion of cadmium to CdS has taken place in this time. Hence, we have made detailed analysis of only the ASP and 60M &lms. In Figs. 26(a) and 27(a), the specular re6ectivity data along with the &t (solid line) for both the ASP and 60M &lms are shown. The re6ectivity pro&les &tted using simpli&ed models (dashed line) are also shown. In the simpli&ed model, the number of slices used to calculate the re6ectivity is minimised. The simpli&ed model in both the &lms indicate the increase in the absolute values of electron densities in each bilayer as a function of depth, possibly resulting from increased disorder and formation of patches due

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N ) pro&le for ASP Fig. 26. (a) Specular re6ectivity, (b) o:-specular longitudinal and (c) obtained electron density (e= A &lm are shown. The dashed line in (a) is obtained by &tting simple model shown in solid line in (c). The solid line in (a) is &t of a detailed model shown in open boxes in (c). The line joining these boxes are provided only as a guide to eye and does not provide roughness. The solid line in (b) is the calculated pro&le for longitudinal scattering as described in text. 3

N ) pro&le for the 60M Fig. 27. (a) Specular re6ectivity, (b) o:-specular scattering and (c) obtained electron density (e= A &lm are shown. Other details are as described in the caption of Fig. 25.

to incomplete &lm coverage [47], towards the top of the &lm [z = 0 in Fig. 26(c) and 27(c)]. The EDP obtained from simpli&ed and detailed model are shown in Fig. 26(c), where the metal sites are indicated by the higher electron density boxes. In detailed model of ASP &lm, N and one slice of thickness 2:5 A, N near the substrate were used. 49 slices of thicknesses 5 A N It is interesting to note For the 60M &lm, we used 36 slices each having a thickness of 7 A. that the simple model for both unexposed and exposed &lms generates the essential features the re6ectivity pro&les while the detailed model takes care of &ner details by variation of electron N which is a quarter of the bilayer densities around the simple model. Using a box size of 14 A, thickness, at the metal sites in the 60M &lm the simple model correctly brings out the most prominent feature in the measured re6ectivity pro&le, i.e. vanishing of the fourth Bragg peak [28]. The values of e: obtained from the &tting of the simpli&ed model for both the &lms are N for the air–&lm interface and 0:3 ± 0:2 A N for the buried interfaces and corresponding 2:1 ± 0:1 A

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N and 0:4 A. N The smearing of EDP across the interfaces occurs true roughness values are 2:4 A N as shown in Fig. 26(c), and due to interdi:usion of CdS nanoparticles con&ned within 14 A, capillary wave roughness. The total &lm thickness (bilayer thickness) for the 9 ML ASP and N and 252 A N (56 A), N respectively. By integrating the N (55 A) 60M &lm were found to be 247:5 A EDP over the total &lm thickness, for both the ASP (Fig. 26(c)) and 60M (Fig. 27(c)) &lms, we notice that the di:erence in the electrons per unit area, for each bilayer, between the ASP and N 2 [64,80] we &nd that there N 2 . By taking the area per molecule to be 18 A 60M &lms is ∼ 1 e= A are e:ectively 18 excess electrons in the 60M &lm as compared to the ASP &lm, per bilayer. This is equal to the total electrons in a H2 S molecule. In view of the above result and the fact that the molecular stacking is not disturbed, due to H2 S exposure, we conclude that the CdS nanostructures formed remain con&ned within the interfacial regions of the LB &lms around the N cylinder of cross-section 18 A N 2, metal sites and that there is still one metal atom within a 56 A as indicated earlier [84]. The three set of transverse scans (each at constant qz ) for the ASP and 60M &lms are shown in Fig. 28. For both the &lms the scans were performed at the &rst, second and third Bragg peaks.

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q x (Å ) Fig. 28. Transverse di:use scattering data for as prepared &lm (upper panel) and for 60-min-exposed &lm (lower panel) are shown along with the &tted curves. Dashed lines in upper panel indicate resolution function at &rst Bragg peak position. Filled circles in lower panel represents the transverse di:use scan of substrate at &rst Bragg peak position. All data of LB &lms are normalised to unity and intensity ratio of substrate and &lm data are shown in proper scale. Bragg peak numbers are indicated along with the respective curves.

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N −1 , respectively, For the ASP &lm, these scans were done at qz values of 0:108; 0:228; 0:340 A N −1 , respectively. The while for the 60M &lm these scans were done at 0.107, 0.219, 0.333 A 2 N obtained value for B was 2:1 ± 0:2 A for both ASP and 60M &lm, giving as 62:8 mN= m ◦ (at 27 C). It may be noted here that the Kummer function in Eq. (68) not only gives proper slope of the asymptotic di:use scattering tail but also the proper branching point from the Gaussian shaped specular peak. This in turn decides the ratio of specular to di:use intensity at a qx o:set. Typically, for a change in ; from 0.01 to 0.1 this ratio changes by an order of magnitude (Fig. 28), and as a result gives us the sensitivity in the obtained value of B. Also N −1 ; and the direct plotted are the transverse scan of the bare substrate taken at qz = 0:107 A N −1 . It may beam pro&le converted to an e:ective transverse resolution function at qz = 0:108 A N −1 is nearly be noted from Fig. 28 that the specular re6ectivity of the substrate at qz = 0:107 A 20 times smaller than that of the &lm, while the di:use intensity is higher than that from the &lm. This is due to the high value of roughness of the bare substrates which was found to be N Clearly, the e:ective roughness of the bare substrates gets modi&ed during the deposition 8:0 A. due to the attachment of the initial monolayers, otherwise we would not have observed clear total &lm oscillations in the re6ectivity data (refer Figs. 26 and 27). In Figs. 26 and 27 along with the specular data, we have shown the longitudinal di:use data taken with a constant angular o:set of 1:0 and 0:7 mrad for the ASP &lm (Fig. 26(b)) and for the 60M &lm (Fig. 27(b)), respectively. The parameters obtained from the &tting of transverse di:use and specular data were used to be calculate longitudinal di:use scattering pro&les shown in the respective &gures to check self-consistency of the analysis scheme. The excellent agreement of the calculated pro&le with the measured data demonstrates the assumed conformality of the interfaces above the critical length scale, rc . The estimated value of rc (=2=qu ) comes N This value in turn gives us an estimate of the compressibility, given out to be about 1000 A. 2 by 2d=rc [157,158], of the &lm as 3 × 105 N= m2 . Thus the highly incompressible and sharp interfaces in LB &lms con&nes CdS molecules, formed by chemical reaction, within a N at each interface, without disturbing the multilayer molecular stack region of thickness ∼14 A, in the &lm. Size of semiconductor nanoparticles are generally estimated from band structure using isolated cluster model. It will be very illustrative to calculate band structure of such conformal interN diameter, N layers of CdS, as we found that CdS nanoparticles of about 60 A faces having 14 A N estimated from isolated cluster calculation cannot exist in 56 A thick ordered bilayers of LB &lms [28].

9. Melting of LB 4lms 9.1. Earlier studies Melting of cadmium arachidate LB &lms was studied using IR spectroscopy and di:erential scanning calorimetry (DSC) [159]. The IR spectra clearly showed a decrease in the relative ◦ intensities of the bands in the CH stretching regions above 65 C although practically no change could be observed for both the symmetric and anti-symmetric COO− stretching bands. This was

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59

also supported by the DSC measurements which showed a weak kink around this temperature. ◦ Signi&cant changes in the COO− bands occurred at around 110 C which was also the melting point obtained from DSC. Hence it was concluded that the LB &lms melted via a two-step ◦ process of pretransitional disordering of the hydrocarbon tails at around 65 C followed by the ◦ disordering of the head group at the melting point at 110 C. The thermal evolution of the structure of one, three and ten bilayers of arachidic acid nested between two bilayers of a polymer was studied [73] using meridional X-ray di:raction over the ◦ temperature range 30 –80 C. Nesting was done in order to overcome the problem of irreversibility of the thermally induced changes due to the 6ow of the &lm material away from the area of deposition. Nesting of the arachidic acid molecules within the polymer material matrix provided a more constant area for the arachidic acid molecules within the accessible temperature range. A direct correlation was found between the average in-plane molecular chain density in an individual monolayer within an arachidic acid bilayer and the mean tilt angle for straight chains from the monolayer normal at lower temperatures with the tilt angle being higher for lower density layers. The addition of thermal energy resulted in a two-stage process consisting of a slowly evolving, continuous untilting of the initially tilted straight chains resulting in increase in total thickness followed by a continuous decrease in total thickness due to the chain melting via the formation of kinks and jogs. It was also found that the initially untilted straight chains in the higher density monolayers melt in a single stage process more abruptly or discontinuously over a much narrower temperature range. It was also observed that the magnitude of these thermally induced structural changes in an individual layer, depending on the in-plane molecular chain density, varies strongly with the number of layers, being much greater for one bilayer than for three or ten bilayers. The thermal behaviour of monolayers of cadmium arachidate, stearate and behenate was studied using electron di:raction [72]. A sharp decrease of the di:raction peak intensities occurs well below the main melting temperature which increases with the increase in chain length from ◦ ◦ ◦ 35 C for stearate to 55 C for arachidate to 75 C for behenate. The hexagonal symmetry of the di:raction patterns does not change up to the pretransition temperature. The angular and radial FWHM increases only slightly as compared to the pronounced decrease in peak intensity. It was concluded that the overall bond orientational and translational order due to the hexatic arrangement of the molecules is preserved and observed decrease in the di:raction peak intensities is caused by thermally induced random tilt orientational disorder or bending of the chains. Small angle X-ray di:raction was used to study the thermal behaviour of cadmium arachidate LB &lms [70]. It was concluded that before melting, the LB &lms underwent a three-step ◦ structural change consisting of thermal annealing up to 60 C, onset of pretransitional disorder◦ ◦ ing between 60 C and 80 C and a conformational change from trans to gauche state in the ◦ ◦ hydrocarbon chains between 80 C and 100 C. The melting process in trilayer cadmium arachidate LB &lms was studied using in situ AFM and IR spectroscopy [74]. They found that the initial 2D orthorhombic crystal phase melted ◦ ◦ to a 2D smectic phase at 91 C followed by a transition to a hexatic phase at 95 C. This result seems to be in agreement with the theoretical predictions of Halperin and Ostlund [160] regarding melting of 2D anisotropic crystals. Moreover, no evidence for co-existing phases, which is a signature of a continuous phase transition as is expected for 2D systems in general, could be found at any stage.

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Recently, a very interesting study [75] was performed on the dependence of thickness, and hence dimensionality, on the nature of melting of fatty acid LB &lms using quartz crystal microbalance and Fourier transform infrared dichroism. It was found that samples with number of layers, n, more than 12 melt as a normal 3D bulk solid, independent of the sample history, defect content, morphology and=or structure. Films with n ¡ 8 melt continuously as is expected for 2D systems. For &lms with 8 ¡ n ¡ 12 the nature of melting seems to be dependent on sample history. The as-deposited samples show a change of the order (from &rst to second) of the melting transition as n is reduced. For the annealed samples, the transition behaviour changes from continuous to discontinuous upon annealing. Clearly the role of defects, interfacial morphology and several other factors are important in deciding the nature of the melting transition in LB &lms. 9.2. X-ray re<ectivity studies of LB melting 9.2.1. Other studies Grazing incidence X-ray and neutron re6ectivity are ideal techniques to probe the lateral and vertical ordering in LB &lms. Recently, several studies have been made on melting of LB &lms using both these techniques. X-ray re6ectivity studies were made on behenic acid mono- and multilayer LB &lms of ◦ various layer numbers deposited on SiO2 substrates, both before and after annealing at 65 C for 20 –30 min [161]. Whereas in the unannealed &lms, three di:erent layer spacings were found ◦ ◦ ◦ corresponding to molecular tilt angles of 21 , 28 and 36 in the annealed &lms the molecular packing changed irreversibly to produce a uniform layer spacing corresponding to a molecular ◦ ◦ tilt of 36 . It may be noted that this temperature is well below the melting temperature (80 C) of behenic acid. For the thicker &lms, the reduced layer spacing after heating and the conservation of material leads to an increase of total number of layers and to islands of bilayer thickness on top of the annealed &lm. This indicates possible vertical motion of molecules. It may be noted that the &rst layer near the substrate is practically una:ected by the annealing process. Neutron and X-ray re6ectivity measurements were used to characterise 20 bilayer LB &lms of barium stearate [162]. To facilitate contrast for neutron measurements, molecules with deuterated aliphatic tails were used for the deposition of every odd-numbered bilayer. Structural changes of ◦ the &lm during temperature annealing at 66 C were studied. The results were interpreted in terms of interlayer di:usion and &lm evaporation. After 90 min of annealing, complete intermixing of deuterated and hydrogenated aliphatic chains were observed. In that time, 35% of the &lm had evaporated and was interpreted as melting of the acid part of the &lm. Using a combination of angle dispersive X-ray re6ectivity and di:use scattering measurements as well as energy dispersive measurements on lead stearate LB &lms, it was shown that melting of these &lms takes place predominantly by lateral melting of domains [163] and not by vertical interdi:usion as was concluded in the earlier studies mentioned above. 9.2.2. Our studies We have investigated melting of 9 ML CdA LB &lms [76,91]. The deposited &lms were subjected to thermal treatment in a sample cell shown schematically in Fig. 7. X-ray

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re6ectivity and di:use scattering data were collected using a 18 kW rotating anode generator described earlier. The melting experiments were performed on two types [29] of LB &lms, to be denoted here as A and B, di:ering in interfacial morphology. Film A has a logarithmic interfacial height–height correlation, characteristic of capillary waves while &lm B has self-aOne interfacial correlation with exponent = 0:5. For the &lm B, the X-ray data were collected in the sample cell in situ as the &lm was heated in &xed temperature steps. Since it was found ◦ that no signi&cant change occurs in the structure of the &lm between room temperature (20 C) ◦ ◦ and 65 C, the &lm was heated directly to 65 C. After raising the temperature, 10 min time was allowed for the sample to equilibrate to a set temperature before X-ray specular re6ectivity data were taken. The temperature was kept &xed while the specular re6ectivity data were taken and ◦ the temperature stability ±0:1 C was ensured by the PID controller. Subsequently, the sample ◦ ◦ ◦ temperature was changed in steps of 5 C up to 110 C. At all temperatures up to 100 C, the ◦ re6ectivity data were collected up to 4:45 () and was repeated once more to observe the changes, if any, taking place during the time (∼ 3:5 h) a particular re6ectivity scan was being performed. Signi&cant changes in the specular re6ectivity data, during the time of one such ◦ ◦ scan, were observed only above 95 C. After taking specular re6ectivity data at 100 C, the &lm was allowed to cool down to room temperature. Specular re6ectivity data of the &lm were taken again at this temperature. The &lm was then taken out of the cell and AFM data were taken in ambient condition. The &lm was put back again into the cell and specular re6ectivity data were collected again to ensure that no change has taken place in the &lm during AFM measurements. The o:-specular di:use data were also collected at this stage. After collecting the specular data ◦ at 110 C, the &lm was allowed to cool down to room temperature before collecting specular and o:-specular data. The &lm was then taken out of the cell again to collect AFM data. Since it was observed that the changes in the &lm structure as a function of temperature was irreversible and remained essentially the same after cooling down to room temperature, the high temperature data for &lm A were taken o: situ. The sample was kept at a &xed temperature for 1 h after the particular temperature was attained and then rapidly cooled to room temperature. X-ray specular and di:use re6ectivity and AFM data on the &lm A were then collected under ambient conditions. All AFM data were collected in contact mode with a silicon nitride cantilever and integrated tip using a 100 m piezo scanner. Scans were performed over several areas of the &lm and at each position several scans of di:erent sizes were taken. The data presented are representative of all these scans. 9.2.3. Data analysis based on EDS parameter It is known that specular re6ectivity is sensitive to both electron density gradation or smearing and roughness both of which results in the reduction of electron density contrast across an interface. It is not possible to estimate separately the value of these components from specular re6ectivity measurements alone. Since di:use scattering is sensitive to roughness alone, it is possible to extract the contribution of interdi:usion or electron density smearing (EDS) [76] in specular re6ectivity data by using the information about roughness from di:use scattering data analysis [43,116]. Fig. 29 shows one set of transverse di:use scattering data for &lms A and B ◦ in the untreated condition and after the &lms were heat treated to 100 C and cooled to room N −1 have been shown here. temperature. Only the pro&les taken about qz = 0:23 A

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All the pro&les for &lm B, before heat treatment, are best &t by Lorentzians (Eq. (66)), with the widths scaling as qz2 indicating that the interfaces in the &lm are self-aOne fractals with exponent = 0:5, as was found earlier [29,31]. The pro&les of the same &lm after being cooled ◦ from 100 C are also Lorentzians implying that the conformal interfacial roughness exponent ◦ does not change even up to 100 C. The only di:erence is in the value of the &t parameter N at room temperature and is 0:018 A N after being cooled from 100◦ C. The A which is 0:014 A ◦ di:use re6ectivity data at 110 C are very weak and hence no meaningful information about the conformal interfacial correlation could be obtained from these data. For the &lm A, the transverse di:use re6ectivity data are &t by Kummer function (Eq. (68)) indicating presence of N2 logarithmic correlation. The only &t parameter [28,29,76] being B which turns out to be 1:6 A ◦ for all the pro&les at room temperature and at 100 C again indicating that the correlation of the ◦ interfaces remain unchanged up to this temperature. The di:use data at 110 C seem to indicate that the conformal interfacial correlation remains unchanged although a de&nitive value of the parameter B could not be obtained. Figs. 30 and 31 show the experimental specular re6ectivity data (◦) at di:erent temperatures for &lms A and B, respectively. The re6ectivity pro&les (solid lines) that were &tted to these data were calculated from Eq. (76) using the scheme based on EDS parameter [76]. In Eq. (76), R(qz ) was calculated on the basis of a model EDP for the &lm with an explicit EDS parameter, H (HS for &lm B and HL for &lm A). This parameter which enters into the EDP as roughness is in addition to the conformal roughness (self-aOne for &lm B and logarithmic for &lm A) of the interfaces, which could be obtained from the transverse di:use scattering measurements either

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explicitly as in case of &lm A or implicitly as for &lm B. For &lm B, the parameters used for &tting [76] the re6ectivity pro&les were all the AEDs and HS at each of the head–tail interfaces ◦ and the top surface of the &lm and &lm–substrate interface for the 65 C data. The EDP obtained ◦ as a result of &tting of the 65 C data is shown in the inset of Fig. 31. The reduction in the overall density in the &lm from the &lm–substrate interface towards the air–&lm interface, as evident from the obtained EDP, has been attributed earlier to the propagation of defects [47,91] during deposition. In the present scheme of analysis, it was found that the reduction in density is not entirely due to this defect propagation but there is an additional contribution from the ◦ parameter H which quanti&es di:usion of molecules. All the other re6ectivity data up to 100 C were &tted using only HS as &t parameters keeping the EDP same as that obtained for the &lm ◦ at 65 C [76]. Any additional density variation at elevated temperatures has been attributed to di:usion of molecules which is incorporated in HS [76]. Fig. 32 shows the plot of the obtained HS values for the various interfaces in the &lm B ◦ (upper panel) as a function of temperature. Initially up to 85 C, HS reduces, as compared to ◦ that at 65 C. The initial decrease in HS is because of ordering due to annealing, whereby the &lm thermalises to low energy con&guration by reducing the frozen disorders which are present ◦ in the &lm due to the non-equilibrium nature of the growth process. Beyond 85 C, HS increases with temperature for all the interfaces. Similar analysis was done on &lm A with the di:erence that for this &lm the reference EDP was taken to be the one obtained for the &lm before heat treatment, as shown in the inset of Fig. 31. It may be noted that we could get a good &t to the data using a simpler model with lesser number of slices used to represent the EDP as compared to the &lm B. The &t parameters for the room temperature data were only the head

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Tem p( C )

Fig. 32. (A) HS vs. temperature for di:erent interfaces for the &lm A. (1) , (2) , (3) , (4) , (5) +, (6) ×. Here, (1) air–&lm interface while (6) &lm–substrate interface. The other interfaces are indicated by respective numbers. The y-axis has been compressed for clarity. The H value of one data point (indicated by value) not accommodated in 2 the scale to improve clarity. (B) Interfacial roughness (2 = e: + HL2 ) vs. temperature for the &lm B. () air–&lm interface; ( ) all internal interfaces.

group AEDs and two roughness parameters; one for all the metal interfaces and one for the air–&lm interface [76]. For the data at higher temperatures, these roughnesses were the only &t parameters keeping all AEDs &xed to that obtained from room temperature data. The variation 2 + H2 ) as a function of temperature for the &lm B is shown in Fig. of roughness (2 = e: L 32 (lower panel). Note that we have not plotted the EDS parameter, HL , separately for &lm A. This is because the variation in total as a function of temperature for this &lm is essentially due to the variation of HL as transverse di:use data indicate no signi&cant variation in e: , at ◦ least up to 100 C. It may be mentioned here that for &lm B, strictly speaking no intrinsic parameter can be de&ned as it has self-aOne interfacial correlation without cuto:. The e:ect of this roughness has been taken into account implicitly in Eq. (76) and hence an explicit separation of the EDS parameter is necessary for &lm B. The EDS parameter H (HS or HL ) could signify di:usion of molecules either in the z direction (longitudinal di:usion) or in x–y plane (lateral di:usion). Longitudinal motion of molecules can only take place through the defects in the &lm since the molecules are in a close packed con&guration in the solid phase and interpenetration of chains is energetically unfavourable. In any case this motion, through the defects, will lead to piling up of molecules on top of the &lm leading to increase in e:ective &lm thickness. Neither the X-ray re6ectivity nor the AFM data, presented later, provides evidence for this. Moreover, longitudinal motion of molecules will lead to deviation of in-plane correlation leading to change in roughness exponent. But we have ◦ found that there is no change in the roughness exponent from room temperature up to 100 C. This is clearly evident on comparison of the di:use scattering data for both the as-grown &lms ◦ and 100 C (cooled) &lms, which have identical line shape (Fig. 29) indicating same interfacial correlation. So in this case the EDS parameter H indicates lateral motion of the molecules. It

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10

-3

10

(a)

-4

10

Reflectivity

65

(b)

-4

10

-5

10

w

-5

0

10

20 C

-6

10

0

20 C -6

10

-7

10

0

100 C

-8

10

-7

10

0

-9

10 -0.002

-8

-0.001

0.000

0.001

10 0.002 -0.002

10 10 10

Reflectivity

10 10 10 10 10 10 10 10 10

3

-0.001

0.000

0.001

0.002

-1

-1

10

100 C

qx (Å )

qx (Å )

2

(c)

1 0

-1 -2 -3 -4 -5 -6 -7 -8 -9

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1

qz (Å ) ◦

Fig. 33. Di:use scattering data () and &t (—) for (a) &lm A (b) &lm B at room temperature (20 C) and after cooling ◦ N −1 . The resolution function at this qz position for 100 m incident and 400 m from 100 C, all taken at qz = 0:23 A detector slits is also shown in both (a) and (b). Experimental specular (lower) and di:use integrated re6ectivity (upper) pro&les for &lm A (—) and B () at room temperature. The di:use integrated pro&les have been shifted with respect to the experimental data for clarity.

was also found earlier [71] that some LB &lms exhibit structural changes prior to melting which was interpreted in terms of lateral melting of domains without disturbance of vertical ordering and is similar to what we &nd here. It appears that the extent of the in-plane expansion due to lateral motion of molecules varies at the di:erent multilayer interfaces as is evident from the di:erent smearing parameters, HS , obtained at each interface and may also be non-uniform within an interface, the expansion being more around the defect sites as compared to the more ordered regions. 9.2.4. Di:use integrated re<ectivity analysis In order to obtain a uni&ed picture of the melting mechanism irrespective of the nature of the interfacial correlation of the &lms, we have also analysed the di:use integrated re6ectivity pro&les for &lms A and B. Fig. 33 (lower panel) shows the e:ect of the scaling (Eq. (78)) on the experimental specular re6ectivity data for &lms A and B, before heat treatment, to convert them to the respective di:use integrated re6ectivity pro&les. In the same &gure, we have also

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J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

3

EDP (El/Å )

0.55

0

20 C

0.50

65 C

0.45

85 C

0.40

95 C

0.35

100 C

0 0 0

0

0.30 0

50

100

Reflectivity

12

10 11 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 10 0.0

Z (Å)

150

200

250

300

1100 C

1100 C 1000 C

0.1

0.2

0.3

0.4

qz (Å)

0.5

0.6

1000 C

950 C

950 C

850 C

850 C

650 C

650 C

200 C

200 C

0.7 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

qz (Å)

Fig. 34. (a) Obtained EDPs at di:erent temperatures for &lm A. Same EDPs were used to match the calculated re6ectivity pro&les (—) with the respective di:use integrated pro&les () for &lm A (b) and B (c). Only a scaled background was ◦ added to the re6ectivity pro&les of &lm B at each temperature (refer text). No meaningful &t to the data at 110 C, for &lms A and B could be obtained. All the pro&les were multiplied with respect to its immediate lower curve by a factor of 100 for clarity.

plotted the direct beam pro&le for the specular re6ectivity geometry with 400 m detector slits (upper panel). It is clear from this &gure that for logarithmic correlation (&lm A) FD FW and hence di:use integration is e:ectively performed during the experimental measurement of re6ectivity while for self-aOne correlation with = 0:5 (&lm B) FD FW and hence scaling had to be done numerically [91]. Both FD and FW were evaluated numerically for the two &lms A and B from the respective set of transverse di:use pro&les (one of which is shown for each &lm in Figs. 29 and 33). The value of this ratio was then interpolated over the entire range of the measured specular re6ectivity data and then the re6ectivity data at all temperatures for &lm B were scaled by RS (qz ) to convert them to the respective di:use integrated pro&les [91], which is equivalent to re6ectivity pro&les of zero roughness. Fig. 34 shows the di:use integrated re6ectivity data, and the &tted pro&les, at di:erent temperN −1 , atures for &lms A and B. Strong Bragg peaks at qz = 0:107, 0.226, 0.339, 0.457 and 0:569 A −1 N ) are observed respectively, and well de&ned separation between the peaks (0:115 ± 0:003 A ◦ in the re6ectivity data, even when the sample is heated up to 85 C. The presence of strong ◦ Kiessig fringes up to 85 C indicates that the &lm has more or less uniform thickness (and hence equal number of bilayers) within the coherence length of our X-ray beam. Beyond this temN −1 ) starts decreasing perature, the amplitude of the &rst observable Kiessig fringe (qz 0:05 A ◦ and vanishes totally at 100 C. This may be an indication that the &lm thickness becomes

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67

◦

inhomogeneous. At 110 C, the bilayer structure for both the &lms A and B breaks down completely as is evident from the disappearance of the bilayer Bragg peaks in the respective X-ray ◦ specular re6ectivity data. No physically meaningful &t to the 110 C data could be obtained and hence the respective &tted pro&les were not presented. We have used Eqs. (75) and (78) to perform the scaling of measured re6ectivity pro&les for &lm A and B and R(qz ) (in Eq. (75)) was calculated on the basis of a model EDP of the &lms consisting of slices of di:erent thickness with each slice having a constant AED to carry out the &tting procedure. The thickness of N and in the low electron the slices in the high electron density head region was taken to be 5 A N (refer Fig. 5). The only &t density tail region the thickness of each slice was taken to be 50 A parameters were the AEDs of the slices representing the head region the size of all the slices and the AED of the slices representing the hydrocarbon tail region being kept &xed. The tail N 3 [37]. The re6ectivity pro&les calculated AEDs were &xed at the theoretical value of 0:3 e= A on the basis of this simple model were &tted to the di:use integrated re6ectivity pro&les at each temperature for &lm A. A clear trend of reduction in the density of the head regions with ◦ increase of temperature up to 100 C is evident from the EDPs obtained at each temperature from independent &tting of the respective di:use integrated data for the &lm A as shown in Fig. 34 (upper panel). Note that we have used a very simpli&ed model for the &lms since it brings out the essential features of the melting process. In order to obtain a uni&ed picture of the melting mechanism irrespective of the nature of the interfacial correlation of the &lms we have used the respective EDPs at each temperature, as obtained for the &lm A, to calculate the di:use integrated re6ectivity pro&les at all temperatures for the &lms B (shown in Fig. 34). Only a scaled background was added to the calculated pro&les to account for the scaling of the background counts (D0 ) during conversion of the re6ectivity data to the di:use integrated pro◦ &les for &lm B [91]. The close matching indicates that the mechanism of melting up to 100 C is independent of the type of interfacial correlation present in the untreated &lm. The trend of density reduction with temperature could either be due to vertical molecular interdi:usion or due to lateral expansion of the &lm. Since di:use scattering measurements have ruled out signi&cant vertical molecular interdi:usion, this trend of density reduction is a signature of lateral motion of molecules leading to expansion of the &lm. Here, we have used a much simpler model to extract the same information. A better &t could be obtained with more parameters, ◦ as was done earlier [76], but the essential information obtained remains unchanged. At 110 C, the bilayer structure of the &lm is destroyed and the &lm disorders irreversibly. It appears that at this stage gauche disorder sets in the chains whereby the chains no longer retain their vertical orientation leading to destruction of the ordered bilayer structure. Also since the molecular packing is considerably relaxed interlayer di:usion could become dominant at this temperature. Hence results of analysis using both the formalisms indicates that melting of LB &lms lateral ◦ takes place predominantly by lateral molecular di:usion, at least up to 100 C, irrespective of the nature of the interfacial morphology of the untreated &lm. 9.3. AFM studies Not much work has been done on extracting morphological information regarding melting of LB &lms since information available from AFM topographs is restricted to the top surface only.

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◦

Fig. 35. AFM images (5 m × 5 m) for &lm A: (a) the as-grown &lm, (b) after cooling from 100 C and (c) after cooling ◦ from 110 C. Typical height pro&les along the lines drawn on the respective images are also shown alongside.

Nevertheless, AFM studies can provide important information which could be related to the mechanism of melting. In addition to the X-ray scattering data, we have also performed AFM measurements on both ◦ ◦ the untreated &lms and also after heat treatment at 100 C and 110 C. Although AFM images are restricted in their information content only to the surface of the &lms, they nevertheless can provide useful information regarding the nature of the local defect structure which cannot be obtained from X-ray measurements which provides a statistically averaged information. AFM ◦ images for the &lm A before heat treatment (as-grown), after being cooled from 100 C and ◦ 110 C are shown in Fig. 35(a), (b) and (c), respectively. The height pro&les drawn beside each topographical image represents the height variation along the line drawn through the respective images, the heights being de&ned relative to the minimum height along the particular line. The ◦ untreated &lm is very smooth with only small pin-hole defects being present. At 100 C, the &lm appears to have been cracked with large craters having opened up. As the line pro&le indicates, the depth of a typical crater is nearly the same as the total &lm thickness. Similar morphology

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

69

◦

Fig. 36. AFM images (1 m × 1 m) for &lm B: (a) the as-grown &lm, (b) after cooling from 100 C and (c) after cooling ◦ from 110 C. Typical height pro&les along the lines drawn on the respective images are also shown alongside.

◦

is also seen for the &lm at 110 C, with a noticeably higher defect density as compared to that ◦ at 100 C. Clearly we do not see pileup of molecules or height di:erence of the order of bilayer or monolayer between di:erent regions which could indicate possible vertical interdi:usion of molecules. AFM images for the &lm B before heat treatment (as-grown), after being cooled ◦ ◦ from 100 C and 110 C are shown in Fig. 36(a), (b) and (c), respectively. The height pro&les drawn beside each topographical image represents the height variation along the line drawn through the respective images, the heights being de&ned relative to the minimum height along the particular line. The as-grown &lm is quite smooth interspersed with pin-hole defects as can be clearly seen from the typical height pro&le shown alongside the topographical image. After ◦ cooling from 100 C, the &lm is found to predominantly consist of domains of di:erent heights in the z direction. Typical size of domains is ∼ 0:5 m. Within a domain the &lm is quite smooth except for the presence of few defects. The height pro&le for the &lm after it is cooled from ◦ 100 C clearly shows that the domain of the largest size in the image (top left corner) presented has thickness close to the bilayer thickness of the as-grown &lm. Moreover, the di:erence in

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J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

the thickness of the other domains as compared to this domain is small which explains why appreciable change in bilayer spacing could not be detected from X-ray measurements, even at this temperature, although the thickness inhomogeneity is suOcient to reduce the Kiessig fringe N intensity [76]. It is known that the translational correlation length of LB &lms is ∼ 100 A. However, long range bond orientational order may exist in these &lms due to hexatic head group arrangement [72,74]. This in turn produces micron size domains [72] as observed here. ◦ Although the presence of domains is clearly evident in the AFM data at 100 C (cooled) it is ◦ likely that the formation of these domains starts from 90 C onwards since at this temperature the ◦ Kiessig fringes tends to become weaker. After the &lm is cooled from 110 C, no such smooth domains are visible and the &lm becomes rougher, as is evident from the rapid 6uctuations in the height pro&le. It is clear that the smooth large domains that were present in the &lm at ◦ N 100 C (cooled) is absent and the &lm becomes disordered at short length scales ( 100 A). This type of morphology is typical of a disordered state, typical of liquids where there is no long range translational or orientational order. Fig. 37 shows the plot of rms roughness against scan length r (actually 2 vs. r) for scans of di:erent sizes for the &lm A at room temperature, after it is cooled to room tem◦ ◦ perature from 100 C and 110 C. The linear &t to the data (in log-linear scale) clearly indicate that the correlation at the surface of the &lm is logarithmic con&rming our earlier observation of the presence of conformal logarithmic interfacial correlation in the &lm. It is ◦ interesting to note that this correlation persists at the surface of the &lm even up to 110 C where X-ray re6ectivity data indicate that the &lm has melted. The slope of the linear &t is proportional to the parameter B and the increase of B with temperature is indicative of decrease in surface tension and hence increase in roughness [15,28]. It is diOcult to understand whether this is purely a surface e:ect (as evident from AFM measurements alone) or takes place within the &lm also since this trend cannot be unambiguously con&rmed from X-ray di:use re6ectivity at elevated sample temperatures due to low di:use signal-to-noise ratio inherent in a lab source. We believe that due to disorder in the &lm, the adhesive force between the &lm and substrate decreases substantially at elevated temperatures and as a result B increases. Fig. 38 shows the plot of rms roughness against scan length r for scans of di:erent sizes ◦ for the &lm B at room temperature and after it is cooled to room temperature from 110 C. The linear &t to the room temperature data clearly indicate that the &lm has self-aOne rough interfaces, with exponent 0.5, up to a certain correlation length ( 15 m) which is beyond the coherence length of our X-ray beam ( 10 m here). Hence in X-ray measurements, the &lm appears to have self-aOne rough interfaces with no cuto: [29]. It may be noted that the ◦ roughness exponent of the &lm at 110 C changes from 0.5 to 0.8. Once again the X-ray di:use ◦ re6ectivity data at 110 C for &lm B, as for &lm A, are very weak and hence no conclusion about this change in exponent, especially whether it is limited to the surface or occurs at other interfaces as well, could be drawn from X-ray scattering measurements. Thus analysis of X-ray re6ectivity, di:use scattering and AFM data on LB &lms of cadmium arachidate with two di:erent types of in-plane height–height correlation, self-aOne and logarithmic, indicates that melting of the &lms occur predominantly through lateral motion of molecules. Our results also suggest that the LB &lms, we have studied, do not behave strictly as a 2D system since the interlayer coupling is important in deciding the melting behaviour.

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

71

100

2000 1800

o

110 C

room temp

1600

0

100 C 1400

10

0

110 C σ (Å)

o

1000

20 C α = 0.8

2

2

σ (Å )

1200

800 α = 0.5

1

600 400 200 0 3 10

10

4

10

r (Å)

5

10

6

10

2

10

3

10

4

10

5

10

6

r(Å) ◦

Fig. 37. 2 vs. scan length, r, from AFM images of &lm A for untreated &lm (), after cooling from 100 ( ) and from ◦ 110 C (♦). The solid lines are the &ts to the linear portion of the respective curves. ◦

Fig. 38. vs. scan length, r, from AFM images of &lm B for untreated &lm () and after cooling from 110 ( ). The dashed lines are the &ts to the linear portion of the respective curves. Slopes are indicated alongside the respective &ts. The solid lines are only a guide to eye.

10. Conclusion Ultrathin organic &lms formed by Langmuir–Blodgett technique are not only important for technological and biological applications but also to understand physics and chemistry in con&ned geometries. However, many of these important research areas have not evolved as expected primarily because our knowledge about the structure of these &lms and its relationship with growth condition is yet to be understood clearly. The advent of X-ray scattering techniques and atomic force microscopy measurements are providing us fresh insight into these aspects and the purpose of this report is to highlight our present understanding in this evolving &eld of research. We have speci&cally focussed our attention on the growth–morphology relationship of LB &lms, nanoclusters formation in these &lms and melting transitions of these quasi-2D systems studied using X-ray re6ectivity and di:use scattering techniques. In the process, we have also reviewed experimental and theoretical aspects of these scattering techniques.

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Acknowledgements We would like to acknowledge the contributions of Dr. Alokmay Datta, Dr. Sangam Banerjee, Dr. Satyajit Hazra and Dr. Manabendra Mukherjee who have been involved in some of the work. Appendix A Here we will only discuss the case of scattering from true self-aOne rough surfaces, i.e. surfaces without cuto: in self-aOne scaling characterised by the correlation function g(r) = Ar . As has been discussed earlier, for such systems the scattering cannot be separated into specular and di:use components and hence total intensity has to be calculated. The expression for total intensity for multilayer systems is given in Eq. (73). The integral in Eq. (73) can be worked out analytically only for =0:5 and 1. We discuss the case =0:5 which is of particular interest to us. Eq. (73) for = 0:5 and for systems with conformal interfaces (gij (r) = g(r)) becomes ∞ 2r02 2 qz (zi −zj ) T'i T'j e d r r e−(1=2)qz Ar J0 (qr r) ⊗ R(qx ; qy ; qz ) : (A.1) I = I0 2 qz sin i ij 0 De&ning S as ∞ 2 S= d r r e−(1=2)qz Ar J0 (qr r) ⊗ R(qx ; qy ; qz ) ; 0

(A.2)

we can write S=√

Aqz2 :( 32 ) (A=2)qz2 ⊗ R (q ; q ; q ) = ⊗ R(qx ; qy ; qz ) : x y z [qr2 + ((A=2)qz2 )2 ]3=2 [qr2 + ((A=2)qz2 )2 ]3=2

(A.3)

In most experimental geometries, the qy component of scattering is integrated out so that S can be written as (A=2)qz2 S = d qy 2 ⊗ R(qx ; qz ) : (A.4) [qr + ((A=2)qz2 )2 ]3=2 Using a standard integral [164] S can be written as S=

Aqz2 ⊗ R(qx ; qz ) : 4[qx2 + ((A=2)qz2 )2 ]

De&ning Iconv (qx ; qz ) as A ⊗ R(qx − qxc ) ; Iconv (qx ; qz ) = 2 qx + ((A=2)qz2 )2

(A.5)

(A.6)

S can be written as, S = Aqz2 Iconv (qx ; qz ) :

(A.7)

Using Eqs. (A.7) and (A.1), the total intensity can be calculated. The qz convolution has been assumed to be performed separately.

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73

Appendix B Here we derive the expression for specular re6ectivity of a thin &lm on a substrate using the DWBA formalism. To arrive at the expression for specular re6ectivity (Eq. (81)) we &rst derive the expressions for specular re6ectance r0 (qz ) and the Fresnel re6ection and transmission coeOcients a(qz ) and b(qz ). The wave function 1(z) satisfying the Helmholtz equation of the form Eq. (15) for the X-rays in a 1D potential V (z) due to the &lm of thickness d on a substrate can be written as  kz z −kz z ; z¡0 ;   e + r0 (qz )e k2z z −k2z z 1(z) = ae (B.1) + be ; 0¡z¡d ;   k3z z z¿d ; Te ; where kz ; k2z and k3z have already been de&ned earlier to be the momentum transfers in air, &lm and substrate, respectively. T is the Fresnel transmission coeOcient in the substrate. It is obvious that by de&nition we take the top of the &lm to be at z = 0 and the direction into the &lm as z positive. Applying the conditions of continuity of the wave function and its derivative at z = 0, we arrive at the following set of equations 1 + r0 (qz ) = a + b

(B.2)

kz (1 − r0 (qz )) = k2z (a − b) :

(B.3)

and Similarly, applying the boundary conditions at z = d, we get aek2z d + be−k2z d = T ek3z d

(B.4)

k2z (aek2z d − be−k2z d ) = Tk3z ek3z d :

(B.5)

and Using Eqs. (B.4) and (B.5), we can write k2z [aek2z d + be−k2z d ] = ; k3z [aek2z d − bek2z d ]

(B.6)

which on further simpli&cation gives b = ar23 : Again using Eqs. (B.2), (B.3) and (B.7), we can write 1 + r23 1 + r0 (qz ) = : 1 − r0 (qz ) (k2z =k3z )[1 − r23 ] It can be easily shown from above that r12 + r23 : r0 (qz ) = 1 + r12 r23

(B.7)

(B.8)

(B.9)

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Using Eqs. (B.2) and (B.9), we can write a=

1 + r0 (qz ) 1 + r23

=

1 + (r12 + r23 )=(1 + r12 r23 ) 1 + r23

=

1 + r12 : 1 + r12 r23

Let us de&ne the    0; V1 = r0 'f ;  r ' ; 0 s and

(B.10)

potential V as V1 + V2 , where z ¡ 0; 0¡z¡d ; z¿d

   0;

z¡0 ; V2 = r0 T'f ; 0 ¡ z ¡ d ;   0; z¿d :

(B.11)

(B.12)

Here, r0 is the classical electron radius for X-ray scattering. Let us also de&ne a time reversed wave function corresponding to the one de&ned in Eq. (B.1) above as  −k z z + r0∗ (qz )ekz z ; z ¡ 0 ;  e ˜ 1(z) = a∗ e−k2z z + b∗ ek2z z ; 0 ¡ z ¡ d ; (B.13)   ∗ −k3z z T e ; z¿d : The specular re6ectivity, R(qz ), in DWBA is given by R(qz ) = |1˜ |V1 | + 1˜ |V2 |1|2 :

(B.14)

Here, | = eikz z . The &rst term can be shown to be equal to ir0 (qz ). The second term can be written as 2r0 2 ∗ 2 ˜ 1|V2 |1 = a T'˜ f (q2z ) + b T'˜ f (q2z ) + 2ab d zT'f (z) k2z =

4r0 2 [a T'˜ f (q2z ) + b2 T'˜ ∗f (q2z )] ; q2z

(B.15)

where d z T'=0 since T' is de&ned about the mean density and T'˜ f is the Fourier transform of T'.

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75

Hence, the specular re6ectivity in DWBA for a &lm on a substrate can be written as 2 4r 0 ∗ 2 2 R(qz ) = ir0 (qz ) + [a T'˜ f (q2z ) + b T'˜ f (q2z )] : qz

(B.16)

Appendix C C.1. Capillary wave theory A liquid interface is commonly described as a layer of thickness int , where the density changes from 'A to 'B , the respective densities of the two bulk phases separated by the interface [165]. This layer is constantly distorted by thermal motion and therefore presents a certain roughness which can be quanti&ed in terms of an additional interfacial width 0 (refer Fig. 39). This is often identi&ed as the width due to the thermally excited capillary waves [165 –167]. Let us denote by z(r) the vertical displacement of the interface at the point r = (x; y) on the 6at dividing surface between the two phases. This can be written in terms of its Fourier components as 1 z(r) = d q eq·r z(q) ˜ : (C.1) (2)2 The work required to perform small amplitude long wavelength 6uctuations of the interface in the presence of an external &eld (commonly gravity) contains two main contributions. The &rst is the change in surface free energy due to change in surface area. This is given by

TFC =

dr

2

1 + [∇z(r)] − 1 ;

(C.2)

Fig. 39. Schematic of a liquid interface separated by the bulk phases A and B. The intrinsic width, int , of the interface and the width due to thermally excited capillary waves, 0 are also indicated.

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J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

which under the assumption of small amplitude 6uctuations can be written as d r[∇z(r)]2 : TFC = 2 as

(C.3)

The second contribution is the work done against the gravitational &eld which can be written 1 TFg = g T' 2

d r z(r) :

Using Eq. (C.1), Eq. (C.3) can be written as d q z(q) ˜ z(q ˜ )(qq ) d r e(q+q )·r TFC = − 4 2(2) d q |z(q) = ˜ |2 q2 ; 2(2)2 where we have used the relations 1 d r exp[(q + q ) · r] ; $(q + q ) = (2)2 and z( ˜ −q) = z˜∗ (q). Similarly, the excess gravitational free energy, TFg , can be written as 1 TFg = g T' d q z(q) ˜ z(q ˜ ) d r e(q+q )·r 4 2(2) 1 = g T' d q |z(q) ˜ |2 : 2(2)2 The total excess free energy, TF, due to the 6uctuation of the interface is 1 2 g T' d |z(q) TF = q q + ˜ |2 : (2)2 2

(C.4)

(C.5)

(C.6)

(C.7)

(C.8)

Since we are dealing with a thermal distribution of independent modes of 6uctuations, the probability of these modes follows the Maxwell–Boltzman statistics and the average energy of each mode can be written in terms of the mean squared amplitude of a mode, |z(q) ˜ |2 , and by 1 applying equipartition theorem this must be equal to 2 kB T so that we can write 1 ˜ |2 : kB T = [q2 + G2 ]|z(q) 2 2

(C.9)

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84

77

Hence, we can write |z(q) ˜ |2 =

1 kB T : 2 q + G2

We can now de&ne the interface width due to capillary waves, z(r)2 (=02 ), as kB T 1 dq 2 02 = 2 : 4 q + G2 Transforming to polar coordinates, we get ∞ kB T 2 q 2 0 = 2 d dq 2 4 0 q + G2 0 kB T ∞ q = dq 2 : 2 0 q + G2

(C.10)

(C.11)

(C.12)

Using a standard integral [164], we get 02 =

kB T [ln(q2 + G2 )∞ 0 ] : 4

(C.13)

Obviously this expression diverges as q → ∞. Hence, an upper cuto: qu ∼ 1=
B kB T K0 (Gr) = K0 (Gr) : 2 2

(C.16)

Here, K0 is the modi&ed Bessel function of second order [109,168]. We can then de&ne a height di:erence correlation function, g(r), for capillary waves on liquid surfaces as g(r) = 202 − 2C(r) = 202 − BK0 (Gr) :

(C.17)

78

J.K. Basu, M.K. Sanyal / Physics Reports 363 (2002) 1 – 84 5.0 4.5 K0 (X)

4.0

-ln(X/2) - 0.5772

f(X) (arb units)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

X(arb units)

Fig. 40. K0 (x) vs. x. For typical values of cuto:s G (∼ 10−8 –10−5 ) and the in-plane length scales probed by X-ray measurements x (=Gr) ∼ 1 so that the approximation in Eq. (C.18) is quite good in most cases of interest.

For small x, K0 (x) can be written as K0 (x) = −ln(x=2) − E ;

(C.18)

while asymptotically K0 (x) → 0 (Fig. 40). For most experimental situations, the argument of K0 is quite small so that Eq. (C.18) is a reasonable approximation. With this assumption, g(r) can then be written as Gr g(r) = 202 + B E + ln : (C.19) 2 Here, E = 0:5772 is the Euler constant. C.2. X-ray scattering from capillary waves We now discuss the scattering of X-rays from multiple interfaces with capillary waves. For multilayers, we can generalise the correlation function de&ned above in two ways. For the case of conformal capillary waves with non-conformal intrinsic interfacial roughness, g(r) can be written as Gr gij (r) = i2 + j2 + B E + ln ; (C.20) 2 2 + 2 . This behaviour has been found earlier in di:erent types of organic &lms where i2 = int 0 like free standing smectic liquid crystal &lms [157,158] and soap &lms [169], etc.

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79

From Eq. (73) the total scattering intensity for such a multilayer system can be written as

r02 ij TNi TNj e−qz (zi −zj ) 2 d X d Y e−(i=2)qz gij (X;Y ) e−(qx X +qy Y ) ⊗ R(qx ; qy ; qz ) : I = I0 qz2 sin i (C.21) For our convenience we de&ne Dij as 2 d X d Y e−(i=2)qz gij (X;Y ) e−(qx X +qy Y ) ⊗ R(qx ; qy ; qz ) : Dij =

(C.22)

Now in usual experimental geometry, the qy component of scattering is integrated out and hence for that geometry Dij can be written as −(i=2)qz2 gij (X;Y ) −qx X dX dY e d qy e−qy Y ⊗ R(qx ; qz ) e Dij = 2 = d X d Y e−(i=2)qz gij (X;Y ) e−qx X $(Y ) ⊗ R(qx ; qz ) 2 = d X e−(i=2)qz gij (X;0) e−qx X ⊗ R(qx ; qz ) : (C.23) Using Eq. (C.20) in the above equation, we get 2 −(1=2)qz2 [i2 +j2 +BE ] d X e−(1=2)Bqz ln(GX=2) e−qx X ⊗ R(qx ; qz ) : Dij = e

(C.24)

De&ning the resolution function in qx as 2

2

R(qx − qx ) = e−6 (qx −qx ) ;

(C.25)

where 6=2=Tqx , Tqx being the resolution half width in qx , and using the convolution theorem we can write Dij as √ −(1=2)qz2 [i2 +j2 +BE ] 2 2 2 2 d X e−(1=2)Bqz ln(GX=2) e−qx X e−( =6 )X : Dij = e (C.26) 6 Using a standard integral [168], Dij can be written as ; 1 1−; 1 − ; 1 −qx2 62 −(1=2)qz2 [i2 +j2 +BE ] 2 √ : : Dij = e ; ; 1 F1 G6 2 2 2 42

(C.27)

Here ; = 12 Bqz2 , :(x) is the gamma function and 1 F1 (x; y; z) is the con6uent hypergeometric function or Kummer function [168]. Hence, from Eq. (C.21), we have R(qz )qz 1 1−; 1 − ; 1 −qx2 62 √ : I = I0 + D0 : (C.28) ; ; 1 F1 2k0 sin i 2 2 2 42

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Here, R(qz ) is the specular re6ectivity and D0 is a constant background, mainly arising from detector dark current and air scattering. For the case of capillary waves at conformal multilayer interfaces the correlation function, g(r), can be written as Gr g(r) = 22 + B E + ln ; (C.29) 2 2 + 2 . In this case, D de&ned earlier is the same for all interfaces (D = D) where 2 = int ij ij 0 and can be written as ; 1 1−; 1 − ; 1 −qx2 62 −(1=2)qz2 [22 +BE ] 2 √ : D=e ; ; 1 F1 G6 2 2 2 42 1 1−; 1 − ; 1 −qx2 62 2 −qz2 e: √ : =e ; (C.30) ; ; 1 F1 2 2 2 42 where 2 e:

1 1 2 = + BE − B ln : 2 2 G6 2

The total intensity can then be written as R(qz )qz 1 1−; 1 − ; 1 −qx2 62 2 2 √ exp[ − qz e: ]: + D0 : I = I0 ; ; 1 F1 2k0 sin i 2 2 2 42

(C.31)

(C.32)

We have used Eq. (C.32) in our analysis of specular re6ectivity and di:use scattering from LB &lms with capillary waves at the interfaces. References [1] B. Lawrie, M.A. Pais, B. Pippard (Eds.), Twentieth Century Physics, Institute of Physics, Philadelphia, PA, 1995; G. Grinstein, G. Mazenko, Directions in Condensed Matter Physics, World Scienti&c, Singapore, 1986. [2] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993. [3] P.-G. de Gennes, Simple Views on Condensed Matter Physics, World Scienti&c, Singapore, 1992. [4] P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1998. [5] M. Daoud, C.E. Williams (Eds.), Soft Matter Physics, Springer, Berlin, 1995. [6] G. Bauer, F. Kuchar, H. Heinrich, Two Dimensional Systems: Physics and New Devices, Springer Series in Solid State Sciences, Vol. 67, Springer, Berlin, 1986. [7] Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, Oxford, 1997. [8] M.A. Reed, W.P. Kirk (Eds.), Nanostructure Physics and Fabrication, Academic Press, New York, 1989. [9] M. Riordan, L. Hoddeson, C. Herring, Rev. Mod. Phys. 71 (1999) S336 and references therein; W.F. Brinkman, D.V. Lang, Rev. Mod. Phys. 71 (1999) S480. [10] D.A. Glocker, S.I. Shah (Eds.), Handbook of Thin Film Process Technology, IOP Publishing, Bristol, 1995. [11] R.F. Bunshah (Ed.), Handbook of Deposition Technologies for Films & Coatings, Noyes Publications, New Jersey, 1994. [12] See for e.g. M. K. Sanyal (Ed.), Special section: Surface Characterization, Current Science 78 (2000) 1445 –1530. [13] See for e.g. B. Ocko (guest editor), Synchrotron Radiation News 12(2) (1999) 1 (special issue).

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Physics Reports 363 (2002) 85 – 171 www.elsevier.com/locate/physrep

On the manifestation of chiral symmetry in nuclei and dense nuclear matter G.E. Browna; ∗ , Mannque Rhob; c a

Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794, USA b Service de Physique Th#eorique, CE Saclay, 91191 Gif-sur-Yvette, France c School of Physics, Korea Institute for Advanced Study, Seoul 13-012, South Korea Received September 2001; editor: W: Weise

Contents 1. Introduction 2. From quark to nucleon 2.1. Modeling the nucleon in QCD 2.2. The chiral bag Lagrangian 3. E9ective :eld theories in nuclear physics 3.1. Chiral symmetry in nuclear processes 3.2. Objectives of EFTs in nuclear physics 3.3. E9ective chiral Lagrangians 3.4. Nucleon–nucleon scattering 3.5. The PKMR approach: more e9ective EFT 4. “Vector manifestation” of chiral symmetry 4.1. Harada–Yamawaki scenario 4.2. Vector manifestation in hot matter 4.3. Vector manifestation in dense matter 5. Landau Fermi liquid from chiral Lagrangians 5.1. Fluctuating around zero-density vacuum 5.2. Skyrmion vs. Q-ball 5.3. E9ective chiral Lagrangian for many-nucleon systems 5.4. Brown–Rho scaling 5.5. Landau mass and BR scaling 6. Indirect evidences for BR scaling 6.1. Indications in :nite nuclei

87 89 89 91 94 94 95 96 98 101 113 113 115 116 119 119 120 121 122 124 126 126

7. 8.

9.

10.

6.2. The anomalous gyromagnetic ratio in nuclei 6.3. Axial charge transitions in heavy nuclei 6.4. Axial-vector coupling constant gA? in dense matter 6.5. Evidences from “On-shell” vector mesons Direct evidence from the quasielastic (e; e ) response functions in nuclei BR scaling in chiral restoration 8.1. Bag constant and scalar :eld energy 8.2. “Nambu scaling” in temperature 8.3. “Nambu scaling” in density Signals from heavy-ion collisions 9.1. Top–down and bottom–up and how they connect 9.2. Distinguishing Rapp=Wambach and Brown=Rho “Broad-band equilibration” of strangeness in heavy-ion collisions 10.1. Kaons and chiral symmetry 10.2. Equilibration vs. dropping kaon mass 10.3. The equilibrium K + = + ratio

∗

Corresponding author. E-mail address: [email protected] (G.E. Brown).

c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 8 4 - 9

126 127 128 130 131 134 135 135 137 139 139 141 142 142 143 146

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10.4. The top–down scenario of K ± production 10.5. Partial decoupling of the vector interaction 10.6. Schematic model 10.7. Discussions 11. E9ective :eld theories for dense QCD 11.1. Color–Kavor locking for NF = 3

147 150 152 154 156 156

11.2. Complementarity of hidden gauge symmetry and color gauge symmetry and BR scaling 11.3. Koan condensation: encore Cheshire Cat Acknowledgements References

161 165 167 167

Abstract This article reviews our view on how chiral symmetry, its pattern of breaking and restoration under extreme conditions manifest themselves in the nucleon, nuclei, nuclear matter and dense hadronic matter. We discuss how :rst-principle (QCD) calculations of the properties of :nite nuclei can be e9ectuated by embedding the “standard nuclear physics approach (SNPA)” into the framework of e9ective :eld theories of nuclei that incorporate chiral dynamics and then exploit the predictive power of the theory to accurately compute such solar neutrino processes as the proton–proton fusion and the “hep” process and such cosmological nucleosynthesis process as thermal neutron–proton capture etc. The Brown–Rho (BR) scaling that implements chiral symmetry property of baryon-rich medium is re-interpreted in terms of “vector manifestation” of hidden local symmetry aM la Harada–Yamawaki. We present a clear direct evidence and a variety of indirect evidences for BR scaling in nuclear processes at normal nuclear matter density probed by weak and electromagnetic :elds and at higher density probed by heavy-ion collisions and compact-star observables. We develop the notion of “broadband equilibration” in heavy-ion processes and sharpen the role of strangeness in the formation of compact stars and their collapse into black-holes. We revisit the Cheshire Cat phenomenon :rst discovered in the skyrmion structure of baryons and more recently revived in the form of “quark–hadron continuity” in mapping low-density structure of hadrons to high-density structure of quarks and gluons and argue once more for the usefulness and power of e9ective :eld theories based on chiral symmetry under extreme conditions. It is shown how color– Kavor locking in terms of QCD variables and hidden local symmetry in terms of hadronic variables can be connected and how BR scaling :ts into this “continuity” scheme exhibiting a novel aspect of the Cheshire Cat c 2002 Elsevier Science B.V. All rights reserved. phenomenon. PACS: 11.30.Rd; 24.85.+p

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1. Introduction In a recent paper, Morgenstern and Meziani [1] reported a remarkable result of their extensive analysis of the longitudinal and transverse response functions in medium-weight and heavy nuclei and concluded that the available world data showed unequivocally the quenching of ∼20% in the longitudinal response function and the only viable way to explain this quenching is to invoke the Brown–Rho (BR) scaling [2] formulated to account for the change of the strong-interaction vacuum induced by matter density. Partly motivated by this development and other indirectly related developments including the most recent one on color–Kavor-locking and color superconductivity, we shall make in this review our overview of evidences that indicate both directly and indirectly that the BR scaling is indeed operative in :nite nuclei and dense (and superdense) nuclear matter. We shall develop our arguments starting with the basic structure of the nucleon, then go to that of nuclei and of nuclear matter and :nally to that of dense matter that one expects to create in relativistic heavy-ion collisions and :nd in the interior of highly compact stars. The principal theme will be that the chiral symmetry of quantum chromodynamics—its spontaneous breaking in free space and partial or full restoration in medium—underlies the common feature from elementary hadrons to complex dense systems. We admit that our views are not necessarily shared by others in all details and that some points may not be fully correct. We are however con:dent that the general theme that we have developed and we shall review here will survive the experimental test that is to come. It was recognized a long time ago that chiral symmetry, now identi:ed as an essential ingredient of QCD associated with the light-mass up (u), down (d) and strange (s) quarks, with its “hidden” realization in Nambu–Goldstone mode plays an important role in nuclear physics. In fact, the emphasis on its role in a variety of nuclear processes predates even the advent of QCD proper and subsequent acceptance as the correct theory of strong interactions [3]. Among the time-tested and established is the role of chiral symmetry in exchange currents in electro-weak processes in nuclei [4,5] and its subsequent importance in probing nuclear structure at various electron machines, now out of operation, throughout the world [6]. Since our early review on the subject [3], there has been continuous and signi:cant evolution in the :eld speci:cally associated with the role of QCD in nuclear physics, more or less unnoticed by workers in other areas of physics. The evolution touched on the structure of the nucleon, the proton and the neutron, starting with the notion of a “little bag” [7], the recognition of the preeminent importance of Goldstone pion clouds in both the nucleon structure and nuclear forces [7,8], followed by the resurrection of the Skyrme soliton model that contains baryons—which are fermions—in a bosonic :eld theory [9,10] and then the emergence of the notion of “Cheshire Cat phenomenon”. At present, the issue of both “spontaneous” breaking and restoration of chiral symmetry occupies one of the central positions in current activities in both nuclear and hadron physics communities. The development up to 1995 has been summarized in a recent monograph by Nowak et al. [11]. It is now becoming clearer in which direction the chiral symmetry of QCD will steer the next generation of nuclear and hadronic physicists, with the advent of new accelerators such as the Je9erson Laboratory (JL) and the RHIC at Brookhaven—both of which are already operating—and ALICE=LHC at CERN—which is to come in a few years. On one hand, the electron machine at the JL will probe deeper into shorter distance properties of the nucleon–nucleon interactions, exploring how the chiral structure of the nucleon in medium changes from hadronic over to quark=gluon picture as shorter distances are probed. As will be repeatedly stressed, one does not expect any abrupt phase

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transitions along the way; as the change will most likely be continuous (for a multitude of reasons we will discuss), it will then be a matter of economy which language will be more eScient for given processes with given conditions, implying that there will be a continuous map between the various descriptions. This continuity will play a central role in our discussion and will later be referred, quite broadly, to as “Cheshire Cat Phenomenon”. On the other hand, the heavy-ion machines at RHIC and ALICE=LHC will create systems at high temperature and density, mimicking the early Universe. At high temperatures, QCD predicts a phase transition from chiral symmetry in the Goldstone mode to that in the Wigner mode with a consequent change in con:nement=decon:nement. This is more or less con:rmed on lattice in QCD. Indications are coming out already from available experimental data that such a transition has been seen. Matter at high density is a completely di9erent matter. Up to date, it has not been feasible to put density on lattice, so there is no model-independent theoretical indication for a similar phase change as density is increased. Theoretical models however do predict that there can be a series of phase changes including the chiral symmetry restoration. As will be discussed in this article, there are a variety of processes involving nuclei and nuclear matter that provide, albeit indirect, evidence for such phase changes, most intriguingly of which are possible signals from compact (neutron) stars. Going to the physics of matter under extreme conditions—high temperature, high density or both— encompasses a multitude of scales. At low density ¡ 0 (where 0 is nuclear matter density), one can rely on the rich phenomenology available in the guise of standard nuclear physics to work with an e9ective Lagrangian :eld theory that is anchored on the premise of quantum chromodynamics (QCD), e.g., chiral perturbation theory. We may call this “phenomenological perturbation approach (PPA)”. However as one approaches the density 0 , an e9ective :eld theory constructed in the matter-free vacuum encounters diSculty and becomes more or less unpredictive: Presently there is no way to derive a nucleus starting from a :rst-principle Lagrangian, not to mention denser hadronic matter. The reason why this is so will be explained in the course of this review. The attitude we will adopt here is then to develop a generic description of dense matter based on global characteristics of hadrons in the environment of surrounding dense medium. For this, we exploit the chiral structure of the QCD vacuum and the excitations thereon (i.e., hadrons) following the notion of “vector manifestation” recently proposed and developed by Harada and Yamawaki [12]. In the Harada– Yamawaki scenario, chiral restoration from the spontaneously broken mode to an unbroken mode takes place with the massless pions (in the chiral limit) coming together with the would-be scalar Goldstone bosons that are the longitudinal components of massive vector mesons, with the massless vector mesons consequently decoupling. This scheme requires that near the chiral phase transition— independently of whatever form it may take, the vector ( and !) meson masses drop as the chiral transition point is approached bottom–up. This, we will see, provides a compelling theoretical support to BR scaling. We will also see that most remarkably, this scenario receives further support from the QCD structure of quark condensates in medium associated with color–Kavor locking. BR scaling as originally formulated in Ref. [2] corresponded to a mean-:eld approximation in a large Nc e9ective chiral Lagrangian theory (where Nc is the number of colors in QCD) with the strong interaction vacuum 1 “sliding” with density or temperature of the system. For a given density, 1

Throughout this article, the terminology vacuum will be used in the generalized sense of a ground state in the presence of matter.

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say, quantum Kuctuations or loop corrections are to be calculated for correlation functions, etc. with a chiral Lagrangian whose parameters are suitably de:ned at the sliding vacuum appropriate to physical processes of given kinematical conditions. How this works out has been explained in the literature but will be summarized in the review at pertinent places. We propose that the way the BR-scaling mean-:eld quantities vary as a function of the medium condition (density or temperature) should roughly match with the change of the vacuum suggested by Harada–Yamawaki’s vector manifestation. As mentioned above, at small external conditions (e.g., at low density), the PPA using phenomenological Lagrangians :xed in free space will be in no disagreement with the notion of BR scaling. However as density increases, Kuctuations are to be built on a vacuum modi:ed from the free-space one. At some point, this e9ect can no longer be accessed adequately by the PPA. This can be seen in what we call “sobar” description as explained in a later section. Eventually a correct theory will have to approach Landau Fermi-liquid :xed point at normal nuclear matter density. We will show how this comes about with BR scaling chiral Lagrangians: BR scaling and a Landau parameter can be identi:ed. One of the major themes of this review is that this picture combining the vector manifestation of chiral symmetry, color–Kavor locking and BR scaling passes several tests in nuclear and dense hadronic systems. This, we believe, points to the validity of the notion that understanding what goes on in many-body systems of nucleons at high density and understanding the chiral structure of the nucleon at the QCD level relies on the same principle, a long standing issue in nuclear and hadronic physics. We shall summarize the evidences accumulated since our earlier publication [3,13]. 2. From quark to nucleon 2.1. Modeling the nucleon in QCD For concrete applications of QCD to physical hadronic processes, the only model that purports to encode QCD and at the same time is versatile enough for a wide range of strong interaction physics has been, and still is, the bag model. All other models of equal versatility can be considered as belonging to the same class. There is a long history in the development of this model which, as we shall mention later, continues even today in the guise of string theory and large N Yang–Mills gauge theory. The :rst such model that incorporates con:nement and asymptotic freedom of QCD is the MIT bag model [14]. However it was realized early on that since this model by construction lacks chiral symmetry, it could not be directly applied to the description of nuclear interactions, the domain of strong interaction physics most thoroughly and accurately studied. A simple stability argument showed that the bag for the nucleon must have a bag size typically of ∼ 1 fm, essentially the con:nement scale implicit in the model. And this is simply too big if one naively applies the picture to nuclear systems. As a possible way to reconcile this simple bag modelling of QCD with the “size crisis”, it was proposed in 1979 [7] to implement the spontaneously broken chiral symmetry and bring in pion cloud to the model. How to introduce pion cloud to the MIT bag producing a “chiral bag” was already known and discussed in the literature [15]. It consisted of adding into the model Goldstone pion :elds outside of the bag and imposing chiral-invariant boundary conditions that assure both chiral invariance and con:nement. The initial idea in Ref. [7] was to use the pressure generated by the external pion cloud to squeeze the bag to a smaller size so as to accommodate the

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standard nuclear physics picture of meson exchange interactions. At the time the idea was proposed, we had no idea how to squash the bag into a “little bag” in a way consistent with the premise of QCD. We simply dialed the parameters of the model such that the size diminished from ∼ 1 fm to ∼ 0:3 fm, a size known to be a spatial cuto9 between two nucleons. We now know that this prescription was not natural and in fact totally unnecessary. How to reconcile the bag size and nuclei can now be put in the form of what is known as “Cheshire Cat Principle” [16]. 2 Apart from the concern with the nucleon size in nuclei and apparent incompatibility with the successful standard meson-exchange potential picture, the “little bag” was not indispensable for understanding the wide variety of nucleon dynamics as has been argued by Thomas and his co-workers [8,18]. In fact, this latter approach where the pion :eld enters only as a Kuctuating :eld at the bag boundary with its size comparable to that of the original MIT one has been reasonably successful in a wide range of applications to those nucleon and nuclear processes that are dominated by pionic perturbative e9ects and in which nonpertubative chiral e9ects play a subdominant role [18]. It is now well understood how both the “big cloudy bag” of Ref. [8] and the “little bag” of Ref. [7] can be accommodated in a uni:ed framework of the chiral bag model. In fact, we now know how to shrink the bag even further from the “little bag” and eventually obtain a “point bag”, the skymion [9,10] which can be considered as the limit that the bag size is shrunk to zero. What has transpired from the development since late 1970s=early 1980s up until now is that the bag size that appears in the simple con:nement model is more like a gauge degree of freedom [19,20] and depending upon the gauge choice, it can be of any size without changing basic physics. That physics should not depend upon the con:nement (bag) size is known as “Cheshire Cat phenomenon” and has been amply reviewed in the literature (for an exhaustive list of references, see Refs. [11,21,22]). Unfortunately there is no known exact formulation of this “Cheshire Cat gauge theory”. This is mainly because there is no exact bosonization of QCD known in four dimensions, the skyrmion limit corresponding to the totally bosonized theory, and the chiral bag partially bosonized. Asking what size of the chiral bag is needed for a particular phenomenology is like asking what gauge choice is optimal in gauge theories that are solved only partially as is the case with QCD. In practice, what bag size is optimal depends upon what physical processes are concerned. Thus far our argument has been phrased in the framework of a bag picture. The notion of Cheshire Cat is however much more general than in this restricted context. It is a reKection of a variety of “duality” in the problem which is a statement that there are a variety of di9erent ways of describing the same physical process. In other areas of physics (such as string theories, condensed matter physics, etc.), such a notion can even be formulated in an exact way. In the strong interaction dynamics, however, the notion is at best approximate and hence its predictive power semi-quantitative. Even so, it is suSciently versatile as to be applied to a variety of problems in nuclear physics. As we will describe in later sections, a remarkable recent development in a related context is the “complementarity” of two descriptions in terms of hadrons and in terms of quark=gluon variables for both dilute matter and dense matter as discussed in a later section of this review. This suggests that the Cheshire Cat notion can be generalized to a wider class of phenomena, going beyond the “con:nement size”.

2

A historical note: Simultaneously and independently of Ref. [16], a similar idea was proposed based on phenomenology by Brown et al. [17].

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2.2. The chiral bag Lagrangian The Cheshire Cat phenomenon in our view is a generic feature in modelling QCD or other gauge theories. One may therefore formulate it in various di9erent ways. The use of the chiral bag is not the only way or maybe not even the best way to exploit it. Since we understand the chiral-bag structure better than others, however, we write here the explicit form of the Lagrangian that the chiral bag model takes and that should capture the essential physics of the nucleon. We shall do this for the Kavor SU (3) (with up, down and strange quarks) although the nucleon structure is mainly dictated by the non-strange quarks, with the strange quark playing a minor and as yet obscure role. It can be written generically in three terms, S = SV + SV˜ + S9V :

(2.1)

The :rst term de:ned in the volume V is the “bag” action that reKects the explicit QCD variables, quarks and gluons Ga : 1 4 W + ··· ; (2.2) d x i D== − tr G G SV = 2 V where the trace goes over the color. Here is the quark :eld with the color, Kavor and PoincarXe indices suppressed, G the gluon :eld tensor and D the covariant derivative. The ellipsis stands for quark mass terms that we will leave unspeci:ed. The action for the outside sector occupying the volume V˜ contains relevant physical hadronic :elds of zero baryon charge representing color-singlet e9ective degrees of freedom. It includes the octet Goldstone bosons and non-Goldstone excitations as well as the singlet which acquires mass through U (1)A anomaly and takes the form f2 1 2 4 † † 2 SV˜ = d x Tr9 U 9 U + m [Tr(ln U − ln U )] + · · · + SWZW ; (2.3) 4 V˜ 4NF where NF = 3 is the number of Kavors, Tr goes over the Kavor index and U = ei =f0 e2i =f ; f0 ≡ NF =2f :

(2.4)

Here the ellipsis stands for mass terms and heavy-mass :elds or higher derivative terms, etc. The SWZW is the Wess–Zumino–Witten term [23] that encodes chiral anomalies present in QCD: i j!" Tr(L L L! L L" ) (2.5) SWZW = −Nc 240 2 D5 [9D5 =V ×[0;1]] with L = g† (x; s) dg(x; s) de:ned as g(x; s = 0) = 1 and g(x; s = 1) = U (x). The inside QCD sector and the outside hadronic sector must be connected by an action that “translates” them. This is the role of the surface term. It has two terms, ) S9V = S9(nV) + S9(an V

(2.6)

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with the “normal” boundary term 1 (n ) S9 V = d$ (n W U &5 ) + · · · 2 9V

(2.7)

with U &5 = ei &5 =f0 e2i &5 =f

and the “anomalous” boundary term g2 (an) S 9V = i d$ K5 (Tr ln U † − Tr ln U ) + · · · 32 2 9V where K5 is the “Chern–Simons current” 2 abc a b c () a a K5 = ' G G() − gf G G( G) : 3

(2.8)

(2.9)

(2.10)

Again irrelevant terms are subsumed in the ellipses. The “normal” term (2.7) assures chiral invariance of the model as well as color con:nement at the classical level. The “anomalous” term—which is not gauge invariant—takes care of the quantum anomaly (Casimir) induced inside the bag that would violate color con:nement [24,11] if left un-eliminated. 3 Up to date, nobody has been able to extract the full content of this model. There are some unresolved technical diSculties that frustrate its solvability. Even so, whatever we have been able to learn from it so far has been found to be fully consistent with Nature. We believe that physics of any other viable chiral model, solitonic or non-solitonic (e.g., [27]), is essentially—though perhaps not in detail—captured in the model (2.1). Since there have been extensive reviews on the matter [11], we shall simply summarize the salient features obtained from the model: • Due to a Casimir e9ect in the bag controlled by the surface term (2.7), the baryon charge is fractionized as a function of the bag radius into the V and V˜ sectors such that the conserved integer baryon charge is recovered. The baryon charge fractionization was :rst noticed in 1980 by Vento et al. [28] with a reasoning based on an analogy to the monopole-isodoublet system studied by Jackiw and Rebbi [29] and established later [30] for the special chiral angle (“magic angle”) of the pionic hedgehog :eld and then generalized in Ref. [31] for arbitrary chiral angle. The result is that the con:nement size or the bag radius has no physical meaning although the baryon size does. The “cloudy bag” model [18] ignores the hedgehog component of the pion :eld, so in that picture, the baryon charge will be entirely lodged by :at inside the bag. This implies, in the framework of the chiral bag, that the cloudy bag is forced to be big to be consistent with Nature. The real question here is whether or not such a big bag can capture the essential physics 3

This form of bulk action plus a surface Chern–Simons term for the bag structure has recently been seen to arise from non-perturbative string theory [25]. It turns out that in four dimensions, the interior of the hadronic bag can be identi:ed as the 3-brane and the boundary of the bag as the world-volume of the Chern–Simons 2-brane in which the dynamics is entirely lodged. This theory does not contain quarks as in the model we are considering but the topological term that is added carries the information on the anomaly present in the theory. We should also point out that the chiral bag structure of the sort we are discussing arises from QCD proper in the large Nc limit [26].

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involved. This does not address the question of consistency. On the other hand, the limit where the bag radius is shrunk to zero with the totality of the baryon charge “leaked” into the “pion cloud” describes the skyrmion, with the pion cloud picking up the topological charge. The only issue here is whether or not the point-bag description captures the essential physics involved. There is no inconsistency in this description of the baryon. That the physics should not depend upon the bag radius is an aspect of the “Cheshire Cat Principle” (CCP) [16]. It is the trading between the topological character of the skyrmion con:guration and the explicit fermionic charge of the quarks that makes it satisfy exactly the CCP. It is possible to formulate this notion as a gauge symmetry. Formulated as a gauge symmetry, picking a particular bag radius R is equivalent to :xing the gauge as formulated by Damgaard et al. [20]. We will call this “Cheshire Cat gauge symmetry”. • While the topological nature of the baryon charge implies an exact Cheshire Cat, the Kavor quantum numbers such as isospin, strangeness etc. are not directly linked to topology. Therefore their conservation is not directly connected with topology. They are quantum variables given in terms of collective coordinates. This means that while conserved Kavor quantum numbers in QCD remain conserved after collective quantization (unlike the classically conserved baryon charge in the skyrmion limit), various static quantities such as axial charges gA (both singlet and non-singlet), magnetic moments as well as energies and form factors may not necessarily manifest the perfect CC property. They must depend more or less on the details of the parameters and dynamics taken into account. A good illustration of this feature is the Nc dependence of the theory (where Nc is the number of colors). While the skyrmion sector is strictly valid in the large Nc limit, the bag sector containing the QCD variables encompasses all orders of 1=Nc . This means that in order to assure the CC for the quantities that depend on Nc , the skyrmion sector would have to have appropriate 1=Nc corrections that map to the quark–gluon sector. In practice this means that other hadronic (massive) degrees of freedom than pionic must be introduced into the outside of the bag. What enters will depend on the processes involved. Most remarkably, however, extensive studies carried out in the :eld, chieKy by Park [32], con:rmed that all nucleon properties, be they static or non-static, topological or non-topological, can be satisfactorily and consistently described in terms of the Cheshire Cat picture. Even more surprisingly, the so-called “proton-spin problem” can also be resolved in a simple way within the model (2.1) [33]. For this, the anomalous surface term (2.9) associated with the U (1)A anomaly and the mass :gures importantly. As in other Cheshire Cat phenomena (e.g., the baryon charge), various radius dependent terms conspire to give a more or less radius independent result: Here the cancellation is between the “matter” contribution (from both the quark and ) and the “gauge :eld” (gluon) contribution, each of which is individually bag-radius dependent and grows in magnitude with bag radius. The resulting Kavor-singlet axial charge a0 for the proton shown in Fig. 1 comes out to be small as observed in the EMC=SMC experiments. We should stress that this is not the only explanation for the small a0 . There are several seemingly alternative explanations but our point here is that the Cheshire Cat emerges even in a case where QCD aspects are highly non-trivial and subtle involving quantum anomalies of the chiral symmetry. • So far we have been dealing with long-distance phenomena only. What about shorter-distance physics? Can Cheshire Cat for instance address such short-distance problems as nucleon structure functions seen in deep inelastic lepton scattering where asymptotic freedom is operative and hence

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Fig. 1. Various contributions to the Kavor singlet axial current of the proton as a function of bag radius and comparison with the experiment: (a) the “matter” (quark plus ) contribution (a0BQ + a0 ), (b) the gauge :eld: the static gluon contribution due to quark source (a0G; stat ) and the gluon vacuum contribution (a0G; vac ), and (c) the total (a0total ). The shaded area corresponds to the range admitted by experiments.

QCD should be “visible”? The answer to this is that as one probes shorter distances, the model (2.1) should be modi:ed with more and more higher-dimension operators implied by the ellipsis in the action. There is nothing that suggests that the skyrmion picture when suitably extended should fail if one works hard enough. It is simply that the calculation will get harder and harder and it will be more economical to switch over to the “big bag” or partonic (quark=gluon) picture. Logically, therefore, a hybrid description that exploits both regimes would have a more predictive power. Thus it is easy to understand that the chiral quark–soliton model, a variation on the model (2.1) formulated in a relativistically invariant form [27], has been successfully applied to the problem. We will see later that even at asymptotic density where the QCD variables are the correct variables, the notion of e9ective :elds is applicable. 3. Eective eld theories in nuclear physics 3.1. Chiral symmetry in nuclear processes The next question we wish to address is: Given the Cheshire Cat freedom, how does one treat nuclei, namely, two- or more-nucleon systems? That is, how is nuclear physics formulated in QCD? For this, it is clear that the most eScient and economical way would be to adopt the “point-bag” description. 4 Once we adopt the zero-radius bag surrounded by meson cloud as a nucleon, we can represent it as a local :eld interacting with pions. How to do nuclear physics in terms of 4

In accordance with the Cheshire Cat principle, one could continue using the hybrid bag-meson description by incorporating relevant (hadron) degrees of freedom [34]. One advantage of this approach might be that the quark-bag introduction can implement more readily short-range physics that enters in many-body systems, albeit at a sub-leading level.

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local nucleon and meson :elds and still do QCD is encoded in what is called “Weinberg theorem” [35]. In the present context, the theorem states that doing in a consistent way an e9ective :eld theory with the nucleon, pion and other meson :elds is no more and no less than doing the gauge :eld theory QCD in terms of quarks and gluons. 5 What is required is that such an approach preserve the necessary symmetries—Lorentz invariance, unitarity, cluster decomposition, etc.—and is quantum mechanical. We are simply to write down the most general such local Lagrangian with all possible terms consistent with the symmetries involved. This strategy is powerfully illustrated in the description of the baryon (octet and decaplet baryon) structure in terms of an e9ective chiral Lagrangian constructed of baryon and Goldstone boson :elds [36]. 6 An initial attempt to implement Weinberg theorem in nuclear physics was made in 1981 [37] (it was incomplete in the chiral counting at the time but was completed in 1991 as mentioned later) but the major development came after Weinberg’s paper in 1990 [38]. 3.2. Objectives of EFTs in nuclear physics There are two major roles of EFT in nuclear physics. One is to establish that nuclei as strongly interacting systems—that have been accurately described in the past in what one would call “standard nuclear physics approach (SNPA)” or alternatively “potential model (PM)” based on phenomenological potentials—can also be understood from the point of view of a fundamental theory, QCD, and the other is to be able to make non-trivial and precise predictions that are important not only for nuclear physics per se but also for other areas of physics such as astrophysics and condensed matter physics. The former renders the impressively rich variety of nuclear processes a respectable domain of research. The latter is to provide an invaluable tool for progress in related areas of physics. In this review some recent developments that are relevant to the general theme of this review will be discussed. The examples of remarkable success are the n + p → d + & at threshold [39]—a classic nuclear physics process—and the inverse & + d → n + p [40]—a process relevant to cosmological baryosynthesis. Going beyond the simplest nuclear systems involving two nucleons, the method can be applied to processes that involve n nucleons where n ¿ 2, such as for instance the solar “hep” process [41– 43] 3 He + p → 4 He + e+ + e which :gures in the solar neutrino problem and the axial charge transitions in heavy nuclei (to be discussed later) which provide evidence for BR scaling. 5 To quote Weinberg [35]: “We have come to understand that particles may be described at suBciently low energies by Celds appearing in so-called eDective quantum Celd theories, whether or not these particles are truly elementary. For instance, even though nucleon and pion Celds do not appear in the Standard Model, we can calculate the rates for processes involving low-energy pions and nucleons by using an eDective quantum Celd theory of pion and nucleon Celds rather than of quark and gluon Celds: : : . When we use a Celd theory in this way, we are simply invoking the general principles of relativistic quantum theories, together with any relevant symmetries; we are not making any assumption about the fundamental structure of physics”. 6 While the Lagrangian (2.1) is a model of QCD, the baryon chiral Lagrangian used here is an e9ective theory of QCD. It is therefore reasonable to expect that being a theory, the more judiciously organized and the more terms computed, the more accurate the calculation will become. There is of course the question of convergence but applied within appropriate kinematics, the theory should work out better the more one works. This is indeed what is being found in baryon chiral perturbation calculations for which the strategy is fairly well formulated. This is not the case with “nuclear chiral perturbation theory” where no systematic strategy has yet been fully worked out as we will stress below.

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An exhaustive and beautiful review on the subject with a somewhat di9erent emphasis has recently been given by Beane et al. [44] where a comprehensive list of references is found. In addressing nuclear EFT, there are currently two complementary—and not diDerent as one might be led to believe—ways of organizing the expansion, one based on Weinberg’s strategy [38] and the other based on Seattle–Pasadena strategy [45]. We will loosely refer to the former as “Weinberg scheme” and to the latter as “KSW (Kaplan–Savage–Wise) scheme”. The Weinberg scheme contains the SNPA=PM (standard nuclear approach=potential model) as a legitimate and essential component of the theory whereas the KSW scheme by-passes it, although the latter can be e9ectively exploited to justify the former in certain kinematic domains that we are interested in. We will focus on the former as we :nd it more natural and straightforward from nuclear physicists’ point of view and for the processes we will discuss, sometimes considerably more predictive. The Weinberg scheme [38,46] that we will use here is the version that has been formulated by Park et al. [47,48] (this will be referred to hereafter as PKMR) and uses the following basic strategy. It starts by recognizing that the SNPA=PM (e.g., Refs. [49,50]) based on solutions of the Schr[odinger equation with the “most realistic” potential available in the market :ne-tuned to a large number of experimental data on one and two nucleon systems, captures most accurately the essential physics of, and hence describes, when implemented by many-body forces, the bulk of properties of few-nucleon systems, in some cases (e.g., low-energy scattering) within ∼ 1% accuracy. 7 We will therefore consider the SNPA=PM results as input (e.g., “counter terms”) in lieu of attempting to compute them from :rst principles. Our aim is not to recalculate them from a more fundamental theory but to correctly interpret and incorporate them as a legitimate leading term in a systematic expansion of a more fundamental framework [51,47], namely, an e9ective theory of QCD. This, we believe, is completely in line with the notion of e9ective theory, e.g., the one aM la Wilson. Our key point is that it is in how to do embed, and how to systematically calculate the quantities that are missing from, the SNPA=PM results into a fundamental theory that EFT has its true power. Since the standard nuclear physics approach=potential model (SNPA=PM) results are to be inputs to the more fundamental theory, it is clearly more pro:table to look at the response functions to external :elds rather than at scattering amplitudes (which the SNPA=PM can address accurately) to make meaningful predictions. Our thesis is that this is where chiral symmetry and its broken mode can play their primary role in nuclear physics, in both quantitatively describing and accurately predicting nuclear phenomena. 3.3. EDective chiral Lagrangians For very low-energy processes that we are dealing with, the nucleons can be considered “heavy”, i.e., non-relativistic while the pions are relativistic. We shall therefore use the heavy-baryon formalism 7 There is a well-known caveat in carrying out this program in general and in assessing the “accuracy” of approximations in particular for n-body systems with n ¿ 2. While there are useful constraints that help in organizing, in Weinberg’s terminology, the “reducible” and “irreducible” terms for two-body systems, this is not usually the case for many-body systems. Together with many-body forces in the potential which are diScult to pin down precisely although numerically small, there can be a variety of organizational ambiguities in classifying “reducible” and “irreducible” contributions, e.g., o9-shell e9ects, that could contaminate the calculations resorting to a variety of approximations one is compelled to make, even though a full consistent :eld theoretic approach should in principle be free from such ambiguities. This is an issue that will require considerable e9ort to sort out. We are grateful to Kuniharu Kubodera for stressing this point in the context of the PKMR approach.

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although a relativistic formulation can be made, particularly for the case where the nucleon mass drops as in BR scaling that we will discuss below. The leading-order chiral Lagrangian that consists only of local nucleon and pion :elds—with other heavy :elds integrated out—takes the form f2 1 L0 = NW [iv · D + 2igA S · 2]N − CA (NW 4A N )2 + f 2 Tr i2 i2 + Tr(5+ ) (3.1) 2 A 4 with D N = (9 + 4 )N ; 1 i i 4 = [6† ; 9 6] − 6† R 6 − 6L 6† ; 2 2 2 1 i i 2 = [6† ; 9 6] + 6† R 6 − 6L 6† ; 2 2 2 5+ = 6† 56† + 65† 6 ; a

=2)(Va

(3.2) Aa )

a

=2)(Va

Aa )

+ and L = (7 − denote, respectively, the left and right where R = (7 external gauge :elds, 5 is proportional to the quark mass matrix and if we ignore the small isospin-symmetry breaking, becomes 5 = m2 in the absence of the external scalar and pseudo-scalar gauge :elds, and √ ˜7 · ˜ 6 = $ = exp i (3.3) 2f is the chiral :eld for the Goldstone bosons. The L0 is the leading order chiral Lagrangian in the sense that it is leading order in derivatives, in pion mass and in “1=M ” where M is the free-space nucleon mass. This is a suitable Lagrangian for (chiral) perturbation near the medium-free vacuum described by the so-called “irreducible graphs”. However for nuclear processes in general, certain “reducible graphs” involve propagators that are infra-red enhanced and require terms higher order in 1=M , namely, nucleon kinetic energy term. For the KSW approach, the ab initio account of this “correction” term is essential. For the PKMR where the reducible graphs are accounted for via Schr[odinger equation or Lippman–Schwinger equation, the chiral expansion is all that matters, so this Lagrangian plays a key role. The second term in (3.1) is the leading four-fermion interaction and contains no derivatives. We will specify the explicit form later. For convenience, we will work in a reference frame in which the four velocity v and the spin operator S are ˜" ˜ : (3.4) v = (1; 0) and S = 0; 2 The next-to-leading order Lagrangian including four-fermion contact terms can be written as v v − g gA g2 D D + {S · D; v · D} + c1 Tr 5+ + 4 c2 − A (v · i2) L1 = NW 2mN mN 8mN 1 [S ; S ] [i2 ; i2 ] N + 4c3 i2 · i2 + 2c4 + 2mN − 4id1 NW S · \N NW N + 2id2 jabc j!: v 2; a NW S ! 7b N NW S : 7c N + · · · ;

(3.5)

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Fig. 2. Graphs contributing to the scattering amplitude

where j0123 = 1, 2 = (7 a =2)2a . Terms that are not relevant for the present discussion are implied in the ellipses. 3.4. Nucleon–nucleon scattering Although the genuine predictivity of the EFT is in response functions, we start with nucleon– nucleon scattering at very low energies as this issue has been the focus of the recent activity in the :eld. If the energy scale involved is much less than the pion mass, m ∼ 140 MeV, then we might integrate out the pion :eld and work with an e9ective theory containing only the nucleon :eld. We will reinstate the pions later since the PKMR strategy relies on the pion degree of freedom explicitly. Since the pionless theory has interesting aspects on its own that merits study, we will consider this before putting the pion :eld into the picture. When the pions are integrated out, the Lagrangian (3.1) further simpli:es to 2 ˜ 1 ∇ L = N † i9t + N + · · · − [C s (N † N )2 + C t (N † N )2 + · · · ] ; (3.6) 2M 2 where the nucleon kinetic energy term (which is formally higher order from the heavy-baryon chiral counting) is reinstated. With the pion :elds removed from the theory, this Lagrangian has no remnant of chiral symmetry. The nucleon :eld is just a matter :eld and does not know anything about the symmetry. This does not mean that the theory is inconsistent with the symmetry of QCD. We shall consider the S-wave channels 1 S0 and 3 S1 (in the notation of 2S+1 SJ ). The C coeScients for these channels are related to the C coeScients in (3.6) by 1

C ( S0 ) = C s − 3C t ;

3

C ( S1 ) = C s + C t :

(3.7)

The scattering amplitude is A given by the sum of the Feynman diagrams given in Fig. 2: A=−

C ; 1 − C(GG)

(3.8)

where C is the four-fermion interaction constant in (3.6) and (GG) is the two-nucleon propagator connecting the two vertices, both written symbolically. In the center-of-mass (CM) frame, this propagator has a linear divergence in dimension D = 4 as one can see from a naive dimensional counting of i i d4 q (GG) = −i (2 )4 E + q0 − q2 =2M + ij −q0 − q2 =2M + ij d3 q 1 = ; (3.9) (2 )3 E − q2 =M + ij

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√ where |k| = ME is the CM momentum. To make the integral meaningful, let us put a momentum cuto9 at say at ?. Then (3.9) is given by M ? + i|k| : (3.10) (GG)? = − 4 This linear divergence poses no fundamental problem here since the theory is an e9ective one with the degrees of freedom above a certain cuto9 having been integrated out. On the contrary, in the Wilsonian sense, this cuto9 dependence has a physical meaning: It delineates the onset of “new physics”. Substituting this into the amplitude formula (3.8), we have −C A= : (3.11) 1 + (CM=4 )(?= + i|k|) Next, we trade in the theoretical constant C for a physical quantity. To do this, the scattering amplitude (for S wave) is written in terms of the phase shift : ≡ :S as 1 4 : (3.12) A= M |k| cot : − i|k| Now using the e9ective range formula 1 1 |k| cot : = − + r0 |k|2 + O(|k|4 ) ; (3.13) a 2 where a is the scattering length and r0 the e9ective range, we have, to the lowest order in momentum, 1 4 : (3.14) A=− M 1=a + i|k| So from (3.11) and (3.14), we :nd 4 1 C(?) = M −?= + 1=a

(3.15)

where we have explicited the cuto9 dependence of the e9ective Lagrangian which shows how an e9ective theory carries the “memory” of the degrees of freedom that have been integrated out. Obviously physical observables should not depend upon what value one takes for the cuto9 ?, since the cuto9 can be chosen arbitrarily. This is a statement of renormalization group (RG) invariance of the physical quantities, which reads for the amplitude as dA ? =0 : (3.16) d? One can easily verify that (3.15) is the solution of this renormalization group equation (RGE). In standard :eld theory calculations in particle physics, one uses dimensional regularization (DR) instead of cuto9. This is because the DR has the advantage that it preserves symmetries, e.g., chiral symmetry in our case, order by order. In the case we are concerned with, with no pions, no symmetry is spoiled by the cuto9 regularization at least up to the order considered. In general when one does higher order calculations, however, it is not a straightforward matter to preserve chiral symmetry with the cuto9 regularization since it requires judicious inclusion of symmetry breaking counter terms so as to restore symmetry. This is why some authors (e.g., the KSW collaboration) prefer using DR. Now what does the above linear divergence mean in DR?

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For this we go back to the propagator (GG), Eq. (3.9), and work out the integral using DR:

4−D d D q i i (GG) = −i 2 (2 )D E + q0 − q2 =2M + i' −q0 − q2 =2M + i'

4−D d D−1 q 1 = 2 (2 )D−1 E − q2 =M + i' (=2)4−D 3−D 2 (D−3)=2 = M (−|k| ) 4 : (3.17) 2 (4 )((D−1)=2) This is regular for D = 3 but singular for D = 3. In renormalizable :eld theories where power divergences are absent in principle, it is legitimate to simply set D = 4, thereby dropping power divergences. But in non-renormalizable theories, one cannot sweep under the rug the singularities that lurk behind the physical dimension. This is because in the Wilsonian renormalization scheme to which our EFT belongs, there are no true singularities and as mentioned, the power divergence structure instead carries physical information. As mentioned, the situation here is quite analogous to the quadratic divergence that needs to be included in hidden local symmetry (HLS) theory mentioned above. 8 The physics behind may be the same. In fact, this must be a generic feature of all e9ective :eld theories of the sort we are dealing with here, that is, the theories that have :xed points nearby. To do a correct regularization with DR in the present case, the linear power divergence— the only power divergence there is in this theory at one loop—has to be subtracted. In this sense, the “power divergence subtraction (PDS)” scheme used by KSW is a proper implementation of Wilsonian e9ective :eld theory in nucleon–nucleon scattering, the unnatural length scale associated with the large scattering length signaling the presence nearby of an infrared :xed point. Subtracting the linear divergence, we have the power-divergence-subtracted (PDS) propagator (GG)PDS = (GG) −

M M = − ( + i|k|) : 4 (D − 3) 4

(3.18)

Comparing with (3.10), we see that the scale parameter in DR is equal to the cuto9 ?= . 1 As noted by Ref. [52], the large scattering length for the S-wave scattering, i.e., a( S0 ) =−23:714 fm ∼ (1=(8 MeV)) reKects that nature is close to a :xed point and conformal invariance. This is seen in (3.15) in the limit 1=a → 0: ?

d (?C)|1=a→0 = 0 : d?

(3.19)

In view of the length scale involved, a → ∞ is close to nature, signaling a quasi-bound state. Note that the linear divergence subtraction plays a crucial role here, a feature analogous to the conformal invariance of hidden local symmetry (HLS) theory at chiral restoration with the quadratic divergence playing a crucial role [12]. The discussion made up to this point is general, clarifying the essential feature of e9ective :eld theories in two-body nuclear systems. The KSW approach has been developed further, including 8

Taking into account the power divergence (i.e., quadratic divergence) in HLS is crucial in arriving at the vector manifestation of chiral symmetry of Harada and Yamawaki [12] discussed in Section 4 and also in Section 11.2.

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pions as perturbation [44]. As we will show below, the PKMR strategy, while faithful to the EFT strategy, departs from this point in that the rigorous counting rule is sacri:ced somewhat in favor of predictivity. 3.5. The PKMR approach: more eDective EFT The faithful adherence to the counting rule has the short-coming in that while post-dictions are feasible provided enough experimental information is available, it is hard if not impossible to make accurate predictions. The reason is that as one goes to higher orders to achieve accuracy, one encounters increasing number of counter terms that remain as unknown parameters and these parameters can at best be :xed by the very process one would like to predict. This is just a :tting, and not a prediction. Thus instead of adhering strictly to the counting rule that requires by-passing wave functions, we shall develop a scheme along the line conceived by Weinberg that would allow us to exploit the wave functions that are “accurately” determined from the standard nuclear physics=PM procedure. The key observation that we will exploit is the following. All nuclear processes involve two sorts of graphs: “reducible” and “irreducible”. In the case of two-body systems, they will be two-particle reducible and two-particle irreducible (2PI). Instead of summing the two classes of graphs order-by-order in strict accordance with a given counting scheme as in KSW, one computes irreducible graphs to a given order in chiral perturbation theory (5PT) and then account for reducible graphs by solving Schr[odinger equation or Lippman–Schwinger equation with the irreducible vertex injected into the equation. This procedure accords to potentials and wave functions a special role as in the standard nuclear physics calculations. What we hope to do is then to combine the accuracy of the standard nuclear physics=potential model approach with the power of 5PT. We might call this more eDective e9ective :eld theory (MEEFT). The advantage of working with wave functions is that one can make predictions in nuclear electro-weak response functions that are diScult to make in an order-by-order calculation because of un-:xed counter terms. The disadvantage is of course that one is forced to sacri:ce the strict adherence to a counting rule. It turns out however that in the processes so far studied with controlled approximations, this sacri:ce is minor numerically. A clear explanation as to how this comes about in two-body systems was given by Cohen and Phillips [53]. 3.5.1. NN scattering To justify the method that we will apply to response functions, we :rst discuss how the method works out in nucleon–nucleon scattering which is well understood in various di9erent ways, :eld theoretical or non-:eld theoretical. Compared with the KSW approach, the present scheme is a bit less elegant but the point of our exercise is to show that it can be at least as accurate as the KSW scheme [53]. For simplicity consider proton-neutron scattering in the 1 S0 channel. 9 Since we are to account for the reducible graphs by Lippman–Schwinger equation, we only need to compute the potential 9

The proton–proton interaction as in the solar proton fusion process requires the Coulomb interaction in addition to the QCD interaction. Although technically delicate, this part is known, so we will not go into the detail. We shall also restrict to the 1 S0 channel. The 3 S1 channel is slightly complicated because of the coupling to the D-wave but involves no new ingredient.

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as the sum of irreducible graphs to a certain chiral order in the Weinberg counting. For this we can work directly with the heavy-baryon chiral Lagrangian (3.1) with the pion exchange put on the same footing as the contact four-fermion interaction. The potential calculated to the next-to-leading order (NLO) is of the form V(q) = −71 · 72 with 3 " =√ 8 ij

gA2 "1 · q "2 · q 4 [C0 + (C2 :ij + D2 "ij )qi qj ] + · · · + 4f 2 q2 + m2 M

:ij "1i "2j + "1j "2i − "1 · " 2 2 3

(3.20)

;

(3.21)

where q is the momentum transferred. The C2 and D2 terms containing quadratic derivatives are NLO contributions that are added to (3.6) since we will work to that order. There are one-loop corrections involving two-pion exchange that appear at the NLO and that can be taken into account but as shown by Hyun et al. [54], in the kinematic regime we are considering here, they are not important for the key point we want to discuss here. As it stands, the potential (3.20) is not regularized, so it has to be regularized just as the Feynman propagator (GG), Eq. (3.9), had to be. One can do this in various di9erent ways; they should all give the same result. A particularly convenient one for solving di9erential equations in coordinate space is to take the following form when one goes to coordinate space d 3 q i q ·r V (r) ≡ e S? (q2 )V(q) ; (3.22) (2 )3 with a smooth regulator S? (q2 ) with a cuto9 ?. For our purpose it is convenient to take the Gaussian regulator, q2 2 (3.23) S? (q ) = exp − 2 : ? In the PKMR formalism, pions play a special role due to what is called “chiral :lter mechanism” (to be de:ned below) but we can illustrate our essential point without the pions, so let us drop the pion exchange term for the moment. Given the Fourier-transformed potential properly regulated as mentioned, the Lippman–Schwinger equation for the S-wave can be solved in a standard way. The resulting wave function can be explicitly written down:

√ √ S(ME=?2 )CE ZC2 2 ZD2 S12 (r) ˆ 9 1 9 ˜ √ r (r) = ’(r) + 4? (r) ; 1− (∇ + ME) − 1 − 4 E CE CE CE 8 9r r 9r (3.24) where ’ is the free wave function and S?2 (p) d3 p ; 4E = 4 (2 )3 ME − p2 + i0+

(3.25)

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4˜ ? (r) = 4

S? (p) d3 p ei p · r ; 3 (2 ) ME − p2 + i0+

Z = (1 − C2 I2 )−2 ; C E = a?

1 1 + a? r? ME 2

103

(3.26) (3.27)

√ + ( ZD2 ME)2 4E ;

(3.28)

with a? ≡ Z[C0 + (C22 + :S; 1 D22 )I4 ] ; r? ≡

2Z [2C2 − (C22 − :S; 1 D22 )I2 ] a2?

where In (n = 2; 4) are de:ned by ?n+1 ∞ d x xn S 2 (x2 ) : In ≡ − −∞

(3.29) (3.30)

(3.31)

The phase shift can be calculated by looking at the large-r behavior of the wave function,

1 − 2 (E) 1 I? (E) − p cot : = 2 ; (3.32) S (ME=?2 ) a? (1 + 12 a? r? ME) √ where the (E) is the D=S ratio, which vanishes for the 1 S0 channel and I? (E) = 4E + i MES 2 × (ME=?2 ) ≡ ?I (ME=?2 ). The expression for the phase shift contains a lot more than what was considered above even at LO since the wave function is computed to all orders in the potential and we are using a Gaussian cuto9. As such it is not obvious that the renormalization group invariance, i.e., the ? independence, is preserved. This may appear to be a shortcoming of the approach and would constitute a defect if the ? dependence were non-negligible. Let us see how this works out with the present pionless example. As before, we use the e9ective range formula (3.13) to :x the coeScients C0; 2 . We have 1 ? 1 1 = + ?I (0) = − √ ; a? a a r ? = re −

2 2I (0) 4S (0) 4 + = re − √ : − 2 ? a? ? a?2

(3.33) (3.34)

At threshold, it can be easily veri:ed that the renormalization-group (RG) invariant form for C0 , (3.15), is recovered. What happens away from the threshold is less trivial. For illustration we take the CM momentum p = 68:5 MeV corresponding to ∼ m =2. The result for the 1 S0 channel is given in Fig. 3; here the phase shift : (in degree) is plotted vs. ?. We learn from this exercise that within the present scheme it is clearly inconsistent with the renormalization group invariance if the potential is calculated only up to LO. It is at NLO and ? & m that the cuto9 independence is recovered. We see that ? ∼ m is the scale at which a “new degree of freedom” enters into the pionless theory, an obvious but important fact.

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Fig. 3. Cuto9 dependence=independence in pionless theory: np 1 S0 phase shift (degrees) vs. the cut-o9 ? for a :xed CM momentum p = 68:5 MeV. The solid curve represents the NLO result, the dotted curve the LO result and the horizontal dashed line represents the experimental value.

Fig. 4. The np 1 S0 phase shift (degrees) vs. the cuto9 ? for :xed CM momenta p = 70 MeV (left) and p = 140 MeV (right). The next-to-leading order (NLO) results are given by the solid (with pions) and dashed (without pions) curves, and the leading-order (LO) results by the dot–dashed (with pion) and dot–dot–dashed (without pion) curves. The horizontal line represents the experimental data obtained from the Nijmegen multi-energy analysis. The NLO result without pions is drawn only up to ? & 300 MeV, above which the theory becomes meaningless, that is, unnatural as well as inconsistent.

Next we incorporate pions. Once pions are included, the cuto9 should be put above m . Furthermore the pion presence should reduce the ? dependence—if any—for given CM momenta. This feature is seen in Fig. 4. We see clearly that the pion presence improves markedly the ability to describe scattering. The interplay between the probe momentum, the pion presence, the cuto9 and the chiral order becomes transparent: The more re:ned the potential and the higher the chiral order, the more consistent and more accurate becomes the prediction. This will be the key point of our next consideration when we look at response functions. 3.5.2. Electroweak response functions: accurate post-dictions and predictions We now turn to our principal assertion that it is in calculating nuclear response functions to the electroweak (EW) external :elds that the PKMR approach has its power. We will :rst discuss two-body systems and then make predictions for n-body systems with n ¿ 2. In doing this, we will put pions to start with since we will exploit what is known as “chiral :lter mechanism” :rst

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introduced in Ref. [51] and elaborated further in Ref. [48]. The chiral :lter mechanism which has been veri:ed in 5PT states that whenever allowed by symmetry and kinematics, one-soft-pion exchange in the electroweak currents dominates, with the corrections to which are in principle controllable by 5PT and, conversely, whenever forbidden by symmetry or suppressed by kinematics, the corrections to the leading order current are uncontrolled by chiral expansion and require going beyond standard low-order 5PT. We will refer to these two phenomena as “two sides of the same coin”. In this formalism, therefore, pions play a prominent role. At a momentum scale pm , one can work with a pionless EFT as described in Ref. [44] in which the role of the chiral :lter would of course be moot. However in this case, one loses the predictive power since one would be left with one or more counter terms that are left undetermined by theory or experiments. The PKMR strategy circumvents this diSculty. The PKMR strategy is quite simple. We take the most sophisticated wavefunctions from the SNP (standard nuclear physics)=PM (potential model) approach with the potential :t to an ensemble of empirical data on two-body systems and compute the EW currents using the chiral Lagrangian (3.1) to as high an order as possible in the chiral counting. Such realistic potentials—which have been extensively studied and con:rmed empirically—can then be interpreted as resulting from high-order 5PT calculations. For two-body systems, we use as the input the most “sophisticated” potential available in the literature, i.e., the Argonne v18 potential—and the resulting wavefunctions [50]. For making predictions in n-body systems with n ¿ 2, we rely on Ref. [49]. The power of these wavefunctions in confronting few-body nuclei is well documented in the literature. In all cases, the scattering and static properties of the nuclei involved are extremely accurately given and more or less justi:ed within 5PT whenever many-body corrections are unimportant. Whenever many-body corrections are important, on the other hand, they can in turn be calculated by 5PT as we shall show using the chiral :lter argument. In all cases studied up to date, it is possible to make error estimates by assessing the deviation from the RG invariance, that is, the ? dependence. Before calculating the full amplitudes to confront nature, we :rst verify that using the “realistic wave function” is indeed consistent with actually calculating the wavefunctions systematically in the chiral counting starting with (3.1) or (3.6). For this purpose, it suSces to look at the single-particle M 1 matrix elements for the process n+p→d+&

(3.35)

and the single-particle Gamow–Teller matrix element in the solar proton fusion process p + p → d + e + + e

(3.36)

and verify that there is a matching between a bona-:de EFT and the hybrid scheme we are using. We are interested 10 in E(M 1) and E(GT ) de:ned by E(M 1) ≡

th − Mv18 MM 1 M1 ; v18 MM 1

E(GT ) ≡

th − Mv18 MGT GT ; v18 MGT

(3.37)

th and Mv18 denote, respectively, the single-particle M 1 transition matrix element of our where MM 1 M1 NLO calculation considered above for the np scattering and that of the Argonne potential, and 10

These are of course not the whole story since there are two-body corrections known as “meson-exchange currents” that we will consider shortly.

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Fig. 5. E(M 1) (upper) and E(GT ) (lower) vs. the cuto9 ?. The solid curves represent the NLO results with pions and the dotted curves without pions. When pions are included, the physically relevant cuto9 must lie between the pion mass m and the lowest resonance, say, m" ∼ 450 MeV.

similarly for E(GT ). Here we are taking the Argonne potential as the most accurate and closest to “experiment”. In Fig. 5 are plotted the deviations from the Argonne result as a function of the cuto9. We see that as in the case of the phase shift for p ∼ m , the inclusion of the pion degree of freedom markedly reduces the cuto9 dependence, in conformity with the stated requirement for a viable e9ective theory. Leaving the details to the literature [48,39], we merely summarize a few concrete results, both post-dictions and predictions, of full calculations to NLO or NNLO including pions. Some technical details including precise de:nitions of the quantities involved can be found in the next subsection immediately following this where we address the question of “hard-core correlations” invoked extensively in standard calculations dealing with short-distance nuclear interactions. • Post-diction The most celebrated case of post-diction is the thermal np capture process (3.35). The unpolarized cross-section is dominated by the M 1 operator—de:ned below—that is protected by the “chiral :lter” and has been calculated in the PKMR approach with the Argonne v18 to the accuracy of ∼ 1% [39], "th = 334 ± 3 mb

(3.38)

to be compared with the experimental value "exp = 334:2 ± 0:5 mb. The error bar on the theoretical value represents the range of uncertainty in the short-distance physics unaccounted for in the NNLO chiral perturbation series that is reKected in the cuto9 dependence. Here the correction to the single-particle M 1 matrix element is dominated by the soft-pion contribution with the remaining corrections amounting to less than 10% of the unsuppressed soft-pion term in conformity with the chiral :lter argument. Since this calculation has no free parameters, it is actually a prediction whereas the KSW calculation without pions has one unknown counter term

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as the leading correction term [55]. 11 The upshot of the 5PT result aM la PKMR is that it con:rms the result of Riska and Brown [5] and in addition supplies the corrections to their result with an error estimate, which is the ultimate power of the approach as far as post-dictions are concerned. Now having explained accurately the threshold np capture, one can then proceed to make predictions for the inverse process d + & → n + p for which cross section data are available as well as for the process e + d → n + p + e as a function of momentum transfer. A good agreement with experiments is obtained [56] for the photodisintegration of the deuteron d + & → n + p. Remarkably the simple theory seems to work also for the latter to a momentum transfer much greater than justi:ed by the cuto9 involved [6]. • Predictions To bring our point home, we cite a few cases of predictions that precede experiments. For these predictions, the 5PT Lagrangian (3.1) is used for calculating the irreducible graphs entering in the electroweak currents. 1. While the unpolarized cross-section for the np capture process is dominated by the isovector M 1 operator that is protected by the chiral :lter mechanism, the process ˜n + p ˜ →d+&

(3.39)

with polarized neutron and proton can provide information of the suppressed isoscalar M 1 and E2 matrix elements. The isoscalar M 1 matrix element turns out to be particularly interesting because a precise calculation of this matrix element requires implementing “hard-core correlations” of standard nuclear physics in terms of regularization in :eld theory. Although the understanding is still poor, the issue presents a :rst indication of how the short-distance prescription encoded in the hard core may be understood in a more systematic way. As noted more precisely below, the matrix elements involved are suppressed by three orders of magnitude with respect to the isovector M 1 matrix element—due to the fact that these are unprotected by the chiral :lter mechanism, so cannot be “seen” directly in the cross section but can be singled out by polarization and anisotropy measurements (see below for de:nitions). The ratios with respect to the isovector M 1 are predicted to be RM 1 = (−0:49 ± 0:01) × 10−3 ;

RE2 = (0:24 ± 0:01) × 10−3 :

(3.40)

There are no data available at the moment; in this sense these are true predictions. The error bars represent the possible uncertainty in the cuto9 dependence, indicating the uncertainty in the short-distance physics. We should mention that the identical results were obtained in the KSW scheme, with however, somewhat bigger error estimates [44]. 2. The next prediction is on the proton fusion process (3.36). Here the dominant matrix element involved, i.e., Gamow–Teller, is unprotected by the chiral :lter but it turns out that the single-particle matrix element which can be precisely calculated in the standard nuclear physics=PM approach with accurate wave two-nucleon wave functions dominates. The remaining correction terms are not protected by the chiral-:lter mechanism and hence can be 11

In order to explain the experimental value in this calculation, the counter term turns out to require to be considerably bigger in magnitude than, and to have a sign opposite to that of, the soft-pion correction of Ref. [39]. This seems to imply that the counter term that appears in this pionless theory describes a di9erent physics than the chiral-:lter protected mechanism.

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subject to some uncertainty although numerically small [57]. It turns out however that to next-to-next-to-leading order (NNLO) it is possible to make an extremely accurate totally parameter-free estimate of the correction [58]. The quantity that carries information on nuclear dynamics is the astrophysical S factor, Spp = "(E)EE 2 where " is the cross section for the process (3:36) and = M(=(2p). The prediction for S is [58] Spp (0) = (3:94 × 10−25 )(1 ± 0:0015 ± 0:001) MeV barn

(3.41)

where the :rst error is due to the uncertainty in the one-body matrix element and the second due to the uncertainty in the two-body matrix element that is truncated at the NNLO. Higher-order corrections are expected to be completely negligible. This probably is the most accurate prediction anchored on the “fundamental principle” achieved in nuclear physics. In the solar neutrino problem, the S factor is an input as a Standard Model property. In view of the important role that the pp fusion process will play in the future measurements of the solar neutrinos, this accurate Standard Model prediction will prove valuable for the issue. 3. So far we have been concerned with two-nucleon systems that can be more or less equally well treated by both the Kaplan–Savage–Wise (KSW) approach and the Park–Kubodera–Min–Rho (PKMR) approach. We now turn to a case where the PKMR can make a possible prediction [42,43] whereas a strict adherence to the counting rule would frustrate the feat due to too many unknown parameters. This is the solar “hep” problem p + 3 He → 4 He + e+ + e :

(3.42)

This process produces neutrinos of the maximum energy E ≈ 19:8 MeV; for this lepton momentum, not only the S wave but also the P wave need to be taken into account. Computing the neutrino Kux as needed for the solar neutrino problem requires both vector and axial-vector matrix elements. While the vector matrix element involves no subtleties—and hence is straightforwardly calculated (including two-body currents), the principal matrix element of the axial-vector current turns out to be extremely intricate. Since the P wave enters non-negligibly, the time component of the axial current cannot be ignored. However this part of the current is under theoretical control since the axial charge operator is chiral-:lter protected [51,47]: the corrections to the known single-particle axial-charge matrix element are dominated by one-soft-pion-exchange two-body matrix elements (with three-body operators absent to the order considered) and are accurately calculable, given the “accurate” wave functions available in the literature. The principal diSculty is in the matrix element of the space component of the axial current, the most important of which is the Gamow–Teller operator. (Three-body contributions are higher-order than those coming from one-body and two-body terms and can be ignored.) Normally the matrix element of the single-particle Gamow–Teller operator gA i 7± (i)"a (i) is of order 1 if the symmetries of the initial and :nal nuclear states are normal since the operator just Kips spin and isospin. However in the “hep” case, the dominant components of the initial state |p3 He and the :nal state |4 He have di9erent spatial symmetries so the overlap is largely suppressed. One :nds, numerically, that to the chiral order that is free of unknown parameters (e.g., next-to-next-to leading order, NNLO, or O(Q3 )), the single-particle GT matrix element comes out to be suppressed by a factor of ∼ 3 relative to the normal matrix element. Now main corrections to the one-body Gamow–Teller are expected to be from two-body terms with three-body terms down by a chiral order. However these

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corrections are chiral-:lter-unprotected as we have explained and hence cannot in general be calculated accurately in chiral perturbation theory unlike the corrections to the single-particle axial charge matrix elements. Furthermore, the two-body corrections typically come with a sign opposite to the single-particle one with a size comparable to the main term, thus causing a drastic further suppression. This shows why it is essential to be able to calculate in a controlled fashion higher chiral order corrections to the GT matrix element. Such a calculation turns out to be feasible given accurate wave functions for three- and four-nucleon systems [41]. The key point is that the calculation of this chiral-:lterunprotected term parallels closely the calculation of the suppressed isoscalar M 1 and E2 matrix elements in the polarized np capture mentioned above, thereby enabling the PKMR approach to pin down the most problematic part of the “hep” process within a narrow range of uncertainty dictated by the range of the cuto9 involved [42]. Referring for details to the literature [42,43], we simply summarize what enters in the calculation. The :rst is that as in the case of the pp case, there are no unknown parameters once the accurate data on triton beta decay are used. The second is that the strategy for controlling the chiral-:lter-unprotected two-body corrections to the Gamow–Teller matrix element parallels that for the pp fusion process, i.e., the renormalization-cuto9 interpretation of what is known in nuclear physics as “hard-core correlation” discussed below. The S factor so obtained [43] is Shep (0) = (8:6 × 10−20 )(1 ± 0:15) MeV barn :

(3.43)

The error bar stands for uncertainty due to the cuto9 dependence which signals the uncertainty inherent in terminating the chiral series at NNLO. The uncertainty is bigger here than in the pp case since one is dealing with four-nucleon systems. Higher-order corrections are expected to considerably reduce the error bar. A similar calculation in a standard nuclear physics approach based on the traditional treatment of exchange currents [4] has been performed by Marcucci et al. [41]. This approach uses the same accurate wave functions as in Ref. [43], without however resorting to a systematic chiral counting. That the standard nuclear physics approach of Ref. [41] agrees with (3.43) within the error bar suggests that the standard approach of Ref. [41] may well be consistent—to the order considered—with the EFT approach to the “short-range correlation”. 3.5.3. The chiral Clter: two sides of the same coin While the processes protected by the chiral :lter, such as the M 1 operator and the axial charge operator [51], are accurately calculable since one-soft-pion exchange dominates with correction terms suppressed typically by an order of magnitude relative to the dominant soft-pion exchange, the situation is not the same for those processes that have no chiral-:lter protection. Even so, the PKMR approach can still make a meaningful statement on such processes. That is the story of the other side of the same coin. If the chiral :lter does not apply, then the role of the pion is considerably diminished and, compounded with higher order e9ects, clean and reliable low-order calculations will no longer be feasible. A prime example of this situation is the process p + p → p + p + 0 . Here one-soft-pion exchange is suppressed by symmetry and a variety of correction terms of comparable importance

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compete in such a way that no simple 5PT description can be obtained. Both the PKMR and the KSW are unsuccessful for this process. The proton fusion process (3.36) presents a di9erent situation at least in the PKMR approach [57]. Since the process is essentially dominated by the Gamow–Teller operator with no accidental suppression, the single particle matrix element in the standard nuclear physics (SNP)=potential model (PM) approach simply dominates. On the other hand, since one is dealing with low-energy (or momentum) probes with the relevant scale much less than the pion mass, one may simply integrate out the pion and work with a pionless e9ective Lagrangian as in Ref. [59]. However the price to pay here is to face higher-order counter terms. We learn from the chiral :lter that these corrections cannot be computed in a low-order expansion. This is because those terms reKect the physics of a distance scale shorter than that can be captured in a few perturbative expansion. While in the PKMR approach, the counter terms of high order that are required are numerically small compared with the leading single-particle term given in the SNP=PM approach, with no help of the chiral :lter, there is nothing that says that subleading corrections are necessarily small compared with the leading-order correction that one may be able to calculate. For example in high-order KSW calculations in the “pionless EFT” for the proton fusion (see Ref. [59]), the corrections occur at :fth order. Now even if the counter terms that appear at that order are—eventually—:xed by the inverse neutrino processes that may be measurable in the laboratories, the error incurred in this :fth-order calculation will not be any smaller than what the PKMR approach can obtain at a lower order. 3.5.4. An interpretation of hard-core correlation in an eDective Celd theory The isoscalar matrix elements of the polarized neutron and proton capture process (3.39), ˜n + p ˜ →d+&

(3.44)

are not chiral-:lter-protected. Nonetheless it was possible to make the predictions (3.40). How this comes about is interesting from the point of view of learning something about short-range correlations in nuclei. We elaborate on this point in the rest of this section. At threshold, the initial nuclear state in (3.44) is in either the 1 S0 (T = 1) or 3 S1 (T = 0) channel, where T is the isospin. The process therefore receives contributions from the isovector M 1 matrix element (M 1V ) between the initial 1 S0 (T = 1) and the :nal deuteron (T = 0) state, along with the isoscalar M 1 matrix element (M 1S) and the isoscalar E2 (E2S) matrix element between the initial 3 S1 (T = 0) and the :nal deuteron 3 S1 − 3 D1 states. Since M 1V is by far the largest amplitude, the spin-averaged cross section "unpol (np → d&) is totally dominated by M 1V . Meanwhile, since the initial 1 S0 state has J = 0, the M 1V cannot yield spin-dependent e9ects, whereas M 1S and E2S can. The T matrix for the process can be written as

ˆ

d ; &(k; !)|T| np

with ˆ !) = M(k;

√

ˆ !) 5np = 5d† M(k;

(3.45)

√ vn ∗ i(kˆ × jˆ! ) · (1 − 2 ) M 1V 4 √ 2 !As

(M 1S) ∗ ˆ 2 · jˆ∗! + 2 · k ˆ 1 · jˆ∗! ) E2S √ − i(kˆ × jˆ! ) · (1 + 2 ) √ + (1 · k 2 2

;

(3.46)

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where vn is the velocity of the projectile neutron, As is the deuteron normalization factor As 0:8850 fm−1=2 , and 5d (5np ) denotes the spin wave function of the :nal deuteron (initial np) state. ˆ the energy ! and the helicity !, The emitted photon is characterized by the unit momentum vector k, ˆ The amplitudes, M 1V , M 1S and E2S, represent and its polarization vector is denoted by 'ˆ! ≡ 'ˆ! (k). the isovector M 1, isoscalar M 1 and isoscalar E2 contributions, respectively, all of which are real at threshold. These quantities are de:ned in such a manner that they all have the dimension of length, and the cross section for the unpolarized np system takes the form "unpol = |M 1V |2 + |M 1S|2 + |E2S|2 :

(3.47)

The isoscalar terms (|M 1S|2 and |E2S|2 ) are strongly suppressed relative to |M 1V |2 —approximately by a factor of ∼ O(10−6 )—so the unpolarized cross-section is practically una9ected by the isoscalar terms. As mentioned above, the isovector M 1 amplitude was calculated [39] very accurately up to O(Q3 ) relative to the single-particle operator. The result expressed in terms of M 1V is: M 1V = exp 5:78 ± 0:03 fm, which should be compared to the empirical value "unpol = 5:781 ± 0:004 fm. Here, we will focus on the isoscalar amplitudes. Now the isoscalar matrix elements can be isolated by spin-dependent observables, namely by the ◦ ◦ ◦ ◦ photon circular polarization P& ≡ (I+1 (0 ) − I−1 (0 ))=(I+1 (0 ) + I−1 (0 )) and the anisotrophy & ≡ ◦ ◦ ◦ ◦ (I (90 ) − I (0 ))=(I (90 ) + I (0 )) where I! (H) is the angular distribution of photons with helicity ! = ±1, with H the angle between kˆ (direction of photon emission) and a quantization axis of nucleon polarization. 12 Measurement of P& and & can determine the empirical values of the ratios RM 1 ≡

M 1S ; M 1V

RE2 ≡

E2S : M 1V

(3.48)

Both M 1S and E2S are suppressed relative to the isovector M 1 matrix element (M 1V ) due to mismatch in symmetries of the wave functions, etc. We will however take the single-particle matrix elements of M 1S and E2S as the leading order (LO) in chiral counting of the higher order terms in calculating the ratios RM 1 and RE2 . Radiative corrections to these leading-order terms come at N 1 LO and are calculated easily given the wave functions. 13 Relative to the leading order (LO) terms, corrections will then be classi:ed by N n NO for O(Qn ). For the chiral-:lter protected M 1V , 12

Explicit forms are not needed for our discussion but we write them down for completeness: √ 2(RM 1 − RE2 ) + 12 (RM 1 + RE2 )2 P& = |˜Pn | ; 2 2 1 + RM 1 + RE2 & = pP

2 2 RM 1 + RE2 − 6RM 1 RE2 ; 2 2 4(1 − pP) + (4 + pP)(RM 1 + RE2 ) + 2pP RM 1 RE2

where pP ≡ ˜Pp · ˜Pn and ˜P is the polarization three-vector. 13 Although not in the main line of discussion, we quote as an aside the numbers to give an idea of the size we are dealing with. With the realistic Av18 wave functions, the numerical results for the sum of the LO and NLO come out to be (M 1S)1B (fm) = (−4:192 − 0:105) × 10−3 = −4:297 × 10−3 ; E2S1B (fm) = (1:401 − 0:007) × 10−3 = 1:394 × 10−3 :

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Fig. 6. Generalized tree diagrams for the two-body isoscalar current. The solid circles include counter-term insertions and (one-particle irreducible) loop corrections. The wiggly line stands for the external :eld (current) and the dashed line the pion. Fig. 7. One-loop graphs contributing to the two-body currents. They come at O(Q4 ) and higher orders relative to the LO one-body term. All possible insertions on the external line are understood.

the leading two-body correction appears at N 1 LO and the one-loops and the counter-terms appear at N 3 LO. For the isoscalar current, not protected by the chiral :lter, the counting rules are quite di9erent. While the E2S receives only negligible contributions from higher order (N 3 LO and N 4 LO) corrections, the situation with M 1S is quite di9erent. The corrections are quite large even though the leading two-body corrections for the M 1S turn out to appear in tree order at N 3 LO while the loop corrections come at N 4 LO and are :nite. Given that the higher corrections separately are quite large as expected from the chiral-:lter argument, there is a problem which does not arise in chiral-:lter-protected processes. It has to do with the zero-range interactions that come as radiative corrections involving the four-Fermi contact interactions (proportional to CA ) in Eq. (3.1) (see Fig. 7c) and as four-Fermi counter terms (see Fig. 6b). It turns out that there is only one term of the latter type that contributes, i.e., a term in the Lagrangian of the form \L ∼ −ig4 (9 B − 9 B )NW [S ; S ]N NW N ;

(3.49)

where B is the isoscalar external :eld and g4 is an undetermined coeScient. When the chiral-:lter mechanism is operative, these zero-range interactions appear at higher order and are expected to be suppressed by “naturalness” conditions and more importantly by “hard-core” correlation functions that are incorporated in the wave functions. In the present case, the chiral :lter is not e9ective and hence it is not a good approximation to let hard-core correlation functions “kill” the contact interactions. One way to resolve this problem is to exploit the cuto9 regularization that was used above in Section 3.4 for NN scattering. Operationally, we can achieve this goal by replacing the delta function in coordinate space attached to zero-range terms by the delta-shell with hard core rc , :3 (˜r) →

:(r − rc ) : 4 rc2

(3.50)

We then have to assure that the total contribution we compute be independent of rc within a reasonable range. This requirement constitutes a sort of renormalization-group invariance. When

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Table 1 The total contributions to M 1S in unit of 10−3 fm vs. rc in fm. Recall the single-particle contribution: M 1S1B ≈ −4:30 × 10−3 fm rc (fm)

0.01

0.2

0.4

0.6

0.8

M 1S

−2:849

−2:850

−2:852

−2:856

−2:861

the chiral :lter mechanism is operative, dropping the zero-range terms is justi:ed since the rc independence is not spoiled. If this requirement is not satis:ed, the e9ective theory is not reliable and the result cannot be trusted. The present case belongs to this class. It turns out fortunately that for M 1S, there enters only one particular linear combination of four-Fermi terms involving g4 and CA ’s and that it is this combination—call it X —that :gures in the magnetic moment of the deuteron. Fitting to the deuteron magnetic moment allows the constant X to be uniquely determined for any given value of rc . The result for the total M 1S for wide-ranging values of rc is given for illustration in Table 1. The remarkable insensitivity to rc may be taken as a sign of RG invariance and a support for the procedure. This is more remarkable considering that the series in the chiral order does not appear to be converging. To exhibit this, we quote how each term of N n LO for n ¿ 3 contributes to the ratio RM 1 in the range of rc rcmin ≡ 0:01 fm 6 rc 6 rcmax ≡ 0:8 fm :

(3.51)

Expressed in the chiral order and in the range (rcmax ; rcmin ), the RM 1 comes out to be RM 1 × 103 = “LO” + “N 3 LO” + “N 4 LO” + “(N 3 LO + N 4 LO)X ” = −0:74 + (−0:48; −0:74) + (0:23; 0:46) + (0:49; 0:54) = (−0:50; −0:49)

(3.52)

where the contribution that depends on the single parameter X :xed by the deuteron magnetic moment is indicated by the subscript X . This is how the value given in (3.40) has been arrived at. The crucial observation which applies directly also to the proton fusion process—and to a somewhat less extent, to the hep process—discussed above is that the :nal result cannot be arrived at by any partial sum of the terms. Also note the importance of the X -dependent term. Needless to say, an experimental check of the result obtained here would be highly desirable. It should be noted that the scale we are talking about in connection with the “hard-core regularization” is of the chiral scale ?5 ∼ 4 f that delineates low-energy degrees of freedom of hadrons from high-energy, and not that of nuclei that enters in low-energy two-body interactions discussed above. 4. “Vector manifestation” of chiral symmetry 4.1. Harada–Yamawaki scenario The major problem that we have to face in going to many-body systems is how the structure of the ground state, i.e., “vacuum” is to be described and how the properties of hadronic degrees

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of freedom get modi:ed in the changed background. This question will be addressed more specifically in the next section in terms of an e9ective in-medium :eld theory. Here we address the generic aspects based on symmetries involved in the problem. We will discuss the problem in terms of density but we believe the general argument applies as well to temperature at least up to the chiral restoration point. Possible di9erences that appear after the critical point will be mentioned in Section 11. To discuss how the vacuum changes as matter density increases and how the properties of baryonic matter are a9ected by the vacuum change, we need to :rst address how chiral symmetry is a9ected by the medium. While it is generally accepted that chiral symmetry endowed with NF -Kavor massless quarks breaks spontaneously in the (zero-density) vacuum as SU (NF )L × SU (NF )R → SU (NF )L+R , it is not fully known how the broken symmetry gets restored in medium under extreme conditions, that is, at high temperature and=or at high density. There are several di9erent scenarios with which the broken symmetry can get restored as temperature and=or density are=is dialed. As we will argue, what happens in nuclei and nuclear matter where matter density is non-zero depends rather crucially on what scenario the chiral symmetry restoration adopts. For concreteness and simplicity, take the two-Kavor case with up and down quarks (NF = 2). 14 The “standard scenario” is the " model one where the broken Nambu–Goldstone mode gets restored to the Wigner–Weyl mode wherein the triplet of pions i with i = ±; 0 and the scalar " (in the chiral limit) merge to an O(4) massless degenerate multiplet at the phase transition. At the phase transition, other hadrons such as nucleons, vector mesons, etc. could be either massless or massive as long as they appear in chiral multiplet structure. Now in zero-density, the scalar " need not be the fourth component of O(4) as in linear " model. It could for instance be a chiral singlet but what is required is that at the phase transition it joins the pions into the O(4) as described by Weinberg in his “mended symmetry” scenario [60,61]. One should note that the "-model scenario is moot on the spin-1 mesons, so, e.g., the vector meson (which will :gure importantly in later discussions) could become either lighter or heavier as one approaches the chiral restoration. This model belongs to the O(4) universality class and has been extensively discussed in the literature [62]. At present, it is not inconsistent with lattice measurements with temperature but it is not the only viable scheme either. There is an alternative scenario to the above standard one recently proposed by Harada and Yamawaki [12] which we believe provides support to the BR scaling to be elaborated below as well as connects with the color–Kavor locking scenario of QCD discussed in Section 11.2. The distinguishing feature of Harada–Yamawaki’s “vector manifestation” is that chiral symmetry with, say, two Kavors (u and d) is restored by the triplet of pions merging with a triplet of the longitudinal components of vector mesons to the representation (3; 1) ⊕ (1; 3). At the chiral point, the vector coupling gV Kows to the :xed point gV = 0; a = 1 corresponding to what Georgi called vector symmetry limit [63] together with the pion decay constant f going to zero. In this case, the vector will become massless together with other matter :elds, i.e., constituent quarks and the vector coupling will vanish, a feature which is an important factor in our discussions to follow.

14

The Harada–Yamawaki argument was originally developed with three Kavors u, d and s. Although it has not been veri:ed explicitly, we see no reason why the same argument should not apply to the two-Kavor case with some minor changes.

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Limiting ourselves to the two-Kavor case (the three-Kavor case can be similarly treated), we consider speci:cally the hidden local symmetry (HLS) theory of Bando et al. [64] with the symmetry group [U (2)L ×U (2)R ]global ×[U (2)V ]local consisting of a triplet of pions, a triplet of -mesons and an !-meson. Motivated by the observation that in the vacuum, the and ! mesons are nearly degenerate and the quartet symmetry is fairly good phenomenologically, we put them into a U (2) multiplet. In this theory, baryons (proton and neutron) do not appear explicitly. They can be considered as having been integrated out. If needed, they can be recovered as solitons (skyrmions) of the theory. The relevant degrees of freedom in the HLS theory are the left and right chiral :elds denoted by 6L; R , 6L; R = ei"=f e∓i =f with "(x) = "a T a and = a T a and the hidden local gauge :elds denoted by 7a 1 V ≡ V( T ( = a + ! 2 2

(4.1)

(4.2)

with Tr(T ( T ) )= 12 :() . If we denote the [U (2)L ×U (2)R ]global ×[U (2)V ]local unitary transformations by † † (gL ; gR ; h), then the :elds transform 6L; R → h(x)6L; R gL; R and V → h(x)(V − i9 )h (x). How chiral symmetry restoration can come about in the HLS framework at large NF was discussed by Harada and Yamawaki [65] and at high temperature by Harada and Sasaki [66]. Since their arguments are quite general, it is highly plausible that they can be applied to high density. This is what we shall do here. 4.2. Vector manifestation in hot matter To motivate our argument to be developed below for density-driven phenomena, we brieKy summarize, without going into speci:c details which we will do in the next subsection, the work of Harada and Sasaki [66] on how the VM manifests itself as the critical temperature Tc is approached. We do this because this work brings out a subtlety in the behavior of vector-meson excitations in medium in general and in the neighborhood of the critical temperature in particular that has not been duly taken into account in other works on the matter available in the literature. There are two distinctive features in the HLS approach to the chiral phase transition. One is that in order to match with QCD while preserving hidden local symmetry at the chiral scale ?5 in the T = n = 0 space, 15 the and ! mesons must be considered as “light” in the sense that the pion mass is light as :rst pointed out by Georgi for a consistent power counting [63]. As noted below, it is in this sense that the HLS theory can be mapped to chiral perturbation theory [67]. This feature :gures importantly in the hot (and=or dense) matter. The second feature is that in hot medium, there are two e9ects that control hadronic properties. One is the “intrinsic” dependence on T through the T -dependence of the QCD condensates (both quark and gluon) of hadronic masses and coupling constants and the other is the hadronic thermal Kuctuations. The former is controlled by the matching to QCD and hence so is the latter through the parameters. In Ref. [68], Harada and Yamawaki studied the renormalization group structure of HLS in terms of the parameters of the HLS Lagrangian that enter at higher orders in the chiral expansion and 15

In this section, the matter density will be denoted n, reserving for the meson.

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showed that e9ective low-energy theories with the given Kavor symmetries can Kow in various di9erent directions as scale is varied but there is only one :xed point that is consistent with QCD, namely the Georgi vector limit. This implies that the intrinsic temperature dependence must be constrained by the Georgi vector limit, which means that the mass parameter MV and the gauge coupling constant gV will approach zero as T → Tc . When T is large, i.e., MV T , 16 the pole mass of the vector () meson mass takes the simple form m2 (T ) = M2 (T ) + gV2 J (T ) :

(4.3)

Here M2 = af 2 gV2 where gV is the (hidden) gauge coupling and a = 1 signals deviation from the Georgi vector limit and J (T ) is a :nite temperature-dependent function. As T → Tc , M → 0 and gV → 0, so the pole mass goes to zero. This is consistent with BR scaling. We should stress that while the mass goes to zero at near Tc , the theory is nonetheless consistent with what one would expect at low T . For T M , the pole mass is found to be of the form [66] m2 (T ) = M2 (T ) + c

T4 + ··· ; F 2

(4.4)

where c is a numerical constant and F 93 MeV is the pion decay constant in the vacuum. The ellipsis stands for terms higher order in T . Note that while this agrees with the low-temperature result of Ref. [69], the pole mass need not increase with temperature as suggested in Ref. [69]. A close look indicates that it can even decrease at about the same rate as f (T ) since M decreases faster than f (T ), suggesting the following—BR scaling—relation: m (T )=m ∼ f (T )=f :

(4.5)

Note also that an early QCD sum-rule calculation by Adami and Brown [70] of the thermal behavior of the mass contains features that are similar to those in the vector manifestation. Although the connection to the Georgi vector limit is yet to be established, the vanishing of the mass in the work of [70] at the critical temperature Tc due to the approach to zero of the Wilson coeScient that multiplies a certain combination of gluon condensates with the vanishing of quark condensate resembles the way the mass goes to zero in the vector manifestation. In the Adami–Brown formulation, the tadpoles which give the quark and antiquark masses in NJL are jammed together in going to deep Euclidean, so that to begin with, it appears that there is qWqqq W condensate. Adami and Brown factorized this, since they believed that before deep Euclidean was reached there were really two qq W condensates. Whereas the question of “factorization” remained in the QCD sum rule approach, the present work of Harada and Sasaki shows that this was exactly right; namely, the scale ? changes smoothly as chiral restoration is reached, with dynamically generated masses going to zero with it. 4.3. Vector manifestation in dense matter In addressing dense matter, we follow the reasoning of [65 – 67]. Here we supply some details left out in the preceding subsection. We take the HLS Lagrangian as an e9ective Lagrangian that results when high-energy degrees of freedom above the chiral scale ? taken to be higher than the 16

This condition makes sense since MV → 0 as T → Tc .

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vector meson mass mV are integrated out. Now the scale ? will in general depend on the number of Kavors NF , density n or temperature T depending upon what system is being considered. This is to be taken as a bare Lagrangian in the Wilsonian sense with the parameters gV (?), a(?) and f (?) which plays the role of order parameter for chiral symmetry with f = 0 signaling the onset of the Wigner–Weyl phase. Ref. [67] describes how these parameters can be determined in terms of QCD condensates by matching—Ma la Wilson—the vector and axial-vector correlators of HLS theory with the ones of QCD at the chiral scale ? and how by following renormalization group (RG) Kows to low energy scales, one can determine low-energy parameters that can be related to those that :gure in chiral perturbation theory (for, e.g., scattering). An important observation here is that the (assumed) equality at chiral restoration (where qq W = 0) of the vector and axial-vector correlators, i.e., LV |qq W =0 = LA |qq W =0 ; where

i i

(4.6)

d 4 x eiq·x 0|TJa (x)Jb (0)|0 = :ab (q q − g q2 )LV (−q2 ) ; a b (x)J5 (0)|0 = :ab (q q − g q2 )LA (−q2 ) d 4 x eiq·x 0|TJ5

(4.7)

implies that at the critical point, independently of how the symmetry change is driven, the HLS theory approaches the Georgi vector limit, namely, gV = 0 and a = 1, in addition to the vanishing of f . That chiral symmetry is restored with f = 0 at the Georgi vector limit follows from a proper account of quadratic divergence in the renormalization group Kow equation for f and a. 17 In terms of baryon density n, we interpret this to imply that at the critical density n = nc , we must have gV (?(nc ); nc ) = 0;

a(?(nc ); nc ) = 1 ;

(4.8)

where we have indicated the density dependence of the cuto9 ?. The reasoning used here is identical to that for the temperature-driven transition. In HLS theory, the vector mass is given by the Higgs mechanism. In free space, it is of the form (4.9) mV ≡ m = m! = a(mV )gV (mV )f (mV ) ; 17

The quadratic divergence is present when the cuto9 regularization is used whereas it is absent at four dimensions D=4 when the dimensional regularization is used. The quadratic divergence present in the cuto9 regularization corresponds to a singularity at D = 2 in the dimensionally regularized integral from one-loop graphs which comes out to be proportional to the 4 function ∼ 4(1 − D=2) =

4(3 − D=2) − 4(2 − D=3) : 1 − D=2

In a renormalizable theory, this singularity (or any other power divergence) does not :gure at D = 4 but in an e9ective (non-renormalizable) theory like HLS, this singularity present at D = 2—which is absent at D = 4—has to be subtracted. This procedure is quite similar to the subtraction of the singularity at D = 3 that was required in the KSW scheme for two-nucleon scattering discussed in Section 3.4 and we suspect that a similar physics is at work here. Note that were it not for this quadratic divergence, the f would not run and hence would not go to zero at the point where the Georgi vector limit is reached. Thus the quadratic term plays a crucial role in specifying the symmetry structure of the chiral restoration phase transition.

