Physics Reports vol.340

L.S. Brown, L.G. Yawe / Physics Reports 340 (2001) 1}164

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EFFECTIVE FIELD THEORY FOR HIGHLY IONIZED PLASMAS

Lowell S. BROWN, Laurence G. YAFFE Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 340 (2001) 1}164

E!ective "eld theory for highly ionized plasmas Lowell S. Brown*, Laurence G. Ya!e Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA Received March 2000; editor: E. Brezin Contents 1. Introduction and summary 1.1. Relevant scales and dimensionless parameters 1.2. Utility of the e!ective theory 2. Classical Coulomb plasmas 2.1. Functional integral for the classical partition function 2.2. Mean "eld theory 2.3. Loop expansion 2.4. Particle densities 2.5. Loop expansion parameter 2.6. Thermodynamic quantities 2.7. Density}density correlators 2.8. Charge correlators and charge neutrality 3. E!ective "eld theory 3.1. Quantum theory 3.2. Classical limit 3.3 Induced couplings 3.4 Renormalization 3.5 Matching 3.6 Non-zero frequency modes 4. Two-loop results 4.1. Number density 4.2. Energy density 4.3. Pressure and free energy density 4.4. Number density correlators 5. Three-loop thermodynamics 5.1. Binary plasma 5.2. One-component plasma 6. Higher orders and the renormalization group

4 7 13 20 22 26 29 31 33 33 35 40 41 41 45 46 49 52 59 63 65 66 66 66 70 76 77 79

6.1. Renormalization group equations and leading logs 6.2. Leading logs to all orders 6.3. &&Anomalous'' virial relation 7. Long distance correlations Acknowledgements Appendix A. Functional methods A.1. Connected generating functional A.2. E!ective action A.3. E!ective potential, thermodynamic quantities A.4 Time-dependent correlations Appendix B. Green's functions and determinants Appendix C. Required integrals C.1 Coulomb integrals C.2 Debye integrals Appendix D. Quantum Coulomb su(1, 1) symmetry exploited D.1 Coulomb su(1, 1) symmetry D.2 Direct contribution D.3 Exchange contribution Appendix E. First quantum correction to classical one-component plasma Appendix F. Some elements of quantum "eld theory F.1 Perturbation theory F.2 Straightforward expansions F.3 Loop parameter Appendix G. Calculations using functional method G.1 Results through one loop G.2 Two-loop e!ective action References

* Corresponding author. 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 6 8 - 5

79 83 88 93 100 100 102 103 108 112 115 125 125 127 134 136 139 144 147 151 151 153 155 156 157 160 163

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Abstract We examine the equilibrium properties of hot, non-relativistic plasmas. The partition function and density correlation functions of a plasma with several species are expressed in terms of a functional integral over electrostatic potential distributions. This is a convenient formulation for performing a perturbative expansion. The theory is made well-de"ned at every stage by employing dimensional regularization which, among other virtues, automatically removes the unphysical (in"nite) Coulomb self-energy contributions. The leading order, "eld-theoretic tree approximation automatically includes the e!ects of Debye screening. No further partial resummations are needed for this e!ect. Subleading, one-loop corrections are easily evaluated. The two-loop corrections, however, have ultraviolet divergences. These correspond to the short-distance, logarithmic divergence which is encountered in the spatial integral of the Boltzmann exponential when it is expanded to third order in the Coulomb potential. Such divergences do not appear in the underlying quantum theory } they are rendered "nite by quantum #uctuations. We show how such divergences may be removed and the correct "nite theory obtained by introducing additional local interactions in the manner of odern e!ective quantum "eld theories. We compute the two-loop induced coupling by exploiting a non-compact su(1, 1) symmetry of the hydrogen atom. This enables us to obtain explicit results for density}density correlation functions through two-loop order and thermodynamic quantities through three-loop order. The induced couplings are shown to obey renormalization group equations, and these equations are used to characterize all leading logarithmic contributions in the theory. A linear combination of pressure plus energy and number densities is shown to be described by a "eld-theoretic anomaly. The e!ective Lagrangian method that we employ yields a simple demonstration that, at long distance, correlation functions have an algebraic fall o! (because of quantum e!ects) rather than the exponential damping of classical Debye screening. We use the e!ective theory to compute, easily and explicitly, this leading long-distance behavior of density correlation functions. The presentation is pedagogical and self-contained. The results for thermodynamic quantities at three-loop [or O(n)] order, and for the leading long-distance forms of correlation functions, agree with previous results in the literature, but they are obtained in a novel and simple fashion using the e!ective "eld theory. In addition to the new construction of the e!ective "eld theory for plasma physics, we believe that the results we report for the explicit form of correlation functions at two-loop order, as well as the determination of higher-order leading-logarithmic contributions, are also original.  2001 Elsevier Science B.V. All rights reserved. PACS: 52.25.Kn; 11.10.Wx Keywords: Non-relativistic plasma thermodynamics; E!ective "eld theory

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1. Introduction and summary Our work applies contemporary methods of e!ective quantum "eld theory to the traditional problem of a multicomponent, fully ionized hot (but non-relativistic) plasma. In this regime, a classical description might appear to su$ce. But the short-distance (1/r) singularity of the Coulomb potential gives rise to divergences in higher-order terms. Taming these divergences requires the introduction of quantum mechanics. Quantum #uctuations smooth out the shortdistance singularity of the Coulomb potential so that the quantum, many-particle Coulomb system is completely "nite. This necessity for including quantum e!ects, even in a dilute plasma, is discussed later in this introduction when the relevant parameters which characterize the various physical processes in the plasma are examined. As we shall see, contemporary e!ective quantum "eld theory methods simplify high-order perturbative computations and generally illuminate the structure of the theory. E!ective quantum "eld theories do, however, utilize a somewhat complicated formal apparatus involving regularization, counter terms, and renormalization. In an e!ort to make our work available to a wider audience, we shall develop the theory in several stages, and attempt to give a largely self-contained presentation. A brief review of some of the basic quantum "eld theory techniques used in our paper is presented in Appendix F. We begin, in Section 2, by casting the purely classical theory in terms of a functional integral. We show how dimensional continuation is convenient even at this purely classical level because it automatically and without any e!ort removes the in"nite Coulomb self-energy contributions of the particles. The simple saddle-point evaluation of the functional integral } known as a tree approximation in quantum "eld theory } immediately gives Debye screening of the long-distance Coulomb potential without any need for the resummations appearing in traditional approaches. The "rst sub-leading, so-called `one-loop,a corrections to the plasma thermodynamics and correlation functions are also evaluated in this section. These lowest-order results are also used to illustrate general relations among correlations functions which are described more formally, and systematically, in Appendix A. The divergence associated with the singular, short-range behavior of the Coulomb potential "rst arises at the subsequent, `two-loopa level of approximation as shown in Section 3. This section explains how the previous purely classical theory is obtained from a limit of the quantum theory, and how the quantum corrections that tame the classical divergences appear in the form of induced couplings that contain compensating divergences. This discussion uses various results on functional determinants and Green's functions contained in Appendix B. Although it is relatively easy to construct the `counter termsa that render the classical theory "nite, it is considerably more di$cult to obtain the "nite pieces in the induced couplings that ensure that a calculation in the e!ective theory correctly reproduces the corresponding result in the full quantum theory. In  Other discussions of e!ective "eld theory techniques, applied to quite di!erent physical problems, may be found in Refs. [1,2] and references therein.  The saddle point of the functional integral de"nes the `mean-"elda solution. The order of the perturbative expansion about this saddle point is commonly referred to as the `loopa order in the quantum "eld literature because contributions at the kth order in perturbation theory are graphically represented by k-loop diagrams. As will become clear later, in thermodynamic quantities such as the pressure, contributions of k-loop order correspond to terms which formally depend on the mean particle density n as O(n>I), up to logarithms of the density.

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the latter part of this section the `matching conditionsa for the leading two-loop induced couplings are derived, and the two-loop induced couplings are explicitly evaluated. The key to this evaluation is the exploitation of an su(1, 1) symmetry of the Coulomb problem, which permits one to derive a simple and explicit representation for the two-particle contribution to the density}density correlation function. This su(1, 1) symmetry, and its consequences, are presented in Appendix D. Section 3 concludes with an examination of the necessary inclusion in the e!ective theory of interactions involving non-zero frequency components of the electrostatic potential. It is worth noting that our determination of the induced couplings is based on examining Fourier transforms of number-density correlation functions at small but non-vanishing wave number. We use this method because these functions } at non-vanishing wave number } may be computed in a strictly perturbative fashion without taking into account the Debye screening that is necessary to make the zero wave number limit of these correlation functions "nite. Matching in this fashion enables us to use the simple pure Coulomb potential for which the exact group-theoretical techniques apply. Our procedure is roughly equivalent to computing the second-order virial coe$cient for a pure Coulomb potential, except that this coe$cient has a long-distance, infrared divergence. This logarithmic divergence is removed by Debye screening, but there is always the di$culty of determining the constant under the logarithm. Our method avoids this di$culty. Years ago, W. Ebeling [and later Ebeling working together with collaborators] computed the secondorder virial coe$cient for a pure Coulomb potential with a long-distance cuto!, and then related this quantity to other ladder approximation calculations so as to obtain results that are, except for one term, equivalent to, and consistent with, our results for the induced couplings. Their work is summarized in Ref. [4]. This seminal work is certainly very impressive and signi"cant, but it is much more complex than our approach, and (at least in our view) is far more di$cult to understand in detail. With the leading-induced couplings in hand, we turn in Section 4 to compute all the thermodynamic quantities and the density}density correlators to two-loop order. As far as we have been able to determine, the two-loop results for the density}density correlation functions obtained in Section 4 are new. Various integrals required for these computations are evaluated in Appendix C, and an alternative derivation of the two-loop thermodynamic results using compact functional methods appears in Appendix G. The thermodynamic results are extended to the next, three-loop, order in Section 5. We give complete, explicit results for the pressure (or equation of state), Helmholtz free energy, and internal energy, as well as the relations between particle densities and chemical potentials in a general multi-component plasma. We also display the specializations of the equation of state for the cases of a binary electron}proton plasma, and a one-component plasma (in the presence of a constant neutralizing background charge density). As discussed at the end of this section, a genuine classical limit exists only for the special case of a one-component plasma. As a check on our results, Appendix E presents an independent, self-contained calculation of the leading O( ) corrections to the equation of state of a one-component plasma in the semi-classical regime.

 The utility of matching at small but non-vanishing wave number, thereby enabling one to ignore the e!ects of Debye screening, has been emphasized by Braaten and Nieto [3].  This is a classic result which may also be found in [5].

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Prior results in the literature, corresponding to our three-loop [or O(n)] level of accuracy, for the free energy and/or the equation of state go back more than 25 years. The book by Kraeft Kremp, Ebeling and RoK pke [6] quotes a result for the Helmholtz free energy which is nearly correct but omits one term. A fairly recent publication by Alastuey and Perez [7] contains the complete, correct expression for the Helmholtz free energy, to three loop order, with which we agree. Recent papers by DeWitt, Riemann, Schlanges, Sakakura and Kraeft [8}10] report results for some, but not all, of the terms contained in the three-loop pressure. These partial results are consistent with our three-loop pressure, once an unpublished erratum of J. Riemann is taken into account. Just as in any e!ective "eld theory, the induced couplings that must be introduced to remove the in"nities of the classical plasma theory obey renormalization group equations. In Section 6 we show how these renormalization group equations may be employed to compute leading logarithmic terms in the partition function } terms involving powers of logarithms whose argument is the (assumed large) ratio of the Debye screening length to the quantum thermal wavelength of the plasma. We show that, in general, logI terms "rst arise at 2k-loop order. We derive a simple recursion relation which determines their explicit form. We explicitly evaluate the coe$cients of the logarithm-squared terms which arise at four and "ve loop order, and the log-cubed terms which appear at six-loop order. The existence of these higher powers of logarithms which "rst appear at four-loop order (where they correspond to n ln n contributions to the pressure) has often not been recognized. Recently, Ortner [12] has examined a classical plasma consisting of several species of positive ions moving in a "xed neutralizing background of negative charge. He computed the free energy to four-loop (or n) order and correctly obtained the n ln n term. Since Planck's constant, which carries the dimensions of action, does not appear in classical physics, fewer dimensionless ratios can be formed in a classical theory than in its quantum counterpart. In particular, the partition function of the classical theory depends upon a restricted number of dimensionless parameters, from which a linear relation between the pressure, internal energy, and average number densities follows. This relation is altered by the necessary quantummechanical corrections. Section 6 also explains how this alteration of the linear relationship is connected to `anomaliesa brought about by the renormalization procedure that makes the classical theory "nite. We conclude our work, in Section 7, with an examination of the long-distance behavior of the density}density correlation function. Despite the presence of Debye screening, it is known that quantum #uctuations cause correlations to fall only algebraically with distance [14}17]. Using the e!ective theory, we compute the coe$cient of the resulting leading power-law decline in particleand charge-density correlators in a very simple and e$cient fashion, and obtain results which agree with previously reported asymptotic forms [16,17]. It should be emphasized that the major purpose of this paper is to introduce the methods of modern e!ective "eld theory into the traditional "eld of plasma physics. Although many of the results that we derive and describe have been obtained previously, the methods that we employ to obtain these results are new, and they substantially reduce the computational e!ort as well as  See Eqs. (2.50)}(2.55) of Ref. [6]. See Footnote 54 for details. The missing `quantum di!ractiona term noted in that footnote was obtained for the electron gas in a neutralizing background by DeWitt. See Ref. [13] and references therein.  For example, Refs. [7,11] imply that the next correction is of the form n ln n rather than the correct n ln n.

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illuminating the general structure of the theory. Although our work may have the length, it is not a review paper; its length results from our desire to make the presentation self-contained so that it may be read by someone who is neither an expert in plasma physics nor quantum "eld theory. Since our work is not a survey of the "eld, we have not endeavored to provide anything resembling a comprehensive bibliography. For a recent review of known results concerning Coulomb plasmas at low density, including rigorous theorems and detailed discussion of the long distance form of correlators, we refer the reader to Ref. [18] and references therein. 1.1. Relevant scales and dimensionless parameters Various dimensionless parameters characterize the relative importance of di!erent physical e!ects in the plasma. Before plunging into the details of our work, we "rst pause to introduce these parameters and discuss their signi"cance. Let e and n denote the charge and number density of a typical ionic species in the plasma. For simplicity of presentation in this qualitative discussion, we shall assume that the charges and densities of all species in the multicomponent plasma are roughly comparable, and shall ignore the sums over di!erent species which should really be present in formulas such as (1.2) below. The subsequent quantitative treatment will, of course, remedy this sloppiness. We shall be concerned with neutral plasmas which are su$ciently dilute so that the average Coulomb energy of a particle is small compared to its kinetic energy. We use energy units to measure the temperature ¹ and write b"1/¹. In the ideal gas limit, the average kinetic energy is equal to ¹. The Coulomb  potential is e/(4pr) in the rationalized units which we shall use. So the typical Coulomb energy is e/(4pd) where d,n\ denotes the mean inter-particle separation. Hence, the dimensionless parameter be e " n C, 4p 4pd¹

(1.1)

is essentially the ratio of the potential to kinetic energy in the plasma, and it is an often used measure of the relative strength of Coulomb interactions in a plasma. However, we shall see that C is not the proper dimensionless parameter which governs the size of corrections in the classical perturbation expansion. A charge placed in the plasma is screened by induced charges. The screening length equals the inverse of the Debye wave number which we denote as i. It is given (to lowest order in a dilute plasma) by i"ben .

(1.2)

 During the "nal preparation of this report, we became aware of a paper by Netz and Orland [19] that employs a functional integral representation for a classical plasma which is similar in spirit to our formulation. That paper considers only the special cases of one-component, or charge symmetric two-component plasmas and, moreover, does not address the inclusion quantum e!ects which we deal with using e!ective "eld theory techniques.

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A di!erent measure of the strength of Coulomb interactions in the plasma is de"ned by bei g" . 4p

(1.3)

This is the ratio of the electrostatic energy of two particles separated by a Debye screening length to the temperature (which is roughly the same as the average kinetic energy in the plasma). Equivalently, it is ratio of the `Coulomb distancea be d , ! 4p

(1.4)

to the screening length i\. The Coulomb distance d is the separation at which the electrostatic ! potential energy of a pair of charges equals the temperature. The number of particles N contained within a sphere whose radius equals the screening length G i\ is inversely related to g, 1 4p N " i\n" . G 3g 3

(1.5)

Hence the weak coupling condition g;1 is equivalent to the requirement that the number of charges within a `screening volumea be large, N <1. In this case, a mean-"eld treatment of Debye G screening holds to leading order, and perturbation theory is a controlled approximation. It is easy to check that the two measures of interaction strength, g and C, are related by g"(4pC. However, we shall show in our subsequent development that g, not C, is the dimensionless parameter whose increasing integer powers characterize the size of successive terms in the classical perturbative expansion for thermodynamic properties of the plasma. As we shall discuss, the classical perturbation series has a convenient graphical representation in which contributions at nth order in perturbation theory are represented by graphs (or Feynman diagrams) with n loops. We shall see that g is the `loop expansiona parameter, such that contributions represented by n-loop graphs are of order gL. Although g and C are directly related as noted above, we emphasize again that it is g which is the correct classical expansion parameter. To bring out this point even more strongly, we note that the screened Debye potential between two charges e and e a distance r apart is given by e e e\GP/(4pr). The modi"cation of the ? @ ? @ self-energy of a particle of charge e when it is brought into the plasma is given by half the di!erence ? between the Debye potential and its Coulomb limit for the case of zero charge separation, (e/4pr)[e\GP!1], which is !ei/(8p). Each particle in the plasma makes this correction lim ?  P ? to the thermodynamic internal energy of the plasma, and so including this leading order correction to the energy density gives









3 3 bei ei u" ¹n ! ? n "¹ n ! ? , ? 2 2 ? 8p ? 8p ? ?

(1.6)

 Dynamically, the Coulomb distance d is also the impact parameter necessary for an O(1) change in direction to ! occur during the scattering of a typical pair of particles in the plasma.

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which shows the appearance of g, or more explicitly bei/(8p), as the correct Coulomb coupling ? constant in this case. This result for the internal energy which we have heuristically obtained agrees with the correct one-loop result (2.86) that is derived below. A classical treatment of a plasma with purely Coulombic interactions is, however, never strictly valid. The classical partition function fails to exist due to the singular short-distance behavior of the Coulomb interaction. This can be seen in an elementary fashion directly from the divergence, for opposite signed charges, of the Boltzmann-weighted integral over the relative separation of two charges,  (dr) exp+be/4pr,. In the perturbative expansion of the classical theory, this problem "rst manifests itself at two-loop order through the diagram

(1.7)

The three lines in this graph correspond to the three factors of the Coulomb interaction energy (e/4pr) that appear in the expansion of the Boltzmann exponential to third order. This graph represents a relative correction to the partition function of 







e  1 bei  dr 1 nb (dr) . " 4pr 3! 4p r 3!

(1.8)

Once screening e!ects are properly included, the large-distance logarithmic divergence of this integral will be cut o! at the classical Debye screening length i\. But no classical mechanism exists to remove the short-distance divergence of the integral. To tame this divergence, one must include quantum e!ects. The non-relativistic quantum-mechanical description of a charged plasma is completely "nite; quantum #uctuations cut o! the short-distance divergences of the classical theory. The de Broglie wavelength for a particle of mass m and kinetic energy comparable to the temperature is of order



j,

2pb . m

(1.9)

This is in accord with the average (rms) momentum of (3m/b for a particle in a free gas at temperature ¹"1/b. We will refer to j as the `thermal wavelengtha. This length sets the scale of the limiting precision with which a quantum particle in the plasma can be localized. Using the thermal wavelength as the lower limit in the integral (1.8), and the Debye length as the upper limit, one obtains a "nite result,



G\ dr "!ln(ji) , r H

 Note that this contribution is indeed of order g, in accordance with its origin as a two-loop graph.

(1.10)

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which replaces the in"nity that would otherwise arise in a purely classical treatment. This logarithm of the ratio of a quantum wavelength to the screening length will necessarily appear in coe$cients of two-loop (and higher order) contributions to all thermodynamic quantities. This quick discussion shows that quantum mechanics must enter into the description of the thermodynamics of a plasma } at least if two-loop or better accuracy is desired. In addition to regularizing the divergences of the classical theory, quantum mechanics also provides `kinematica corrections via the in#uence of quantum statistics. To estimate the size of these e!ects, we recall that for a free Bose (!) or Fermi (#) gas, the partition function is given by










(dp) p ln Z "Gg ln 1G exp !b !k 1 (2p ) 2m V

.

(1.11)

Here V is the volume containing the system, g "2S#1 is the spin degeneracy factor, k is the 1 chemical potential of the particle, and z,e@I

(1.12)

is the corresponding fugacity. The limit of classical Maxwell}Boltzmann statistics is obtained when !bk<1 so that the fugacity z;1. Near this regime, the logarithm in Eq. (1.11) may be expanded in powers of the fugacity, and the resulting Gaussian integrals then yield





ln Z z z "g j\z 1$ # #2 . 1 V 2 3

(1.13)

The corresponding number density de"ned by nV"* ln Z/*(bk) is given by





z z n"g j\z 1$ # #2 . 1 2 3

(1.14)

We shall always assume that the plasma is dilute, nj ;1 , (1.15) g 1 so that a fugacity expansion is appropriate. This condition that the plasma be dilute can be stated in another way. If all single-particle states in momentum space were "lled up to a (Fermi) momentum p , the density would take on the value n"g p /(6p ) corresponding to a non$ 1 $ interacting Fermi gas at zero temperature. The diluteness condition is equivalent to the requirement that the Fermi energy E "p /2m corresponding to the given density be small in comparison $ $  A one-component plasma (with an inert, uniform background neutralizing charge density) has only repulsive Coulomb interactions. In this special case, the Boltzmann factor exp+!be/4pr, itself provides a short-distance cuto! at the Coulomb distance d "be/4p, resulting in logarithmic terms of the form ln(d i)"ln g. [In this regard, see Eq. (3.48) ! ! and its discussion.] However, if the quantum thermal wavelength is larger than the Coulomb distance, j'd , then this ! purely classical removal of the would-be short-distance divergence is physically incorrect, for the quantum e!ects already come into play at larger distances, and the correct logarithmic term has the form ln(ji). The neutrality of a binary or multicomponent plasma requires that they have attractive as well as repulsive Coulomb interactions. These plasma thus always require quantum-mechanical #uctuations to remove their potential short-distance divergences.

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with the temperature,







9p  E b  6pn  $" " ;1 . (1.16) z 16 2m g ¹ 1 However, it is the fugacity z, not this ratio, that is the appropriate expansion parameter. Once quantum mechanics enters the analysis, another dimensionless parameter involving the ratio of two energies appears. This is the Coulomb potential energy for two particles separated by one thermal wavelength, divided by the temperature, be . g" 4pj

(1.17)

Recalling the de"nition (1.9) of the thermal wavelength and noting that the average (rms) particle velocity in a free gas is given by v "(3/bm, this ratio may equivalently be expressed as



g"

3 e . 2p 4p v

(1.18)

This parameter is also related to the ratio of temperature to binding energy of two particles in the plasma with equal and opposite charge e and reduced mass m. The hydrogenic ground state of two such particles has a binding energy of




e"

e  m . 4p 2 

(1.19)

The ratio of this energy to the temperature is just g (up to a factor of p), 1 g" be . p

(1.20)

Note that the quantum parameter g becomes small at su$ciently high temperature, but that it diverges at low temperatures or in the formal P0 or mPR limits. We should also remark that the quantum e!ects measured by g only appear in two-loop and higher-order processes. Thus these e!ects are suppressed by a factor of g. The quantum parameter g, together with the particle densities, also provides an estimate of how many bound atoms are present in a dilute plasma. The Saha equation, which is simply the condition for chemical equilibrium between bound atoms and ionized particles, states that the fraction of bound atoms in the plasma is nje@C"njepE .

(1.21)

 For the general case of opposite but unequal charges, e is replaced by the product of charges !e e . ? @  This is just the requirement that the chemical potential plus binding energy of the lowest bound state equal the sum of the chemical potentials of the bound state constituents. Since an atom in free space has an in"nite number of bound levels, and the presence of the surrounding particles in the plasma produces screening e!ects, the Saha equation only provides a rough indication of the numbers of bound atoms present. Indeed, the fraction of bound atoms in a plasma is intrinsically only an approximately de"ned concept.

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Here j refers to the thermal wavelength corresponding to the reduced mass of the two charges. Thus, for a dilute plasma to be (nearly) fully ionized, the parameter pg, for opposite signed charges, must be small compared to !ln nj. If the plasma is su$ciently dense that the Debye screening length becomes comparable to the size of isolated atoms, then the Saha equation } which neglects interactions with the plasma } breaks down. Such plasmas can remain essentially fully ionized, even when the Saha equation predicts a substantial number of bound atoms, because Debye screening shortens the range of attractive interactions and e!ectively prevents the formation of bound states. The perturbative treatment which we shall develop applies only to the case of well ionized plasmas. Underlying any e!ective "eld theory, such as the one that we develop in this paper, is a separation between the length scales of interest and the scales of the underlying dynamics. Our length scales of interest will be of order of the Debye screening length i\ or longer. The relevant microscopic scales are the Coulomb distance d "be/4p and the thermal wavelength j. The ! condition that the screening length i\ be much larger that the Coulomb distance d is just the ! statement that the classical loop expansion parameter must be small, d bei " ! ;1 . g" i\ 4p

(1.22)

As noted above, the thermal wavelength j will provide the short-distance cuto! in expressions, such as Eq. (1.10), which diverge in the purely classical theory. We assume that ji;1 ,

(1.23)

so that there is a large separation between the scales of interest and this short distance cuto!. The quantum theory will generate additional corrections suppressed by powers of (ji) which, since j is proportional to Planck's constant , represent an ascending series in powers of , in contrast to the ln e!ects arising from the short-distance cuto!. The diluteness parameter nj is not independent of our other dimensionless parameters since (ij) (ij) " , nj" 4pg (bei)

(1.24)

(ij) g (ij) " " . nj" 4pg 4pg (be/j)

(1.25)

or

In order to have a systematic expansion in which the size of di!erent e!ects can be easily categorized, we will treat the Coulomb parameter g"be/4pj as a number that is formally of order one. Consequently, if we regard ij as the basic small parameter which justi"es the use of an e!ective "eld theory, then g"bei/4p"g(ij) is O(ij), while the diluteness parameter nj is O[(ij)], thus formally justifying the inequalities g;1 and nj;1. The highly ionized plasma at the core of the Sun provides an example of astrophysical interest. This plasma is mostly composed of electrons and protons. We take the nominal values for the central temperature as ¹"1.5;10 K, and the electron and proton densities as n "n "5.0; C N 10/cm. Since this temperature is to be compared to atomic energies, electron volts are far more convenient units, with ¹"1.3 KeV. It is also convenient to think of distances and densities in

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terms of the atomic length unit, the Bohr radius a "5.3;10\ cm. Thus n "n "7.4/a . Since  C N  e/4pa "27 eV, and a "4p /m e, it is easy to "nd that the Debye wave number at the Sun's   C center is given by i"2.0/a and that the electron's quantum thermal wavelength is j "0.36a ,  C  with the proton wavelength a factor of (1840 smaller, j "8.4;10\a . Hence, at the center of the N  Sun, the classical loop expansion parameter is quite small, g"bei/4p"0.042. For the proton, be "2.4 , 4pj N so the inequalities ij;1 and nj;1 are also well satis"ed. For the electron, ij "0.017, N

n j"4.4;10\, N N

(1.26)

be "0.058 . (1.27) 4pj C While the proton fugacity is tiny, z "2.2;10\, the electron fugacity z "exp(bk )" N C C n j/2"0.17 is small but not insigni"cant, which means that the Fermi}Dirac correction to C C Maxwell}Boltzmann statistics for the electron are a few percent. Although the Saha equation predicts that there are 20% or so neutral hydrogen atoms in the core of the sun, this is wrong since the Debye screening length is half the Bohr radius. The core of the Sun is essentially completely ionized. The fact that ij is only slightly less than one means that the utility of the e!ective theory C (for describing electron contributions to the thermodynamics at the core of the Sun) cannot really be judged until one knows whether ij or, for example, ij/2p appears as the natural expansion parameter. And there is only one way to "nd out } one must compute multiple terms in the perturbative expansion and examine the stability of the series for the actual parameters of interest. ij "0.72, C

n j"0.35, C C

1.2. Utility of the ewective theory For a su$ciently dilute ionized plasma, all corrections to ideal gas behavior are negligible. As the plasma density increases, the leading corrections are very well known and come from either the inclusion of quantum statistics for the electrons or the "rst-order inclusion of Debye screening. At this `triviala level of e!ort, the resulting equation of state is easy to write down: n i bp "1# C [2\z #2(2\!3\)z#2]! . C C n 24pn n

(1.28)

Here n is the total particle density (ions plus electrons), and z is the electron fugacity, which is C related to the electron number density as shown in Eq. (1.14). The electron fugacity corrections just come from combining Eqs. (1.13) and (1.14) [and noting that in the thermodynamic limit bp"(ln Z)/V], while the Debye screening correction will be derived in Section 2 [Eq. (2.83)]. Since the ions are so much more massive than the electrons, their fugacity will be very small, and their quantum statistics corrections may be neglected. The e!ective theory we construct incorporates systematically higher-order interaction e!ects not contained in the trivial equation of state (1.28). In Sections 4 and 5 we will give explicit forms for the complete second- and third-order corrections to the equation of state expanded in powers of the loop expansion parameter g"bei/4p. These results are valid provided the temperature and

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density are not in a regime where (1) The electron density is so large that an expansion in electron fugacity is useless. This occurs when the electrons are nearly degenerate and their quantum degeneracy pressure becomes a dominant e!ect. (2) The temperature is so low that the loop expansion of the e!ective theory is useless. This happens when the plasma ceases to be nearly fully ionized. (3) The temperature is so high that a non-relativistic treatment is inadequate. This requires that the temperature be small in comparison with the electron rest energy of 511 KeV. As a concrete test of the utility of our e!ective theory, one may insert the numerical values of the density and temperature quoted above as characteristic of the solar interior (¹,1.3 KeV, n,15a\) into the third-order result (5.24) for the equation of state. Displaying  the "rst-, second-, and third-order corrections separately, one "nds that bp/n"1!0.00693#0.01429#0.00074#2 .

(1.29)

All corrections to the ideal gas limit are small, but the second-order correction is larger than the "rst. However, it is important to understand that our expansion of the e!ective theory is based on formally treating the Coulomb parameters g"be/4pj of all species as numbers of order one. As indicated in Eq. (1.25), this means that quantum statistics corrections proportional to the kth power of fugacity (or nj) are automatically included at 2k-loop order in the e!ective theory. For the solar plasma, because the electron fugacity is small, but larger than the plasma coupling g, the dominant correction to ideal gas behavior comes from quantum statistics, not from Debye screening. Consequently, a more instructive comparison is to examine the size of corrections generated by the e!ective theory after removing (or resuming) the non-interacting quantum statistics corrections. This comparison gives bp bp ! n n



"!0.006930!0.001516#0.000736#2 ,

(1.30)



where (bp/n) " denotes the equation of state for non-interacting particles, but with quantum  statistics for the electrons. Expanding in electron fugacity, as in (1.28), and inserting the same characteristic parameters gives bp n



"1#0.01581#0.00105#0.00010#2 .