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where the cuto9 dependence is understood. Here the parameter a(mV ), etc. means that it is the value at the scale mV determined by an RG Kow from the √ bare quantity a(?). Note that (4.9) is similar, but not identical, to the KSRF relation m(KSRF) = 2g f (0). 18 Now in medium with n = 0, this mass formula will remain the same except that it will depend upon density, 19 ? ? ? ? m? ≡ m = m = a(m? (4.10) V ! V )gV (mV )f (mV ) : The density dependence is indicated by the star. As in the case of NF discussed in Ref. [12], the cuto9 ? will depend upon density, say, ?? implicit in (4.10). The Harada–Yamawaki argument (or theorem) suggests that at n = nc where qq W = 0 and hence LV? (nc ) = LA? (nc ), the Georgi’s vector limit gV = 0, a = 1 is reached together with f = 0. This means that both the mass parameter (4.10) and the pole mass vanish, m? V (nc ) = 0 :

(4.11)

At the point where the vector meson mass vanishes, the quartet scalars will be “de-Higgsed” from the vector mesons and form a degenerate multiplet with the triplet of massless pions (and “” 20 ) with the massless vectors decoupled. This assures that the vector correlator is equal to the axial-vector correlator in the HLS sector matching with the QCD sector. In this scenario, dictated by the renormalization group equations, the vector meson masses drop as density (or T) increases. Note that this scenario is distinct from the “standard” (as yet to-be-established) picture in which the and a1 come together as do the pions and a scalar ". In the standard scenario, there is nothing 18

However numerically they are very close. We explain as a side remark how this comes about. While Eq. (4.9) for mV = MV (Q = mV ) is approximate with numerically small corrections of order gV2 =(4 )2 ignored there, the corresponding formula for Q = 0, i.e., MV2 (0) = a(0)g2 (0)f 2 (0) is an exact low-energy theorem [71]. The small corrections arise in going from the o9-shell point Q = 0 to the on-shell point Q = m . Noting that the KSRF relation is given in terms of the on-shell constant g and the pion on-shell constant f (0), we :rst extract the former from HLS Lagrangian given in terms of on-shell quantities: g =

g(m ) f"2 (m ) ; 2 f 2 (0)

where f" is the decay constant of the scalar that goes into the longitudinal component of the by Higgs mechanism (which becomes equal to f at the Georgi vector limit). The pion decay constant f runs with pion loops below the scale but f" does not run below m since the decouples. Using f"2 (m ) = a(m )f 2 (m ), we can rewrite the KSRF mass formula as m2(KSRF) = M2 (m )

a(m ) f 2 (m ) : 2 f 2 (0)

By the renormalization group equations given by Harada and Yamawaki, we have a(m ) 1 and f 2 (m )=f 2 (0) 2. Thus, we get m2(KSRF) M (m ) ≡ m2 : 19

Note that the mass in this formula is a parameter in the Lagrangian and not necessarily a pole mass except at the phase transition point. The pole mass will generically have additional density-dependent terms which will go to zero as the coupling gV reaches the Georgi vector limit. 20 We assume that the fourth pseudoscalar “” gets a mass by U (1)A anomaly, so can be excluded from our consideration.

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which forces the vector mesons to become massless and decouple. They can even become more massive at chiral restoration than in the vacuum as discussed by Pisarksi [72]. 21 Thus the vanishing of the vector-meson mass is a prima facie signal for the phase transition in the Harada–Yamawaki picture. Here using the same reasoning as for large NF and high T , we have arrived at the conclusion that the same vector manifestation can take place at the critical density. 22 As in the case of temperature, the “running” in density of the parameters gV , a and f implies that away from the critical point, the vector meson (pole) masses drop as density is increased. In the next section we will interpret this phenomenon in terms of BR scaling. In this section as well as in what follows, we are interpreting the vector-meson “mass” in the sense of the BR-scaling mass. Of course in the presence of a medium with non-zero temperature and=or density, the lack of Lorentz invariance gives rise to di9erent components and it will be necessary to specify which component gets “de-Higgsed” at the phase transition. The meaning of the vector mass as used here is not the same as the standard de:nition as we will try to clarify in the following sections.

5. Landau Fermi liquid from chiral Lagrangians 5.1. Fluctuating around zero-density vacuum We have developed the thesis that chiral symmetry can be suitably implemented into nuclear problems involving a few nucleons. Furthermore EFT can be formulated in such a way that the traditional nuclear physics approach with realistic potentials (i.e., PM) can be identi:ed as a legitimate part of a consistent EFT supplying the leading term in the expansion. This means that what nuclear physicists have been doing up to today—and their highly successful results—can be considered as an essential part of the modern structure of strong interactions based on QCD. For the few-nucleon processes we have been dealing with, e.g., NN scattering, n + p ↔ d + &, solar p + p → d + e+ + e , etc., what enters into the theory is the chiral symmetric Lagrangian determined in free space, that is, at zero density. The starting point there is an e9ective chiral Lagrangian that describes QCD in the matter-free vacuum, with masses and coupling constants all determined in particle experiments in medium-free space. The few-nucleon problems are then treated by Kuctuating around the matter-free vacuum near a “:xed point” (at in:nite scattering length) discussed above. The machinery for doing this is the well-established chiral perturbation theory (5PT) involving baryons. Since the processes involved occur near the :xed point of the theory, one can successfully do the perturbation. 21

An important point to note here is that Pisarski’s argument hinges on the vector dominance picture. However Harada and Yamawaki show that at the vector manifestation, the vector dominance is strongly violated [68]. This is one of the basic di9erences in treating vector mesons. 22 There is a puzzle here. The analysis of Ref. [73] for the NF -driven phase transition is that the top–down phase transition from NF ¿ NFc to NF ¡ NFc is neither :rst-order nor second-order. The order parameter is found to be continuous at the phase transition, so it is not :rst-order. However the correlation length does not diverge at the phase transition, so it is not second-order either. If correct, this phase transition may appear to be di9erent from that expected in temperature or density.

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Now how do we go to heavier nuclei, nuclear matter and denser matter which must live near a di9erent :xed point, say, the Fermi-liquid :xed point to be described below which is not necessarily near the above :xed point? This is the principal question that we wish to address in this review. If there are no phase changes along the way, one may attempt to build many-nucleon systems in a perturbative scheme starting with the free-space Lagrangian mentioned above. Indeed there have been several such attempts in 5PT to reach at nuclear matter [74,75]. It is not obvious though that this can be done beyond the normal matter density since a strong-coupling expansion is involved and large anomalous dimensions arise. This means that it may be simply too hard if not impossible to handle various phase changes (kaon condensation, chiral restoration, color superconductivity, etc.) that are conjectured to occur as one increases density toward chiral restoration as demanded for relativistic heavy-ion processes and for describing compact stars. A di9erent but perhaps more precise way of saying the same thing is in terms of the skyrmion picture for dense matter which is closest to QCD if Nc is taken to be large. In the skyrmion description, a lump of baryonic matter with a given baryon number B is characterized by a “topological vacuum” associated with a conserved winding number B. A system with density B B means that B B for a given volume. Now the point is that the B vacuum may not be simply connected to the B vacuum by a small perturbation. In what follows, therefore, we shall take a di9erent route and formulate a chiral Lagrangian :eld theory using a “sliding” vacuum with parameters of the Lagrangian running with matter density. This leads to the notion of Brown–Rho scaling and in-medium e9ective :eld theory for a system with Fermi seas, i.e., Landau Fermi-liquid theory. 5.2. Skyrmion vs. Q-ball Consider bound many-nucleon systems corresponding to nuclei. If the number of colors can be taken to be very large, then one may approach them starting with a skyrmion-type Lagrangian constructed of Goldstone boson :elds (and heavy mesons) and looking for solutions of the winding number W equal to the baryon number B or the mass number A in a nucleus. The Skyrme Lagrangian which consists of the current algebra term and the quartic term f2 1 Tr(9 U 9 U † ) + Tr[U 9 U † ; U 9 U † ]2 + · · · (5.1) 4 32e2 which may be viewed as an approximate zero bag size limit of the Lagrangian (2.1) has been extensively studied mathematically for baryon number up to B = 22 with a fascinating result [76]. (The ellipsis stands for higher derivative and=or heavy-meson and chiral-symmetry breaking terms that are to be added as needed.) The Lagrangian of the type (5.1) suitably extended with higher derivatives and=or massive boson :elds is accurate only at large-Nc limit, so it is not clear what the classical solution of this Lagrangian for B ¿ 2 represents since in nature, Nc ∞. Nonetheless this development is quite exciting since it is closest to QCD at least for Nc = ∞. As stated above, the ground states with baryons numbers B and B with BB cannot be connected by small perturbation because they represent di9erent topological “vacua”. A direction we will adopt here is not as ambitious as the above but we will attempt to account for the di9erent vacuum structure for dense systems (heavy nuclei) than for dilute systems (light nuclei). We shall start with a theory for a system with a Fermi sea. It is possible that such a system emerges as a sort of nontopological soliton or “Q-ball” from a theory based on chiral perturbation Lsk =

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theory de:ned at zero density as suggested :rst by Lynn [74] and recently re:ned by Lutz et al. [75]. Although there is no proof, we believe that the Q-ball system which is non-topological is essentially equivalent to the skyrmion system which is topological. This equivalence is somewhat like in the nucleon structure where the two pictures work equally well. The essential point we shall accept here is that a non-topological soliton solution exists possessing the liquid structure that is supposed to be present in nuclear matter. We will later identify this with what we do know but for the moment let us assume that we have a Fermi sea :lled with nucleons. It is fortunate for our purpose that such a system is accessible to an elegant and powerful e9ective :eld theory treatment as shown by various authors [77] that leads to Landau’s Fermi liquid theory applied to nuclear systems [78]. 5.3. EDective chiral Lagrangian for many-nucleon systems What we wish to do in our case is to arrive, following the developments in Refs. [79 –81], at the Landau Fermi liquid theory starting from an e9ective chiral Lagrangian that we have been discussing above. Treating the nucleon in terms of a local :eld denoted by N and the Goldstone boson :eld √ ? by 6 = U = ei ·7=2f , we can write a simple Lagrangian of the form L = NW [i& (9 + iv + gA? &5 a ) − M ? ]N − Ci? (NW 4i N )2 + · · · ; (5.2) i

where the ellipsis stands for higher dimension nucleon operators and the 4i ’s Dirac and Kavor matrices as well as derivatives consistent with chiral symmetry. The star aSxed on the masses and coupling constants will be de:ned precisely below. The induced vector and axial vector “:elds” are given by v = − 2i (6† 9 6 + 69 6† ) and a = − 2i (6† 9 6 − 69 6† ). 23 In (5.2) only the pion ( ) and nucleon (N ) :elds appear explicitly: all other :elds have been integrated out. The e9ect of massive degrees of freedom will be lodged in higher-dimension and=or higher-derivative interactions. The external electro-weak :elds if needed are straightforwardly incorporated by suitable gauging. When applied to symmetric nuclear matter in the mean :eld approximation, the Lagrangian (5.2) is known to be equivalent [82,83] to the Lagrangian that contains just the degrees of freedom that :gure in a linear model of the Walecka-type [84] 24 1 2 1 m?2 m?2 L = NW (i& (9 + igv? ! ) − M ? + h? ")N − F + (9 ")2 + ! !2 − " "2 + · · · ; (5.3) 4 2 2 2 where the ellipsis denotes higher-dimension operators. We should stress that (5.3) is consistent with chiral symmetry since here both the ! and " :elds are chiral singlets. In fact the " here has nothing to do with the chiral fourth-component scalar :eld of the linear sigma model except near the chiral phase transition density; it is a “dilaton” connected with the trace anomaly of QCD and is supposed to approach the chiral fourth component in the “mended symmetry” way aM la Weinberg [60,61] near chiral restoration. In practice, depending upon the problem, one form is more convenient than the other. In what follows, both (5.2) and (5.3) will be used interchangeably. In (5.2) and (5.3), the mass and coupling parameters aSxed with stars depend upon the “sliding” vacuum de:ned at a given “density”. (Putting density into the parameters of a Lagrangian is subtle The 6 :eld appearing here is the “gauge-:xed” one, 6L = 6†R = 6. Here we are dealing with the matter far away from the chiral phase transition point, so we are taking the unitary gauge. 24 For asymmetric nuclear matter, isovector :elds (e.g., ; a1 , etc.) must be included in a chirally symmetric way. 23

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because of chiral symmetry and thermodynamic consistency. We will specify more precisely how the “density” is to be de:ned.) We are thinking of the situation where an external pressure (e.g., gravity for compact stars) is exerted and hence we can think of the parameters of the Lagrangian varying as a function of density. They need not be associated with quantities de:ned at a minimum of an e9ective potential of the theory without the external pressure. The next important observation we shall exploit is that by Matsui [85] that a Lagrangian of the type (5.2) can give in the mean :eld the result of Landau Fermi liquid. This means that in the same approximation, the chiral Lagrangian (5.2) can be mapped to Landau Fermi liquid. From the works in Ref. [77] which tell us that Landau theory is a :xed-point theory with a Fermi-liquid Cxed point which is distinct from that of the e9ective Lagrangian used for two-nucleon systems discussed above, we learn that the theory (5.2) in the mean :eld has two :xed-point quantities, viz., the e9ective quasiparticle (or Landau) mass m? N and the Landau parameters F for quasiparticle interactions. The full interaction between two quasiparticles p1 and p2 at the Fermi surface of symmetric nuclear matter written in terms of spin and isospin invariants is de:ned by 1 F(cos H12 ) + F (cos H12 )1 · 2 + G(cos H12 )1 · 2 Fp 1 " 1 7 1 ; p 2 " 2 7 2 = N (0) + G (cos H12 )1 · 2 1 · 2 + +

q2 H (cos H12 )S12 (q) ˆ 1 · 2 kF2

q2 H (cos H12 )S12 (q) ˆ kF2

(5.4)

where H12 is the angle between p1 and p2 and N (0) = (4kF2 =2 2 )(dp=d')F is the density of states at the Fermi surface. Also, q = p1 − p2 and S12 (q) ˆ = 31 · q ˆ 2 · qˆ − 1 · 2 , where qˆ = q=|q|. The tensor interactions H and H are important for the axial charge to be considered later. The functions F; F ; : : : are expanded in Legendre polynomials, F(cos H12 ) = ‘ F‘ P‘ (cos H12 ), with analogous expansion for the spin- and isospin-dependent interactions. The :xed point m? N is associated with a given Fermi momentum kF or density of the given system, so that there will be a set of :xed points F for each (given) kF or . Thus we will have the Landau parameters that depend upon density once we accept that the Landau mass depends on density as is obvious from the Landau mass formula −1 1 1 m? N = 1 + F1 = 1 − F˜ 1 ; (5.5) M 3 3 where F˜ 1 ≡ (M=m? N )F1 and F1 is the l = 1 component of the interaction F in (5.4). 5.4. Brown–Rho scaling How the :xed point quantities vary with density cannot be determined from the e9ective Lagrangian alone. It must be tied to the vacuum property characterized by the chiral condensate W where is the quark :eld. The dependence must therefore be derived from the fundamental theory, QCD. In the past QCD sum rules have been used to extract information on this matter. For our purpose, it is more convenient to take the Skyrme Lagrangian (5.1) as a starting point and try

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to deduce the necessary information. We expect it to give the leading behavior corresponding to a mean :eld theory in the sense to be de:ned more precisely later. This is the argument presented in the original BR paper [2]. Implementing the scalar dilaton :eld 5 associated with the trace anomaly of QCD to the Lagrangian extended to take explicitly into account the vector mesons V = ; ! , it has been found that in the mean :eld the following scaling holds, e.g., in the Lagrangian (5.3): 25 P() ≈

f ? () m? () m? () M ? () : ≈ " ≈ V ≈ f m" mV M

(5.6)

Here f is the pion decay constant, the f ? being the in-medium one. 26 We caution that the M ? is the nucleon in-medium mass (or BR scaling mass) which is not the same as the Landau mass m? W ? =qq) W a N as we will see shortly. The quantity P(), related to the quark condensate P ≈ (qq where a is a constant which we will :nd in Section 8 equals ∼ 1, is the scaling factor that needs to be determined from (fundamental) theory or experiments. Let us now examine more closely what the “density dependence” in general (not necessarily in the context of BR scaling) means in the Lagrangian (5.2) or (5.3). For this, let us take (5.3) and introduce the chirally invariant operator u ^ ≡ NW & N (5.7) √ where u = ( 1 − C2 )−1 (1; C) = ( 2 − j 2 )−1 (; j) is the Kuid 4-velocity. Here j = NW N is the baryon current density and = N † N = i ni the baryon number density. The expectation value of ^ yields the baryon density in the rest-frame of the Kuid. Using ^ it is easy to construct a Lorentz invariant, chirally invariant Lagrangian with density dependent parameters. Now a density dependent mass parameter in the Lagrangian should be interpreted as m? = m? (). ^ This means that the model (5.3) is no longer linear. It is highly non-linear even at the mean :eld level. The Euler–Lagrange equations of motion for the bosonic :elds are the usual ones but the nucleon equation of motion is not. This is because of the functional dependence of the masses and coupling constants on the nucleon :eld: ^ − M ? + h? "]N = 0 [i& (9 + igv? ! − iu $)

(5.8)

with 9L $^ = 9^ 2 = m? !!

25

? 9M ? 9m? 9g? ! 2 9m" − m? − NW ! & N v − NW N : "" 9^ 9^ 9^ 9^

(5.9)

The scalar " as well as the vector V are absent in (5.2) since they have been integrated out. Their scaling will appear therefore in the coeScients representing those degrees of freedom, namely, Ci? ’s. The connections between the two are given in Refs. [79 –81]. 26 Since Lorentz invariance is lost in medium, the constants f ? , gA? , etc. contain two components, space part and time part, that are in general di9erent. The parameters that appear in BR scaling are mean-:eld quantities and are de:ned more precisely in the sense described above for a generalized Walecka model (5.3). The same remark applies to the in-medium vector coupling we will discuss in later sections.

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Here we are assuming that (9=9)h ^ ? ≈ 0. This can be justi:ed using NJL model [120] but we will ^ which may be related to what is referred to in simply assume it here. 27 The additional term $, many-body theory as “rearrangement terms”, is essential in making the theory consistent. This point has been overlooked in the literature. The crucial observation is that when one computes the energy–momentum tensor with (5.3), one :nds, in addition to the canonical term (which is obtained when the parameters are treated as constants), a new term proportional to $^ ^ NW u · &N )g : T = Tcan + $(

(5.10)

The pressure is then given by 13 Tii C=0 . The additional term in (5.10) matches precisely the terms that arise when the derivative with respect to acts on the density-dependent masses and coupling constants in the formula derived from T00 : p=−

9E 9E= = 2 = − E ; 9V 9

(5.11)

where E = T 00 :

(5.12)

This matching assures energy–momentum conservation and thermodynamic consistency. This veri:es that interpreting “density-dependent” parameters in our Lagrangian as the dependence on ^ with ^ being the density is consistent with both chiral symmetry and thermodynamics. 5.5. Landau mass and BR scaling We end this section by writing down the relations between the parameters of the Lagrangian (5.2) or equivalently (5:3) and Landau parameters. We will use the former and take (including the isovector channel) −

i

C?2 C ?2 ˜ (NW & 7N )2 + · · · : Ci? (NW 4i N )2 ≈ − !˜ (NW & N )2 − 2 2

(5.13)

This may be considered as the leading terms that result when the vectors ! and are integrated out. But it is more than just that. In fact, the subscripts can be taken to represent not only the vector mesons ! and that nuclear physicists are familiar with but also all vector mesons of the same quantum numbers (which account for the appearance of the tilde on the vector mesons), so the two “counter terms” on the right-hand side subsume the full short-distance physics of the given chiral order. Since the Lagrangian (5.2) in tree order must correspond to the mean :eld of (5.3), one can simply compute the relevant Feynman graphs with (5.2) and match them to the Landau Fermi liquid formula. One obtains [79] −1 m? 1˜ N −1 = P − F 1 ( ) ; (5.14) M 3 27

In fact apart from the phenomenological requirement, there is no reason to assume this. All our arguments that follow would go through without diSculty even if the coupling constant were to scale with density.

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where F˜ 1 ( ) is the pion contribution to F1 which is completely :xed by chiral symmetry so the only parameter that enters in (5.14) is the BR scaling P ≈ M ? =M . Now using the Landau mass formula (5:5) and F˜ 1 = F˜ 1 (!) ˜ + F˜ 1 ( )

(5.15)

we :nd ˜ = 3(1 − P−1 ) ; F˜ 1 (!)

(5.16)

˜ includes contributions from all excitations of the ! meson quantum numbers. This is where F˜ 1 (!) the main result of the connection between the chiral Lagrangian with BR scaling and Landau Fermi liquid theory. This is a highly non-trivial relation which says that BR scaling—which in the context of the QCD vacuum reKects modi:cation due to matter density—is tied to short-range quasiparticle interactions. This means that what may be considered to be a basic feature of QCD and nuclear interactions via nuclear forces are inter-related, indicating another level of the Cheshire Catism. • Determining P() If one accepts the BR scaling (5.6), there are several sources to use for determining the scaling factor P(). For instance, one can determine it from the QCD sum rule 28 for the mass ratio m? =m [88] or from the ratio f ? =f using the Gell–Mann–Oakes–Renner (GMOR) formula for pion mass in medium 29 or a :t to nuclear matter properties [79]. The result comes out to be about the same at nuclear matter density P( = 0 ) ≈ 0:78 :

(5.17)

In terms of the parameterization of the form P() = (1 + y=0 )−1 ;

(5.18)

the result corresponds to y( = 0 ) ≈ 0:28. This gives the Landau mass m? N =M ≈ 0:70 which is to be compared—with a due consideration of the caveat mentioned above on the diSculty with QCD sum rules in nuclear physics—with the presently available QCD sum-rule value 0:69+0:14 −0:07 [87]. This value is also consistent with the properties of heavy nuclei.

28 There is a great deal of controversy in the application of QCD sum rules in dense and=or hot medium. As it stands, the situation is totally confused and it is perhaps too dangerous to take seriously any result so far obtained. For a recent discussion on this matter, see Ref. [86]. Our proposal is that the value (5.17) which will result from the :t to a variety of nuclear processes discussed below be taken as the value to be obtained from the QCD sum rule approach. 29 Using the in-medium GMOR formula to arrive at the value (5.17), it is assumed that the pion mass remains una9ected by density up to nuclear matter density. This assumption may have to be re-examined in view of the “pion mass” measured in deeply bound pionic states which indicate a mild increase as a function of density as discussed in Section 8. This issue is not so simple to address since it is not clear that the mass that appears in the GMOR relation is the same quantity that has been “measured”.

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6. Indirect evidences for BR scaling 6.1. Indications in Cnite nuclei Before the publication on BR scaling in Ref. [2], Brown and Rho proposed [89,90] that some of the missing strength in the longitudinal response function in nuclei could be explained by the drop? ping masses in medium m? =m ≈ mN =mN ≈ 0:75. The same dropping masses (of both the nucleon and the ) could also explain that within certain range of kinematics relevant to experiments, the transverse response function does not scale in medium. This is essentially the picture that is con:rmed by the Morgenstern–Meziani analysis mentioned in the introduction and described in detail in Section 7. In addition to the electromagnetic response functions, there were further predictions in Ref. [90] that followed from the dropping masses. One was the anomalous gyromagnetic ratio :gl in nuclei which gets enhanced by the scaling factor. This will be discussed below in terms of a more modern formulation with chiral Lagrangians and Landau Fermi liquid theory although the physics is essentially the same as in this work. The second is that the dropping mass implies that as density increases, the tensor force in nuclei gets weakened. The reason is that the tensor force is given by two agencies, one-pion exchange and one- exchange coming with opposite signs. At low density, the pion exchange dominates, so the net e9ect is more or less controlled by the pion tensor. 2 However as density increases, the tensor gets enhanced by the factor (mN =m? N ) apart from the increased range of the -exchange force. Therefore the tensor tends to cancel more and more the pion tensor as density increases, thereby suppressing the net tensor force in the outer region where pion exchange dominates. There are some indications that the suppressed tensor force is consistent with nuclear spectra as well as with Gamow–Teller transitions in nuclei to which we shall return shortly. There is, however, a possible caveat to this as we will see later and that has to do with the ? possibility that the vector coupling gNN will also drop as density increases, eventually decoupling as dictated by the “vector manifestation” discussed above. It is not clear at what density this decoupling sets in but there are empirical indications that the tensor force weakening is operative at least up to nuclear matter density. The phenomenology in nuclei discussed above also indicates that at least at ? the low densities encountered in nuclei, the ratio gNN =m? does increase. It should be noted that even when the vector decoupling takes place, with the Kavor symmetry in the hadronic sector yielding to the color gauge symmetry in the quark–gluon sector, 30 some vector repulsion remains coming from (Fierzed) gluon exchange [13]. 6.2. The anomalous gyromagnetic ratio in nuclei The development given above in Section 5 of the e9ective chiral Lagrangians with BR scaling and Fermi liquid :xed point theory can provide a more sophisticated and rigorous explanation for the enhanced gyromagnetic ratio gl in nuclei mentioned above. We discuss this as a strong, though indirect, evidence for BR scaling. The nuclear gyromagnetic moment gl in the convection current 30

The way this change-over takes place is discussed in Section 11.

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for a particle sitting on top of the Fermi sea is de:ned as k gl : (6.1) M We have de:ned gl ≡ 1 + :gl such that the bare nucleon mass M , not the e9ective (Landau) mass m? N , appears in the formula. This is because we want to preserve charge conservation associated with the factor 1 in gl ; :gl being purely isovector. The :gl has been determined from experiments. The measurement that is most relevant to our theory is the one on giant dipole resonances on heavy nuclei, in particular in the lead region. The result of Ref. [91] on 209 Bi gives the proton value J=

:glp = 0:23 ± 0:03 :

(6.2)

This can also be determined from magnetic moment measurements but the analysis is somewhat more complex and hence less precise. We shall refer to (6.2). Given the e9ective chiral Lagrangian (5.2), with the connections we have established so far, it is easy to compute in the mean :eld order the nuclear gyromagnetic ratio gl . It suSces to calculate all terms of the same order to assure that charge is conserved or what is equivalent, “Kohn theorem” is satis:ed. The result is the Migdal formula [78] given in terms of the parameters of the Lagrangian, that is, the BR scaling P(), 1 + 73 + :gl gl = (6.3) 2 with 1 ˜ 4 1˜ −1 ˜ P − 1 − F 1 ( ) 73 : (6.4) :gl = (F 1 − F 1 )73 = 6 9 2 As mentioned above, F˜ 1 ( ) is completely :xed by chiral symmetry for any density—possibly up to chiral restoration—so the only quantity that appears here is the BR scaling factor. At nuclear matter density, we have 13 F˜ 1 ( )|=0 = −0:153. Now taking P(0 ) ≈ 0:78 from (5.17), we predict from (6.4) :glth = 0:22773 :

(6.5)

This agrees with the experiment (6.2) providing a quantitative check of the BR scaling. 6.3. Axial charge transitions in heavy nuclei Another process where the Lagrangian (5.2) with BR scaling together with the PKMR’s EFT can make a prediction is in the axial charge beta transition of the type A(J ± ) → A (J ∓ ) + e− (e+ ) + () W

\T = 1 :

(6.6)

This process was studied by several experimentalists with the objective of exhibiting strong mediumenhancement of the matrix element of the axial charge operator Aa0 that governs the process (6.6) in heavy nuclei. Such an enhancement was :rst observed by Warburton in the lead region [92] and since then several authors con:rmed Warburton’s :nding in various other heavy nuclei. Warburton focused

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Table 2 “Empirical values” for jMEC Mass number A

jMEC

Reference

12 50 205 208

1:64 ± 0:05 1:60 ± 0:05 1:95 ± 0:05 2:01 ± 0:10

[93] [94] [95] [92]

on the quantity (that we shall refer to as “Warburton ratio”) jMEC = Mexp =Msp ;

(6.7)

where Mexp is the measured matrix element for the axial charge transition and Msp is the theoretical single-particle matrix element for a nucleon without BR scaling. There are several theoretical uncertainties involved in extracting the Warburton ratio. First, one has to extract from given beta decay data what corresponds to the axial charge matrix element. This involves estimating accurately other terms than the axial charge that contribute. The second is what one means precisely by Msp which is a theoretical entity. For these reasons, an unambiguous conclusion is diScult to arrive at. However what is signi:cant is Warburton’s observation that in heavy nuclei, jMEC can be substantially larger than the possible uncertainties involved: Heavy Nuclei

jMEC

= 1:9–2:0 :

(6.8)

Furthermore more recent measurements and their detailed analyses in di9erent nuclei [93–95] as shown in Table 2 quantitatively con:rm this result of Warburton. The prediction from the Lagrangian (5.2) with BR scaling in the same approximation as in the gyromagnetic ratio case is [96,79] 5th ˜ ; jMEC = P−1 (1 + 2)

(6.9)

where 2˜ is, in accordance with the chiral :lter mechanism, dominated by the pionic contribution with small controllable corrections from vector degrees of freedom, 31 the magnitude and the density dependence of which are again totally controlled by chiral symmetry, with the BR scaling P appearing as an overall factor. For = 0 =2 and = 0 , we :nd [79] 5th jMEC |=0 =2 ≈ 1:63;

5th jMEC |=0 ≈ 2:06 :

(6.10)

This is in an overall agreement with all the available Warburton ratios (see Table 2). 6.4. Axial-vector coupling constant gA? in dense matter One can make a rather simple statement on how the axial-vector coupling constant gA? scales in dense matter on the basis of BR scaling and the Fermi liquid theory. In terms of the Landau–Migdal :xed-point parameters, 2˜ = G1 =3 − 10H0 =3 + 4H1 =3 − 2H2 =15 with the dominant contribution coming from the soft pions [51]. 31

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The story of gA? in nuclei is a long story and rather involved with nuclear structure e9ects compounded with strong nuclear tensor correlations, excitations of nucleon resonances and possible e9ects of the vacuum change [97–100]. Gamow–Teller transitions in complex nuclei, particularly giant resonances thereof, play an important role in presupernova stage of the collapsing stars and the e9ective coupling constant gA? :gures importantly in their description [101]. What we will present here is probably not the unique explanation of what is going on but it is a version that is immediate from, and consistent, with the general theme of this review. 32 We have arrived above at the Landau mass formula (5.14) given in terms of the BR scaling parameter P. The Landau mass di9ers from the BR scaling mass M ? due to the pion contribution. From the Skyrme-type Lagrangian from which we have deduced the scaling behavior, this means that there must be contributions from other than the property of f . The only other factor that enters in the skyrmion description is the coeScient of the Skyrme quartic term e or in terms of a physical quantity, gA . When this is taken into account, the resulting expression is found to be [79] 33 m? gA? N = P (6.11) M gA from which we get 2 −2 1 1˜ gA? = 1 + F1 ( ) = 1 − F 1 ( )P : gA 3 3

(6.12)

At nuclear matter density, this predicts gA? (0 ) ≈ 1 :

(6.13)

What does this mean physically? If we accept that the Landau mass is a :xed point of the Fermi liquid theory, then the gA? must be a constant deCned at the :xed point. It therefore must correspond to decimating all the way down to the Fermi surface, that is to say, sending the infrared cuto9 kc in the renormalization-group equation to zero. In this limit, gA? must be proportional to the Landau parameter G0 . This connection in in:nite (translationally invariant) matter is being investigated in Ref. [102]. What this means in practice in :nite nuclei in which the transition is measured is not yet fully understood. But it appears reasonable to suppose that applied to :nite nuclei, what we obtained here is an e9ective constant that should be used when the Gamow–Teller matrix element is computed within the valence shell with the single-particle states undergoing the Gamow–Teller 32

It is perhaps appropriate at this point to clarify how the absence of Lorentz symmetry manifests itself in our treatment of gA? , an issue we brieKy alluded to earlier. As the two (preceding and this) subsections clearly show, the e9ective axial coupling constant gA in matter is di9erent for the space and time components of the axial current. We see that the time component of the axial charge (measured through axial-charge transitions) is enhanced in nuclear matter whereas the space component of the axial charge (measured through Gamow–Teller transitions as here) is quenched. Although we did not make the distinction in discussing BR scaling, it is evident that the gA? that appears in the relation between the BR scaling mass M ? and the Landau mass m? N involves the space component measured by Gamow–Teller transitions. A similar separation can be made for instance for the pion decay constant f ? which will be discussed in Section 8. 33 Since the middle expression of (6.12) involves the pionic contribution F1 ( ), one might think that gA? di9ers from gA merely due to the presence of the pion. This interpretation is not correct. The Landau parameter F1 ( ) contains the 2 ? density of states N (0) = 2kF m? N = which carries the Landau mass mN and hence does involve BR scaling. This is clearly seen in the last equality.

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transition restricted to the lowest-lying excitations on top of the Fermi sea. In other words, the gA? so obtained is a :xed-point quantity. An apt illustration of how this reasoning seems to be working is given by the complete 0f1p-shell Monte Carlo calculation of Gamow–Teller response functions by Radha et al. [103] and also big shell-model diagonalization by Caurier et al. [104]. Remarkably, it is found that gA? ≈ 1 is universally needed in these calculations [103,105,101] which may be taken as a support for the :xed-point notion of the gA? . 6.5. Evidences from “On-Shell” vector mesons The indirect evidences we have discussed above dealt with the properties of hadrons that are way o9-shell, with q2 ≈ 0. The masses and coupling constants that are probed in such processes correspond to the parameters of the e9ective Lagrangian and they would be near their physical parameters only if the tree approximations were valid. One would of course like to confront with measurements of the “physical hardrons” propagating in dense matter. In order to do this quantitatively with a theory that implements BR scaling, one would have to compute to higher orders, say, in chiral perturbation suitably formulated to incorporate the “sliding” vacuum structure associated with BR scaling. This would of course give rise to widths of the hadrons involved and to a further shift in mass from that of BR scaling which is mean :eld. Such a program has not yet been clearly formulated. Hence there are no predictions in our approach. For low density, there are several calculations that start from the zero density vacuum [106] that predict that while the meson may or may not be shifted in mass, the ! will. The will broaden in width in matter but the ! will remain narrow and may even be bound in nuclear matter [107]. The experimental :nding of this property of ! will con:rm unambiguously the observations made in the preceding and following sections. Experiments are presently being performed to check all this at various laboratories, notably at GSI and Je9erson Lab. Speci:cally, the signals from heavy-ion collisions to be discussed below address this issue although with results that are compounded with temperature e9ects. Here we brieKy comment on some recent experiments that seem to indicate that the vectormeson masses do decrease with density in nuclear matter or that at least the dropping masses are compatible with the observations. One is the TAGX collaboration result [108] on the process 3 He(&; 0 )ppn with the tagged photons in the range 800 –1120 MeV. The mass shift of the 0 meson reported was :m = 160 ± 35 MeV. This is nearly the amount predicted for nuclear matter, i.e., m (1 − P) ≈ 170 MeV for P = 0:78 used above. This seems to be a bit too much a shift for such a small nucleus. It seems to involve other than just BR scaling. The other experiment is a more recent measurement [109] at KEK on the invariant mass spectra of the e+ e− pairs in the target rapidity region of 12 GeV p + A reactions with the nucleus A = C and Cu. The experimental result is given in Fig. 8. This collaboration “sees” the mass shift of the vectors in the heavier nucleus indicative of a BR scaling. Although the precise value of the mass shift is not determined in this work, it is consistent with the expectation. We learned from private communication [110] that with their kinematic distribution, about 60% of the mesons and about 10% of the ! mesons are estimated to decay inside Cu nucleus, so most of the excess on the low-mass side of the ! peak is from the mesons. The authors of Ref. [109] considered the possibility of an in-medium increase in ! mass, but this did not give a statistically signi:cant e9ect.

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Fig. 8. Distribution with invariant mass of e+ e− pairs [109]: (a) from a carbon target; (b) from a copper target. The di9erence between histogram and thin solid line represents the e9ect of medium (density) on vector-meson mass, the e+ e− invariant mass being determined by the in-medium vector-meson mass.

More data and their analyses are forthcoming and will give a clearer insight into the properties of the hadrons on the “mass-shell”, thereby checking the BR scaled e9ective Lagrangian in more details. 7. Direct evidence from the quasielastic (e; e ) response functions in nuclei The evidences for BR scaling given so far are more or less indirect. They are based on consistency with what we already know of nuclear interactions. In this section, we discuss what we consider to be the most direct evidence for BR scaling, namely, the longitudinal and transverse electromagnetic response functions in medium-heavy and heavy nuclei. Although the process involves o9-shell properties of the vector meson, we believe that this evidence is cleaner and more direct than what we can extract from presently available heavy-ion data—including the celebrated CERES data—described below. Electron scattering which knocks nucleons out of nuclei have been used for many years to tell us about the binding energy of protons in nuclei and also to pin down the e9ective mass that the proton has in the nucleus. To the extent that the electron couples through vector mesons to the nucleons, these experiments can also tell us about the in-medium properties of the vector mesons. This was worked out :rst in a paper by Soyeur et al. [111] following the initial suggestion in Refs. [89,90]. We follow here a more schematic treatment [112] which brings out the main ideas more clearly, and which is in semi-quantitative agreement with the more detailed calculation. Before going further, let us remark that the experimental analysis has been thoroughly confused during the 1990s and only now is being clari:ed, chieKy due to e9orts of Joseph Morgenstern.

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A global analysis by Jourdan [113] which relied on three-dimensional numerical calculation of Onley et al. [114] indicated that medium e9ects appeared to be absent, or at least the situation was unclear. The analyses involved therein were unnecessarily complicated in view of the fact that Yennie et al. [115] had shown that the chief distortion by the Coulomb :eld could be reliably taken into account by shifting the momentum q to an e9ective momentum qe9 = (1 − Vc =|q|)

(7.1)

where Vc is an e9ective Coulomb potential of the electron. This is a semiclassical correction and had to be accompanied by higher order corrections for a quantitative description of the elastic electron scattering o9 nuclei [115] since the wavefunctions oscillate rapidly. Even there the qe9 of Eq. (7.1) gave a semiquantitative description of the distortion. Quasielastic scattering is very smooth as function of position in the integrand, and one would expect the introduction of qe9 should be adequate there. For a more recent discussion on this matter, see Triani [116]. Before the linear accelerator was shut down in Saclay, quasielastic experiments were performed on 12 C and 208 Pb with a positron beam. In this case the sign of Vc changes, and the correction (7.1) to the momentum is of opposite sign. Using the e9ective momentum approximation of Eq. (7.1), it has been shown [117] that both electron and positron scattering of 12 C and 208 Pb could be well described in a consistent way. Introduction of qe9 handles what is called the lowest order focusing e9ect in the theory. Whereas Onley et al. [114] found higher-order focusing e9ects to be important, introduction of these would ruin the consistency between the electron and positron scattering. Of course without substantial numerical work, it is not possible to check the Onley et al.’s results. • Experimental situation Although as theorists we cannot go into the experimental situation in detail, we can see that the early disagreement between the MIT-Bates data and Saclay data changed when MIT replaced their old scattering chamber, where background was corrected for by simulation, by a new one. The updated MIT-Bates data and the Saclay data are now in substantial agreement. The analysis of the positron scattering introduced a new element. It could be straightforwardly explained by the e9ective momentum approximation (EMA). Kim et al. [118] found that inclusion of higher order screening corrections in an approximate treatment of electron Coulomb distortion in quasielastic (e; e ) reaction was important. Indeed for forward electron angles the low-energy side of the DWBA peak looks similar to the plane-wave result. The authors suggest that the phase factors e9ectively cancel the e9ect of the e9ective momentum on the low ! side of the quasielastic peak. However, if so, the next correction beyond the EMA would involve the square of the projectile charge which would be the same for electron and positron, negating the simple relation (7.1). This would also mean that the di9erence between plane wave and DWBA for positrons would be large. In any case, these matters can be only conjectured because the Ohio University collaborations have not been extended to positrons although it seems obvious that this should be done. It is our belief that the whole situation, especially the theoretical analysis, has been a real shambles, for which we take partial responsibility. The Yennie, Boos and Ravenhall work was so convincing that theorists should have insisted that experiments were analyzed in the e9ective momentum approximation.

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• Theory with BR scaling The mechanism for the change in quasielastic scattering can be quite simply understood by considering the vector dominance model in which the electron couples to the nucleon through the vector meson. The vector mass enters into this coupling through D(mV ) =

m2V 1 = : 2 2 mV + q 1 + q2 =m2V

(7.2)

Usually the numerator enters into the electron– meson coupling and the denominator into the propagator, but the terms should be put together as in Eq. (7.2) because D(mV ) must go to 1 as ? q → 0 in order to conserve the electron charge. If now mV → m? V ∼ 0:8mV then D(mV ) ¡ D(mV ). The longitudinal response measures the charge density for which the operator is OL =

1 1 + 73 : 2 2

(7.3)

The !-meson couples to the 12 and the -meson to the 12 73 . The scattering via the !-meson exchange will be chieKy longitudinal in nature, since the isoscalar magnetic moment, which gives the transverse response, is small. Thus, the suppression in the longitudinal response is essentially F=

1 + q2 =m2! : 1 + q2 =m?2 !

(7.4)

Here what :gures is the BR scaling in the ! meson mass. For the transverse response the main operator is O T = V

· [ × q] 73 ; 2mN

(7.5)

this response coupling chieKy to the isovector magnetic moment. Now in medium mN → m? N as well as the propagator changing, so the relevant factor is 1 + q2 =m2 mN FT = : (7.6) 1 + q2 =m?2 m? N ? What enters here is the BR scaling in the nucleon mass. Now taking m? =m ∼ mN =mN according to the BR scaling, one :nds that the two factors in (7.6) nearly cancel each other in the range of momentum transfers q ∼ several hundred MeV in which the longitudinal and transverse responses have been measured. In Fig. 9, we show comparison of longitudinal data with the theory of Soyeur, Brown and Rho. Also a point from Jourdan’s analysis using the Onley et al. theory is shown. The drop of the dash–dot line, which included the Soyeur et al. medium e9ects, below the Fabrocini and Fantoni solid line, which does not include these e9ects, is very clear, an ∼ 20% e9ect. Both predicted and experimental e9ects on the transverse scattering are small. This is an example where BR scaling has a visible e9ect. Clearly the separation of longitudinal and transverse components was essential to expose BR scaling.

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Fig. 9. The longitudinal response function SL as a function of qe9 using (a) Saclay data only and (b) Saclay data plus SLAC NE3 and Bates data. The SLAC NE9’s 56 Fe result is shown by a cross and that of the Jourdan analysis by a star in (b). The theoretical curves are: the solid black curve for nuclear matter without BR scaling and the dashed curve for 4 He; the long dashed curve is the same as the nuclear matter one integrated within the experimental limits of !. The dot-dashed curve is with BR scaling. See Section 7 for details.

• Comments The Soyeur–Brown–Rho treatment in Ref. [111] was more complicated than our above schematic model because it built in the quark structure of the nucleon at short distances, essentially through inclusion of the chiral bag model. Since the latter was not scaled, the e9ect of scaling in our schematic model was somewhat diminished. Nonetheless they were quite large as seen from the di9erence between the dashed line and the “Jourdan analysis” and Soyeur et al.’s dash–dot line. It should be noted that Saito et al. [119] obtained within the quark–meson (QMC) model results for the quasielastic electron scattering similar to those of Soyeur et al. In Section 9, we will develop the evidence of medium-dependent vector-meson masses from the dilepton and photon experiments carried out at the SPS of CERN. Whereas the excess dilepton can be as simply produced in the Rapp–Wambach scenario as in the Brown–Rho scenario, the photons take us to higher densities and temperature, beyond the chiral restoration transition. We will develop a notion of complementarity between the BR dropping mass scenario and quark–gluon plasma (QGP) description for the photon treatment. As for quasielastic electron scattering we need not go into QGP since the hadron language is amply adequate. The dileptons we consider next :t in also quite simply with the quasielastic electron scattering. Our treatment in the next section will, however, allow us to unite the low-energy sector with the chiral restoration region of energies. 8. BR scaling in chiral restoration One can approach the phenomenology of the chiral restoration from a variety of di9erent angles. We de:ne ours by introducing the role of BR scaling by a simple construction of the chiral

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restoration transition as mean :eld in the Nambu–Jona–Lasinio model. More precisely we consider the transition as mean :eld up to both c and Tc , the critical density and temperature, although there may well be a density discontinuity at the transition, making it :rst order. Most likely the transition is second order for = 0, with a tri-critical point on the phase boundary going towards :nite density. What is important for our discussion, however, is that masses of mesons other than the pion go smoothly to zero in the chiral limit. The simplest possible model for this is mean :eld NJL. We follow the construction of Brown et al. [120]. Relevant parts of this paper were reviewed in Brown and Rho [13]. The present description in NJL agrees with the vector manifestation scenario of Harada–Yamawaki’s hidden local symmetry theory (see Section 4) and with the quark–hadron complementarity picture given in Section 11. 8.1. Bag constant and scalar Celd energy Brown et al. showed that the Walecka nuclear mean :eld theory at nuclear matter density could be connected with NJL mean :eld theory at higher densities; e.g., the "-:eld coupling the quarks in NJL can be taken to be 1=3 of that to nucleons in Walecka theory, g"QQ = g"NN =3 10=3 :

(8.1)

Furthermore, the scalar :eld energy 12 m2" "2 plays the role of the bag constant B in the chiral restoration transition. On the quark–gluon side of the transition the appearance of the bag constant corresponds to the disappearance of the scalar :eld energy going upwards from below through the transition. The magnitude of the e9ective bag constant is only ∼ 1=2 of that would be given by the trace anomaly: B=−

)(g) a 2 0|(G ) |0 8g

(8.2)

which to one loop (for NF = 3) is B=

9(s a 2 0|(G ) |0 = (245 MeV)4 8

= 469 MeV=fm3 2Be9

(8.3)

where Be9 = 12 m2" "2 . As found by Miller [121], only about 50% of the glue is “uncovered” in the chiral restoration transition. The other 50% remains as “covered” glue. In Shuryak et al. [122], the soft glue disappears with the dynamically generated quark masses whereas the hard glue (“epoxy”) is made up of instanton molecules which do not break chiral symmetry. The amount of trace anomaly which disappears at the transition gives Be9 235 MeV=fm3 . 8.2. “Nambu scaling” in temperature From the Gell–Mann–Oakes–Renner relation 2 m2 = 2mq qq=f W

(8.4)

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Fig. 10. Comparison of entropy derived by Koch and Brown from the lattice data on energy density (dashed line) and from dropping masses (full lines) calculated in the Nambu scaling and in the naive Brown–Rho scaling. In the lower dashed line for W the bare quark mass has been taken out.

and taking m not to scale with density, Brown and Rho [13] found hadron masses scaling as f ? with f ? ˙ |qq W ? |1=2 . 34 It was then with some surprise that Koch and Brown [123] found that the temperature dependence of the entropy in the many-body system as calculated in lattice gauge calculations [124] could be reproduced in the hadron sector best with the scaling qq W ? m? ≈ m qq W

(8.5)

for hadron masses other than the pion mass. In Fig. 10 is shown how Nambu scaling behaves in the phase transition region. This scaling follows from the mean :eld NJL model and is referred to as “Nambu scaling” although it is a generic feature of the linear sigma model. The entropy is particularly suitable to study, since quark mean :elds drop out so it tells us directly about the number of degrees of freedom.

34 In Ref. [2], using Skyrme’s Lagrangian and looking at pions in medium, the scaling f ? ˙ |qq W ? |1=3 was obtained. We now know that because of chiral symmetry broken only slightly in nature, the in-medium properties of a pion cannot be given correctly in the same mean :eld argument as used for other non-Goldstone bosons. Therefore it is safe to say that there is no accurate description of the pionic property based on model considerations. Future experiments will help in this direction.

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The hadron language is suitable for low temperatures. As stressed above and further elaborated later, we believe that our description is “dual” to a quark–gluon language as temperature moves towards the chiral restoration temperature. In particular, Koch and Brown limited their number of hadrons to 24, the number of light quarks. Rather than just cutting the number at this value, the same number can be consistently imposed by introducing excluded volume e9ects [125]. In any case the restriction that the hadronic description with dropping masses go over, in terms of the number of degrees of freedom, to the quark picture at higher temperature, must be built in. Otherwise it would not be possible to go beyond the Hagedorn temperature. The scale at which the change-over from hadron to quark language is made is one of convenience in accordance with the Cheshire Cat principle. Presumably it is in the region of the chiral restoration transition. This aspect becomes even more evident with the “hadron–quark continuity” discussed in Section 11. Finally note that the Nambu scaling that :ts the entropy in the hadronic description comes from mean :eld, i.e., the coupling constant g"QQ is not scaled. As long as we deal with the ", the Kuctuation :eld connected with chiral symmetry restoration, mean :eld seems to work very well, perhaps at least approximately up to the chiral restoration transition. In fact it was noticed in Ref. [79] that the mean :eld description of nuclear matter in terms of chiral Lagrangians implemented with BR scaling required that the scalar coupling be unscaled at least up to nuclear matter density. This will also be observed in the problem of “strangeness equilibration” discussed in Section 10. 8.3. “Nambu scaling” in density Given that there is nothing which dictates that the scaling in temperature and that in density be the same, the question is whether the Nambu scaling also works in density, at variance with those found in Refs. [2,13] which are di9erent from the Nambu one. To answer this question let us :rst address the question: How must the pion mass m? change with density in order to have Nambu scaling; i.e., hadron masses other than that of the pion to scale as W ? m? f ? qq = ? = m f qq W

(8.6)

From Gell–Mann–Oakes–Renner equation 2 f 2 = 2mq qq=m W

(8.7)

and since the quark mass does not change with density, we have f ? = f

qq W ? qq W

1=2

m : m?

(8.8)

Now, to linear order in density we have qq W ? " N = 1 − 2 2 ≈ 0:63 qq W f m

for = 0 :

(8.9)

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Setting (8.6) equal to (8.7) at = 0 and using (8.8), we :nd that in order for the Nambu scaling to apply to density, the pion mass should scale up with density by a factor 1=2 m? qq W = ≈ 1:26 for = 0 : (8.10) m qq W ? It turns out that this increase seems to be supported empirically although it is not clear whether the “measured” mass is the same quantity that appears in the in-medium GMOR relation. Experiments which pinned down the increase in pion mass from bare mass by a factor of 1.20 –1.24 were performed recently at GSI [126] in the deeply bound states of 207 Pb. Since the outer part of the pion wave functions is at the densities rather less than nuclear matter density, these factors may represent the lower limits and a factor of 1.26 for = 0 seems quite reasonable. Thus the Nambu scaling may hold at least near nuclear matter density. We should mention however two caveats to this. One is that two-loop chiral perturbation calculations :nd smaller values for the mass shift. The calculation by Park et al. [127] predicts the pion mass at nuclear matter density to be at most 6 –7% higher than that at zero-density. This is a calculation where the o9-shell ambiguity associated with the pion in medium being an o9-shell quantity is minimized. For a pion :eld that contains substantial o9-shell ambiguity, one can easily obtain the mass factor 1.20 –1.24 but this seems to be unreasonable. Another two-loop calculation by Kaiser and Weise [128] for the pion self-energy in asymmetric nuclear matter obtains about 10% increase in mass of the pion which is consistent with the result of Park et al. The other caveat is that the scaling (8.6) with the value (8.9) at nuclear matter density would disagree with the value expected from the experimental value (6.2) for :gl , Eq. (6.4), for which we have P(0 ) = 0:78. In the above we have established that NJL theory at mean :eld [120] works well in describing the scaling of masses. In Ref. [83] it was shown that the Walecka mean :eld theory in terms of nucleons can be carried over with smooth change of parameters to a mean :eld theory of constituent quarks at higher densities. The constituent quark goes massless in the chiral limit with chiral restoration. In Section 11.2, we will suggest how this can be understood in terms of color-isospin locking and hidden local symmetry. In low-energy nuclear physics we are used to introducing form factors with couplings. These decrease with increasing scale. In modern language, the form factors are there as a sort of regularization that plays an important role in e9ective :eld theories as discussed in Section 3. Why do we do so well here without form factors? We believe that the reason lies in the special role that the scalar :eld " has in chiral restoration. A priori it would seem reasonable to continue the linear approximation for "-exchange between quarks −1 W 2 − m?2 Q(q " ) Q ;

(8.11)

where Q is the constituent quark, down to small m? " where higher-order or nonlinear e9ects must enter. We believe that these e9ects must be counterbalanced, to a large extent, by form factors. As we develop in the next section, this does not seem to be so with vector mesons, where form factors really do not cut down the exchanges. As formulated more precisely in Section 11.2, the vector meson exchange at low densities (i.e., Kavor symmetry) changes into gluon exchange at higher density (QCD symmetry) and asymptotic freedom assures that their e9ects go to zero with increasing scale.

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9. Signals from heavy-ion collisions Extreme conditions of high temperature and=or high density are expected to be created in relativistic heavy-ion collisions being probed in RHIC at Brookhaven and to be probed in CERN’s LHC. In this section, we discuss how BR scaling manifests itself in heavy-ion processes, leaving highdensity zero-temperature situation to later sections. Before going into the details, we :rst de:ne precisely the meaning of BR scaling in the context that we shall use in this section. 9.1. Top–down and bottom–up and how they connect There are e9ectively two ways of approaching BR scaling. The BR scaling as :rst suggested in Ref. [2] is a “top–down” version starting, using quark mean :eld, from a high temperature=density regime wherein constituent quarks (or quasiquarks) are relevant degrees of freedom, i.e., near the chiral transition point and then extrapolating down to a low temperature=density regime wherein hadrons are appropriate degrees of freedom. In contrast, Rapp and Wambach and others [129] calculated medium dependent hadron properties in hadronic variables in a strong-coupling perturbation approach. We shall call this a “bottom–up” approach. The advantage of the latter approach is that one can resort to standard nuclear physics information using phenomenological Lagrangians at (generalized) tree order with more or less known coupling constants. This approach has been found to be successful in reproducing the medium-dependent -meson properties deduced from the CERN CERES dilepton experiments. In these experiments most of the -mesons come from densities less than nuclear matter density. In the low-density regime, the phenomenological approach is justi:ed by construction in a well-prescribed way. The disadvantage of this approach is that as density increases beyond nuclear matter density, the possible vacuum change makes the generalized tree order calculation unreliable because of the strong coupling or more precisely a large anomalous dimension which signals that one is Kuctuating at a wrong ground state and that it is preferable to shift to a di9erent vacuum even though no phase transitions are involved. This aspect was associated above with di9erences in topological vacua. In Refs. [130 –132], the top–down BR scaling and the bottom–up Rapp–Wambach approach were combined into a uni:ed description. We will follow this description. We consider the most important component of the Rapp–Wambach theory, i.e., the introduction of the “-sobar” which is the excitation of the I = 1, J = 3=2 N ? (1520) coupled together with the nucleon–hole to the quantum numbers of the . There are of course other N ? -hole states of the same quantum numbers which would come in but here we shall focus on the dominant component only. At zero density this isobar-hole state is at 580 MeV. The phenomenological Lagrangian coupling the “elementary ” and the isobar-hole ˆ that we shall use 35 is -wave = LsN ?N

35

f m

†

s N ∗ (q0˜

· ˜a − 0a˜s · ˜q)7a

N

+ h:c: ;

The non-covariant form of the Lagrangian is not unique as pointed out in Ref. [133].

(9.1)

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2 where fN q, q0 ) are the -meson momenta. ? N =4 5:5 from :ts to photodisintegration data. Here (˜ We shall keep only the q0 term in our dilepton discussion. The two branches of the spectral function can be located by solving self-consistently the real part of the -meson dispersion relation (taking ˜q = 0)

q02 = m2 + Re $N ? N (q0 ) :

(9.2)

The natural width of the N ? (1520) is supplemented at = 0 by an additional medium-dependent width of 250 MeV. Including also the backward-going graph, the N ? (1520)N −1 excitation contributes to the self-energy of the -meson at nuclear matter density n0 (in this section we denote density by n to avoid confusion with the -meson) $N ? N (q0 ) =

2 fN ?N

8 q02 n0 3 m2 4

(\E)2 (q0 + i4tot =2)2 − (\E)2

;

(9.3)

where \E 1520 − 940 = 580 MeV and 4tot = 40 + 4med is the total width of the N ? (1520). If we neglect any in-medium correction to the width of the N ? (1520) we :nd two solutions, located at q0− 540 MeV and q0+ 895 MeV. As n increases, $ will become larger and initially the q0 of the lower branch will go down. However, this downward movement will be halted once q0 has decreased substantially. We note that already at n = 0, the N ? (1520) decays ∼ 15% of the time to N indicating that there is already mixing there. The mixing becomes stronger with increasing n. As noted, the Rapp–Wambach theory works well for the low densities n . n0 important for the CERES experiments. We show in Fig. 11 :ts to the CERES spectra with both Rapp–Wambach and Brown–Rho theories. In the BR approach, however, the vacuum is “sliding”, that is, for each new density, one has a new “vacuum” to Kuctuate around. Recalling the skyrmion structure for given baryon number, we observe that “vacua” are solitonic in nature and cannot be connected by perturbation theory. Thus, the vacuum at n = 0 has no continuous connection with that at n = 0. This means that with the Lagrangian (9.1), the Rapp–Wambach theory can never approach the BR theory as density increases. One can see from (9.3) that the self-energy will stop bringing down the energy of the lower branch of the as (q0 =m )2 becomes smaller. This situation is somewhat like the usual Walecka mean :eld theory where the nucleon self-energy has a scalar density s = † &0 in it. As s goes to zero, the self-energy which involves s linearly does likewise, so the nucleon mass can go to zero only at n = ∞. However with the introduction of negative energy states in Nambu–Jona–Lasinio theory, the mass of the constituent quark goes smoothly to zero at a density nc ∼ 3n0 [120]. As argued in Refs. [131,132], this can happen with (9.3) only if we replace 2 there the factor (q0 =m )2 by ∼ j(q0 =m? ) . The unknown factor ' can be adjusted to give the same critical density as in NJL model. It is plausible that this change-over is related to the requirement in the Harada–Yamawaki scheme which is not present in few-order perturbation theory that the phase transition coincide with the VM limit :xed point. As mentioned, the Harada–Sasaki work [66] in temperature indeed suggests that the Wilsonian matching with QCD with hidden gauge invariance requires the parametric mass to have an “intrinsic” T=n dependence. Once one accepts

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Fig. 11. Dilepton data from central Pb (158 GeV=u)+Au collisions compared with Rapp=Wambach (labelled as “in-medium ”) and Brown=Rho (labelled as “dropping m ”) for qtee ¡ 0:5 GeV (upper) and qtee ¿ 0:5 GeV (lower).

this, what follows turns out to be quite reasonable: with j = 1; m? drops to zero at the same density as in the NJL model, nc ∼ 3n0 . 9.2. Distinguishing Rapp=Wambach and Brown=Rho The modi:ed sobar picture obtained by replacing m by m? as described above still di9ers in detail from the BR scaling picture of 1991 even though they describe more or less equally well the m? → 0 limit, because at any density less than nc there are two branches of the quantum number. However as n → nc the width of the lower branch goes to zero and all of the spectral strength goes into it so we do recover the original BR scaling. We will encounter this feature once more in Section 11.2 from the consideration of hidden local symmetry and color–Kavor-locking in QCD. At densities n . n0 , the lower branch most likely becomes very wide because of the large number of open channels that it couples to, which has led Rapp and Wambach to interpret their :t to the dilepton data from the broadening of the , the lower parts of the spectral function contributing importantly to the dilepton production because of the greater Boltzmann factors. So how do we distinguish between Rapp=Wambach and Brown=Rho; equivalently between broadening and movement towards chiral restoration? Formally we do not violate gauge invariance or any other invariance by using m? rather than m . It is really up to nature to decide.

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In a recent paper, Alam et al. [134] have suggested that nature has decided for Brown=Rho. 36 This work is extended in Alam et al. [137]. In these papers, the calculations were based on mesonic processes such as → &; → &, etc. leading to thermal production of photons. The observation then was that when increasing the width of the meson to ∼ 1 GeV in order to take into account medium e9ects, the photon yield decreases as compared to the case using the free width, thus underestimating the spectra. However these calculations neglect the contributions to the photons coming from some of the processes that increase the width, e.g., the → N ? (1520)N −1 excitation (the collective “-sobar”) which provides the lower-mass excitation in the Rapp=Wambach calculation. This contributes to the photon through the N ? (1520) → N& decay. Calculations in progress (by C. Gale, M.A. Hadasz and R. Rapp) show that these baryonic processes contribute importantly so the conclusions of [134,137] may be premature. The bottom–up approach of Rapp and Wambach should be reliable at low density since it is based on phenomenological Lagrangians constructed in standard nuclear physics with known properties. Our principal argument is that this picture should cede to the Brown-Rho picture at higher densities replacing m by m? in scaling the Lagrangian required to be consistent with Harada–Yamawaki’s vector manifestation. Of course much more detailed experiments will be needed to con:rm this scenario. 10. “Broad-band equilibration” of strangeness in heavy-ion collisions 10.1. Kaons and chiral symmetry The strange quark is massive compared with the light (u; d) quarks, with ms ∼ 150 MeV. This gives the kaon a mass of ∼ 1=2 GeV. This is a manifestation of both explicit and spontaneous chiral 36

In fact Alam et al. suggest two scenarios that can describe the photons, one being the hadronic description with BR scaling and the other one in which quark–gluon–plasma (QGP) is initially produced at a temperature Ti = 196 MeV. One has a mixed phase of QGP and hadronic matter which persists down to a transition to hadronic matter at Tc = 160 MeV. In a later paper [135], Sarkar et al. :nd that for relatively central collisions, the dilepton yield can be described by either of the two scenarios, BR scaling or one in which the QGP is originally produced as above. These authors :nd that “as of yet, it has not been possible to explain the observed low-mass enhancement of dileptons measured in the Pb + Au collisions as well as in the S + Au collisions at the CERN SPS in a scenario which does not incorporate in-medium e9ects in the vector meson mass”. As noted earlier, we believe that two descriptions, BR scaling and the one in which the QGP is initially produced in a mixed phase, to be “dual” as long as the number of degrees of freedom in the former is constrained to equal that in the latter. The mixed phase in the scenario with the QGP formation could, however, be just the part of a second-order phase transition in the BR scaling scenario. In the latter the energy is used up, going down in temperature through the phase transition, in building up the scalar :eld energy as the hadrons gain back their masses. Most of the entropy decrease will come from the higher mass hadrons disappearing as their mass is increased, due to decreasing Boltzmann factors, rather than through expansion. It seems therefore reasonable that the various freeze-out species exhibit thermal equilibration at the chiral restoration temperature, the one at which hadron masses go to zero in the chiral limit in the BR scaling scenario, since the system spends a long time there. Although the dropping masses with BR scaling and production of QGP in mixed phase at T ∼ 200 MeV may be dual around the phase transition, it is excluded that the latter scenario describe the dropping masses found in ∼ 300 MeV inelastic electron scattering, which we described in the previous section, or the many indirect evidences we described earlier in our review. In other words, the duality is useful only at higher scales, near and above the chiral restoration transition. One should remember the BXeg–Shei theorem, that the nature of symmetry realization (i.e., Wigner–Weyl vs. Nambu–Goldstone) is irrelevant to discussing the short-distance symmetry [136].

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symmetry breaking. Unlike the previous cases where the decondensing of the condensate qq—that W is, restoring the spontaneously broken chiral symmetry—enters directly into the phenomena we were looking at, here it is the “rotating away” of the explicit chiral symmetry breaking intricately tied in with the decondensing of qq W that is principally manifested in the behavior of kaons in dense medium. This section deals with how this aspect is probed in the kaon production experiments at GSI. Based on work done recently [138], we will develop, in conjunction with BR scaling, the notion of “broadband equilibration” in heavy-ion processes and suggest the vector decoupling in dense medium. 10.2. Equilibration vs. dropping kaon mass Following work by Hagedorn [139] on production of anti-3 He, Cleymans et al. [140] have shown that for low temperatures, such as found in systems produced at GSI, strangeness production is strongly suppressed. The abundance of K + mesons, in systems assumed to be equilibrated, is given by [141], d 3 p −EKW =T d 3 p −(E? −B )=T −EK + =T : (10.1) V gKW e + g? e nK + ∼ e (2 )3 (2 )3 Here the g’s are the degeneracies. Because strangeness must be conserved in the interaction volume V , assumed to be that of the equilibrated system for each K + which is produced, a particle of “negative strangeness” 37 containing s, say, KW or ?, must also be produced, bringing in the KW or ? phase space and Boltzmann factors. The K + production is very small at GSI energies because of the low temperatures which give small Boltzmann factors for the KW and ? in addition to the small Boltzmann factor for the K + . Note the linear dependence on interaction volume which follows from the necessity to include KW or ? phase space. In an extensive and careful analysis, Cleymans et al. [141] show that measured particle multiplicity ratios + =p; − = + ; d=p; K + = + , and K + =K − —but not = 0 —in central Au–Au and Ni–Ni collisions at (0.8–2.0)A GeV are explained in terms of a thermal model with a common freeze-out temperature and chemical potential, if collective Kow is included in the description. In other words, a scenario in which the kaons and anti-kaons are equilibrated appears to work well. This result is puzzling in view of a recent study by Bratkovskaya et al. [142] that shows that the K + mesons in the energy range considered would take a time of ∼ 40 fm=c to equilibrate. We remark that this is roughly consistent with the estimate for higher energies in the classic paper by Koch et al. [143] that strangeness equilibration should take ∼ 80 fm=c. Such estimates have been applied at CERN energies and the fact that emergent particle abundances are described by Boltzmann factors with a common temperature ∼ 165 MeV [144] has been used as part of an argument that the quark=gluon plasma has been observed. We interpret the result of Ref. [141] as follows. Since free-space masses are used for the hadrons involved, Cleymans et al. [141] are forced to employ a B substantially less than the nucleon mass mN in order to cut down ? production as compared with K − production, the sum of the two being equal to K + production. This brings them to a di9use system with density of only ∼ 0 =4 at chemical freeze-out. But this is much too low a density for equilibration. 37

By “negative strangeness” we are referring to the negatively charged strange quark Kavor. The positively charged anti-strange quark will be referred to as “positive strangeness”.

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We shall :rst show how this situation can be improved by replacing the K − mass by the K − energy at rest !K?− ≡ !− (k =0) ¡ mK . (The explicit formula for !± is given later, see Eq. (10.25).) In doing this, we :rst have to reproduce the K + to K − ratio found in the Ni + Ni experiments [145]: nK + =nK − 30 :

(10.2)

Cleymans et al. reproduce the earlier smaller ratio of 21 ± 9 with B = 750 MeV and T = 70 MeV. How this or rather (10.2) comes out is easy to see. The ratio of the second term on the RHS of Eq. (10.1) to the :rst term is roughly the ratio of the exponential factors multiplied by the phase space volume g? pW ? 3 e−(E? −B )=T m? 3=2 e−(m? −B )=T R= ≈ ≈ 21 (10.3) gKW pW KW e−EK − =T mKW e−mK − =T where we have used g? ≈ gKW 2; (p) W 2 =2m 32 T; E? = 1115 MeV and EKW = 495 MeV. Inclusion of the $ and R hyperons would roughly increase this number by 50% with the result that the ratio of K − to ?; $; R production is 38 nK − 1=32 : (10.4) n?+$+R Since a K + must be produced to accompany each particle of one unit of strangeness (to conserve strangeness Kavor), we then have nK + =nK − ∼ 33 :

(10.5)

This is consistent with the empirical ratio (10.2). It should be noted that had we set B equal to mN , we would have had the K + to K − ratio to be ∼ 280 because it costs so much less energy to make a ? (or $) rather than K − in this case. In other words the chemical potential B is forced to lie well below mN in order to penalize the hyperon production relative to that of the K − ’s. One can see from Fig. 5 in Li and Brown [146] that without medium e9ects in the K − mass, the + K =K − ratio is ∼ 100, whereas the medium e9ect decreases the ratio to about 23. This suggests how to correctly redo the Cleymans et al.’s analysis, namely, by introducing the dropping K − mass into it. In the next subsection we show that positive strangeness production takes place chieKy at densities greater than 20 . As the :reball expands to lower densities the amount of positive strangeness remains roughly constant. The number of K + ’s is such as to be in equilibrium ratio K + = + with 3 the equilibrated number density of pions at T = 70 MeV; n ≈ 0:37 T197 fm−3 . Only in this sense do the K + ’s equilibrate. It will be noted, however, that with the T of 70 MeV; B is chosen so that the empirical number of hyperons are produced. Since the number of K + is one greater than that of the hyperons (including the K − in the negative strangeness), it will also be the apparent equilibrated ratio for this B and T . Thus, with these two latter parameters the ; K + and K − can be put into apparent equilibration. It is fairly easy for the nucleons to equilibrate with the pions at the given temperature because of the strong interaction. 38

In order to reproduce this result with B = 750 MeV and T = 70 Mev within our approximation, we have assumed only the $− and $0 hyperons to equilibrate with the ?. This may be correct because the $+ and R couple more weakly. Inclusion of the latter could change our result slightly. Probably they should be included in analysis of the AGS experiments at higher energies where they would be more copiously produced.

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It is amusing to note that the “equilibrated ratio” of ∼ 30 for the nK + =nK − holds over a large range of densities for T = 70 MeV, once density-dependent K − masses are introduced, in that the ratio R of (10.3) is insensitive to density. (Remember that because of the small number of K − ’s, the number of K + ’s must be nearly equal to the number of hyperons, ?; $− and R, in order to conserve strangeness.) This insensitivity results because !K?− decreases with density at roughly the same rate as B increases. We can write R of (10.3), neglecting possible changes in T and mKW in our lowest approximation, as 39 m? 3=2 (B +!?− )=T −m? =T K R= e e : (10.6) mKW As will be further stressed later, the most important point in our arguments is that the B + !K?− is nearly constant with density. This is because whereas B increases from 860 to 905 MeV as goes from 1:20 to 2:10 ; !K?− decreases from 380 to 332 MeV, the sum B + m? K − decreasing very slightly from 1240 to 1237 MeV. Indeed, even at = 0 =4; B + m? ∼ 1218 MeV, not much − K smaller. We believe that the temperature will change only little in the region of dropping masses because in a consistent evolution (which we do not carry out here) the scalar :eld energy m2" "2 =2 in a mean :eld theory plays the role of an e9ective bag constant. In Ref. [120] (which was reviewed in Section 8) this is phrased in terms of a modi:ed Walecka theory, 1 1 Be9 = m2" "2 ⇒ m2" (MN =g"NN )2 ; (10.7) 2 2 the " going to MN =g"NN as the nucleon e9ective mass goes to zero with chiral restoration. Most of the energy with compression to higher densities goes with this e9ective bag constant, estimated [120] to be ∼ 240 MeV=fm3 , rather than heat, mocking up the behavior of a mixed phase with constant temperature. Moreover at = 20 where m? N may be ∼ 0:5mN , only about 25% of the bag constant B may have been achieved, so there may be some increase in temperature. We shall, at the same level of accuracy, have to replace mKW in the prefactor of Eq. (10.6) by !K?− . We adjust the increase in temperature so that it exactly compensates for the decrease in prefactor so that the K + =K − ratio is kept the same, as required by experiment [145]. We then :nd that the temperature at = 20 must be increased from 70 to 95 MeV. This is roughly the change given by that in inverse slopes of K − and K + transverse momentum distributions found in going from low multiplicities to high multiplicities [145]. In any case, we see that R will depend but little on density. This near cancellation of changes in the factors is fortunate because the K − + p ↔ ? reaction, operating in the negative strangeness sector, is much stronger than the positive strangeness reactions, so the former should equilibrate to densities well below 20 and we can see that the “apparent equilibration” might extend all the way down to ∼ 0 =4. The near constancy of R with density also explains the fact the K + =K − ratio does not vary with centrality. 40 Although R is the ratio of ?’s to K − ’s, both of which are in the negative strangeness To be fully consistent, we would also have to consider the medium modi:cations of the ? and K + properties. These and other improvements are left out for our rough calculation. 40 We are grateful to Helmut Oeschler for pointing this out to us. 39

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Fig. 12. Calculations by Bratkovskaya and Cassing (private communication) which show the density of origin and that of the last interaction of the K − mesons.

sector, nonetheless, the number of K + ’s must be equal to the sum of the two and since the ?’s are much more abundant than the K − ’s, R essentially represents the K + =K − ratio. Detailed transport results of Bratkovskaya and Cassing (see Fig. 12) show the last scattering of the detected K − to be spread over all densities from 0 =2 to 30 , somewhat more of the last scatterings to come from the higher density. This seems diScult to reconcile with a scenario of the K − numbers being decided at one de:nite density and temperature, but given our picture of dropping masses, one can see that the K + =K − ratio depends little on density, that is, on B at which the K − last interacts. In any case we understand from our above argument that the apparent density of equilibration can be chosen to be very low in a thermal description and still get more or less correct K + =K − ratio. 10.3. The equilibrium K + = + ratio We show here how our argument that gives a correct K + =K − ratio can reproduce the K + = + ratio. Let us leave T = 70 MeV and choose ∼ 20 as educated guesses. We are thereby increasing the equilibration density by a factor ∼ 8. We then calculate the baryon density for this B and T = 70 MeV and :nd = 20 which checks the consistency. According to Brown et al. [147] the K + production under these conditions will come chieKy from BB → N?K, with excited baryon states giving most of the production. From the solid curve for =0 = 2 in Fig. 9 of Ref. [147] we :nd "v ∼ 2 × 10−3 mb = 2 × 10−4 fm2 . The rate equation reads 1 BYK 1 1 vBB )n2B (2 × 10−4 fm2 )2B = dn=dt = × 10−4 fm−4 :SK = ("BB 2 2 9

(10.8)

where B; Y and K stand, respectively, for baryon, hyperon and kaon. For this, we have taken BYK "BB vBB from Fig. 9 of [147] and = 20 = 13 fm−3 . Choosing a time t = 10 fm=c we obtain nK + :SK t =

2 × 10−3 fm−3 : 9

(10.9)

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147

Now equilibrated pions have a density n = 0:37 (T=197 MeV)3 fm−3 = 0:016 fm−3

(10.10)

for T = 70 MeV. From (10.9) and (10.10) we get nK + =n + 0:0069

(10.11)

which is slightly below the “equilibrated value” of 0.0084 of Table 1 of Cleymans et al. [141]. Production of K + by pions may increase our number by ∼ 25% [148]. Our discussion of K + production is in general agreement with earlier works by Ko and collaborators [148,149]. In fact, if applied at the quark level, the vector mean :eld is conserved through the production process in heavy-ion collisions, so it a9ects only the strangeness condensation in which there is time for strangeness non-conservation. These earlier works establish that at the GSI energies the K + content remains roughly constant once the :reball has expanded to ∼ 20 , so that in this sense one can consider this as a chemical freeze-out density. It should be noted that in the papers [147–150] and others, the net potential—scalar plus vector— on the K + -meson is slightly attractive at ∼ 20 even though the repulsive vector interaction is not decoupled. A hint for such change-over was noted already at nuclear matter density by Friedman et al. [151]. Since in our top–down description the vector interaction can be thought of as applied to the quark (matter) :eld in the K + , the total vector :eld on the initial components of a collision is then the same as on the :nal ones, so the vector mean :elds have e9ectively no e9ect on the threshold energy of that process. Our proposal here is that the vector mean :eld on the K + should be below the values used by the workers in Refs. [147–150] due to the decoupling. However, in comparison with Ref. [150], our total potential on the K + at = 20 is ∼ −85 MeV, as compared with their ∼ −30 MeV. We have not redone the calculation of Ref. [147] to take into account this di9erence. 10.4. The top–down scenario of K ± production Brown and Rho [83] discussed Kuctuations in the kaon sector in terms of a simple Lagrangian :LKN =

−6i W $KN W W W + · · · ≡ L ! + L" + · · · (N N )KK 2 (N &0 N )K9t K + ? 8f f ?2

(10.12)

It was suggested there that at high densities, the constituent quark or quasi quark description can be used with the !-meson coupling to the kaon viewed as a matter Celd (rather than as a Goldstone boson). Such a description suggests that the ! coupling to the kaon which has one non-strange quark is 1=3 of the ! coupling to the nucleon which has three non-strange quarks. The L! in the Lagrangian was obtained by integrating out the !-meson as in the baryon sector. We may therefore replace it by the interaction 1 V K ± ≈ ± VN : 3

(10.13)

In isospin asymmetric matter, we shall have to include also the -meson exchange [83] with the vector-meson coupling treated in the top–down approach.

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Fig. 13. Projection onto the "; K plane. The angular variable H represents Kuctuation toward kaon mean :eld.

For the top–down scenario, we should replace the chiral Lagrangian (10.12) by one in which the 2 “heavy” degrees of freedom :gure explicitly. This means that 1=2f? in the :rst term of (10.12) 2 2 2 2 ?2 should be replaced by g? =m? in the second term by 23 mK g"? =m? ! and $KN =f " assuming that 2 2 both ! and " are still massive. We will argue below that while the ! mass drops, the ratio g? =m? ! should stay constant or more likely decrease with density and that beyond certain density above nuclear matter, the vector :elds should decouple. On the other hand, g" is not scaled in the mean :eld that we are working with; the motivation for this is given in Brown et al. [120] who construct the chiral restoration transition in the mean :eld in the Nambu–Jona–Lasinio model. Thus $KN 2 g"2 m ≈ K 2 2 : 3 f? m? "

(10.14)

M la BR scaling [2]: In this framework, m? " is the order parameter for chiral restoration which drops a 1 m? " ≡ P() m" 1 + y=0

(10.15)

with y 0:28, at least for . 0 . 41 Once the vector is decoupled, a simple way to calculate the in-medium kaon e9ective mass, equivalent to using the L" , is to consider the kaon as Kuctuation about the “"”-axis in the V-spin formalism [152] as depicted pictorially in Fig. 13. The Hamiltonian for explicit chiral symmetry breaking is 1 2 H5SB = $KN NW N cos(H) + m2K f? sin2 (H) 2 2 1 H 2 + m2K f? H 2 $KN NW N 1 − 2 2

41

(10.16)

y may well be di9erent from this value for ¿ 0 . In fact the denominator of P() could even be signi:cantly modi:ed from this linear form. At present there is no way to calculate this quantity from :rst principles.

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where the last expression is obtained for small Kuctuation H. Dropping the term independent of H, we :nd $KN NW N ?2 2 mK = m K 1 − : (10.17) f?2 m2K Using Eq. (10.14) we obtain 2 g"2 NW N ?2 2 mK = mK 1 − : 2 3 m? " mK

(10.18)

In accord with Brown and Rho [83] we are proposing that Eq. (10.17) should be used for low densities, in the Goldstone description of the K ± , and that we should switch over to Eq. (10.18) for higher densities. It is possible that the mK appearing in (10.18) should be replaced by m? K for ?2 ? self-consistency but the dropping of m" makes the mK of (10.18) decrease more rapidly than that of (10.17) so that Eq. (10.17) with NW N set equal to the vector density , 42 a much used formula valid to linear order in density $KN ?2 2 mK ≈ mK 1 − 2 2 (10.19) f mK obviously gives too slow a decrease of m? K with density. Although the above are our chief points, there are two further points to remark. One, even without scaling, our vector interaction on the kaon is still too large. Two, more importantly, there is reason to believe in the large $KN term, $KN ∼ 400 MeV :

(10.20)

This comes from scaling of the pion sigma term $KN ≡

W (mu + ms )N |uu + ss|N $ N : W (mu + md )N |uu + dd|N

(10.21)

W W ∼ 13 N |dd|N from lattice Taking ms ∼ 150 MeV; mu + md ∼ 12 MeV; $ N = 46 MeV and N |ss|N calculations [153], one arrives at (10.20). Other authors, in adjusting the $ term to :t the kaon-nucleon scattering amplitudes, have obtained a somewhat smaller $KN . This can be understood in the chiral perturbation calculation of C.-H. Lee [154] where the only signi:cant e9ect of higher chiral order terms can be summarized in the “range term”; 43 namely $KN is to be replaced by an e9ective $, ($KN )e9 = (1 − 0:37!K?2 =m2K )$KN : 42

(10.22)

The correction to this approximation which may become important as the nucleon mass drops comes as a “1=m” correction in the heavy-fermion chiral perturbation theory (as for the “range term” mentioned below) and can be taken into account systematically. It can even be treated fully relativistically using a special regularization scheme being developed in the :eld. Our approximation does not warrant the full account of such terms, so we will not include this correction here. 43 As mentioned, in the language of heavy-baryon chiral perturbation theory, this corresponds to a “1=m” correction term.

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It should be pointed out that although the $KN is important at low densities, !K decreases with m? K, this “range-term” correction becomes less important at higher densities. This e9ect—which is easy to implement—is included in the realistic calculations. 10.5. Partial decoupling of the vector interaction 10.5.1. Theoretical and experimental evidences In Section 4, a general argument for the vector decoupling in hot=dense matter was presented based on Wilsonian matching with QCD of HLS. Here we supply more speci:c theoretical and empirical reasons why we believe that the vector interaction should decouple at high density. 1. We :rst give the theoretical arguments why the vector coupling g!? should drop with density. • The :rst is the observation by Song et al. [155] that describing nuclear matter in terms of chiral Lagrangian in the mean :eld requires the ratio g!? =m? ! to at least be roughly constant or even decreasing as a function of density. In fact to quantitatively account for non-linear terms in a mean-:eld e9ective Lagrangian, a dropping ratio is de:nitely favored. 44 For instance, as discussed in Ref. [155], the in-medium behavior of the !-meson :eld is encoded in the “FTS1” version of the non-linear theories of Ref. [156]. In fact, because of the attractive quartic ! term in the FTS1 theory, the authors of Ref. [156] have (for the parameter = −1=2 favored 2 2 by experiments) g!?2 =m?2 ! 0:8g! =m! as modi:cation of the quadratic term when rewritten in our notation. In other words, their vector mean :eld contains a partial decoupling already at ≈ 0 although they do not explicitly scale g! as we do. Historically, Walecka-type mean :eld theories with only quadratic interactions (i.e., linear Walecka model) gave compression modulus K ∼ 500 MeV, about double the empirical value. This is cured in nonlinear e9ective :eld theories like FTS1 by higher-dimension non-renormalizable terms which e9ectively decrease the growth in repulsion in density. As suggested in Ref. [155], an e9ective chiral Lagrangian with BR scaling can do the same (by the increase in magnitude of the e9ective scalar :eld with density) but more economically and eSciently. This decrease of g!? =m? ! as density increases toward nuclear matter density contrasts with the ? increase of gNN =m? within the same density range discussed in Section 6.1. This can be explained as follows. In free space g! ≡ g!NN = 3gNN whereas near chiral restoration, we ? ? expect (as in the case of the quark number susceptibility mentioned below) that g!NN gNN . This implies that at low density, the ! coupling must fall faster than the coupling by “shedding o9” the factor of 3. • Implementing baryons into the HLS Lagrangian, Kim and Lee [157] showed explicitly that both gV? and m? V (where V stands for hidden gauge bosons) are predicted to fall very rapidly with baryon chemical potential [157]. 45 The main agent for this behavior is found in Ref. [157] to be the pionic one-loop contribution linked to chiral symmetry which is lacking in 44 Since the non-linear terms—though treated in the mean :eld—are Kuctuation e9ects in the e9ective :eld theory approach, this represents a quantum correction to the BR scaling. 45 Kim and Lee found that even the ratio gV? =m? V fell rapidly. This is at variance with what one would expect if the vector manifestation aM la Harada–Yamawaki were realized. As noted, this may be due to the incompleteness of the renormalization group analysis of Ref. [157].

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the mean-:eld treatment for BR scaling [2]. The Kim–Lee arguments were made for the which has a simple interpretation in terms of hidden gauge symmetry but it will apply to the ! if the nonet symmetry is assumed to hold in nuclear medium. • Finally, close to chiral restoration in temperature, there is clear evidence from QCD for an equally rapid drop, speci:cally, from the quark number susceptibility that can be measured on the lattice [13]. The lattice calculation of the quark number susceptibility dealt with quarks and the large drop in the (isoscalar) vector mean :eld was found to be due chieKy to the change-over from hadrons to quarks as the chiral restoration temperature is approached from 2 2 below. The factor of 9 in the ratio g!NN =g!QQ (where Q is the constituent quark) should disappear in the change-over. Now since the electroweak properties of a constituent quark (quasiquark) are expected to be the same as those of a bare Dirac particle with gA = 1 and no anomalous magnetic moment (i.e., the QCD quark) [158] with possible corrections that are suppressed as 1=Nc [159], there will be continuity between before and after the chiral transition. This is very much in accordance with the “Cheshire Cat picture” developed elsewhere [11]. In fact, it is possible to give a dynamical (hadronic) interpretation of the above scenario. For instance in the picture of Ref. [132] sketched in Section 9, this may be understood as the “elementary” ! strength moving downwards into the “nuclear” !, the [N ? (1520)N −1 ]J =1; I =0 isobar-hole state involving a single-quark spin Kip [133]. The mechanism being intrinsic in the change-over of the degrees of freedom, we expect the same phenomenon to hold in density as well as in temperature. The upshot of this line of argument is that the suppression of the vector coupling is inevitable as density approaches the critical density for chiral transition. In Section 11.2, we will have an additional support for this from color-isospin locking in QCD both in low density and in high density. We believe that the di9erent behavior of vector and scalar mean :elds, the latter to be discussed below, follows from their di9erent roles in QCD. With the vector this is made clearer in the lattice calculations of the quark number susceptibility which involves the vector interactions. In Brown and Rho [13], it is shown that as the description changes from hadronic to quark=gluonic at T ∼ Tc , the critical temperature for chiral restoration, the vector interaction drops by an order of magnitude, much faster than the logarithmic decrease due to asymptotic freedom. We expect a similar feature in density, somewhat like in the renormalization-group analysis for the isovector vector meson of Kim and Lee [157]. The scalar interaction, on the other hand, brings about chiral restoration and must become more and more important with increasing density as the phase transition point is approached. 2. From the empirical side, the most direct indication of the decoupling of the vector interaction is from the baryon Kow [160] which is particularly sensitive to the vector interaction. The authors of Ref. [160] :nd a form factor of the form fV (p) =

?2V − 16 p2 ?2V + p2

(10.23)

with ?V =0:9 GeV is required to understand the baryon Kow. Connecting momenta with distances, one :nds that this represents a cuto9 at √ 6 Rcuto9 ∼ ∼ 0:5 fm : (10.24) ?V

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Furthermore it is well known that the vector mean :eld of the Walecka model must be modi:ed, its increase as E=mN removed, at a scale of ∼ 1 GeV. The reason for this is presumably that inside of R ∼ 0:5 fm, the :nite size of the solitonic nucleon must be taken into account. A repulsion still results, but it is scalar in nature as found in Ref. [161] and for which there are direct physical indications [162]. 10.5.2. Kaons at GSI The K + and K − energies in the top–down scenario are given by !± !± = ± VK + k 2 + m?2 K mK

(10.25)

with VK given in (10.27) below. Although at high densities it will decouple, the term linear in VK that :gures in the range correction in ($KN )e9 will give a slightly di9erent e9ective mass for K + and K − before decoupling. Although the large distance vector mean :eld must arise from vector meson exchange, this must be cut o9 at a reasonably large distance, say, ∼ 0:5 fm as indicated by the baryon-Kow mentioned above. For the GSI experiments with temperature ∼ 70 MeV, the nucleon and kaon momenta are |pN | ∼ 444 MeV and |pK | ∼ 322 MeV, respectively, and fV (p) ∼ 0:82 :

(10.26)

We therefore propose to use VK 1 ≈ fV2 VN (p = 0)=mK ∼ 0:15 : mK 3

(10.27)

This is small. Furthermore we assume it to be constant above 0 . This assumption amounts to taking the vector coupling to drop as ∼ 1=. The above arguments could be quanti:ed by a speci:c model. For example, as alluded to above, the low-lying - and !-excitations in the bottom–up model can be built up as N ? -hole excitations [132]. At higher densities, these provide the low-mass strength. One might attempt to calculate the coupling constants to these excitations in the constituent-quark (or quasi–quark) model, which as we have suggested would be expected to be more applicable at densities near chiral restoration. Riska and Brown [133] :nd the quark model couplings to be a factor ∼ 2 lower than the hadronic ones [163]. 10.6. Schematic model 10.6.1. First try with the simplest form On the basis of our above considerations, a :rst try in transport calculation might use the vector potential with the Song scaling [155] as g!? =m? ! = constant and the e9ective mass ($KN )e9 2 (10.28) m?2 K ≈ mK 1 − f2 m2K

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with $KN = 400 MeV and ($KN )e9 given by Eq. (10.22). While as argued above the vector coupling will decouple at very high densities, as !K drops, the vector potential will become less important even at moderate densities since the factor !K =mK comes into the coupling of the vector potential to the kaon. Our schematic model (10.28) gives roughly the same mass as used by Li and Brown [146] to predict kaon and antikaon subthreshold production at GSI. For ∼ 30 it gives m? K − ∼ 230 MeV, − less than half the bare mass. We predict somewhat fewer K -mesons because the (attractive) vector interaction is largely reduced if not decoupled. Cassing et al. [150] have employed an m? K − somewhat lower than that given by Eq. (10.28) to describe a lot of data. The experimental data verify that our description is quite good up to the densities probed, i.e., ∼ 30 . In order to go higher in density, we switch to our top–down description which through Eq. (10.14) involves g"2 =m?2 " . Although the scalar interaction could have roughly the same form factor as the vector, cutting it o9 at ∼ 0:5 fm as mentioned above, we believe that this will be countered by the dropping scalar mass m? " which must go to zero at chiral restoration (viewed as an order parameter). Treating the scalar interaction linearly as a Kuctuation (as in (10.14)) cannot be expected to be valid all the way to chiral restoration but approaching the latter the "-particle becomes the “dilaton” in the sense of Weinberg’s “mended symmetry” [60,61] with mass going to zero (in the chiral limit) together with the pion. At high densities at which the vector interaction decouples, the K + and K − will experience nearly the same very strong attractive interactions. This can be minimally expressed through the e9ective mass m? K . At low densities where the vector potential not only comes into play but slightly predominates over the scalar potential, the K + will have a small repulsive interaction with nucleons. It is this interaction, extrapolated without medium e9ects by Bratkovskaya et al. [142] which gives the long equilibration time of 40 fm=c. However, clearly the medium e9ects will change this by an order of magnitude. 10.6.2. Implications on kaon condensation and maximum neutron-star mass While in heavy-ion processes, we expect that taking m? " to zero (or nearly zero in the real world) is ? a relevant limiting process, we do not have to take m" to zero for kaon condensation in neutron stars, since the K − -mass m? K must be brought down only to the electron chemical potential e EF (e), the approximate equality holding because the electrons are highly degenerate. Arguments based on e9ective chiral Lagrangians that are consistent with low-energy phenomenology in kaon–nucleon and kaon–nuclear systems typically give the critical density in neutron-star matter of c ∼ 30 [164,165,154]. We should however mention that it has been suggested that the electron chemical potential e could be kept low by replacing electrons plus neutrons by $− hyperons (or more generally by exploiting Pauli exclusion principle with hyperon introduction) in neutron stars [166]. In this case, the e might never meet m? K. Hyperon introduction may or may not take place, but even if it does, the scenario will be more subtle than considered presently. To see what can happen, let us consider what one could expect from a naive extrapolation to the relevant density, i.e., ∼ 30 , based on the best available nuclear physics. The replacement of neutron plus electron will take place if the vector mean :eld felt by the neutron is still high at that density. The threshold for that would be 2 EFn + VN + e M$− + VN + S$− ; (10.29) 3

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where EFn is the Fermi energy of the neutron, M$− the bare mass of the $− and the S$− the scalar potential energy felt by the $− . Here we are simply assuming that the two non-strange quarks of the $− experience 2=3 of the vector mean :eld felt by the neutron. Extrapolating the FTS1 theory [156] 46 and taking into account in VN the e9ect of the -meson using vector dominance, we :nd EFn + VN ∼ 1064 MeV at ≈ 30 . From the extended BPAL 32 equation of state with compression modulus 240 MeV [167], the electron chemical potential comes out to be e 214 MeV. So the left-hand side of (10.29) is EFn + VN + e ∼ 1278 MeV. For the right-hand side, we use the scalar potential energy for the $− at ≈ 30 estimated by Brown et al. [168] to :nd that M$− + 2 V + S$− ∼ 1240 MeV. The replacement of neutron plus electron by $− looks favored but only 3 N slightly. What is the possible scenario on the maximum neutron star mass if we continue assuming that the calculation we made here can be trusted? A plausible scenario would be as follows. K − condensation supposedly occurs at about the same density and both the hyperonic excitation (in the form of $N −1 —where N −1 stands for the nucleon hole—component of the “kaesobar” [168]) and K − condensation would occur at T ∼ 50 MeV relevant to the neutron-star matter. Now if as is likely the temperature is greater than the di9erence in energies between the two possible phases, although the hyperons will be more important initially than the kaons, all of the phases will enter more or less equally in constructing the free energy of the system. In going to higher density the distribution between the di9erent phases will change in order to minimize the free energy. Then it is clear that dropping from one minimum to another, the derivative of the free energy with density—which is just the pressure—will decrease as compared with the pressure from any single phase. This would imply that the maximum neutron star mass calculated with either hyperonic excitation or kaon condensation alone must be greater than the neutron star mass calculated with inclusion of both. The story will be quite di9erent if the vector :eld decouples. We showed in Section 10.5.1 that the isoscalar vector mean :eld must drop by a factor & 9 in the change-over from nucleons to quasiquarks as variables as one approaches the chiral restoration density. Hyperons will disappear during this drop. It is then highly likely that the kaon will condense before chiral restoration and that the kaon condensed phase will persist through the relevant range of densities which determine the maximum neutron star mass. 10.7. Discussions In this section, we have given arguments to support that not only the kaon mass but also its coupling to vector mesons should drop in matter with density. In particular, with the introduction of medium e9ects the apparent equilibration found in strangeness production at GSI can be increased from the baryon number density of ∼ 14 0 up to the much more reasonable ∼ 20 . These properties do not appear at :rst sight to be connected to BR scaling per se. But they must be connected in an albeit indirect way. From the baryon Kow analysis we have direct indications that the vector interaction decouples from the nucleon at a three-momentum of |p| ∼ 0:9 GeV=c or at roughly 0:5mN c. In colliding heavy 46

There is nothing that would suggest that the e9ective Lagrangian valid up to ∼ 0 will continue to be valid at ∼ 30 without addition of higher mass-dimension operators, particularly if the chiral critical point is nearby. So this exercise can be taken only as indicative.

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ions this is reached at a kinetic energy per nucleon of ∼ 18 mN c2 which means a temperature of 78 MeV when equated to 32 T . This is just the temperature for chemical freeze-out at GSI energies. We have given several theoretical arguments why the vector coupling should drop rapidly with density. Once the vector mean :eld, which acts with opposite signs on the K + and K − mesons is decoupled, these mesons will feel the same highly attractive scalar meson :eld. Their masses will fall down sharply; e.g., from Eq. (10.28) with proposed parameters, m? K ∼ 0:5 mK

(10.30)

+ at ≈ 30 and possibly further because of the dropping m? and " . The di9ering slopes of K − K with kinetic energy will then develop after chemical freeze-out, as suggested by Li and Brown [146]. In this section we have focused on the phenomenon at GSI energies. Here the chief role that the dropping K − -mass played was to keep the combination B + !K?− nearly constant so that low freeze-out density in the thermal equilibration scenario became irrelevant for the K + =K − ratio. We suggest that the same scenario applies to AGS physics, where the freeze-out density in the thermal equilibration picture comes out to be ∼ 0:350 [169]. In fact, there is no discernible dependence on centrality in the K − =K + ratios measured at 4A GeV; 6A GeV; 8A GeV and 10:8A GeV [170]. From this it follows either that the ratio of produced K − to hyperons is nearly independent of density or that the negative strangeness equilibrates down to a lower freeze-out density and then disperses. Given the relative weakness of the strange interactions, we believe the former to be the case. In fact, we suggest that near constancy with multiplicity of the K − =K + ratio found experimentally be used to determine temperature dependence of !K?− in the region of temperatures reached in AGS physics. As was done for GSI energies, the temperature can be obtained from the inverse slopes of the kaon and antikaon distributions of p⊥ and then the temperature dependence of !K?− can be added to the density dependence as in Ref. [147], in such a way that B + !K?− stays roughly constant as function of density. At least this can be done in the low-density regime considered in Ref. [169] when the approximation of a Boltzmann gas is accurate enough to calculate B . Our “broad-band equilibration”; i.e., the production of the same, apparently equilibrated ratio of K − -mesons to hyperons over a broad band of densities, avoids complications in the way in which the K − degree of freedom is mixed into other degrees of freedom at low density [171]. Most of the K − -production will take place at the higher densities, as shown in Fig. 12, where the degrees of freedom other than K − have been sent up to higher energies by the Pauli principle. Unless the electron chemical potential in dense neutron star matter is prevented from increasing with density (as might happen if the repulsive $-nuclear interaction turns to attraction at ¿ 0 ), kaon condensation will take place before chiral restoration. This has several implications at and beyond chiral restoration. For instance, its presence would have inKuence on the conjectured color superconductivity at high density, in particular regarding its possible coexistence with Overhauser e9ects, skyrmion crystal and other phases with interesting e9ects on neutron star cooling. The phenomenon of vector decoupling, if con:rmed to be correct, will have several important spin-o9 consequences. The :rst is that it will provide a refutation of the recent claim [172] that

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in the mean-:eld theory with a kaon-nuclear potential given by the vector-exchange (Weinberg– Tomozawa) term—both argued to be valid at high density—kaon condensation would be pushed up to a much higher density than that relevant in neutron-star matter. Our chief point against that argument is that the vector decoupling and the di9erent role of scalar :elds in QCD (e.g., BR scaling) described in this article cannot be accessed by the mean :eld reasoning used in Ref. [172] or by any other standard nuclear physics potential models. The second consequence that could be of a potential importance to the interpretation of heavy-ion experiments is that if the vector coupling rapidly diminishes with density, the strong-coupling perturbation calculation of the vector response functions used in terms of “melting” vector mesons to explain [129], e.g., the CERES dilepton data must break down rapidly as density increases. This could provide a speci:c mechanism for the quasiparticle description of BR scaling found to be successful in nuclear matter [78]). It is of course diScult to be quantitative as to precisely where this can happen. Finally we should stress that the scenario presented in this section—which is anchored on Brown– Rho scaling—should not be considered as an alternative to a possible quark–gluon scenario currently favored by the heavy-ion community. It is once again more likely a sort of complementarity of the same physics along the line that the Cheshire Cat Principle [11] embodies that would continue to apply at higher energies. 11. Eective eld theories for dense QCD Thus far we have been climbing up in density in what may be aptly called “bottom–up” approach. In this section, we will start at some asymptotic density and come down to the density regime relevant to dense compact stars and possibly in certain regimes of heavy-ion collisions. This will be a “top–down” approach exploiting once more how chiral symmetry in nature manifests itself through (Nambu– )Goldstone mode. There is an intense activity on this matter, comprehensively reviewed in the recent literature [173], so we will not go into most of the topics. Here we will focus on the most intriguing aspect of dense QCD that reKects an intriguing repeat of the Cheshire Cat phenomenon. We wish to highlight the seemingly continuous role of chiral symmetry from low-density strong-coupling regime to high density weak-coupling regime in the form of a skyrmion replica that maps high-density excitations to low-density excitations. In view of the rapid development on this matter, our discussion will necessarily be incomplete and primitive but the general feature of the Cheshire Cat phenomenon is likely to survive the test of the time. 11.1. Color-Javor locking for NF = 3 11.1.1. Instability at high density Asymptotic freedom of QCD suggests that at very high density with → ∞, the QCD gauge coupling becomes very weak and so interactions disappear. However if there is an attraction in a certain channel, now matter how weak, the same renormalization group consideration discussed above in connection with Landau Fermi liquid state [77] tells us that the system will be unstable against a phase change. Indeed in the weak coupling limit, one gluon exchange that survives asymptotic

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freedom has an attraction in the channel that involves diquarks sitting on the opposite side of the Fermi sphere, namely, the BCS channel and this inevitably leads to the Cooper pair condensation as in superconductivity. That this must occur at asymptotic density is without doubt as dictated by the renormalization group argument. Whether or not—and in what manner—this will occur in a physically relevant regime of density cannot be addressed by the asymptotic QCD and it appears most plausible that the scenario is much richer in various di9erent ways than just the Cooper pairing type as one approaches top–down to the density regime of relevance. This matter is amply discussed in the literature [173] but the situation is totally unclear. It is not our aim to dwell on the multitude of possibilities. We will focus on one particular aspect of this phenomenon, that is, the situation where color and Kavor of QCD get locked, i.e., the color–Kavor-locked (CFL) phase which renders the discussion simple and transparent. 11.1.2. Symmetry breaking and excitations We shall discuss :rst the case of Nc = NF = 3 which is relevant at asymptotic density, returning to the case of Nc = 3; NF = 2 in the next subsection. Gluon exchange between two quarks is attractive in the color 3W channel and repulsive in the 6 channel. So one expects diquarks would condense in the 3W channel. We shall ignore a small mixing to the 6 channel which is allowed by symmetry. The diquark condensate would be in the form ia jb ia jb qL( qL) = −qR( qR) = Vjij jabI j()I ;

(11.1)

where V is a constant, i; j are SL(2; C) indices, a; b are color indices and (; ) are Kavor indices. This condensation breaks the symmetry G→H

(11.2)

G = SU (3)c × SU (3)L × SU (3)R × U (1)B ;

(11.3)

H = SU (3)c+L+R × Z2 :

(11.4)

with

At superdensity, instantons are suppressed, but the anomaly never disappears, so the U (1)A symmetry is never truly restored. We shall not be concerned with it here. The color is completely broken and since the vectorial color is locked to both L and R, chiral symmetry is also broken so the a b & : resulting invariance group is SU (3)C+L+R . Since qL( qL) qWRa qWRb ∼ V2 j()4 j&:4 is gauge invariant, it can be used as an order parameter but while it breaks chiral symmetry, it leaves Z2 invariant, thus its appearance in H . The spontaneous symmetry breaking induces Goldstone bosons: eight from the breaking of the color SU (3)C , eight from chiral symmetry breaking and one from U (1)B → Z2 . We shall forget the last one since it does not concern us here. The striking feature we would like to focus on here is the uncanny resemblance of the spectrum in the CFL phase to that of low density in terms of a hidden local symmetry [64]. The scalar Goldstones are eaten up by the gluons thereby the vector bosons becoming massive. The octet of the massive gluons in the CFL are the analogs to the (octet) massive light-quark vector mesons in the low-density vacuum. The octet pseudoscalar Goldstones in the CFL are the analog to the octet pions in the low density vacuum, the dynamics of which can be described by an analogous chiral Lagrangian. The quarks in the system are gapped—and

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hence massive—and bear the quantum numbers of the octet baryon of the zero-density vacuum. We shall show below that this can be understood by looking at the solitons in the chiral Lagrangian [174]. Since the soliton corresponds to a gapped colored quark, we call it “qualiton” in analogy to the qualiton considered by Kaplan for the constituent quark [175]. (The breaking of U (1)B leads to superKuidity which resembles the same in nuclear matter, another uncanny correspondence.) The one-to-one matching between the low density spectrum in terms of hadronic variables and the high density spectrum in terms of quark–gluon variable is referred to as “quark–hadron continuity” [176]. We will identify this as belonging to the class of the Cheshire Cat phenomenon in the strong interaction physics [11]. 11.1.3. Chiral Lagrangians and qualitons As suggested in Refs. [174,177], the dynamics of the surviving Goldstone modes can be described by an e9ective chiral :eld theory as in the zero-density situation. We introduce the chiral e9ective :eld 47 b) c& 6La( (x) ∼ jij jabc j()& qLi (−CF ; x)qLj (CF ; x)

(11.5)

corresponding to the map SU (3)c × SU (3)L =SU (3)C+L and similarly for R, b) c& (−CF ; x)qRj (CF ; x) 6Ra( (x) ∼ jij jabc j()& qRi

(11.6)

corresponding to the map SU (3)c × SU (3)R =SU (3)c+R . Under an SU (3)c × SU (3)L × SU (3)R transformation by unitary matrices (gc ; gL ; gR ), 6L transforms as 6L → gc∗ 6L gL† and 6R transforms as 6R → gc∗ 6R gR† . In the ground state of the CFL superconductor, 6L and 6R take the same constant value. QCD symmetries with (11.1) imply 6La( = −6Ra( = V:a( :

(11.7)

The Goldstone bosons are the low-lying excitations of the condensate, given as unitary matrices 6L (x)=gcT (x)gL (x) and 6R (x)=gcT (x)gR (x). For the present decomposition, we note the extra invariance under the (hidden) local transformation gc+L+R (x)—which is an analog to the hidden gauge transform h(x) of [64]—within the diagonal SU (3)c+L+R through † (x) gcT (x) → gcT (x)gc+L+R

gL; R (x) → gc+L+R (x)gL; R (x) :

(11.8)

Hence, the spontaneous breaking of SU (3)c × (SU (3)L × SU (3)R ) → SU (3)c+L+R can be realized non-linearly through the use of 6L; R (x) or linearly through the use of gc; L; R (x) with the addition of an octet vector gauge :eld transforming inhomogeneously under local gc+L+R (x). This can be identi:ed

47

One has to be careful with a singularity in the product of two fermion :elds at one point. See Ref. [174] for a more proper de:nition.

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with the hidden local symmetry [64]. 48 Clearly the composites carry color–Kavor in the unbroken subgroup, with a mass of the order of the superconducting gap. The current associated with det 6L (det 6R ) is the left (right)-handed U (1)L (U (1)R ) baryon number current. Because of the U (1) axial anomaly, the :eld det 6L =det 6R is massive due to instantons. We decouple the massive :eld from the low energy e9ective action by imposing det 6L =det 6R = 1. The Goldstone boson associated with spontaneously broken U (1)B symmetry is described by det 6L ·det 6R and responsible for baryon superKuidity. But since it is not directly relevant for our problem, we further choose det 6L · det 6R = 1 to isolate the Goldstone bosons resulting from the spontaneous breaking of chiral symmetry, from the massive ones eaten up by the gluons. We now parameterize the unitary matrices 6L; R as 6L (x) = exp(i LA T A =f);

6R (x) = exp(i RA T A =f) ;

(11.9)

48

We comment as a side remark on a technical detail which is somewhat outside of the scope of this review but may be helpful to those interested in subtleties involved in hidden-gauge-symmetry aspects of the problem. As mentioned, when the color is completely broken, the octet gluons become massive by Higgs mechanism with the mass proportional to the chemical potential . These were found to have the same quantum numbers as those of the light-quark vector mesons present at zero density. The hidden gauge bosons excited at high density discussed here belong to the unbroken diagonal subgroup SU (3)c+L+R . How are these vectors related to the massive gluons? In Ref. [178], the hidden gauge bosons ∈ SU (3)c+L+R were identi:ed with the spin-1 two-quark bound states with the properties that match with the Georgi vector limit at some non-asymptotic density. At the leading order in 1=, these states seem to have nothing to do with the Higgsed gluons: The overlap is zero. But this cannot be correct. In nature, the low-lying vector excitations at large density must be a coherent mixture of the two but at the leading order, they must be equivalent. This is analogous to the complementary description of the pion as a Goldstone mode and as a zero-mass bound state of a quark and antiquark. Indeed this complementarity has also been veri:ed in the pion channel in dense QCD in terms of bound diquarks [179,178]. To see this aspect in the vector channel, we follow the CCWZ (Callan– Coleman–Wess–Zumino) formalism [180]. Consider the symmetry group G = SU (3)L × SU (3)R × SU (3)c broken down to H = SU (3)L+R+c . Using a notation slightly di9erent from what is used in this subsection, let the G transformation be represented in the block diagonal form g = diag(L; R; C) corresponding to the left, right and color transformations. Let the generator of the diagonal subgroup be given by h = diag(U; U; U ). Now we would like to write R(x) ∈ G, the rotation matrix that transforms the standard vacuum conCguration to the local Celd conCguration. One choice consistent with the CCWZ prescription is R(x) = diag (6L (x); 6R (x); 1) which transforms as gRh−1 = diag(L6L U −1 ; R6R U −1 ; CU −1 ) : It is clear that we are required to take CU −1 = 1 which means that U = C and that 6L; R transform as 6L → L6L C −1 , 6R → R6R C −1 : Now U is a non-linear function of L, R, 6L and 6R , and so must be C. In HGS, this is the group that is gauged. Therefore the “hidden gauge symmetry” can be equated to the color symmetry of QCD as was done by Casalbuoni and Gatto [177]. This can be seen also in Eq. (11.10) given below.

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where T A are SU (3) generators, normalized as Tr(T A T B )=:AB =2. The Goldstone bosons A transform non-linearly under SU (3)c × SU (3)L × SU (3)R but linearly under the unbroken symmetry group SU (3)c+L+R . The e9ective Lagrangian for the CFL phase at asymptotic densities follows by integrating out the ‘hard’ quark modes at the edge of the Fermi surface [181]. The e9ective Lagrangian for UL; R is a standard non-linear sigma model in D = 4 dimensions including the interaction of Goldstone bosons with colored but “screened” gluons G. Expanding in powers of derivative, the e9ective Lagrangian for the Goldstone bosons is then fT2 f2 Tr(90 6L 90 6†L ) − S Tr(9i 6L 9i 6†L ) + gs G · JL + (L → R) + · · · : (11.10) 4 4 Here the ellipsis stands for higher derivative terms including the Wess–Zumino–Witten term [23] that is needed to account for anomalies. The vectorial coupling to the gluon :eld G via the color current J indicates that the left–right chiral symmetry is also locked to each other. In the presence of a chemical potential, the SL(2; C) breaks down to O(3), so the “decay constant” f has two di9erent values for the time and space components. The temporal and spatial decay constants fT; S are :xed by the ‘hard’ modes at the Fermi surface. Their exact values are determined by the dynamics at the Fermi surface. They go like fS ∼ fT ∼ . The e9ective Lagrangian (11.10) in the CFL phase bears much in common with Skyrme-type Lagrangians [11], with the screened color mediated interaction analogous to the exchange of massive vector mesons. Indeed in the CFL phase the ‘screened’ gluons and the ‘Higgsed’ gauge composites are the analog of the massive vector mesons in the low density phase. In the CFL phase the WZW term is needed to enforce the correct Kavor anomaly structure. The ‘Higgsed’ gluons are very massive in the large limit, so can be integrated out from the Lagrangian (11.10). This is very much like the hidden gauge symmetry theory wherein the vector mesons can be integrated out to give rise to a non-linear sigma model. The theory then can have a soliton as suggested by Hong et al. [174] if there are higher derivative terms that stabilize it. The qualiton will be colored. However the pair-condensed ground state is colored also, so combined with the background, the excitation will be e9ectively color-singlet. In other words, the chiral :eld from which the soliton arises can be considered as color-singlet Kuctuation. This means that one can rewrite the e9ective :eld theory in terms of the color-singlet Celd L=+

Uij ≡ 6Lai 6∗Raj

(11.11)

which transforms U → gL UgR† . The Lagrangian (11.10) can then be rewritten in the form familiar from low density f2 fT2 Tr(90 U 90 U † ) − S Tr(9i U 9i U † ) + O(94 ) + O(M2 ) + · · · : (11.12) 4 4 The resulting theory is essentially identical to the familiar non-linear sigma model giving rise to the Skyrme soliton [11], except that the mass terms which we did not specify here should be quadratic in the quark mass matrix M because of the Z2 invariance. The qualiton can be collective-quantized in a way paralleling the low-density skyrmions. This was worked out recently in Ref. [182] in which the authors argue that since a qualiton is a quark on top of the condensed state, it would correspond to the quasiparticle of the particle–hole complex and the qualiton mass that is obtained as a soliton L=

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solution must therefore correspond to 22 plus the ground-state (or condensation) energy where 2 is the gap. 11.2. Complementarity of hidden gauge symmetry and color gauge symmetry and BR scaling The Nc = 3, NF = 2 case is a bit more intricate and involved than the Nc = NF = 3 CFL discussed above. It turns out, however, that color–Kavor locking in this case [183] is more relevant in the density regime closer to nature, both in high density and in low density. It can also be mapped as we shall argue to hidden local symmetry discussed in the previous sections. 11.2.1. Quark number susceptibility It was observed in Refs. [13,184] that the quark number susceptibility 5± =(9=9u ±9=9d )(u ±d ) where u; d and u; d are respectively u; d-quark number density and chemical potential measured on lattice as a function of temperature [185] exhibited a smooth change-over from a Kavor-gauge symmetry or hidden gauge symmetry to QCD color-gauge symmetry at the chiral phase transition critical temperature Tc . It was suggested [13] that at the phase transition, the Kavor gauge symmetry—which is induced and hence not fundamental—gets converted directly to the color gauge symmetry—which is fundamental, implying that they could be related in an intricate way. In this subsection, following Ref. [186], we suggest how this can be realized in terms of color–Kavor-locked (CFL) quark– antiquark and diquark condensates and “vector manifestation” (VM) of chiral symmetry explained in Section 4. It will be seen how BR scaling can be :t into the general scheme that results from these developments: Its proof will be the most convincing case for the validity of BR scaling. Our argument relies on two recent remarkable developments that come from seemingly unrelated sectors. One is the suggestion by Harada and Yamawaki [12] that the phase transition from the Nambu–Goldstone phase to the Wigner–Weyl phase involves “vector manifestation” of chiral symmetry, that is, at the phase transition, the longitudinal components of the light-quark vector mesons (i.e., the triplet in the 2-Kavor case) and the triplet pions ( a ) come together becoming massless in the chiral limit, the vectors decoupling aM la Georgi’s vector limit [63] but with the pion decay constant f vanishing at that point. The other important development is the proposal by Berges and Wetterich [187,183] that color and Kavor get completely locked in the Nambu–Goldstone phase (I) for three Kavors (Nf = 3) [187] by the quark–antiquark condensate in the color-octet (8) channel 3 a ab b (11.13) (7i )() (!i ) q) 5 = qW( i=1

and (II) for two Kavors (NF = 2) [183] by (11.13) together with the diquark condensate in the W channel color-antitriplet (3) 2 = q(a (72 )() (!2 )ab q)b :

(11.14)

In (11.13) and (11.14), the indices (; ) denote the Kavors and a; b the colors. In the Berges–Wetterich scenario, the chiral phase transition and decon:nement occur through the melting of the condensates 5 and 2.

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11.2.2. Color-isospin locking The scenario for color–Kavor-locking (CFL) is a bit di9erent depending on the number of Kavors NF . For our purpose, the case of two Kavors is more relevant, so we shall focus on this case. In Ref. [183], Berges and Wetterich argue that both the 5 and 2 condensates can be nonzero in the vacuum, thereby completely breaking the color. Should one of the condensates turn out to be zero, then color would be only partially broken (see Ref. [173] for a simple explanation for this and references). This pattern of color breaking and color–Kavor locking renders the octet gluons and six quarks massive by the Higgs mechanism and generates three Goldstone pions due to the broken chiral symmetry. All of the excitations are integer-charged. Among the eight massive gluons, three of them are identi:ed with the isotriplet ’s with the mass m = Vgc 5 ;

(11.15)

where V is an unknown constant and gc the color gauge coupling. The fourth vector meson is identi:ed with the isosinglet ! with the mass m ! = V gc 2 ;

(11.16)

where V is another constant. The remaining four vector mesons turn out to have exotic quantum numbers and are presumably heavy. We assume that they decouple from the low-energy regime. As for the fermions, there are two baryons with the quantum numbers of the proton and neutron with their masses proportional to the scalar condensate W, W = qWa( q(a :

(11.17)

The four remaining fermions are also of exotic quantum numbers with zero baryon number and heavier, so we assume that they also decouple from the low-energy sector. What concerns us here is therefore the three pions, the proton and neutron, the -mesons and the !-meson. 11.2.3. Implications of the lattice measurements of quark number susceptibility It was argued in Ref. [83] that the “measured” singlet and non-singlet QNS’s [185] indicate that both the and ! couplings vanish at the transition temperature Tc . This meant that the !NN coupling which is ∼ 3 times the NN coupling at zero temperature became equal to the latter at near the critical temperature. This also meant that the both vector mesons became massless and decoupled. Viewed from the CFL point of view, it follows from (11.15) and (11.16) that the condensates 5 and 2 “melt” at that point. This is consistent with the observation by Wetterich [187] that for three-Kavor QCD, the phase transition—which is both chiral and decon:ning—occurs at Tc with the melting of the color-octet condensate 5. The transition is :rst-order for NF = 3 in agreement with lattice calculations, so the vector meson mass does not go to zero smoothly but makes a jump from a :nite value to zero. We expect however that in the case of NF = 2 the transition will be second-order with the vector mass dropping to zero continuously. Noting that both the CFL formulas (11.15) – (11.16) and the vector manifestation result (4.9) are of the Higgsed type, we invoke the lattice results to arrive at agV f ≈ Vgc 5 ≈ V gc 2 :

(11.18)

We admit that this relation follows neither from the group-theoretical considerations of Berges and Wetterich [183,187] nor from the vector manifestation of HLS [12]. We are proposing that this

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is indicated by the lattice data, at least near the phase transition point. It holds empirically at zero temperature and zero density, so we are led to assume that it holds at least approximately from T =0 up to T = Tc . 49 It should be noted that the vanishing of the hidden gauge coupling gV corresponds to the vanishing of the condensates 5 and 2 with the color gauge coupling remaining non-vanishing, gc = 0. Next, we have shown in Ref. [83] that above the chiral transition temperature T & Tc , the QNS’s can be well described by perturbative gluon exchange with a gluon coupling constant gc2 =4 ≈ 0:19 and argued that the Kavor gauge symmetry cedes to the fundamental QCD gauge symmetry. Now the HLS theory is moot on what it could be beyond the chiral restoration point since the theory essentially terminates at Tc . We propose that this is where the color–Kavor locking of Refs. [187,183] phrased in the QCD variables takes over by supplying a logical language for crossing-over from below Tc to above Tc . Indeed (11.18) describes the relay that must take place in terms of the hidden Kavor gauge coupling gV on one side and the color gauge coupling gc on the other side. Now above Tc , the color and Kavor must unlock, with the gluons becoming massless and releasing the scalar Goldstones. The dynamics of quarks and gluons in this regime will then be given by hot QCD in the proper sense. The way the two condensates melt as temperature is increased is a dynamical issue which seems to be diScult to address unambiguously within the present scheme. It will have to be up to lattice measurements to settle this issue. Our chief point here is that their melting is intricately connected. 11.2.4. Link to BR scaling and Landau parameter F1 The situation appears to be quite di9erent in dense medium. Since there is no guidance from lattice as it is impossible at present to put density on lattice except for unphysical cases of two colors or adjoint quarks, we shall simply assume that the above scenario holds in density up to n = nc . 50 Certain models indicate that the phase structure near chiral restoration could be quite involved and complex. As suggested by Sch[afer and Wilczek [176], an intriguing possibility is that the three-Kavor color–Kavor locking operative at asymptotic density continues all the way down to the “chiral transition density” (nc ) in which case there will be no real phase change since there will then be a one-to-one mapping between hadrons and quark=gluons, e.g., in the sense of “hadron–quark continuity”. However the non-negligible strange-quark mass is likely to spoil the ideal three-Kavor consideration. One possible alternative scenario is that viewed from “bottom–up”, one gets into the phase where 5 = " = 0 and 2 = 0 corresponding to the two-Kavor color superconducting (2csc) phase [173]. Unless 2 goes to zero at c , this would mean that the mesons become massless but the ! meson remains massive. One cannot say that this is inconsistent with the vector manifestation since the HLS does not require that U (2) symmetry hold at the chiral restoration point or in medium 49 This would imply that the nonet symmetry for NF = 3 or the quartet symmetry NF = 2 is a good symmetry not only at T = 0 but also for T = 0. Why this symmetry should hold at any non-zero temperature or density is not obvious either in CFL QCD or in HLS e9ective theory. In discussing color–Kavor locking in two-Kavor QCD, Berges and Wetterich [183] entertain among others the possibility that the two condensates 5 and 2 could be di9erent at the critical point. As for the HLS theory, at one-loop order, the RG Kows are expected to be di9erent for the and ! properties, so it is not obvious that the and ! mesons would reach the chiral restoration with the Georgi vector limit at the same temperature. Nonetheless if our interpretation of QNS is correct, it seems most plausible and appealing that the nonet or quartet symmetry does hold at the phase transition. Proving this conjecture remains as a theoretical challenge. 50 In this subsection, we again denote density by n reserving for the vector meson.

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in general. But this seems unlikely. On the other hand, it is highly plausible that both 5 and 2 approach zero (or near zero if it is :rst-order) from below nc and then 2 picks up a non-zero value at or above nc in which case we will preserve the mass formula (4.9) as one approaches nc . This is the scenario that we favor. Among the scaling relations implied by BR scaling [2,83], the one most often discussed in the literature is the dropping of the -meson mass in medium. This relation has been extensively discussed recently in connection with the CERN-CERES data on dilepton production in heavy-ion collisions. The simplest explanation for the observed dilepton enhancement at an invariant mass ∼ 400 MeV is to invoke BR scaling for the excitations relevant in the process [188]. It turns out however that this explanation is not unique. One could explain it equally well if the meson “melted” in dense medium with a broadened width [129]. Since the process is essentially governed by a Boltzmann factor, all that is needed is the shift downward of the strength function: the expanding width simply does the job as needed for the dilepton yield. If one calculates the current-current correlation function in low-order perturbation theory with a phenomenological Lagrangian, it is clear, because of the strong coupling of the meson with the medium, that the meson will develop a large width in medium and “melt” at higher density. The upshot of the dilepton experiments then is that they cannot distinguish the variety of scenarios that probe average properties of hadrons in the baryon density regime—which is rather dilute—encountered in the experiments. In the vector manifestation scenario, the width should become narrower, decreasing like ∼ gV2 . Then the vector mesons become more a quasiparticle at high density than at lower density. This is the underlying picture of BR scaling. Thus far, we have made a link between the color–Kavor-locked condensates and hidden gauge symmetry. We can go even one step further and via BR scaling, make an intriguing connection between QCD “vacuum” properties and many-body nuclear interactions. This connection was discussed in Section 5. Simply put, it comes about because nuclear matter owes its stability to a Fermi-liquid :xed point [77]. Certain interesting nuclear properties were found to be calculable in terms of the Fermi-liquid :xed point parameters [79]. Among others, it was shown that the Landau parameter F1 —which is a component of quasiparticle interactions—can be expressed in terms of the BR scaling factor P(n) ≡ m? (n)=m (0) F˜ 1 = 3(1 − P−1 ) + F˜ 1 ( ) ;

(11.19)

where F˜ 1 ( ) is the contribution from the pion which is completely :xed by chiral dynamics. One of the most remarkable prediction was Eq. (6.4) for the anomalous gyromagnetic ratio :gl in nuclei 4 1 P−1 − 1 − F˜ 1 ( ) 73 : (11.20) :gl = 9 2 At nuclear matter density n = n0 , we have F˜ 1 ( )|n=n0 = −0:153. Note that (11.20) depends on only one parameter, P. This parameter can be extracted from various sources and all give about the same value, P(n0 ) ≈ 0:78. Given that this is not very accurately determined, it is probably a better strategy to determine P from the data on :gl . In any event, given P at nuclear matter density, Eq. (11.20) makes a simple prediction, :gl = 0:2373

(11.21)

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which should be compared with the measurement in the Pb region [91], :glp = 0:23 ± 0:03 :

(11.22)

We believe this to be a good agreement within the theoretical uncertainty involved. It is intriguing that what is an intrinsic QCD quantity can be related to what appears to be a standard many-body nuclear interaction summarized in the Landau quasiparticle parameter F1 . 11.3. Kaon condensation: encore Cheshire Cat In addition to the “continuity” in excitations and chiral phase transitions between the hadronic phase and the quark phase, meson condensations can occur in high density matter with n ¿ nc and in hadronic matter with n ¡ nc where nc is the presumed critical density for the so-called “chiral restoration”—whatever that may be. Pion condensation is unlikely in the density regime that is relevant for laboratory or astrophysical observations, but the negatively charged kaon K − can condense at a relatively low density as well as at an asymptotically high density providing yet another support for “continuity” or Cheshire Cat. For completeness we brieKy describe this phenomenon although the story is by no means :nal. • K− condensation in the hadronic sector Since this matter was discussed already in Section 10, we shall simply summarize the pertinent feature in a slightly di9erent language. As originally pointed out by Kaplan and Nelson [189] and reinterpreted by Brown et al. [152], the S-wave condensation of K − ’s is driven by “rotating away” of the kaon mass associated with both the explicitly broken chiral symmetry and spontaneously broken chiral symmetry. As is widely discussed, hyperons can also participate through P-wave coupling with the kaons but near the condensation transition, the P-wave coupling would be “irrelevant” in contrast to the S-wave that has to do with “relevant” terms [190] and hence may be ignored in this qualitative discussion. The condensation therefore occurs when the kaon mass is eaten up by attractive interactions. Negative kaons have attractive interactions with nucleons by exchanging the vector mesons and ! as well as scalar mesons, e.g., the " meson. The vector exchange is dictated by the vector current conservation and is given by the Weinberg–Tomozawa term and the " exchange by the $KN —the KN sigma term. When many-body interactions through many-Fermi contact terms in the Lagrangian are implemented through BR scaling, one then has a Lagrangian of the form (10.12). One can generate this e9ective Lagrangian in chiral perturbation theory as reviewed e.g. by Lee [154]. The condensation occurs when these relevant terms drive the system to instability [190]. In neutron star matter, the kaon energy !K need not go all the way to zero. It suSces to drop to the electron chemical potential e . Since the e increases as a function of density in nonrelativistic nuclear systems and the !K must fall, the crossing is bound to occur at some density. However exactly at what density it will occur will depend upon details of the dynamics and this is still a controversial issue [172]. In Section 10, we have proposed that the critical density is rather low, Kc ∼ 30 . • K− condensation in the CFL phase The e9ective chiral Lagrangian discussed above, (11.12), implemented with mass terms predicts [179,191–194] at high density where the color–Kavor locking sets in that the kaon mass is of the

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form mK ± = cmd (mu + ms ) ;

(11.23) 2

2

f 2 )

where 2 is where c is a constant that can be evaluated in the weak-coupling QCD, c = 32 =( the pairing gap and the subscripts u; d and s stand for up, down and strange quarks, respectively. Only the pairing gap 2 appears in the mass formula despite that an antiquark is involved. This is because the Majorana mass of the antiquark is equal to the gap [195]. Because the Goldstone bosons in this phase are of q2 qW2 con:guration, the kaon is less massive than the pion which has the mass m ± = cms (mu + md ) :

(11.24)

It is found numerically that for ∼ 500–1000 MeV, the kaon mass ranges from ∼ 5 MeV to ∼ 1 MeV. The reason for this small mass is easy to understand. First of all, because of the Z2 symmetry, the Goldstone mass is quadratic in the quark mass and secondly, the mass is proportional to (2=)2 that goes to zero as → ∞. For large density, one can ignore the attractive kaon–quark interaction analogous to the kaon– nuclear since the interaction is suppressed by (1=)2 . Considering the Fermi sea :lled with noninteracting massless up, down and strange quarks, the electron chemical potential e is found in this case to drop as ∼ ms =kF . This contrasts with the increasing e found in the nonrelativistic hadronic system. Sch[afer [196] found that for a reasonable range of parameters involved for the CFL phase, the kaon mass is always less than the electron chemical potential. However this does not lead to a K − condensation of the type seen in the hadronic sector since as argued by Rajagopal and Wilczek [197], the CFL phase is charge-neutral more or less independently of the mass of the strange quark and hence no electrons need to be present in the phase. Kaon condensation can however occur in superdense matter, not just because of the small mass of the kaons (which helps) but because of the presence of the chiral symmetry breaking mass term. Though perhaps di9erent in character, this is analogous to the case of kaon condensation in the hadronic sector where the phase change is essentially e9ectuated by “rotating away” the large $ term arising from the chiral symmetry breaking due to the strange-quark mass. Indeed in superdense regime, a term of the form − M 2 =2 ;

(11.25)

where M is the quark-mass matrix plays the role of a Kavor chemical potential that provides an e9ective attraction favoring neutral kaon condensation [198]. Although interesting purely from the theoretical point of view, the physical relevance of this observation to the physics of compacts stars is yet to be established. Whereas the phenomenon lends itself to a simple and elegant analysis for asymptotic densities, the relevant process for the formation and structure of compact stars requires instead a “bottom–up” approach starting from low density at the stage of supernovae explosion and climbing up to high but non-asymptotic density as the matter is compressed in the interior of compact stars. This must involve, along the way, such hadronic phases as kaon (or pion) condensation, hyperon presence etc. for which QCD is intractable and it is not clear that the simple picture based on QCD at asymptotic density is directly relevant to what actually takes place in the stellar matter. Working out the change-over from hadronic variables to QCD variables necessary for a realistic description of the process—which we believe involves a “Cheshire cat” mechanism—remains a challenge for nuclear theorists in the sense discussed in this review.

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Heuristic eld theory of Bose–Einstein condensates Stig Stenholm Physics Department, Royal Institute of Technology (KTH), SCFAB, Roslagstullsbacken 21, SE-10691 Stockholm, Sweden Received October 2001; editor: G:E: Brown Contents 1. Introduction 2. Bose–Einstein condensation 2.1. A condensate of bosons 2.2. The in0uence of the interaction 3. Field theory description 3.1. Many-particle eld theory 3.2. Symmetry breaking 4. Collective modes in the Bose system 4.1. The density operator 4.2. Phase operator of the condensate 4.3. Dynamics of the phase 4.4. The Goldstone modes

175 176 176 180 183 183 186 189 189 192 196 197

5. Perturbation theory 5.1. An ensemble with condensate 5.2. Graphical perturbation theory 5.3. Renormalization of the eld theory 6. Theory of trapped condensates 6.1. The Thomas–Fermi regime 6.2. Elementary excitations in the condensate 7. Conclusion Acknowledgements Appendix A. Many-body formalism References

198 198 200 204 206 206 209 211 212 212 216

Abstract This paper reviews the basic ideas of the eld theoretic approach to Bose–Einstein condensation. The central concepts are presented in a historical way, the most important results are given and they are justied by heuristic physical ideas instead of detailed derivations. The mathematical level of rigour is low, and the intuitive relations between the concepts and their physical origin is used to justify the various results. I present the original Bose–Einstein conception of bosons condensing into a macroscopically occupied state. The special features deriving from this situation are discussed, and the concept of long-range order is introduced. Historically the concept of broken symmetry has played a central role in the description of Bose condensates. In this approach the system is described by states with broken particle conservation, which justies the introduction of a physical phase factor. When interactions are present, this acquires its own dynamic behaviour, which gives rise to the low-lying collective excitations of the system. They can be interpreted

E-mail address: [email protected] (S. Stenholm). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 9 5 - 3

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as collisionless oscillations of the particle density. The condensate acting as a particle reservoir introduces a coupling between these collective excitations and the single-particle ones, which makes the collective excitations dominate the physics of the system. This justies the hydrodynamic view of a condensed Bose system, which has been central to both the theoretical approaches and the interpretations of experiments on super0uid helium. The results are derived using both algebraic and perturbative methods; some of the tools from formal many-body theory are summarized in the appendix. The treatment approaches most issues from a bulk material point of view, but I also keep the possibility open to apply the results to the topical eld of trapped condensates. The equation for the order parameter, the Gross–Pitaevski equation, is presented and discussed, but its detailed applications to trapped alkali atoms is not covered by the present review. Here basic ideas, their logical connections and their physical consequences are presented. For more detailed treatments of the formal results the reader has to consult the original papers or the monographs referred to. No attempt is made c 2002 to cover the most recent experimental and theoretical results for alkali atoms condensed in traps. Elsevier Science B.V. All rights reserved. PACS: 05.30.−d; 03.75.Fi

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1. Introduction The recent experimental success in achieving quantum condensation in a trapped cloud of alkali–metal atoms has spurred a new interest in quantum condensates [1,2]; for a recent compilation of experimental and theoretical results see Ref. [3]. This is a manifestation of the bosonic character of the particles and presents the physicists with a quantum system of totally novel character. Compared with super0uid 4 He, the atomic phase transition is much closer to the Bose– Einstein condensation predicted for an ideal gas by Einstein in 1924–25 [4]. The particle interaction is not small when compared with the potential energy in the trap, but the system is essentially a weakly interacting gas and not a liquid. Thus, many of the features discussed in the early theories of super0uid helium have a much better validity in the trapped alkali condensates. Consequently the theoretical results have been in excellent agreement with the rapidly progressing experiments [3]. The theoretical understanding and its physical basis have been discussed in two recent reviews [5,6]. In many ways, these oFer points of view complementary to the present article. Even the theory of the ideal gas is, however, rather sophisticated. As the condensate provides an invisible source of particles, the ordinary proof of the equivalence between the canonical and the microcanonical ensembles becomes dubious [7]. This tends to obscure the eFects of even very weak interactions. The very order of the transition is not self-evident; this question has been claried only recently [8–11]. However, even if the dense structure of 4 He prevents any computations from rst principles, the theoretical activity describing its physics has been lively and comprehensive, most of the features of super0uidity are well understood on both a formal and a physical level. These works have a long history, they are based on advanced many-body techniques and many results can be proved on a mathematically sophisticated level. The works are well documented; in addition to the numerous original papers, a multitude of lecture notes and text books are available [12–14]. The concepts and methods of these investigations have been taken over by the recent research on Bose–Einstein condensates, even if the weak interaction allows a much simpler theoretical basis for the understanding. Thus, recent investigations do not require the most sophisticated many-body technology of the earlier endeavors. Actually, the modern eFort has developed its own tradition, which is to some extent independent of the earlier one. One may even claim that it is a waste of time to penetrate the many-body literature with all its technical complications. However, the development spans nearly a century of theoretical research and it may be useful to attempt a survey of the central results. In addition, the central concept of broken symmetry plays such an essential role in modern eld theory, that it is of interest to see how this has been applied to bosonic systems. It is the aim of the present paper to review these developments, present the main results and some heuristic derivations based on the underlying physical situation. I believe that these considerations add to our intuition about condensed Bose systems, even if one may claim that “the utility of the idea is outweighted by its dangers” [6]. Bogoliubov carried out the earliest successful attempt to incorporate particle interactions in Bose systems [15]. He introduced the transformation which has played such a central role in the discussion ever since. His results are summarized in the lecture notes [16]. A more direct perturbative approach was developed by Huang and his collaborators, see [17,18]. Bohm and Salt presented an early description of the collective excitations in a Bose system in [19].

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Penrose and Onsager realized that the condensed phase carries a correlation they termed long-range order [20,21], see also the article by Yang [22]. This allows the possibility to introduce an order parameter in the sense of Landau [23]. Following the successful application of this concept to superconductivity, Pitaevski [24] and Gross [25] introduced their well-known equation for super0uidity. In liquid helium this was not very successful, but it has proved to be of essential importance in the eld of condensed atoms. The condensate forms a reservoir of particles whose number needs not to be specied because it does not contribute to the thermodynamic functions of the system. Thus, it may inject an unlimited number of particles into the system. Consequently a careless application of perturbation theory leads to divergences as the particle 0uctuations become too large. This was the basis of the approach introduced by Bogoliubov, but it was systematically incorporated into a many-body perturbation method by Beliaev [26] and Hugenholtz and Pines [27]; see also Ref. [28]. The rst approach to the excitation spectrum of the condensed Bose system was provided by Landau [29], which gave an excitation spectrum that could be veried by neutron scattering. It turned out that the results could be given an interpretation in terms of hydrodynamic models [30,31]. The many-body approach was adapted to this point of view, and rather complete understanding of the super0uid dynamics emerged [32–34]; see also [35]. This paper aims at a review of the main results obtained from the eld theoretic treatment of bosonic systems. The approach is heuristic and non-mathematical; it is trying to give physical derivations and logical connections between the ideas. For exact proofs, if and when such exist, the reader is referred to the literature. The motivation is that modern research can benet from the discussion in the earlier works and the researchers can prot from an understanding of the older conceptual arguments and their justication within the theory. If this can be achieved without too much of formal technology, the time may not be wasted. The present article aims at providing the material for this process.

2. Bose--Einstein condensation 2.1. A condensate of bosons In 1924 Einstein translated Bose’s paper on the statistical consequences of symmetry and forwarded it to publication in Zeitschrift fMur Physik. He then proceeded to discover the phase transition now bearing the names of Bose and Einstein. His argument was based on the following considerations: In a system of noninteracting bosons, we choose a set of basis functions for the single-particle states as {uk (r)} , where k labels the states and r is the position in real space. 1 For free particles, a suitable set of states are dened by box normalization and periodic boundary conditions. This is known to give satisfactory results in the thermodynamic limit, when both the volume V and the total particle number N go to innity. The particle momentum can then be identied with ˝k and

1

Throughout the paper we consider this space to be three-dimensional. I do not, however, indicate the vector character of r except when confusion may arise. As alternative three-dimensional position variables I will use x or y:

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the energy is ˝2 k 2 ; (2.1) 2m where m is the particle mass. If the state labelled by k has the occupation nk , we nd the total number of particles to be given by d3 k 1 1 +V ; (2.2) nk = − N= e −1 (2)3 e (j(k)− ) − 1 j(k) =

k

where is the chemical potential and −1 = kB T as usual. Assuming the thermodynamic limit, we have replaced the terms with k = 0 by an integral. From (2.2), we see that 6 0 to insure convergence of the integral. However, when = 0; the integral converges to a xed number of particles N ; if their total number N is larger, the rest of the particles have to condense into the ground state, which has the occupation N0 = N − N ≈

−kB T :

(2.3)

In the thermodynamic limit, a xed fraction of the particles will stay in the condensate, and this implies that has to approach zero from below. In the limit of innite volume, the value becomes exactly zero at the temperature giving N = N and stays there for all lower temperatures. Einstein found this to give a phase transition of the rst order [4], and the thermodynamic properties can be evaluated in a straightforward way. The argument above is based on the grand canonical ensemble, and it has proved singularly diPcult to carry out Einstein’s argument in the canonical ensemble. Ordinarily this would not matter, because there is a proof that the results are equivalent in the thermodynamic limit [18]. This, however, is based on the assumption that (QN=N )1, i.e. that the particle 0uctuations are a small fraction of the total number only. In a Bose–Einstein condensate, it is easily proved that the 0uctuations of particle number in the grand canonical ensemble is of the order of the particle number itself. The vanishing chemical potential of the condensate allows its occupation number to 0uctuate greatly without penalty in energy variations. The Einstein result can be projected on the xed particle number subensemble [7], and then the particle 0uctuations become regular, but the situation invalidates the ordinary proof of the equivalence of the ensembles. Recent calculations [8,9,11] indicate that the phase transition may be of second order in the canonical ensemble. Ref. [11] also discusses the relations between the 0uctuations in the two ensembles in some detail. The transition has been found to be of second order in the weakly interacting Bose gas [17], and it has been interpreted as a discontinuous change of the order of the transition when the interaction is switched on. If it is of second order in the canonical ensemble also for the ideal gas case, the disagreement disappears, and the paradox arises from the anomalous 0uctuations of condensed systems in the grand canonical ensemble. The assembly of condensed particles does act as an uncontrollable reservoir of particles. We now introduce second quantized boson creation and annihilation operators obeying [aˆk ; aˆ†k ] = k; k :

(2.4)

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They create particles in the basis states chosen so that x|aˆ†k |vac = uk (x) ;

(2.5)

where |vac is the true vacuum of the system, with no particles. With N particles, the ground state |0 contains a multitude of condensed particles, which puts the operator aˆ†0 in a special position because √ aˆ†0 |0 ˙ N0 |0 ˙ V ; (2.6) where V is the volume of the system and N0 is dened in Eq. (2.3). In the thermodynamic limit of V → ∞, operator (2.6) contains a component which becomes singular. This was realized by Bogoliubov [15], who utilized the observation to introduce the operator a˜†0 by setting √ aˆ†0 = a˜†0 + ∗ V : (2.7) As we see, this is a canonical transformation preserving the commutation relations, but the thermo√ dynamic behaviour of the condensate state is now captured in the c-number contribution ˙ V , which keeps track of the behaviour in the thermodynamic limit. 2 With this denition we nd aˆ†0 aˆ0 = V (||2 + O(V −1=2 )) :

(2.8)

We can set the particle density of the condensate to aˆ†0 aˆ0 (2.9) ⇒ 0 = ||2 : V In quantum mechanics, we know how to eFect the transformation (2.7). It implies replacing the vacuum by the coherent state √ (2.10) | = exp[ V (aˆ†0 − ∗ aˆ0 )]|vac : The overlap between two such states is given by | | |2 = exp(−V | − |2 ) :

(2.11)

In the thermodynamic limit, the states with diFerent parameters become orthogonal, and the canonical transformation mediating between them ceases to exist in a strict mathematical sense. The diFerent ground states are said to give inequivalent representations of the operator algebra [36]. The argument has been applied also to superconductors [37], where states with diFerent amounts of paired electrons become inequivalent. In our case too, the diFerent ground states (2.10) are taken to describe diFerent physical situations, with the fraction of particles in the condensed state determined by the parameter (2.9) being dened even in the thermodynamic limit. When it is nonvanishing, the condensate exists, and the parameter is an order parameter characterizing the ground state. It is clear that the state (2.10) contains admixtures of various particle numbers. Thus, it is a quantum equivalent of the grand canonical ensemble treated in thermodynamics. Utilizing it in the dynamics, Bogoliubov found, that with interactions, the Hamiltonian does no longer conserve particle number. The processes where the condensate absorbs or emits particles dominate, and the phase of the parameter has to be kept xed; if it is given a random phase, the dynamic eFects go away. 2

In the many-body literature the volume of the system is often put equal to unity. In addition to causing dimensional havoc, this can be very confusing when, in the thermodynamic limit, this unity goes to innity.