(1.31)



Both the quantum statistics series (1.31), and the e!ective theory expansion (1.30) are now quite well behaved. For these parameter values, it appears that the three-loop e!ective theory result

 For this comparison, we assume that the plasma contains only protons and electrons. This is not realistic very near the center of the sun, where a signi"cant abundance of helium is also present.

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15

(1.30), combined with the "rst three terms in the fugacity expansion (1.31), will correctly predict the equation of state to within an accuracy of a few parts in 10. Missing from the above quantitative results, and from our analysis in subsequent sections, are relativistic corrections. The leading `kinematica relativistic e!ects may be obtained by inserting the relativistic kinetic energy E(p)"((pc)#(mc)!mc into the ideal gas partition function (1.11). The dominant e!ects come from the electrons, due to their small mass. Expanding E(p) in powers of momentum, one "nds that











¹  15 ¹ z 15 ¹ 2z ! C 1# #O(z)#O , (1.32) bp"n # C 1# C G m c 8 m c 2 16 m c j C C C C where n denotes the total density of ions. The electron density, n ,z *(bp)/*z , receives exactly G C C C the same ¹/m c correction. Hence, this correction (plus all further corrections to bp which are  C linear in z ) cancels in the equation of state. However, other thermodynamic quantities, such as C the internal energy, do receive relative O(¹/m c) relativistic corrections. For the equation of state, C the "rst relativistic correction which does contribute comes from the O(¹/m c) perturbation to the C O(z) quantum statistics term, and one "nds that C bp ¹ 15 n z C C " . (1.33) D  n 16 n 2 m c C For the characteristic solar parameters used above, this correction is less than a part in 10,





bp D "0.000036 .  n

(1.34)

A hot plasma also contains black body radiation. The contribution to the pressure arising from this photon gas is given by the familiar formula







bp p ¹ 1 p 2p¹  1 " " D  n nj 45 c n 45 m c C C ¹  1 " . 6.1 KeV na  For the solar parameters that we have adopted,




bp D "0.00063 ,  n

(1.35)

(1.36)

 Adding the quadratic electron fugacity correction [that is, the O(z) term in (1.28), or the 10\ term in (1.31)] to the C three-loop e!ective "eld theory result is entirely reasonable since this quantum statistics correction is in fact the dominant part of the complete four-loop contribution of the e!ective theory when the Coulomb parameter for the electron is small, be/4pj ;1, as it is in the Sun. The last term of Eq. (1.31) is the free-particle limit of the six-loop contribution in our C expansion of the e!ective theory. This cubic fugacity correction, for our characteristic solar parameters, makes only a 10\ correction to the equation of state.  We remind the reader that portions of the solar neutrino spectrum are exceptionally sensitive to the central temperature. So a very small change in the equation of state can potentially produce a measurable change in the solar neutrino #ux.

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which is the size of our third-order correction. The relative importance of this photon gas correction increases rapidly as the temperature is increased, and it must be included in some of the regions discussed at the end of this section. The transverse photons also interact with the charged particles to alter the thermodynamic relations. This e!ect is dominated by the coupling with the light electrons. It may be easily obtained by using the radiation gauge to compute the "rst-order perturbation arising from the `seagulla interaction Hamiltonian density (e/2m c)tRtA and taking the j ) A interaction to second order. C Since the current involves e*/c, one expects that the second-order j ) A contribution is suppressed by (v /c)&¹/m c relative to the `seagulla term. This is con"rmed by a detailed computation. C C A simple calculation expresses the (leading order in ¹/m c) `seagulla contribution as C ap ¹ e [1A(0)2 !1A(0)2 ]n "! n , (1.37) D (bp)"!b 2 2 C   3 m c C 2m c C C where a"e/(4p c)"1/1372 is the "ne structure constant. Note that the vacuum, or ¹P0, contributions are subtracted as they are completely absorbed by renormalization of the bare electron parameters. Since n "*(bp)/*(bk ), this correction is equivalent to a shift in the electron C C chemical potential of dk "!(p/3)(a¹/m c). It modi"es the chemical potential } electron C C density relation and thus has no e!ect on the equation of state. However, the correction does a!ect other thermodynamic quantities such as the internal energy. The leading corrections to the equation of state involving the interactions of transverse photons are actually of relative order az (¹/m c). One "nds that C C bp ¹  ap n z C C . (1.38) "! D   n 3 n 2 m c C For the characteristic solar parameters used above, this is utterly negligible even at the part in 10 level,










bp "1.5;10\ . (1.39) D   n C Depending on the mass and composition of a star, the electron fugacity in stellar interiors may be relatively small (as in the Sun), or may be large enough to completely invalidate a quasi-classical treatment (as in white dwarfs or very massive stars). Figs. 1}4 represent an attempt to delineate the region of validity of the e!ective theory in the temperature}density plane for the case of a pure Z"1 proton}electron plasma [Fig. 1], a pure Z"2 (ionized helium) plasma [Fig. 2], a pure Z"6 (ionized carbon) plasma [Fig. 3], and a pure Z"13 (ionized aluminum) plasma [Fig. 4]. The solid line shows where the second- and third-order corrections in the fugacity expansion for electrons become equal in size. This occurs before any of the individual "rst-, second-, or third-order fugacity corrections exceed unity, and provides a convenient signal that the fugacity expansion is no longer well-behaved. The dashed line shows where the size of e!ective "eld theory

 A recent paper [20] has attempted to argue that radiative corrections are far larger than this relative O(a¹/m c) C e!ect. The conclusions of this paper are not correct.

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Fig. 1. Region of validity of the e!ective theory for the case of a pure Z"1 ionized hydrogen plasma. On the bottom axis, density denotes the total particle density (electrons plus protons) in units of the Bohr radius, while the top axis shows the corresponding mass density. The solid line shows where the fugacity expansion breaks down. The dashed line shows where the size of `non-triviala e!ective "eld theory corrections to the equation of state "rst exceed unity. (See the text for more precise descriptions.) The e!ective "eld theory is valid only in the region above both of these lines.

Fig. 2. Region of validity for the e!ective theory for a pure Z"2 ionized helium plasma. The curves have the same meaning as in Fig. 1.

corrections to the equation of state "rst exceed unity. This is taken as an indication that the perturbative expansion of the e!ective "eld theory has broken down. The e!ective "eld theory is valid only in the region above (or to the left of) both of these lines. In Fig. 4, the temperature range extends into the relativistic domain. The horizontal dotted line in this "gure shows where the  More precisely, this line shows where any of the one-, two-, or three-loop corrections "rst exceed unity. To match the earlier discussion, the non-interacting quantum statistics portion of the two-loop correction is not included.

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Fig. 3. Region of validity for the e!ective theory for a pure Z"6 ionized carbon plasma. The curves have the same meaning as in Fig. 1.

Fig. 4. Region of validity for the e!ective theory for a pure Z"13 ionized aluminum plasma. The solid and dashed curves have the same meaning as in Fig. 1. The dotted horizontal line shows where relativistic corrections to the electron pressure exceed unity; our non-relativistic treatment is valid only below this line.

(¹/m c) relative correction to the electron pressure exceeds unity, and provides an indication of  C where relativistic corrections invalidate our non-relativistic treatment. For a given density (and composition), if the e!ective "eld theory is to be useful, then the temperature must be high enough so that the perturbative expansion of the theory is valid, but not so high so that all corrections to ideal gas behavior generated by the e!ective theory are too small to be relevant. In other words, the size of the e!ects produced by the e!ective theory must be large enough to be interesting. Figs. 5}10 show log plots of the size of corrections to the equation of state for various compositions and two di!erent densities of the plasma. In these plots, the solid line shows the ideal gas result, including quantum statistics for the electrons but no interactions. The long dashed line shows the one-loop Debye screening correction, the medium dashed line shows

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19

Fig. 5. Corrections to the equation of state, bp/n, as a function of temperature for a pure Z"1 plasma with a total density (electrons plus protons) of 1 a\. Here, and in the following related "gures, the solid line shows the ideal gas  result, including quantum statistics for the electrons but no interactions. The long dashed line shows the one-loop Debye screening correction, the medium dashed line shows the two-loop correction (minus its non-interacting quantum statistics piece), and the short dashed line shows the three-loop e!ective "eld theory correction. The absolute values of the various corrections are plotted. On the two- and three-loop curves, the `cuspsa pointing downward show where these corrections cross zero and change sign. For this density, the e!ective "eld theory is only useful for temperatures above about 0.06 KeV. Below this temperature, the three-loop correction exceeds the size of the one-loop correction (and exceeds unity at temperatures below about 0.04 KeV), clearly showing that the perturbative expansion of the e!ective theory has ceased to be reliable.

Fig. 6. Same as Fig. 5, but at a total particle density of 10a\. 

the two-loop correction (minus its non-interacting quantum statistics piece), and the short dashed line shows the three-loop e!ective "eld theory correction. Plotted are the absolute values of the various corrections. The one-loop Debye screening correction is always negative. The `cuspsa pointing downward on the two- and three-loop curves show where these corrections cross zero and change sign. Asymptotically, for large temperature, the (non-trivial part of the) two-loop correction is negative for Z"1 and positive for Z52, while the three-loop correction is asymptotically

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Fig. 7. Same as Fig. 5, but for a pure Z"6 plasma at a particle density of 1a\. 

Fig. 8. Same as Fig. 5, but for a pure Z"6 plasma at a particle density of 10a\. 

positive in all these plots. Each plot begins at temperatures which are too low for the e!ective theory to be valid, includes the region where the e!ective theory can be useful, and ends at temperatures su$ciently high that all corrections to ideal gas behavior are tiny. 2. Classical Coulomb plasmas We consider a plasma of A di!erent species of charged particles (ions and electrons) and use the letters a, b,2"1,2, A to denote a speci"c species with charge e and mass m . In the classical ? ? limit, the particle mass only appears in the thermal wavelength 2pb  , (2.1) j" ? m ? where b is the inverse temperature measured in energy units, and the thermal wavelength itself only serves to de"ne the free-particle density n in terms of the chemical potential k and spin ? ?

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Fig. 9. Same as Fig. 5, but for a pure Z"13 plasma at a particle density of 1a\. 

Fig. 10. Same as Fig. 5, but for a pure Z"13 plasma at a particle density of 10a\. 

degeneracy factor g of the given species: ? n"g j\ e@I? . (2.2) ? ? ? The grand canonical partition function for a free gas composed of these species is given by



Z " dp, 2dp, ,    + , ,? where dp, is the N-particle measure for species a, ? 1 dp,, (dr )n2(dr )n . ? N! ? ? ?, ?

(2.3)

(2.4)

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The factors of j\ hidden in the n free-particle densities in this measure come from performing the ? ? momentum integrals in the equilibrium phase-space distribution,



(dp) j\" exp+!bp/2m , , ? ? (2p )

(2.5)

and the remaining parts of n arise from the degeneracy (g ) and fugacity (e@I? ) factors that enter ? ? into the de"nition of the grand canonical ensemble. Introducing the total volume of the system



V, (dr) ,

(2.6)

which we shall always assume is arbitrarily large, and carrying out the summations, we get





 (Vn),?  ? "exp V n . Z "“  ? N ! ? ? ,? ?

(2.7)

2.1. Functional integral for the classical partition function The corresponding grand canonical partition function for a plasma with Coulomb interactions between all the charged particles is







b Z" dp, 2dp, exp ! e e < (r !r ) . (2.8)   I J ! I J 2 + , I$J ,? Here the indices k, l in the exponential run over all particles of all the various types; r and e denote I I the coordinates and charge of any given particle, respectively. We employ rational units, so that the Coulomb potential for unit charges is given by 1 . < (r)" ! 4pr

(2.9)

We choose to work with the grand canonical ensemble because, as we shall see, it has a simple functional integral representation which leads to a very convenient diagrammatic form for perturbation theory and allows easy use of e!ective "eld theory techniques. However, we are ultimately interested in calculating physical quantities as a function of the particle densities, not chemical potentials, of the various species. Since the presence of interactions between particles will modify the particle density}chemical potential relation, we will need to compute particle densities as a function of chemical potential, and then invert this relation (order-by-order in perturbation theory) to re-express results in terms of particle densities. The physical particle densities, which we will denote as n , satisfy charge neutrality, ? 1Q2 "V e n "0 , (2.10) @ ? ? ? as required for a sensible thermodynamic limit. It will be useful to regard the chemical potentials as temporarily having arbitrary spatial variation, k (r). This extends the partition function to be a functional of these generalized chemical ? potentials, ZPZ[k], which is then the generating functional for number density correlation

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23

functions. The free-particle number density}chemical potential relation (2.2) is now generalized to n(r),g j\ e@I? r , ? ? ? with the variational derivative

(2.11)

d n(r)"d d(r!r) n(r) . (2.12) ?@ ? dbk (r) ? @ Here, and henceforth, variations in bk , and in b, will be regarded as independent. In other words, ? bk is to be varied while holding b "xed, and vice versa. The density of particles of species a is given ? by the variational derivative of ln Z[k] with respect to the corresponding generalized chemical potential,





d ln Z[k] , (2.13) 1n (r)2 , d(r!r ) " ?G ? @ dbk (r) @ ? G while two functional derivatives yield the connected part of the density}density correlator, K (r!r),1n (r)n (r)2 !1n (r)2 1n (r)2 ?@ ? @ @ ? @ @ @



 









d(r!r ) " d(r!r )d(r!r ) ! d(r!r ) ?G @H ?G @H @ @ H @ GH G d d ln Z[k] . (2.14) " dbk (r) dbk (r) @ ? After the functional derivatives have been taken, it will be assumed that the spatially-dependent, generalized chemical potentials k (r) revert to the usual constant chemical potentials k . A A The cumbersome form of the grand partition functional (2.8) can be replaced by a much leaner functional integral representation by using the Gaussian integral relation



[d ] exp b (dJr)[ (r)  (r)#io(r) (r)] 

 



b "Det\[b(! )] exp ! (dJr)(dJr)o(r) < (r!r) o(r) , J 2

(2.15)

which follows from completing the square in the functional integral on the left. The auxiliary "eld

(r) is nothing but the electrostatic scalar potential. The relation above has been written in

 The derivation of the results (2.13) and (2.14) from the spatially varying chemical potential extension of the standard form (2.8) of the partition function requires a little thought. These results are obvious however if one imagines the classical partition function to be given by the classical limit of the quantum form Z[k]"Tr exp+!bH#  (dr) bk (r)n (r),, with all operators commuting in this classical limit. ? ? ?  The use of Gaussian integral relations such as this has a very long history in statistical physics, going back at least as far as Hubbard [21] and Stratonovitch [22].  More precisely, !i is the normal electrostatic potential. Inserting an i (or rotating the contour of the functional integral) is necessary to obtain an absolutely convergent functional integral.

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l spatial dimensions with



(dJk) e k r\rY < (r!r), J (2p)J k

(2.16)

the Coulomb potential in l dimensions. We choose to make a continuation in spatial dimensions at this juncture because it automatically removes in"nite particle self-interactions. Dimensional continuation is a regularization procedure which introduces no external or extraneous dimensional constants. Hence, since there is nothing available to make up the correct dimensional quantity, in dimensional continuation < (0)"0 , (2.17) J and particle self-interactions vanish. We shall see how this works out in practice as our development unfolds. We shall also need the technique of dimensional continuation to deal with the short-distance divergences of the classical Coulomb theory } the divergences that are removed by quantum #uctuations which we shall later handle using e!ective "eld theory methods. Hence one might as well get accustomed to dimensional continuation at an early stage. At the end of our computations we shall, of course, take lP3. In view of the functional formula (2.15), it follows that the grand canonical partition function may be written as



  



b Z[k]"Det [b(! )] [d ] exp ! (dJr) (r)(! ) (r) 2

 

 ;exp (dJr)n(r)e @C? (r . (2.18) ? ? Since !  is a positive operator, the "rst, Gaussian, part of the integrand gives a well-de"ned and convergent functional integral. Expanding the second exponential in a power series in the free-particle densities n, and using the functional integration formula (2.15), it is easy to see that the ? result (2.18) does indeed reproduce the Coulomb plasma generating functional (2.8). Note that this equivalence requires that the self-interaction terms vanish, which is the case with our dimensional regularization [Eq. (2.17)]. Combining the two exponentials of (2.18), one may write the partition function in the concise form



Z[k]"N [d ] e\1 (_I , 

(2.19)

 The dimensional regularization method is widely employed in relativistic quantum "eld theory calculations, and is discussed in many texts. For example, the book [23] contains a detailed treatment. The conclusion that < (0)"0 may be J justi"ed more explicitly by starting from the integral representation (2.58) with i "0, which shows that < (r)Jr\J.  J Dimensional regularization is de"ned by the prescription that one "rst go to a region of spatial dimensions in which the quantity being examined is well de"ned, and thereafter analytically continue to the dimensionality of interest. In the present case, this requires going to l(2 where < (0)"0. Then one continues this result to arbitrary dimension, with J zero of course remaining zero as l varies, including l"3.

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25

with an `actiona functional de"ned by

 



b  S [ ; k], (dJr) [

(r)]! n(r) e @C? (r , ?  2 ? and the overall normalization factor

(2.20)

N ,Det [b(! )] . (2.21)  Varying the functional integral representation (2.19) with respect to the chemical potential k (r) ? yields the representation 1n (r)2 "11n(r) e @C? (r22 (2.22) ? @ ? for the density of particles of type a, where in general 11222 denotes a functional integral average,



11O22,Z[k]\N [d ]e\1 ( O . 

(2.23)

With the generalized chemical potentials restricted to constant values, Eq. (2.22) gives the functional integral representation for the usual grand canonical average of the number density of particles of species a. A second variation with the chemical potentials then restricted to constant values yields the representation of the density}density correlation function (2.14), K (r!r)"11n e @C? (rn e @C@ (rY22!11n e @C? (r2211n e @C@ (rY22 @ ? @ ?@ ? r ? #d d(r!r)11n e @C ( 22 . (2.24) ?@ ? The "nal contact term proportional to d(r!r) appears (when a"b) because the functional integral naturally generates correlators involving distinct particles,





11n e @C? (rn e @C? (rY22" d(r!r ) d(r!r ) . (2.25) ? ? ?G ?H @ G$H This di!ers from the corresponding term in (2.14) precisely by the single-particle contact term 1 d(r!r ) d(r!r )2 "d(r!r) 1n 2 . G ?G ?G @ ? @ Since the functional integral of a total derivative vanishes,



0" [d ]

d e\1 (_I , d (r)

(2.26)

the "eld equation 11dS [ ; k]/d (r)22"0 is an exact identity. For the action (2.20), this is the  Poisson equation

11i (r)22"11o(r)22

(2.27)

 In the special case of a binary plasma with equal fugacities, note that the action (2.20) reduces to the much-studied Sine-Gordon theory. See, for example, Ref. [24] and references therein.

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with the charge density  (2.28) o(r), e n(r) e @C? (r . ? ? ? Integrating both sides of (2.27) over all space yields the condition of total charge neutrality,



 (2.29) 0"1Q2 " e (dJr)1n (r)2 . ? @ @ ? ? This identity holds for any choice of the generalized chemical potentials k (r), in essence because ? the average value of the electrostatic potential will always adjust itself to produce a charge neutral equilibrium state. The fact that the chemical potentials enter the action (2.20) only through the combination ne @C? ( (with nJe@I? ) means that the theory is completely unchanged if the electrostatic potential ? ? is shifted by an arbitrary constant, i Pi #c ,

(2.30)

provided the chemical potentials are correspondingly adjusted, k Pk !e c . (2.31) ? ? ? Consequently, the values of the chemical potentials are not uniquely determined by the physical particle densities. This is also re#ected in the fact that the conditions n "1n 2 , a"1,2, A , (2.32) ? ? @ only give A!1 linearly independent constraints on the chemical potentials } precisely because charge neutrality (2.29) is an automatic identity. To obtain uniquely de"ned chemical potentials (when they revert back to their normal constant values), one must remove the (physically irrelevant) freedom (2.30)}(2.31) to shift the mean value of the electrostatic potential. We will make the obvious choice, and demand that the thermal average of the electrostatic potential vanish, 11 22,0 ,

(2.33)

to "x the chemical potentials uniquely. 2.2. Mean xeld theory Saddle-points of the functional integral (2.19) correspond to solutions of the "eld equation dS [ ; k]  "0 , d (r)

(2.34)

which, for the action (2.20), is just the Debye}HuK ckel equation  !  (r)"i e n(r) e @C? (r . ? ? ?  Assuming, of course, that the plasma contains both positively and negatively charged species.

(2.35)

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The leading saddle-point approximation corresponds to neglecting all #uctuations in away from the saddle point, so that ln Z [k]"!S [ ; k] , (2.36)   with solving the "eld equation (2.35). In quantum "eld theory, this approximation is commonly called the tree approximation because the classical action is the generating functional of connected tree graphs. In statistical mechanics it is known as the mean "eld approximation. In Appendix A we shall describe the e!ective action functional C[ ; k] which is the generalization of the classical action S [ ; k] that takes account of the thermal #uctuations about the mean "eld which are  described by the functional integral and thus provides an exact description of the plasma. As will be shown in Appendix A, the e!ective action method can be used to derive general properties of the plasma physics. Our work now with the mean "eld approximation will provide an introduction to the later use of the more general e!ective action as well as illustrating basic plasma properties. For constant chemical potentials, the "eld equation reduces to the (lowest-order) charge neutrality condition, e n e C? @("0 , ? ? ?

(2.37)

and ln Z [k]"V n e C? @( .  ? ? The mean-"eld number density}chemical potential relation is given by





(2.38)



*

* ln Z  "ne C? @(#ib e ne C? @( "ne C? @( , (2.39) n "V\ ? ? ? ? ? *bk *bk ? ? @ ? with the last equality following from the charge neutrality condition (2.37). If the free-particle densities satisfy `barea charge neutrality, 0" e n , (2.40) ? ? ? then the saddle-point condition (2.37) has the trivial solution (r)"0, the physical densities n , ? within this mean "eld approximation, will equal the free-particle densities n, and the mean-"eld ? partition function equals the usual ideal gas result,





 Z "exp V n . ?  ?

(2.41)

 Note that this constraint does not have a perturbative solution that can be be expanded in powers of the electric charge. This lack of a perturbative solution occurs because appears only in the combination e . Moreover, the lack of ? a perturbative solution and consequent condition of overall charge neutrality is related to the in"nite range of the Coulomb potential. If, for example, the Coulomb potential were replaced by a Yukawa potential with range 1/m, the classical "eld equation for constant "elds would become !im " e n e C? @(, which imposes no constraint on the ? ? ? total charge and which does have a perturbative solution for .

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The average energy of our grand canonical ensemble is the thermodynamic internal energy, * ln Z ;"EM "1E2 "! @ *b



.

@I Since n&b\, varying the neutrality condition (2.37) with respect to b gives ? i *(b ) 3 "0 , ! b\ e ne C? @(# i ? ? b  *b 2 ? where

(2.42)

(2.43)

(2.44) i , ben e C? @(  ? ? ? will be seen to be the lowest-order (squared) Debye wave number. The "rst term of (2.43) again vanishes by virtue of charge neutrality (2.37), and so *(b ) "0 . *b

(2.45)

Hence, to lowest order the average energy 3 3 (2.46) EM " b\V n " ¹ NM , ? 2 ? 2 ? ? which is just the familiar formula for an ideal gas. Second derivatives of ln Z produce correlators. The second derivative of ln Z with respect to the  inverse temperature gives the lowest-order result for the mean square #uctuation in energy, *EM 15 1(E!EM )2 "! " ¹EM . @ *b 4

(2.47)

Mixed temperature}chemical potential derivatives yield the correlation between energy and particle number #uctuations, 3 *NM 1(E!EM ) (N !NM )2 "! ? " ¹NM . ? ? ? @ 2 *b

(2.48)

These are again just the results for a free gas. But for #uctuations in particle numbers, given by second derivatives with respect to the chemical potentials, one must account for the fact that varying the chemical potentials will cause the mean "eld to vary. Since the charge neutrality constraint (2.37) holds for arbitrary chemical potentials, varying it with respect to the chemical potentials yields *

"0 . e n #ii ? ?  *bk ?

(2.49)

L.S. Brown, L.G. Yawe / Physics Reports 340 (2001) 1}164

29

Hence, *N *

1(N !NM ) (N !NM )2 " ? "d NM #NM ie b ? ? @ @ @ *bk ?@ ? ? ? *bk @ @ b "d NM !e NM e n . (2.50) ?@ ? ? ? i @ @  The physical implications of this result, which di!ers from the ideal gas result, will be discussed below in Section 2.7. 2.3. Loop expansion The saddle-point (or `loopa) expansion of the functional integral (2.19), incorporates corrections beyond mean "eld theory and systematically generates the perturbative expansion for physical quantities of interest. In the development that follows, we shall assume that all of the desired functional derivatives with respect to the generalized, spatially varying chemical potentials which produce the insertions in the functional integral, as shown in the previous number density (2.22) and density}density correlator (2.24), have already been taken. Thus, we henceforth restrict our considerations to constant chemical potentials. In the lowest-order approximation, the freeparticle densities n will equal the physical densities n , which are charge neutral (2.10). However, ? ? perturbative corrections to the chemical potential}number density relation will shift the freeparticle densities away from the physical densities, and therefore displace the true saddle point away from "0. Even though the bare neutrality constraint (2.40) no longer holds in higher orders, it will be most convenient to expand the functional integral about "0 instead of the true saddle-point value. At each stage of this (loop) expansion, further corrections to the bare (tree approximation) charge neutrality constraint (2.40) appear which alter the relation amongst the chemical potentials that arises from charge neutrality. Expanding the action in powers of and separating the quadratic and constant terms gives S [ ; k]"S [ ; k]#*S[ ; k] ,   where

 

(2.51)



 b S [ ; k], (dJr) ! n# (r) [! #i ] (r) , ? 2   ?

(2.52)

and

 

 *S[ ; k],! (dJr) n+e @C? (r!1# be (r), ?  ? ?  "! (dJr) n+[ibe (r)]# [ibe (r)]# [ibe (r)]#2, . (2.53) ? ? r ? r ? ? In Eq. (2.52), i is the lowest-order Debye wave number previously de"ned in Eq. (2.44). Since the  bare neutrality condition is modi"ed by loop corrections, e n will not vanish beyond the mean ? ? ?

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"eld approximation. Consequently, *S contains a piece linear in the "eld and "0 does not remain a saddle point in higher orders. Evaluating the action at "0 gives the ideal gas partition function The "rst (`one-loopa) correction is obtained by neglecting *S and integrating over #uctuations in with just the quadratic action S . This gives the Gaussian functional integral  b Z "Z Det[b(! )] [d ] exp ! (dJr) (r)(! #i ) (r)    2





 



1 "Z Det\ 1# i .  !  



(2.54)

The product of the determinant produced by the Gaussian integration with the prefactor (which may be written as the inverse determinant of the operator inverse) produces the determinant shown on the second line. This functional determinant will be evaluated shortly. The correlation function of potential #uctuations 11 (r) (r)22, to lowest order, is given by Green's function for the linear operator (! #i ) appearing in S ,   N (2.55) b11 (r) (r)22"  [d ]e\1 b (r) (r)"G (r!r) . J Z  Here G (r!r) denotes the Debye Green's function (in l-dimensions), which satis"es J [! #i ]G (r!r)"d(r!r) , (2.56)  J and has the Fourier representation





(dJk) e k r\rY . (2.57) G (r!r)" J (2p)J k#i  Expanding the functional integral (2.19) in powers of *S will lead to Feynman diagrams in which each line represents a factor of this Debye Green's function times 1/b, with vertices joining k lines representing factors of n (ibe )I. ? ? ? A convenient integral representation for the Debye Green's function in a space of arbitrary dimensions is obtained by writing the denominator in (2.57) as a parameter integral of an exponential, interchanging the parameter and wave number integrals, and completing the square to perform the wave number integral:









(dJk) k k r  e\ Qe  " (2.58) ds (4ps)\Je\G Q\rQ . (2p)J   The coincident limit of the Debye Green's function G (0) will be needed in the following sections. In J this limit, the representation (2.58) becomes the standard representation of the Gamma function, G (r)" J

ds e\G Q

 As discussed in the next subsection, the term in *S linear in the "eld may be counted as being of one-loop order. However, because it is odd in , its "rst-order contribution to the functional integral vanishes (just like the  term) and so it does not contribute to one-loop result (2.54).

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31

and we have




l iJ\ G (0)"  C 1! . J 2 (4p)J

(2.59)

Since C(!)"!2(p, the lP3 limit of G (0) is perfectly "nite and yields J  i lim G (0)"!  . J 4p J Comparing this with the Debye Green's function "xed at three dimensions, e\G P , G (r)"  4pr

(2.60)

(2.61)

one sees that i 1 [e\G P!1]"!  . (2.62) lim 4p 4pr P In other words, the dimensional regularization method automatically deletes the vacuum selfenergy contribution that comes from the pure Coulomb potential. 2.4. Particle densities Although the densities of the various particle species may be obtained simply by di!erentiating the partition function with respect to the corresponding chemical potential } which we shall do subsequently } one may directly evaluate these densities using diagrammatic perturbation theory. We shall do this through one-loop order to illustrate the working of the perturbation theory and charge neutrality. In perturbation theory, the density of particles of a given species is evaluated by expanding the exponential in (2.22) in powers of yielding, to one loop order, 1n 2"11n e @C? (r22 ? @ ? "n [1#ibe 11 (r)22!be11 (r)22] ? ?  ? "n [1#ibe 11 22!b e G (0)]. (2.63) ? ?  ? J In the tree approximation with "0, the charge neutrality condition (2.37) requires that the chemical potentials are arranged such that e n"0. Thus, this sum should be considered to start ? ? out at one-loop order. The one-legged vertex, the coe$cient of the term in the interaction part of the action (2.53) linear in , is proportional to this sum, and hence it also should be considered to start at one-loop order. Thus computing the expectation value of to one-loop order requires expanding e\ 1 in powers of and keeping the linear and cubic terms. This expansion, shown in the graphs of Fig. 11, gives





N 11 22"  [d ]e\1 (0) (dJr) n[(ibe (r))# (ibe (r))] ? ? r ? Z  ? i  " e n[1! beG (0)]. ? ?  ? J i  ?

(2.64)

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Fig. 11. One-loop order contributions to 1 2 . Unlabeled blobs (or vertices) represent insertions of !*S taken to some @ order in ; a vertex joining k lines stands for a factor of n (ibe )I. Each line represents a factor of the Debye Green's ? ? ? function divided by b, and the contribution of each diagram is to be multiplied by the appropriate symmetry factor which, for the diagram above containing a loop, the `tadpole graph,a is 1/2. [The basic rules for diagrammatic perturbation theory, corresponding to a saddle-point expansion of the functional integral, are discussed in virtually all textbooks on quantum "eld theory. See, for example, Refs. [23,25,26].] The condition 11 22"0 taken to one-loop order implies that the one-legged vertex ( ) must cancel the one-loop `tadpolea. Hence this one-legged vertex should be counted here as being a one-loop contribution. Two-loop diagrams (and beyond) generate further higher-order corrections to the one-legged vertex ib e n. ? ? ?

This calculation is spelled out in greater detail in the derivation of Eq. (F.21) in Appendix F. Note that the "rst term in Eq. (2.64), the tree approximation, is obtained by expanding the tree level neutrality condition (2.37) to zeroth and "rst order in . Imposing condition (2.33) that the mean electrostatic potential vanish now requires, to this order, that   e n"b G (0) e n , (2.65) ? ?  J ? ? ? ? which alters the tree level neutrality constraint (2.40) on the chemical potentials, making the sum on the left-hand side of Eq. (2.65) equal to the one-loop contribution on the right-hand side. This con"rms the statement above that the sum on the left-hand should be considered to start out at one-loop order. With the imposition of the one-loop constraint (2.65), the expression (2.63) for the one-loop densities simpli"es to (2.66) 1n 2"n [1!beG (0)]. ? @ ?  ? J The discussion of the density that we have just given is illustrated in Fig. 12. Inverting the one-loop density relation (2.66) to express the bare density n in terms of the physical density n gives ? ? (2.67) n"n [1! beG (0)]\"n [1# beG (0)] , ?  ? J ? ?  ? J to one-loop order. Note that eG (0)/2 is the self-energy of a charge e in the Debye screened ? J ? plasma, and so the right-hand side of Eq. (2.66) may be recognized as the "rst-order expansion of the Boltzmann factor exp+!beG (0)/2,. Other e!ects besides this simple exponentiation of ? J course appear in higher orders. Also note that the mean charge density (computed to one-loop order) vanishes, as it must, even before the imposition of the constraint (2.65), for it follows from Eqs. (2.63) and (2.64) and de"nition (2.44) of the lowest-order Debye wave number that  1o2" e 1n 2 @ @ @ @ @   " e n [1! beG (0)]# ben11i 22 @ @  @ J @ @ @ @

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33

Fig. 12. One-loop order contributions to the mean particle density 1n 2 . Labeled blobs ( 䢇) refer to insertions of the ? @ ? number density ne @C? ( for a given species; a labeled blob radiating k lines stands for a factor of n (ibe )I. The condition ? ? ? 11 22"0 implies that the second and third diagrams cancel. More generally, the condition 11 22"0 implies that such `tadpolea diagrams cancel in the expansion of any quantity, and such diagrams may simply be neglected. This cancellation is described more fully in Appendix A.




  ben @ @ " e n [1! beG (0)] 1! ? ?  ? J i ? @  "0.



(2.68)

2.5. Loop expansion parameter We have just seen that the size of one-loop corrections is measured, in l dimensions, by the dimensionless parameter beG (0)&beiJ\, which reduces to bei in three dimensions. This J   parameter is the essentially the ratio of the Coulomb energy for two particles separated by a Debye screening distance to their typical kinetic energy in the plasma. Since i &be/d, where d is the  average interparticle spacing, this expansion parameter is also [be/d] } the 3/2 power of the ratio of the average Coulomb energy in the plasma to the kinetic energy in the plasma. At higher orders in the perturbative expansion, the relative contribution of any Feynman diagram containing l loops will be suppressed by [beiJ\]l, or in three dimensions, by [bei ]l.   A detailed proof of this appears in Section 3 of Appendix F. In other words, the loop expansion parameter is [bei ] (up to some O(1) numerical factor). In fact, we shall "nd in our explicit  calculations that [bei/4p] appears as the most natural loop expansion parameter. 2.6. Thermodynamic quantities All thermodynamic quantities may be derived from the grand canonical partition function. In particular, the internal energy density u is given by * ln Z , (2.69) uV"! *b @I where, as indicated the partial derivative is taken with all the bk "xed, while the chemical ? potential}number density relation is given by





* ln Z n V" , ? *bk ? @

(2.70)

 Here is a brief version. The rescaling " I /(be), r"r /i in the functional integral (2.19) conveniently reveals the  dimensionless loop expansion parameter g"beiJ\: the integrand acquires the canonical form e\1I (I E, with all  dependence on the dimensionless parameter g isolated in the explicit prefactor which controls the validity of a saddlepoint expansion.

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where now b is held "xed in the partial di!erentiation. The grand potential X(V, ¹, +k ,) is related ? to the partition function of the grand canonical ensemble by Z"e\@X .

(2.71)

The grand potential is extensive for a macroscopic volume, and it is simply related to the pressure, X"!pV, or ln Z p" . bV

(2.72)

The Legendre transform of the grand potential gives the Helmholtz free energy, F(V, ¹, +N ,)"X(V, ¹, +k ,)# k N . Hence the free energy density is given by ? ? ? ? ? f"!p# k n . (2.73) ? ? ? The previous zeroth order and one-loop results (2.41) and (2.54) express the partition function through one-loop order as









 1 Z "exp V n Det\ 1# i .  ? !   ? To evaluate the determinant, one may apply the general variational formula d ln Det X"Tr X\dX

(2.75)

to a variation of i , to show that  1 i " (dJr)G (0) di . d ln Det 1# J  !  



 

Since this is homogeneous in i of degree l!2, it implies that  1 2 ln Det 1# i " G (0) i V ,   !  l J



(2.74)



(2.76)

(2.77)

and thus



Z "exp 



1 n! i G (0) V , ? l  J ?

(2.78)

 This result assumes that the chemical potentials (and temperature) are constrained so that 11 22"0 (to one loop order). If this constraint is violated, as it apparently is in varying b to obtain the internal energy by Eq. (2.69) or varying bk to obtain the density of particles of species a by Eq. (2.70), then additional terms are present in the complete one-loop ? result. These additional terms do not contribute to the "rst variations yielding the energy or number densities and hence may be neglected for these terms, but they do contribute to second or higher variations that de"ne correlation functions. This is discussed more fully in Appendix A; see in particular Sections 1 and 3.

L.S. Brown, L.G. Yawe / Physics Reports 340 (2001) 1}164

Let us now go over to the physical limit lP3. Using Eq. (2.60) for G (0), we have  i Z "exp n#  V , ? 12p  ? Since









*n *i ? "n,  "be n , ? ? ? *bk *bk ?@ ?@ it follows from Eq. (2.79) that the number density to one-loop order is given by





bei n "1n 2"n 1# ?  , ? ? @ ? 8p

35

(2.79)

(2.80)

(2.81)

in agreement with the physical lP3 limit of the previous direct calculation (2.66). To one-loop order, the pressure is given by





be i (2.82) p "¹V\ ln Z "¹ n 1# ?  .   ? 12p ? Re-expressing the one-loop pressure in terms of physical particle densities using Eq. (2.81) produces





be i p "¹ n 1! ?  .  ? 24p ? This is the equation of state of the plasma to one-loop order. Using *n ! ? *b



" ¹n,  ?

*i !  *b



"¹i ,  

(2.83)

(2.84)

@I @I it follows from Eq. (2.79) that the internal energy to one-loop order is given by





(2.85)





(2.86)

3 be i u "¹ n # ?  .  ? 2 16p ? or, in terms of the physical density n , ? 3 bei ! ?  . u "¹ n  ? 2 8p ? And "nally, the Helmholtz free energy density, to one-loop order, is





bei f "¹ n !1#ln(n j/g )! ?  . ? ? ?  ? 12p ?

(2.87)

2.7. Density}density correlators We now compute the density}density correlator K (r!r) through one loop order. Expanding ?@ about "0, the "rst non-vanishing (`treea graph) contribution appears when *S is neglected and the explicit exponentials in (2.24) are expanded to linear order, yielding K(r!r)"d d(r!r) n!bnne e G (r!r) . ?@ ?@ ? ? @ ? @ J

(2.88)

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Fourier transformation produces the density}density correlation as a function of wave number, e n e n KI (k)"d n!b ? ? @ @ . (2.89) ?@ ?@ ? k#i  Multiplying this result by V and taking the limit kP0 gives the tree or mean-"eld approximation to the total particle number #uctuations for the various species: b 1(N !NM )(N !NM )2"d NM !e NM e n , (2.90) ? ? @ @ @ ?@ ? ? ? i @ @  in agreement with the previous result (2.50). The second term on the right-hand side of this equality is a consequence of charge neutrality. It involves the ratio of charges, and shows that one cannot naively expand in powers of charges. It causes the number #uctuations to depart from Poisson statistics even in this lowest-order approximation. Its presence ensures that 1(N !NM )Q2" e 1(N !NM )(N !NM )2"0 , ? ? @ @ ? ? @ @ @ @ where in the "rst equality we made use of total average charge neutrality,

(2.91)

1Q2 " e NM "0 . (2.92) @ ? ? ? Multiplying Eq. (2.91) by e and summing over a shows that ? 1Q2"0 . (2.93) @ Thus, at least at tree level, there is no #uctuation in the total charge of the ensemble described by our functional integral. The usual grand canonical ensemble is modi"ed by the long-range Coulomb potential so that only subsectors of totally neutral particle con"gurations appear in the sum over con"gurations. The general structure of the number density correlation function described below [in particular Eq. (2.117)] shows that the vanishing of charge #uctuations (2.93) holds to all orders, and thus, in general, only neutral con"gurations contribute to the ensemble. Finally, we note that, to lowest order, charge neutrality also ensures that the #uctuation of the total number of particles N" N in the grand canonical ensemble is Poissonian, ? ? 1(N!NM )2" 1(N !NM )(N !NM )2"NM . (2.94) @ ? ? @ @ @ ?@ As shown in Eq. (2.115) below, higher-order corrections alter this result. One-loop corrections to the density correlator are obtained by expanding both e\ 1 and the exponentials in the density operator insertions of (2.14) in powers of , and retaining all next-to In this regard, it is worth noting that KI (0) is a symmetrical, real, positive, semi-de"nite matrix whose only ?@ vanishing eigenvalue appears for the eigenvector whose components are the electric charges e (provided all densities ? n are non-zero). These properties are easily demonstrated explicitly. First de"ne the matrix N ,d (n and then the ? ?@ ?@ ? matrix L,N\KI (0)N\, so that L "d !v v with v ,e (bn/i . The claimed properties hold because v is ?@ ?@ ? @ ? ? ?  a unit vector.

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37

Fig. 13. One-loop diagrams contributing to the connected density}density correlation function K (r!r)" ?@ 1n (r)n (r)2. Diagrams a}f are all tadpole diagrams which cancel and hence can be neglected. Diagrams g}i merely ? @ @ serve to correct the bare densities appearing in the lowest-order result. Diagrams j}m involve the essentially new contribution C discussed in the text. ?@

leading order corrections. This leads to the one-loop contributions shown graphically in Fig. 13. There are three classes of diagrams: those which cancel, those which simply serve to replace bare densities by the physical densities (to one-loop order), and the rest. Diagrams a and b cancel, as do c & d, and e & f, because their sum is proportional to 1 2 ,0. Here, as well as in higher orders, all @ such `tadpolea diagrams can simply be neglected. That these single-particle reducible graphs cancel to all orders is proven in Appendix A. Diagrams g and h correct the explicit bare densities in (2.88) by *n"1n 2!n , ? ? @ ? giving the one-loop contribution

(2.95)

!b[*n n#n *n]e e G (r!r) . (2.96) ? @ ? @ ? @ J Diagram i corrects the Debye wave number which appears in the Green's function G (r!r); J explicitly it produces !bnne e ? @ ? @

*G (r!r) J be*n , ? ? *i  ?

(2.97)

 A graph is `single-particle reduciblea if it can be separated into two disjoint pieces by cutting a single line.

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or in Fourier space, bnne e ? @ ? @ be*n . (2.98) ? ? (k#i )  ? The net e!ect of these two classes of diagrams (plus the one-loop correction to the d 1n (r)2 ?@ ? @ contact term) is to replace, through one-loop order, the particle densities and Debye wave number appearing in (2.89) with their physical values, e n e n KI (k)PKI (k),d n !b ? ? @ @ . ?@ ?@ ?@ ? k#i

(2.99)

Here i is the Debye wave number computed with physical particle densities, i, ben . ? ? ? The second part of Eq. (2.99) involves b\ GI (k)" , k#i

(2.100)

(2.101)

which is just the Fourier transform of the tree level electrostatic potential correlator 11 (r) (r)22 as given in Eq. (2.55), but with the physical Debye wave number i. Understanding the general structure of the number density correlation function will be facilitated if (2.99) is rewritten in the form KI (k)"d n !(be n ) GI (k) (be n ) . (2.102) ?@ ?@ ? ? ? @ @ The remaining graphs j}m give non-trivial corrections. Diagram j may be viewed as generating a correction to the "rst, `contacta term part of (2.102), d n PCI (k) ?@ ? ?@ where, to one-loop order, CI (k)"d n #(ben ) D(k) (ben ) , @ @ ?@ ?@ ?  ? ? J

(2.103)

(2.104)

with



D(k), (dJr)e\ k  r G (r) . J J

(2.105)

This function represents the loop which is common to diagrams j}m. Graphs k and l correspond to making the corrections e n P e CI (k), e n P e CI (k) , (2.106) ? ? A A? @ @ A A@ A A in the factors #anking GI (k) in Eq. (2.102). Physically, these diagrams may be viewed as generating corrections to the coupling between the particle density operators and #uctuations in the electrostatic potential. The "nal graph m is a one-loop polarization (or `self-energya) correction

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39

to the electrostatic potential correlator G(r!r)"11 (r) (r)22 .

(2.107)

This graph, together with higher-order graphs in which the same `bubblea is inserted two or more times, produce a change in the (Fourier transformed) potential correlator given by





GI (k)\PGI (k)\,b k#b e e CI (k) , (2.108) ? @ ?@ ?@ with the same one-loop result (2.104) for CI (k). Note that, according to Eq. (2.104), ?@ b (2.109) b e e CI (k)"i# (ben ) D(k) (ben ) , ? ? J @ @ ? @ ?@ 2 ?@ ?@ showing that this `self-energya contribution includes the previous squared Debye wave number i as well as the loop contribution described by graph m. Putting the pieces together, we "nd that the one-loop corrections conform to the general structure



 



KI (k)"CI (k)! b e CI (k) GI (k) b e CI (k) . (2.110) ?@ ?@ A A? A A@ A A That this form holds to all orders is proven in Appendix A, with this result given in Eq. (A.57). This Appendix shows that CI (k) is a single-particle irreducible function, symmetric in a and b, and ?@ provides its de"nition in terms of an e!ective action functional. Section G.1 of that appendix also demonstrates how the complete one-loop calculation may be easily performed using somewhat more sophisticated functional techniques. The explicit form of the one-loop function D(k) is easily evaluated in three dimensions since J 1  e\IP e\G P " dk . (2.111) G (r)"  4pr (4pr) 4p  G Thus taking the Fourier transform and interchanging integrals yields the dispersion relation representation





1  dk D (k)" ,  4p  k#k G which is readily evaluated to give

(2.112)

k D (k)"(4pk)\ arctan . (2.113)  2i  The k"0 limit of KI (k) characterizes the #uctuations in particle numbers, ?@ 1(N !NM )(N !NM )2 "VKI (0) . (2.114) ? ? @ @ @ ?@ The one-loop result for KI (0) is easily generated by inserting (2.113) into (2.104) and thence into ?@ (2.110). In particular, for the total particle number N" N , one "nds to one-loop order ? ? Vi , (2.115) 1(N!NM )2 "NM # @ 16p

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which explicitly shows that the Coulomb interactions generate non-Poissonian statistics for #uctuations in total particle number. 2.8. Charge correlators and charge neutrality As noted earlier, the charge neutrality condition (2.29) holds in the presence of arbitrary chemical potentials k (r). Consequently, a corollary of (2.29) is an identity for the correlator of the number ? density of some species a with the total charge:



 d 1Q2 " (dJr)K (r!r) e 0" @ ?@ @ dk (r) ? @  " KI (0) e . (2.116) ?@ @ @ It follows from the general structure (2.110) of the density correlator and the form (2.108) of the inverse Green's function that







KI (k) e " CI (k) e 1!b GI (k) b e e CI (k) ?@ @ ?@ @ A B AB @ @ AB

" CI (k) e bk GI (k) , ?@ @ @ which does indeed vanish in the limit kP0 in accordance with Eq. (2.116). The charge density}charge density correlation function is given by KI (k)" e KI (k) e ? ?@ @ ?@ k[ e CI (k)e ] ?@ ? ?@ @ , " k#b[ e CI (k)e ] ?@ ? ?@ @ or equivalently KI (k)"k ¹!k GI (k) ,

(2.117)

(2.118)

(2.119)

where ¹"1/b is the temperature in energy units. It has the small wave number limit KI (k)"k¹#O(k) .

(2.120)

This relation is known as the Stillinger}Lovett sum rule [27,28]. This limit, which follows directly from the structure (2.110) that is established in Appendix A, also follows from examining the coupling of the plasma to a static external electric potential. The static dielectric function of the plasma e(k) is related to the charge density correlation function by





1 k 1! . KI (k)" e(k) b

(2.121)

This will be derived in the following section [cf. Eq. (3.22)]. Thus, the small wave number limit (2.120) implies that e(k)PR as kP0. But this is just the statement that the plasma is a conductor

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41

} when an external uniform electric "eld is applied to the plasma, charges move and the plasma becomes polarized in such a way as to completely screen the constant external "eld. The small wave number behavior of the static dielectric function is made explicit by inserting Eq. (2.118) in Eq. (2.121) to obtain





b e CI (k) e . e(k)"1# ? ?@ @ k ?@

(2.122)

3. E4ective 5eld theory We have just worked out the statistical mechanics of a classical, multicomponent plasma through one-loop order. One cannot go to higher order in this purely classical theory. Ultraviolet divergences appear at two-loop order and beyond. For example, the pressure in two-loop order receives a contribution from the diagram

(3.1)

which is proportional to the integral of the cube of the Debye Green's function,  (dJr)G (r). In J three dimensions, the short-distance part of this integral behaves as  (dr)/r, which is logarithmically divergent. This divergence can be seen in an elementary fashion directly from the divergence (for opposite signed charges) of the Boltzmann-weighted integral over the relative separation of two charges,  (dr) exp+!be e < (r),. Diagram (3.1) is just the third-order term in the expansion of ? @ ! this integral in powers of the charges. These ultraviolet divergences of the classical theory are tamed by quantum mechanics}quantum #uctuations smear out the short distance singularities. To reproduce the e!ects of this quantum mechanical smearing, we must augment our previous dimensionally regulated classical theory with additional local interactions which both serve to cancel the divergences present in diagrams such as (3.1), and reproduce quantum corrections which are suppressed by powers of (or equivalently ij). The coe$cients of some of these induced interactions will diverge in the lP3 limit. The "nite parts of these coe$cients (or `induced couplingsa) will then be determined by matching predictions of this e!ective quasi-classical theory with those of the underlying quantum mechanical theory. 3.1. Quantum theory The full (non-relativistic) many-body quantum theory generates the grand canonical partition function } extended to be a number density generating functional Z [k] by the introduction of /+ the generalized, spatially varying chemical potentials k (r) } as a trace over all states, ?  (3.2) Z [k]"Tr exp !b H! (dr)k (r)n (r) , ? ? /+ ?

 





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where n (r) is the number density operator for particles of species a. The multiparticle Hamiltonian ? of the complete system has the structure   H" K # H! , (3.3) ? ?@ ? ?@ where K represents the kinetic energy of all particles of species a and H! is the Coulomb energy ? ?@ between particles of types a and b. In second-quantized notation,



1 (dr) tK (r)R ) tK (r) , K " ? ? ? 2m ?

(3.4)

and



e e H! " ? @ (dr)(dr)tK (r)RtK (r)R< (r!r)tK (r)tK (r) . ? @ ! @ ? ?@ 2

(3.5)

The quantum-mechanical partition function Z [k] may be expressed as a functional integral /+ involving A pairs of "elds tH(r, q), t (r, q) de"ned on the imaginary time interval [0, b]. Just as in ? ? the previous section, the Coulomb interaction between charges can be written in terms of a Gaussian functional integral over an auxiliary electrostatic potential. Therefore,



      



1 @ dq (dJr)(

(r, q)) Z [k]"Det[! ] [d ] exp ! /+ 2  @  ; “ [dtH dt ] exp ! dq (dJr)L , ? ? ?  ? where





(3.6)



*

 ! !k (r)!ie (r, q) t (r, q) . (3.7) L "tH(r, q) ? ? ? ? ? *q 2m ? The integrations are now over l(3 spatial dimensions, since we work with the dimensionally regulated theory. As explained earlier, the dimensionally continued Coulomb potential vanishes at vanishing spatial separation [Eq. (2.17)], and so there are no in"nite particle self-energies with this regularization scheme.

 These "elds may be either complex "elds satisfying periodic boundary conditions, t (r, q#b)"t (r, q), or ? ? anti-commuting Grassmann algebra valued "elds satisfying antiperiodic boundary conditions, t (r, q#b)"!t (r, q). ? ? The "rst case describes the quantum mechanics of Bosons, while the second describes Fermions. The following discussion is applicable to either case.  It should be emphasized that our functional integral representation (3.6) involves an integral over the Gaussian distributed electrostatic potential together with functional integrals over matter "elds that generate the ideal gas partition function of a second-quantized theory in the presence of the background "eld . This is in marked contrast to approaches using "rst-quantized representations in which one writes a path integral (i.e., Feynman}Kac formula) over the trajectories of individual particles. For a short discussion of this Feynman}Kac approach see, for example, Section 3.3 of Ref. [29], and references therein.

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43

If the generalized chemical potentials have arbitrary variation in both space and imaginary time, then ln Z [k] is the generating functional for connected time-ordered correlation functions of the /+ density operators n (r, q)"tH(r, q)t (r, q) . (3.8) ? ? ? These correlation functions are periodic in the imaginary time q with period b. Thus they have a Fourier series representation with frequencies u ,2pn/b"2pn¹/ , where in the last equality L we have restored Planck's constant . In the P0 classical limit, all these frequencies run o! to in"nity save for the static n"0 mode. Thus the classical limit involves zero-frequency correlators and, correspondingly, generalized chemical potentials that are independent of the imaginary time q. If one is only interested in thermodynamic quantities, or zero frequency correlators, then one may restrict the generalized chemical potentials to be time independent. Since the extended Hamiltonian of the system including the chemical potential terms is then time independent, the ensemble averages remain time-translationally invariant. Thus 1n (r, q)2 is independent of q, and it ? @ may be replaced by the q"0 form 1n (r)2 . Accordingly, ? @ 1 @ d dq1n (r, q)2 "1n (r)2 . (3.9) ln Z [k]" ? @ ? @ /+ b dbk (r)  ? The variational derivative of this result now yields



d d K (r, r)" ln Z [k] ?@ /+ dbk (r) dbk (r) ? @ @ "b\ dq[1n (r, q) n (r, 0)2 !1n (r)2 1n (r)2 ] . (3.10) ? @ @ ? @ @ @  If every chemical potential is shifted by an amount proportional to the corresponding charge, k (r)Pk (r)#e j(r), then derivatives of the partition function with respect to j(r) generate ? ? ? correlation functions of the charge density o(r, q), e n (r, q), ? ? ? d , (3.11) ln Z [k#ej] 1o(r)2 " e 1n (r)2 " /+ @ ? ? @ dbj(r) H ?





 Generalized chemical potentials that depend upon both space and real time do, however, have a role to play in the classical theory since they may be used to probe the response to time-dependent disturbances.  We will only need to use time-dependent chemical potentials when discussing the di!erence between zero-frequency and equal time correlators in Section 7.  The "nal form shown for the second variation (3.10) involves an integral over imaginary time of the time-ordered correlation function d d K (r, q; r, q)" ln Z[k] ?@ dk (r, q) dk (r, q) ? @ which is symmetric, K (r, q; r, q)"K (r, q; r, q), periodic in imaginary time, K (r, q; r, q)"K (r, q!b; r, q)" ?@ @? ?@ ?@ K (r, q; r, q!b), and (when evaluated at constant chemical potentials), time-translation invariant, K (r, q; r, q)" ?@ ?@ K (r, q!q; r, 0). Since the integral in (3.10) has q'0, the product of density operators appearing in the integrand is ?@ trivially time ordered.

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and



d d ln Z [k#ej] K(r, r)" e e K (r, r)" /+ ? @ ?@ dbj(r) dbj(r) H ?@ @ "b\ dq [1o(r, q)o(r, 0)2 !1o(r)2 1o(r)2 ]. (3.12) @ @ @  Alternatively, if one makes a compensating change of variables P #ij in the functional integral (3.6), all dependence on j disappears from the charged "eld Lagrangian L , and the net ? e!ect is merely to shift the Gaussian measure for the electrostatic potential,



 



   



1 1 exp ! dq(dJr)[

] Pexp ! dq (dJr)[ ( #ij)] 2 2



1 "exp ! dq (dJr)[(

)!2i

j!( j)] . 2

(3.13)

Hence,  O Jr ( H @Jr H 22 , 11e Z [k#ej]"Z [k] e /+ /+ and consequently



d 1o(r)2 " ln Z [k#ej] @ dbj(r) /+

" e11 (r)22 ,

(3.14)

(3.15)

H

and



d d ln Z [k#ej] K(r, r)" /+ dbj(r) dbj(r)

H "!b\ r d(r!r)! r r G(r, r) , Y where G is the zero-frequency correlator of #uctuations in the electrostatic potential,

(3.16)



@ dq [11 (r, q) (r, 0)22!11 (r)2211 (r)22] . (3.17)  The relation (3.15) is just the Poisson equation (now derived in the full quantum theory). When the chemical potentials have no spatial variation, 11 (r)22 is constant, the charge density 1o(r)2 @ vanishes, and the correlation functions K(r, r) and G(r, r) depend only on r!r. In this case, Eq. (3.16) becomes a simple relation between the Fourier transformed correlators: G(r, r),b\

KI (k)"b\k!(k) GI (k) .

(3.18)

Because of screening, GI (k) is bounded as kP0. Hence, KI (k)"¹k#O(k) ,

(3.19)

and we have an alternative derivation of the Stillinger}Lovett relation (2.120) discussed in the previous section.

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The charge density correlator KI (k) is directly related to the static dielectric function of the plasma. To see this, note that Z[k#ej] is precisely the partition function in the presence of an applied electrostatic potential !j(r). The variation of charge density with respect to j is just the charge density correlator times b, d1o(r)2 /dj(r)"bK(r, r). Hence, the Fourier transform of the @ charge density induced by this applied potential, to "rst order in the applied "eld, is o (k)"bKI (k) jI (k), or equivalently the induced electric "eld is  bKI (k) ik DI (k) , (3.20) EI (k)"! o (k)"!  k k  where D(r)"j(r) is the applied "eld. The ratio of the applied "eld to the total "eld (at a given wave number) de"nes the static dielectric function e(k), DI (k)"e(k)+DI (k)#EI



(k), .

(3.21)

Thus





1 bKI (k) \ " . e(k)" 1! kGI (k) k

(3.22)

The "rst equality is equivalent to Eq. (2.121) asserted previously. Condition (3.19) implies that e(k) diverges as kP0. This, of course, re#ects the fact that the plasma is a conducting medium which exactly screens uniform applied electric "elds. Finally, expressing the correlator GI (k) in terms of the self-energy (or polarization tensor), GI (k)\"b[k#P(k)], shows that P(k)/k and e(k) are related by P(k) . e(k)!1" k

(3.23)

Appendix A (as quoted in Eq. (2.108)) shows that the self-energy P(k)"b e e CI (k). Inserting ?@ ? @ ?@ this form yields the previously quoted relation (2.122) between the dielectric function and CI (k). ?@ 3.2. Classical limit In the limit in which the thermal wavelength j is much smaller that the scale of spatial variation ? in the electrostatic potential, j " ln (r)";1, the functional integral over the charged "elds tH and ? ? t may be performed explicitly. Appendix B presents this calculation in detail. Neglecting ? corrections suppressed by powers of j , one "nds that ?



  

[dtH dt ] exp ! ? ?

@





dq (dJr)L "exp ?

@

C?  O (rO (dJr)n(r)e  . ?



(3.24)

 This is just the classical limit of the quantum partition function for particles moving in a background potential !i (r, q). Here n(r) is the free-particle density of species a (in l dimensions), ? m J ? n(r)"g e@I? r "g j\Je@Ir , (3.25) ? ? ? ? 2pb




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(which reduces to (2.22) when lP3). Notice that the result (3.24) only depends on the time integral of the electrostatic potential. Consequently, it is useful to make a Fourier series expansion of the electrostatic potential on the imaginary time interval 0(q(b. We separate out the zero frequency mode by writing

(r, q)" (r)# L(r) e\ SL O , L$ where

(3.26)

2pn . u , L b

(3.27)

Since (r, q) is real, the zero mode part is real, (r)H" (r), while \L(r)" L(r)H. The non-zero frequency modes do not contribute to the functional integral result (3.24). Hence, in this classical limit, the non-zero frequency modes only appear in the initial Gaussian functional integral in Eq. (3.6), and they may be trivially integrated out. Their only e!ect is to change the determinantal prefactor in Eq. (3.6) from its implicit (l#1)-dimensional form to an l-dimensional form which just normalizes the Gaussian functional integral of the zero modes to unity if there were no other factors. Hence, in the classical limit one "nds that



Z[k]"N [d ]e\1 (_I ,  where

 

(3.28)



b  S [ ; k]" (dJr) (r)[! ] (r)! n(r) e @C? (r . (3.29)  ? 2 ? This is precisely the representation (2.18) for the classical partition function derived in the preceding section. We have just seen that this form emerges naturally as the limit of the quantum partition function. 3.3. Induced couplings But this `derivationa of (3.29) as the classical limit of the quantum partition function (3.6) is wrong! As emphasized earlier, the classical partition function (3.29) is singular when lP3, while the quantum partition function (3.6) is completely regular in three dimensions. It is impossible for the classical partition function (3.29) to equal the quantum partition function up to negligible corrections. What went wrong was the use of Eq. (3.24), which is valid for a su$ciently slowly varying background (r, q), inside a functional integral over #uctuations in } which includes #uctuations on scales comparable to the typical de Broglie wavelengths of the charged particles. In other words, the contributions of short distance #uctuations in were mangled when going from the quantum partition function (3.6) to the classical partition function (3.29). To "x this error one may, in principle, integrate exactly over the charged "elds t(r, q) together with the non-zero frequency modes (r) of the electrostatic potential, to produce a non-local, e!ective action S [ ; k] for the /+  This will not be true when sub-leading terms are included, as discussed later in this section.

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47

remaining zero-frequency mode (r) such that



Z[k]"N [d ]e\1/+ (_I . 

(3.30)

However, explicitly constructing or dealing with this non-local action is impossible. Our aim is to construct a local approximation to S which retains those parts of the complete non-local action /+ which must be added to the classical theory to obtain "nite, correct results to a given order in powers of the ratio of scales ij. To do this, the "rst step is to regulate the theory by working in l(3 dimensions and then add to the classical action (2.20) additional local terms, referred to as induced interactions, which both serve to "x the incorrect short-distance behavior of the classical theory, and incorporate quantum e!ects suppressed by powers of , S [k]PS [ ; k],S [ ; k]#S [ ; k] . (3.31)     The induced interactions S may, in general, include arbitrary combinations of the "eld (r) and  its derivatives at a point r, integrated over all space. However, only terms which are consistent with the symmetries of the original underlying theory can appear. Of particular importance is the invariance P !ic and k Pk !e c. The discussion of Eq. (2.30) shows that this is an ? ? ? invariance of the classical theory. In view of the structure (3.7) of the quantum Lagrangian, this shift is also an invariance of the full quantum theory. In addition, a constant shift in any chemical potential by an integer multiple of 2ni/b causes no change in the grand canonical partition function, and hence must also not change the e!ective theory. (Because the total particle number operators NK have only integer eigenvalues, a shift in bk by an integer multiple of 2ni leaves ? ? unchanged the exponential of bk NK appearing in the grand canonical partition function.) Conse? ? quently, only the combination n e @C? (, which is invariant under this combined shift of and k , ? ? plus spatial derivatives of k (r)#ie (r), will appear in S . As a result, the induced interactions ? ?  have the general structure S [ ; k]"  N ? 2?N



(dJr)g 2 N bN\n (r)e @C? (r2nN (r)e @C?N (r ? ? ? ?