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Thus, breaking of the particle conservation symmetry equips the condensate phase with a meaning not usually ascribed to phases of quantum states. We will nd below, that this theme will play a recurrent role in our subsequent discussion. The wave function of a many-body system is a complicated function of all the particle coordinates 3 {ri | i = 1; : : : ; N }, when N is the number of particles. In the case of bosons, the function must be totally symmetric, and hence we can write the reduced one-body density matrix as (2.12) (r1 ; r1 ) = · · · (r1 ; r2 ; : : : ; rN )∗ (r1 ; r2 ; : : : ; rN ) d3 ri : i=1

Because the wave function is symmetric, it does not matter which variable is left for the density matrix, symmetrizing the expression over all possible variables would give no diFerent result. For the case of noninteracting particles the ground state wave function is (r1 ; r2 ; : : : ; rN ) =

N

u0 (ri ) :

(2.13)

i=1

In this case, the density matrix (2.12) becomes (r1 ; r1 ) = u0 (r1 )u0∗ (r1 ) ;

(2.14)

this remains = 0 when the distance goes to innity, |r1 − r1 | → ∞. The correlations between diFerent parts of the system does not depend on the distance. Penrose [20] generalized this to be a general character of condensed Bose systems; see also [21,22]. The one-body density matrix is then written in the form r1 ||r1 = (r1 )∗ (r1 ) + (r1 ; r1 ) ;

(2.15)

where lim

|r1 −r1 |→∞

(r1 ; r1 ) = 0 :

(2.16)

The function (r1 ) contains the information about the quantum coherence inherent in the ground state, and it generalizes the concept of a complex order parameter as suggested by the treatment of the previous section. Because N r||r ≡ N(r) is the particle density, we nd from the normalization of the density matrix that |(r)|2 d 3 r 6 1 : (2.17) In the present representation, the number of particles participating in forming a coherent condensate can be written as (2.18) N0 = N |(r)|2 d 3 r even in the interacting case. 3

In physical condensates of alkali atoms, the angular momentum state of the atoms plays a central role. In this article, however, we concentrate on issues of principles, and hence only scalar spinless atoms are discussed. This is suPcient to treat a system with a single condensed species.

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There are good physical reasons to think that, if the system forms a condensate with long-range order, this can take place only into one function (r), even if there seems to be no conclusive proof of this statement. We are thus prevented from nding a density matrix having r1 ||r1 = 1 (r1 )1∗ (r1 ) + 2 (r1 )2∗ (r1 ) + (r1 ; r1 )

(2.19)

with 1 (r1 ) = 2 (r1 ). In liquid 4 He this implies that there cannot be two independent super0uid components formed by the same liquid. In 3 He the situation allows a more complicated result because of the internal quantum numbers of the condensing fermions; then a multi-component order parameter is possible. When the condensing systems of alkali atoms comprise several angular momentum substates, the condensation into several possible ordered components can occur. This is usually referred to as fragmentation of the condensate. If on the other hand, we look at the total wave function of the system, there are arguments against the individual bosons occupying diFerent states when the particle–particle interaction is repulsive, see [6]. It is not clear, that such an “attraction in momentum space” can be used to argue against the existence of states of the form (2.19). 2.2. The in2uence of the interaction When the particles are made to interact, the system deviates from the ideal case. We rst consider this in a classical way by adding to the free particle Hamiltonian the energy 1 Eint = V (x − y)(x)(y) d 3 x d 3 y : (2.20) 2 In the low-energy regime, the interaction depends only on the long-wavelength part of the quantum state, and then the interaction can be approximated by an eFective local interaction of the form V (x − y) = VU (x − y) ;

(2.21)

where the coePcient [38] is given in terms of a scattering length a as 4˝2 a VU = : m In the limit of a homogeneous system, the energy becomes 2 1 N Eint = VU = N U ; 2 V

(2.22)

(2.23)

where U is the average energy per particle. Following Kadomtsev and Kadomtsev [39], we are going to explore the consequences of this result for the homogeneous Bose gas. If we take (2.23) to be the total energy of the system, we can calculate the pressure directly from 9E 2˝2 a 2 = U 0= 0 ; 9V m where we have set the particle density to N : 0 = V p=−

(2.24)

(2.25)

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From this, we can obtain the velocity of sound in the system by calculating VU 0 4˝2 a 1 9p 2 = = 0 : vs = m 90 m m2 If we calculate the energy of this excitation for wave vector k, we nd VU 0 = 2VU 0 (k) ; ˝!(k) = ˝kvs = ˝k m

181

(2.26)

(2.27)

where (k) is the free particle excitation energy according to (2.1). This result is exactly the long-wavelength limit of the excitation energy as derived by Bogoliubov based on the assumption of condensate domination as in Eq. (2.8). The average energy per particle is given by , U but the chemical potential is given by the energy needed to add one particle to the system. In addition to the average energy required, this assumes that we have to make room to insert the particle. As the average volume per particle is (V=N ) this gives the expression p

= U + = 2U = VU 0 : (2.28) 0 If we try to impose a spatial distortion on the quantum state of a condensed system, we have to keep the ensuring energy smaller than the typical particle energy, which we take to be given by . The corresponding length scale is given by ˝2 2 : (2.29) = 2mVU 0 The parameter is called the healing length, because it is the upper limit to spatial structures which can be imposed on the quantum system without creating energies which perturb the stability of the condensate. It describes the stiFness of the system to distorting forces. We can see that in the noninteracting case VU → 0, the healing length diverges, and we conclude that such nite stiFness is a consequence of the particle interactions. We are now introducing an external potential U (x), and we assume this to be such that the local conditions are similar to the innite bulk case. We can then draw some further conclusions from the results above. We write the Euler equation [18] as dv(x) 1 m + ∇p(x) = −∇U (x) : (2.30) dt (x) This hydrodynamic description can be justied by an approach brie0y outlined in Section 6.2, where it is shown that the Eq. (2.30) follows when certain quantum corrections are omitted. In steady state, we nd a relation between the local density and the external potential. From (2.24) we obtain ∇p(x) = VU (x)∇(x) ; which allows us to integrate (2.30) to (U0 − U (x)) (x) = : VU

(2.31)

(2.32)

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When we assume the harmonic potential U (x) = 12 m#2 x2

(2.33)

the trap frequency # determines the length scale of the noninteracting ground state as ˝ : (2.34) a0 = m# Then density (2.32) becomes

x 2 (x) = 0 1 − ; (2.35) R where the density at the trap center is U0 0 = ; (2.36) VU and the radius of the condensate is given by 2VU R2 = 0 : (2.37) m#2 The parameter U0 can be determined if we assume the density to vanish for x ¿ R and determine the total number of particles from R

x 2 8R3 : (2.38) 1− x2 d x = N = (x) d 3 x = 40 R 15 0 Introducing the harmonic oscillator parameter (2.34) we obtain 1=5 a 1=5 R = 15 a0 N 1=5 (2.39) a0 and 152=5 a0 3=5 N 2=5 : (2.40) 0 = 8 a a30 Thus, we see that with interactions, the size of the condensate scales as N 3=5 and the central density as N 2=5 . The concept of long-range order is diPcult to formulate in the case of a nite trap. Outside the range of R; all particle variables disappear, so we may try to generalize relation (2.15) to states in a trap, by looking at positions such that |r1 | ¡ R; |r2 | ¡ R; and require lim r1 ||r2 = (r1 )∗ (r2 ) :

|r1 −r2 |

(2.41)

Here the healing distance is introduced as the distance over which correlations can be observed, i.e. a correlation length. However, the validity and physical meaning of condition (2.41) is not clear at the present time. It does rest on the assumption that R ;

(2.42)

which states that locally the system looks much like a bulk one. As the derivations we have performed above rest on such an assumption, relation (2.41) may well be relevant under the conditions of this section.

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3. Field theory description 3.1. Many-particle 4eld theory In Section 1:1 we introduced the single-particle modes {uk (r)} which we assume to be complete uk (r)uk∗ (r ) = (r − r ) (3.1) k

and orthonormal uk (r)uk∗ (r) d 3 r = k; k ;

(3.2)

in the thermodynamic limit the index k becomes continuous. Using these modes and their corresponding annihilation operators (2.4) we dene a quantum eld operator ˆ (r) = aˆk uk (r) : (3.3) k

Utilizing the commutation relations between the creation and annihilation operators, we derive † ˆ [(r); ˆ (r )] = (r − r ) ;

(3.4)

which is sometimes imposed directly as the quantization condition of a boson eld. † The operator ˆ (r) can be understood to create a particle at the position r, even if such an interpretation poses diPcult mathematical problems. The corresponding single-particle wave function is namely † y|ˆ (x)|vac = y | x = (y − x) ;

(3.5)

which is far from being a proper wave function. The result is, however, in agreement with the normalization condition † † ˆ ˆ y | x = vac|(y) ˆ (x)|vac = vac|[(y); ˆ (x)]|vac = (x − y) :

(3.6)

If one wants to present a proper mathematical treatment of such elds, much sophistication is needed [40], but here I want to present the main results only and try to convey some intuitive feeling for the physical features behind the formal results. The particle number operator can now be written as † † ˆ ˆ N = ˆ (x)(x) d3 x = aˆk aˆk ; (3.7) k

where (3.2) has been used. The operators nˆk = aˆ†k aˆk give the population of states labelled by k, and the local particle density is given by † ˆ : (x) ˆ = ˆ (x)(x)

(3.8)

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Using the bosonic eld operators, we can write the many-particle Hamiltonian with two-body interaction through the potential V (r) in the form

˝2 † ˆ d3 x Hˆ = ˆ (x) − ∇2 + U (x) (x) 2m 1 † † ˆ ˆ ˆ (x)ˆ (y)V (x − y)(y) (x) d3 y d3 x ; (3.9) + 2 where U (x) is an external potential where the bosons are conned [40]. Hamiltonian (3.9) is independent of the particle number, it is thus a suitable form for descriptions where this is not xed. However, if we introduce the subspace of the total Hilbert space, where the particle number is conserved, the formulation is totally identical with the one usually given in quantum mechanics. The basis states for this subspace are the states (aˆ† )nk √k |vac |n1 ; n2 ; : : : ; ni ; : : : n = nk ! nk =n

aˆ† = √i |n1 ; n2 ; : : : ; ni − 1; : : : n−1 ni

(3.10)

for any i. This space is usually called the Fock space [41]. The multi-particle wave function in this space can be written in the form n+1 (x1 ; x2 ; : : : ; x n ; x n+1 | n1 ; n2 ; : : : ; ni + 1; : : :) n+1 = r=1

ui (xr ) n (x1 ; x2 ; : : : ; xr −1 ; ; xr+1 ; : : : x n | n1 ; n2 ; : : : ; ni ; : : :) ; (ni + 1)(n + 1)

(3.11)

where we have indicated both the total number of particles as subscripts and their distribution over the states. As the function n is totally symmetric, we generate a new totally symmetric state. This is the bosonic version of the Slater determinant, which has a set of alternating signs and no doubly occupied single-particle states ui (x). If we introduce an operator Px which permutes the coordinate variables x, we can write the state as 1 Px (u1 (x1 )u2 (x2 ) : : : un (x n )) ; (3.12) n (x1 ; x2 ; : : : ; x n | n1 ; n2 ; : : : ; ni ; : : :) = n! ni ! Px where the sum goes over all the permutations of the position variables, and the state ui (x) occurs ni times. The normalization is found to be correct, because of the multiple occupancy of some states the factors ni ! are needed. Such a wave function is called a permanent. In the n-particle Hilbert space spanned by the basis functions (3.12), the multi-particle Hamiltonian (3.9) is equivalent with the ordinary one. This is well known, but its justication is not commonly given. The details are rather cumbersome, but I nd it instructive to outline the basis of the demonstration. The details are presented in Ref. [41]. The approach we use is based on the interpretation of (3.5) as giving a basis function for the position representation ux (y) = (y − x) :

(3.13)

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We write the position state as † ˆ (xi ) |vac |x1 ; x2 ; : : : ; x n = i √ n! and the normalization condition is given as n 1 y1 ; y2 ; : : : ; yn | x1 ; x2 ; : : : ; x n = Px (yi − xj ) ; n P i=1

185

(3.14)

(3.15)

x

which generalizes relation (3.6). We note that these equations do not allow double occupancy of the same spatial position. This is a manifestation of the mathematical singularity of the single-particle wave function, and it can only be justied within a more detailed mathematical formulation of the basis for our eld theory. The treatment must then be based on the mathematical entity called a distribution, which lies beyond the scope of the present paper. Because from Eq. (3.14) we have † ˆ (x n+1 ) |x1 ; x2 ; : : : ; x n+1 = √ |x1 ; x2 ; : : : ; x n n+1 we can take the scalar product with an arbitrary state | and directly write √ ˆ n+1 )| = n + 1x1 ; x2 ; : : : ; x n+1 | : x1 ; x2 ; : : : ; x n | (x

(3.16)

(3.17)

Subtracting a particle from the state | thus gives a simple expression. To add a particle, on the other hand, is more complicated because of the requirements by symmetry. Using interpretation (3.13) and expression (3.11), we write n+1 (x − xr ) † x1 ; x2 ; : : : ; x n+1 |ˆ (x)| = (3.18) x1 ; x2 ; : : : ; xr −1 ; xr+1 ; : : : x n | ; (n + 1) r=1 note again that no double occupancy of positional states are allowed. After having obtained the representation of the eld operators in terms of the n-particle states, we need to proceed to nd expressions for other operators in order to be able to write down Hamiltonian (3.9). We start by considering a general one-particle operator † (1) ˆ Hˆ = ˆ (x)(x) ˆ (x) d3 x (3.19) in the above case we have

˝2 2 (x) ˆ = − ∇ + U (x) : 2m Using Eq. (3.17) we obtain ˆ (x) ˆ (x) (n + 1)(x) ˆ n (x1 ; x2 ; : : : ; x n ) = n+1 (x; x1 ; x2 ; : : : ; x n ) :

(3.20)

(3.21)

Using (3.18) we nd † ˆ ˆ (x)(x) ˆ (x) n (x1 ; x2 ; : : : ; x n ) =

n (x − xr ) (n + 1) (x) ˆ (n + 1) r=1

×n (x; x1 ; : : : ; xr −1 ; xr+1 ; : : : ; x n ) :

(3.22)

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Here we have 3only n terms, because the term (x − x) must be excluded. Finally, performing the integration d x we obtain n (1) Hˆ n (x1 ; x2 ; : : : ; x n ) = (x ˆ r )n (x1 ; x2 ; : : : ; x n ) ; (3.23) r=1

which result emerges when one realizes that all diFerent orders of the arguments can be rearranged to give the same wave function because of symmetry. With (3.20) this is exactly the SchrModinger equation for n noninteracting particles. Using the same procedure, but going one step further to n+2 , we obtain for the two-body interaction term the expression 1 (2) V (xi − xj )n (x1 ; x2 ; : : : ; x n ) : (3.24) Hˆ n (x1 ; x2 ; : : : ; x n ) = 2 i=j

We have thus shown that the eld theoretic formulation of the many-particle problem is fully identical with the ordinary SchrModinger approach if the particle number is xed. Our treatment is sketchy and intuitive, but the details can be looked up from Ref. [41]. The method outlined in this section is often called second quantization. This is misleading, as it suggests that we are doing something beyond ordinary quantum mechanics. Our discussion above should convince you that the treatment is only a formal reformulation of the conventional quantization; no new fundamental assumptions are added and no new physics should be included. However, in the case of Bose systems, the eld theoretic treatment oFers clear advantages. Some of the aspects of this problem are denitely more easily discussed within the eld theory than in the ordinary formulation. We turn to these questions in the next section. 3.2. Symmetry breaking Following the lead from the treatment in Section 1:1, we expect the assembly of condensed particles to form a reservoir of particles, which will lead to a treatment where the particle number is not determined in the ground state of the system. The system Hamiltonian, however, conserves the particle number, which means that this symmetry in only broken in the state attained by the system. The situation is well known from other phase transitions. A prototype example is the full rotational symmetry of the Hamiltonian of a magnetic substance, which is broken in the ordered ferromagnetic state appearing at low temperatures. It must, of course, be possible to treat the ground state of the boson system with a xed number of particles, i.e. in a canonical manner, and then this type of symmetry breaking does not occur. The treatment becomes much more diPcult, however, and most results have not been derived within such an approach. As in Eq. (2.7) we shift the eld operators by setting ˆ ˜ (x) = (x) + 0 (x) ;

(3.25)

˜ which preserves the commutation relations for the operators (x), as required by a canonical transformation. The particle number is now given by † 3 ˆ Nˆ = ˆ (x)(x) d x = |0 (x)|2 d 3 x + O(N 1=2 ) ≡ )N + O(N 1=2 ) ; (3.26)

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where )N is the fraction N0 of particles in the condensate. Using expansion (3.3) and the coherent ground state (2.10), we nd ˆ 0 (x) = 0 |(x)| k uk (x) ; (3.27) 0 = k

where |0 is the ground state of the system. For the noninteracting particles, only the lowest energy state is occupied and we have 0 (x) = 0 u0 (x) ; √ in this case 0 = )N . As with the coherent states, we introduce the operator † ˆ ˆ * = d 3 x[0 (x)ˆ (x) − 0∗ (x)(x)]

(3.28)

(3.29)

and calculate ˆ ˆ −*ˆ ˜ ˆ (x) = e* (x)e = (x) − 0 (x)

as we want. The corresponding ground state can be written as

1 † *ˆ 2 3 3 ˆ |0 = e |vac = exp − |0 (x)| d x exp d x0 (x) (x) |vac : 2 Using (3.26) we nd the normalization coePcient to be

1 1 2 3 exp − |0 (x)| d x = exp − )N ; 2 2

(3.30)

(3.31)

(3.32)

which does not survive in the thermodynamic limit. This is the re0ection of the result (2.11) in the present formulation of the bosonic theory. The results above are easily justied in the case of noninteracting particles. When interactions are present, we expect the multi-particle ground state |0 to retain the property of symmetry breaking. This is then characterized by the nonvanishing of the expectation value ˆ 0 |(x)| 0 ≡ 0 (x) ;

(3.33)

which is the order parameter signalling the breaking of the symmetry involved. Landau [23] was the rst to state that phase transitions involving a change of symmetry are characterized by the appearance of an order parameter. In the magnetic system this is the macroscopic magnetization, which vanishes in the disordered phase. In order to avoid the singularity of the derivatives of the order parameter at the phase transition, the system can be subjected to a small external potential which enforces the symmetry breaking. In the magnetic case this is an external magnetic eld, which makes the dipoles point in its direction. If the magnetization remains nonvanishing when the eld goes to zero, the system is in the ordered phase. Here we achieve the same by taking the modied Hamiltonian † ∗ ˆ ˜ ˆ H = H + [(x)J (x) + ˆ (x)J (x)] d 3 x : (3.34) As long as J (x) = 0, the expectation value in (3.33) will not vanish. The energy of the system is now a functional of J (x), which corresponds to the external eld. As in that case, we can obtain

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the order parameter, the eld, from the functional derivative 0 (x) =

E[J (x)] J ∗ (x)

(3.35)

if this is nonzero the symmetry is broken. We may, however, assume that relation (3.35) can be inverted to give J (x) as a functional of the order parameter 0 (x). When this is inserted into the energy, we obtain a new functional F[0 (x)], which we call a free energy. In thermodynamics our procedure corresponds to a Legendre transformation from one set of variables to another. We set (3.36) F = E − [0 (x)J ∗ (x) + 0∗ (x)J (x)] d 3 x : Calculating the functional derivative we nd ∗ E J (x) 3 J (x) 3 E F = d x+ d x ∗ ∗ ∗ 0 (y) J (x) 0 (y) J (x) 0∗ (y) J (x) 3 J ∗ (x) 3 −J (y) − 0∗ (x) ∗ d x − 0 (x) ∗ d x: 0 (y) 0 (y)

(3.37)

Applying (3.35) and its conjugate relation, we nd the inversion of the variable relation to be J (x) = −

F[0 (x)] : 0∗ (x)

(3.38)

For vanishing external eld J (x) = 0, this says that F[0 (x)] is an extremum, a minimum, with respect to the variation of the order parameter. This is just the functional introduced by Landau to describe second-order phase transitions. In the method called density functional theory [42] it is shown, that there is a one-to-one correspondence between the ground state and the particle density † ˆ (x) = ˆ (x)(x) . For the situation in a condensed Bose system, this theory has been discussed by GriPn [43]. He shows that the energy functional depends on the order parameter as well as the particle density. In eld theory contexts the functional F[0 (x)] is called the eFective potential. To obtain an expression for F[0 (x)] is far from trivial. However, if we assume the ground state of the many-particle system to be of the character (3.31) we immediately nd † ˆ = 0∗ (x)0 (x) ˆ (x)(x)

(3.39)

and †

†

ˆ ˆ (x) = 0∗ (x)0∗ (y)0 (y)0 (x) : ˆ (x)ˆ (y)(y) Then the expectation value of the Hamiltonian (3.9) becomes

˝2 2 ∗ ˆ H = 0 (x) − ∇ + U (x) 0 (x) d 3 x 2m 1 0∗ (x)0∗ (y)V (x − y)0 (y)0 (x) d 3 y d 3 x : + 2

(3.40)

(3.41)

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After introducing local potential (2.21), we identify this with the functional

˝2 2 ∗ F[0 (x)] = 0 (x) − ∇ + U (x) − 0 (x) d 3 x 2m 1 (3.42) + VU 0∗ (x)0∗ (x)0 (x)0 (x) d 3 x : 2 Here we have subtracted the term N in order to be able to use the system as a grand canonical ensemble. Functional (3.42) is just the one introduced by Ginzburg and Landau in the theory of superconductivity. In the bosonic case it is called the Gross–Pitaevski functional, and applying variational (3.38) we obtain the Gross–Pitaevski equation [24,25], which has been widely used in calculations of the properties of trapped alkali atoms in a condensed state. It should be pointed out, that the Gross–Pitaevski equation can be obtained in several ways, many of which do not need to abandon particle conservation. In addition to the discussion in [6], detailed consideration are presented by Gardiner [44] and Castin and Dum [45] and Girardeau [46], who treat the Bose condensing system in the canonical ensemble. The question of broken symmetry and the related question of the phase variable of the condensate are, however, rather subtle; for a discussion see [47]. 4. Collective modes in the Bose system 4.1. The density operator From Eq. (3.8), we learn that the operator corresponding to the density of particles is given by † ˆ ˆ (x)(x). Utilizing relation (3.22) with (x)=1, ˆ we nd its representation in the n-particle subspace to be n (x) ˆ = (x − xs ) ; (4.1) s=1

where the operator character resides in the position variables of the particles {xs }: This is obviously the right expression; integrating it over a volume V0 ⊂ V we nd the number of particles residing inside that volume. In the homogeneous system, we may use a set of basis functions exp(iqx) uq (x) = √ ; (4.2) V which with periodic boundary conditions are the eigenfunctions of momentum. Fourier transforming density (4.1) we obtain the momentum representation density n 1 1 d 3 x exp(−iqx)(x) ˆ =√ exp(−iqxs ) = ˆ†−q : (4.3) ˆq = √ V V s=1 Introducing the density in terms of the creation and annihilation operators ˆ aˆk uk (r) (r) = k

(4.4)

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we nd also 1 ˆq = √ V

1 † 1 † † ˆ d 3 x exp(−iqx)ˆ (x)(x) =√ aˆk −q aˆk = √ aˆk aˆk+q : V k V k

Using expansion (4.4) in the interaction expression we nd 1 † † ˆ ˆ ˆ ˆ (x)ˆ (y)V (x − y)(y) (x) d3 y d3 x V= 2 1 = Vq aˆ†k2 +q aˆ†k1 −q aˆk1 aˆk2 : 2V

(4.5)

(4.6)

k1 ; k2 ;q

Here Vq is the Fourier transform of the two-particle interaction V (x − y). Using the commutation relations between the boson operators, we can cast this in the form 1 ˆ 1 Vq ˆq ˆ−q − Vq N : (4.7) Vˆ = 2 q 2V q The last term is a Hartree interaction term giving the energy involving the averaged interaction V (0). In spite of the fact that this is often divergent, we can assume it incorporated into the chemical potential and hence omit it from our further considerations. In Section 1:2, we saw that the interacting Bose system can sustain long-wavelength excitations of a collective nature. We are now going to nd that these are closely related to the density operator, thus allowing the interpretation that the excitations are periodic modulations of the particle density, i.e. of a character similar to the ordinary sound waves. These, however, occur in the hydrodynamic regime where collisions are frequent, whereas in a condensed system, the long-range order introduced by symmetry breaking makes the low-energy excitation of the system possess a quantum character. We start by assuming the n-particle ground state wave function to be given in the momentum representation ˜ 0 (p1 ; p2 ; : : : ; pn ). In the noninteracting case, this would be a product of the single-particle states corresponding to the individual particle momenta {pi }. If we want to kick a particular particle to momentum pi + ˝q, we apply the operator 9 ˜ 0 (p1 ; p2 ; : : : ; pn ) = exp(−iqxi )˜ 0 (p1 ; p2 ; : : : ; pn ) : (4.8) exp ˝q 9pi This way to kick one single particle must cost a considerable energy needed to move it against all the other particles. The same situation is known from the ferromagnetic case; to turn one single spin against the ordered array of all other spins is costly. Here we know what to do, we bend the array of all spins gently, in such a way that neighbouring spins point in very nearly the same direction, we create a long-wave-length spin wave. We can try the same trick in the case of bosons, we apply an operator which kicks particles by almost the same amount if they are situated near each other. Thus we form the quantum state 4 n 1 √ exp(−iqxi )˜ 0 (p1 ; p2 ; : : : ; pn ) = ˆq ˜ 0 (p1 ; p2 ; : : : ; pn ) : (4.9) V i=1 4

Note that if we applied the product operator 0i exp(−iqxi ), we would kick all atoms equally and only cause a translation of the state.

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Trying to nd out the energy of the state ˆq |0 generated by the density operator, we use it as a variational state to estimate the corresponding excitation energy E(q) =

0 |ˆ−q Hˆ ˆq |0 : 0 |ˆ−q ˆq |0

(4.10)

The appendix shows that ˝2 q 2 1 0 |ˆ−q Hˆ ˆq |0 = 0 |[ˆ−q [Hˆ ; ˆq ]]|0 = 0 : 2 2m We also set 0 |ˆ−q ˆq |0 = 0 S(q)

(4.11)

(4.12)

in the appendix, I show that the function S(q) is closely related to the quantity (4.11). With these results, the excitation energy (4.10) becomes E(q) =

˝2 q 2 : 2mS(q)

(4.13)

This expression has rst been derived by Feynman [48] and shows the correct qualitative behaviour for the excitation spectrum. For long-wavelength excitation, q → 0, we expect the phonon-like behaviour E(q) = ˝vs q ;

(4.14)

where vs is the velocity of the long-wavelength sound. This implies the relation lim S(q) =

q→0

˝q ; 2mvs

(4.15)

which can, indeed be proved to hold. The appendix shows that S(q) is the Fourier transform of the density–density correlation function

1 3 3 d r (r )(r + r) : S(q) = d r exp(−iqr) (4.16) N The density–density correlation function is called the structure factor, and, at high enough particle density, it is expected to have a maximum when the variable r is close to the average interparticle spacing Qr. Its Fourier transform is then expected to have the corresponding maximum at q = (2=Qr). This will give a dip in the energy E(q), which is called the roton minimum in the spectrum of liquid 4 He. This is because it is believed to be the limiting case of a small vortex ring deriving from the back 0ow of the liquid around the emerging nearly free particle excitation [49]. The behaviour of the Bose system reminds us of the 0ow of a 0uid. If we use the Hamiltonian

2

1 ˝ 3 2 3 3 ˆ d x d yV (x − y)(x) Hˆ = d x + ˆ (y) ˆ (4.17) |∇(x)| 2m 2 to calculate the equation of motion for the density operator i˝

d (x) ˆ = [(x); ˆ Hˆ ] dt

(4.18)

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we obtain the result d (x) ˆ + ∇—(x) ˆ =0 : (4.19) dt This is clearly the operator form of a continuity equation with the current density operator given by i˝ † † ˆ ˆ —(x) ˆ = − [ˆ (x)∇(x) − ∇ˆ (x)(x)] 2m † ˝ 1 ˆ† ˝ ˆ ˆ ˆ (x) −i ∇ (x) (x) = + −i ∇(x) 2 m m ↔ ˆ

p ˆ = ˆ (x) (x) : (4.20) m Thus, the 0ow of the quantum 0uid can be related to the 0ow of an ideal 0uid but with quantum long-range coherence. †

4.2. Phase operator of the condensate Let us consider the particle number operator † ˆ Nˆ = ˆ (x)(x) d3 x :

(4.21)

This variable is conserved, because it commutes with the Hamiltonian. Its commutator with the eld is particularly simple ˆ ˆ [(y); Nˆ ] = (y) ;

(4.22)

which implies that † † Nˆ ˆ (x) = ˆ (x)(Nˆ + 1) :

(4.23) †

Applied to states with xed particle number, this tells that after operating with ˆ (x) we have a state with one more particle than we had before. This agrees with our interpretation of the creation eld operator. We consider the object ˆ ’) ≡ exp(i’Nˆ )(x) ˆ (x; exp(−i’Nˆ ) ; where ’ is a parameter. Object (4.24) obeys the equation 9 ˆ ˆ ’)] = −i(x; ˆ ’) ; (x; ’) = i[Nˆ ; (x; 9’ which integrates to ˆ ’) = e−i’ (x) ˆ (x; :

(4.24) (4.25)

(4.26)

The particle number operator thus generates phase shifts of the quantum eld. ˆ In the system with broken symmetry, we have argued that the phase of (x) acquires special signicance because it is related to the dynamics of the system. We are now going to investigate

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ˆ the possibility to consider a phase operator related to the quantum eld (x). To this end, we write the eld operator in the form ˆ ; ˆ (x) = W exp(i5)

(4.27)

where we, for the moment, assume that the operator character of W can be neglected. We are thus putting all the dynamics of the system into the phase. Relations (4.24) and (4.26) are now written in the form ˆ exp(−i’Nˆ ) = exp[i(5ˆ − ’)] : exp(i’Nˆ ) exp(i5)

(4.28)

This should remind us of a similar, well-known relation from quantum mechanics ˆ exp(iP) ˆ = exp[i(Pˆ − a)] : ˆ exp(−iaQ) exp(iaQ)

(4.29)

von Neumann and Weyl [50] have shown that a relation of this type guarantees the existence of ˆ P}, ˆ and that they have the property canonical variables {Q; ˆ P] ˆ =i : [Q;

(4.30)

Polynomials in these variables are said to form a Heisenberg algebra, and this is, of course, the subject matter of quantum mechanics as we usually know it. If we now interpret relation (4.28) in the spirit of a Heisenberg algebra, it implies that we can set ˆ =i : [Nˆ ; 5]

(4.31)

This suggests that the particle number and the eld phase form a pair of conjugate variables. In a phase-representation, the number operator can then be taken as 9 Nˆ = i : (4.32) 95 We can cast additional light on the number–phase relationship by looking at the ground state with broken particle conservation symmetry. This can be expressed in terms of the functions {uk (r)}. These basis functions are to some degree arbitrary, we obtain the same elds by shifting their phases by ’, and then Eq. (3.3) becomes ˆ (r) = ei’ aˆk (uk (r)e−i’ ) : (4.33) k

This change of uk (r) induces a phase change also of the annihilation and creation operators. If particle conservation is not obeyed, the quantum state of the system has to be taken as a linear superposition of states with diFerent particle numbers. Its state can then be written as 5 n ∞ ∞ aˆ†0 (4.34) |0 (’) ≡ Cn |n = Cn √ exp(−i’n)|vac : n! n=0 n=0 5

This state can only have a suggestive meaning because strictly speaking no physical system may be in such a state. This derives formally from the observation that in the nonrelativistic regime, the Galilean invariance depends on the mass of the state. As the diFerent terms have diFerent masses, the state is not invariant under this transformation. Obviously this remark does not apply to the massless photons or the relativistic case of Lorentz invariance.

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Applying the operator 9 Nˆ = i 9’

(4.35)

directly gives |0 (’) ≡

∞

Cn n|n :

(4.36)

n=0

This shows that (4.35) acts exactly like the number operator on the states |0 (’) . Thus, our adoption of the commutator relation (4.31) receives further support. The result does, however, depend on the ˆ assumption that |(r)| = W can be treated like a c-number. This becomes questionable in regions where W ∼ 0. Then the phase becomes harder to dene, and the dynamic description in terms of phase only is questionable. The same shortcoming aFects all phase variables in quantum mechanics; near the vacuum the quantum 0uctuations in amplitude mix strongly with the dynamics of the phase. In quantum optics, this problem has attracted considerable interest, and various solutions have been put forward [51]. In the case of a condensate of bosons, we have assumed that the condensate density 0 = W 2 is large, and we expect the phase description to hold with a high degree of validity. In discussions of Bose–Einstein condensates, it has recently been argued, see e.g. [52–55], that the phase variable is not a genuine parameter of the system, because it can only acquire value when two systems are compared experimentally. Indeed, only the phase relative to some reference has got a physical meaning, but this does not argue away its value as a physical variable. In the same way, the position variable can only refer to relative positions; no absolute frame of space is available to dene it. Relations of the type (4.35) have found wide use in the theory of superconductivity [56]. Here the phase of the electron–hole pairing amplitude acts as a dynamical variable, and its dynamic behaviour determines the many physically interesting properties of the Josephson eFects. If one wants to have a state with xed particle number, it is easy to project out this from (4.34). We only need to write 2 1 |0 n = ein’ |0 (’) d’ : (4.37) 2 0 Assuming the eld to be of the form (4.27), we calculate the particle current (4.20) and nd —(x) ˆ =−

ˆ i˝ ˆ † ˝∇5(x) † ˆ ˆ [ (x)∇(x) 0 ; − ∇ˆ (x)(x)] = 2m m

(4.38)

where the mean particle density is written as 0 = W 2 . This demonstrates that the velocity eld is given by the expression ˆ ˝∇5(x) : m From this follows immediately the fact that the velocity eld is irrotational v(x) ˆ =

(4.39)

∇ × v(x) ˆ =0 ;

(4.40)

which forms the basis for many of the super0uid characteristics of 4 He.

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We can obtain further insight into the character of the number-phase commutator, by looking at ˆ some implications of the assumption that a local operator 5(x) exists. From the continuity equation (4.19), we obtain by a double Fourier transform the relation !(!; k) = kj(!; k) :

(4.41)

This serves to relate correlation functions between the density operator and the current operator. Following the formalism presented in the appendix, we introduce the spectral density functions AXY to be derived from the correlation between the operators X and Y . Because AXY depends linearly on each observable, we can, with (4.41) write Ajj =

! !2 Aj = 2 A : k k

(4.42)

From (4.39) we nd j(k) = −i

˝k5k 0 : m

(4.43)

Using this in (4.42) we nd m! A5 = i 2 A : ˝k 0

(4.44)

From the appendix we have m d! d! ˆ =i 2 : !A [(−k); ˆ 5(k)] = A5 2 ˝k 0 2

(4.45)

Now using the sum-rule (A.27) from the appendix, we obtain the result ˆ [(−k); ˆ 5(k)] =i :

√

(4.46)

Taking the limit k → 0, noting that (k ˆ =0)=(Nˆ = V ) and dening the corresponding phase operator as 1 ˆ ˆ d 3 x 5(x) 5= (4.47) V we nd the relation ˆ 0 = i ; 0 |[Nˆ ; 5]|

(4.48)

which agrees with our earlier result (4.31). We especially note that the relation (4.48) involves a contradiction if the state |0 is an eigenstate of the particle number. This can be seen as an indication that the introduction of a phase operator is conditioned on the use of states with broken particle conservation. Our derivation of the result (4.48) is heuristic and nonrigorous. Its convincing power is thus not more than the earlier arguments in this section, but it adds insight into the consistency of the assumption of dynamics residing in the phase. Within a more exact framework of many-body theory, a similar result is derived by Forster in [30].

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4.3. Dynamics of the phase Inserting into the Hamiltonian (4.17) the form (4.27) we nd

2

˝ 1 3 2 3 3 ˆ ˆ |∇5(x)| 0 + d x d yV (x − y)(x) ˆ (y) ˆ ; H= d x 2m 2

(4.49)

where the kinetic energy term now is clearly of the form of a velocity eld. Fourier transforming this we obtain the Hamiltonian ˝ 2 0 2 ˆ ˆ 1 Hˆ = k 5 k 5 −k + Vk ˆk ˆ−k ; (4.50) 2m 2 k

k

see equation (4.7)! Assuming that the canonical relation (4.46) between particle number and phase can be used in the operator form, we write [ˆ−k ; 5ˆk ] = ikk :

(4.51)

With these assumptions, the equations of motion for the operators become 9 ˆ 5k = −i[5ˆk ; Hˆ ] = −Vk ˆk 9t 9 ˝2 k 2 ˆ ˆk = −i[ˆk ; Hˆ ] = 0 5 k : (4.52) 9t m These equations give the oscillator result 92 ˆ = −˝2 !(k)2 ˆk ; (4.53) 9t 2 k with the frequency ˝2 k 2 Vk 0 → 2V0 0 (k) : (4.54) ˝2 !(k)2 = m In the limit of small wave vectors, this gives the phonon type excitation spectrum derived earlier in (2.27). This result is thus in full agreement with our interpretation of the collective excitations deriving from the introduction of the phase operator as a dynamic variable. We may say that the character of the low-lying collective excitations in the condensed phase, is most easily treated by using states with broken symmetry. A treatment in terms of states with xed particle number must, in principle, be possible, but it lacks the simplicity and clarity of the one with broken symmetry. Hamiltonian (4.50) still mixes the variables with k and −k. This is, however, a situation known from many other boson systems, and we can diagonalize the Hamiltonian by introducing the variables 1 bˆk = (4.55) (!(k)5ˆk + iˆ−k ) : 2!(k) These are boson annihilation operators, and the Hamiltonian is found to become 1 † † ˝!(k)(bˆk bˆk + bˆk bˆk ) ; Hˆ = 2 k

as we expect.

(4.56)

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4.4. The Goldstone modes In many-body physics it is known that, when broken symmetry emerges, when the order parameter becomes nonzero, there appears new long-wavelength excitations in the system. The appearance of a macroscopic magnetization in a given direction breaks the rotational symmetry of the Hamiltonian. By gently tilting the spins over a large distance, we create an excitation, which has nearly zero energy, namely the spin wave. In the limit of innite wavelength, this is a tilt of the whole spin system. Thus, it carries out the symmetry operation under which the original Hamiltonian was invariant; it is said to restore the symmetry broken by the phase transition. The long-wavelength excitation with zero energy is called a Goldstone mode [30]. In the condensed Bose system, we have found that breaking the symmetry introduces a new dynamical variable, the phase operator. This describes an oscillation which has phonon character for long wavelengths, and it is supposed to represent the Goldstone mode in the system. The ordinary proof of the existence of such a mode can be adapted to the case of a Bose system, and I will brie0y indicate how this is done. We utilize relations (3.33) and (4.22) to write ˆ t); Nˆ ]|0 = 0 | exp(iHˆ t=˝)(x) ˆ 0 |[(x; exp(−iHˆ t=˝)|0 = 0 (x) :

(4.57)

The time evolution is here taken to be induced by the Hamiltonian; the result follows trivially because the particle number operator is conserved and the ground state is assumed to have energy zero. Fourier transforming this we obtain 3 ˆ t); (y)]| d y ei!t 0 |[(x; ˆ (4.58) 0 dt = 20 (x)(!) : We now insert a complete set of eigenstates {|n } of the Hamiltonian with energies En . We nd ˆ t); (y)]| dt ei!t 0 |[(x; ˆ 0 =

dt ei!t

ˆ [0 |(x)| ˆ n n |(y)| 0 exp(−iEn t=˝)

n

ˆ −0 |(y)| ˆ n n |(x)|0 exp(iEn t=˝)] : Because all En ¿ 0, only the rst term contributes when ! ¿ 0, and we obtain ˆ t); (y)]| ˆ dt ei!t 0 |[(x; ˆ 0 |(x)| ˆ 0 = 2 n n |(y)| 0 (! − En ) :

(4.59) (4.60)

n

Integrating this over y and using result (4.58) we obtain ˆ 2 d 3 y 0 |(x)| ˆ n n |(y)| 0 (! − En ) = 20 (x)(!) :

(4.61)

n

For this to be possible, there must exist a continuum of modes with En → 0: It is not enough to have E0 = 0, because this term cancels exactly in Eq. (4.59). We thus conclude that there must exist low-energy modes |n in the Bose system. These are ˆ† excited by the eld operator or the density operator. The states (y)| ˆ 0 and (x)|0 both contain

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a nonvanishing component on these excited states. This conclusion agrees with our understanding † of the role of the operator ˆ (x) and the variational trial function used in (4.10). This closes the argument, and suggests that there is an important connection between the single particle and the collective excitations in the condensed system. The relation between the phase operator and the Goldstone modes is further elucidated in [57]. 5. Perturbation theory 5.1. An ensemble with condensate In many-body perturbation theory, we expand the physical quantities around the noninteracting state, and hence the free particle creation and annihilation operators (2.4) are used. The interaction term is a product of several such operators (4.6). In the presence of the particle condensate, the operators aˆ0 become cumbersome, because their powers grow with the volume as V 1=2 , see (2.6). This makes it impossible to handle the perturbation series directly, and one must resort to special arrangements. These were rst introduced by Bogoliubov [15] and developed further by Beliaev [26] and Hugenholtz and Pines [27]. The idea derives from the fact that the condensate acts as a reservoir for an arbitrary number of particles, which can absorb and emit without undergoing any appreciable change. I have repeatedly stressed that this is a propitious way to consider the situation even in an interacting system. It turns out that, once the condensate reservoir is introduced, the perturbation expansion can be derived and it may be used to obtain interesting and nontrivial results. It is obvious that, with such an approach to the condensate, particle conservation is not possible. We start by introducing a Hamiltonian, which allows us to treat varying particle number, that is a quantum equivalent of the grand canonical ensemble. We write as usual Hˆ ≡ Hˆ − Nˆ :

(5.1)

The parameter is introduced to x the average particle number by imposing Nˆ ( ) = N ;

(5.2)

which determines = (N ). With the condensate present, we now assume that we have N0 particle in the condensate and that the ground state is a function of this number and ; i.e. |0 (N0 ; ) . Thus, we regard the Hamiltonian as a function of the condensed particles Hˆ (N0 ). We then use a modied total Hamiltonian Hˆ (N0 ; ) = Hˆ (N0 ) − Nˆ ;

(5.3)

where Nˆ now refers only to the particles above the condensate. When we omit the particles in the condensate, the particles above the condensate are given by N = N − N0 = 0 (N0 ; )|Nˆ |0 (N0 ; ) :

(5.4)

Thus, the number of particles in the condensate is treated as a parameter only. Relation (5.4) now determines as function of N0 and N . As the particles in the condensate are assumed not to contribute to the energy, we have E0 = E0 + N ;

(5.5)

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199

where E0 is the ground state energy of the modied Hamiltonian (5.1). Thus, there is no explicit dependence on the parameter N0 . It can, however, be determined by demanding that the energy is minimized with respect to it, namely d E0 = 0 : (5.6) dN0 Using the Feynman–Hellmann theorem, we can calculate the derivative 9Hˆ 9E0 = 0 (5.7) = −0 |Nˆ |0 = −N : 9 0 9 We now carry out the derivation in Eq. (5.6) 9 9E0 9E0 9 9N 9E 9N d E0 = + + + N = 0 + =0 : dN0 9N0 9 9N0 9N0 9N0 9N0 9N0 Because 9N =9N0 = −1, this gives 9E0 :

= 9N0

(5.8)

(5.9)

This shows that the parameter serves as a condensate chemical potential in the modied ensemble. It is the derivative of the ground state energy with respect to the particles in the condensate, which serve as the particle reservoir here. The particles in the ground state N0 act as a parameter only labelling the ensemble we use. We can, however, show that coincides with the correct chemical potential for the system. To this end we remember that the ground state energy is E0 = E0 [N0 (N ); (N0 (N ); N )] : We thus have 9E0 9E0 9E0 (9E0 =9 )N0 dN0 d = + = 9N 9N0 dN 9 N0 dN (dN=d ) because of (5.6). Because of (5.4) we obtain dN dN = d d and because of (5.5) and (5.7) we have 9E0 9E0 dN dN = +N + = : 9 9 d d Thus Eq. (5.11) gives 9E0 = ; 9N

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

which proves that also is the real chemical potential of the physical system. We have thus proved that the condensate can be treated as a particle reservoir, and that its physical characteristics may be identied with those of the real system. It is then possible to proceed and make a perturbation expansion in this modied ensemble.

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5.2. Graphical perturbation theory The fact that the condensate can be regarded as a reservoir of particles may be utilized to replace the condensate creation and annihilation operators by c-numbers. According to (2.7) this becomes exact in the thermodynamic limit. The density operator (4.5) then becomes 1 † ˆq = √ aˆk −q aˆk = (∗ aˆq + aˆ†−q + O(V −1=2 )) : (5.15) V k We introduce the canonically conjugate pair of variables 1 Qˆ = (∗ aˆq + aˆ†−q ) 2 2|| i ( aˆ†q + ∗ aˆ−q ) : Pˆ = 2||2

(5.16)

Thus, we see that ˆq ˙ Qˆ and represents the coordinate for creating bosonic excitations in the system. This follows also from our discussion in Section 4.4. In the limit of large volume, the operators {aˆ0 ; aˆ†0 } can be replaced by c-numbers, and the corresponding terms are dominating the dynamics. This was the observation made by Bogoliubov [16] and it has signicant implications for the perturbation approach in the modied ensemble introduced in the previous section. In many-body theory, the perturbation expansion is represented by series of graphs, which correspond to the various terms in the expansion. This approach was developed originally in quantum electrodynamics and high-energy physics, but it has turned out to be highly successful in many-particle systems and statistical mechanics too [58]. It serves to classify the various eFects of the interaction, and it allows one to resum innite series of graphs to obtain new and nonperturbative results. We are going to consider the application of the graphical perturbation series to the system of interacting bosons in the presence of the condensate. We do not go into the technical details of generating the expansion; these can be learned from the texts referred to. Only the heuristic image conveyed by the method will be presented. The Hamiltonian is

˝2 2 † ˆ ˆ ˆ H = (x) − ∇ + U (x) − (x) d3 x 2m 1 U † † ˆ (x) ˆ d3 x ; (5.17) + V ˆ (x)ˆ (x)(x) 2 where we retain the chemical potential to keep track of the particle number as explained above. The equation of motion for the eld variable then follows from the commutation relations to be

˝2 2 d ˆ † ˆ ˆ ˆ U : (5.18) i˝ (x) = − ∇ + U (x) − + V (x)(x) (x) dt 2m The propagation of a noninteracting (bare) particle is represented by G0 and a single line showing the direction of propagation, see Fig. 1a. With the contact interaction (2.21), four particles interact at the same point in space, two go in and two go out as indicated in Fig. 2a, where the dot denotes

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201

(a) (a)

(b)

(b)

(c)

(c)

Fig. 1.

Fig. 2.

the interaction vertex (VU =˝) given in frequency units. When the bare particle propagator is modied by interaction events, it becomes the full (dressed) propagator indicated by G and a double line, Fig. 1b. These interactions are, however, modied by the reservoir of condensed particles. We indicate the exchange of one condensate particle by a wavy line ending in a blob, Fig. 1c, and represented by a condensate amplitude ’0 . This implies that one or more of the particles involved in the interaction may derive from the condensate. It may absorb or emit particles, thus the two graphs of Fig. 2b enter. They represent the possibility that one particle scatters into two or two particles merge to one; the particle conservation is taken care of by the condensate. Finally the Fig. 2c describes the √ scattering of a pair of particles against the condensate. In the spirit of our earlier discussion, ’0 ˙ V and the graphs containing condensate terms are dominating the dynamics. The original Bogoliubov treatment rests on the assumption that graphs of the type Fig. 2c determine the dynamics. When the external potential U (x) is not present, it is expedient to use a momentum eigenstate expansion of the eld variables. This has been the method used the most in the case of bulk many-particle systems, but with recent interest in condensates in traps, the translational invariance is lacking and the graphical expansion is more convenient in the position representation. We do not here make detailed distinction between the two cases, but write the graphs the same way. In the many-body theory, the exact relationship between the two methods is derived. As explained above, the collective excitations in the system are related to the dynamics of the density operator. In addition to the single-particle propagator, we thus need to consider the dynamics of the density–density propagator which we denote by D. This is proportional to correlations between the density operators D ˙ ˆ ˆ :

(5.19)

This implies that the propagator D involves operators absorbing two particles and re-emitting two. We thus represent it by the symbol in Fig. 3. We now have all the ingredients to proceed to write down graphical perturbation expressions for various quantities. In the homogeneous system, the equations are most easily interpreted in the momentum representation, and the time dependence can be replaced

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=

+

+ Fig. 3.

Fig. 4.

by the corresponding frequency dependence. The excitations in the system are given by the poles in these !-dependent expressions. The single-particle excitations occur in G and the collective modes in D. For the noninteracting particles, the single-particle energy is given by + ˝2 (k) ˜ and hence we assume 1 G0 (!) = : (5.20) ! − ˝(k) ˜ Here we use frequency units and set k2 : (5.21) 2m If we now follow the general argument and assume that the interaction with the condensate dominates, i.e. graphs containing the parameter ’0 are the only ones of relevance, we may calculate several interesting results. The density–density propagator can be seen to obey the relation in Fig. 4. The corresponding equation is VU ’20 VU (G0 + G0T )D : D= + (5.22) ˝ ˝ The backward propagator G0T is the time reversal of the propagator G0 which for the frequency representation gives G0T (!) = G0 (−!). With these results, the density–density propagator becomes (!2 − ˝2 (k) ˜ 2) VU : (5.23) D(!) = ˝ (!2 − ˝2 (k) ˜ 2 − 2VU ’20 (k)) ˜ (k) ˜ =

The pole giving the collective excitation is determined to be VU ’20 + O(k 2 ) : ˜ + ˝2 (k) ˜ 2) ≈ k !(k) = (2VU ’20 (k) m

(5.24)

This is the result derived by Bogoliubov, and in the long-wavelength limit, it agrees with our earlier results (2.27) and (4.54). Here we nd it to emerge directly from the assumption that the condensate dominates the dynamics.

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= +

=

Fig. 5.

Fig. 6.

However, the interaction vertices in Fig. 2b show that there will be direct coupling between the single-particle propagator and the density–density two-particle one. This is seen to play a central role in the graphs depicted in Fig. 4. The pole in the density–density propagator will thus appear also in the single-particle propagator aFecting the singularities of this. That fact was originally stressed by Gavoret and Nozieres [32] and followed up by Huang and Klein [33] and Hohenberg and Martin [34]. The dressed single-particle propagator can easily be calculated in an approximation consistent with Eq. (5.22). The interactions with the condensate are supposed to dominate and the propagator is given by the graphs in Fig. 5 giving the equation G = G0 + ’20 G0 DG0 : Using expressions (5.20) and (5.23), we nd

VU ’20 (! + ˝(k)) ˜ 1 1+ G(!) = ! − ˝(k) ˜ ˝ (!2 − !(k)2 ) VU ’20 ! + ˝(k) ˜ + ˝ : = 2 2 2 (! − ˝ (k) ˜ − 2VU ’20 (k)) ˜

(5.25)

(5.26)

Thus, we see that the single-particle propagator has got exactly the same singularity !(k) as the density–density propagator, i.e. the collective mode. Note especially how elegantly the single-particle pole is eliminated in the present approximation. The same phenomenon is characteristic of condensed systems more generally; the same excitation poles appear in both the density–density and the single-particle propagators. For a more detailed discussion see GriPn [14]. Because our modied ensemble does not assume particle conservation, we also have anomalous propagators. One of these is shown in Fig. 6 giving VU ’20 ˝ F(!) = ’20 G0 DG0T = − 2 : (5.27) 2 2 (! − ˝ (k) ˜ − 2VU ’20 (k)) ˜ The expression for the single-particle propagator (5.26) and the anomalous propagator (5.27) agree with those derived by Beliaev [26] and later used widely to describe bulk 4 He. Here we have provided a heuristic derivation based on our assumption that the interaction with the condensate dominates the dynamics.

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ψ

ψ Γ

ψ†

= ψ†

+

+ Fig. 7.

5.3. Renormalization of the 4eld theory The eld theory is based on the Hamiltonian (5.17) from which we assume that we may derive the Gross–Pitaevski functional (3.42). Then, however, we assume that the mass parameter m is the same as the free mass and the interaction vertex VU remains unchanged. As we know from eld theory, these parameters will be renormalized in a theory of the present type, in eld theory contexts called a 4 -theory after the interaction [59]. The graphical perturbation theory allows us to calculate corrections to the bare quantities in perturbation theory, and we proceed to discuss this without going into the technical details. Only the structure of the equations will be given, and the spirit of the argument will be presented. It is most understandable that the potential is not modied, because the introduction of the contact form (2.21) already assumes a nonperturbative representation of the low-energy scattering amplitude. If, however, we strive to renormalize the interaction, we have to replace the bare four-boson interaction by the eFective four-boson vertex. Using for this the second-order perturbation terms of Fig. 7, we can write the expansion of the free energy as a function of the order parameter in the form 2 VU F = F0 + d 4 x d 4 yG(x; y)G(y; x)∗ (x)∗ (x)(y)(y) 2 2 VU d 4 x d 4 yG(x; y)G(y; x)∗ (x)∗ (y)(y)(x) ; (5.28) + 2 where the four-dimensional integrals are carried out over the three space and the one time dimension. Here F0 is the functional (3.42). Comparing the interaction term in F0 with the terms added in (5.28) we nd that the correction to the potential is proportional to 2 QV = VU G(x; y)G(y; x) d 4 y : (5.29)

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205

In the bulk system, such integrals are divergent, see [14], but for the nite trapped systems we may conjecture that they are nite. If needed any regularization process will work. We here propose to use the fully dressed propagators G, which we do not know. If we, without any real justication, assume these to go to zero over a distance equal to the correlation length 6 and the correlation time diFerence Q>, we can estimate the expression as 2 QV = VU g02 3 Q> :

(5.30)

For the corrections to the potential VU to be small we must require VU g02 3 Q>1 :

(5.31)

In addition to the assumption about the correlation distances, this contains a constant g0 which is not known to us (it may even be divergent). For the moment we do, however, accept this as a requirement on the parameters of the theory. Next we want to look at the corrections to the mass term which has a form ˝2 EKIN = d 3 x |∇(x)|2 : (5.32) 2m In order to nd a term of this type, we have to relate the double integrals to a single one in the terms in (5.28). To achieve this we expand the products ∗ (x)(y) = ∗ (x)(x) + 12 ∗ (x)(x − y)2 ∇2 (x) ; which contributes a term of the type ˝2 QEKIN = K d 3 x |∇(x)|2 ; 2m where 2m 2 K = 2 VU ˝

G(x; y)G(y; x)(x − y)2 |(x)|2 d 4 y :

(5.33)

(5.34)

(5.35)

We estimate this as the integral above; the factor (x − y)2 gives two more factors of the correlation length and the order parameter is replaced by its average value to give 2 −1 ˝ 2m U 2 2 2 5 VU g02 3 Q> ≈ VU g02 3 Q> : (5.36) K = 2 V g0 ’0 Q> = VU ’20 ˝ 2m 2 The rst factor approximates unity, because VU ’20 approximates the chemical potential and this determines the healing length according to (2.29). The resulting expression is small because of the requirement (5.31). We have thus shown that assuming the corrections to the interaction vertex to be negligible, we are allowed to use the free particle mass in the Gross–Pitaevski energy functional. 6

When this is limiting the spatial extension of a quantum propagator it is usually referred to as the correlation length. It is understood to be the same as the healing length introduced in Section 2.2.

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The details of the calculation are only suggested in the presentation above; I have developed the argument in some detail in [60] but it still rests on assumptions about the correlation lengths and correlation times which have not been proved. The algebraic part of the argument does, however, emerge as in the present highly simplied version. Thus the main conclusion remains, having negligible corrections to the interaction vertex suggests that the Gross–Pitaevski theory can use the free mass of the particles. 6. Theory of trapped condensates 6.1. The Thomas–Fermi regime The recent magnicent progress in BEC in trapped alkali atoms has initiated a large activity. It is not my intention to review this here, but I do want to connect the eld theoretic approach to the most essential parts of the topical activity in the eld. For a more complete overview, e.g. the recent article [5] may be consulted. In the case of trapped alkali atoms, the crudest theory is based on an approximation termed the Thomas–Fermi assumption. To simplify the treatment, we assume the trap to be symmetrically harmonic with the potential 2 2 1 x ˝ 2 2 U (x) = m# x = ; (6.1) 2 2 a0 2ma0 where the length (2.34) has been inserted. Performing the variation with respect to ∗ (x) of the functional (3.42), we nd the Gross–Pitaevski equation

˝2 2 2 U − ∇ + U (x) + V |0 (x)| 0 (x) = 0 (x) : (6.2) 2m The potential is supposed to be given by the contact form (2.22) 4˝2 a ; VU = m

(6.3)

which introduced the scattering length a. We estimate the average density of the condensate to be 0 ∼ |0 (x)|2 ∼

N : R3

(6.4)

Here R is the size of the condensate which is assumed to be much larger than the size a0 of the ground state of the oscillator, because of the assumed repulsive interaction. Expanding the condensate decreases the particle–particle interaction energy but increases the potential energy by the particles having to climb the walls of the trap. The size can be estimated [61] by requiring these two terms to compensate 1 N m#2 R2 = VU 0 = VU 3 : 2 R

(6.5)

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This xes the radius of the condensate to be 8a 1=5 1=5 R= N a0 = )a0 ; a0 where )5 = 8

a a0

207

(6.6)

N :

(6.7)

In the strong interaction limit, Naa0 , the parameter )−1 can be assumed to be a small expansion parameter of the problem. From this it follows that the volume is proportional to R3 ∼ N 3=5 . The particle density is given by N 1 a0 3=5 0 = 3 = (8N )2=5 : (6.8) R 8a30 a Except for the numerical factors, results (6.6) and (6.8) agree exactly with those derived in (2.39) and (2.40). Thus, the present assumptions are equivalent to the gas approximation used there. We now introduce the scaled position variable x x (6.9) y= = R )a0 into equation (6.2) giving

1 2 2VU

2 ∗ − 4∇ + y + 0 (y) 0 (y) 0 (y) = 0 (y) : ) m#2 R2 E 0 )2

(6.10)

where E0 = 12 ˝# is the ground state energy of the oscillator. The chemical potential is found to scale as

∼ E0 )2 = 12 m#2 R2

(6.11)

and the kinetic energy is down by a factor )−4 with respect to the other energy terms. If we assume that this justies us to neglect the kinetic energy, we obtain the density prole ( − 12 m#2 x2 ) m#2 2 (R2 − x2 ) ; = (6.12) |0 (y)| = 2VU VU where we used estimate (6.11). This result agrees result (2.35), when we observe that m#2 R2 = 0 2VU

(6.13)

as seen from Eq. (6.5). The approximation when the kinetic energy is neglected in the Gross–Pitaevski equation is usually called the Thomas–Fermi approximation. This derives from a corresponding approximation in fermionic systems, where various approximations lead to the same expressions. In particular, it implies that the kinetic energy is expressed in terms of the local particle density in the same way as in the ideal Fermi–Dirac gas. In the boson system, the various criteria for the Thomas–Fermi situation

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may lead to diFerent expression [62,63], but it seems that the most obvious one is to replace the kinetic energy by that of the noninteracting gas. As this is fully condensed at zero temperature, its kinetic energy vanishes and the approximation agrees with that which is generally referred to by the name Thomas–Fermi. It is clear, however, that it cannot be valid near the edge of the condensate. The abrupt cut-oF in the density needs to be treated separately [64,65] in order to give the smooth disappearance of the condensate density here. This can be obtained from a numerical solution of the Gross–Pitaevski equation, but as the healing length grows to innity here, one may even question the justication to use that equation for the edge region with extremely low density. The healing distance follows from 2 2 1 ˝ 2 a20 ˝ 2 = = = a20 : (6.14) 2m

2m ˝#)2 )2 The order parameter of the liquid can sustain bending over at most the scale of , which hence has to be much longer than the scattering length a which represents the spatial shift of the phase of a wave function. In the Thomas–Fermi regime, we thus expect the various length scales to form a hierarchy Ra0 a :

(6.15)

Under such conditions we expect the Thomas–Fermi approximation and the Gross–Pitaevski equation to give a good description of the physics of condensed alkali atoms. It is not denitely proved that these results hold for such systems, but the theoretical success in describing the observed physical results indicates that this may be valid. There is still some need to justify the use of this theoretical approach. Finally I want to comment on the relation of the Gross–Pitaevski approach to the Hartree–Fock approximation. If we use an ansatz of type (2.13), the single-particle ground state is found to satisfy an equation of the type

˝2 2 2 U − ∇ + U (x) + V |u0 (x)| u0 (x) = u0 (x) : (6.16) 2m This equation is the only one that needs to be solved if the interaction is so weak that a single state only contributes to the permanent, see Section 3.1. The form (6.16) indeed, agrees in form with Eq. (6.2), but its physical meaning is diFerent from the equation derived by breaking of the symmetry in eld theory. Solving Eq. (6.16) we have to use the function u0 (x) in the state function (2.13) only, its phase is no dynamical variable and the kinetic energy term does not correspond to any rigidity of the order parameter. In the limit of a homogeneous system, the Hartree–Fock spectrum retains a gap, and no collective excitations appear. The Hartree–Fock approach is based on minimising the energy, and thus applies to the ground state only. No vortex-type excitations are relevant, and when the particle interaction goes to zero, the states are the eigenstates of the potential U (x). If this has rotational symmetry, it supports angular momentum eigenstates, but these do not describe velocity 0ow around a vortex core, but correspond to the states known from atomic physics. Recently the derivation of the Gross–Pitaevski equation has been based on a diFerent philosophy. One assumes that, even in the interacting case, the bosonic character of the particles justies the

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209

approximation that they all reside in the same single-particle states. Thus, also in the time-dependent situation, one takes for the many-particle wave function the ansatz (r1 ; r2 ; : : : ; rN ; t) =

N

’(ri ; t) :

(6.17)

i=1

We then assert that the actual time evolution extremizes [66] the functional 9 A= dt(t) i˝ − Hˆ (t) ; 9t

(6.18)

where Hˆ is the N-body Hamiltonian; this implies that |(t) satises the correct SchrModinger equation. When ansatz (6.17) is inserted into the functional A and we take the variation with respect to the single-particle state ’∗ (ri ; t) we do, indeed, obtain the time-dependent Gross–Pitaevski equation. In Ref. [6] it is shown that this approximation is not entirely consistent, and it is only the rst of a series of approximations where the Bogoliubov approach is the next and in many ways better approach. The exact relationship of this approach and the broken symmetry one is not clear. With properly introduced renormalization and without any truncation of the expansion in the order parameter, the expression of type (5.28) should have a broader validity than ansatz (6.17). 6.2. Elementary excitations in the condensate Much theoretical eFort has gone into the description of small excitations of the boson condensate. These are the fundamental collective modes of the system, they describe the dynamics of the interacting particles and they are susceptible to experimental observation. I cannot here go into the details of these works, the calculations depend on the shape and size of the trap, and their evaluation has required some detailed numerical work. Here I can only outline the general idea of one approach and refer the reader to the references for the results. We use the generalized Gross–Pitaevski equation (6.2) generalized to become time dependent

˝2 2 9 2 U − ∇ + U (x) − + V |(x)| (x) = i˝ (x) ; (6.19) 2m 9t which may be justied by the variational approach of the previous section. Here we, however, assume this to be taken as a direct consequence of the equation of motion (5.18) when we apply approximations like (3.39) and (3.40). We now separate the amplitude and the phase of the order parameter as (x) = (x) exp[i5(x)] : (6.20) Inserting this into (6.19) and separating real and imaginary parts we obtain rst the equation ˝ 9 (x) = − ∇((x)∇5(x)) = −∇((x)v(x)) ; 9t m where we have introduced the velocity eld v(x) =

˝ ∇5(x) : m

(6.21)

(6.22)

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This is the c-number version of the particle conservation equation (4.19). We also nd the equation ∇2 (x) ˝ ˝ 1 9 2 5(x) + (∇5(x)) − (6.23) + (U (x) − + VU (x)) = 0 : 9t 2m 2m ˝ (x) Multiplying this equation by m((˝=m)∇) we nd 2 2 ∇ (x) m 9 ˝ v(x)2 + U (x) − + VU (x) − m v(x) + ∇ =0 : 9t 2 2m (x)

(6.24)

If we leave out the last term ˙ ˝2 we recognize the Euler equation from classical hydrodynamics. In the semiclassical approximation, this quantum potential can be omitted, and we have reduced our problem to one in classical hydrodynamics. In order to describe the elementary excitations, we can linearize the hydrodynamic equations in a well-known manner. The velocity v(x) is supposed to be small without a steady component, but the density is written as (x; t) = 0 (x) + (x; t) :

(6.25)

The continuity equation now gives 9 (x) = −∇(0 (x)v(x)) ; 9t and the Euler equation becomes 9 ∇ v(x) = − [U (x) − + VU 0 (x) + VU (x)] : 9t m From these equations we obtain

92 0 (x) U U ∇(U (x) − + V 0 (x) + V (x)) : (x) = ∇ 9t 2 m

(6.26)

(6.27)

(6.28)

In the Thomas–Fermi approximation, we have VU 0 (x) = − U (x) ;

(6.29)

which gives the equation 92 (x) = ∇(vs2 (x)∇(x)) ; 9t 2 where the local velocity of sound is given by

(6.30)

[ − U (x)] : (6.31) m The existence of such a local velocity assumes that it is slowly varying. As the spatial dependence derives from the external potential U (x) we expect this to imply vs2 (x) =

A

A ∇vs (x) ∼ ; vs (x) a0

(6.32)

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211

where A is the wavelength of the excitation. Thus, only excitations which distort the condensate shape more than the trap size are relevant. Their energy is expected to be above those of the trapping potential. This implies that we should have A ¡ a0 R :

(6.33)

It is easy to see the consistency of Eq. (6.30). In the homogeneous case we may set U (x) = 0 and = VU 0 . Thus we obtain VU 0 2 2 ; (6.34) ! = k m which gives the spectrum VU 0 = 2VU 0 (k) : ˝!(k) = ˝k m

(6.35)

This agrees with our earlier results (2.27) and (4.54). Wave equation (6.30) can be solved in various nite geometries [67]. For the spherically symmetric case, we may utilize the eigenfunctions of angular momentum (x) ˙ Ylm and introducing an integer ns ; one has derived the spectrum (6.36) !(ns ; l) = # (2n2s + 2ns l + 3ns + l) : √ For the lowest symmetric mode, l = 0 called a breathing mode, we obtain the result ! = 5#, which satisfactorily far exceeds the harmonic oscillator frequency. This frequency, however, is always found in the oscillation spectrum, because the center of mass of the condensate can oscillate undistorted at the bare trap frequency. For the details of the calculations of the excitation mode spectrum, I refer to the literature and especially the recent review [67].

7. Conclusion I have recapitulated the progress in shaping a theoretical understanding of the physics of a trapped condensate. The eld theoretic approach is emphasized and its various methods and results have been presented. The level of mathematical rigour is low; many of the results can be obtained more precisely, but mainly at the expense of the intuitive character of the reasoning. As the full many-body apparatus is not needed to describe atomic condensates, a physical understanding of the earlier results seems more useful than a full 0edged technical mastery of the formalism. I have referred to the key papers of the historical development and some specic papers which relate to special topics of relevance for the narrative. However, in most cases text books and review articles have been given. It is an unmanageable task to give a full list of references. The eld of Bose–Einstein condensation is developing so fast that such an attempt is doomed to fail. There are also many reviews of the present status of the experiments, and they have to be consistently updated. No attempt has been made to give justice to the many beautiful experiments continually forthcoming in this eld.

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Acknowledgements The main part of this work was carried out during the Workshop on Bose–Einstein Condensation at the Lorentz Center in Leyden. I thank Professor H.T. Stoof for inviting me to this Workshop and for helpful exchange of information. I have also enjoyed discussions with many colleagues over the years. My original interest in the eld was kindled by D. ter Haar and W.E. Parry. I especially M want to thank the following persons for useful inspiration: M. Wilkens, K.-A. Suominen, P. Ohberg, K. Burnett, J. Javanainen, M. Holthaus, A.L. Fetter, A. GriPn, M. Lewenstein, T.W. HMansch, C.J. Pethick and G. Kavoulakis. I also want to thank C.E. Wieman, E.A. Cornell and W. Ketterle for keeping me informed about the experimental progress. I owe thanks to many more persons, who have contributed to my joy and understanding in following the development of the fascinating eld of BEC. They are, however, too numerous to be mentioned here. Appendix A. Many-body formalism This paper is not going into the highly developed eld of many-body formalism. However, for completeness, I summarize the main features and results needed in the text. The results and their detailed derivations are found in [68,69]. Many probes of the physical properties of complicated systems rest on weak perturbations whose response is mainly determined by the state of the system before the probing. The linear eFect of a weakly perturbing cause, can be the quantity of interest in many situations. Thus we assume that the system, characterized by the Hamiltonian Hˆ is coupled to an operator Yˆ by the time-dependent function f(t) according to Hˆ Y = Hˆ − Yˆ f(t) :

(A.1)

The state of the system is supposed to be described by a density matrix, which we write as the steady-state part ˆ0 and a small deviation caused by the perturbation ˆ = ˆ0 + ˆ ;

(A.2)

where we assume ˆ0 to be stationary, i.e. to commute with the Hamiltonian Hˆ . The linearized equation of motion for ˆ is now d ˆ = [Hˆ ; ] ˆ − [Yˆ ; ˆ0 ]f(t) dt with the solution i t (t) ˆ = dt exp[ − iHˆ (t − t )=˝][Yˆ ; ˆ0 ] exp[iHˆ (t − t )=˝]f(t ) : ˝ −∞ i˝

(A.3)

(A.4)

Calculating the expectation value of some other observable Xˆ using the perturbation solution, we get the linear response +∞ ˆ = DXY (t − t )f(t ) dt ; (A.5) Xˆ − Xˆ 0 = Tr(Xˆ (t)) −∞

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where the susceptibility function is dened as DXY (t − t ) = i5(t − t )Tr{[Xˆ (t); Yˆ (t )]ˆ0 } :

(A.6)

Note that this is zero for t ¡ t and the operators are in the Heisenberg representation Xˆ (t) = exp[iHˆ t=˝]Xˆ exp[ − iHˆ t=˝] :

(A.7)

Function (A.6) only depends on the time diFerence, if no external time dependences occur. We proceed by dening the spectral density of the susceptibility function. This is given by +∞ AXY (!) = d(t − t ) exp[i!(t − t )][Xˆ (t); Yˆ (t )] : (A.8) −∞

Inverting this Fourier transform, and setting t = t , we nd +∞ d! ˆ ˆ AXY (!) : [X ; Y ] = −∞ 2

(A.9)

If we Fourier transform the susceptibility function (A.6) and add the convergence factor exp(−t) we obtain +∞ DXY (!) = DXY (t − t ) exp[i!(t − t )] d(t − t ) −∞

=i

0

=i

+∞

0

+∞

exp[i(! + i)>][Xˆ (>); Yˆ (0)] d> exp[i(! + i)>]

+∞

−∞

This gives the expression +∞ AXY (! ) d! XY D (!) = : ! − ! − i −∞ 2

d! AXY (! ) exp(−i! >) d> : 2

(A.10)

(A.11)

The frequency form of the linear response susceptibility is thus the Hilbert transform of the spectral density. The function has a branch cut along the real axis everywhere where AXY (! ) = 0 and the term i tells that we have to choose the branch which has got ! in the upper half-plane. Using the properties of the function in the complex plane, we can derive relations between the real and imaginary parts; these are usually called dispersion relations. As we have seen, the susceptibility is a special type of correlation function between two operators. In the main part of the paper, we have been interested in the properties of the density operator, and hence we dene the function 1 S(k; !) = dt ei!t ˆ−k (t)ˆk (0) : (A.12) 20 This function is also found to describe the scattering of neutrons from a bulk system with the change of momentum being ˝k and the energy ˝!: Using such experiments one has been able to determine

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the spectrum of the elementary excitations in many systems [12], in particular liquid 4 He has been investigated this way. In the present paper, we consider only zero temperature, i.e. the averages are taken with respect to the ground state |0 . The Hamiltonian is written formally as Hˆ = |n En n | (A.13) n

and we set the ground state energy to E0 = 0: When this is introduced into Eq. (A.12) we nd 1 dt |n | ˆk | 0 |2 exp[i(! − En =˝)t] S(k; !) = 20 n =

1 |n | ˆk | 0 |2 (! − En =˝) : 0 n

(A.14)

As we assume the ground state to be invariant under time reversal, a change of the sign of the momentum k does not change the result. We dene the structure factor by setting +∞ 1 S(k) = S(k; !) d! = 0 | ˆ−k ˆk | 0 : (A.15) 0 −∞ This contains the information about the static structure of the physical system, the liquid. Dening the correlation function ikr d! e−i!t S(k; !) G(r; t) = e (A.16) k

we nd directly G(r; t = 0) =

k

eikr S(k) =

1 ikr e 0 | ˆ−k ˆk | 0 : 0

(A.17)

k

Inserting expansion (4.5), we nd 1 d 3 x0 | (x) G(r; t = 0) = ˆ (x ˆ + r) | 0 : 0 Inverting the Fourier sum in Eq. (A.17) we nally obtain 1 d 3 rG(r; t = 0) S(k) = V 1 3 d r exp(−ikr) d 3 x0 | (x) ˆ (x ˆ + r) | 0 : = N

(A.18)

(A.19)

This shows that the structure factor is the Fourier transform of the spatial correlation function in the system. The function S(k; !) satises simple sum rules. The most important one generalizes the f-sum rule of atomic physics, and we derive this one here for the case of zero temperature. First we invert

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relation (A.12) to obtain 0 e−i!t S(k; !) d! = 0 | ˆ−k (t)ˆk (0) | 0 :

(A.20)

Applying the operator (i˝ d=dt) and setting t = 0, we nd 0 ˝ !S(k; !) d! = 0 | [ˆ−k ; Hˆ ]ˆk | 0 = 0 | ˆ−k Hˆ ˆk | 0 :

(A.21)

Utilizing the time reversal symmetry of the state, we can write 0 | ˆ−k Hˆ ˆk | 0 = 12 [0 | ˆ−k Hˆ ˆk | 0 + 0 |ˆk Hˆ ˆ−k | 0 ] = 12 0 | [ˆ−k; [Hˆ ; ˆk ]] | 0 : All these relations hold because Hˆ | 0 = 0: We use representation (4.3) for the density operator n 1 √ ˆk = exp(−ikxs ) : V s=1 The only part of the Hamiltonian which does not commute with this is the kinetic energy n pˆ 2r Hˆ KIN = : 2m r=1

(A.22)

(A.23)

(A.24)

The result in (A.21) is thus independent of the interaction as long as this is a function of positions only. We nd n 1 exp(−ikxs ) 2 2 [Hˆ KIN ; ˆk ] = √ (A.25) (˝ k − 2˝k pˆ s ) : 2m V s=1 Only the second term contributes to the next commutator, and hence we nd n ˝k ˝2 k 2 ˝2 k 2 [ˆ−k; [Hˆ ; ˆk ]] = − √ exp(−ikxs )[exp(ikxs ); pˆ s ] = 1= 0 : mV s=1 m m V Introducing this into (A.21) we obtain ˝k 2 !S(k; !) d! = : 2m

(A.26)

(A.27)

This is the f-sum rule in many-body theory. We can use this to obtain one more relation we need. We use the denition (A.8) of the spectral density for the density–density correlation function +∞ A (k; !) = dt exp[i!t]0 | [ˆ−k (t); ˆk (0)] | 0 : (A.28) −∞

In the same way as we derived (A.14) we can use (A.13) to write A (k; !) = 2 (|n | ˆk | 0 |2 (! − En =˝) − |0 | ˆk | n |2 (! + En =˝)) : n

(A.29)

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Because we can see that, at zero temperature, S(k; !) = 0 for ! ¡ 0, we write A (k; !) = 20 [S(k; !) − S(k; −!)] : Using the sum-rule (A.27) we can now write

+∞

0 +∞ ˝k 2 d! = 0 : !A (k; !) !S(k; !) d! − !S(k; −!) d! = 0 2 m 0 −∞ −∞

(A.30)

(A.31)

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Renormalization group theory in the new millennium. IV edited by Denjoe O’Connor, C.R. Stephens editor: I. Procaccia Contents D. O’Connor, C.R. Stephens, Renormalization group theory in the new millennium. IV J. Berges, N. Tetradis, C. Wetterich, Non-perturbative renormalization ﬂow in quantum ﬁeld theory and statistical physics

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D. Kreimer, Combinatorics of (perturbative) quantum ﬁeld theory D. O’Connor, C.R. Stephens, Renormalization group theory of crossovers

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Editorial

Renormalization group theory in the new millennium. IV 1. Introduction This volume constitutes the fourth in a series of reviews based loosely on plenary talks given at the conference “RG2000: Renormalization Group Theory at the Turn of the Millennium” held in Taxco, Mexico in January 1999. The chief purpose of the conference was to bring together a group of people who had made signi1cant contributions to RG Theory and its applications, especially those who had contributed to the development of the subject in quantum 1eld theory=particles physics and statistical mechanics=critical phenomena, i.e. the high- and low-energy regimes of RG theory. In the last half century renormalization group (RG) theory has become a central structure in theoretical physics and beyond, though it is not always clear that di5erent authors mean the same thing when they speak about it. The aim of these reviews is to try and convey some of the power and scope of RG theory and its applications and in the process hopefully convey the underlying unity of the set of ideas involved. Although RG theory has had a major impact it has tended to be viewed as a tool rather than as a subject in and of itself. Being presented principally in terms of its applications has therefore meant a lack of contact between practitioners from di5erent 1elds. An important exception to this tendency is the series of RG conferences organized by Dimitri Shirkov and others of the Joint Institute for Nuclear Research, Dubna theoretical physics community. The Taxco conference was in the same spirit. The advent in recent years of conferences on the “exact” RG has also provided an opportunity for practitioners to come together. The only criticism one might have of this latter series is the large emphasis on 1eld theory. This small criticism notwithstanding we hope that there will be continued opportunity to bring together RG practitioners from di5erent 1elds. In obtaining contributions for these reviews we did not restrict ourselves to speakers from the conference. A major concern was to avoid producing yet another typical conference proceedings. Hence, the remit given to the contributors was to write as extensively and comprehensively as they saw 1t. Naturally, with such a liberal regime the length of article varies signi1cantly. Our goal was to try and review the state of the art of RG theory given that it could now be considered mature enough to warrant a large scale overview. We believe that we were to some extent defeated in our purpose by the very size and range of applicability of the RG. Although we have managed to cover a large gamut we know there are glaring omissions. Nevertheless, we feel it is of great bene1t to have reviews by leading practitioners all brought together in the same place even if the range of coverage is suboptimal. A possible remedy to this would be for specialists in areas not adequately c 2001 Published by Elsevier Science B.V. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 9 7 - 7

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covered here to submit articles which would naturally fall into the present series. We particularly wanted to emphasize the idea that although mature enough to warrant a major review, RG theory is young enough, and more signi1cantly, deep enough, that such a review would still only barely scratch its surface. We hope that young researchers will get the feeling that it is still very much an emerging 1eld with a large number of open problems associated with the understanding of RG theory itself and an even larger number associated with its applications. 2. Introduction to the fourth volume Bagnuls and Bervillier gave a nice introduction to “exact” RG equations in the second volume of this series. The article of Berges, Tetradis and Wetterich is a thorough revue of a particular type of such equation. Based on the “average e5ective action” they review the theory and applications of this concept in both quantum 1eld theory and statistical physics. The associated RG falls, in spirit at least, into the category of 1eld-theoretic coarse graining RGs. However, in contrast to a strict Wilsonian approach, where an ultraviolet cuto5 is present in the form of a non-zero lattice spacing, in this approach an e5ective infrared cuto5 suppressing Auctuations in the infrared is introduced by modifying the propagators of the theory. An RG equation is generated by requiring that physics be independent of this cuto5 and physical results are obtained in the limit that the cuto5 is sent to zero. The advantages and disadvantages of this approach are similar to its other coarse grained cousins (see for example the article of Bagnuls and Bervillier in Vol. II of this series). It is not as wedded to perturbation theory as reparametrization RG-based methods. However, for the same reason the development of systematic approximation methods is in a more primitive state. For this reason the theory is at its best when considering more complicated objects than critical exponents and amplitude ratios, such as the equation of state and other scaling functions. The method is versatile and fairly easy to implement, though results are almost inevitably numerical. The authors, after brieAy introducing and discussing the foundations of the method, cover an impressive array of applications both in statistical and particle physics. The most interesting results are from particle physics based models, application to furthering the understanding of the QCD phase diagram being a particularly important area. In statistical physics the methodology has to compete against tried and trusted methods for calculating critical exponents and, to a lesser extent, equations of state or crossover functions. The methodology seems to be at its most useful not so much in the area of precision estimates of already known quantities but rather in illuminating the physics in problems where more traditional perturbative RG methods have problems. This is potentially of great importance in problems such as QCD where a description of the crossover between bound state and fundamental degrees of freedom still challenges us. An interesting example is the extra insight obtained from coarse graining in the context of nucleation in phase transitions. Without doubt the method will continue to be one of importance in the years to come. In the second article Kreimer presents a review of recent discoveries in the combinatorics of perturbative quantum 1eld theory. The principal and surprising discovery here is that the combinatorics of the forest formula are governed by a Hopf algebra, in particular the Hopf algebra of decorated rooted trees. The intricacies of the perturbative renormalization process lie in its iterative nature. The divergences of one loop arise as subdivergences of higher order diagrams. It is the task of

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a proof of renormalizability to establish that all these divergences can be removed by a small number of counter terms. Perhaps the reason the intricate combinatorics behind this forest formula went unrecognised for so long is that the notion of Hopf algebras is not particularly familiar to most physicists. However, now that the structure has been exposed we expect it will slowly be absorbed into the lore of perturbative 1eld theory. It would seem that this Hopf structure should also play an important role in the application of RG ideas to dynamical systems. It remains to be seen what further secrets perturbative renormalization theory still holds. In this review Kreimer goes on to show the connection of the renormalization problem with the Riemann–Hilbert problem, thus giving a nice interpretation of the -expansion renormalization prescription, establishing that the prescription can be interpreted as a di5eomorphism of the couplings of the theory. Since the Hopf algebra product is commutative and the coproduct is not the Hopf algebra is associated with a group. This is the di5eomorphism group on the space of couplings. This makes rather nice contact with the reparametrization approach. The third review, by the present authors, is devoted to the RG associated with crossover phenomena. Many physical systems exhibit qualitatively di5erent e5ective degrees of freedom at di5erent scales. In the context of critical phenomena a characteristic of such systems is the existence of more than one asymptotic scaling regime, and therefore of more than one set of critical exponents, with an associated crossover scaling function that interpolates between them. The most notable success of RG theory to date has been the description of second order phase transitions and the precise calculation of the associated critical exponents. In RG terms this description requires the calculation of the eigenvalues of the RG linearized around the RG 1xed point that characterizes the transition. In contrast, a crossover requires a global, non-linear analysis that captures the presence of more than one 1xed point, while at the same time allowing for systematic approximation procedures. This is a tall order and, as emphasized in the articles of Binder and Zinn-Justin in the 1rst volume of this series, even in the simplest cases, such as crossover in the equation of state and the crossover to mean 1eld theory, relatively little is known when contrasted with the case of universal quantities associated with the dominant 1xed point. O’Connor and Stephens give an overview of RG theory of crossover scaling, concentrating principally on 1eld-theoretic reparametrization RG methods and, in particular, one variant known as “environmentally friendly” renormalization. They show how choice of renormalized parameters can crucially a5ect the validity of calculations of crossover scaling functions and give a prescription for choosing suitable parameters in a wide class of crossovers. Focusing on the appropriate RG generalization of a 1xed point to a crossover—the separatrix curve that joins two 1xed points—taking as fundamental building blocks the Wilson functions associated with this separatrix and integrating renormalized vertex functions along contours in the phase diagram, they show how essentially any universal crossover scaling function may be systematically calculated, in principle with as much precision as a critical exponent. Of particular interest is the ability to calculate critical temperature shifts and rounding and, consequently, equations of state in a naturally parametric representation that satis1es all analyticity requirements. The general formalism sets up a calculational algorithm within which crossover scaling functions may be calculated with the same precision as critical exponents. The authors consider several illustrative examples, concentrating on 1nite size scaling in a 1lm geometry as a canonical example, calculating crossover scaling functions in a PadIe-resummed perturbation expansion up to two loops.

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Acknowledgements We take this opportunity to thank our coorganizers of the Taxco conference, Alberto Robledo, Riccardo Capovilla and Juan Carlos D’Olivo, and the conference secretaries, Trinidad Ramirez and Alejandra Garcia. We thank the conference sponsors for their signi1cant 1nancial support: CONACyT, MIexico; NSF, USA; ICTP, Italy; the Depto de FILsica, Cinvestav, MIexico; Instituto de Ciencias Nucleares, UNAM, MIexico; Instituto de FILsica, UNAM, MIexico; Fenomec, UNAM, MIexico; Cinvestav, MIexico; DGAPA, UNAM, MIexico and the CoordinaciIon de InvestigaciIon CientIL1ca, UNAM, Mexico. It is fair to say that without this generous support a conference of such caliber could not have taken place. We also take this opportunity to express our gratitude, for their advice and assistance, to the international advisory committee comprised of: A.P. Balachandran, Syracuse University, USA; K. Binder, Mainz, Germany; M.E. Fisher, University of Maryland, USA; N. Goldenfeld, University of Illinois, USA; B.L. Hu, University of Maryland, USA; D. Kazakov, Dubna, Russia; V.B. Priezzhev, Dubna, Russia; I. Procaccia, Weizmann Institute, Israel; M. Shifman, University of Minnesota, USA; D.V. Shirkov, Dubna, Russia; F. Wegner, Heidelberg, Germany; J. Zinn-Justin, Saclay, France. We express our special thanks to Michael Fisher for his cogent advice and organizational help, to Bei Lok Hu for helping organize the US component of the conference and to Itamar Procaccia for organizing an appropriate forum in which to present this overview. Denjoe O’Connor Dept. de Fisica, CINVESTAV, A. Postal 14740, 07360 Mexico DF, Mexico E-mail address: [email protected] C.R. Stephens Insituto de Ciencieas Nucleares, A. Postal 70-543, 04510 Mexico DF, Mexico E-mail address: [email protected]

Physics Reports 363 (2002) 223 – 386 www.elsevier.com/locate/physrep

Non-perturbative renormalization "ow in quantum $eld theory and statistical physics J'urgen Bergesa; ∗ , Nikolaos Tetradisb; c , Christof Wetterichd a

Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA b Scuola Normale Superiore, 56126 Pisa, Italy c Nuclear and Particle Physics Sector, University of Athens, 15771 Athens, Greece d Institut f)ur Theoretische Physik, Universit)at Heidelberg, 69120 Heidelberg, Germany Received September 2001; editor: I. Procaccia This work is dedicated to the 60th birthday of Franz Wegner

Contents 1. Introduction 1.1. From simplicity to complexity 1.2. Fluctuations and the infrared problem 2. Non-perturbative "ow equation 2.1. Average action 2.2. Exact "ow equation 2.3. Truncations 2.4. Flow equation for the average potential 2.5. A simple example: the quartic potential 3. Solving the "ow equation 3.1. Scaling form of the exact "ow equation for the potential 3.2. Threshold functions 3.3. Large-N expansion 3.4. Graphical representation and resummed perturbation theory

225 225 230 235 235 241 244 247 248 252 252 254 257 260

3.5. Exact "ow of the propagator 3.6. Approach to the convex potential for spontaneous symmetry breaking 4. O(N )-symmetric scalar models 4.1. Introduction 4.2. The running average potential 4.3. Universal critical equation of state 4.4. Gas–liquid transition and the Ising universality class 4.5. Universal and non-universal critical properties 4.6. Equation of state for $rst-order transitions 4.7. Critical behavior of polymer chains 4.8. Two-dimensional models and the Kosterlitz–Thouless transition

263 266 270 270 271 276 281 286 292 296 297

∗

Corresponding author. Tel.: +1-617-253-6268; fax: +1-617-253-8674. E-mail addresses: [email protected] (J. Berges), [email protected] uni-heidelberg.de (C. Wetterich).

(N.

c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 9 8 - 9

Tetradis),

c.wetterich@thphys.

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5. Scalar matrix models 5.1. Introduction 5.2. Scalar matrix model with U (2) × U (2) symmetry 5.3. Scale dependence of the eEective average potential 5.4. Renormalization group "ow of couplings 5.5. Phase structure of the U (2) × U (2) model 5.6. Universal equation of state for weak $rst-order phase transitions 5.7. Summary 6. Spontaneous nucleation and coarse graining 6.1. Introduction 6.2. Calculation of the nucleation rate 6.3. Region of validity of homogeneous nucleation theory 6.4. Radiatively induced $rst-order phase transitions 6.5. Testing the approach through numerical simulations

303 303 305 307 312 316 321 324 326 326 330 335 339 343

7. Quantum statistics for fermions and bosons 7.1. Quantum universality 7.2. Exact "ow equation for fermions 7.3. Thermal equilibrium and dimensional reduction 7.4. The high-temperature phase transition for the 4 quantum $eld theory 8. Fermionic models 8.1. Introduction 8.2. Linear quark meson model 8.3. Flow equations and infrared stability 8.4. High-temperature chiral phase transition 8.5. Renormalization "ow at non-zero chemical potential 8.6. High-density chiral phase transition Acknowledgements Appendix A. Threshold functions Appendix B. Anomalous dimension in the sharp cutoE limit References

345 345 347 350 351 358 358 359 362 365 369 372 374 374 376 379

Abstract We review the use of an exact renormalization group equation in quantum $eld theory and statistical physics. It describes the dependence of the free energy on an infrared cutoE for the quantum or thermal "uctuations. Non-perturbative solutions follow from approximations to the general form of the coarse-grained free energy or eEective average action. They interpolate between the microphysical laws and the complex macroscopic phenomena. Our approach yields a simple uni$ed description for O(N )-symmetric scalar models in two, three or four dimensions, covering in particular the critical phenomena for the second-order phase transitions, including the Kosterlitz–Thouless transition and the critical behavior of polymer chains. We compute the aspects of the critical equation of state which are universal for a large variety of physical systems and establish a direct connection between microphysical and critical quantities for a liquid–gas transition. Universal features of $rst-order phase transitions are studied in the context of scalar matrix models. We show that the quantitative treatment of coarse graining is essential for a detailed estimate of the nucleation rate. We discuss quantum statistics in thermal equilibrium or thermal quantum $eld theory with fermions and bosons and we describe the high-temperature symmetry restoration in quantum $eld theories with spontaneous symmetry breaking. In particular, we explore chiral symmetry breaking and the high-temperature or high-density chiral phase transition c 2002 Elsevier Science in quantum chromodynamics using models with eEective four-fermion interactions. B.V. All rights reserved. PACS: 11.10.Gh; 05.10.Cc

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1. Introduction 1.1. From simplicity to complexity A few fundamental microscopic interactions govern the complexity of the world around us. The standard model of electroweak and strong interactions combined with gravity is a triumph for the way of uni$cation and abstraction. Even though some insuIciencies become apparent in astrophysical and cosmological observations—the oscillations of neutrinos, the missing dark matter and the need for cosmological in"ation and baryogenesis—there is little doubt that no further basic interactions are needed for an understanding of “everyday physics”. For most phenomena the relevant basic interactions are even reduced to electromagnetism and gravity. Still, for many common observations there is a long way to go before quantitative computations and predictions from the microscopic laws become feasible. How to go back from the simplicity of microphysics to the complexity of macrophysics? We will deal here only with very simple systems of many particles, like pure water or vapor, where the interactions between molecules are reasonably well understood. We also concentrate on the most simple situations like thermal equilibrium. Concerning dynamics, we only touch on properties that can be calculated in equilibrium, whereas we omit the complicated questions of the time evolution of statistical systems. Nevertheless, it remains a hard task to compute quantitatively such simple things as the phase transition from water to vapor, starting from the well-known microscopic interactions. How can we calculate the pressure dependence of the transition temperature from atomic physics or the van der Waals interactions between molecules? How do the optical properties change as we approach the endpoint of the critical line? What would be the rate of formation of vapor bubbles if we heat extremely pure water in space at a given temperature T slightly above the critical temperature Tc ? Similar questions may be asked about the temperature dependence of the density of super"uid He4 or the magnetization in ferromagnets. One often takes a higher level of abstraction and asks for the properties of simpli$ed theoretical models, like the two-dimensional Hubbard model, which is widely believed to describe high Tc -superconductivity, or the Heisenberg model for the description of ferromagnetism or antiferromagnetism. Despite the considerable simpli$cations of these models compared to real physics they remain very diIcult to solve. Common to all these questions is the role of "uctuations in statistical many-body systems. Statistical physics and thermodynamics oEer a powerful framework for the macroscopic behavior of systems with a very large number of degrees of freedom. The statistical treatment makes the predictions about the behavior of stationary many-body systems independent of many irrelevant details of the microscopic dynamics. Nevertheless, one needs to bridge the gap between the known microscopic interactions and the thermodynamic potentials and similar quantities which embody the eEective macroscopic laws. This is the way to complexity. For a thermodynamic equilibrium system of many identical microscopic degrees of freedom the origin of the problems on the way to complexity is threefold. First, often there is no small parameter that can be used for a systematic perturbative expansion. Second, the correlation length can be substantially larger than the characteristic distance between the microscopic objects. Collective eEects become important. Finally, the relevant degrees of freedom which permit a simple formulation of the macroscopic laws may be diEerent from the microscopic ones. A universal theoretical method for the transition from micro- to macrophysics should be able to cope with these generic features.

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We propose here a version of non-perturbative "ow equations based on an exact renormalization group equation. This theoretical tool acts like a microscope with variable resolution. The way from a high-resolution picture of an eEectively small piece of surface or volume (microphysics) to a rough resolution for large volumes (macrophysics) is done stepwise, where every new step in the resolution only uses information from the previous step [1–7]. We use here a formulation in terms of the eEective average action [8,9], which permits non-perturbative approximations for practical computations. Our method may be considered as an analytical counterpart of the often successful numerical simulation techniques. Modern particle physics is confronted with precisely the same problems on the way from the beautiful simplicity of fundamental interactions to a “macroscopic” description. Basically, only the relevant length scales are diEerent. Whereas statistical physics may have to translate from Angstr'oms to micrometers, particle physics must build a bridge from the Fermi scale (∼ (100 GeV)−1 ) to a fermi (1 fm =10−15 m=(197:33 MeV)−1 ). For interactions with small couplings, particle physics has mastered this transition as far as the vacuum properties are concerned—this includes the dynamics of the excitations, e.g. the particles. The perturbative renormalization group [10] interpolates between the Fermi scale and the electron mass 1 or even between a grand uni$cation scale ∼ 1016 GeV and the Fermi scale. The dynamics of electrons, positrons and photons in vacuum can be predicted with extremely high accuracy. This extends to weak interactions between leptons. The situation is very diEerent for strong interactions. The running gauge coupling grows large at a scale below 1 GeV and the generic problems of statistical physics reappear. Whereas quantum electrodynamics may be compared with a dilute gas of weakly interacting molecules, strong interactions are analogous to dense systems with a large correlation length. At microscopic distances, e.g. the Fermi scale, quantum chromodynamics (QCD) can give precise predictions in terms of only one gauge coupling and the particle masses. At the “macroscopic scale” of around 1 fm numerical simulations approach only slowly the computation of the masses and interactions of the relevant degrees of freedom, namely baryons and mesons, and no analytical method has achieved this goal yet. The smallness of microscopic couplings is no guarantee for a simple transition to “macrophysics”. The vacuum "uctuations may enhance considerably the relevant eEective coupling at longer distances, as in the case of QCD. In the presence of thermal "uctuations a similar phenomenon happens even for the electroweak interactions. In fact, at high-temperature the electroweak interactions have a large eEective gauge coupling and show all properties of a strongly interacting model, very similar to QCD [11,12]. The evidence for this behavior is striking if one looks at the recently computed spectrum of quasiparticles for the standard model at high-temperature [13]. In the language of statistical physics the hot plasma in thermal equilibrium is dense at suIciently high-temperature, with a correlation length (typically given by the inverse magnetic mass of the gauge bosons) substantially larger than the inverse temperature. The high-temperature behavior of strong or electroweak interactions has attracted much interest recently. It is relevant for the hot plasma in the very early universe. Possible phase transitions may even have left observable “relics” behind that could serve as observational tests of cosmology before nucleosynthesis. From a statistical physics viewpoint the particle physics systems are extremely 1

We employ the word scale for distance as well as momentum or mass scales. In our units ˝ = c = kB = 1 they are simply the inverse of each other.

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pure—no dirt, dotation with other atoms, impurities, seeds of nucleation, gravitational eEects etc. need to be considered. In principle, tests of particle physics in thermodynamic equilibrium would be ideal experiments for statistical physics. 2 Unfortunately, these “ideal experiments” are diIcult to perform—it is not easy to prepare a high-temperature plasma of particles in equilibrium in a laboratory. Nevertheless, an impressive experimental program is already under way, aiming at a test of high-temperature and high-density QCD and possible phase transitions [14]. This highlights the need of a theoretical understanding of the QCD phase diagram at high temperature and density, including such interesting issues as color superconductivity and the possibility of (multi-) critical points with observable eEects from long-range correlations [15 –17]. Renormalization group methods should be an important tool in this attempt. From a theoretical point of view there is no diEerence between thermal quantum $eld theory and quantum–statistical systems. In a modern language, both are formulated as Euclidean functional integrals with “Euclidean time” wrapped around a torus with circumference T −1 . (For dynamical questions beyond the equilibrium properties the time coordinate has to be analytically continued to real Minkowski time.) The only special features of the particle physics systems are the very precisely known microscopic interactions with their high degree of symmetry in the limit of vanishing temperature. This concerns, in particular, the Lorentz symmetry or its Euclidean counterpart of four-dimensional rotations and Osterwalder–Schrader positivity [18]. In this line of thought the recent high-precision numerical simulations of the high-temperature electroweak interactions [19,20] can be considered as a $ne example of a quantitative (theoretical) transition from microphysics to macrophysics, despite the presence of strong eEective interactions and a large correlation length. They have con$rmed that the high-temperature $rst-order phase transition which would be present in the standard model with modi$ed masses turns into a crossover for realistic masses, as has been suggested earlier by analytical methods [11,12,21,22]. Beyond the identical conceptual setting of particle and statistical physics there is also quantitative agreement for certain questions. The critical exponents at the endpoint of the critical line of the electroweak phase transition (the onset of crossover) are believed to be precisely the same as for the liquid–gas transition near the critical pressure or for magnetic transitions in the Ising universality class. 3 This reveals universality as a powerful feature for the transition to complexity. Indeed, the transition to macrophysics does not involve only complications. For certain questions it can also bring enormous simpli$cations. Because of partial $xed points in the renormalization "ow of eEective couplings between short and long distances much of the microscopic details can be lost. If the "uctuation eEects are strong enough, the long-distance behavior loses memory of the microscopic details of the model. This is the reason why certain features of high-temperature QCD may be testable in magnets or similar statistical systems. For example, it is possible that the temperature and density dependence of the chiral condensate in QCD can be approximated in a certain range by the critical equation of state of the O(4) Heisenberg model [24 –28] or by the Ising model at a non-zero density critical endpoint [29,30]. Exact renormalization group equations describe the scale dependence of some type of “eEective action”. In this context an eEective action is a functional of $elds from which the physical properties at a given length or momentum scale can be computed. The exact equations can be derived as formal 2 3

The microwave background radiation provides so far the most precise test of the spectrum of black-body radiation. Numerical simulations [23] are consistent with this picture.

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identities from the functional integral which de$nes the theory. They are cast in the form of functional diEerential equations. DiEerent versions of exact renormalization group equations have already a long history [1–7]. The investigation of the generic features of their solutions has led to a deep insight into the nature of renormalizability. In particle physics, the discussion of perturbative solutions has led to a new proof of perturbative renormalizability of the 4 -theory [6]. Nevertheless, the application of exact renormalization group methods to non-perturbative situations has been hindered for a long time by the complexity of functional diEerential equations. The $rst considerable progress for the description of critical phenomena has been achieved using the scaling-$eld method [31]. In this approach the exact renormalization group equation is transformed into an in$nite hierarchy of non-linear ordinary diEerential equations for scaling $elds [32]. Estimates for non-trivial critical exponents and scaling functions, e.g. for three-dimensional scalar O(N )-models, are obtained from a truncated expansion around the trivial (Gaussian) $xed point and certain balance assumptions constraining what operators to include in the approximation [31]. Another very fruitful approach is based on evaluating the eEective action functional for constant $elds and neglecting all non-trivial momentum dependencies. This so-called local potential approximation, originally considered in [33], was $rst employed in [34,7] in order to compute critical exponents for three-dimensional scalar O(N )-models. Unfortunately, the formulation used in that work could not be used for a systematic inclusion of the neglected momentum dependencies. Some type of expansion is needed if one wants to exploit the exactness of the functional diEerential equation in practice—otherwise any reasonable guess of a realistic renormalization group equation does as well. Since exact solutions to functional diEerential equations seem only possible for some limiting cases, 4 it is crucial to $nd a formulation which permits non-perturbative approximations. These proceed by a truncation of the most general form of the eEective action and therefore need a qualitative understanding of its properties. The formulation of an exact renormalization group equation based on the eEective average action [8,9] has been proven successful in this respect. It is the basis of the non-perturbative "ow equations which we discuss in this review. The eEective average action is the generating functional of one-particle irreducible correlation functions in the presence of an infrared cutoE scale k. Only "uctuations with momenta larger than k are included in its computation. For k → 0 all "uctuations are included and one obtains the usual eEective action from which appropriate masses and vertices can be read oE directly. The k dependence is described by an exact renormalization group equation that closely resembles a renormalization group improved one-loop equation [9]. In fact, the transition from classical propagators and vertices to eEective propagators and vertices transforms the one-loop expression into an exact result. This close connection to perturbation theory for which we have intuitive understanding is an important key for devising non-perturbative approximations. Furthermore, the one-loop expression is manifestly infrared and ultraviolet $nite and can be used directly in arbitrary number of dimensions, even in the presence of massless modes. The aim of this report is to show that this version of "ow equations can be used in practice for the transition from microphysics to macrophysics, even in the presence of strong couplings and a large correlation length. We derive the exact renormalization group equation and various non-perturbative truncations in Section 2. In particular, we demonstrate in Section 2.5 that already an extremely simple truncation gives a uni$ed picture of the phase transitions in O(N )-symmetric scalar theories in arbitrary dimensions d, including the critical exponents for d = 3 and the Kosterlitz–Thouless 4

An example is the O(N )-model for N → ∞ discussed in Section 3.3.

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phase transition for d = 2, N = 2. In Section 3 we discuss the solutions to the "ow equation in more detail. We present an exact solution for the limit N → ∞. We also propose a renormalization group improved perturbation theory as an iterative solution. We show how the eEective potential becomes convex in the limit k → 0 in the case of spontaneous symmetry breaking when the perturbative potential is non-convex. Section 4 discusses the universality class of O(N )-symmetric scalar models in more detail. We derive the universal critical equation of state. For the special example of carbon dioxide, we connect explicitly the microphysics with the macrophysics. In addition to the universal part this also yields the non-universal critical amplitudes in the vicinity of the second-order phase transition at the endpoint of the critical line. After a short discussion of the critical behavior of polymer chains we turn to the Kosterlitz–Thouless phase transition for two-dimensional models with a continuous Abelian symmetry. First-order phase transitions are discussed in Sections 4.6 and 5. The matrix model investigated in Section 5 gives an example of a radiatively induced $rst-order transition, such as the one appearing in the Abelian Higgs model that is relevant for low Tc superconductivity. We discuss under which conditions $rst-order transitions are characterized by universal behavior. In particular, we present a universal critical equation of state for $rst-order transitions which involves two scaling variables and we discuss its range of applicability. In Section 6 we turn to the old problem of spontaneous nucleation in $rst-order transitions. We show that a detailed understanding of coarse graining is crucial for a quantitative computation of the nucleation rate. We also discuss the limits of validity of spontaneous nucleation theory, in particular for radiatively induced $rst-order transitions. Our results agree well with recent numerical simulations. Section 7 is devoted to quantum statistics and quantum $eld theory in thermal equilibrium. The "ow equation generalizes easily to the Matsubara formalism and one sees that dimensional reduction arises as a natural consequence. Whereas for momenta larger than the temperature the (four-dimensional) quantum statistics is relevant, the momenta below T are governed by classical (three-dimensional) statistics. Correspondingly, the "ow changes from a four-dimensional to a three-dimensional "ow as k crosses the temperature. This section also contains the "ow equation for fermions. We show how the renormalization "ow leads to a consistent picture of a second-order phase transition for the high-temperature 4 -quantum $eld theory and introduce the notion of quantum universality. Finally, Section 8 deals with the chiral phase transition in QCD. We discuss both the high-temperature and the high-density chiral phase transition within eEective fermionic models with multi-quark interactions. In particular, we relate the universal critical behavior at the high-temperature phase transition or crossover to particle masses and decay constants in the vacuum for QCD with two light quarks. After reading Section 2 all sections are essentially self-contained, with the exception of Section 8 relying on results from Section 7. Section 3 contains some more advanced topics that are not mandatory for a $rst understanding of the concrete models in Sections 4 –8. In the second part of the introduction, we brie"y review the basics of the "uctuation problem in statistical physics and quantum $eld theory. This section may be skipped by the experienced reader. Several results in statistical and particle physics have been obtained $rst with the method presented here. This includes the universal critical equation of state for spontaneous breaking of a continuous symmetry in Heisenberg models [36], the universal critical equation of state for $rst-order phase transitions in matrix models [37], the non-universal critical amplitudes with an explicit connection of the critical behavior to microphysics (CO2 ) [38], a quantitatively reliable estimate of the rate of spontaneous nucleation [39,40], a classi$cation of all possible $xed points for (one component) scalar

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theories in two and three dimensions in case of a weak momentum dependence of the interactions [41], the second-order phase transition in the high-temperature 4 quantum $eld theory [42], the phase diagram for the Abelian Higgs model for N charged scalar $elds [43,44], the prediction that the electroweak interactions become strong at high-temperature, with the suggestion that the standard model may show a crossover instead of a phase transition [11,12], in strong interaction physics the interpolation between the low-temperature chiral perturbation theory results and the high-temperature critical behavior for two light quarks in an eEective model [28]. All these results are in the non-perturbative domain. In addition, the approach has been tested successfully through a comparison with known high-precision results on critical exponents and universal critical amplitude ratios and eEective couplings. Our main conclusion can be drawn already at this place: the method works in practice. If needed and with suIcient eEort high-precision results can be obtained in many non-perturbative situations. New phenomena become accessible to a reliable analytical treatment. We do not attempt to give an overview over all relevant results. Rather, we concentrate on a systematic development which should enable the reader to employ the method by himself. For an extensive review on work for scalar $eld theories and a comprehensive reference list we refer the reader to Bagnuls and Bervillier [45]. For a review including the basis and origins of renormalization group ideas in statistical physics and condensed matter theory we refer to Fisher [46]. We have omitted two important issues: the formulation of the "ow equations for gauge theories [47,11], [48–74] and for composite operators [76]. The latter is important in order to achieve a change of eEective degrees of freedom during the "ow. 1.2. Fluctuations and the infrared problem (This introductory subsection may be skipped by experienced readers.) The basic object in statistical physics is the (canonical) partition function Z = Tr e−H

(1.1)

with H the Hamiltonian and = T −1 . The trace involves an integration over all microscopic degrees of freedom. For classical statistics it typically stands for a (generalized) phase space integral and the classical Hamiltonian is simply a function of the integration variables. In most circumstances it can be decomposed as H =H1 +H2 with H2 quadratic in some momentum-type variable on which H1 does not depend. The Gaussian integration over the momentum-type variable is then trivial and usually omitted. We will be concerned with many-body systems where the remaining degrees of freedom a (x) can be associated with points x in space. For simplicity, we discuss in this introduction only a single real variable (x). The partition function can then be written in the form of a “functional integral” Z = D e−S[] ; (1.2) where S = H1 . If the space points x are on some discrete lattice with $nite volume, the functional measure is simply the product of integrations at every point D ≡ d(x) : (1.3) x

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This can be generalized to the limits of continuous space (when the lattice distance goes to zero) and of in$nite volume. In this review we concentrate on continuous space with the appropriate translation and rotation symmetries. (Our formalism is, however, not restricted to this case.) As a typical example one may associate (x) with a density $eld n(x) by (x) = a + bn(x) and consider a Hamiltonian containing local and gradient interactions 5 m2 2 1 4 3 ∇(x)∇(x) + (x) + (x) : S= d x (1.4) 2 2 8 The mean values or expectation values can be computed as weighted integrals, e.g. −1 D (x)(y)e−S[] : (x)(y) = Z

(1.5)

A few generalizations are straightforward: In the presence of a chemical potential for some conserved quantity N we will consider the grand canonical partition function Z = Tr e−(H −N )

(1.6)

The -dependent part can either be included in the de$nition of S = (H1 − N ) or, if linear in , be treated as a source (see below and Section 2.1). For quantum statistics H is an operator acting on the microphysical states. Nevertheless, the partition function can again be written as a functional integral, now in four dimensions. We will discuss this case in Section 7. Furthermore, the relevant physics may only involve degrees of freedom on a surface, a line or a single point. Classical statistics is then given by a D-dimensional functional integral and quantum statistics by a (D + 1)-dimensional functional integral, with D = 2; 1; 0, respectively. Particle physics can be derived from a four-dimensional functional integral, the Feynman path integral in Euclidean space. Except for the dimensionality, we therefore can treat particle physics and classical statistical physics on the same footing. There is no diEerence 6 between the formulation of particle physics—i.e. quantum $eld theory—and the theory of many-body quantum–statistical systems, besides the symmetries particular to particle physics. The thermodynamic potential W [J ] = ln Z[J ]

(1.7)

is related to an extension of the partition function in the presence of arbitrary inhomogeneous “sources” or “external $elds” J (x) that multiply a term linear in : d Z[J ] = D exp −S[] + d x (x)J (x) : (1.8)

5

The constants a; b can be chosen such that there is no cubic term ∼ 3 (x) and the gradient term has standard normalization. A linear term will be included below as a “source” term. 6 In the modern view, quantum $eld theory is not considered to be valid up to arbitrarily short distances. Similarly to statistical physics, a given model should be considered as an eEective description for momenta below some ultraviolet cutoE . This cutoE typically appears in all momentum integrals. For the standard model, it indicates the onset of new physics like grand uni$cation or uni$cation with gravity.

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W and Z are functionals of J (x). The (functional) derivatives of W with respect to J (x) generate the connected correlation functions, e.g. the average density or the density–density correlation W = (x) ≡ (x) ; J (x) 2 W = (x)(y) − (x)(y) : J (x)J (y)

(1.9) (1.10)

Here the mean values depend on the sources, e.g. ≡ [J ]. The eEective action [] is another thermodynamic potential, related to W [J ] by a Legendre transform [] = −W [J ] + d d x(x)J (x) (1.11) with J [] obtained by the inversion of [J ] from Eq. (1.9). [] is easier to compute than W [J ] and the physical observables can be extracted very simply from it (see Section 2.1). The eEective action can be written as an implicit functional integral in the presence of a “background $eld” d (1.12) exp{−[]} = D exp −S[ + ] + d x (x) (x) : A perturbative expansion treats the "uctuations around the background in a saddle-point approximation. In lowest order (tree approximation) one has (0) [] = S[]. For the one-loop order we expand 1 d S d d x d dy (x)S (2) (x; y) (y) + · · · (1.13) S[ + ] = S[] + d x (x) (x) + 2 with S (2) (x; y) =

2 S : (x)(y)

(1.14)

The linear terms cancel in this order and one $nds from the Gaussian integral [] = S[] + 12 Tr ln S (2) [] + · · · :

(1.15)

For constant this yields the one-loop eEective potential U0 = =Vd U0 () = 12 m2 2 + 18 4 + U0(1) () + · · · ; dd q 1 3 2 2 2 U0(1) () = (1.16) ln q + m + 2 (2")d 2 with Vd = d d x. A typical Landau theory results from the assumption that the "uctuation eEects induce a change of the “couplings” m2 and (“renormalization”) without a modi$cation of the quartic form of the potential.

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Let us consider classical statistics (d = 3) and perform the momentum integration with some ultraviolet cutoE 7 q2 6 2 3 1 1 (m2 + 3#)3=2 ; # = 2 : U0(1) (#) = #− (1.17) 2 4" 12" 2 Here we have neglected corrections that are suppressed by powers of (m2 + 3#)=2 . De$ning renormalized couplings 8 m2R =

9U0 (0); 9#

R =

92 U0 (0) ; 9#2

(1.18)

one obtains (m2 ¿ 0) 3 3 m; − m2R = m2 + 4"2 8" 92 R = − : (1.19) 16"m As expected, the corrections are large for large couplings . In addition, the correction to diverges 1 as m → 0. Since the correlation length % is given by m− R , we see here the basic reason for which the transition from microphysics to macrophysics becomes diIcult for large couplings and large correlation length. Because of the linear “ultraviolet divergence” in the mass renormalization 9 ∼ 3=4"2 , a large 1 correlation length m− actually requires a negative value of m2 . On the other hand, we note that R the saddle point expansion is valid only for m2 + 3# ¿ 0 and breaks down at # = 0 if m2 becomes negative. The situation can be improved by using in the one-loop expression (1.17) the renormalized parameters mR and R instead of m and . For the second term in Eq. (1.17) the justi$cation 10 for the “renormalization group improvement” arises from the observation that the momentum integral is dominated by momenta q2 ≈ m2 . Through an iterative procedure corresponding to the inclusion of higher loops, the physical infrared cutoE should be replaced by mR . We will see later in more detail how this renormalization improvement arises through the formulation of "ow equations. Writing U0 in terms of renormalized parameters 1 3R mR 9R2 1 2 #+ R + (m2 + 3R #)3=2 #2 − (1.20) U0 = mR + 8" 2 16"mR 12" R we can compute the deviations from the Landau theory, i.e. 27R3 9 3 U0 (0) = : 9#3 32"m3R

(1.21)

We can also formulate a criterion for the validity of the renormalization group improved saddle-point approximation, namely that the one-loop contributions to mR and R should not dominate these 7 On a cubic lattice, would be related to the lattice distance a by = "=a and the inverse propagator is obtained through the replacement q2 → (2=a2 ) (1 − cos aq ). 8 We omit here for simplicity the wave function renormalization. 9 For $nite there is of course no divergence. We employ here the language of particle physics which refers to the limit → ∞. 10 For the correction to m2 ∼ this replacement is not justi$ed.

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couplings. This yields R 16" : ¡ mR 9

(1.22)

The renormalized coupling R is not independent of mR . If we take the renormalization group improvement literally, we could solve the relation (cf. Eq. (1.19)) = R +

9R2 16"mR

for $xed and $nd

8" 9 R = 1+ −1 : mR 9 4"mR

(1.23)

(1.24)

We see that for an arbitrary positive condition (1.22) breaks down in the limit of in$nite correlation length mR → 0. For a second-order phase transition this is exactly what happens near the critical temperature, and we encounter here the infrared problem for critical phenomena. One concludes that "uctuation eEects beyond the Landau theory become important for 3 : (1.25) 4" If one is close (but not too close) to the phase transition, a linear approximation mR = A(T − Tc ) remains a good guide. This provides a typical temperature interval around the critical temperature for which the Landau theory fails, namely mR .

|T − Tc | 3 . : Tc 4"ATc

(1.26)

The width of the interval depends on the dimensionless quantities A and =Tc . Inside the interval (1.26) the physics is governed by the universal critical behavior. In fact, we should not trust relation (1.23) for values of mR for which (1.22) is violated. The correct behavior of R (mR ) will be given by the renormalization group and leads to a $xed point for the ratio limmR →0 (R =mR ) → const. As an example of universal behavior we observe that the 6 -coupling (1.21) is completely determined by the ratio R =mR , independently of the value of the microphysical coupling . This is generalized to a large class of microscopic potentials. We see how universality is equivalent to the “loss of memory” of details of the microphysics. Similar features in four dimensions are the basis for the impressive predictive power of particle physics. In this report we present a version of the renormalization group equation where an infrared cutoE k is introduced for the momentum integral (1.16). In a rough version, we use a sharp cutoE k 2 ¡ q2 ¡ 2 in order to de$ne the scale-dependent potential Uk [8,9]. The dependence of the “average potential” Uk on k is simply computed (d = 3) as 2 k2 k + V + 2#V 9k Uk (#) = − 2 ln ; (1.27) 4" k2 where we have introduced the classical potential V = m2 # + 12 #2

(1.28)

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235

and subtracted an irrelevant #-independent constant. (Primes denote derivatives with respect to #.) The renormalization group improvement 11 replaces V (#) by Uk (#) and therefore leads to the non-linear partial diEerential equation k2 Uk (#) 2#Uk (#) : (1.29) 9k Uk (#) = − 2 ln 1 + + 4" k2 k2 This simple equation already describes correctly the qualitative behavior of U0 = limk →0 Uk . We will encounter it later (cf. Eq. (3.30)) as an approximation to the exact renormalization group equation. The present report motivates this equation and provides a formalism for computing corrections to it. 12 We also generalize this equation to other forms of the infrared cutoE. In Section 2.5 we show how this type of equation, even in the simple quartic approximation for the potential, gives a uni$ed picture of the phase transitions for O(N )-symmetric scalar theories in arbitrary 13 dimensions. 2. Non-perturbative ow equation 2.1. Average action We will concentrate on a "ow equation which describes the scale dependence of the eEective average action k [9]. The latter is based on the quantum $eld theoretical concept of the eEective action , i.e. the generating functional of the Euclidean one-particle irreducible (1PI) correlation functions or proper vertices (cf. Eq. (1.11)). This functional is obtained after “integrating out” the quantum "uctuations. The scattering amplitudes and cross sections follow directly from an analytic continuation of the 1PI correlation functions in a standard way. Furthermore, the $eld equations derived from the e:ective action are exact as all quantum eEects are included. In thermal and chemical equilibrium includes in addition the thermal "uctuations and depends on the temperature T and chemical potential . In statistical physics is related to the free energy as a functional of some space-dependent order parameter (x). For vanishing external $elds the equilibrium state is given by the minimum of . More generally, in the presence of (spatially varying) external $elds or sources the equilibrium state obeys = J (x) (x) and the precise relation to the thermodynamic potentials like the free energy F reads F = Teq + N − T d x eq (x)J (x) :

(2.1)

(2.2)

Here eq (x) solves (2.1), eq = [eq ], and N is the conserved quantity to which the chemical potential is associated. For homogeneous J = j=T the equilibrium value of the order parameter is 11

For a detailed justi$cation see Refs. [8,9]. Eq. (1.29) can also be obtained as the sharp cutoE limit of the Polchinski equation [6] and was discussed in Ref. [7]. However, in this approach it is diIcult to include the wave function renormalization and to use Eq. (1.29) as a starting point of a systematic procedure. 13 For the Kosterlitz–Thouless phase transition for d = 2; N = 2 the wave function renormalization must be included. 12

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often also homogeneous. In this case the energy density j, entropy density s, “particle density” n and pressure p can be simply expressed in terms of the eEective potential U () = T=V , namely j=U −T n=−

9U ; 9

9U 9U − ; 9T 9

s=−

9U j + ; 9T T

p = −U = −T=V :

(2.3)

Here U has to be evaluated for the solution of 9U=9 = j, n = N=V and V is the total volume of (three-dimensional) space. Evaluating U for arbitrary yields the equation of state in the presence of homogeneous magnetic $elds or other appropriate sources. 14 More formally, the eEective action follows from a Legendre transform of the logarithm of the partition function in the presence of external sources or $elds (see below). Knowledge of is in a sense equivalent to the “solution” of a theory. Therefore, is the macroscopic quantity on which we will concentrate. In particular, the eEective potential U contains already a large part of the macroscopic information relevant for homogeneous states. We emphasize that the concept of the eEective potential is valid universally for classical statistics and quantum statistics, or quantum $eld theory in thermal equilibrium or the vacuum. 15 The average action k is a simple generalization of the eEective action, with the distinction that only "uctuations with momenta q2 & k 2 are included. This is achieved by implementing an infrared (IR) cutoE ∼ k in the functional integral that de$nes the eEective action . In the language of statistical physics, k is a type of coarse-grained free energy with a coarse graining length scale ∼ k −1 . As long as k remains large enough, the possible complicated eEects of coherent long-distance "uctuations play no role and k is close to the microscopic action. Lowering k results in a successive inclusion of "uctuations with momenta q2 & k 2 and therefore permits to explore the theory on larger and larger length scales. The average action k can be viewed as the eEective action for averages of $elds over a volume with size k −d [8] and is similar in spirit to the action for block-spins on the sites of a coarse lattice. By de$nition, the average action equals the standard eEective action for k = 0, i.e. 0 = , as the IR cutoE is absent in this limit and all "uctuations are included. On the other hand, in a model with a physical ultraviolet (UV) cutoE we can associate with the microscopic or classical action S. No "uctuations with momenta below are eEectively included if the IR cutoE equals the UV cutoE. Thus the average action k has the important property that it interpolates between the classical action S and the eEective action as k is lowered from the ultraviolet cutoE to zero: ≈ S;

14

lim k = :

k →0

(2.4)

For the special case where (x) corresponds to the density of a conserved quantity and j = one has F = Teq . The thermodynamic appropriate for this case are speci$ed in Section 4.4. Our notation to classical relations is adapted statistics where d x ≡ d 3 x. For quantum statistics or quantum $eld theory one has to use d x ≡ d 4 x where the “Euclidean time” is a torus with circumference 1=T . Relations (2.3) remain valid for a homogeneous source J = j. 15 The only diEerence concerns the evaluation of the partition function Z or W = ln Z = −(F − N )=T . For classical statistics it involves a D-dimensional functional integral, whereas for quantum statistics the dimension in the Matsubara formalism is D + 1. The vacuum in quantum $eld theory corresponds to T → 0, with V=T the volume of Euclidean “spacetime”.

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237

The ability to follow the evolution to k → 0 is equivalent to the ability to solve the theory. Most importantly, the dependence of the average action on the scale k is described by an exact non-perturbative "ow equation presented in the next subsection. Let us consider the construction of k for a simple model with real scalar $elds a ; a = 1; : : : ; N , in d Euclidean dimensions with classical action S. We start with the path integral representation of the generating functional for the connected correlation functions in the presence of an IR cutoE. It is given by the logarithm of the (grand) canonical partition function in the presence of inhomogeneous external $elds or sources Ja d a Wk [J ] = ln Z[J ] = ln D exp −S[] − TSk [] + d xJa (x) (x) : (2.5) −S In classical statistical physics S is related to the Hamiltonian H by S = H=T , so that e is the usual Boltzmann factor. The functional integration D stands for the sum over all microscopic states. In turn, the $eld a (x) can represent a large variety of physical objects like a (mass-) density $eld (N = 1), a local magnetization (N = 3) or a charged order parameter (N = 2). The only modi$cation compared to the construction of the standard eEective action is the addition of an IR cutoE term TSk [] which is quadratic in the $elds and reads in momentum space (a (−q) ≡ a∗ (q)) dd q 1 TSk [] = Rk (q)a (−q)a (q) : (2.6) 2 (2")d

Here the IR cutoE function Rk is required to vanish for k → 0 and to diverge for k → ∞ (or k → ) and $xed q2 . This can be achieved, for example, by the exponential form Rk (q) ∼

q2 : e −1 q2 =k 2

(2.7)

For "uctuations with small momenta q2 k 2 this cutoE behaves as Rk (q) ∼ k 2 and allows for a simple interpretation: Since TSk [] is quadratic in the $elds, all Fourier modes of with momenta smaller than k acquire an eEective mass ∼ k. This additional mass term acts as an eEective IR cutoE for the low momentum modes. In contrast, for q2 k 2 the function Rk (q) vanishes so that the functional integration of the high momentum modes is not disturbed. The term TSk [] added to the classical action is the main ingredient for the construction of an eEective action that includes all "uctuations with momenta q2 & k 2 , whereas "uctuations with q2 . k 2 are suppressed. The expectation value of , i.e. the macroscopic $eld , in the presence of TSk [] and J reads Wk [J ] : (2.8) a (x) ≡ a (x) = Ja (x) We note that the relation between and J is k-dependent, = k [J ] and therefore J = Jk []. In terms of Wk the average action is de$ned via a modi$ed Legendre transform k [] = −Wk [J ] + d d xJa (x)a (x) − TSk [] ; (2.9) where we have subtracted the term TSk [] in the r.h.s. This subtraction of the IR cutoE term as a function of the macroscopic $eld is crucial for the de$nition of a reasonable coarse-grained free energy with the property ≈ S. It guarantees that the only diEerence between k and is the eEective infrared cutoE in the "uctuations. Furthermore, it has the consequence that k does not need

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to be convex, whereas a pure Legendre transform is always convex by de$nition. The coarse-grained free energy has to become convex [77,78] only for k → 0. These considerations are important for an understanding of spontaneous symmetry breaking and, in particular, for a discussion of nucleation in a $rst-order phase transition. In order to establish the property ≈ S we consider an integral equation for k that is equivalent to (2.9). In an obvious matrix notation, where J ≡ d d xJa (x)a (x) = (d d p=(2")d )Ja (−p)a (p) and Rk; ab (q; q ) = Rk (q)ab (2")d (q − q ), we represent (2.5) as 1 exp(Wk [J ]) = D exp −S[] + J − Rk : (2.10) 2 As usual, we can invert the Legendre transform (2.9) to express J=

k + Rk :

(2.11)

It is now straightforward to insert de$nition (2.9) into (2.10). After a variable substitution = − one obtains the functional integral representation of k k 1 − Rk : (2.12) exp(−k []) = D exp −S[ + ] + 2 This expression resembles closely the background $eld formalism for the eEective action which is modi$ed only by the term ∼ Rk . For k → ∞ the cutoE function Rk diverges and the term exp(− Rk =2) behaves as a delta functional ∼ [ ], thus leading to the property k → S in this limit. For a model with a sharp UV cutoE it is easy to enforce the identity = S by choosing 2 2 2 2 a cutoE function Rk which diverges for k → , like Rk ∼ q2 (eq =k − eq = )−1 . We note, however, that the property = S is not essential, as the short-distance laws may be parametrized by as well as by S. For momentum scales much smaller than universality implies that the precise form of is irrelevant, up to the values of a few relevant renormalized couplings. Furthermore, the microscopic action may be formulated on a lattice instead of continuous space and can involve even variables diEerent from a (x). In this case one can still compute in a $rst step by evaluating the functional integral (2.12) approximately. Often a saddle-point expansion will do, since no long-range "uctuations are involved in the transition from S to . In this report we will assume that the $rst step of the computation of is done and consider as the appropriate parametrization of the microscopic physical laws. Our aim is the computation of the eEective action from —this step may be called “transition to complexity” and involves "uctuations on all scales. We emphasize that for large the average action can serve as a formulation of the microscopic laws also for situations where no physical cutoE is present, or where a momentum UV cutoE may even be in con"ict with the symmetries, like the important case of gauge symmetries. A few properties of the eEective average action are worth mentioning: 1. All symmetries of the model which are respected by the IR cutoE TSk are automatically symmetries of k . In particular, this concerns translation and rotation invariance. As a result the approach is not plagued by many of the problems encountered by a formulation of the block-spin action on a lattice. Nevertheless, our method is not restricted to continuous space. For a cubic lattice with lattice distance a the propagator only obeys the restricted lattice translation and rotation

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symmetries, e.g. a next neighbor interaction leads in momentum space to 2 dd q S= 2 (1 − cos aq )∗ (q)(q) + · · · : a (2")d

239

(2.13)

The momentum cutoE |q | 6 , = "=a also re"ects the lattice symmetry. A rotation and translation symmetric cutoE Rk which only depends on q2 obeys automatically all possible lattice symmetries. The only change compared to continuous space will be the reduced symmetry of k . 2. In consequence, k can be expanded in terms of invariants with respect to these symmetries with couplings depending on k. For the example of a scalar O(N )-model in continuous space one may use a derivative expansion (# = a a =2) 1 d a k = d x Uk (#) + Z; k (#)9 a 9 + · · · (2.14) 2 and expand further in powers of # Uk (#) = 12 Uk (# − #0 (k))2 + 16 -Uk (# − #0 (k))3 + · · · ; Z; k (#) = Z; k (#0 ) + Z; k (#0 )(# − #0 ) + · · · :

(2.15)

Here #0 denotes the (k-dependent) minimum of the eEective average potential Uk (#). We see that k describes in$nitely many running couplings. 3. Up to an overall scale factor, the limit k → 0 of Uk corresponds to the eEective potential U = T=V , from which the thermodynamic quantities can be derived for homogeneous situations according to Eq. (2.3). The overall scale factor is $xed by dimensional considerations. Whereas the dimension of Uk is (mass)d the dimension of U in Eq. (2.3) is (mass)4 (for ˝ = c = kB = 1). For classical statistics in d = 3 dimensions one has Uk = k =V and U = T limk →0 Uk . For two-dimensional systems, an additional factor ∼ mass appears, since Uk = k =V2 = Lk =V implies U = TL−1 limk →0 Uk . Here L is the typical thickness of the two-dimensional layers in a physical system. In the following we will often omit these scale factors. 4. The functional ˜ k [] = k [] + TSk [] is the Legendre transform of Wk and therefore convex. This implies that all eigenvalues of the matrix of second functional derivatives (2) + Rk are positive semi-de$nite. In particular, one $nds for a homogeneous $eld a and q2 = 0 the simple exact bounds for all k and # Uk (#) ¿ − Rk (0) ; Uk (#) + 2#Uk (#) ¿ − Rk (0) ;

(2.16)

where primes denote derivatives with respect to #. Even though the potential U () becomes convex for k → 0 it may exhibit a minimum at #0 (k) ¿ 0 for all k ¿ 0. Spontaneous breaking of the O(N )-symmetry is characterized by limk →0 #0 (k) ¿ 0. 5. For a formulation that respects the reparametrization invariance of physical quantities under a rescaling of the variables a (x) → /a (x) the infrared cutoE should contain a wave function

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renormalization, e.g. Rk (q) =

Zk q 2 : eq2 =k 2 − 1

(2.17)

One may choose Zk = Z; k (#0 ). This choice guarantees that no intrinsic scale is introduced in the inverse average propagator 2

2

Zk q + Rk = Zk P(q) = Zk q p

6. 7.

8.

9.

10.

16 17

q2 k2

:

(2.18)

This is important in order to obtain scale-invariant "ow equations for critical phenomena. There is no problem incorporating chiral fermions, since a chirally invariant cutoE Rk can be formulated [79,80] (cf. Section 7.2). Gauge theories can be formulated along similar lines 16 [47,11,48–74] even though TSk may not be gauge invariant. 17 In this case the usual Ward identities receive corrections for which one can derive closed expressions [50]. These corrections vanish for k → 0. On the other hand, they appear as “counterterms” in and are crucial for preserving the gauge invariance of physical quantities. For choice (2.18) the high momentum modes are very eEectively integrated out because of the exponential decay of Rk for q2 k 2 . Nevertheless, it is sometimes technically easier to use a cutoE without this fast decay property, e.g. Rk ∼ k 2 or Rk ∼ k 4 =q2 . In the latter cases one has to be careful with possible remnants of an incomplete integration of the short-distance modes. An important technical simpli$cation can also be achieved by a sharp momentum cutoE [3]. This guarantees complete integration of the short distance modes, but poses certain problems with analyticity [8,81–84]. In contrast, a smooth cutoE like (2.18) does not introduce any non-analytical behavior. The results for physical quantities are independent of the choice of the cutoE scheme Rk . On the other hand, both and the "ow with k are scheme-dependent. The scheme dependence of the $nal results is a good check for approximations [8,85 –88]. Despite a similar spirit and many analogies, there is a conceptual diEerence to the Wilsonian effective action SW . The Wilsonian eEective action describes a set of diEerent actions (parametrized by ) for one and the same model—the n-point functions are independent of and have to be computed from SW by further functional integration. In contrast, k can be viewed as the eEective action for a set of diEerent “models”—for any value of k the eEective average action is related to the generating functional of 1PI n-point functions for a model with a diEerent action Sk = S + TSk . The n-point functions depend on k. The Wilsonian eEective action does not generate the 1PI Green functions [89]. Because of the incorporation of an infrared cutoE, k is closely related to an eEective action for averages of $elds [8], where the average is taken over a volume ∼ k −d .

See also [75] for applications to gravity. For a manifestly gauge invariant formulation in terms of Wilson loops see Ref. [67].

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241

2.2. Exact ;ow equation The dependence of the average action k on the coarse graining scale k is described by an exact non-perturbative "ow equation [9,90 –92] − 1 9 1 9 (2) k [] = Tr k [] + Rk Rk : (2.19) 9k 2 9k The trace involves an integration over momenta or coordinates as well as a summation over internal indices. In momentum space it reads Tr = a d d q=(2")d , as appropriate for the unit matrix 1 = (2")d (q − q )ab . The exact "ow equation describes the scale dependence of k in terms of the inverse average propagator k(2) , given by the second functional derivative of k with respect to the $eld components (k(2) )ab (q; q ) =

2 k : a (−q)b (q )

(2.20)

It has a simple graphical expression as a one-loop equation 9k 1 = 9k 2 with the full k-dependent propagator associated to the propagator line and the dot denoting the insertion 9k Rk . Because of the appearance of the exact propagator (k(2) + Rk )−1 , Eq. (2.19) is a functional diEerential equation. It is remarkable that the transition from the classical propagator in the presence of the infrared cutoE, (S (2) +Rk )−1 , to the full propagator turns the one-loop expression into an exact identity which incorporates eEects of arbitrarily high loop order as well as genuinely non-perturbative eEects 18 like instantons in QCD. The exact "ow equation (2.19) can be derived in a straightforward way [9]. Let us write k [] = ˜ k [] − TSk [] ; where, according to (2.9), ˜ k [] = −Wk [J ] + d d x J (x)(x)

(2.21)

(2.22)

and J =Jk (). We consider for simplicity a one-component $eld and derive $rst the scale dependence ˜ of : 9Wk 9J (x) 9 ˜ d Wk 9J (x) k [] = − [J ] − d x + d d x (x) : (2.23) 9k 9k J (x) 9k 9k

18

We note that anomalies which arise from topological obstructions in the functional measure manifest themselves already in the microscopic action . The long-distance non-perturbative eEects (“large-size instantons”) are, however, completely described by the "ow equation (2.19).