# N ? 2?N

 

(dJr)h 2 N bN>[ (k  (r)#ie  (r))]n (r)e @C? (r2nN (r) e @C?N (r ? ? ? ? ? ?

(dJr)k 2 N bN>[ (k  (r)#ie  (r)) ) (k  (r)#ie  (r))] # ? ? ? ? ? ? N ? 2?N ;n (r) e @C? (r2nN (r) e @C?N (r#2 , (3.32) ? ? where the "nal ellipsis 2 stands for similar terms with four or more derivatives. For calculations to a given loop order, only a "nite number of the induced interactions are needed. The classi"cation of the various terms according to the order in which they "rst contribute will be spelled out below.  It will also be necessary to include non-linear interactions involving the non-zero frequency modes (r). This will be discussed at the end of this section.

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Interactions involving only a single density (that is, the classical !ne @C? ( interaction, the ? two-derivative term proportional to h, and corresponding higher derivative terms) have ? coe$cients which are "nite in three dimensions, and are simply determined by expanding the charged "eld functional integral (3.24) as described in Appendix B. The result (B.54) of this appendix, also shown in Eq. (3.91), gives

 j h" ? " . (3.33) ? 48pb 24bm ? The induced couplings g 2 N (as well as h 2 N , etc.) multiplying two or more densities will contain ? ? ? ? poles in l!3 which serve to cancel poles at l"3 generated by two-loop and higher order graphs generated by the classical interaction (or the single-density induced interactions). The `in"nitea parts of S [ ; k] (that is, the residues of these pole terms) are relatively easy to calculate } they are  precisely the terms needed to make the complete theory "nite (as it must be). This will be illustrated explicitly in the following sub-section. The remaining `"nitea parts of these induced couplings, the non-pole terms, can only be obtained by matching results for some physical quantity computed in this e!ective theory with corresponding results for the same quantity computed in the original (full quantum) theory. The "rst such matching for an induced coupling will be performed at the end of this section. Once the required matching is done, to a given loop order, the e!ective theory may then be used to calculate any other physical quantity. To ascertain the loop order of the various induced interactions, we note that since  (dJr)n is ? dimensionless, g 2 N times the remaining p!1 factors of bn must be dimensionless. In the ? ? ? physical lP3 limit, the particle density n&1/d, where d is the interparticle spacing, e/d has the ? dimensions of energy, and be/d is dimensionless. Hence g 2 N must be a pure number times ? ? eN\ (where, by e we mean six factors of the various charges e ), so that each of the p!1 ? densities is accompanied by a factor of be. Equivalently, each of the p!1 densities appears in the form be(ben)&[bei ]. Recalling that bei is just the loop-counting parameter, we see that ?   the g 2 N interaction with no derivatives and p densities will "rst contribute at 2(p!1) loop order. ? ? Similarly, for the interactions with two derivatives, h 2 N and k 2 N must both be dimensionless ? ? ? ? functions of the quantum parameters times eN\ in l"3 dimensions. This is because each particle density is again accompanied by a factor of be, so that the p-density two-derivative interactions involve the dimensionless quantity (bei )N  (dr)b(

). Consequently, the induced couplings  h 2 N and k 2 N "rst contribute to correlation functions at 2p loop order. Induced interactions ? ? ? ? with four or more derivatives, which were not displayed explicitly in (3.32), are only needed for calculations at four-loop order or beyond. Note that there are no induced couplings which "rst contribute at any odd-loop order. The multiple-density induced couplings have poles at l"3, and so the dimensionality l must be kept away from three until all terms of a given order have been combined. The extra dimensional factors needed away from l"3 have the form of factors of j\J, where j stands for a characteristic thermal wavelength of particles in the plasma. Since the Coulomb potential in l dimensions has the coordinate dependence r\J, an extension of the analysis in the previous paragraph shows that the induced coupling g 2 N Jj\N\J\ while both h 2 N and k 2 N are proportional to j\NJ\. ? ? ? ? ? ?  More precisely, a dimensionless function of the various quantum parameters be/j . ? ?

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49

Because the interactions depend on the chemical potentials, physical particle densities in the e!ective theory (3.31) are not equal to the functional integral average of n e @C? (, as in the original ? classical theory. Rather,

  



d ln Z dS dS  1n 2 , "! " ne @C? (!  ? @ ? dbk dbk dbk ? ? ? and similarly for the density}density correlator, d ln Z K (r, r), ?@ dbk (r)dbk (r) ? @ dS dS   " dbk (r) dbk (r) ? @





,

(3.34)



!1n 2 1n 2 ! ? @ @ @

dS  dbk (r)dbk (r) ? @



.

(3.35)

3.4. Renormalization The residue of a pole in an induced coupling may be determined by calculating a suitable n-point density correlator to a given loop order, and requiring that the result be "nite as lP3. Once this has been done for all the couplings that appear in a given order, then any other process will be "nite to this order. In addition to the pole terms in the induced couplings, there are, of course, "nite remainders. These "nite terms are determined by matching a result computed in our e!ective theory to the same result computed in the full quantum theory. We shall take up the matching problem later. Here we shall exhibit the nature of the (in"nite) pole terms by examining several examples. At two-loop order, the induced coupling g contributes through the last term in (3.35) to the ?@ irreducible part CI (k) of the density}density correlator. The only other contributions at this order ?@ which are singular as lP3 are the diagrams: (3.36)

[The full set of two-loop diagrams contributing to CI (k) is shown in Fig. 16 of the following ?@ section.] The second diagram of (3.36) generates a contact term proportional to d and indepen?@ dent of the external momentum k. The contribution to CI (k) of these diagrams, plus the ?@ g interaction, is ?@ ee  ee CI  (k)"bnn !2g ! ? @ D(k) #bd nn !2g ! ? A D(0) , ?@ ? @ ?@ ?@ ? A ?A 3! J 3! J A (3.37)









where D(k) denotes the Fourier transform of the cube of the Debye potential, J



D(k), (dJr)e\ k  rG (r) . J J

(3.38)

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The function D(k) has a simple pole in l!3, which arises from the existence, in 3 dimensions, of J a non-integrable 1/r short-distance singularity in the integrand. The long-distance behavior of the integral is e!ectively cut o! by the larger of the Debye wave number i or the external wave vector k. This function is evaluated explicitly in Section C.2 of Appendix C [cf. Eq. (C.31)] but for our present purposes all we need is the residue of the pole in l!3. Since this pole arises solely from the short-distance behavior, its residue does not depend on whether k or i controls the long distance behavior. Using the result (C.31) and neglecting pieces that are non-singular when l"3 gives 1 1 (i)J\[1#O(l!3)] . D(k)" J 2(4p) 3!l

(3.39)

Note that it makes no di!erence whether the factor which provides the correct dimensions is written as (i)J\, as (k)J\, or as a power of some arbitrary wave vector k, since di!erent choices merely correspond to a change in the non-singular part of D(k). For example, J




iJ\ kJ\ k " #ln #O(l!3) . 3!l i 3!l

(3.40)

As will be seen explicitly later on, it is generally very convenient to make use of this arbitrary scale in the pole residue and write all such divergent quantities in terms of a single, standard, but arbitrary parameter k with the dimensions of wave number or inverse length, a parameter that is also used to exhibit the extra dimensions that arise when the parameters are extended beyond l"3 dimensions. Thus we write the induced coupling g as ?@





1 (e e ) 1 ? @ #g (k) . g "kJ\ ! ?@ ?@ 4! (4p) 3!l

(3.41)

In view of the result (3.39), the "rst pole term in this expression cancels the singular contributions arising from D(k) in Eq. (3.37). The prefactor kJ\ absorbs the variation in dimension when J l departs from l"3, and so the remaining "nite coupling g (k) always retains its l"3 dimen?@ sions. This "nite (or `renormalizeda) coupling must depend upon k in such a way as to ensure that the bare coupling g is independent of the arbitrary value of k. Thus we have de"ned the "nite ?@ coupling g to be a scale-dependent `#oatinga coupling, and we shall later exploit the renormaliz?@ ation group results that follow from the arbitrary character of k. For now, we simply note that g (k) will soon be determined by matching the e!ective theory to the underlying microscopic ?@ theory. Similar considerations apply to the induced couplings of higher loop order. At four-loop order, the irreducible correlator CI (k) receives contributions from the h and k derivative interactions ?@ ?@ ?@ which are proportional to k. Therefore, to determine the pole parts of these couplings, it is su$cient to focus just on those contributions to CI (k) (at four-loop order) which are also ?@ proportional to k and singular as lP3. [There are additional singular contributions to CI (k) at ?@ four-loop order which are proportional to i. The renormalization of these terms requires the four-loop coupling g in addition to h and k . The determination of g is discussed below.] ?@A ?@ ?@ ?@A

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51

There is only one four-loop diagram constructed from the classical interaction which contributes to CI (k) and contains a term singular as lP3 that is proportional to k: ?@ (3.42) In addition, the following four-loop order diagrams involving the classical interaction plus the "nite h-induced interaction contain terms proportional to k which are singular as lP3: ? (3.43)

Here, the circled `Xa denotes the vertex generated by the h induced interaction. Since h itself ? ? counts as a two-loop factor, these diagrams contribute to the correlator at four-loop order. The contributions of the h and k interactions, plus the above graphs, give ?@ ?@ ee ee CI  (k)"bnn 2kk !k ? @ (h#h)D(k)! ? @ D(k) ?@ ? @ ?@ ? @ J 3! 5! J










 ee # kd b nn 2h !2h ? A D(0) #2 , ?@ ? A ?A ? 3! J A where only the pieces proportional to k have been displayed, and where

(3.44)



D(k), (dJr)e\ k  rG (r) . J J

(3.45)

This integral may be evaluated explicitly using the methods of Appendix C. However, the part of the integral which is proportional to k and singular as lP3 arises solely from the short-distance singularity in the integrand. To extract just this portion of the integral, it is su$cient to use unscreened Coulomb potentials instead of the Debye potential. The resulting Fourier transform of < (r) is evaluated in Appendix C [cf. Eq. (C.14)] where it is shown that J k J\ 1 k (dJr)e\ k  r< (r)"! #"nite . (3.46) J 3!l 4!(4p) 4p






Using this result plus (3.39), it is easy to see that the four-loop O(k) part of CI (k) will be "nite as ?@ lP3 provided
















p e e  1 ? @ h "kJ\ !h #h (k) , ?@ ? 3 4p ?@ 3!l

(3.47)

and



1 2p e e  1 p e e  ? @ ? @ (h#h) # #k (k) . k "kJ\ ! ? @ 3!l ?@ ?@ 3!l 3! 4p 4! ) 5! 4p

(3.48)

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Just as before with g (k), the "nite renormalized couplings h (k) and k (k) can only be determined ?@ ?@ ?@ by matching with the full quantum theory. The "nal four loop induced coupling g multiplies three factors of bare particle densities. The ?@A most convenient way to determine the poles in this coupling is to consider the (irreducible part of the) triple density correlator,



d ln Z . KI (k, q), (dJr)(dJr)(dJr)e k  r>q  rY\k>q r ?@A dbk (r)dbk (r)dbk (r) ? @ A

(3.49)

This correlator receives a contribution of !3!g bnnn from the g induced coupling. It also ?@A ? @ A ?@A receives contributions, which are singular as lP3, from the four-loop diagrams:

(3.50)

(plus 5 other versions of the second diagram, and 2 other versions of the third diagram, in which the labels are permuted in various ways). In addition, there is a singular four-loop contribution involving the two-loop coupling g . Provided the external momenta k and q are non-zero, one ?@ may replace the Debye potentials in all these diagrams by unscreened Coulomb potentials without changing the residue of the 1/(l!3) poles. In order for the sum of these contributions to be "nite, the four-loop coupling g must have both single and double poles in l!3. The resulting structure ?@A for g , and yet higher-order couplings, will be discussed further in Section 6. ?@A 3.5. Matching The most direct approach to determine the "nite part of the two-loop coupling g given in Eq. ?@ (3.41) is to compare the density}density correlator KI (k) in the e!ective theory and the original ?@ quantum theory. Because the induced coupling g makes a contribution to CI (k) (and hence to the ?@ ?@ full correlator KI (k)) proportional to nn, it is su$cient to retain in both the e!ective and ?@ ? @ fundamental theories only those contributions with the same nn dependence on the bare ? @ densities. Since it is the short-distance contributions which must be correctly matched, Debye screening may be completely ignored [3] if one compares the correlator evaluated at a non-zero wave number k. Consequently, to determine the two-loop coupling g it is su$cient to work just to ?@ second order in the fugacity expansion. And because the induced coupling g makes ?@  There are also `contacta terms proportional to d , d or d , analogous to the second term appearing in (3.37). ?@ @A ?A However, the required pole terms in g may be entirely inferred from the non-contact terms in KI (k, q) which are ?@A ?@A proportional to nnn. The resulting value of g necessarily also renders the contact terms "nite, just as seen explicitly ? @ A ?@A at two-loop order in (3.37).

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53

a momentum-independent contribution to the correlator (3.37), it is also su$cient to work in the limit of small momentum k;j\ and neglect all contributions which vanish as kP0. The tree and one-loop contributions to the correlator are given by Eqs. (2.104) and (2.108)}(2.110). Two-loop contributions to KI (k) which are proportional to nn arise in two ways. ?@ ? @ The one-particle irreducible part CI (k) receives such a contribution from the "rst term in (3.37). In ?@ addition, there is a one-particle reducible contribution arising from the two-loop h interaction ? appearing in Eq. (3.32). This may be seen as follows. The h interaction generates, through the last ? term of (3.35), a two-loop contribution of !2bd hnk to the irreducible correlator CI (k). ?@ ? ? ?@ A two-loop reducible contribution to the full correlator KI (k) of  ?@ 2bnne e (h#h)kGI (k) (3.51) ? @ ? @ ? @ is then generated by the two-loop cross-term in Eq. (2.110) which results from this irreducible contribution together with the lowest-order piece contained in Eq. (2.104). As noted above, for this matching calculation (only), we may neglect Debye screening by sending iP0. In this limit, the electrostatic potential correlator G(k) is, to lowest order, just 1/(bk). In other words, the 1/k of the (Fourier transformed) Coulomb potential cancels the k appearing from the two derivatives in the h interaction, leading to result which (with the neglect of Debye screening) is non-vanishing as kP0. ? Consequently, the relevant portion of the complete correlator KI (k) in the e!ective theory is ?@ be e (be e ) (be e ) KI (k)"nn ! ? @ # ? @ D(k)! ? @ D(k) J J ?@ ? @ k 2 3!





!2bg #2be e (h#h) #2 ?@ ? @ ? @

(3.52)

where 2 denotes irrelevant terms with di!erent dependence on the bare densities. With the neglect of Debye screening, the integrals DK(k) reduce to Fourier transforms of powers of the J  To carry out the matching for the four-loop derivative coupling k , one would need to evaluate and compare the ?@ O(k) terms in the density}density correlator.  This term may equivalently be described as arising from the "rst term of (3.35) when one variation acts on the

) k part of the h interaction in Eq. (3.32) and the other variation acts on the classical interaction. A A  An independent way to derive the h terms in the result (3.52), which illuminates the character of the theory, is as ? follows. In our construction of the interaction terms in the e!ective theory (3.32), we "xed the meaning of the functional integration "eld by requiring that the invariance P !ic, k Pk !e c be maintained, implying that this "eld and ? ? ? the chemical potentials always appear in the combination k #ie . This requirement casts the theory in its most useful ? ? form. However, since is simply a dummy integration variable, one is free to make "eld rede"nitions that violate this restriction, and it is sometimes convenient to do so temporarily. Since n(r)Jexp+bk (r),, the cross term in the ? ? h interaction involving

) k may equivalently be written as 2b ie he @C? (

) n. To "rst order in h, which is all ? ? ? ? ? ? ? that concerns us, the "eld rede"nition

P !2b ie hne @C@ ( @ @ @ @ in the kinetic term (b/2)(

) removes the cross-term. The e!ect of the same "eld rede"nition on the classical interaction term ! ne @C? (, again to leading order in h, is a change in the action that is equivalent to the induced coupling ? ? ? alteration g Pg !e e (h#h) . ?@ ?@ ? @ ? ? This combination is precisely what appears in Eq. (3.52), and serves as an independent check on the validity of that result.

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original Coulomb potential,



CK(k)"lim DK(k)" (dJr)e\ k  r < (r)K . J J J G The iP0 limit of (2.113) immediately gives

(3.53)

1 C(k)" ,  8k

(3.54)

and in the "rst part of Appendix C it is shown [cf. Eq. (C.13)] that


 



k J\ 1 1 #3!c#O(l!3) , C(k)" J 3!l 2(4p) 4p

(3.55)

where c"0.577212 is Euler's constant. Inserting g from (3.41), and using ?@ k J\ k 1 (k)J\! "ln lim 4p 4pk 3!l J to take the physical lP3 limit yields




 






 

(3.56)



k be e (be e ) p be e  ? @ ln !3#c KI (k)"nn ! ? @ # ? @ # ?@ ? @ 4pk k 16k 3 4p be e j !2bg (k)# ? @ ?@ #2 ?@ 24p

(3.57)

for the nn piece of the density}density correlator, neglecting Debye screening, to two-loop order. ? @ In writing the last term of (3.57) we have made use of the de"nition (3.33) of h to express ? j j#j @ " ?@ , (3.58) h#h" ? ? @ 48pb 48pb in which j is the thermal wavelength for the reduced mass m "[1/m #1/m ]\. ?@ ?@ ? @ We write the corresponding result in the underlying quantum theory as the Fourier transform of the density}density correlator K (r , r ),1n (r )n (r )2 !1n (r )2 1n (r )2 ?@   ?  @  @ ?  @ @  @ "Z\ Tr [e\@&> ? @I? ,? n (r )n (r )]!1n (r )2 1n (r )2 . ?  @  ?  @ @  @

(3.59)

 We are glossing over a subtlety here, for Eq. (3.59) involves the equal time expectation value 1n (r , 0)n (r , 0)2 , ?  @  @ whereas our desired correlator is the zero-frequency correlation function (3.10). The di!erence between these two is just the sum of correlations at all non-zero Matsubara frequencies u . However, as discussed at the end of this section, L non-zero frequency correlators are proportional to k (due to current conservation), and hence do not a!ect the matching for the g interaction in the e!ective classical theory, which may be extracted from the kP0 behavior of the ?@ density}density correlator. On the other hand, the di!erence between the equal-time and zero-frequency correlators would be essential for the matching of the four-loop couplings k since these depend on the O(k) terms of the ?@ zero-frequency correlator.

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55

The subtraction of 1n 2 1n 2 removes what would otherwise be a delta function contribution to ? @ @ @ the Fourier transform at k"0, and is completely ignorable when working at kO0. We speci"cally want the second-order contribution in the fugacity expansion of K (r , r ). For our purposes, this ?@   is most conveniently obtained by using an (old-fashioned) expansion of the trace in terms of ordinary quantum-mechanical multi-particle states rather than using many-body quantum "eld theory. The desired second-order terms in the fugacity expansion come from the two-particle subspace of the thermodynamic trace over all particle states, so that



1 K(r , r )" e@IA e@IB (dr)(dr)1rc, rd"e\@&AB n (r )n (r ) ?  @  ?@   2 AB ;["rc, rd2$"rd, rc2] .

(3.60)

Here, "rc, rd2 denotes the (un-symmetrized) two-particle basis ket with one particle of species c at r and one of species d at r, H is the ("rst-quantized) two-particle Hamiltonian, AB p p ee A B H "  #  # , (3.61) AB 2m 2m 4p"r !r " A B   and the $ sign in the "nal combination of ket vectors accounts for Bose (#) or Fermi (!) statistics. To avoid a clutter of notation, we temporarily use the indices a, b to denote spin components as well as species labels. Now n (r )n (r )["rc, rd2$"rd, rc2] ?  @  "+d d(r !r)#d d(r !r),+d d(r !r)#d d(r !r),["rc, rd2$"rd, rc2] ?A  ?B  @A  @B   P+d d(r !r )d d(r !r)#d d(r !r)d d(r !r),["rc, rd2$"rd, rc2] , (3.62) ?@   ?A  ?A  @B  where in the last line terms which become equivalent when inserted into (3.60) have been combined. Since we are only interested in terms proportional to nn (or equivalently e@I? e@I@ ) the contact term ? @ involving d d(r !r ) may be neglected. The density operators n (as well as the Hamiltonian ?@   ? H ) are spin independent, so that the sum over particle spins just produces the spin degeneracies AB g and g . Hence, reverting to the previous notation in which the indices a, b label only di!erent ? @ species without regard to spin, the required piece of the quantum mechanical density}density correlator is given by (3.63) K(r , r )"g e@I? g e@I@ [1r , r "e\@&?@ "r , r 2$(d /g )1r , r "e\@&?@ "r , r 2] . @     ?@ ?     ?@   ? At this point, it is convenient to write the two-particle Hamiltonian in terms of center-of-mass R and relative r coordinates, H "H #H , ?@ ?@ ?@

(3.64)

 As it stands, this contact term is infrared divergent since Debye screening, which involves an arbitrary number of particles, is needed to provide the long-distance cuto! which makes the contact term infrared "nite. It is precisely because the required value of the induced coupling g can be deduced solely from the non-contact part of the correlator that it is ?@ permissible to ignore Debye screening in this matching calculation and just use a fugacity expansion.

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with P , (3.65) H " ?@ 2M ?@ the Hamiltonian for center-of-mass motion with total mass M ,m #m , and ?@ ? @ e e p # ? @ (3.66) H " ?@ 2m 4p"r" ?@ the Hamiltonian for relative motion with reduced mass m\,m\#m\. The Fourier transform ?@ ? @ now reads



  ?@ "R2[1r"e\@&?@ "r2$(d /g )1r"e\@&?@ "!r2] . K I (k)" (dr)e\ k  rg e@I? g e@I@ 1R"e\@& ? @ ?@ ? ?@

(3.67) Our goal is to compare K I (k) with the e!ective theory result (3.57), and to adjust the "nite ?@ coupling g (k) in the e!ective theory so that both results coincide up to corrections that vanish as ?@ kP0. The center-of-mass matrix element is just



?@ "R2" 1R"e\@&





(dP) P exp !b "K\ , ?@ (2p) 2M ?@

(3.68)

where




2pb  (3.69) K "

?@ M ?@ is the thermal wavelength of the center-of-mass motion. We shall also make use of the thermal wavelength of the relative motion,




2pb  j "

. (3.70) ?@ m ?@ Note that since the product of the reduced mass m and the total mass M is just the product of ?@ ?@ the separate masses, m M "m m , the corresponding relation also holds for the thermal ?@ ?@ ? @ wavelengths, K j "j j . Hence ?@ ?@ ? @ (3.71) g e@I? g e@I@ K\"nnj , @ ?@ ? @ ?@ ? and we may write K I (k)"nnj [F (k)$(d /g )F (k)] , ?@ ? @ ?@ > ?@ ? \

(3.72)

with



?@ "$r2 . F (k)" (dr)e\ k  r1r"e\@& !

(3.73)

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57

As shown in Appendix D, an explicit representation for the matrix elements F (k) may be found by ! expressing the relative Hamiltonian in terms of the generators of an su(1, 1) algebra. The result for the direct term, given in Eq. (D.67), is



be e (be e ) F (k)"j\ ! ? @ # ? @ > ?@ k 16k








j k be e p be e  8pj ? @ ? @ ?@ #f # ln ?@ !3#c# 4p 4pj 3 4p (be e ) ?@ ? @ where the function f (y) has the (convergent) power series expansion



#O(k) ,

(3.74)

3(p  f(n#1) 3 (!(py)L . (3.75) f (y)"! ! 2 C((n#5)/2) 4y L The asymptotic behavior of this function as yP$R is spelled out in detail in Eqs. (D.71) and (D.74). Here we note that in the case of strong repulsive interactions corresponding to yP#R, f (y) increases only as ln y, with 1 8 #O(y\) . f (y)&ln(4py)#3c! ! 3 4py

(3.76)

For the case of strong attractive interactions with the resulting deeply bound Coulombic states when yP!R, f (y) grows very rapidly, 3 exp+py, . f (y)& py

(3.77)

(This asymptotic result corresponds exactly to the contribution of the hydrogen-like intermediate ground state term in Eq. (3.67).) The corresponding result for the exchange piece, given in Eq. (D.98), is





be  be p ? fI ? #O(k) , F (k)" j\ \ 4pj 3 ?? 4p ?? where the function fI (y) has the (convergent) expansion

(3.78)





f(n#2) 3 3 ln 2 3(p  1 3 (!(py)L 1! ! # ! . fI (y)" 2 2y 2>L C((n#5)/2) 8py 2py L The yPR asymptotic behavior is given in Eq. (D.97), and yields the strong decrease





3p 2(3 exp ! (2y) . fI (y)& yp 2

(3.79)

(3.80)

Note that since the j were de"ned in terms of the reduced mass of the a}b system, j "(2j . ?@ ?? ? Inserting the results (3.74) and (3.78) into Eq. (3.72), and comparing to the result (3.57) computed in the e!ective quasi-classical theory, we see that the two results coincide provided that




p e e  ? @ [ln(kj )#C ] , g (k)"! ?@ ?@ ?@ 6 4p

(3.81)

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where C ,f (g )$(d /g ) fI (g ) , (3.82) ?@ ?@ ?@ ? ?? with g denoting the quantum parameter for species a and b, ?@ be e g , ? @, (3.83) ?@ 4pj ?@ and where, as usual, the exchange term in (3.82) comes in with a plus (minus) sign if species a is a Boson (Fermion). In the limit of strong repulsion, g P#R, Eqs. (3.76) and (3.80) inform us that ?@ 8 beek p e e  ? @ ? @ ln #3c! . (3.84) g (k)&! ?@ 3 4p 6 4p









Note that this limit does not involve Planck's constant : The argument of the logarithm entails the classical ratio of the Coulomb energy of two charges a distance k\ apart to the temperature. (When this coupling is inserted in physical quantities, it will appear with a ln(i/k) term which turns the arbitrary distance k\ into the Debye length i\.) In view of Eq. (3.76), the "rst correction to this classical limit is of order . On the other hand, in the limit of strong attraction, g P!R, the ?@ exponential blowup exhibited in Eq. (3.77) shows that our perturbative development breaks down, as it must, since in this limit the ionized plasma must condense into neutral atoms. This is, of course, a highly quantum-mechanical regime. Finally, for small g , the exchange term (3.79) ?@ dominates and, with j "(2j , one has ? ?? 4p  3j d ? . (3.85) C &$ ?@ ?@ g be 2(2p ? ?




 Previous work [4,6,7] makes use of dimensionless parameters m (also called x ) related to our notation by ?@ ?@ m "!(4pg , and functions Q(m ) , E(m ) of these parameters. To establish contact with this prior work (which also ?@ ?@ ?@ ?@ does not use our rationalized Gaussian electrostatic units), we note that



 



6 4p  j  1 d ?@ C(g )#c#ln 9!1" m #Q(m )$ ?@ E(m ) . ?@ ?@ ?@ ?@ be e 2p 6 g (2 ? @ ? Here, j " [2pb/m ] is our de"nition of the thermal wavelength; various previous work uses the same symbol to ?@ ?@ denote either (b/m or (b/2m . Note that our e e /(4p) becomes just e e when converting to unrationalized ?@ ?@ ? @ ? @ electrostatic units.  Writing the result in terms of the Coulomb distance (1.4), but for the speci"c charges e , e , d "be e /4p, gives ? @ ?@ ? @ 1 4p e e  ? @ [ln(d k)#2] . g (k)&! ?@ ?@ 2 3! 4p




This form is in precise accord with the remarks made in Footnote 10. Namely, the coe$cient of ln(d k) exactly ?@ corresponds to the two-particle part of the partition function, with the exponential of the Coulomb interaction expanded to third order and the integration over the relative coordinate cut o! at the short distance d and at the long distance ?@ k\.

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59

Noting that this multiplies (n)e, the result appears as an exchange term independent of the ? ? particle's charge. Indeed, we shall shortly see in the following Section 4 that this is just the usual free particle exchange correction that is quadratic in the fugacity. The next term of order g\ in the ?? exchange contribution fI (g ) gives the familiar order e exchange correction to the plasma. ?? With the single two-loop coupling g completely determined by Eqs. (3.41) and (3.81), one may ?@ now use the quasi-classical e!ective theory to compute thermodynamics, or other quantities of interest, to two- or three-loop order. Before four-loop calculations of physical quantities can be performed, the "nite renormalized parts of the four-loop couplings g , h , and k would need to ?@A ?@ ?@ be determined by an analogous higher-order matching calculation. This we have not attempted to do. However, it should be noted that determining the "nite part of the g couplings requires a fully ?@A quantum-mechanical three-body calculation which (almost certainly) is not possible to do analytically. Only when this three-body matching is accomplished will it be possible to extend the perturbative analysis of Coulomb plasmas to four (or higher) loop order. 3.6. Non-zero frequency modes Up to this point, the e!ects of the non-zero frequency components of the potential (r, q), de"ned by the Fourier series (3.26) and repeated here for convenience, 2pm ,

(r, q)" (r)# K(r) e\ SK O, u " K b K$ have been ignored. These components, which obey the reality constraint

(3.86)

K(r)H" \K(r) ,

(3.87)

characterize quantum #uctuations in the electrostatic potential. They decouple from the zerofrequency degrees of freedom and could be trivially integrated out in the leading-order classical limit. But in higher orders, this is no longer true. To examine the e!ects which result from non-zero frequency #uctuations, we return to the full quantum theory whose functional integral representation (3.6) may be rewritten as

 

      

1 @ dq (dJr)(

(r, q)) exp+!S [ ; k], Z [k]"N [d ] exp !  /+  2  b "N [d ] “ [d K] exp ! (dJr) "

(r)"# "

K(r)"  2 K$ K$ ;exp+!S [ ; k], . (3.88)  In the "rst line, the integration measure [d ] represents functional integration over the space}time dependent "eld (r, q). In the second line, [d ] now stands for functional integration over just the



 This same issue arises regardless of whether one is using our e!ective "eld theory approach, or more traditional methods. Results for some contributions to the pressure at four-loop order [or O(n)] have recently been reported [11]. However, these partial results do not include the most di$cult contributions which are sensitive to the short-distance behavior of three-body Coulomb systems.

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time-independent (static mode) (r), while in the following product [d K] denotes functional integration over the remaining non-zero frequency modes. The prefactor N involves the square  root of the functional determinant of the Laplacian operator for all of the modes, N "  Det[!b ]. The "nal factor of e\1  denotes the product of Gaussian functional integrals for each charged species,



  



 @ exp+!S [ ; k],, “ [dtH dt ] exp ! dq (dJr)L ,  ? ? ?  ?

(3.89)

with





*

 L "tH(r, q) ! !k (r, q)!ie (r, q) t (r, q) . (3.90) ? ? ? ? ? *q 2m ? We are now allowing arbitrary temporal, as well as spatial, variation in the chemical potentials so that the resulting partition function may be used to generate time-dependent number density correlators. This will be of use in Section 7. In Appendix B, we derive the large mass asymptotic expansion of S , and give the explicit form for both "rst- and second-order corrections in j (or  inverse mass). For our present purposes, only the classical term and the "rst-order corrections contained in Eq. (B.54) are required. They are





 bj S [ ; k]" (dJr)n(r) e C? @(r !1# ? [ k (r)#ie

(r)]  ? ? ? 48p ? 1 bj ? ! [ kK(r)#ie

K(r)] ) [ k\K(r)#ie

\K(r)]#2 . (3.91) ? ? ? ? m 16p K$ The zero-frequency parts which appear in (3.91) have already been included in the e!ective theory. These are precisely the classical e C? @( interaction, plus the "rst derivative interaction in (3.32) involving h[ k (r)#ie

(r)]. Because of the presence of the exponential factor e C? @(r, note ? ? ? that the third term in (3.91) generates a coupling between the non-zero frequency modes of and the static mode. The "nal ellipsis denotes higher-order terms containing at least four spatial derivatives acting on various factors of b(k#ie ) [in such a way that every "eld is di!erentiated at least once], with each derivative accompanied by a factor of some thermal wavelength j. The four-derivative O(j) terms, which are formally of four-loop order, are exhibited explicitly in Eq. (B.54). The expansion (3.91) is valid if the potential (r, q) varies slowly on the scale of the thermal wavelength j . As discussed earlier, inserting this expansion (truncated at some order) into the ? functional integral (3.89) completely mangles the e!ects of short-distance #uctuations in . However, as with any e!ective "eld theory, the resulting errors are compensated, to any given order in ij, by including the requisite induced interactions and suitably adjusting their coe$cients. At this point, one may contemplate completely integrating out the non-zero frequency modes of

in order to generate an e!ective theory containing only the static potential (r). But doing so would be a mistake. Integrating out the non-zero frequency modes is no longer trivial because of the coupling between the static and non-zero frequency modes. More importantly, the resulting functional of (r) could not be adequately approximated by any set of local interactions. Correlations of the non-zero frequency components of only decrease like 1/r (due to the long-range



L.S. Brown, L.G. Yawe / Physics Reports 340 (2001) 1}164

61

nature of Coulomb interactions), and are not Debye screened. This will be demonstrated below. In physical terms, the absence of Debye screening in the potential correlations at non-zero (Matsubara) frequencies re#ects the e!ect of inertia on the response of charges in the plasma. Consequently, if one completely integrates out the non-zero frequency components of , then the resulting theory will contain complicated non-local interactions which decrease only algebraically with distance. To produce a useful e!ective theory, that can be approximated by local interactions, one must explicitly retain in the e!ective theory all degrees of freedom with long distance correlations } including the non-zero frequency components of . In other words, the complete e!ective theory must have the form



[ , K; k], , Z[k]"N [d ] “ [d K] exp+!S [ ; k]!S [ ; k]!S   U    K$ where S and S are given in Eqs. (3.29) and (3.32), respectively, and   b b "

K"! jne C? @( S [ , K; k]" (dJr) ? ? U   2 8pm K$ ?







;[ kK#ie

K] ) [ k\K#ie

\K]#2 . ? ? ? ?

(3.92)

(3.93)

Here kK(r) denotes the non-zero frequency components of the chemical potential, ? (3.94) k (r, q),k (r)# kK(r) e\ SK O . ? ? ? K$ The "nal ellipsis in (3.93) denotes yet higher-order terms involving four or more derivatives, as well as non-zero frequency induced interactions involving "

K" multiplying products of two or more densities. At leading order (when all interaction terms in S are neglected), the correlator of the U   non-zero frequency components of (times b) is given by an unscreened Coulomb potential, 1 , bG(r, r),b1 K(r) K(r)H2 " K  4p"r!r"

(3.95)

since this is the Green's function of ! . In other words, the Fourier transformed correlator is given by bGI (k)"1/k . (3.96) K Because the sub-leading interaction in (3.93) involves the gradient

K(r), and not K(r) itself, this interaction does not cause non-zero frequency correlations to develop a "nite correlation length. Rather, it merely produces an O[(ij)] change in the residue of the 1/k pole of GI (k). Recalling K that u "2pm/b (which has units of energy in our notation) and noting that the (lowest-order) K plasma frequency u is de"ned by . ejn en (3.97) u " ? ? " ? ? ? , . 2pb  m ? ? ?

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we "nd that with this correction,





u . . (3.98) [bGI (k)]\"k 1# K u K This same result is obtained from the kP0 limit of the sum of ring diagrams contributing to this correlator, which generates a denominator involving the one-loop polarization function P(k, u). This well-known function is presented in Eq. (B.40) of Appendix B, as well as in many textbooks (such as Ref. [30]). The kP0 limit corresponds to the classical limit, and the continuation u / Pi(u!ie), eP0>, further produces the classical retarded response function. The resulting K pole at u"u corresponds to the propagation of classical longitudinal plasma waves, waves . whose resonant frequency is independent of their wave number. The lack of Debye screening of the non-zero frequency #uctuations in the electrostatic potential is an exact result. It is a consequence of electromagnetic current conservation, do/dt# ) j"0, or equivalently gauge invariance. The fundamental quantum theory (3.6) is, in particular, invariant under time dependent, but space independent, gauge transformations, ds(q) ,

(r, q)P (r, q)# dq

t (r, q)Pe C? QOt (r, q) . ? ?

(3.99)

The e!ective theory must necessarily share this invariance. But in the e!ective theory, where the charged "elds have been integrated out, these gauge transformations reduce to arbitrary constant shifts in the non-zero frequency components of ,

K(r)P K(r)#iu sK , (3.100) K where sK are the Fourier components of s. This means that the e!ective theory (3.93) can never depend on the non-zero frequency "elds K(r) other than through their gradients. And this implies that arbitrarily long wavelength #uctuations in the non-zero frequency components of must have arbitrarily low action, which in turn implies that the Fourier transform of the correlation function 1 K(r) K(r)H2 will diverge as kP0. In other words, the interactions of the e!ective theory cannot cause the pole in the non-zero frequency correlator to shift away from k"0. A detailed explanation of these points is given in Section A.4. The "rst interaction term in S is formally O[(ij)] smaller than the leading "

K" term U   and thus is of two-loop order. As noted above, this term produces a relative change of this size in non-zero frequency correlators. However, it does not a!ect thermodynamic quantities, or static correlators, at two-loop order because



11

K(r) )

\K(r)22"b\

(dJk) "0 (2p)J

(3.101)

 This is completely analogous to Goldstone's theorem proving the presence of long-range #uctuations in any theory with a spontaneously broken continuous symmetry. Since the symmetry (3.100) shifts K, it is impossible for the expectation values 1 K2 to be invariant under this symmetry. Consequently, K must have long-range correlations.

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63

Fig. 14. First non-vanishing correction to ln Z involving the non-zero frequency modes of (r, q). The dashed lines represent the long-range, unscreened Coulomb Green's functions of the non-zero frequency modes. Each vertex represents one insertion of S (speci"cally that part which is linear in the static mode (r)). The relative size of the     resulting contribution is O[(ji)(bei)], since each vertex contains an overall factor of j, and the two loops of the diagram generate two powers of the loop expansion parameter bei. Because we are treating the quantum parameters be/j as "xed numbers of order one, the net result is a contribution of six-loop order. ? ?

in our dimensional continuation regularization. In fact, the non-zero frequency interaction term "rst a!ects the thermodynamic quantities at six-loop order, through the diagram illustrated in Fig. 14. However, even though the non-zero frequency interactions are suppressed by numerous powers of ij, they fundamentally alter the long-distance behavior of the static density}density correlator. Instead of exhibiting classical Debye-screened exponential decay, the correlator acquires a long-distance tail which decreases only algebraically with distance. This happens at "ve-loop order as shown in Section 7, where a simple but explicit evaluation of the resulting long-distance limit is given.

4. Two-loop results The two-loop contributions to ln Z are given by the diagrams shown in Fig. 15. They correspond to the analytic expression

 



ln Z 1 ee ln Z  # n[beG (0)]! bnn ? @ (dJr)G (r)#g . " (4.1) ? ? J ? @ 2 ) 3! J ?@ V 2 V ? ?@ As noted earlier in Eq. (2.60), the lP3 limit of G (0) is "nite and equals !i /(4p). The integral of J  the cube of the Debye Green's function,



D(0)" (dJr)G (r) , J J

(4.2)

is the vanishing wave number limit of the corresponding Fourier transform (which is the reason for the notation used here). It is computed in Section C.2 [cf. Eq. (C.32)] and shown to be


 



1 9i J\ 1 1 D(0)" #1!c#O(l!3) . J (4p) 4p 2 3!l

(4.3)

 This identity would not hold if we had chosen to employ a di!erent regularization scheme, such as a wave-number cuto!, or a hard core interaction, in de"ning the e!ective theory. Had we done so, it would be necessary to adjust the coe$cient of the classical n e @C? ( interaction in order to compensate for cuto!-dependent e!ects resulting from ? #uctuations of the non-zero frequency modes.

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Fig. 15. Two-loop diagrams contributing to ln Z. The circled &X' denotes the -independent part of the two-loop g induced interaction. ?@

The two-loop induced coupling g , given in (3.41), cancels the lP3 pole of D(0). It is convenient ?@ J to write the "nal bracket in (4.1) as

 



ee ? @ (dJr)G (r)#g "  eeD(0; k)#g (k) , J ?@  ? @ 0 ?@ 2 ) 3!

(4.4)

where, in view of Eqs. (3.41) and (4.3), 1 D(0; k)" 0 32p








9i J\ 1  #1!c#O(l!3) . !(k)J\ 4p 3!l

The physical lP3 limit is "nite, as it must be, and gives






!1 9i  !1#c . D(0; k)" ln 0 32p 4pk

(4.5)

(4.6)

Note that the coe$cient of the induced interaction that produces this "nite result was determined from a di!erent physical quantity, the density}density correlator. Nevertheless the structure of the e!ective theory [in particular, the shift symmetry (2.30)] guarantees that the single two-loop g interaction removes the cuto! dependence in any physical quantity computed to either two- or ?@ three-loop order. Putting the pieces together, including the previous one-loop result (2.82) and the value (3.81) for the renormalized coupling g (k), produces ln Z to two-loop order (as a function of bare particle ?@ densities),









 1 bei 1 bei  ln Z ?  # ?   " n 1# ? 3 4p 8 4p V ? be e  9j i p  ? @ ?@  !1#c#C . nn # ln (4.7) ? @ ?@ 4p 4p 6 ?@ As was remarked above, the leading term in C when g becomes small comes from the exchange ?@ ?@ contribution. Using the limiting form (3.85) of C gives the exchange correction ?@ ln Z 1   e@I?  "$ , (4.8) n(jn/g )"$ n ? 2 V 4(2 ? ? ? ? ? ? which is just the "rst quantum statistics correction shown in Eq. (1.13).



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65

4.1. Number density The particle number density of species a is given by * ln Z , 1n 2 " ? @ *bk V ? where the partial derivative is computed at "xed b. Inserting the result (4.7) and using *n *i ben @ "d n, " ? ?, ?@ ? *bk *bk 2i ? ?  and *g /*(bk )"0, yields @A ? bei be  1 bei  1  ?  # ?  @ # (nbe) n 1n 2 "n 1# @ 4p ? @ ? 8p 2 8p 8 ? ? @ p nbe  be e  @ A # ? ? nn @ A 4p 6 i  @A be e  9j i p  ? @ ?@  !1#c#C . # n n ln @ 4p ?@ 4p 3 ? @ The "rst bracket contains the "rst three terms in the expansion of the exponential




exp










(4.9)


 



bei ?  "exp[!beG (0)] ,  ?  8p

(4.10)

(4.11)

which is just the Boltzmann factor for the polarization correction to the self-energy of a species a particle when it is placed in the plasma. The next term is just






bedi 1 ? n , 2 ? 4p

(4.12)

where




be  di 1  @ (4.13) " n @ 4p 4p 4 @ is the change in the lowest-order Debye wave number induced by the "rst-order density correction. Thus, through the order we have computed, our result is equivalent to











 

be i be  be  ? @ # ? n n "1n 2 "n exp @ 4p ? ? @ ? 2 4p 4!i  @ p  be e  9j i ? @ ?@  !1#c#C # n ln , (4.14) ?@ @ 3 4p 4p @ where i"[ ben ] is the Debye wave number computed with the physical particle densities. ? ? ? Inverting this result to express the bare densities in terms of the physical densities is now easy since, to this order, the bare quantities in the remaining two-loop terms may simply be replaced by



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physical quantities,




 


 

be i b e ? ! n"n exp ! ? ? ? 2 4p 4! i

 be  @ n @ 4p @ p  be e  9j i ? @ ?@ ! n ln !1#c#C ?@ @ 4p 3 4p @

.

(4.15)

4.2. Energy density The internal energy density in the plasma is given by

 

u"

H V

R ln Z , "! Rb V

(4.16)

@ where now bk is kept "xed for all a. Noting that nJj\Jb\, i Jb\, and g Jb, ? ? ?  ?@ one "nds at two-loop order









 p  3 1 bei be e  ?  ! ? @ [1#g C ] , bu" n # nn ? 2 4 4p ? @ ?@ ?@ 12 4p ? ?@ or in terms of the physical densities,





(4.17)





i p  be e  9j i 3  ? @ ?@ ln !#c#C #g C . bu" n ! ! n n ? ? @  ?@  ?@ ?@ 8p 2 4p 4p 2 ? ?@ Here (and henceforth), C ,f (g )$d fI (g ) , ?@ ?@ ?@ ?@ with the functions f and fI given in Eqs. (3.75) and (3.79), respectively.

(4.18)

(4.19)

4.3. Pressure and free energy density The pressure, re-expressed in terms of physical densities, is the equation of state. To two-loop order









be e  9j i  i p  ln Z ? @ ?@ " n ! ! n n ln #c#C . bp" ? @ 4p ? 24p 6 ?@ 4p V ?@ ? And the two-loop Helmholtz free energy density is







(4.20)



 i p  be e  9j i ? @ bf" n [!1#ln(n j/g )]! ! n n ln ?@ #c#C !1 . ? ? ? ? ?@ ? @ 12p 6 4p 4p ? ?@ (4.21) 4.4. Number density correlators The two-loop diagrams contributing to the irreducible part CI (k) of the number density ?@ correlation function are shown in Fig. 16. Diagrams a}d represent (momentum independent)

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67

Fig. 16. Two-loop diagrams contributing to the irreducible part CI (k) of the density}density correlator. The arrows ?@ merely serve to indicate where external momentum #ows in and out of each diagram; the vertices where momentum #ows in and out should be regarded as having attached species labels a and b, respectively. The circled &X' in diagram l denotes the contribution from the two-loop interactions proportional to either g or h. Diagrams a}d are two-loop contribu?@ ? tions to the contact term d 1n 2 . These terms, together with part of diagram l, simply provide the two-loop correction ?@ ? @ to the number density 1n 2 . Not shown are re#ected versions of diagrams e and h which di!er only by an interchange of ? @ incoming and outgoing vertices. Diagram e and its re#ection are one-loop corrections to the densities that appear in the vertex factors in the one-loop result for CI (k). Diagram f is a one-loop density correction to the Debye wave number in ?@ the Debye Green's function appearing in the one-loop CI (k). Thus, the e!ect of diagrams a through f (plus a part of l) is ?@ merely to put the correct physical densities, to two-loop order, in the previous one-loop CI (k). Only diagrams g through ?@ k (plus the remaining part of l) give non-trivial, two-loop corrections to CI (k). ?@

contributions to the contact term d 1n 2 . As noted earlier, only diagrams d and k (plus the ?@ ? @ induced coupling contribution l) are singular as lP3; all the other diagrams may be evaluated directly in three dimensions. The explicit contributions of these diagrams to CI (k) are given by ?@ 1 R "d bne G (0) , (4.22) ? ?@ ? ? 2  1 R "!d bnnee D(0)G (0) , @ ?@ ? A ? A 2   A 1 R "d bnnneee D(0) , A ?@ ? A B ? A B 2  AB

(4.23) (4.24)

 There are two versions each of diagrams e and h di!ering only by the interchange a  b and kP!k. The contributions of both diagrams of each pair are included in R and R . C F

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1 R "!d bnnee D(0) , B ?@ ? A ? A 3! J A 1 R "!bnn(ee#ee) D(k)G (0) ,  C ? @ ? @ ? @ 2  1 R " bnnneee D(k)G (0) ,  D ? @ A ? @ A2  A 1 R " bnnneee D(k) , E ? @ A ? @ A 2  A 1 R " bnnn(ee#ee)e D(k) , F ? @ A ? @ ? @ A2  A 1 R "! bnnnneeee D(k) , G ? @ A B ? @ A B2  AB 1 R "! bnnnneeee D (k) , H ? @ A B ? @ A B2 ( AB 1 R "!bnnee D(k) , I ? @ ? @ 3! J R "!2bg nn!2d bg nn!2d bhnk , J ?@ ? @ ?@ ?A ? A ?@ ? ? A where the required integrals are



DL(k), (dJr)e\ k  rG (r)L , J J

 

(4.25) (4.26) (4.27) (4.28) (4.29) (4.30) (4.31) (4.32) (4.33)

(4.34)

DJKL(k), (dJr)(dJr )e\ k  rG (r!r )JG (r )KG (r)L ,  J  J  J J

(4.35)

DIJKL(k), (dJr)(dJr )(dJr )e\ k  rG (r!r )IG (r !r )JG (r )KG (r)L ,   J  J   J  J J

(4.36)



D (k), (dr)(dr )(dr )e\ k  rG (r!r )G (r!r )G (r !r )G (r )G (r ) .              (

(4.37)

These integrals are evaluated in Section C.2 (with help from Ref. [32]). By examining the graphical structure, it is easy to see that the irreducible density correlator has the form CI (k)"d n FI (k)#(ben )FI (k)(ben ) , (4.38) ?@ ?@ ? ?  ? ? ?@ @ @ which generalizes the one-loop result (2.104). The derivative interaction involving the induced coupling h is responsible for generating the k-dependence in the d contact term, ? ?@ FI (k)"1!2bhk . (4.39) ? ?

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69

The terms R through R , together with the second part of the renormalization term R , just give ? B J the two-loop corrections to the number density 1n 2 that appears in d n in the general form ? @ ?@ ? above. It is straightforward to show that these terms are just the two-loop parts in the previous result (4.10) for the number density. The one-loop correction, given in Eq. (2.104), involves FI (k)"D(k) . ?@  Recalling that the one-loop density correction reads

(4.40)

(4.41) d1n 2 "!beG (0)n , ? ? @  ?  we see that the two-loop term R gives the one-loop correction for each of the two explicit density C factors appearing in the second term of (4.38), with FI (k) taking on its one-loop value D(k). ?@  Writing out D(k) in terms of Fourier integrals [as is done explicitly in Eq. (C.39)], it is easy to  see that 1 dD(k)  . (4.42) D(k)"!  2 di  Therefore R accounts for the correction to the one-loop FI (k) brought about by replacing the D ?@ bare Debye wave number i by its one-loop corrected value. In summary, the two-loop terms  R through R , plus the second piece of R , just provide simple density corrections to the one-loop ? D J CI (k), and all of these terms may be omitted if the correct physical densities n are used in the ?@ ? construction of CI (k). ?@ To assemble the remaining terms in the two-loop, irreducible correlator, we "rst use the explicit form (3.41) of the induced coupling g to write the sum of R and the "rst piece of R as ?@ I J 1 R (k)!2bg nn"!bnn (ee)D(k; k)#2g (k) (4.43) I ?@ ? @ ? @ 3! ? @ 0 ?@





where 1 kJ\ D(k; k),D(k)! . 0 J 2(4p) 3!l

(4.44)

This is the generalization to non-vanishing wave number of D(0; k) previously introduced in 0 Eq. (4.5). Using the result (C.30) for D(k) in dispersion relation form and taking the physical lP3 J limit yields











1 1 9i  k 1 D(k; k)"! ln !3#c#2 ds #3i . (4.45) 0 (4p) 2 4pk s k#s G Alternatively, using the result (C.31) for D(k) evaluated in terms of elementary functions gives the J l"3 limit






9i 1 1 6i k k D(k; k)"! ln !3#c# arctan #ln 1# 0 4pk (4p) 2 k 3i 9i



.

(4.46)

Combining the one-loop result (4.40) with this renormalized contribution of R plus the other I non-trivial two-loop terms R through R , and recalling the de"nition (6.3) of the coupling g (k), E H ?@

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leads to





1 FI (k)"D(k) 1# bn eD(k) # bn (e #e )eD(k) A A  A ? @ A  ?@  2 A A ! bn n ee[D(k)#D (k)] A B A B  ( AB (4.47) !be e [D(k)!(4p)\(ln kj #C )] .  ?@ ?@  ? @ 0 The functions D(k), D(k), and D(k) are given in dispersion relation form in Eqs. (C.21),    (C.51) and (C.52) of Appendix C. The same functions are also expressed in terms of elementary functions and the Euler dilogarithm (or Spence function) in Eqs. (C.21), (C.56), and (C.57). The function D (k) is not so tractable. However, it has been expressed in terms of a one-dimensional ( integral by Rajantie [32]. His result is quoted in Eq. (C.73) of Appendix C. The kP0 limit of the irreducible correlator is related to the particle number #uctuations. Using the results (C.22), (C.53), (C.54), and (C.74) for D(0), D(0), D(0) and D (0), plus Eq. (4.6)    ( for D(0), we have 0 1 bn e 1 1 b(e #e )e A A # n ? @ A 1# FI (0)" A (4p)i ?@ 2 8pi 6 8pi A A 1 1 be e 9j i bee ? @ ln ?@ A B# ! n n !1#c#C . (4.48) A B ?@ 12 4p (4p)i 6 (4p) AB To check this result, we note that, as is shown in Eq. (A.68) of Appendix A, there is a simpler way to obtain the same result, namely:












R1n 2 R1n 2 ? @ "! @ @ , CI (0)"! (4.49) ?@ Rbk Rbk @ ? where the partial derivatives are to be computed at "xed b. It is a straightforward matter to take the bk derivative of the two-loop result (4.10) for the density 1n 2 and con"rm [via Eq. (4.38)] that @ ? @ the result (4.48) is indeed correct.

5. Three-loop thermodynamics There are nine diagrams, shown in Fig. 17, which contribute to ln Z (or equivalently the pressure) at three-loop order. Diagrams a}f are "nite as lP3 and may be computed directly in 3 dimensions. The explicit contributions of these diagrams to (ln Z)/V are:





1 1 C #C " bnnnneeee D # D(0) , ? @ ? @ A B ? @ A B 4! + 2  ?@AB 1 1 D(0)G (0)# D(0) , C #C "! bnnneee  A B ? @ A ? @ A 2  2  ?@A 1 C "! bne G (0) C ? ? 2 ) 3!  ?





(5.1) (5.2) (5.3)

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71

Fig. 17. Three-loop diagrams contributing to the ln Z. The circled &X' denotes the insertion of the part of the two-loop g and h interactions which are quadratic in . ?@ ?





1 1 C #C " bnnee D(0)G (0)# D(0) , D E ? @ ? @ 2   2 ) 4! J ?@ 1 D(0)G (0) , C " bnnee J F ? @ ? @ 2 ) 3! J ?@ C " bg nn(e #e )G (0)! bhnei G (0) . G ?@ ? @ ? @  J ? ? ?  J ?@ ? The last term of Eq. (5.6) comes from i b11

(r)22"!i G (0) &
 ,  J 4p J which easily follows using the same method that lead to Eq. (2.60). The integral



D , (dr)(dr)(dr)G (r)G (r)G (r)G (r!r)G (r!r)G (r!r) +      

(5.4) (5.5) (5.6)

(5.7)

(5.8)

corresponds to the `Mercedesa graph a of Fig. 17, and the other required integrals are de"ned in Eqs. (4.34)}(4.36). These integrals are evaluated in Section C.2 (with help from Refs. [32,36]). Once again, the "nal contribution C involving the single two-loop coupling g removes the shortG ?@ distance singularities in both the three-loop graphs C and C that diverge in three dimensions. E F Note that this cancelation of divergences involves the detailed structure of the induced coupling

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interaction with its exponential dependence upon the potential . As discussed earlier, the basic (`primitivea) divergences which the induced couplings must cancel appear only in even loop order. The subsidiary divergences at this three-loop order are canceled by the non-trivial potential dependence of the two-loop induced coupling. Inserting the explicit results for these integrals produces the physical lP3 limit of ln Z to three loop order:

 

ln Z ln Z 1 " # V V 4!(4p)

C C bnnnneeee  # bnnneee  ? @ A B ? @ A B i ? @ A ? @ A i   ?@AB ?@A 8pj ? # bnei C # ? ?   be ? ? 9j i ?@ # bnneei C #ln #C ? @ ? @   ?@ 4p ?@ 9j i ?@ # bnneei C #ln #C , ? @ ? @   ?@ 4p ?@

 







 

 

(5.9)

with p C "C #Li (!)#ln # "0.1191312 ,   16  +  

(5.10)

p C "!3Li (!)#! "!1.040302,     4

(5.11)

C " ,   C "c!!2 ln "!1.097422 ,    C "c!1"!0.4227842 ,  where

(5.12) (5.13) (5.14)



X dt ln(1#t) . (5.15) t  is Euler's dilogarithm, C is given in Eq. (C.70), and C was de"ned in Eq. (3.82). + ?@ Di!erentiating Eq. (5.9) with respect to b yields the internal energy, in terms of bare parameters, to three-loop order: Li (!z),! 












 C bei  3 1 bei 1 i j ?  !  ?  #  ? bu" n # ? 2 4 4p 32 4p 48 4p ? 1 bee  ? @ [1#g C ] ! nn ? @ 48 (4p) ?@ ?@ ?@

bei ?  4p



 The closed form expression (5.11) for C involving the Euler dilogarithm was obtained previously by Kahlbaum  [11].

L.S. Brown, L.G. Yawe / Physics Reports 340 (2001) 1}164





1 beei ? @  # (4p) 32 1 beei ? @  # 32 (4p)


 
 
 
 C ##ln  

9j i ?@ #C #g C ?@  ?@ ?@ 4p

C ##ln  

9j i ?@ #C #g C ?@  ?@ ?@ 4p

73

 

C beee  ? @ A ! nnn  ? @ A 32 i (4p)  ?@A C beeee  ? @ A B ! nnnn  , (5.16) ? @ A B 32 i (4p)  ?@AB while di!erentiating with respect to the chemical potentials yields the particle densities (in terms of bare parameters) to three-loop order:













  




  

 


n "1n 2 "n 1# Q Q @ Q

bei 1 bei 1 bei  2C bei  Q  # Q  #  Q  # Q  8p 3! 8p 12 4p 2 8p


 i j  Q 4p

 1 bee 1 bee Q ? # Q ? [c!1#ln(j i /p)#C ] # n ? 8 (4p)  ?Q  ?Q 12 (4p) ? i beej C beei Q ?  #  Q ? ? #  (4p) (4p) 8 16 i bee Q ? [C #ln(j i/p)#C ] #    ?Q ?Q 12 (4p)

i b[ee#ee] Q ? Q ? [C #ln(j i/p)#C ] #    ?Q ?Q 24 (4p)

1 beee  C beee C beee Q ? @ #  Q ? @ #  Q ? @ # nn ? @ 24 i (4p) 24 i (4p) 12 i (4p)    ?@ 1 beee Q ? @ [C #2#ln(j i/p)#C ] #   ?@ ?@ 48 i (4p)  1 beee Q ? @ [C #2#ln(j i/p)#C ] #   ?@ ?@ 48 i (4p)  C beeee  C beeee Q ? @ A !  Q ? @ A ! nnn  ? @ A 48 i (4p) i (4p) 6   ?@A C beeeee  Q ? @ A B ! nnnn  . ? @ A B 16 i (4p)  ?@AB



(5.17)

 Graphs with n `clover leafsa produce a factor of [G (0)]L. The "rst graph shown in Fig. 15 is a two-loop clover leaf  graph; graph e of Fig. 17 is a three-loop clover leaf graph, and it yields the value C " given in Eq. (5.12). Formula (5.17)   shows that these graphs form part of the generic density correction factor exp+bei /8p,, extending the result quoted in  Eq. (4.11).

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Inverting the relation between physical and bare densities, and inserting the explicit values for C ,  C and C (because this simpli"es the subsequent results), yields   1 bei  1 bei  1 bei ij bei Q # Q Q Q Q ! ! n"n 1! Q Q 2 8p 8p 3! 8p 6 8p 4p






 



 



  

 


1 bee  Q ? [c!1#ln(j i/p)#C ] ! n ? 12 (4p)  ?Q ?Q ? i bee Q ? [c!3#ln(4j i/p)#C ] # ?Q ?Q 12 (4p)

i beej i bee Q ? [c!1#ln(j i/p)#C ]# Q ? ? !  ?Q ?Q 8 (4p) 24 (4p)

 C !1 beee C ! beee 1 beee Q ? @ #  Q ? @ #   Q ? @ ! n n ? @ 24 i(4p) 24 i(4p) i(4p) 12 ?@ 1 beee Q ? @ [c!1#ln(4j i/p)#C ] # ?@ ?@ 48 i(4p)



C ! beeee  C beeee   Q ? @ A !  Q ? @ A # n n n ? @ A 48 i(4p) i(4p) 6 ?@A C beeeee   Q ? @ A B # n n n n . (5.18) ? @ A B 16 i(4p) ?@AB Using this result to express the pressure in terms of physical densities gives the equation of state






 1 bei 1 bei ln Z ? ! ? " n 1! bp" ? 6 4p 8 4p V ?

ij ? 4p

 In Footnote 41 it was remarked that a "eld rede"nition could be performed that removes the k )

cross-term in ? the h interaction which contributes to the reducible part of the density}density correlator. For thermodynamic A quantities, it is the (

) term in the h coupling of the induced interactions (3.32) that contributes, since the chemical ? potentials are now constants. To independently check the h contributions to the pressure or free energy, one may follow ? the logic of Footnote 41 in a slightly di!erent way. To leading order in h, the (

) part of this interaction is removed by ? the "eld rede"nition P !b ie hne @C@ (. The e!ect of this rede"nition on the classical interaction, again to @ @ @ @ leading order in h, is equivalent to the alteration of the induced coupling g Pg !e e (h#h). In view of the ? ?@ ?@  ? @ ? @ evaluation (3.33) of h, and relation (3.81) between g and C , this substitution is equivalent to the change ? ?@ ?@ p e e  p e e  1 e e ? @ C P ? @ C # ? @ (j#j) . @ ?@ 6 4p ?@ 2 48pb ? 6 4p










Due to the charge neutrality condition e n "0, this change has no e!ect at two-loop order. It is easy to check that the ? ? three-loop terms in the equation of state (5.19) and the Helmholtz free energy (5.21) are in agreement with the corrections produced by this rede"nition. It is worth noting that the e!ect of the h interaction for the original partition function ? written in terms of the bare densities n is not produced by the change given above. The "eld rede"nition changes the ? dependence of the partition function on the bare densities, and also changes the relation between physical and bare densities. However, when these modi"ed results are re-expressed in terms of the physical densities, the same physical equation of state emerges, as it must.