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With (x) = Wk =J (x) the last two terms in (2.23) cancel. The k-derivative of Wk is obtained from its de$ning functional integral (2.5). Since only Rk depends on k this yields 9 ˜ 9 1 9 d d k [] = TSk [] = d x d y (x) Rk (x; y)(y) ; (2.24) 9k 9k 2 9k where Rk (x; y) ≡ Rk (ix )(x − y) and −1 DA[] exp(−S[] − 1k S[] + d d x J (x)(x)) : A[] = Z

(2.25)

Let G(x; y) = 2 Wk =J (x)J (y) denote the connected two-point function and decompose (x)(y) = G(x; y) + (x)(y) ≡ G(x; y) + (x)(y) :

(2.26)

After plugging this decomposition into (2.24), the scale dependence of ˜ k can be expressed as 9 ˜ 1 9 9 k [] = dd x dd y Rk (x; y)G(y; x) + (x) Rk (x; y)(y) 9k 2 9k 9k 9 9 1 (2.27) ≡ Tr G Rk + TSk [] : 2 9k 9k The exact "ow equation for the average action k follows now through (2.21) − 1 9 9 1 1 9 (2) k [] = Tr G Rk = Tr k [] + Rk Rk : 9k 2 9k 2 9k

(2.28)

(2) For the last equation, we have used that ˜ k (x; y) ≡ 2 ˜ k =(x)(y) = J (x)=(y) is the inverse of G(x; y) ≡ 2 Wk =J (x)J (y) = (x)=J (y): d d yG(x; y)(k(2) + Rk )(y; z) = (x − z) : (2.29)

It is straightforward to write the above identities in momentum space and to generalize them to N components by using the matrix notation introduced above. Let us point out a few properties of the exact "ow equation: 1. For a scaling form of the evolution equation and a formulation closer to the usual -functions, one may replace the partial k-derivative in (2.19) by a partial derivative with respect to the logarithmic variable t = ln(k=). 2. Exact "ow equations for n-point functions can be easily obtained from (2.19) by diEerentiation. The "ow equation for the two-point function k(2) involves the three- and four-point functions, k(3) and k(4) , respectively. One may write schematically 9 (2) 9 92 k = 9t k 9t 99 9Rk 9 1 ([k(2) + Rk ]−1 k(3) [k(2) + Rk ]−1 ) = − Tr 2 9t 9

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9Rk (2) [k + Rk ]−1 k(3) [k(2) + Rk ]−1 k(3) [k(2) + Rk ]−1 = Tr 9t 9Rk (2) 1 −1 (4) (2) −1 [k + Rk ] k [k + Rk ] : − Tr 2 9t

243

(2.30)

By evaluating this equation for = 0, one sees immediately the contributions to the "ow of the two-point function from diagrams with three- and four-point vertices. Below we will see in more detail that the diagrammatics is closely linked to the perturbative graphs. In general, the "ow equation for k(n) involves k(n+1) and k(n+2) . 3. As already mentioned, the "ow equation (2.19) closely resembles a one-loop equation. Replacing k(2) by the second functional derivative of the classical action, S (2) , one obtains the corresponding one-loop result. Indeed, the one-loop formula for k reads k [] = S[] + 12 Tr ln(S (2) [] + Rk )

(2.31)

and taking a k-derivative of (2.31) gives a one-loop "ow equation very similar to (2.19). The “full renormalization group improvement” S (2) → k(2) turns the one-loop "ow equation into an exact non-perturbative "ow equation. Replacing the propagator and vertices appearing in k(2) by the ones derived from the classical action, but with running k-dependent couplings, and expanding the result to lowest non-trivial order in the coupling constants, one recovers standard renormalization group improved one-loop perturbation theory. 4. The additional cutoE function Rk with a form like the one given in Eq. (2.17) renders the momentum integration implied in the trace of (2.19) both infrared and ultraviolet $nite. In particular, for q2 k 2 one has an additional mass-like term Rk ∼ k 2 in the inverse average propagator. This makes the formulation suitable for dealing with theories that are plagued with infrared problems in perturbation theory. For example, the "ow equation can be used in three dimensions in the phase with spontaneous symmetry breaking despite the existence of massless Goldstone bosons for N ¿ 1. We recall that all eigenvalues of the matrix (2) + Rk must be positive semi-de$nite (cf. Eq. (2.16)). We note that the derivation of the exact "ow equation does not depend on the particular choice of the cutoE function. Ultraviolet $niteness, however, is related to a fast decay of 9t Rk for q2 k 2 . If for some other choice of Rk the r.h.s of the "ow equation would not remain ultraviolet $nite this would indicate that the high momentum modes have not yet been integrated out completely in the computation of k . Unless stated otherwise we will always assume a suIciently fast decaying choice of Rk in the following. 5. Since no in$nities appear in the "ow equation, one may “forget” its origin in a functional integral. Indeed, for a given choice of the cutoE function Rk all microscopic physics is encoded in the microscopic eEective action . The model is completely speci$ed by the "ow equation (2.19) and the “initial value” . In a quantum $eld theoretical sense the "ow equation de$nes a regularization scheme. The “ERGE”-scheme is speci$ed by the "ow equation, the choice of Rk and the “initial condition” . This is particularly important for gauge theories where other regularizations in four dimensions and in the presence of chiral fermions are diIcult to construct. For gauge theories has to obey appropriately modi$ed Ward identities. In the context of perturbation theory a $rst proposal for how to regularize gauge theories by use of "ow equations can be found in [48]. We note that in contrast to previous versions of exact renormalization

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6.

7. 8.

9.

10.

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group equations there is no need in the present formulation to construct an ultraviolet momentum cutoE—a task known to be very diIcult in non-Abelian gauge theories. As for all regularizations the physical quantities should be independent of the particular regularization scheme. In our case diEerent choices of Rk correspond to diEerent trajectories in the space of eEective actions along which the unique infrared limit 0 is reached. Nevertheless, once approximations are applied not only the trajectory but also its end point may depend on the precise de$nition of the function Rk . As mentioned above, this dependence may be used to study the robustness of the approximation. Extensions of the "ow equations to gauge $elds [47,11], [48–74] and fermions [79,80] are available. We emphasize that the "ow equation (2.19) is formally equivalent to the Wilsonian exact renormalization group equation [2–7]. The latter describes how the Wilsonian eEective action SW changes with an ultraviolet cutoE . Polchinski’s continuum version of the Wilsonian "ow equation [6] 19 can be transformed into Eq. (2.19) by means of a Legendre transform, a suitable $eld rede$nition and the association = k [90,94,95]. Although the formal relation is simple, the practical calculation of SkW from k (and vice versa) can be quite involved. 20 In the presence of massless particles the Legendre transform of k does not remain local and SkW is a comparatively complicated object. We will argue below that the crucial step for a practical use of the "ow equation in a non-perturbative context is the ability to device a reasonable approximation scheme or truncation. It is in this context that the close resemblence of Eq. (2.19) to a perturbative expression is of great value. In contrast to the Wilsonian eEective action no information about the short-distance physics is eEectively lost as k is lowered. Indeed, the eEective average action for $elds with high momenta q2 k 2 is already very close to the eEective action. Therefore, k generates quite accurately the vertices with high external momenta. More precisely, this is the case whenever the external momenta act eEectively as an independent “physical” IR cutoE in the "ow equation for the vertex. There is then only a minor diEerence between k(n) and the exact vertex (n) . An exact equation of the type (2.19) can be derived whenever Rk multiplies a term quadratic in the $elds, cf. (2.6). The feature that Rk acts as a good infrared cutoE is not essential for this. In particular, one can easily write down an exact equation for the dependence of the eEective action on the chemical potential [96]. Another interesting exact equation describes the eEect of a variation of the microscopic mass term for a $eld, as, for example, the current quark mass in QCD. In some cases an additional UV-regularization may be necessary since the UV-$niteness of the momentum integral in (2.19) may not be automatic.

2.3. Truncations Even though intuitively simple, the replacement of the (RG improved) classical propagator by the full propagator turns the solution of the "ow equation (2.19) into a diIcult mathematical problem: The evolution equation is a functional diEerential equation. Once k is expanded in terms of 19

For a detailed presentation see e.g. [93] If this problem could be solved, one would be able to construct an UV momentum cutoE which preserves gauge invariance by starting from the Ward identities for k . 20

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invariants (e.g. Eqs. (2.14) and (2.15)) this is equivalent to an in$nite system of coupled non-linear partial diEerential equations. General methods for the solution of functional diEerential equations are not developed very much. They are restricted mainly to iterative procedures that can be applied once some small expansion parameter is identi$ed. They cover usual perturbation theory in the case of a small coupling, the 1=N -expansion or expansions in the dimensionality 4 − d or d − 2. They may also be extended to less familiar expansions like a derivative expansion which is related in critical three-dimensional scalar theories to a small anomalous dimension [97]. In the absence of a clearly identi$ed small parameter one needs to truncate the most general form of k in order to reduce the in$nite system of coupled diEerential equations to a (numerically) manageable size. This truncation is crucial. It is at this level that approximations have to be made and, as for all non-perturbative analytical methods, they are often not easy to control. The challenge for non-perturbative systems like critical phenomena in statistical physics or low momentum QCD is to $nd "ow equations which (a) incorporate all the relevant dynamics so that neglected eEects make only small changes, and (b) remain of manageable size. The diIculty with the $rst task is a reliable estimate of the error. For the second task, the main limitation is a practical restriction for numerical solutions of diEerential equations to functions depending only on a small number of variables. The existence of an exact functional diEerential "ow equation is a very useful starting point and guide for this task. At this point the precise form of the exact "ow equation is quite important. Furthermore, it can be used for systematic expansions through enlargement of the truncation and for an error estimate in this way. Nevertheless, this is not all. Usually, physical insight into a model is necessary in order to device a useful non-perturbative truncation. Several approaches to non-perturbative truncations have been explored so far (# ≡ 12 a a ): (i) Derivative expansion. We can classify invariants by the number of derivatives 1 1 d a 4 k [] = d x Uk (#) + Zk (#)9 9 a + Yk (#)9 #9 # + O(9 ) : 2 4

(2.32)

The lowest level only includes the scalar potential and a standard kinetic term. The $rst correction includes the #-dependent wave function renormalizations Zk (#) and Yk (#). The next level involves then invariants with four derivatives, etc. One may wonder if a derivative expansion has any chance to account for the relevant physics of critical phenomena, in a situation where we know that the critical propagator is non-analytic in the momentum. 21 The reason why it can work is that the non-analyticity builds up only gradually as k → 0. For the critical temperature a typical qualitative form of the inverse average propagator is k(2) ∼ q2 (q2 + ck 2 )−6=2

(2.33) 2

with 6 the anomalous dimension. Thus the behavior for q → 0 is completely regular. In addition, the contribution of "uctuations with small momenta q2 k 2 to the "ow equation is suppressed by the IR cutoE Rk . For q2 k 2 the “non-analyticity” of the propagator is already manifest. The contribution of this region to the momentum integral in (2.19) is, however, strongly suppressed by the derivative 9k Rk . For cutoE functions of the type (2.17) only 21

See [98,99] for early applications of the derivative expansion to critical phenomena. For a recent study on convergence properties of the derivative expansion see [100].

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a small momentum range centered around q2 ≈ k 2 contributes substantially to the momentum integral in the "ow equation. This suggests the use of a hybrid derivative expansion where the momentum dependence of k − d d x Uk is expanded around q2 = k 2 . Nevertheless, because of the qualitative behavior (2.33), also an expansion around q2 = 0 should yield valid results. We will see in Section 4 that the $rst order in the derivative expansion (2.32) gives a quite accurate description of critical phenomena in three-dimensional O(N ) models, except for an (expected) error in the anomalous dimension. (ii) Expansion in powers of the >elds. As an alternative ordering principle one may expand k in n-point functions k(n)   ∞ n 1  d d xj [(xj ) − 0 ] (n) (x1 ; : : : ; x n ) : k [] = (2.34) k n! n=0 j=0 If one chooses [8] 22 0 as the k-dependent expectation value of , series (2.34) starts eEectively at n = 2. The "ow equations for the 1PI Green functions k(n) are obtained by functional diEerentiation of (2.19). Similar equations have been discussed $rst in [5] from a somewhat diEerent viewpoint. They can also be interpreted as a diEerential form of Schwinger–Dyson equations [106]. (iii) Expansion in the canonical dimension. We can classify the couplings according to their canonical dimension. For this purpose we expand k around some constant $eld #0 1 d k [] = d x Uk (#0 ) + Uk (#0 )(# − #0 ) + Uk (#0 )(# − #0 )2 + · · · 2 1 1 2 Zk (#0 ) + Zk (#0 )(# − #0 ) + Zk (#0 )(# − #0 ) + · · · a 9 9 a − 2 2 1 + (Z˙ k (#0 ) + Z˙ k (#0 )(# − #0 ) + · · ·)a (9 9 )2 a 2 1 − Yk (#0 )#9 9 # + · · · : 4

(2.35)

The $eld #0 may depend on k. In particular, for a potential Uk with minimum at #0 (k) ¿ 0 the location of the minimum can be used as one of the couplings. In this case, #0 (k) replaces the coupling Uk (#0 ) since Uk (#0 (k)) = 0. In three dimensions one may start by considering an approximation that takes into account only couplings with positive canonical mass dimension, i.e. Uk (0) with mass dimension M 2 and Uk (0) with dimension M 1 in the symmetric regime (potential minimum at # = 0). Equivalently, in the spontaneously broken regime (potential minimum for # = 0) we may take #0 (k) and Uk (#0 ). The $rst correction includes then the dimensionless parameters Uk (#0 ) and Zk (#0 ). The second correction includes Uk(4) (#0 ), Zk (#0 ) and Yk (#0 ) with mass dimension M −1 and so on. Already the inclusion of the dimensionless couplings gives a very satisfactory description of critical phenomena in three-dimensional scalar theories (see Section 4.2). 22

See also [101,102] for the importance of expanding around = 0 instead of = 0 and Refs. [103–105].

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2.4. Flow equation for the average potential For a discussion of the ground state, its preserved or spontaneously broken symmetries and the mass spectrum of excitations the most important quantity is the average potential Uk (#). In the absence of external sources the minimum #0 of Uk →0 determines the expectation value of the order parameter. The symmetric phase with unbroken O(N ) symmetry is realized if #0 (k → 0) = 0, whereas spontaneous symmetry breaking occurs for #0 (k → 0) ¿ 0. Except for the wave function renormalization to be discussed later the squared particle masses M 2 are given by M 2 ∼ U (#0 = 0) for the symmetric phase. Here primes denote derivatives with respect to #. For #0 = 0 one $nds a radial mode with M 2 ∼ U (#0 ) + 2#0 U (#0 ) and N − 1 Goldstone modes with M 2 ∼ U (#0 ). For vanishing external sources the Goldstone modes are massless. Therefore, we want to concentrate on the "ow of Uk (#). The exact "ow equation is obtained by evaluating Eq. (2.19) for a constant value of ’a , say ’a (x) = ’a1 ; # = 12 ’2 . One $nds the exact equation d d q 9Rk N − 1 1 1 9t Uk (#) = (2.36) + 2 (2")d 9t M0 M1 with M0 (#; q2 ) = Zk (#; q2 )q2 + Rk (q) + Uk (#) ; M1 (#; q2 ) = Z˜ k (#; q2 )q2 + Rk (q) + Uk (#) + 2#Uk (#) ;

(2.37)

parametrizing the (a; a) and (1,1) element of k(2) + Rk (a = 1). As expected, this equation is not closed since we need information about the # and q2 -dependent wave function renormalizations Zk and Z˜ k for the Goldstone and radial modes, respectively. The lowest order in the derivative expansion would take Z˜ k = Zk = Zk (#0 ; k 2 ) independent of # and q2 , so that only the anomalous dimension 9 6 = − ln Zk (2.38) 9t is needed in addition to the partial diEerential equation (2.36). For a $rst discussion let us also neglect the contribution ∼ 9t Zk in 9t Rk and write 9 Uk (#) Uk (#) + 2#Uk (#) d d d Uk (#) = 2vd k (N − 1)l0 + l0 (2.39) 9t Zk k 2 Zk k 2 with −1

vd = 2

d+1 d=2

"

d ; 2

v2 =

1 ; 8"

v3 =

1 ; 8"2

v4 =

1 : 32"2

Here we have introduced the dimensionless threshold function dd q 9t (Rk (q)=Zk ) 1 ld0 (w) = vd−1 k −d : 4 (2")d q2 + Zk−1 Rk (q) + k 2 w

(2.40)

(2.41)

It depends on the renormalized particle mass w = M 2 =(Zk k 2 ) and has the important property that it decays rapidly for w1. This describes the decoupling of modes with renormalized squared mass

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M 2 =Zk larger than k 2 . In consequence, only modes with mass smaller than k contribute to the "ow. The "ow equations ensure automatically the emergence of eEective theories for the low-mass modes. The explicit form of the threshold functions (2.41) depends on the choice of Rk . We will discuss several choices in Section 3.2. For a given explicit form of the threshold functions Eq. (2.39) turns into a non-linear partial diEerential equation for a function U depending on the two variables k and #. This can be solved numerically by appropriate algorithms [107] as is shown in later sections. Eq. (2.39) was $rst derived [8] as a renormalization group improved perturbative expression and its intuitive form close to perturbation theory makes it very suitable for practical investigations. Here it is important to note that the use of the average action allows for the inclusion of propagator corrections (wave function renormalization eEects) in a direct and systematic way. Extensions to more complicated scalar models or models with fermions [79] are straightforward. In the limit of a sharp cutoE (see Section 3.2) and for vanishing anomalous dimension Eq. (2.39) coincides with the Wegner–Houghton equation [3] for the potential, $rst discussed in [33] (see also [35,34,7,108]). Eq. (2.39) can be used as a practical starting point for various systematic expansions. For example, it is the lowest order in the derivative expansion. The next order includes q2 -independent wave function renormalizations Zk (#); Z˜ k (#) in Eq. (2.37). For N = 1 the $rst order in the derivative expansion leads therefore to coupled partial non-linear diEerential equations for two functions Uk (#) and Zk (#) depending on two variables k and #. We have solved these diEerential equations numerically and the result is plotted in Fig. 1. The initial values of the integration correspond to the phase with spontaneous symmetry breaking. More details can be found in Section 4. 2.5. A simple example: the quartic potential Before we describe the solutions of more sophisticated truncations of the "ow equation (2.36), we consider here a very simple polynomial approximation to Uk , namely Uk (#) = 12 Uk (# − #0 (k))2 :

(2.42)

In four dimensions the two parameters #0 (k) and Uk correspond to the renormalizable couplings. It is not surprising that the "ow of Uk will reproduce for d = 4 the usual one-loop -function. Compared to standard perturbation theory, one gets in addition a "ow equation for the k-dependent potential minimum #0 (k) which re"ects the quadratic renormalization of the mass term. It is striking, however, that with the inclusion of the anomalous dimension 6 the simple ansatz (2.42) also describes [8] the physics for lower dimensions d = 3 or 2. This includes the second-order phase transition for d = 3 with nontrivial critical exponents and even the Kosterlitz–Thouless transition for d = 2, N = 2. In consequence, we obtain a simple uni$ed picture for the 4 -model in all dimensions. This is an excellent starting point for more elaborate approximations. All characteristic features of the "ow can be discussed with approximation (2.42). We use this ansatz to rewrite Eq. (2.39) as a coupled set of ordinary diEerential equations for the minimum of the potential #0 (k) and the quartic coupling Uk . This system is closed for a given anomalous dimension 6 (2.38). The "ow of the potential minimum can be inferred from the identity 0=

d U (#0 (k)) = 9t Uk (#0 (k)) + Uk (#0 (k))9t #0 (k) : dt k

(2.43)

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249

0.02

Uk ( )

0.015

0.01

0.005

0

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Fig. 1. Average potential Uk () for diEerent scales k = et . The shape of Uk is displayed at smaller intervals Tt = −0:02 after the minimum has settled at a constant value. This demonstrates the approach to convexity in the “inner region”, while the “outer region” becomes k-independent for k → 0.

Here 9t Uk (#) is the partial t-derivative with # held $xed, which is computed by diEerentiating Eq. (2.39) with respect to #. De$ning the “higher” threshold functions by 9 d l (w) 9w 0 1 9 d l (w); ldn+1 (w) = − n 9w n

ld1 (w) = −

n¿1

and constants ldn = ldn (0) one obtains U 2# 0 + (N − 1)ld1 : 9t #0 = 2vd k d−2 Zk−1 3ld1 Zk k 2

(2.44)

(2.45)

We conclude that #0 (k) always decreases as the infrared cutoE k is lowered. For d=2 and N ¿ 2 only a Z-factor increasing without bounds can prevent #0 (k) from reaching zero at some value k ¿ 0. We will see that this unbounded Z-factor occurs for N =2 in the low-temperature phase. If #0 (k) reaches zero for ks ¿ 0 the "ow for k ¡ ks can be continued with a truncation Uk (#) = m2k # + 12 Uk #2 with m2k ¿ 0. This situation corresponds to the symmetric or disordered phase with N massive excitations. On the other hand, the phase with spontaneous symmetry breaking or the ordered phase is realized for limk →0 #0 (k) ¿ 0.

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It is convenient to introduce renormalized dimensionless couplings as < = Z k k 2− d #0 ;

= Zk−2 k d−4 U :

(2.46)

For our simple truncation, one has u = 12 (#˜ − <)2

(2.47)

and the "ow equations for < and read 9t < = < = (2 − d − 6)< + 2vd {3ld1 (2<) + (N − 1)ld1 } ;

(2.48)

9t = = (d − 4 + 26) + 2vd 2 {9ld2 (2<) + (N − 1)ld2 } :

(2.49)

In this truncation the anomalous dimension 6 is given by [8] 16vd 2 d 6= (2.50) <m2; 2 (0; 2<) ; d where md2; 2 is another threshold function de$ned in Section 4.2. It has the property 1 6= for d = 2; <1 : (2.51) 4"< The universal critical behavior of many systems of statistical mechanics is described by the $eld theory for scalars with O(N ) symmetry. This covers the gas–liquid and many chemical transitions described by Ising models with a discrete symmetry Z2 ≡ O(1), super"uids with continuous Abelian symmetry O(2), Heisenberg models for magnets with N = 3, etc. In 2 ¡ d 6 4 dimensions all these models have a continuous second-order phase transition. In two dimensions one observes a second-order transition for the Ising model, a Kosterlitz–Thouless phase transition [109] for N =2 and no phase transition for non-Abelian symmetries N ¿ 3. It is known [110] that a continuous symmetry cannot be broken in two dimensions in the sense that the expectation value of the unrenormalized scalar $eld vanishes in the limit of vanishing sources, a = a (x) = 0. We want to demonstrate that the two diEerential equations (2.48) and (2.49) describe all qualitative features of phase transitions in two or three dimensions correctly. A quantitative numerical analysis using more sophisticated truncations will be presented in Section 4. In four dimensions the anomalous dimension and the product < become rapidly small quantities as k decreases. We recognize then in Eq. (2.49) the usual perturbative one-loop -function for the quartic coupling N +8 2 9t = : (2.52) 16"2 Eqs. (2.48) and (2.49) also exhibit the well-known property of “triviality” which means that the quartic coupling vanishes for k → 0 for the massless model. For d = 3, Eqs. (2.48) and (2.49) exhibit a $xed point (<∗ ; ∗ ) where < = = 0. This is a $rst example of a “scaling solution” for which all couplings evolve according to an eEective dimension composed from their canonical dimension and anomalous dimension, i.e. Z k ∼ k − 6∗ ;

#0 ∼ k d−2+6∗ ; 2

U ∼ k 4−d−26∗ :

(2.53)

From the generic form = − + (c1 + c2 (<)) one concludes that essentially corresponds to an infrared stable coupling that is attracted towards its $xed point value ∗ as k is lowered. On the other

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251

hand, < = −< + c3 + c4 (<) shows that < is essentially an infrared unstable or relevant coupling. For given , starting with < = <∗ ( ) + < , < = characterizes the 1 −> with m2 =lim U −1 divergence of the correlation length for T → Tc , i.e., % ∼ m− k →0 2#0 Zk . R R ∼ |T −Tc | It corresponds to the negative eigenvalue of the “stability matrix” Aij = (9i =9j )(<∗ ; ∗ ) with i ≡ (<; ). (This can be generalized to more than two couplings.) The critical exponent 6 determines the long-distance behavior of the two-point function for T = Tc . It is given by the anomalous dimension at the $xed point, 6 = 6(<∗ ; ∗ ). It is remarkable that already in a very simple polynomial truncation the critical exponents come out with reasonable accuracy [8,111]. In three dimensions the anomalous dimension comes out to be small and can be neglected for a rough treatment, further simplifying the "ow equations (2.48) and (2.49). As an example, for the critical exponent > for N = 3 and 6 = 0 one $nds > = 0:74, to be compared with the known value > = 0:71. (See also [85] for a discussion of the N = 1 case in three dimensions and Section 4.) In two dimensions the term linear in < vanishes in < . This changes the $xed point structure dramatically as can be seen from N −2 ; (2.54) lim < = <→∞ 4" where l21 = 1 was used. Since < is always positive for < = 0 a $xed point requires that < becomes negative for large <. This is the case for the Ising model [8,67,112] where N = 1. On the other hand, for a non-Abelian symmetry with N ¿ 3 no $xed point and therefore no phase transition occurs. The location of the minimum always reaches zero for some value ks ¿ 0. The only phase corresponds to a linear realization of O(N ) with N degenerate masses mR ∼ ks . It is interesting to note that the limit < → ∞ describes the non-linear sigma model. The non-Abelian coupling g of the non-linear model is related to < by g2 = 1=(2<) and Eq. (2.54) reproduces the standard one-loop beta function for g 9g2 N −2 4 =− g ; (2.55) 9t 2" characterized by asymptotic freedom [113]. The “con$nement scale” where the coupling g becomes strong can be associated with ks . The strongly interacting physics of the non-linear model $nds a simple description in terms of the symmetric phase of the linear O(N )-model [8]. This may be regarded as an example of duality: the dual description of the non-linear A-model for large coupling is simply the linear 4 -model. Particularly interesting is the Abelian continuous symmetry for N =2. Here < vanishes for < → ∞ and < becomes a marginal coupling. As is shown in more detail in Section 4.8 one actually $nds [114] a behavior consistent with a second-order phase transition with 6 0:25 near the critical trajectory. The low-temperature phase (< ¿ <∗ ) is special since it has many characteristics of the phase with spontaneous symmetry breaking, despite the fact that #0 (k → 0) must vanish according to the Mermin–Wagner theorem [110]. There is a massless Goldstone-type boson (in$nite correlation length) and one massive mode. Furthermore, the exponent 6 depends on < or the temperature (cf. Eq. (2.51)), since < "ows only marginally. These are the characteristic features of a Kosterlitz– Thouless phase transition [109]. The puzzle of the Goldstone boson in the low-temperature phase

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despite the absence of spontaneous symmetry breaking is solved by the observation that the wave function renormalization never stops running with k: −6 k : (2.56) Zk = ZU √ Even though the renormalized $eld R = Zk1=2 acquires a non-zero expectation value R = 2<, for k → 0 the unrenormalized order parameter vanishes because of the divergence of Zk , 1=4"< 2< k : (2.57) (k) = ZU Also the inverse Goldstone boson propagator behaves as (q2 )1−1=(8"<) and circumvents Coleman’s no-go theorem [115] for free massless scalar $elds in two dimensions. It is remarkable that all these features arise from the solution of a simple one-loop-type equation without ever invoking non-perturbative vortex con$gurations. 3. Solving the ow equation 3.1. Scaling form of the exact ;ow equation for the potential In this section we discuss analytical approaches to the solution of the exact "ow equations for the average potential and the propagator. They prove to be useful guides for the numerical solutions of truncated partial diEerential equations. After writing the exact equation for the potential in a scale-invariant form and discussing explicitly the threshold functions, we present an exact solution of the "ow equation in the limit N → ∞. A renormalization group improved perturbation theory is developed as the iterative solution of the "ow equations. This incorporates the usual gap equation and can be used as a systematic procedure with the gap equation as a starting point. We write down the exact "ow equation for the propagator as a basis for a systematic computation of the anomalous dimension and related quantities. Finally, we show how the average potential approaches a convex form for k → 0 in the non-trivial case of spontaneous symmetry breaking. Let us $rst come back to the exact "ow equation (2.36) and derive an explicitly scale-invariant form of it. This will be a useful starting point for the discussion of critical phenomena in later sections. It is convenient to use a dimensionless cutoE function x R (x) k rk 2 = ; x ≡ q2 ; Zk = Zk (#0 ; k 2 ) (3.1) k Zk x and write the "ow equations as ∞ x N − 1 9 1 d=2 d x x sk 2 + Uk (#) = vd 9t k M0 =Zk M1 =Zk 0 with sk

x 9 x x x 9 x r r = − 6r = −2x − 6r : k k k k k2 9t k2 k2 9x k2 k2

(3.2)

(3.3)

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We parametrize the wave function renormalization by Zk (#; k 2 ) Z˜ k (#; k 2 ) − Zk (#; k 2 ) d−2 zk (#) = ; zk (#0 ) ≡ 1; #y˜ k (#) = k ; Zk Zk2 x Z (#; x) − Z (#; k 2 ) k k ; Tzk (#; 1) = 0 ; Tzk #; 2 = k Zk x Z˜ (#; x) − Z˜ (#; k 2 ) x k k Zk k 2−d # Ty˜ k #; 2 = − Tzk #; 2 ; Ty˜ k (#; 1) = 0 ; (3.4) k Zk k so that Uk Uk + 2#Uk 9 d d d 2− d Uk = 2vd k (N − 1)l0 + TCk k d ; ; 6; zk + l0 ; 6; zk + Zk #y˜ k k 9t Zk k 2 Zk k 2 (3.5) where ld0 (w; 6; z) is a generalized dimensionless threshold function (y = x=k 2 ) 1 ∞ ld0 (w; 6; z) = dyyd=2 sk (y)[(z + rk (y))y + w]−1 : 2 0

(3.6)

The correction TCk contributes only in second order in a derivative expansion. Finally, we may remove the explicit dependence on Zk and k by using scaling variables u k = Uk k − d ;

#˜ = Zk k 2−d # :

(3.7) ˜ u = 9u=9#,

Evaluating the t-derivative at $xed #˜ and using the notation form of the exact evolution equation for the average potential

etc., one obtains the scaling

9t u|#˜ = −du + (d − 2 + 6)#u ˜ + Ck ;

˜ ; 6; z + #˜y)} ˜ + TCk ; Ck = 2vd {(N − 1)ld0 (u ; 6; z) + ld0 (u + 2#u

(3.8)

with TCk

= − vd +

0

∞

dyy

d=2+1

sk (y)

(N − 1)Tz(y) [(z + rk (y))y + u ][(z + Tz(y) + rk (y))y + u ]

Tz(y)+#T ˜ y(y) ˜ [(z+#˜y+r ˜ k (y))y+u +2#u ˜ ][(z+#˜y+Tz(y)+ ˜ #T ˜ y(y)+r ˜ ˜ ] k (y))y+u +2#u

: (3.9)

All explicit dependence on the scale k or the wave function renormalization Zk has disappeared. Reparametrization invariance under $eld scaling is obvious in this form. For # → /2 # one also has Zk → /−2 Zk so that #˜ is invariant. This property needs the factor Zk in Rk . This version is therefore most appropriate for a discussion of critical behavior. The universal features of the critical behavior for second-order phase transitions are related to the existence of a scaling solution. 23 This scaling solution solves the diEerential equation for k-independent functions u(#); ˜ z(#), ˜ etc., which results 23

More precisely, 9t u = 0 is a suIcient condition for the existence of a scaling solution. The universal aspects of $rst-order transitions are also connected to exact or approximate scaling solutions.

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from (3.8) by setting 9t u = 0. For a constant wave function renormalization the scaling potential can be directly obtained by solving the second-order diEerential equation 9t u = 0. It has been shown that, of all possible solutions, the physical $xed point corresponds to the solution u(#) ˜ which is non-singular in the $eld [105,116 –118]. The scaling form of the evolution equation is the best starting point for attempts of an analytical solution. In fact, in the approximation where Ck can be expressed as a function of u and 6 one may $nd the general form of the solution by the method of characteristics. For this purpose we consider the #-derivative ˜ of Eq. (3.8) 9t u = −(2 − 6)u + (d − 2 + 6)#u ˜ −

ku

;

k

=−

We further assume for a moment that the dependence of k (u

9Ck : 9u

(3.10)

; 6)

on 6 has (approximately) the form

k (u

; 6) = (2 − 6)|u |d=(2−6) ˆ k (u ) :

(3.11)

For u ¿ 0 one $nds the solution

#˜ (u )1−d=(2−6) + G(u ) = F+ u exp 2t −

0

t

dt 6(t )

(3.12)

with G(u ) obeying the diEerential equation 9G 1 (u )−d=(2−6) = 9u 2−6

k

= ˆ k (u )

(3.13)

and F+ (w) an arbitrary function. Similarly, for u ¡ 0 one has t 1−d=(2−6) #(−u ˜ ) + H (−u ) = F− u exp 2t − dt 6(t ) 0

(3.14)

with 9H 1 = (−u )−d=(2−6) 9(−u ) 2 − 6

k

= ˆ k (u )

(3.15)

and F− (w) again arbitrary. For known ˆ k one can now solve the ordinary diEerential equations for G and H . The initial value of the microscopic potential u $xes the free functions F± through the evaluation of (3.12) and (3.14) for t = 0. We $nally note that the solution for constant 6 can be obtained from the solution for 6 = 0 by the replacements d → d6 = 2d=(2 − 6), t → t6 = t(2 − 6)=2, G → G6 = 2G=(2 − 6); H → H6 = 2H=(2 − 6). We will see below that this solution becomes exact in the large N limit. For $nite N it may still be used if the functions #u ˜ ; z; #˜y˜ appearing in Eq. (3.8) can be expressed in terms of u and 6. Condition (3.11) may be abandoned in regions of k where 6 varies only slowly. If ˆ k depends on 6, the corrections to the generic solution (3.12) and (3.14) are ∼ (9 ˆ k =96)(9t 6). 3.2. Threshold functions In situations where the momentum dependence of the propagator can be approximated by a standard form of the kinetic term and is weak for the other 1PI correlation functions, the

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“non-perturbative” eEects beyond one loop arise to a large extent from the threshold functions. We will therefore discuss in this subsection their most important properties and introduce the notation dd q n + n; 0 −1 2n−d vd k 9t Rk (q)(Zk zq2 + Rk (q) + wk 2 )−(n+1) ; ldn (w; 6; z) = 4 (2")d 9 d 1 l (w; 6; z) ; ldn+1 (w; 6; z) = − n + n; 0 9w n (3.16) ldn (w; 6) = ldn (w; 6; 1); ldn (w) = ldn (w; 0; 1) ldn = ldn (0) : The precise form of the threshold functions depends on the choice of the cutoE function Rk (q). There are, however, a few general features which are independent of the particular scheme: 1. For n = d=2 one has the universal property l2n n =1 :

(3.17)

This is crucial because it guarantees the universality of the perturbative -functions for the quartic coupling in d = 4 or for the coupling in the nonlinear A-model in d = 2. 2. If the momentum integrals are dominated by q2 ≈ k 2 and Rk (q) . k 2 , the threshold functions obey for large w ldn (w) ∼ w−(n+1) :

(3.18)

We will see below that this property is not realized for sharp cutoEs where ldn (w) ∼ w−n . 3. The threshold functions diverge for some negative value of w. This is related to the fact that the average potential must become convex for k → 0. It is instructive to evaluate the threshold function explicitly for a simple cutoE function of the form Rk = Zk k 2 E(k 2 − q2 ) ;

(3.19)

2

where (x = q ) ld0 (w)

=k

2− d

0

k2

dx x

d=2−1

2 −1

[x + (w + 1)k ]

+k

4− d

0

∞

d x xd=2−1

(x − k 2 ) : x + k 2 E(k 2 − x) + k 2 w (3.20)

The second term in the expression for ld0 has to be de$ned and we consider Eq. (3.19) as the limit - → ∞ of a family of cutoE functions 2 - 2 - −1 q q 222 Zk k exp −1 : (3.21) Rk = 1+k2 1 + - k2 This yields the threshold functions 1 1 d d + lˆ0 (w); lˆ0 (w) = ld0 (w) = ln 1 + dy yd=2−1 (y + 1 + w)−1 ; 1+w 0 √ 1 3 ˆ ˆl20 (w) = ln 1 + 1 ; l0 (w) = 2 − 2 1 + w arctg √ 1+w 1+w 1 2 4 d+2 d ; lˆ0 (w) = − (1 + w)lˆ0 (w) : (3.22) lˆ0 (w) = 1 − (1 + w) ln 1 + 1+w d

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In order to get a more detailed overview, it is useful to discuss some other limiting cases for the choice of the cutoE. For - = 1 the family of cutoEs (3.21) represents the exponential cutoE (15), for - → 0 one $nds a masslike cutoE Rk = Zk k 2 and for - → ∞ we recover the step-function cutoE (3.19). One may modify the step-function cutoE (3.19) to Rk = Zk k 2 E(/k 2 − q2 ) with ld0 (w)

= ··· +

0

/

dy yd=2−1 (y + 1 + w)−1 :

(3.23)

(3.24)

In the limit / → ∞ one approaches again a masslike cutoE Rk = Zk k 2 . We observe that ld0 does not remain $nite in this limit. This re"ects the fact that for a masslike cutoE the high momentum modes are not yet fully integrated out in the computation of k . However, for low enough dimensions suitable diEerences remain $nite √ l30 (w) − l30 (0) = "( 1 + w − 1) : (3.25) For d ¡ 4 the masslike cutoE can be used except for an overall additive constant in the potential. Another interesting extension of the step cutoE (3.19) is Rk = Zk k 2 E(k 2 − q2 ) ;

(3.26)

such that +1+w d + lˆ0 (w + − 1) : (3.27) 1+w It is instructive to consider the limit → ∞ which corresponds to a sharp momentum cutoE, where rk (y ¿ 1) → 0 and rk (y ¡ 1) → ∞. Although a sharp momentum cutoE leads to certain problems with analyticity, it is useful because of technical simpli$cations. The momentum integrals are now dominated by an extremely narrow range q2 ≈ k 2 . (This holds except for a $eld-independent constant in 9t k .) We can therefore evaluate the two-point function in the r.h.s. of 9t Uk by its value for q2 = k 2 . In consequence, the correction TCk vanishes in this limit. Furthermore, the threshold functions take a very simple form and one infers 24 from (3.22), (3.27) 2 1 1 + ln + ; ld1 (w) = : (3.28) ld0 (w) = ln 1+w d 1+w ld0 (w) = ln

Actually, the constant part in ld0 depends on the precise way that the sharp cutoE is de$ned. Nevertheless, this ambiguity does not aEect the $eld-dependent part of the "ow equation and the sharp cutoE limit obeys universally 1 : (3.29) ld1 (w; 6; z) = z+w One ends in the sharp cutoE limit with a simple exact equation (up to an irrelevant constant) 9t u = −du + (d − 2 + 6)#u ˜ − 2vd {(N − 1) ln(z + u ) + ln(z + #˜y˜ + u + 2#u ˜ )} : 24

(3.30)

The functions ld1 (w) coincide with the sharp cutoE limit of a diEerent family of threshold functions considered in [8].

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In leading order in the derivative expansion (z = 1; y˜ = 0) and neglecting the anomalous dimension (6 = 0), this yields for N = 1; d = 3 1 ln(u + 2#u ˜ ) ; 4"2 which corresponds to the Wegner–Houghton equation [3] for the potential [33–35,7,108]. 9t u = −3u + #u ˜ −

(3.31)

3.3. Large-N expansion In the limit N → ∞ the exact "ow equation for the average potential (3.8) can be solved analytically [111,119]. 25 In this limit the quantities 6; y˜ and TCk vanish and z = 1. Furthermore, the correction from #u ˜ in the second term in Eq. (3.8) is suppressed by 1=N . In consequence, the right-hand side of the "ow equation for the potential only depends on u (and #) ˜ 9t u = −du + (d − 2)#u ˜ + 2vd Nld0 (u ) :

(3.32)

We can therefore use the exact solution (3.12) and (3.13) with k (u

) = 2vd Nld1 (u ) :

(3.33)

In particular, for the sharp cutoE (3.29) the functions G, H obey the diEerential equation 9G(w) = vd Nw−d=2 (1 + w)−1 ; 9w 9H (w) = vd Nw−d=2 (1 − w)−1 : 9w Consider the three-dimensional models. For d = 3 the general solution reads √ #˜ − 2v3 N √ − 2v3 N arctg( u ) = F+ (u e2t ) for u ¿ 0 ; u √ 1 + −u #˜ − 2v3 N √ √ + v3 N ln = F− (u e2t ) for u ¡ 0 : −u 1 − −u

(3.34)

(3.35)

The functions F± (w) are arbitrary. They are only $xed by the initial conditions at the microscopic scale k = (t = 0). As an example, consider the microscopic potential u = 12 (#˜ − < )2 : Insertion into Eq. (3.35) at t = 0 yields √ w 1 F+ (w) = √ + < − 2v3 N − 2v3 N arctg( w) ; w √ √ 1 + −w −w < − 2v3 N √ F− (w) = − + √ + v3 N ln ; −w 1 − −w 25

(3.36)

(3.37)

See also Refs. [3,117,108,120] for the large N limit of the Wegner–Houghton [3] and Polchinski equation [6].

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where we note the consistency condition u ¿ − 1;

< ¡ 1 :

(3.38)

˜ t) obeys for u ¿ 0 In consequence, the exact solution for u (#; √ √ u t e #˜ − 2v3 N = 2v3 N u arctg u + √ √ + (< − 2v3 N )e−t − 2v3 N u arctg ( u et ) :

(3.39)

Going back to the original variables # = #k; ˜ U = 9U=9# = u k 2 and using v3 = 1=(8"2 ), this reads √ √ U N √ U U − arctg + # − #0 (k) = 2 U arctg 4" k with

N #0 (k) = < − 2 4"

+

N k : 4"2

(3.40)

One sees how the average potential interpolates between the microscopic potential U = 12 (# − < )2

(3.41)

and the eEective potential for k → 0 1 8" 2 N 3 U= (# − #0 ) ; #0 = < − 2 : 3 N 4"

(3.42)

Here the last formula is an approximation valid for U =2 (N =8")2 and U =2 ("=2)2 which can easily be replaced by the exact expression in the range of large U . For vanishing source the model is in the symmetric phase for #0 ¡ 0 with masses of the excitations given by 2 8" N 2 < − 2 2 : M 2 = U (0) = (3.43) N 4" For #0 ¿ 0 spontaneous symmetry breaking occurs with order parameter (for J = 0) N 1=2 = 2#0 = < − 2 (2)1=2 : 4"

(3.44)

If we associate the deviation of < from the critical value <; c = N=(4"2 ) with a deviation from the critical temperature Tc N A < = 2 + (Tc − T ); #0 = A(Tc − T ) ; (3.45) 4" it is straightforward to extract the critical exponents in the large N approximation M ∼ (T − Tc )> ;

= (Tc − T ) ;

> = 1;

= 0:5 :

(3.46)

Eq. (3.42) constitutes the critical equation of state for the dependence of the magnetization on a homogeneous magnetic $eld J with 2 4" 9U = U = (2 − 2#0 )2 : (3.47) J= 9 N

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259

At the critical temperature one has #0 = 0 and J ∼ ;

=5 ;

(3.48)

whereas the susceptibility 2 N 9 = = (2 − 2#0 )−1 (52 − 2#0 )−1 9J 4"

(3.49)

obeys for #0 ¡ 0 at J = 0 ∼ (T − Tc )−- ;

-=2 :

(3.50)

We note that the critical amplitudes (given by the proportionality constants in Eqs. (3.46), (3.48) and (3.50)) are all given explicitly by Eq. (3.42) once the proportionality constant A in Eq. (3.45) is $xed. Universal amplitude ratios are those which do not depend on A. In the large N approximation the explicit solution U (#; T ) contains only one free constant instead of the usual two. This is related to the vanishing anomalous dimension. Finally, we may de$ne the quartic coupling R = U (0)

for #0 6 0 ;

R = U (#0 )

for #0 ¿ 0

(3.51)

and observe the constant critical ratio in the symmetric phase R 16" : (3.52) = M N All this agrees with diagrammatic studies of the large N approximation [121] and numerical solutions of the "ow equation [111,107,36] for the exponential cutoE function (2.17). The solution for the region u ¡ 0 can be found along similar lines √ √ + −U U N √ k + −U √ √ : (3.53) + # − #0 (k) = 2 −U ln − ln 8" − −U k − −U Up to corrections involving inverse powers of this yields the implicit relation √ √ 8"2 (#0 (k) − #) √ : (3.54) −U = k − (k + −U ) exp − N −U √ One infers that −U is always smaller than k and maximal for #=0. In the phase with spontaneous symmetry breaking where #0 (k = 0) = #0 ¿ 0 the behavior near the origin is given for small k by 2 8"2 (#0 − #) 2 : (3.55) U (#) = −k 1 − 2 exp − Nk For k → 0 the validity of this region extends to the whole region 0 6 # ¡ #0 . The “inner part” of the average potential becomes "at, 26 in agreement with the general discussion of the approach to convexity in Section 3.6. 26

The exponential approach to the asymptotic form U = −k 2 may cause problems for a numerical solution of the "ow equation. The analytical information provided here can be useful in this respect.

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It is interesting to compare these results with the large N limit of the scaling solution which obeys N 1 1 #˜ − 2 u : (3.56) u = 2 4" 1 + u For u (<) = 0 this yields for the minimum of u N <= 2 : 4" Taking #-derivatives of Eq. (3.56) and de$ning = u (<); - = u (<), we $nd
(3.57)

(3.58)

2 32"4 1 4"2 = ; - = 2 = : (3.59) < N 3 3N 2 We emphasize that we have chosen the quartic potential (3.36) only for the simplicity of the presentation. The general solution can be evaluated equally well for other microscopic potentials (provided u ¿ − 1). This can be used for an explicit demonstration of the universality of the critical behavior. It also may be employed for an investigation of tricritical behavior which can happen for more complicated forms of the microscopic potential. In summary, the exact solution of the "ow equation for the average potential in the limit N → ∞ provides a very detailed quantitative description for the “transition to complexity”. =

3.4. Graphical representation and resummed perturbation theory One is often interested in the "ow equation for some particular n-point function k(n) . Here k(n) is de$ned by the nth functional derivative of k evaluated for a $xed $eld, for example = 0. Correspondingly, the "ow equation for k(n) can be obtained by taking n functional derivatives of the exact "ow equation (2.19). We give here a simple prescription for how the r.h.s. of the "ow equation can be computed from the usual perturbative Feynman graphs. Only one-loop diagrams are needed, but additional vertices are present and the propagators and vertices in the graphs correspond to full propagators and full vertices as derived by functional diEerentiation of k . Our purpose is achieved by writing Eq. (2.19) formally in the form 9t k = 12 Tr 9˜t ln(k(2) + Rk ) ;

(3.60)

where the derivative 9˜t acts only on Rk and not on k , i.e. 9˜t = (9Rk =9t)9=9Rk . If we forget for a moment possible problems of regularization, we may place 9˜t in front of the trace so that 9t k = 9˜t ˆ k(1) ;

(3.61)

where ˆ k(1) is the one-loop expression with “renormalization group improvement”, i.e. full vertices and propagators instead of the classical ones. Functional derivatives commute with 9t and 9˜t . In consequence, the right-hand side of the exact "ow equation for k(n) can be evaluated by the following procedure: (1) Write down the one-loop Feynman graphs for k(n) . (2) Insert “renormalized” couplings instead of the classical ones. This also introduces a momentum dependence of the vertices which may not be present in the classical couplings. There may also be contributions from higher vertices

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261

(e.g. six-point vertices) not present in the classical action. (The classical couplings vanish, the renormalized eEective ones do not.) All renormalized vertices in the graphs correspond to appropriate functional derivatives of k . (3) Replace the propagators by the full average propagator (k(2) +Rk )−1 (evaluated at a $xed $eld). (4) Apply the formal diEerentiation 9˜t . More precisely, the derivative 9˜t should act on the integrand of the one-loop momentum integral. This makes the expression $nite so that regularization is of no worry. The result is the exact "ow equation for k(n) . An example is provided by Eq. (2.30). Standard perturbation theory can easily be recovered from an iterative solution of the "ow equation (3.60). Starting from the leading or “classical” contribution k(0) ≡ one may insert this instead of k in the r.h.s. of (3.60). Performing the t-integration generates the one-loop contribution k − = 12 Tr {ln((2) + Rk ) − ln((2) + R )} ;

(3.62)

where we recall that Rk → 0 for k → 0. We observe that the momentum integration in the r.h.s. of (3.62) is regularized in the ultraviolet through subtraction of ln((2) + R ). This is a type of implicit Pauli–Villars regularization with the heavy mass term replaced by a momentum-dependent piece R in the inverse propagator. With suitable chirally invariant R [79] this can be used for a regularization of models with chiral fermions. Also gauge theories can, in principle, be regularized in this way, but care is needed since has to obey identities re"ecting the gauge invariance [47,11,48–74]. Going further, the two-loop contribution is obtained by inserting the one-loop expression for k(2) as obtained from Eq. (3.62) into the r.h.s. of (3.60). It is easy to see that this generates two-loop integrals. Only the classical inverse propagator (2) and its functional derivatives appear in the nested expressions. They are independent of k and the integration of the approximated "ow equation is straightforward. It is often useful to replace the perturbative iteration sketched above by a new one which involves full propagators and vertices instead of the classical ones. This will amount to a systematic resummed perturbation theory [122]. We start again with the lowest order term k(0) [] = [] ;

(3.63)

where is now some conveniently chosen scale (not necessarily the ultraviolet cutoE). In the next step, we write equation (3.60) in the form 9t k = 12 Tr 9t ln(k(2) + Rk ) − 12 Tr{9t k(2) (k(2) + Rk )−1 } :

(3.64)

Here 9t k(2) can be inferred by taking the second functional derivative of Eq. (3.60) with respect to the $elds . Eq. (3.64) can be taken as the starting point of a systematic loop expansion by counting any t-derivative acting only on k or its functional derivatives as an additional order in the number of loops. From (3.60) it is obvious that any such derivative indeed involves a new momentum loop. It will become clear below that in the case of weak interactions it also involves a higher power in the coupling constants. The contribution from the $rst step of the iteration k(1) can now be de$ned by k = k(0) + k(1)

(3.65)

k(1) = 12 Tr{ln(k(2) + Rk ) − ln(k(2) + R )} :

(3.66)

with

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J. Berges et al. / Physics Reports 363 (2002) 223 – 386

In contrast to Eq. (3.62) the r.h.s. involves now the full ($eld-dependent) inverse propagator k(2) , and, by performing suitable functional derivatives, the full proper vertices. Putting k =0 the resummed one-loop expression (3.66) resembles a Schwinger–Dyson [106] or gap equation, but in contrast to those only full vertices appear. For example, for k = 0 the $rst iteration to the inverse propagator is obtained by taking the second functional derivative of Eq. (3.36) (2) −1 1 2 (2) (2) ((1) )ab (q; q ) = Tr 2 a (q)U b (q ) (2) −1 (2) (2) −1 (2) 1 − Tr 2 a (q) U b (q ) − regulator terms

(3.67)

(2) and and involves the proper three- and four-point vertices. By adding the lowest order piece (0) approximating the vertices by their lowest order expressions, Eq. (3.67) reduces to the standard gap equation for the propagator in a regularized form. Assuming that this is solved (for example, numerically by an iterative procedure) we see that the resummed one-loop expression (3.67) already involves arbitrarily high powers in the coupling constant, and, in particular, contains part of the perturbative two-loop contribution. The remaining part of the perturbative two-loop contribution appears in the resummed two-loop contribution. A systematic resummed perturbation theory (SRPT) can be developed along these lines [122]. Systematic resummed perturbation theory is particularly convenient for a computation of ultraviolet $nite n-point functions (as the 6 coupling) or diEerences of n-point functions at diEerent momenta. In this case the momentum integrals in the loop expansion are dominated by momenta for which renormalized vertices are appropriate. 27 For → ∞ all dependence on the eEective ultraviolet cutoE is absorbed in the renormalized couplings. In this limit, the regulator terms vanish as well. In this context, one can combine SRPT with approximate solutions of the "ow equation. In fact, the “closure” of the "ow equation by SRPT instead of truncation constitutes an interesting alternative: Eq. (3.60) is a functional diEerential equation which cannot be reduced to a closed system for a $nite number of couplings (for $nite N ). For example, the beta function for the four-point vertex (the fourth functional derivative of the r.h.s. of Eq. (3.60)) involves not only two-, three- and four-point functions, but also up to six-point functions. Approximate solutions to the "ow equation often proceed by truncation. For example, contributions involving the $ve- and six-point function could be neglected. As an alternative, these higher n-point functions can be evaluated by SRPT. We observe that the momentum integrals relevant for the higher n-point functions (also for diEerences of lower n-point functions at diEerent momenta) are usually dominated by the low momentum modes with q2 ≈ k 2 . This motivates the use of SRPT rather than standard perturbation theory for this purpose. A successful test of these ideas is provided by a computation of the -function for the quartic scalar coupling for d = 4 via an evaluation of the momentum dependence of the vertices appearing in the "ow equation by SRPT [158]. For a small coupling this lead to the universal two-loop -function, without ever having to compute two-loop momentum integrals.

27

A direct use of the “gap equation” (3.67) for the mass term is no improvement compared to the standard Schwinger– Dyson equation since high momentum modes play an important role in the mass renormalization.

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3.5. Exact ;ow of the propagator We now turn to the exact "ow equation for the propagator. This can be derived either by the graphical rules of the last subsection or by the functional derivatives sketched in Eq. (2.30). We concentrate here on the inverse propagator in the Goldstone direction in a constant background $eld (see Eq. (2.38)) G −1 (#; q2 ) = Zk (#; q2 )q2 + Uk (#) = M0 (#; q2 ) − Rk (q) :

(3.68)

According to Eq. (2.30), its "ow involves the three- and four-point functions. We notice that the three-point function k(3) vanishes for # = 0. We parametrize the most general form of the inverse propagator in an arbitrary constant background a by dd q 1 (2)k = {(Uk (#) + Zk (#; q2 )q2 ) a (q)a (−q) 2 (2")d 1 + a b (2Uk (#) + Yk (#; q2 )q2 ) a (q)b (−q)} (3.69) 2 and note that Eq. (3.69) speci$es all 1PI n-point functions with at most two non-vanishing momenta. Similarly the eEective interactions which involve up to four $elds with non-vanishing momentum are d d q1 d d q2 1 (3)k = {a k(1) (#; q1 ; q2 ) a (q1 )b (q2 )b (−q1 − q2 ) 2 (2")d (2")d 1 + a b c -(1) k (#; q1 ; q2 ) a (q1 )b (q2 )c (−q1 − q2 )} ; 3 d d q1 d d q2 d d q3 (2) 1 { (#; q1 ; q2 ; q3 ) a (q1 )a (q2 )b (q3 )b (q4 ) (4)k = 8 (2")d (2")d (2")d k + 2a b -(2) k (#; q1 ; q2 ; q3 )a (q1 )b (q2 )c (q3 )c (q4 ) 1 + a b c d Fk (#; q1 ; q2 ; q3 ) a (q1 )b (q2 )c (q3 )d (q4 )} 3

(3.70)

with q4 = −(q1 + q2 + q3 ); # = 12 a a . The couplings k(i) ; -(i) k and Fk are appropriately symmetrized in the momenta (including symmetrization in the momentum of the last $eld, i.e. q4 ). The eEective vertices are connected to Uk ; Zk and Yk by continuity k(1) (#; q1 ; q2 ) = Uk (#) + (q2 (q1 + q2 ))Zk (#; (q2 (q1 + q2 ))) + 12 q12 Yk (#; q12 ) + Tk(1) (#; q1 ; q2 ) k(2) (#; q1 ; q2 ; q3 ) = Uk (#) − (q1 q2 )Zk (#; −(q1 q2 )) − (q3 q4 )Zk (#; −(q3 q4 )) + 12 (q1 + q2 )2 Yk (#; (q1 + q2 )2 ) + Tk(2) (#; q1 ; q2 ; q3 ) 1 -(1) k (#; q1 ; q2 ) = Uk (#) − 2 {(q1 q2 )Yk (#; −(q1 q2 ))

+ (q1 → −(q1 + q2 )) + (q2 → −(q1 + q2 ))} + T-(1) k (#; q1 ; q2 )

264

J. Berges et al. / Physics Reports 363 (2002) 223 – 386 -(2) k (#; q1 ; q2 ; q3 ) = Uk (#) − (q3 q4 )Zk (#; −(q3 q4 ))

+ 12 {(q1 (q1 + q2 ))Yk (#; (q1 (q1 + q2 )) + (q1 → q2 )} − 12 (q1 q2 )Yk (#; −(q1 q2 )) + T-(2) k (#; q1 ; q2 ; q3 )

(3.71)

Fk (#; q1 ; q2 ; q3 ) = U (4) (#) − 12 {(q1 ; q2 )Yk (#; −(q1 q2 )) + 5 permutations} + TFk (#; q1 ; q2 ; q3 ) : In fact, we require that Eq. (3.70) coincides with Eq. (3.69) if only two of the momenta are non-vanishing, without the corrections T-; T; TF being involved. In particular, Tk(1) and T-(1) k vanish for q1 = 0; q2 = 0 or (q1 + q2 ) = 0, and, similarly, Tk(2) ; T-(2) and TF vanish if two of the k k four momenta q1 ; q2 ; q3 or q4 are zero. In terms of these couplings the exact "ow equation for G −1 (#; q2 ) reads dd p 1 −1 2 9t Rk (p)[4#{M1−2 (#; p2 )M0−1 (#; (p + q)2 )(k(1) (#; p; q))2 9t G (#; q ) = d 2 (2") + M0−2 (#; p2 )M1−1 (#; (p + q)2 )(k(1) (#; −q − p; q))2 } − M0−2 (#; p2 ){(N − 1)k(2) (#; q; −q; p) + 2k(2) (#; q; p; −q)} − M1−2 (#; p2 ){k(2) (#; q; −q; p) + 2#-(2) k (#; p; −p; q)}] dd p ˜ 1 9t {4#(k(1) (#; −q − p; q))2 M0−1 (#; p2 )M1−1 ((q + p)2 ) =− d 2 (2") − [(N − 1)k(2) (#; q; −q; p) + 2k(2) (#; q; p; −q)]M0−1 (#; p2 ) −1 2 − [k(2) (#; q; −q; p) + 2#-(2) k (#; p; −p; q)]M1 (#; p )} :

(3.72)

Subtraction of the mass term in Eq. (3.68) yields 9t Zk (#; q2 ) =

q2 1 −1 2 #; Zk (9 G (#; q ) − 9 U (#)) ≡ −% t t k k q2 k2

(3.73)

and the exact expression for the anomalous dimension is d ln Zk (#0 (k); k 2 ) dt 2k 2 9 9#0 Zk (#0 ; k 2 ) 2 2 =k 2 − : = %k (#0 ; 1) − Z (# ; q )| k 0 q Zk 9q2 9t Zk (#0 ; k 2 )

6=−

(3.74)

Alternative de$nitions of 6 related to the "ow of Zk (#0 ; q2 → 0) will be discussed in later sections. Expressed in terms of the scaling variables the anomalous dimension can be extracted equivalently from condition (3.4), i.e. d zk (<)=dt = 0 or 9t z(<) = −zk (<) 9t <

(3.75)

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with < = Zk k 2−d #0 the minimum of u(#). ˜ The "ow of zk (#) ˜ obeys 9 ˜ y)|y=1 : 9t z(#) ˜ = 6z(#) ˜ + (d − 2 + 6)#z ˜ (#) ˜ − %k (#; ˜ 1) + 2 Tz(#; 9y

265

(3.76)

We note that for the scaling solution the r.h.s. of Eq. (3.75) vanishes because 9t < = 0. An exact expression for %k is computed for a sharp cutoE in Appendix B. Here we evaluate %k in $rst order in the derivative expansion. In this order the momentum dependence of Z ; Z ; Y and Y is neglected and we can omit the terms Tk(i) ; T-(i) k and TFk . Inserting Eq. (3.71) into (3.72) yields d d p −1 ˜ q2 1 Z 9t #(2U + p2 Y )2 M0−1 (p2 )[M1−1 ((p + q)2 ) − M1−1 (p2 )]=q2 %k #; 2 = k 2 (2")d (pq) (pq) 2 Y −2 2 Z + # (4U + 2p Y ) 1+2 2 q q ! (pq) (pq) 2 2 1+2 2 Y −2 2 Z M0−1 (p2 )M1−1 ((p + q)2 ) +q q q " − [(N − 1)Z + Y ]M0−1 (p2 ) − (Z + 2#Z )M1−1 (p2 ) : (3.77) In this approximation we can also neglect the term 9Tz=9y in Eq. (3.76), so that for the scaling solution (9t z(#) ˜ = 0) the anomalous dimension is given by 6y6 = %k (<; y6 ) − (d − 2 + 6y6 )
(3.78)

Here we show the freedom in the de$nition of 6 by the subscript y6 which indicates the value of q2 =k 2 for which Zk is de$ned. The optimal choice is presumably y6 = 1 (see above) which corresponds to the hybrid derivative expansion. An algebraic simpli$cation occurs, however, for y6 = 0, corresponding to the “direct” derivative expansion. For a smooth cutoE, the propagators M0−1 ((p + q)2 ) and M1−1 ((p + q)2 ) can be expanded for q2 → 0 M0−1 ((p + q)2 ) = M0−1 (p2 ) − (q2 + 2pq)(Z + R˙ k (p))M0−2 (p2 ) # $ + (q2 + 2(pq))2 (Z + R˙ k (p))2 M0−3 (p2 ) − 12 R' k (p)M0−2 (p2 ) + · · ·

(3.79)

with R˙ k = 9Rk =9p2 and Z replaced by Z + #Y in a similar expression for M1−1 ((p + q)2 ). Since integrals over odd powers of p vanish, one concludes that 60 is well de$ned for a smooth cutoE. On the other hand, one observes [8] in the sharp cutoE limit a divergence in the derivative expansion limy6 →0 6y6 ∼ (y6 )−1=2 . We emphasize that the dependence of 6 on the choice of y6 is a pure artifact of the truncation. If we go beyond the derivative expansion the scaling solution is characterized by the same anomalous dimension 6 independently of y6 . Nevertheless, for practical calculations some type of derivative expansion is often crucial. For the direct derivative expansion it has been argued [100] that only smooth and rapidly falling cutoEs (like the exponential cutoE (2.16)) and only the "ow equations for k (as opposed to the one for the Wilsonian eEective action) have satisfactory convergence properties. However, the constraints on the precise formulation seem much less restrictive for the

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hybrid derivative expansion. This is strongly suggested by the computation of 6 for a sharp cutoE in Appendix B. A detailed investigation is presented in [123]. The result for %k (#; 0) in $rst order in the derivative expansion can be found in [8,42]. Here we present explicitly only the lowest order in the derivative expansion for which all terms ∼ Z ; Z and Y are neglected. One $nds 16vd 2 < md2; 2 (0; 2<) ; 60 = %k (<; 0) = d 1 2(n1 +n2 −1)−d n1 +n2 −2 ∞ d Zk d x xd=2 9˜t mn1 ;n2 (w1 ; w2 ) = − k 2 0 2

×{P˙ (x)(P(x) + Zk k 2 w1 )−n1 (P(x) + Zk k 2 w2 )−n2 } ; (3.80) where P(x) = Zk x + Rk (x) and P˙ = 9P=9x. In particular, the limit w2 1 is simple for all cutoE functions for which R˙ k (x) does not exceed Zk by a large factor in some region of x, namely lim mdn1 ;n2 (w1 ; w2 ) = w2−n2 mdn1; 0 (w1 ) :

w2 →∞

For d = 2 and neglecting terms ∼ 6 one has the identity 1 ∞ 2 2 m2; 0 (0) = − d x x9˜t {P˙ (x)=P 2 (x)} 2 0 2 ∞ d P˙ (x)x2 =1 : dx = dx P 2 (x) 0

(3.81)

(3.82)

For all cutoEs in this class one concludes that 2v2 1 lim 60 = = : (3.83) <→∞ < 4"< This independence on the precise choice of the cutoE is directly related to the universality of the one-loop -function for the two-dimensional non-linear A-models (cf. Eq. (2.55)). 3.6. Approach to the convex potential for spontaneous symmetry breaking For k → 0 the eEective potential U is a convex function of [124], i.e. U0 (#) ¿ 0. This is not only a formal property due to the Maxwell construction of thermodynamic potentials. The convexity re"ects directly the eEect of "uctuations. Indeed, as long as all "uctuations are not included the average potential Uk for k ¿ 0 does not need to be convex—and it is actually not convex in case of spontaneous symmetry breaking. One therefore has to understand quantitatively how the "uctuations lead to an approach to convexity for k → 0. The “"attening” of the non-convex “inner region” of the potential is crucial to the computation of the nucleation rate in case of $rst-order phase transitions. This rate receives an exponential suppression factor from the free energy of the saddle point solution which interpolates between two local minima of Uk and corresponds to tunneling through a (non-convex) potential barrier. We discuss this issue in detail in Section 6. The discussion of this subsection is relevant for average potentials Uk () with several local minima for all k ¿ 0, and not for the case where only a single minimum survives for small k (e.g. the symmetric phase of O(N ) models).

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267

DiEerent pictures can describe how "uctuations lead to a "attening of the potential. In a $rst approach [77] the average potential has been computed in a saddle-point approximation. In the inner (non-convex) region of the potential the relevant saddle point does not correspond to a constant $eld but rather to a spin wave (for N ¿ 1) or a kink (for N = 1). Because of the existence of these non-trivial extrema the average potential shows in case of spontaneous symmetry breaking a generic behavior Uk (#) = V (k) − pk 2 #

(3.84)

for small # and k. (Here p is a positive constant and V (k) is independent of #.) Similar non-trivial saddle points have been discussed [125] in the context of the exact Wegner–Houghton [3] equation. This “classical renormalization” con$rms the generic behavior (3.84). Since the "ow equation (2.35) for the average potential is exact, there is, in principle, no need to investigate special non-perturbative saddle point solutions. The eEects of all non-perturbative solutions like spin waves, kinks or instantons in appropriate models are fully included in the exact equation. For an appropriate truncation the solution of Eq. (2.35) should therefore exhibit directly the approach to convexity. A $rst discussion of the "ow of the curvature of Uk around the origin has indeed shown [78] the generic behavior (3.84). We extend this discussion here to the whole “inner region” of the potential. In this region the poles of the threshold functions for negative arguments dominate for small k. This leads indeed to a "attening of the average potential and to convexity for k → 0, with the universal behavior (3.84). The approach to convexity depends only on a few characteristics of the infrared cutoE Rk and is otherwise model independent. It is dominated by poles of the threshold functions for negative arguments. Therefore, we need the behavior of the threshold functions for negative w. For our discussion to become independent of truncations we generalize ld0 for a momentum-dependent wave function renormalization 28 1 ∞ d dyyd=2 sk (y)(p(#; ˜ y) + w)−1 ; lU0 (w) = 2 0 p(#; ˜ y) = (z(#; ˜ y) + rk (y))y :

(3.85)

We consider a class of infrared cutoEs with the property that the function p(#; ˜ y) has a minimum at y0 ¿ 0 for a given #, ˜ with p(#; ˜ y0 ) = p0 (#). ˜ In a range of (#; ˜ w) for which j = p0 + w ¿ 0

(3.86)

and for small values of j, integral (3.85) is dominated by the region y ≈ y0 . In the vicinity of the minimum of p(y) we can expand p(y) = p0 + a2 (y − y0 )2 + · · ·

(3.87)

d We have not indicated in our notation here the dependence of lU0 on 6 and the function z(y; #) ˜ = zk (#) ˜ + Tzk (#; ˜ y). d d U The exact "ow equation for the potential is obtained by the replacement l0 → l0 ; TCk = 0. 28

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and approximate the threshold function 29 (for sk (y0 ) ¿ 0) by ∞ Uld0 (w) = 1 dyy0d=2 sk (y0 )[a2 (y − y0 )2 + j]−1 2 0 " = y0d=2 sk (y0 )(a2 j)−1=2 : (3.88) 2 d One observes that lU0 has a singularity 30 ∼ j−1=2 . This is the generic behavior 31 for smooth threshold functions with suIciently large Rk (0). For N ¿ 1 the contribution of the radial mode cannot be stronger than that from the Goldstone modes and the exact evolution equation for the potential can be approximated in the vicinity of the singularity at u = −p0 by ˜ = 2c(p ˜ 0 + u )−1=2 : 9t u + du − (d − 2 + 6)#u

(3.89)

In the range of its validity we can solve Eq. (3.89) by the method of characteristics, using Eqs. (2.56) and (2.57) with k (u

) = c(p ˜ 0 + u )−3=2 :

(3.90)

For 6 = 0 the most singular terms yield 1 1 (p0 + u )− 2 + p0d=2 #˜ (−u )1−d=2 = f(u e2t ) : c˜ ˜ at t and reads Here f is $xed by the “initial value” u0 (#) 1 1 ˆ e2(t −t0 ) )(−u )1−d=2 e(2−d)(t −t0 ) : f(u e2t ) = (p0 + u e2(t −t0 ) )− 2 + p0d=2 #(u c˜ ˜ The function #ˆ is obtained by inversion of u0 (#): ˜ = #˜ : #(u ˆ 0 (#)) For simplicity, we consider a linear approximation 0 ¡ j0 1, or 1 #(u ˆ e2(t −t0 ) ) = u e2(t −t0 ) + <0 : 0

(3.91)

(3.92)

(3.93) ˜ u (#) 0

= 0 (#˜ − <0 ) with 0 <0 = p0 − j0 ; (3.94)

For d ¿ 2 the approximate expression for Eq. (3.91) in the limit kk0 reads (with e2(t −t0 ) =k 2 =k02 ) d−2 d−2 −2 p < −2 k 2(d−2) k k # ˜ c ˜ 0 0 1− u = −p0 + + p0−3=2 : (3.95) c˜ k0 <0 k0 <0 k0 We see that u +p0 remains always positive. The singularity is approached, but never crossed. (These features hold for generic u0 (#) ˜ ¿ − p0 .) In consequence, the derivative of the average potential is 29 See Ref. [78] for a detailed discussion. The quantities p0 and a2 depend on #. ˜ If w(#) ˜ is monotonic in the appropriate range of #˜ we can consider them as functions of w and expand p0 (w) = p0 (−p0 ) + jp0 + · · ·, a2 (w) = a2 (−p0 ) + ja2 + · · ·. Up to higher orders in j we can neglect the #-dependence ˜ of p0 and a2 and use constants p0 ≡ p0 (−p0 ); a2 ≡ a2 (−p0 ). 30 Near the singularity the correction is ∼ ln j or smaller. 31 It is, however, not realized for a sharp cutoE for which the threshold function diverges only logarithmically.

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always larger than −p0 k 2 9Uk ¿ − p0 k 2 9#

269

(3.96)

and the potential becomes indeed convex for k → 0. Furthermore, this implies that the region where Eq. (3.96) is valid extends towards the potential minimum for k → 0. For small k, a reasonable approximate form for the phase with spontaneous symmetry breaking and # ¡ #0 (k) is Uk (#) = V˜ (k) − p0 k 2 # + C#20 (k)k 2(d−1) (#0 (k) − #)−1 ;

(3.97)

where the constant C can be extracted by comparing with Eq. (3.95). This agrees with Eq. (3.84). The validity of approximation (3.97) breaks down in a vicinity of #0 (k) which shrinks to zero as k → 0. In this vicinity the behavior for # ¡ #0 (k) is essentially determined by analytic continuation from the region # ¿ #0 (k). For a non-vanishing constant anomalous dimension 6 we have to replace 2d 2−6 2 d → d6 = ; t → t6 = t; c˜ → c˜6 = c˜ : (3.98) 2−6 2 2−6 This is important for d=2; N =2 where the anomalous dimension governs the approach to convexity. Indeed, at the Kosterlitz–Thouless phase transition or in the low-temperature phase the anomalous dimension remains strictly positive for all values 32 of k. The above discussion (3.95) of the approach to convexity remains valid, with (k=k0 )d−2 replaced by (k=k0 )6 . We $nally turn to the case N = 1 which we only discuss for 6 = 0. Near the pole at j = u + 2#u ˜ + p0 → 0 the evolution equation for the potential is now approximated by ˜ + 2c(p ˜ 0 + u + 2#u ˜ )−1=2 : 9t u = −du + (d − 2)#u

(3.99)

For d ¿ 2 and small k an iterative solution can be found for |2#u ˜ ||p0 + u |. In lowest order it is given by (3.95) d−2 !−2 p < −2 k 2(d−2) k #˜ 0 0 1− p0 + u = : (3.100) c˜ k0 <0 k0 One $nds that 4#˜ 2#u ˜ = p0 + u <0

k k0

d−2

#˜ 1− <0

k k0

d−2 !−1

;

(3.101)

becomes indeed negligible for k → 0. For this solution one obtains the same approach to convexity as for N ¿ 1. It is instructive to compare the “power-law approach” (3.97) towards the asymptotic behavior for the “inner region” Uk = −p0 k 2 with the “exponential approach” (3.55). It is obvious that the $rst is much easier to handle for numerical solutions. Indeed, already a tiny error in the numerical computation of (3.55) may lead to vanishing or negative values of p0 k 2 +Uk for which the threshold functions are ill de$ned. This is often a source of diIculties for numerical solutions of the partial 32

This is in contrast to d = 3 or 2; N = 1 where the anomalous dimension vanishes for k → 0, except for the critical hypersurface of the phase transition.

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diEerential equations. For numerical investigations of the approach to convexity it is advantageous to use smooth threshold functions like (2.17) or (3.21) multiplied by ¿ 1, say = 4. Then a simple truncation of the momentum dependence of the propagator like (2) ∼ Zk q2 + const. obeys the conditions for a power-law approach (3.97), namely a minimum of p(y) for y0 ¿ 0 (3.85) and sk (y0 ) ¿ 0 (3.89). On the other hand, the above simple truncation fails to describe the appropriate approach to convexity for cutoE (2.17). This is due to the fact that the minimum of p(y) occurs for y0 =0 in this case. One learns that for this cutoE the simple truncation of the propagator is insuIcient for the inner region of the potential. A correct reproduction of the exact bound Uk ¿ − Zk k 2 (2.16) needs an extension of the truncation for the momentum dependence. For practical investigations of problems where the precise approach to convexity is not relevant one may use directly the knowledge of the exact result (2.16). Neglected eEects of an insuIcient truncation for the “inner region” may then be mimicked by a modi$cation “by hand” of the threshold function in the immediate vicinity of the pole, for example by imposing form (3.88). Let us summarize the most important result of this subsection. In the case of spontaneous symmetry breaking the solution of the exact "ow equation for the average potential leads to a universal form Uk (#) ≈ −p0 k 2 for small # and k. As k → 0 the region of validity of this behavior extends towards the minimum of the potential. Eq. (3.84) becomes valid in a range 0 6 # ¡ #0 (k) − 1(k) with 1(k) ¿ 0 and limk →0 1(k) = 0. The potential becomes convex, in agreement with general properties and the exact bound (2.16). 4. O(N )-symmetric scalar models 4.1. Introduction In this section we study the N -component scalar model with O(N )-symmetry in three and two dimensions. The case of four-dimensional quantum $eld theories will be considered in Section 8. The O(N ) model serves as a prototype for investigations concerning the restoration of a spontaneously broken symmetry at high-temperature. For N = 4 the model describes the scalar sector of the electroweak standard model in the limit of vanishing gauge and Yukawa couplings. It is also used as an eEective model for the chiral phase transition in QCD in the limit of two quark "avors [24 –27], which will be discussed in Section 8. In condensed matter physics N = 3 corresponds to the Heisenberg model used to describe the ferromagnetic phase transition. There are other applications like the helium super"uid transition (N = 2), liquid–vapor transition (N = 1) or statistical properties of long polymer chains (N = 0). In three dimensions we will concentrate on the computation of the equation of state near the critical temperature of the second-order phase transition. The equation of state for a magnetic system is speci$ed by the free energy density as a function of arbitrary magnetization and temperature T . All thermodynamic quantities can be derived from the function U (; T ). For example, the response of the system to a homogeneous magnetic $eld H follows from 9U=9 = H . This permits the computation of for arbitrary H and T . There is a close analogy to quantum $eld theory at non-vanishing temperature. Here U corresponds to the temperature dependent eEective potential as a function of a scalar $eld . For instance, in the O(4) symmetric model for the chiral phase transition in two "avor QCD the meson $eld has four components. In this picture, the average light quark

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271

mass mˆ is associated with the source H ∼ mˆ and one is interested in the behavior during the phase transition (or crossover) for H = 0. The temperature and source dependent meson masses and zero momentum interactions are given by derivatives of U (cf. Section 8). The applicability of the O(N )-symmetric scalar model to a wide class of very diEerent physical systems in the vicinity of the critical temperature Tc is a manifestation of universality of critical phenomena. There exists a universal scaling form of the equation of state in the vicinity of the second-order phase transition. The quantitative description of this scaling form will be the main topic here [36,107]. The calculation of the eEective potential U (; T ) in the vicinity of the critical temperature of a second-order phase transition is an old problem. One can prove through a general renormalization group analysis [2,3] the Widom scaling form [126] of the equation of state 33 ˜ H = f((T − Tc )=1= ) :

(4.1)

Only the limiting cases → 0 and ∞ are described by critical exponents and amplitudes. For classical statistics in three dimensions we present in Section 4.2 a computation of the eEective potential U0 = limk →0 Uk = U=T from a derivative expansion of the eEective average action with a uniform wave function renormalization factor. The approximation takes into account the most general $eld dependence of the potential term. This will allow us to compute the non-analytic behavior of U in the vicinity of the second-order phase transition. From U the universal scaling form of the equation of state is extracted in Section 4.3. We demonstrate in Section 4.4 that the non-universal aspects can be described by these methods as well. The example of carbon dioxide is worked out in detail. Going beyond the lowest order in a derivative expansion the approximation used in these sections takes into account the most general $eld dependence of the wave function renormalization factor. Section 4.7 describes an application to the critical swelling of long polymer chains (N = 0). For two dimensions, we present in Section 4.8 a quantitative description of the “Kosterlitz–Thouless” transition (N = 2). 4.2. The running average potential In this section we compute the eEective average potential Uk (#) directly in three dimensions [36,107]. Here # = 12 a a and a denotes the N -component real scalar $eld. For k → 0 one obtains the eEective potential U0 (#) ≡ U (#)T −1 . We omit the factor T −1 in the following. It is related to the (Helmholtz) free energy density fH by fH =T = U0 − 2#9U0 =9# = U0 − H . In the phase with spontaneous symmetry breaking the minimum of the potential occurs for k = 0 at #0 = 0. In the symmetric phase the minimum of Uk (#) ends at #0 = 0 for k = 0. The two phases are separated by a scaling solution for which Uk =k 3 becomes independent of k once expressed in terms of a suitably rescaled $eld variable and the corresponding phase transition is of second order. Our truncation is the lowest order in a derivative expansion of k 1 d a : (4.2) k = d x Uk (#) + Zk 9 a 9 2 33

In our notation, we frequently suppress an appropriate power of a suitable “microscopic” length scale −1 which is used to render quantities dimensionless.

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For the potential term we keep the most general O(N )-symmetric form Uk (#), whereas the wave function renormalization is approximated by one k-dependent parameter. We study the eEects of a $eld-dependent Zk for the Ising model in Section 4.4. Next order in the derivative expansion would be the generalization to a #-dependent wave function renormalization Zk (#) plus a function Yk (#) accounting for a possible diEerent index structure of the kinetic term for N ¿ 2. Going further would require the consideration of terms with four derivatives and so on. We employ in this section the exponential infrared cutoE (2.17). For a study of the behavior in the vicinity of the phase transition it is convenient to work with dimensionless renormalized $elds 34 #˜ = Zk k 2−d # ; ˜ = k −d Uk (#) : uk (#)

(4.3)

The scaling form of the evolution equation for the eEective potential has been derived in Section 3.1. With the truncation of Eq. (4.2) the exact evolution equation for uk ≡ 9uk =9#˜ reduces to the partial diEerential equation 9uk ˜ k = (−2 + 6)uk + (d − 2 + 6)#u 9t − 2vd (N − 1)uk ld1 (uk ; 6) − 2vd (3uk + 2#u ˜ k )ld1 (uk + 2#u ˜ k ; 6) ;

(4.4)

where t = ln(k=), v3 = 1=8"2 , primes denote derivatives with respect to #˜ and is the ultraviolet cutoE of the theory. The “threshold” functions ldn (w; 6) ≡ ldn (w; 6; z = 1) are discussed in Section 3.2. We evaluate these functions numerically for cutoE (2.17). Finally, the anomalous dimension is de$ned here by the q2 -derivative of the inverse propagator at q2 = 0. According to Section 3.5 it is given in our truncation by 35 [42,111,107] 16vd 2 d < m2; 2 (2<) 6(k) = (4.5) d with < the location of the minimum of uk and the quartic coupling uk (<) = 0 ; uk (<) = :

(4.6)

The function md2; 2 is given by ∞ md2; 2 (w) = dy yd=2−2

1 + r + y 9r=9y (1 + r)2 [(1 + r)y + w]2 0 9 2 1 1 9r 9r 9r 2 +2 y + : 2y r − 2y 1 + r + y 9y 9y 9y 9y (1 + r)y (1 + r)y + w (4.7)

34

We keep the number of dimensions d arbitrary and specialize only later to d = 3. We neglect here for simplicity the implicit, linear 6-dependence of the function md2; 2 . We have veri$ed numerically that this approximation has only a minor eEect on the value of 6. 35

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273

1 k

0

u'k

0.5

0 k scaling solution

-0.5

0

0.02

0.04

~ ρ

0.06

0.08

0.10

Fig. 2. The evolution of uk (#) ˜ as k is lowered from to zero for N = 1. The initial conditions (bare couplings) have been chosen such that the scaling solution is approached before the system evolves towards the symmetric phase with uk (0) ¿ 0. The concentration of lines near the scaling solution ("at diagonal line) indicates that the model is close to criticality. The scaling solution for u(#) ˜ has a minimum for #˜ ≈ 0:04.

We point out that the argument 2< turns out generically to be of order one for the scaling solution. Therefore, < ∼ −1 and the mass eEects are important, in contrast to perturbation theory where they are treated as small quantities ∼ . At a second-order phase transition there is no mass scale present in the theory. In particular, one expects a scaling behavior of the rescaled eEective average potential uk (#). ˜ This can be studied by following the trajectory describing the scale dependence of uk (#) ˜ as k is lowered from to zero. Near the phase transition the trajectory spends most of the “time” t in the vicinity of the k-independent scaling solution of Eq. (4.4) given by 9t u∗ (#) ˜ = 0. Only at the end of the running the “near-critical” trajectories deviate from the scaling solution. For k → 0 they either end up in the symmetric phase with < = 0 and positive constant mass term m2 so that uk (0) ∼ m2 =k 2 ; or they lead to a non-vanishing constant #0 indicating spontaneous symmetry breaking with < → Z0 k 2−d #0 . The equation of state involves the potential U0 (#) for temperatures away from the critical temperature. Its computation requires the solution for the running away from the critical trajectory which involves the full partial diEerential equation (4.4). In Fig. 2 we present the results of the numerical integration of Eq. (4.4) for d = 3 and N = 1. The function uk (#) ˜ is plotted for various values of t = ln(k=). The evolution starts at k = (t = 0) where the average potential is equal to the classical potential (no eEective integration of modes has been performed). We start with a quartic classical potential parameterized as u (#) ˜ = (#˜ − < ) :

(4.8)

We arbitrarily choose = 0:1 and $ne tune < so that a scaling solution is approached at the later stages of the evolution. There is a critical value
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the scaling solution. For the results in Fig. 2 a value < slightly smaller than
(4.9)

˜ stays near the scaling solution, the smaller the resulting value of #0 (k) when the The longer uk (#) system deviates from it. As this value determines the mass scale for the renormalized theory at k = 0, the scaling solution governs the behavior of the system very close to the phase transition, where the characteristic mass scale goes to zero. Another important property of the “near-critical” trajectories, which spend a long “time” t near the scaling solution, is that they become insensitive to the details of the classical theory that determine the initial conditions for the evolution. After uk (#) ˜ has evolved away from its scaling form u∗ (#), ˜ its shape is independent of the choice of for the classical theory. This property gives rise to the universal behavior near-second-order phase transitions. For the solution depicted in Fig. 2, uk (#) ˜ evolves in such a way that its minimum runs to zero with uk (0) subsequently increasing. Eventually the theory settles down in the symmetric phase with a positive constant renormalized mass term m2 = k 2 uk (0) as k → 0. Another possibility is that the system ends up in the phase with spontaneous symmetry breaking. In this case < grows in such a way that #0 (k) approaches a constant value for k → 0. The approach to the scaling solution and the deviation from it can also be seen in Fig. 3. The evolution of the running parameters <(t), (t) starts with their initial classical values, leads to $xed point values <∗ , ∗ near the scaling solution, and $nally ends up in the symmetric phase

Fig. 3. The scale evolution of <, and 6 for the initial conditions of Fig. 2. The plateaus correspond to the scaling solution.

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275

(< runs to zero). Similarly the anomalous dimension 6(k), which is given by Eq. (4.5), takes a $xed point value 6∗ when the scaling solution is approached. During this part of the evolution, the wave function renormalization is given by Zk ∼ k − 6∗

(4.10)

according to Eq. (2.38). When the parts of the evolution towards and away from the $xed point become negligible compared to the evolution near the $xed point—i.e. very close to the phase transition—Eq. (4.10) becomes a very good approximation for suIciently low k. This indicates that 6∗ can be identi$ed with the critical exponent 6. For the solution of Fig. 2 (N = 1) we $nd <∗ = 4:07 × 10−2 , ∗ = 9:04 and 6∗ = 4:4 × 10−2 . As we have already mentioned the details of the renormalized theory in the vicinity of the phase transition are independent of the classical coupling . Also the initial form of the potential does not have to be of the quartic form of Eq. (4.8) as long as the symmetries are respected. The critical theory can be parameterized in terms of critical exponents [127], an example of which is the anomalous dimension 6. These exponents are universal quantities that depend only on the dimensionality of the system and its internal symmetries. For our three-dimensional theory they depend only on the value of N and can be easily extracted from our results. We concentrate here on the exponent >, which parameterizes the behavior of the renormalized mass in the critical region. Other exponents are computed in the following subsections along with the critical equation of state. The other exponents are not independent quantities, but can be determined from 6 and > through universal scaling laws [127]. We de$ne the exponent > through the renormalized mass term in the symmetric phase 1 dUk (0) m2 = (4.11) = k 2 uk (0) for k → 0 : Zk d# The behavior of m2 in the critical region depends only on the distance from the phase transition, which can be expressed in terms of the diEerence of < from the critical value is determined from the relation m2 ∼ |< |2> = |< − :

(4.12)

Assuming proportionality < ∼ Tc − T , this yields the critical temperature dependence of the correlation length % = m−1 . For a determination of > from our results we calculate m2 for various values of < near from the constant slope. Our numerical solution of the partial diEerential equation (4.4) corresponds to an in$nite level of truncation in a Taylor expansion around the “running” minimum of the potential. This in$nite system may be approximately solved by neglecting #-derivatives ˜ of uk (#) ˜ higher than a given order. The apparent convergence of this procedure can be observed from Table 1. We present results obtained through the procedure of successive truncations and through our numerical solution of the partial diEerential equation for N = 3. We give the values of <, , uk(3) (<) for the scaling solution and the critical exponents 6, >. We observe how the results stabilize as more #-derivatives ˜ of uk (#) ˜ at #˜ = < and the anomalous dimension are taken into account. The last line gives the results of our numerical solution of Eq. (4.4). By comparing with the previous line we conclude that the inclusion of all the #-derivatives ˜ higher than uk(6) (<) and the term ∼ 6 in the “threshold” functions generates an improvement of less than 1% for the results. This is smaller than the error induced by the omission

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Table 1 Truncation dependence of the scaling solution and critical exponentsa

a b c d e f g

<∗

∗

u∗(3)

6:57 × 10−2 8:01 × 10−2 7:86 × 10−2 7:75 × 10−2 7:71 × 10−2 7:64 × 10−2 7:765 × 10−2

11.5 7.27 6.64 6.94 7.03 7.07 6.26

52.8 42.0 43.5 43.4 44.2 39.46

6

>

3:6 × 10−2 3:8 × 10−2 3:8 × 10−2 3:8 × 10−2 4:9 × 10−2

0.745 0.794 0.760 0.753 0.752 0.747 0.704

a The minimum < of the potential uk (#) ˜ and the quartic and six-point couplings = u (<), uk(3) (<) are given for the scaling solution. We also display the critical exponents 6 and >, in various approximations: (a) – (e) from Refs. [42,111] and (f) from the present section [107,36]. N = 3. (a) Quartic truncation where only the evolution of < and is considered and higher derivatives of the potential and the anomalous dimension are neglected (cf. Section 2.5). (b) Sixth-order truncation with <, , uk(3) (<) included. (c) All couplings with canonical dimension ¿ 0 are included and 6 is approximated by Eq. (4.5). (d) Addition of uk(4) (<) which has negative canonical dimension. (e) Additional estimate of uk(5) (<), uk(6) (<). (f) The partial diEerential equation (4.4) for uk (#) ˜ is solved numerically and 6 is approximated by Eq. (4.5). (g) First-order derivative expansion with $eld-dependent wave function renormalizations z and y [128].

of the higher derivative terms in the average action, which typically generates an uncertainty of the order of the anomalous dimension. A systematic comparison [101] between the expansion around < presented here and an expansion around #˜ = 0 reveals that only the $rst procedure shows this convergence. In Table 2, we compare our values for the critical exponents obtained from the numerical solution of the partial diEerential equation (4.4) and (4.5) with results obtained from other methods (such as the j-expansion, perturbation series at $xed dimension, lattice high-temperature expansions, Monte Carlo simulations and the 1=N -expansion). As expected 6 is rather poorly determined since it is the quantity most seriously aEected by the omission of the higher derivative terms in the average action. The exponent > is in agreement with the known results at the 1–5% level, with a discrepancy roughly equal to the value of 6 for various N . Our results compare well with those obtained by similar methods using a variety of forms for the infrared cutoE function [140,108,102,85,81]. In conclusion, the shape of the average potential is under good quantitative control for every scale k. This permits a quantitative understanding of the most important properties of the system at every length scale. We will exploit this in the following in order to extract the scaling form of the equation of state. 4.3. Universal critical equation of state In this section we extract the Widom scaling form of the equation of state from a solution [36] of Eqs. (4.4) and (4.5) for the three-dimensional O(N ) model. Its asymptotic behavior yields the universal critical exponents and amplitude ratios. We also present $ts for the scaling function for N = 3 and 4. A detailed discussion of the universal and non-universal aspects of the Ising model (N = 1) is given in Section 4.4.

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Table 2 Critical exponents > and 6 for various values of N a N

>

0

0.589f

0.5882(11)a 0.5875(25)b 0.5878(6)c1 0.5877(6)e

0.040f

0.0284(25)a 0.0300(50)b

1

0.643f 0.6307g

0.6304(13)a 0.6290(25)b 0.63002(23)c2 0.6294(9)e

0.044f 0.0467g

0.0335(25)a 0.0360(50)b 0.0364(4)c2 0.0374(14)e

2

0.697f 0.666g

0.6703(15)a 0.6680(35)b 0.67166(55)c2 0.6721(13)e

0.042f 0.049g

0.0354(25)a 0.0380(50)b 0.0381(3)c2 0.042(2)e

3

0.747f 0.704g

0.7073(35)a 0.7045(55)b 0.716(2)c1 0.7128(14)e

0.038f 0.049g

0.0355(25)a 0.0375(45)b

0.741(6)a 0.737(8)b 0.759(3)c1 0.7525(10)e

0.034f 0.047g

4

0.787f 0.739g

6

0.041(2)e 0.0350(45)a 0.036(4)b 0.038(1)e

10

0.904f

0.894(4)c1 0.877d

0.019f

100

0.990f

0.989d

0.002f

0.025d 0.003d

a

For comparison we list results obtained with other methods as summarized in [129 –132]: (a) From perturbation series at $xed dimension including seven-loop contributions. (b) From the j-expansion at order j5 . (c) From high-temperature expansions (c1: [131], c2: [132], see also [133,134]). (d) From the 1=N -expansion at order 1=N 2 . (e) From lattice Monte Carlo simulations [135,136] (see also [137,138]). (f) Average action in lowest order in the derivative expansion (present section). (g) From $rst order in the derivative expansion for the average action with $eld-dependent wave function renormalizations (for N = 1 see [38] and Section 4.4, and [128] for N ¿ 1).

Eq. (4.1) establishes the scaling properties of the equation of state. The external $eld H is related to the derivative of the eEective potential U = 9U=9# by Ha = U a . The critical equation of state, relating the√temperature, the external $eld and the order parameter, can then be written in the scaling form ( = 2#) −< U = f(x); x = 1= (4.13) − 1 with critical exponents and . A measure of the distance from the phase transition is the diEerence < = < −
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Fig. 4. Universal critical equation of state, symmetric phase: Logarithmic plot of f and df=d x for x ¿ 0. Fig. 5. Universal critical equation of state, spontaneous symmetry breaking: Logarithmic plot of f and df=d x for x ¡ 0.

the deviation from the critical temperature, i.e. < =A(T )(Tcr −T ) with A(Tcr ) ¿ 0. For → ∞ our numerical solution for U obeys U ∼ −1 with high accuracy. The inferred value of is displayed in Table 4, and we have checked the scaling relation = (5 − 6)=(1 + 6). The value of the critical exponent 6 is obtained from Eq. (4.4) for the scaling solution. We have also veri$ed explicitly that f depends only on the scaling variable x for the value of given in Table 2. In Figs. 4 and 5 we plot log(f) and log(df=d x) as a function of log|x| for N = 1 and N = 3. Fig. 4 corresponds to the symmetric phase (x ¿ 0), and Fig. 5 to the phase with spontaneous symmetry breaking (x ¡ 0). One can extract easily the asymptotic behavior from the logarithmic plots and compare with known values of critical exponents and amplitudes. The curves become constant, both for x → 0+ and x → 0− with the same value, consistently with the regularity of f(x) at x = 0. For the universal function one obtains lim f(x) = D

x→0

(4.14)

and H = D on the critical isotherm. For x → ∞ one observes that log(f) becomes a linear function of log(x) with constant slope -. In this limit the universal function takes the form lim f(x) = (C + )−1 x- :

x→∞

(4.15)

The amplitude C + and the critical exponent - characterize the behavior of the “unrenormalized” squared mass or inverse susceptibility 2 9U 2 −1 mU = = lim = (C + )−1 |< |- −1−-= : (4.16) →0 92 We have veri$ed the scaling relation -= = − 1 that connects - with the exponents and appearing in the Widom scaling form (4.1). One observes that the zero-$eld magnetic susceptibility, or equivalently the inverse unrenormalized squared mass mU −2 = , is non-analytic for < → 0 in the symmetric phase: = C + |< |−- . In this phase we $nd that the correlation length % = (Z0 )1=2 , which is equal to the inverse of the renormalized mass mR , behaves as % = %+ |< |−> with > = -=(2 − 6).

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279

Table 3 Universal critical exponents for N = 3; 4 N

-

>

6

3 4

0.388 0.407

1.465 1.548

4.78 4.80

0.747 0.787

0.038 0.0344

Table 4 Universal amplitude ratios R ; R˜ % and %+ E for N = 3; 4. The amplitudes C + ; D; B; %+ and E are not universal N

C+

D

B

%+

E

R

R˜ %

%+ E

3 4

0.0743 2.79

8.02 1.82

1.180 7.41

0.263 0.270

0.746 0.814

1.11 1.02

0.845 0.852

0.196 0.220

The spontaneously broken phase is characterized by a non-zero value 0 of the minimum of the eEective potential U with H = (9U=9)(0 ) = 0. The appearance of spontaneous symmetry breaking below Tc implies that f(x) has a zero x=−B−1= where one observes a singularity of the logarithmic plot in Fig. 5. In particular, according to Eq. (4.1) the minimum behaves as 0 = B(< ) . Below the critical temperature, the longitudinal and transversal susceptibilities L and T are diEerent for N ¿1 x 92 U 1 9U − 1 f = −1 f(x) L−1 = f(x) − = (x) ; T−1 = (4.17) 92 9 (with f = df=d x). This is related to the existence of massless Goldstone modes in the (N − 1) transverse directions, which causes the transversal susceptibility to diverge for vanishing external $eld. Fluctuations of these modes induce the divergence of the zero-$eld longitudinal susceptibility. This can be concluded from the singularity of log(f ) for N = 3 in Fig. 5. The $rst x-derivative of the universal function vanishes as H → 0, i.e. f (x = −B−1= ) = 0 for N ¿ 1. For N = 1 there is a non-vanishing constant value for f (x = −B−1= ) with a $nite zero-$eld susceptibility = C − (< )−- , where (C − )−1 = B−1−1= f (−B−1= )=. For a non-vanishing physical infrared cutoE k, the longitudinal susceptibility remains $nite also for N ¿ 1: L ∼ (k#0 )−1=2 . For N = 1 in the ordered phase, the correlation length behaves as % = %− (< )−> , and the renormalized minimum #0R = Z0 #0 of the potential U scales as #0R = E(< )> . The amplitudes of singularities near the phase transition D; C ± ; %± ; B and E are given in Table 4. They are not universal. All models in the same universality class can be related by a multiplicative rescaling of and < or (Tc − T ). Accordingly there are only two independent amplitudes and exponents, respectively. Ratios of amplitudes which are invariant under this rescaling are universal. We display the critical exponents and the universal combinations R =C + DB−1 ; R˜ % =(%+ )=> D1=(+1) B and %+ E for N = 3; 4 in Tables 3 and 4. The asymptotic behavior observed for the universal function can be used in order to obtain a semi-analytical expression for f(x). We $nd that the following two-parameter $t reproduces the numerical values for both f and df=d x with 1–2% accuracy for N = 3: f$t (x) = D(1 + B1= x)2 (1 + Ex)1 (1 + cx)-−2−1

(4.18)

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Fig. 6. Critical equation of state for the three-dimensional O(4) Heisenberg model. We compare our results for the scaling function, denoted by “average action”, with results of other methods. We have labeled the axes in terms of the expectation value AU0 and the source — relevant for the chiral phase transition in QCD discussed in Section 8. In this context they describe the dependence of the chiral condensate ∼ AU0 on the quark mass ∼ — for two "avors of quarks. The constants B and D specify the non-universal amplitudes of the model. The curve labeled by “MC” represents a $t to lattice Monte Carlo data. The second-order epsilon expansion [142] and mean $eld results are denoted by “j” and “mf”, respectively. Apart from our results the curves are taken from Ref. [141].

with c = (C + DB2= E1 )−1=(-−2−1) . The $tting parameters are chosen as E = 1:312 and 1 = −0:595. For N = 4 we $nd the following $t: f$t (x) = 1:816 × 10−4 (1 + 136:1x)2 (1 + 160:9Kx)1 (1 + 160:9(0:9446K 1 )−1=(-−2−1) x)-−2−1 (4.19) with K = 0:625 (0:656); 1 = −0:490 (−0:550) for x ¿ 0 (x ¡ 0) and - as given in Table 3. The universal properties of the scaling function can be compared with results obtained by other methods for the three-dimensional O(4) Heisenberg model. In Fig. 6 we display our results for N =4 along with those obtained from lattice Monte Carlo simulation [141], second-order epsilon expansion [142] and mean $eld theory. In summary, our numerical solution of Eq. (4.3) gives a very detailed picture of the critical equation of state of the three-dimensional O(N ) model. The numerical uncertainties are estimated by comparison of results obtained through two independent integration algorithms [107,37]. They are small, typically less than 0:3% for critical exponents. The scaling relations between the critical exponents are ful$lled within a deviation of 2 × 10−4 . The dominant quantitative error stems from the relatively crude approximation of the kinetic term in (4.2) and is related to the size of the anomalous dimension 6 4%. We emphasize that in contrast to most other analytical methods no scaling hypothesis is used as an input and no resummations of series are needed. The scaling behavior is simply a property of the solution of the "ow equation. In the following we improve on this approximation for the Ising model (N = 1). We allow for a most general $eld dependence of the wave function renormalization factor, and compare with the

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281

results of this section. The Ising model is analyzed with a particular short-distance action relevant for carbon dioxide. This allows us to compare the non-universal aspects as well as the universal aspects of the liquid–gas phase transition in carbon dioxide with experiment. 4.4. Gas–liquid transition and the Ising universality class36 Field theoretical description. Near the critical temperature many phase transitions are described by a one-component (N =1) scalar $eld theory without internal symmetries. A typical example is the water-vapor transition where the $eld (x) corresponds to the average density $eld n(x). At normal pressure one observes a $rst-order transition corresponding to a jump in from high (water) to low (vapor) values as the temperature T is increased. With increasing pressure the $rst-order transition line ends at some critical pressure p∗ at an endpoint. For p ¿ p∗ the phase transition is replaced by an analytical crossover. This behavior is common to many systems and characterizes the universality class of the Ising model. As another example from particle physics, the high-temperature electroweak phase transition in the early universe is described by this universality class if the mass of the Higgs particle in the standard model is near the endpoint value MH ∗ ≈ 72 GeV [143]. An Ising-type endpoint should also exist if the high temperature or high-density chiral phase transition in QCD or the gas–liquid transition for nuclear matter are of $rst order in some region of parameter space. Very often the location of the endpoint—e.g. the critical T∗ ; p∗ and n∗ for the liquid–gas transition—is measured quite precisely. The approach to criticality is governed by universal scaling laws with critical exponents. Experimental information is also available about the non-universal amplitudes appearing in this scaling behavior. These non-universal critical properties are speci$c for a given system, and the question arises how they can be used to gain precise information about the underlying microscopic physics. This problem clearly involves the diIcult task of an explicit connection between the short-distance physics and the collective behavior leading to a very large correlation length. So far renormalization group methods [1–7], [121] have established the structure of this relation and led to a precise determination of the universal critical properties. The non-perturbative "ow equation (2.19) allows us to complete the task by mapping details of microscopic physics to non-universal critical quantities. A demonstration is given for the liquid–gas transition in carbon dioxide. In the following, we will work with a truncation which includes the most general terms containing up to two derivatives, 1 k [] = d 3 x Uk ((x)) + Zk ((x))9 9 : (4.20) 2 In contrast to ansatz (4.2) we now include the $eld dependence of the wave function renormalization factor Zk ((x)). Our aim is the computation of the potential U0 ≡ Uk →0 =U=T and the wave function renormalization Z ≡ Zk →0 for a vanishing infrared cutoE. For the liquid gas transition the source J is linear in the chemical potential . Therefore, for a homogeneous situation U0 T corresponds to the free energy density. Indeed, expressing U0 as a function of the density one $nds for the liquid–gas 36

Sections 4.4 – 4.6 are based on a collaboration with Seide [38].

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system at a given chemical potential 9U0 = : (4.21) 9n T Equivalently, one may also use the more familiar form of the equation of state in terms of the pressure p, U0 p 2 9 = : (4.22) n 9n n T (Here the additive constant in U0 is $xed such that U0 (n=0)=0:) The wave function renormalization Z() contains the additional information needed for a determination of the two-point correlation function at large distance for arbitrary pressure. The computation of thermodynamic potentials, correlation length, etc., is done in two steps: The $rst is the computation of a short-distance free energy . This does not involve large length scales and can be done by a variety of expansion methods or numerical simulations. Our main emphasis here is not on this step and we will use a relatively crude approximation for the gas–liquid transition. The second step is more diIcult and will be addressed here. It involves the relation between and 0 , and has to account for possible complicated collective long-distance "uctuations. For a large infrared cutoE k = one may compute perturbatively. For example, the lowest order in a virial expansion for the liquid–gas system yields MT n b1 () 2 3 n + c : + n ln − (4.23) U (n) = −n 1 + ln g + ln 2 2"2 (1 − b0 ()n)3 T Here −1 should be of the order of a typical range of intermolecular interactions, M and g are the mass and the number of degrees of freedom of a molecule and b0 , b1 parameterize the virial coeIcient B2 (T )=b0 −b1 =T . 37 (The (mass) density # is related to the particle density n by #=Mn.) We emphasize that the convergence of a virial expansion is expected to improve considerably in the presence of an infrared cutoE that suppresses the long-distance "uctuations. The $eld (x) is related to the (space-dependent) particle density n(x) by (x) = K (n(x) − n) ˆ

(4.24)

with nˆ some suitable $xed reference density. We approximate the wave function renormalization Z ˆ evaluated at some reference density by a constant. It can be inferred from the correlation length %, nˆ and temperature Tˆ away from the critical region, through % 2 % −2 − 1 9 U0 % ˆ % = Z : (4.25) 2 9 % ˆ ; Tˆ

For a suitable scaling factor ˆ 1 b0 (2 − b0 n) 2b1 1=2 ˆ + K = − %; nˆ (1 − b0 n) ˆ 2 Tˆ one has Z = 1. 37

(4.26)

The Van der Waals coeIcients b0 ; b1 for real gases can be found in the literature. These values are valid for small densities. They also correspond to k = 0 rather than to k = . Fluctuation eEects lead to slightly diEerent values for bi () and bi (k = 0) even away from the critical line. We $nd that these diEerences are small for nn∗ . Similarly, a constant c should be added to U so that U0 (0) = 0.

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283

We observe that the terms linear in n in Eq. (4.23) play a role only for the relation between n and . It is instructive to subtract from U the linear piece in and to expand in powers of : m2 - (4.27) U () = 2 + 3 + 4 + · · · 2 6 8 with 1 b0 (2 − b0 n) 2b1 ˆ + ; m2 = K−2 − nˆ (1 − b0 n) ˆ 2 T 2 b0 (3 − b0 n) 1 ˆ - = K−3 ; − (1 − b0 n) ˆ 3 nˆ2 ˆ 2 −4 b30 (4 − b0 n) 1 = K (4.28) + 3 : 3 (1 − b0 n) ˆ 4 nˆ 8 )b1 =b0 one has - = 0 and For a convenient choice nˆ = 1=3b0 ; Tˆ = ( 11 27 b1 −2 −4 ˆ %ˆ ; K = 2b01=2 %; = 243 b %ˆ : (4.29) m2 = − 128 0 16 2b0 T Carbon dioxide. In order to be speci$c, we will discuss the equation of state for carbon dioxide near the endpoint of the critical line. Typical values of the parameters are m2 =2 =−0:31; ==6:63 for −1 = 5 × 10−10 m; %ˆ = 0:6−1 . In limit (4.27) one obtains a 4 -model. Our explicit calculations for carbon dioxide will be performed, however, for the microscopic free energy (4.23). The linear piece in the potential can be absorbed in the source term so that the equation of state reads 38 3 MT 9U0 −1 = j; j = K + 1 + ln g + ln : (4.30) 9 T 2 2"2 We emphasize that a polynomial microscopic potential (4.27) with equation of state 9U0 =9 = j is a good approximation for a large variety of diEerent systems. For the example of magnets corresponds to the magnetization and jT to the external magnetic $eld. For - = 0 and → ∞, with $nite negative m2 = , this is the Z2 -symmetric Ising model. For values of for which the mass term m2 () = (1=Z)92 U=92 is much larger than 2 the microscopic approximation to k remains approximately valid also for k → 0, i.e. U () ≈ U (). The contribution of the long wavelength "uctuations is suppressed by the small correlation length or large mass. In the range where m2 ()2 , however, long-distance "uctuations become important and perturbation theory looses its validity. Beyond the computation of universal critical exponents and amplitude ratios we want to establish an explicit connection between the universal critical equation of state and the microscopic free energy . In Fig. 7 we plot the results for the equation of state near the endpoint of the critical line for carbon dioxide. For the microscopic scale we have chosen −1 = 0:5 nm. For %ˆ = 0:6−1 ; b0 () = 34 cm3 mol−1 ; b1 () = 3:11 × 106 bar cm6 mol−2 one $nds the location of the endpoint at T∗ = 307:4 K; p∗ = 77:6 bar; #∗ = 0:442 g cm−3 . This compares well with the experimental values T∗ = 304:15 K; p∗ = 73:8 bar; #∗ = 0:468 g cm−3 . Comparing with literature values bi (0)ld for low 38

Note that the source term is independent of k. Therefore, the linear piece in the potential can be added easily to Uk→0 once all "uctuation eEects are included.

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Fig. 7. Liquid–vapor transition for carbon dioxide. We display p–#-isotherms for T = 295 K; T = 300 K; T = 303 K; T = 305 K; T = 307 K and T = 315 K. The dashed lines represent the virial expansion (at the scale k = ) in second order with b0 = 34 cm3 mol−1 , b1 = 3:11 × 106 bar cm6 mol−2 . The solid lines are the results at the scale k = 0 (%ˆ = 0:6).

density this yields b0 ()=b0 (0)ld = 0:8; b1 ()=b1 (0)ld = 0:86. We conclude that the microscopic free energy can be approximated reasonably well by a van der Waals form even for high densities near n∗ . The coeIcients of the virial expansion are shifted compared to this low density values by 15 –20%. The comparison between the “microscopic equation of state” (dashed lines) and the true equation of state (solid lines) in the plot clearly demonstrates the importance of the "uctuations in the critical region. Away from the critical region the "uctuation eEects are less signi$cant and could be computed perturbatively. Flow equations. Our aim is a numerical solution of the "ow equation for Uk with given initial conditions at the scale k ≈ . We introduce a dimensionless renormalized $eld ˜ = k (2−d)=2 Z0;1=2k

(4.31)

with wave function renormalization Z0; k = Zk (0 (k)) taken at the global potential minimum 0 (k). We do not choose the $eld squared as a variable in order to faciliate the discussion of $rst-order transitions below, where terms cubic in the $eld appear. 39 We also use ˜ = k −d Uk () ; uk () ˜ = Z −1 Zk () z˜k () 0; k 39

(4.32)

A combination of Eq. (3.8) with y˜ = 0; TCk = 0 with a suitable "ow equation for z(#) ˜ has led to identical results.

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285

˜ This yields the scaling form of the "ow and denote here by u ; z˜ the derivatives with respect to . equation for uk : 1 9 ˜ ˜ + 1 (d − 2 + 60 )˜ · u () ˜ uk () = − (d + 2 − 60 ) · uk () k 9t 2 2 ˜ · ld+2 (u (); ˜ 60 ; z˜k ()) ˜ − 2vd u () ˜ · ld (u (); ˜ 60 ; z˜k ()) ˜ : − 2vd z˜k () k k 1 k 1

(4.33)

Similarly, the evolution of z˜k is described in the truncation (4.20) by 9 ˜ = 60 · z˜k () ˜ + 1 (d − 2 + 60 )˜ · z˜ () ˜ − 4 vd · u () ˜ 2 · md (u (); ˜ 60 ; z˜k ()) ˜ z˜k () k k 4; 0 k 9t 2 d 8 ˜ z˜ () ˜ · md+2 (u (); ˜ 60 ; z˜k ()) ˜ − vd · uk () k k 4; 0 d 4 ˜ 2 · md+4 (u (); ˜ 60 ; z˜k ()) ˜ − 2vd · z˜ () ˜ · ld (u (); ˜ 60 ; z˜k ()) ˜ − vd · z˜k () k k 1 k 4; 0 d ˜ () ˜ · ld (u (); ˜ 60 ; z˜k ()) ˜ + 4vd · z˜ ()u k

k

2

k

2 ˜ 2 · ld+2 (u (); ˜ 60 ; z˜k ()) ˜ : + (1 + 2d)vd · z˜k () k 2 d

(4.34)

Here the mass threshold functions are ∞ 1 1 2n−d n d d=2−1 ˜ ; ˜ =− k · Z0; k · d xx ln (u ; 60 ; z) 9t 2 (P(x) + Z0; k k 2 u )n 0 1 mdn; 0 (u ; 60 ; z) ˜ = − k 2(n−1)−d · Z0;n−k 2 · 2

0

∞

d xxd=2 9˜t

2 P˙ (x) (P(x) + Z0; k k 2 u )n

(4.35)

(with P(x) = zZ ˜ 0; k x + Rk (x); P˙ ≡ dP=d x and 9˜t acting only on Rk ). The anomalous dimension % % d d0 −1 9 −1 9Zk % (4.36) · 60; k ≡ − ln Z0; k = −Z0; k Zk (0 ) − Z0; k · dt 9t 9 %0 dt is determined by the condition d z( ˜ ˜ 0 )=dt = 0. It appears linearly in the threshold functions because of 9˜t acting on Z0; k in Rk , Rk (x) =

Z0; k x : exp(x=k 2 ) − 1

(4.37)

For a computation of 60 we need the evolution of the potential minimum 0 (k), which follows from the condition (d=dt)(9Uk =9(0 (k)) = 0, namely z˜ (˜ ) u (˜ ) d ˜ 0 1 = (2 − d − 60 )˜ 0 + 2vd k 0 · ld+2 (uk (˜ 0 ); 60 ; 1) + 2vd k 0 · ld1 (uk (˜ 0 ); 60 ; 1) : 1 dt 2 uk (˜ 0 ) uk (˜ 0 ) (4.38)

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One infers an implicit equation for the anomalous dimension 60; k , 4 8 ˜ 60 = vd · uk (˜ 0 )2 · md4; 0 (uk (˜ 0 ); 60 ; 1) + vd · uk (˜ 0 )z˜k (˜ 0 ) · md+2 4; 0 (uk (0 ); 60 ; 1) d d 4 ˜ ˜ d ˜ + vd · z˜k (˜ 0 )2 · md+4 4; 0 (uk (0 ); 60 ; 1) + 2vd · z˜k (0 ) · l1 (uk (0 ); 60 ; 1) d 2 ˜ − 4vd · z˜k (˜ 0 )uk (˜ 0 ) · ld2 (uk (˜ 0 ); 60 ; 1) − (1 + 2d)vd · z˜k (˜ 0 )2 · ld+2 2 (uk (0 ); 60 ; 1) d z˜ (˜ ) ˜ ˜ d ˜ − 2vd k 0 · {z˜k (˜ 0 )ld+2 (4.39) 1 (uk (0 ); 60 ; 1) + uk (0 )l1 (uk (0 ); 60 ; 1)} ; uk (˜ 0 ) that can be solved by separating the threshold functions in 60 -dependent and 60 -independent parts (c.f. Eq. (4.35)). Since 60 will turn out to be only a few percent, the omission of contributions from higher derivative terms not contained in (4.20) induces a substantial relative error for 60 , despite the good convergence of the derivative expansion. We believe that the missing higher derivative contributions to 60 constitute the main uncertainty in the results. For given initial conditions U (); Z () the system of partial diEerential equations (4.33), (4.34), (4.38), (4.39) can be solved numerically. A description of the algorithm used can be found in [37]. 4.5. Universal and non-universal critical properties In order to make the discussion transparent we present $rst results for polynomial initial conditions ˜ = 1. The term linear in is considered as a source j. The special value - = 0 (4.27) with z˜ () realizes the Z2 -symmetric Ising model. We start with the results for the universal critical behavior for this case. For this particular purpose, we hold $xed and measure the deviation from the critical temperature by m2 = m2 − m2; crit = S(T − Tc ) :

(4.40)

For the liquid–gas system one has S = 2b1 =(K2 Tc2 ). The anomalous dimension 6 determines the two-point function at the critical temperature and equals 60; k for the scaling solution where 9t u = 9t z˜ = 0. The results for the critical exponents are compared with those from other methods in Table 5. We observe a very good agreement for > whereas the relative error for 6 is comparatively large as expected. Comparison with the lowest order of the derivative expansion (f), used in Section 4.2, shows a convincing apparent convergence of this expansion for >. For 6 this convergence is hidden by the fact that in (4.5) a diEerent determination of 6 was used. Employing the present de$nition would lead in lowest order of the derivative expansion to a value 6 = 0:11. As expected, the convergence of the derivative expansion is faster for the very eEective exponential cutoE than for the powerlike cutoE (g) which would lead to unwanted properties of the momentum integrals in the next order. In order to establish the quantitative connection between the short-distance parameters m2 and and the universal critical behavior one needs the amplitudes C ± ; %± , etc. For = = 5 we $nd C + = 1:033; C − = 0:208; %+ = 0:981; %− = 0:484; B = 0:608; E = 0:208. Here and in the following all dimensionful quantities are quoted in units of . The amplitude D is given by 9U0 =9 = D · on

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287

Table 5 Critical exponents of the (d = 3)-Ising model, calculated with various methods >

-

6

(a) (b) (c) (d)

0:6304(13) 0:6290(25) 0:63002(23) 0:6294(9)

0:3258(14) 0:3257(25)

1.2396(13) 1.2355(50) 1:2371(4)

0:0335(25) 0:0360(50) 0.0364(4) 0:0374(14)

(f) (g) (h)

0:643 0:6181 0:6307

0:336

1:258

0:3300

1:2322

0:044 0:054 0:0467

(i)

0:625(6)

0.316 – 0.327

1.23–1.25

(a) From perturbation series at $xed dimension d = 3 including seven-loop contributions [121,129]. (b) j-expansion in $ve-loop order [121,129]. (c) High-temperature series [132] (see also [131,133,134]). (d) Monte Carlo simulation [136] (see also [144 –146]). (f) – (h): “Exact” renormalization group equations. (f) EEective average action for the O(N )-model, N → 1, with uniform wave function renormalization [36] (see also [111]). (g) Scaling solution of equations analogous to (4.33) and (4.34) with powerlike cutoE [118]. (h) EEective average action for one-component scalar $eld theory with $eld-dependent wavefunction renormalization (present section). (i) Experimental data for the liquid–vapor system quoted from [121]. Table 6 Universal amplitude ratios and couplings of the (d = 3)-Ising model C + =C −

%+ =%−

a b c d

4:79 ± 0:10 4:73 ± 0:16 4:77 ± 0:02 4:75 ± 0:03

1:96 ± 0:01 1:95 ± 0:02

f h

4.29 4.966

1.86 2.027

i

4.8–5.2

R

R˜ %

%+ E

1:669 ± 0:018 1:648 ± 0:036 1:662 ± 0:005 1.61 1.647

0.865 0.903

0.168 0.204

R =mR

ˆR = mˆ R

7:88 9:33 7.9 –8.15 7:76

5:27

9:69 8:11

5:55 4:96

1:69 ± 0:14

(a) Perturbation theory at $xed dimension d = 3 [129,147]. (b) j-expansion [129,147]. (c) High-temperature series. Amplitude ratios from [132,121], R =mR from [133,134,150]. (d) Monte Carlo simulations. Amplitude ratios from [146], ˆR = mˆ R from [145], R =mR from [144]. (f) EEective average action for the O(N )-model, N → 1, with uniform wave function renormalization [36]. (h) Present section with $eld-dependent wave function renormalization. (i) Experimental data for the liquid–vapor system [148].

the critical isotherm and we obtain D = 10:213. In Table 6, we present our results for the universal amplitude ratios C + =C − ; %+ =%− ; R = C + DB−1 ; R˜ % = (%+ )=> D1=(+1) B. The critical exponents and amplitudes characterize the behavior of U0 () only in the limits → 0 and ∞. Our method allows us to compute U0 () for arbitrary . As an example, the quartic coupling R = 13 (94 U0 =94R )(0) = (92 U0 =9#2R )(0); ˆR = (92 U0 =9#2R )(0R ); #R = 12 2R , becomes in the critical region proportional to mR . Our results for the universal couplings R =mR in the symmetric

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Fig. 8. Widom scaling function f(x) of the (d=3)-Ising model. (The present curve is generated for a quartic short-distance potential U with = = 5). The dashed lines indicate extrapolations of the limiting behavior as given by the critical exponents.

and ˆR = mˆ R in the ordered phase can also be found in Table 6. Here mR = 92 U0 =92R |R =0 in the symmetric and mˆ R = 92 U0 =92R |0R in the ordered phase. We should emphasize that the shape of the potential in the low-temperature phase depends on k in the “inner” region corresponding to || ¡ 0 . This is due to the "uctuations which are responsible for making the potential convex in the limit k → 0 [77,78,125]. We illustrate this by plotting the potential for diEerent values of k in Fig. 1. Our results for the scaling function f(x) are shown in Fig. 8, together with the asymptotic behavior (dashed lines) as dictated by the critical exponents and amplitudes. A diEerent useful parameterization of the critical equation of state can be given in terms of non-linearly rescaled $elds ˆ R , using a -dependent wave function renormalization Z() = Zk →0 (), ˆ R = Z()1=2 :

(4.41)

Our numerical results can be presented in terms of a $t to the universal function 2 1=2 ˆ 9U Z() 0 − 5=2 R ˆ s) F( ˆ = mR ; sˆ = 1=2 = ; mR 9ˆ R mR ˆ = (a0 sˆ + a1 sˆ3 + a2 sˆ5 + a3 sˆ7 ) · f/ (s) ˆ + (1 − f/ (s)) ˆ · a4 sˆ5 : Fˆ Fit (s)

(4.42) (4.43)

The factors f/ and (1−f/ ) interpolate between a polynomial expansion and the asymptotic behavior for large arguments. We use f/ (x) = /−2 x2 ·

exp(−x2 =/2 ) : 1 − exp(−x2 =/2 )

(4.44)

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289

Fig. 9. Universal rescaled wave function renormalization z˜ in the symmetric and ordered phase of the (d = 3)-Ising model. In the low-temperature phase the plots are for diEerent k with (1=k 2 )(92 U=92R )(R; max ) = c (c = −0:9; −0:95; −0:99). Here R; max is the location of the local maximum of the potential in the inner (non-convex) region. In the graph for the high-temperature phase we have inserted Monte Carlo results by M. Tsypin (private communication).

A similar $t can be given for limk →0 Zk () z(s) ˜ = ; Z0

s=

R mR1=2

=

Z 0 2 mR

1=2

= z˜−1=2 sˆ ;

z˜Fit (s) = (b0 + b1 s2 + b2 s4 + b3 s6 + b4 s8 ) · f (s) + (1 − f (s)) · b5 |s|−26=(1+6) :

(4.45) (4.46)

In the symmetric phase one $nds (with 6 = 0:0467) / = 1:012; a0 = 1:0084; a1 = 3:1927; a2 = 9:7076; a3 =0:5196; a4 =10:3962 and =0:5103; b0 =1; b1 =0:3397; b2 =−0:8851; b3 =0:8097; b4 = −0:2728; b5 = 1:0717, whereas the $t parameters for the phase with spontaneous symmetry breaking are / = 0:709; a0 = −0:0707; a1 = −2:4603; a2 = 11:8447; a3 = −1:3757; a4 = 10:2115 and = 0:486; b0 = 1:2480; b1 = −1:4303; b2 = 2:3865; b3 = −1:7726; b4 = 0:4904; b5 = 0:8676 (our $t parameters are evaluated for this phase for (92 Uk =92R )(R; max )=k 2 = −0:99). One observes that the coeIcients a2 and a4 are large and of comparable size. A simple polynomial form Fˆ =a˜0 s+ ˆ a˜1 sˆ3 +a˜2 sˆ5 is not too far from the more precise result. We conclude that in terms of the rescaled $eld ˆ R (4.41) the potential is almost a polynomial 6 -potential. In Fig. 9 we show z˜ as a function of s both for the symmetric and ordered phase. Their shape is similar to the scaling solution found in [118]. Nevertheless, the form of z˜ for k = 0 which expresses directly information about the physical system should not be confused with the scaling solution which depends on the particular infrared cutoE. For the low-temperature phase one sees the substantial dependence of z˜k on the infrared cutoE k for small values s ¡ s0 . Again this corresponds to the “inner region” between the origin (s = 0) and the minimum of the potential (s0 = 0:449) where the potential $nally becomes convex for k → 0.

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Fig. 10. The critical equation of state in the ordered phase: (1) mean $eld approximation, (2) eEective average action with uniform wave function renormalization [36], (3) Monte Carlo simulation [145], (4) resummed j-expansion in O(j3 ), $ve loop perturbative expansion and high-temperature series [147], (5) results of the present section.

Knowledge of U0 and z˜ permits the computation of the (renormalized) propagator for low momenta with arbitrary sources j. It is given by −1 2 9 U0 (R ) 2 2 + z( ˜ R )q (4.47) G(q ) = 92R for zq ˜ 2 . 92 U0 =92R . Here R obeys 9U0 =9R = Z0−1=2 j. We emphasize that the correlation length %(R ) = z˜1=2 (R )(92 U0 =92R )−1=2 at given source j requires information about z. ˜ For the gas–liquid transition %(R ) is directly connected to the density dependence of the correlation length. For magnets, it expresses the correlation length as a function of magnetization. The factor z˜1=2 is often omitted in other approaches. From ! 1 9 ln z˜ 1 9 ln z˜ 2 1 9 ln z˜ 2 9Fˆ 92 ln z˜ Fˆ −2 2 + + (4.48) 1+ + % = mR 2 29 ln s 9sˆ 2 9 ln s 2 9 ln s (9 ln s)2 sˆ one can extract the behavior for || → ∞ for the high- and low-temperature phases % = L± ||− :

(4.49)

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291

Fig. 11. The critical equation of state in the symmetric phase: (1) Monte Carlo simulation [149], (2) j-expansion [147], (3) eEective average action with uniform wave function renormalization [36], (4) Monte Carlo simulation [144], (5), (6) high-temperature series [150,134], (7) results of the present section.

Critical equations of state for the Ising model have been computed in the past with several methods. 40 They are compared with our result for the phase with spontaneous symmetry breaking in Fig. 10 and for the symmetric phase in Fig. 11. For this purpose we use F(s) ˜ = mR−5=2 9U0 =9R with s˜ = R =0R in the phase with spontaneous symmetry breaking (note s˜ ∼ s). The constant cF is chosen such that (1=cF )(9F=9s)( ˜ s˜ = 1) = 1. In the symmetric phase we take instead s˜ = R =mR1=2 so that (9F=9s)( ˜ s˜ = 0) = 1. One expects for large s˜ some inaccuracy in our results because of the error in 6. Summarizing this subsection we conclude that the non-perturbative "ow equations in second order in a derivative expansion lead to a critical equation of state which is well compatible with high-order expansions within other methods. In addition, it establishes an explicit connection between the parameters appearing in the microscopic free energy and the universal long-distance behavior. For a quartic polynomial potential this involves in addition to the non-universal amplitudes the value of m2; crit . We have listed these quantities for diEerent values of in Table 7. Finally, the temperature scale is established by S = 9m2 =9T |Tc .

40

For recent studies see Refs. [132,138,139].

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Table 7 The critical values m2; crit and the non-universal amplitudes C + ; D as a function of the quartic short-distance coupling (all values expressed in units of ). Other non-universal amplitudes can be calculated from the universal quantities of Table 6

= 0:1 = 1 = 5 = 20

m2; crit

C+

D

−6:4584 × 10−3 −5:5945 × 10−2 −0:22134 −0:63875

0:1655 0:485 1:033 1:848

5:3317 7.506 10:213 16:327

4.6. Equation of state for >rst-order transitions Our method is not restricted to a microscopic potential with discrete Z2 -symmetry. The numerical code works for arbitrary initial potentials. We have investigated the polynomial potential (4.27) with - = 0. The numerical solution of the "ow equations (4.33) and (4.34) shows the expected $rst-order phase transition (in the case of vanishing linear term j). Quite generally, the universal critical equation of state for $rst-order transitions will depend on two scaling parameters (instead of one for second-order transitions) since the jump in the order parameter (or the mass) introduces a new scale. This has been demonstrated in [37] for a scalar matrix model and is discussed in more detail in Section 5. The degree to which universality applies depends on the properties of a given model and its parameters. For a 4 -model with cubic term (4.27) one can relate the equation of state to the Ising model by an appropriate mapping. This allows us to compute the universal critical equation of state for arbitrary $rst-order phase transitions in the Ising universality class from the critical equation of state for the second-order phase transition in the Ising model. For other universality classes a simple mapping to a second-order equation of state is not always possible—its existence is particular to the present model. By a variable shift - A=+ ; (4.50) 3 we can bring the short-distance potential (4.27) into the form 2 (4.51) U (A) = −J- A + A2 + A4 + c 2 8 with - 2 -3 m − 2 ; J- = 3 27 2 2 = m2 − : (4.52) 6 Now we can solve the "ow equations in terms of A and re-express the result in terms of by Eq. (4.50) at the end. The exact "ow equation (2.19) does not involve the linear term ∼ J- A in the r.h.s. (also the constant c is irrelevant). Therefore, the eEective potential (k = 0) is given by - - Z2 Z2 U0 = U (A) − J- A + c = U + − + J- + c ; (4.53) 3 3

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293

where U Z2 is the eEective potential of the Ising-type model with quartic coupling and mass term 2 . As a result, the equation of state 9U=9 = j or, equivalently % 9U Z2 %% = j + J- ; (4.54) 9 %+- =3 is known explicitly for arbitrary m2 ; - and (cf. Eq. (4.13) for the universal part). This leads immediately to the following conclusions: (i) First-order transitions require that the combination U0 () − j has two degenerate minima. This 2 happens for J- + j = 0 and 2 ¡ ; crit or - -2 j=− m2 − ; (4.55) 3 9 2 m2 ¡ ; crit +

-2 : 6

(4.56)

2 Here ; crit is the critical mass term of the Ising model. (ii) The boundary of this region for 2 m2 = ; crit +

-2 6

(4.57)

is a line of second-order phase transitions with vanishing renormalized mass or in$nite correlation length. For j = 0 (e.g. magnets with polynomial potential in the absence of external $elds) Eqs. (4.55), (4.57) have the solutions -;1 = 0;

2 1=2 -;2; 3 = ±(−18 ; : crit )

(4.58)

The second-order phase transition for - = 0 can be described by Ising models for shifted $elds A. For a given model, the way a phase transition line is crossed as the temperature is varied follows from the temperature dependence of j; m2 ; - and . For the gas–liquid transition both j and m2 depend on T . In the vicinity of the boundary of the region of $rst-order transitions the long-range "uctuations play a dominant role and one expects universal critical behavior. The detailed microscopic physics is only re"ected in two non-universal amplitudes. One re"ects the relation between the renormalized and unrenormalized $elds as given by Z0 . The other is connected to the renormalization factor for the mass term. Expressed in terms of renormalized $elds and mass the potential U looses all memory of the microphysics. The critical equation of state of the non-symmetric model (- = 0) follows from the Ising model. With 9U Z2 =9| = || f(x), the scaling form of the equation of state j = 9U=9 for the model with cubic coupling can be written as % % % - - %% - 2 -3 % − j = % + f(x) − ; crit + 2 ; (4.59) % 2 3 3 3 54

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2 where x = 2 =| + - =3 |1= and 2 = m2 − -2 =6 − ; crit . One may choose % % % - %%− -2 - 2 2 % ; crit + + % + y= 3 18 3 %

(4.60)

as the second scaling variable. For a small symmetry breaking cubic coupling - one notes that y ∼ - . The scaling form of the equation of state for the non-symmetric model reads % % % - %% % {f(x) − y} : (4.61) j = % + 3 % This universal form of the equation of state is relevant for a large class of microscopic free energies, far beyond the special polynomial form used for its derivation. It is often useful to express the universal equation of state in terms of renormalized $elds and masses. We use the variables s˜ =

R ; 0R

v=

mR ; mZR2

(4.62)

where mR = (92 U=92R |0R )1=2 is the renormalized mass at the minimum 0R of U (R ), whereas mZR2 is the renormalized mass at the minimum of the corresponding Z2 -symmetric eEective potential obtained for vanishing cubic coupling - = 0. Then the critical temperature corresponds to v = 1. In this parameterization the universal properties of the equation of state for the Ising type $rst-order transition can be compared with transitions in other models—e.g. matrix models [37]—where no simple mapping to a second-order phase transition exists (see Section 5). A convenient universal function G(s; ˜ v) for weak $rst-order transitions can be de$ned as G(s; ˜ v):=

U0 (R ) : 60R

(4.63)

We plot G(s; ˜ v) in Fig. 12 as a function of s˜ for diEerent values of v. For the present model all information necessary for a universal description of $rst-order phase transitions is already contained in Eqs. (4.53) or (4.54). The function G(s; ˜ v) can serve, however, for a comparison with other models, for which a simple relation to a second-order phase transition does not exist. We discuss the function G for matrix models in Section 5.6. At this place we mention that we have computed the potential U both by solving the "ow equations with initial values where - = 0 and by a shift from the Ising model results. We found good agreement between the two approaches. In conclusion, we have employed non-perturbative "ow equations in order to compute explicitly the equation of state. First, we studied models in which the microscopic free energy can be approximated by a polynomial approximation with terms up to quartic order. This covers second-order as well as $rst-order transitions, both for the universal and non-universal features. The same method can be used away from the critical hypersurface, allowing for an explicit connection between critical and non-critical observations. The ability of the method to deal also with a microscopic free energy which is not of a polynomial form is demonstrated by a particular example, namely the equation of state for carbon dioxide. In the vicinity of the endpoint of the critical line we can give an explicit formula for the free energy

J. Berges et al. / Physics Reports 363 (2002) 223 – 386

295

Fig. 12. Universal critical equation of state for a $rst-order transition. We display G(s;˜ v) for v = 1; 1:037; 1:072; 1:137 and 1.195.

ˆ s) density U (n; T )T . Using $ts (4.43) and (4.46), for F( ˆ and z(s) ˜ one $nds U0 (n; T ) = U Z2 (ˆ R (n; T );

mR (T )) + J (T )(n − n∗ ) − K(T ) ;

2 4 6 U Z2 = 12 a˜0 m2R ˆ R + 14 a˜1 mR ˆ R + 16 a˜2 ˆ R

with a˜i ≈ ai and ˆ R (n; T ) = z˜

R (n; T ) mR1=2 (T )

(4.64) (4.65)

R (n; T ) ;

% % % T − T∗ %−6> % % (n − n∗ ) ; R (n; T ) = H± % T∗ % % % % T − T∗ % > % % : mR (T ) = %T ± % T∗ %

(4.66)

The two non-universal functions J (T ) and K(T ) enter in the determination of the chemical potential and the critical line.

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J. Berges et al. / Physics Reports 363 (2002) 223 – 386

In particular, the non-universal amplitudes governing the behavior near the endpoint of the critical line can be extracted from the equation of state: In the vicinity of the endpoint we $nd for T = T∗ 1= −1 |p − p∗ | #¿ − # ∗ = # ∗ − # ¡ = D p (4.67) p∗ with Dp =2:8 g−1 cm3 , where #¿ ¿ #∗ and #¡ ¡ #∗ refer to the density in the high- and low-density region, respectively. At the critical temperature Tc ¡ T∗ and pressure pc ¡ p∗ for a $rst-order transition one $nds for the discontinuity in the density between the liquid (#l ) and gas (#g ) phase p∗ − p c T∗ − T c = BT (4.68) T# = #l − #g = Bp p∗ T∗ with Bp = 0:85 g cm−3 ; BT = 1:5 g cm−3 . This relation also de$nes the slope of the critical line near the endpoint. There is no apparent limitation for the use of the "ow equation for an arbitrary microscopic free energy. This includes the case where U has several distinct minima and, in particular, the interesting case of a tricritical point. At present, the main inaccuracy arises from a simpli$cation of the q2 -dependence of the four-point function which is re"ected in an error in the anomalous dimension 6. The simpli$cation of the momentum dependence of the eEective propagator in the "ow equation plays presumably only a secondary role. In summary, the non-perturbative "ow equation appears to be a very eIcient tool for the derivation of an explicit quantitative connection between the microphysical interactions and the long-range properties of the free energy. 4.7. Critical behavior of polymer chains The large scale properties of isolated polymer chains can be computed from the critical behavior of the O(N )-symmetric scalar theory using a variant of the so-called replica limit: the statistics of polymer chains can be described by the N -component $eld theory in the limit N → 0 [151]. In polymer theory critical behavior occurs when the size of an isolated swollen polymer becomes in$nite. The size of a chain can be de$ned by its mean square distance (˜r() − ˜r(0))2 . Here denotes the distance along the chain between the end points with position ˜r() and ˜r(0). The scaling behavior of the mean square distance is characterized by the exponent >, (˜r() − ˜r(0))2 ∼ 2> :

(4.69)

The exponent > for the polymer system corresponds to the correlation length exponent of the N -component $eld theory in the limit N → 0. There are two independent critical exponents for the O(N ) model near its second-order phase transition. In polymer theory, the exponent - characterizes the asymptotic behavior of the number of con$gurations N () of the self-avoiding chains N () ∼ -−1 :

(4.70)

For a chain with independent links one has -=1. The exponent - corresponds to the critical exponent that describes the behavior of the magnetic susceptibility in the zero component $eld theory. We compute the critical exponents > and -, as well as the anomalous dimension 6, for the O(N ) model in the limit N → 0 using the lowest order of the derivative expansion of the eEective average action (4.2). The "ow equations for the scale dependent eEective potential Uk and wave function

J. Berges et al. / Physics Reports 363 (2002) 223 – 386

297

Table 8 Critical exponents for 0 6 N 6 1 N

>

-

6 (10−2 )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.589 0.594 0.600 0.605 0.610 0.616 0.621 0.626 0.632 0.637 0.643

1.155 1.165 1.175 1.185 1.195 1.205 1.216 1.226 1.237 1.247 1.258

4.06 4.12 4.18 4.22 4.26 4.30 4.32 4.34 4.36 4.37 4.37

Table 9 Critical exponents for polymer chains: comparison between average action, epsilon expansion [129], perturbation series at $xed dimension [129], lattice Monte Carlo [135,153], high-temperature series [131] and experiment [154] N =0

>

-

Average action j-expansion d = 3 expansion Lattice MC HT series Experiment

0.589 0.5875 (25) 0.5882 (11) 0.5877 (6) 0.5878 (6) 0.586 (4)

1.155 1.1575 1.1596 1.1575 1.1594

6 (60) (20) (6) (8)

0.0406 0.0300 (50) 0.0284 (25)

renormalization Zk (or equivalently the anomalous dimension 6) are derived for integer N . The results are given in Eqs. (4.4) and (4.5). We analytically continue the evolution equations to non-integer values of N . We explicitly verify that the limit N → 0 is continuously connected to the results for integer N by computing the exponents for numbers of components N between one and zero. The case N = 1 is also useful in polymer theory. It belongs to the universality class that describes the point at the top of the coexistence curve of a polymer solution [151]. The $rst Table 8 below shows the results for the critical exponents >; - and 6 for several (non-integer) values of N [152]. In the following Table 9, we compare our results for N = 0 with the epsilon expansion [129], perturbation series at $xed dimension [129], lattice Monte Carlo [135], high-temperature series [131] and experiment [154]. The comparison shows a rather good agreement of these results. 4.8. Two-dimensional models and the Kosterlitz–Thouless transition We investigate the O(N )-symmetric linear A-model in two dimensions. Apart from their physical relevance, two-dimensional systems provide a good testing ground for non-perturbative methods. Let us consider $rst the O(N ) model in the limit N → 0 motivated in the previous section. In this limit the two-dimensional model exhibits a second-order phase transition. The critical exponent > describes

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the critical swelling of long polymer chains [151]. The value of this exponent is known exactly, >exact = 0:75 [121]. To compare with the exact result, we study the N = 0 model using the lowest order derivative expansion (4.2) of the eEective average action. We obtain the result > = 0:782 [152] which already compares rather well to the exact result. It indicates that the present techniques allow us to give a uni$ed description of the O(N ) model in two and in three dimensions, as well as four dimensions which will be discussed in Section 8. In the following, we extend the discussion in two dimensions and show that for N = 2 one obtains a good picture of the Kosterlitz–Thouless phase transition. 41 To evaluate the equation for the potential we make a further approximation and expand around the minimum of uk for non-zero squared $eld #˜ = < up to the quadratic order in #: ˜ uk (#) ˜ = uk (<) + 12 (#˜ − <)2 :

(4.71)

= 0 holds independently of t and gives us the evolution equation for the The condition 9u=9#| ˜ #=< ˜ location of the minimum of the potential parametrized by <. A similar evolution equation can be derived [111] for the symmetric regime where < = 0 and an appropriate variable is 9u=9#| ˜ #=0 ˜ . The "ow equations for < and read d< < ≡ = −(d − 2 + 6)< + 2vd (N − 1)ld1 (0) + 6vd ld1 (2<) ; dt d = (d − 4 + 26) + 2vd (N − 1)2 ld2 (0) + 18vd 2 ld2 (2<) ; ≡ (4.72) dt where the “threshold functions” are de$ned in Section 3.2. The theory is in the symmetric phase if <(0) = 0. This happens if < reaches zero for some non-vanishing ks ¿ 0. On the other hand, the phase with spontaneous symmetry breaking corresponds to #0 (0) ¿ 0 where #0 (k) = k d−2 Zk−1 <(k). A second-order phase transition is characterized by a scaling solution corresponding to $xed points for < and . For small deviations from the $xed point, there is typically one infrared unstable direction which is related to the relevant mass parameter. The phase transition can be studied as a function of <() with a critical value <() =
For investigations in two dimensions using similar methods see also Refs. [155,156].