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75

1 bee  ? @ [c#ln(j i/p)#C ] ! n n ? @ 24 (4p)  ?@ ?@ ?@ 1 beei ? @ [c!#ln(4j i/p)#C ] #  ?@ ?@ 16 (4p)  1 beee ? @ A [C !] ! n n n ? @ A 16 i(4p)   ?@A 1 beeee  ? @ A B [C ] . ! n n n n ? @ A B 16 i(4p)  ?@AB The internal energy expressed in terms of the physical densities is given by




 

 
  
 

(5.19)

 7 bei ij 3 1 bei ? ! ? ? bu" n ! ? 2 2 4p 24 4p 4p ?  1 bee ? @ [c!#ln(j i/p)#C #g C ] ! n n ? @ 8 (4p)   ?@ ?@  ?@ ?@ ?@ 3 beei ? @ [c!#ln(4j i/p)#C #g C ] #  ?@ ?@  ?@ ?@ 16 (4p)



3 beee  ? @ A [C !] ! n n n ? @ A 16 i(4p)   ?@A 3 beeee  ? @ A B [C ] , ! n n n n ? @ A B 16 i(4p)  ?@AB while the Helmholtz free energy is



(5.20)






1 bei 1 bei ij  ? ! ? ? bf" n !1#ln(n j/g )! ? ? ? ? 4p 3 4p 12 4p ? 1 bee  ? @ [c!1#ln(j i/p)#C ] ! n n ? @ 24 (4p)  ?@ ?@ ?@ 1 beei ? @ [c!3#ln(4j i/p)#C ] # ?@ ?@ 24 (4p)



  
  
 

1 beee  ? @ A [C !] ! n n n ? @ A 24 i(4p)   ?@A 1 beeee  ? @ A B [C ] . ! n n n n ? @ A B 24 i(4p)  ?@AB

(5.21)

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This result agrees with the corresponding result in Alastuey and Perez [7]. It also agrees, except for some small misprints and the omission of one quantum-mechanical term, with the result to be found in Ref. [6]. Note that the pressure is related to the Helmholtz free energy by the thermodynamic identity p"!RF/RV" . It is easy to apply this identity and verify that Eq. (5.19) 2, follows from (5.21). As explained in the next section, the combination 3  A"b(3p!u)! n (5.22) ? 2 ? would vanish identically if the plasma could be treated entirely in a classical manner. However, the results above give









1  1 be e i 1  n n bee[1!g C ] 1# ? @ # n beij . A"! (5.23) ? @ ? @  ?@ ?@ ? ? ? (4p) 24 4p 12 ?@ ? The next section shows how A may be independently computed from quantum corrections to the virial theorem for a classical Coulomb plasma, and discusses how A plays a role analogous to the anomalies which appear in relativistic quantum "eld theory. 5.1. Binary plasma These expressions simplify considerably for a two-component plasma such as an electron} proton plasma where !e "e ,e, and charge neutrality requires that n "n ,n/2. For C N C N example, the three-loop equation of state becomes



 


bp 1 bei "1! n 3 8p

 

3 i(j#j ) C N 1# 4p 8



1 bei  4m m N C ln #C #C !2C CC NN CN 24 8p (m #m ) N C 1 bei  +4c!#2 ln[j j j (4i/p)]#C #C #2C , . !  CC NN CN CC NN CN 8 8p !

(5.24)

This expression simpli"es a bit more if the very small electron/proton mass ratio is neglected, which is to say that the formal m PR limit is taken. In this limit, Eqs. (3.76), (3.80), (3.82), and (3.83) N  In particular, we agree with the result for the Helmholtz free energy given in Eq. (7.3) of Ref. [7]. Their result involves two constants, also called C and C . The relations between their and our parameters reads C "!p[C !], and   .   C "!4pC . Using our numbers gives C "15.2021, to be compared with their value 15.201$0.001, and .  . C "!14.7752, to be compared to their !14.734$0.001. Also, as will be shown explicitly in the next section, the . error estimate given at the end of their equation, which reads O(n ln n) in our notation, should actually be O(n ln n).  The last term in the "rst line of our Eq. (5.21) is often referred to as a `quantum di!raction terma. It is missing from Eq. (2.52) of Ref. [6]. In addition, the coe$cient of the term involving (1!ln ) should be 1/6, not p/3. [Finally, there is  a typographical error in the free energy for a non-interacting gas, which lacks the spin degeneracy factor inside the logarithm appearing in Eq. (2.50).]

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yield



and






4m m be  8 N C #C &
ln 8p #3c! , NN (m #m ) 4pj 3 N C C KN 

ln

(5.25)




bei  8 #3c! , (5.26) 2 ln j i#C &
ln 4p NN NN 4p 3 KN  so that bp be  g g 8 "1! ! ln 8p #3c! #C !2C lim CC CN n 4pj 6 96 3 C KN  g g 16pj C , ! ln (4ij ) #7c!12#C #2C # (5.27) C CC CN 2p 64 (be) where g"bei/(4p) is the dimensionless Coulomb coupling parameter. We note that the proton mass m disappears and this limit is well-behaved. N

 
  





5.2. One-component plasma Another special case is the `jelliuma model, in which a single charged particle species moves in the presence of a neutralizing, uniform background charge density. This is the one-component plasma (OCP) which is much discussed in the literature. It may be obtained by taking a limit of a plasma containing two species: one of charge e, number density n, and mass m; the other `spectatora species of charge e ,!ze, number density n ,n/z, and mass m , with zP0. The charge of each spectator particle becomes vanishingly small, but their density diverges, so as to preserve total charge neutrality. The net result (for static equilibrium properties) is that the spectator particles act like an smooth inert background charge density. The ideal gas pressure of the spectator particles diverges as zP0, and must be subtracted from the total pressure before sending z to zero. If the background, spectator particles are not taken to have a very large mass, m PR, then they will also make quantum, exchange contributions to the pressure. To the three-loop order to which we compute, these unwanted exchange contributions, in the zP0 limit, are given by p"Ge n n PG 16g



 



1 be  i be  1 # fI (g ) 24 4p 16 4p g


 

1 4pb   b  b   !4e #3(pei , bz m m m

(5.28)

where g is the spin degeneracy of the spectator particles. These terms are also to be subtracted from the total pressure. The resulting one-component equation of state, to three-loop order, is given by bp n



-!.



   



9ij g g C #C (g ) c#ln "1! ! CC CC 2p 6 24





8ij g 1 C #C (g )# ! c#ln !#C #C . CC CC    p 16 4pg CC

(5.29)

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Here i"(nbe) is the Debye wavelength due to the single charge species. Once again, we have written the result in terms of ascending powers of the dimensionless parameter g"bei/(4p) which characterizes the strength of Coulomb interactions in the plasma. On the other hand, our result entails no restriction on the size of the quantum parameter be , (5.30) g " CC 4pj CC which, together with j "2j , has been used to re-express the order gj term in terms of 1/g . CC C C CC An often treated special case of the one-component plasma is its classical limit. As already alluded to in Footnotes 10 and 45, in this limit the Boltzmann factor with the repulsive potential provides damping at the Coulomb distance d "be/4p, and the quantum-mechanical #uctuations ! are not required to obtain a "nite theory. (And, moreover, d is the correct, physical cuto! if ! d 'j .) The P0 limit takes g PR, and Eqs. (3.76), (3.80) and (3.82) give ! C CC 8 1 C (g )"ln(4pg )#3c! ! #O(1/g ) . (5.31) CC CC CC CC 3 4pg CC Thus, we see explicitly that the short-distance cuto! in the logarithm now involves




1 be  1 jg " " d , C CC 2 4p 2 !

(5.32)

and so, including the O( ) corrections which come from the 1/g terms, we "nd that CC g g 8 1 bp "1! ! 2 ln(3id )#4c! ! ! 6 24 3 4pg n -!.   CC g ! +2 ln(4id )#4c!#C #C , . !    16







(5.33)

To obtain an independent check on this result, the order  quantum correction for the canonical partition function of the classical one-component plasma is independently derived in Appendix E. There it is shown that there are no  corrections in three and higher loop orders } in agreement with the lack of an  correction to the order g term here } while the two-loop  correction given in Eq. (E.21) of that appendix agrees exactly with that in the g term in Eq. (5.33), the term involving 1/g . CC Riemann, Schlanges, DeWitt and Kraeft [8] report an equation of state for a one-component plasma. The terms in their formula which we classify as being of tree, one-, and two-loop order } the terms of order g, g, and g which appear in the "rst line of Eq. (5.29) } agree precisely with our result. They do not, however, present all the terms of three-loop, g order, but rather only include terms `up to the order (ne)a. We note that such a statement has only a formal signi"cance since ne bears dimensions, and hence there is no physical signi"cance in assuming that it is small. The terms retained by Riemann et al. are only those parts of the three-loop results which involve leading inverse powers of the quantum parameter g , namely CC g 1 3 1 1 bp -E-C "! $ ! . (5.34) 32p 2g g 4g g n -!. CC CC C CC









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79

Here, g "2 is the spin degeneracy of the electron, and we have chosen to separate the exchange C contributions so as to facilitate comparison with Ref. [8]. Our result does not altogether agree with formula (23) given by Riemann et al. [8] in that our exchange term of order g/g "O(e) is CC a factor 1/2 than theirs. The earlier paper by DeWitt et al. [9] contains, in its Eq. (15), the same three-loop contributions, with the same discrepancies. Their two-loop terms are correct as far as they go, but in this paper the two-loop terms also stop at the formal order of (ne) rather than containing the full dependence on be/j as in the later paper. Recently, we received an unpublished erratum from J. Riemann in which the coe$cient of the O(g/g ) exchange term is corrected by CC a factor of two, and now all results are in agreement.

6. Higher orders and the renormalization group 6.1. Renormalization group equations and leading logs In Section 3.4 we introduced an arbitrary scale k in order to separate induced couplings into pole terms and `renormalizeda "nite contributions. For the "rst induced coupling this amounted to writing





1 (e e ) 1 ? @ #g (k) . g "kJ\ ! ?@ ?@ 4! (4p) 3!l

(6.1)

However, since the theory in general, and the bare coupling g in particular, knows nothing about ?@ the arbitrary value of the scale k, we must have dg /dk"0. This requires that ?@ (e e ) d g (k)"(3!l)g (k)! ? @ , (6.2) k ?@ 4!(4p) dk ?@ which is the renormalization group equation for the renormalized coupling g (k). In the physical ?@ limit l"3, the solution of the renormalization group equation may be expressed as




k 1 (e e ) ? @ ln ?@ . g (k)" ?@ k 4! (4p)

(6.3)

In other words, the form of the renormalized (or `runninga) coupling g (k) is completely dictated ?@ by the pole terms, which in turn depend only on the form of the e!ective theory. It is only the integration constants, which we have expressed as k , that must be determined by matching the ?@ e!ective theory to the underlying quantum theory. The wave numbers k provide the quantum ?@ damping or cuto! to the classical theory and hence must be proportional to inverse thermal wave lengths. The result of the matching (6.3) shows that (6.4) k\"j eC?@ , ?@ ?@ where C , de"ned in Eq. (3.82), depends only the quantum parameter g ,be e /(4pj ). ?@ ?@ ? @ ?@ In this section we are interested not in the precise results but rather in exhibiting the leading logarithmic parts, that is, those contributions that acquire arbitrarily large logarithms in the limit of small thermal wave lengths. Thus we introduce j to denote a characteristic thermal wave length

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in the plasma, and write k "c /j where c are dimensionless numbers that depend on the ?@ ?@ ?@ species and on the quantum parameters g , but formally are O(1) and "xed. Thus ?@ ln(k /k)"!ln(jk)#ln c , and the extra logarithm involving c is negligible in the formal ?@ ?@ ?@ jP0 limit. The "rst contribution to ln Z involving a potentially large logarithm arises at two-loop order. From Eq. (4.7), the relevant two-loop part of ln Z,









be e  9j i ln Z p  ? @ ?@  !1#c#C #2 , " nn ln (6.5) ? @ ?@ 4p 4p 6 V ?@ exhibits a term which depends logarithmically on the ratio of scales (ji). If the plasma is su$ciently dilute, then ln(ji) will be large compared to one, and the logarithmic term will provide the dominant part of the entire two-loop correction. The terms shown in (6.5) come from the sum of the induced coupling g contribution and the two-loop graph ?@ (6.6)

which together contribute





(6.7)



(6.8)

ee ! bnn ? @ D(0; k)#g (k) ? @ 12 0 ?@ ?@ to (ln Z)/V, with




!1 9i D(0; k)" ln !1#c , 0 32p 4pk

as shown in Eqs. (4.1), (4.4) and (4.6). The renormalization group equation (6.2) ensures that the sum (6.7) does not depend upon the arbitrary scale k. It is, however, convenient to choose k"i/4p, for then the entire logarithmic term in ln Z comes from the induced coupling g (k)& ?@ !ln(ji/4p), rather than from the two-loop graph (6.6). Thus, the leading logarithmic piece of the two-loop partition function may be expressed as




ji 1 (be e ) ln Z  "! nnbg (i/4p)" nn ? @ ln . (6.9) ? @ ?@ ? @ (4p) 4p 4! V ?@ ?@ The virtue of this simple observation is that it easily generalizes to higher orders, and allows one to determine the leading logarithmic contributions to the pressure at any order with very little work. To be concrete, we "rst consider four-loop contributions to ln Z. [Logarithmic contributions at odd-loop orders are discussed below.] Pole terms in l!3 arise from (a) divergent four-loop graphs (shown below), (b) the induced coupling g which "rst contributes at four-loop order, and (c) the ?@A  We could equally well have chosen k to equal 9i/4p or just i, instead of i/4p. Such O(1) changes in the scale k have no e!ect on the following discussion of higher-order leading-log results.

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two-loop induced coupling g inserted into the two-loop graph ?@ (6.10)

in which the left vertex represents the usual classical interaction while the cross on the right vertex denotes the insertion of the induced interaction with coupling g . The contribution of the ?@ four-loop induced coupling g to (ln Z)/V is just ?@A ,!b nnng . (6.11) I ? @ A ?@A E?@A ?@A Using Eq. (4.3), the leading divergence in the contribution of diagram (6.10) to (ln Z)/V is easily seen to be









1 1 i J\ b(e #e )e ? @ A bg [1#O(3!l)] . I , (6.12) n n n E?@ 12 3!l 4p ?@ ? @ A (4p) ?@A Since the bare coupling g itself contains a single pole in l!3, this contribution has a double pole. ?@ Various four-loop graphs also yield double poles in l!3. There is, however, no need to compute the double pole terms of these four loop graphs because they are completely determined by the renormalization group. To prove this, we "rst note that g is completely symmetrical in the ?@A indices abc, has the dimensions of kJ\ times 12 powers of charges, and must cancel the double poles (as well as lower-order single poles) in both diagram (6.10) and the divergent four-loop graphs. Consequently, g must have the form ?@A 1 1 +g (e #e )e#g (e #e )e#g (e #e )e, g "kJ\ @ A @A @ A ? A? A ? @ ?@A (3!l) (24p) ?@ ?



# kJ\



R R ?@A # ?@A #g (k) . ?@A (3!l) (3!l)

(6.13)

The "rst set of terms removes the divergence in Eq. (6.12), while the R and R terms cancel the ?@A ?@A double and single poles generated by four-loop graphs, respectively. The remaining "nite `renormalizeda coupling is g (k). Now dg /dk"0 and dg /dk"0, while dg?@A(k)/dk must be ?@A ?@A ?@ "nite. Therefore, the single pole terms that result when k in Eq. (6.13) is varied must cancel, 2R 1 +g (e #e )e#g (e #e )e#g (e #e )e," #kJ\ ?@A , 0" @ A @A @ A ? A? A ? @   3!l (24p) ?@ ?

(6.14)

 Strictly speaking, the R and R terms cancel primitively divergent four-loop graphs. Four-loop graphs ?@A ?@A containing divergent two-loop sub-graphs are rendered "nite by insertions of the g interaction in "nite two-loop ?@ graphs.

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or, inserting the explicit form (6.1) for g , ?@ 1 eee ? @ A +(e #e )#(e #e )#(e #e ), . R " @ @ A A ? ?@A 72 4!(4p) ?

(6.15)

This result is easily con"rmed by direct computation. The graph

(6.16) produces a double pole contribution to (ln Z)/V involving the square of Eq. (4.3):




 

1 1 i J\  b nnn(eee) . ? @ A ? @ A 32p 3!l 4p 2(3!) ?@A The graph

(6.17)

(6.18) also produces a double pole contribution to the partition function. It is not di$cult to show that the double pole in this graph, without vertex and symmetry factors, is just 1/2 times the square of the single pole contribution (4.3) of the two-loop graph. A easy exercise now shows that the double pole contribution of this graph is given by




 

1 i J\  b 1 1 nnn(eee). (6.19) ? @ A ? @ A 2 ) 3! 2 32p 3!l 4p ?@A It is a simple matter to verify that the double pole divergences in Eqs. (6.17) and (6.19) are indeed canceled by the contribution (6.15) to the g coupling term. ?@A The renormalization group equation for the "nite coupling g?@A(k) may now be obtained by returning to the condition that dg /dk"0. Since the single pole terms in dg /dk have been ?@A ?@A shown to cancel, this condition reduces to 1 +g (e #e )e#g (e #e )e#g (e #e )e, 0" @ A @A @ A ? A? A ? @ 36(4p) ?@ ? d #2R #2(3!l)g !k g . ?@A ?@A dk ?@A

(6.20)

 An outline of the proof is as follows. Choose the left-hand 5-point vertex in (6.18) to be the origin. Assign the upper 4-point vertex the coordinate r , and the lower 3-point vertex the coordinate r . The double pole contribution comes   from the most singular integration region where "r ";"r ";i\. In computing the leading contribution from this   region, the right-hand line running between r and r can be replaced by a line that runs between r and the origin. Thus,    as far as the leading singularity is concerned, the graph reduces to the graph (6.16) except that the condition "r "("r "   must be imposed. Since the graph (6.16) is symmetrical under the interchange of these two coordinates, imposing this condition merely multiplies the result by 1/2.

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Using Eq. (6.3) for g and taking the physical limit l"3, gives ?@ d 8 eee k k k ? @ A (e #e ) ln ?@ #(e #e ) ln @A #(e #e ) ln A? . k g?@A"2R # ?@A 27 (16p) ? @ @ A A ? dk k k k





(6.21) The integration of this renormalization group equation yields












k k k 4 eee ? @ A (e #e ) ln ?@ #(e #e ) ln @A #(e #e ) ln A? g?@A(k)"! @ @ A A ? k k k 27 (16p) ?




k !2R ln ?@A , ?@A k

(6.22)

where the integration constant has been written as a scale k which, once again, will be of order of ?@A (the inverse of) a typical thermal wavelength j\, but whose precise value can only be determined by matching to the underlying quantum theory. Note that the single pole residue R in the ?@A renormalization group equation (6.21) gives rise to single log terms in the running coupling (6.22), which are subleading compared to the double log terms when k is much much less than j\. The residue R is determined by the less singular single-pole terms of the previous double pole ?@A contributions, plus the single pole produced by the graph

(6.23) which has no double pole contribution. Since our purpose here is just to illustrate the character of the theory, we shall not bother to compute R explicitly. ?@A Recalling that the divergent terms in the classical theory have all their non-integral dimensional dependence appearing as integer powers of iJ\, we see that, just as in the previous two-loop discussion, choosing k"i/4p in the induced couplings not only removes the poles in these classical loop graphs, it also prevents the appearance of any additional large logarithms in the resulting "nite contributions of four-loop graphs. Thus taking k"i/4p and inserting (6.22) into (6.11) immediately yields the leading logarithmic contribution to the partition function at four-loop order:








ji 1 ln Z  " bnnn [eee#3eee] ln #O[e ln(ji)] . ? @ A  ? @ A ? @ A 4p (16p) V ?@A

(6.24)

6.2. Leading logs to all orders Exactly the same approach may be used to determine the leading log contributions at higher orders. Consider "rst the situation at an even loop order. The induced coupling g 2 N> makes its ? ? "rst contribution to (ln Z)/V at 2p-loop order. This contribution is IN,!bN n 2nN> g 2 N> .  ? ? ? ? ? 2?N>

(6.25)

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For later convenience, we will refer to g 2 N> as the rank-p coupling. Poles in l!3 up to order ? ? p are generated at 2p-loop order and must be canceled by the rank-p induced coupling. In particular, order p poles are generated by diagrams of the form (6.26)

in which the left and right vertices represent insertions of the rank k and rank p!k!1 couplings g 2 I> and gI> 2 N> , respectively. The contribution of these diagrams to (ln Z)/V is ? ? ? ? bN i J\ 1 N\ n 2nN> g 2 I> gI> 2 N> IN"! ? ? ? ? ? ? 3!l 4!(4p) 4p 2 I ? ?N> ;(e  2 I> )(e I> 2 N> ) , (6.27) ? ? ? ? where we have introduced the shorthand abbreviation




(6.28) e  2 I ,e  #2#e I . ? ? ? ? By de"ning g,!1, and including the terms where k"0 and k"p!1, this expression also ? includes the case where either vertex in the diagram (6.26) represents the original classical interaction. Since the rank-k coupling g 2 I> contains poles in l!3 up to order k (and ? ? gI> 2 N> has poles of order p!k!1), this contribution does generate order-p poles. To cancel ? ? these poles, the rank-p coupling must have the form 1 1 N\ S+g 2 I> gI> 2 N> (e  2 I> )(e I> 2 N> ), kJ\ g 2 N> "! ? ? ? ? ? ? ? ? ? ? 4!(4p) 3!l I N R I2 (6.29) # 2#kNJ\ ? ?N> #g  2 N> (k) . ? ? (3!l)I I Here, S denotes a symmetrization operator which averages over all permutations of the indices a 2a . The RI terms cancel the poles of order k generated by (primitively divergent) 2p-loop  N> graphs, and the unwritten `2a pieces denote terms proportional to kJ\ which cancel the poles generated by induced couplings totaling rank p!2 inserted into divergent four-loop graphs, terms proportional to kJ\ which cancel the poles generated by induced couplings of rank p!3 inserted into divergent six loop graphs, etc. The renormalization group condition dg 2 N> /dk"0 must hold as an exact identity. The ? ? variation of (6.29) with respect to k has poles in l!3 up to order p!1. The coe$cients of each order pole must cancel independently. As discussed before, this means that the residues RI, for k"2,2, p, are completely "xed by the structure of the lower-order diagrams. The renormalization group equation for the remaining "nite terms, evaluated at l"3, becomes



k



1 N\ d S+g  2 I> g I> 2 N> (e  2 I> )(e I> 2 N> ),#2 . g  2 N> "! ? ? ? ? ? ? ? ? 4!(4p) dk ? ? I

(6.30)

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The key point is that when k is chosen to be of order i, the only source of large logarithms are the induced couplings themselves; the rank-k renormalized coupling g  2 I is of order [ln(jk)]I\. ? ? Consequently, the terms shown explicitly on the right-hand side of (6.30) are proportional to lnN\(jk), while all the unwritten `2a terms have at most p!2 powers of ln(jk). Integrating (6.30), neglecting the sub-leading pieces, gives ln jk N\ S+g  2 I> g I> 2 N> (e  2 I> )(e I> 2 N> ), . (k)"! (6.31) ? ? ? ? ? ? ? ? 4!(4p)p I Eq. (6.31), together with the starting condition g ,!1, provides a simple recursive recipe for ? determining the leading-log contribution to the rank-p induced coupling g  2 N> . The leading-log ? ? contribution to ln Z at 2p-loop order is then just g

? 2?N>

ln ZN  "!bN n 2nN> g  2 N> , (6.32) ? ? ? ? V 2 ? ?N> where the renormalized induced couplings are to be evaluated at k"O(i). The resulting contribution is O[(bei)N lnN(ji)] in magnitude. The leading-log contributions at odd-loop orders are also easily determined. When the scale k is O(i), so that the only source of large logarithms are the induced couplings themselves, the largest number of logarithms at a given odd loop order will result from diagrams where the maximal number of induced couplings are inserted into a graph with only one explicit loop. At loop order of 2p#1, this means a single insertion of the rank p induced coupling, or two insertions of rank k and rank p!k couplings, or more generally, the insertion of any collection of induced couplings whose ranks total p, as illustrated in Fig. 18. Rather than following the cookbook method and struggling to get the proper combinatorial factors to evaluate these diagrams, it is much easier to simply return to the original functional integral representation (3.30). The sum of the graphs in question just corresponds to the contribution of the order (r) terms in the "rst line of the induced interaction (3.32) to the action



b (dJr) (r) , S "*i  2

(6.33)

where  *i"! bN\n 2nN g  2 N (e  2 N ) . ? ? ? ? ? ? N ? 2?N

(6.34)

 This has been shown explicitly for k"2 and 3. The current section may be regarded as an inductive proof of this assertion to all orders.  The alert reader will have noticed that we have ignored the induced couplings for derivative interactions, h 2 N , ? ? k 2 N , etc., in this discussion. The four-loop couplings h and k in the induced action (3.32) give rise to only a single log ? ? ?@ ?@ at four-loop order, not a double log, and so it does not contribute to the leading log result. Moreover, just as in the previous case of the g  2 N couplings, these two-derivative couplings generate a sequence of higher powers of logs, but ? ? each member in this sequence of contributions is suppressed by one power of ln ij compared to the corresponding leading-log contribution. In the same manner, the four-derivative or higher terms schematically denoted by the ellipsis 2 in Eq. (3.32) give rise to still further sub-dominant logs.

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Fig. 18. Leading-log contributions to ln Z at odd-loop order. Circled vertices labeled k denote the insertion of the rank-k induced interaction proportional to g 2 I> . For order 2p#1 contributions, the ranks of all the insertions around the ? ? loop must sum to p.

Thus, the total e!ect of these terms is to simply shift the unperturbed (squared) Debye wave number, i Pi ,i #*i . (6.35)   Referring back to the one-loop correction (2.79), we see the sum of these odd-loop order leading logarithms plus the original one-loop contribution is given by  i (6.36) ln ZN> "! V . 12p N A straightforward exercise expanding (6.36) and (6.34), and iterating (6.32), will yield the explicit leading-log contributions at any given order. The results up to order 6 are: ln Z  "L b nnee , ? @ ? @ V ?@ ln Z  i L "  bnnee(e #e ) , ? @ ? @ ? @ 8p V ?@ ln Z  "L bnnneee(e #e ) , ? @ A ? @ A @ A V ?@A L ln Z  " bnnnneeee ? @ A B ? @ A B 32pi V  ?@AB ;[4(e #e )(e #e #e )#e (e #e )(e #e )] , @ A ? @ A B ? @ A B ln Z  L " bnnnneeee[(e #e )#2(e #e #e )](e #e ) , ? @ A B ? @ A B ? @ @ A B A B 3 V ?@AB where




ji 1 ln . L, 4p 4!(4p)

(6.37) (6.38) (6.39)

(6.40) (6.41)

(6.42)

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87

The expression for the partition function in leading logarithmic orders reads  i (6.43) ln Z "V n#V  # ln ZI . ? 12p I ? The derivatives of this form with respect to bk give the leading logarithmic relations between the ? physical densities n and the `barea densities n. Solving for the bare densities in terms of the ? @ physical quantities expresses the leading-log partition function in the form  i # ln ZM I  , ln Z "V n !V ? 24p I ? in which

(6.44)

ln ZM   "!L bn n ee , (6.45) ? @ ? @ V ?@ 3iL ln ZM   "! bn n ee , (6.46) ? @ ? @ 8p V ?@ ln ZM   "!12L bn n n eee , (6.47) ? @ A ? @ A V ?@A ln ZM   5L "! bn n n n eeee[ee #12e e e #16e e#12e e] , (6.48) ? @ A B ? @ A B ? @ @ A B @ B A B V 16pi ?@AB ln ZM   "!12L bn n n n eeee ? @ A B ? @ A B V ?@AB ;[3e e e #3ee #3ee #3e e#5e e#14e e e ] . (6.49) ? @ A @ A A B A B @ B @ A B Note that for even-loop orders these leading-logarithmic contributions always include a sum of particle densities weighted by an odd power of the charge. Consequently, the leading-logarithmic contributions at even-loop order vanish in the special case of a neutral symmetric binary plasma, such as a pure electron}proton plasma, where the charges of the two species are equal and opposite and the physical densities are necessarily equal due to charge neutrality. This is a general result, which follows from the recursive structure of (6.31) and the vanishing of its initial term. The corresponding leading logarithmic expansion of the free energy F is easily obtained from the thermodynamic relation bp"ln Z/V"!R(bF)/RV. Using this relation, it is easy to con"rm that the overall coe$cients of the "rst three logarithmic terms, the two-, three-, and four-loop terms, (6.45), (6.46), and (6.47), agree with the corresponding free-energy terms computed by Ortner [12] and displayed in his Eq. (93). As we have noted before, Ortner works with a plasma of various species of positive ions moving in a "xed background of neutralizing negative charge. This model allows him to use a purely classical description, in which the e!ective short-distance cuto! is provided by the essential singularities of the Boltzmann factors with purely repulsive Coulomb interactions. Thus, as was explained before in the discussion of Eq. (5.32), the quantum length j in the argument of the logarithm in Eq. (6.42) is replaced by the Coulomb length d "be/(4p) in ! Ortner's results.

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6.3. `Anomalousa virial relation The grand canonical partition function may be regarded as a function of the temperature and the (bare) density n, charge e , and thermal wavelength j of each species. If one de"nes an average ? ? ? density n which is the geometric mean of the bare densities n and thus entails b\ times the ? exponential of the average of the chemical potentials, then one may alternatively express a speci"c density in terms of the average density and a relative density ratio x , ? n,nx . (6.50) ? ? The charges of each species may similarly be written in terms of some mean charge e and a relative charge ratio y , ? e ,ey . (6.51) ? ? Any dependence on the thermal wavelength j may be re-expressed as dependence on the ? dimensionless quantum parameter g "be/4pj . Consequently, any n-loop contribution will ? ? ? equal (bei )L times some function of the dimensionless variables +x ,, +y , and +g ,. This is  ? ? ? a precise version of the statement that the loop expansion parameter (in the physical limit of three dimensions) is bei . The point to be emphasized is that the parameter bei captures the overall   powers of the inverse temperature, charge, and densities that appear in a given loop order. Therefore, the grand canonical partition function has the functional form






(6.52) ln Z"F(bei , x, y, g) V n . @  @ Let us pretend, for the moment, that F does not depend on the quantum parameters g } or that ? the purely classical theory exists. We note that the di!erential operator













R 3 R ! Rb 2 Rbk ? ? annihilates the density ratios x , the charge ratios y , and also bei because ? ?  R R R 3 R 3 b ! bei & b ! b e@I@ "0 .  Rb 2 Rbk Rb 2 Rbk ? ? ? ? @ Since b





R R 3 ! n"!3n , @ @ Rb 2 Rbk ? ? this shows that b





(6.54)






R 3 R ! F(bei , x, y) V n "0 .  @ Rb 2 Rbk ? ? @ In other words, a purely classical partition function must satisfy 3#b



3#b



R R 3 ! ln Z"0 . Rb 2 Rbk ? ?