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! = 2< of the radial mode. The threshold functions vanish with powers of !−1 and for N ¿ 1 the leading contributions to the -functions are those from the Goldstone modes. Therefore this limit is called the Goldstone regime. In this approximation the -functions can be expanded in powers of !−1 . In particular, the leading order of the anomalous dimension can be extracted immediately from (4.2): 1 6= + O(<−2 ) : (4.73) 4"< Inserting this result in (4.72) we have (N − 2) (4.74) < = + O(<−1 ) 4" and the leading order of is (N − 1) ln 2 2 + O(<−1 ) : = −2 + (4.75) 2" Eq. (4.75) has a $xed point solution ∗ = 4"=(N − 1) ln 2 ≈ 18:13=(N − 1). For N ¿ 2 there exists a simple relation between the linear and the non-linear A-model: The eEective coupling between the Goldstone bosons of the non-Abelian non-linear A-model can be extracted directly from (4.2) and reads in an appropriate normalization [157] 1 g2 = : (4.76) 2< The lowest order contribution to < (4.74) coincides with the one-loop expression for the running of g2 as computed in the non-linear A-model. We emphasize in this context the importance of the anomalous dimension 6 which changes the factor (N − 1) appearing in (4.72) into the appropriate factor (N − 2) in (4.74). In correspondence with the universality of the two-loop -function for g2 in the non-linear A-model we expect the next to leading term ∼ <−1 in < (4.74) to be also proportional to (N − 2). In order to verify this, one has to go beyond truncation (4.2) and systematically keep all terms contributing in the appropriate order of <−1 . (This calculation is similar to the extraction of the two loop -function of the linear A-model in four dimensions by means of an “improved one loop calculation” using the "ow equation (2.19) [158].) We have calculated the expansion of < up to the order O(<−1 ) for the most general two derivative action, i.e. neglecting only the momentum dependence of Zk and Yk . The result agrees with the two-loop term of the non-linear A-model within a few per cent, and the discrepancy should be attributed to the neglected momentum dependence of the wave function renormalization. The issue of the contribution to < in order <−2 is less clear: Of course, the direct contribution of the Goldstone bosons (combined with their contribution to 6) should always vanish for N = 2 since no non-Abelian coupling exists in this case. The radial mode however, could generate a contribution which is not proportional to (N − 2). This contribution is possibly nonanalytic in <−1 and would correspond to a non-perturbative contribution in the language of the non-linear A-model. Let us now turn to the two-dimensional Abelian model (d = 2; N = 2) for which we want to describe the Kosterlitz–Thouless phase transition. In the limit of vanishing < for large enough < the location of the minimum of uk (#) ˜ (4.71) is independent of the scale k. Therefore the parameter <, or, alternatively, the temperature diEerence Tc − T , can be viewed as a free parameter. If we go beyond the lowest order estimate (4.75) the $xed point for remains, but ∗ becomes dependent on

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Fig. 13. The scale-dependent anomalous dimension 6(t) for a series of initial potentials U approaching the phase transition.

<. This implies that the system has a line of $xed points which is parametrized by < as suggested by results obtained from calculations with the non-linear A-model [159]. In particular, the anomalous dimension 6 depends on the temperature Tc − T (4.73). Even if this picture is not fully accurate for non-vanishing < , it is a very good approximation for large <: The possible running of < is extremely slow, especially if < vanishes in order <−1 . We associate the low temperature or large < phase with the phase of vortex condensation in the non-linear A-model. The correlation length is always in$nite because of the Goldstone boson. Since 6 ¿ 0 we expect the inverse propagator of this Goldstone degree of freedom ∼ (q2 )1−6=2 , thus avoiding Coleman’s no go theorem [115] for free massless particles in two dimensions. On the other hand, for small values of < the threshold functions can be expanded in powers of <. The anomalous dimension is small and < is driven to zero for ks ¿ 0. This corresponds to the symmetric phase of the linear A-model with a massive complex scalar $eld. We associate this high-temperature phase with the phase of vortex disorder in the picture of the non-linear A-model. The transition between the behavior for large and small < is described by the Kosterlitz–Thouless transition. In the language of the linear A-model it is the transition from a special type of spontaneous symmetry breaking to symmetry restoration. Finally, we give a summary of the results obtained from the numerical integration of the evolution equations (4.72) and (4.5) for the special case d=2; N =2. 42 We use a Runge–Kutta method starting at t =0 with arbitrary initial values for < and and solve the "ow equations for large negative values of t. Results are shown in Figs. 13–16 where we plot typical trajectories. The distance between points corresponds to equal steps in t, so that very dense points or lines indicate the very slow running in the vicinity of $xed points. The understanding of the trajectories needs a few comments: The work of Kosterlitz and Thouless [109] suggests that the correlation length is divergent for all temperatures below a critical temperature 42

This study has also been done for d = 2; N = 1. There we $nd a $xed point which corresponds to the second-order phase transition in the Ising model.

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Fig. 14. Flow of the quartic coupling and the minimum < for diEerent t. One observes the (pseudo) critical line (<) as an (approximate) line of $xed points.

Fig. 15. Flow of the anomalous dimension 6 and the minimum <. One observes (approximate) scaling solutions with the universal function 6(<).

Tc and that the critical exponent 6 depends on temperature. The consequence for our model is that above a critical value for < all -functions should vanish for a line of $xed points parametrized by <. From the results in the Goldstone regime and from earlier calculations [159] we conclude that < should vanish faster than <−1 for large <. Our truncation (4.2), however, yields a function < which vanishes only like <−1 . The consequence is that even if the system reaches the supposed line of $xed

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Fig. 16. The beta function for the quartic coupling for diEerent initial < (see text).

points the parameter < decreases very slowly until the transition to the symmetric regime is reached. The anomalous dimension $rst grows with decreasing < (4.73) until the critical value is reached. Then the system runs into the symmetric regime and 6 vanishes. So we expect that 6 reaches a maximum near the phase transition. We use this as a criterion for the critical value
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depends also on (). This can be seen from the curves with <() = 1.) After hitting the line of $xed points the trajectories stay for a large t-interval at very close to zero. Subsequently, the “outgoing curve” indicates the transition to the symmetric phase. In conclusion, both the analytical and the numerical investigations demonstrate all important characteristics of the Kosterlitz–Thouless phase transition for the linear A-model. This belongs to the same universality class as the non-linear A-model and we have demonstrated a close correspondence between the linear and the non-linear A-model with Abelian symmetry. In particular, the phase with vortex disorder in the non-linear A-model corresponds simply to the symmetric phase of the linear A-model. We emphasize that we have never needed the explicit investigation of vortex con$gurations. The exact non-perturbative "ow equation includes automatically all con$gurations. Its ability to cope with the infrared problems of perturbation theory is con$rmed by the present work. Despite the simple and clear qualitative picture arising from truncation (4.2) this study only constitutes a $rst step for a quantitative investigation. It is not excluded that the coincidence of our critical 6c ≈ 0:24 with 14 is somewhat accidental. In order to answer this question one needs to go beyond truncation (4.2). In view of the relatively large value of 6c we expect that the momentum dependence of the wave function renormalization Zk (or the deviation of the inverse propagator from q2 ) could play an important role at the phase transition. This eEect should be included in a more detailed quantitative investigation. 5. Scalar matrix models 5.1. Introduction Matrix models are extensively discussed in statistical physics. Beyond the O(N )-symmetric Heisenberg models (“vector models”), which we have discussed in the previous sections, they correspond to the simplest scalar $eld theories. There is a wide set of diEerent applications, such as the metal insulator transition [160] or liquid crystals [161] or strings and random surfaces [162]. The universal behavior of these models in the vicinity of a second-order or weak $rst-order phase transition is determined by the symmetries and the $eld content of the corresponding $eld theories. We will consider here [37] models with U (N ) × U (N ) symmetry with a scalar $eld in the (NU ; N ) representation, described by an arbitrary complex N × N matrix O. 43 We do not impose non-linear constraints for O a priori but rather use a “classical” potential. This enforces non-linear constraints in certain limiting cases. Among those, our model describes a non-linear matrix model for unitary matrices or one for singular 2 × 2 matrices. The universal critical behavior does not depend on the details of the classical potential and there is no diEerence between the linear and non-linear models in the vicinity of the limiting cases. We concentrate here on three dimensions, relevant for statistical physics and critical phenomena in high-temperature $eld theory. The cases N = 2; 3 have a relation to high-temperature strong interaction physics. At vanishing temperature the four-dimensional models can be used for a description of the pseudoscalar and 43

The methods presented here have recently been applied also to the principle chiral model with SO(3)×O(3) symmetry [165] and to Heisenberg frustrated magnets with O(N ) × O(2) symmetry [166].

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scalar mesons for N quark "avors. For N = 3 the antiHermitean part of O describes here the pions, kaons, 6 and 6 whereas the Hermitean part accounts for the nonet of scalar 0++ mesons. 44 For non-zero temperature T the eEects of "uctuations with momenta p2 . ("T )2 are described by the corresponding three-dimensional models. These models account for the long distance physics and are obtained by integrating out the short distance "uctuations. In particular, the three-dimensional models embody the essential dynamics in the immediate vicinity of a second-order or weak $rst-order chiral phase transition [24 –27]. The four-dimensional models at non-vanishing temperature have also been used for investigations of the temperature dependence of meson masses [163,164]. The simple model investigated in this section is not yet realistic for QCD—it neglects the eEect of the axial anomaly which reduces the chiral "avor symmetry to SU (N ) × SU (N ) × U (1). In the present section, we also neglect the "uctuations of fermions (quarks). They play no role for the universal aspects near the phase transition. They are needed, however, for a realistic connection with QCD and will be included in Section 8. For simplicity, we will concentrate here on N = 2, but our methods can be generalized to N = 3 and the inclusion of the axial anomaly. The case N = 2 also has a relation to the electroweak phase transition in models with two Higgs doublets. Our model corresponds to the critical behavior in a special class of left–right symmetric theories in the limit where the gauge couplings are neglected. Even though vanishing gauge couplings are not a good approximation for typical realistic models one would like to understand this limiting case reliably. For the present matrix models one wants to know if the phase transition becomes second order in certain regions of parameter space. In the context of "ow equations this is equivalent to the question if the system of running couplings admits a $xed point which is infrared stable (except one relevant direction corresponding to T − Tc ). We $nd that the phase transition for the investigated matrix models with N = 2 and symmetry breaking pattern U (2) × U (2) → U (2) is always ("uctuation induced) $rst order, except for a boundary case with enhanced O(8) symmetry. For a large part of parameter space, the transition is weak and one $nds large renormalized dimensionless couplings near the critical temperature. If the running of the couplings towards approximate $xed points (there are no exact $xed points) is suIciently fast the large distance physics looses memory of the details of the short distance or classical action. In this case, the physics near the phase transition is described by a universal equation of state. This new universal critical equation of state for $rst-order transitions involves two (instead of one for second-order transitions) scaling variables. In Section 5.2 we de$ne the U (2)×U (2) symmetric matrix model and we establish the connection to a matrix model for unitary matrices and to one for singular complex 2 × 2 matrices. There we also give an interpretation of the model as the coupled system of two SU (2)-doublets for the weak interaction Higgs sector. The evolution equation for the average potential Uk and its scaling form is computed in Section 5.3. A detailed account on the renormalization group "ow is presented in Section 5.4. Section 5.5 is devoted to an overview over the phase structure and the coarse-grained eEective potential Uk for the three-dimensional theory. We compute the universal form of the equation of state for weak $rst-order phase transitions in Section 5.6 and we extract critical exponents and the corresponding index relations.

44

See Refs. [167,168] for a phenomenological analysis.

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5.2. Scalar matrix model with U (2) × U (2) symmetry We consider a U (2) × U (2) symmetric eEective action for a scalar $eld O which transforms in the (2; 2) representation with respect to the subgroup SU (2) × SU (2). Here O is represented by a complex 2 × 2 matrix and the transformations are O → UOV † ; ’† → VO† U † ;

(5.1)

where U and V are unitary 2 × 2 matrices corresponding to the two distinct U (2) factors. We classify the invariants for the construction of the eEective average action by the number of derivatives. The lowest order is given by ∗ ab k = d d x {Uk (O; O† ) + Zk 9 Oab 9 O } (a; b = 1; 2) : (5.2) The term with no derivatives de$nes the scalar potential Uk which is an arbitrary function of traces of powers of O† O. The most general U (2) × U (2) symmetric scalar potential can be expressed as a function of only two independent invariants, # = Tr (O† O) F = 2 Tr (O† O − 12 #)2 = 2 Tr (O† O)2 − #2 :

(5.3)

Here we have used for later convenience the traceless matrix O† O − 12 # to construct the second invariant. Higher invariants, Tr (O† O − 12 #)n for n ¿ 2, can be expressed as functions of # and F [169]. For the derivative part we consider a standard kinetic term with a scale-dependent wave function renormalization constant Zk . The $rst correction to the kinetic term would include $eld-dependent wave function renormalizations Zk (#; F) plus functions not speci$ed in Eq. (5.2) which account for a diEerent index structure of invariants with two derivatives. The wave function renormalizations may be de$ned at zero momentum or for q2 = k 2 in the hybrid derivative expansion. The next level involves invariants with four derivatives and so on. We de$ne here Zk at the minimum #0 ; F0 of Uk and at vanishing momenta q2 , Zk = Zk (# = #0 ; F = F0 ; q2 = 0) :

(5.4)

The factor Zk appearing in the de$nition of the infrared cutoE Rk in Eq. (2.7) is identi$ed with (5.4). The k dependence of this function is given by the anomalous dimension 6(k) = −

d ln Zk : dt

(5.5)

If ansatz (5.2) is inserted into the "ow equation for the eEective average action (2.19) one obtains "ow equations for the eEective average potential Uk (#; F) and for the wave function renormalization constant Zk (or equivalently the anomalous dimension 6). This is done in Section 5.3. These "ow equations have to be integrated starting from some short-distance scale and one has to specify U and Z as initial conditions. The short-distance potential is taken to be a quartic potential which is parametrized by two quartic couplings U1 ; U2 and a mass term. We start in the spontaneously

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broken regime where the minimum of the potential occurs at a non-vanishing $eld value and there is a negative mass term at the origin of the potential (U 2 ¿ 0), U (#; F) = −U 2 # + 12 U1 #2 + 14 U2 F

(5.6)

and Z = 1. The potential is bounded from below provided U1 ¿ 0 and U2 ¿ − 2U1 . For U2 ¿ 0 one observes the potential minimum for the con$guration Oab = Oab corresponding to the spontaneous symmetry breaking down to the diagonal U (2) subgroup of U (2) × U (2). For negative U2 the potential is minimized by the con$guration Oab = Oa1 ab which corresponds to the symmetry breaking pattern U (2) × U (2) → U (1) × U (1) × U (1). In the special case U2 = 0 the theory exhibits an enhanced O(8) symmetry. This constitutes the boundary between two phases with diEerent symmetry breaking patterns. The limits of in$nite couplings correspond to non-linear constraints in the matrix model. For U1 → ∞ with $xed ratio U 2 = U1 one $nds the constraint Tr (O† O)= U 2 = U1 . By a convenient choice of Z (rescaling of O) this can be brought to the form Tr (O† O) = 2. On the other hand, the limit U2 → +∞ enforces the constraint O† O = 12 Tr(O† O). Combining the limits U1 → ∞; U2 → ∞ the constraint reads O† O=1 and we deal with a matrix model for unitary matrices. (These considerations generalize to arbitrary N .) Another interesting limit is obtained for U1 = − 12 U2 + 1 ; 1 ¿ 0 if U2 → −∞. In this case the non-linear constraint reads (Tr O† O)2 = Tr (O† O)2 which implies for N = 2 that det O = 0. This is a matrix model for singular complex 2 × 2 matrices. One can also interpret our model as the coupled system of two SU (2)-doublets for the weak interaction Higgs sector. This is simply done by decomposing the matrix Oab into two two-component complex fundamental representations of one of the SU (2) subgroups, Oab → O1b ; O2b . The present model corresponds to a particular left–right symmetric model with interactions speci$ed by # = O1† O1 + O2† O2 ;

(5.7)

F = (O1† O1 − O2† O2 )2 + 4(O1† O2 )(O2† O1 ) :

(5.8)

We observe that for a typical weak interaction symmetry breaking pattern the expectation values of O1 and O2 should be aligned in the same direction or one of them should vanish. In the present model this corresponds to the choice U2 ¡ 0. The phase structure of a related models without the term ∼(O1† O2 )(O2† O1 ) has been investigated previously [170] and shows second- or $rst-order transitions. 45 Combining these results with the outcome of this work leads already to a detailed qualitative overview over the phase pattern in a more general setting with three independent couplings for the quartic invariants (O1† O1 + O2† O2 )2 ; (O1† O1 − O2† O2 )2 and (O1† O2 )(O2† O1 ). We also note that the special case U2 = 2U1 corresponds to two Heisenberg models interacting only by a term sensitive to the alignment between O1 and O2 , i.e. a quartic interaction of the form (O1† O1 )2 + (O2† O2 )2 + 2(O1† O2 )(O2† O1 ). The model is now completely speci$ed and it remains to extract the "ow equations for Uk and Zk . 45

First-order phase transitions and coarse graining have also been discussed in a multi-scalar model with Z2 symmetry [171].

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5.3. Scale dependence of the e:ective average potential To obtain Uk we evaluate the "ow equation for the average action (2.19) for a constant $eld with k =PUk where P denotes the volume. With the help of U (2)×U (2) transformations the matrix $eld O can be turned into a standard diagonal form with real non-negative eigenvalues. Without loss of generality, the evolution equation for the eEective potential can therefore be obtained by calculating the trace in (2.19) for small $eld "uctuations ab around a constant background con$guration which is real and diagonal, Oab = Oa ab ;

Oa∗ = Oa :

√

(5.9)

We separate the "uctuation $eld into its real and imaginary part, ab = (1= 2)(Rab + iIab ) and perform the second functional derivatives of k with respect to the eight real components. For the constant con$guration (5.9) it turns out that k(2) has a block diagonal form because mixed derivatives with respect to real and imaginary parts of the $eld vanish. The remaining submatrices 2 k =Rab Rcd and 2 k =Iab Icd can be diagonalized in order to $nd the inverse of k(2) + Rk under the trace occurring in Eq. (2.19). Here the momentum-independent part of k(2) de$nes the mass matrix by the second functional derivatives of Uk . The eight eigenvalues of the mass matrix are (M1± )2 = Uk + 2(# ± (#2 − F)1=2 )9F Uk ; (M2± )2 = Uk ± 2F1=2 9F Uk

(5.10)

corresponding to second derivatives with respect to I and (M3± )2 = (M1± )2 ; (M4± )2 = Uk + #Uk + 2#9F Uk + 4F9F Uk + 4#F92F Uk ± {F(Uk + 49F Uk + 4#9F Uk + 4F92F Uk )2 + (#2 − F)(Uk − 29F Uk − 4F92F Uk )2 }1=2 ;

(5.11)

corresponding to second derivatives with respect to R . Here the eigenvalues are expressed in terms of the invariants # and F using O12 = 12 (# + F1=2 );

O22 = 12 (# − F1=2 )

(5.12)

and we adopt the convention that a prime on Uk (#; F) denotes the derivative with respect to # at $xed F and k and 9nF Uk ≡ 9n Uk =(9F)n . The "ow equation for the eEective average potential is simply expressed in terms of the mass eigenvalues dd q 9 9 1 Uk (#; F) = Rk (q) 9t 2 (2")d 9t 2 2 1 × + + Pk (q) + (M1+ (#; F))2 Pk (q) + (M1− (#; F))2 Pk (q) + (M2+ (#; F))2 1 1 1 : + + + Pk (q) + (M2− (#; F))2 Pk (q) + (M4+ (#; F))2 Pk (q) + (M4− (#; F))2 (5.13)

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In the r.h.s. of the evolution equation appears the (massless) inverse average propagator Pk (q) = Zk q2 + Rk (q) =

Zk q2 ; 1 − e−q2 =k 2

(5.14)

which incorporates the infrared cutoE function Rk given by Eq. (2.17). The only approximation so far is due to the derivative expansion (5.2) of k which enters into the "ow equation (5.13) through the form of Pk . The mass eigenvalues (5.10) and (5.11) appearing in the above "ow equation are exact since we have kept for the potential the most general form Uk (#; F). Spontaneous symmetry breaking and mass spectra. In the following we consider spontaneous symmetry breaking patterns and the corresponding mass spectra for a few special cases. For the origin at Oab = 0 all eigenvalues equal Uk (0; 0). If the origin is the absolute minimum of the potential we are in the symmetric regime where all excitations have mass squared Uk (0; 0). Spontaneous symmetry breaking to the diagonal U (2) subgroup of U (2)×U (2) can be observed for a $eld con$guration which is proportional to the identity matrix, i.e. Oab =Oab . Invariants (5.3) take on values #=2O2 and F=0. The relevant information for this symmetry breaking pattern is contained in Uk (#) ≡ Uk (#; F = 0). In case of spontaneous symmetry breaking there is a non-vanishing value for the minimum #0 of the potential. With Uk (#0 )=0 one $nds the expected four massless Goldstone bosons with (M1− )2 = (M2± )2 = (M3− )2 = 0. In addition there are three massive scalars in the adjoint representation of the unbroken diagonal SU (2) with mass squared (M1+ )2 =(M3+ )2 =(M4− )2 =4#0 9F Uk and one singlet with mass squared (M4+ )2 = 2#0 Uk . The situation corresponds to chiral symmetry breaking in two "avor QCD in absence of quark masses and the chiral anomaly. The Goldstone modes are the pseudoscalar pions and the 6 (or 6 ), the scalar triplet has the quantum numbers of a0 and the singlet is the so-called A-$eld. Another interesting case is the spontaneous symmetry breaking down to a residual U (1) × U (1) × U (1) subgroup of U (2) × U (2) which can be observed for the con$guration Oab = Oa1 ab (#=O2 ; F=O4 =#2 ). Corresponding to the number of broken generators one observes the $ve massless Goldstone bosons (M1± )2 =(M2+ )2 =(M3± )2 =0 for the minimum of the potential at Uk +2#0 9F Uk =0. In addition there are two scalars with mass squared (M2− )2 = (M4− )2 = Uk − 2#0 9F Uk and one with (M4+ )2 = Uk + 2#0 Uk + 6#0 9F Uk + 8#20 9F Uk + 8#30 92F Uk . We $nally point out the special case where the potential is independent of the second invariant F. In this case there is an enhanced O(8) symmetry instead of U (2) × U (2). With 9nF Uk ≡ 0 and Uk (#0 ) = 0 one observes the expected seven massless Goldstone bosons and one massive mode with mass squared 2#0 Uk . Scaling form of the ;ow equations. For the O(8) symmetric model in the limit U2 = 0 one expects a region of the parameter space which is characterized by renormalized masses much smaller than the ultraviolet cutoE or inverse microscopic length scale of the theory. In particular, in the absence of a mass scale one expects a scaling behavior of the eEective average potential Uk . The behavior of Uk at or near a second-order phase transition is most conveniently studied using the scaling form of the evolution equation. This form is also appropriate for an investigation that has to deal with weak $rst-order phase transitions as encountered in the present model for U2 ¿ 0. The remaining part of this subsection is devoted to the derivation of the scaling form (5.18) of the "ow equation (5.13). In the present form of Eq. (5.13) the r.h.s. shows an explicit dependence on the scale k once the momentum integration is performed. By a proper choice of variables, we cast the evolution equation

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into a form where the scale no longer appears explicitly. We introduce a dimensionless potential uk = k −d Uk and express it in terms of dimensionless renormalized $elds #˜ = Zk k 2−d # ; F˜ = Zk2 k 4−2d F :

(5.15)

The derivatives of uk are given by 9nF˜ uk(m) (#; ˜ F) ˜ = Zk−2n−m k (2n+m−1)d−4n−2m 9nF Uk(m) (#; F) :

(5.16)

(Note that uk(m) denotes m derivatives with respect to #˜ at $xed F˜ and k, while Uk(m) denotes m derivatives with respect to # at $xed F and k.) With 9 uk (#; ˜ F) ˜ |#;˜ F˜ = −duk (#; ˜ F) ˜ + (d − 2 + 6)#u ˜ k (#; ˜ F) ˜ + (2d − 4 + 26)F9 ˜ F˜uk (#; ˜ F) ˜ 9t 9 + k −d Uk (#(#); ˜ F(F)) ˜ |#; F 9t

(5.17)

one obtains from (5.13) the evolution equation for the dimensionless potential. Here the anomalous dimension 6 arises from the t-derivative acting on Zk and is given by Eq. (5.5). Using the notation ld0 (w; 6; z = 1) = ld0 (w; 6) for the threshold functions de$ned in Section A (see also Section 3.2) one obtains 9 uk (#; ˜ F) ˜ = −duk (#; ˜ F) ˜ + (d − 2 + 6)#u ˜ k (#; ˜ F) ˜ + (2d − 4 + 26)F9 ˜ F˜uk (#; ˜ F) ˜ 9t + 4vd ld0 ((m+ ˜ F)) ˜ 2 ; 6) + 4vd ld0 ((m− ˜ F)) ˜ 2 ; 6) + 2vd ld0 ((m+ ˜ F)) ˜ 2 ; 6) 1 (#; 2 (#; 1 (#; ˜ F)) ˜ 2 ; 6) + 2vd ld0 ((m+ ˜ F)) ˜ 2 ; 6) + 2vd ld0 ((m− ˜ F)) ˜ 2 ; 6) ; + 2vd ld0 ((m− 4 (#; 2 (#; 4 (#;

(5.18)

where the dimensionless mass terms are related to (5.11) according to ˜ F)) ˜ 2= (m± i (#;

˜ F(F))) ˜ 2 (Mi± (#(#); : Zk k 2

(5.19)

Eq. (5.18) is the scaling form of the "ow equation we are looking for. For a F-independent ˜ potential it reduces to the evolution equation for the O(8) symmetric model [111,118] given by Eq. (3.8) with z =1; y˜ =0; TCk =0. The potential uk at a second-order phase transition is given by a k-independent (scaling) solution 9uk =9t = 0 [111,118]. For this solution all the k-dependent functions in the r.h.s. of Eq. (5.18) become independent of k. For a weak $rst-order phase transition these functions will show a weak k dependence for k larger than the inherent mass scale of the system (cf. Section 5.4). There is no particular advantage of the scaling form of the "ow equation for strong $rst-order phase transitions. Eq. (5.18) describes the scale dependence of the eEective average potential uk by a non-linear partial diEerential equation for the three variables t, #˜ and F. ˜ We concentrate in the following on spontaneous symmetry breaking with a residual U (2) symmetry group. As we have already pointed out in Section 5.2 this symmetry breaking can be observed for a con$guration which is proportional to the identity and we have F˜ = 0. In this case, eigenvalues (5.10) and (5.11) of the mass matrix

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with (5.19) are given by ± 2 − 2 2 (m− 1 ) = (m2 ) = (m3 ) = uk ; − 2 2 + 2 (m+ ˜ F˜uk ; 1 ) = (m3 ) = (m4 ) = uk + 4#9 2 (m+ ˜ k 4 ) = uk + 2#u

(5.20)

˜ ≡ uk (#; ˜ F˜ = 0) only the functions and in the r.h.s. of the partial diEerential equation (5.18) for uk (#) uk (#); ˜ uk (#) ˜ and 9F˜uk (#) ˜ appear. We determine these functions through the use of "ow equations which are obtained by taking the derivative in Eq. (5.18) with respect to #˜ and F˜ evaluated at F˜ = 0. Since we are interested in the #-dependence ˜ of the potential at F˜ = 0 we shall use a truncated expansion in F˜ with 9nF˜ uk (#; ˜ F˜ = 0) = 0

for n ¿ 2 :

(5.21)

In three space dimensions the neglected (#-dependent) ˜ operators have negative canonical mass dimension. We make no expansion in terms of #˜ since the general #-dependence ˜ allows a description of a $rst-order phase transition where a second local minimum of uk (#) ˜ appears. Approximation (5.21) only aEects the "ow equations for 9F˜uk . The form of the "ow equation for uk is not aEected by the truncation. From uk we obtain the eEective average potential uk by simple integration. We have tested the sensitivity of our results for uk to a change in 9F˜uk by neglecting the #-dependence ˜ of the F-derivative. ˜ We observed no qualitative change of the results. We expect that the main truncation error is due to the derivative expansion (5.2) for the eEective average action. To simplify notation we introduce j(#) ˜ = uk (#; ˜ F˜ = 0) ; ˜ = uk (#; ˜ F˜ = 0) ; 1 (#) ˜ = 49F˜uk (#; ˜ F˜ = 0) : 2 (#)

(5.22)

Higher derivatives are denoted by primes on the #-dependent ˜ quartic “couplings”, i.e. 1 = uk ; 2 = 9F˜uk , etc. It is convenient to introduce two-parameter functions ldn1 ;n2 (w1 ; w2 ; 6) [169]. For n1 = n2 = 1 their relation to the functions ldn (w; 6) can be expressed as ld1; 1 (w1 ; w2 ; 6) =

1 [ld (w1 ; 6) − ld1 (w2 ; 6)] w2 − w 1 1

for w1 = w2 ;

ld1; 1 (w; w; 6) = ld2 (w; 6)

(5.23)

and 1 9 d l (w1 ; w2 ; 6) ; ldn1 ;n2 (w1 ; w2 ; 6) = ldn2 ;n1 (w2 ; w1 ; 6) : n1 9w1 n1 ;n2 With the help of these functions the scale dependence of j is described by 9j = (−2 + 6)j + (d − 2 + 6)# ˜ 1 − 6vd (1 + 2 + # ˜ 2 )ld1 (j + # ˜ 2 ; 6) 9t ldn1 +1;n2 (w1 ; w2 ; 6) = −

− 2vd (31 + 2# ˜ 1 )ld1 (j + 2# ˜ 1 ; 6) − 8vd 1 ld1 (j; 6)

(5.24)

(5.25)

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311

and for 1 one $nds 91 = (d − 4 + 26)1 + (d − 2 + 6)# ˜ 1 9t + 6vd [(1 + 2 + # ˜ 2 )2 ld2 (j + # ˜ 2 ; 6) − (1 + 22 + # ˜ 2 )ld1 (j + # ˜ 2 ; 6)] ˜ 1 )2 ld2 (j + 2# ˜ 1 ; 6) − (51 + 2# ˜ 1 )ld1 (j + 2# ˜ 1 ; 6)] + 2vd [(31 + 2# + 8vd [(1 )2 ld2 (j; 6) − 1 ld1 (j; 6)] :

(5.26)

Similarly the scale dependence of 2 is given by 92 = (d − 4 + 26)2 + (d − 2 + 6)# ˜ 2 − 4vd (2 )2 ld1; 1 (j + # ˜ 2 ; j; 6) 9t + 2vd [3(2 )2 + 121 2 + 8# ˜ 2 (1 + 2 ) + 4#˜2 (2 )2 ]ld1; 1 (j + # ˜ 2 ; j + 2# ˜ 1 ; 6) − 14vd 2 ld1 (j + # ˜ 2 ; 6) − 2vd (52 + 2# ˜ 2 )ld1 (j + 2# ˜ 1 ; 6) + 2vd [(2 )2 ld2 (j; 6) − 42 ld1 (j; 6)] :

(5.27)

For the numerical solution we evaluate the above "ow equations at diEerent points #˜i for i = 1; : : : ; l and use a set of matching conditions that are described in Ref. [107]. If there is a minimum of the potential at non-vanishing < ≡ #˜0 , the condition j(<) = 0 can be used to obtain the scale dependence of <(k): % % d< −1 9j % = −[1 (<)] dt 9t %#=< ˜ 2 (<) + <2 (<) d l1 (<2 (<); 6) = − (d − 2 + 6)< + 6vd 1 + 1 (<) 2<1 (<) d l1 (2<1 (<); 6) + 8vd ld1 (0; 6) : (5.28) + 2vd 3 + 1 (<) To make contact with -functions for the couplings at the potential minimum < we point out the relation % 91;(m)2 %% d1;(m)2 (<) d< = : (5.29) + 1;(m+1) (<) % 2 dt 9t % dt #=< ˜

It remains to compute the anomalous dimension 6 de$ned in (5.5) which describes the scale dependence of the wave function renormalization Zk . We consider a space dependent distortion of the constant background $eld con$guration (5.9) of the form Oab (x) = Oa ab + [Oe−iQx + O∗ eiQx ]Rab :

(5.30)

Insertion of the above con$guration into parametrization (5.2) of k yields % % 9 k 1 1 2 % lim : Zk = Zk (#; F; Q = 0) = ∗ 2 ∗ 2 2 R Rab Q →0 9Q (OO ) % ab

O=0

(5.31)

To obtain the "ow equation of the wave function renormalization one expands the eEective average action around a con$guration of form (5.30) and evaluates the r.h.s. of Eq. (2.19). This computation

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has been done in Ref. [169] for a “Goldstone” con$guration with Rab = a1 b2 − a2 b1

(5.32)

and Oa ab = Oab corresponding to a symmetry breaking pattern with residual U (2) symmetry. The result of Ref. [169] can be easily generalized to arbitrary $xed $eld values of #˜ and we $nd vd 2 d ˜ ˜ 1 ; 6) + (2 )2 md2; 2 (j; j + # ˜ 2 ; 6)] : (5.33) 6(k) = 4 #[4( 1 ) m2; 2 (j; j + 2# d The de$nition of the threshold function md2; 2 (w1 ; w2 ; 6) = md2; 2 (w1 ; w2 ) − 6mˆ d2; 2 (w1 ; w2 ) ;

(5.34)

can be found in Appendix A. For vanishing arguments the functions md2; 2 and mˆ d2; 2 are of order unity. They are symmetric with respect to their arguments and in leading order md2; 2 (0; w) ∼ mˆ d2; 2 (0; w) ∼ w−2 for w1. According to Eq. (5.4) we use #˜ = < to de$ne the uniform wave function renormalization Zk ≡ Zk (<) :

(5.35)

We point out that according to our truncation of the eEective average action with Eq. (5.33) the anomalous dimension 6 is exactly zero at #˜ = 0. This is an artifact of the truncation and we expect the symmetric phase to be more aEected by truncation errors than the spontaneously broken phase. We typically observe small values for 6(k) = −d(ln Zk )=dt (of the order of a few per cent). The smallness of 6 is crucial for our approximation of a uniform wave function renormalization to give quantitatively reliable results for the equation of state. For the universal equation of state given in Section 5.6, one has 6 = 0:022 as given by the corresponding index of the O(8) symmetric “vector” model. 5.4. Renormalization group ;ow of couplings To understand the detailed picture of the phase structure, which is presented in Section 5.5, we will consider the "ow of some characteristic quantities for the eEective average potential as the infrared cutoE k is lowered. The short-distance potential U given in Eq. (5.6) is parametrized by quartic couplings, (5.36) U1 ; U2 ¿ 0 and the location of its minimum is given by #0 = U 2 = U1 :

(5.37)

We integrate the "ow equation for the eEective average potential Uk for a variety of initial conditions #0 ; U1 and U2 . In particular, for general U1 ; U2 ¿ 0 we are able to $nd a critical value #0 =#0c for which the system exhibits a $rst-order phase transition. In this case, the evolution of Uk leads at some scale k2 ¡ to the appearance of a second local minimum at the origin of the eEective average potential and both minima become degenerate in the limit k → 0. If #0 (k) ¿ 0 denotes the k-dependent outer minimum of the potential (Uk (#0 ) = 0, where the prime on Uk denotes the derivative with respect to # at $xed k) at a $rst-order phase transition one has lim (Uk (0) − Uk (#0 )) = 0 :

k →0

(5.38)

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313

A measure of the distance from the phase transition is the diEerence < = (#0 − #0c )=. If U 2 and therefore #0 is interpreted as a function of temperature, the deviation < is proportional to the deviation from the critical temperature Tc , i.e. < = A(T ) (Tc − T ) with A(Tc ) ¿ 0. We will always consider in this subsection the trajectories for the critical “temperature”, i.e. < = 0, and we follow the "ow for diEerent values of the short-distance parameters 1 and 2 . The discussion for suIciently small < is analogous. In particular, we compare the renormalization group "ow of these quantities for a weak and a strong $rst-order phase transition. In some limiting cases, their behavior can be studied analytically. For the discussion, we will frequently consider the "ow equations for the quartic “couplings” 1 (#); ˜ 2 (#), ˜ Eqs. (5.26), (5.27) and for the minimum <, Eq. (5.28). In Figs. 17, 18 we follow the "ow of the dimensionless renormalized minimum < and the radial mass term m˜ 2 = 2<1 (<) in comparison to their dimensionful counterparts #0R = k< and m2R = k 2 m˜ 2 in units of the momentum scale . We also consider the dimensionless renormalized mass term m˜ 22 = <2 (<) corresponding to the curvature of the potential in the direction of the second invariant F. ˜ 3 The height of the potential barrier UB (k) = k uk (#˜B ) with uk (#˜B ) = 0; 0 ¡ #˜B ¡ <, and the height of the outer minimum U0 (k) = k 3 uk (<) is also displayed. Fig. 17 shows these quantities as a function of t = ln(k=) for 1 = 2; 2 = 0:1. One observes that the "ow can be separated into two parts. The $rst part ranging from t = 0 to t −6 is characterized by < const: and small m˜ 22 . It is instructive to consider what happens in the case m˜ 22 ≡ 0. In this case 2 ≡ 0 and the "ow is governed by the Wilson–Fisher $xed point of the O(8) symmetric theory. At the corresponding second-order phase transition the evolution of uk leads to the scaling solution of (5.18) which is obtained for 9uk =9t = 0. As a consequence uk becomes a k-independent function that takes on constant ($xed point) values. In particular, the minimum < of the potential takes on its $xed point value <(k)=
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Fig. 17. Running couplings for a weak $rst-order transition. We show the scale dependence of the dimensionless renormalized masses m˜ 2 ; m˜ 22 , minimum < and dimensionful counterparts m2R = k 2 m˜ 2 ; #0R = k< in units of . We also show UB (k) and U0 (k), the value of the potential at the top of the potential barrier and at the minimum #0R , respectively. The short-distance parameters are 1 = 2; 2 = 0:1 and < = 0. The right panel shows the approximate scaling solution.

the short-distance potential u . As a consequence we are able to observe universal behavior as is demonstrated in Fig. 23. To discuss the case 2 =1 1 we consider the "ow equations for the couplings at the minimum < = 0 of the potential given by (5.28) and (5.26) with (5.26), (5.27). In the limit of an in$nite mass term m˜ 22 = <2 (<) → ∞ the -functions for 1 (<) and < become independent from 2 (<) due to the threshold functions, with l3n (<2 ) ∼ (<2 )−(n+1) for large <2 (<). As a consequence 1 and < equal the -functions for an O(5) symmetric model. We argue in the following that in this large coupling limit "uctuations of massless Goldstone bosons lead to an attractive $xed point for 2 (<). We take the "ow equations (5.29) and (5.27) for 2 (<) keeping only terms with positive canonical mass dimension for a qualitative discussion. (This amounts to the approximation 1(n) (<) = 2(n) (<) = 0 for n ¿ 1.) To be explicit, one may consider the case for given 1 = 2. The critical cutoE value for the potential minimum is < 0:2 for 2 1. For <2 (<)1 and taking 6 0 the -function for 2 (<) is to a good approximation given by (d = 3) d2 (<) = −2 (<) + 2v3 (2 (<))2 l32 (0) : dt

(5.39)

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315

Fig. 18. Running couplings for a strong $rst-order transition. We plot the same couplings as in Fig. 17 for 1 = 0:1 and 2 = 2. No approximate scaling solution is reached.

The second term in the r.h.s. of Eq. (5.39) is due to massless Goldstone modes which give the dominant contribution in the considered range. The solution of (5.39) implies an attractive $xed point for 2 (<) with a value 1 (5.40) 4"2 : 2? (<) = 2v3 l32 (0) Starting from 2 one $nds for the “time” |t| necessary to reach a given 2 (<) ¿ 2? (<) 2 (<) − 2? (<) : |t| = −ln 2 (<)(1 − 2? (<)=2 )

(5.41)

This converges to a $nite value for 2 → ∞. The further evolution therefore becomes insensitive to the initial value for 2 in the large coupling limit. The "ow of 1 (<) and < is not aEected by the initial running of 2 (<) and quantities like T#0R = or mR =T#0R become independent of 2 if the coupling is suIciently large. This qualitative discussion is con$rmed by the numerical solution of the full set of equations presented in Figs. 22 and 23 of Section 5.5. For the $xed point value we obtain 2? (<)=38:02. We point out that an analogous discussion for the large coupling region of 1 cannot be made. This can be seen by considering the mass term at the origin of the short-distance potential (5.6) given by u (0; 0) = −< 1 . Because of the pole of l3n (w; 6) at w = −1 for n ¿ 12 [37] one obtains the constraint < 1 ¡ 1 :

(5.42)

In the limit 1 → ∞, the mass term 2< 1 at the minimum < of the potential at the critical temperature therefore remains $nite.

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Fig. 19. The eEective average potential Uk=kf as a function of the renormalized $eld OR . The potential is shown for various values of < ∼ Tc − T . The parameters for the short-distance potential U are (1) < = −0:03, (2) < = −0:015, (3) < = 0, (4) < = 0:04, (5) < = 0:1 and 1 = 0:1, 2 = 2.

5.5. Phase structure of the U (2) × U (2) model We study the phase structure of the U (2) × U (2) symmetric model in three space dimensions. We concentrate here on the spontaneous symmetry breaking with a residual U (2) symmetry group. We consider in the following the eEective average potential Uk for a nonzero scale k. This allows to observe the non-convex part of the potential. As an example we show in Fig. 19 the eEective average potential Uk=kf for 1 = U1 = = 0:1 and 2 = U2 = = 2 as a function of the renormalized $eld OR = (#R =2)1=2 with #R = Zk=kf #. The scale kf is some characteristic scale below which the location of the minimum #0 (k) becomes essentially independent of k. Its precise de$nition is given below. We have normalized Ukf and OR to powers of the renormalized minimum O0R (kf ) = (#0R (kf )=2)1=2 with #0R (kf ) = Zkf #0 (kf ). The potential is shown for various values of deviations from the critical temperature or < . For the given examples < =−0:03; −0:015 the minimum at the origin becomes the absolute minimum and the system is in the symmetric (disordered) phase. Here O0R denotes the minimum in the metastable ordered phase. In contrast, for < = 0:04; 0:1 the absolute minimum is located at OR =O0R = 1 which characterizes the spontaneously broken phase. For large enough < the local minimum at the origin vanishes. For < = 0 the two distinct minima are degenerate in

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317

Fig. 20. The minimum #0R and the mass term mR in units of the momentum scale as a function of −< or temperature (1 = 0:1; 2 = 2, k = kf ). For < = 0 one observes the jump in the renormalized order parameter T#0R and mass TmR .

height. 46 As a consequence the order parameter makes a discontinuous jump at the phase transition which characterizes the transition to be $rst order. It is instructive to consider some characteristic values of the eEective average potential. In Fig. 20 we consider for 1 = 0:1; 2 = 2 the value of the renormalized minimum #0R (kf ) and the radial mass term as a function of −< or temperature. In the spontaneously broken phase the renormalized radial mass squared is given by m2R (kf ) = 2Zk−f 1 #0 Ukf (#0 ) ;

(5.43)

whereas in the symmetric phase the renormalized mass term reads m20R (kf ) = Zk−f 1 Ukf (0) :

(5.44)

At the critical temperature (< = 0) one observes the discontinuity T#0R = #0R (kf ) and the jump in the mass term TmR = mR (kf ) − m0R (kf ) = mcR − mc0R . (Here the index “c” denotes < = 0). The ratio T#0R = is a rough measure for the “strength” of the $rst-order transition. For T#0R =1 the phase transition is weak in the sense that typical masses are small compared to . In consequence, the long-wavelength "uctuations play a dominant role and the system exhibits universal behavior, 46

We note that the critical temperature is determined by condition (5.38) in the limit k → 0. Nevertheless for the employed non-vanishing scale k = kf the minima of Uk become almost degenerate at the critical temperature.

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Fig. 21. Lines of constant jump of the renormalized order parameter T#0R = at the phase transition in the ln(1 ); ln(2 ) plane. The curves correspond to (1) ln(T#0R =) = −4:0, (2) ln(T#0R =) = −4:4, (3) ln(T#0R =) = −10:2, (4) ln(T#0R =) = −14:3.

i.e. it becomes largely independent of the details at the short-distance scale −1 . We will discuss the universal behavior in more detail below. In order to characterize the strength of the phase transition for arbitrary positive values of 1 and 2 we consider lines of constant T#0R = in the 1 ; 2 plane. In Fig. 21 this is done for the logarithms of these quantities. For $xed 2 one observes that the discontinuity at the phase transition weakens with increased 1 . On the other hand for given 1 one $nds a larger jump in the order parameter for increased 2 . This is true up to a saturation point where T#0R = becomes independent of 2 . In the plot this can be observed from the vertical part of the line of constant ln(T#0R =). This phenomenon occurs for arbitrary non-vanishing T#0R = in the strong 2 coupling limit as discussed in Section 5.4. In the following we give a detailed quantitative description of the $rst-order phase transitions and a separation in weak and strong transitions. We consider some characteristic quantities for the eEective average potential in dependence on the short-distance parameters 1 and 2 for < = 0. We consider the discontinuity in the renormalized order parameter T#0R and the inverse correlation lengths (mass terms) mcR and mc0R in the ordered and the disordered phase, respectively. Fig. 22 shows the logarithm of T#0R in units of as a function of the logarithm of the initial coupling 2 . We have connected the calculated values obtained for various $xed 1 = 0:1, 2 and 1 = 4 by straight lines.

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Fig. 22. Strength of the $rst-order phase transition. We plot the logarithm of the discontinuity of the renormalized order parameter T#0R = as a function of ln(2 ). Data points for $xed (1) 1 = 0:1, (2) 1 = 2, (3) 1 = 4 are connected by straight lines. One observes a universal slope for small 2 which is related to a critical exponent.

For 2 =1 . 1 the curves show constant positive slope. In this range T#0R follows a power-law behavior T#0R = R (2 )K ;

K = 1:93 :

(5.45)

The critical exponent K is obtained from the slope of the curve in Fig. 22 for 2 =1 1. The exponent is universal and, therefore, does not depend on the speci$c value for 1 . On the other hand, the amplitude R grows with decreasing 1 . For vanishing 2 the order parameter changes continuously at the transition point and one observes a second-order phase transition as expected for the O(8) symmetric vector model. As 2 =1 becomes larger than one the curves deviate substantially from the linear behavior. The deviation depends on the speci$c choice of the short-distance potential. For 2 =1 1 the curves "atten. In this range T#0R becomes insensitive to a variation of the quartic coupling 2 . In addition to the jump in the order parameter we present the mass terms mcR and mc0R which we normalize to T#0R . In Fig. 23 these ratios are plotted versus the logarithm of the ratio of the initial quartic couplings 2 =1 . Again values obtained for $xed 1 = 0:1, 2 and 1 = 4 are connected by straight lines. The universal range is set by the condition mcR =T#0R const. (equivalently for mc0R =T#0R ). The universal ratios are mcR =T#0R = 1:69 and mc0R =T#0R = 1:26. For the given curves universality holds approximately for 2 =1 . 0:5 and becomes “exact” in the limit 2 =1 → 0. In this range we obtain ˜ 2 )K : mcR = S(2 )K ; mc0R = S( (5.46)

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Fig. 23. The inverse correlation lengths mcR and mc0R in the ordered and the disordered phase, respectively. They are normalized to T#0R and given as a function of ln(2 =1 ). Data points for $xed (1) 1 = 0:1, (2) 1 = 2, (3) 1 = 4 are connected by straight lines. One observes universality for small ratios 2 =1 .

The universal features of the system are not restricted to the weak coupling region of 2 . This is demonstrated in Fig. 23 for values up to 2 2. The ratios mcR =T#0R and mc0R =T#0R deviate from the universal values as 2 =1 is increased. For $xed 2 a larger 1 results in a weaker transition concerning T#0R =. The ratio mcR =T#0R increases with 1 for small $xed 2 whereas in the asymptotic region, 2 =1 1, one observes from Fig. 23 that this tendency is reversed and mcR =T#0R , mc0R =T#0R start to decrease at about 1 2. In summary, the above results show that though the short-distance potential U indicates a second-order phase transition, the transition becomes $rst order once "uctuations are taken into account. This "uctuation induced $rst-order phase transition is known in four dimensions as the Coleman–Weinberg phenomenon [172]. The question of the order of the phase transition of the three-dimensional U (2) × U (2) symmetric model has been addressed also using the j-expansion [173,24], in lattice studies [174] and in high-temperature expansion [175]. All studies are consistent with the $rst-order nature of the transition and with the absence of non-perturbative infrared stable $xed points. Our method gives here a clear and unambiguous answer and allows a detailed quantitative description of the phase transition. The universal form of the equation of state for weak $rst-order phase transitions is presented in Section 5.6.

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In the following, we specify the scale kf for which we have given the eEective average potential Uk . We observe that Uk depends strongly on the infrared cutoE k as long as k is larger than the scale k2 where the second minimum of the potential appears. Below k2 the two minima start to become almost degenerate for T near Tc and the running of #0 (k) stops rather soon. The non-vanishing value of k2 induces a physical infrared cutoE and represents a characteristic scale for the $rst-order phase transition. We stop the integration of the "ow equation for the eEective average potential at a scale kf ¡ k2 which is determined in terms of the curvature (mass term) at the top of the potential barrier that separates the two local minima of Uk at the origin and at #0 (k). The top of the potential barrier at #B (k) is determined by Uk (#B ) = 0

(5.47)

for 0 ¡ #B (k) ¡ #0 (k) and for the renormalized mass term at #B (k) one obtains m2B; R (k) = 2Zk−1 #B Uk (#B ) ¡ 0 :

(5.48)

We $x our $nal value for the running by kf2 − |m2B; R (kf )| kf2

= 0:01 :

(5.49)

For this choice the coarse-grained eEective potential Ukf essentially includes all "uctuations with momenta larger than the mass |mB; R | at the top of the potential barrier. It is a non-convex function which is the appropriate quantity for the study of physical processes such as tunneling or in"ation. 5.6. Universal equation of state for weak >rst-order phase transitions We presented in Section 5.5 some characteristic quantities for the eEective average potential which become universal at the phase transition for a suIciently small quartic coupling 2 = U2 = of the short-distance potential U (5.6). The aim of this section is to generalize this observation and to $nd a universal scaling form of the equation of state for weak $rst-order phase transitions. The equation of state relates the derivative of the free energy U = limk →0 Uk to an external source, 9U=9O = j. Here the derivative has to be evaluated in the outer convex region of the potential. For instance, for the meson model of strong interactions the source j is proportional to the average quark mass [24,176] and the equation of state permits us to study the quark mass dependence of properties of the chiral phase transition. We will compute the equation of state for a nonzero coarse graining scale k. It therefore contains information for quantities like the “classical” bubble surface tension in the context of Langer’s theory of bubble formation which will be discussed in Section 6. In three dimensions the U (2) × U (2) symmetric model exhibits a second-order phase transition in the limit of a vanishing quartic coupling 2 due to an enhanced O(8) symmetry. In this case, there is no scale present in the theory at the critical temperature. In the vicinity of the critical temperature (small |< | ∼ |Tc − T |) and for small enough 2 one therefore expects a scaling behavior of the eEective average potential Uk and accordingly a universal scaling form of the equation of state. At the second-order phase transition in the O(8) symmetric model there are only two independent scales corresponding to the deviation from the critical temperature and to the external source or O. As a consequence the properly rescaled potential U=#3R or U=#(+1)=2 (with the usual critical exponent ) can only depend on one dimensionless ratio. A possible set of variables to represent

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the two independent scales are the renormalized minimum of the potential O0R = (#0R =2)1=2 (or the renormalized mass for the symmetric phase) and the renormalized $eld OR = (#R =2)1=2 . The rescaled potential will then only depend on the scaling variable s˜ = OR =O0R [36]. Another possible choice is the Widom scaling variable x = −< =O1= [126]. In the U (2) × U (2) symmetric theory 2 is an additional relevant parameter which renders the phase transition $rst order and introduces a new scale, e.g. the non-vanishing value for the jump in the renormalized order parameter TO0R = (T#0R =2)1=2 at the critical temperature or < = 0. In the universal range, we therefore observe three independent scales and the scaling form of the equation of state will depend on two dimensionless ratios. 6 The rescaled potential U=O0R can then be written as a universal function G U = G(s; ˜ v) ˜ ; (5.50) 6 O0R which depends on the two scaling variables TO0R OR ; v˜ = : (5.51) s˜ = O0R O0R Relation (5.50) is the scaling form of the equation of state we are looking for. At a second-order phase transition the variable v˜ vanishes and G(s; ˜ 0) describes the scaling equation of state for the model with O(8) symmetry [36]. The variable v˜ accounts for the additional scale present at the $rst-order phase transition. We note that s=1 ˜ corresponds to a vanishing source and G(1; v) ˜ describes the temperature dependence of the free energy for j = 0. In this case v˜ = 1 denotes the critical temperature Tc whereas for T ¡ Tc one has v˜ ¡ 1. Accordingly, v˜ ¿ 1 is obtained for T ¿ Tc and O0R describes here the local minimum corresponding to the metastable ordered phase. The function G(s; ˜ 1) accounts for the dependence of the free energy on j for T = Tc . We consider the scaling form (5.50) of the equation of state for a non-zero coarse-graining scale k with renormalized $eld given by OR = Zk1=2 O. As we have pointed out in Section 5.5 there is a characteristic scale k2 for the $rst-order phase transition where the second local minimum of the eEective average potential appears. For weak $rst-order phase transitions one $nds #0R ∼ k2 . To observe the scaling form of the equation of state the infrared cutoE k has to run below k2 with kk2 . For the scale kf de$ned in Eq. (5.49) we observe universal behavior to high accuracy (cf. Fig. 23 6 for small 2 =1 ). The result for the universal function Ukf =O0R = Gkf (s; ˜ v) ˜ is presented in Fig. 24. For v˜ = 1 one has O0R (kf ) = TO0R (kf ) which denotes the critical temperature. Accordingly v˜ ¿ 1 denotes temperatures above and v˜ ¡ 1 temperatures below the critical temperature. One observes that Gkf (s; ˜ 1) shows two almost degenerate minima. (They become exactly degenerate in the limit k → 0.) For the given examples v˜ = 1:18 and 1:07 the minimum at the origin becomes the absolute minimum and the system is in the symmetric phase. In contrast, for v˜ = 0:90 and 0:74 the absolute minimum is located at s˜ = 1 which characterizes the spontaneously broken phase. For small enough v˜ the local minimum at the origin vanishes. We have explicitly veri$ed that the universal function Gkf depends only on the scaling variables s˜ and v˜ by choosing various values for < and for the quartic couplings of the short-distance potential, 1 and 2 . We have seen in Section 5.5 that the model shows universal behavior for a certain range of the parameter space. For given 1 and small enough 2 one always observes universal behavior. For 1 = 0:1, 2 and 4 it is demonstrated that (approximate) universality holds for 2 =1 . 12 (cf. Fig. 23). For 1 around 2 one observes from Figs. 22 and 23 that the system is to a good accuracy described by its universal properties for even larger values of 2 . The

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Fig. 24. Universal shape of the coarse-grained potential (k =kf ) as a function of the scaling variable s=O ˜ R =O0R =(#R =#0R )1=2 1=2 for diEerent values of v˜ = TO0R =O0R = (T#0R =#0R ) . The employed values for v˜ are (1) v˜ = 1:18, (2) v˜ = 1:07, (3) v˜ = 1, (4) v˜ = 0:90, (5) v˜ = 0:74. For vanishing sources one has s˜ = 1. In this case v˜ = 1 denotes the critical temperature Tc . Similarly v˜ ¿ 1 corresponds to T ¿ Tc with O0R denoting the minimum in the metastable ordered phase. The function G is symmetric for s˜ → −s˜ and one notes the qualitative diEerence with Fig. 12.

corresponding phase transitions cannot be considered as particularly weak $rst order. The universal function Gkf therefore accounts for a quite large range of the parameter space. We emphasize that the universal form of the eEective potential given in Fig. 24 depends on the scale kf where the integration of the "ow equations is stopped (cf. Eq. (5.49)). A diEerent prescription for kf will, in general, lead to a diEerent form of the eEective potential in the “inner region”. We may interpret this as a scheme dependence describing the eEect of diEerent coarse-graining procedures. This is fundamentally diEerent from non-universal corrections since Gkf is independent of details of the short distance or classical action and in this sense universal. Also the “outer region” for s˜ ¿ 1 is not aEected by the approach to convexity and becomes independent of the choice of kf . A more quantitative discussion of this scheme dependence will be presented in Section 6. Since "uctuations on scales k ¡ kf do not in"uence substantially the location of the minima of the coarse-grained potential and the form of Uk (OR ) for OR ¿ O0R one has 9Ukf =9O = j(kf ) with j(kf ) ≈ limk →0 j(k) = j. 47 47

The role of massless Goldstone boson "uctuations for the universal form of the eEective average potential in the limit k → 0 has been discussed previously for the O(8) symmetric model [36].

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Let us consider the renormalized minimum O0R in two limits which are denoted by TO0R = 0 O0R (< = 0) and O0R = O0R (2 = 0). The behavior of TO0R is described in terms of the exponent K according to Eq. (5.45), TO0R ∼ (2 )K=2 ;

K = 1:93 :

(5.52)

0 The dependence of the minimum O0R of the O(8) symmetric potential on the temperature is characterized by the critical exponent >, 0 ∼ (< )>=2 ; O0R

> = 0:882 :

(5.53)

The exponent > for the O(8) symmetric model is determined analogously to K. 48 We can also introduce a critical exponent C for the jump of the unrenormalized order parameter TO0 ∼ (2 )C ;

C = 0:988 :

(5.54)

With O00 ∼ (< ) ;

= 0:451 ;

(5.55)

it is related to K and > by the additional index relation K > = = 1:95 : C

(5.56)

We have veri$ed this numerically. For the case < = 2 = 0 one obtains j ∼ O :

(5.57)

The exponent is related to the anomalous dimension 6 via the usual index relation =(5−6)=(1+6). From the scaling solution of Eq. (5.33) we obtain 6 = 0:0224. With the help of the above relations one immediately veri$es that for 2 = 0 s˜ ∼ (−x)− ;

v˜ = 0

(5.58)

and for < = 0 s˜ ∼ y−C ;

v˜ = 1 :

(5.59)

Here we have used the Widom scaling variable x and the new scaling variable y given by x=

−< ; O1=

y=

2 : O1=C

(5.60)

5.7. Summary We have presented a detailed investigation of the phase transition in three-dimensional models for complex 2 × 2 matrices. They are characterized by two quartic couplings U1 and U2 . In the limit U1 → ∞, U2 → ∞ this also covers the model of unitary matrices. The picture arising from this study is unambiguous: 48

0 For the O(8) symmetric model (2 = 0) we consider the minimum O0R at k = 0.

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(1) One observes two symmetry breaking patterns for U2 ¿ 0 and U2 ¡ 0, respectively. The case U 2 = 0 denotes the boundary between the two phases with diEerent symmetry breaking patterns. In this special case, the theory exhibits an enhanced O(8) symmetry. The phase transition is always $rst order for the investigated symmetry breaking U (2) × U (2) → U (2) (U2 ¿ 0). For U2 = 0 the O(8) symmetric Heisenberg model is recovered and one $nds a second-order phase transition. (2) The strength of the phase transition depends on the size of the classical quartic couplings U 1 = and U2 =. They describe the short distance or classical action at a momentum scale . The strength of the transition can be parametrized by mcR = with mcR a characteristic inverse correlation length at the critical temperature. For $xed U2 the strength of the transition decreases with increasing U1 . This is analogous to the Coleman–Weinberg eEect in four dimensions. (3) For a wide range of classical couplings the critical behavior near the phase transition is universal. This means that it becomes largely independent of the details of the classical action once everything is expressed in terms of the relevant renormalized parameters. In particular, characteristic ratios like mcR =T#0R (critical inverse correlation length in the ordered phase over discontinuity in the order parameter) or mc0R =T#0R (same for the disordered phase) are not in"uenced by the addition of new terms in the classical action as far as the symmetries are respected. (4) The range of short-distance parameters U1 , U2 for which the phase transition exhibits universal behavior is not only determined by the strength of the phase transition as measured by mcR =. For a given U1 = and small enough U2 = one always observes universal behavior. In the range of small U1 = the essential criterion for universal behavior is given by the size of U2 = U1 , with approximate universality for U2 ¡ U1 . For strong couplings universality extends to larger U2 = U1 and occurs for much larger mcR =. (5) We have investigated how various characteristic quantities like the discontinuity in the order parameter T#0 or the corresponding renormalized quantity T#0R or critical correlation lengths depend on the classical parameters. In particular, at the critical temperature one $nds universal critical exponents for not too large U2 , T#0R ∼ (U2 )K ;

K = 1:93 ;

T#0 ∼ (U2 )2C ;

C = 0:988 :

(5.61)

These exponents are related by a scaling relation to the critical correlation length and order parameter exponents > and of the O(8) symmetric Heisenberg model according to K=C = >= = 1:95 (> = 0:882, = 0:451 in our calculation for U2 = 0). Small values of U2 can be associated with a perturbation of the O(8) symmetric model and K; C are related to the corresponding crossover exponents. On the other hand, T#0R (T#0 ) becomes independent of U2 in the in$nite coupling limit. (6) We have computed the universal equation of state for the $rst-order transition. It depends on two scaling variables, e.g. s˜ = (#R =#0R )1=2 and v˜ = (T#0R =#0R )1=2 . The equation of state relates the derivative of the free energy U to an external source, 9U=9O = j. From there one can extract universal ratios e.g. for the jump in the order parameter (T#0R =mcR = 0:592) or for the ratios of critical correlation lengths in the disordered (symmetric) and ordered (spontaneously broken) phase (mc0R =mcR =0:746). It speci$es critical couplings (1R =mcR =0:845; 2R =mcR =15:0). The universal behavior of the potential for large $eld arguments ##0 is U ∼ #3=(1+6) provided #R is suIciently small compared to . Here the critical exponent 6 which characterizes the dependence of the potential on the unrenormalized $eld # is found to be 6 = 0:022. For large # the universal equation of state

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equals the one for the O(8) symmetric Heisenberg model and 6 speci$es the anomalous dimension or the critical exponent = (5 − 6)=(1 + 6). In contrast to the Ising universality class (Section 5.6) the $rst-order universal equation of state cannot be reduced to the universal equation of state for the O(8) model for general #. Finally, we should mention that our approach can be extended to systems with reduced SU (N ) × SU (N ) symmetry. They obtain by adding to the classical potential a term involving the invariant % = det O + det O† . (Note that % is not invariant with respect to U (N ) × U (N ).) This will give an even richer pattern of phase transitions and permits a close contact to realistic meson models in QCD where the axial anomaly is incorporated. Finally one can extend the three-dimensional treatment to a four-dimensional study of $eld theories at non-vanishing temperature. How this can be used to approach the chiral phase transition in QCD is presented in Section 8. 6. Spontaneous nucleation and coarse graining49 6.1. Introduction Let us consider a slow change with time of the “parameters” of the model that describes a physical system. This concerns, for example, the change in temperature in the early universe or a variation of the magnetic $eld in an experiment with ferromagnets. We assume that the time scale of the parameter change is much larger than the characteristic equilibration time teq of the system, so that the system can follow adiabatically in (approximate) local equilibrium. (For the example of the early universe the ratio of time scales involves the age of the universe H −1 , i.e. the characteristic small quantity is Hteq .) A second-order phase transition can proceed under these circumstances without major non-equilibrium eEects. In this section we consider $rst-order phase transitions. Because of the discontinuity in the order parameter no continuous equilibrium evolution through the phase transition is possible. Near the phase transition, the eEective average potential Uk is characterized by two separate local minima. In the course of the evolution the minimum corresponding to the “true vacuum” (for late times) becomes lower than the one corresponding to the “false vacuum”. However, the system may not adapt immediately to the new equilibrium situation, and we encounter the familiar phenomena of supercooling or hysteresis. As vapor is cooled below the critical temperature, local droplets form and grow until the transition is completed. The inverse evolution proceeds by the formation of vapor bubbles in a liquid. The transition in ferromagnets is characterized by the formation of local Weiss domains with the magnetization corresponding to the late time equilibrium. The formation of “bubbles” of the new vacuum is similar to a tunneling process and typically exponentially suppressed at the early stages of the transition. The reason is the “barrier” between the local minima. The transition requires at least the action of the saddle point corresponding to the con$guration with lowest action on the barrier. One therefore encounters a Boltzmann factor involving the action of this “critical bubble” that leads to exponential suppression. A quantitative understanding of this important process is diIcult both from the experimental and theoretical side. The theory deals mainly with pure systems, whereas in an experiment the 49

This section is based on a collaboration with Strumia [39,40,192,193].

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exponentially suppressed rate of “spontaneous nucleation” has to compete with processes where impurities act as seeds for the formation of bubbles. As long as the exponential suppression is substantial, the theoretical treatment may separate the dynamics (which involves the growth of bubbles, etc.) from the computation of the exponential suppression factor. The latter can be computed from equilibrium properties. Its quantitative determination is by itself a hard theoretical problem for which we propose a solution in this section. We also discuss carefully the range of applicability of this solution. The problem of calculating nucleation rates during $rst-order phase transitions has a long history. (For reviews with an extensive list of references, see Refs. [177,178].) Our present understanding of the phenomenon of nucleation is based largely on the work of Langer [179]. His approach has been applied to relativistic $eld theory by Coleman [180] and Callan [181] and extended by A]eck [182] and Linde [183] to $nite-temperature quantum $eld theory. The basic quantity in this approach is the nucleation rate I , which gives the probability per unit time and volume to nucleate a certain region of the stable phase (the true vacuum) within the metastable phase (the false vacuum). The calculation of I relies on a semiclassical approximation around a dominant saddle point that is identi$ed with the critical bubble. This is a static con$guration (usually assumed to be spherically symmetric) within the metastable phase whose interior consists of the stable phase. It has a certain radius that can be determined from the parameters of the underlying theory. Bubbles slightly larger than the critical one expand rapidly, thus converting the metastable phase into the stable one. The nucleation rate is exponentially suppressed by a suitable eEective action of the critical bubble. Possible deformations of the critical bubble generate a static pre-exponential factor. The leading contribution to it has the form of a ratio of "uctuation determinants and corresponds to the $rst-order correction to the semiclassical result. Apart from the static prefactor, the nucleation rate includes a dynamical prefactor that takes into account the expansion of bubbles after their nucleation. In this review we concentrate only on the static aspects of the problem and neglect the dynamical prefactor. Its calculation requires the extension of our formalism to real time nonequilibrium dynamics. For a four-dimensional theory of a real scalar $eld at temperature T , the nucleation rate is given by [179 –183] %−1=2 % E0 b 3=2 %% det [2 =2 ]=b %% I= exp(−b ) : (6.1) % % % det[2 =2 ]=0 % 2" 2" Here is the eEective action (see Sections 1.2 and 2.1) of the system for a given con$guration of the $eld that acts as the order parameter of the problem. The action of the critical bubble is b = [b (r)] − [0], where b (r) is the spherically symmetric bubble con$guration and = 0 corresponds to the false vacuum. The "uctuation determinants are evaluated either at =0 or around = b (r). The prime in the "uctuation determinant around the bubble denotes that the three zero eigenvalues of the operator [2 =2 ]=b have been removed. Their contribution generates the factor (b =2")3=2 and the volume factor that is absorbed in the de$nition of I (nucleation rate per unit volume). The quantity E0 is the square root of the absolute value of the unique negative eigenvalue. In $eld theory, the rescaled free energy density of a system for homogeneous con$gurations is identi$ed with the temperature-dependent eEective potential. This is often evaluated through some perturbative scheme, such as the loop expansion [172]. In this way, the pro$le and the action of

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the bubble are determined through the potential. This approach, however, faces three fundamental diIculties: (a) The eEective potential, being the Legendre transform of the generating functional for the connected Green functions, is a convex function of the $eld. Consequently, it does not seem to be the appropriate quantity for the study of tunneling, as no structure with more than one minima separated by a barrier exists. 50 (b) The "uctuation determinants in the expression for the nucleation rate have a form completely analogous to the one-loop correction to the potential. The question of double-counting the eEect of "uctuations (in the potential and the prefactor) must be properly addressed. This point is particularly important in the case of radiatively induced $rst-order phase transitions. These are triggered by the appearance of a new vacuum state in the theory as a result of the integration of (quantum or thermal) "uctuations [172]. A radiatively induced $rst-order phase transition takes place in theories for which the tree-level potential has only one minimum, while a second minimum appears at the level of radiative corrections. 51 (c) Another diIculty concerns the ultraviolet divergences that are inherent in the calculation of the "uctuation determinants in the prefactor. An appropriate regularization scheme must be employed in order to control them (for other approaches see Refs. [187–191]). Moreover, this scheme must be consistent with the one employed for the absorption of the divergences appearing in the calculation of the potential that determines the action of the critical bubble. In Ref. [184] it was argued that all the above issues can be resolved through the implementation of the notion of coarse graining in the formalism, in agreement with Langer’s philosophy. The problem of computing the diEerence of the eEective action between the critical bubble and the false vacuum may be divided into three steps: In the $rst step, one only includes "uctuations with momenta larger than a scale k which is of the order of the typical gradients of b (r). For this step one can consider approximately constant $elds and use a derivative expansion for the resulting coarse-grained free energy k []. The second step searches for the con$guration b (r) which is a saddle point of k . The quantity b in Eq. (6.1) is identi$ed with k [b ]−k [0]. Finally, the remaining "uctuations with momenta smaller than k are evaluated in a saddle-point approximation around b (r). This yields the ratio of "uctuation determinants with an ultraviolet cutoE k. Indeed, Langer’s approach corresponds to a one-loop approximation around the dominant saddle point for "uctuations with momenta smaller 50

It has been argued in Ref. [185] that the appropriate quantity for the study of tunneling is the generating functional of the 1PI Green functions (calculated perturbatively), which diEers from the eEective potential in the non-convex regions. However, as we discuss in the following, the consistent picture must rely on the notion of coarse graining and on the separation of the high-frequency "uctuations that may be responsible for the non-convexity of the potential, from the low-frequency ones that are relevant for tunneling. Such notions cannot be easily implemented in the context of perturbation theory. 51 In Ref. [186] an alternative procedure was suggested for the treatment of radiatively induced $rst-order phase transitions: The $elds whose "uctuations are responsible for the appearance of the new vacuum are integrated out $rst, so that an “eEective” potential with two minima is generated for the remaining $elds. Our philosophy is diEerent: We integrate out high-frequency "uctuations of all $elds, so that we obtain an eEective low-energy action which we use for the calculation of the nucleation rate. Our procedure involves an explicit infrared cutoE in the calculation of the low-energy action. This prevents the appearance of non-localities arising from integrating out massless $elds, which may be problematic for the approach of Ref. [186]. For example, the $elds that generate the new vacuum in radiatively induced ("uctuation-driven) $rst-order phase transitions are usually massless or very light at the origin of the potential.

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than a coarse-graining scale k. We solve here the problem of how to determine the coarse-grained free energy k in a consistent way. This is crucial for any quantitative treatment of the nucleation rate since k appears in an exponential. In the following subsections, we review studies of nucleation based on the formalism of the eEective average action k , which can be identi$ed with the free energy, rescaled by the temperature, at a given coarse-graining scale k. In the simplest case, we consider a statistical system with one space-dependent degree of freedom described by a real scalar $eld (x). For example, (x) may correspond to the density for the gas=liquid transition, or to a diEerence in concentrations for chemical phase transitions, or to magnetization for the ferromagnetic transition. Our discussion also applies to a quantum $eld theory in thermal quasi-equilibrium. As we will see in Section 7, an eEective three-dimensional description applies for a thermal quantum $eld theory at scales k below the temperature T . We assume that k0 has been computed (for example perturbatively) for k0 = T [42,44] and concentrate here on the three-dimensional (eEective) theory. We compute k by solving the "ow equation between k0 and k. For this purpose we approximate k by a standard kinetic term and a general potential Uk . This corresponds to the $rst level of the derivative expansion of Eq. (2.14), where we set ZO; k (#) = 1 and neglect the higher derivative terms. Our approximation is expected to be a good one for the models we consider, because the deviations of ZO; k (#) from 1 and the size of the higher-derivative terms are related to the anomalous dimension of the $eld, and this is small (6 0:04). The long-range collective "uctuations are not yet important at a short-distance scale 52 k0−1 = T −1 . For this reason, we assume here a polynomial potential Uk0 () = 12 m2k0 2 + 16 -k0 3 + 18 k0 4 :

(6.2)

The parameters m2k0 ; -k0 and k0 depend on T . This potent ial has the typical form relevant for $rst-order phase transitions in statistical systems with asymmetric potentials or in four-dimensional quantum $eld theories at high temperature [42,44]. The two $rst-order critical lines are located at 2 2 2 -2k0 = 9k0 m2k0 and -k0 = 0, with endpoints at m2k0 = −2cr , -2k0 = −18k0 cr and m2k0 = cr , -k0 = 0, 2 2 2 respectively. Here cr is the critical mass term of the Ising model (cr =k0 ≈ −0:0115 for k0 =k0 =0:1). We point out that, for $xed m2k0 and k0 , opposite values of -k0 result in potentials related through ↔ −. Also a model with m2k0 ¡ 0 can be mapped onto the equivalent model with mk20 ¿ 0 by the shift → + c, k0 c2 + -k0 c = −2m2k0 , where mk20 = −2m2k0 − 12 -k0 c, -k0 = -k0 + 3k0 c. As we have seen in Section 4.6 a diEerent shift → + c˜ can eliminate the cubic term in favor of a term linear in . Therefore, potential (6.2) also describes statistical systems of the Ising universality class in the presence of an external magnetic $eld. For a Hamiltonian ˆ C (2 − 1)2 − B + 9i 9i ; H = d3 x (6.3) 8 2 ˆ 2 −1)=2C, -k0 =3T ˆ 1=2 y=C3=2 , k0 =T=C ˆ 2 , with y given by y(y2 −1)=2B= . ˆ the parameters are m2k0 =(3y For real magnets k0 must be taken somewhat below the inverse lattice distance, so that eEective rotation and translation symmetries apply. Correspondingly, and H are the eEective normalized spin $eld and the eEective Hamiltonian at this scale. We emphasize that our choice of potential encompasses a large class of $eld-theoretical and statistical systems. The universal critical behavior 52

In a three-dimensional picture, k0 plays the role of the microscopic scale .

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of these systems has been discussed extensively in Section 4. In a diEerent context, our results can also be applied to the problem of quantum tunneling in a (2+1)-dimensional theory at zero temperature. In this case k0 ; m2 ; - and bear no relation to temperature. We compute the form of the potential Uk at scales k 6 k0 by integrating the evolution equation (2.37) with Zk (#; q2 ) = Z˜ k (#; q2 ) = 1 in Eqs. (2.38). The form of Uk changes as the eEect of "uctuations with momenta above the decreasing scale k is incorporated in the eEective couplings of the theory. We consider an arbitrary form of Uk which, in general, is not convex for non-zero k. Uk approaches the convex eEective potential only in the limit k → 0. In the region relevant for a $rst-order phase transition, Uk has two distinct&local minima, where one is lower than the other away

from the phase transition at -k0 = 0 or |-k0 | = 3 |k0 m2k0 |. The nucleation rate should be computed for k larger than or around the scale kf at which Uk starts receiving important contributions from $eld con$gurations that interpolate between the two minima. This happens when the negative curvature at the top of the barrier becomes approximately equal to −k 2 [77,78] (see Section 3.6). Another consistency check for the above choice of k is the typical length scale of a thick-wall critical bubble which is & 1=k for k ¿ kf . The use of Uk at a non-zero value of k resolves the $rst fundamental diIculty in the calculation of bubble–nucleation rates that we mentioned earlier. The other two diIculties are overcome as well. In our approach the pre-exponential factor in Eq. (6.1) is well-de$ned and $nite, as an ultraviolet cutoE of order k must be implemented in the calculation of the "uctuation determinants. The cutoE must guarantee that "uctuations with characteristic momenta q2 & k 2 do not contribute to the determinants. This is natural, as all "uctuations with typical momenta above k are already incorporated in the form of Uk . The choice of the ultraviolet cutoE must be consistent with the infrared cutoE procedure that determines k and, therefore, Uk . In the following subsection we show how this is achieved. It is clear that our approach resolves then automatically the problem of double-counting the eEect of the "uctuations. As a test of the validity of the approach, the result for the rate I must be independent of the coarse-graining scale k, because the latter should be considered only as a technical device. In the following we show that this is indeed the case when the expansion around the saddle point is convergent and the calculation of the nucleation rate reliable. Moreover, the residual k dependence of the rate can be used as a measure of the contribution of the next order in the saddle-point expansion. 6.2. Calculation of the nucleation rate In all our calculations of bubble–nucleation rates we employ a mass-like infrared cutoE k for the "uctuations that are incorporated in k . This corresponds to the choice Rk =k 2 for the cutoE function de$ned in Eq. (2.6). The reason for our choice is that the evaluation of the "uctuation determinants is technically simpli$ed for this type of cutoE. In three dimensions and for our approximation of 3 neglecting the eEects of wave √ function renormalization, the threshold function l0 (w), de$ned in 53 3 Eqs. (3.16), is l0 (w) = " 1 + w. The evolution equation (3.5) for the potential can now be 53

We have neglected an in$nite w-independent contribution to the threshold function that aEects only the absolute normalization of the potential. As we are interested only in relative values of the potential for various $eld expectation values, this contribution is irrelevant for our discussion.

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written as & & 1 9 2 + U () − [U [ () − U (0)] = − k k 2 + Uk (0)] : (6.4) k k k 9k 2 8" In this entire section primes denote derivatives with respect to , similar to Sections 4.4 – 4.6. In order to avoid confusion with the notation of other previous sections, in which primes denote derivatives with respect to # = 2 =2, we display explicitly the argument of the function with respect to which we diEerentiate. In order to implement the appropriate ultraviolet cutoE ∼ k in the "uctuation determinant, let us look at the $rst step of an iterative solution for Uk , discussed in Section 3.4 Uk(1) () − Uk(1) (0) = Uk0 () − Uk0 (0) 1 det[ − 92 + k 2 + Uk ()] det[ − 92 + k02 + Uk (0)] : + ln 2 det[ − 92 + k02 + Uk ()] det[ − 92 + k 2 + Uk (0)]

(6.5)

For k → 0, this solution is a regularized one-loop approximation to the eEective potential. Due to the ratio of determinants, only momentum modes with k 2 ¡ q2 ¡ k02 are eEectively included in the momentum integrals. The form of the infrared cutoE in Eq. (6.4) suggests that we should implement the ultraviolet cutoE for the "uctuation determinant in the nucleation rate (6.1) as I ≡ Ak exp(−Sk ) %−1=2 % E0 Sk 3=2 %% det [ − 92 + Uk (b (r))] det[ − 92 + k 2 + Uk (0)] %% Ak = ; % det[ − 92 + k 2 + U (b (r))] det[ − 92 + U (0)] % 2" 2" k k

(6.6)

where we switch the notation to Sk =k [b ]−k [0] instead of b in order to make the k-dependence in the exponential suppression factor explicitly visible. A comparison between Eqs. (6.5) and (6.6) shows that the explicitly k-dependent regulator terms drop out for the combination Sk − ln Ak . Our computation of Uk also includes contributions beyond Eq. (6.5). The residual k-dependence of the nucleation rate will serve as a test for the validity of our approximations. The critical bubble con$guration b (r) is an SO(3)-invariant solution of the classical equations of motion which interpolates between the local maxima of the potential −Uk (). It satis$es the equation d 2 b 2 db = Uk (b ) + (6.7) dr 2 r dr with the boundary conditions b → 0 for r → ∞ and db =dr = 0 for r = 0. The bubble action Sk is given by ! ∞ 1 db (r) 2 Sk = 4" + Uk (b (r)) − Uk (0) r 2 dr ≡ Skt + Skv ; (6.8) 2 dr 0 where the kinetic and potential contributions, Skt and Skv , respectively, satisfy Skv =Skt = −1=3: The computation of the "uctuation determinants Ak is more complicated. The diEerential operators that appear in Eq. (6.6) have the general form W
(6.9)

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where m2< ≡ Uk (0) +
∞

(det W‘
‘=0 −∇2‘ +

m2< + /Wk (r) :

(6.11)

We recall that det W‘
det W‘<1 det[ − ∇2‘ + m2< + 1 · Wk (r)] y‘<1 (r → ∞) ; = = det W‘<0 det[ − ∇2‘ + m2< + 0 · Wk (r)] y‘<0 (r → ∞)

where y‘
(6.12)

(6.13)

with the behavior y‘
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Fig. 25. Dependence of eEective potential, critical bubble and nucleation rate on the coarse-graining scale k. The parameters are k0 =0:1k0 , m2k0 =−0:0433k02 , -k0 =−0:0634k03=2 (a–c) and m2k0 =−0:013k02 , -k0 =−1:61×10−3 k03=2 (d–f). All dimensionful quantities are given in units of kf , equal to 0:223k0 in the $rst series and to 0:0421k0 in the second series.

Our results for the nucleation rate are presented in Fig. 25c. The horizontal axis corresponds to k= Uk (t ), i.e. the ratio of the scale k to the square root of the positive curvature (equal to the mass of the $eld) at the absolute minimum of the potential located at t . Typically, when k crosses below this mass, the massive "uctuations of the $eld start decoupling. The evolution of the convex parts of the potential slows down and eventually stops. The dark diamonds give the values of the action Sk of the critical bubble. We observe a strong k dependence of this quantity, which is expected from the behavior in Figs. 25a and b. The stars in Fig. 25c indicate the values of ln(Ak =kf4 ). Again a substantial decrease with decreasing k is observed. This is expected, because k acts as the eEective ultraviolet cutoE in the calculation of the "uctuation determinants in Ak . The dark squares give our results for −ln(I=kf4 ) = Sk − ln(Ak =kf4 ). It is remarkable that the k dependence of this quantity almost disappears for k= Uk (t ) . 1. The small residual dependence on k can be used to estimate the contribution of the next order in the expansion around the saddle point. It is reassuring that this contribution is expected to be smaller than ln(Ak =kf4 ). This behavior con$rms our expectation that the nucleation rate should be independent of the scale k that we introduced as a calculational tool. It also demonstrates that all the con$gurations plotted in Fig. 25b give equivalent descriptions of the system, at least for the lower values of k. This indicates that the critical bubble should not be associated only with the saddle point of the semiclassical

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approximation, whose action is scale dependent. It is the combination of the saddle point and its possible deformations in the thermal bath that has physical meaning. For smaller values of |m2k0 | the dependence of the nucleation rate on k becomes more pronounced. We demonstrate this in the second series of Figs. 25d–f where k0 =(−m2k0 )1=2 = 0:88 (instead of 0:48 for Figs. 25a–c). The value of k0 is the same as before, whereas -k0 = −1:61 × 10−3 k03=2 and kf =k0 = 0:0421. The strong k dependence is caused by the larger value of the dimensionless renormalized quartic coupling for the second parameter set [184]. Higher-loop contributions to Ak become important and the expansion around the saddle point does not converge any more. There are two clear indications of the breakdown of the expansion: (a) The values of the leading and subleading contributions to the nucleation rate, Sk and ln(Ak =kf4 ), respectively, become comparable. (b) The k dependence of ln(I=kf4 ) is strong and must be canceled by the higher-order contributions. We point out that the discontinuity in the order parameter at the phase transition is approximately 5 times smaller in the second example than in the $rst one. As a result, the second phase transition can be characterized as weaker. Typically, the breakdown of the saddle-point approximation is associated with weak $rst-order phase transitions. It is apparent from Figs. 25c and f that the leading contribution to the pre-exponential factor increases the total nucleation rate. This behavior, related to the "uctuations of the $eld whose expectation value serves as the order parameter, is observed in multi-$eld models as well. The reason can be traced to the form of the diEerential operators in the prefactor of Eq. (6.6). This prefactor involves the ratio det [ − 92 + Uk (b (r))]=det [ − 92 + Uk (0)] before regularization. The function Uk (b (r)) always has a minimum away from r = 0 where it takes negative values (corresponding to the negative curvature at the top of the barrier), while Uk (0) is always positive. As a result the lowest eigenvalues of the operator det [−92 +Uk (b (r))] are smaller than those of det [−92 +Uk (0)]. The elimination of the very large eigenvalues through regularization does not aEect this fact and the prefactor is always larger than 1. Moreover, for weak $rst-order phase transitions it becomes exponentially large because of the proliferation of low eigenvalues of the $rst operator. In physical terms, this implies the existence of a large class of $eld con$gurations of free energy comparable to that of the saddle point. Despite the fact that they are not saddle points of the free energy (they are rather deformations of a saddle point) and are, therefore, unstable, they result in an important increase of the nucleation rate. This picture is very similar to that of “subcritical bubbles” of Ref. [197]. In Figs. 25c and f we also display the values of ln(Ak =kf4 ) (dark triangles) predicted by the approximate expression ∞ 1=2 Ak "k 3 − ln 4 ≈ r [Uk (b (r)) − Uk (0)] dr : (6.14) 2 kf 0 This expression is based on the behavior of the ratio of determinants (6.12) for large ‘, for which an analytical treatment is possible [39,40]. It is apparent from Figs. 25c and f that Eq. (6.14) gives a good approximation to the exact numerical results, especially near kf . It can be used for quick checks of the validity of the expansion around the saddle point.

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6.3. Region of validity of homogeneous nucleation theory It is useful to obtain some intuition on the behavior of the nucleation rate by using the approximate expression (6.14). We assume that the potential has a form similar to Eq. (6.2) even near kf , i.e. Ukf () ≈ 12 m2kf 2 + 16 -kf 3 + 18 kf 4 :

(6.15)

(Without loss of generality we take m2kf ¿ 0.) For systems not very close to the endpoint of the $rst-order critical line, our assumption is supported by the numerical data, as can be veri$ed from Fig. 25. The scale kf is determined by the relation kf2 ≈ max|Ukf ()| =

-2kf 6kf

− m2kf :

(6.16)

˜ = ˜ 2 =2− Through the rescalings r = r=m ˜ kf , = 2˜ m2kf =-kf , the potential can be written as U˜ () 3 4 ˜ =3 + h ˜ =18, with h = 9kf m2kf =-2kf . For h ≈ 1 the two minima of the potential have approximately equal depth. The action of the saddle point can be expressed as 4 mk f ˜ Skf = hS(h) ; (6.17) 9 k f ˜ Similarly, the pre-exponential factor can ˜ where S(h) must be determined numerically through U˜ (). be estimated through Eq. (6.14) as Akf " 3 ˜ − 1A(h) ; ln 4 ≈ 2 2h kf ∞ 2 ˜ A (h) = [U˜ (˜ b (r)) ˜ − 1] r˜3 d r˜ ; (6.18) 0

˜ with A(h) computed numerically. Finally, the relative importance of the "uctuation determinant is given by ˜ ln (Akf =kf4 ) kf k A(h) 9" 1 3 −1 R= ≈ = T (h) f : (6.19) ˜ Sk f 8 h 2h mkf m kf S(h) The ratio R can be used as an indicator for the validity of the saddle-point expansion, which is valid only for R . 1. In Fig. 26 we plot T (h) as a function of h in the interval (0, 1). It diverges for h → 0. For h → 1, our estimate of the prefactor predicts T (h) → 0. The reason is that, for our approximate potential of Eq. (6.15), the $eld masses at the two minima are equal in this limit. As a result, the integrand in Eq. (6.14) vanishes, apart from the surface of the bubble. The small surface contribution is negligible for h → 1, because the critical bubbles are very large in this limit. This behavior is not expected to persist for more complicated potentials. Instead, we expect a constant value of T (h) for h → 1. However, we point out that the approximate expression (6.14) has not been tested for very large critical bubbles. The divergence of the saddle-point action in this limit results in low accuracy for ˜ our numerical analysis. Typically, our results are reliable for S(h) less than a few thousand. Also, Eq. (6.14) relies on a large-‘ approximation and is not guaranteed to be valid for large bubbles. We have checked that both our numerical and approximate results are reliable for h . 0:9.

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20

T (h)

15 10 5 0 0

0.2

0.4

0.6

0.8

1

h Fig. 26. The parameter T (h), de$ned in Eq. (6.19), as a function of h.

The estimate of Eq. (6.19) suggests two cases in which the expansion around the saddle point is expected to break down: (a) For $xed kf =mkf , the ratio R becomes larger than 1 for h → 0. In this limit the barrier becomes negligible and the system is close to the spinodal line. (b) For $xed h, R can be large for suIciently large kf =mkf . This is possible even for h close to 1, so that the system is far from the spinodal line. This case corresponds to weak $rst-order phase transitions, as can be veri$ed by observing that the saddle-point action (6.17), the location of the true vacuum

mkf t 2√ ˜ = ht (h) ; (6.20) √ mkf 3 kf and the diEerence in free-energy density between the minima mk TU 4 = h TU˜ (h) f 3 9 k f mkf

(6.21)

go to zero in the limit mkf =kf → 0 for $xed h. This is in agreement with the discussion of Fig. 25 in the previous subsection. The breakdown of homogeneous nucleation theory in both the above cases is con$rmed through the numerical computation of the nucleation rates [40]. In Fig. 27, we show contour plots for I=kf4 and for R = ln(Akf =kf4 )=Skf in the (m2k0 ; -k0 ) plane for $xed k0 =k0 = 0:1. One can see the decrease of the rate as the $rst-order critical line -k0 = 0 is approached. The spinodal line (end of the shaded region), on which the metastable minimum of Uk becomes unstable, is also shown. The nucleation rate becomes large before the spinodal line is reached. For −ln(I=kf4 ) of order 1, the exponential suppression of the nucleation rate disappears. Langer’s approach can no longer be applied and an alternative picture for the dynamical transition must be developed [198]. In the region between the contour I=kf4 = e−3 and the spinodal line, one expects a smooth transition from nucleation to spinodal decomposition. The spinodal and critical lines meet at the endpoint in the lower right corner, which

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Fig. 27. Contour plots of the nucleation rate I=kf4 and of R = ln(Akf =kf4 )=Skf in the plane (m2k0 ; -k0 ), for k0 =k0 = 0:1. Regions to the right of the spinodal line (only one minimum) are shaded. The dashed lines correspond to R = {0:2; 0:3; 0:5; 1} and the solid lines to I=kf4 .

corresponds to a second-order phase transition. The $gure exhibits an increasing rate as the endpoint is approached at a $xed distance from the critical line. The ratio R is a better measure of the validity of the semiclassical approximation. For R ≈ 1 the "uctuation determinant is as important as the “classical” exponential factor e−Sk . There is no reason to assume that higher loop contributions from the "uctuations around the critical bubble can be neglected anymore. Near the endpoint in the lower-right corner, Langer’s semiclassical picture breaks down, despite the presence of a discontinuity in the order parameter. Requiring I=kf4 . e−3 , R . 1, gives a limit of validity for Langer’s theory. For a $xed value of the nucleation rate (solid lines in Fig. 27), the ratio R grows as the endpoint in the lower-right corner is approached. This indicates that Langer’s theory is not applicable for weak $rst-order phase transitions, even if the predicted rate is exponentially suppressed. The concept of nucleation of a region of the stable phase within the metastable phase may still be relevant. However, a quantitative estimate of the nucleation rate requires taking into account "uctuations of the system that are not described properly by the semiclassical approximation [197]. The parameter region discussed here may be somewhat unusual since the critical line of the phase transition is approached by varying -k0 from negative or positive values towards zero. We have chosen it only for making the graph more transparent. However, the results of Fig. 27 can be mapped by a shift → + c to another region with m2k0 ¿ 0, for which the $rst-order phase transition can be approached by varying m2k0 at $xed -k0 . As opposite values of -k0 result in potentials related by ↔ −, we can always choose -k0 ¡ 0. Then the phase transition proceeds from a metastable minimum at the origin to a stable minimum along the positive -axis (as in Fig. 25a). Potentials with m2k0 ¿ 0, -k0 ¡ 0 are relevant for cosmological phase transitions, such as the electroweak phase transition.

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Fig. 28. Nucleation rate as function of thermodynamic variables. We show the contour plots of Fig. 27 in the plane (mf ; TU ), where m−1 is the correlation length in the metastable phase and TU is the diEerence in free-energy density f between the metastable and stable phase. The spinodal line corresponds to the vertical axis.

The microscopic parameters at the scale k0 are often not known in statistical systems. In order to facilitate the interpretation of possible experiments where the nucleation rate would be measured together with the correlation length and the latent heat, we also give I as a function of renormalized parameters. In Fig. 28 we depict the region of validity of homogeneous nucleation theory in terms of parameters of the low-energy theory at the scale kf . The contours correspond to the same quantities as in Fig. & 27. They are now plotted as a function of the renormalized mass at the false vacuum mf = Ukf (0) in units of k0 and the diEerence in free-energy density between the two vacua in 1 3 units of m3f . Here m− f corresponds to the correlation length in the false vacuum and TU=mkf can be related to observable quantities like the jump in the order parameter or the latent heat if k0 =k0 is kept $xed. Furthermore, for given kf =mkf , we can relate TU=m3kf to h in the approximation of Eq. (6.15) using Eq. (6.21) and compute the observables from the explicit form of the free energy density (6.15). The spinodal line corresponds to the vertical axis, as for mf = 0 the origin of the potential turns into a maximum. The critical line corresponds to the horizontal axis. The origin is the endpoint of the critical line. All the potentials we have studied have an approximate form similar to Eq. (6.15) with h . 0:9. From our discussion in the previous subsection and Fig. 26 we expect that R is approximately given by Eq. (6.14) with T (h) & 0:3 for h . 0:9. This indicates that R & 1 for mf =k0 . 0:05 even far from the spinodal line. This expectation is con$rmed by Fig. 28. Even for theories with a signi$cant exponential suppression for the estimated nucleation rate we expect R ∼ 1 near mf =k0 ≈ 0:05. Finally, we point out that realistic statistical systems often have large dimensionless couplings k0 =k0 ∼ 10. Our results indicate that Langer’s homogeneous nucleation theory breaks down for

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such systems even for small correlation lengths in the metastable phase (mf =k0 ∼ 1). For a large correlation length the universal behavior of the potential has been discussed in Section 4.6. One obtains a large value kf =mf ≈ 5, independently of the short-distance couplings [38]. Therefore, a saddle-point approximation for the "uctuations around the critical bubble will not give accurate results in the universal region. 6.4. Radiatively induced >rst-order phase transitions We now turn to a more complicated system, a theory of two scalar $elds. It provides a framework within which we can test the reliability of our approach in the case of two "uctuating $elds. The evolution equation for the potential resembles very closely the ones appearing in gauged Higgs theories, with the additional advantage that the approximations needed in the derivation of this equation are more transparent. We expect the qualitative conclusions for the region of validity of Langer’s picture of homogeneous nucleation to be valid for gauged Higgs theories as well. The most interesting feature of the two-scalar models is the presence of radiatively induced $rst-order phase transitions. Such transitions usually take place when the mass of a certain $eld is generated through the expectation value of another. The "uctuations of the $rst $eld can induce the appearance of new minima in the potential of the second, resulting in $rst-order phase transitions [172]. As we have already discussed, the problem of double-counting the eEect of "uctuations is particularly acute in such situations. The introduction of a coarse-graining scale k resolves this problem, by separating the high-frequency "uctuations of the system which may be responsible for the presence of the second minimum through the Coleman–Weinberg mechanism, from the low-frequency ones which are relevant for tunneling. Similarly to the one-$eld case, we approximate the eEective average action as 1 3 k = (9 1 9 1 + 9 2 9 2 ) + Uk (1 ; 2 ) : d x (6.22) 2 The evolution equation for the potential can be written in the form [40,170] & & 1 9 2 + M 2 ( ; ) − [ [U ( ; ) − U (0; 0)] = − k k 2 + M12 (0; 0) k 1 2 k 1 2 1 9k 2 8" & & + k 2 + M22 (1 ; 2 ) − k 2 + M22 (0; 0)] ;

(6.23)

where M1;2 2 (1 ; 2 ) are the two eigenvalues of the $eld-dependent mass matrix, given by & 2 ] M1;2 2 (1 ; 2 ) = 12 [U11 + U22 ± (U11 − U22 )2 + 4U12

(6.24)

with Uij ≡ 92 Uk =9i 9j . The only neglected corrections to Eq. (6.23) are related to the wave function renormalization of the $elds. We expect these corrections to be small, as the anomalous dimension is 6 ≈ 0:035–0:04. We consider models with the symmetry 2 ↔ −2 throughout this paper. This means that the expressions for the mass eigenvalues simplify along the 1 -axis: M12 = 92 Uk =921 ; M22 = 92 Uk =922 . We always choose parameters such that minima of the potential are located along the 1 -axis. The saddle-point con$guration satis$es Eq. (6.7) along the 1 -axis and has 2 =0. The bubble–nucleation

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Fig. 29. Nucleation rate for multicomponent models. We show the scale dependence of various quantities necessary for the computation of the nucleation rate. The microscopic potential is given by Eq. (6.26) with m22k0 = −m21k0 = 0:1k02 , k0 = gk0 = 0:1k0 , Jk0 = 0:6k05=2 . The coarse graining scale k varies between ki = e−0:8 k0 and kf = e−1:2 k0 . All dimensionful quantities are given in units of kf . In f we plot the saddle-point action (diamonds), the two prefactors ln(A1k =kf4 ) (stars) and ln(A2k ) (triangles), and the nucleation rate ln(I=kf4 ) (squares) as a function of k= U11 (t ; 0).

rate is derived in complete analogy to the one-$eld case and is given by I = A1k A2k exp(−Sk ); %−1=2 % E0 Sk 3=2 %% det [ − 92 + U11 (b (r))] det[ − 92 + k 2 + U11 (0)] %% ; A1k = % det[ − 92 + k 2 + U11 (b (r))] det[ − 92 + U11 (0)] % 2" 2" % % % det[ − 92 + U22 (b (r))] det[ − 92 + k 2 + U22 (0)] %−1=2 % % A2k = % : det[ − 92 + k 2 + U22 (b (r))] det[ − 92 + U22 (0)] %

(6.25)

The calculation of the various determinants proceeds very similarly to the previous subsection. The details are given in Ref. [192]. In Fig. 29, we present results for a class of models de$ned through the potential Uk0 (1 ; 2 ) = −Jk0 1 + 12 m21k0 21 + 12 m22k0 22 + 18 k0 (41 + 42 ) + gk0 21 22 :

(6.26)

The term linear in 1 can be removed through an appropriate shift of 1 . This would introduce additional terms ∼31 and ∼1 22 . In Fig. 29a, we present the evolution of Uk (1 ) ≡ Uk (1 ; 0) for m21k0 = −0:1k02 , m22k0 = 0:1k02 , k0 = gk0 = 0:1k0 and Jk0 = 0:6k05=2 . We always shift the location of

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the false vacuum to zero. The evolution of U22 (1 ) ≡ 92 Uk =922 (1 ; 0) is displayed in Fig. 29b. The solid lines correspond to ki =k0 = e−0:8 , while the line with longest dashes (that has the smallest barrier height) corresponds to kf =k0 = e−1:2 . The potential and the $eld have been normalized with respect to kf , so that they are of order 1. The pro$le of the critical bubble b (r) is plotted in Fig. 29e in units of kf for the same sequence of scales. The quantities W1k (r) = U11 (b (r)) − U11 (0) and W2k (r) = U22 (b (r)) − U22 (0) are plotted in Figs. 29c and d, respectively. Our results for the nucleation rate are presented in Fig. 29f. The horizontal axis corresponds to k= U11 (t ), i.e. the ratio of the scale k to the square root of the positive curvature of the potential along the 1 -axis at the true vacuum. The latter quantity gives the mass of the $eld 1 at the absolute minimum. Typically, when k crosses below this mass the massive "uctuations of the $elds start decoupling (in all the examples we present the mass of 2 is of the same order or larger than that of 1 at the absolute minimum) and the evolution of the convex parts of the potential slows down and eventually stops. The dark diamonds give the negative of the action Sk of the saddle-point at the scale k. We observe a strong k dependence of this quantity. The stars in Fig. 29d indicate the values of ln(A1k =kf4 ) and the triangles those of ln(A2k ), where the two prefactors A1k , A2k are de$ned in Eqs. (6.25). Again a signi$cant k dependence is observed. The dark squares give our results for ln(I=kf4 ) = −Sk + ln(A1k A2k =kf4 ). This quantity has a very small k dependence, which con$rms our expectation that the nucleation rate should be independent of the scale k. The small residual dependence on k can be used to estimate the contribution of the next order in the expansion around the saddle point. This contribution is expected to be smaller than the $rst-order correction ln(A1k A2k =kf4 ). We now turn to the discussion of radiatively induced $rst-order phase transitions. An example can be observed in a model with a potential Uk0 (1 ; 2 ) =

k0 gk [(21 − 20k0 )2 + (22 − 20k0 )2 ] + 0 21 22 8 4

(6.27)

with 20k0 = 1:712k0 ; k0 = 0:01k0 and gk0 = 0:2k0 . Since Jk0 = 0, in this case the “classical potential” Uk0 only shows second-order transitions independence on the classical parameters 20k0 ; k0 ; gk0 . Our results for this model are presented in Fig. 30. In Fig. 30a we plot a large part of the evolution of Uk (1 ). The initial potential has only one minimum along the positive 1 -axis (and the equivalent ones under the symmetries 1 ↔ −1 , 2 ↔ −2 , 1 ↔ 2 ) and a maximum at the origin. In the sequence of potentials depicted by dotted lines we observe the appearance of a new minimum at the origin at some point in the evolution (at k=k0 ≈ e−4:4 ). This minimum is generated by the integration of "uctuations of the 2 $eld, whose mass depends on 1 through the last term in Eq. (6.27) (the Coleman–Weinberg mechanism). In Fig. 30b it can be seen that the mass term of the 2 $eld at the origin turns positive at the same value of k. This is a consequence of the 1 ↔ 2 symmetry of the potential. We calculate the nucleation rate using the potentials of the last stages of the evolution. The solid lines correspond to ki =k0 = e−4:7 , while the line with longest dashes corresponds to kf =k0 = e−5:2 . In Figs. 30b–e we observe that the mass of the 2 "uctuations in the interior of the critical bubble is much larger than the other mass scales of the problem, which are comparable to kf . This is a consequence of our choice of couplings g==20. Such a large ratio of g= is necessary for a strongly $rst-order phase transition to be radiatively induced. Unfortunately, this range of couplings also leads to large values for the 2 mass and, as a result, to values of | ln(A2k )| that are comparable or larger than the saddle-point action Sk , even though ln(A1k =kf4 ) remains small.

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Fig. 30. Nucleation rate for a "uctuation-induced $rst-order transition. We show the same quantities as in Fig. 29 for a model with initial potential given by Eq. (6.27) with 20k0 = 1:712k0 ; k0 = 0:01k0 and gk0 = 0:2k0 . The calculation is performed between the scales ki = e−4:7 k0 and kf = e−5:2 k0 . All dimensionful quantities are given in units of kf . The strong scale dependence of the nucleation rate indicates the limits of this calculation.

As a result, the saddle-point approximation breaks down and the predicted nucleation rate I=kf4 is strongly k dependent. One may wonder if it is possible to obtain a convergent expansion around the saddle point by considering models with smaller values of g. This question was addressed in Ref. [192]. For smaller values of the ratio g=, a weaker $rst-order phase transition is observed. The expansion around the saddle point is more problematic in this case. Not only | ln(A2k )| is larger than the saddle-point action Sk , but the prefactor ln(A1k =kf4 ), associated with the "uctuations of the 1 $eld, becomes now comparable to Sk . No region of the parameter space that leads to a convergent saddle-point expansion for the nucleation rate was found. The above results are not surprising. The radiative corrections to the potential and the preexponential factor have a very similar form of "uctuation determinants. When the radiative corrections are large enough to modify the bare potential and generate a new minimum, the pre-exponential factor should be expected to be important also. More precisely, the reason for this behavior can be traced to the form of the diEerential operators in the prefactor. The prefactor associated with the $eld 2 involves the ratio det(−92 + m22 + W2k (r))=det(−92 + m22 ), with m22 = U22 (0) and W2k (r) = U22 (b (r)) − U22 (0). In units in which b (r) is of order 1, the function W2k (r) takes very large positive values near r = 0 (see Figs. 30). This is a consequence of the large values of g that are required for the appearance of a new minimum in the potential. As a result, the lowest eigenvalues of the operator det(−92 + m22 + W2k (r)) are much larger than those of det(−92 + m22 ).

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This induces a large suppression of the nucleation rate. In physical terms, this implies that the deformations of the critical bubble in the 2 direction cost excessive amounts of free energy. As these "uctuations are inherent to the system, the total nucleation rate is suppressed when they are taken into account properly. The implications for these $nding for cosmological phase transitions, such as the electroweak, were discussed in Refs. [192,194]. 6.5. Testing the approach through numerical simulations As a $nal application of our formalism we consider (2 + 1)-dimensional theories at non-zero temperature. These theories provide a test of several points of our approach that depend strongly on the dimensionality, such as the form of the evolution equation of the potential, the nature of the ultraviolet divergences of the "uctuation determinants, and the k dependence of the saddle-point action and prefactor. The complementarity between the k dependence of Sk and Ak is a crucial requirement for the nucleation rate I to be k independent. Another strong motivation stems from the existence of lattice simulations of nucleation for (2 + 1)-dimensional systems [199]. As before, we work within an eEective model after dimensional reduction. In two dimensions the evolution equation for the potential takes the form [193] 9 Uk () Uk (0) 1 ln 1 + − ln 1 + : (6.28) [Uk () − Uk (0)] = − 9k 2 8" k2 k2 An approximate solution of this equation is given by Uk () ≈ V () +

2 1 1 2 k + V () V () − (k + V ()) ln : 8" 8" m2

(6.29)

The potential V () is taken V () = with

m2 2 - 3 4 + + ; 2 6 8

√ = − K; 2 m

1 = K ˆ ; 2 m 3

(6.30)

(6.31)

in order to match the parameters K and ˆ of the renormalized theory simulated in Ref. [199]. The potential of the simulated model is calculated through lattice perturbation theory. The dimensionless ˆ coupling that controls the validity of the perturbative expansion is Kˆ=3. As a result, this expanˆ ˆ sion is expected to break down for K & 3= . Similarly, Eq. (6.29) is an approximate solution of ˆ Eq. (6.28) only for Kˆ & 3= . The calculation of bubble–nucleation rates proceeds in complete analogy to the (3+1)-dimensional case at non-zero temperature. The technical details are given in Ref. [193]. In Fig. 31 we present a comparison of results obtained through our method with the lattice results of Fig. 1 of Ref. [199]. For each of several values of ˆ we vary the parameter K and determine the couplings -, according to Eqs. (6.31). The coarse-grained potential is then given by Eq. (6.29) for k ¿ kf . The diamonds ˆ K we determine Sk at two scales: 1:2kf and denote the saddle-point action Sk . For every choice of ; 2kf . The light-gray region between the corresponding points gives an indication of the k dependence Sk . The bubble–nucleation rate −ln(I=m3 ) is denoted by dark squares. The dark-gray region between

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Fig. 31. Comparison of our method with lattice studies: Diamonds denote the saddle point action Sk and squares the bubble–nucleation rate −ln(I=m3 ) for k =1:2 kf and 2 kf (shaded regions). Dark circles denote the results for the nucleation rate from the lattice study of Ref. [199]. Finally, the dashed straight lines correspond to the action of the saddle point computed from the potential of Eq. (6.30).

the values obtained at 1:2kf and 2kf gives a good check of the convergence of the expansion around the saddle-point. If this region is thin, the prefactor is in general small and cancels the k dependence of the action. The dark circles denote the results for the nucleation rate from the lattice study of Ref. [199]. The dashed straight lines correspond to the action of the saddle point computed from the ‘tree-level’ potential of Eq. (6.30). For ˆ = 0 the values of −ln(I=m3 ) computed at 1:2kf and 2kf are equal to a very good approximation. This con$rms the convergence of the expansion around the saddle point and the reliability of the calculation. The k dependence of the saddle-point action is canceled by the prefactor, so that the total nucleation rate is k independent. Moreover, the prefactor is always signi$cantly smaller than the saddle-point action. The circles indicate the results of the lattice simulations of Ref. [199]. The agreement with the lattice predictions is good. More speci$cally, it is clear that the contribution of the prefactor is crucial for the correct determination of the total bubble–nucleation rate. Similar conclusions can be drawn for ˆ = 0:1 and 0:2. For larger values of ˆ the lattice simulations have been performed only for K signi$cantly larger than 1. For smaller K, nucleation events become too rare to be observable on the lattice. Also the matching between the lattice and the renormalized actions becomes imprecise for large K. This indicates that we should expect deviations of our results from the lattice ones. These deviations start becoming apparent for the value ˆ = 0:25, for which the lattice simulations were performed with K ∼ 10–20. For ˆ ¿ 0:3 the lattice results are in a region in which the expansion around the saddle point is not reliable any more.

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For ˆ = 0:32 the breakdown of the expansion around the saddle point is apparent for 1=K . 0:12. The k dependence of the predicted bubble–nucleation rate is strong. 54 The prefactor becomes comparable to the saddle-point action and the higher-order corrections are expected to be large. The k dependence of Sk is very large. For this reason, we have not given values of Sk in this case. The comparison of our results with data from lattice simulations constitutes a stringent quantitative test of our method. In the region where the renormalized action for the lattice model is known, the data of Fig. 31, provide a strong con$rmation of the reliability of our approach. Another test has been carried out as well. In Ref. [195] a comparison has been made with the results of the thin-wall approximation in three dimensions. Very good agreement has been found, which provides additional support for the validity of the method. 7. Quantum statistics for fermions and bosons 7.1. Quantum universality At low temperature the characteristic energies are near the ground state energy. Only a few states are important and one expects the eEects of quantum–mechanical coherence to become important. The low-temperature region is therefore the quantum domain. In particular, the limit T → 0 projects on the ground state and single excitations of it. In particle physics this is the vacuum, and the single excitations correspond to the particles. The opposite is classical statistics. For large T many excitations contribute in thermal equilibrium. These thermal "uctuations destroy the quantum–mechanical coherence. The high-temperature region is the classical domain. This also holds for quantum $eld theories. Their properties in thermal equilibrium are dominated by classical aspects for high T . For bosons (and vanishing chemical potential ) there is another interesting limit. If the characteristic length scale of a physical process becomes very large, many individual local modes must be involved. Again, the quantum mechanical coherence becomes unimportant and classical physics should prevail. Translated to characteristic momenta ˜q one concludes that the limit |˜q| → 0 belongs to the classical domain, typically well described by classical $elds. In thermodynamic equilibrium a characteristic length scale is given by the correlation length 1 % = m− R . This provides us with a dimensionless combination %T for an assessment of the relevance of quantum statistics. For %T 1 a classical treatment should be appropriate, whereas for %T 1 quantum statistics becomes important. For second-order phase transitions the correlation length diverges. The universal critical phenomena are therefore always described by classical statistics. This extends to $rst-order transitions with small enough %T . On the other hand the temperature may be much smaller than the microphysical momentum scale . The modes with momenta ("T )2 ¡ q2 ¡ 2 are governed by quantum statistics. Correspondingly, the renormalization "ow in the range ("T )2 ¡ k 2 ¡ 2 is determined by the partial $xed points of the quantum system. This introduces a new type of universality since much of the microscopic information is lost in the running from to "T . Subsequently, the diEerence in the "ow for k ¿ "T and k ¡ "T leads to a crossover phenomenon 55 to classical 54

The additional squares for 1=K = 0:1 correspond to results from the numerical integration of Eq. (6.23). Such crossover phenomena have also been discussed in the framework of “environmentally friendly renormalization” [204]. 55

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statistics. For second-order phase transitions and mR T not only the standard universal critical exponents and amplitude ratios can be computed. Also the (classically non-universal) critical amplitudes can be predicted as a consequence of the new type of universality. One may call this phenomenon “quantum universality”. More formally, one can express the quantum trace in the partition function (1.1) by a functional integral in D + 1 dimensions, with D the number of space dimensions of the classical theory. The basic ingredient is Feynman’s path integral in Euclidean space. Using insertions of a complete set of eigenstates 1= dn |n n | ; (7.1) one has (for a single degree of freedom) N −1 −H dn n |e−(=N )H |n+1 = Z = tr e

(7.2)

n=0

with N ≡ 0 . Introducing the Euclidean time F = n=N; n ≡ (F); () = (0) one obtains for N → ∞; dF = =N a functional integral representation of Z Z= D e−S[] ; (F)|e−HdF |(F + dF) : (7.3) e−S[] = n

This formulation is easily generalized to the case where carries additional indices or depends on coordinate or momentum variables. If is a $eld in D dimensions, we encounter a (D +1)-dimensional functional integral. The additional dimension corresponds to the Euclidean time F and is compacti$ed on a torus with circumference = 1=T . The action S[] can be evaluated by standard methods for the limit dF → 0. For the example of a Hamiltonian depending on coordinate and momentum-type operators Q; P with [Q(˜x); P(˜y)] = i(˜x − ˜y) 1 2 1˜ 3 ˜ H= d x P (x) + V (Q(x)) + ∇Q(x)∇Q(x) ; (7.4) 2 2 one $nds 1 4 9 (x)9 (x) + V ((x)) S= d x (7.5) 2 ˜ Upto the dimensionality this is identical to Eq. (1.4) if V is a quartic with x = (F;˜x); 9 = (9F ; ∇). polynomial. In particular, Eqs. (7.4) and (7.5) describe the Hamiltonian and the action for a scalar quantum $eld theory. Bosonic $elds obey the periodicity condition (˜x; ) = (˜x; 0) and can be expanded in “Matsubara modes” with j ∈ Z (˜x; F) = j (˜x)e(2"i=) jF : (7.6) j

Correspondingly, the zero components of the momenta q =(q0 ;˜q) are given by the discrete Matsubara frequencies q0 = 2"jT :

(7.7)

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This discreteness is the only diEerence between quantum $eld theory in thermal equilibrium and in the vacuum. (For the vacuum T → 0 and q0 becomes a continuous variable.) Eq. (7.7) is the only point where the temperature enters into the quantum $eld theoretical formalism. Furthermore, the discreteness of q0 constitutes the only diEerence between quantum statistics for a system in D space dimensions and classical statistics for a corresponding system in D + 1 dimensions. In thermal equilibrium a characteristic value of q0 may be associated with mR = %−1 , where mR is the smallest renormalized mass of the system. For T=mR 1 the discreteness of q0 may be neglected and the quantum system is determined essentially by the ground state. According to our general discussion the opposite limit T=mR 1 should be dominated by classical statistics, at least for the “infrared sensitive” quantities (i.e. except for the relevant parameters in critical phenomena). For action (7.5) this is easily seen in perturbation theory by realizing that the propagator (q q + m2R )−1 = (˜q2 + (2"jT )2 + m2R )−1 ;

(7.8)

has a mass-like term ∼ T 2 for all j = 0. For large T the contributions from the j = 0 Matsubara modes are strongly suppressed in the momentum integrals for the "uctuation eEects and may be neglected. In the limit where only the j = 0 mode contributes in the functional integral (7.3), we are back to a classical functional integral in D dimensions. (The F-integration in Eq. (7.8) results in a simple factor that can be reabsorbed by a rescaling of .) This “dimensional reduction” [205] from D + 1 to D dimensions is characteristic for the crossover from the quantum domain to the classical domain as the characteristic momenta given by mR fall below T . We will see in Section 7.3 how the "ow equations realize this transition in a simple and elegant way. We also want to discuss the quantum statistics of fermionic systems. They can be treated in parallel to the bosonic system, with one major modi$cation: The classical $elds (˜x) are replaced by anticommuting Grassmann variables (˜x) { a (˜x);

y)} b (˜

=0 :

(7.9)

Correspondingly, they are antiperiodic in , i.e. (˜x; ) = − (˜x; 0). In the next Section 7.2, we generalize the exact renormalization group equation to Grassmann variables. 7.2. Exact ;ow equation for fermions The exact "ow equation (2.19) can be extended to fermions in a straightforward way except for one important minus sign [79,80]. We write the fermionic infrared cutoE term as dd q U (F) TSk = (7.10) C(q)RkF (q)C(q) ; (2")d U C are Euclidean Dirac or Weyl spinors in even dimensions and all spinor indices have been where C, suppressed. (There appears an additional factor of 12 in the r.h.s. of (7.10) if CU and C are related or obey constraints, as for the case of Majorana and Majorana–Weyl spinors in those dimensions where this is possible. For Dirac spinors or the standard 2d=2−1 component Weyl spinors this factor is absent.) In complete analogy to the case of bosonic $elds (cf. Sections 2.1, 2.2) we add the infrared cutoE term (7.10) to the classical action S U C] = S[; C; U C] + TSk [] + TS (F) [C; U C] : Sk [; C; (7.11) k

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The generating functional of connected Green functions Wk [J; 6; U 6] is now a generating functional of the Grassmann valued sources 6U and 6 in addition to the scalar sources J already contained in (2.5). The eEective average action is then de$ned as U U 6] + d d x[J (x)(x) + 6(x) U (x) − U (x)6(x)] k [; ; ] = −Wk [J; 6; − TSk [] − TSk(F) [ U ; ]

(7.12)

with (x) =

U 6] Wk [J; 6; ; 6(x) U

U 6] U (x) = Wk [J; 6; : 6(x)

(7.13)

The matrix k(2) [; U ; ] of second functional derivatives of k has a block substructure in boson– U antifermion–fermion space which we abbreviate as B–F–F:   (2) k; BB k;(2)BFU k;(2)BF    (2) (2) (2)  k(2) =  k; FB U U  k; FU FU k; FF   (2) (2) (2) k; FB k; F FU k; FF where k;(2)BB =

2 k ;

k;(2)BFU =

2 k ; U

k;(2)FF U =−

2 k ; U

etc:

(7.14)

It is now straightforward to generalize the exact "ow equation (2.19) to include fermions and one $nds 9 1 (2) −1 9 U U k [; ; ] = S Tr [k [; ; ] + Rk ] Rk 9k 2 9k 1 (2) −1 9 U = Tr [k [; ; ] + Rk ]BB Rk 2 9k (2) −1 9 U (7.15) − Tr [k [; ; ] + Rk ]F FU RkF 9k U where the block structure of Rk in B–F–F space is given by   Rk 0 0 Rk =  0 0 RkF  0 −RkF 0

(7.16)

and the minus sign in the fermionic trace arises from the anticommutation of fermions in the step corresponding to Eq. (2.27). We observe that the infrared cutoE does not mix bosons and fermions. For purely bosonic background $elds the inverse propagator k(2) is also block diagonal, and one obtains separate contributions from fermions and bosons. This is not true any more for fermionic background $elds since the inverse propagator has now oE-diagonal pieces. Again, the right-hand side of the "ow equation can be expressed as the formal derivative 9˜t acting on the IR-cutoE in a one-loop diagram.

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The infrared cutoE for fermions has to meet certain requirements for various reasons. First of all, chiral fermions do not allow a mass term. In order to remain consistent with chiral symmetries (a necessity for neutrinos, for example), the infrared cutoE must have the same Lorentz structure as the kinetic term, i.e. RkF ∼ - q [79] or, at least, not mix left- and right-handed fermions, e.g. 2 Rk ∼ - -5 q . 56 On the other hand, for q → 0 the infrared cutoE should behave as RkF ∼ k, e.g. RkF ∼ k- q = q2 . The nonanalyticity of q2 may then be a cause of problems. We will discuss here a few criteria for a chirally invariant infrared cutoE and present an explicit example which is suitable for practical calculations. First of all, the fermionic infrared cutoE term TSk(F) should be quadratic in the fermion $elds as in (7.10). We next require that TSk(F) should respect all symmetries of the kinetic term for free fermions. We will include here chiral symmetries and Lorentz invariance. (Gauge symmetries may be implemented by covariant derivatives in a backgroundgauge $eld [47].) The symmetry requirement implies in a momentum space representation ( (x) = (d d q=(2")d )eiqx (q)) 2 q dd q U (F) TSk = − (q) ; (7.17) (q)Z ; k q= rF d (2") k2 i.e. RkF ≡ −Z ; k q= rF , where U , are Dirac spinors and q= = q - . The wave function renormalization Z is chosen for convenience such that it matches with a fermion kinetic term of the form dd q U kin U k [ ; ] = − (7.18) (q)Z ; k q= (q) : (2")d The third condition requires that TSk(F) acts eEectively as an infrared cutoE. This means that for k → ∞ the combination Z rF (q2 =k 2 ) should diverge for all values of q2 . This divergence should also occur for $nite k and q2 =k 2 → 0 and be at least as strong as (k 2 =q2 )1=2 . As a fourth point we remark that k becomes the eEective action in the limit k → 0 only if limk →0 hTSk(F) = 0. This should hold for all Fourier modes separately, i.e. for limk →0 Z ; k = const: one requires 2 q lim rF q= = 0 : (7.19) k →0 k2 We furthermore request that (7.19) also holds in the limit q2 → 0. Together with the third condition, this implies exactly 2 2 −1=2 q q lim rF ∼ : (7.20) 2 2 2 k k2 q =k →0 The requirement (7.20) implies a smooth behavior of RkF for q2 =k 2 → 0. However, the non-analyticity of rF at q2 = 0 requires a careful choice of rF in order to circumvent problems. We note that rF appears in connection with the fermion propagator from k . The combination which will appear in calculations is PF = q2 (1 + rF )2 :

(7.21)

Up to the wave function renormalization, PF corresponds to the squared inverse propagator of a free massless fermion in the presence of the infrared cutoE. We will require that PF and therefore 56

We use a Euclidean convention with {- ; -> } = 2> and -5 = −-0 -1 -2 -3 in four dimensions.

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(1 + rF )2 is analytic in q2 for all q2 ¿ 0. A reasonable choice is PF =

q2 : 1 − exp{−q2 =k 2 }

(7.22)

We will employ this choice in the treatment of Nambu–Jona–Lasinio (NJL)-type models in Section 8, where we also discuss another choice for PF . 7.3. Thermal equilibrium and dimensional reduction The extension of the "ow equations to non-vanishing temperature T is straightforward [42]. The (anti-)periodic boundary conditions for (fermionic) bosonic $elds in the Euclidean time direction [206] leads to the replacement d d−1˜q dd q 2 f(q ) → T f(q02 (j) + ˜q 2 ) (7.23) (2")d (2")d−1 j ∈Z

in the trace of the "ow equation (2.36) when represented as a momentum integration. One encounters a discrete spectrum of Matsubara frequencies for the zero component q0 (j) = 2j"T for bosons and q0 (j) = (2j + 1)"T for fermions. Hence, for T ¿ 0 a four-dimensional QFT can be interpreted as a three-dimensional model with each bosonic or fermionic degree of freedom now coming in an in$nite number of copies labeled by j ∈ Z (Matsubara modes). Each mode acquires an additional temperature-dependent eEective mass term q02 (j) except for the bosonic zero mode for which q02 (0) vanishes. At high temperature all massive Matsubara modes decouple from the dynamics of the system. In this case, one therefore expects to observe an eEective three-dimensional theory with the bosonic zero mode as the only relevant degree of freedom. One may visualize this behavior by noting that for a given characteristic length scale l much larger than the inverse temperature the compact Euclidean “time” dimension cannot be resolved anymore. This phenomenon of dimensional reduction can be observed directly through the non-perturbative "ow equations. Replacement (7.23) in (2.36) manifests itself in the "ow equations only through a change to T -dependent threshold functions. For instance, the dimensionless threshold functions ldn (w; 6 ) de$ned in Eq. (3.16) are replaced by 1 d d−1˜q T n + n; 0 −1 2n−d 1 9Rk (q2 ) d ln w; ; 6 ≡ T ; vd k d − 1 2 k 4 (2") Z; k 9t [P(q ) + k 2 w]n+1 j ∈Z (7.24) where q2 = q02 +˜q 2 and q0 = 2"jT . In the limit kT the sum over Matsubara modes approaches the integration over a continuous range of q0 and we recover the zero temperature threshold function ldn (w; 6 ). In the opposite limit kT the massive Matsubara modes (j = 0) are suppressed and we expect to $nd a (d − 1)-dimensional behavior of ldn . In fact, one obtains from (7.24) ldn (w; T=k; 6 ) ldn (w; 6 ) for T k ; T vd−1 d−1 l (w; 6 ) for T k : ldn (w; T=k; 6 ) k vd n

(7.25)

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For the choice of the infrared cutoE function Rk Eq. (2.17) for bosons and Eq. (7.22) for fermions the contribution of the temperature-dependent massive Matsubara modes to ldn (w; T=k; 6 ) is exponentially suppressed for T k. Nevertheless, all bosonic threshold functions are proportional to T=k for T k whereas those with fermionic contributions vanish in this limit. This behavior is demonstrated [28] in Fig. 32 where we have plotted the quotients l41 (w; T=k)=l41 (w) and l1(F)4 (w; T=k)=l1(F)4 (w) of bosonic and fermionic threshold functions, respectively. One observes that for kT both threshold functions essentially behave as for zero temperature. For growing T or decreasing k this changes as more and more Matsubara modes decouple until $nally all massive modes are suppressed. The bosonic threshold function l41 shows for kT the linear dependence on T=k derived in Eq. (7.25). In particular, for the bosonic excitations the threshold function for w1 can be approximated with reasonable accuracy by l4n (w; 6 ) for T=k ¡ 0:25 and by (4T=k)l3n (w; 6 ) for T=k ¿ 0:25. The fermionic threshold function l1(F)4 tends to zero for kT since there is no massless fermionic zero mode, i.e. in this limit all fermionic contributions to the "ow equations are suppressed. On the other hand, the fermions remain quantitatively relevant up to T=k 0:6 because of the relatively long tail in Fig. 32b. The formalism of the average action automatically provides the tools for a smooth decoupling of the massive Matsubara modes as the momentum scale k is lowered from kT to kT . Therefore, it permits the direct link of the four-dimensional quantum $eld theory at low T to the eEective three-dimensional high-T -theory. Whereas for kT the theory is most eIciently described in terms √ of standard four-dimensional $elds a choice of rescaled three-dimensional variables 3 = = T becomes better adapted for kT . Accordingly, for high temperatures one will use the rescaled dimensionless potential u3 (t; #˜3 ) =

k u(t; #); ˜ T

#˜3 =

k #˜ : T

(7.26)

For numerical calculations at non-vanishing temperature one can exploit the discussed behavior of the threshold functions by using the zero temperature "ow equations in the range, say, k ¿ 10T . For smaller values of k one can approximate the in$nite Matsubara sums (cf. Eq. (7.24)) by a $nite series so that the numerical uncertainty at k = 10T is better than a given value. This approximation becomes exact in the limit k10T . 7.4. The high-temperature phase transition for the 4 quantum >eld theory The formalism of the previous sections can be applied to the phase transition of the fourdimensional O(N )-symmetric 4 theory at non-vanishing temperature [42,207] (see also [204,208– 211] for studies using similar techniques). We consider models with spontaneous symmetry breaking at zero temperature, and investigate the restoration of symmetry as the temperature is raised. We specify the action together with some high momentum cutoE T so that the theory is properly regulated. We then solve the evolution equation for the average potential for diEerent values of the temperature. For k → 0 this solution provides all the relevant features of the temperature-dependent eEective potential. In order not to complicate our discussion, we neglect the wave function renormalization and consider a very simple ansatz for the potential, in which we keep only the quadratic and quartic terms. Improved accuracy can be obtained by using more extended truncations of the average action. The "ow equations for the rescaled minimum of the potential <(k; T ) = #0 (k; T )=k 2

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Fig. 32. EEective dimensional reduction: The plot shows the temperature dependence of the bosonic (a) and the fermionic (b) threshold functions l41 (w; T=k) and l(F)4 (w; T=k), respectively, for diEerent values of the dimensionless mass term w. 1 We have normalized them to the T = 0 threshold functions. The solid line corresponds to w = 0, whereas the dotted ones correspond to w = 0:1; 1 and 10 with decreasing size of the dots. For T k the bosonic threshold function becomes proportional to T=k, whereas the fermionic one tends to zero. In this range the theory with properly rescaled variables behaves as a classical three-dimensional theory.

J. Berges et al. / Physics Reports 363 (2002) 223 – 386

and the quartic coupling (k; T ) are T T 1 4 4 + (N − 1)l1 ; 3l1 2<; 9t < = < = −2< + 2 16" k k T T 1 2 4 4 + (N − 1)l2 : 9 t = = 9l2 2<; 2 16" k k

353

(7.27) (7.28)

The initial conditions are determined by the “short-distance values” #0 (k = ) and (k = ) that correspond to the minimum and the quartic coupling of the bare potential. We then have to compute the evolution, starting at k = and following the renormalization "ow towards k = 0. This procedure has to be followed for T = 0 and then to be repeated for T = 0 in order to relate the zero and non-zero temperature eEective potential of the same theory. Since the running of the parameters is the same in the zero and non-zero temperature case for kT we do not need to compute the evolution in this range of k. Our strategy is equivalent to the following procedure: We start with the zero-temperature theory at k = 0 taking the renormalized parameters as input. We subsequently integrate the zero-temperature "ow equations “up” to k = T=K1 , where K1 is chosen such that l4n (w; T=k) = l4n (w) to a good approximation for k ¿ T=K1 . For bosons K1 . 0:1 is suIcient. We can now use the values of the running parameters at k = T=K1 as initial conditions for the non-zero temperature "ow equations and integrate them “down” to k = 0. In this way we obtain the renormalized parameters at non-zero temperature in terms of the renormalized parameters at zero temperature. As we discussed in the previous subsection, the threshold functions simplify considerably for k ¡ T=K2 , with K2 0:4 for bosons. In this range l4n (w; T=k) = 4l3n (w)T=k to a good approximation and we expect an eEective three-dimensional evolution. We can de$ne the dimensionless quantities # (k; T ) #0 (k; T ) = ; k kT ˜ T ) = (k; T ) = (k; T ) T ; (k; k k

<(k; ˜ T) =

(7.29)

where # and have the canonical three-dimensional normalization. In terms of these quantities the "ow equations read for k ¡ T=K2 : d <˜ 1 ˜ ; = −<˜ + 2 {(N − 1)l31 + 3l31 (2˜<)} dt 4"

(7.30)

1 2 d ˜ = −˜ + 2 ˜ {(N − 1)l32 + 9l32 (2˜<)} ˜ : dt 4"

(7.31)

The main qualitative diEerence of the last equations from those of the zero-temperature theory arises from the term −˜ in the r.h.s. of Eq. (7.31), which is due to the dimensions of . In consequence, the dimensionless quartic coupling ˜ is not infrared free. Its behavior with k → 0 is characterized by an approximate $xed point for the region where <˜ varies only slowly. Taken together, the pair ˜ <) of diEerential equations for (; ˜ has an exact $xed point (<˜ ∗ , ˜∗ ) corresponding to the phase transition.

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Fig. 33. The evolution of the minimum of the potential #0 (k; T ) at various temperatures. For T ¿ Tcr the evolution of the mass term m2 (k; T ) in the symmetric regime is also displayed. N = 1 and R = 0:1.

U with KU between K1 By using l4n (w; T=k)=l4n (w) for k ¿ T= KU and l4n (w; T=k)=4l3n (w)T=k for k ¡ T= K, and K2 , our procedure simpli$es even further. It may be summarized as “run up in four dimensions, U KU ≈ run down in three dimensions”, with a matching of the k-dependent couplings at the scale T= K; 0:25. In practice, we take the “threshold correction” from the diEerent running for T=K2 ¡ k ¡ T=K1 into account numerically. In the case that #0 (k; T ) becomes zero at some non-zero ks we continue with the equations for the symmetric regime with boundary conditions m2 (ks ; T ) = 0 and (ks ; T ) given by its value at the end of the running in the spontaneously broken regime. The results of the numerical integration of the "ow equations for N =1 are presented in Figs. 33 and 34 for a zero-temperature theory with renormalized quartic coupling R =0:1. The solid line in Fig. 33 displays the “quadratic renormalization” of the minimum of the zero-temperature average potential. At non-zero temperature (dashed lines) we notice the deviation from the zero-temperature behavior. For low temperatures, in the limit k → 0; #0 (k; T ) reaches an asymptotic value #0 (0; T ) ¡ #0 (0) ≡ #0 . This value corresponds to the vacuum expectation value of the non-zero temperature theory and we denote it by #0 (T ) = #0 (0; T ). At a speci$c temperature Tcr ; #0 (T ) becomes zero and this signals the restoration of symmetry for T ¿ Tcr . The running of (k); (k; T ) is shown in Fig. 34. We observe the logarithmic running of (k) (solid line) which is stopped by the mass term. For non-zero temperatures (k; T ) deviates from the zero temperature running and reaches a non-zero value in the limit k → 0. We observe that (k; T ) runs to zero for T → Tcr [203,42]. For T ¿ Tcr

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Fig. 34. Scale dependence of the quartic coupling (k; T ) at various temperatures. The strong drop near the critical temperature is characteristic for the critical behavior.

the running in the spontaneously broken regime ends at a non-zero ks , at which #0 (ks ; T ) equals zero. From this point on we continue the evolution in the symmetric regime. The running of the mass term at the origin m2 (k; T ) is depicted in Fig. 33, while the evolution of (k; T ) proceeds continuously in the new regime as shown in Fig. 34. The procedure of “running up in four dimensions” and “running down in three dimensions” provides the connection between the renormalized quantities at zero and non-zero temperature. We de$ne the zero-temperature theory in terms of the location of the minimum #0 and the renormalized quartic coupling R = (0). Through the solution of the evolution equations we obtain #0 (T ) and R (T ) = (0; T ) for non-zero temperatures T ¡ Tcr . For T ¿ Tcr the symmetry is restored (#0 (T ) = 0) and the non-zero temperature theory is described in terms of m2R (T ) = m2 (0; T ) and R (T ). In Fig. 35 we plot #0 (T )=#0 ; R (T ) and m2R (T )=T 2 as a function of temperature for N = 1 and R = 0:1. As the temperature increases towards Tcr we observe a continuous transition from the spontaneously broken to the symmetric phase. This clearly indicates a second-order phase transition. The renormalized quartic coupling R (T ) remains close to its zero temperature value R for a large range of temperatures and drops quickly to zero at T = Tcr . Recalling our parametrization of the average potential in terms of its successive # derivatives at the minimum, we conclude that, at Tcr , the $rst non-zero term in the expression for the eEective potential is the 6 term (which we have neglected in our truncated solution). For T Tcr the coupling R (T ) quickly grows to approximately its zero temperature value R , while m2R (T ) asymptotically becomes proportional to T 2 as T → ∞. In the temperature range where R (T ) . 0:5R (0), the "uctuations are important and the universal critical behavior becomes dominant (cf. Section 1.2).

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Fig. 35. High-temperature symmetry restoration: we show the temperature dependence of the order parameter #0 (T ), the renormalized mass m2R (T ) and the quartic coupling R (T ) near Tcr . The critical behavior of a second-order phase transition is apparent.

The value of the critical temperature Tcr in terms of the zero-temperature quantities has been calculated in the context of “naive” perturbation theory [200 –202]. It was found that it is given by Tcr2 = 24#0 =(N + 2), independently of the quartic coupling in lowest order. This prediction was con$rmed in Ref. [42] for various values of N . Another parameter which can be compared with the perturbative predictions is m2R (T ) in the limit T → ∞. In Ref. [42] it was shown that the quantity [m2R (T )=R (T )(N + 2)T 2 ]−1 becomes equal to 24 for T 2 =#0 → ∞ and small R , again in agreement with the perturbative result. The most important aspect of our approach is related to the infrared behavior of the theory for T → Tcr . The temperature dependence of #0 (T ); R (T ); m2R (T ) near Tcr is presented in Fig. 35 (R = 0:1). We have already mentioned that all the above quantities become zero at T = Tcr . What becomes apparent in this $gure is a critical behavior which can be characterized by critical exponents. Following the notation of statistical mechanics, we parametrize the critical behavior of #0 (T ) and m2R (T ) as #0 (T ) ˙ (Tcr2 − T 2 )2 ; m2R (T ) ˙ (T 2 − Tcr2 )2> :

(7.32)

We also de$ne a critical exponent C for R (T ) in the symmetric regime: R (T ) ˙ (T 2 − Tcr2 )C :

(7.33)

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Fig. 36. Temperature-dependent eEective critical exponents ; > and C. As the critical temperature is approached (T → Tcr ) they become equal to the critical exponents of the zero-temperature three-dimensional theory.

These exponents are plotted as function of the logarithm of |T 2 − Tcr2 | in Fig. 36. We notice that in the limit T → Tcr the critical exponents approach asymptotic values. These are independent of R and therefore fall into universality classes determined only by N . They are equal to the critical exponents of the zero-temperature three-dimensional theory. This fact can be understood by recalling that the evolution in the high-temperature region is determined by an eEective three-dimensional theory whose phase diagram has a $xed point corresponding to the phase transition. For T → Tcr the evolution of #0 (k; T ); (k; T ) in the high-temperature region is given by a line in the phase diagram very close to the critical line. In this case, #0 (k; T ); (k; T ) spend an arbitrarily long “time” t close to the $xed point and, as a result, lose memory of their “initial values” #0 (T=K2 ; T ); (T=K2 ; T ). The critical behavior is determined solely by the $xed point. Our crude truncation for the potential results in the values = 0:25; > = C = 0:5, which can be compared with the more accurate ones presented in Section 4 (cf. Tables 2 and 5). Notice that these values satisfy the correct scaling law > = 2 in the limit of zero wave function renormalization. The critical behavior of R (T ) is related to the resolution of the problem of the infrared divergences which cause the breakdown of the “naive” perturbative expansion in the limit T → Tcr [200,202]. The infrared problem is manifest in the presence of higher-order contributions to the eEective potential which contain increasing powers of R (T )T=k, where k is the eEective infrared cutoE of the theory. If the evolution of (k; T ) is omitted and R (T ) is approximated by its zero-temperature value R , these contributions diverge and the perturbative expansion breaks down. A similar situation appears for the zero-temperature three-dimensional theory in the critical region [203]. In this case the problem results from an eEective expansion in terms of the quantity u=[M 2 − Mcr2 ]1=2 , where u is the bare

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three-dimensional quartic coupling and [M 2 −Mcr2 ]1=2 is a measure of the distance from the point where the phase transition occurs as it is approached from the symmetric phase. The two situations can be seen to be of identical nature by simply remembering that the high-temperature four-dimensional coupling corresponds to an eEective three-dimensional coupling T and that the eEective infrared cutoE in the symmetric phase is equal to mR (T ). In the three-dimensional case the problem has been resolved [203] by a reformulation of the calculation in terms of an eEective parameter 3 =m, where 3 is the renormalized 1PI four-point function in three dimensions (the renormalized quartic coupling) and m the renormalized mass (equal to the inverse correlation length). It has been found [203,121] that the above quantity has an infrared stable $xed point in the critical region m → 0: No infrared divergences arise within this approach. Their only residual eEect is detected in the strong renormalization of 3 . In our scheme the problem is formulated in terms of the eEective ˜ T ) = (k; T )T=k (see Eqs. (7.29)), for which a dimensionless parameters <(k; ˜ T ) = #0 (k; T )=kT; (k; $xed point corresponding to the phase transition is found. The critical behavior is determined by this $xed point in the limit k → 0. Everything remains $nite in the vicinity of the critical temperature, and the only memory of the infrared divergences is re"ected in the strong renormalization of R (T ) near Tcr . We conclude that the infrared problem disappears if formulated in terms of the appropriate renormalized quantities. When expressed in the correct language, it becomes simply a manifestation of the strong renormalization eEects in the critical region. In Ref. [42] the quantity R (T )T=mR (T ) was calculated in the limit T → Tcr . It reaches a universal asymptotic value depending only on N . For N = 1 we $nd R (T )T=mR (T ) = 6:8 within our crude truncation, to be compared with the more accurate result R (T )T=mR (T ) ≈ 8 shown in Table 6. Moreover, the existence of this asymptotic value explains the equality of the critical exponents > and C. Finally, we point out that the “non-universal quantities” as the critical temperature or the nonuniversal amplitudes are completely determined by the renormalized zero-temperature couplings R ; #0 . No additional free parameters (amplitudes) appear in Section 7.1. The microphysics at the scale may not be known precisely, similarly to the situation often encountered in statistical mechanics. Nevertheless, most of the memory of the microphysics is already lost at the scale k ≈ T , except from the relevant parameter #0 and the marginal coupling R . This predictive power for the amplitudes is an example of quantum universality. 8. Fermionic models 8.1. Introduction Shortly after the discovery of asymptotic freedom [212], it was realized [213] that at suIciently high temperature or density the theory of strongly interacting elementary particles, quantum chromodynamics (QCD), diEers in important aspects from the corresponding zero temperature or vacuum theory. A phase transition at some critical temperature Tc or a relatively sharp crossover may separate the high- and low-temperature physics [214]. At non-zero baryon density QCD is expected to have a rich phase structure with diEerent possible phase transitions as the density varies [15 –17,30]. We will concentrate in the following on properties of the chiral phase transition in QCD and consider an application of the average action method to an eEective fermionic model. The vacuum of QCD contains a condensate of quark–antiquark pairs, U = 0, which spontaneously breaks the

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(approximate) chiral symmetry of QCD and has profound implications for the hadron spectrum. At high temperature this condensate is expected to melt, i.e. U 0, which signals the chiral phase transition. We will investigate the chiral phase transition within the linear quark meson model [169] for two quark "avors. Truncated non-perturbative "ow equations are derived at non-zero temperature [28] and chemical potential [215]. Whereas the renormalization group "ow leads to spontaneous chiral symmetry breaking in vacuum, the symmetry gets restored in a second-order phase transition at high temperature for vanishing quark mass. The description [28] covers both the low-temperature chiral perturbation theory domain of validity as well as the high temperature domain of critical phenomena. In particular, we obtain a precise estimate of the universal equation of state in the vicinity of critical points. We explicitly connect the physics at zero temperature and realistic quark mass with the universal behavior near the critical temperature Tc and the chiral limit. An important property will be the observation that certain low-energy properties are eEectively independent of the details of the model even away from the phase transition. This behavior is caused by a strong attraction of the renormalization group "ow to approximate partial infrared $xed points [169,28]. Within this approach at high density we $nd [215] a chiral symmetry restoring $rst-order transition. The results imply the presence of a critical endpoint in the phase diagram in the universality class of the three-dimensional Ising model [29,30]. The universal properties of this endpoint have been discussed in Section 4.4. For details of the QCD aspects of this approach see the reviews [216,17] for a discussion of the phase diagram. Similar non-perturbative renormalization group studies of QCD motivated models can be found in [217–219]. Field theories with scalars and fermions have been investigated using similar techniques in [220 –223]. 8.2. Linear quark meson model We consider a Nambu–Jona–Lasinio-type model for QCD in which quarks interact via eEective fermionic interactions, with a; b; c; d = 1; 2 "avors and i; j = 1; 2; 3 colors. The model is de$ned at some “high” momentum scale kO ≈ 600–700 MeV i U [ U; ] (8.1) kO [ ; ] = d 4 x{ U a (x)Z ; kO [i- 9 + m(x)-5 ] ia (x)} + k(int) O with given fermion wave function renormalization constant Z ; kO . The curled brackets around the fermion bilinears in (8.1) indicate contractions over the Dirac spinor indices which are suppressed. For later purposes, we allow for a non-constant mass term m(x)-5 and we concentrate at the end on equal constant current quark masses m = 12 (mu + md ). 57 The fermions interact via a four-fermion interaction which in momentum space is given by 4 4 1 d p 2 l k(int) (2")4 (p1 + p2 − p3 − p4 )hUkO G(pl ) [ U; ] = − O 4 8 (2") l=1

×[{

U i (−p1 )i(Fz )a b (p2 )}{ U j (p4 )i(Fz )c d (−p3 )} b i d j a c i

+ { U a (−p1 )-5 57

a 5 b Uj i (p2 )}{ b (p4 )- j (−p3 )}]

:

(8.2)

We use chiral conventions in which the (real) mass term is multiplied by -5 . The more common version of the fermion mass term can be obtained by a chiral rotation.

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We consider a momentum-dependent four-fermion interaction G −1 = mU 2kO + ZO; kO (p1 + p2 )2 :

(8.3)

The action is invariant under the chiral "avor group SU (2)L × SU (2)R except for the mass term. Let us de$ne composite $elds d4 p U U i 1 z U h (p)i(Fz )ab ib (q + p) ; O [ ; ; q] = − 2 (2")4 a d4 p U U i 1 O(5) [ U ; ; q] = − (8.4) h (p)-5 ia (q + p) 2 (2")4 a and rewrite kO [ U ; ] = −

d4 q {Z (2")4

; kO

U i (q)- q a

a i (q)

−1

+ 2hU Z

; kO m(q)O(5) [

U ; ; −q]

1 + G(q2 )(Oz [ U ; ; −q]Oz [ U ; ; q] + O(5) [ U ; ; −q]O(5) [ U ; ; q])} : (8.5) 2 It is advantageous to consider an equivalent formulation which introduces bosonic collective $elds with the quantum numbers of the fermion bilinears appearing in (8.5). This amounts to replace the eEective action (8.1) by d4 q −1 U U ˆ kO [ ; ; s; "] ≡ kO [ ; ] + (s(−q) − 2Z ; kO hU m(−q) − O(5) (−q)G(q2 )) (2")4 ×(s(q) − 2Z +

U−1 m(q) − O(5) (q)G(q2 ))

; kO h

1 2G(q2 )

1 d4 q z : (" (−q) − Oz (−q)G(q2 ))("z (q) − Oz (q)G(q2 )) 4 (2") 2G(q2 )

(8.6)

The scalar $elds "z and s have the quantum numbers of the pions and the A-$eld. The terms added to kO [ U ; ] are quadratic in the fermion bilinears. They cancel the original four-fermion interaction (8.2) and introduce a Yukawa interaction between fermions and collective $elds, as well as a propagator term for the collective $elds: d4 q i U ˆ kO [ ; ; s; "] = −Z ; kO U a (q)- q ia (q) 4 (2") 1 + [ZO; kO q2 + mU 2kO ](s(−q)s(q) + "z (−q)"z (q)) − —(−q)s(q) 2 U 5 d4 p U i ha z a 5 b (s(q)b + i" (q)(Fz )b - ) i (p − q) ; + (p) (2")4 a 2

(8.7)

where we dropped $eld independent terms in (8.6). The above replacement of kO [ U ; ] by ˆ kO [ U ; ; s; "] corresponds to a Hubbard–Stratonovich transformation in the de$ning functional integral for the eEective action, in which the collective $elds are introduced by inserting identities into the functional integral. The introduction of collective $elds in the context of "ow equations is

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discussed in Refs. [76,28]. The equivalence of ˆ kO [ U ; ; s; "] with the original formulation in terms of fundamental $elds only is readily established by solving the $eld equations for s and " ˆ kO [ U ; ; s; "] −1 = 0; s0 = GO(5) [ U ; ] + 2Z ; kO hU m ; s ˆ kO [ U ; ; s; "] = 0; "0z = GOz [ U ; ] (8.8) "z and inserting the solution in the eEective action ˆ kO [ U ; ; s0 ; "0 ] = kO [ U ; ]. The SU (2)L × SU (2)R symmetry is most manifest in a 2 × 2 matrix notation O ≡ 12 (s + i"z Fz ) :

(8.9)

The quark mass term in the original fermionic description appears now as a source term which is proportional to m −1 —(q) ≡ 2hU Z

U 2kO ; kO m(q)(m

+ ZO; kO q2 ) :

(8.10)

We will be interested mainly in momentum-independent sources —(q ≡ 0) or constant fermion masses. Since the chiral symmetry breaking is linear in O we can de$ne a chirally symmetric eEective action U ˆ (8.11) k [ ; O] = k [ ; ; s; "] + d 4 x— tr O : The "ow of k will conserve the chiral symmetries. The explicit chiral symmetry is now re"ected by the $eld equation k =j : (8.12) s Knowledge of 0 for an arbitrary constant $eld O = diag(AU 0 ; AU 0 ) contains information on the model for arbitrary quark masses. Spontaneous chiral symmetry breaking manifests itself by AU 0 = 0 for j → 0. As a result, our approach allows for a simple uni$ed treatment of spontaneous and explicit chiral symmetry breaking. The eEective action at the scale kO speci$es the “initial condition” for the renormalization "ow of the average action k . For scales k ¡ kO we allow for a more general form and consider a truncation (# = (s2 + "z "z )=2 = tr O† O) 1 + -5 a 1 − -5 † a b i i 4 a U U U Ob − (O )b i k [ ; O] = d x i Z ; k a - 9 i + hk a 2 2 † ba + ZO; k 9 (O )ab 9 O + Uk (#) ; (8.13) which takes into account the most general $eld dependence of the O(4)-symmetric average potential Uk . Here ZO; k and Z ; k denote scale dependent wave function renormalizations for the bosonic $elds and the fermionic $elds, respectively. We note that in our conventions the scale-dependent Yukawa coupling hUk is real. In terms of the renormalized expectation value A0 = ZO1=2 AU 0 ;

(8.14)

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we obtain the following expressions for quantities as the pion decay constant f" , chiral condensate U , constituent quark mass Mq and pion and sigma mass, m" and mA (d = 4) [28] f"; k = 2A0; k ; −1=2 U k = −2mU 2kO [ZO; k A0; k − m] ;

Mq; k = hk A0; k ; −1=2 m2"; k = ZO; k

mU 2kO m −1=2 — = ZO; ; k A0; k 2A0; k

−1=2 m2A; k = ZO; k

mU 2kO m + 4k A0;2 k : A0; k

(8.15)

Here we have de$ned the dimensionless, renormalized couplings −2 k = ZO; k

92 U k (# = 2AU 20; k ) ; 9#2

−1=2 −1 U hk = ZO; k Z ; k hk :

(8.16)

We are interested in the “physical values” of quantities (8.15) in the limit k → 0 where the infrared cutoE is removed, i.e. f" = f"; k=0 ; m2" = m2"; k=0 , etc. 8.3. Flow equations and infrared stability The dependence of the eEective action k on the infrared cutoE scale k is given by the exact "ow equation (7.15) or (2.19) for fermionic $elds (quarks) and bosonic $elds O (mesons) [9,79] (t = ln(k=kO )) 9RkB (2) 9RkF (2) 1 9 −1 −1 k [ ; O] = Tr (k [ ; O] + Rk ) (k [ ; O] + Rk ) − Tr : (8.17) 9t 2 9t 9t Here k(2) is the matrix of second functional derivatives of k with respect to both fermionic and bosonic $eld components. The $rst trace in the r.h.s. of (8.17) eEectively runs only over the bosonic degrees of freedom. It implies a momentum integration and a summation over "avor indices. The second trace runs over the fermionic degrees of freedom and contains in addition a summation over Dirac and color indices. The infrared cutoE function Rk has a block substructure with entries RkB and RkF for the bosonic and the fermionic $elds, respectively (cf. Section 7.2). We compute the "ow equation for the eEective potential Uk from Eq. (8.17) using ansatz (8.13) for k . The bosonic contribution to the running eEective potential corresponds exactly to Eq. (2.36) for the scalar O(4) model in lowest order of the derivative expansion. The fermionic contribution to the evolution equation for the eEective potential can be computed without much additional eEort from (8.13) since the fermionic $elds appear only quadratically. The respective "ow equation is obtained by taking the second functional derivative evaluated at = U = 0.

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For the study of phase transitions it is convenient to work with rescaled, dimensionless and renormalized variables. We introduce (with a generalization to arbitrary dimension d) u(t; #) ˜ ≡ k −d Uk (#);

#˜ ≡ ZO; k k 2−d #;

−1=2 −1 d−4 U hk : hk = ZO; k Z ;kk

(8.18)

Combining the bosonic and the fermionic contributions one obtains the "ow equation [28] 9 u = −du + (d − 2 + 6O )#u ˜ 9t d 1 2 +1 (F)d d d : + 2vd 3l0 (u ; 6O ) + l0 (u + 2#u ˜ ; 6O ) − 2 2 3l0 #h ˜ ;6 2

(8.19)

Here vd−1 ≡ 2d+1 "d=2 (d=2) and primes denote derivatives with respect to #. ˜ The bosonic, ld0 , and (F)d the fermionic, l0 , threshold functions are de$ned in Section 3.2 and Appendix A. The $rst two terms of the second line in (8.19) denote the contributions from the pions and the A $eld, and the last term corresponds to the fermionic contribution from the u; d quarks. Eq. (8.19) is a partial diEerential equation for the eEective potential u(t; #) ˜ which has to be supplemented by the "ow equation for the Yukawa coupling hk and expressions for the anomalous dimensions, where d d (8.20) 6O = (ln ZO; k ); 6 = (ln Z ; k ) : dt dt Here the wave function renormalizations are evaluated for a k-dependent background $eld #0; k or < ≡ k 2−d ZO; k #0; k determined by the condition —0 −1=2 u (t; <) = √ k −(d+2)=2 ZO; (8.21) k ≡ jg 2< with —0 some $xed source. For a study of realistic quark masses the optimal choice is given by (8.10), whereas an investigation of the universal critical behavior for mq → 0 should employ j0 = 0. Eq. (8.21) allows us to follow the "ow of < according to < 9 d <= [6O − d − 2]jg − 2 u (t; <) (8.22) dt jg + 2< 9t with ≡ u (t; <). We also de$ne the Yukawa coupling at #=< ˜ and its "ow equation reads [169,224] d 2 h = (d − 4 + 26 + 6 )h2 dt 1 2 1 2 (FB)d (FB)d 4 h <; jg ; 6 ; 6O − l1; 1 h <; jg + 2<; 6 ; 6O − 2vd h 3l1; 1 2 2 1 2 h + 4vd h4 < 3l(FB)d <; j ; 6 ; 6 g O 1; 2 2 1 2 (FB)d − (3 + 2
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Similarly, the scalar anomalous dimension is infered from d vd 4<2 md2; 2 (jg ; jg + 2<; 6O ) 6O ≡ − ln ZO; k = 4 dt d d d 1 2 (F)d (F)d 1 2 − 1 2 4 h <; 6 + 2 2 Nc h <m2 ( 2 h <; 6 ) + 2 2 Nc h m4 2

(8.24)

and the quark anomalous dimension reads 1 2 1 2 d vd 2 (FB)d (FB)d h <; jg ; 6 ; 6O + m1; 2 h <; jg + 2<; 6 ; 6O 6 ≡ − ln Z ; k = 2 h 3m1; 2 : dt d 2 2 (8.25) This constitutes a linear set of equations for the anomalous dimensions. The threshold functions (F)d d ; m4(F)d and m(FB)d l(FB)d n1 ;n2 ; mn1 ;n2 ; m2 n1 ;n2 are speci$ed in Appendix A. Most importantly for practical applications to QCD, the system of "ow equations for the eEective potential Uk (#), the Yukawa coupling hk and the wave function renormalizations ZO; k ; Z ; k exhibits an approximate partial $xed point [169,28]. For a small initial value of the scalar wave function renormalization, ZO; kO 1 at the scale kO , one observes a large renormalized meson mass term −1=2 U −2 ZO; kO UkO and a large renormalized Yukawa coupling hkO = ZO; kO hkO (for Z ; kO = 1). In this case, for the initial running one can neglect in the "ow equations all scalar contributions with threshold functions involving the large meson masses. This yields the simpli$ed equations [28,169] for the rescaled quantities (d = 4; v4−1 = 32"2 ) Nc (F)4 1 2 9 u = −4u + (2 + 6O )#u #h ˜ ; ˜ − 2 l0 9t 2" 2 d 2 h = 6 O h2 ; dt Nc 6O = 2 h2 ; 8" 6 =0 :

(8.26)

Of course, this approximation is only valid for the initial range of running below kO before the (dimensionless) renormalized scalar mass squared u (t; #˜ = 0) approaches zero near the chiral symmetry breaking scale. System (8.26) is exactly soluble and we $nd [28] h2I h2 (t) = ZO−1 (t) = ; 1 − (Nc =8"2 )h2I t t 2 Nc −4t 2t h (t) −4r (F)4 1 2 2r − 2 : (8.27) dre l0 ˜ h (t)#e u(t; #) ˜ = e uI e #˜ 2 2" 0 2 hI Here uI (#) ˜ ≡ u(0; #) ˜ denotes the eEective average potential at the scale kO and h2I is the initial value 2 of h at kO , i.e. for t = 0. To make the behavior more transparent we consider an expansion of the initial value eEective potential uI (#) ˜ in powers of #˜ around #˜ = 0 ∞ (n) uI (0) n #˜ : uI (#) ˜ = (8.28) n! n=0

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Expanding also l0(F)4 in Eq. (8.27) in powers of its argument one $nds for n ¿ 2 (n) u(n) (t; 0) Nc (−1)n (n − 1)! (F)4 2(n−2)t uI (0) = e l (0)[1 − e2(n−2)t ] : + h2n (t) "2 2n+2 (n − 2) n h2n I

(8.29)

For decreasing t → −∞ the initial values uI(n) become rapidly unimportant and u(n) =h2n approaches a $xed point. For n = 2, i.e. for the quartic coupling, one $nds u(2) (t; 0) 1 − uI(2) (0)=h2I ; = 1 − h2 (t) 1 − (Nc =8"2 )h2I t

(8.30)

leading to a $xed point value (u(2) =h2 )∗ =1. As a consequence of this $xed point behavior the system looses all its “memory” of the initial values uI(n¿2) at the compositeness scale kO . Furthermore, the attraction to partial infrared $xed points continues also for the range of k where the scalar "uctuations cannot be neglected anymore. On the other hand, the initial value of the bare dimensionless mass parameter uI (0) mU 2kO = 2 h2I kO

(8.31)

is never negligible. In other words, for hI → ∞ the infrared behavior of the linear quark meson model will depend (in addition to the value of the compositeness scale kO and the quark mass m) only on one parameter, mU 2kO . One can therefore add higher scalar self-interactions to kO in Eq. (8.7) without changing the result much. We have veri$ed numerically this feature by starting with diEerent values for the quartic scalar self-interaction uI(2) (0). Indeed, the diEerences in the physical observables were found to be small. For de$niteness, the numerical analysis of the full system of "ow equations [28] is performed with the idealized initial value uI (#) ˜ = uI (0)#˜ in the 2 limit hI → ∞. Deviations from this idealization lead only to small numerical deviations in the infrared behavior of the linear quark meson model as long as hI & 15. 8.4. High-temperature chiral phase transition Strong interactions in thermal equilibrium at high-temperature T deviate in important aspects from their well-tested vacuum or zero temperature properties. A phase transition at some critical temperature Tc or a relatively sharp crossover may separate the high- and low-temperature physics [214]. It was realized early that the transition should be closely related to a qualitative change in the chiral condensate according to the general observation that spontaneous symmetry breaking tends to be absent in a high-temperature situation. A series of stimulating contributions [24,26,27] pointed out that for suIciently small up and down quark masses, mu and md , and for a suIciently large mass of the strange quark, ms , the chiral transition is expected to belong to the universality class of the O(4) Heisenberg model. It was suggested [26,27] that a large correlation length may be responsible for important "uctuations or lead to a disoriented chiral condensate. A main question we are going to answer using non-perturbative "ow equations for the linear quark meson model is: How small mu and md would have to be in order to see a large correlation length near Tc or could this scenario be realized for realistic values of the current quark masses? In order to solve our model we need to specify the “initial condition” kO for the renormalization "ow of k . We will choose in the following a normalization of ; O such that Z ; kO = hUkO = 1.

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We therefore need as initial values at the scale kO the scalar wave function renormalization ZO; kO and the shape of the potential UkO . We will make here the important assumption that ZO; k is small at the compositeness scale kO (similarly to what is usually assumed in Nambu–Jona–Lasinio-like models) ZO; kO 1 :

(8.32)

This results in a large value of the renormalized Yukawa coupling hk = ZO; k Z ; k hUk . A large value of hkO is phenomenologically suggested by the comparably large value of the constituent quark mass Mq . The latter is related to the value of the Yukawa coupling for k → 0 and the pion decay constant f" = 92:4 MeV by Mq = hf" =2 (with h = hk=0 ), and Mq 300 MeV implies h2 =4" 3:4. For increasing k the value of the Yukawa coupling grows rapidly for k & Mq . Our assumption of a large initial value for hkO is therefore equivalent to the assumption that the truncation (8.13) can be used up to the vicinity of the Landau pole of hk . The existence of a strong Yukawa coupling enhances the predictive power of our approach considerably. It implies a fast approach of the running couplings to partial infrared $xed points as shown in Section 8.3 [169,28]. In consequence, the detailed form of UkO becomes unimportant, except for the value of one relevant parameter corresponding to the scalar mass term mU 2kO . In this work we $x mU 2kO such that f" = 92:4 MeV for m" = 135 MeV. The value f" = 92:4 MeV (for m" = 135 MeV) sets our unit of mass for two "avor QCD which is, of course, not directly accessible by observation. Besides mU 2kO (or f" ) the other input parameter used in this work is the constituent quark mass Mq which determines the scale kO at which hkO becomes very large. We consider a range 300 MeV . Mq . 350 MeV and $nd a rather weak dependence of our results on the precise value of Mq . The results presented in the following are for Mq = 303 MeV. We $rst consider the model at nonzero temperature T . The case for non-vanishing baryon number density will be presented in Section 8.5. Fig. 37 shows our results [28] for the chiral condensate U as a function of the temperature T for various values of the average quark mass m = (mu + md )=2. Curve (a) gives the temperature dependence of U in the chiral limit m = 0. We $rst consider only the lower curve which corresponds to the full result. One observes that the order parameter U goes continuously (but non-analytically) to zero as T approaches the critical temperature in the massless limit Tc = 100:7 MeV. The transition from the phase with spontaneous chiral symmetry breaking to the symmetric phase is second order. The curves (b) – (d) are for non-vanishing values of the average current quark mass m. The transition turns into a smooth crossover. Curve (c) corresponds to mphys or, equivalently, m" (T = 0) = 135 MeV. The transition turns out to be much less dramatic than for m = 0. We have also plotted in curve (b) the results for comparably small quark masses 1 MeV, i.e. m = mphys =10, for which the T = 0 value of m" equals 45 MeV. The crossover is considerably sharper but a substantial deviation from the chiral limit remains even for such small values of m. For comparison, the upper curves in Fig. 37 use the universal scaling form of the equation of state of the three-dimensional O(4)-symmetric Heisenberg model which has been computed explicitly in Section 4.3. The scaling equation of state in terms of the chiral condensate for the general case of a temperature and quark mass dependence is 1= —=Tc3 2 U = −mU kO Tc +— (8.33) f(x) −1=2 −1

as a function of T=Tc = 1 + x(—=Tc3 f(x))1= . The curves shown in Fig. 37 correspond to quark masses m = 0; m = mphys =10; m = mphys and m = 3:5mphys or, equivalently, to zero temperature pion masses m" = 0; m" = 45 MeV; m" = 135 MeV and m" = 230 MeV, respectively. We see perfect agreement of

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Fig. 37. Chiral equation of state for a phase transition or crossover in two-"avor QCD. The plot shows the chiral condensate U as a function of temperature T . Lines (a) – (d) correspond at zero temperature to m" =0; 45; 135; 230 MeV, respectively. For each pair of curves the lower one represents the full T -dependence of U , whereas the upper one shows for comparison the universal scaling form of the equation of state for the O(4) Heisenberg model (cf. Fig. 6). The critical temperature for zero quark mass is Tc = 100:7 MeV. The chiral condensate is normalized at a scale kO 620 MeV. Table 10 Critical and “pseudocritical” temperature for various values of the zero-temperature pion mass. Here Tpc is de$ned as the in"ection point of U (T ) m" =MeV Tpc =MeV

0 100:7

45 110

135 130

230 150

both curves in the chiral limit for T suIciently close to Tc which is a manifestation of universality and the phenomenon of dimensional reduction. In particular, we reproduce the critical exponents of the O(4)-model given in Table 3 of Section 4.3. Away from the chiral limit we $nd for a realistic pion mass that the O(4) universal equation of state provides a reasonable approximation for U in the crossover region T = (1:2 − 1:5)Tc . In order to facilitate comparison with lattice simulations which are typically performed for larger values of m" we also present results for m" (T =0)=230 MeV in curve (d). One may de$ne a “pseudocritical temperature” Tpc associated to the smooth crossover as the in"ection point of U (T ). Our results for Tpc are presented in Table 10 for the four diEerent values of m or, equivalently, m" (T = 0). The value for the pseudocritical temperature for m" = 230 MeV compares well with the lattice results for two "avor QCD. This may be taken as an indication that the linear quark meson model gives a reasonable picture of the chiral properties in two-"avor QCD. An extension of the

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Fig. 38. Temperature dependence of the pion mass: The plot shows m" as a function of temperature T for three diEerent values of the average light current quark mass m. The solid line corresponds to the realistic value m = mphys , whereas the dotted line represents the situation without explicit chiral symmetry breaking, i.e., m = 0. The intermediate, dashed line assumes m = mphys =10.

truncation for the linear quark meson model may lead to corrections in the value of Tc , but we do not expect qualitative changes of the overall picture. One should mention, though, that a determination of Tpc according to this de$nition is subject to sizeable numerical uncertainties for large pion masses as the curve in Fig. 37 is almost linear around the in"ection point for quite a large temperature range. A problematic point in lattice simulations is the extrapolation to realistic values of m" or even to the chiral limit. Our results may serve here as an analytic guide. The overall picture shows the approximate validity of the O(4) scaling behavior over a large temperature interval in the vicinity of and above Tc once the (non-universal) amplitudes are properly computed. We point out that the link between the universal behavior near Tc and zero current quark mass on the one hand, and the known physical properties at T = 0 for realistic quark masses on the other hand, is crucial to obtain all non-universal information near Tc . A second important result is the temperature dependence of the space-like pion correlation length 1 m− " (T ). (We will often call m" (T ) the temperature dependent pion mass since it coincides with the physical pion mass for T = 0.) Fig. 38 shows m" (T ) and one again observes the second-order phase transition in the chiral limit m = 0. For T ¡ Tc the pions are massless Goldstone bosons whereas for T ¿ Tc they form with the sigma a degenerate vector of O(4) with mass increasing as a function of temperature. For m=0 the behavior for small positive T −Tc is characterized by the critical exponent >, i.e. m" (T ) = (%+ )−1 Tc ((T − Tc )=Tc )> and we obtain > = 0:787; %+ = 0:270. For m ¿ 0 we $nd that m" (T ) remains almost constant for T . Tc with only a very slight dip for T near Tc =2. For T ¿ Tc

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the correlation length decreases rapidly and for T Tc the precise value of m becomes irrelevant. We see that the universal critical behavior near Tc is quite smoothly connected to T = 0. The full functional dependence of m" (T; m) allows us to compute the overall size of the pion correlation length near the critical temperature and we $nd m" (Tpc ) 1:7m" (0) for the realistic value mphys . This correlation length is even smaller than the vacuum (T = 0) one and gives no indication for strong "uctuations of pions with long wavelength. 58 We will discuss the possibility of a tricritical point with a massless excitation in the two-"avor case at non-zero baryon number density in Section 8.6. We also point out that the present investigation for the two "avor case does not take into account a possible “eEective restoration” of the axial UA (1) symmetry at high temperature [24,225]. 8.5. Renormalization ;ow at non-zero chemical potential At non-zero temperature and chemical potential associated to the conserved quark number we consider the following ansatz for k : 1=T 5 1 − -5 † b i a a 1+Oab − (O )a b d x0 d 3 x i U i (- 9 + -0 ) ai + hUk U i k = 2 2 0 ∗ ab + ZO; k 9 Oab 9 O + Uk (#; ; T ) : (8.34) a

A non-zero to lowest order results in the term ∼ i U -0 a appearing in the r.h.s. of (8.34). We neglect here the running of the fermionic wave function renormalization constant and the dependence of ZO; k and hUk on and T . The temperature dependence of ZO; k and hUk , which has been taken into account for the results presented in Section 8.4, is indeed small [28]. We also neglect a possible diEerence in normalization of the quark kinetic term and the baryon number current. There is a substantial caveat concerning the approximation (8.34) at non-zero density. At suf$ciently high-density diquark condensates form, opening up a gap at the quark Fermi surfaces [226,227]. In order to describe this phenomenon in the present framework, the above ansatz for k has to be extended to include diquark degrees of freedom. However, a non-zero diquark condensate aEects the equation of state for the chiral condensate [29] only marginally. In particular, in the two "avor case the inclusion of diquark degrees of freedom hardly changes the behavior of the chiral condensate at non-zero density and the order of the transition to the chirally symmetric phase. For NJL-type models diquark condensation is suppressed at low density by the presence of the chiral condensate [29,228]. We therefore expect ansatz (8.34) to give a good description of the restoration of chiral symmetry within the present model. It is straightforward to generalize our method to include diquark degrees of freedom. For simplicity we concentrate here on the chiral properties and neglect the superconducting properties at high densities. We employ the same exponential infrared cutoE function for the bosonic $elds RkB (2.17) as in the previous section at non-zero temperature. At non-zero density a mass-like fermionic infrared cutoE simpli$es the computations considerably compared to an exponential cutoE like (7.22) because of the trivial momentum dependence. In the presence of a chemical potential we use RkF = −(- q − i-0 )rkF : 58

(8.35)

For a QCD phase transition far from equilibrium long-wavelength modes of the pion $eld can be ampli$ed [26,27].

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The eEective squared inverse fermionic propagator is then of the form PkF = [(q0 − i)2 + ˜q2 ](1 + rkF )2 = (q0 − i)2 + ˜q2 + k 2 E(kO2 − (q0 − i)2 − ˜q2 ) ;

(8.36)

where the second line de$nes rkF and one observes that the fermionic infrared cutoE acts as an additional mass-like term ∼ k 2 . We compute the "ow equation for the eEective potential Uk from Eq. (8.17) using ansatz (8.34) for k . The only explicit dependence on the chemical potential appears in the fermionic contribution to the "ow equation for Uk , whereas the derivation of the bosonic part strictly follows Section 8.3. It is instructive to consider the fermionic part of the "ow equation in more detail and to perform the summation of the Matsubara modes explicitly for the fermionic part. Since the "ow equations only involve one momentum integration, standard techniques for one-loop expressions apply [206] and we $nd 9 UkF (#; T; ) 9k ∞ 4 ∞ 3 d q kE(kO2 − q2 ) d ˜q k = − 8Nc + 4Nc 2 4 3 2 2 2 ˜q + k 2 + h2k #=2 −∞ (2") q + k + hk #=2 −∞ (2") 1 1 + : × exp[( ˜q2 + k 2 + h2k #=2 − )=T ] + 1 exp[( ˜q2 + k 2 + h2k #=2 + )=T ] + 1 (8.37) For simplicity, we sent here kO → ∞ in the ; T -dependent second integral. This is justi$ed by the fact that in the ; T -dependent part the high momentum modes are exponentially suppressed. For comparison, we note that within the present approach one obtains standard mean $eld theory results for the free energy if the meson "uctuations are neglected, 9UkB =9k ≡ 0, and the Yukawa coupling is kept constant, hk =h in (8.37). The remaining "ow equation for the fermionic contribution could be integrated then easily with the (mean $eld) initial condition UkO (#)= mU 2kO #. In the following we will concentrate on the case of vanishing temperature. We $nd (see below) that a mean $eld treatment yields relatively good estimates only for the -dependent part of the free energy U (#; ) − U (#; 0). On the other hand, mean $eld theory does not give a very reliable description of the vacuum properties encoded in U (#; 0). The latter are important for a determination of the order of the phase transition at = 0. In the limit of vanishing temperature one expects and observes a non-analytic behavior of the -dependent integrand of the fermionic contribution (8.37) to the "ow equation for Uk because of the formation of Fermi surfaces. Indeed, the explicit -dependence of the "ow equation reduces to a step function ∞ 4 9 d q kE(kO2 − q2 ) UkF (#; ) = −8Nc 2 4 2 2 9k −∞ (2") q + k + hk #=2 ∞ 3 & d ˜q k E( − ˜q 2 + k 2 + h2k #=2) : (8.38) + 4Nc 3 ˜q 2 + k 2 + h2k #=2 −∞ (2")

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The quark chemical potential enters the bosonic part of the "ow equation only implicitly through the meson mass terms Uk (#; ) and Uk (#; ) + 2#Uk (#; ) for the pions and the A-meson, respectively. For scales k ¿ the E-function in (8.38) vanishes identically and there is no distinction between the vacuum evolution and the = 0 evolution. This is due to the fact that our infrared cutoE adds to the eEective quark mass (k 2 + h2k #=2)1=2 . For a chemical potential smaller than this eEective mass the “density” −9Uk =9 vanishes whereas for larger one can view = [˜qF2 (; k; #) + k 2 + h2k #=2]1=2 as an eEective Fermi energy for given k and #. A small infrared cutoE k removes the "uctuations with momenta in a shell close to the physical Fermi surface 59 2 − h2k #=2 − k 2 ¡ q2 ¡ 2 − h2k=0 #=2. Our "ow equation realizes the general idea [229] that for = 0 the lowering of the infrared cutoE k → 0 should correspond to an approach to the physical Fermi surface. For a computation of the meson eEective potential the approach to the Fermi surface in (8.38) proceeds from below and for large k the eEects of the Fermi surface are absent. By lowering k one “$lls the Fermi sea”. As discussed in Section 8.3 the observed $xed point behavior in the symmetric regime allows us to $x the model by only two phenomenological input parameters and we use f" = 92:4 MeV and 300 MeV . Mq . 350 MeV. The results for the evolution in vacuum [169,28] show that for scales not much smaller than kO 600 MeV chiral symmetry remains unbroken. This holds down to a scale of about kSB 400 MeV at which the meson potential Uk (#) develops a minimum at #0; k ¿ 0 even for a vanishing source, thus breaking chiral symmetry spontaneously. Below the chiral symmetry breaking scale the running couplings are no longer governed by the partial $xed point. In particular, for k . kSB the Yukawa coupling hk and the meson wave function renormalization ZO; k depend only weakly on k and approach their infrared values. At = 0 we will follow the evolution from k = kSB to k = 0 and neglect the k-dependence of hk and ZO; k in this range. According to the above discussion the initial value UkSB is -independent for ¡ kSB . We solve the "ow equation for Uk numerically as a partial diEerential equation for the potential depending on the two variables # and k for given [215]. Nonzero current quark masses result in a pion mass threshold and eEectively stop the renormalization group "ow of renormalized couplings at a scale around m" . In the fermionic part (8.38) of the "ow equation the vacuum and the -dependent term contribute with opposite signs. This cancellation of quark "uctuations with momenta below the Fermi surface is crucial for the restoration of chiral symmetry at high density. 60 In vacuum, spontaneous chiral symmetry breaking is induced in our model by quark "uctuations which drive the scalar mass term Uk (#=0) from positive to negative values at the scale k =kSB . (Meson "uctuations have the tendency to restore chiral symmetry because of the opposite relative sign, cf. (8.17).) As the chemical potential becomes larger than the eEective mass (k 2 + h2k #=2)1=2 quark "uctuations with momenta smaller than ˜qF2 (; k; #) = 2 − k 2 − h2k #=2 are suppressed. Since ˜qF2 is monotonically decreasing with # for given and k the origin of the eEective potential is particularly aEected. We will see in the next section that 59

If one neglects the mesonic "uctuations one can perform the k-integration of the "ow equation (8.38) in the limit of a k-independent Yukawa coupling. One recovers (for kO2 k 2 + h2 #=2; 2 ) mean $eld theory results except for a shift in the mass, h2 #=2 → h2 #=2 + k 2 , and the fact that modes within a shell of three-momenta 2 − h2 #=2 − k 2 6˜q2 6 2 − h2 #=2 are not yet included. Because of the mass shift the cutoE k also suppresses the modes with q2 ¡ k 2 . For k ¿ 0 no infrared singularities appear in the computation of Uk and its #-derivatives. 60 The renormalization group investigation of a linear sigma model in 4 − j dimensions misses this property [230].

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Fig. 39. High-density chiral phase transition in the NJL model. The zero-temperature eEective potential U (in MeV4 ) as a function of A = (ZO; k=0 #=2)1=2 (in MeV) is shown for diEerent chemical potentials. One observes two degenerate minima for a critical chemical potential c =Mq = 1:025 corresponding to a $rst-order phase transition at which two phases have equal pressure and can coexist (Mq = 316:2 MeV).

for large enough this leads to a second minimum of Uk=0 (#; ) at # = 0 and a chiral symmetry restoring $rst-order transition. 8.6. High-density chiral phase transition In vacuum or at zero density the eEective potential U as a function of A = ZO; k=0 #=2 has its minimum at a non-vanishing value f" =2 corresponding to spontaneously broken chiral symmetry. As the quark chemical potential increases, U can develop diEerent local minima. The lowest minimum corresponds to the state of lowest free energy and is favored. In Fig. 39 we plot the free energy as a function of A for diEerent values of the chemical potential = 322:6; 324:0 and 325:2 MeV. Here the eEective constituent quark mass is Mq = 316:2 MeV. We observe that for ¡ Mq the potential at its minimum does not change with . Since 9U (8.39) nq = − 9 |min we conclude that the corresponding phase has zero density. In contrast, for a chemical potential larger than Mq we $nd a low-density phase where chiral symmetry is still broken. The quark number density as a function of is shown in Fig. 40. One observes clearly the non-analytic behavior at = Mq which denotes the “onset” value for nonzero density. From Fig. 39 one also notices the appearance of an additional local minimum at the origin of U . As the pressure p = −U increases in the low-density phase with increasing , a critical value c is reached at which there are two degenerate potential minima. Before can increase any further the system undergoes a $rst-order phase transition at which two phases have equal pressure and can coexist. In the high-density phase

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Fig. 40. Density discontinuity in the NJL model at T = 0. The plot shows n1=3 q , where nq denotes the quark number density as a function of in units of the eEective constituent quark mass (Mq = 316:2 MeV).

chiral symmetry is restored as can be seen from the vanishing order parameter for ¿ c . We note that the relevant scale for the $rst-order transition is Mq . For this reason we have scaled our results for dimensionful quantities in units of Mq . For the class of quark meson models considered here (with Mq =f" in a realistic range around 3–4) the $rst-order nature of the high-density transition has been established clearly. In particular, these models comprise the corresponding Nambu–Jona–Lasinio models where the eEective fermion interaction has been eliminated through the introduction of auxiliary bosonic $elds. In summary, we $nd that the linear quark meson model exhibits in the chiral limit a high-temperature second-order chiral transition at zero chemical potential (cf. Section 8.4) and a $rst-order high-density chiral transition at zero temperature. By continuity these transitions meet at a tricritical point in the (; T )-plane. Away from the chiral limit, the second-order chiral transition turns into a smooth crossover. The $rst-order line of transitions at low temperatures now terminates in a critical endpoint in the Ising universality class with long-range correlations [29,30]. It is an interesting question to what extent the phase diagram for the NJL-type models re"ects features of two-"avor QCD. The prominent property of QCD which is missing in the NJL-models is con$nement which binds the quarks to color-neutral baryons. As a consequence one should use for QCD baryons instead of quarks as the relevant fermionic degrees of freedom for low momenta or low k. For the high-temperature behavior at zero density the eEects of con$nement are presumably not too important. The reason is the eEective decoupling of the quarks for k . 300 MeV because of their constituent mass. This decoupling would only be enhanced by the binding to nucleons and

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we do not expect qualitative changes. 61 The situation is diEerent for the high-density behavior at zero temperature [96]. The mass of the fermions has an important in"uence on their Fermi surface. The high-density transition in the NJL-type models describes in the chiral limit a transition from a “constituent quark liquid” at low density to a “massless quark liquid” at high-density. Especially, the low-density constituent quark liquid has no direct correspondence in QCD where one rather encounters a gas of nucleons and a liquid of nuclear matter. Acknowledgements The work of J.B. is supported in part by funds provided by the US Department of Energy (DOE) under cooperative research agreement # DE-FC02-94ER40818. The work of N.T. is supported by the EC under Contract Nos. ERBFMRXCT960090 and ERBFMBICT983132. The work of C.W. is supported in part by the Deutsche Forschungsgemeinschaft and by the EC contract ERBFMRX-CT97-0122. Appendix A. Threshold functions In this appendix we list the various de$nitions of threshold functions appearing in the "ow equations and the expressions for the anomalous dimensions. They involve the inverse scalar average propagator for the IR cutoE (2.17) −1 P(q) = q2 + Z; k Rk (q) =

q2 1 − exp{−q2 =k 2 }

(A.1)

and the corresponding fermionic function PF which can be chosen as (cf. Section 7.2) PF (q) = P(q) ≡ q2 (1 + rF (q))2 :

(A.2)

We abbreviate x = q2 ;

9 ˙ P(x) ≡ P(x); 9x

P(x) ≡ P(q);

9 9˜t P˙ ≡ 9˜t P ; 9x

(A.3)

etc., and de$ne 2 PF 9[Z ; k rF ] 9 1 9Rk 9 9˜t ≡ + : Z; k 9t 9P Z ; k 1 + rF 9t 9PF

(A.4)

The bosonic threshold functions read d

ldn (w; 6 ) = ldn (w) − 6 lˆn (w) n + n; 0 2n−d ∞ 1 9Rk d=2−1 k (P + wk 2 )−(n+1) ; = dx x 2 Z; k 9t 0 61

There may be some quantitative in"uence of this eEect on the value of Tc , however.

(A.5)

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d ldn1 ;n2 (w1 ; w2 ; 6 ) = ldn1 ;n2 (w1 ; w2 ) − 6 lˆn1 ;n2 (w1 ; w2 ) ∞ 1 = − k 2(n1 +n2 )−d d x xd=2−1 9˜t {(P + w1 k 2 )−n1 (P + w2 k 2 )−n2 } ; 2 0

where n; n1 ; n2 ¿ 0 is assumed. For n = 0 the functions ldn may also be written as ∞ 1 ldn (w; 6 ) = − k 2n−d d x xd=2−1 9˜t (P + wk 2 )−n : 2 0 (F)d The fermionic integrals ln(F)d (w; 6 ) = ln(F)d (w) − 6 lan (w) are de$ned analogously as ∞ 1 PF 9[Z ; k rF ] (F)d 2n−d ln (w; 6 ) = (n + n; 0 )k d x xd=2−1 (P + wk 2 )−(n+1) : Z 1 + r 9t ; k F 0

(A.6)

(A.7)

Furthermore, one has (FB)d a (FB)d ˆ(FB)d l(FB)d n1 ;n2 (w1 ; w2 ; 6 ; 6 ) = ln1 ;n2 (w1 ; w2 ) − 6 ln1 ;n2 (w1 ; w2 ) − 6 ln1 ;n2 (w1 ; w2 ) 1 2(n1 +n2 )−d ∞ 1 d=2−1 ˜ =− k 9t dx x 2 [PF (x) + k 2 w1 ]n1 [P(x) + k 2 w2 ]n2 0

mdn1 ;n2 (w1 ; w2 ; 6 ) ≡ mdn1 ;n2 (w1 ; w2 ) − 6 mdn1 ;n2 (w1 ; w2 ) ∞ ˙ ˙ 1 P(x) P(x) = − k 2(n1 +n2 −1)−d d x xd=2 9˜t 2 [P(x) + k 2 w1 ]n1 [P(x) + k 2 w2 ]n2 0 m2(F)d (w; 6 ) = m2(F)d (w) − 6 ma 2(F)d (w) 2 P˙ F (x) 1 6− d ∞ d=2 ˜ d x x 9t =− k 2 [PF (x) + k 2 w]2 0 m4(F)d (w; 6 ) = m4(F)d (w) − 6 ma 4(F)d (w) ∞ 9 1 + rF (x) 2 1 = − k 4− d d x xd=2+1 9˜t 2 9x PF (x) + k 2 w 0 (FB)d m(FB)d a n(FB)d (w1 ; w2 ) − 6 m(FB)d n1 ;n2 (w1 ; w2 ; 6 ; 6 ) = mn1 ;n2 (w1 ; w2 ) − 6 m n1 ;n2 (w1 ; w2 ) 1 ;n2 ∞ ˙ 1 1 + rF (x) P(x) = − k 2(n1 +n2 −1)−d : d x xd=2 9˜t 2 [PF (x) + k 2 w1 ]n1 [P(x) + k 2 w2 ]n2 0 (A.8)

The dependence of the threshold functions on the anomalous dimensions arises from the t-derivative acting on Z; k and Z ; k within Rk and Z ; k rF , respectively. We furthermore use the abbreviations ldn (6 ) ≡ ldn (0; 6 ); ldn (w) ≡ ldn (w; 0);

ln(F)d (6 ) ≡ ln(F)d (0; 6 ) ; ldn ≡ ldn (0; 0) ;

(A.9)

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etc., and note that in four dimensions the integrals (F)4 l42 (0; 0) = l2(F)4 (0; 0) = l(FB)4 (0) = m(FB)4 1; 1 (0; 0) = m4 1; 2 (0; 0) = 1

(A.10)

are independent of the particular choice of the infrared cutoE. Appendix B. Anomalous dimension in the sharp cuto1 limit It is instructive to evaluate %k as de$ned by Eq. (3.73) in the sharp cutoE limit. In this limit one has M0−1 (#; q2 ) = (Zk (#; q2 )q2 + Uk (#))−1 E(q2 − k 2 )

(B.1)

9t Rk (p) 2 = (z(#) + u (#))−1 (p2 − k 2 ) M02 (#; p2 ) Zk

(B.2)

and

with similar expressions for M1−1 and 9t Rk =M12 . The momentum integration in 9t G −1 (3.72) reduces to an angular integration for the angle between p and q; (pq) = |p| |q| cos K. For an evaluation of 9t G −1 (#; k 2 ) one has q2 = p2 = k 2 and M0−1 (#; (p + q)2 ) = [2k 2 Zk (#; 2k 2 (1 + cos K)) (1 + cos K) + Uk ]−1 E(1 + 2 cos K) : 2

2

2

(B.3)

2

Let us de$ne for p = q = k ; (pq) = k cos K (1) ˜k (#; s) = Zk−2 k d−4 k(1) (#; p; q);

(2) ˜k (#; s) = Zk−2 k d−4 k(2) (#; q; −q; p) ;

(3) ˜k (#; s) = Zk−2 k d−4 k(2) (#; q; p; −q);

−3 2d−6 (2) -˜(2) -k (#; p; −p; q)) ; k (#; s) = Zk k

(B.4)

where we have introduced the variable s = 2(1 + cos K) :

(B.5)

With (1)

(1) (#; −q − p; q) = Zk2 k 4−d ˜ (#; 2 − one $nds

62

9t G

−1

√

s) ;

(B.6)

for d = 3

4 1 (1) 2 (#; k ) = vd k Zk 4#˜ ds{(˜ (#; ˜ s))2 (z + #˜y˜ + u + 2#u ˜ )−1 2 1 √ (1) [s(z + Tz(#; ˜ s)) + u ]−1 + (˜ (#; ˜ 2 − s))2 (z + u )−1 2

[s(z + #˜y˜ + Tz(#; ˜ s) + #T ˜ y( ˜ #; ˜ s)) + u + 2#u ˜ ]−1 } 4 ) (2) (3) ds (z + u )−1 ((N − 1)˜ (#; ˜ s) + 2˜ (#; ˜ s)) − 0

*

(2) ˜ ) (˜ (#; ˜ s) + 2#˜-˜(2) (#; ˜ s)) + (z + #˜y˜ + u + 2#u

62

−1

For d = 3 the integration measure contains an additional Jacobian J (d) (s).

:

(B.7)

J. Berges et al. / Physics Reports 363 (2002) 223 – 386

377

Next, one uses relations (3.66) s 1 s (1) (1) z (#) + y( ˜ #) ˜ + T˜ (#; ˜ s) = u (#) ˜ + ˜ + Tz #; ˜ s) ; ˜ ˜ (#; 2 2 2 (2) (2) ˜ s) = u (#) ˜ + 2z (#) ˜ + T˜ (#; ˜ s) ; ˜ (#; (3) (3) ˜ s) = u (#) ˜ + (2 − s)(z (#) ˜ + Tz (#; ˜ 2 − s)) + 12 s(y( ˜ #) ˜ + Ty( ˜ #; ˜ s)) + T˜ (#; ˜ s) ; ˜ (#;

-˜ (2) (#; ˜ s) = u (#) ˜ + z (#) ˜ + 12 y˜ (#) ˜ + T-˜(2) (#; ˜ s) ;

(B.8)

where

1 4 (1) (2; 3) (2; 3) ˜ ˜ ˜ 0) = 0; T (#) ˜ = ds T˜ (#; ˜ s) ; T (#; 4 0 1 4 ˜ = ds T-˜(2) (#; ˜ s) ; T-˜ (2) (#) 4 0 and $nally $nds the exact expression 2 4 1 s (1) s ˜ z + Tz + y˜ + T (s) ˜ 1) = −2vd #˜ ds u + %k (#; 2 2 2 1 √ s −1 −1 ×[s(z + Tz(s)) + u ] [z + #˜y˜ + u + 2#u ˜ ] + u + 1− 2 2 √ √ 1 (1) × (z + Tz (2 − s)) + y˜ + T˜ (2 − s) [s(z + #˜y˜ + Tz(s) 2

(B.9)

˜ ]−1 [z + u ]−1 + 2vd {(z + u )−1 [2u + (N + 3 − 2s)z + #T ˜ y(s)) ˜ + u + 2#u (2) (3) +2(2 − s)Tz (2 − s) + sy˜ + sTy(s) ˜ + (N − 1)T˜ + 2T˜ ]

˜ )−1 [2u − z − 2#z ˜ + y˜ − T˜ − (z + #˜y˜ + u + 2#u

(2)

− 2#T ˜ -˜(2) ]

:

(B.10)

The $rst order in the hybrid derivative expansion neglects the momentum-dependent corrections (i) ˜ This yields the evolution equation for z(#) ˜ T˜ ; T-˜ (2) ; Tz and Ty. 2 1 ˜ + 2vd #˜ u + Az z + y˜ 9t z = 6z + (d − 2 + 6)#z 2 4z + u 1 ln (z + #˜y˜ + u + 2#u × ˜ )−1 z z + u 1 ˜ 4(z + #˜y) ˜ + u + 2#u −1 (z + u + ) ln z + #˜y˜ z + #˜y˜ + u + 2#u ˜ 1 −1 2u + (N + 2)z + y˜ − 2vd (z + u ) 2 −1 ˜ ) [2u − z − 2#z ˜ + y] ˜ ; (B.11) − (z + #˜y˜ + u + 2#u

378

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where we have replaced for simplicity in the third term the correct s-integration of the terms ∼ z (z )2 by an approximate expression with Az ≈ 0:5–1. The anomalous dimension reads for < ¿ 0

2 1 6 = (1 +

where we have de$ned = u (<);

z0 = z (<);

z0 = z (<);

y˜ 0 = y(<) ˜ :

(B.13)

˜ remain We observe that Eq. (B.11) is well de$ned as long as z; z + u ; z + #˜y˜ and z + #˜y˜ + u + 2#u all positive. These are the same conditions as those required for a consistent "ow of the potential. The problems that a sharp cutoE engenders for a de$nition of the anomalous dimension at q2 = 0 are avoided by the use of the hybrid derivative expansion with de$nition (3.74). For large N the characteristic scaling #˜ ∼ N; u ∼ 1; u ∼ 1=N implies for the solution of Eq. (B.11) z ∼ 1=N 2 ; z ∼ a=N 3 . On the other hand the evolution equation for the inverse radial −1 propagator G˜ (#; q2 ) = M1 (#; q2 ) − Rk (q) leads to y˜ ∼ 1=N and y˜ contributes in order 6 ∼ 1=N . Nevertheless, taking y˜ = 0 for a $rst discussion, one $nds the anomalous dimension 2

6 = 2vd <

4 ln 4 4 + 2< + 2vd Nz0 −
(B.14)

For d = 3 one may insert for the scaling solution the leading expression < = 1 (cf. (3.59)) so that 1 6= N

0:178 4 1 − ln 4 − ln 2 = : 3 3 N

(B.15)

For a computation of the exact expression for 6 in order 1=N one needs to include eEects from y˜ 0 = 0. Also the contribution −2(9Tz(<; y)=9y)(y = 1) has to be added in order 1=N . In the same approximation as above the scaling solution for z(#) ˜ obeys the diEerential equation 4 + u 4 + u + 2#u ˜ −1 #z ˜ + 2vd #(u + (1 + u ) ln ˜ ) (1 + u + 2#u ˜ ) ln 1 + u 1 + u + 2#u ˜ 2u + Nz 2u = −6z : − − 2vd 1 + u 1 + u + 2#u ˜

2

−1

(B.16)

J. Berges et al. / Physics Reports 363 (2002) 223 – 386

379

By diEerentiation with respect to #, ˜ Eq. (B.16) implies for the scaling regime z0 (1 + 2vd N) + z0 (< − 2vd N ) 4 + 2< 2 ( + 2-< − <( − 2-<)) ln 4 + ( + 2-< − <) ln = − 2vd (1 + 2<)2 1 + 2< 15 3 + 2-< 2(3 + 2-<) 1 22 2 + 2-< − + 2 − 2- + − − < : 1 + 2< 4 4 + 2< 1 + 2< (1 + 2<)2 (B.17) For d = 3; 2vd N = <; < = 1; -< = 2=3, this yields N
89 216

−

4 27

ln 4 − 23 ln 2

(B.18)

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Physics Reports 363 (2002) 387 – 424 www.elsevier.com/locate/physrep

Combinatorics of (perturbative) quantum $eld theory D. Kreimera; b;∗;1 a

Center for Mathematical Physics, Boston University, Boston, MA 02215, USA b IHES, F-91440 Bures-sur-Yvette, France Received September 2001; editor: I: Procaccia

Contents 1. Introduction 2. The Hopf algebra structure: trees and graphs 2.1. Basic considerations 2.2. Sector decomposition and rooted trees 2.3. The Hopf algebra of undecorated rooted trees 2.4. The Hopf algebra of graphs 3. Rescalings and renormalization schemes 3.1. Chen’s Lemma 3.2. Automorphisms of the Hopf algebra 4. The insertion operad of Feynman graphs 5. The Lie algebra structure 6. The Birkho9 decomposition and the renormalization group

388 388 388 391 394 400 403 403 403 405 408 410

6.1. Minimal subtraction: the Birkho9 decomposition 6.2. The -function 6.3. An example 7. Conclusions and outlook 7.1. Numbers and Feynman diagrams 7.2. Gauge symmetries 7.3. The exact renormalization group and the non-perturbative regime 7.4. Further aspects 7.5. Conclusions Acknowledgements References

410 411 415 417 417 419 420 420 421 421 422

Abstract We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann–Hilbert problem. Finally, we c 2002 Elsevier outline how these structures relate to the numbers which we see in Feynman diagrams. Science B.V. All rights reserved. PACS: 11.10.Gh; 11.10.Hi; 11.15.Bt

hep-th=0010059, MZ-TH=00-42. ∗ Corresponding author. Tel.: +49-6131-392358; fax: +49-6131-394611. E-mail address: [email protected] (D. Kreimer). 1 Heisenberg Fellow at Mainz Univ., D-55099 Mainz, Germany. c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 9 9 - 0

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1. Introduction Renormalization (see [1] for a classical textbook treatment) has been settled as a self-consistent approach to the treatment of short-distance singularities in the perturbative expansion of quantum $eld theories thanks to the work of Bogoliubov, Parasuik, Hepp, Zimmermann, and followers. Nevertheless, its intricate combinatorics went unrecognized for a long time. In this review, we want to describe the results in a recent series of papers devoted to the Hopf algebra structure of quantum $eld theory (QFT) [2–16]. These results were obtained during the last three years, starting from $rst papers on the subject [2– 4] and Lourishing in intense collaborations with Connes [5 –9] and Broadhurst [10 –12]. We will review the results obtained so far in a fairly informative style, emphasizing the underlying ideas and concepts. Technical details and mathematical rigor can be found in the above-cited papers, while it is our present purpose to familiarize the reader with the key ideas. Furthermore, we intend to spell out lines for further investigation, as it more and more becomes clear that this Hopf algebra structure provides a very $ne tool for a better understanding of a correct mathematical formulation of quantum $eld theory (QFT) as well as for applications in particle and statistical physics. Nevertheless, we will use one concept for the $rst time in this paper: we will introduce an operad of Feynman graphs, as it is underlying many of the operations involved in the Hopf and Lie algebras built on Feynman graphs. 2. The Hopf algebra structure: trees and graphs Let us start right away with the consideration of how rooted trees and Feynman graphs are connected in perturbative QFT. 2.1. Basic considerations There are two basic operations on Feynman graphs which govern their combinatorial structure as well as the process of renormalization. The question to what extent they also determine analytic properties of Feynman graphs is one of these future lines of investigations, with $rst results in [13,14]. We will comment in some detail on this aspect later on. These two basic operations are the disentanglement of a graph into subgraphs, and the opposite operation, plugging a subgraph into another one. Let us consider the disentanglement of a graph $rst. We consider the following three-loop vertex-correction = We regard it as a contribution to the perturbative expansion of 3 theory in six spacetime dimensions, where this theory is renormalizable. 2 contains one interesting subgraph, the one-loop 2

External lines are amputated, but still drawn, in a convenient abuse of notation. In the massless case considered here, no further notation is needed for insertions into propagators. In the general case (massive theories, spin), the external structures de$ned in [5] are a suNcient tool.

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389

self-energy graph :

=

We are interested in it because it is the only subgraph which provides a divergence, and the whole UV-singular structure comes from this subdivergence and from the overall divergence of itself. Let 0 := = = be the graph where we shrink to a point. From the analytic expressions corresponding to , to 0 and to , we can form the analytic expression corresponding to the renormalization of the graph . It is given by − R() − R()0 + R(R()0 ) ;

(1)

where we still abuse, in these introductory remarks, notation by using the same symbol for the graph and the analytic expression corresponding to it. We do so as we want to emphasize for the moment that the crucial step in obtaining this expression is the use of the graph and its disentangled pieces, and 0 = =. The analytic expressions will come as characters on these Hopf algebra elements, and we will discuss these characters in detail below. Diagrammatically, the above expression reads − R(

) − R(

)

+ R(R(

)

):

The unavoidable arbitrariness in the so-obtained expression lies in the choice of the map R which we suppose to be such that it does not modify the short-distance singularities (UV divergences) in the analytic expressions corresponding to the graphs. This then renders the above combination of four terms $nite. If there were no subgraphs, a simple subtraction − R() would suNce to eliminate the short-distance singularities, but the necessity to obtain local counterterms forces us to $rst subtract subdivergences, which is achieved by Bogoliubov’s famous RO operation [1], which delivers here: O → R() = − R()0 :

(2)

This provides two of the four terms above. Amongst them, these two are free of subdivergences and hence provide only a local overall divergence. The projection of these two terms into the range of R provides the other two terms, which combine to the counterterm Z = −R() + R(R()0 )

(3)

of , and subtracting them delivers the $nite result above by the fact that the UV divergences are not changed by the renormalization map R. 3 3

Locality is connected to the absence of subdivergences: if a graph has a sole overall divergence, UV singularities only appear when all loop momenta tend to in$nity jointly. Regarding the analytic expressions corresponding to a graph as a Taylor series in external parameters like masses or momenta, powercounting establishes that only the coeNcients of the $rst few polynomials in these parameters are UV singular. Hence, they can be subtracted by a counterterm polynomial in $elds and their derivatives. The argument fails as long as one has not eliminated all subdivergences: their presence can force each term in the Taylor series to be divergent.

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Fig. 1. A decorated rooted tree with two vertices, each decorated by a graph without subdivergences (assuming this is an example in 3 theory in six dimensions). The root (by our convention the uppermost vertex) is decorated by the graph 0 = 1 = = 2 = which we obtain when we shrink the subdivergence to a point in either 1 or 2 . The vertex decorated by the one-loop self-energy corresponds to this subdivergence, and the rooted tree stores the information that this divergence is nested in the other graph. The information at which place the subdivergence is to be inserted is not stored in this notation. The hierarchy which determines the recursive mechanism of renormalization is independent of this information. It can easily be restored allowing marked graphs as decorations, or one could directly formulate the Hopf algebra on graphs as we do below. Fig. 2. This graph has a hierarchy of divergences given by two disjoint subdivergences, the self-energy and a one-loop vertex-correction , ˜ so that its divergent structure represents the decorated rooted tree indicated. As a parenthesized word, the graph corresponds to (()() ˜ 0 ). There are, by the way, 5 × 6 = 30 graphs which are all equivalent in the sense that they represent this rooted tree or parenthesized word, generated by the 5 internal vertices and 6 internal edges which provide places for insertion in 0 .

The basic operation here is the disentanglement of the graph into pieces and =, and this very disentanglement gives rise to a Hopf algebra structure, as was $rst observed in [2]. This Hopf algebra has a role model: the Hopf algebra of rooted trees. We $rst want to get an idea about this universal Hopf algebra after which all the Hopf algebras of Feynman graphs are modelled. Consider the two graphs 1 =

;

2 =

They have one common property: both of them can be regarded as the graph 0 = 1 = = 2 = = into which the subgraph = is inserted, at two di9erent places though. But as far as their UV-divergent sectors go they both realize a rooted tree of the form given in Fig. 1, in the language of [2] both graphs 1 ; 2 correspond to a parenthesized word of the form ((

)

):

In [2] such graphs were considered to be equivalent, as the combinatorial process of renormalization produces exactly the same terms for both of them. We will formulate this equivalence in a later section using the language of operads.

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391

The combinatorics of renormalization is essentially governed by this bookkeeping process of the hierarchies of subdivergences, and this bookkeeping is what is delivered by rooted trees. They are just the appropriate tool to store the hierarchy of disjoint and nested subdivergences. Another example given in Fig. 2 might be better suited than any formalism to make this clear. At this stage, the reader should wonder what to make out of graphs which have overlapping divergences. This can be best understood when we turn to the other basic operation on graphs: plugging them into each other. On the one hand, for the non-overlapping graphs 1 ; 2 above there is a unique way to obtain them from 0 = 1 = = 2 = = and the self-energy . We plug into the vertex-correction at an appropriate internal line to obtain these graphs. This operation will be considered in some more detail in a later section. On the other hand, for a graph which contains overlapping divergences we have typically no unique manner, but several ways instead, how to obtain it. For example,

= can be obtained in the two ways indicated in Fig. 3. Each of these ways corresponds to a rooted tree [4], and the sum over all these rooted trees bookkeeps the subdivergent structures of a graph with overlapping divergences correctly. The resolution of overlapping divergences into rooted trees corresponds to the determination of Hepp sectors, and amounts to a resolution of overlapping subsets into nested and disjoint subsets generally [4]. 4 One remark is in order: the very fact that overlapping divergences can be reduced to divergences which have a tree-hierarchy has a deeper reason: the short-distance singularities of QFT result from confronting products of distributions which are well-de$ned on the con$guration space of vertices located at distinct space–time points, but which become ill-de$ned along diagonals [17,18]. But then, the various possible ways how an ensemble of distinct points can collapse to (sub-)diagonals is known to be strati$ed by rooted trees [19], and this is what essentially ensures that the Hopf algebra structure of these trees can reproduce the forest formulas of perturbative QFT. Let us then have a closer look at the connection between graphs and rooted trees. 2.2. Sector decomposition and rooted trees Consider the Feynman graph once more, as given in Fig. 4. It corresponds to a contribution to the perturbative expansion in the coupling constant g of the theory to order g4 . It has short-distance (UV) singularities which are apparent in the following sectors: I1 := {1; 2; 3};

I2 := {2; 3; 4};

I = {1; 2; 3; 4} ;

4 The remarks above are speci$c to theories which have trivalent couplings. In general, the determination of divergent sectors still leads to rooted trees [4]. A concrete example how the Hopf algebra structure appears in 4 -theory can be found in [20]. Also, resolving the overlapping divergences in terms of decorated rooted trees determines the appropriate set of primitive elements of the Hopf algebra, which can for example be systematically achieved by making use of Dyson–Schwinger equations [2,15], see also [21].

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Fig. 3. Finding the two ways of getting the overlapping graph . There are two vertices in the one-loop self-energy into which the one-loop vertex correction can be inserted. Both result in the same graph. The short-distance singularities in

arise from two sectors, described by two decorated rooted trees. Fig. 4. A di9erent way of looking at the graph . We label its vertices by 1; : : : ; 4. Then, the set of vertices {1; 2; 3} belongs to a vertex subgraph, as does the set {2; 3; 4}. The fact that both are proper subsets of the set of all vertices {1; 2; 3; 4} is again reLected in a tree-like hierarchy. A short-distance singularity appears when these labelled vertices of a divergent graph collapse to a point. This point (a diagonal) constitutes one vertex of a rooted tree, with the root corresponding to the collapse of all vertices to the overall diagonal jointly. Again, the graph gives rise to two rooted trees, which corresponds to the two divergent sectors along two di9erent diagonals. When we blow up the vertices {1; 2; 3; 4} of the graph to vertical lines, we can represent the edges of the Feynman graph as horizontal chords, and we regain the graph by shrinking the vertical lines to a point. The Hopf algebra structure operates on the bold black rooted trees, as they store the information which diagonals contain short-distance singularities in the graph under consideration.

which give the label of the vertices participating in the divergent (sub-)graphs. Note that the sectors overlap: I1 ∩ I2 = ∅. The singularities are strati$ed so that they can be represented as rooted trees, as described in Fig. 4. In this strati$cation of sectors, each node at the rooted tree corresponds to a Feynman graph which connects the vertices attached to the node by propagators in a manner such that it has no subdivergences. We call such graphs primitive graphs. Each primitive graph is only overall divergent. Now, where do singularities reside? Typically, if we write down analytic expressions in terms of momentum integrals, UV-divergences appear when the loop momenta involved in a primitive graph tend to in$nity jointly, and this can be detected by powercounting over edges and vertices in the graph. On the other hand, we can consider Feynman rules in coordinate space. Then, the UV-singular integrations over momenta become short-distance singularities. Again, they creep in from the very fact that closed loops, cycles in the graph, force the integration over the positions of vertices to produce ill-de$ned products of distributions with coinciding support. Powercounting amounts to a check of the scaling degree of the relevant distributions and ultimately determines the appearance of a short-distance singularity at the diagonal under consideration.

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The short-distance singularities of Feynman graphs then come solely from regions where all vertices are located at coinciding points. One has no problem to de$ne the Feynman integrand in the con$guration space of vertices at distinct locations, while a proper extension to diagonals is what is required. In the above, the two divergent subgraphs are ill-de$ned along the diagonals x1 = x2 = x3 and x2 = x3 = x4 while the overall divergence corresponds to the main diagonal x1 = x 2 = x3 = x4 . Due to the Hopf algebra structure of Feynman graphs, we can de$ne the renormalization of all such sectors without making recourse to any speci$c analytic properties of the expressions (Feynman integrals) representing those sectors. The only assumption we make is that in a suNciently small neighborhood of such an ultralocal region (the neighborhood of a diagonal) we can de$ne the scaling degree, the powercounting, in a sensible manner. Apart from this assumption our approach is purely combinatorial and in particular independent of the geometry of the underlying spacetime manifold. Fig. 4 also gives a $rst idea why the Hopf algebra of undecorated rooted trees is the universal object underlying the Hopf algebras of Feynman graphs. The essential combinatorics needed to obtain local counterterms will solely use cuts on these rooted trees which are drawn in bold black lines in the $gure, with no further operation on decorations. Di9erent theories just di9er by having di9erent types of chords and vertices, while to each chord and vertex in the $gure we assign the appropriate scaling degree, the weight with which they contribute to the powercounting. One further remark is in order: the existence of a purely combinatorial solution coincides with the result of Brunetti and Fredenhagen [22], who showed that the renormalization mechanism is indeed unchanged in the context of curved manifolds in a detailed local analysis using the Epstein– Glaser mechanism. To my mind, quite generally, the Hopf algebra can be used to make sense out of extensions of products of distributions to diagonals of con$guration spaces even before we decide by which class of (generalized) functions we want to realize these extensions. While consistency of the Hopf algebra approach to renormalization with the Epstein–Glaser formalism was settled once the Hopf algebra was directly formulated on graphs [4,8], it was also addressed at a notational level making use of con$guration space Feynman graphs in [20]. Still, one should regard the splitting of distributions itself as the $rst instance where a representation of the Hopf algebra is realized, so that properties like Lorentz covariance appear as properties of the representation alone, maintaining a proper separation of the combinatorics of the Bogoliubov recursion from the analytic properties of the functions de$ned over the con$guration space, enabling also a direct formulation on the level of time-ordered products instead of Feynman graphs. Once more, that the Hopf algebra structure coming in is one of the rooted trees should be no surprise: limits to diagonals in con$guration spaces are strati$ed by rooted trees [19], and it is the Hopf algebra structure of these rooted trees which describes the combinatorics of renormalization, as we will see. The Hopf algebra of rooted trees will be the role model for all the Hopf algebras of Feynman graphs for a speci$cally chosen QFT, a classifying space in technical terms (see Theorem 2, Section 3 in [5]), while each such chosen QFT probes the short-distance singularities according to its Feynman graphs. The resulting iterative procedure gives rise to the Hopf algebra of rooted trees which was $rst described, in the equivalent language of parenthesized words, in [2] and then in its $nal notation in [5]. It is now time to describe this Hopf algebra of rooted trees in some detail.

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Fig. 5. The action of B− on an undecorated rooted tree. Fig. 6. The action of B+ on a monomial of trees.

2.3. The Hopf algebra of undecorated rooted trees We follow Section II of [5] closely. A rooted tree t is a connected and simply connected set of oriented edges and vertices such that there is precisely one distinguished vertex which has no incoming edge. This vertex is called the root of t. Further, every edge connects two vertices and the fertility f(v) of a vertex v is the number of edges outgoing from v. The trees being simply connected, each vertex apart from the root has a single incoming edge (we could attach, if we like, an extra edge to the root as well, for a more common treatment). Each vertex in such a rooted tree corresponds to a divergent sector in a Feynman diagram. The rooted trees store the hierarchy of such sectors. We will always draw the root as the uppermost vertex in $gures, and agree that all edges are oriented away from the root. As in [5], we consider the (commutative) algebra of polynomials over Q in rooted trees, where the multiplication m(t; t ) of two rooted trees is their disjoint union, so we can draw them next to each other in arbitrary order, and the unit with respect to this multiplication is the empty set. 5 Note that for any rooted tree t with root r which has fertility f(r) = n ¿ 0, we have trees t1 ; : : : ; tn which are the trees attached to r. Let B− be the operator which removes the root r from a tree t, as in Fig. 5: B− : t → B− (t) = t1 t2 : : : tn :

(4)

We extend the action of B− to a product of rooted trees by a Leibniz rule, B− (XY ) = B− (X )Y + XB− (Y ). We also set B− (t1 ) = 1; B+ (1) = t1 , where t1 is the rooted tree corresponding to the root alone. Let B+ be the operation which maps a monomial of n rooted trees to a new rooted tree t which has a root r with fertility n which connects to the n roots of t1 ; : : : ; tn : B+ : t1 : : : tn → B+ (t1 : : : tn ) = t :

(5)

This is clearly the inverse to the action of B− on single rooted trees. One has B+ (B− (t)) = B− (B+ (t)) = t

(6)

for any rooted tree t. Fig. 6 gives an example. 5

We restrict ourselves to one-particle irreducible diagrams for the moment. Then, the disjoint union of trees corresponds to the disjoint union of graphs. One could also set up the Hopf algebra structure such that one-particle reducible graphs correspond to products of rooted trees [2].

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All the operations described here have a straightforward generalization to decorated rooted trees, in which case the operator B+ carries a further label to indicate the decoration of the root [5]. We will not use decorated rooted trees later, as we will directly formulate the Hopf algebras of speci$c QFTs on Feynman graphs. The Hopf algebra of undecorated rooted trees is the universal object [5] for all those Hopf algebras, and hence we describe it here in some detail. Note that while [B+ ; B− ](t) = 0 for any single rooted tree t, this commutator is non-vanishing on products of trees. Obviously, one always has id = B− B+ , while B+ B− acts trivially only on single rooted trees, not on their product. 6 We will introduce a Hopf algebra on our rooted trees by using the possibility to cut such trees in pieces. For the reader not familiar with Hopf algebras, let us mention a few very elementary facts $rst. An algebra A is essentially speci$ed by a binary operation m : A × A → A (the product) ful$lling the associativity m(m(a; b); c) = m(a; m(b; c)) so that to each two elements of the algebra we can associate a new element in the algebra, and by providing some number $eld K imbedded in the algebra via E : K → A, k → k1. In a coalgebra we do the opposite, we disentangle each algebra element: each element a is decomposed by the coproduct : A → A × A in a coassociative manner, ( × id)(a) = (id × )(a). Further, the unit 1 of the algebra, m(1; a) = m(a; 1) = a, is dualized to the counit eO in the coalgebra, (eO × id)(a) = (id × e)(a) O = a. If the two operations m; are compatible (the coproduct of a product is the product of the coproducts), we have a bialgebra, and if there is a coinverse, the celebrated antipode S : H → H , as well, we have a Hopf algebra. While in the algebra the unit, the inverse and the product are related by m(a; a−1 ) = m(a−1 ; a) = 1, the counit, the coproduct and the coinverse are related by m(S × id) = E ◦ e. O A thorough introduction can be found for example in [23]. To de$ne a coproduct for rooted trees we are hence looking for a map which disentangles rooted trees. We start with the most elementary possibility. An elementary cut is a cut of a rooted tree at a single chosen edge, as indicated in Fig. 7. By such a cutting procedure, we will obtain the possibility to de$ne a coproduct, as we can use the resulting pieces on either side of the coproduct. It is this cutting operation which corresponds to the disentanglements of graphs discussed before. Still before introducing the coproduct we introduce the notion of an admissible cut, also called a simple cut [5]. It is any assignment of elementary cuts to a rooted tree t such that any path from any vertex of the tree to the root has at most one elementary cut, as in Fig. 8. An admissible cut C maps a tree to a monomial in trees. If the cut C contains n elementary cuts, it induces a map C : t → C(t) =

n+1

tji :

(7)

i=1

Note that precisely one of these trees tji will contain the root of t. Let us denote this distinguished tree by RC (t). The monomial which is delivered by the n −1 other factors is denoted by P C (t). In graphs, P C (t) corresponds to a set of disjoint subgraphs i i which we shrink to a 6

This has far reaching consequences and is closely connected to the fact that logarithmic derivatives (with respect to the log of some scale say) of Z-factors are $nite quantities. Indeed, Z-factors can be regarded as formal series over Feynman diagrams graded by the loop number starting with 1, and their logarithm de$nes a series in graphs which typically demands that commutators like [B+ ; B− ](t1 t1 ) are a primitive element in the Hopf algebra, and hence provide only a $rst order pole [16,12]. This is a $rst instance of a ’t Hooft relation to which we turn later when we review the results of [9].

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Fig. 7. An elementary cut c splits a rooted tree t into two components. We remove the chosen edge and get two components. Both are rooted trees in an obvious manner: one contains the vertex which was the old root and the root of the other is provided by the vertex which was at the endpoint (edges are oriented away from the root) of the removed edge. Fig. 8. An admissible cut C acting on a tree t. It produces a monomial of trees. One of the factors, RC (t), contains the root of t.

point and take outof the initial graph corresponding to t, while RC (t) corresponds to the remaining graph =( i i ). Admissibility means that there are no further disentanglements in the set i i . Hence, a sum over all such sets provides a sum over all unions of subgraphs, as we will discuss below. Arbitrary non-admissible cuts correspond to the notion of forests in the sense of Zimmermann [2,5]. Let us now establish the Hopf algebra structure. Following [2,5], we de$ne the counit and the coproduct. The counit eO : H → Q is simple: e(X O )=0 for any X = 1, e(1) O =1 : The coproduct is de$ned by the equations (1) = 1 ⊗ 1 ; (t1 : : : tn ) = (t1 ) : : : (tn ) ; (t) = t ⊗ 1 + (id ⊗ B+ )[(B− (t))] ;

(8)

which de$nes the coproduct on trees with n vertices iteratively through the coproduct on trees with a lesser number of vertices (see Fig. 9). The coproduct can be written in a non-recursive manner as [2,5] P C (t) ⊗ RC (t) : (9) (t) = 1 ⊗ t + t ⊗ 1 + adm: cuts C of t

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S( )= − S( )= − + S( )= − + 2 S( )= −

+2

− −

Fig. 9. The coproduct. We work it out for the trees t1 ; t2 ; t31 ; t32 , from top to bottom. Fig. 10. The antipode. Again we work it out for the trees t1 ; t2 ; t31 ; t32 .

Up to now we have established a bialgebra structure. It is actually a Hopf algebra. Following [2,5] we $nd the antipode S as S(1)

=1;

S(t1 : : : tk ) = S(t1 ) : : : S(tk ) ; S(t) = −t −

(10) C

C

S[P (t)]R (t) :

adm: cuts C of t

Fig. 10 gives examples for the antipode. Let us give yet another formula to write the antipode, which one easily derives using induction on the number of vertices [2,5]: S(t) = −

(−1)nC P C (t)RC (t) ;

(11)

all cuts C of t

where nC is the number of elementary cuts in C. This time, we have a non-recursive expression, summing over all cuts C, relaxing the restriction to admissible cuts. By now we have established a Hopf algebra H on rooted trees, using the set of rooted trees, the commutative multiplication m for elements of this set, the unit 1 and counit e, O the coproduct and antipode S. Still following [2,5], we allow to label the vertices of rooted trees by Feynman graphs without subdivergences, in the sense described before. Quite general, if Y is a set of primitive elements providing labels, we get a similar Hopf algebra H (Y ). The determination of all primitive graphs which can appear as labels corresponds to a skeleton expansion and is discussed in detail in [4]. Instead of using the language of a decorated Hopf algebra, we use directly the corresponding Hopf algebra of graphs below.

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Let us also mention again that m[(S ⊗ id)(t)] = E ◦ e(t) O (=0 for any non-trivial t = 1) :

(12)

As the divergent sectors in Feynman graphs are strati$ed by rooted trees, we can use the Hopf algebra structure to describe the disentanglement of graphs into pieces, and it turns out that this delivers the forest formulas of renormalization theory. Let us now come back to the graph and its representation in Fig. 4. We want to look at the relevant Hopf algebra operations in some detail, which we describe in Fig. 11. The operations described in this $gure go through for any QFT whose ultraviolet divergences are local, strati$ed by rooted trees that is. A renormalizable $eld theory will only demand a $nite number of counterterms in the action, while an e9ective theory is $nite in the number of needed counterterms only for a $nite loop order, but the number will actually increase with the loop order. A superrenormalizable theory gives only a truncated representation of rooted trees: higher orders in the perturbative expansion do not deliver new short-distance singularities, and hence the existent divergences are strati$ed by rooted trees with a restricted number of vertices. Each short-distance singularity corresponds to a sector which can be described by a rooted tree, which itself notates the hierarchy of singularities. We have a coproduct which describes the job-list [10] of renormalization: we use it to disentangle the singularities located at (sub-)diagonals. The Feynman rules are then providing a character : H → V on this Hopf algebra. They map a Hopf algebra element to an analytic expression, typically evaluating in a suitable ring V of Feynman integrands or Laurent polynomials in a regularization parameter. These maps being characters, we have (1 2 ) = (1 )(2 ) :

(13)

Then, renormalization comes from the very simple Hopf algebra property Eq. (12), as we now explain. Let us describe carefully how to use the Hopf algebra structure in the example of Fig. 11. The $rst thing which we have to introduce, together with our Feynman rules, is a map R : V → V which is essentially determined by the choice of a renormalization scheme. The freedom in this choice is essentially what makes up the renormalization group. The presence of the antipode S allows to consider, for each , its inverse character −1 = ◦ S. Actually, we have a group structure on characters: to each two characters ; we can assign a new character ? = mV ◦ ( ⊗ ) ◦ ; and a unit of the ?-product is provided as ?) = )? = and the inverse is indeed provided by the antipode: − 1 ? = ? − 1 = ) ; where ) comes from the counit and is uniquely de$ned as ) = ◦ E ◦ eO so that )(1) = 1V , )(X ) = 0, ∀X = 1, and for any arbitrarily chosen character (any character ful$lls (1) = 1V ).

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Fig. 11. The graph gives rise to two rooted trees corresponding to its two (overlapping) divergent sectors. Each of the two rooted trees allows for a single admissible cut. We implement it in each case by the gray curve which encircles the one vertex which constitutes P C (t) and the whole chord diagram attached to this subtree. It hence corresponds to a subgraph which is a three-point graph, as three chords are crossed by this gray curve. The cut at the rooted tree then corresponds to shrinking the subgraph to a point, which is a vertex in the remaining graph (a one-loop self-energy). This vertex we have decorated by {2; 3; 4} or {1; 2; 3}. It amounts to a local polynomial insertion in the self-energy. If the vertices so-generated always give rise to polynomial insertions which are part of the action already, we have a renormalizable theory. For a general theory, one will have a variety of di9erent chords represented by di9erent propagators, and a variety of vertices as well. For a renormalizable theory, there will be only a $nite number of each. It may happen that there are various di9erent vertices into which a graph can shrink, in which case a sum over the corresponding external structures is involved [8].

The next thing to do is to use and R to de$ne a further character SR : H → V by SR (t )(t ) ; SR = −R (t) + where we used the notation (t) = t ⊗ 1 + 1 ⊗ t + t ⊗ t . By construction, if we choose R = id V , the identity map from V → V , we have SidV = ◦ S.

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Now, consider SR ?. We have SidV ? = ◦ m ◦ (S ⊗ id) ◦ = ) by the Hopf algebra property Eq. (12) above. This guarantees that from regions where R becomes the identity map id V : V → V , we get a vanishing contribution from any non-trivial sector t realized in a Feynman graph , as )(t) = 0. So if we demand that R leaves short-distance singularities unaltered, so that R = id V for large loop momenta, we automatically have a vanishing contribution of those singularities to SR ?. 7 What we see at work here is a general principle of multiplicative subtraction [5]: while for a primitive Hopf algebra element t; (t) = t ⊗ 1 + 1 ⊗ t, SR ? amounts simply to the additive operation (t) − R[(t)] ; for a general Hopf algebra element the coproduct provides a much more re$ned multiplicative subtraction mechanism, which can obviously be considered for a wide class of Hopf algebras. This principle can certainly be applied in the future not only in the problem of short-distance singularities, but in a much wider class of problems, with asymptotic expansions coming to mind immediately. Fig. 12 describes how the Hopf algebra is realized on the sectors of the graph and how this relates to the Hopf algebra of Feynman graphs to which we now turn. 2.4. The Hopf algebra of graphs As we already have emphasized the Hopf algebra of rooted trees is the role model for the Hopf algebras of Feynman graphs which underly the process of renormalization when formulated perturbatively at the level of Feynman graphs. The following formulas should be of no surprise after our previous discussions. First of all, we start considering one-particle irreducible graphs as the linear generators of the Hopf algebra, with their disjoint union as product. We then de$ne a Hopf algebra by a coproduct () = ⊗ 1 + 1 ⊗ + ⊗ = ; (14) ⊂

where the sum is over all unions of one-particle irreducible (1PI) super$cially divergent proper subgraphs and we extend this de$nition to products of graphs so that we get a bialgebra [8]. The above sum should, when needed, also run over appropriate external structures to specify the appropriate type of local insertion [8] which appear in local counterterms, which we omitted in the above sum for simplicity. 8 The counit eO vanishes, as before, on any non-trivial Hopf algebra element. At this stage we have a commutative, but typically not cocommutative bialgebra. It actually is a Hopf algebra as the 7

That R leaves short-distance singularities unaltered typically requires that the $rst few Taylor coeNcients in the Feynman integrands, as determined by powercounting, are left unaltered. 8 A simple example exhibited in [8] is the self-energy in massive 3 theory in six dimensions. It provides two external structures, corresponding to local insertions of counterterms for the m2 2 and for the (9* )2 term.

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Fig. 12. The result of the operation SR ?( ) graphically, where an application of the operation R is indicated by encircling the graph whose corresponding analytic expression is to be mapped to the range of R by a thick gray line. In the upper row, we see the result in terms of the decorated rooted trees of Fig. 11 while in the second row we see the result directly expressed in terms of Feynman integrals. Again, the map is not explicitly written out. The gray boxes indicate the full and normal forests of classical renormalization theory [1] and are in one-to-one correspondence with the cuts at the corresponding rooted trees if we incorporate the empty and the fullcut in the sum over cuts, so that the two terms T ⊗ 1 + 1 ⊗ T which appear in any coproduct (T ) = T ⊗ 1 + 1 ⊗ T + adm:C P C (T ) ⊗ RC (T ) can be regarded as generated by the full (T ⊗ 1) and the empty cut (1 ⊗ T ) [5].

antipode in such circumstances comes almost for free as S()= : S() = − − ⊂

(15)

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The next thing we need are Feynman rules, which we regard as maps : H → V from the Hopf algebra of graphs H into an appropriate space V . Over the years, physicists have invented many calculational schemes in perturbative quantum $eld theory, and hence it is of no surprise that there are many choices for this space. For example, if we want to work on the level of Feynman integrands in a BPHZ scheme, we could take as this space a suitable space of Feynman integrands (realized either in momentum space or con$guration space, whatever suits). An alternative scheme would be the study of regularized Feynman integrals, for example the use of dimensional regularization would assign to each graph a Laurent-series with poles of $nite order in a variable , near , = 0, and we would obtain characters evaluating in this ring. In any case, we will have (1 2 ) = (1 )(2 ). Then, with the calculational scheme chosen and the Feynman rules providing a canonical character , we will have to make one further choice: a renormalization scheme. This is a map R : V → V , and we demand that it does not modify the UV-singular structure: in BPHZ language, it should not modify the Taylor expansion of the integrand for the $rst couple of terms divergent by powercounting. In dimensional regularization, we demand that it does not modify the pole terms in ,. Finally, the principle of multiplicative subtraction works as before: we de$ne a further character SR which deforms ◦ S slightly and delivers the counterterm for :   SR ()(=) (16) SR () = −R[()] − R  ⊂

which should be compared with the undeformed ◦ S()(=) : ◦ S = −() −

(17)

⊂

Then, the classical results of renormalization theory follow suit [2,4,5]. We obtain the renormalization of by the application of a renormalized character → SR ?() and the RO operation as O SR ()(=) ; R() = () +

(18)

⊂

so that we have O + SR () : SR ?() = R()

(19)

In the above, we have given all formulas in their recursive form. Zimmermann’s original forest formula solving this recursion is obtained when we trace our considerations back to the fact that the coproduct of rooted trees can be written in non-recursive form, and similarly the antipode. It is not diNcult to see that the sum over all cuts corresponds to a sum over all forests, and the notion of full and normal forests of Zimmermann [1] gives rise to appropriate sums over cuts [2,5], making use of the graphical implementation of cuts as for example in Fig. 12.

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3. Rescalings and renormalization schemes Let us come back to unrenormalized Feynman graphs, and their evaluation by some chosen character , and let us also choose a renormalization scheme R. The group structure of such characters on the Hopf algebra can be used in an obvious manner to describe the change of renormalization schemes. This has very much the structure of a generalization of Chen’s Lemma [3]. 3.1. Chen’s Lemma Consider SR ?. Let us change the renormalization scheme from R to R . How is the renormalized character SR ? related to the renormalized character SR ?? The answer lies in the group structure of characters: SR ? = [SR ?SR ◦ S]?[SR ?] :

(20)

We inserted a unit ) with respect to the ?-product in form of ) = SR ◦ S ?SR ≡ SR−1 ?SR , and can now read the renormalization, switching between the two renormalization schemes, as composition with the renormalized character SR ?SR−1 . 9 Similar considerations apply to a change of scales which determine a character [3]. If - is a dimensionful parameter which appears in a character = (-), 10 then the transition - → - is implemented in the group by acting on the right with the renormalized character -;- := (-) ◦ S ?(- ) on (-), (- ) = (-)?

-; -

:

(21)

Let us note that this Hopf algebra structure can be eNciently automated as an algorithm for practical calculations exhibiting the full power of this combinatorics [10]. Now, assume we compute Feynman graphs by some Feynman rules in a given theory and decide to subtract UV singularities at a chosen renormalization point *. This amounts, in our language, to saying that the map SR is parametrized by this renormalization point: SR = SR (*). Then, let .(*; -) be the ratio .(*; -)=SR (*)?(-). We then have the groupoid law generalizing the before-mentioned Chen’s Lemma [3] .(*; ))?.(); -) = .(*; -) :

(22)

While this looks like a groupoid law, the product of two unrelated ratios .(*1 ; *2 ) ? .(*3 ; *4 ), as any other product of characters, is always well-de$ned in the group of characters of the Hopf algebra. 3.2. Automorphisms of the Hopf algebra In the setup discussed so far, the combinatorics of renormalization was attributed to a Hopf algebra, while characters of this Hopf algebra took care of the speci$c Feynman rules and chosen 9

SR ?SR ◦ S is a renormalized character indeed: if R; R are both self-maps of V which do not alter the short-distance singularities as discussed before, then in the ratio SR ?SR ◦ S those singularities drop out. 10 Typically, it could be a scale which dominates the process under consideration.

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renormalization schemes. Renormalized quantities appear as the ratio of two characters, while divergences drop out in this ratio SR ?. Typically, such characters introduce a renormalization scale (cut-o9, the ’t Hooft mass * in dimensional regularization), and we can use these parameters to describe the change of schemes in a fairly uni$ed manner, as discussed in [3]. These considerations of changes of renormalization schemes are related to another interesting aspect discussed in [3]. So far, we regarded the map R as a self-map in a certain space V . We will not have R(XY ) = R(X )R(Y ) (for example, minimal subtraction cannot possibly ful$ll that the poleterms of a product is the product of the poleterms), but R obeys the multiplicativity constraints R(XY ) + R(X )R(Y ) = R(XR(Y )) + R(R(X )Y ) ;

(23)

which ensure that SR (1 2 ) = SR (1 )SR (2 ) [3,8]. This leads to the Riemann–Hilbert problem to be discussed below. We now want to investigate to what extent the map R : V → V can be lifted to an automorphism /R : H → H of the Hopf algebra. We regard V as the space in which Feynman graphs evaluate by the Feynman rules, as discussed above. Let again the Feynman rules be implemented by . The map SR is then a character constructed with the help of , so we should write SR ≡ SR to be exact. The question is if one can construct, for any R, an automorphism /R : H → H of the Hopf algebra such that one has O ≡ SR ◦ S = ◦ /R ;

(24)

so that (using S 2 = id which is true in any commutative Hopf algebra) 1 SR ? = O ◦ S ? = ◦ /R ◦ S ? = ◦ [/− R ?id] :

(25)

The answer is aNrmative [3]. Following [3] and the use of one-parameter group of automorphisms in the renormalization group [9] to be discussed below, we make the following Ansatz for /R : /R () = e−, deg()-R () ;

(26)

where, in the context of dimensional regularization or any other analytic regularization, -R () will be a character evaluating in the ring of Taylor series in , regular at , = 0 and deg() = n if has n loops. 11 Then, one determines SR ◦ S() −1 log ; (27) -R () = , deg() () so that indeed -R () is free of poleterms, as one easily shows SR ◦ S() = 1 + O(,) ; () 11

It is convenient but not necessary to work with dimensional regularization here. In BPHZ, one could work for example with the ratio of Taylor series in external parameters.

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for arbitrary graphs . This gives a unifying approach to the treatment of renormalization schemes and changes between them. 12

4. The insertion operad of Feynman graphs In this section, we want to describe an operad structure on Feynman graphs. This operad was implicitly present in many results in [5,8,9], and so it is worth to describe it shortly at this stage, also with regard to the fact that it will prove to be a useful construct to investigate the number-theoretic aspects of Feynman graphs [13,14] to be discussed below. While the previous two sections discussed the process of disentangling a Feynman graph into subgraphs according to the presence of UV singularities, we now turn to the process of plugging graphs into each other. This will lead us in the next section to Lie algebras of Feynman graphs. Here, we want to study the most basic operation: plugging one graph 1 into another graph 2 . Typically, there are various places in 2 , provided by edges and vertices of 2 , which can be replaced by 1 . To obtain a sensible notion of this operation we should ful$ll operad laws in this process. These operad laws can be described as follows. Operad laws are concerned with rules which should be ful$lled when we insert several times. First, assume we have graphs 1 ; 2 and want to plug both of them into di9erent places of a graph . Then, the result should be independent of the order in which we do it. Next, when we plug 1 into 2 at some place, and insert the result into , the result should be the same as inserting 2 at the same place in , and then 1 into the corresponding relabelled place of 2 . Finally, the permutation of places should be compatible with the composition (see for example [25] for a formal de$nition of these requirements). We only describe the operad in the context of massless 3 theory in six dimensions, the generalizations to more general cases are obvious and will be discussed elsewhere. A Feynman graph provides vertices and edges connecting these vertices. The operad essentially consists of regarding these vertices and edges as places into which other graphs can be inserted. Naturally, a vertex correction can replace a vertex of a similar type, and a propagator-function can replace a line which represents a free propagator of a similar type. In massless 3 theory, we only have one type of lines and one type of vertices. First, we note that the overall divergent Feynman graphs in this theory are given by 1PI graphs with two or three amputated external lines. Thus, vertices in the graphs are either internal three-point vertices, or two-point vertices resulting from the amputation of an external leg from a three-point vertex. Hence, self-energies can be described as graphs which precisely have two two-point vertices, while three-point graphs, vertex corrections, have precisely three two-point vertices. Propagator-functions then have two external edges. When we want to replace an internal vertex, we just replace it by a vertex correction. When we want to replace an internal edge, a free propagator, we replace it by a propagator-function, as described by Figs. 13 and 14. 12

From here, one can start considering categorical aspects of renormalization theory and in particular address, the question posed in [2] if a modi$ed coproduct R = (/R ⊗ id) ◦ is (weak-)coassociative in dependence of R, with $rst results upcoming in a recent thesis [24].

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3 2

1

11

5

4 6

2

∗6ε

8

11 1

3

6

5 7

8 10 9

4

7 10 9

4

3

=

15

2

1

5

14 13 9 11 12 10 8 6 7

21

16

17

20

18 19

Fig. 13. We consider a propagator-graph and a vertex-function and as an example their concatenation ?6 . The propagator function replaces the line with label 6 in the vertex-function. The propagator-function provides four vertices (labelled 1; 3; 4; 8) and seven edges (labelled 2; 5; 6; 7; 9; 10; 11). Two of the edges, 10 and 11, are external. The vertex-function provides $ve vertices (labelled 1; 3; 5; 7; 9) and six edges (labelled 2; 4; 6; 8; 10; 11). The vertices 1; 5; 9 are external, they connect to edges which are not part of the vertex function. We still indicated them by open-ended lines at those vertices, but one should regard vertices 1; 5; 9 as two-point vertices. Note that each internal edge ends in two labelled vertices. We replace the edge labelled 6 by the propagator-function, connecting the external edges 10 and 11 of the latter to the vertices 5 and 7 of the vertex-function. We glue the edge with the lower label (10) to the vertex with the lower label (5). Relabelling is done in the obvious way: labels 1–5 in the vertex-function remain unchanged, the labels at the inserted propagator function become labels 6 –16, and labels 7–11 become labels 17–21, increasing their labels by 4 + 7 − 1 = 10.

How many places are there? Let (p1 ; p2 ) be a 1PI vertex function given by a three point graph with l loops, which then provides 2l + 1 vertices and 3l internal lines, hence 5l + 1 places for insertion altogether. Let 2(p) be a propagator function given by a (not necessarily one-particle irreducible) two-point graph 2 with l loops, it then provides 2l vertices and 3l + 1 lines, hence again 5l + 1 places (we not necessarily have to label all edges and vertices, for example dropping the label at an external edge of the propagator function takes into account quite naturally the fact that self-energies are proportional to an inverse propagator, and, in a massless theory, cancel one of the external lines).

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Fig. 14. To explain the insertion of a vertex-function, we replace in this example vertex 3 of a vertex-function by the very same vertex-function, so we describe ?3 . We do it by connecting edges 2; 4; 8 which are attached to vertex 3 to the three two-point vertices 1; 5; 9, respecting the order: edge 2 connects to vertex 1, edge 4 to vertex 5, edge 8 to vertex 9. Relabelling is done in the obvious way: labels 1 and 2 in the vertex-function remain unchanged, the labels at the inserted vertex-function become labels 3–13, and labels 4 –11 become labels 14 –21.

We label all edges and vertices in arbitrary order, and the composition laws described in the captions of Figs. 13 and 14 ful$ll the operad laws (the before-mentioned requirements are ful$lled), so that Figs. 13 and 14 de$ne this operad by way of example. So, with these rules for insertion (we also understand that insertion of a propagator-function at a vertex place or a vertex-function at an edge vanishes trivially by de$nition), one gets indeed an (partial) operad. Note further that insertion of a free propagator or vertex leaves the result unchanged. One easily extends this construction to the case that one has vertices of other valencies and with di9erent sorts of lines coming in.

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This operad can be conveniently used to study the Lie algebraic structure of diagrams as well as for the investigation of number-theoretic aspects as we will see below. Also, the operad viewpoint is helpful in understanding the equivalence classes discussed in [2]. For example, the two graphs 1 and 2 of Section 2.1 belong to the same equivalence class, 1 ∼ 2 , given by the parenthesized word (()0 ), and are distinguished only by the place into which we insert . In general, two graphs are equivalent if one is obtained via a permutation of concatenation labels of the other, while maintaining the tree structure of its subdivergences: all Feynman graphs which represent the same rooted tree or parenthesized word can be obtained from each other by the change of labels of places where we insert the primitive graphs into each other. Also, typical equations in $eld theory like Schwinger–Dyson equations are naturally formulated by this operad, using the fact that the sum over all diagrams can be written as a sum over all primitive ones into which all diagrams are plugged in all possible places. Details will be given in future work. 5. The Lie algebra structure In [5,8,9], the reader $nds various Lie algebra structures which appear in the dual of the Hopf algebra which is the universal enveloping algebra of a Lie algebra. Here, we describe the Lie algebra of Feynman graphs. There is also one for rooted trees, which can be found in [5]. Study of these Lie algebras is a very convenient way of understanding the structure of Feynman graphs. These Lie algebras play a crucial role when one wants to understand the connection between the group of di9eomorphisms of physical parameters like coupling constants with the group of characters of the Hopf algebra, to which we will turn in the next section. It is also quite useful in determining the Hopf algebra structure of a chosen QFT correctly, because, once it is found, the corresponding enveloping algebra will be the dual of a commutative non-cocommutative Hopf algebra (by the celebrated Milnor–Moore theorem [5,8]) whose coproduct gives us the forests formulas of renormalization. 13 To $nd these Lie algebras, one de$nes a Lie-bracket of two 1PI graphs 1 ; 2 by plugging 1 into 2 in all possible ways and subtracts all ways of plugging 2 into 1 . These Lie algebras all arise from a pre-Lie structure which we can describe in Fig. 15. The operation of inserting one graph 1 in another graph 2 in all possible ways is a pre-Lie operation 2 ?1 , which means that it ful$lls 3 ?(2 ?1 ) − (3 ?2 )?1 = 3 ?(1 ?2 ) − (3 ?1 )?2 : Antisymmetrization then gives automatically a bracket [1 ; 2 ] = 1 ?2 − 2 ?1 , which ful$lls the Jacobi identity. This operation of inserting one graph in another in all possible ways can obviously written with the help of the operad structure of the previous section as a sum over all places where to insert (plus a sum over all permutations of the labels of identical external vertices of the graph 13

For example, one easily determines the Lie algebra of QED, having one type of vertex connecting to two di9erent type of lines for fermion and photon propagators. This then con$rms the corresponding Hopf algebra structure of 1PI graphs to be commutative non-cocommutative. One-particle reducible graphs can be treated as in [16]. In the literature, there are other attempts to describe the renormalization of QED by binary rooted trees [26]. But the singularities of QED are strati$ed along diagonals as in any local QFT, and the rather arti$cial restriction to binary rooted trees ultimately runs into trouble [27].

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409

Fig. 15. The (pre-)Lie algebra structure of Feynman graphs. The fact that the operation of plugging a graph into another one in all possible ways is pre-Lie is essentially due to the fact that the ways of plugging (in all possible ways) 1 into 2 , and the result into 3 , subtracted from the ways of plugging (in all possibly ways) 1 into the result of plugging (in all possible ways) 2 into 3 is the sum over all possible ways to plug 1 ; 2 disjointly into 3 .

which is to be inserted) and the operad laws then guarantee that the pre-Lie property is ful$lled, making use of the intimate connection between rooted trees, operads and pre-Lie algebras [28]. Once this Lie algebra is found, one knows that dually one obtains a commutative, non-cocommutative Hopf algebra which is the basis of the forest formulas of renormalization as discussed in the previous section. It is not diNcult to work out the corresponding pre-Lie structure for QED for example, and indeed, reading the graphs of Fig. 15 as QED graphs in the obvious possible manners only demands to cancel a few of the terms in that $gure, because a photon propagator can only replace a photon line, and not a fermion line. Similarly, for any local QFT, one can determine the corresponding Hopf and Lie algebras, incorporating external structures whenever necessary as in [8]. The resulting Lie algebras of Feynman graphs play a fundamental role in understanding how the combinatorial properties of renormalization connect to the renormalization group, to the running of physical parameters. We now turn to study these results of [7–9].

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6. The Birkho' decomposition and the renormalization group In [7–9] the reader $nds an amazing connection between the Riemann–Hilbert problem and renormalization. This result was $rst announced in [7]. It is based on the use of a complex regularization parameter. Typically, dimensional regularization provides such a parameter as the deviation , from the relevant integer dimension of spacetime, but for example analytic regularization would do as well. With such a regularization parameter, the Feyman rules map a Feynman graph to a Laurent series with poles of $nite order in this regularization parameter, hence the Feynman rules provide a character from the Hopf algebra of Feynman graphs to the ring of Laurent polynomials with poles of $nite order in ,. As mentioned before, the multiplicativity constraints [3,7,8] R[xy] + R[x]R[y] = R[R[x]y] + R[xR[y]] ensure that the corresponding counterterm map SR is a character as well, SR [xy] = SR [x]SR [y]

∀x; y ∈ H :

We now study how this set-up leads to the Riemann–Hilbert problem and the Birkho9 decomposition. 6.1. Minimal subtraction: the Birkho; decomposition To make contact with the Riemann–Hilbert problem, the crucial step is to recognize that, for R = MS being chosen to be projection onto these poles of $nite order (the minimal subtraction scheme MS), = SMS ◦ S ?[SMS ?] is a decomposition of the character into a part which is holomorphic at , = 0 : SMS ? ≡ + is a character evaluating in the ring of functions holomorphic at , = 0, while SMS ≡ − maps to polynomials in 1=, without constant term, it delivers, when evaluated on Feynman graphs, the MS counterterms for those graphs. This corresponds to a Birkho9 1 decomposition = − − + . For an introduction to the Riemann–Hilbert problem and the associated Birkho9 decomposition we refer the reader to [29]. SuNces it here to say that the Riemann–Hilbert problem is a type of inverse problem. For a given complex di9erential equation Ai y (z) = A(z)y(z); A(z) = z − zi i with given regular singularities zi and matrices Ai , one can determine monodromy matrices Mi integrating around curves encircling the singularities. The inverse problem, $nding the di9erential equation from knowledge of the singular places and monodromy matrices, is the Riemann–Hilbert problem. A crucial role in its solution plays the Birkho9 decomposition: for a closed curve C in the Riemann sphere, and a matrix-valued loop : z → (z) well-de$ned on C, decompose it into parts ± well-de$ned in the interior=exterior of C. Thus, renormalization in the MS scheme can be summarized in one sentence: with the character given by the Feynman rules in a suitable regularization scheme and well-de$ned on any small curve around , = 0, $nd the Birkho9 decomposition + (,) = − , where now and in the following the product in expressions like − is meant to be just the convolution product − ? of characters used before.

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411

The unrenormalized analytic expression for a graph is then [](,), the MS-counterterm is SMS () ≡ − [](,) and the renormalized expression is the evaluation + [](0). Once more, note that the whole Hopf algebra structure of Feynman graphs is present in this group: the group law demands the application of the coproduct, + = − ≡ SMS ?. The transition from here to other renormalization schemes can be achieved in the group of characters in accordance with our previous considerations in Section 3. But still, one might wonder what a huge group this group of characters really is. What one confronts in QFT is the group of di9eomorphisms of physical parameter: low and behold, changes of scales and renormalization schemes are just such (formal) di9eomorphisms. So, for the case of a massless theory with one coupling constant g, for example, this just boils down to formal di9eomorphisms of the form g → (g) = g + c2 g2 + · · · : The group of one-dimensional di9eomorphisms of this form looks much more manageable than the group of characters of the Hopf algebras of Feynman graphs of this theory. Thus, it would be very nice if the whole Birkho9 decomposition could be obtained at the level of di9eomorphisms of the coupling constants, and this is what was achieved in [9]. 6.2. The -function Following [8], in the above we have seen that perturbative renormalization is a special case of a general mathematical procedure of extraction of $nite values based on the Riemann–Hilbert problem. The characters of the Hopf algebra of Feynman graphs form a group whose concatenation, unit and inverse are given by the coproduct, the counit and the antipode. So we can associate to any given renormalizable quantum $eld theory an (in$nite dimensional) complex Lie group G of characters of its Hopf algebra H of Feynman graphs. Passing from the unrenormalized theory to the renormalized one corresponds to the replacement of the loop , → (,) ∈ G (obtained by restricting the character to an arbitrarily chosen curve C around , = 0) of elements of G obtained from dimensional regularization (still, , = 0 is the deviation from the integer dimension of space–time) by the value + (,) of its Birkho9 decomposition, (,) = − (,)−1 + (,). In [9], it was shown how to use the very concepts of a Hopf and Lie algebra of graphs to lift the usual concepts of the -function and renormalization group from the space of coupling constants of the theory to the complex Lie group G. We now exhibit these results. The original loop , → (,) not only depends upon the parameters of the theory but also on the additional unit of mass *, the ’t Hooft mass in dimensional regularization, required by dimensional analysis. But although the loop (,) does depend on the additional parameter *, * → (,; *) ; the negative part *− in the Birkho9 decomposition, the character delivering the MS counterterms, (,; *) = − (,; *)−1 + (,; *)

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is actually independent of *, 9 − (,; *) = 0 : 9*

(28)

This is a remnant of the fact that our Hopf algebra is constructed so as to achieve local counterterms: is a character which can be easily shown to be a series in log(q2 =*2 ) so that a remaining *2 dependence in MS counterterms would be accompanied by a remaining q2 dependence, and would hence violate locality. 14 The Lie group G turns out to be graded, with grading, 8- ∈ AutG;

-∈R ;

inherited from the grading of the Hopf algebra H of Feynman graphs given by the loop number, deg() = loop number of

(29)

for any 1PI graph , so that 8- () = e- deg() . 15 This leads to (j; e- *) = 8-, ((,; *))

∀- ∈ R ;

so that the loops (*) associated to the unrenormalized theory have the property that the negative part of their Birkho9 decomposition is unaltered by the operation, (,) → 8-, ((,)) : if we replace (,) by 8-, ((,)) we do not change the negative part of its Birkho9 decomposition. A complete characterization of the loops (,) ∈ G ful$lling this invariance can be found in [9]. This characterization only involves the negative part − (,) of their Birkho9 decomposition which by hypothesis ful$lls, − (,)8-, (− (,)−1 )

is convergent for , → 0 :

(30)

It is then easy to see that this de$nes in the limit , → 0 a one-parameter subgroup, F- ∈ G;

-∈R :

(31)

Now, the role of the -function is revealed: the generator := (9=9 -F- )-=0 of this one-parameter group is related to the residue of the loop

1 9 − Res = − (32) ,=0 9u u u=0 14 A similar argument applies when the Feynman rules provide a character parametrized by several scales. Again, by a group action which is a $nite renormalization, we can reduce the unrenormalized theory to a dependence on a single scale. This reduction can constrain the renormalization group Low to a submanifold though, in which case an explicit group action is needed to switch from mass-independent to mass-dependent renormalization group functions, as it is well-known [30]. 15 Here - is to be regarded as a constant. If we promote it to a character evaluating in the ring of functions holomorphic at , = 0 we obtain the automorphisms used in Section 3 to lift the renormalization map R to automorphisms of the Hopf algebra. Note that a constant - is suNcient to describe momentum schemes for example, using that one only has to use - = , log(*2 =q2 ) to compensate for the canonical q2 -dependence [2,3,16,10].

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413

by the simple equation, = Y Res ;

(33)

where Y = (9=9-8- )-=0 is the grading. In a moment, we will see how this generator relates to the common -function of physics. All this is a simple consequence of the setup described so far and is worked out in detail in [9] (essentially, at the moment we quote a summary of the results of that paper), while the central result of [9] gives − (,) in closed form as a function of . Let us use an additional generator in the Lie algebra of G (i.e. primitive elements of H ∗ ) implementing the grading such that [Z0 ; X ] = Y (X )∀X ∈ Lie G: Then, the loop − (,) corresponding to the MS counterterm evaluated on any close curve around , = 0 can be written by a scattering type formula for − (,) as − (,) = lim e−t((=,)+Z0 ) etZ0 : t →∞

(34)

Both factors in the right-hand side belong to the semi-direct product, G˜ = G o R 8

of the group G by the grading, but their product belongs to the group G. As a consequence, the higher pole structure of the divergences is uniquely determined by the residue and this gives a strong form of the ’t Hooft relations, which come indeed as an immediate corollary. 16 The most fundamental result of [9] is obtained though when considering two competing Hopf algebra structures: di9eomorphisms of physical parameters carry, being formal di9eomorphisms, with them the Hopf algebra structure of such di9eomorphisms. This structure was recognized for the $rst time by Connes and Moscovici in [31]. On the other hand, a variation of physical parameters induced by a variation of scales is a renormalization, which directly leads to the Hopf algebra of Feynman graphs. Let us $rst describe the Hopf algebra structure of the composition of di9eomorphisms in a fairly elementary way, while mathematical detail can be found in [31]. Assume you have formal di9eomorphisms ; in a single variable x → (x) = x + ck x k ; (35) k¿1

and similarly for . How do you compute the Taylor coeNcients ck◦ for the composition ◦ from the knowledge of the Taylor coeNcients ck ; ck ? It turns out that it is best to consider the Taylor coeNcients <k = log( (x))(k) (0) 16

(36)

The explicit formulas in [9] allow to $nd the combinations of primitive graphs into which higher order poles resolve. The weights are essentially given by iterated integrals which produce coeNcients which generalize the tree-factorials obtained for the undecorated Hopf algebra in [3,16,10]. Iterated application of this formula allows to express inversely the $rst-order poles contributing to the -function as polynomials in Feynman graphs free of higher-order poles.

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instead, which are as good to recover as the usual Taylor coeNcients. The answer lies then in a Hopf algebra structure: ˜ ◦ CM (
˜
(37)

(where both gZ1 = g +
HCM → H ; from the Hopf algebra HCM of coordinates on the group of formal di9eomorphisms of C (i.e. such that ’(0) = 0; ’ (0) = id as in Eq. (35)) to the Hopf algebra H of the massless theory. 18 Having this Hopf algebra homomorphism from HCM to H , dually one gets a transposed group homomorphism -, a homomorphism from the huge group of characters of the Hopf algebra to the group of di9eomorphism of physical parameters [9]. We $nally recover the usual -function: the image by - of the previously introduced generator = Y Res is then the usual -function of the coupling constant g. While this might sound rather abstract, it can be easily translated into the standard notions of renormalization theory (see, for example, [32]). While in [9] the physical parameter under consideration was a single coupling, similar considerations apply to other physical parameters which run under the renormalization group, making use of the Hopf algebraic description of composition of di9eomorphisms in general. As a corollary of the construction of -; one gets an action by (formal) di9eomorphisms of the group G on the space X of (dimensionless) coupling constants of the theory. One can then in particular formulate the Birkho9 decomposition directly in the group Di9 (X ) of formal di9eomorphisms of the space of coupling constants. 17

Taking the
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415

Fig. 16. The geometric picture of [9] allows for the construction of a complex bundle, P = (S + × X ) ∪< (S − × X ) over the > sphere S = P1 (C) = S + ∪ S − , and with $ber X , X → P →S; where X is a complex manifold of physical parameters. The transition in this $ber are di9eomorphisms. <(,) delivers a di9eomorphism of X for any , ∈ C, where C is the boundary of the two half-spheres S + ; S − . It extends to the interiors of the half-spheres via its Birkho9 decomposition. The meaning of this Birkho9 decomposition, <(,) = <+ (,)<− (,)−1 is then exactly captured by an isomorphism of the bundle P with the trivial bundle, S × X: Note that <− (∞) is well-de$ned due to the fact that SMS has no constant term in ,, which characterizes a minimal subtraction scheme.

The unrenormalized theory delivers a loop <(,) ∈ Di9 (X );

, = 0 ;

whose value at , = 0 is simply the unrenormalized e9ective coupling constant. The Birkho9 decomposition <(,) = <+ (,)<− (,)−1 of this loop gives directly <− (,) = bare coupling constant and <+ (,) = renormalized e9ective coupling constant : This result is now stated in a manner independent of our group G or the Hopf algebra H , its proof makes heavy use of these ingredients though. Finally, the Birkho9 decomposition of a loop, <(,) ∈ Di9 (X ) admits a beautiful geometric interpretation [9], described in Fig. 16. 6.3. An example In [9], the reader can $nd explicit computational examples up to the three-loop level, and a complete proof to all loop orders, for the group and Hopf algebra homomorphisms described above. We only want to check the Hopf algebra homomorphism HCM → H up to two loops here. We regard g0 as a series in a variable x (which can be thought of as a physical coupling) up to

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order x4 , making use of g0 = xZ1 Z3−3=2 and the expression of the Z-factors in terms of 1PI Feynman graphs. The challenge is then to con$rm that the coordinates
∞

z1; 2k x2k ;

k=1

Z3 = 1 −

∞

z3; 2k x2k ;

k=1

and Zg = Z1 Z3−3=2 ;

zi; 2k ∈ Hc ; i = 1; 3 ;

as formal series in x2 . Using

∞
which de$nes
1 g0 < 2! 2

≡ <˜2 = 3z1; 2 + 92 z3; 2 ;

1 g0 < 4! 4

≡ <˜4 = 5[z1; 4 + 32 z3; 4 ] − 92 z1;2 2 − 6z1; 2 z3; 2 − 34 z3;2 2 ;

g0

(38)

The algebra homomorphism HCM → H is e9ected by expressing the zi; 2k in Feynman graphs, with 1PI graphs with three external legs contributing to Z1 , and 1PI graphs with two external legs, self-energies, contributing to Z3 . Explicitly, we have z1; 2 = z3; 2 =

; 1 2

z1; 4 = 1 z3; 4 = [ 2

; +

+ +

1 + [ 2 ]:

+

+

]+

1 2

;

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On the level of di9eomorphisms, we have the coproduct CM [<4 ] = <4 ⊗ 1 + 1 ⊗ <4 + 4<2 ⊗ <2 ;

(39)

where we skip odd gradings (in 3 theory, adding a loop order increases the order in the coupling by g2 ). We have to check that the coproduct of Feynman graphs reproduces these results. Applying to the rhs of (39) gives, using the expressions for zi; k in terms of Feynman graphs, 9 27 (<˜4 ) = 6 ⊗ + [ ⊗ + ⊗ ]+ ⊗ + <˜4 ⊗ 1 + 1 ⊗ <˜4 : 2 8 This has to be compared with <˜4 ⊗ 1 + 1 ⊗ <˜4 + 2!2! 4<˜2 ⊗ <˜2 , which matches nicely, as <˜2 ⊗ <˜2 = 9

⊗

27 + [ 4

4!

⊗

+

⊗

]+

81 16

⊗

:

7. Conclusions and outlook In this $nal section, we mainly want to comment on some more future lines of investigation, which in part are already work in progress. We start with the connection between Feynman diagrams and the numbers which we see in their coeNcients of ultraviolet divergence, which is a rich source of structure [15]. 7.1. Numbers and Feynman diagrams There is an enormous amount of interesting number theory in Feynman diagrams [33,34,15]. In particular, the primitive elements in the Hopf algebra, those graphs which have no subdivergences and provide a renormalization scheme independent coeNcient of ultraviolet divergence, show remarkable and hard to explain patterns. These coeNcients evaluate in Euler–Zagier sums (generalized polylogs evaluated at (suitable roots of) unity so that they generalize multiple zeta values (MZVs) [15,33,34]), numbers which have remarkably fascinating algebraic structure [35 –38]. These algebraic structures are believed to be governed by shuUe algebras, and by the much more elusive Grothendieck–TeichmVuller group (see, for example, [39] for an introduction to the Grothendieck–TeichmVuller group which is close in spirit to the consideration of short-distance singularities). The coeNcients of UV-divergence in Feynman diagrams typically evaluate, up to the six-loop level, in terms of these Euler–Zagier sums, but the question if this will always be so remains open in light of the failure to identify all these coeNcients in this number class at the seven loop level [33,34,15]. While the embarrassingly successful heuristic approach summarized in [15], providing a knot-to-number dictionary for those numbers, only emphasizes the need for a more thorough understanding, the algebraic structures in Feynman graphs hopefully lead to such an understanding in the future. It is already remarkable that shuUe products can be detected in Feynman graphs [13], but there are hints for much more structure [14]. But while the existence of shuUe algebras in Feynman graphs can essentially be straightforwardly addressed due to the fact that a shuUe algebra makes use of the B+ ; B− operators in a natural way

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Fig. 17. These two Feynman graphs (with their distinct topologies indicated on the rhs of each: the topology of the upper graph is that of disjoint one-loop insertions, the lower is a ladder topology) in massless Yukawa theory have a remarkable relation: their di9erence is a primitive Hopf algebra element. When evaluating the character SMS on both, one $nds a Laurent series with poles of fourth order from both of them. In the di9erence, the highest three-pole terms drop out, and the remaining term is ∼ ?(3)=,. Similar phenomena happen at higher loop orders [40]: higher poleterms are invariants under the permutation of places where we insert subgraphs.

[13], these remaining algebraic relations between Feynman graphs will be harder to address. 19 But the very fact that Feynman graphs realize their short-distance singularities in tree-like hierarchies suggests that they can be understood along lines similar to what is known for Euler–Zagier sums. In particular, Feynman graphs whose subdivergences realize the same rooted tree but with subgraphs inserted at di9erent internal lines provide remarkable number-theoretic features [40]. As mentioned before, in the operad picture, such di9erences are given by permutations @(i) = j of places i at which we compose: ◦i → ◦@(i) : Note that, if we let U be the di9erence of the two expressions, we get a primitive element in the Hopf algebra (if the two graphs and are both primitive), (U ) = U ⊗ 1 + 1 ⊗ U . Quite often, one $nds that these di9erences are even $nite, which means that the coeNcients of ultraviolet divergence are the same and drop out in the di9erence: short distance singularities are invariant under the above permutations. Fig. 17 gives an example of such an invariance observed in [40]. We insert a one-loop bubble at di9erent places i; j in the graph. We do not have to worry that in one case it is a one-loop fermion self-energy, in the other case a one-loop boson self-energy. In massless Yukawa theory, they both evaluate to the same analytic expression. This makes it very easy to study the e9ect of a subdivergence being inserted at di9erent places in a larger graph. In this four-loop example, the di9erence becomes a primitive element and hence delivers only a $rst order pole ∼ ?(3)=,, signalling the di9erence in topology between the two diagrams [15]. The ladder 19

But note that these shuUe algebras and shuUe identities only hold for the coeNcients of ultraviolet singularity: they hold up to $nite parts, up to $nite renormalizations that is.

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diagram evaluates to rational coeNcients in the poleterms of its MS counterterm, while the other diagram has the same rational part, but also has ?(3) in the 1=, pole. In the di9erence, only this $rst order pole ∼ ?(3)=, remains. Comparing the two three-loop subgraphsof each diagram, one $nds their di9erence to be $nite 3 i and ∼ ?(3), so that the three coeNcients i=1 ci =, are invariant under an exchange of the place where we insert the subgraph: the morphism sending one graph to the other, and thus sending one con$guration of internal vertices with its characteristic short-distance singularities to another, is a $nite one. Similar observation hold for higher loop orders [40]. A systematic understanding of such phenomena, and a possible relation to $nite-type invariants, seems crucial to understand the algebraic relations in Feynman graphs completely. Ultimately, one hopes for a geometric understanding of the analytic challenge posed by Feynman diagrams. Meanwhile, similar relations have been observed in QED [41]. A requirement on the way to such an understanding is the question how in the geometric picture of Fig. 16, one can relate an in$nitesimal variation in the base space to a variation in the $ber, i.e. the quest for a connection? For the <− part of the Birkho9 decomposition, this leads to an investigation as to how a derivative with respect to the regularization parameter , is related to the insertion of a further graph. First results at low-loop orders to be discussed elsewhere indicate that this is a source for relations between the coeNcients of ultraviolet divergence similar but not quite like the four-term relations discussed in the study of $nite-type invariants [15]. This is not impossible: while all higher poleterms are $xed in terms of the residue by the scattering type formula Eq. (34) of the previous section, this formula can by its very nature not deliver relations between residues of graphs. 7.2. Gauge symmetries Clearly, one of the most urgent and fascinating questions is the role of symmetries in quantum $eld theories. Having, with the Hopf algebra structures reported here, discovered such a wonderful machinery which encapsulates the quest for locality, one should expect interesting structure when considering local gauge symmetries, in particular also with respect to the role which foliations play naturally in non-commutative geometry [31,42]. There are many aspects which can hopefully be addressed in the near future. • To what extent can Ward and Slavnov–Taylor identities be incorporated in this picture? Do these identities form something like an ideal in the algebra of graphs? Note that the language of external structures allows nicely to formulate concepts like the longitudinal and transversal part of a vertex-correction for example, and is hence well-adopted to address such questions. • Has BRST cohomology a natural formulation in this context? • Gauge theories provide number-theoretical miracles in abundance, with the most signi$cant observation being Rosner’s observation [43] of the vanishing of ?(3) from the -function of quenched QED. While this can be understood heuristically [44,15], eventually the role between internal symmetries and number-theoretic properties must be properly understood. For the practitioner of quantum $eld theory, the real challenge lies in the treatment of the perturbative expansion in circumstances when there is no regularization available which preserves the

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symmetries of the initial theory. A notorious and famous problem at hand is the 5 problem in dimensional regularization [45]. In realistic circumstances like the Standard Model, this already demands a formidable e9ort at the one-loop level if one uses a calculational scheme which violates the BRS symmetry even in the absence of anomalies (see [46] for such an example), which then is an unavoidable e9ort dictated by the demand to restore the BRS symmetry using the quantum action principle. There is one obvious useful role for the Hopf algebra: the analysis at the one-loop level would in many ways not change when extended to any other primitive element of the Hopf algebra, which, being primitive, all share with the one-loop graphs that they have no subdivergences. From there, the Hopf algebra structure governs the iteration of graphs into each other. But then, the prominent role and natural role which $eld-theoretic ingredients like the Dirac propagator and 5 itself, a volume form on four-dimensional space essentially, play in non-commutative geometry [6,42], gives hope for a more profound understanding of this problem in the future. 7.3. The exact renormalization group and the non-perturbative regime Ultimately, the renormalization group is a non-perturbative object, and can indeed be addressed without necessarily making use of the usual concepts of graph-theoretic expansions [47,48]. This is nicely reLected by the fact that the transition from the perturbative to the non-perturbative just amounts, in the picture outlined here, to a Birkho9 decomposition of an actual instead of a formal di9eomorphism. Integrating out high-frequency modes in the functional integral step-by-step produces a sequence of di9eomorphisms of the correlation function under consideration. 20 The Hopf algebra of rooted trees, thanks to its universality, provides the relevant backbone in any case, and indeed rooted trees underly any iterative equation, like, for example, the Wilson equation 9SB = F(SB ) ; 9B for some action parametrized by some cut-o9 B and some suitable functional F. Integrating this functional F now plays the role of the operator B+ in the universal setting of the Hopf algebra of rooted trees [5]. Rooted trees are deeply built into solutions of (integro-) di9erential equations [49,50]. It is no miracle then that on the other hand one $nds that the understanding of the Hopf and Lie algebras of Feynman graphs not only enables high-loop order calculations [10,11,51] which allow to analyze PadXe–Borel resummations [11,51,52] but also allows to $nd exact non-perturbative solutions in new problems. A $rst result can be found in [51]. 7.4. Further aspects Combinatorially, rooted trees are very fundamental objects, and their Hopf and Lie algebra structure underlies not only the combinatorial process of renormalization, but can hopefully be used in the future in other expansions in perturbation theory, starting from a disentanglement of infrared divergent sectors [53] to more general applications in asymptotic expansions [54]. Its universal nature already 20

The fact, emphasized by Polchinski [47], that in such an approach one does not see the graph-theoretical notions emphasized in textbook approaches to renormalization theory is a mere reLection of the fact that one can formulate the Birkho9 decomposition directly on the level of di9eomorphisms of physical observables [9], as exhibited in the previous section.

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allowed to use it in a straightforward formulation of block spin transformations, coarse graining and the renormalization of spin networks [55]. Eventually, one hopes, this basic universal combinatorial structure $nds its way into other approaches to QFT, from the constructive approach [56] which in its nature is very tree-like from a start [57], to the algebraic school [58,59], which all have to handle the basic combinatorial step that we can address a problem only after we addressed its subproblems. 21 Note also that applications of forest formulas in the context of non-commutative $eld theory and string $eld theory (see [62] for a detailed graphical analysis) naturally change the criteria for the subgraphs over which a sum ⊗ = () = ⊗ 1 + 1 ⊗ +

runs, while the results in [4] underline that a Hopf algebra structure can still be established when we vary these criteria. There is no space here to comment in detail on some other mathematical developments which are related to the discovery of the Hopf algebra structure of renormalization. We can only address the interested reader to [63– 65,28]. But note that such mathematical investigations are often very useful for a practitioner of QFT: clearly, the classi$cation of all primitive Hopf algebra elements is of importance even for the case of the undecorated Hopf algebra of rooted trees, and leads for example to the notion of a bigrading which characterizes potential higher divergences algebraically [12,65]. 7.5. Conclusions Rooted trees and Feynman graphs are familiar objects for anybody working on the perturbative expansion of a functional integral, and as familiar are forest formulas and the Bogoliubov recursion. What is new is that there is a universal Hopf algebra on rooted trees, devoted to the problem of singularities along diagonals in con$guration spaces and providing a principle of multiplicative subtraction, which reproduces just these recursions and forest formulas. That Feynman graphs, with all their external structure, form a Lie algebra is a very nice consequence which hopefully gives a new and strong handle for the understanding of QFT in the future. The consequences of the connection to the Riemann–Hilbert problem and the Birkho9 decomposition of di9eomorphisms, the connection between short-distance singularities in perturbation theory and polylogarithms, all this indicates what a rich source of mathematical structure and beauty is imposed on a quantum $eld theory by its in$nities. Acknowledgements A large body of the work presented here was done in past and ongoing collaborations with David Broadhurst and Alain Connes. Helpful discussions with Jim Stashe9 on operads are gratefully 21

The universality of the Hopf algebra can be used to describe e9ective actions in a unifying manner, which was indeed one of the main points of [8,9], while the connection to integrable models promoted in [60,61] can hopefully be substantiated further in the future.

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acknowledged. This work was done in part for the Clay Mathematics Institute. Also, the author thanks the DFG for a Heisenberg fellowship. References [1] J. Collins, Renormalization, Cambridge University Press, Cambridge, 1984. [2] D. Kreimer, On the Hopf algebra structure of perturbative quantum $eld theories, Adv. Theor. Math. Phys. 2 (1998) 303[q-alg=9707029]. [3] D. Kreimer, Chen’s iterated integral represents the operator product expansion, Adv. Theor. Math. Phys. 3 (2000) 627 [hep-th=9901099]. [4] D. Kreimer, On overlapping divergences, Commun. Math. Phys. 204 (1999) 669 [hep-th=9810022]. [5] A. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998) 203 [hep-th=9808042]. [6] A. Connes, D. Kreimer, Lessons from quantum $eld theory: Hopf algebras and spacetime geometries, Lett. Math. Phys. 48 (1999) 85 [hep-th=9904044]. [7] A. Connes, D. Kreimer, Renormalization in quantum $eld theory and the Riemann–Hilbert problem, JHEP 9909 (1999) 024 [hep-th=9909126]. [8] A. Connes, D. Kreimer, Renormalization in quantum $eld theory and the Riemann–Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210 (2000) 249 [hep-th=9912092]. [9] A. Connes, D. Kreimer, Renormalization in quantum $eld theory and the Riemann–Hilbert problem. II: The beta-function, di9eomorphisms and the renormalization group, Commun. Math. Phys. 216 (2001) 215. [10] D.J. Broadhurst, D. Kreimer, Renormalization automated by Hopf algebra, J. Symb. Comput. 27 (1999) 581 [hep-th=9810087]. [11] D.J. Broadhurst, D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade–Borel resummation, Phys. Lett. B 475 (2000) 63 [hep-th=9912093]. [12] D.J. Broadhurst, D. Kreimer, Towards cohomology of renormalization: Bigrading the combinatorial Hopf algebra of rooted trees, Commun. Math. Phys. 215 (2000) 217 [13] D. Kreimer, ShuUing quantum $eld theory, Lett. Math. Phys. 51 (2000) 179 [hep-th=9912290]. [14] D. Kreimer, Feynman diagrams and polylogarithms: ShuUes and pentagons, Nucl. Phys. Proc. Suppl. 89 (2000) 289 [hep-th=0005279]. [15] D. Kreimer, Knots and Feynman Diagrams, Cambridge University Press, Cambridge, 2000. [16] D. Kreimer, R. Delbourgo, Using the Hopf algebra structure of QFT in calculations, Phys. Rev. D 60 (1999) 105025 [hep-th=9903249]. [17] H. Epstein, V. Glaser, The role of locality in perturbation theory, Ann. Inst. H. PoincarXe 19 (1973) 211. [18] R. Stora, Renormalized perturbation theory: a theoretical laboratory, talk given at Mathematical Physics in Mathematics and Physics, Siena, June 2000. [19] W. Fulton, R. MacPherson, A compacti$cation of con$guration spaces, Ann. Math. 139 (1994) 183. [20] J.M. Gracia-Bondia, S. Lazzarini, Connes–Kreimer–Epstein–Glaser renormalization, hep-th=0006106. [21] T. Krajewski, R. Wulkenhaar, On Kreimer’s Hopf algebra structure of Feynman graphs, Eur. Phys. J. C 7 (1999) 697 [hep-th=9805098]. [22] R. Brunetti, K. Fredenhagen, Microlocal analysis and interacting quantum $eld theories: renormalization on physical backgrounds, Commun. Math. Phys. 208 (2000) 623 [math-ph=9903028]. [23] C. Kassel, Quantum Groups, Springer, Berlin, 1995. V [24] M. Mertens, Uber die Rolle von Hopfkategorien in stVorungstheoretischer Quantenfeldtheorie, Ph.D. Thesis, Mainz University, Fall 2000 (in German). [25] J.L. Loday, La renaissance des opXerades (The rebirth of operads) SXeminaire Bourbaki, Vol. 1994=95. AstXerisque No. 237 (Exp. No. 792, 3) 1996, p. 47 (in French). [26] C. Brouder, On the trees of quantum $elds, hep-th=9906111; C. Brouder, A. Frabetti, Renormalization of QED with planary binary trees, hep-th=0003202. [27] C. Brouder, Private communication.

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Physics Reports 363 (2002) 425 – 545 www.elsevier.com/locate/physrep

Renormalization group theory of crossovers Denjoe O’Connora; ∗ , C.R. Stephensb b

a Departmento de F sica, Cinvestav, Apartado Postal 14-740, M exico D.F. 07000, Mexico Instituto de Ciencias Nucleares, UNAM, Circuito Exterior, Apartado Postal 70-543, M exico D.F. 04510, Mexico

Received September 2001; editor: I: Procaccia We dedicate this review to the memory of Lochlainn O’Raifeartaigh Contents 1. Introduction 2. Phenomenology of crossovers 3. The O(N ) Landau–Ginzburg–Wilson Model 4. Exactly solvable models 4.1. The Ising model on a torus 4.2. The large N limit of the O(N ) model 5. Renormalization and the RG 5.1. The coarse graining approach 5.2. Coarse graining and crossovers 6. Renormalization as a coordinate transformation 6.1. Which coordinates should be reparametrized? Conceptual arguments 6.2. Coordinate transformations should be environment dependent 7. Environmentally friendly renormalization 7.1. Which coordinates should be reparametrized? Perturbative arguments 7.2. What :ow parameter should be used? 7.3. An analytic parametrization via normalization conditions 7.4. Fixing the sliding scale 7.5. Integrating along contours in the phase diagram 7.6. E=ective exponents 7.7. Formal scaling forms

427 433 438 442 443 446 452 453 455 456 459 460 462 462 466 469 474 475 488 489

8. Perturbative results for crossover between the critical >xed point and the strong coupling discontinuity >xed point 8.1. One-loop equation of state for an O(N ) model 8.2. Crossover between >rst and second order transitions 9. Perturbative results for >nite size scaling 9.1. Dimensional crossover in a >lm geometry with periodic boundary conditions: the disordered phase 9.2. Dimensional crossover in a >lm geometry with periodic boundary conditions: the ordered phase 9.3. The e=ect of boundary conditions on the crossover 10. Quantum to classical or >nite temperature crossover 10.1. Ising model in a transverse magnetic >eld 10.2. Finite temperature renormalization 11. Uniaxial dipolar ferromagnet 12. Bicritical crossover 13. Kinematic crossovers 13.1. Asymmetric scattering and the Regge limit 14. Conclusions and comments Acknowledgements References

∗

Corresponding author. E-mail addresses: denjoe@>s.cinvestav.mx (D. O’Connor), [email protected] (C.R. Stephens).

c 2002 Published by Elsevier Science B.V. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 1 0 0 - 4

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Abstract Crossover phenomena are both ubiquitous and important. In this report, we review the foundations of renormalization group theory as applied to crossover behavior and consider several paradigmatic applications. We con>ne ourselves to situations where the crossover can be described in terms of an e=ective >eld theory, in particular concentrating on the prototypical example of an O(N ) model in a constrained geometry or at >nite temperature. Calculation of universal crossover scaling functions is considered where we show how the renormalization group can in principle be applied to the latter to obtain expressions as accurate as those of standard universal quantities, such as critical exponents and amplitude ratios. Particular attention is paid to the scaling equation of state for an O(N ) model. ? 2002 Published by Elsevier Science B.V. PACS: 64.60.Ak; 05.70.Fh; 11.10.Hi; 11.10.Wx Keywords: Renormalization group; Crossover; Phase transitions; Dimensional reduction; Finite size scaling

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1. Introduction Many of the most exciting and diJcult problems in the physical sciences, and indeed many other >elds, have a common element—they are associated with systems that exhibit radically di=erent e=ective degrees of freedom (EDOF) at di=erent scales. Each area of physics has its own paradigmatic examples. For instance, in particle physics, in the strong interactions, it is the change between a description in terms of quarks and gluons at high energies and mesons and baryons (bound states of quarks and gluons) at low energies. In statistical mechanics, and in particular in critical phenomena, such qualitative changes in the EDOF can give rise to di=erent scaling behavior in di=erent asymptotic regimes. In the literature, such a change is known as a crossover. Crossovers usually occur in a small region of the parameter space and are important both theoretically and experimentally. Although crossover phenomena are ubiquitous they are usually treated on a case by case basis rather than in terms of a unifying framework. Field theory is a tool applicable to a host of physical systems that span many di=erent >elds of interest and o=ers a formal framework within which a large class of crossover phenomena may be studied. Our aim here is to review applications of the renormalization group (RG) to crossover phenomena describable within a >eld theoretic formalism, viewing the RG as a reparametrization of the parameters (coordinates), e.g. temperature, coupling constant, etc., with which we describe the theory, rather than as a coarse graining. 1 We will for the most part keep the formalism as general as possible, emphasizing the generality of the approach and specializing to speci>c crossovers of interest in order to make concrete illustrations of the basic methodology. Our reason for doing this is that the formalism is applicable to a very wide class of crossovers. By presenting the general results, the reader is then at liberty to treat a crossover of particular interest to him=her merely by inserting into the appropriate parts of the formalism the particular details speci>c to that crossover. We will use a language that is hopefully accessible to people with only a little >eld theory background, be it in either statistical physics or particle physics where the applications of the formalism are equally applicable. We will, somewhat unashamedly, concentrate more on the particular methodology “environmentally friendly” renormalization [3–12] and will compare and contrast it with other known methods. The de>ning characteristic of a crossover problem—the non-trivial scale dependence of the EDOF— is in marked contrast to that of scale-invariant systems, many of which have yielded to theoretical analysis and for which there now exist a variety of theoretical tools for handling them. The principal such tool is the RG, but in two dimensions conformal symmetry and the study of conformal >eld theory has led to deep insights and many results about such systems [13]. However, much less is known, even in two dimensions, about crossover. The description of crossover behavior is an important problem since ultimately real physical systems are not scale invariant. In fact, the world is full of important length scales. An useful initial step in the study of scale non-invariant systems is to understand where scale invariant systems >t into the greater scheme of things. Scale invariance, in RG terms, is associated with the existence of >xed points of the RG :ow equations. A linearized neighborhood of such 1

See the reviews of Jona-Lasinio [1] and Shirkov [2] in this series for a discussion of the relationship between the two formulations.

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a >xed point determines “universal properties” of the system, such as critical exponents and amplitude ratios. Such universal properties are insensitive to microscopic detail, depending only on robust parameters such as dimensionality and order parameter symmetry, both of which characterize the >xed point. On the contrary, a crossover is associated with more than one set of exponents and amplitude ratios and depends on some of the parameters of the system. In this case, a linearized analysis is inadequate as there exists more than one >xed point. In fact, there may be several such >xed points. However, one may always consider :ows between pairs of them. A particular crossover can fruitfully be thought of as being induced by an “anisotropy” parameter, g; the two end >xed points being associated with the isotropic, g = 0, system and the anisotropic, g = ∞, system respectively. Typically, standard approximations will be capable of accessing the non-perturbative features of one of these >xed points but not simultaneously the other. Scaling analysis indicates that the di=erent sets of exponents and amplitude ratios should >t together into a crossover scaling function. The most basic of these scaling functions, from a statistical mechanics point of view, is the scaling equation of state. Even this simple function however, has proved to be notoriously diJcult to calculate, and only in recent years have reasonably precise determinations emerged (see [14] for a recent review). A particularly interesting set of scaling functions are the e=ective exponents, which interpolate between the >xed constant values of the corresponding isotropic and anisotropic critical exponents. Also of great interest are critical temperature shifts and rounding due to the presence of the anisotropy. It is not our intention to give a review of all that has been done on crossover phenomena here—it is far too vast a subject—consequently, we will be somewhat selective in our treatment, and, more apologetically, in our references. In critical phenomena the subject has received much more attention than in particle physics (for an early review see [15]), even though many important problems there are essentially crossover problems. In principle, basically any laboratory system will exhibit crossover behavior in some regime. Some of the more experimentally accessible ones are: uniaxial dipolar ferromagnets [16], systems exhibiting a bicritical point [17], dimensional crossover in liquid 4 He [18], quantum–classical crossover in quantum ferromagnets ([19] and references therein), bulk=surface crossovers [20,21], crossover in random systems [22] and mean >eld to critical crossover [23]. The latter is particularly important as there are large quantities of accurate data available both from experiments and numerical simulations with which one can compare one’s theoretical predictions. Dimensional crossover, one of the chief concerns of this paper, has been studied mainly in the context of >nite size scaling (see [24,25] for reviews) though it plays a role in a vast array of phenomena. In numerical simulations of critical behavior, the intrinsically >nite size of the lattice ensures that eventually >nite size e=ects will play a predominant role and so must be thoroughly understood in order to unambiguously infer bulk critical properties from the simulation [26 –30]. Equally, in the physics of thin >lms and single crystals >nite size e=ects appear in a very direct way, dependent on the typical size ‘L’ of the system. Apparent changes in dimension can also be associated with other less obvious causes. 2 A wellknown one is >nite temperature quantum >eld theory, which is equivalent to >eld theory on a cylinder, the radius of the cylinder being ˝, the inverse absolute temperature 1=T . Finite temperatureinduced dimensional crossovers play an important role in phase transitions in relativistic >eld theory, 2

A de>nition of a “>nite” size system associated with a dimensional crossover can be found in [31].