(6.53)

(6.55)

(6.56)

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Recalling that the pressure p appears as ln Z[k]"bpV ,

(6.57)

the thermodynamic, internal energy density u is given by R ln Z[k] "uV , ! Rb

(6.58)

and the number density n of species a by ? R ln Z[k] "n V , ? Rbk ? the identity (6.56) for a purely classical plasma is equivalent to the relation

(6.59)

3 (6.60) 3bp!bu! n "0 . ? 2 ? Of course, the purely classical plasma (with oppositely charged particles) does not exist. The induced couplings necessary to render the theory "nite give rise to additional dependence on the quantum parameters g . Hence, in fact, ? 3 A,3bp!bu! n O0 . (6.61) ? 2 ? The non-vanishing of A arises from the `anomalousa dependence on the underlying quantum physics. This behavior shows that A is akin to the anomalies encountered in relativistic quantum "eld theories. To "nd an expression for the anomaly A, which may be evaluated without separately computing the pressure, internal energy, and densities, we turn to the functional integral representation of the grand canonical partition function. It proves convenient for this speci"c application to use a scaled potential I (r)"b (r) so that the interaction terms now involve (6.62) ne C? @("ne C? (I , ? ? with no explicit appearance of the inverse temperature b (although it does reside in the densities n). ? Thus the functional integral takes the form



 



1 (dJr)(

I (r))!S [ I ; k] . Z[k]"Det[!b\ ] [d I ] exp !  2b

(6.63)

Although the method that we shall outline is valid for S [ I ; k] taken to arbitrary order, to keep  the notation simple, we shall consider only those pieces that contribute to the three-loop order to which we have calculated,

 

S [ I ; k]" (dJr) ! ne C? (I r# g bne C? (I rne C@ (I r ? ?@ ? @  ? ?@



! hbe(

I (r))ne C? (I r#2 . ? ? ? ?

(6.64)

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We shall "rst "nd an expression for the pressure using its identi"cation with the response to a change in the volume, d ln Z[k]"bpdV ,

(6.65)

with the change in volume realized by a dilation transformation of the spatial coordinates within the functional integral. To do this in a conceptually simple way, we temporarily introduce general coordinates xI and a metric tensor g , so that the physical distance between neighboring points is IJ given by J ds" g dxI dxJ . (6.66) IJ IJ The (

I ) part of the action in the functional integral now takes on the generally covariant form







1 (dJx)(det g gIJR I (x)R I (x) ! hbene C? (I V . (6.67) I J ? ? ? 2b ? For the terms in S which do not involve derivatives, the introduction of generalized coordinates  is e!ected by simply including the factor (det g in the spatial integration measure. To e!ect a dilation or scale change, we take g "eNd . (6.68) IJ IJ In view of the distance interval (6.66), this has the e!ect of the length alteration ¸P¸eN. With this metric, the determinantal factor and inverse metric are simply (det g"eJN,

gIJ"e\Nd . (6.69) IJ Finally, taking the constant parameter p to be in"nitesimal, pPdp, we have a volume change dV"l dp V. Thus, the variation of the functional integral (6.63) brought about by the volume change in the pressure de"nition (6.65) gives

 

lbp"!(l!2)



(

I )

 

1 ! hbene C? (I ? ? ? 2b ?

ne C? (I ! g bne C? (I ne C@ (I . (6.70) ? ?@ ? @ ? ?@ Using the functional integral representation (6.63) to evaluate the de"nitions (6.58) and (6.59) of the energy and particle number yields #l



1 !buV" 11(

I )22V! 2b

b



R S [ I ; k] Rb 

(6.71)

and ! n V" ? ? ?



R S [ I ; k] Rbk  ?



.

(6.72)

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91

Since each of the quantities that make up the anomaly (6.61) is well de"ned, we may write it as a lP3 limit in a form that will prove to be convenient, l A"lbp!bu! n . ? 2 ? The results above express this as

 

(6.73)





1 R l R (

I )# l!2#b ! hbe(

I )ne C? (I ? ? ? 2b Rb 2 Rbk A ? A R l R # l#b ! ne C? (I ! g bne C? (I ne C@ (I . ? ?@ ? @ Rb 2 Rbk A A ? ?@ The commutation relation A"

(3!l)











 (6.74)



R R R l R l ! n"n b ! (6.75) ? Rb 2 Rbk Rb 2 Rbk ? A A A A implies that the classical action part of S , proportional to ne C? (I , does not contribute to the  ? ? anomaly A (as required). Moreover, l#b









R l R R l R ! bn"bn (3!l)#b ! . @ @ Rb 2 Rbk Rb 2 Rbk A A A A Hence we have, to our three-loop order of accuracy, b



A"

(6.76)



1 Rh (

I )# b ? en(

I )e C? (I 2b Rb ? ? ? R (3!l)#b g b11ne C? (I ne C@ (I 22 . ? @ Rb ?@

(3!l)




 

! (6.77) ?@ To compute the two and three loop contributions to the anomaly A, we "rst note that since to these orders the number densities are given by n "n11e C? (I 22#hben11(

I )e C? (I 22!2 bg nn11e C? (I e C@ (I 22 , ? ? ? ?@ ? @ ? ? @ we may write the pressure (6.70) as lbp"!(l!2)

(6.78)

1 11(

I )22# [ln #2enhb11(

I )e C? (I 22] ? ? ? ? 2b ?

(6.79) #l bg nn11e C? (I e C@ (I 22 . ?@ ? @ ?@ The pressure and densities are, of course, well de"ned in the lP3 limit. Hence, in the above expression, the pole in l!3 in the "nal term, coming from g , must be canceled by a similar pole, ?@ with opposite residue, in 11(

I )22. Since the contribution of the coupling h has a coe$cient that ?

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is already of two-loop order, to the order to which we work,





1 lim (3!l) 11(

I )22"3 b lim (3!l)g nn11e C? (I e C@ (I 22 . ?@ ? @ 2b J ?@ J Recalling the result (6.1) for g , we have ?@ 1 (e e ) ? @ . lim (3!l)g "! ?@ 4! (4p) J Since j &b, Eqs. (6.3), (6.4) and (3.82) also inform us that, in the lP3 limit, ?@ 1 (e e ) Rg ? @ [1#g C ] , b ?@ "!  ?@ ?@ 4! (4p) Rb where C ,dC /dg is given in Eq. (4.19). Finally, Eq. (3.33) gives ?@ ?@ ?@ j R b h"! ? . 48p Rb ?

(6.80)

(6.81)

(6.82)

(6.83)

Thus, reverting to the conventionally normalized "eld , and discarding terms of higher order, (e e ) 1 A"! bnn ? @ 11exp+ib(e #e ) (r),22[3!2!g C ] ? @ (4p) ? @  ?@ ?@ 4! ?@ j ! ? ebn11(

)22 . ? 48p ? ? In the last line we use the result (5.7), i b11(

)22" . 4p

(6.84)

(6.85)

For the remaining terms, we expand the exponential involving to second order to generate the sub-leading (three loop) contribution. It involves, in the physical lP3 limit i (6.86) b11 22"lim G (0)"!  . J 4p J Expanding (e #e ), the terms involving e and e just provide the one-loop corrections that alter ? @ ? @ the bare densities n and n to the physical densities n and n . Hence, only the cross-term provides ? ? ? @ a non-trivial correction, and we have






ji bei be e i 1 1 ? n . n n (be e )[1!g C ] 1# ? @ ! ? A"! ? @ ? @  ?@ ?@ 48p 4p ? 4p 4! (4p) ?@ ? This agrees with Eq. (5.23).

(6.87)

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7. Long distance correlations As noted in Section 3, the interaction (3.93) which couples the static and non-zero frequency modes of the electrostatic potential only a!ects thermodynamic quantities at six-loop order. However, this term does generate some qualitatively new e!ects in correlation functions. In particular, it destroys the exponential screening of the quasi-classical theory [14}16]. This e!ect is easy to calculate using the e!ective theory as given in Eqs. (3.92) and (3.93). We will "rst examine the workings of this e!ect on the single-particle irreducible part C (r!r) ?@ of the number density correlation function. The graph of Fig. 19 is produced if each of the in variational derivatives in the de"nition (3.35) of the number density correlator act on S     the functional integral (3.92). The non-zero-frequency potentials that this brings down from the exponential become tied together into the product of two unscreened, long-ranged Coulomb Green's functions. Since the result is single-particle irreducible, it de"nes an O[(bei)(ij)] correction to C (r!r). Explicitly, the calculation that we have just described gives the ?@ long-ranged contribution 2 1 n n eejjb [ < (r!r)] . *C (r!r)" ?@ (2pm) I J ! (4p) ? @ ? @ ? @ K$

(7.1)

f(4) 1 2 " " , (2pm) 4p 360 K$

(7.2)

6 , [ < (r!r)]" I J ! (4p)"r!r"

(7.3)

Here

and

and so 1 (ben j)(ben j) ? ? ? @ @ @ . *C (r!r)" ?@ 60 (4p)"r!r"

(7.4)

Consequently, density}density correlations do not, in fact, decay exponentially but rather have long-distance 1/r tails. We may use the relation given in Eq. (2.108) connecting the electrostatic Green's function G(r!r)"11 (r) (r)22 and C , ?@ \ , (7.5) b GI (k)" k#b e e CI (k) ? @ ?@ ?@ to "nd the long-distance tail in G(r!r). Treating *C (r!r) as a perturbation and noting that ?@ the long-distance limit of the unperturbed Green's function is given by the Debye screened function,





 More precisely, a correction of relative size O[(bei)(ij)] for wave numbers of O(i).

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Fig. 19. Correction to the irreducible density}density correlator C generated by the induced interaction S in?@ U   volving the non-zero frequency modes of (r, q). The dashed lines represent the long-range, unscreened Coulomb Green's functions of the non-zero frequency modes.

we have



e\Gr\r  e\Gr \rY b*G(r!r)"! (dr )(dr ) b e e *C (r !r ) . (7.6)   4p"r!r " ? @ ?@   4p"r !r"  ?@  Since the #anking Debye Green's functions that appear here are of short range, to obtain the long-distance behavior of b*G(r!r) we may replace *C (r !r ) by *C (r!r) and use ?@   ?@ e\GP 1 " (7.7) GI (0)" (dr) 4pr i



to "nd that the potential correlator also acquires a 1/r tail, (ben j)(ben j) 1 ? ? ? @ @ @ . *G(r!r)&! (7.8) (4pi)"r!r" 60 ?@ Comparing the magnitude of this 1/r tail to the original e\GP/4pr Debye potential, one "nds that the cross-over from exponential to power-law decay occurs at the parametric scale ir&!ln[(ji)(bei)] .

(7.9)

This characterizes the number of e-foldings over which exponential Debye screening could, in principle, be observed before the power-law tail takes over. The function C (r!r) describes the `single-particlea irreducible part of the density}density ?@ correlation function. The (Fourier transform) of the complete correlator is given, according to Eq. (2.110), by



 



KI (k)"CI (k)! b e CI (k) GI (k) b e CI (k) . (7.10) ?@ ?@ A A? A A@ A A Recalling [Eq. (2.103)] that C (k)"d n in lowest order, and employing the reasoning just used ?@ ?@ ? to "nd the long-distance behavior of the potential correlator, we see that the leading long-distance

 Treating *C (r!r) as a perturbation is legitimate, even though it determines the leading long-distance behavior. ?@ One may show this rigorously by noting that *CI (k)&"k" for small k, and that this controls the discontinuity of GI (k) ?@ when k is small and negative.

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95

behavior of the complete density}density correlation function is given by K (r!r)&*C (r!r)!be n *GI (r!r)be n ?@ ?@ ? ? @ @ 1 1 ! *C (r!r)e be n #be n e *C (r!r) , ?A A i @ @ ? ? i A A@ A





(7.11)

or







b be n e be n e 1 K (r!r)& d ! ? ? A d ! @ @ B (en j)(en j) . (7.12) ?@ ?A @B A A A B B B 60 (4p)"r!r" i i AB The square brackets appearing in (7.12) function as projection operators into the sub-space orthogonal to the charge vector +e ,. Hence, the number-density}charge-density correlator con? structed from this result by multiplying by e and summing over b (or multiplying by e and @ ? summing over a) vanishes. Consequently, the number-density}charge-density correlation function must vanish more rapidly than 1/r at large distances. The leading behavior is obtained by returning to the general forms (7.10) and (7.5) to write





KI (k)e "k b e CI (k) GI (k) . (7.13) ?@ @ A A? @ A Again using the same reasoning to secure the asymptotic form, with the factor of k in the Fourier transform becoming !  in the spatial form, gives





b be n e 1 d ! ? ? A (en j)(ejn ) . (7.14) e K (r!r)&! ?A A A A B B B @ ?@ i 2 (4p)i"r!r" AB @ The charge-density}charge-density correlation function K(r!r) formed from this result (by multiplying by e and summing over a) again vanishes. The charge-density}charge-density correla? tor again vanishes more rapidly at in"nity. This "nal correlation function may be obtained by multiplying Eq. (7.13) by e and summing over a. The result is equivalent to the previous relation ? Eq. (3.18), and we "nd that the charge-density}charge-density correlator K(r!r) acquires a 1/r tail,





ben j  28 (7.15) ? ? ? . K(r!r)& (4p)i "r!r" ? When specialized to the case of a one-component plasma in the presence of a constant neutralizing background, Eq. (7.15) becomes K(r)&28




e j 7e b   1 " . (4p) r (4p) m r

(7.16)

The asymptotic results (7.12), (7.14) and (7.15) for the correlation functions agree with the results of the calculations of Cornu and Martin [16] and Cornu [17]. Note that, once the appropriate e!ective "eld theory is constructed, the single key result (7.4) is obtained in only a few lines. The fact that the charge-density}particle-density correlators fall o! as 1/r, or two powers of r faster than the 1/r tail of the potential correlator G, and that the charge-density}charge-density correlator falls o! yet faster by another two powers, is a direct consequence of the Poisson equation

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i  (r)"o(r) relating the charge density and electrostatic potential. The positive overall sign of the asymptotic charge}charge correlator (7.15) re#ects the fact that a charge #uctuation of a given sign will attract oppositely charged particles and repel similarly charged particles. Therefore, the long-distance tail of the charge}charge correlator shows a positive correlation. Essentially the same argument may also be used to con"rm the overall sign of the charge-density}number-density correlator (7.14). If the charge e of species a is arbitrarily small (so that it is, in e!ect, a test charge), ? then Eq. (7.14) becomes





ben j  1 bn e ? ? A A A #O(e) , (7.17) e K (r!r)& ? @ ?@ (4p)i 2 "r!r" A @ showing that, in linear response, a test charge of species a creates an induced charge density whose long-distance tail is positively correlated with the sign of the test charge. It should be emphasized that the above results for long distance tails were derived for zerofrequency correlator functions. One may wish to consider equal-time correlators instead. The di!erence between the two is determined by the behavior of non-zero frequency correlations. If K (r, u ) denotes the density}density correlator at spatial separation r and (Matsubara) frequency ?@ L u , then the equal-time density}density correlator is L 1 K (r, t"0)" K (r, u ) , (7.18) ?@ ?@ L b L which di!ers from the zero-frequency correlator K (r)"b\K (r, u"0) by the sum over all ?@ ?@ non-zero frequency components, K (r, t"0)!K (r)" b\K (r, u ) . (7.19) ?@ ?@ ?@ L L$ However, these non-zero frequency number density correlation functions decrease with increasing spatial separation faster than the zero frequency correlation (7.12), and consequently the equal-time and zero-frequency correlators have the same leading long-distance behavior. To understand the long-distance behavior of the non-zero frequency correlators, it is convenient to begin with the de"nition of the correlation function at (Matsubara) frequency u as a second variational K derivative with respect to the mth Fourier component of the (time-dependent) chemical potentials, d ln Z b\K (r!r, u ), ?@ K dbkK(r)dbk\K(r) ? @ dS dS U   U   " dbkK(r) dbk\K(r) ? @



  !

dS U   dbkK(r)dbk\K(r) ? @



.

(7.20)

 For the charge-density}particle-density correlator, this argument relies on the relation connecting the particledensity}potential correlator 11n (r) (r)22 with G and C , namely 11n (r) (r)22" (dr) ibe C (r!r)G(r!r). ? ?@ ? @ @ ?@ This relation may be easily derived graphically, and also follows from Eq. (A.43) and the immediately following discussion in Appendix A. Since G and C both have 1/r tails, the same analysis used above, applied to this relation, ?@ shows that a 1/r tail is also present in the particle-density}potential correlator. And hence the Laplacian of this result, which yields the particle-density}charge-density correlator, will have a 1/r tail.  A variant of this same argument appears in [17].

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97

Fig. 20. Contributions to the non-zero frequency components of the density}density correlator K (r!r, u ). Dashed ?@ K lines represent long-range, unscreened propagator G of non-zero frequency modes, while solid lines represent the K zero-frequency propagator G. Some non-zero Matsubara frequency u #ows in at the vertex labeled r and out at r. The K dashed lines in diagrams (a) and (b) must carry frequency u ; in diagrams (c) and (d) the non-zero frequencies carried by K each of the internal lines must sum to u . K

The leading terms of S are shown in Eq. (3.93) and further corrections are presented in U   Eq. (B.54). The resulting tree-level, one-loop, and multiloop contributions are illustrated schematically in Fig. 20. Determining the long-distance behavior of each contribution is straightforward by regarding r and r as arbitrarily far apart and evaluating each diagram in coordinate/frequency space. The leading long-distance behavior is determined by the minimal number of propagators which run from the vicinity of r to near r, together with the minimal number of spatial derivatives which act on these propagators. As noted in Eq. (3.95) of Section 3, each non-zero frequency propagator is given by 1 . G (r, r)"1 K(r) \K(r)2" K 4pb"r!r"

(7.21)

However, the structure of S (or ultimately, the gauge invariance of the underlying quantum U   theory) requires that only the gradient

K of the non-zero frequency components of appear in the e!ective theory. Consequently, each non-zero frequency propagator G (r, r) will be acted upon, K on either end, by one or more spatial derivatives, and so each non-zero frequency propagator which runs from near r to near r will contribute a factor which decreases at least as fast as "r!r"\ to the result. Taking account of the earlier discussion of Eq. (7.8), the zero frequency propagator G(r!r) shows exponential Debye screening out to the cross-over distance (7.9), beyond which it falls o! like "r!r"\. Furthermore, the contribution of every diagram will also contain at least one overall gradient with respect to r, and one with respect to r, coming from the fact that the variational derivatives in Eq. (7.20) can only act on gradients of the (non-zero frequency components of the) chemical potential appearing in Eq. (3.93) [augmented by Eq. (B.54)]. Therefore, diagrams in which k non-zero frequency propagators, and l zero frequency propagators run from (a neighborhood of ) r to (a neighborhood of) r will have long distance behavior which falls at least as fast as "r!r"\I\J\. If k is even, the fallo! must actually be at least two powers faster. This is because such contributions can only come from terms in S which have an odd number of factors of U    In other words, each line of a diagram represents a propagator G (r, r) which is Fourier transformed in imaginary K time, but not in space. Each vertex is labeled by a spatial position, and the sum of all discrete frequencies (including the external frequency) coming into each vertex must vanish.

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(kK#ie K). Such terms must actually involve at least one more derivative acting on these "elds, ? ? since rotation invariance implies that only even numbers of gradients can appear in any term in the action. This is shown explicitly in the "nal term shown in Eq. (B.54). Consequently, in these diagrams the long-range propagators are acted upon by at least one more derivative on either end of the diagram. Since the total frequency u #owing through each diagram is, by assumption, non-zero, every K diagram must have at least one non-zero frequency propagator crossing from r to r. But in any single-particle reducible contribution where only a single non-zero frequency propagator crosses from r to r, such as diagram (a) of Fig. 20, the gradients appearing in the non-static action (3.93) necessarily generate Laplacians acting on either end of the propagator so the net contribution is only a local contact term proportional to d(r!r). Consequently, single particle reducible diagrams such as (a) do not contribute at all to the long-distance behavior. Diagram (b) containing one zero-frequency, and one non-zero frequency propagator running from r to r has "r!r"\ long-distance behavior, while diagram (c), with two long-distance non-zero frequency propagators, has "r!r"\ fallo!, since there must be a total of eight spatial derivatives acting on the two non-zero frequency propagators. Multiloop diagrams, such as (d), with three or more long-distance propagators, necessarily decrease like "r!r"\ or faster (up to possible logarithmic factors). Therefore, the leading long-distance behavior of the irreducible part C (r!r, u ) of the ?@ K non-zero frequency number-density correlation function is generated by diagram (c), and is order "r!r"\. A simple computation using the vertex (B.57) presented in Appendix B, together with the non-zero frequency propagator (7.21), shows that diagram (c) represents the contribution

  

je ? ? *C (r, u )"bn n ?@ K ? @ (2p)











je 1 @ @ H(u )

K I J IY JY r (2p)



1





, I J IY JY r

(7.22)

where 1  p 1 3 1 " ! H(u ), K 2mn(m!n) (2p) 3m m (2p) L\ L$K "  (bu )\!3(bu )\ .  K K

(7.23)

With the (continuing) neglect of delta function terms which do not a!ect the long-range tail,




1





I J IY JY r






1 3;5;7 1





P r r rr



I J IY JY r JY IY J I I J IY JY r r 3;5;7 " 4! , r

 A convenient generating function for evaluating this sum is  1 cot pb!cot pa "p . (n!a)(n!b) a!b \

(7.24)

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99

where in the last equality repeated use was made of the dimension-counting properties of the Euler operator, namely r ) r\L"!nr\L. Thus

  

n n je *C (r, u )"b ? @ ? ? ?@ K r (2p)

je @ @ 3;5;7;4! H(u ) . K (2p)

(7.25)

This result for the long-range tail of the irreducible, non-zero frequency density}density correlator neglects additional corrections involving higher powers of 1/r, and relative corrections to the coe$cient of the r\ tail which are suppressed by further powers of the loop expansion parameter bei (or equivalently ij). The Fourier transform of the result (7.25) gives

  

b je ? ? *CI (k, u )""k"n n ?@ K ? @ 2 (4p)

je @ @ H(u ) (mO0) . K (4p)

(7.26)

Note that this result involves the non-analytic term "k""(k)(k. The action (3.93) for the non-zero frequency modes gives the lowest order, analytic contribution to the irreducible correlation function, n j k CI (k, u )"d b ? ? ?@ K ?@ 2p (2pm)

(mO0) .

(7.27)

Using the exact relation (A.50) of Appendix A, and the same type of analysis employed above, the result (7.26) for the irreducible correlator may be converted into corresponding results for the long-distance behavior, at non-zero frequency, of the full number-density}number-density correlator KI (k, u ). It is easy to check that the leading non-analytic part of the single-particle reducible ?@ K contribution to KI (k, u ), the second set of terms in Eq. (A.50), is also of order "k", and thus gives ?@ K an additional contribution to the 1/r tail. However, these contributions always involve at least one extra factor of CI (k, u )(e e /k) which is a correction of relative order benj&(ij), or two ?A K A B powers of the loop expansion parameter. Hence this correction has the same size as terms that have already been omitted in the calculation of the irreducible part of the correlator, and thus to our leading order, the complete correlator at non-zero frequency has the same long-distance behavior as its irreducible part,



315 b n je ? ? ? K (r, u )&*C (r, u )& ?@ K ?@ K p 8 r





n je @ @ @ H(u ) . K p

(7.28)

Since the non-zero frequency number-density}number-density correlator (7.28) falls faster than the zero-frequency component (7.12), this shows that the equal-time and zero-frequency number

 The representation



1  r\" ds se\QP 4!  makes the evaluation of this Fourier transform easy.

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density correlators have the same long-distance behavior. This is equally true for the chargedensity}number-density correlator (7.14). But, from Eq. (7.28), the charge-density}charge-density correlator K(r, u )" e e K (r, u ) has a 1/r tail at non-zero frequency, K ?@ ? @ ?@ K n je  315 b (7.29) ? ? ? H(u ) , K(r, u )& K K p 8 r ? just like its zero-frequency counterpart (7.15). However, the size of the non-zero frequency tail (7.29) is a factor of (ji) smaller than the zero-frequency tail (7.15). Consequently, the equal time and zero-frequency charge-density}charge-density correlators have the same long-distance behavior up to relative corrections suppressed by four powers of the loop expansion parameter. When specialized to the case of a one-component plasma, the expression (7.26) for the leading non-analytic piece of the irreducible correlator at non-zero frequency agrees to leading order with the corresponding result of Cornu and Martin [16]. [We have made no e!ort to retain further corrections to the "k" coe$cient, some of which are included in the result (3.25) of [16], which are higher order in our expansion parameter ij.]





Acknowledgements The interest of one of the authors (L.S.B) in classical plasma physics was piqued by R.F. Sawyer. His work on this paper began while visiting the Los Alamos National Laboratory and continued at the Aspen Center for Physics and was largely completed during another visit to the Los Alamos National Laboratory. We would like to thank H. De Witt for several clarifying discussions and particularly for making us aware of various related prior results which were helpful in resolving interim discrepancies. Communications with W.-D. Kraeft and M. Schlanges were also helpful in this regard. We would also like to thank T. Kahlbaum for informative communications and useful comments. This work was supported, in part, by the U.S. Department of Energy under Grant No. DE-FG03-96ER40956.

Appendix A. Functional methods In this appendix, we de"ne, in the context of our plasma theory, the generating function of connected correlation functions and its Legendre transform, the e!ective action. We review relevant properties of these functionals that are well known in quantum "eld theory, and then describe how number densities and density}density correlation functions are related to them. In particular, we show how the density}density correlator may be expressed in terms of a `single-particle irreduciblea function in a way that explicitly exhibits its structure, particularly its small wave-number behavior. We also show how the mean-square #uctuations in energy, and

 Our discussion of the e!ective action for a plasma parallels that given for quantum "eld theory in Sections 4 and 5 of Brown [23, Chapter 6], which contains many more details.

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particle numbers, may be expressed in terms of the same single-particle irreducible functions. This formalism is applied to compute the number densities, density}density correlators, and equation of state to two loop order in an particularly e$cient manner in a "nal appendix so as to illustrate methods complimentary to those employed in the text. The partition function of our theory, in either its original quantum form (3.6), or re-expressed as an e!ective theory (3.92), has a functional integral representation



Z[k]"N [d ] exp +!S[ ; k], ,

(A.1)

where [d ] denotes functional integration over a space and time dependent potential (r, q) which is periodic, (r, b)" (r, 0). The action S[ ; k] has the form

 

1 @ S[ ; k]" dq (dJr)[

(r, q)]#S [ ; k] .  2 

(A.2)

In the original quantum theory the interaction part of the action S is (minus the logarithm of) the  functional integral over all charged "elds,



  



@ exp+!S [ ; k],"“ [dtH dt ] exp ! dq (dJr)L , ? ? ?   ?

(A.3)

with L the charged "eld Lagrangian de"ned in (3.90), but with the chemical potentials now ? extended to be functions of imaginary time as well as space, k (r)Pmu (x)"k (r,q). In the e!ective ? ? ? theory, S is the sum of the classical interaction and the various induced interactions, 



[ , K; k] , S [ ; k]"!b (dJr) n? e @C? (#S [ ; k]#S   U    ?

(A.4)

with S and S given in Eqs. (3.32) and (3.93), respectively, and + K(r), denoting the Fourier  U   components of (r, q), as de"ned in Eq. (3.26). In the following formal discussion, we will allow the generalized chemical potentials k(r, q) to vary both in space and (imaginary) time. The only feature of the interaction terms which will be relevant is the fact that i (r, q) couples to the total charge density via the dependence of S [ ; k]  on the generalized chemical potentials k (r, q), or ? d d S [ ; k]"i e S [ ; k] . ? dk (r, q)  d (r, q)  ? ?

(A.5)

This is a re#ection of the invariance (2.30) of the theory under the combined shifts P !ic and k Pk !e c. ? ? ? In the following discussion, for notational convenience, we will use single symbols x, y, etc., to denote a (Euclidean) space}time coordinate so that, for example, (x), (r, q). And we will write  as shorthand for @ dq(dJr). V 

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A.1. Connected generating functional The addition of an external charge density or source p(x) coupled to the "eld (x) de"nes a functional =[p; k] which is the generating functional for connected "eld correlation functions } correlators whose graphical representations have no disconnected parts. The de"nition is







exp =[p; k]" [d ] exp !S[ ; k]#



(x)p(x) . V In the presence of the source, a normalized thermal expectation value is de"ned by









1F[ ] 2N "e\5 N_I [d ]F[ ] exp !S[ ; k]# (x)p(x) , @ V and in terms of this expectation value d=[p; k] "1 (x)2N . @ dp(x)

(A.6)

(A.7)

(A.8)

The insertion in the functional integrand of the functional derivative of !S [ ; k] with respect to  a generalized chemical k (x) produces the average particle number density, up to an overall factor ? of e5 N_I . Thus the properly normalized particle number density of species a is given by d=[p; k] . (A.9) 1n (x)2N " ? @ dk (x) ? This is the number density in the presence of both spatially (or temporally) varying chemical potentials k (x) and the external charge density p(x). With the chemical potentials +k , taken to be @ ? constants and p taken to vanish, Eq. (A.9) reduces to the constant number density n "1n 2 of ? ? @ particles of species a. We shall denote this limit by a vertical bar with a subscript 0. Thus,



d=[p; k] . n " ? dk (x)  ? The total charge density in the presence of all the sources is given by

(A.10)

d=[p; k] . (A.11) 1o(x)2N " e 1n (x)2N " e @ ? ? @ ? dk (x) ? ? ? The partial derivative of =[p; k] with respect to the inverse temperature de"nes the average energy in the presence of the source p, *=[p; k] . 1E2N "! @ *b

(A.12)

 When varying b, the Fourier components k (r, u ),@ k (r, q) e SK O dq and p(r, u ),@ p(r,q) e SK O dq (with ? K  ? K  u ,2pm/b) are to be held "xed. K

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In the limit of vanishing source and constant chemical potentials, this reduces to the thermodynamic internal energy,



*=[p; k] ;"1E2 "! . (A.13) @ *b  Second variations with respect to the chemical potentials produce the number density correlation function,



d=[p; k] . (A.14) K (x!x)" ?@ dk (x)dk (x)  ? @ The static correlator discussed in the text is just the time average of this space}time dependent correlator,



K (r!r)"b\ ?@

@

dq K (r!r,q!q) . ?@  We shall also have occasion to use the "eld correlation function de"ned by



d=[p; k] . G(x!x)" dp(x)dp(x)  Since the functional integral of a total functional derivative vanishes,









d exp !S[ ; k]# (y)p(y) , d (x) W the functional integral with an extra factor of 0" [d ]

dS[ ; k] !p(x) d (x)

(A.15)

(A.16)

(A.17)

(A.18)

included in the integrand vanishes. Hence, in view of the form (A.2) of the action and the result (A.5), the expectation value of the "eld equation is an exact identity: d=[p; k] ! 1 (x)2N "i e #p(x)"i1o(x)2N #p(x) . @ ? dk (x) @ ? ?