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where they have been treated in the context of environmentally friendly renormalization [32–39], as well as using other techniques [40,41], and in condensed matter physics [42– 44]. External electric and magnetic >elds can also induce dimensional crossovers, a paradigmatic example being that of a superconductor in an external magnetic >eld, H [45 –50] where :uctuations are con>ned to Landau orbits, leading to an e=ective dimensional reduction from d → (d − 2). Some other interesting and non-trivial examples are the cosmological constant in De Sitter space [51] and the Regge limit in quantum >eld theory [52,53]. In all these cases it is useful to think of the anisotropy parameter that induces the crossover as an “environmental” variable. 3 In spite of the fact that it was devised to treat scale-invariant systems the RG was used from a very early stage of its development in the calculation of crossover scaling functions. For a typical crossover, induced by a generic anisotropy parameter g, scaling formulations have often been based on RGs that are g independent. This automatically leads to critical and crossover exponents that are de>ned with respect to the isotropic >xed point. The crossover to the anisotropic >xed point is taken into account by corrections to scaling around the isotropic one. However, given that g is a relevant parameter with respect to the latter, corrections to scaling become large and perturbatively uncontrollable in the anisotropic regime. Hence, generically, such RGs are incapable, in and of themselves, of bridging the crossover to the anisotropic >xed point since the resultant scaling functions must encode the exponents of the latter. Put another way: the >xed points of isotropic RGs will not yield all the possible points of scale invariance of the system. In particular, the anisotropic scale invariant system will be inaccessible. In this sense, even the crossover between the Gaussian and Wilson–Fisher >xed points requires an RG that is dependent on the parameter associated with the leading irrelevant operator (irrelevant with respect to the Wilson–Fisher >xed point but relevant with respect to the Gaussian one). The desire to make accessible another >xed point besides the isotropic one has often entailed the matching of asymptotic expansions around the anisotropic and isotropic >xed points, or the use of high-temperature expansions in conjunction with an ansatz for the scaling function [54 –56]. Exact models have also played a role. For instance, the two dimensional Ising model [57] and the spherical model [58– 60] have been the testing laboratories in the context of >nite size scaling. An old RG approach, due to Riedel and Wegner [61], and used by others, utilized “model recursion relations” to access the two >xed points of the crossover and assumed that they were known. Subsequently, RG equations for the scaling >elds were postulated that interpolate between the two >xed points. However, there are many di=erent crossover systems that exhibit crossover between exactly the same two >xed points. Therefore, a scheme that makes no reference to the underlying microscopic Hamiltonian can only o=er qualitative information. Universality implies that two systems, e.g. a simple :uid and an Ising ferromagnet, that lie in the same universality class can, via a suitable rescaling of variables, be shown to have exactly the same infrared (IR) properties in a suJciently small temperature neighborhood of the critical point. For crossovers the asymptotic >xed points do not determine “crossover universality classes” as the scaling functions associated with di=erent crossovers between the same two >xed points cannot generally be transformed into one another by a change of scaling variable. However, there does exist a concept of two systems being in the same crossover universality class. A pertinent example is that of a (d − 1)-dimensional quantum Ising 3

Such a nomenclature is intuitively obvious in the case of, say, >nite size or quantum classical crossover. By a slight abuse of language we shall refer to any parameter that induces a crossover as an environmental parameter.

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model in a transverse magnetic >eld and a d-dimensional Ising-like ferromagnetic thin >lm with periodic boundary conditions. Two systems will be in the same crossover universality class if all of the universal scaling functions associated with the crossover of both systems are identical up to a constant rescaling. Wilsonian RGs of the “momentum shell integration” type have been frequently employed in the analysis of crossover problems, with varying success. One of the earliest and most noteworthy formulations, directly applicable to crossover problems, is Nelson’s “trajectory integral” method [62] which calculates the free energy as a line integral along a RG trajectory. Nelson and others used the methodology to calculate crossover scaling functions for bicritical systems in the disordered phase [63,64] and isotropic dipolar ferromagnets [65]. Coarse graining methods in general are diJcult to extend to higher orders. Also, in the case of a crossover it is easy to use too rough an approximation in the shell integration. An example of this is the case of quantum magnets [42], where the temperature independent coarse graining implemented made the quantum–classical dimensional crossover inaccessible. Given that many physically di=erent systems can crossover between the same asymptotic >xed points it is important to be able to pick out the details of the crossover curves in order to distinguish between them. By using an approximation on the momentum shell integration, it is easy to blur such distinctions. Another related methodology is that of the ‘exact renormalization group’ (see the reviews of Bagnuls and Bervillier [66] and of Berges et al. [67]). Unfortunately, even the ‘exact RG’ needs approximation, since very few systems yield exact solutions. Various “non-perturbative” approximation techniques have been developed, such as the derivative expansion. The best understood approximation techniques however, are still those associated with linearization about a Gaussian theory with the non-linearities being developed in a perturbation series of Feynman diagrams ordered by the number of loops. In this context a resummation technique is required. There are several RG formulations that may be used to achieve this. We believe the most powerful is still the original RG of StTuckelberg–Peterman, Gell–Mann Low and Bogoliubov–Shirkov [68–70], which we refer to as the reparametrization RG. Although discovered in perturbative studies of quantum electrodynamics in the process of removing the ultraviolet (UV) divergences from measured quantities, it was pointed out very early in the development of the subject, by Blank Bonch–Bruervich and Shirkov [71], that the RG is not dependent on the existence of such UV divergences and that it could be a useful tool in a variety of >elds (they mention condensed matter physics). An old but elegant review of the principles of the RG method was given by Ginzburg and Shirkov [72] in 1966 where the discussion makes this especially clear. Combined with the scaling ideas developed in the theory of phase transitions this original >eld theoretic RG evolved in the 1970s into a powerful tool in both statistical and particle physics. One of the main aims of this review is to explain the modi>cations necessary within this formalism in order to be able to adequately describe crossover behavior. From a modern perspective, the original >eld theoretic renormalization can now be seen to be nothing more than a coordinate change from original bare parameters to renormalized parameters. A coordinate transformation, of itself, does not change the physics, but, as we shall see and demonstrate, one coordinate system may be vastly superior to another when doing perturbative calculations, especially when combined with the notion of a “sliding scale” for the renormalization point. It turns out that one can make powerful use of this freedom in a crossover problem by choosing the coordinate transformation to be one that depends on the “environment”, in other words

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depends on all the relevant scales present. This is especially useful in the perturbative treatments typically necessary in >eld theory, where the coordinate change is adjusted with the perturbation order. When compared to coarse graining RGs, less attention has been paid to describing crossover behavior from a purely >eld theory RG point of view. This is mainly because of the, now rapidly disappearing, prejudice that >eld theoretic renormalization is for treating UV divergences. In the language of perturbative >eld theory the principal symptom of a crossover is the existence of divergences within the Feynman diagrams that are not of UV origin. Amit and Goldschmidt [73] in a signi>cant paper introduced the concept of generalized minimal subtraction (GMS) 4 wherein a subtraction scheme is implemented, or counterterms added, that remove not only UV divergences but also divergences associated with other asymptotic limits. Importantly, the counterterms that implement such renormalizations have to depend on the parameter that induces the crossover. Amit and Goldschmidt considered crossover in the context of crossover at a bicritical point in the disordered phase. Their results for the susceptibility e=ective exponent, e= , however, di=er from those found in [63] and [64] using momentum shell integration, in particular in that the latter >nd a characteristic “dip” in the curves. We believe this to be due to a de>ciency of GMS which fails to capture the true behavior of the leading irrelevant operator. GMS was also applied to bicritical systems in the ordered phase using a non-linear -model representation [75]. Other applications include: uniaxial dipolar ferromagnets [76]; isotropic dipolar ferromagnets [77] and the crossover between isotropic and directed percolation [74], while Schmeltzer [78], in a relatively unnoticed work, used it to calculate e= to one loop for three-dimensional quantum ferroelectrics. Lawrie [79] considered dimensional crossover for d-dimensional quantal and (d + 1)-dimensional >nite-sized Ising models for 3¡d¡4. The expansion used there could not capture the crossover between two non-trivial >xed points since the deviation from the upper critical dimension is di=erent at the ends of the crossover and the respective critical exponents are sensitive to this di=erence. Nemirovsky and Freed ([80] and references therein) used minimal subtraction techniques but failed to be able to access the full crossover. Field theoretic results for dimensional crossover in a fully >nite geometry or a cylinder were >rst obtained in [81,82] and extended by Dohm and coworkers [83]. These techniques are RG “hybrids” in that they isolate the most important IR modes, then use the RG to calculate the parameters of an e=ective Hamiltonian for these lowest modes, the integration of which corresponds to either a zero or one-dimensional functional integral. In the case where the >nite system itself exhibits critical behavior however, such as a thin >lm, this methodology fails, as now the lowest modes themselves correspond to a non-trivial >eld theory. Additionally, there are signi>cant discrepancies between these “lowest mode theory” [84,85] results and numerical simulations [30] above the upper critical dimension. Another important area of application has been the study of surface e=ects. For systems with surfaces the interactions in the surface may be “enhanced” relative to those of the bulk. Hence, crossovers may occur as a function of the relative strength, c, of this surface enhancement. Generically, there are three lines of continuous phase transition—the ordinary, extraordinary and the surface. The ordinary and extraordinary transitions can be accessed via standard expansion techniques and corresponding crossover scaling functions calculated [86]. Interestingly, the surface transition has still not yielded to a theoretical analysis. In this case the crossover involves a dimensional reduction 4

For a recent review of GMS see [74].

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as scaling behavior in the “bulk” is that of a d-dimensional system, while at the surface it is that of a (d − 1)-dimensional one. The crossover between the two, due to the presence of the surface as an inhomogeneity, is governed by the distance from the surface and hence an environmentally friendly renormalization will involve a position-dependent renormalization. Although some progress has been made along these lines [87–89] it remains very much an open and diJcult problem. The crossover between classical and critical behavior, i.e. the crossover to mean >eld theory, has also served as an important test problem for RG calculations of crossover scaling functions ranging from early work [90,91] on calculating the susceptibility scaling function to O(2 ) to more sophisticated recent calculations. The most salient feature of this problem is that even though it is probably the simplest crossover imaginable, the anisotropic >xed point being trivial, it is still not particularly well understood. As mentioned, one of the most interesting, and most ubiquitous, crossovers is that induced by >nite size e=ects. For >nite size systems the environmental variable which sets the crossover scale is L, the system size, while in quantum systems it is ˝, the inverse absolute temperature. The crossover in both cases is a dimensional one and it is this physical setting of dimensional crossover that we will take as our paradigm. The simplest setting for this that involves crossover between two non-trivial >xed points is >eld theory on a >lm geometry with periodic boundary conditions, or a quantum system at >nite temperature. In both cases the >eld theory is that on a topologically non-trivial manifold Rd−1 ×S 1 , i.e. a d-dimensional cylinder. When d is large enough (typically ¿ 2) the crossover is non-trivial and can admit critical singularities for any value of the radius of the cylinder. The interesting point here is that the asymptotic ends of the crossover are scale invariant, being associated with massless >eld theories in d dimensions and d − 1 dimensions, respectively. A perturbative treatment of a near massless >eld theory on Rd−1 ×S 1 is plagued by the presence of IR divergences. These are in many ways reminiscent of the UV divergences discovered in quantum electrodynamics and other relativistic quantum >eld theories and in fact are more similar than one might expect. The solution to the original UV divergences problem was renormalization and the consequent RG. Similarly, in the case of dimensional crossover, as we shall see, an appropriately chosen renormalization prescription and its consequent RG solves the problem of IR divergences and reveals the underlying physics. Another interesting crossover we shall consider in some detail is that of the equation of state for an O(N ) model. RG calculations of this quantity date back to the early 1970s [92] where the expansion was used. Although giving reasonable results for small x = t= ’V 1= , the large x limit was ill de>ned. An expansion in z = ’t V − could be carried out to access the large x (small z) limit but no uniform approximation was available. Implementing an expansion within a parametric representation [93,94] ameliorated the problem but left the case N ¿1 problematical due to the presence of Goldstone bosons. High order calculations have been carried out at >xed dimension but once again only for N = 1 due to the problems of having two length scales, transverse and longitudinal. These >xed dimension calculations also have the problem that they use the disordered phase correlation length to investigate the ordered phase. Various methods have been used for the case N ¿1. Many emphasize the analogy between this situation and that of a bicritical point [95 –97] but none seem to be readily extendible to higher orders. The format of the paper will be as follows: in Section 2 we will give a brief exposition of some phenomenological elements of crossovers introducing key concepts such as shifts and rounding,

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e=ective exponents and considering generic properties of crossover scaling functions. The motivation here is to introduce a set of physically interesting quantities generic to a large class of crossovers. In Section 3 we will introduce some basic elements of >eld theory and perturbation theory showing how various physical quantities of interest can be related to the vertex functions of the theory viewed as building blocks. In Section 4 we utilize some exact models to illustrate some of the quantities discussed in Section 2. In Sections 5 and 6 we discuss the two principle threads of RG theory: the coarse graining approach and the reparametrization approach. In Section 7 we present the basic methodology of “environmentally friendly” renormalization, seen as a variant of the latter, showing that it can be used to systematically calculate any of the scaling functions of interest associated with a large class of crossovers, including temperature shifts and rounding. In Section 8 we use the formalism to examine the crossover in the equation of state itself and, in particular, the crossover between a >rst order and second order phase transition. In Section 9 we consider environmentally friendly renormalization applied to >nite size systems and the problem of dimensional reduction, concentrating mostly on thin >lms. In Section 10 we consider quantum=classical crossover both in its relativistic and non-relativistic settings. Sections 11–13 are associated with various other applications of interest. Finally, in Section 14 we draw some conclusions.

2. Phenomenology of crossovers The purpose of this section is to lay out, chie:y in the context of scaling theory, the principal phenomena that are associated with crossovers. We will consider the physical setting to be critical phenomena in the presence of an anisotropy parameter, g. The extension to more than one anisotropy parameter is straightforward. A useful framework in which to understand crossover phenomena is in terms of symmetry. For g = 0 we assume there is a symmetry, S, and call the system “isotropic”. The anisotropy g breaks this symmetry S → S where S ⊂ S, so that the full symmetry is never strictly present when g is non-zero. For a crossover we always require two scales and the crossover occurs as the ratio of these two scales changes. A very natural scale in a near-critical system is the correlation length. In this case, one may speak of isotropic and anisotropic correlation lengths, 0 and g , respectively. If g is also a physical mass scale, i.e. a non-linear scaling >eld, then the crossover naturally occurs as a function of the scaling variable x = g and is typically centered around x∼1. If g is a linear scaling >eld then the appropriate crossover variable is x = g =t, where t = 2 (T − Tc )=T , with being a microscopic scale. The exponent is referred to as the crossover exponent and is determined by the isotropic system. Small values of correspond to slow crossovers. If we consider a generic scaling function F(x; y; z; : : :) then the canonical scaling behavior in the isotropic (x → 0) limit is x → 0;

F → xa A(y; z; : : :) :

(1)

On the other hand, in the anisotropic limit the scaling function exhibits a zero or singularity, xs , so that in the vicinity of this zero, x → xs , one >nds x → xs ;

F → (x − xs )ap Ap (y; z; : : :) :

(2)

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The exponents a and ap are critical exponents associated with the two di=erent asymptotic limits, where the symmetries are S and S , respectively, while A and Ap are universal scaling functions that are x independent. Throughout the paper, unless otherwise stated, we will use the notation that critical exponents without indices are those of the isotropic >xed point while the subscript p will denote an exponent associated with the anisotropic >xed point. Of course, the scaling variables y; z, etc., may depend on the anisotropy variable g. To obtain such di=erent asymptotic scaling behavior it is clear that the scaling function F itself must be non-analytic. Some pertinent examples of anisotropy parameters associated with crossovers are: (i) in our paradigmatic example of dimensional crossover, g = 1=L for a thin >lm, where L is the >lm thickness. When g = 0 the system will exhibit d-dimensional symmetries whereas for g = 0 it will exhibit (d − 1)-dimensional symmetries asymptotically. Accordingly, a and ap will be d and (d−1)-dimensional exponents, respectively; (ii) for a uniaxial dipolar ferromagnet g=1=2 , where is the strength of the dipolar coupling. In this case a and ap correspond to d and (d + 1)-dimensional exponents; (iii) for the crossover to mean >eld behavior g = , where is the strength of the self-interaction in a standard Landau–Ginzburg–Wilson Hamiltonian. In this case, a and ap will be the exponents associated with the Gaussian (non-interacting) and Wilson–Fisher (interacting) >xed points; (iv) for surface to bulk crossover g = z −1 , where z is the distance from the surface in a semi-in>nite system. In this case, there can be a dimensional crossover from bulk to surface behavior as a function of z, with ap and a representing the surface and bulk exponents; and (v) spin anisotropy for a Heisenberg system, where g is a measure of the strength of the anisotropy term in the Hamiltonian Haniso = −Jg i; j (siz sjz − (1=2)(six sjx + siy sjy )). For g¿0 the strength of the “parallel” (i.e. along z) coupling is J (1 + g) while for the “perpendicular” coupling it is J (1 − g). In this case, the system orders into an Ising-like state. For g¡0, on the contrary, the system exhibits XY order, while for g = 0, O(3) order. In (i), (ii) and (iv) g enters as a non-linear scaling >eld while in (iii) and (v) it corresponds to a linear scaling >eld. Other important phenomenon associated with crossovers are shifts in the critical temperature and rounding of the critical singularities. If the anisotropic system exhibits critical behavior then we de>ne the shift 5 as &g ≡ 2

Tc (0) − Tc (g) ; Tc (g)

(3)

where Tc (0) is the critical temperature of the isotropic system and Tc (g) is the critical temperature of the anisotropic system. When the anisotropic system does not possess a true critical point then a shift can still be de>ned relative to a suitably de>ned “pseudo-critical” point. A natural de>nition of Tc (g) in this case would be “that temperature at which some thermodynamic quantity such as the susceptibility or speci>c heat has a maximum”. For the shift one >nds that &g ≡ 2

Tc (0) − Tc (g) ∼bg Tc (g)

for g → 0 ;

where b is an overall non-universal amplitude and 5

(4) is the shift exponent.

We have chosen this de>nition for consistency with our de>nition of t in later sections. One could of course de>ne the shift as 2 (Tc (0) − Tc (g))=Tc (0). The two de>nitions are equivalent in the critical regime. Note that we have chosen & to be the negative of that which appears in some of the literature.

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When the critical singularities disappear, for example in the case of a >nite size sample, due to the anisotropy, i.e. they are rounded, one can de>ne a rounding temperature as “that temperature where the curve associated with a thermodynamic quantity begins to deviate signi>cantly from its asymptotic isotropic form”. This notion is naturally more diJcult to quantify. For instance, in >nite size scaling it can be de>ned as that temperature at which the speci>c heat curve for the anisotropic system begins to deviate signi>cantly from the isotropic one. Often rounding is de>ned relative to Tc (g) but can equally well be de>ned with respect to Tc (0). In any case, scaling arguments lead to the conjecture that & r ≡ 2

T ∗ (g) − Tc (0) ∼ cg) T ∗ (g)

for g → 0 ;

(5)

where c is, once again, a non-universal amplitude and ) is the rounding exponent. In the above case of a single crossover, the amplitudes of the shift and rounding are constants. However, in the case of more than one crossover they can be scaling functions in their own right that can also exhibit universal properties. A simple example of this is the shift for a >nite size system, such as an Ising-like thin >lm, with uniaxial dipolar interactions as well as the usual short-range interactions. In this case the >nite size shift is a function of 1=2 L. Let us now consider crossover in the context of some explicit scaling functions. Consider a susceptibility scaling function, *(x; V y; z; : : :), scaled with respect to the g = 0 asymptotic behavior so that * = t − *(x; V y; z; : : :) ;

(6)

where t = 2 (T − Tc (0))=T , Tc (0) being the bulk (“isotropic”) critical temperature and is the bulk susceptibility exponent. For x → 0 *V → *(0; V y; z; : : :) while for x → xs , *V possesses a singularity such that the susceptibility exhibits leading scaling behavior characteristic of the anisotropic >xed point, ap i.e. *∼(x−x V s ) , where ap =− p . Note that *V may always be expanded in x around the isotropic >xed point, thus providing the leading corrections to scaling due to g. Such corrections cannot however, at any >nite order, provide the non-analytic corrections needed to access the anisotropic >xed point. As a more explicit example, consider the susceptibility of a >nite size system *(t; L) = t − *(x) V ;

(7)

where x = t + L and + is the bulk correlation length exponent. In the limit x → ∞ *V → const thus yielding the characteristic isotropic critical behavior. If there is a crossover to an anisotropic critical point associated with an exponent p then in the limit t + L → xs *V → A(x − xs )− p with A an universal amplitude 6 in order to yield the correct anisotropic critical behavior. This implies that &L ∼xs =L1=+ . If the sample were of >nite size then there would be no singularity and one could, for example, de>ne the shift with respect to the maximum of the susceptibility in which case F (x) must have a zero at x = xc , which once again implies that &L ∼xc =L1=+ .

6

Since the scaling function *V is universal once x is chosen, then all constants derived from it must be universal amplitude ratios.

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Besides obvious functions of interest, such as the crossover free energy and equation of state, a very natural and intuitive set of scaling functions to examine are the e=ective exponents. One natural de>nition of these functions for magnetic systems, with H the external magnetic >eld, is +e=

d ln (g) =− ; d ln t H =0

-e=

d ln G(r) =2−d− ; d ln r T =Tc (g)

e=

d ln *(g) =− ; d ln t H =0 e=

d ln C(g) =− ; d ln t H =0

(8)

(9)

above Tc and 0e=

d ln H (g) = d ln t T =Tc (g)

+e=

d ln (g) =− ; d ln |t| H =0

d ln C(g) =− ; d ln |t| H =0

e=

(10)

e=

d ln *(g) =− ; d ln |t| H =0

(11)

d ln ’(g) V = ; d ln |t| H =0

(12)

e=

below Tc , where is the correlation length, * the susceptibility, G(r) the two-point correlation function as a function of separation r, and C is the speci>c heat. We are here considering the e=ective exponents to be associated with certain contours in the phase diagram. Obviously, analogous quantities could be de>ned for other contours. For instance, one can approach the coexistence curve along an isotherm other than the critical one, in which case it is natural to generalize 0e= . Any such i generalized e=ective exponent, Ee= , is itself a crossover scaling function and, appropriately de>ned, interpolates between the asymptotic values E i and Epi associated with the isotropic and anisotropic theories, respectively. There is one last concept we will introduce here that will be found to play a special role in dimensional crossover and conventionally only enters in corrections to scaling. Consider the vertex function associated with the four point coupling, 2(4) (see Section 3 for notation); its behavior for dimension d near the isotropic >xed point is given by d ln 2(4) = (4 − d − 2-)+ : d ln t

(13)

This quantity can be used to obtain information about the dimensionality, d, of the system. In many crossovers the e=ective dimension can change and therefore it is natural to introduce the concept of an e=ective dimensionality, de= . Schematically, one can think of de>ning de= through an

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analog 7 of (13) d ln 2(4) = (4 − de= − 2-e= )+e= : d ln t g

437

(14)

This e=ective dimension can be thought of as a measure of the importance of the leading correction to scaling due to the ’4 operator. In the case of a crossover these corrections to scaling actually interpolate between the isotropic and anisotropic critical points. In fact, for crossovers where the di=erent asymptotic regimes correspond to systems with di=erent upper critical dimensions, de= , in the absence of transients, correctly gives the change in upper critical dimension as one interpolates between the two >xed points. The concept of an e=ective dimension also arises naturally in the context of numerical simulations of >nite size scaling [98,100] where it has been used to describe crossover scaling functions in two- and three-dimensional >nite Ising models. An important notion associated with the vicinity of a critical point, or more generally, when the length scales of the physics of interest are a, where a is a microscopic scale such as the lattice spacing, is that of universality classes. It is possible to generalize the standard notion of a universality class to that of a crossover universality class, where two systems will be said to be in the same crossover universality class if all of their universal scaling functions are the same (up to constant rescalings). This is a much stronger requirement than that they possess the same critical exponents. An example of two systems that possess the same asymptotic exponents but are not in the same crossover universality class is that of a d-dimensional ferromagnet with uniaxial dipolar interactions and a ferromagnet in a (d+1)-dimensional >lm geometry with periodic boundary conditions. Another example is a (d+1)-dimensional Ising >lm with periodic boundary conditions and one with anti-periodic boundary conditions. All these systems are in the same asymptotic universality classes—those of a (d + 1)- and d-dimensional Ising model—in that they yield the same asymptotic exponents in the limits g → 0 and g → ∞, where for the Ising thin >lm g = 1=L and for the uniaxial system g = 1=2 . In contrast a d-dimensional quantum ferromagnet and a (d + 1)-dimensional ferromagnetic thin >lm with periodic boundary conditions is an example of two distinct systems that are in the same crossover universality class. An explicit model of a quantum ferromagnet is the transverse >eld quantum Ising model where g = T , T being the temperature. Given the strong correspondence between standard universality and critical exponents and crossover universality and e=ective exponents it is natural to ask if there exist analogs of the standard scaling laws for the e=ective exponents. Such e=ective exponent laws have sometimes been conjectured based on experimental analyses [101] and it has been demonstrated to lowest order in that the relation e= +e= + e= =2 is valid for corrections to scaling to the standard critical exponents. Chang and Houghton [102] however have shown that the relation is violated at higher orders. And there is evidence [30] that they are violated in the crossover between critical and mean >eld behavior. In + + = 2, along with some other interesting [103] it was shown that the exponent law e= e= e= universal relations, holds exactly for corrections to scaling in the planar Ising model. In [12] it was explicitly demonstrated that in the large N limit of an O(N ) Landau–Ginzburg–Wilson model that the e=ective exponents satis>ed all the standard scaling relations, including hyperscaling. All the e=ective exponents were in fact found to be a function of a single universal function which could 7

We will see later that renormalization ideas provide the correct natural generalization of de= to be 4 − , where is the anomalous dimension of the leading irrelevant operator.

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be identi>ed as an e=ective dimensionality de= . Although there is some evidence from simulations [100,104] on Ising models that such laws are obeyed it is now clear that this is the exception rather than the rule. It is probable that such apparent evidence for scaling laws is due to the fact that corrections to them are small and vanish asymptotically.

3. The O(N ) Landau–Ginzburg–Wilson model In this section we will introduce some basic >eld theoretic quantities and show how they are related to physical quantities such as the susceptibility, speci>c heat, etc. A simple, and yet relatively generic, model one can study is the Landau–Ginzburg–Wilson (LGW) N vector model with internal O(N ) symmetry described by the continuum “microscopic” Hamiltonian 8 √ 1 1 ∇3 ’a g3+ ∇+ ’a + r(x)’a ’a + (’a ’a )2 + gO(’) : H[’] = dd x g (15) 2 2 4! M The general methodology espoused in this review is not con>ned to systems described by Hamiltonian (15), though the latter is general enough for most of our considerations and serves to put things on a more concrete footing while retaining a high degree of generality. This model describes a scalar >eld, 9 ’, in a d-dimensional space M with metric g3+ (this could, for example, be a >lm geometry of thickness L). The term gO(’) abstractly refers to an anisotropy in the system, O(’) being an operator conjugate to the anisotropy coupling g (not to be confused with the metric in the present case). An interesting example of such an “anisotropy” term is the modi>cation used in the e=ective average action [67] where O(’) is given by O(’) = 12 ’a Rk (&)’a with for example Rk (&) =

& −1

e&=k 2

and & the Laplacian. This involves a modi>cation of the propagator by the “environmental” function Rk . Another example, which we will return to in Section 12, is where O(’) = 12 ’a 3ab ’b ;

(16)

in which 3ab is a symmetry-breaking matrix of mass parameters. In the special case where 3ab breaks into two diagonal blocks, the associated crossover can be between two O(N ) models with di=erent values of N , i.e. a bicritical crossover. The anisotropy term associated with >nite size systems can also be written in the general form above if a and b run over the discrete Fourier modes and the anisotropy term takes the form (16). 8

When dealing with a Euclidean Field theory we prefer to use the term Hamiltonian rather than Lagrangian. In this section as we will not consider renormalization, we will drop the conventional B subscript understanding that all quantities in this section are bare, which in statistical physics at least are the physical quantities of interest. 9

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The generator of connected correlation functions is given by W [r; ; g; g3+ ; Ja ] = ln Z ;

(17)

where Z is given by performing the functional integral over the order parameter >elds, ’a , with Hamiltonian (15) and an external source Ja (x) a√ d Z[J ] = [d’] e−H[’]+ M Ja ’ g d x : (18) The connected correlation functions of the theory, obtained by repeated functional di=erentiation of W with respect to r(xi ) and J (yj ), at non-coincident points are (N; M ) (x1 ; : : : ; xN ; y1 ; : : : ; yM ) : G(a 1 :::aN )

(19)

(N; M ) are obtained by functional di=erentiation of the e=ective action, Similarly the vertex functions 2(a 1 :::aN ) 2[r; ’] V given by the Legendre transform √ 2[r; ’] V = −W [r; J ] + Ja (x)’V a (x) g d d x (20)

M

(N; M ) are the objects of primary interest with respect to ’V a (xi ) and r(xj ). The vertex functions 2(a 1 :::aN ) to us as once these are known all the correlation functions of the theory can be reconstructed from them. In statistical mechanics applications the partition function ZLGW is obtained by evaluating the generating function Z at Ja (x) = Ha (x), where Ha (x) is the applied external >eld. The free energy density of the physical system is then F = −(T=V ) ln Z = F b − (T=V ) ln ZLGW , where V is the volume of M (e.g. V = L0 : : : Ld for a toroidal geometry with periods L0 ; : : : ; Ld ) and F b is the background free energy density 10 obtained after coarse graining from the underlying microscopic degrees of freedom to those of the e=ective >eld theory description in terms of the LGW Hamiltonian (15). F b is assumed to be an analytic function of the thermodynamic variables. The internal energy density is

U =F −T

9F 9T

(21)

and the speci>c heat, by de>nition 9U=9T , is given by C = −T 2

92 F : 9T 2

(22)

The standard assumption in working with the LGW Hamiltonian (15) is that the only one of its parameters to retain a dependence on temperature is the mass parameter r. Thus the internal energy 10

In critical phenomena the >eld theory should be viewed as an e=ective ‘mesoscopic’ description of the system, obtained by coarse graining to some UV cuto= scale . Thus there is a contribution to the total free energy which comes from the degrees of freedom that have been integrated out to get to this mesoscopic description. Also, terms involving couplings whose dimensions are negative powers of the cuto= scale are dropped in Hamiltonian (15).

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density is given by T2 U =U − 2V b

dd x

9r(x) (0; 1) G 9T

(23)

and the speci>c heat by 1 9 2 9r(x) T2 9r(x) (0; 2) 9r(y) b d (0; 1) d d x G d x dd y C =C − (x) + (x; y) T G ; 2V 9T 9T 4V 9T 9T

(24)

where G (0; 1) (x) = ’2 (x) and

G (0; 2) (x; y) = ’2 (x)’2 (y) − ’2 (x) ’2 (y) :

(25)

In terms of the vertex functions of the theory, we have 2(0; 1) (x) = 12 G (0; 1) (x)

and

2(0; 2) (x; y) = − 14 G (0; 2) (x; y) :

(26)

So, if the sources r and Ha are taken to be homogeneous, then, for a translationally invariant system, ’V a is also homogeneous and in the direction of Ha . In this case U = Ub − T2 C = Cb −

9r (0; 1) 2 ; 9T

9 2 9r T 9T 9T

(27) 2(0; 1) − T 2

9r 9T

2

2(0; 2) ;

(28)

where 2(0; 1) and 2(0; 2) are to be evaluated at zero external momentum. If we wish to incorporate all of the non-analytic dependence of the internal energy and the speci>c heat into 2(0; 1) and 2(0; 2) , respectively, then a natural choice of the temperature dependence of r is given by r = rc + t where rc is the value of r at the critical temperature Tc and t = 2

(T − Tc ) : T

(29)

In the vicinity of the critical temperature, the results with this choice of temperature dependence will be the same as those obtained with the linear temperature dependence (2 =Tc )(T − Tc ). However, one may capture a wider window of temperature with choice (29), a dependence we will assume in what follows. The internal energy density therefore becomes U = U b − 2 Tc 2(0; 1)

(30)

and the speci>c heat is given by C = Cb −

2 (0; 2) 2 : T2

(31)

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For an O(N ) model 2(0; 2) is manifestly negative and for a bulk system either diverges or goes to zero at the bulk critical temperature according to the sign of the critical exponent (which depends on the value of N ). Therefore, the determination of the singular part of the speci>c heat is equivalent to a calculation of 2(0; 2) . For the O(N ) model there are two types of modes: Those along the direction picked out by the >eld, Ha , and those perpendicular to it. If we choose na for the direction picked out by the external >eld then using the two projectors Plab = na nb ;

Ptab = 0ab − na nb

(32)

we can decompose a general vertex function into block diagonal form. We denote a generic ver(N; M ) tex function by 2l:::lt:::t , where the number of l and t subscripts indicates whether a longitudinal or a transverse propagator is to be attached to the vertex at the corresponding point. When all sub(N; M ) scripts are either l or t we will use a single l or t, for example 2t:::t will be abbreviated 2t(N; M ) . Furthermore, when there are no ’2 insertions (i.e. M = 0) the second index will be left o= e.g. 2(N ) indicates 2(N; 0) . Due to the Ward identities of the model it is suJcient to know only the 2t(N; M ) as all the other vertex functions can be reconstructed from these. Thus, for example, the equations of state, 2a(1) =Ha become 2t(1) = 0; 2l(1) = H . Furthermore, using the Ward identity 2l(1) = 2t(2) ’, V we obtain 2t(2) ’V = H;

2t(1) = 0 :

(33)

(2) yields 2l(2) , 2t(2) ; and 2lt(2) . Ward identities imply that Decomposing 2(ab)

2l(2) = 2t(2) +

2t(4) 2 ’V 3

and

2lt(2) = 0 :

(34)

More generally, we >nd the decomposition 2l(N )

=

N k=0

N! 2t(N +k) ’V k ; k!(N − k)!!(N + k − 1)!!

(35)

where 2t(K) = 0 for K odd and 0!! = 1. One can obtain similar expressions for all the other vertex functions in terms of the 2t(N; M ) , thus the more elementary building blocks are the 2t(N; M ) . The vertex function 2t(2) is the nontrivial building block of the equation of state (33). It plays a central role in determining if there is a phase transition in the model or not and its zeros determine the coexistence curve. As depicted in Fig. 1, we see that if H = 0 there are three possibilities: either (i) ’V = 0, (ii) 2t(2) = 0, or (iii) both ’V = 0 and 2t(2) = 0. Case (i) is the normal situation, while (ii) implies that there is a spontaneous magnetization and 2t(2) = 0 speci>es the co-existence curve, while (iii) occurs at the critical point and so determines whether the model admits a critical point or not. An important point here is that it is possible for the model to admit a critical point but never admit a spontaneous magnetization. In other words we can have critical :uctuations without an actual transition. Of course, one can equally well have a transition without critical :uctuations. This is the more familiar situation of a discontinuous, or >rst order, transition.

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Fig. 1. The phase diagram for the O(N ) model: the solid line is the co-existence curve, the di=erent branches of which are connected by tie lines. The horizontal dashed line is a line of constant ’V and will play an important role in the RG method of reconstructing the equation of state. The point on the t-axis marked &(d; L) is the location of the bulk critical point.

The equation of state is written in terms of the vertex function 2t(2) while other physical quantities, such as the longitudinal and transverse susceptibilities, are associated with the inverses of 2l(2) and 2t(2) , respectively. These also de>ne the longitudinal and transverse correlation lengths, where we use the second moment de>nition of the correlation length, via 2 − l =

2 − t =

2l(2)

(92l(2) =9k 2 )|k 2 =0 2t(2)

(92t(2) =9k 2 )|k 2 =0

;

(36)

:

(37)

In our construction of a parametric representation of the equation of state we will see that t plays an especially important role. 4. Exactly solvable models Having discussed the phenomenology of crossovers, i.e. what one expects to see resulting from a theoretical calculation, and having discussed the >eld theoretic setting for our canonical crossover example, we will now brie:y review some exact results associated with crossovers in exactly solvable models. Such models provide a useful testing ground for ideas on phase transitions and quantum >eld theory in general, and in the present case on crossover in particular. Perhaps surprisingly, there are very few crossovers for which an exact solution is known. The only model (to our knowledge) that permits an exact solution in all dimensions is the spherical model of a ferromagnet [105], or equivalently the large N limit of the O(N ) model [106,107]. In two dimensions various models admit exact solutions for various quantities; the most soluble of these being the Ising model. In what follows, we will concentrate on just these two examples. Additionally, both of these models

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are solvable at the lattice level and therefore permit us to study the crossover associated with the continuum limit. 4.1. The Ising model on a torus The Ising model on a two dimensional torus provides a particularly interesting arena for considering crossover phenomena as shape-dependent crossovers are readily accessible. Here we will consider a generic skew torus and then pass to the cylinder limit. The special case of the Ising model on a square lattice with equal couplings and with periodic boundary conditions which, as we will see below, corresponds to @0 = 0 and @1 = k, was studied in detail by Ferdinand and Fisher [57] while the general case of an arbitrary modular parameter @ = @0 + i@1 was studied by Nash and O’Connor [108,109] on a triangular lattice. The general form of the two-dimensional Ising partition function (for ferromagnetic couplings) in the absence of an external magnetic >eld on a toroidal lattice is 1

1

1 1

Z Ising = 12 eWB {∓e(1=2)WF (0; 0) + e(1=2)WF (0; 2 ) + e(1=2)WF ( 2 ; 0) + e(1=2)WF ( 2 ; 2 ) }

(38)

with + referring to T ¡Tc and − to T ¿Tc . The function WB is −kB TFB , FB being the bulk free energy, and is given by 1 2A d+1 d+2 WB = ln[2{0 − cos(+1 ) − cos(+2 ) − cos(+1 − +2 )}] : (39) 2 0 (2A)2 Denoting nearest-neighbor spin–spin couplings by J1 , J2 and J3 we have , , and 0 given by 2J2 2J3 2J1 ; = sinh ; = sinh ; = sinh kB T kB T kB T 2J1 2J2 2J3 2J1 2J2 2J3 0 = cosh cosh cosh + sinh sinh sinh : (40) kB T kB T kB T kB T kB T kB T The functions WF (u1 ; u2 ) are complicated and we do not repeat them here as our interest is in the continuum limit. Detailed expressions for them can be found in [109]. The scaling or continuum limit for the Ising model corresponds to the constrained thermodynamic limit achieved by taking the number of lattice sites, K1 and K2 , in the respective directions to √ in>nity while simultaneously approaching the critical point with ((T − Tc )=T √ ) K0 K1 , k = K1 =K0 and the nearest-neighbor couplings held >xed. Here the microscopic scale 2 = K0 K1 . √ With V = gK0 K1 , 1=g = + + the scaling limit of WB is given by 2 m2 V mV m2 V D− ln lim WB = K0 K1 B + ; (41) scaling 8A 8A K0 K1 where D = ln(4=A) − 34 ln g − 12 ln(( + )( + )( + )) and √ g 1 2A d+1 d+2 ( − cos(+1 ) − cos(+2 ) − cos(+1 − +2 )) B = ln 2 2 0 (2A) A and is the value of WB at the critical point, m = 0, and = + + .

(42)

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The bulk contribution (42) can be split into a singular part that is modular invariant, WB , and a regular contribution, W reg . However, this decomposition is not unique, therefore we introduce a free parameter, , into the decomposition where is simply a numerical scale that can be chosen at our convenience. The two terms of the decomposition are then W reg = K0 K1 B +

m2 V D; 8A

D = D − ln

and m2 V ln WB = − 8A

m2 V : K 0 K1

(43)

When we take the scaling limit WF (u0 ; u1 ) → WF (u0 ; u1 ) the >nite size contribution to the partition function is unambiguous and automatically in universal form being given by 1

1

1 1

ZFIsing = 12 {∓e(1=2)WF (0; 0) + e(1=2)WF (0; 2 ) + e(1=2)WF ( 2 ; 0) + e(1=2)WF ( 2 ; 2 ) } where WF (u0 ; u1 ) is WF (u0 ; u1 ) = @1 +

∞

−∞ ∞

(44)

√ 2 2 dp ln|1 − e− p +(m V=@1 )+2Aiu0 |2 2A ln|1 − e−2A@1

√

(n+u0 )2 +(m2 V=4A2 @1 )+2Ai[u1 −@0 (n+u0 )] 2

|

(45)

n=−∞

The modular parameter @ = @0 + i@1 and the combination m2 V can be identi>ed in terms of the Ising couplings (vi∗ = tanh(Ji =kB Tc )) and the temperature as

2 2 k(1 − v2∗ ) 1 − v3∗ ∗ and m2 V = A(T − Tc =T )2 k0 k1 ; v3 + i @= ∗ (46) 2 2 ∗ ∗ ∗ 2 v2 (1 − v3 ) + v3 (1 − v2 ) where A=8

2

J2 J3 J1 b(v2∗ ; v3∗ ) + b(v3∗ ; v1∗ ) + b(v∗ ; v∗ ) kB Tc kB Tc kB Tc 1 2

(47)

and ∗

∗

b(v1 ; v2 ) =

(1 − v1∗ v2∗ )(v1∗ + v2∗ ) : 2 2 (1 − v1∗ )(1 − v2∗ )

When we restrict i.e. J3 = 0 and J1 = J2 we have √ to the square lattice with symmetric couplings, √ v3∗ = 0 and v2∗ = 2 − 1 and therefore @ = ik and A = {2 ln [1 + 2]}2 , and we reduce to the Ferdinand and Fisher result [57].

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4.1.1. The cylinder limit The crossover to the Ising model on a cylinder starting from the more general toroidal model, can be obtained from (45) by taking the limit k → ∞. In this limit WF (u0 ; u1 ) becomes ∞ √ 2 2 dp ln|1 − e− p +(m V=@1 )+2Aiu0 |2 : (48) WF = @1 −∞ 2A We see that only the dependence on the phase u0 survives. Note that when u0 is zero the contribution to the log of the partition function is negative, while when u0 = 12 it is positive. Hence, since @1 → ∞ only the u0 = 12 contributions survive. With 1 reg − 2Ising ln ZIsing = −2Ising V we therefore get the complete cylinder scaling function 11 as (L2 = V=@1 ) 2 ∞ √ 2 2 mV dp 1 m2 ln ln|1 + e− p +(m V=@1 ) |2 : − 2 2Ising = 8A K 0 K1 2L −∞ 2A

(49)

(50)

At the critical point m=0, 2Ising L2 tends to the >nite negative value 12 −A=12. As has been argued in [110,111], this value should be −cA=6, where c is the central charge of the model, and hence we deduce that c = 1=2 for the Ising model. In the above, we have found the universal scaling function for the free energy in a generic >nite size box. The resulting expression (44) with (45) gives the full, zero external magnetic >eld, two-dimensional >nite size free energy scaling function. We see that it is a function of m2 V , and the toroidal shape, governed by, L0 =L1 and the angle ) between the sides. For a square box, the expression is simply a function of mL where m = At + . For the two-dimensional model + = 1 and so derivatives with respect to m can be used to obtain the internal energy and speci>c heat scaling functions. In the cylinder limit the free energy density scaling function takes the very simple form ∞ √ 2 2 1 dp m2 m 2 (51) G= ln 2 − 2 ln|1 + e− p +z |2 ; 8A 3 2L −∞ 2A where z = mL. Again derivatives with respect to m yield 2(0; 1) and 2(0; 2) and therefore the corresponding scaling functions for the internal energy and speci>c heat. The constant 3 is a residue of the microscopic physics and cannot be avoided due to the logarithmic dependence associated with the lower critical dimension and re:ects the fact that = 0. For L >nite, one can see that the speci>c heat has a maximum at ∞ √ 2 2 d3 z 3 dp ln(1 + e p +z ) = 1 : (52) d z −∞

11 12

The relative minus sign between the two terms arises since the >nite size contribution to 2Ising is −WF (0; 12 ). We could equally take L0 large if we interchange (u0 ; L0 ) with (u1 ; L1 ).

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This is more easily seen by noting that 2Ising in the cylinder limit is given by a single Feynman diagram corresponding to a free massive Majorana fermion. It can therefore be expressed as 2Ising = − 12 a (m; L). Where a (m; L) is the independent contribution to

2 1 dp 2A(n + 1=2) = ln p2 + (53) + m2 : a m; 2 2A L n∈ Z − We therefore obtain 13 (0; 1) 2Ising = −m (0; 2) =− 2Ising (0; 3) = 6m 2Ising

(m; L) ; (m; L) + 2m2 (m; L) − 8m3

(m; L) ; (m; L) :

(0; 3) = 0 which yields the expression The maximum occurs for 2Ising 2 z 1 = : 2 2 2 5 2 2 (z + A (2n + 1) ) =2 n∈Z (z + A (2n + 1)2 )3 =2 n∈ Z

(54) (55)

(56)

We therefore see that the critical singularity is both rounded and shifted by the >nite size e=ect of the cylinder. Furthermore we see that the shift and rounding are consistent with the remarks of Section 2, i.e. that they scale with the exponent +. This appears somewhat trivially here since + = 1. However, this general feature applies also when + = 1. In fact, even in the general case, we will see that the natural parameterization is in terms of an inverse correlation length and so the scaling variable that will naturally arise from the >eld theory discussion will be mL. The qualitative features of the fully >nite case are similar, the principal additional feature of the scaling functions is their dependence on the >nite size shape. 4.2. The large N limit of the O(N ) model Shortly after Stanley [106] established the equivalence for an in>nite lattice of the partition functions of the N = ∞ limit of the O(N ) non-linear -model and the spherical model, the large N limit of the O(N ) LGW model was discussed from a >eld theoretic point of view by Wilson [107]. This analysis led to subsequent developments where this model at its >xed point was shown to be equivalent to the previous two and the model served as the starting point of a perturbative expansion in 1=N (see [112] for a detailed set of reprints on this topic). The original lattice spherical model was solved for both strictly >nite geometries and geometries exhibiting a dimensional crossover by Barber and Fisher [58]. Many generalizations have also received attention. For example, a generalization that includes long-range interactions has been studied for fully >nite and cylindrical geometries 13

We use the diagrammatic notation of [10], where (−1)k−1 =(k − 1)! times the kth derivative with respect to m of a circle with no dots will be represented by a circle with k dots, the dots representing the point at which each derivative acts.

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[113–115]. (See Rudnick [59] for a more recent discussion of the model in a purely >nite geometry.) While Allen and Pathria [116,117] have recently studied the model’s two-point correlation function. No attempt is made to derive the results presented here, we rather refer the reader to the literature [12]. The O(N ) “microscopic” LGW Hamiltonian (15) admits an exact solution in the large N limit. The limit is taken such that N is held >xed as N → ∞. In this setting it is possible to obtain exact expressions for the vertex functions of the theory. One can do this either by a direct resummation of the Feynman diagrams, either using the RG or a direct method, or via a saddle point approximation. For the translationally invariant case with homogeneous H , the equations of state (33) become m2t ’V = H;

2t(1) = 0 ;

(57)

where 2t(2) =m2t , mt being the transverse mass. The free energy is then given by 2[’; V m2t ]+2reg [r; ]= 2 W + H ’V where ’V = −9W=9H and 2 has been split into a singular part 2[’; V mt ], which vanishes at the bulk critical point, and a remaining regular part 2reg [r; ]. The function 2[’; V m2t ] determines the singular part of the free energy density to which it is related by kB T . The key to the solution of the model is the spherical constraint which arises from the method of steepest descent. This constraint takes the form 14 6mtd−2 = @ + ’˜ 2 + ˜

;

(58)

where ˜ = N mtd−4 and @ = 6t=N . This is really a special case of the constitutive relation between the mass, or correlation length, and the mass parameter, or temperature variable, @, which we will discuss more completely in a later section. Using (58), the singular part of the free-energy per component 2˜ can be expressed in the form d

1 3m t 2 2˜ = ; (59) + − mt ˜ 2 while the regular part is given by 2

3r + cuto= -dependent terms : 2˜ reg [r; ] = − 2N

(60)

All vertex functions of interest can be obtained by di=erentiation of (59) and, given the Ward identities (35), we need only specify the even transverse vertex functions 2t(N ) . In any case the longitudinal ones will be suppressed by orders of 1=N . For a d-dimensional >lm (d¡4) with periodic boundary conditions the basic diagram is ∞ 1 2(−d=2)mdt 2 qd √2 2 (mt ; L) = − − dq : (61) d=2 (d − 1)=2 d 2 2 (4A) (4A) 2((d + 1)=2)L 0 q + z e q +z − 1

14

As in the previous section, we use the diagrammatic notation of [10], and any cuto= dependence of the diagram has been canceled against part of r.

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At mt = 0 we have (0; L) = −ad =Ld where ad is a universal number ˜ Tc (∞) − 2| ˜ Tc (L) = 22(d=2)G(d) : ad = 2Ld 2| Ad=2 In the small ˜ limit we obtain mean >eld results, while the universal scaling form, governed by the limit ˜ → ∞, is 1 G(d; z) − ad 2˜ = ( − m2t ) = (62) 2 2Ld where z = mt L, and the scaling function G(d; z) vanishes at z = 0 and is given by ∞ 2 2 d−2 2 d d−2 (q + ((d − 1)=2)z ) √

d z − G(d; z) − ad = dqq 2 2 d (4A)(d−1)=2 2((d + 1)=2) 0 q2 + z 2 (e q +z − 1) and d = −2((2 − d)=2)=(4A)d=2 . For d = 3 the result simpli>es to G(3; z) = z 3 =12A + 1=2A (ey − 1) with a3 = G(3)=A. The tadpole, (mt ; L), has the useful decomposition (mt ; L) = −mtd−2 F(d; z) +

bd ; Ld − 2

z 0

(63)

y2 dy=

(64)

where bd =

2((d − 2)=2)G(d − 2) 2Ad=2

and mtd−2 F vanishes at mt = 0 with

F(d; z) = d − z

2− d

2 (d − 1)=2 (4A) 2((d − 1)=2)

(65) 0

∞

q d− 2

1

√ dq − bd : q2 + z 2 e q2 +z2 − 1

(66)

The critical temperature is determined by the zero of the right-hand side of (58) with ’V = 0. Since we have chosen the origin for the parameter r to be the bulk critical temperature, Tc (∞), it is convenient to introduce an alternative parameter tL whose origin is the >lm critical temperature. In general the two di=er by a shift, &L (L), so that tL =t+&L (L) and (64) implies &L (L)=(N =6)bd =Ld−2 . Since bd ¿0 for d¿3, we see that the >lm critical temperature is suppressed relative to the bulk one and this shift scales with the shift exponent d − 2 = 1=+(d), +(d) being the bulk correlation exponent, all of which is in agreement with the lattice results of Barber and Fisher [58] and later calculations of [118–120]. Furthermore since bd diverges at d = 3 we see that for a three dimensional >lm the critical temperature Tc (L) is driven to zero and more careful analysis is appropriate. In the universal ( ˜ → ∞) limit mt (@; ’; ˜ L) is determined by 15 bd −1 2 d− 2 2 w = Q (d; z ) = z F(d; z); where w ≡ @ + d−2 + ’˜ Ld−2 : (67) L 15

More generally if we do not take the universal limit we have the more general two variable scaling form z 2 =Q(d; v; w) where v = N L4−d and the functional inverse is with d and v >xed.

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In terms of the basic scaling variables x˜ = (@ + bd =L1=+(d) )|’| ˜ −1= and y˜ = L|’| ˜ +(d)=(d) , with +(d) = 1 1=(d − 2) and (d) = 2 being the bulk d-dimensional exponents, w = (1 + x) ˜ y˜ 1=+(d) . In the large N limit, irrespective of whether we consider the universal limit or not, only the combination @ + |’| ˜ 1= plays a role. This has signi>cant consequences for the e=ective exponents to be considered later, since there is a reduction from two variables to one in the scaling functions. The equation of state is given by Q(d; w)’L ˜ −1=(d) = H˜ ; √ where H˜ = H= N , the asymptotic forms of which are ˜ (d) ’˜ 0(d) = H˜

d− (d) (1 + x)

L

d

(d )

(68)

for z → ∞ ;

(1 + x) ˜ (d ) ’˜ 0(d ) = H˜

(69)

for z → 0 ;

(70)

where 0(d) = (d + 2)=(d − 2) and d = d − 1. Both limiting forms agree with the usual universal form of the equation of state [121] aside from the factors of d− (d) and (L= d ) (d ) which could be absorbed into a rede>nition of ’˜ and H˜ . We choose not to absorb dimension- or L-dependent factors into our variables as we are interested in a problem involving two dimensions at once with L the interpolating physical variable. Similarly, 2˜ has the asymptotic forms 2˜ = Dd (1 + x) ˜ 2−(d) ’˜ (2−(d))=(d)

for z → ∞ ;

ad 2˜ = L1−(d ) Dd (1 + x) ˜ 2−(d ) ’˜ (2−(d ))=(d ) + d 2L

for z → 0 ;

(71)

where Dd = (d) d(d)−1 =2 and (d) = (d − 4)=(d − 2). For d approaching three, the >lm critical temperature Tc (L) is driven to zero as bd diverged with a simple pole at d = 3. There is a similar pole in d−1 which cancel in (m2t ; L) and (67) becomes (@ + ’˜ 2 )L =

1 z + ln[1 − e−z ] 4A 2A

(72)

in agreement with [58], who restricted their considerations to the zero >eld case, H = 0. It is convenient to de>ne w = (@ + ’˜ 2 )L which with (72) implies 2

√ e2Aw + e4Aw + 4 Q(3; w) = 2 ln : (73) 2 Then (73), together with (68), speci>es the universal equation of state. Similarly, (62) with the ˜ change of variables (67) in the speci>c case (73) speci>es 2. For L → ∞ we have w → ∞ and we recover the three-dimensional scaling function (69) discussed 1 above, and for >xed L with L = m− → ∞, the two dimensional critical regime which is governed t by @ → −∞ whereupon (72) gives 2

L = Le−2A(@+’˜ )L

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in agreement with the H = 0 result of Singh and Pathria [60]. The limiting form of the equation of state becomes 2 e4A(@(L)+’˜ ) ’˜ = H˜ ;

where @(L) = @ −

1 ln L ; 2AL

in agreement with [58]. The other special dimension of interest is d = 4 where has an UV divergence. In this case it is necessary to send to in>nity in such a way as to cancel the divergent contribution from and render 2t(4) >nite. We then >nd the constraint retains logarithmic corrections to scaling and becomes 2 ∞ 1 1 z2 z q2 √2 2 (@ + ’˜ 2 )L2 = − − ln dq ; 2 2 2 q +z (4A)2 2A2 0 z0 q +z e −1 where z0 =IL with I a remnant microscopic scale, such that mt I. Similarly the free energy scaling function is given by ∞ (q2 + 32 z 2 ) z2 1 1 z4 A2 2 √ + ln (74) − G(4; z) = − dqq + 2 2 32A2 2 3A2 0 z02 q2 + z 2 (e q +z − 1) 45 where we see that a4 = A2 =45. For mt → 0 with >xed L we recover the three-dimensional results above, and for L → ∞ the constraint becomes 2 (@ + ’˜ 2 ) 1 m2t mt ; =− ln 2 2 2 I (4A) I I2 while 2˜ becomes m4 2˜ = − t 2 64A

1 m2 + ln 2t 2 I

:

4.2.1. E8ective exponents in the large N limit In the above we calculated the free energy and equation of state for the large N limit. Any scaling function can be in principle be deduced from these expressions, in particular the e=ective exponents introduced in Section 2. The relevant expression for +e= is 1 + 6F= ˜ +e= = ; (75) d − 2 + d ln F=d ln z + 12F= ˜ where we have used (58) and the scaling function F is given by (66). This scaling function captures the crossover between three di=erent >xed points: the Gaussian >xed point, the d-dimensional Wilson–Fisher >xed point and the (d − 1)-dimensional Wilson–Fisher >xed point. It is the coupling ˜ which governs the crossover from universal to mean >eld-like behavior when ˜ → 0, in which −1 limit +e= → 12 . In the critical regime, where z ˜ and 1 ˜, the terms proportional to ˜ may be neglected thus obtaining a true universal scaling function. From (75) we see that as z → 0 then +e= → 1=(d − 3) whereas for z → ∞, +e= → 1=(d − 2). Thus +e= interpolates between the two exact asymptotic values associated with the spherical model in d and (d − 1) dimensions.

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The e=ective exponent e= is 1 + 6F= ˜ : e= = 2 d − 2 + d ln F=d ln z + 12F= ˜

451

(76)

In the mean >eld limit e= → 1, whereas in the universal limit ( ˜ → ∞) we have e= → 2=(d − 3) as z → 0 and e= → 2=(d − 2) as z → ∞. Thus e= also captures both the dimensional crossover and the mean >eld one. For T ¡Tc (L) and H = 0, +e= and e= are ill de>ned due to Goldstone modes. However, the e=ective exponent e= is well de>ned. From the saddle point equation (58), due to the vanishing of the transverse mass on the coexistence curve, we see that ’V 2 = 6t= , which implies that e= = 12 , i.e. there is no crossover as one proceeds along the coexistence curve. This is in strong distinction to the >nite N case where there is a crossover between the critical point and the strong coupling discontinuity >xed point at T = 0. For the approach to the critical point as a function of >eld, H , on the critical isotherm, T = Tc (L), the e=ective exponent 0e= is 2t(4) ’V 2 d + 2 + d ln F=d ln z + 36F= ˜ (77) = 0e= = 1 + d − 2 + d ln F=d ln z + 12F= ˜ 32t(2) which interpolates between the mean >eld and the respective d and d -dimensional critical exponents. For the speci>c heat and the energy density we may de>ne e= , as in Section 2, via the speci>c heat, as is conventional, or, alternatively, we can use the internal energy via e e= =1−

d2(0; 1) ; 2(0; 1) dt t

(78)

where 2(0; 1) is the singular part of the internal energy density, and we evaluate it at H = 0. From e e the de>nition of e= we have e= = 1 − t2(0; 2) =2(0; 1) which gives d − 4 + d ln F=d ln z e : (79) e= = d − 2 + d ln F=d ln z + 12F= ˜ The speci>c heat e=ective exponent is more cumbersome. However, both exponents vanish in the mean >eld limit and in the universal limit yield e=ective exponents that interpolate between (d) and (d ) as z ranges from zero to in>nity. For T ¡Tc on the coexistence curve the singular parts of the energy density and the speci>c heat are identically zero implying the associated amplitudes are zero and the associated e=ective exponents are ill de>ned. Finally, the e=ective exponent -e= is identically zero in the large N limit being associated with terms ∼O(1=N ). 4.2.2. E8ective exponent scaling laws We now make some observations concerning certain algebraic relations between the e=ective exponents. Eqs. (75), (76) and the fact that -e= ≡ 0 due to the vanishing of ’ in the large N limit, yield the relation e= = +e= (2 − -e= )

(80)

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e and the expressions for e= , e= and e= imply e + 2e= + e= = 2 : e=

(81)

Note that these relations hold even for the more general scaling functions that include the crossover to mean >eld theory. The exponent e= (derived from the speci>c heat), however, does not satisfy this relation. We see then that direct analogs of the normal scaling laws between exponents hold, however, here the relations are between entire scaling functions. It is natural to ask if there are analogs of other scaling laws, in particular hyperscaling, where 2 − = +d. This cannot be achieved for >xed d. However, one can de>ne the notion of an e=ective e dimensionality, dee= , such that an e=ective hyperscaling law is valid. De>ning dee= = (2 − e= )=+e= one >nds ˜ d + d ln F=d ln z + 24F= : (82) dee= = 1 + 6F= ˜ In the mean >eld limit dee= → 4, i.e. the upper critical dimension as one might expect. In the limit z → ∞, one >nds dee= → d, whilst in the limit z → 0 for >xed L, dee= → (d − 1). Of course, one s s could also de>ne an e=ective dimensionality via e= as dse= = (2 − e= )=+e= . Again it interpolates s between 4, d and (d − 1) in the appropriate asymptotic limits. However, since the exponent e= s did not satisfy the scaling law (81) neither does de= . We might enquire now as to the validity of other scaling laws that involve the dimensionality explicitly. Noticing that dee= = 2 + 1=+e= (i.e. +e= = 1=(dee= − 2)) one arrives at the scaling relations e de= + 2 − -e= +e= e (d − 2 + -e= ) and 0e= = : (83) e= = 2 e= dee= − 2 + -e= The standard exponent relations, including hyperscaling, tell us that two exponents and the dimensionality are suJcient to specify all other exponents. In the N → ∞ limit the dimensionality alone is suJcient to determine all exponents. What we have found in this limit is that the six scaling e functions e= , +e= , e= , -e= , 0e= and e= can analogously all be expressed in terms of the one funce tion, de= , by natural analogs of the scaling laws. In this case, we can easily trace the origin of the validity of these scaling laws to the fact that the two scaling variables x˜ and y˜ entered in the combination w = (1 + x) ˜ y˜ 1=+(d) and so there is a collapse from a two-variable scaling function to a one-variable one. This is not something we expect to occur with any generality and therefore we do not expect e=ective exponent scaling laws to be generally valid either, though as mentioned previously deviations will generally be small.

5. Renormalization and the RG There are two distinct but related points of view about the RG. One has as its basis the notion of “coarse graining” (the Wilson–Kadano= approach [122–124]) and the other that of reparametrization, or coordinate, invariance. Many aspects of the relationship between them are discussed elsewhere [1,2] in other reviews of this series. Approaches related to the former are discussed in [66,67,125] and

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453

to the latter in [2,126,127]. Each formulation has its own particular advantages and disadvantages, yet both are associated with the fundamental problem of how to treat the physics of systems composed of many non-linearly coupled degrees of freedom as a function of “scale”. The Wilsonian RG is intuitive and well adapted for truly non-perturbative approaches such as discussed in [66] and as manifest in Wilson’s famous solution of the Kondo problem [128]. It can also be implemented at a perturbative level and it is in that guise that it most resembles the reparametrization approach. As mentioned, each methodology has its advantages and disadvantages. It is not easy to imagine the reparametrization approach tackling the Kondo problem. On the other hand it is hard to imagine the Wilsonian approach giving non-perturbative results by way of improved asymptotics for the solution of non-linear partial or ordinary di=erential equations [129]. Here we will emphasize the reparametrization point of view but at the same time try to point out similarities and di=erences with the coarse graining approach. 5.1. The coarse graining approach One of the deep insights of Wilson was to realize that the natural arena in which to consider systems with many non-linearly coupled degrees of freedom was the space of parameters, M, or space of Hamiltonians as it is more often termed. This, actually, is equally true irrespective of whether one adopts the reparametrization point of view or the coarse graining approach, although this does not seem to have been realized in QFT before the advent of Wilson’s work. Given the existence of M, how does a coarse graining manifest itself? It is well known that there are many di=erent constructions of coarse graining RGs, e.g. momentum shell integration, block spinning, majority rule, etc. They are realizable on some suJciently large space of probability distributions, and correspond to maps from measures to measures. Usually, however, the Wilsonian RG is taken to be a map from Hamiltonians to Hamiltonians, and realizable as a :ow on M. 16 The mappings (almost invariably approximate) are between di=erent e=ective degrees of freedom, represented by an e=ective Hamiltonian associated with di=erent scales. If, for a particular system, each di=erent scheme could be implemented, then universal quantities ought to be independent of the particular scheme used. A simple, exactly solvable example of coarse graining is the one-dimensional Ising model on a chain of lattice spacing a with N spins (see for example [131] for a full discussion of this model from the RG point of view). The Hamiltonian H ≡ H(T; H; E) has the simple functional form H = −(J=2T ) i; j i j − (H=T ) i i + E, where we have included the additive constant E since it runs under the renormalization transformation. Taking the Wilson point of view that for a theory with many coupled degrees of freedom calculations can be carried out using a philosophy of “divide and conquer” we will calculate the partition function Z ≡ Z(T; H; E) iteratively. Any coarse graining prescription can be used, the simplest being to >rst sum over the values ±1 of every odd spin. Naturally, the choice of coarse graining cannot a=ect the partition function Z = { } exp(−H). However, by a suitable reparametrization of T , H and E, T = T (T; H ), H = H (T; H ) and E = E (T; H; E) the partition function can be rewritten in the form Z = { } exp(−H (T ; H ; E )) where 16

Some of the subtle di=erences between the two di=erent notions and some of the pathologies associated with the existence of RG transformations below the critical temperature are discussed in [130].

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H = −(J=2T ) i; j i j − (H =T ) i + E and the sum is only over the remaining spins, i.e. the number of degrees of freedom has been halved. This is now the partition function of a system at temperature T , magnetic >eld H and with lattice spacing 2a. Note that a physical quantity such as the correlation length is invariant under this procedure, we would simply be calculating it in terms of new “coordinates” T and H . If we now perform a rescaling 2a → a then the correlation length → =2 in units of the new lattice spacing. So the full coarse graining here consists of a reparametrization and a rescaling. Under the coarse graining transformation Rb , here with rescaling factor b = 2, we have Z(T; H; E; a; N ) = Z(T ; H ; E ; a; N=2) ; F(T; H; a; N ) = F(T ; H ; a; N=2) + G(T; H ) ; 2(N ) (T; H; a; N ) = 2(N ) (T ; H ; a; N=2) ;

(84)

where G(T; H ) = E − E is a spin-independent constant. These coarse graining transformations, or renormalizations, satisfy the group multiplication law Ra Rb = Rab , i.e. they possess a semi-group structure. 17 These equations relate properties of a system at temperature T and magnetic >eld H with N degrees of freedom to those of the same system but at temperature T , magnetic >eld H and with N=2 degrees of freedom. In statistical physics this observation is just the “law of corresponding states” (see for example [132] for a discussion). The crucial point is that the coarse graining allows one to relate the physics of one physical system to that of another. Of course, one may keep on coarse graining. For n iterations one will relate the physics of a system of correlation length to that of a system of correlation length =2n . For the correlation functions one may iterate until one reaches a point where the correlation length is suJciently small that one may do a “matching” to a known calculable limit such as mean >eld theory (this would not however be true if one implemented a non-linear RG where the correlation functions do not map onto themselves). Neither is it possible at the level of the free energy, as successive iterations lead to singular behavior in the “additive renormalization” term G(T; H ). In this context however, if Ltn→∞ 2−n F(Tn ; Hn ) = 0 where Tn = R2n T and Hn = R2n H then F(T; H; a; N ) =

∞

2−n G(Tn ; Hn ) :

(85)

n=1

In the limit of a continuous system one can write b = e0 and l = n0 and take the limit 0 → 0 to >nd ∞ e−dl G0 (T (l); H (l)) dl ; (86) F(T; H ) = 0

where G0 (T; H ) = (9G(T; H; b)=9b)|b=1 . This is the “trajectory integral”, developed by Nelson [62], that expresses the free energy as a line integral along an RG trajectory. 17

Unlike its reparametrization counterpart as there are no inverse transformations one strictly speaking talks of a renormalization semi-group rather than a RG. Here we will use the familiar abuse of language and talk of an RG.

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The RG transformation Rb generates a one-parameter :ow in M. Of particular interest are the >xed points of the transformation. In the vicinity of a >xed point, where T = T ∗ and H = H ∗ , with t = T − T ∗ and h = H − H ∗ , a linearization of the RG transformation around the >xed point yields t

h

Rb (T; H ) ≡ (t ; h )∼(by t; by h) ; t

(87) h

where the eigenvalue exponents y and y are related to the critical exponents for the >xed point. For instance, the correlation length exponent + = 1=yt . The variables t and h here are linear scaling >elds with scaling dimensions yt and yh ; respectively, which are the eigenvalues of the RG transformation Rb linearized around the >xed point. In the case of the one-dimensional Ising model the >xed point corresponds to T = 0, H = 0. The linear scaling >elds t and h satisfy (87) only in a linearized neighborhood of a >xed point. However, one may de>ne non-linear scaling >elds [133], gt and gh , such that t

h

Rb (gt ; gh ) = (by gt ; by gh )

(88)

is satis>ed exactly, where gt (t; h) = t + O(t 2 ; th; h2 ) and gh (t; h) = h + O(t 2 ; th; h2 ). In the vicinity of the >xed point the non-linear scaling >elds reduce to the linear ones. As we will see, they play an important role in crossover behavior. To recapitulate then: a Wilsonian RG transformation can be viewed as a :ow in the (potentially) in>nite dimensional space of parameters M (or the space of all Hamiltonians). The transformations are associated with a one-parameter :ow with respect to a quantity which can be interpreted as a change in lattice spacing or some UV or IR cuto=. The RG transformation can be linearized around its >xed points, the resulting eigenvalues yielding the critical exponents associated with that >xed point. From a calculational point of view it throws the emphasis onto calculating the parameter :ows, the idea being that one may wish to calculate the physics of a system at T and H , where an approximate calculation is extremely diJcult by relating it to the system at T and H where the approximation is more reliable, the two systems being connected by an RG :ow. Being able to describe the original system then depends on being able to calculate the transformation Rb . 5.2. Coarse graining and crossovers If the action of the RG operator is to be a “good” representation of the action of the dilatation operator then the >xed points of the RG transformation should be coincident in M with the points of scale invariance. That is not to say that they must have the same coordinates, but that the intrinsic geometry of the RG :ows and the dilatation :ows should be the same. Additionally, the properties of the >xed points should be independent of the “coarse graining” procedure one uses, i.e. of the particular choice of RG. Crossover systems, generically, can exhibit scale invariance at more than one point of M. Hence, a crossover often interpolates between two or more scale invariant >eld theories and hence in an RG setting between more than one >xed point. If one considered a “coarse graining” procedure, such as momentum shell integration, for a system which possesses another length scale g−1 , other than the correlation length , it is clear that the qualitative nature of the iterations of the RG may well change as one passes from momentum shells with kg to kg. If one starts iterating at a scale g, then typically, as the iteration proceeds into the IR, the RG :ow will pass close to the isotropic, g = 0, >xed point, S, (it will only actually hit it for g = 0) before proceeding on to the anisotropic, g = ∞, >xed point, P. What is manifest

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is that a “physically sensible” coarse graining procedure will show up the qualitative change in the e=ective degrees of freedom as one considers di=erent regimes of the iteration. In terms of our simple example of a coarse graining RG with two relevant couplings T and H ; in the neighborhood of S t

h

Rb (T; H ) ≡ (t ; h )∼(by t; by h) ;

(89)

where t =T −T ∗ and h=H −H ∗ are deviations from the isotropic >xed point S which has coordinates H ∗ and T ∗ . In the neighborhood of P t

h

Rb (T; H ) ≡ (tp ; hp )∼(byp tp ; byp hp ) ;

(90)

where tp =a11 T +a12 H −Tp∗ and hp =a21 T +a22 H −Hp∗ . In this case the variables tp and hp are linear scaling >elds with respect to the anisotropic >xed point P with ypt and yph being the eigenvalues of Rb linearized around P. Of course, here we are restricting attention to a crossover with two >xed points. A system may well possess several. However, each pair of these will exhibit behavior analogous to the above. There may also exist lines of >xed points. In general, the isotropic >xed point will have more unstable directions than the anisotropic one. For every two >xed points there exists a special trajectory, the separatrix, that joins the two >xed points. Flows along the separatrix go from the isotropic to the anisotropic >xed point. In the case where the crossover is associated with an anisotropy parameter g, and the only other relevant parameter is T , then the scaling variable which naturally appears in scaling functions is x = t =g, where is the crossover exponent, and, importantly, is a property of the isotropic >xed point. If g is a non-linear scaling >eld, such as >nite size L, then = 1=yt while in the case that g is a linear scaling >eld = yg =yt , where yg is the eigenvalue exponent associated with the relevant operator Og , conjugate to g, at the isotropic >xed point. The above discussion quite clearly tells us that one cannot expect to describe a crossover in terms of a linear, local RG and this is the nub of the problem. One needs a global, non-linear RG that is capable of encompassing more than one >xed point. In terms of scaling >elds what are required are non-linear scaling >elds gt and gh , as introduced in the previous section, which are eigenvalues of the dilatation operator. Scaling functions would then naturally be expected to be generalized homogeneous functions of these scaling >elds rather than the linear scaling >elds which naturally appear in a linearized analysis of a particular >xed point. Obviously, such global solutions are diJcult to >nd, though calculations do exist. In [134,135], for example, non-linear scaling >elds were explicitly calculated to >rst order in for the crossover between the Gaussian and Wilson– Fisher >xed points and also for a system with cubic anisotropy. See also Orrick et al. [136] for recent calculations of nonlinear scaling >elds, gt and gh in the two-dimensional Ising model.

6. Renormalization as a coordinate transformation We will now consider the reparametrization notion of renormalization. We will assume that the partition function depends on a set of parameters {3i }. In the 1D Ising model above one would have

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457

31 = T − T ∗ and 32 = H − H ∗ . We consider a one-parameter set of reparametrizations depending on an arbitrary ‘sliding’ scale I 3i = Zi (I; {3i (I)})3i (I) ;

(91)

where the {3i (I)} are the renormalized parameters. Note that for some parameters one may have Z = 1, i.e. there is no renormalization of the corresponding parameter. One must now ask how do things transform under this coordinate transformation? The answer is, as with any type of coordinate transformation, that the physics is invariant. This can be encapsulated in the statement I

d2(N; M ) ({3i }) = 0: dI

(92)

If we assume further that the reparametrized, or “renormalized” vertex functions 18 themselves transform under the coordinate transformation as 2(N; M ) ({3i }) = Z’−N=2 (I; {3i (I)})Z’−2M (I; {3i (I)})2(N; M ) ({3i (I)}) ; then one obtains d2(N; M ) ({3i (I)}) = I dI

N 2 ’ (I; {3i (I)}) − M ’ (I; {3i (I)}) 2(N; M ) ({3i (I)}) ; 2

(93)

(94)

where ’ = d ln Z’ =d ln I and ’2 = −d ln Z’2 =d ln I are Wilson functions and at a >xed point correspond to the anomalous dimensions of the operators ’ and ’2 , respectively. In certain cases, there may also be a term independent of 2(N; M ) on the right-hand side when an additive renormalization of the corresponding vertex function is necessary. One thing we wish to strongly emphasize here is the following: strictly speaking ansatz (93) can never be wrong, as it is simply a relabeling of a physical quantity. Traditionally, the “validity” of an ansatz of this sort for a particular theory has been checked perturbatively to all orders by invoking all the full apparatus for investigating renormalizability (see for example [137]). However, “validity” in this context has a speci>c meaning: that the renormalization constants Z’ , Z’2 and Z can be chosen such that UV divergences associated with the perturbative series for all quantities can be absorbed into the Zi and rc . One of the key characteristics of a crossover is that term by term in perturbation theory there arise divergences other than UV ones. If ansatz (93) is such that divergences of one type or another are not ameliorated then rather than being wrong the ansatz simply indicates that we have implemented a coordinate change or relabeling that perturbatively is not very useful. So, any coordinate change or relabeling is allowable. Whether or not the relabeling is useful for a perturbative or other approximate calculation, however, is another matter. The in>nitesimal generator of RG transformations, Id=dI, can be written in a coordinate basis as 9 9 d =I + I i ; (95) dI 9I 93i i 18

We adopt the notation in the rest of the paper that vertex functions without a B subscript are the renormalized functions.

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where the -functions are de>ned by i =I d3i =dI and related to the corresponding Wilson functions, i , by i =3i = i . Note that all derivatives are at constant bare theory. The :ow equations for the renormalized parameters can be formally solved to yield 3i (I) = 3i (I0 )e

I

I0

i (x) d x=x

(96)

which allow for an “exponentiation” of perturbation theory if perturbative expressions for the corresponding i can be found. Similarly RG equation (94) can be formally integrated to yield 2(N; M ) ({3i (I0 )} = e

−N=2

I

I0

’ (x; {3i (x)}) d x=x M

e

I

I0

’2 (x;{3i (x)}) d x=x

2(N; M ) ({3i (I)}) :

(97)

Very often the scale I0 can be chosen such that the vertex functions at that scale correspond to the original bare functions which are, ultimately, the physical functions of interest. Note that Eq. (97) absolutely does not relate correlation functions of one physical system to those of another. In this sense, there is no underlying law of corresponding states at work, this formulation being based on a di=erent principle to that of a coarse graining RG, or indeed, to other >eld theoretic relatives such as the Callan–Symanzik equation. Rather, it is a rewriting of the correlation function of a physical system in terms of the parameters=coordinates associated with an appropriately chosen ;ducial system which may or may not be strongly related to the system of interest. We emphasized that the di=erence between a coarse graining Wilsonian RG and a reparametrization RG is that the latter represents a passive coordinate transformation, unrelated to any physics, whereas the former, represents an active transformation, that actually changes the microscopic physics. Given the fact that the physical content of the RG equation (94) is zero it is legitimate to ask what is the point of a reparametrization? Although all coordinate systems are formally equivalent for calculating invariant or covariant objects we will see that this is not so when one is reduced to doing approximate calculations, such as using perturbation theory. In this case, for reasons that will become more transparent later on, the choice of coordinate system plays a crucial role. In particular, we will see that there are four crucial questions to be answered (1) Which coordinates should be reparametrized? (2) What I should be used as a >ow parameter? (3) What “Normalization conditions” should be used to ;x Z factors? (4) What “Gauge ;xing” condition should be used to ;x I? The >rst question points to the fact that a reparametrization can depend on a variable numbers of parameters. For instance, for a massive >eld theory it is possible to have renormalized parameters that depend on, or are independent of, the mass parameter. The second question is associated with the fact that there are many possible choices of :ow parameter in the reparametrization approach. Neither of these points have direct analogs in a coarse graining RG. The >rst because under a coarse graining all parameters generically change, and the second because coarse graining is inevitably associated with an IR or UV cuto=. The third question determines how much information is “exponentiated” by controlling how much information is included in the Wilson functions. The fourth has some aspects common to both types of RG and is related to the concept of matching, but the term ‘gauge >xing’ is more naturally associated with the reparametrization point of view. In both cases, a value for I,

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Im , is sought such that at that scale an approximate calculation can be successfully carried out. Of course, as the physical parameters of the system of interest change so must the corresponding Im , i.e. generally, Im ≡ Im ({3i }). Once a speci>c “gauge” choice is made RG invariance is naturally lost and (97) becomes 2(N; M ) ({3i (I0 )}) = e

−N=2

Im I0

’ (x; {3i (x)}) d x=x M

e

Im I0

’2 (x;{3i (x)}) d x=x

2(N; M ) ({3i (Im )}) ;

(98)

where the 3i (Im ) are expressible as a general coordinate transformation of the {3I0 }. Now the two sides of this equation represent two di=erent physical systems as in the coarse graining approach. This “gauge >xing” (or >xing of the sliding scale) of I turns the original passive coordinate transformation into an active one [138]. 6.1. Which coordinates should be reparametrized? Conceptual arguments The question of which coordinates to reparametrize can best be illustrated with a concrete example. Consider a massive ’4 theory in three dimensions with the >eld theoretic machinery of Section 3 set up as shown therein. If one wishes to investigate the theory under changes in scale, and capture the points, in M, of scale invariance, then one should look for >xed points of a suitable RG. The question is: which RG? If renormalization is solely thought of as a way of accounting for UV divergences then there is a great deal of latitude as to what parameters counterterms should depend on as they are then e=ectively released from any dependence on IR scales. The idea of a minimal scheme is to >nd the simplest form of (93) which eliminates the UV divergences by making use of this freedom. The ultimate version of this type of approach is minimal subtraction, where one chooses as counterterms only those parts of diagrams that survive as poles in the extreme UV limit. For Hamiltonian (15), for instance, one knows that in this limit counterterms that are mass independent are as good as mass-dependent ones, if one’s purpose is solely to perturbatively take the UV cuto= to in>nity. This is because mass is an irrelevant coupling in the UV. In our three-dimensional example, only two >xed points are perturbatively accessible using mass independent schemes, the Gaussian and Wilson–Fisher >xed points with zero mass. If one uses mass dependent schemes, another >xed point becomes accessible—the in>nite mass Gaussian >xed point [134,135]. In fact these statements go beyond perturbation theory—if mass-independent renormalization is used then the in>nite mass Gaussian >xed point cannot be seen from the RG :ow of the corresponding Wilson functions. If mass-dependent renormalization is used then it can. In other words, in the mass-independent renormalization the in>nite mass Gaussian >xed point is not captured by (94), but relegated to a secondary, e.g. perturbative, analysis. Of course, one can do such straight perturbation theory near the in>nite mass limit, and in that sense there was no necessity to implement a mass dependent RG. The reason one can get away with it, in the case of a mass operator, is that the critical exponents associated with the in>nite mass Gaussian >xed point are mean >eld exponents, and therefore there was nothing extra to exponentiate. However, one is not always so fortunate. Typically, if one tries to track a theory back into the IR, having used a minimal subtraction scheme, in the presence of an additional mass scale, g, perturbation theory will break down, and some other procedure will be necessary. It is only the g-dependent RG which is capable of giving globally valid perturbative information.

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More generally, if one has a >eld theory parametrized by a set of parameters, P ≡ {gi }, corresponding to a point in M, it might occur that di=erent subsets of the parameters, relevant for describing the theory at di=erent scales, are taken into one another by the RG :ow on M. If the renormalization depends only on a subset, K, of the parameters the associated RG :ows, RGT , are restricted to a subspace T of M. If any of the remaining P–K parameters are relevant in the RG sense, then some of the complete set of RG :ows of the theory, RGM , thought of as true scale changes, will tend to :ow o= T into M. However, the use of RGT does not allow for such :ows. Such a state of a=airs would be shown up by the perturbative unreliability of the results based on RGT . If none of the parameters K are relevant then there should be no problem. However, one can only say what parameters are relevant when the full >xed point structure of the theory is known! In principle, it is better to work with RGM . If a certain parameter is important, then one ensures that its e=ects are treated properly, and if it was not then that will come out of the analysis. There can be no danger, except for extra work, from keeping a parameter in, but there can be severe problems if it is left out. The message then is that in the >eld theoretic context the choice of renormalization can be quite crucial, some points of scale invariance in M being inaccessible with respect to certain RGs. This may sound somewhat disturbing, given that physics should be renormalization scheme independent. One must be careful to make a distinction between the points in M, where the system under consideration is scale invariant, and those points which are >xed points of a particular RG. They are not necessarily the same. If an RG is chosen which is a good representation of scale changes then they will be. To be clear on this point: We are not saying that there are scale invariant systems that can only be accessed utilizing certain RGs. Rather, we are saying that there exist RGs which are suJcient in and of themselves to describe the system. On the other hand, there are others that must be supplemented by extra perturbative or non-perturbative information. For our massive >eld theory, the Wilson functions derived using mass-dependent renormalization are suJcient on their own to access the mean >eld >xed point, whereas for mass-independent schemes one has to supplement the Wilson functions with extra information from some perturbation theory. It should be added that coarse graining methods are not immune to these problems. If, for instance, one implements a momentum shell integration for say a quantum ferromagnet, but argues that to access the zero temperature quantum transition it is suJcient to ignore T in the coarse graining, then it will be impossible using the resultant Wilson functions to access the true T = 0 critical behavior. 6.2. Coordinate transformations should be environment dependent In the above we discussed the fact that there are inequivalent >eld theoretic RGs, in the sense that their >xed points were not the same. Intuitively why is this, and why, in particular, does it happen in the >eld theoretic RG? Consider a physical system as we observe it at di=erent scales: If we start at a very small scale, −1 , characterizing the system by a set of microscopic degrees of freedom and bare parameters, then try to describe the e=ective physics at some much larger scale I−1 , it is almost invariably true, in a system with many degrees of freedom, that the physics at the scale I is very complicated in terms of the physics at the scale . More often than not a better prescription is in terms of e=ective degrees of freedom, more appropriate for scales ∼I. This is the whole raison d’ˆetre behind e=ective >eld theory. Of course, experimentally one >nds that

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the e=ective degrees of freedom are very di=erent at di=erent scales. Clearly one would like one’s calculational schemes to capture this experimental fact. A “good” coarse graining procedure will naturally track changes in the e=ective degrees of freedom. Similarly, we will show that a “good” >eld theoretic, or “environmentally friendly”, RG can also follow the changing e=ective degrees of freedom. One of the early lessons of >eld theory, con>rmed experimentally in, for example, the Casimir e=ect, is that the :uctuations of a >eld theory are “background”, or in the terminology we will adopt here, “environment” dependent. We naturally consider the space itself as part of the environment. Thus, in in>nite :at space in three dimensions, R3 , or in a “box” of size L (L3 ), the :uctuations are di=erent and L is the environmental variable. If one’s only concern is UV divergences then certainly :uctuations with momenta kL−1 are qualitatively the same on either R3 or L3 , however, for momenta k∼L−1 the opposite is true. One can remove the UV divergences on L3 using counterterms obtained for R3 , e.g. using minimal subtraction. However, the information obtained from the consequent RG will be information appropriate to the >eld theory on R3 , not L3 . Hence, we will say that minimal subtraction is not environmentally friendly on L3 . More generally, for an anisotropic box with sides of di=erent size, the overall size of the box and the ratio of the lengths of the sides (and possibly angles) enter as environmental variables. A proper description of the :uctuations from an RG standpoint would demand that the RG depended on all such variables. In a space with curvature (constant or not), this would add additional environment dependence. This latter case could be of some relevance in the early universe. More generally, any background >eld can be considered as an environmental variable. A gravitational background >eld is obviously related to the above question of what space one is working in. In the case of the :uctuations of a charged >eld coupled to a background electric or magnetic >eld, the spectrum of :uctuations is completely di=erent to that in the absence of the >eld. A uniform background magnetic >eld, B, sets a length scale, and :uctuations with wavelengths very large or small relative to it are very di=erent. In fact one can think of the B >eld as having the e=ect of putting the system in a kind of “box”, so that the :uctuations are those in this “box” [46 –50]. In some cases, the particular background of interest is much more complicated, such as an Abrikosov :ux lattice. In others, such as QCD, there are indications that the relevant background is a condensate of monopoles. The way quarks and gluons behave in a background of monopoles will be very di=erent to their behavior in the absence of this background. The standard renormalization of QCD, using minimal subtraction, is really only appropriate for the situation where there is no background, i.e. at high energy where quarks and gluons are a good representation of the e=ective degrees of freedom. The environmental variables can often be thought of as setting a >xed length scale, or set of scales. In practice, one might not be able to make a clean distinction between :uctuations and environment as they interact dynamically. Not only does the environment in:uence the :uctuations but also the :uctuations feed back into the environment. When this feedback is not negligible the RG should capture this dynamic. A simple example of the static case is ’4 below the critical point. In such a situation the :uctuations in:uence the background magnetization and one can investigate this explicitly via the equation of state. It is the latter which determines how the :uctuations are reacting back on the environment, the environment in this case being the background magnetization. For more complicated systems there will be more “equations of state”, these must be solved in conjunction with the environment dependent RG equations to get a closed system.

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7. Environmentally friendly renormalization In the previous section, we argued that a successful renormalization should track the evolving nature of the e=ective degrees of freedom as a function of scale and that as the latter depend on the “environment” the reparametrization chosen should also depend on it. We call such a renormalization and its associated RG “environmentally friendly” if it leads to perturbatively well-de>ned answers. The purpose of this section is to discuss, in the context of the reparametrization formulation, the properties of such choices. As discussed in Section 6 there are four key elements in the choice of coordinates. We shall consider each one in turn and show how it can a=ect the outcome of a perturbative calculation of the vertex functions of a theory. 7.1. Which coordinates should be reparametrized? Perturbative arguments We have stated that di=erent coordinate choices lead to di=erent RGs and given some explicit examples. Here we wish to see this emerging in the context of perturbation theory. We will consider our paradigmatic example of crossover associated with an environmental variable g, taken, for simplicity, to be a non-linear scaling >eld, where there exist nontrivial isotropic and anisotropic scaling regimes. Bare perturbation theory automatically depends on the parameter g in that it will either appear in Feynman diagrams or contribute additional diagrams. However, the bare series almost never gives a perturbatively trustworthy account of the physics due to large dressings in the UV and=or IR. The role of the RG is to account for such large dressings by taking advantage of its sliding scale to map correlation functions to a region of M where a reliable approximation may be carried out. We will use the ’4 -function, with the anisotropy entering the propagator, as an explicit example of how di=erent RGs and, importantly, di=erent RG >xed points, emerge in a crossover. Once Z depends on g (or gV = g=I) the perturbative structure of the -function for the dimensionless coupling V = Id−4 has the form ∞

d V(I) 2 n I V V (I) + an (g) V V (I) : = ( V; g) V = − V(I) + a2 (g) dI n=3

(99)

The corresponding g-independent -function corresponds to using the coeJcients an (0). We can now examine the di=erence in >xed point structure. For the g-independent equation the an are numbers, so, besides the trivial Gaussian >xed point, one >nds a non-trivial >xed point, which to lowest order ∗ is V = (4 − d)=a2 (0) (the higher orders can be found in the standard fashion [139,140]). Thus, the only nontrivial >xed point emerging is the isotropic one. Clearly no anisotropic >xed point is accessible. In this sense the anisotropic critical behavior cannot be accessed by this environmentally unfriendly, g independent, RG. If one utilizes this RG then one will >nd corrections to RG improved perturbation theory of the form ∼O((g) V n ) for some n ¿ 0 (n = 0 includes logarithmic corrections) which represent corrections to scaling to the leading g = 0 asymptotic scaling behavior. For g1 V these corrections are small. However, as g is a relevant coupling (its conjugate operator being a relevant operator) with respect to the isotropic >xed point, as one proceeds into the critical region these corrections to scaling become large and this isotropic RG improved perturbation theory breaks down.

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Turning now to the g-dependent RG, the >xed point structure is more easily analyzed in terms of the rescaled coupling or :oating coupling, h, de>ned as h = a2 (g) V V(I). The corresponding -function is ∞

I

dh(I) bn (g)h V n (I) ; = (h; g) V = −(g)h(I) V + h2 (I) + dI n=3

(100)

where (g) V = 4 − d − (d ln a2 =d ln I). To analyze the asymptotic behavior it is useful to consider the :ow equation for gV itself and of the temperature parameter, tg (g) V = 2 (Tc (0)=Tc (g))(T − Tc (g))=T . As g is a non-linear scaling >eld the relevant equation for the dimensionless anisotropy, gV = g=I, is very simple I

d gV = −gV ; dI

(101)

while for tVg = tg =I2 : I

d tVg = −(2 − ’2 )tVg : dI

(102)

Thus, the >xed points of all three equations are to lowest order (we omit consideration of the trivial Gaussian >xed points, both with tV = 0 and ∞) gV = 0;

h = (0);

tVg (0) = 0

(103)

and gV = ∞;

h = (∞);

tVg (∞) = 0 :

(104)

The >rst one is the isotropic >xed point accessible to the g-independent RG. The second one, however, is new. If under favorable circumstances one has a good idea of what the e=ective degrees of freedom associated with the g=∞ >xed point are, then one might impose normalization conditions at g = ∞. Once again the consequent RG would be g independent, only this time the >xed point accessed would be (104), not (103) as in the case of the g = 0 RG. The RG that is capable of encapsulating both >xed points is the environmentally friendly gdependent one. It is, of course, by no means unique. Just as in the non-crossover case, where di=erent renormalization schemes can exhibit the same physics, e.g. minimal subtraction and massless normalization conditions, so here there will be di=erent schemes. The key is that they will di=er only by relatively unimportant (perhaps non-constant) reparametrizations of the crossover variables. These di=erences are the >eld theoretic analog of the di=erences, in a coarse graining RG, between block spinning, decimation, blockspinning on di=erent lattices, etc. In the space of coupling constants, t, h and g, the physical system exhibits many points of scale invariance, two of which are (103) and (104). One of these is accessible to a g = 0 RG, one to a g = ∞ RG, and only for a g-dependent RG are both accessible simultaneously. In this sense the g-dependent RG is global in the space of couplings whereas the others are not. Once again, the reader should understand that we are not saying that using a g-independent RG precludes accessing the other >xed point, but rather that the g-independent RG on its own cannot access it—the points of scale invariance and the >xed points of

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the g-independent RG are not the same. If one is able to supplement the g-independent RG with extra information then perhaps the other >xed point could become accessible. Almost certainly this will require the extra information to be non-perturbative with respect to the g = 0 >xed point. The solution of the di=erential equation (100), h = h(I; g; h0 ; I0 ), depends on the initial condition imposed on the :oating coupling at the initial >ducial scale I0 . For g=I0 1 RG :ows will >rst approach the isotropic >xed point before swinging over to the anisotropic one. In the limit I0 → ∞ 19 we can choose our initial condition h0 → h∗ , where h∗ is the isotropic >xed point, and obtain, h = h(g), V independent of the initial condition. h(g) V is the unique separatrix trajectory that passes between the two >xed points and in a crossover is the analog of a >xed point. All RG :ows eventually fall onto the separatrix. Finding this separatrix solution helps to isolate the universal part of scaling functions. 7.1.1. In what should we perturb? Having shown that only a g-dependent environmentally friendly RG may perturbatively access all >xed points it behooves us to think about the implications of this for perturbative calculations in general, and, in particular, higher order calculations where the validity of perturbation theory is naturally called into question. Put simply, we must address the question—in what should we perturb? The use of g independent renormalization to eliminate UV problems has two direct implications: >rstly, that direct perturbation theory fails badly in the regime g1. New “divergences” appear, which at a given order in perturbation theory are generically of the form (g)n as g → ∞. Secondly, one >nds a running coupling which, in the same regime, becomes very strong. For example, in the symmetric phase of ’4 theory on S 1 ×R3 one >nds (LI) =

1 + (3 =16A2 ) ln LI

;

where I is the momentum=mass scale of interest. Clearly in the IR regime, LI → 0, one is entering a strongly coupled regime. The words “divergences” and “strong coupling” lead invariably to the invocation of “non-perturbative” techniques. It sometimes seems that what is meant by the latter is something which cannot be formulated perturbatively in terms of any coupling, such as doing a lattice simulation. Quite often the case is overstated “divergences” and “strong coupling” being symptomatic of the fact that one is implementing a perturbation theory which is not capturing the correct qualitative nature of the e=ective degrees of freedom. In a crossover standard expansion parameters are not readily applicable. For instance, in the case of a theory with an O(N ) symmetry, if one uses a 1=N expansion, one >nds that if a g independent renormalization is used, the 1=N expansion breaks down in the g → ∞ limit, except when N → ∞ before g does. This is the spherical model limit which is exactly solvable. Additionally, for a crossover where the e=ective symmetry of the order parameter changes, such as at a bicritical point where the e=ective symmetry changes from O(N ) to O(N − M ), an expansion in 1=N will miss the crossover altogether. Similarly, if we think of an expansion in the context of a dimensional 19

We assume that this limit may be taken here. Sometimes, such as in the upper critical dimension, there is a residual dependence that cannot be eliminated.

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crossover, d → (d − 1), an expansion around the upper critical dimension with = (4 − d) will fail badly in the dimensionally reduced regime where = (4 − (d − 1)) is the natural expansion parameter. Thus we see that canonical expansion parameters will generically fail in the case of a crossover. One could also consider summing up sets of Feynman diagrams, such as is done in summing up the daisy diagrams (Hartree=Fock approximation) in >nite temperature >eld theory [141]. Naturally, this is always a tricky proposition, as certain diagrams that are dominant in one regime might not be so dominant in another. One must still also address the question of how to renormalize. Additionally, one faces the problem that any crossover accessed by the resummation might not be the one of interest. This is in fact what happens in >nite temperature >eld theory where the resummation of daisy diagrams in an attempt to get an improved description of a phase transition in a relativistic >eld theory merely accesses a mean >eld >xed point instead of the >xed point associated with the transition. The key to understanding whether a particular perturbation parameter is suitable or not is to see how it gets dressed by :uctuations. If an e=ective coupling is “small” in one regime there is no guarantee that it will be small in some other. We mentioned above the failure of standard “small” parameters such as 1=N and ’ when considering crossovers. The :oating coupling, and in particular the separatrix, have all the desired properties of capturing all the >xed points of the crossover, etc. However, one has no guarantee that they will be small throughout the crossover, or even at its asymptotic ends. This will be the case for example when the coupling ∼1. In this case, as in the >xed dimension expansion, a resummation procedure should be used. When we consider explicit two-loop results for the symmetric phase of ’4 during a dimensional crossover, in Section 9.1.2 we will use a [2,1] PadZe approximant [139] to >nd (h) = −( )h +

h2 ; 1 + Fh

(105)

where 4((5N + 22)A2(4−d) ( − 12 2 ) + (N + 2)A2(4−d) F= ((N + 8)A4−d )2

)

;

(106)

where A4−d = I d=dI − (4 − d) and the diagrammatic notation is as explained in footnote 12. In the line through the diagram denotes (d=dp2 )|p2 =0 . The explicit coupling obviously depends on the crossover in question. Higher order perturbative calculations can be carried out using, as above, PadZe or PadZe–Borel resummation techniques. There is nothing, in principle, in the present formalism, besides tedious computation, to prevent this from being done. As we shall see, the precision of our perturbative, resummed expansions at the asymptotic ends of the crossover is exactly the same as that of the >xed dimension expansion at the same >xed points. One may con>dently expect that the precision of an entire scaling function will be the same. Basically, the essence of an environmentally friendly renormalization is to calculate the Wilson and -functions that crossover correctly; resum them using PadZe or PadZe–Borel resummation techniques; solve the resummed -function equation and substitute the solution into the Wilson functions.

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To two loops there is no resummation of ’ . The [2,1] PadZe resummed expression for ’2 is h N +2 : (107) ’2 = 2 N +8 A2(4−d) ( − 12 ) 1 A2(4−d) h 1 − 6=(N + 8) +3 (A4−d )2 (A4−d )2 For small values of N one >nds that the di=erences between answers derived from the [2,1] PadZe resummed ’2 and the unresummed one are small for small values of N . Correlation functions are related to exponentials of these functions. It would be quite against the spirit of the RG to start perturbatively expanding the exponentials, otherwise there would be little point in using the RG in the >rst place. This is one of the problems with trying to use expansion methods to calculate scaling functions. The question is always: if something is to a power of should one expand it? In (98), for example, one should use the PadZe resummed Wilson function ’2 evaluated on the solution of the PadZe resummed -function and one should not expand the exponential itself. The theory has a natural structure in terms of basic building blocks, ’ , ’2 , and amplitudes. There in no particular reason why one could not attempt to reconstruct the vertex functions 2(N; M ) from their logarithmic derivatives. This would appear in the present formulation to be the most natural approach. We will not investigate this aspect further and leave the best method open to future developments. 7.2. What >ow parameter should be used? Another crucial di=erence between the coarse graining and reparametrization approaches is the wide degree of :exibility at one’s disposal in the choice of :ow parameter. In the coarse-graining approach, the RG :ows with respect to some UV or IR cuto=, while in the reparametrization approach it can be physically motivated, or something quite abstract. Remember that the ethos of the reparametrization approach is to describe the physics of a given system using coordinates associated with another, >ducial one. One typically chooses a one parameter family of such coordinate transformations denoted by a parameter I. The choice of this :ow parameter is closely related to the choice of >ducial system. A convenient and useful choice is to take I−1 to be a reference correlation length. This has several advantages: >rst of all it is a physical quantity that governs when IR e=ects are strong and secondly it is intrinsically a non-linear scaling >eld. As we have pointed out however, for crossover systems there is more than one correlation length that may be used. If the isotropic correlation length is used, then the associated RG will use the parameters of a >ducial, isotropic system to parametrize the properties of the anisotropic one. But the isotropic RG cannot access the anisotropic >xed point and therefore the deeply anisotropic regime must be accessed via direct perturbation theory, or some other technique. Such a perturbation theory, as seen, is ill behaved when corrections to scaling become large. When using the anisotropic correlation length the associated >ducial system is anisotropic but generically at a di=erent temperature than that of the actual system of interest. In this case a suitable choice, or gauge >xing, of the sliding scale I, along with suitable normalization conditions, can exponentiate all of the perturbative information. Another possible choice of >ducial scale is momentum, where the system of interest is associated with a momentum scale k while the >ducial system, isotropic or ansiotropic, is associated with

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a momentum scale I. This choice of >ducial scale would be useful when working directly at the critical point or when wishing to examine crossovers directly as a function of momentum. 7.2.1. “Running” the environment Above we considered RG :ows generated by mass=temperature or momentum. Here we will consider brie:y the concept of RG :ows generated by the environment itself. Previously, the environment was held >xed, in the sense that the anisotropy g was >xed, as far as the RG :ow was concerned. One can examine what happens when the environment is changed by examining RG :ows with respect to the new environment. One cannot however go directly from one environment to another by the action of the :ow itself. For the generic anisotropy g this means that a particular value of g >xes a curve, one then examines RG :ow along this curve. Another value of g yields another, distinct curve along which one can also examine RG :ows. It is possible however, to also generate an RG :ow in the “direction” of the environment itself. This can generically be done by implementing normalization conditions with respect to some >ducial environment. We will assume as usual that the parameter describing the environment does not itself renormalize and therefore there is no distinction between bare and renormalized g. If the >ducial environment is associated with a value, g, ˜ of g, then physical quantities, and in particular bare vertex functions, are independent of g˜ and one has in analogy with (94) d2(N; M ) ({3i (g)}) N ˜ g˜ = ˜’ (g; ˜ {3i (g)}) ˜ − M ˜’2 (g; ˜ {3i (g)}) ˜ 2(N; M ) ({3i (g)}) ˜ ; (108) d g˜ 2 where ˜’ = d ln Z’ =d ln g˜ and ˜’2 = −d ln Z’2 =d ln g˜ are Wilson functions which at a >xed point correspond to the anomalous dimensions of the operators ’ and ’2 , respectively. Once again, the in>nitesimal generator of RG transformations, gd=d ˜ g, ˜ can be written in a coordinate basis with -functions de>ned by i = gd3 ˜ i =d g˜ with corresponding Wilson functions, i = i =3i . The :ow equations for the renormalized parameters can be formally solved to yield ˜ = 3i (g˜0 )e 3i (g)

g˜

g˜0

i (x) d x=x

(109)

and the RG equation (108) integrated to yield 2(N; M ) ({3i (g˜0 )} = e

−(N=2)

g˜

g˜0

’ (x; {3i (x)}) d x=x M

e

g˜

g˜0

’2 (x;{3i (x)}) d x=x

2(N; M ) ({3i (g)}) ˜ :

(110)

The beauty of running the environment is that, by an appropriate choice of initial condition for the characteristic equations, one can relate parameters associated with the environment g to those associated with the environment g = 0. Thus, one can answer questions about the “shift” in a perturbatively controllable manner using RG techniques. One can also naturally access non-universal quantities, such as the critical temperature in a quantum–classical crossover, using purely RG methods. We shall see this in more detail in Section 10 where the critical temperature will be determined for a relativistic ’4 theory by integrating RG equations as a function of a >ducial temperature, @. The non-universality naturally enters from the dependence of the solution of the :ow equation on the initial condition at @ = 0.

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7.2.2. Two RGs are better than one In the previous two sections we saw that there is ample scope for choosing di=erent :ow parameters in the reparametrization approach. The question then naturally arises as to whether it is possible to implement coordinate transformations that simultaneously depend on more than one :ow parameter, and if so what are the advantages, if any, of this? Multiple RGs have been considered before, mainly in the context of quantum >eld theory at >nite temperature 20 [36 –38,142,143] and that of an O(N ) model below the critical temperature where there exist two correlation lengths—transverse and longitudinal. An analysis of this situation was carried out in [145,146] in four dimensions using RGs based on modi>cations of minimal subtraction. Up to now, we have viewed the RG as a one parameter group of reparametrizations of the parameters that describe a physical system. If there are n such independent parameters, one of which will be used as :ow parameter, and therefore n -functions, some of which may of course be trivial, then in order to solve the :ow equations it is necessary to specify n − 1 initial conditions per :ow line. Geometrically, once an initial condition has been chosen one is restricted to the :ow line corresponding to that initial point, i.e. the RG cannot be used to get from one :ow line to another. Hence, to span the space of parameters by the RG :ows one needs an (n − 1)-dimensional set of initial conditions. For instance, considering 2(4) (u; s; t) at the critical point (where u, s and t are the Mandelstam variables) as a function of the momentum variables s and t. If one chooses a >ducial value of u as :ow parameter, I, then it is possible to use the RG associated with I to generate the behavior of 2(4) as a function of u. However, one must use as initial condition 2(4) (I0 ; s; t) at some initial value of I0 , and at some values of s and t. The latter will be constant along a particular :ow line. This means that in order to generate 2(4) (u; s; t) one needs as physical input 2(4) (I0 ; s; t). One cannot generate 2(4) (u; s; t) purely using this RG with input 2(4) (I0 ; s ; t ) where s = s. Of course, one can change initial condition but this requires further input, i.e. knowledge of 2(4) (I0 ; s; t) for di=erent s and t. As we will see, in many systems such further information might not be readily available. If one extends the fundamental relation between bare and renormalized vertex functions to the case of k (k¡n) RG :ows one generates k RG equations for the N -point vertex functions based on the reparametrization invariance of the bare theory Ij

d2(N ) N ({gi ({Ij })}; {Ij }) = j’ ({gi ({Ij })}; {Ij })2(N ) ({gi ({Ij })}; {Ij }) ; dIj 2

(111)

where 1 6 j 6 k; j’ = d ln Z’ =d ln Ij , and the derivatives are taken at >xed values of the bare parameters. One now has a set of k vector >elds which are assumed to be independent. As the bare theory is invariant under a Lie dragging along any of these :ows then it is also invariant under the action of the vector >eld [Ij d=dIj ; Ii d=dIi ]. This integrability condition leads to non-trivial relations between the -functions of the theory. In a coordinate representation each vector >eld can be decomposed as Ij 20

d 9 9 = Ij + ji i ; dIj 9Ij 9g

(112)

Though see the article by Kovalev and Shirkov in this series [2] for an interesting application to laser optics, where two RGs are used to access a higher dimensional singularity.

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where ji = Ij dgi =dIj . Integrability requires that Ii

dji dj = Ij i : dIi dIj

(113)

In the case of two RGs for example, the solution of Eq. (111) is 2

(N )

(I10 ; I20 )

=e

−(N=2)

I1 I0 1

1’ (x1 ; I20 ) d x1 =x1 −

N I2 2 (I ; x ) d x2 =x2 2 I20 ’ 1 2

2(N ) (I1 ; I2 ) :

(114)

With k RGs, in principle, one need only specify n − k initial conditions and therefore one has access to a k-dimensional subspace of M using only one initial condition, as by specifying only one initial condition in this k-dimensional space one can :ow to any other point purely through solving the RG equations. So, the advantages of using multiple RGs are that with enough independent :ows any point in M can be reached from any other purely with RG :ows thereby restricting the physical input necessary in the form of initial conditions. It also allows one to access non-universal quantities, normally thought to be outside the realm of RG methods, using only RG :ows. 7.3. An analytic parametrization via normalization conditions In the previous sections, we saw that there is a great deal of latitude as to which parameters will be renormalized in the reparametrization method, and that this can play a crucial role in the implementation of an approximation scheme such as perturbation theory. In this section we will consider how to >x a coordinate system formally. One of the most common ways of doing this, and perhaps the most intuitive, is by using normalization conditions. By >xing the entire coordinate transformation we will see that normalization conditions play a multipurpose role, being capable of >xing the particular RG and, simultaneously, the :ow parameter. We will illustrate the above in a concrete context. Consider Hamiltonian (15). We now perform a coordinate change from bare to renormalized coordinates. We indicate by t(m) that temperature parameter which yields the correlation length = m−1 . Whether this corresponds to the isotropic or anisotropic or g dependent correlation length we will specify on a case by case basis. The reparametrization RG method is to change from the original bare parameters to new renormalized ones given by t(m; I) = Z’−21 (I)tB (m) ; (I) = Z (I)

B

;

’(I) V = Z’−1=2 (I)’V B ;

(115) (116) (117)

where I is the arbitrary sliding scale and, at this moment, we do not specify if the coordinate transformation depends on g. The subscript B is the usual one for denoting bare quantities. Note that we are here assuming a simple multiplicative renormalization. Frequently, due to operator mixing for instance, it is necessary to implement a renormalization which unlike the above is not diagonal in the space of parameters. This adds some degree of technical complication but does not change any of the basic results we will derive. Neither does the basic formalism change in the case where

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there are more parameters to be renormalized or more anisotropy parameters. The Wilson functions corresponding to the coordinate transformation (115) – (117) age given by d ’2 (I) = − I ln Z’2 ; (118) dI c d (I) = I ln Z ; (119) dI c d ’ (I) = I ln Z’ ; (120) dI c where the derivative is along an appropriately chosen curve in the phase diagram which we here denote by c. The coordinate transformation (115) – (117) implies a renormalization of the vertex functions such that 2(N; M ) (I) = Z’N=2 (I)Z’M2 (I)2(N; M ) + 0N0 0Mn A(n) (I);

n = 0; 1; 2 ;

which obey the RG equation N d (N; M ) + M ’2 − ’ 2(N; M ) = 0N 0 0Ln B(n) : I 2 dI 2

(121)

(122)

The equation is inhomogeneous for the three vertex functions 2; 2(0; 1) and 2(0; 2) , where the “source” term is B(n) = I

dA(n) + n ’2 A(n) : dI

(123)

The solution of this equation is 2(N; M ) (t(m; I); ’(I); V (I); I) = e

I

((N=2) ’ −M ’2 ) d x=x

2(N; M ) (t(m; I0 ); ’(I); V (I0 ); I0 ) I I d x (n) n (x) d x=x −n x (y) dy=y + 0N 0 0Mn e I1 ’2 : B (x)e I1 ’2 I0 x

The renormalized coordinates t(I);

I0

(124)

(I) and ’(I) V satisfy

I

dt(I) = ’2 (I)t(I) ; dI

(125)

I

d (I) = (I) (I) ; dI

(126)

I

1 d ’(I) V = − ’ (I)’(I) V : dI 2

(127)

For the generic set of crossovers considered here, associated with (15), the three Wilson functions ’2 ; and ’ are the fundamental building blocks, perturbative knowledge of which allows for the construction of any scaling function of interest. The coordinate transformation speci>ed by

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471

(115) – (117) has yet to be >xed. To specify a particular coordinate change, we must >x the transformation functions Z’2 ; Z’ and Z using normalization conditions. 7.3.1. Disordered phase with H = 0 For simplicity we consider >rst the disordered phase, ’V = 0. For the canonical system (15), where the correlation length is m−1 as given by (36), we impose the following normalization conditions which explicitly depend on the anisotropy parameter g 9p2 2(2) (p; t(I; I); (I); g; I)p2 =0 = 1 ; (128) 2(2; 1) (0; t(I; I); (I); g; I) = 1 ;

(129)

2(4) (0; t(I; I); (I); g; I) = :

(130)

These three conditions >x the three normalization factors Z’ ; Z’2 and Z and hence correspond to reparametrization of the coordinates t, and ’. V A fourth condition, I2 = 2(2) (0; t(I; I); (I); g; I)

(131)

serves as a “gauge” >xing condition that relates the sliding scale I to the physical temperature t and will be discussed more fully in the next section. With the above normalization conditions one >nds the following expressions for the Z factors, which in their turn de>ne the coordinate transformation between bare and renormalized parameters Z’−1 = 9p2 2B(2) (p; tB (I); B ; g) 2 ; (132) p =0

Z’−1 Z’−21 = 2B(2; 1) (0; tB (I); Z’−2 Z =

2B(4) (0; tB (I); B

B ; g) ;

B ; g)

(133)

;

(134)

where the relation between tB (I) and I is speci>ed by I2 =

2B(2) (0; tB (I);

B ; g) (2) 9p2 2B (p; tB (I); B ; g)|p2 =0

:

(135)

Note that the Zi are obtained from the vertex functions of the system speci>ed at an arbitrary, >ducial correlation length I−1 , as opposed to the correlation length of interest, m−1 . The coordinate transformation we have speci>ed is from bare coordinates to new ones corresponding to the observable quantities associated with certain vertex functions for a >ducial system identical to the one of interest except at a di=erent temperature. If one chooses a ;xed value for g then the resulting renormalization constants and renormalized parameters are no longer sensible to changes in the environment. Some particular choices of interest are g = 0 and ∞ which correspond to the isotropic and anisotropic limits, respectively. Hence, there are at least three correlation lengths of interest for a given environmental scale g; (g), (∞) and (0). In the case of a >nite size system the isotropic limit L = ∞ is that of the bulk theory.

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We will now discuss the relative advantages and disadvantages of one set of normalization conditions versus another. As an example of this consider 2(4) (t; g; ; I). Using the g-dependent conditions (128) – (131) one >nds 2(4) (t(I; I0 ); (I0 ); g; I0 ) = (I0 )e

I

I0

( (x; g)−2 ’ (x; g)) d x=x

:

(136)

In this case, by suitable normalization conditions we have been able to totally exponentiate the four-point vertex function. By contrast, using g = 0 conditions one will >nd 2(4) (t(I; I0 ); (I0 ); g; I0 ) =

0 (I0 )e

I

I0

( (x; 0)−2 ’ (x; 0)) d x=x

F (4) ;

(137)

where F (4) is a scaling function that must be determined perturbatively, or otherwise, around the isotropic >xed point and 0 (I0 ) is the initial coupling in the g = 0 scheme. Let us highlight the key di=erences between (136) and (137): I in the former represents a >ducial correlation length of the g-dependent system whereas in the latter it is a >ducial correlation length of the isotropic g=0 system. In (136) the fully exponentiated form means that a perturbative calculation of the environmentally friendly Wilson functions (g) and ’ (g) is all that is required. On the other hand, for (137) one requires the additional calculation of the scaling function F4 , which perturbatively encounters problems in the isotropic regime (g=I)1, typically leading to corrections to scaling that are a power series in (g=I)n for n ¿ 0. Such large corrections to scaling are symptomatic of the existence of the anisotropic scaling regime. Of course, the normalization conditions (128) – (131) are not the only environmentally friendly ones. A popular choice is that of GMS [73] which takes as its inspiration minimal subtraction. In some sense, diagramatically, the essence of a crossover is the existence of perturbative divergences in the limit g → ∞ di=erent in nature to those of the isotropic >xed point. For instance, in dimensional crossover in a three-dimensional >lm geometry in the limit L=L → 0, L → ∞ one >nds IR divergences characteristic of a two-dimensional system whereas in the limit L=L → ∞, L → ∞ they are those of a three-dimensional system. In the language of counterterms any renormalization that removes one by addition of a counterterm without being able to treat the other will be incapable of capturing the crossover. What is required is to use a subtraction scheme that uses a function of LI, f(IL), as a counterterm such that in one limit it is equivalent to minimal subtraction in a three-dimensional system while in the other limit to minimal subtraction of a two-dimensional system. GMS will be environmentally friendly in that remnant scaling functions such as F4 will be perturbatively well de>ned across the entire crossover. However, this perturbative information will not be exponentiated and can lead to quantitative di=erences. The normalization conditions (128) – (131) also allow for a complete exponentiation of the vertex function 2(2; 1) and of 9p2 2(2) |p2 =0 . As we will see, a suitable gauge >xing of I also allows for a complete exponentiation of the susceptibility. One could in principal exponentiate all vertex function by calculating − M ’2 +

N 1 d ’ + N; M = (N; M ) I 2(N; M ) 2 2 dI

where N; M is the additional contribution from the scaling function.

(138)

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473

7.3.2. Ordered phase In the previous section we saw how in the disordered phase, by a judicious choice of normalization conditions, it was possible to exponentiate perturbation theory for a subset of vertex functions in terms of the non-linear scaling >eld I(t; g). An analogous procedure may be followed below the critical temperature. In this case, the appropriate normalization conditions are 9p2 2t(2) (p; t(I; I); (I); g; ’(I); V I) 2 = 1 ; (139) p =0

V I) = 1 ; 2t(2; 1) (0; t(I; I); (I); g; ’(I);

(140)

V I) = ; 2t(4) (0; t(I; I); (I); g; ’(I);

(141)

Note that in this case we impose the normalization conditions on the transverse correlation functions introduced in Section 3. These conditions serve to >x the three Z functions associated with ’; V t and while the condition I2 = 2t(2) (0; t(I; I); (I); g; ’(I); V I) ;

(142)

as in the disordered phase, serves as a gauge >xing condition that relates the sliding scale I to the physical temperature t and the physical magnetization ’. V Physically, I is a >ducial value of the non-linear scaling >eld mt which is the inverse transverse correlation length. It is legitimate to ask why the transverse functions are considered and not the longitudinal ones? As stated in Section 3, the transverse functions are the more fundamental building blocks, given that the longitudinal vertex functions can be built from them using the Ward identities. This is even true for N = 1; 0 where an analytic continuation of the Ward identities is used [38]. The Wilson functions that correspond to the above Z factors also depend on ’. V As in the disordered case the normalization conditions should be optimally chosen to “exponentiate” as much perturbative information as possible. Here, it is the transverse vertex functions 2t(2) ; 2t(4) ; 2t(2; 1) and 9p2 2t(2) that exponentiate naturally. For instance, we consider the transverse four point function V I). Using the g-dependent conditions (139) – (142) one >nds 2t(4) (t; (I); g; ’(I); V 0 ); I0 ) = (I0 )e 2t(4) (t(I; I0 ); (I0 ); g; ’(I

I

I0

( −2 ’ ) d x=x

:

(143)

Once again, by suitable normalization conditions we have been able to totally exponentiate the transverse four-point vertex function. By contrast, using g = 0 conditions one will >nd 2t(4) (t(I; I0 );

V 0 ); I0 ) 0 (I0 ); g; ’(I

=

0 (I0 )e

I

I0

( −2 ’ ) d x=x

Ft(4) ;

(144)

where Ft(4) is a scaling function that must be determined perturbatively around the symmetric >xed point. Note that as a consequence of the Ward identities certain longitudinal vertex functions also naturally ‘exponentiate’. For example, 2l(2) = 2t(2) + 13 2t(4) ’V 2 , exponentiated as V g; I0 ) = I2 e 2l(2) (t(I; I0 ); (I0 ); ’;

−

I

I0

’2 d x=x

+

(I0 )’V 2 (I0 ) II ( e 0 3

− ’ ) d x=x

:

(145)

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The integrations in (143) – (145) are along the appropriately chosen contour used to de>ne the Wilson functions. In analogy with the disordered phase I in (142) represents a >ducial transverse correlation length of the anisotropic system, whereas if we set g=0 in this condition it is a >ducial transverse correlation length of the isotropic system. In distinction to the disordered phase however, in both cases I will depend on ’. V In (143) the fully exponentiated form means that a perturbative calculation of the environmentally friendly Wilson functions and ’ is all that is required. In this case however they also depend on ’. V On the other hand, for (144) such an environmentally friendly renormalization requires the additional calculation of the scaling function Ft(4) . In the present case due to the extra length scale, ’, V the Wilson, and other scaling functions are functions of two scaling variables, x = g=I and y = ’V 2 =Id−2 , not one. This means that in the space of parameters there is a new asymptotic regime associated with the limit y → ∞ which corresponds to approaching the g-dependent coexistence curve. A ’V independent but g-dependent renormalization would be capable of accessing the crossover between the isotropic and anisotropic >xed points associated with the g-dependent critical point at the end of the coexistence curve. However, it would not be capable of accessing the y-dependent crossover to the strong coupling >xed point that governs the coexistence curve. In this sense an environmentally friendly renormalization must be both g and ’V dependent. An RG that is independent of both x and y will perturbatively encounter problems in the anisotropic regimes x1 or y1, typically leading to corrections to scaling that are a power series in xn or ym for n; m ¿ 0. Such large corrections to scaling are, once again, symptomatic of the existence of anisotropic scaling regimes. 7.4. Fixing the sliding scale The choice of “gauge >xing”, or matching condition, on the arbitrary sliding scale I cannot a=ect any physical result in an exact formulation. However, when an approximation scheme is implemented the choice turns out to be quite crucial. In both coarse graining and reparametrization approaches the choice of I is made so as to match on to a physical limit where a non-RG approximation technique may be used. What we mean by a choice of I is the imposition of a condition of type (142), that relates it to the parameters of the physical system under consideration. Once we implement such a gauge >xing condition then RG invariance is lost. The motivation here is exactly the same as mentioned for a coarse graining RG in reference to the law of corresponding states. If one can relate the correlation function of interest, where there are many non-linearly coupled degrees of freedom within a correlation length, to another, where there are few, then the latter may be calculated using various standard techniques. A most natural place to map, or match, to is mean >eld theory where one can map, or match, the correlation functions onto known answers. One may also match the functions onto ones where a perturbative expansion about the Gaussian >xed point is valid. Such a mean >eld gauge >xing condition is implicit in the normalization conditions of Sections 7.3.1 and 7.3.2, where the motivation in the context of the stated vertex functions is to absorb all :uctuations into rede>nitions of the parameters t; ’; V . In the disordered phase the >ducial correlation length is given by (131) while in the ordered phase it corresponds to (142), which now involves the background >eld ’. V Hence, in this case conditions

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475

(131) and (142) relate the non-linear scaling >eld I to the linear scaling >eld t and, in the latter case also to ’. V The >ducial correlation length, I−1 , in this case depends on both linear scaling >elds. Other gauge >xing conditions are of course possible as long as they lead to a well de>ned environmentally friendly perturbation theory. For instance, we could have replaced (129) by the condition t(I; I) = I2 ;

(146)

a condition used in [73] and some of our earlier work. This condition together with (131) determines a multiplicative renormalization of t, and of ’2 insertions, via a renormalization function Zt . The two renormalization functions Zt and Z’2 are di=erent, the latter being determined by (129). The quantity t = −d ln Zt =d ln I is an analog of ’2 , however, the problem with implementing a condition such as (146) in perturbation theory is that the resulting Zt involves diagrams with massless propagators, some of which are strictly in>nite even after the introduction of an UV cuto=. 7.4.1. Fixing the sliding scales for more than one RG The principle function of a gauge >xing choice of the sliding scale is to map to a place in M where a direct perturbative, or other non-RG approximation, treatment of the vertex functions is feasible. This is often done by choosing the sliding scale such that in units of this scale the inverse correlation length is of order one, thereby circumventing IR diJculties. However, in crossover problems there are generally many length scales of interest, and in particular there may exist more than one relevant correlation length that may diverge. Thus, >xing the RG scale such that one correlation length is of order unity will not necessarily help in the regime where the other correlation lengths are large. Such a circumstance has, for instance, caused diJculties in the description of the ordered phase in ferromagnets with N ¿1 due to the fact that both the transverse and longitudinal correlation lengths may be large. When correlation functions depend on many parameters it may occur that perturbation theory is badly behaved as a function of more than one parameter. The use of a single RG scale might be suJcient to map to a subspace of M which is amenable to a perturbative treatment, but generically there will exist other regions that are not treatable. Using more than one RG o=ers more :exibility in >nding a region that is perturbatively accessible. For instance, considering an analogue of the case considered in Section 7.2.2 where vertex functions of interest depend on two momentum variables, p1 , and p2 ; assume that the correlation functions exhibit di=erent singular behavior in the limits p1 → 0 and p2 → 0. A single RG may be used to control the singular behavior in one limit or the other but not both. Having two RGs enables one to choose one scale, I1 , to control the singular p1 behavior and the other to control the singular p2 behavior. It is worth emphasizing that this example is not as contrived as it sounds. In deep inelastic scattering in QCD there are logarithms that need to be summed that are functions of the two Bjorken variables x and Q2 . A single RG is capable of summing one set but not the other, the use of two RGs gives the opportunity of controlling both sets of logs. 7.5. Integrating along contours in the phase diagram The Wilson functions are associated with multiplicative renormalizations of the vertex functions and are the natural building blocks of the theory. In the previous sections we renormalized by

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choosing our sliding scale, I to be a >ducial inverse correlation length. This inverse correlation length, m, is naturally a non-linear scaling ;eld. In terms of this non-linear scaling >eld and the Wilson functions many quantities have a natural exponential form. For example, in the disordered phase with H = 0 we have *−1 (m; (I); g; I) = I2 e 2(2; 1) (m; (I); g; I) = e

m I

m I

(2− ’ (x; g)) d x=x

;

( ’2 (x;g)− ’ (x;g)) d x=x

2(4) (m; (I); g; I) = (I)e

m I

(147) ;

( (x; g)−2 ’ (x; g)) d x=x

(148) ;

(149)

m ≡ m(t; g) being the actual correlation length in the system of interest. In the case of a superrenormalizable theory, such as ’4 in three dimensions, we can take the limit I → ∞ whereupon all diagrams go to zero and so i (I) → 0 and we recover expressions for the vertex functions in terms of the bare parameters. In the zero momentum case we note that, for example, the susceptibility exponentiates in a very natural way in terms of m. However, if we require a description in terms of the linear scaling >elds t; ’, V etc., we have to determine m in terms of them. Thus, we have to develop a formalism such that the relation between linear and non-linear scaling >elds is manifest. This in itself is a coordinate transformation. 7.5.1. Relating non-linear and linear scaling ;elds in the disordered phase The initial variable in the Hamiltonian is rB = rc + tB , where rc is the value of rB at the critical point for a >xed B . It also depends on the UV cuto= necessary to de>ne the theory. It can naturally be interpreted in statistical physics terms as rc ˙ Tc − Tm where Tm is the critical temperature as predicted by mean >eld theory, or critical temperature in the absence of :uctuations. As is well known, an additive renormalization is >rst necessary to compensate for the critical temperature shift, then a further multiplicative renormalization of tB is needed. It is the >rst renormalization that complicates direct exponentiation in terms of the Wilson functions. However, an expression can be found by using the di=erential relation d2t(2) = 2t(2; 1) dt + 16 2t(4) d ’V 2 ;

(150)

where the exterior derivative “d” acts in the (t; ’) V plane. To change variables from t to m we >rst restrict our consideration to the disordered phase T ¿Tc ; ’V = 0 i.e. to the >ducial curve of constant ’V = 0 in Fig. 1. In this case the longitudinal and transverse vertex functions are identical. With de>nition (132) of the function Z’ and the Wilson function (127) one obtains dt = (2 − ’ )

2(2) dm : 2(2; 1) m

(151)

The above relation is generically true for the isotropic or anisotropic system. On renormalization at a >ducial sliding scale I, using the normalization conditions (128) and (129), together with (125) and (127) and the gauge >xing condition (131), one >nds, integrating along a line of constant

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

’V = 0, that tg =

m(tg ;g)

0

(2 − ’ (x; g))e−

x I

’2 (x ;g)d x =x

x dx :

477

(152)

This equation tells us how the non-linear scaling >eld m(tg ; g), the inverse correlation length of the anisotropic system, depends on the temperature parameter tg and the anisotropy g. The interpretation of the temperature variable, ‘t’, depends on the initial conditions one uses for the integration. In (152) we are assuming that the anisotropic system exhibits a critical point; the contrary case will be considered later. Hence, tg (I) ˙ (T − Tc (g))=T where Tc (g) is the critical temperature of the anisotropic system. The philosophy here is to use as initial condition that point at which the corresponding correlation length diverges, i.e. ‘t’ measures deviations from the anisotropic critical temperature. If we had chosen as temperature variable t(I) = Z’−21 (I)2 (T − Tc (0))=T where Z’−21 is speci>ed by (129), one would have m(t;g) x (2 − ’ (x; g))e− I ’2 (x ;g)d x =x x d x ; (153) t= m(0;g)

where m(0; g) is the inverse anisotropic correlation length at the isotropic critical point. Eq. (153) tells us how m(t; g) depends on the temperature parameter t, the anisotropy g and m(0; g). It is natural to enquire as to how tg and t are related. Simply, tg = t + &g , where the shift &g = Z’−21 (I)2 (Tc (0) − Tc (g))=Tc (g). Note that the amplitude Z’2 is g dependent but once I is chosen such that g=I this g dependence disappears. From (152) and (153) one may derive an expression for the critical temperature shift m(0;g) x 2 &g = (2 − ’ (x; g))e− I ’ (x ;g)d x =x x d x : (154) 0

This equation will be discussed in more detail shortly in a more general context. In all these cases the Wilson functions explicitly depend on g as they were de>ned from environmentally friendly normalization conditions. Recall that the renormalized phase diagram is the bare diagram rescaled by Z factors. We see from (152) that I

dtg (m; I) = ’2 tg (m; I) dI

(155)

as we expect. Furthermore, if we take I → ∞, then, in the super-renormalizable case (d¡4) all :uctuations are suppressed in that limit and are zero for I = ∞. This implies that the Z factors become unity so that (152) becomes m(tg ;g) x (2 − ’ (x; g))e− ∞ ’2 (x ;g) d x =x x d x : (156) tg = 0

While in the universal limit when the Wilson functions are evaluated on the separatrix solution of the -function and depend only on the scaling variable zg = m=g we have zg x dx -’2 1=+ ; (157) tg = I g (2 − ’ (x))e− ∞ [ ’2 (x ) d x =x x1=+ x 0

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where [ ’2 (zg ) = ’2 (zg ) − -’2 , -’2 is the anomalous dimension at the isotropic >xed point (i.e. limx→∞ ’2 (x) = -’2 ) and + = 1=(2 − -’2 ). Furthermore, (154) can be put in a similar scaling form &g = −I-’2 g1=+ w0 ; where

w0 = −

0

G

(2 − ’ (x))e−

x

∞

[ ’2 (x ) d x =x 1=+

x

dx ; x

(158)

where G is a universal amplitude ratio. Since tg = t + &g , Eq. (157) establishes that in the universal limit m takes the scaling form m = g*g (w) with the scaling variable w = I−-’2 tg−1=+ and the zero of the scaling function *g is located at w0 . It is important to draw a distinction between the use of the functions ’ ; ’2 and as used in relations such as (152) and as abstract objects that enter the RG equation. Had we chosen conditions other than the environmentally friendly set (128) – (130) for the speci>cation of the Z factors (such as minimal subtraction for example) then the associated Wilson functions would not be the objects that enter (152); rather they would enter in conjunction with appropriately subtracted vertex functions which would have to be determined separately. The most important and innovative feature of the above is that the determination of 2(2) by integrating 2(2; 1) allows us to bypass the need to determine rc perturbatively. We will see this to be a crucial advantage when it comes to calculating shifts, as a perturbative calculation without this arti>ce leads inexorably to the appearance of IR divergences. 7.5.2. Relating non-linear and linear scaling ;elds in the ordered phase In the above we considered problems with two non-linear scaling >elds m and g and performed a coordinate transformation to relate m and g to the linear scaling >eld t, assuming that we were interested in scaling functions as functions of g and not of any, if they exist, associated linear scaling >elds. This philosophy can be repeated for the ordered phase. The di=erence now, is that given that there are two linear scaling >elds, t and ’, V we will have to determine two corresponding non-linear scaling >elds. We have seen that mt is one logical candidate. For the other, we will choose m2’ =

1 2t(4) ’V 2 : 3 9p2 2t(2) |p2 =0

(159)

Note that this quantity acquires no multiplicative or additive renormalizations and is therefore an RG invariant. It represents the anisotropy in the masses of the longitudinal and transverse modes and in mean >eld theory is m2’ = ’V 2 =3. It is related to the sti=ness constant Ds = ’V 2 9p2 2t(2) |p2 =0

via m2’ =

1 3

Ds :

(160)

In fact we can think of m’ as the non-linear scaling >eld associated with the anisotropy of the system in the ordered phase. In distinction to the anistropy g however, it is clearly of interest to

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479

know how m’ depends on its constituent linear scaling >eld ’V as it is the magnetization that appears most naturally in the equation of state. The coordinate transformation of interest will be that between (t; ’) V and (mt ; m’ ). The normalization conditions which >x the particular parameterization we will use to describe physical quantities are those of Eqs. (139) – (142), where the relevant non-linear scaling >elds are the inverse transverse correlation length, mt (t; ’), V and m’ (t; ’). V Previously our environmentally friendly renormalizations were carried out at >xed anisotropy non-linear scaling >eld g with constant bare parameters and at a >ducial anisotropic correlation length I. This can be repeated here keeping m’ >xed and varying mt . It is also possible and, perhaps, thermodynamically more natural to keep ’V >xed, rather than m’ , and vary mt . The two possibilities, ’V >xed and m’ >xed, lead to two distinct RGs. The associated Wilson functions are d i = I ln Zi (161) dI m’

for the m’ constant RG and d Vi = I ln Zi dI ’V

for the ’V constant RG. The relation between the in>nitesimal generators of the RGs is d d 1 d I = I + ( V − V’2 )m’ : dI ’V dI m’ 2 dm’ I

(162)

(163)

For constant m’ the relevant :ow equations are dm2’ =0 (164) dI dV (165) = −(4 − d) V + (I; m’ ; g; V) V : I dI Obviously, only the :ow for the renormalized coupling is non-trivial. In the constant ’V case there are two non-trivial coupled :ow equations for m’ and V. These are I

dm2’ = ( V (I; m’ ; g; V) − V’ (I; m’ ; g; V))m2’ ; (166) dI dV (167) I = −(4 − d) V + V (I; m’ ; g; V) V : dI From the de>nition of m’ , (159), and using the normalization conditions (139) and (141) we have I

(I)’V 2 (I) mt ( (x; m’ ; g)− ’ (x; m’ ; g)) d x=x eI ; (168) 3 where ’(I) V is the physical magnetization renormalized at I. This is consistent with the di=erential equation (164) on >xing the sliding scale to be mt . Here, the Wilson functions and ’ are integrated along contours of constant m’ and g, which is the curve along which they were originally m2’ =

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de>ned via Eqs. (119) and (120). With (168) we see that ’V satis>es a constitutive relation of the form ’V = f(mt ; m’ ). For the ’V >xed case the solution to (166) takes the form m2’ (I) = m2’ (I0 )e

I

I0

( V (x)− V’ (x)) d x=x

:

(169)

Here the Wilson functions are integrated along contours of constant ’V so that m’ is varying and the solution is only formal since a nontrivial set of di=erential equations need to be solved. For consistency we require that m’ (I0 ) = (I0 )’V 2 (I0 )=3. Furthermore, when we >x the sliding scale I to m (and by convention, as throughout the paper, rename I0 to I) we >nd the relationship between m’ and ’V is m2’ =

(I)’V 2 (I) mt ( Vl (x; ’;V g)− V’ (x; ’;V g)) d x=x eI ; 3

(170)

where, as mentioned, the integral is performed along a line of constant ’V and anisotropy g. In either case, Eqs. (170) and (168) give the constitutive relation between m’ and ’. V So, integrating along a curve of constant ’V we have mt (t;’;g) V x V d x =x (2 − V’ (x; ’; V g))e− I V’2 (x ;’;g) x dx ; (171) tg; ’V = 0

V is a measure of the temperature deviation from the anisotropic where tg; ’V ˙ (T − Tc (g; ’))=T coexistence curve where mt = 0. If we had chosen as temperature variable t = Z’−21 (I)2 (I)(T − Tc (0; 0))=T one would have mt (t;’;g) V x V d x =x (2 − V’ (x; ’; V g))e− I V’2 (x ;’;g) x dx : (172) t= mt (0;’;g) V

Eq. (172) tells us how mt (t; ’; V g) depends on the temperature parameter t, the magnetization ’V and the anisotropy g and mt (0; ’; V g) where mt (0; ’; V g) is the inverse transverse correlation length at the isotropic critical temperature. Once again tg; ’V and t are related via a “shift” relation tg; ’V = t + &g; ’V , where the shift &g; ’V = Z’−1 2 (Tc (0; 0) − Tc (g; ’))=T V V measures the temperature deviation of a c (g; ’) point on the anisotropic coexistence curve from the isotropic critical point. From (171) and (172) one derives an expression for this “shift” m(0;’;g) V x V d x =x (2 − V’ (x; m’ ; g))e− I V’2 (x ;’;g) x dx : (173) &g; ’V = 0

As before, in the super-renormalizable case (d¡4), if we take I → ∞, then all :uctuations are suppressed and we recover the bare parameters. Returning to the constant m’ representation, we have mt (t;’;g) V x (2 − V’ (x; m’ (x); g))e− I ’2 (x ;m’ (x);g) d x =x x d x ; (174) tg; ’V = 0

where V’ (x; m’ ; g) is a somewhat complicated function of ’ , and integrals of their derivatives with respect to m’ . Here, in the >nal integration over x, m’ varies as this integration has to be done along a curve of constant ’V not m’ .

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481

In analogy with the previous section it is useful to introduce dimensionless ratios of nonlinear scaling >elds, z = mt =m’ and zg = mt =g. In the universal limit, when the Wilson functions are evaluated on the separatrix solution of the coupled equations and interpolate between di=erent >xed point values, we >nd in terms of the z and zg variables that zg

d z2 = (2 − V (zg ; z; V) + V’ (zg ; z; V))z 2 ; d zg

(175)

zg

dV = −(4 − d) V + V (zg ; z; V) V : d zg

(176)

∗ ∗ The separatrix solution of (166) and (167) is obtained with asymptoting to V = V , where V is d−2+2 the isotropic >xed point coupling, and z has the asymptotic form z (zg ) = (1=y)zg as zg goes to in>nity. Here y is a constant which can be determined by the normalization conditions. We >nd

y=

V∗ 3

I- ’V 2 g−2=+ ;

(177)

where = (+=2)(d − 2 + -). The Wilson functions Vi on the separatrix are then functions of zg and y. Furthermore, we deduce that z = z(zg ; y). We can also see this from the scaling form of the solution for z which takes the form z2 =

1 d−2+- − zg ([ V (x; y)−[ V’ (x; y)) d x=x z e ∞ : y g

We further obtain the analog of (157) to be zg x dx -’2 1=+ tg; ’V = I g (2 − V’ (x; y))e− ∞ [ V’2 (x ;y) (d x =x ) x1=+ : x 0

(178)

(179)

In further anology with (158) the shift &g; ’V has the limiting form lim I

I→∞

−-s’2

&g; ’V (’; V g; I) = g1=+ F&g (y) ;

where the universal scaling function F&g (y) is G(y) x dx & : (2 − V’ (x; y))e− ∞ [ V’2 (x ;y) d x =x x1=+ Fg (y) = x 0

(180)

(181)

Note that G has now been promoted to a universal scaling function of y as opposed to a universal constant as in the previous section. Again with tg; ’V =t +&g; ’V , we see that (179) implies the universal limit of mt takes the scaling form mt = g*g; ’V (w; y)

(182)

with the scaling variable w = I-’2 tg−1=+ as before. The zero of the scaling function *g; ’V (w; y) is located at w0 = −F&g (y).

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Up to this point, all our scaling functions have been parametrized in terms of an ‘initial condition’ mt (0; ’; V g) (or in scaling form G(y)). An expression for mt (0; ’; V g) can be determined (another method will be shown in the next sections) by integrating the di=erential relation (150) along the critical isotherm to obtain ’V2 (4) 2t (x; g) (2) 2t (0; ’; V g) = dx : (183) 6 0 By choosing normalization conditions de>ned on the critical isotherm such that the transverse correlation length is I−1 , we can express 2t(4) in the form 2t(4) = (I)e

mt (0; ’;V g) I

( ˜ (x; ’; V g)−2 ˜’2 (x;’;g)) V d x=x

;

(184)

V g) and ˜’2 (x; ’; V g) are the resulting Wilson functions from this prescription. Inverting where ˜ (x; ’; (183) one >nds V x 6 mt (0;’;g) 2 (2 − ˜’ (x; ’; V g))e− I ( ˜ (x ; ’;V g)− ˜’ (x ; ’;V g)) d x =x x d x ; (185) ’V = 0

V g) as a function of ’V and subsequently substituted into (172) which can be used to solve for mt (0; ’; and (173). Note that (172) and (173) give the full anisotropic coexistence curve that, as a function of g, will interpolate between the isotropic and anisotropic coexistence curves. 7.5.3. The universal shift and rounding in the disordered phase We now consider how to calculate a critical temperature shift or rounding due to the presence of anisotropy parameters. We start once again with the di=erential statement (150). In the previous section integration of this equation yielded the relation between the non-linear scaling >eld mt and the temperature like linear scaling >eld. We saw that the meaning of this linear scaling >eld depended on the choice of lower limit of integration. For example in (153) by choosing the lower limit to be mt (0; ’; V g) we recovered the deviation from the isotropic critical point t while in (152) choosing the lower limit to be zero gave tg , a measure of the deviation from the anisotropic coexistence curve. In some instances, for example in a >nite box, no true critical temperature exists, hence, more generally, we choose tr ˙ (T − Tr (g))=T , Tr (g) being some g-dependent reference temperature which can be chosen to suit various physical requirements. For instance, in the case where the anisotropic system exhibits a critical point then Tr (g) = Tc (g), the ansiotropic critical temperature. In the contrary case, it can represent some suitably chosen “pseudocritical” temperature. An example of this, once again, is a completely >nite system where Tr (g) is de>ned as that temperature where the speci>c heat is a maximum. In the case of a rounding temperature Tr (g) can be de>ned via some rounding condition such as where the anisotropic and isotropic speci>c heat curves cross. Integrating (150) over temperature along a line of constant g, from Tr (g) to T and assuming we are in the disordered phase of the anisotropic system, ’V = 0, and using the coordinate transformation induced by the normalization conditions (128) – (131) one obtains m(t;g) x (2 − ’ (x; g))e− I ’2 (x ;g) d x =x x d x ; (186) t + &g = m(−&g ;g)

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

483

where &g = Z’−21 (I)2 (Tc (0) − Tr (g))=Tr (g) is the shift or rounding, depending on the circumstance, and, importantly, is temperature independent. m(−&g ; g) is the inverse correlation length in the anisotropic system at T = Tr (g). Note that we are using an environmentally friendly RG to obtain result (186). If we take I suJciently large so that Z’2 (I) is independent of g, then we can isolate the shift &g by noting that using the isotropic, g = 0, RG we have m x t= (2 − ’ (x; 0))e− I ’2 (x ;0) d x =x x d x ; (187) 0

where ’ (x; 0) and ’2 (x; 0) are the corresponding Wilson functions of the isotropic RG. m(t; 0) is the inverse isotropic correlation length at temperature T . Again for d¡4 (the super-renormalizable case) we can take the limit I → ∞ and express things in terms of bare quantities. By inserting (187) into (186) one >nds m(t;g) x (2 − ’ (x; g))e− I ’2 (x;g) d x =x x d x &g = m(−&g ;g)

−

m(t;0)

0

(2 − ’ (x; 0))e−

x I

’2 (x;0) d x =x

x dx :

(188)

Now, given that by de>nition &g is temperature independent we can freely choose to evaluate (188) at any value of t. As previously, a particularly convenient choice is t = ∞, i.e. the mean >eld limit where IR :uctuations will be completely suppressed. In this limit the leading asymptotic form of m(t; g) we write as m2 (t; g) → t + N(g) where N(g) takes into account any mass shifts due to that anisotropy that exists at the mean >eld level. For instance, in a box of sides L1 ; L2 : : : Ld ; N(Li ) = d 1 2 (2Au =L i i ) where ui = 0; 2 for periodic and antiperiodic boundary conditions, respectively. Thus, i=1 21 one >nds ∞ x x &g = [(2 − ’ (x; g))e− I ’2 (x ;g) d x =x − (2 − ’ (x; 0))e− I ’2 (x ;0) d x =x ]x d x m(−&g ;g)

+

0

m(−&g ;g)

(2 − ’ (x; 0))e−

x I

’2 (x;0) d x =x

x dx + N :

(189)

Now, in the universal limit where V(I) is chosen to lie on the separatrix, the dependence on the initial condition drops out of the anisotropic Wilson functions and they become functions of the single argument zg = m=g, while the isotropic ones are constants, which we denote - and -’2 . In this case, the shift takes the form &g = A(I)g1=+ + N where + = 1=(2 − -’2 ) is the correlation length exponent of the isotropic >xed point. For I → ∞ the amplitude A(I) becomes very large, increasing like I-’2 and we can neglect the mean >eld term N(g). This makes a contribution of &MF = N to the shift or rounding, which is subdominant below the upper critical dimension. 21

In certain cases it is necessary to introduce an arbitrary division at some x0 of the integration range [0; ∞] of the >rst term of (189) in order to avoid spurious IR divergences that can occur under certain circumstances, e.g. the small mass limit in a totally >nite box. Of course, &g is independent of x0 .

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We will see cases later, in the context of >nite size scaling above the upper critical dimension, where it plays an important role and must be retained. However, there are many cases, such as a thin >lm with periodic boundary conditions, where it is zero. When the anisotropic system possesses a critical point then m(−&g ; g) = 0 and only the >rst term in (189) contributes. Note further that in the I → ∞ limit one can write lim

I→∞

&g = A(a)g1=+ ; I -’ 2

(190)

where a is a universal constant associated with de>ning the rounding temperature or pseudocritical temperature given by a = m(−&g ; g)=g. The amplitude A(a) is given by ∞ x dx + a1=+ ; A(a) = [(2 − ’ (x))e− ∞ [ ’2 (x ) d x =x − (2 − -)]x2−-’2 (191) x a where [ ’2 = ’2 − -’2 , and = +(2 − -). Note for a true critical point that A(0) = −w0 , the zero of the scaling function *g (w). An important result we have just derived here is that the shift and rounding exponents are equal, i.e. ) = = +, and are properties of the isotropic >xed point. Note that this is a robust result. Irrespective of how we choose the reference temperature Tr the RG is telling us that the scaling behavior with g is the same. Additionally, the above formulae show that shifts and rounding may be determined using environmentally friendly renormalization in exactly the same way as other scaling functions, using as building blocks the Wilson functions ’ , ’2 and . The prediction is that below the upper critical dimension both the shift and rounding scale as &g ∼g1=+ for shifts caused by an anisotropy non-linear scaling >eld. This is in agreement with the scaling arguments of Section 2 with the additional ingredients that the amplitudes and exponents are determined. There is however more information contained in these RG predictions, as with little extra diJculty we can extract the dependence of the shift on the leading irrelevant operator. Hence we may obtain shifts as crossover scaling functions in their own right. 7.5.4. The universal shift in the ordered phase In the ordered phase of the anisotropic system, as we have seen, another linear scaling >eld, ’, V must be taken into account and therefore, likewise, its corresponding non-linear scaling >eld, m’ . Integrating (150) along a contour of constant ’V = 0 and constant g one may write mt (t;’;g) V x V d x =x t + &g; ’V = (2 − V’ (x; ’; V g))e− I V’2 (x ;’;g) x dx ; (192) 0

V g) where the lower limit of integration is zero as tg; ’V = t + &g; ’V uses as reference temperature Tc (’; which is on the coexistence curve. It is useful to decompose the shift &g; ’V into its two physically distinct components &g; ’V = &g + &’V , where &g (g) is the previously de>ned shift of critical temperature due to g in the disordered phase of the anisotropic system, and &’V = Z’−21 2 Tc (g)(Tc (g) − V g))=Tc (g)Tc (’; V g), is the shift from the anisotropic critical point along the anisotropic coexisTc (’; tence curve due to ’. V Once again the individual shifts can be isolated by using (187). Note that we already have an expression for &g (g), Eq. (190), so it just remains to calculate &g; ’V .

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

485

The shift &g; ’V can be obtained using the technique of the previous section. Explicitly, one starts with (192) substitutes in t from (187) and takes the limit t → ∞ >nding ∞ x (I)’V 2 (I) V d x =x + &g; ’V = N + [(2 − V’ (x; ’; V g))e− I V’2 (x ;’;g) 6 0 −(2 − V’ (x; 0; 0))e−

x I

V’2 (x ;0;0) d x =x

]x d x :

(193)

For I → ∞ the mean >eld term becomes negligible and, on changing variables to zg , we can extract a factor of g1=+ as in the disordered phase. The asymptotic form of (193) in the universal limit gives (180), where in distinction to the earlier section we have an explicit expression for F&g (y), which in turn implies an expression for G(y). Explicitly, we have ∞ x dx & : (194) Fg (y) = [(2 − V’ (x; y))e− ∞ [ V’2 (x ;y) d x =x − (2 − -)]x1=+ x 0 The equation for the coexistence curve, t + &g; ’V = 0 is now in scaling form w + F&g (y) = 0 :

(195)

Therefore, from the above we see that the coexistence curve will interpolate between those of the isotropic and anisotropic systems. This can be put into a more standard form, that allows the g → 0 limit to be extracted more easily, by dividing by y1=2 to >nd x + F&g (y)=y1=2 = 0 ; where x=I

− -’ 2

3 ∗ I V

1=2

(196)

t : |’| V 1=

(197)

7.5.5. The universal equation of state In Section 7.5.2 we derived an expression (182), for mt by integrating the RG along contours of constant ’V in the phase diagram. We then found an expression for the shift &g; ’V , analyzed its dependence on both ’V and an anisotropy g, then subsequently extracted the universal form of the coexistence curve. Given the basic form of the equation of state (33), using our RG construction this now takes the form H= ’V = m2t e−

mt I

V’ (x; ’; V g) d x=x

:

(198)

Extracting the universal limit and passing to the variable zg we >nd H= ’V = I- g2−- zg2−- e−

zg I

[ V’ (x; y) d x=x

:

(199)

This is a parametric representation in terms of zg whose relationship to the underlying physical variables is given by (182). Multiplying by y(1−0)=2 we obtain C2 (I)

H = f(x; y) ; |’| V0

(200)

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D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

where the metric factor ∗ (0−1)=2 I- V −C2 (I) = I 3 and the universal scaling function

f(x; y) = y(0−1)=2 *g;2 ’V (xy1=2 ; y) exp −

*g;’V

I

[ V’ (x; y)

dx x

:

(201)

If we absorb the metric factors into rede>nitions of our parameters so that 2

@ = tI−-’ ;

M2 =

∗

I- V 2 ’V ; 3

h2 =

∗

I −- V 2 H ; 3

then the equation of state takes the standard form h @ |M | : =f ; |M |0 |M |1= g=+

(202)

(203)

The equation of state has a zero on the coexistence curve as discussed at the end of Section 7.5.4. In the one variable case, where g = 0, we cannot use zg as a variable for integration. In this case there exists only one variable, z. Of course, we could in the previous sections have used z rather than zg as our variable of integration since they are related by the di=erential equation (175) and we have d zg dz = J (zg ; z; V) ; z zg

(204)

where J (zg ; z; V) =

2 : (2 − V (zg ; z; V) + V’ (zg ; z; V))

(205)

In the one variable case (where we are on the separatrix) the Jacobian J → J (z). With this formulation our parametric representation of the equation of state is given by z h = G(z) = z 0−1 e((0−1)=2) ∞ J (x)([ V −[ V’ ) d x=x ; 0 |M |

(206)

where z is given by z = M−1 (x − x0 )

(207)

with the universal zero of the scaling function being given by ∞ x J (x)[(2 − V’ )e− ∞ J (x )[ V’2 d x =x − (2 − -)] x0 = − 0

×e1=2

x

∞

J (x )([ V −[ V’ ) d x =x 1=

x

dx x

(208)

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

and, >nally, M(z) is a universal scaling function given by z 2[ V’2 −[ V +[ V’ d x =x 1= d x − x J (x ) 2 : J (x)(2 − V’ )e ∞ x M(z) = x 0

487

(209)

Thus we have derived the equation of state h = f(x) ; |M |0

(210)

where x =@=|M |1= and the universal scaling function f is given parametrically as f(x)=G(M−1 (x)) with M−1 the inverse function of M. It is relatively easy to check that this representation of the equation of state has all of the desired properties, i.e. GriJth’s analyticity and the correct asymptotic behavior for large and small x. 7.5.6. The free energy, internal energy and speci;c heat The free energy can be obtained in a manner similar to that used above. To reconstruct the free energy 2 = 2(0; 0) it proves best to work with the system of di=erential relations d2 = 2(0; 1) dt + 12 2t(2) d ’V 2

(211)

d2(0; 1) = 2(0; 2) dt + 12 2t(2; 1) d ’V 2

(212)

d2(0; 2) = 2(0; 3) dt + 12 2t(2; 2) d ’V 2

(213)

Beginning with (213), integrating along a >ducial line of constant ’V in Fig. 1, and with a change of integration variable from t to m, one obtains mt x (0; 3) (0; 2) (0; 2) (t; g; ’) V =2 (mi2 ; g) + (2 − ’ (x))e2 I ’2 (x ) d x =x 2V (x)xd−5 d x ; (214) 2 mi2

where we are free to choose mi2 to correspond to a suitable value of 2(0; 2) (m), and 2(0; 3) (2(2) )3 (0; 3) = 2V (2(2; 1) )3 md

(215)

is a RG invariant, where all the Zi factors cancel. A similar integration of (212) yields 2(0; 1) (t; ’; V g) = 2(0; 1) (mi1 ) + (mt − mi1 )2(0; 2) (mi2 ) mt x + (2 − ’ (x))e− I ’2 (x ) d x =x 2(0; 2) (x)x d x ; mi1

(216)

while one further integration yields the free energy 1 2(t; ’; V g) = (m − mi0 )2(0; 1) (mi1 ) + (m − mi0 )(m − 2mi1 + mi0 )2(0; 2) (mi2 ; g) 2 mt x V g) + (2 − ’ (x))e− I ’2 (x ) d x =x 2(0; 1) (x)x d x : +2(mi0 ; ’; mi0

(217)

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The prescription above prescribes the A(n) and B(n) of (122) and (123) in terms of the respective 2(0; n) (mi ). The advantage of integrating up from the co-existence curve is that we can extract the singular part by requiring that both 2(0; 1) and 2(0; 2) vanish there. We can impose such boundary conditions only at the critical point in the special case of ¡0 which is the case for the large N limit. A boundary condition at some other point is also possible but the formulas are more complicated. 7.6. E8ective exponents Having derived formal expressions for the free energy, equation of state and shift and rounding we will now turn to deriving formal expressions for the e=ective exponents. 7.6.1. Disordered phase First of all with the fundamental expression (152) relating m and t in the disordered phase, we >nd x m (2 − ’ )e− I ’2 d x =x x d x 0 m +e= = : (218) (2 − ’ )me− I ’2 d x=x This is naturally a function of m but can be converted to a function of t by using (152). As the other e=ective exponents naturally appear as scaling functions that depend on m it is natural to consider d=d ln t = (d ln m=d ln t)d=d ln m = +e= d=d ln m, to >nd for example e= = d ln2(2) =d ln t = (d ln 2(2) =d ln m)+e= . From equation (147) we see that (d ln 2(2) =d ln m) = 2 − ’ hence m − Ix ’2 d x =x (2 − )e x dx ’ e= = 0 : (219) − Im ’2 d x=x me Here the two e=ective exponents +e= and e= were de>ned from an examination of the temperature dependence of the correlation length and susceptibility, respectively. The e=ective exponent -e= was de>ned in Section 2 by examining the momentum dependence of the two-point correlator at T =Tc (g). In this case, the relevant anomalous dimension of ’ is ˜’ =d ln Z(I)=d ln I where this time I is a >ducial momentum rather than a >ducial correlation length. Using normalization conditions analogous to those of (128) – (131), but now with I as a momentum, one >nds that k -e= = ˜’ : (220) g However, ˜’ (k=g) = ’ (m=g). First of all they are functions of di=erent scaling variables and secondly, even as abstract functions they are di=erent given that the crossovers associated with ˜’ and ’ are physically distinct. It is unreasonable to expect an e=ective exponent law such as e= = +e= (2 − -e= ) when the scaling functions associated with the di=erent exponents relate to physically di=erent crossovers. For the temperature-dependent crossover, given that -e= = ˜’ in the momentum crossover case, it is natural to de>ne -e= = ’ . In this case then the e=ective exponent law is satis>ed. The speci>c heat e=ective exponent, e= , can be calculated directly from the expressions of Section 7.5.6.

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The >nal exponent of interest in the disordered phase is de= de>ned in Section 2 to take into account the crossover in the leading irrelevant operator. From its de>nition we see that de= is in fact simply related to , de= = 4 − . In the vicinity of a particular >xed point plays very little role, merely governing the corrections to scaling about that >xed point. In a small neighborhood of this >xed point these are negligible. In the case of a crossover, however, the corrections to scaling interpolate from one >xed point to another. This is why takes on a more important role. Simpler expressions can be found by transforming to a temperature variable t de>ned such that (d ln m=d ln t ) = (2 − ’2 )−1 . Asymptotically t = A(g)t, where A(g) is a constant at the isotropic and anisotropic >xed points. Hence, the asymptotic values of the e=ective exponents de>ned with respect to this temperature variable are the same as those de>ned with respect to t, i.e. the isotropic and anisotropic critical exponents themselves. In terms of t , +e= =(2− ’2 )−1 and e= =(2− ’ )=(2− ’2 ). Hence, de>ning -e= = ’ one >nds e= = +e= (2 − -e= ). 7.6.2. Ordered phase A similar process to the above leads to expressions for the e=ective exponents in the ordered phase. For instance, for 0e= , one >nds on the critical isotherm x m (0;’) V (2 − ˜’ )e− I ( ˜ − ˜’ ) d x =x x d x 2 0t 0e= = 1 + : (221) mt (0; ’) V ( ˜ − ˜’ ) d x =x e− I V → More generally, 0e= can be de>ned on any isotherm, the only di=erence being that mt (0; ’) mt (t; ’). V In this case a more useful expression is 0e= (t) = 0 −

t ’V 1= z 1=+

e

z 2([ V’2 −[ V =2+[ V’ =2) d x=x I 2− V + V’

:

(222)

For the critical isotropic isotherm t = 0 and 0e= → 0. For any other isotherm, as z → 0, i.e. as the coexistence curve is approached, 0e= → ∞. The e=ective exponent e= can be de>ned and calculated directly from the equation for the coexistence curve to >nd V g) d ln &g; ’V (’; 1 = ; (223) e= d ln ’V H =0

where the explicit expression for &g; ’V is given by (180). For N = 1 e=ective exponents e= and +e= can be calculated from the expressions for the equation of state already given (for N ¿1 they are ill de>ned). 7.7. Formal scaling forms We will now discuss how generic, universal scaling forms can be derived using the RG formulation we have presented here. As a typical example, we will consider the vertex function 2(1) ≡ 2(1) (t; g; ’; V ), which is just H and therefore yields the equation of state. The generalization to N -point vertex functions is straightforward. We will also, as before, restrict to the case of one explicit environmental parameter, g, though once again the modi>cations necessary for a generalization are straightforward. Solving the RG equation for 2(1) in terms of the non-linear scaling >elds mt ,

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g and m’ yields e−1=2 2(1) = m(d=2+1) t

mt I

’

F(zg ; z; mtd−4 ) ;

(224)

where F is an universal scaling function and the dimensionless scaling variables zg and z are as de>ned previously. In fact, using the normalization conditions (139) – (142) this can be simpli>ed by considering the transverse susceptibility which, as we have shown, naturally exponentiates in the coordinate system we have chosen. On the separatrix solution of the -function the dependence of F on mtd−4 disappears. It now remains to write F in terms of the physical linear scaling >elds and to write the vertex function (equation of state in this case) in manifestly universal form. The relation between non-linear and linear scaling >elds was found in Section 7.5.1 yielding Eq. (201). For the isotropic case, to put the equation of state into manifestly universal scaling form one rede>nes the scaling variables x, given by (197), and H= ’V 0 , such that at x = 0 one has H= ’V 0 = 1 and that on the coexistence curve x = −1 (analogous arguments can be formulated for the anisotropic case, e.g. see [8] for the case of >nite size scaling). The result is given by z h 0−1 ((0−1)=2) 1 J (z )([ V −[ V’ ) d z =z = G(z) = z e ; |M |0

(225)

where we have used (206), z is given by (207), and where, by a rede>nition of H , the critical isotherm is now located at z = 1. The equation of state is parametrized in terms of the two non-linear scaling >elds mt and m’ . Contrary to the case of a Josephson–Scho>eld parametrization, these two parameters are physically intuitive non-linear scaling >elds, mt being just the transverse correlation length. The fact that we are using mt rather than the disordered phase mass, as is canonically done in the >xed dimension approach, means our expressions are globally well de>ned on the entire phase diagram. It is valid for all values of N and also satis>es all GriJths analyticity properties. Note that the only unknowns that have to be calculated perturbatively are the Wilson functions ’2 , ’ and . In this sense the entire equation of state has been exponentiated, including all its >xed points. On the coexistence curve mt = 0, therefore z = 0, and so from (207) we see that M(z) = 0. Hence, |t| = A(N )’V 1= ;

(226)

where A(N ) is a constant, which is the equation of the coexistence curve. Turning now to the asymptotic properties of the equation of state we will consider the limits x → 0 and ∞. For small x, expanding around the critical point, known expansion results are obtained. It is a simple matter to derive the leading large x behavior. One divides up the integration range in (209) by introducing a constant B1 such that in the range [B; z] t is large and hence [ ’2 and [ ’ → 0. Thus, C(1 + x) → Dz (d−2)=2 + E ;

(227)

where D and E are constants. Inverting this expression and substituting into the scaling function G one >nds G → Kx ; as required by GriJth’s analyticity.

(228)

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The generic manner in which to relate non-linear and linear scaling >elds was outlined for both the disordered and ordered phases previously. In general, a scaling function, G, will be written as a function of the independent non-linear scaling >elds of the problem, 1 ; 2 ; : : : ; n G ≡ G(1 ; 2 ; : : : ; n ) ;

(229)

where the non-linear scaling >elds are related to the linear scaling >elds x1 ; x2 ; : : : ; x n via a set of “equations of state” x1 = F1 (1 ; 2 ; : : : ; n ) ;

(230)

x2 = F2 (1 ; 2 ; : : : ; n )

(231)

.. . x n = Fn (1 ; 2 ; : : : ; n ) ;

(232)

where in the universal limit the Fi are also universal scaling functions. In the case of the equation of state, in the presence of the anisotropy parameter g, we have three characteristic non-linear scaling >elds 1 = (mt =m’ ), 2 = mt =g and 3 = =mt4−d . In the universal limit → ∞ there is a reduction to the >rst two >elds. They are related to the linear scaling >elds x1 = t= ’V 1= and x2 = ’V 2 =gd−2+- , where tg; ’V = (T − Tc (’; V g))=T , via t mt m’ ; (233) = F1 ; m’ g ’V 1= ’V 2 gd−2+-

= F2

mt m’ ; m’ g

;

(234)

where F1 and F2 are given by Eqs. (170) and (192). Essentially, scaling functions are being calculated in a coordinate system of non-linear scaling >elds. The “equations of state” (230) – (232) then provide the coordinate transformation necessary to pass to the linear scaling >elds. The general arguments do not change in the case where the anisotropy is described by a linear scaling >eld. The only change is that there is an extra equation of state relating g as a non-linear scaling >eld to g as a linear scaling >eld. Obviously, this relation will usually involve the other non-linear scaling >elds too. As long as gI and I−1 , where I is a typical microscopic scale, these exponent relations will be universal, below the upper critical dimension. Just as in the standard case where the scaling relations imply that there are really only two independent critical exponents, so here there are really only two independent e=ective exponents. The di=erence with the crossovers considered here is that it is necessary to know one more function, = 4 − de= , which represents the e=ects of the leading irrelevant operator. In the case of a crossover for which the dimensionality does not change, e.g. crossover in a bicritical system, the e=ects of the leading irrelevant operator can be subsumed into an Ne= , which is a measure of the e=ective number of components of the order parameter [7].

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Thus using the e=ective exponents one can derive natural non-linear scaling >elds (we use the term non-linear here to refer to the fact that they are non-linearly related to the linear scaling >elds associated with the individual >xed points not as in the sense of Wegner that they are eigenfunctions of the dilatation operator). The t natural non-linear scaling >elds that appear above are the various correlation lengths, gt = exp 1 +e= (x) d x=x, g’V , etc., and g. Note that in the disordered phase the non-linear scaling >eld gt , interpolates between the linear scaling >elds t −+ and t −+p associated with the g = 0 and ∞ >xed points in the limits g → 0, gt → ∞ (ggt → 0) and ggt → ∞ ’V respectively. In the ordered phase there is another non-linear scaling >eld, t exp(− 1 (1=e= ) d ’V = ’V ), which interpolates between the two linear scaling >elds t= ’V 1= and t= ’V 1=p in the limits g → 0, ’V → ∞ (g’V → 0) and g’V → ∞, respectively. We hope the general pattern is clear. The natural non-linear scaling >elds for the crossover can be found by taking the natural linear scaling >elds for one of the >xed points, writing it in exponential t t form, e.g. t + = exp 1 + d x=x, then replacing the exponent with the e=ective one, e.g. t + → exp 1 +e= . Note the e=ective exponent cannot be taken outside the integral to recover a form t +e= . In this way one also arrives at the concept of e=ective crossover exponents. Standardly, a crossover exponent involves the ratio of two eigenvalues of the RG operator linearized around the isotropic >xed point. The generalization of this involves exponentiating the integral of the di=erence of two e=ective exponents. Naturally, one must derive what these e=ective exponents are. Finally, in this section we would like to discuss universality in the context of crossover behavior. We have stated that in the regime where all length scales are much bigger than the lattice spacing a, that the scaling functions F, G, etc., and the e=ective exponents will all be universal functions. The meaning of universality here is just the standard one. For example, two layered Ising models with di=erent lattice structures will exhibit precisely the same crossover curves (up to a trivial constant rescaling) as long as the >nite size of the system La. However, the crossover curves for a layered Ising model of size L but with di=erent boundary conditions, for instance, will not be the same. Neither can they be made the same by any rescaling. The environment a=ects the IR behavior of the theory and therefore the universality class. Universality in the non-crossover case is generically governed only by the dimensionality of the system and the symmetry of the order parameter. The environment gives us extra “labels” for delineating di=erent crossover universality classes. The label “boundary condition” for instance allows us to classify, say, three-dimensional Ising models with one >nite dimension into di=erent classes. Logarithmic corrections to scaling are naturally incorporated in the e=ective exponents, as indeed are power law corrections coming from the leading irrelevant operator. 8. Perturbative results for crossover between the critical -xed point and the strong coupling discontinuity -xed point Probably the most ubiquitous anisotropy is that associated with spontaneous symmetry breaking, where in the case of an O(N ) model a spontaneous magnetization results in a breaking of the O(N ) group into two subgroups—Z2 for the longitudinal component and O(N − 1) for the transverse components. The resulting equation of state exhibits di=erent asymptotic regimes as a function of the scaling variable x = t= ’V 1= , x → 0 and ∞. As is well known expansion methods cannot access both regimes in one uniform expansion and so a parametrized formulation must be invoked. Traditional

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493

>xed dimension expansions also encounter problems as they parametrize the scaling functions in terms of the disordered phase correlation length. In the case of N ¿1 there is a crossover between the >xed point associated with the critical point and a >xed point associated with the coexistence curve, where the Goldstone bosons are massless. This crossover has been treated [95 –97] previously with varying degrees of success. None seem to be capable of calculating the equation of state systematically with as much precision as the case N = 1. Additionally, in the case N = 1 there is a crossover to the strong coupling discontinuity >xed point [147,148]. Although understood in terms of a coarse-graining RG linearized around this >xed point it has essentially not been studied in terms of reparametrization RGs, or in terms of >eld theory. In the following we will use the techniques of environmentally friendly renormalization to consider both the above discussed crossovers. 8.1. One-loop equation of state for an O(N ) model From Section 7.5.5 we see that at one loop the universal equation of state is z H = z 4=(d−2) e(2=(d−2)) ∞ [ V (x)=(2− V (x)) d x=x 0 ’V

(235)

where the non-linear scaling >eld z = mt =m’ and is related to the linear scaling >eld x = t= ’V 1= via x z 2 dx e ∞ [ V’2 (x )=(2− V (x )) d x =x x1= (236) x − x0 = 2 2 − V (x) x 0 where, to one loop, = (d − 2)(N + 8)=2(N (d − 2) + 2(d + 4)). To determine V’2 (z) and V (z) we begin with the -function for the dimensionless coupling V which is to one loop given by z

2 d V − V + 32 (4 − d)cd ((1 + 1=z 2 )(d−6)=2 + (N − 1)=9) V = ; dz 2 − 32 (4 − d)cd ((1 + 1=z 2 )(d−6)=2 + (N − 1)=9) V

(237)

where cd =

2((4 − d)=2) : (4A)d=2

The one-loop expression in the constant m’ case is given by (d−6)=2 1 (N − 1) dV 3 2 ; 1+ 2 + z = − V + V (4 − d)cd dz 2 z 9 the separatrix solution of which is −1 (d−4)=2 3 1 (N − 1) V(z) = cd 1+ 2 + : 2 z 9

(238)

(239)

For N ¿1 there are three di=erent >xed points (this is irrespective of which of the two above schemes ∗ is used): (i) the trivial Gaussian >xed point, V = 0; (ii) the Wilson–Fisher >xed point in the limit ∗ z → ∞, V = 6=(N + 8)cd ; and (iii) a “strong coupling” >xed point, which controls the coexistence ∗ curve, in the limit z → 0, V = 6=(N − 1)cd . Standard expansion results [92] are associated with

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D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545 lambda

lambda

175 N =1 150

N =2

N =2

N =1 150 125

125

100

100 N =3

75

N =4

50

N =3

75

N =4

50 25

25 -20

-10

10

20

30

log z

-20

-10

10

20

30

log z

Fig. 2. Graph of solution of (238) as function of z, including the crossover to the Gaussian >xed point. Fig. 3. Graph of separatrix solution of (238) as function of z.

expansions around the Wilson–Fisher >xed point and so are incapable of tracking the crossover to the strong coupling >xed point. In the case of a system with Ising symmetry there are only two >xed points; (i) and (ii). The behavior of the running coupling for d = 3, N = 1, 2, 3 and 4 and including the crossover to mean >eld theory is given in Fig. 2. Note the crossover to the Gaussian >xed point for large z. The universal separatrix is shown in Fig. 3. Note that the N = 1 coupling apparently diverges. This is a “>ction” of this representation as can be seen by passing to the :oating coupling, h, which satis>es z

dh = −(z)h + h2 ; dz

(240)

where (z) = +

(6 − d) (1 + 1=z 2 )(d−6)=2 z2 ((1 + 1=z 2 )(d−6)=2 + (N − 1)=9)

(241)

and is well behaved across the entire crossover, which implies that the :oating coupling is >nite across the entire crossover as well. The universality of the crossover can be observed in Fig. 4. Once again the graph is for d = 3 and N = 1, 2, 3 and 4. Note how all the crossover curves end up on the universal separatrix. It is important to note that as we are using the running parameter, z, then the limit z → 0 corresponds to the entire coexistence curve, i.e. a line of critical points for N ¿1. At each point on the curve the >xed point value of V is 6=(N − 1)cd . This is analogous, as we shall see, to the case of a d-dimensional >lm geometry of thickness L where the >xed point coupling is the same (d − 1)-dimensional >xed point (in the case where the >lm exhibits critical behavior) for all L, i.e. T = Tc (’) is analogous to T = Tc (L). From (238) one >nds the Wilson function to be 1 (d−6)=2 (N − 1) 3V ; (242) = (4 − d)cd 1+ 2 + 2 z 9

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495

lambda N =2

N =1 150 125 100

N =3

75

N =4

50 25

-20

-10

10

20

30

log z

Fig. 4. Graph of solution of (238) as function of z including initial condition.

which on the separatrix is ∗ = (4 − d) :

(243)

The corresponding V has to be determined numerically. Curiously, though it is equal to 4 − d at the end points it deviates from this at intermediate values. This behavior is also what is expected for the m’ group at higher loop order. The only other function needed to determine the equation of state at one loop is ’2 (or V’2 ) which is given by (d−6)=2 1 (N − 1) : (244) 1+ 2 + ’2 = 2 V(4 − d)cd z 9 At the three above >xed points ’2 is given by: (i) ’2 = 0; (ii) ’2 = (N + 2)=(N + 8); (iii) ’2 = 1. On the separatrix solution we have (1 + 1=z 2 )(d−6)=2 + (N − 1)=3 ∗ : (245) ’2 = (4 − d) 3(1 + 1=z 2 )(d−4)=2 + (N − 1)=3 ’2 as a crossover function is plotted in Fig. 5 for d = 3 and N = 1, 2, 3 and 4. Note the interesting “dips” for N ¿1 and the behavior for N = 1. In this case, in the absence of Goldstone bosons, all :uctuations are massive, hence in the IR ’2 → 0 and the theory is mean >eld-like on approaching the coexistence curve. With these building blocks in hand we take the universal limit for d¡4 VI04−d → ∞ to >nd (d−6)=2 N +2 1 (N − 1) V − ; (246) 1+ 2 + [ ’2 = 2 (4 − d)cd z 9 N +8 which interpolates between the values 0 and 1 − (N + 2)=(N + 8) at the critical and strong coupling >xed points, respectively. Substituting into Eq. (236) one obtains the universal scaling function M. Its inverse gives us the universal scaling function f(x). In Fig. 6 we plot this function to one loop

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D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545 4

gammaphi2 1

f

3

0.8 0.6

2 N =4

0.4

N =2 N =1

1

0.2

-20

-10

10

20

30

log z

x

0 -1

-0.5

0

0.5

1

1.5

2

Fig. 5. Graph of ’2 as function of z. Fig. 6. Graph of the universal scaling function to one loop and comparison with Monte Carlo data (top (black) line for large x). 5

t 20

4

15

5 -2

-1

2

1 -5

f

3

10

2

phi 1

-10

x

0

-15

-1

-0.5

0

0.5

1

1.5

2

Fig. 7. Graph of the universal equation of state for N = 1, d = 3 to one loop in the t–’V plane. Fig. 8. Graph of the universal scaling function f(x) for N = 4, d = 3, and with matched exponents, and comparison with Monte Carlo data (bottom (black) line for large x).

for N =2 and compare with the Monte Carlo results discussed in [149]. Given that our representation is uniformly valid in both the small and large x regimes we see that our approximation is still quite good out to larger values of x. A representation in terms of a power-law expansion of f(x) would obviously fail. The equation of state is H= ’V 0 = G(M−1 (t= ’V 1= )). In Fig. 7 we see the phase diagram for N = 1, d = 3 to one loop in the t, ’V plane, the contours being contours of constant H . One other utility of our methodology is the facility with which it can give a >tting function which can easily be compared with experiment or simulation. We see this in Fig. 8 for N = 4, d = 3 where we took the one-loop -function and introduced a constant such that asymptotically, z → ∞, the >xed point when substituted into the one-loop expression for ’2 led to an expression for + which was then matched by adjusting the constant to the most accurate calculation known in three dimensions.

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8.2. Crossover between ;rst and second order transitions In the above we considered the equation of state. This equation exhibits both >rst and second order transitions for N = 1 hence it should be feasible to study both these types of transition in the context of >eld theory and an environmentally friendly RG. Generally, in terms of the RG >rst order phase transitions have not been as extensively studied as their second order counterparts. The widely held coarse graining RG picture of >rst order transitions [147,148] is that the phase boundary is controlled by a “strong coupling” >xed point that is diJcult to access using the standard controlled approximations. The signal for a >rst order >xed point is the existence of RG exponents equal to the dimension of the system, d, other than the standard one associated with volume changes. These other independent eigenvalues re:ect the existence of more than one phase. A scaling analysis of such transitions [148] that can be related to the aforementioned RG analysis can be made. One >nds exponents associated with the strong coupling >xed point: 0=∞, -=2−d, +=1=d, =1, =1 and =0 in the case where both magnetic and thermal eigenvalue exponents attain their thermodynamically allowed limiting values. The values are certainly very di=erent to those associated with the critical point, hence it is of interest to ask if >eld theoretic techniques can give expressions for the scaling functions, e=ective exponents, etc. that interpolate between these values and that capture the crossover between >rst and second order transitions. 8.2.1. Crossover to the strong coupling ;xed point for N = 1 The exponents associated with the strong coupling >xed point look forbidding from a >eld theory point of view. For example, we are used to thinking of - as being proportional to 2 not being of order one. In this naive sense the >xed point that leads to these values does truly look as though it is associated with strong coupling. In a >eld theoretic formalism it is the presence of strong :uctuations that makes calculations problematical without a tool such as the RG. However, as one proceeds down the coexistence curve the ’-dependent V e=ective mass of :uctuations gets larger, suppressing IR :uctuations. In this sense, in the >eld theoretic formalism it should be easier to describe the “strong coupling” regime than the critical regime. These apparently contradictory >ndings can be reconciled by thinking about the physical meaning of the strong coupling exponents. The most problematic seems to be - = 2 − d. The canonical meaning of - is that it is a measure of the fall o= of correlations at the critical point, i.e. the connected correlation function, G (2) (x; x )∼1=|x − x |d−2+- and is related to the disconnected Greens function Gd(2) (x; x ) via Gd(2) (x; x ) = G (2) (x; x ) + ’(x) V ’(x V ) :

(247)

An important distinction between the two types of Green’s function is that the connected Green’s functions cannot distinguish whether or not the two points x and x are in the same phase or two di=erent phases. On the contrary, the disconnected Green’s function can distinguish and thus is a more appropriate quantity for probing a multiphase system. One naturally de>nes an e=ective exponent - associated with Gd(2) (x; x ) via d ln Gd(2) (˜r) ≡ −(d − 2 + -e= ) ; d ln r

(248)

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which can be related to the exponent -e= de>ned in Section 2. -e= =

V r)’(0) V d ln ’(˜ V r)=d ln r (2 − d)’(˜ V r)’(0) V + -e= G (2) (˜r) − ’(˜ : G (2) (˜r) + ’(˜ V r)’(0) V

(249)

The scaling function -e= exhibits a crossover: as ’V → 0 -e= → -e= ;

(250)

while, asymptotically, as r → ∞ we have ’V → const and G

(2)

→ 0. In this limit

-e= → (2 − d) :

(251)

Turning now to the e=ective exponent 0e= (t), generalized to the case of any isotherm, using the equation of state it can be written as 0e= (t) =

2‘(2) 2t(2)

:

(252)

As one approaches the coexistence curve for t’V = 0, then 2t(2) → 0 while 2‘(2) remains >nite and non-zero. Hence, as H → 0 0coex → ∞ :

(253)

Another quantity of interest is how the transverse correlation length changes as a function of H as the coexistence curve is approached. We de>ne d ln mt H +e= = ; (254) d ln H t which leads to −1 (t) 1 − 0 e= +He= = : 2 − ’

(255)

As the coexistence curve is approached 0e= (t) → ∞ hence +He= → (2 − ’ )−1 . If we assume that ’ = - and that - → (2 − d) then +He= → 1=d, a fundamental characteristic of the strong coupling discontinuity >xed point as found in [147,148]. 9. Perturbative results for -nite size scaling 9.1. Dimensional crossover in a ;lm geometry with periodic boundary conditions: the disordered phase 9.1.1. One-loop results: correlation length, susceptibility and coupling We consider an O(N ) model de>ned on a d-dimensional >lm of thickness L satisfying periodic boundary conditions (or in particle physics language ’4 theory on Rd−1 ×S 1 ). We will be concerned here with T ¿Tc (L), where Tc (L) is the critical temperature of the >nite size system, and d 6 4. Here, we will restrict attention to the case where the >nite system exhibits a critical point. The results below are equally applicable to relativistic >nite temperature after the trivial substitution L = 1=T

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499

and L = 1=mT for the case where the >nite temperature mass for >xed temperature is the running parameter. The case where the “environment” is run, i.e. the temperature, will be considered in Section 10. We begin with the one-loop expressions (h) = −(IL)h + h2 + O(h3 ) ;

(256)

(N + 2) h + O(h2 ) ; (N + 8)

(257)

’2 =

’ = 0 to one loop. The solution of (256) is h(IL) =

e−

1 h− 0 −

I=Ii 1

I=Ii

(ILx) d x=x x − 1 (Ii Lx ) d x =x

1

e

d x=x

;

(258)

where with the normalization conditions (128) – (130) I is the inverse transverse correlation length in the >lm. Thus we have h in terms of z = L=L , and an initial coupling at the scale Ii , or equivalently z0 = Ii L. Substituting this into the one-loop contribution for ’2 we have −

z

(x) d x=x

e z0 (N + 2) x ’2 (z) = : (N + 8) h(z0 )−1 − z e− z0 (x) d x =x d x=x z0

(259)

For d¡4 the theory is superenormalizable, hence if we impose the initial condition on the RG :ow at z0 = ∞, i.e. at zero correlation length, and choose a >nite renormalized coupling h(∞), we will be probing the universal part of the crossover. One >nds h(z) = −

A 4− d

;

(260)

which depends only on the variable z, and interpolates between the bulk and dimensionally reduced >xed points. This is the equation of the separatrix between the two >xed points in the one-loop approximation. With periodic boundary conditions, the universal one-loop :oating coupling is ∞ 2 (d−7)=2 −∞ (1 + (2An=z) ) h(z) = (5 − d) n= : (261) ∞ 2 (d−5)=2 n=−∞ (1 + (2An=z) ) Conveniently for d = 3, (z) and h(z) are expressible in terms of elementary functions. We have explicitly 1 sinh z0 + z0 sinh z z sinh(z=2)2 coth(z0 =2) −1 : (262) + −2 h (z) = 2 sinh z + z h(z0 ) z0 sinh(z0 =2) z0 sinh z + z Focusing on the universal part of the crossover by choosing z0 → ∞, with h(z0 ) >nite we >nd z : (263) h(z) = 1 + sinh z

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As only N = 1 exhibits a critical point in the dimensionally reduced limit we restrict attention to that case. We thus >nd z 1 ’2 (z) = 1+ : (264) 3 sinh z Hence, tL5=3 = F(z) ;

(265)

where the universal scaling function F(z) is given by z x2=3 dx : F(z) = 2 1=3 0 (tanh(x=2))

(266)

Inverting (265) gives z = mL = F−1 (tL5=3 ). Hence, +e= =

d ln F−1 (tL5=3 ) : d ln t

Explicitly, +e= =

(267) mL

x2=3 0 (tanh(x=2))1=3 (mL)5=3

(tanh(mL=2))1=3

dx

;

(268)

which varies between 0:6 for Lm → ∞, m → 0 and 0:75 for m → 0 for >xed L. These, of course are just the well known one-loop values of + in three and two dimensions, respectively, in the >xed dimension expansion. Our expression for e= is mL x2=3 (tanh(mL=2))1=3 0 (tanh(x=2)) 1=3 d x e= = 2 (269) 5=3 (mL) which varies between 1:2 and 1:5. To one loop de= = 3 − z=sinh z and varies between 3 and 2 as z varies between ∞ and 0. For d = 4 ∞ 2 2 2 2 2 2 −5=2 n=−∞ 4A n =z (1 + 4A n =z ) (z) = 1 − 3 (270) ∞ 2 2 2 −3=2 n=−∞ (1 + 4A n =z ) and the one-loop solution to the -function is given by (258). It is now not possible to set the initial value of the coupling to a >nite value and take the limit z0 → ∞. The logarithmic corrections to scaling in d = 4 mean that the running coupling will retain a dependence on the initial condition. 9.1.2. Two-loop results for the Wilson functions At two-loop order the Wilson functions are given by [10] (h) = −(z)h + h2 −

4 ((5N + 22)f1 (z) − (N + 2)f2 (z))h3 + O(h4 ) ; (N + 8)2

(271)

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ε(x) 1

0.8

0.6

0.4

f1(x)

0.2

f2(x)

x -2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 9. Graph of (z), f1 (z) and f2 (z).

’ 2

(N + 2) (N + 2) h−6 = (N + 8) (N + 8)2

’ = 2

1 f1 (z) − f2 (z) h2 + O(h3 ) ; 3

(N + 2) f2 (z)h2 + O(h3 ) ; (N + 8)2

(272) (273)

where the functions (z), f1 (z) and f2 (z) are all independent of N in this particular case. Explicit expressions for the functions f1 and f2 applicable for general d for this crossover can be found in [10]. For a three-dimensional >lm geometry (R2 ×S 1 ) (z) interpolates between 2 and 1, f1 (z) interpo4 lates between 0:28 and 13 and f2 (z) interpolates between 0:23 and 27 as z varies from 0 to ∞. In 3 1 the four-dimensional case (R ×S ), (z) ranges monotonically from 1 for z = 0, to 0 for z = ∞, and takes the value 12 for z ≈ 3:3. The function f1 (z) ranges from 13 , for z = 0 to 12 for z = ∞; f2 (z) 4 for z = 0 to 14 for z = ∞. Thus, Eqs. (271) – (273) interpolate between those for R3 ranges from 27 and R4 as z ranges from 0 to ∞. We provide plots of (z), f1 (z) and f2 (z) for N = 1 in Fig. 9 for the layered four-dimensional geometry with periodic boundary conditions (R3 ×S 1 ). As the separatrix, or :oating, coupling varies between 0 and 1, and 1 and 2 for the four- and three-dimensional >lms, respectively, just as in the >xed dimension case it is necessary to implement a resummation of the series. Here we use PadZe approximants. In particular we use a [2; 1] PadZe resummation. In the non-crossover case as this agrees well with high temperature series and expansion results there is every reason to have con>dence in it as a good method of capturing the true nature of the resummed expressions. For three dimensions the two-loop PadZe resummed results are in excellent agreement with six-loop resummed results [150 –153], where the Callan–Symanzik equation, in distinction to our homogeneous RG, was used, and the best high temperature series (for a recent review see Pelissetto and Vicari [149]). Borel resummation is also possible, however, the present series are still too short to make this worthwhile. Calculations to higher orders could be implemented, and it should not be too diJcult in the case of a three-dimensional layered geometry

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Table 1 Asymptotic critical exponents N

’

’ 2

h

e=

+e=

e=

−2 −1 0 1 2 3 4 ∞

0a 0.0200 0.0295 0.0329 0.0332 0.0322 0.0305 0a

0a 0.145 0.277 0.388 0.479 0.552 0.611 1a

1.800 1.820 1.785 1.732 1.675 1.621 1.573 1a

1a 1.088 1.175 1.257 1.330 1.395 1.451 2a

0.5a 0.550 0.596 0.639 0.676 0.709 0.737 1a

0.5a 0.351 0.211 0.083 −0:029 −0:126 −0:211 −1a

a

These values are exact.

to adopt the numerical techniques of [150,151], in conjunction with our diagrammatic approach, to obtain numerical curves for the expressions to a similar accuracy. The series for ’2 should, in principle, also be PadZe resummed in terms of the PadZe resummed solution of (271). The di=erences between these results and those obtained when ’2 is not resummed are small for small values of N . By adopting the philosophy that all the characteristic functions should be treated on an equal footing, in other words they should all be PadZe resummed one retains the spherical model limit as an exact limit of the PadZe resummed expressions. To summarize then: one solves the PadZe resummed di=erential equation z dh=d z = (h; z) for h(z) and substitutes it into the PadZe resummed expressions for ’2 and ’ . The three PadZe resummed functions (h), ’2 and ’ are the building blocks with which any scaling function can be calculated to two-loop PadZe resummed accuracy in terms of the non-linear scaling >eld mL. To write a scaling function in terms of the linear scaling >eld t one simply uses Eq. (152) or (156), which once again only depend on the three primary building blocks. Explicit two-loop results, using PadZe resummed Wilson functions, for h, ’ and ’2 and the e=ective exponents +e= , e= and e= in the disordered phase were presented in [10] for a four-dimensional >lm geometry. The asymptotic value of these quantities in the three-dimensional regime are shown in Table 1. All these values are in very good agreement with corresponding high-temperature series and experimental results. We believe the entire crossover curves are of similar accuracy. Fig. 10 shows the [2; 1] PadZe resummed :oating coupling for N = 0, 1, 2, 3, ∞ and −2 as a function of L=L . One can clearly see the four-dimensional logarithmic corrections to scaling. All curves are with the boundary condition h = 1 at ln(L =L) = −20. The value of h at the initial scale parameterizes di=erent possible crossover curves but all curves asymptote to the same form. In Fig. 11, we plot -e= . Interestingly, this exponent is not a monotonic function of N but attains a maximum for some value of N between N = −2 and N = ∞, where it is identically zero. This is the least accurate of our exponents and the peak appears to be at N = 1, though more accurate values for this exponent suggest it occurs at higher values, probably N = 3. In Fig. 12, we plot = 4 − de= , which governs corrections to scaling from the leading irrelevant operator and which also gives information about the e=ective dimensionality of the system. Notice that is very robust to changes in N , varying very little across the entire range of N , [ − 2; ∞].

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h

503

N=_1

1.75

N=3 1.5

1.25

1

N=∞ 0.75

0.5

0.25

x -15

-10

-5

5

10

15

Fig. 10. Graph of the :oating coupling h as a function of x = L =L for four-dimensional >lm geometry with boundary conditions.

ηeff

N=1 N=3

0.03

N=0 0.025

0.02

0.015

0.01

0.005

x -15

-10

-5

5

10

15

Fig. 11. Graph of -e= as a function of x = L =L for four-dimensional >lm geometry with periodic boundary conditions.

Further calculational simpli>cation results from using the :oating >xed point [10], de>ned by the condition (h) = 0. The :oating >xed point, as a solution of an algebraic and not a di=erential equation, is manifestly independent of z0 . In a comparison of the running and :oating e=ective exponents, as the values of z may di=er by a constant scale di=erence, one curve may be shifted along relative to another. There is also of course some mismatch between the solution of the di=erential equation and the :oating >xed point in the crossover region even after this shift is accounted for. This is because solving the -function di=erential equation is not the same as solving the associated algebraic equation. They do, however, track one another closely (when the solution of the

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1

N=∞ N=_2 0.8

0.6

0.4

0.2

x -15

-5

-10

5

10

15

Fig. 12. Graph of (x) as a function of x = L =L for four-dimensional >lm geometry with periodic boundary conditions.

di=erential equation is dominated by a non-trivial >xed point) but not exactly. The fact that the :oating >xed point is independent of initial conditions means there is no dependence on transients, hence logarithmic corrections to scaling are suppressed. However, these can be recovered by examining corrections to scaling around the :oating >xed point. 9.1.3. One-loop results: the speci;c heat To determine the behavior of the speci>c heat during a dimensional crossover we use (123) and (124) and the results of Section 7.5.6. As mentioned there, some of the subtleties of additive renormalization can be overcome by relating 2(0; 2) to the correlation function 2(0; 3) , the advantage of this approach being that the latter is multiplicatively renormalizable in d¡6. We normalize the speci>c heat such that it vanishes in the mean >eld limit, i.e. 2(0; 2) (t(I; I); (I); L; I) = 0 ;

(274)

which is equivalent to A(2) (I) = −Z’22 2B(0; 2) (tB (I);

B ; L)

(275)

hence the e=ect of :uctuations is captured in the inhomogeneous term of (124). B(2) = −2NI2

|n:p: ;

where the subscript denotes that the diagram is evaluated at the normalization point. Thus we see that the one- and two-loop expressions for B(2) in terms of renormalized quantities are identical. Explicitly to one loop for periodic boundary conditions, one >nds (d−7)=2 ∞ N 2((7 − d)=2) Id−5 4A2 n2 (2) B =− 1+ 2 2 : (276) L (2A)(d−1)=2 LI n=−∞

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2(0; 2) is thus found by substituting (276) and (257) into (124) to obtain 2(0; 2) (t; ; L; I) N 2((7 − d)=2)Id−4 = 2LI (2A)(d−1)=2

1

m I

(d−7)=2 ∞ x 4A2 n2 d x d− 5 x 1+ 2 2 2 e2 1 ((N +2)=(N +8))h dy=y ; x LI x n=−∞ (277)

where m is the inverse correlation length. Thus, we calculate the speci>c heat and other physical quantities directly in terms of the >nite size correlation length. The non-linear scaling >eld L =L can be related to t using the results of Section 7. In the limit m → 0 only the n = 0 term in the sum is important and one >nds N (N + 8) 2((5 − d)=2)Id−4 m d−5+2(5−d)((N +2)=(N +8)) 2(0; 2) → − : 2(4 − N )LI (2A)(d−1)=2 I In the same limit one >nds (m=I) → (t=I2 )+d−1 , +d−1 = (2 − ((N + 2)=(N + 8)) (5 − d))−1 being the (d − 1)-dimensional correlation length exponent. Hence N (N + 8) 2((5 − d)=2)Id−4 t −d−1 2(0; 2) → − ; 2(4 − N )LI (2A)(d−1)=2 I2 where 5 − d − 2(5 − d) ((N + 2)=(N + 8)) d− 1 = 2 − ((N + 2)=(N + 8)) (5 − d) is the (d − 1)-dimensional-speci>c heat exponent. Similarly, in the limit Lm → ∞, m → 0 the sum can be converted to an integral and one >nds that N (N + 8) 2((4 − d)=2)Id−4 m d−4+2(4−d)((N +2)=(N +8)) 2(0; 2) → − 2(4 − N ) (2A)d=2 I and (m=I) → (t=I2 )+ where + is the d-dimensional correlation length exponent. In the bulk limit N (N + 8) 2((4 − d)=2)Id−4 t − 2(0; 2) → − 2(4 − N ) (2A)d=2 I2 where 4 − d − 2(4 − d) ((N + 2)=(N + 8)) = 2 − ((N + 2)=(N + 8)) (4 − d) is the d-dimensional-speci>c heat exponent. Thus, we see that the speci>c heat crosses over precisely between the expected d- and (d − 1)-dimensional asymptotic forms. Note that the amplitude of 2(0; 2) in the above expressions appears to diverge at N = 4. This is an artifact of the one-loop approximation. What actually happens is that for d between two and four there is some value of N for which (N; d) = 0. At this value of N and d we expect the speci>c heat to have a logarithmic dependence on t. For N = 1 this occurs at d = 2. However, at one loop the value appears to be independent of d and occurs at N = 4, which is the relevant value for d = 4. A plot of the speci>c heat as a function of correlation length is shown in Fig. 13 for a threedimensional Ising >lm with periodic boundary conditions. The e=ective speci>c heat exponent de>ned as (9) is plotted in Fig. 14 for the same model. Note that in this approximation the asymptotic two-dimensional value of e= is 0:5 as opposed to the exact value of zero, obtained from the

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Fig. 13. Logarithmic plot of the speci>c heat C against x = L =L for three-dimensional Ising >lm with periodic boundary conditions. Fig. 14. The e=ective exponent e= as a function of x = L =L for a three-dimensional Ising >lm with periodic boundary conditions.

Fig. 15. Logarithmic plot of the speci>c heat C against x = L =L for four-dimensional Ising >lm with periodic boundary conditions. Fig. 16. The e=ective exponent e= for a four-dimensional Ising >lm with periodic boundary conditions.

solution of the two-dimensional Ising model. This is a weakness of the perturbative approach which e=ects the speci>c heat exponent in a particularly acute manner. In the next section by matching to the known asymptotic exponents of the model we investigate a more realistic behavior. In the case of a four-dimensional O(N ) >lm, in the limit L=L → ∞, L → ∞, one >nds that N + 8 1 N t (4−N )=(N +8) (0; 2) →− 2 ln 16A2 4 − N 2 I2 in accordance with known results. Figs. 15 and 16 show plots of the speci>c heat and e= for the four-dimensional Ising >lm. Note the presence in the >gures of logarithmic tails at the fourdimensional end.

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Fig. 17. The e=ective exponent e= for a three-dimensional Ising >lm with periodic boundary conditions and with matching to the asymptotic exponents. Fig. 18. The e=ective exponent e= for a three-dimensional Ising >lm with antiperiodic and Dirichlet boundary conditions and with matching to the asymptotic exponents.

9.1.4. Crossover to logarithmic behavior in a three-dimensional Ising ;lm In this section we will consider the crossover between three and two dimensions for an Ising model in a way that is capable of accessing the logarithmic behavior characteristic of the two-dimensionalspeci>c heat. For the two-dimensional Ising model = 2 − +d = 0. The consequent logarithmic behavior of the speci>c heat is thus due to a competition between + and d. For d = 2 the correlation length exponent + = 1, hence = 0. Now, for a three-dimensional Ising >lm with periodic boundary conditions, at one loop the crossover is governed by the :oating coupling h = 1 + z=sinh z. This implies a crossover for +e= between 16 and 13 . By far the biggest error involved in evaluating crossover functions is associated with the values of the asymptotic exponents themselves. With this in mind one is inclined to try to match the scaling function to the asymptotic exponents. This can very simply be done in the case at hand by writing h = A + (Bz=sinh z) where now the constants A and B will be determined by demanding that as z → 0, +e= → 1 and that as L → ∞, z → 0 one >nds +e= → 0:630. The values 1 and 0:630 are the exact two-dimensional and three-dimensional six-loop Borel resummed [150,151] exponents, respectively. Thus one >nds that A = 1:238 and B = 1:762. In Fig. 17 we plot e= as a function of ln z by substituting our ansatz for h into (277). Note there exists a logarithmic tail as the two-dimensional critical region is approached. More interestingly, there is a pronounced bump in the curve which is absent in the one-loop approximation. This arises due to a competition between the e=ects of +e= and the e=ective dimensionality de= [154]. The bump remains even if one uses a completely di=erent interpolating function such as h = A + (Bz=1 + z), though its amplitude and width vary somewhat. In Fig. 18 we plot analogous results for the case of Dirichlet and antiperiodic boundary conditions. Once again the bump is clearly present. In the case of Dirichlet conditions however there is also a dip before the bump is reached. Based on previous experience of the behavior of e=ective exponents with Dirichlet boundary conditions [10] this is not totally unexpected. In Figs. 19 and 20 we have used instead of the universal :oating, or separatrix, coupling the non-universal coupling. If one uses the non-universal :oating coupling then there is

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Fig. 19. The e=ective exponent e= to one-loop for a three-dimensional Ising >lm with periodic boundary conditions with non-universal crossover in h(x). Fig. 20. The e=ective exponent e= for a three-dimensional Ising >lm with periodic boundary conditions, with matching of the asymptotic exponents and non-universal crossover in h(x).

now a double crossover; >rstly between mean >eld theory and the three-dimensional asymptotic exponent and then to the asymptotic behavior of the two-dimensional exponent. In Fig. 19 we plot the result for the case where we do not match to the exact two-dimensional exponent and in Fig. 20 the result with matching. The asymptotic three-dimensional regime would most probably be much narrower than that shown. This can be very easily modeled by adjusting the initial condition for the RG :ow. In the case at hand, we have, for the sake of clarity, and to emphasize the double crossover, left it large. It is clear from the >gure how the e=ective exponent would be modi>ed as the well developed three-dimensional universal regime is narrowed. 9.2. Dimensional crossover in a ;lm geometry with periodic boundary conditions: the ordered phase 9.2.1. The non-linear -model Here we will consider dimensional crossover in the ordered phase in the context of the non-linear sigma model. Or rather, we will consider the non-linear sigma model as an approximation to the classical model with O(N ) symmetry. The chief reason for considering the -model is to consider how environmentally friendly renormalization functions in the context of a low-temperature expansion as opposed to an expansion around the critical point. Crossover behavior in the context of the non-linear -model has been considered previously. In particular, Amit and Goldschmidt extended their original treatment of bicritical systems above the critical temperature using Generalized Minimal Subtraction [73] to below the critical temperature [75] thus describing the crossover between a system exhibiting an O(N ) symmetry to that of an O(M ) symmetry. In this case, expansion methods were perfectly feasible due to the fact that the upper critical dimension of the two >xed points was the same. In the context of >nite size scaling BrZezin and Zinn-Justin described crossover of the non-linear -model in the context of a box or a cylinder [81] by treating the lowest IR modes of the system non-perturbatively while treating other modes in a perturbative expansion. However, this method does not work in the context of

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a dimensional crossover such as in a thin >lm where the reduced dimension system also exhibits a non-trivial >xed point. More recently, the quantum version of the non-linear -model [43] has generated a great deal of attention as it can describe the long-wavelength, low-temperature behavior of a two-dimensional quantum Heisenberg ferromagnet, which in turn has been proposed as a model of high-temperature superconductors. Given the close analogy between a (d+1)-dimensional layered system with periodic boundary conditions and a d-dimensional quantum system it is certainly of interest from this point of view to investigate further dimensional crossover in the context of the non-linear -model. We begin with the Hamiltonian for a Heisenberg model with O(N ) symmetry L 1 H [’B ] = d d x(∇3 ’iB ∇3 ’iB − HBi (x)’iB ) ; (278) 2TB 0 where i ∈ [1; N ], 3 ∈ [1; d], TB is proportional to the temperature of the system and ’iB is subject to the constraint ’iB ’iB = 1 :

(279)

We will consider the system in a >lm geometry of size L in d dimensions, d¡4, considering only the case of periodic boundary conditions. The partition function Z is obtained by performing the path integral over the order parameter >elds, ’iB , with Hamiltonian (278) subject to constraint (279). Choosing the direction of symmetry breaking to be along the N th direction we de>ne ’N = and ’i = Ai , (i = N ). The constraint implies that (x) = ±(1 − A2 )1=2 . Thus, the partition function becomes 2 2 1 d 1 1 dAB −T d x( 2 (∇ABi )2 + 2 (∇(1−ABi )1=2 )2 −JBi ABi −HB (1−ABi )1=2 ) B Z[H; J ] = e : (280) 2 (1 − ABi )1=2 Clearly this theory is highly non-polynomial. The non-trivial measure term, which ensures the O(N ) invariance of the theory, can of course be exponentiated and expanded in powers of A2 . These terms are necessary to cancel corresponding O(N ) non-invariant terms that arise in perturbation theory. We will assume that such terms have been canceled and not consider them further. Rotations are implemented linearly in the (N − 1)-dimensional Ai -subspace and non-linearly in the Ai – directions. A rotation by an in>nitesimal !i induces the changes 2

0Ai (x) = (1 − Ai (x))1=2 !i ; 2

0(1 − Ai (x))1=2 = −!i Ai (x): As long as |Ai |¡1 the symmetry will remain broken. As T → 0, (x) → 1. From the way in which T appears in (280) we can see that an expansion in terms of temperature is equivalent to an expansion in the number of loops, the only subtlety being that the measure term is then linear in T and therefore contributes to a higher order in T than the other two terms. The free propagator for the A >eld in the absence of a magnetic >eld is GAA (k) =

TB : k2

(281)

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The magnetic >eld coupled to the >eld acts as an IR cuto=. This can be seen by expanding the 2 term HB (1 − ABi )1=2 in powers of A. The resulting two-point vertex function is 2A(2) (k) =

k 2 + HB : TB

(282)

From the form of the Hamiltonian, in terms of an expansion in A, there are interactions of arbitrary order. However, interactions with more than four powers of A contribute at higher than one-loop order, i.e. more powers of the “small” coupling T . Consequently to >rst order in T , i.e. one loop, one need only consider the four-point interaction 1 2 (k + HB )ABi (k1 )ABi (k − k1 )ABj (−k2 )ABj (k2 − k) : (283) 8TB k1 k 2 k

Here, for the purposes of illustration, we will restrict our attention to O(T ) results and therefore will not consider higher order interactions any further. 9.2.2. Renormalization of the non-linear -model In spite of the fact that the theory is non-polynomial, as is well known [81], it is renormalizable using only two renormalization constants ZT and ZA associated with the temperature and the >eld, respectively. The relation between the bare and renormalized parameters is TB = ZT T;

ABi = ZA−1=2 A :

(284)

To preserve the rotational invariance of the renormalized constraint the >eld must renormalize in the same way as A. Invariance of the term HB B =TB thereby yields the renormalization of H HB = ZT ZA−1=2 H :

(285)

The bare and renormalized vertex functions are related via (N ) 2A(N ) (ki ; T; H; L; I) = ZAN=2 2AB (ki ; TB ; HB ; L; ) ;

(286)

where I is an arbitrary renormalization scale and an ultraviolet cuto=. The RG equation follows immediately on di=erentiating (286) with respect to I 9 9 9 N I (287) + t + H − A 2A(N ) (ki ; T; H; L; I) = 0 ; 9I 9t 9H 2 where we have introduced a dimensionless temperature =TId−2 and A = d ln ZA =d ln I is the anomalous dimension of the >eld. The two -functions are t = (d − 2)t − t H =

d ln ZT ; d ln I

d ln ZA d ln ZT − : d ln I d ln I

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

511

The two renormalization constants must be >xed by normalization conditions. The environmentally friendly normalization conditions we will use are T2A(2) (k = 0; t(I; LI); H (I; LI) = I2 ; L; I) = I2 ;

(288)

9 T2(2) (k; t(I; LI); H (I; LI) = I2 ; L; I)|k=0 = 1 : 9k 2 A

(289)

Note that T2(2) is just the inverse susceptibility associated with the A >eld. 9.2.3. Explicit results We now proceed to examine the crossover perturbatively. To one loop ∞ d d− 1 p HB (N − 1) HB 1 (2) 2AB (k = 0) = + : d − 1 2 TB 2 L n=−∞ (2A) p + HB + 4A2 n2 =L2 Using the normalization conditions (288) and (289) one >nds ∞ d d− 1 y 1 (N − 1) t ZA = 1 − d − 1 2 2LI n=−∞ (2A) y + 1 + 4A2 n2 =L2 I2 and

∞ d d− 1 y 1 (N − 2) ZT = 1 − : t 2LI n=−∞ (2A)d−1 y2 + 1 + 4A2 n2 =L2 I2

(290)

(291)

(292)

The -function, t , is thus given by

∞ d d− 1 y 1 2(N − 2) 2 t (t; LI) = (d − 2)t − : t d−1 (y 2 + 1 + 4A2 n2 =L2 I2 )2 LI (2A) n=−∞

(293)

There are three di=erent >xed points associated with (293), a d-dimensional UV >xed point in the limit LI → ∞, I → ∞; a (d − 1)-dimensional UV >xed point in the limit LI → 0 I → ∞; and >nally a zero temperature IR >xed point in the limit I → 0. However, the approach to t = 0 depends on whether we consider I → 0 for >xed L or LI → ∞, I → 0. In terms of the :oating coupling, h, one >nds I where

dh = (LI)h − h2 ; dI

(294)

∞

(LI) = d − 3 + 4 ∞

n=−∞ n=−∞

d d− 1 y 4A2 n2 =L2 I2 (2A)d−1 (y2 +1+4A2 n2 =L2 I2 )3 d d− 1 y 1 (2A)d−1 (y2 +1+4A2 n2 =L2 I2 )2

:

(295)

The quantity de= =2+(LI) can be interpreted as a measure of the e=ective dimension of the system interpolating between d and d − 1 in the limits LI → ∞ and 0, respectively, where in both cases

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D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545 2

1

N=3 0.8

1.5

gf

N=4

h

0.6

N=5 1

0.4

0.5 0.2

Ln kL

Ln kL -5

5

10

15

5

10

15

Fig. 21. Graph of separatrix solution of (294) as function of ln IL. Fig. 22. Graph of gf A on the separatrix solution of (294) as function of ln IL.

we are considering I → ∞, i.e. the behavior near the critical point. correspondingly interpolates between (d − 2) and (d − 3) in the limits LI → ∞ and 0, respectively. The corresponding >xed points for h are: d − 2, d − 3 and 0. In Fig. 21 we see a plot of h as a function of ln z = ln IL. So we see a dimensional crossover manifest in both the t and h couplings which is controllable in the low-temperature expansion. It is in fact the solution to (294) that we use as a “small” parameter in the perturbative expansion of all other quantities. It is the fact that h captures the crossover between the di=erent >xed points that gives us a uniform expansion parameter and therefore perturbative control of the crossover. Of course, when d − 2 is not small one really needs to work to higher order and attempt some resummation method. It should be clear however that there is no impediment to continuing this calculation to arbitrary order in the loop expansion. Turning now to the anomalous dimension of the >eld, A , we >nd ∞ d d− 1 y 2(N − 1) 1 t A = (296) d − 1 2 LI (2A) (y + 1 + 4A2 n2 =L2 I2 )2 n=−∞ or in terms of the :oating coupling N −1 h: (297) A = N −2 The anomalous dimension also exhibits a dimensional crossover, as can be seen in Fig. 22, for the case d = 3, N = 3; 4; 5, interpolating between the values (N − 1)=(N − 2)(d − 2) and (N − 1)=(N − 2) ×(d − 3) in the limits LI → ∞ and 0, respectively, where once again we are considering the behavior near the critical point. Note that in contrast to the case of an expansion around the critical point using a ’4 Landau–Ginzburg–Wilson Hamiltonian A is not simply the critical exponent -. This is due to the fact that the canonical dimension of the >elds A and here is zero. The bulk, L → ∞, value of A is A (∞) = (d − 2 + -) where the critical exponent - = (d − 2)=(N − 2). In the limit LI → 0, I → ∞ we see that A → (d − 2 + - ) where d = d − 1 and - = (d − 2)=(N − 2) is the critical exponent of the dimensionally reduced system. In the large-N limit, N → ∞, note that A → h = dee= , where dee= is the e=ective dimension that was found in considerations of dimensional crossover in the large-N limit of a ’4 theory [12].

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9.2.4. Critical temperature shift The -function equation is easily integrated to >nd I 1 I t ; LI = −1 ; Ii ti (Ii =I)d−2 + Ii f(LI ) (I =I)d−2 dI =I

513

(298)

where Ii and ti are the initial arbitrary renormalization scale and temperature. The function f is ∞ ∞ ds 2(N − 2) 2 2 2 e−s e−n L I =4s : f(LI) = d=2 (d − 2)=2 (4A) s n=−∞ 0

In the limit L → ∞, Eq. (298) becomes I 1 I t ; ∞ = −1 ; d − 2 Ii ti (Ii =I) + I f(∞)(I =I)d−2 dI =I

(299)

i

where f(∞) simply picks out the n = 0 term in the sum in f(LI). Choosing the initial temperature and initial scale in (298) and (299) to be the same one obtains I 1 1 dI − = (f(LI) − f(∞))LId−2 ; (300) T (L) T (∞) I Ii where a factor of I2−d has been absorbed into the dimensionless temperature. The interpretation of Eq. (300) is that given a particular renormalization scale I in two systems of size L and of in>nite size then the corresponding temperatures in the two systems are related as above. Given that the limit I → ∞ corresponds to the approach to the critical point we can take this limit in (300) to >nd 1 1 1 = (f(LIi ) − f(∞)) : (301) − Tc (L) Tc (∞) 2 Taking the limit Ii → 0 corresponds to choosing the initial dimensionless temperature to be zero. The shift then becomes 1 1 bd − = d− 2 ; (302) Tc (L) Tc (∞) L where the dimension dependent constant bd is (N − 2) d−2 G(d − 2) : bd = 2 2Ad=2 2 Result (302) is fully in agreement with the expectations of >nite size scaling, with the exponent + = 1=(d − 2). In the limit d → 3 bd → ∞ as there is a divergence in the G function at d = 2. This corresponds to the fact that the the shift is ill de>ned due to the non-existence of a critical point in

514

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545 0.5

N=3 -1

0.4

d

N=4 0.3 N=5 0.2

0.1

Ln kL -5

5

10

15

20

Fig. 23. Graph of 1=d = 1=0e= on the separatrix solution of (294) as function of ln IL.

two dimensions as discussed by Barber and Fisher [58] and as discussed in Section 4 in the context of the spherical model. 9.2.5. E8ective exponents Here, given that the non-linear -model is restricted to the broken phase we concentrate on the two e=ective exponents e= and 0e= de>ned as d ln d ln −1 0e= = ; e= = ; (303) d ln H d ln (tc (L) − t) tc (L)

H =0

where 0e= is de>ned along the critical isotherm of the >nite size system and e= on the coexistence curve of the >nite size system. To derive e= we solve the RG equation for the magnetization, ’, V with initial condition ’(I V = 0) = 1. We then substitute the anomalous dimension (297) and consider the limit LI → ∞ to >nd d 1 ∞ dx e= = − (tc (L) − t) ; (304) [ A (x; t) dt 2 0 x where = +(d − 2 + -)=2 is the bulk exponent and [ A = A − (d − 2 + -). In the limit (tc (L) − t) → 0, Ld−2 (tc (L) − t) → ∞ one >nds that e= → +(d − 2 + -)=2, while in the limit (tc (L) − t) → 0, Ld−2 (tc (L) − t) → 0 one obtains e= → + (d − 2 + - )=2. At one-loop order, using Eq. (297), e= interpolates between the above asymptotic values, where now + = 1=(d − 2) and - = (d − 2)=(N − 2) and + = 1=(d − 2), - = (d − 2)=(N − 2) and d = d − 1. Similarly, one >nds for 0e= that A 1 ; (305) 0− e= = (2d − A − 2 t ) where t ≡ (t =t). In the limit (tc (L) − t) → 0, L1=+ (tc (L) − t) → ∞ one has A → (d − 2 + -) and t → 0. Hence, we see that 0e= → (d+2−-)=(d−2+-). In the limit (tc (L)−t) → 0, L1=+ (tc (L)−t) → 0 one >nds that A → (d − 2 + - ) and t → 1. Thus, in this limit 0e= → (d + 2 − - )=(d − 2 + - ), where d and - are as above. At the one-loop level A is as given by (297) and t = (LI) − h. 1 In Fig. 23 we see a graph of 0− e= as a function of ln IL for d = 3 and N = 3; 4; 5 where the full dimensional crossover is evident.

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515

9.2.6. Equation of state in a thin ;lm In this section we consider the ordered phase for the dimensional crossover associated with a thin >lm from the point of view of a perturbative expansion based on the Landau–Ginzberg–Wilson Hamiltonian (15) rather than the -model approximation (278). As has been emphasized throughout Section 7 the calculation of universal crossover scaling functions, such as the equation of state, require only the calculation of the environmentally friendly Wilson functions ’2 , ’ and and the solution of the -function. Here, we will give the one-loop results for ’2 and on the separatrix, and the separatrix coupling itself, for an O(N ) model in the broken phase on a d-dimensional thin >lm. With these results in hand the reader may substitute the expressions into the relevant expressions of Section 7, as was done to one loop in the case of the equation of state without dimensional crossover in Section 8. The results are presented in terms of the two scaling variables introduced in Section 7, z = mt =m’ and zg = mt =g. In this case the anisotropy non-linear scaling >eld is g = 1=L. Explicitly, the separatrix solution of the function equation is V(z; zg )

=

3cd−1 2zg

∞ n=−∞

1 4A2 n2 1+ 2 + z zg2

(d−5)=2

(d−5)=2 ∞ 4A2 n2 (N − 1) 1+ + 9 zg2 n=−∞

−1

: (306)

In this more complicated situation there are more >xed points: For N ¿1 there are >ve di=erent ∗ >xed points: (i) the trivial Gaussian >xed point, V = 0; (ii) a d-dimensional Wilson–Fisher >xed ∗ point in the limit z → ∞, zg → ∞, V = 6=(N + 8)cd ; (iii) a (d − 1)-dimensional Wilson–Fisher >xed ∗ point in the limit z → ∞, zg → 0, V = 6=(N + 8)cd−1 ; (iv) a d-dimensional “strong coupling” >xed ∗ point in the limit z → 0, zg → ∞, V = 6=(N − 1)cd and >nally, (v) a (d − 1)-dimensional “strong ∗ coupling” >xed point in the limit z → 0, zg → 0, V = 6=(N − 1)cd−1 . For the case N = 1 only (i), (ii) and (iii) exist. Standard expansions would be incapable of accessing (iii), (iv) or (v). With the separatrix coupling in hand it is a simple matter to calculate the Wilson functions on the separatrix. We >nd ∗’2 (z; zg )

∞ 2 2 2 2 (d−7)=2 2 2 2 (d−7)=2 + (N − 1)=3 ∞ n=−∞ (1 + 1=z + 4A n =zg ) n=−∞ (1 + 4A n =zg ) ∞ = (5 − d) 2 2 2 2 (d−5)=2 + (N − 1)=3 2 2 2 (d−5)=2 3 ∞ n=−∞ (1 + 1=z + 4A n =zg ) n=−∞ (1 + 4A n =zg ) (307) and ∗ (z; zg )

∞ 2 2 2 (d−7)=2 3 n=−∞ (1 + 1=z 2 + 4A2 n2 =zg2 )(d−7)=2 + (N − 1)=3 ∞ n=−∞ (1 + 4A n =zg ) ∞ = (5 − d) : 2 2 2 2 (d−5)=2 + (N − 1)=3 2 2 2 (d−5)=2 3 ∞ n=−∞ (1 + 1=z + 4A n =zg ) n=−∞ (1 + 4A n =zg ) (308)

Considering the asymptotic limits of ∗’2 at the di=erent >xed points: for the >xed point (i) ’2 → 0; for (ii) in the limit z → ∞, zg → ∞, ∗’2 → (4 − d) (N + 2)=(N + 8); for (iii) where z → ∞, zg → 0

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D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

one sees that ∗’2 → (5 − d) (N + 2)=(N + 8); for (iv) ∗’2 → (4 − d) and >nally for (v) ∗’2 → (5 − d). Similarly, for ∗ : for >xed point (i) → 0; for (ii) ∗ → (4 − d); for (iii) ∗ → (5 − d); for (iv) ∗ → (4 − d) and >nally for (v) ∗ → (5 − d). Thus we see that the Wilson functions crossover correctly between their expected asymptotic one-loop values in a >xed dimension expansion. This in turn implies that the scaling exponents associated with these >xed points will be as expected. As mentioned the full crossover scaling equation of state can be obtained by substituting these expressions into the formulae found in Section 8 and performing the relevant numerical integrations. In the limit considered here the full crossover equation of state is universal. 9.3. The e8ect of boundary conditions on the crossover Let us now consider the case of other boundary conditions. We will restrict our attention to antiperiodic boundary conditions, the generalization to translationally non-invariant boundary conditions, such as Dirichlet, generically requires a position dependent renormalization. Di=erent boundary conditions constitute an environmental probe which can be used to di=erentiate between systems as they fall into di=erent crossover universality classes. In the case of anti-periodic boundary conditions the extremal values of the functions h(y) and (y), where y = (z 2 − A2 )1=2 , are those of the d and (d − 1)-dimensional >xed points as in the case of periodic boundary conditions. However, the functions in the two cases di=er in the crossover region. Explicitly, for d = 3, to one loop, one >nds h(y) = (1 + A2 =y2 )(1 − y=sinh y). At the one-loop level the expressions for general d, for antiperiodic boundary conditions are ∞ A2 n(n + 1)=z 2 (1 + A2 n(n + 1)=z 2 )(d−9)=2 (309) (z) = 5 − d − (7 − d) n=−∞∞ 2 2 (d−7)=2 n=−∞ (1 + A n(n + 1)=z ) and >nally for the separatrix ∞ 2 2 (d−7)=2 −∞ (1 + (A n(n + 1))=z ) : h(z) = (5 − d) n= ∞ 2 2 (d−5)=2 n=−∞ (1 + (A n(n + 1))=z )

(310)

Plots comparing the e=ects of di=erent boundary conditions for a three-dimensional layered Ising system can be found in Figs. 24 and 25. In the case of antiperiodic boundary conditions for d = 3 the crossover to two dimensions is completed at larger values of L=L when compared to periodic boundary conditions. At the three-dimensional end, there are persistent tails relative to periodic boundary conditions. These are due to the power law as opposed to exponential decay of the boundary e=ect. In Figs. 24 and 25 we include the case of Dirichlet boundary conditions, however, additional care is needed in this case due to the breaking of translational invariance, and the results shown are applicable only for quantities such as the speci>c heat. 10. Quantum to classical or -nite temperature crossover 10.1. Ising model in a transverse magnetic ;eld The real world is quantum mechanical, although this is not always a dominant feature governing the behavior of systems, the macroscopic world typically being governed by classical laws. However,

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

517

y 0.66

Ant ip

erio dic

0.64

0.62

x -5

-4

-2

-1

1

2

Pe

rio d ic

-3

0.58

Dirichlet

Fig. 24. Graph of y = (2 − ’2 (x))−1 at one loop as a function of x = L =L for four-dimensional Ising >lm geometry with periodic and antiperiodic boundary conditions.

deff

Dirichlet

3

odic

Peri

iodi

iper

Ant

2.8

2.6

c 2.4

2.2

x -5

-4

-3

-2

-1

1

2

Fig. 25. Graph of de= at one loop as a function of x = L =L for four-dimensional Ising >lm geometry with periodic and antiperiodic boundary conditions.

at low temperatures, quantum e=ects play an important role, and typically quantum :uctuations need to be taken into account. When one is dealing with >nite temperature systems there are also thermal :uctuations. An interesting question therefore, especially in the theory of phase transitions, is how does one crossover between the two types of :uctuations in a realistic system. For very low temperatures, one expects quantum :uctuations to dominate, and at higher temperatures classical thermal :uctuations.

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D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

An explicit example of a system that exhibits a quantum–classical crossover is an Ising ferromagnet in a transverse magnetic >eld 2, described by the lattice Hamiltonian H=−

1 Kij Siz Sjz − 2 Six ; 2 ij i

(311)

where Sia are the components of spin in the a direction at lattice site i. The spins Siz and Sjx do not commute and it is this non-commutation that gives the additional complications of the model over that of the usual Ising model. The model is known to have a second order transition, at a temperature Tc (2) for 2 6 2c , while for 2¿2c no ordered phase occurs at any temperature. As we are only interested in the critical properties of this model here we will use the LGW Hamiltonian derived by Hertz using the Stratonovich–Hubbard transformation, but now a time ordered version where time ordering is with respect to imaginary times between 0 and . The explicit Hamiltonian is 1 2 d 2 2 2 H= d x ∇’n ∇’−n + (mB + gB + 4A n T )’n ’−n 2 n +

T 4!

0(n1 + n2 + n3 + n4 )’n1 ’n2 ’n3 ’n4

(312)

n1 ; n 2 ; n 3 ; n4

where we have dropped operators irrelevant in the critical region. m2B + gB = 2 − 2c , where 2c is the critical transverse >eld in the mean >eld approximation. The sum is over the Matsubara frequencies √ and T = =, where = 1=43a2 2 tanh(2) and is the inverse temperature. We have also used Kq = K0 − 3a2 q2 + O(q4 ) for the Fourier transform of the Ising spin–spin coupling. Observe that is dimensionless, once and a are assigned the units of length, since from the original Hamiltonian we deduce that J , 3 and 2 have the dimensions of inverse length. Note that in this guise the problem is in exact correspondence with that of >nite temperature quantum >eld theory the only real di=erence being the di=erent interpretation of some of the parameters in the model and whether the >eld theory model is considered “fundamental” or not. In this problem, the anisotropy parameter, or environmental variable, is T , which is essentially just the thermodynamic temperature. Note that it is entering here in a very di=erent fashion to the dimensional crossover considered previously. There the temperature entered in the standard way in the quadratic term of the LGW Hamiltonian, i.e. as T − Tc (L). We now wish to renormalize in an environmentally friendly manner. We use conditions (128) – (130). Condition (131) implies, with the multiplicative renormalization of g, that g is proportional to 2 − 2c (T ), the deviation from the critical line. In the case of quantum >eld theory at >nite temperature of course the only observable is the mass that appears as a pole in the propagator not its representation in terms of a linear scaling >eld, such as 2 − 2c (T ). Note that in the renormalization used here T is a non-linear scaling >eld. Our task now is to calculate the solution of the -function and to subsequently compute the e=ective exponents using the “black box” expressions of Section 7. The only thing we need to remember is that the environmentally friendly propagator that enters these expressions is G(k; !) = 1=(k2 + !2 + g) where ! = 2AnT .

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519

Consider >rst a three-dimensional model. From (105) with N =1, and using the above propagator in the diagrams, one >nds a separatrix that interpolates between the classical and quantum >xed points. The quantum >xed point in this case is found to be zero, but there exist important logarithmic corrections to scaling which are captured by the :oating coupling. An examination of de= reveals that it interpolates between four and three. This implies, as is well known, that the e=ects of quantum :uctuations are such as to increase the e=ective dimensionality of the system in the low-temperature regime. The classical >xed point is h(g = 0) = 1:732. The e=ective exponents can be found from the expressions of Section 7 for the coupling constant. Using the two-loop PadZe resummed coupling the Wilson functions ’2 , ’ and may be derived. Once again they lead to excellent agreement at the three-dimensional end with the results of [151]. We >nd for instance +(g = 0) = 0:639. The explicit two loop PadZe resummed expressions for a two-dimensional Ising model in a transverse >eld can also be straightforwardly found. Note that here we are deriving the e=ective exponents at >xed temperature, i.e. we are approaching the critical line from the g direction. We could equally well have looked at exponents de>ned with respect to the “environmental” T direction. As far as scaling is concerned the relevant correlation length here for scaling purposes is T the physical correlation length of the system. If we had used a temperature-independent renormalization the relevant correlation length would have been the zero temperature quantum correlation length. One can write the scaling functions in universal scaling form 2(N ) = TNd=2−N −d F(N ) (TT ) : 2 (2) In particular for the susceptibility *−1 =− T F (TT ). Strictly speaking the above functions are only truly universal for d¡3. For d = 3 the logarithmic corrections to scaling evident in the e=ective exponents preclude a true scaling form. However, the scaling form in terms of the non-linear scaling >elds is “universal” as the corrections to scaling are captured by these >elds in their dependence on initial conditions. For >xed initial conditions all scaling functions are universal even in the presence of corrections to scaling. Turning now to the two-dimensional model, at one loop we can give some nice simple analytic forms. The crossover coupling in terms of z = TT is z h(z) = 1 + : sinh(z)

One >nds that ’2 (z) = 13 h(z). The e=ective dimensionality de= = ( ) is given by (z) = 1 +

z 2 coth(z=2) ; sinh z + z

(313)

which once again clearly indicates a change in e=ective dimensionality. In this case from two up to three in the deep quantum regime. If one compares any of the two- or one-loop expressions for e=ective exponents here with the corresponding expressions for a layered Ising model with periodic boundary conditions, one >nds that they have the same functional form. In fact, after the replacements g → t, T → L−1 they are identical. Hence, we would conclude that the (d − 1)-dimensional Ising model in a transverse >eld and the d-dimensional layered Ising model with periodic boundary conditions lie in the same crossover universality class.

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Suppose that at a theoretical level we knew nothing about the quantum >xed point, but that experiment had told us that the critical exponents at zero temperature were very di=erent to those at the classical >xed point. How would one go about >nding a theoretical description of the other >xed point, and indeed the entire crossover? Within our environmentally friendly formalism the answer is clear: one starts at some “microscopic” energy scale Em , one writes down a Hamiltonian for the “microscopic degrees of freedom”, including all relevant environmental parameters, one cranks the handle of the environmentally friendly RG and sees what comes out. If there exists another >xed point which is reached via the e=ects of a relevant environmental parameter, then it will be seen in the environmentally friendly RG :ow. If it does not exist then it will not appear. The point is we do not need to know a priori about its properties, or even of its existence, these can all be deduced from the RG. What is equally important is that one can see when one is not implementing an environmentally friendly renormalization because perturbation theory within the context of the unfriendly group will break down. This is one’s pointer as to the fact that one had missed some important environment dependence. One can extend the results given to a four-dimensional quantum Ising model. Obviously for a four-dimensional model there is no crossover as noted in [155] in the context of the “spherical” quantum Ising model. All e=ects then are due to corrections to scaling, though these can be of great importance for particle physics models in the context of Kaluza–Klein theory [156]. 10.2. Finite temperature renormalization We consider a ’4 theory in equilibrium with a thermal bath at temperature T = 1=. It is described by the Euclidean action 1 1 2 2 B 4 d− 1 2 (314) dt d x (∇’B ) + MB ’B + ’B : S[’B ] = 2 2 4! 0 The e=ective potential V , or e=ective action (2) per unit volume, is a function of the renormalized >eld expectation value and provides a convenient summary of many of the thermodynamic properties of the model. To access an arbitrary value of ’V it is necessary to couple in a constant external current J , which will produce this expectation value. In fact the e=ective potential is a function of ’V 2 and the relation between ’V and J is given by the equation of state 2(1) = 2t(2) ’V = J ;

(315)

which also serves to de>ne 2t(2) again as a function of ’V 2 . We further de>ne 2t(4) through 2

(2)

=

2t(2)

2t(4) 2 ’V + 3

(316)

and more generally each n-point vertex function admits a decomposition in terms of vertex functions 2t(n) which are functions of ’V 2 . For J = 0 there are two possible solutions of (315): Either ’V = 0 with 2t(2) = 0 or ’V = ’V 0 = 0 and 2t(2) = 0. For >xed temperature it is natural to consider this model, and hence the e=ective potential, as a function of both ’V and MB2 . In the entire (MB2 ; ’) V plane both branches of (315) are realized.

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

521

If we locate our origin at the point where the model switches from one branch to another, then we can describe the plane in terms of new coordinates (tB ; ’) V where tB = MB2 + &(T ). It turns out that it is necessary to multiplicatively renormalize tB in a similar fashion to ’V by de>ning t = Z’−21 tB . V plane, In fact to control the IR divergences that necessarily arise close to the origin of the (tB ; ’) which corresponds to the location of a second order phase transition, it is necessary to have both Z’ and Z’2 temperature dependent. If one changes T then the picture remains qualitatively the same but the origin and detailed shape of the coexistence curve change somewhat. We de>ne the Z factors above by the following normalization conditions: 9 (2) 2t (p; ’(@; V I); t(@; I); (@; I); @) = 1; (317) 2 9p p=0 2(2) (0; ’(@; V I); t(I; @); (@; I); @) 2

1 + (’V

=3) (9=9p2 ) 2t(4) (p; @; I)|p=0

= I2 (@) ;

(318)

2t(2; 1) (0; ’(@; V I); t(@; I); (@; I); @) = 1 ;

(319)

2t(4) (0; ’(@; V I); t(@; I); (@; I); @) = (@; I) :

(320)

These normalization conditions allow us to present the two di=erent schemes that we have used, i.e. the running the mass versus running the environment schemes. The >rst corresponds to :owing I while @ is set to the physical temperature T . The >rst prescription has been considered in detail in Section 9, in the context of >nite size scaling, and in the previous section for a quantum Ising model. The principal quantity that is missed in this approach is the critical temperature or equivalently the quantity &(T ) described above. 10.2.1. Flowing the ;nite temperature “environment” The second prescription involves :owing the “environment”, @, at >xed MB and allows us to predict the critical temperature in terms of the zero temperature parameters, precisely as one would wish to do in the case of the standard model. In this prescription we choose to follow the :ow of M 2 = 2(2) |p=0 . The di=erential equations which describe an in>nitesimal change in normalization point with >xed bare parameters are d ln Z’ (@) @ = ’ ; d@

@

d ln Z’−21 d@

= ’2 ;

@

d M 2 (@) = M ; d@

@

d (@) = : d@

(321)

For J = 0 the :ow functions M , and ’ take di=erent functional forms above and below the branching point which corresponds to T =Tc , the critical temperature of the model. The :ow functions to one loop, are given by ’ = 0 ; ’2 = −

(322) d 1 @ 2 d@

;

(323)

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 9   @¿Tc  2 @ 9@ ; M = 9 9   − @ + 32 M 2 @ ; @¡Tc ; 9@ 9@ =−

3 2

2

@

d d@

:

(324)

(325)

As before, the symbol with k dots or with a k in the circle stands for the one-loop diagram with k propagators, without vertex factors, and zero external momentum. The di=erential equation for the coupling is easy to solve, since it contains a total derivative, and to the order we are working takes the same form in both phases. The solution is −1

(@) =

−1

(@0 ) + 32 [ (M (@); @) −

(M (@0 ); @0 )] :

(326)

This expression is now manifestly >nite in four dimensions. To ensure that the same initial conditions are imposed on both sides of Tc one may use the requirement that the bare coupling B is the same −1 in both phases. Eliminating B from the two dimensionally regulated expressions, ± (@) = B−1 + 3 (M± (@); @), then gives + in terms of − . To one loop we >nd 2 −1

+

(@+ ) =

−1 − (@− )

+ 32 [ (M+ ; @+ ) −

(M− ; @− )] :

(327)

This solution may now be substituted into the di=erential equation for M (@), which is solved numerically. After solving the :ow equations we are free to choose the reference temperature @ equal to the actual temperature T of interest. In fact this is essential if one wishes to obtain perturbatively sensible results for physical quantities [157]. Because of the renormalization conditions (318) and (320) the parameters M (T ) and (T ) therefore describe the behavior of the vertex functions 2(2) and 2t(4) at zero momentum. 10.2.2. One-loop solutions: critical temperature and amplitude ratios Fig. 26 shows the behavior of m(T ) and (T ), from zero temperature in the broken phase up to temperatures above the critical temperature. At high temperatures M (T )=AT , where A is a constant. In this regime :uctuations are unimportant and a mean >eld description is adequate. Of course, the vanishing of l as the critical temperature is approached does not imply the theory becomes non-interacting. As seen, in dimensional crossover a more natural dimensionless coupling is the :oating coupling. Using M as a parameter measuring the distance from Tc we obtain for h the :ow equation (for the symmetric phase) M 9h 2 = −h + h + O : (328) M 9M T This equation has, in the limit M → 0, a familiar >xed point structure, with a stable, non-trivial >xed point at h∗ = 1, while in the zero temperature limit h is proportional to the zero-temperature coupling.

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Fig. 26. The mass m(T ) and coupling (T ) as a function of the temperature T . The graphs are obtained by solving the :ow functions with initial (zero temperature) coupling (0) = 1, starting in the broken phase. T and m(T ) are given in units of m(0). The critical temperature Tc = 4:080m(0) separates the two phases.

With these equations one is also able to determine the critical temperature in terms of the zero-temperature parameters m(0) and (0). As may be expected on dimensional grounds Tc is proportional to m(0), the constant of proportionality being a function of (0). To facilitate comparison with the literature we have plotted t := Tc (0)=M (0) versus √ (0) in Fig. 27. The original result of Dolan and Jackiw [141] is then a horizontal line at t = 12 in this graph, where their renormalized parameters are interpreted as our zero-temperature parameters M (0) and (0). Our √ critical temperature is larger than theirs and the value t = 12 is approached in the zero-coupling limit. Amelino-Camelia and Pi [160], who found the transition to be >rst order, have a value of t = 4:901 at (0) = 0:05, which is even larger than our equivalent value t = 3:607. We have also plotted the explicit result given by Lawrie [158]. His curve for t starts o= in the negative direction, irrespective of the linear term of which the prefactor was not determined. Additionally, we compare with “coarse-graining” type RGs such as used in [159]. There the critical temperature is found to increase as a function of (0) but at a rate substantially less than we >nd. Also their results are dependent on the particular type of cuto= function used. If we take values for (0) and M (0) to be those associated with estimates for the equivalent parameters in the Higgs sector of the standard model [161] we >nd, for (0) = 1:98 and M (0) = 200 GeV that Tc = 613 GeV. By comparison Dolan and Jackiw’s result [141] gives 492:4 GeV. For (0) = 3:00 and M (0) = 246 GeV the corresponding results are Tc = 639 GeV and 492:0 GeV, respectively.

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Fig. 27. The critical temperature Tc as a function of the zero-temperature parameters (0) and M (0). Tc is proportional to M (0). Graph (a) is obtained from our numerical solution, (b) is the early result of Dolan and Jackiw [141], and (c) shows the expression obtained by Lawrie [158] (up to a linear term). The crosses and stars represent values obtained by Tetradis and Wetterich [159] for two speci>c choices of coarse-graining, corresponding to their Tables 2 and 6, respectively.

Knowing the behavior of M (T ) and (T ) we are able to plot in Fig. 28 the >eld expectation value versus T . At the critical temperature V 0 is seen to vanish continuously, con>rming that the phase transition is second order. Since M (T )T near Tc the >nite temperature four-dimensional theory reduces there to a threedimensional Landau–Ginzburg model, in accord with >nite size scaling. In the neighborhood of the critical temperature the general vertex functions have the form (n) +(dc −n(dc −2+-)=2) = (n) ; 2± ± |T − Tc |

(329)

where dc =d−1 is the reduced dimension at the critical point, + and - are critical exponents and (n) ± are amplitudes. The appearance of the critical exponent + is unusual in particle physics. It re:ects the need for composite operator renormalization and the physical dependence of m(T ) on temperature. This ensures that the exponent + is physically accessible in >nite temperature >eld theory, whereas it is usually not experimentally observable in a particle physics context as the dependence of the renormalized mass on the bare mass is experimentally inaccessible. As we see here, however, the exponent + in fact plays a highly signi>cant role near the critical point. From the solutions of (325) by noting that the function behaves like T=8AM we see that as the critical point is approached, the coupling vanishes [4,5,33]. Hence, near the critical temperature we have M±2 = (C ± )−1 |T − Tc | with ± = l± |T − Tc |+ and m± = (f1± )−1 |T − Tc |+ , where we use the notation of Liu and Fisher [162] for the amplitudes, and = +(2 − -). Our temperature :ow

D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

525

Fig. 28. The >eld expectation value V as function of the temperature T corresponding to the solution of Fig. 26.

equations give the one loop values for the exponents, +=1 and -=0, which are not as good as those obtained by :owing the mass parameter where our two-loop PadZe results [4,5] for these exponents are + = 0:639 and - = 0:0329. We expect, however, that the agreement between the two schemes should improve at higher orders. The temperature :ow scheme gives the amplitude ratios l+ C+ 12 f1+ 1 ≈ 1:92; (330) = 4; = 2 = : − − C− 13 l 2 f1 The fact that these are not one is indicative of a cusp in the graphs of mass and coupling versus temperature as the theory passes through the critical temperature. These ratios are universal numbers analogous to the critical exponents. The best estimates for the amplitude ratios are the high- and low-temperature series expansion results of Liu and Fisher [162] who >nd C+ = 4:95 ± 0:15; C−

f1+ = 1:96 ± 0:01 ; f1−

(331)

which our results are in good agreement with. By comparison: at tree level (mean >eld theory) C + =C − =2 and f1+ =f1− =1:41, whilst in the expansion at order 2 , assuming dimensional reduction, C + =C − = 4:8 and f1+ =f1− = 1:91 [163,164]. We see here that the amplitude ratios are substantially better than the results for critical exponents. This indicates a complimentarity between the current approach of :owing the environment, temperature, and that of [4,5,9,10] where the :ow parameter was the >nite temperature mass. At one loop the latter group gives better results for amplitudes, whereas the former gives better

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Fig. 29. The minimum of the e=ective potential U as a function of the temperature T , corresponding to the solution of Fig. 26, normalized by the ideal gas value.

results for exponents, but both schemes should converge to the same results as one goes to higher orders. 10.2.3. Finite temperature e8ective potential We now construct the e=ective potential. We determine its minimum, U , in the absence of external currents, H = 0, by using the :ow equation @

dU (@) 1 9 = U = @ + ::: : d@ 2 9@

(332)

Fig. 29 shows the minimum U (T ) of the e=ective potential relative to the minimum at zero temperature U (0). In the setting of the early universe, normalizing the latter to zero would correspond to a vanishing cosmological constant. In the high-temperature limit, T is much larger than both M (0) and Tc and the e=ective potential U is relatively close to that of an ideal gas −A2 T 4 =90 [141]. Since the mass of the e=ective excitations increases approximately linearly with T further increase of temperature will not give rise to a regime where dimensional reduction is valid, rather what will happen is that quantum corrections will be suppressed due to a large mass in the diagrams. However, the e=ective potential should still have the form T 4 with some non-ideal Stefan–Boltzmann constant . The >rst few of the remaining vertex functions involved are plotted in Fig. 30. Near the critical temperature 2(3) ∼|T − Tc |3=2 and 2(5) ∼|T − Tc |1=2 for T ¿Tc , but are zero for T ¡Tc , whilst

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527

Fig. 30. The vertex functions 2(2) ; : : : ; 2(6) to one loop at zero momentum, corresponding to the solutions of Fig. 26.

2(4) ∼|T − Tc | in accordance with (329). The vertex function 2(6) is the >rst one that remains non-zero at the critical temperature, however, this is an artifact of the fact that at this order - = 0. At higher orders all vertex functions 2(n) for n ¿ 6 will in fact diverge as the critical temperature is approached. As one approaches the critical temperature we >nd to this order (4) 28AM− 2 2− = with + = ; (4) 3T 7 − 2 2 (6) 8 (6) (6) −2 640A −2 1520A 2+ = Tc ; 2− = −Tc with + =− : (6) 9 9 19 − (4) 2+

16AM+ ; = 3T

(4)

(333) (334)

In the broken phase, the six-point function turns out to be negative, but one should remember that there are an in>nite number of vertex functions and a truncation of the e=ective potential at a low order is only appropriate for suJciently small [V and for temperatures suJciently far from the critical temperature. The complete shape of the e=ective potential can be thought of as that of a boat where the lower hull has been sawn o=. The :at base of the boat has a side projection whose characteristic shape divided by T 4 is shown in Fig. 29. The :at region slopes downwards as the temperature is increased and ends at the critical temperature. A top projection of the bottom of the boat can be seen from Fig. 28 when the curve V 0 is thought of as being re:ected around the T axis. One sees that as a positive reference external current H is sent to zero for T ¡Tc the minimum of the e=ective potential

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D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545

occurs at a non-zero positive value of the >eld expectation V 0 (T ). If the >eld were reversed in sign and sent to zero the >eld expectation would be −V 0 (T ), the width of the boat across the base being 2V 0 (T ). The bottom of the boat represents the co-existence region of the phase diagram and V 0 (T ) should be connected with −V 0 (T ) for the same value of T by a “tie-line”. The critical point is the front of the boat and is sewn on in a highly non-analytic way, as indeed is the entire base. We retain all these features in our approach. 11. Uniaxial dipolar ferromagnet Another model that falls into the same class as the considerations of this section is that of a uniaxial dipolar ferromagnet [6,76]. In this case, the propagator takes the form p 2 + 0

pz2 : p2

(335)

Here the environmental factor that is followed in the crossover is 0 , the coupling strength of the long-range dipole–dipole interaction between the spins. Once again the formal structure of the diagrammatic series is the same as in Section 9. The diagrams di=er in their dependence on z −1 = 0 01=2 now, hence (z), f1 (z) and f2 (z) are di=erent functions, interpolating however, between the 4 values 1, 13 and 27 , respectively for z = 0, to >nite values close to those of the four-dimensional problem treated above. The numbers are di=erent however, re:ecting the absence of some angular integration variables. Here, we will restrict our considerations to one loop. One >nds that z2 : 1 + z2 The solution of the running coupling equation is given by (z0 is the initial value) 1 √ : h= √ −1 2 2 1 + z (h0 = 1 + z0 + ln(z0 (1 + 1 + z 2 )=z(1 + 1 + z02 ))) (z) =

(336)

(337)

There is an interesting distinction between this situation and that of the dimensional crossover, in that in the case of the uniaxial problem there is a universal form for the crossover from the three-dimensional >xed point to the pseudo-four-dimensional >xed point. The reason for this is that the true microscopic structure of the model is three dimensional, and therefore the logarithmic growth into the UV does not continue ad in>nitum. The universal part of the crossover curve can be obtained as in the case of three-dimensional problems by taking the limit z0 → ∞. We >nd this universal coupling is given by 1 √ h(z) = √ : (338) 2 1 + z ln((1 + 1 + z 2 =z)) We present a comparison of y = (2 − ’2 (z))−1 for the four-dimensional dimensional crossover in a layered geometry with periodic boundary conditions with that of the uniaxial crossover in Fig. 31. The initial coupling for the layered geometry is h0 = 1 for ln(1=z0 ) = −20 and the curve for the uniaxial problem is the universal crossover exponent derived from (338). Fig. 31 clearly shows that the uniaxial crossover and that of the layered geometry are in di=erent crossover universality classes.

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529

Fig. 31. Comparison of y = (2 − ’2 (x))−1 at one-loop for a uniaxial dipolar ferromagnet, three-dimensional quantal Ising model and four-dimensional Ising >lm with periodic boundary conditions.

12. Bicritical crossover Bicritical crossovers have been one of the most studied, both by coarse graining and by reparametrization techniques. The microscopic Hamiltonian we will consider is (15), with the anisotropy term being given by (16) and describing a crossover O(N ) → O(M ) such that the anisotropy operator is 1 O = (Mm21 + (N − M )m22 ) ; (339) N where m21 = (T − Tm (M )) and m22 = (T − Tm (N − M )), Tm (M ) and Tm (N − M ) being the mean >eld critical temperatures for the O(M ) and O(N − M ) components of the order parameter, respectively. The anisotropy parameter, g = (1=2)(m21 − m22 ) · r in this case ˙ (T − Tm (N )), where Tm (N ) is the mean >eld critical temperature of the isotropic system. Crossover scaling theory indicates that a typical scaling function, such as the susceptibility along an easy axis, should have the form *(t0 ; g) = t − F(g=t ) ;

(340)

where t = (T − Tc (N ))=T , is the isotropic susceptibility exponent and is the crossover exponent, also a function of the isotropic >xed point. As g → 0 then F → const. For g¿0 F(z) exhibits a zero, zM , which de>nes a critical line g = zM t associated with the O(M ) theory, while for g¡0 the corresponding zero de>nes a critical line for the O(N − M ) theory. These two critical lines meet at the bicritical point T = Tc (N ), g = 0. g is here a linear scaling >eld being naturally associated with a renormalization gB = ZB g where gB is the bare anisotropy. The associated anomalous dimension is B which at the bicritical >xed point, to >rst order in , is B = 2=(N + 8) from which the crossover exponent can be calculated to be = 1 + (N=2(N + 8)). These results for B and can be obtained

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using any simple renormalization scheme such as minimal subtraction. However, if g independent, such schemes are not environmentally friendly and hence the crossover to the critical “wings” of the phase diagram will be inaccessible. To treat this crossover we >rst pass to a formulation where the anisotropy is represented by a non-linear scaling >eld. In this case, one uses as anisotropy parameter the mass, m, of the (N − M ) components of the order parameter associated with the O(N − M ) symmetry and as temperature parameter tg =(T −Tc (M )). Note that Tc (M ) will be anisotropy dependent, tg measuring the deviation from the critical O(M ) “wing”. One could of course also use the mass of the components of the order parameter associated with the O(M ) symmetry as anisotropy parameter. The two formulations are equivalent. One now implements an environmentally friendly m-dependent renormalization using the normalization conditions (128) – (131) for the corresponding vertex functions of the O(M ) theory. Restricting to three dimensions for the :oating coupling one >nds, using an expansion, that (h) = −(m=I)h + h2 + O(h3 ) ;

(341)

where (I=m) = 1 − I

d ln (N (m=I) + 8) dI

2(N − M ) = 1− 2 ((M + 8)(1 + m =I2 ) + (N + 8)(1 + I2 =m2 )) and Ne= ≡ N

m I

=M +

(N − M ) ; 1 + m2 =I2

(342)

(343)

which, analogous to de= , de>nes an e=ective number for the components of the order parameter. Ne= interpolates between N and M . As I → 0 for >xed m; h → 1 and as I → 0; m2 =I2 → ∞, h → 1 again. So we see interestingly that the crossover in terms of the coupling h is a closed path. The reason for this is that both the O(N ) and O(M ) models have the same upper critical dimension. Of course, in terms of the crossover interpolates between ∗ = (N + 8)=48A and (M + 8)=48A. The Wilson function ’2 is N (m=I) + 2 h + O(h2 ) : ’2 = (344) N (m=I) + 8 With h and ’2 in hand any soft scaling function or shift can be calculated to one loop by substituting these building blocks into the expressions of Section 7. With a bit of extra e=ort hard scaling functions can also be calculated [165]. If the crossover between the O(N − M ) critical line and the bicritical point is required one simply repeats the procedure with m being the mass of the O(M ) hard components and t = (T − Tc (N − M )). 13. Kinematic crossovers In this section we will consider some crossover systems where the crossover is as a function of kinematic variables. Such crossovers have not been attacked using RG methods as their solution

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531

requires several rather novel elements that do not occur in the more standard type of crossover we have considered up till now. So, how do kinematic variables a=ect the e=ective degrees of freedom? Consider the case of a cubic >eld theory in d dimensions. For on shell processes where s ∼ t, with s and t being the standard Mandelstam variables, the two-point scattering amplitude is that characteristic of a d-dimensional >eld theory. However, in the Regge limit s=t → ∞ the scaling form of the scattering amplitude shows an apparent dimensional reduction, or perhaps better said, a dimensional “factorization”. This is manifest, as we shall see below, in the dimensionally reduced form of the dominant ladder diagrams in this limit the dimensional reduction being d → (d − 2). A similar phenomenon occurs in gravity and could well play an important role in quantum gravity as has been emphasized by ’t Hooft [166] and especially in the question of black hole formation as has been advocated in Ref. [167] where the use of an RG that is capable of coarse graining between “hard” particles (that contribute to the background metric of the black hole) and “soft” particles (that represent particle production on the background metric) has been proposed as a method for understanding the physics of gravitational collapse and proving that it is governed by a unitary evolution. An asymmetric or kinematic RG of the type we shall discuss could o=er a >rst step in this direction. In work related to the above, ’t Hooft [168] also showed that certain Planck scale processes can be calculated using known laws of physics. Speci>cally, Regge limit graviton-graviton scattering amplitudes with st can be calculated, the resultant amplitudes possessing certain features in common with string theory amplitudes. This work was extended by the Verlindes [169] who showed that in the “Regge” regime quantum gravity separates into strongly coupled (longitudinal) and weakly coupled (transverse) sectors. This separation into weak and strong coupling sectors is very characteristic of kinematic crossover. Another important area where an extreme kinematic limit plays an important role is that of turbulence. The structure functions of velocity di=erences there can be regarded as the fused limits of basic correlation functions. As emphasized by L’vov and Procaccia and coworkers [170 –173] the calculation of these fused limits leads to anomalous scaling independent of any IR or UV cuto= in the theory but have their origin in the non-perturbative resummation of the classes of ladder diagrams that are seen to dominate in this fused kinematic limit. This mechanism is very similar to the mechanism we shall show below for obtaining anomalous scaling in the Regge limit. 13.1. Asymmetric scattering and the Regge limit In this section we will consider the Regge limit within the context of a cubic scalar >eld theory. This is done for reasons of sheer simplicity, and because cubic couplings also appear in gauge theories. Although these bosonic theories su=er from vacuum instability, perturbation theory around the “perturbative” vacuum is well-de>ned order by order. Furthermore, they have been used extensively as model theories to investigate Regge behavior in perturbation theory (see, for instance, Ref. [174]). In principle, one may always think of adding a stabilizing 4 term but with an arbitrarily small coupling such that it plays no role in what we are about to discuss. The general notation we will use will be the following: an n-point connected (bare) Greens function will be denoted, GBi1 :::in (p1 ; : : : ; pn ) where we use the superscript notation i1 : : : in to denote the external legs, i.e. i1 ; : : : ; in can take di=erent values depending on the >eld content of the theory. For instance, in 3 theory, ik = for all k. For a theory with interaction term † , ik can represent

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D. O’Connor, C.R. Stephens / Physics Reports 363 (2002) 425 – 545 †

†

, † or . Thus GB represents propagation of the charged -particle and GB propagation of the corresponding antiparticle, whilst GB represents propagation of the neutral particle . Rather than deal with the connected Greens functions we will >nd it useful to consider the i1 :::in quantities 2˜ B (p1 ; : : : ; pn ) which are closely related to the S-matrix elements and are obtained from the connected Greens functions by removing the external legs, i1 :::in 2˜ B (p1 ; : : : ; pn ) =

n

2Bik ik (pk )GBi1 :::in (p1 ; : : : ; pn ) :

(345)

k=1

The relation to fully one-particle irreducible Greens functions is found using the standard “tree theorem”. For instance, for the four-point function introduced in the last section one has the decomposition ijkl 2˜ B (s; t; u) = 2Bijkl (s; t; u) + 2Bijm (s)GBmm (s)2Bmkl (s)

+ 2Bikm (t)GBmm (t)2Bmjl (t) + 2Bilm (u)GBmm (u)2Bmjk (u) :

(346)

ijkl On mass shell 2˜ B depends on the momenta only through the Mandelstam variables s; t; u where

s = −(p1 + p2 )2 ;

t = −(p1 − p3 )2 ;

u = −(p1 − p4 )2

(347)

and we consider the particles k and l as incoming and i and j as outgoing. For the pure cubic scalar theory in four dimensions, to one loop the only UV divergence occurs in the mass correction which diverges logarithmically. This UV divergence can be removed in the standard fashion via a normalization condition, for instance that the renormalized two-point vertex function have a zero on the mass shell, i.e. 2˜ (p2 = −m2 ; m; g) = 0 :

(348)

After this mass renormalization, then as far as UV divergences are concerned the theory is totally >nite. However, we will now consider the large momentum behavior of the theory. In particular we will consider two-particle scattering in the asymmetric limit t → ∞ for >xed s. In this limit and considering the on-shell amplitude we can classify the one-loop diagrams according to whether they contain a factor of ln t, a factor of ln(−t), or are “>nite”. This separation can in principle be carried out to all orders leading to the following decomposition for 2Bijkl : ijkl ijkl 2Bijkl (s; t; u) = Aijkl B; t (s; t) + AB; u (s; u) + AB; s (t; u) ;

(349)

where Aijkl B; t includes the one-particle irreducible diagrams with powers of ln(−t) (“t-contributions”), ijkl AB; u includes the diagrams with powers of ln t (“u-contributions”), and Aijkl B; s includes the remaining diagrams (“s-contributions”). There is of course an analogous decomposition associated with the limits s → ∞ and u → ∞. For the 3 theory at one loop Aijkl B; t is given by a planar box diagram, ijkl Aijkl by the u-channel crossed box and A by the s-channel crossed box. B; u B; s

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533

ijkl

At the level of the functions 2˜ B the natural combinations for investigating the large-t behavior ijkl ijkl are the functions BB; t for the t-contributions, and BB; u for the u-contributions, de>ned as ijkl ijkl njl ikm mn BB; t (s; t) = AB; t (s; t) + 2B (t)GB (t)2B (t) ;

(350)

ijkl ijkl njk ilm mn BB; u (s; u) = AB; u (s; u) + 2B (u)GB (u)2B (u)

(351)

ijkl with an analogous de>nition for BB; s for the s-contributions. We hence have the decomposition ijkl ijkl ijkl ijkl 2˜ B (s; t; u) = BB; t (s; t) + BB; u (s; u) + BB; s (t; u) :

(352)

ijkl BB; s

contains no large logarithms and does not play an important role in the asympThe function ijkl ijkl totic t behavior. However, in the large-t limit both BB; t and BB; u become perturbatively uncontrollable due to large logarithmic corrections. One way to ameliorate this problem is to try to sum up the large logarithmic terms thus generating a non-perturbative result. In simple cases this can be done ijkl by hand. Here, however, we will use an RG approach: consider the function BB; t (s; t) in the large-t 3 limit. In the case of theory, we have explicitly to one loop (omitting the trivial upper indices on BB; t ) 4 gB4 gB gB2 + K(s) ln(−t) + O ; (353) BB; t (s; t) = −t −t t where the >rst term on the r.h.s. stems from the t-channel tree diagram and the second from the planar box diagram, while the contributions of the other one-particle reducible diagrams are relatively suppressed in the large-t limit. The function 1 1 d K(s) = ; (354) 2 2 16A 0 m − (1 − )s corresponds to a two-dimensional loop integration owing to the dimensional reduction associated with the Regge limit. It is clear that in the t → ∞ limit the above perturbative expansion is ill de>ned. Two-loop diagrams lead to terms ∼ gB6 t −1 K 2 (s)(ln t)2 and ∼ gB6 t −1 K (s) ln t where K is t independent. Similarly, the function BB; u (s; u) exhibits the behavior 4 gB4 gB gB2 + K(s) ln t + O : (355) BB; u (s; u(t; s)) = t t t We will use a reparametrization RG to access the Regge limit. What will be new in the case at hand, relative to the other crossovers we have treated, is that in order to capture Regge behavior we will be forced to implement a very novel reparametrization. This is already manifest in the fact that there are two sets of large logarithms, one associated with BB; t and another set with BB; u , and thus an overall multiplicative renormalization of the connected four-point function will not be suJcient to make the perturbation series well de>ned. Normally we associate renormalization with a renormalization of a “coupling constant”, i.e. a parameter in the original Lagrangian, or an overall renormalization of a Greens function or vertex function. One may think of this as being associated with the fact that in a certain asymptotic limit there is only one singularity. For instance, in critical phenomena in the massless theory terms of the form n (ln k)n sum to give k −- . In the present case however there is more than one singularity as we shall see. Hence, to access these singularities one needs to get inside the Greens functions and identify the parts that will renormalize “naturally”.

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In one sense our reparametrizations can be seen as just e=ecting a reorganization of a perturbative series. No reorganization is wrong, but as we have constantly emphasized, in perturbation theory one type of reorganization may capture much more of the actual behavior of the entire function than another one. To achieve an adequate renormalization we will de>ne multiplicative renormalizations of BB; t and BB; u separately. Explicitly for BB; t Bt (s; t; g(I); m(I); I) = Zt BB; t (s; t; gB ; mB ; ) ;

(356)

where the precise form of Zt of course depends on the speci>c normalization condition chosen. Hence, Bt satis>es an RG equation that follows naturally from the I-independence of the bare theory, d (357) I Bt (s; t; g(I); m(I); I) = t Bt (s; t; g(I); m(I); I) ; dI where t = d ln Zt =d ln I is the anomalous dimension of Bt . There is of course the question of what are we going to use as our sliding scale? In the large-t regime there is a natural decoupling between the transverse and longitudinal sectors whose consequence is an e=ective dimensional reduction. In other words the e=ective degrees of freedom of the system are four dimensional for small t and two dimensional for large t. As the physics changes as a function of t, it is natural to choose an arbitrary, >ducial value of t as the RG parameter as it is this “environmental” parameter which induces the crossover. We have assumed in the above renormalizations of the coupling constant and mass. This may or may not be necessary. In the current scalar theory in four dimensions it is not essential as there are no large t-logarithms in the vertex or mass that require exponentiation. For instance, the one-loop vertex correction is ∼ gB3 (ln t)2 =t, whilst the one-loop mass correction, after removing the ultraviolet divergence by minimal subtraction, is ∼ gB2 ln t. The former tells us that asymptotically gB is a very good approximation to the vertex function. The latter when inserted into the four-point function gives a term of O(t −2 ) which is negligible compared to O(t −1 ) terms. In six dimensions however, where the theory is renormalizable, a renormalization of the coupling is essential. In this respect it is more akin to the case of QCD. With the above choice of the RG scale we have t = t (s; g(I); m(I); I) which we will abbreviate in the following as t (s; I). Integrating Eq. (357) yields Bt (s; t; g(I0 ); m(I0 ); I0 ) = e

−

I

I0

t (s; x) d x=x

Bt (s; t; g(I); m(I); I)

(358)

with an analogous equation for Bu . As the renormalization scale I is arbitrary we choose I = t, hence Bt (s; t; g(I0 ); m(I0 ); I0 ) = e

−

t

I0

t (s; x) d x=x

Bt (s; t; g(t); m(t); t) :

(359)

Now, in the limit of large t and I there exists a “>xed point” wherein g(t) → g(∞) and t is purely a function of s, i.e. the total dependence of t on I through g(I), m(I) and I disappears. In that the anomalous dimension, t , depends continuously on s it might be better to speak of a “line” of >xed points as in the well-known case of the two-dimensional XY -model. The fact that the anomalous scaling dimension of Bt for I → ∞ is independent of I is a direct result of the term by term factorization in perturbation theory in this limit. Note that t will depend on g(∞), which is a non-universal parameter. In this sense the >xed point has more in common with the “in>nite mass”, or mean >eld >xed point in critical phenomena. The fact that :uctuation corrections to g(t)

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and m(t) also vanish in the large-t limit lends further support to this analogy. For I and t suJciently large we have that s g(t) m(t) g2 (t) t (s) ; Bt (s; t; g(I); m(I); I) = ; (360) Ft ; −I I t t 1=2 t 1=2 where (s) = −1 − t (s) and Ft is a three-variable scaling function. Note that we have removed a factor g2 (t)=t from Bt . As a consequence of the irrelevance of the operators associated with g(t) and m(t) with respect to the t → ∞ “>xed point”, the t-dependence of the scaling function Ft disappears to leave an amplitude Gt ((s=g(∞); s=m(∞)), i.e. in terms of momentum variables the amplitude is only a function of s. 22 Just as the “critical” exponent, , depends on a continuous parameter, s, so too does the “critical amplitude” Gt . Thus using the RG we can recover the generic form expected via Regge theory. To obtain the correct ratio of t- and u-contributions, we have to consider the role played by the Z factors. Let us >rst recall that it is the bare function BB; t that forms part of the scattering amplitude. According to the above, it is given by t (s) Gt (s) g2 (t) BB; t (s; t; gB ; mB ; ) = ; (361) −I Zt (s; g(I); m(I); I) I where we have reexpressed Zt in terms of the renormalized parameters. The factor Zt−1 is obviously necessary to cancel the unphysical I-dependence. Mathematically, Zt is represented by a formal power series. To be able to extract the signature given by the ratio of u- and t-contributions, we have to make sure that Zu = Zt , at least in a formal sense. As we will see below and in the following section in more detail, this implies considering the large-u limit for the u-contributions. Proceeding in exactly the same way as for the t-contributions above (just replacing t by u everywhere) we >nd for the function Bu Bu (s; u; g(I); m(I); I) = e−

u I

u (s; x) d x=x

Bu (s; u; g(u); m(u); u) :

(362)

Once again in the limit of large u and I for >xed s, u is independent of I and the corresponding scaling function Fu → Gu (s). To one loop in the large-t limit, as we will see, t = u and Gt = Gu . It is apparent that this holds to all orders in the case of the simple theory under consideration, since for every diagram that contributes to t and Gt there is a crossed diagram with the incoming or outgoing legs interchanged that contributes identically in the large-u limit to u and Gu . We can now replace u by −t, to be interpreted as eiA t (see below), and add the t- and ucontributions to >nd (with Z ≡ Zt = Zu and similarly for G) g2 (t) G(s)(1 + eiA(s) ) t (s) 2˜ B (s; t) = + Bs (t; u) ; (363) −I Z(s; g(I); m(I); I) I where Bs (t; u) contains the >nite (in the large-t limit) s-contributions. Rather than thinking of calculating Z in the denominator we should use an experimental result on the two-point scattering amplitude at some value of t for >xed s to determine it. Thus we see it is possible using the RG 22

It is instructive to verify the statements made in this general discussion explicitly for the full crossover function in theory presented in Ref. [52]. 3

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to produce signatured amplitudes. In the present case as we are dealing with identical bosons only positive signature states enter. Note that in the above we are only considering the leading Regge trajectory associated with the tree level value (gB → 0) = −1. There are, as mentioned, subleading trajectories associated with = −n, n¿1. One can in fact regard them as corrections to the dominant scaling given by the leading Regge trajectory. In principle the RG techniques we develop here are capable of accessing these “correction to scaling” trajectories as well. The problem of achieving it is the rather technical one of being able to project out from the Feynman diagrams the contributions of O(t −n ) and their associated logarithms. This can be done by analysing the Mellin transforms of the diagrams. It should be clear that our renormalization is completely crossing symmetric. If we consider the limit s → ∞ at >xed t one identi>es the function Bs (s; t) such that it contains large logarithms of the type ln(−s) which will subsequently require a multiplicative renormalization. Similarly, the function Bu (t; u), this time in distinction to the function Bt (s; u), will contain large logarithms of the type ln s. A renormalization procedure totally analogous to the above will yield g2 (s) G(t)(1 + eiA(t) ) s (t) 2˜ B (s; t) = + Bt (s; u) : (364) −I Z(t; g(I); m(I); I) I The procedure is identical in the large-u limit. Naturally in the di=erent asymptotic limits the diagrams that contribute to the renormalization are di=erent. For instance, a three-rung ladder diagram in the s-direction in the large-s limit ∼ gB6 s−2 K (t) ln s, whereas in the large-t limit it is ∼ gB6 t −1 K 2 (s)(ln t)2 . On the contrary, a three-rung ladder in the t-direction varies asymptotically as ∼ gB6 s−1 K 2 (t)(ln s)2 for large s and as ∼ gB6 t −2 K (s)K (s) ln t for large t. Thus a three-rung s-ladder contributes to the leading Regge trajectory in the large-t limit but to a subleading trajectory in the large-s limit. Renormalization then is capable of accessing the dual asymptotic limits, t → ∞ and s → ∞, characteristic of the Veneziano formula and string-like behavior. To calculate explicitly to one loop the scattering amplitude one needs to >x the renormalization constant Zt by choosing a normalization condition. We choose Bt (s; t = I; gB ; m; I) =

gB2 ; −I

(365)

whereby all :uctuations in the large-t limit are absorbed into the renormalization factor Zt and then exponentiated. It is thus an optimum condition in terms of obtaining the maximum amount of non-perturbative information in the large-t limit. We renormalize the mass as in (348), now for 2˜ B . Note that we have chosen not to implement a coupling constant renormalization. At one loop, the large-t limit of BB; t is as given in (353), where we have to interpret the −t as − e iA t, because the continuation from t on the upper edge of the threshold cut to eiA t in the Euclidean region has to yield a real function there (for s¡0). With the normalization condition (131) one >nds for large t and I Zt (s; I) = 1 − gB2 K(s) ln(e−iA I) ;

(366)

t (s) = −gB2 K(s) :

(367)

hence

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Using the normalization condition again, the result for Bt is g2 t (s) ; Bt (s; t; gB ; m; I) = B −I I

537

(368)

where the Regge trajectory, (s), is given by (s) = gB2 K(s) − 1 :

(369)

As for the u-contributions, we can take the perturbative expression (355) with t = e−iA u and proceed as above replacing t by u everywhere. The corresponding normalization condition reads Bu (s; u = I; gB ; m; I) =

gB2 : −I

(370)

Then clearly Zu = Zt , and in particular we >nd the same expression for the Regge trajectory. We then have to continue from u on the upper edge of the u-threshold cut to u = eiA t, giving the correct phase factor for the u-contributions. Note that a di=erent prescription to determine the u-contributions would have led to Zu = Zt , so that the phase factor would have been hidden in the (undetermined) ratio Zu =Zt . The >nal result for 3 theory in the large-t limit is g2 1 + eiA(s) t (s) g2 (371) + 2B : 2˜ B (s; t) = B −I Z(s; I) I m −s Only the function Z(s; I) in the above is not perturbatively well de>ned. Within the normal “philosophy” of renormalization this means that one has to measure the two-point scattering amplitude at a >ducial scale for t and at a >xed value of s. This will >x the value of Z for that value of s. The two-point scattering amplitude is then explicitly calculable for all other values of t. If one wishes to know what happens at another value of s then an appropriate experiment must be carried out at that value of s for a >ducial value of t, whereupon once again one can calculate the amplitude at all other asymptotic values of t. Note that here a one-parameter family of “initial conditions”, as functions of s, must be supplied for the RG. This could be circumvented by implementing two RGs: one associated with t and another with s. For s = 0 one >nds the Regge intercept (0) = −1 + gB2 =(16A2 m2 ). For gB2 ¿32A2 m2 the resulting cross-section will violate the Froissart bound. However, at two loops just by dimensional analysis one can see that a contribution to the Regge trajectory of the form gB4 =m4 will arise. Clearly the unitarity violating region is beyond the reach of perturbation theory. An analysis of this region requires a further renormalization—of the Regge trajectory itself. After this second renormalization (0) may continue to violate the unitarity bound. However, (0)¿0 implies that (s) = 0 at a negative value of s corresponding to a tachyonic “bound state”. This indicates the presence of a vacuum condensate necessitating a shift of the perturbative to the true vacuum of the theory. The above methodology is generalizable to much more complicated cases. Interestingly, even the simple case of an interaction † leads to a quite complicated renormalization scenario, where a matrix renormalization is necessary [53] which leads to an extremely rich set of Regge trajectories. As Regge trajectories also give important information about the bound states of a theory the results of this section show that it is possible to use an environmentally friendly renormalization to access some aspects of the crossover between boundstates and unbound states.

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14. Conclusions and comments The principal theme of this review has been the use of RG techniques in the treatment of crossover problems. As emphasized crossovers are ubiquitous, hence to make the problem more manageable we have concentrated on those that may be described >eld theoretically and have restricted ourselves thereby to the description of “near”-critical systems and quantum >eld theory. However, this restriction still leaves a vast array of phenomena to be considered. Crossovers are intrinsically “infrared” phenomena in that they involve a characteristic scale, g. Momentum scales far removed from g in the UV are insensitive to it and therefore insensitive to the crossover. A paradigmatic crossover involves more than one asymptotic scaling regime as a function of an anisotropy variable, or “environmental” parameter, g. If both g−1 and g are much larger than any microscopic length scale then the entire crossover may be universal. As a function of x = gg an isotropic or anisotropic scaling regime can be accessed in the limits x → 0 and ∞, respectively. Scaling functions, f(x), that interpolate between these di=erent regimes exhibit more than one singularity, or non-analyticity; in distinction to the case of a single scaling regime. In RG terms this is, in fact, the nub of the matter: In contrast to the case of critical exponents, where a linearized analysis around a single RG >xed point suJces, the calculation of crossover scaling functions generically requires a global, non-linear RG capable of encompassing two or more >xed points. This is particularly challenging in that standard approximation techniques, such as perturbation expansions in or 1=N , will not work. For instance, in problems where the upper critical dimension or the symmetry of the order parameter change, respectively. It is well worth emphasizing how primitive the state of the art is in calculating crossover scaling functions relative to calculating simpler universal quantities such as exponents or amplitude ratios, even in the simplest of cases such as the crossover to mean >eld theory or the standard equation of state. This has been emphasized in several of the reviews in this series [30,14] and stands as a clear challenge to present and future workers in the >eld. We discussed, though by no means exhaustively, the di=erent RG approaches that have been implemented to study crossover behavior, drawing a broad distinction between “coarse-graining” and “reparametrization”-type RGs. Although quite di=erent in their conceptual framework, in terms of concrete calculations the two broad branches of RG theory very often lead to similar, if not identical, formulations; especially when implemented diagramatically. We discussed extensively how accessing a crossover successfully required an “environmentally friendly” RG, in that the RG should depend on the environmental parameter that induces the crossover. We elaborated on how coarse graining RGs, such as block spinning, are naturally environmentally friendly but highlighted some of the pitfalls of such methodologies. We highlighted the advantages of reparametrization RGs and presented in some detail a reparametrization-based RG methodology—“environmentally friendly” renormalization—that we believe o=ers the best chance for extending the success of perturbative >eld theoretic RG-based calculations of exponents and amplitude ratios to that of crossover scaling functions. We concentrated on a generic class of crossover phenomena, within the context of an O(N ) Landau–Ginzburg–Wilson Hamiltonian, while emphasizing that the concepts and techniques we developed were equally applicable outside of this framework. Environmentally friendly renormalization was seen to have several key elements. Some, such as g-dependent counterterms, are closely allied to elements that appear in other methodologies, such

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as GMS. Others, such as integrating along thermodynamic contours, in order to obtain critical temperature shifts and rounding, we believe are novel to our approach. We showed how the ultimate building blocks for the calculation of scaling functions in the class of crossover problems we considered are the Wilson functions ’ , ’2 and associated with renormalizations of the transverse vertex functions. We saw that a suitable choice of normalization condition allowed for a maximal exponentiation of the perturbation theory. In conjunction with integrations along constant ’V or t contours in the phase diagram we saw how, basically, any thermodynamic scaling function could be reliably calculated. This covered not only the free energy and the equation of state but a set of e=ective exponents and, interestingly, critical temperature shifts and rounding both in the disordered and ordered phases. An important question in the calculation of crossover functions, as in the case of critical exponents, is: can the functions be calculated in a systematic approximation scheme to higher orders? Given that neither nor 1=N expansions are globally valid, the perturbative paradigm to which environmentally friendly renormalization reduces at the asymptotic ends of the crossover is the >xed dimension expansion. As is well known, in this case, given that often the >xed point coupling is not “small”, a resummation technique must be implemented at higher orders. The machinery for this is both well known and tried and trusted. As it is a separatrix that is analogous to a >xed point in a crossover, environmentally friendly renormalization takes as expansion parameter the separatrix crossing, or more generally, the solution of the beta function for the coupling. As in the non-crossover case a resummation technique should generically be implemented. We have shown how this may be done in the context of a [2,1] PadZe resummation of two-loop environmentally friendly RG-improved perturbation theory. Other techniques may be used. By construction, the results of environmentally friendly renormalization, implemented to a given loop order and with a given resummation technique, are guaranteed, asymptotically at the ends of the crossover, to reduce to the corresponding results of the >xed dimension expansion. In this sense, the only apparent barrier preventing the calculation of crossover scaling functions to the same degree of accuracy as critical exponents is the calculation of the corresponding diagrammatic series for the Wilson functions. Although conceptually this is simple, given that for a crossover the Feynman diagrams are functions rather than numbers, it is tedious and somewhat complicated, both algebraically and numerically. We exempli>ed the eJcacy of the basic formalism by both one and two-loop calculations of diverse scaling functions in various crossover problems. Our paradigmatic problem was taken to be dimensional crossover and, in particular, >nite size scaling. Although, the potential of environmentally friendly renormalization, and related RG techniques, is clear, once again, we wish to emphasize that this is a very open >eld where what is not known is far greater than what is known. In particular, we wish to delineate some problem areas which we think o=er particular challenges to RG techniques in the >eld theoretic setting. Although we have divided the problem areas into four di=erent categories there are many problems that cut across these boundaries. For instance, the crossover between bound and unbound states naturally occurs as a function of certain momentum invariants and hence also falls into the category of Asymmetric Momentum RGs. (i) Functional RGs. A paradigmatic example of this is a semi-in>nite system with boundary. This problem serves as a testing ground for ideas and techniques for even more challenging problems such as wetting and a description of the formation of non-translationally invariant systems such as crystals or the Abrikosov :ux lattice. The problem of >eld theory with a boundary is of interest

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in both statistical systems and in relativistic >eld theory. In real statistical systems the presence of boundaries is unavoidable. In relativistic quantum >eld theory there are two di=erent areas where a boundary plays an important role. The >rst is in the Casimir e=ect, where the force due to quantum :uctuations, in the presence of a wall, is a measurable e=ect. The second is in the SchrTodinger picture of relativistic >eld theory. The functional integral is now not over all time but over time up to the present, in other words it involves the presence of a boundary and will give the functional wavefunction in this picture. In the example of a semi-in>nite system physical properties of the system are sensitive to the distance, z, from the boundary. At large distances relative to the correlation length bulk d-dimensional physics should dominate while close to the boundary the physics should appear to be (d − 1) dimensional. Thus, there may occur a dimensional reduction. Contrary to the case of periodic boundary conditions however in this case the dimensional crossover is naturally position dependent. This implies that an environmentally friendly RG description, whether it appears in the context of a coarse-graining or a reparametrization RG, should be position dependent. The resulting RG equations will then be functional equations. The diJculties of dealing with functional RG equations, such as looking for >xed points of the beta function equations, are immense. (ii) Asymmetric momentum RGs. The physics of systems as a function of momenta, both in critical phenomena and in quantum >eld theory, have traditionally been associated with RGs that depend on only one momentum invariant, e.g. momentum associated with the symmetric point. However, there are many phenomena that cannot be easily accessed by RG methods with this restriction due to the fact that they occur as a function of very asymmetric ratios of momenta. The paradigmatic example we considered was that of physics in the Regge limit. We showed that an environmentally friendly renormalization capable of accessing this limit introduced several novel features such as having to renormalize subparts of Greens or vertex functions rather than the functions themselves. This extreme kinematic limit also appears naturally in stellar collapse and black hole formation and, importantly, in turbulence where it has been identi>ed as a possible mechanism for anomalous scaling. We believe that to a large degree the lack of progress in attacking turbulence using RG techniques is due to the use of symmetric rather than asymmetric momentum RGs. (iii) Systems with more than one diverging length scale. One of the key elements of the RG methodology is to use it to map to a region of parameter space where a reliable calculation may be carried out to one where standard approximation techniques are invalid. Physically, this often entails mapping from a region with a diverging length scale, such as the correlation length near a critical point, to a region where the length scale is small in an appropriate sense. A particular choice of sliding scale, or number of iterations of the RG map for a Wilsonian RG, is made to achieve this. In the case where a system may exhibit more than one diverging length scale however this arti>ce may become somewhat problematical as a matching can be made to one of the scales but the others are left as potentially dangerous. In the context of a reparametrization RG such problems can in principle be attacked by implementing more than one RG. The principal advantages of using more than one RG are: access to complementary information, as shown in the example of comparing >nite temperature >eld theory with a running >nite temperature mass RG and with a running temperature RG; less physical input and therefore more predictive power. This was illustrated in the case of >nite temperature QCD where with two RG’s speci>cation of the coupling at one momentum and one temperature was suJcient for the RG to be able to calculate the coupling at any other momentum or temperature. In the case of one RG it would be necessary to have a line of initial conditions; more :exibility in >nding a perturbatively treatable region of parameter

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space as in the case of a surface=bulk crossover where the bulk and surface correlation lengths may diverge separately. (iv) Crossover between bound states and unbound states. A description of QCD in the IR starting with the QCD Lagrangian in terms of quarks and gluons remains an unsolved problem. This is principally due to the fact that the e=ective degrees of freedom in the IR are baryons, mesons and glueballs—bound states of gluons and quarks. This type of crossover in the e=ective degrees of freedom also appears in other important problems such as superconductivity. In terms of a coarse-graining RG what is required is an RG that coarse grains bound states at low energies and their constituents at high energies. As of yet such an RG has not been developed. For a reparametrization RG we saw that some progress could be made by exploiting the information on bound states contained in the Regge limit. A direct access, however, remains very much an open problem. We believe these di=erent problem categories describe particularly challenging arenas for the application of RG techniques, progress in which would lead to signi>cant developments in many important areas of application. We hope that such progress will be forthcoming in the near future. Acknowledgements We are grateful to many colleagues with whom we have discussed at length the issues of this paper over the years. We especially thank JosZe Antonio Santiago for help with preparing the graphs. This work was supported by CONACyT grants 32729-E, 32399-E and 30422-E. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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547

CONTENTS VOLUME 363 J.K. Basu, M.K. Sanyal. Ordering and growth of Langmuir–Blodgett ﬁlms: X-ray scattering studies G.E. Brown, M. Rho. On the manifestation of chiral symmetry in nuclei and dense nuclear matter

1 85

S. Stenholm. Heuristic ﬁeld theory of Bose–Einstein condensates

173

D. O’Connor, C.R. Stephens. Renormalization group theory in the new millennium. IV

219

J. Berges, N. Tetradis, C. Wetterich. Non-perturbative renormalization ﬂow in quantum ﬁeld theory and statistical physics

223

D. Kreimer. Combinatorics of (perturbative) quantum ﬁeld theory

387

D. O’Connor, C.R. Stephens. Renormalization group theory of crossovers

425

Contents of volume 363

547

Forthcoming issues

548

PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 1 0 2 - 3

548

FORTHCOMING ISSUES* T. Yamazaki, N. Morita, R. Hayano, E. Widmann, J. Eades. Antiprotonic helium C.A.A. de Carvalho, H.M. Nussenzveig. Time delay C. Chandre, H.R. Jauslin. Renormalization-group analysis for the transition to chaos in Hamiltonian systems G. Baur, K. Hencken, D. Trautmann, S. Sadovsky, Yu. Kharlov. Coherent gg and gA interactions in very peripheral collisions at relativistic ion colliders R. Durrer, M. Kunz, A. Melchiorri. Cosmic structure formation with topological defects T. Peitzmann, M.H. Thoma. Direct photons from relativistic heavy-ion collisions H. Raﬁi-Tabar, A. Chirazi. Multi-scale computational modelling of solidiﬁcation phenomena R.N. Lee, A.L. Maslennikov, A.I. Milstein, V.M. Strakhovenko, Yu.A. Tikhonov. Photon splitting in atomic ﬁelds S. Chang, V. Minogin. Density-matrix approach to dynamics of multilevel atoms in laser ﬁelds

*The full text of articles in press is available from ScienceDirect at http://www.sciencedirect.com. PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 1 0 3 - 5

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