(A.19)

A.2. Ewective action The e!ective action functional C[ M ; k] is de"ned by a Legendre transform of the generating functional =[p; k]. It generalizes the mean "eld theory described in Section 2.2 to include the e!ects of thermal and quantum #uctuations. The e!ective action functional has two important properties: Not only does it contain only connected graphs (as does =), it contains no singleparticle reducible graphs } graphs which can be cut into two disjoint pieces by cutting a single line. This is shown explicitly to two-loop order in Appendix G.2 below. This property simpli"es calculations. For example, when C is used to compute the free energy, one can simply delete all `tadpolea graphs. Moreover, as we shall see, the use of the e!ective action together with the

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functional relations that we are developing reveals the basic structure of the theory in a very useful form. The e!ective action functional is obtained by setting 1 (x)2N " M (x) , (A.20) @ that is, by considering the "eld expectation value rather than the source to be the independent variable. The e!ective action is then de"ned by the Legendre transformation



p(x) M (x)!=[p; k] . V Because of Eqs. (A.8), (A.9) and (A.12), C[ M ; k],



dC[ M ; k]"

+p(x)d M (x)!1n (x)2N dk (x),#1E2N db . ? @ ? @

(A.21)

(A.22)

V Thus we may consider C to be a functional of the independent variables M (x), k (x), and b, with the ? (partial) functional derivatives dC[ M ; k] "p(x) , d M (x) dC[ M ; k] ! "1n (x)2N , ? @ dk (x) ? and the ordinary partial derivative *C[ M ; k] "1E2N . @ *b

(A.23) (A.24)

(A.25)

In view of Eq. (A.23), evaluating the e!ective action at a stationary point, a point where dC/d M "0, is the same as setting the source p to zero. With constant chemical potentials and a vanishing source, the last equalities reduce to the ordinary number density and internal energy. As remarked in the text, the grand canonical partition function is related to the grand potential by Z"exp+!bX,, and so the grand potential is given by the e!ective action evaluated at its stationary point in the limit of constant chemical potentials, bX"C[ M ; k]" . (A.26)  We return momentarily to consider p(x) and k (x) as independent variables so as to compute the ? source functional derivative of Eq. (A.23):



dC[ M ; k] d M (y) "d(x!x) , d M (x)d M (y) dp(x)

(A.27)



dC[ M ; k] d=[p; k] "d(x!x) . d M (x)d M (y) dp(y)dp(x)

(A.28)

W or, in view of Eq. (A.8),

W

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In the limit of constant chemical potentials and a vanishing source, this becomes





dC[ M ; k] G(y!x)"d(x!x) . d M (x)d M (y) W 

(A.29)

Thus



dC[ M ; k] "G\(x!x) (A.30) d M (x)d M (x)  is the operator inverse to the potential correlation function G(x!x). After a Fourier transform [in space and (periodic) time],

 

@ dq (dJr) e\ SK O> k  rG(r, q) ,  the linear integral equation (A.29) reduces to the algebraic relation: GI (k, u ), K

(A.31)

GI \(k, u )GI (k, u )"1 . (A.32) K K To uncover the structure of the potential correlation function, we "rst write the "eld equation (A.19) for the expectation value in terms of the e!ective action functional. This is done by using Eqs. (A.20), (A.23) and (A.24), to obtain dC[ M ; k] dC[ M ; k] !  M (x)"!i e # . (A.33) ? dk (x) d M (x) ? ? Taking the functional derivative of this relation with respect to M (x) and then setting the chemical potentials constant and the source to zero produces ! d(x!x)"!i e c (x!x)#G\(x!x) , ? ? ? where we have de"ned a two-point vertex or coupling by



dC c (x!x), . ? dk (x)d M (x) ?  Thus in wave number/frequency space

(A.34)

(A.35)

GI \(k, u )"k#i e c (k, u ) . (A.36) K ? ? K ? The structure of the potential correlation function is intimately connected to that of the number density correlation function. Hence it is useful to examine the relationship between the e!ective action and the number density correlator. Recalling the expression (A.14) for this function in terms of =[p; k] and then the fact [Eq. (A.9)] that one functional derivative de"nes the number density, we see that we may write



d K (x!x)" 1n (x)2N . ?@ @ dk (x) ?  @

(A.37)

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In this equation, p and k are taken to be the independent variables, with M a function of these ? independent variables. Thus, using Eq. (A.24) to express the number density in terms of the e!ective action, we obtain

 



dC d M (y) dC K (x!x)"! ! . ?@ dk (x)d M (y) dk (x) dk (x)dk (x)  W ?  @ ? @ We de"ne



dC , C (x!x),! ?@ dk (x)dk (x)  ? @ and recall the de"nition (A.35) to write Eq. (A.38) as



(A.38)

(A.39)



d M (y) K (x!x)"C (x!x)! c (x!y) . (A.40) ?@ ?@ ? dk (x) W  @ To deal with the "nal variational derivative which appears here, we note that with p and the k taken to be the independent variables, ? dp(z) "0 , (A.41) dk (x) @ and so Eq. (A.23) implies that



dC d M (y) dC # "0 . (A.42) dk (x)d M (z) dk (x) d M (y)d M (z) @ W @ In the limit of constant chemical potentials and vanishing source, the "rst term here is just c (x!z) and the second factor in the integrand is G\(y!z). Hence @ d M (y) "! G(y!z) c (x!z) , (A.43) @ dk (x)  X @ and Eq. (A.40) becomes







K (x!x)"C (x!x)# ?@ ?@

WX or, in wave number/frequency space,

c (x!y)G(y!z) c (x!z) , ? @

(A.44)

KI (k, u )"CI (k, u )#c (k, u )GI (k, u )c (k, u ) . (A.45) ?@ K ?@ K ? K K @ K Since the function CI (k, u ) is a double variational derivative of the e!ective action functional ?@ K C[ M ; k ], it is single-particle irreducible. On the other hand, GI (k, u ), the potential correlation ? K function, is not single-particle irreducible. We have yet to express the number density and potential correlation functions in the simplest terms. To do so, we return to the expectation of the "eld equation (A.19). With k and p taken to be ? independent variables, the functional derivative of this equation with respect to a generalized

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chemical potential, with the chemical potentials then set constant and the source to zero, gives



d M (x) !  "i e K (x!x) . (A.46) ? ?@ dk (x)  @ ? With the use of Eqs. (A.43) and (A.45), the Fourier transform of this constraint may be put in the form





(A.47) ! k#i e c (k, u ) GI (k, u )c (k, u )"i e CI (k, u ) . ? ? K K @ K ? ?@ K ? ? The factor in square brackets on the left-hand side of this result is, according to Eq. (A.36), just G\(k, u ). Hence, K c (k, u )"!i e CI (k, u ) . @ K ? ?@ K ? Accordingly,

(A.48)

G\(k, u )"k# e e CI (k, u ) , K ? @ ?@ K ?@

(A.49)

and









(A.50) KI (k, u )"CI (k, u )! e CI (k, u ) GI (k, u ) e CI (k, u ) . A A? K K A A@ K ?@ K ?@ K A A We have found that both the potential and number density correlation functions are determined by the single-particle irreducible function CI (k, u ). We should note that the de"nition (A.39) of this ?@ K function, plus rotation and time reversal invariance, implies the symmetry CI (k, u )"CI (k, u ) (A.51) ?@ K @? K which thus carries over to the number density correlation function KI (k, u ). ?@ K The above results may also be used to reveal the structure of correlation functions involving the charge density. The correlation function of the charge density with the number density of species a is given by







KI (k, u )e " CI (k, u )e 1!GI (k, u ) e e CI (k, u ) K A B AB K ?@ K @ ?@ K @ AB @ @

" CI (k, u )e kG(k, u ) . (A.52) ?@ K @ K @ The kP0 limit gives the correlator of the number density with the total charge. This vanishes, as it must for the neutral plasma. Finally, the charge density}charge density correlation function is

 See, for example, Brown [23, Chapter II, Problem 5].

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given by KI (k, u )" e KI (k, u )e K ? ?@ K @ ?@ k [ e CI (k, u )e ] ?@ ? ?@ K @ . " k# e CI (k, u ) e ?@ ? ?@ K @ This form exhibits explicitly the small wave number behavior

(A.53)

KI (k, u )&k as kP0 . (A.54) K Static correlators, which are the focus of attention in the main text, are related to the zero frequency component of the corresponding time-dependent correlator functions by a factor of b\: KI (k)"b\KI (k, 0) , (A.55) ?@ ?@ and similarly for CI (k), GI (k), etc. Consequently, the static versions of Eqs. (A.49), (A.50), (A.53) and ?@ (A.54) are G\(k)"bk#b e e CI (k) , ? @ ?@ ?@



 

(A.56)



KI (k)"CI (k)! b e CI (k) GI (k) b e CI (k) , A A? A A@ ?@ ?@ A A k[ e CI (k)e ] ?@ ? ?@ @ , KI (k)" k#b e CI (k)e ?@ ? ?@ @

(A.57) (A.58)

and KI (k)&b\k

as kP0 .

(A.59)

A.3. Ewective potential, thermodynamic quantities In quantum "eld theory, the e!ective potential (times the space}time volume) is de"ned to be the restriction of the e!ective action to spatially (and temporally) uniform "elds. We have already remarked that the further restriction to the stationary point yields the grand potential (times b). With constant chemical potentials, the stationarity condition dC/d M "0 is just the condition that charge neutrality hold for a given value of M . For convenience, we will assume that physical chemical potentials are chosen such that this stationary point lies at M "0, so that . (A.60) bX"C[ M ; k]" M ( This is the function that we have computed to three loops. However, the charge neutrality constraint is never used in our computations, and so, in fact, the function C[ M "0; k] has been calculated for arbitrary (constant) chemical potentials k . This extension of the grand potential is ? needed for the computation of thermodynamic average numbers and energy and for the correlators of these quantities. Just as in our previous work, to derive general relationships it is convenient

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temporarily to work with C[ M ; k] for arbitrary constant M and k . The results of these derivations, ? however, will depend only upon the M "0 functions that have been computed. With uniform "elds, Eq. (A.23) reduces to *C( M ; k) "pbV . * M

(A.61)

As we have remarked before, the restriction to a vanishing source, p"0, determines

M " M (b, +bk ,) , (A.62) ? and inserting this value of M in C yields the physical grand potential bX. With arbitrary chemical potentials, M is non-vanishing so as to keep a zero charge density in the plasma. The previous expressions (A.24) and (A.25), evaluated with M at the stationary point (A.62), gives the physical particle numbers and energy, *C( M ; k) NM "1n V2 "! , ? ? @ *bk ?

(A.63)

*C( M ; k) ;"EM "1E2 " . @ *b

(A.64)

and

To obtain relations for the #uctuations of these quantities, we "rst need two results. The derivative of Eq. (A.61) with respect to the inverse temperature keeping p"0 so that M is determined by Eq. (A.62) gives, just as in the previous analogous calculation of the chemical potential functional derivative (A.42), *C( M ; k) * M #VGI \(0) "0 . *b* M *b

(A.65)

Note that, from (A.56), GI \(0)"b e CI (0)e , ? ?@ @ ?@

(A.66)

and 1 *C( M ; k) CI (0)" (A.67) ?@ V *bk *bk ? @ may be computed directly at M "0 with chemical potentials set to values which satisfy charge neutrality (for M "0) after the derivatives have been performed. Thus CI (0) can be obtained from ?@ the computation of the grand potential bX. We may simply write *1n 2 *1n 2 ? @ "! @ @, CI (0)"! ?@ *bk *bk @ ?

(A.68)

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where the partial derivatives are taken at constant temperature or "xed b. In a similar fashion, the derivative with respect to the inverse temperature of the charge neutrality condition *C( M ; k) "0 e ? *bk ? ? produces



(A.69)



*C( M ; k) *C( M ; k) * M # "0 . e ? *b*bk * M *bk *b ? ? ? We use Eqs. (A.35), (A.48) and (A.66) to write this as

(A.70)

* M ib *C( M ; k) "! GI (0) e . (A.71) ? *b V *b*bk ? ? After the derivatives in the relations above have been taken, we may again assume that the chemical potentials are chosen to give charge neutrality at M "0. With these results in hand, we can examine the #uctuations of energy and particle numbers. The energy #uctuations in the grand canonical ensemble are given by * *C( M ; k) 1(E!EM )2 "! @ *b *b *C( M ; k) *C( M ; k) * M "! ! . *b *b* M *b

(A.72)

We can make use of Eq. (A.65) to write this as *C( M ; k) *C( M ; k) 1 *C( M ; k) 1(E!EM )2 "! # GI (0) , @ *b *b* M V *b* M

(A.73)

or alternatively use Eqs. (A.65) and (A.71) to write









*C( M ; k) b *C( M ; k) *C( M ; k) ! e GI (0) e . (A.74) 1(E!EM )2 "! ? ? @ *b *b *bk V *b *bk ? ? ? ? This latter form may be evaluated at M "0 with the chemical potentials chosen to give charge neutrality after their derivatives have been taken. Thus, this latter form is determined by the quantities calculated for the grand potential bX. The energy}particle number correlation is given by * *C( M ; k) *NM 1(E!EM ) (N !NM )2 "! ? " ? ? @ *b *bk *b ? *C( M ; k) *C( M ; k) * M " # . *b*bk * M *bk *b ? ? With the use of Eqs. (A.35), (A.48) and (A.71), this becomes *C( M ; k) *C( M ; k) 1(E!EM ) (N !NM )2 " ! e GI (0)b CI (0)e . ? ? @ @ *b*bk ?A A *b*bk ? @ @ A

(A.75)

(A.76)

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Again, this result depends only upon quantities involved in the construction of bX. Note that, in view of Eq. (A.66), the charge neutrality condition is obeyed, 1(E!EM ) (N !NM )2 e "0 . (A.77) ? ? @ ? ? Finally, we note that the Fourier transform (A.57) evaluated at zero wave number yields the particle number}particle number correlators,





 



(A.78) 1(N !NM ) (N !NM )2 "V CI (0)! b e CI (0) GI (0) b e CI (0) . ?@ A A? A A@ ? ? @ @ @ A A The results that we have obtained may be used to compute the speci"c heat at constant volume. This is simply related to the derivative of the average energy with respect to the inverse temperature at constant particle numbers,



*EM . C "!b 4 *b + M ? , ,

(A.79)

Thus the chemical potentials must change as the temperature is varied in order to maintain constant numbers. That is, we have *EM *EM d(bk ) dEM " db# ? *bk *b ? ?









 

*C( M ; k) *C( M ; k) b *C( M ; k) # e GI (0) e db ? ? *b V *b*bk *b*bk ? ? ? ? *C( M ; k) *C( M ; k) ! e GI (0)b CI (0)e d(bk ) , # @ b*bk ?A A ? *b*bk ? @ @ A ? with the chemical potential changes constrained by "









(A.80)

*NM *NM ? d(bk ) 0"dNM " ? db# ? @ *bk *b @ @



"!



   

 

*C( M ; k) *C( M ; k) ! e GI (0)b CI (0)e @ ?A A *b*bk *b*bk ? @ @ A





# V CI (0)! b e CI (0) GI (0) b e CI (0) ?@ A A? A A@ @ A A Introducing the inverse matrix CI \(0), CI \(0) CI (0)"d , ?@ @A ?A @

db

d(bk ) . @

(A.81)

(A.82)

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which is a symmetric matrix since ""CI (0)"" is symmetric, we may rewrite Eq. (A.81) as ?@ 1 *C( M ; k) d(bk )" CI \(0) db ? ?@ V *b*bk @ @ 1 *C( M ; k) e GI (0)b db!GI (0)b e CI (0) d(bk ) . (A.83) !e @ *b*bk @ @A A ? V @ @ @A Because of the charge neutrality condition (A.77), a change d(bk ) proportional to e does not alter ? ? Eq. (A.80). Hence

 







*EM *C( M ; k) 1 *C( M ; k) *C( M ; k) " # CI \(0) . ?@ *b*bk *b + ? , *b V *b*bk , ? @ ?@

(A.84)

A.4. Time-dependent correlations We noted in Section 3.6 that although the static two-point potential correlation function, the zero frequency part of the general correlator, describes a Debye screened potential (except for the very long-distance tail elucidated in Section 7), the non-zero frequency parts of this correlation function are not Debye screened. Recalling the general result (A.49): G\(k, u )"k# e e CI (k, u ) , (A.85) K ? @ ?@ K ?@ this lack of Debye screening for u O0 is the statement that, for this case, K kP0: CI (k, u )&k , (A.86) ?@ K which implies that G\(r!r, u ) behaves as "r!r"\ for large "r!r". In this section we shall K show how this follows from the conservation of the number currents or, equivalently, from the gauge invariance of the coupling of the basic theory to a set of [(l#1)-dimensional] vector potentials. Number}current correlation functions are generated by coupling a vector potential A? (x),(A? (x), A?(x)) for each particle species a. This is done by augmenting the Lagrangian (3.90) I  for each basic charged "eld to read





* 1 !A? (r, q)! [ !iA?(r, q)]!k !ie (r, q) t (r, q) . (A.87) L "tH(r, q)  ? ? ? ? ? *q 2m ? Connected correlation functions of n space}time currents J? (x),(n (r, q), J (r, q)) are produced by I ? ? n functional derivatives d/dA? (x) acting on the generating functional =. In particular, the I connected number}density correlation function (A.14) is now extended to the space}time correlation function



d=[p; A] KIJ(x!x)" . (A.88) ?@ dA? (x) dA@ (x)  I J The corresponding connected, single-particle irreducible function is given by the same functional derivatives of the Legendre transform of =, the e!ective action C. This extension of Eq. (A.39)

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reads



dC CIJ(x!x)"! . (A.89) ?@ dA? (x) dA@ (x)  I J The actions formed from the extended Lagrangians (A.87) are invariant under local phase rotations of the charged "elds tH(x)PtH(x) exp+!ij?(x),, t (x)Pexp+ij?(x),t (x) , ? ? ? ? coupled with the gauge transformations of the external potentials *j? , A? (x)PA? (x)#i   *q

*j? A? (x)PA? (x)# . I I *xI

(A.90)

(A.91)

The integration measures of the charged "eld functional integrals are unchanged by the phase rotation (A.90). Hence the connected generating functional =[A] is invariant under the gauge transformation (A.91). This invariance carries over to the e!ective action C[A] since the Legendre transformation which relates it to =[A] involves only neutral "elds that are not altered by the phase rotation or gauge transformation. In the limit of an in"nitesimal transformation, the gauge invariance gives functional di!erential statements d d =[A]"0"* C[A] , I dA? (x) I dA? (x) I I where we have adopted the shorthand notation *






* * * " i , . I *q *xI

(A.92)

(A.93)

Taking additional functional derivatives of Eqs. (A.92) shows that any number current correlation function has a transverse form. In particular, one additional functional derivative yields * KIJ(x!x)"0"* KIJ (x!x) , I ?@ J ?@

(A.94)

* CIJ(x!x)"0"* CIJ(x!x) . I ?@ J ?@ In terms of Fourier components,

(A.95)

and

!iu CI J(k, u )#kJCI JJ (k, u )"0 . (A.96) K ?@ K ?@ K We are now in a position to demonstrate that the potential correlation function at non-zero frequency has no Debye screening. Three paragraphs ago, we remarked that this correlator is determined by CI (k, u )"CI (k, u ). Because of rotational invariance, CI J (k, u )"kJf (k, u ) ?@ K ?@ K ?@ K ?@ K and the l"4 component of the Fourier form (A.96) of the divergence condition becomes !iu CI (k, u )#k f (k, u )"0 . (A.97) K ?@ K ?@ K This demonstrates the assertion (A.86) that CI (k, u )&k as kP0 when u O0 and thus that ?@ K K there is no Debye screening in the u O0 potential correlation function G(k, u ). K K

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The fact that, for small k, CI (k, u )"O(k) when u O0 but CI (k, 0)"O(1) might appear ?@ K K ?@ a bit odd since CI (k, u ) is equal to an analytic function of u, F (k, u), evaluated at discrete ?@ K ?@ points on the imaginary axis, u"iu "i2pm/b. Thus one might expect a uniform behavior in K u which would require that CI (k, 0)"O(k) for small k and no Debye screening. In fact, the ?@ behavior of the analytic function F (k, u) is not uniform in k when u is small. This non-uniform ?@ behavior is illustrated by the simple one-loop contribution of the charged "elds to CI (k, u ). To ?@ K further simplify the result, we also take the P classical limit but with the frequency u / kept K "xed to obtain



* (dJp) 1 k ) n(p) , CI (k, u )"CI (k, u )"d ?@ K ?@ K ?@ (2p)J i(u / )!(p ) k/m ) *p ? K ? where

 

(A.98)



p n(p)"g exp !b !k (A.99) ? ? ? 2m ? is the Maxwell}Boltzmann density of particles in momentum space. This result is obtained by taking the indicated limits of Eq. (B.40) in the following appendix. Taking u "0 gives K (dJp) 1 (!bp ) k/m )n(p) CI (k, O)"d ? ? ?@ ?@ (2p)J (!p ) k/m ) ? "d bn, (A.100) ?@ ? which produces the leading-order contribution to the Debye wave number,



e e CI (k, O)" ben"i , ? @ ?@ ? ?  ?@ ? yielding for small k [cf. Eq. (A.85)]

(A.101)

G\(k, 0)+k#i , (A.102)  On the other hand, for u O0, the linear term in k in Eq. (A.98) vanishes, and expanding the K denominator to "rst order in k together with a partial integration of k ) */*p gives the small k limit




k CI (k, u )"d n . ?@ K ?@ u m ? K ? The corresponding small k contribution to the potential correlator produces





u . , G\(k, u )+k 1# K u K

 See, for example, the discussion in Problem 4 of Chapter II of Brown [23].

(A.103)

(A.104)

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where en u " ? ? (A.105) . m ? ? is the lowest order contribution to the plasma frequency. This is the result (3.98) given in the text. To see how the non-uniform behavior of the one-loop correlator (A.98) is in accord with the conservation (A.96) of the number current correlators, we note that the classical limit of a one-loop calculation also gives



* (dJp) pJ 1 CI J (k, u )"d k ) n(p) . (A.106) ?@ K ?@ (2p)J m i(u / )!(p ) k/m ) *p ? ? K ? We see that these contributions to !iu CI (k, u )#kJ CI J (k, u ) combine to form the integral of K ?@ K ?@ K a total derivative which vanishes, and so the current conservation is con"rmed. We also note the non-uniform limits CI J (k, 0)"0 , ?@ while for u O0, K

(A.107)

i n ? kJ . CI J (k, u )"d (A.108) ?@ K ?@ u m K ? For the sake of completeness, we note that the calculations leading to Eq. (A.50) are easily generalized to relate the number current correlation functions to their single-particle irreducible counterparts. The result is kP0:









KI IJ(k, u )"CI IJ(k, u )! e CI I(k, u ) GI (k, u ) e CI J(k, u ) . (A.109) ?@ K ?@ K A A? K K A A@ K A A We also note that time-reversal and spatial-rotation invariance together with the current conservation imply the symmetries CI IJ(k, u )"CI JI(k, u )"CI IJ (k, u ) , ?@ K ?@ K @? K

(A.110)

CI IJ(k, u )H"CI IJ(!k,!u ) . ?@ K ?@ K

(A.111)

and

Appendix B. Green's functions and determinants The result (3.6) in the text involves a product of path integrals of the form



  

Z[<]" [dtH dt] exp !

@



dq (dJr) tH (r, q)







*

 ! !k#<(r, q) t(r, q) , *q 2m

(B.1)

 We use the notation Z[<] because, when < is independent of imaginary time q, this functional integral is a representation the grand canonical partition function for a gas of particles with no mutual interactions but moving in the external potential <.

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with m one of the masses +m , and <(r, q)"!ie (r, q), with e the corresponding charge e . When ? ? the chemical potential is generalized to contain spatial or temporal variation, so as to generate number density correlation functions, its spatially or temporally varying part will be implicitly included in the potential <. The "eld t(r, q) is either periodic (for Bosons) or antiperiodic (for Fermions) in q with period b. The external potential <(r, q) is initially de"ned in the interval 0(q(b, but may be extended to all real q by regarding it as a periodic function with period b. The functional integral produces an inverse determinant in the Bose case and a determinant in the Fermi case, Z[<]"Det8





 * ! !k#<(r, q) . *q 2m

(B.2)

In this appendix, we shall show how the determinant Z[<] is related to a sum of ordinary, single-particle quantum-mechanical amplitudes. We shall then make use of this result to derive approximate evaluations of Z[<] that become valid in the limit in which the dynamics may be treated classically, approximations that are used in the calculations of the text. These needed results could perhaps be obtained more quickly with other methods, but the development given here hopefully illuminates the character of the theory and the intermediate results that are obtained may be useful in other contexts. The determinant can be constructed by integrating its variation. The familiar form for the variation of the determinant gives



d ln Z[<]"G

@

(dJr)G (r, q; r, q#0)d<(r, q) , @

 in which the thermal Green's function G is de"ned by @ *

 ! !k#<(r, q) G (r, q; r q)"d(q!q)d(r!r) , @ *q 2m





(B.3)

(B.4)

together with the boundary conditions that it be periodic for Bosons and antiperiodic for Fermions with a period of b, G (r, q#b; rq)"G (r, q; r, q#b)"$G (r, q; r, q) . (B.5) @ @ @ The coincident time limit used in the variation (B.3), in which qPq from above, is needed to give the proper operator ordering tRt that represents the density operator. To construct the thermal Green's function, it is convenient to introduce the quantum-mechanical transformation function in imaginary time 1r, q " r, q2 whose dynamics is governed by the external potential. It is de"ned by





*

 ! #<(r, q) 1r, q " r, q2"0 , *q 2m

(B.6)

together with the boundary condition 1r, q " r, q2"d(r!r) .

(B.7)

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We now assert that the thermal Green's function in the interval !b4q, q4b has the construction (akin to an image construction in electrostatics)  G (r, q; r, q)"h(q!q)1r, q " r, q2# ($1)LeIO\OY>L@1r, q#nb"r, q2 , @ L where h(q) is the unit step function. The proof is as follows. Since

(B.8)

G (r, q#0; r, q)!G (r, q!0; r, q)"1r, q"r, q2"d(r!r) , (B.9) @ @ Eq. (B.6) implies that the inhomogeneous Green's function equation (B.4) is obeyed. And the construction is easily seen to satisfy the periodicity condition (B.5). The coincident time limit of Green's function which enters into the variation (B.3) thus has the representation  G (r, q; r, q#0)" ($1)LeL@I1r, q#nb"r, q2 . @ L

(B.10)

Thus

 

@  d ln Z[<]"! ($1)L>eL@I dq (dJr)1r, q#nb"r, q2d<(r, q) .  L Since the potential is periodic, <(r, q#kb)"<(r, q) ,

(B.11)

(B.12)

so is the transformation function in the presence of this potential, 1r, q#kb#nb"r, q#kb2"1r, q#nb"r, q2 .

(B.13)

Hence, since we may add n equal copies if we divide by n, we may write

 

 eL@I L@ (B.14) d ln Z[<]"! ($1)L> dq (dJr)1r, q#nb"r, q2d<(r, q) . n  L To integrate this variational statement, we introduce a complete set of intermediate states and write



1r, q#nb"r, q2" (dJ r )1r, q#nb"r , nb21r , nb"r, q2 ,

(B.15)

and again use the periodicity of the external potential to write 1r, q#nb"r , nb2"1r, q"r , 02 .

(B.16)

Hence, the variational statement may be expressed as

 

 eL@I L@ dq (dJr)(dJr )1r , nb"r, q2d<(r, q)1r, q"r , 02 d ln Z[<]"! ($1)L> n  L  eL@I (dJr )d1r , nb"r , 02 , " ($1)L> n L



(B.17)

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where the second equality recognizes that this is just the variation of the transformation function when the potential is varied. Hence,



 eL@I (dJr)1r, nb"r, 02 , (B.18) ln Z[<]" ($1)L> n L which expresses the determinant in terms of an expansion in powers n of the fugacity z"e@I whose coe$cients are traces of single-particle transformation functions over the imaginary time interval (0, nb). To con"rm that the correct integration constant has been secured, we note that when the external potential <(r, q) vanishes, this form immediately gives the free-particle partition function since in this case



1r, nb"r, 02"

(dJp) e\L@NK , (2p)J

(B.19)

and so

 

 ($1)L> (dJp) eL@I (dJr) e\L@NK ln Z[<]" n (2p)J L (dJr)(dJp) ln [1Ge@Ie\@NK] , "G (2p)J



(B.20)

which is the well-known result for the quantum-statistical free-particle partition function. The single-particle transformation functions that appear here have a convenient path integral representation





  



L@ m dr dr ) #<(r(q), q) . (B.21) dq 2 dq dq  Here the functional integral is over all paths that begin and end at position r, r(0)"r"r(nb), with r then integrated over the large spatial volume V. In other words, the integral is over all paths which are periodic with period nb . In the limit in which the quantum-mechanical aspects of the particle's dynamics is not important, the classical limit for the dynamics which is equivalent to the large mass m limit, the dominant path is just the constant path r(q)"r so that, in this limit, (dJr) 1r, nb"r, 02" [dr] exp !





 



@

(B.22) dq <(r, q) ,  where the overall constant is determined by the free-particle limit (B.19), and the periodicity of the potential has been used to write the integral from 0 to nb as n times the integral from 0 to b. Placing this approximation in the general result (B.18) gives (dJr)1r, nb"r, 02"1r, nb"r, 02 (dJr) exp !n











p (dJr)(dJp) @ ln 1Gexp bk!b ! dq <(r, q) . (B.23) 2m (2p)J  In this expression, the quantum Bose}Einstein or Fermi}Dirac statistics are treated exactly, but the dynamics is treated entirely classically. In the limit of classical statistics, !bk<1, and only the "rst term in the expansion of the logarithm is signi"cant. Replacing < by !ie and remembering ln Z[<]"G

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the de"nition of the bare particle density puts this classical limit in the form







@

(B.24) dq (r, q) .  This is the formula used in the text and derived there so as to obtain the correct Coulomb classical partition function. Here we have obtained it as the classical limit of the many-particle, quantum mechanical system. To "nd sub-leading corrections to the large mass limit, it is convenient "rst to derive an exact series representation. The representation is obtained by placing the Fourier transform representation of the potential ln Z[<]" (dJr) n exp ie



<(r, q)"

(dJk)
(B.25)

in the exponent of the functional integral (B.21) and expanding the exponential in powers of the potential. Interchanging the orders of integration then yields



(dJr) 1r, nb"r, 02

 

 

(dJk ) L@ (dJk )  (!1)J L@ 


  

z [F]" [dr] exp ! L



L@ m dr dr ) #F(q) ) r(q) dq 2 dq dq 

,

(B.26)

(B.27)

with J F(q)"!i k d(q!q ) . (B.28) ? ? ? The remaining path integral (B.27) describes free-particle motion (in imaginary time) between `kicksa introduced by the impulsive force F(q). To evaluate this path integral explicitly, we write the path r(q) as a constant mean position r plus a deviation whose integral over the interval (0, nb) vanishes. The integration measure factors into an ordinary integral over the mean position (dJr) and a constrained measure [dr] which denotes integration over the space of periodic functions with vanishing mean. The integration over the mean position produces a delta-function,












J J (dJr) exp i k ) r "(2p)Jd k , (B.29) ? ? ? ? re#ecting the spatial translational invariance of the theory. Hence the time integral of the impulsive force must vanish,



L@ dq F(q)"0 . 

(B.30)

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The remaining functional integral can be evaluated by `completing the squarea. This is done with the aid of a Green's function f (q!q) de"ned in the space of periodic functions with vanishing L mean. We take this function to be dimensionless so that it obeys nb

d f (q)"d(q)!(nb)\ , dq L

(B.31)

together with the periodicity condition f (q#nb)"f (q) . L L The solution, when !nb4q4nb, is




(B.32)



"q" "q" 1! , f (q)" L nb 2nb

(B.33)

up to an additive constant. For the formulas below, it is convenient to choose the particular solution (B.33) which vanishes at q"0. The square is completed by shifting the functional integration variable to *r(q),r(q)!r (q), where



nb L@ r (q), dq f (q!q) F (q) . (B.34) L m  Since the Green's function f (q!q) is periodic, r (q) is periodic, and since r(q) is periodic, so is *r(q). L Moreover, since f (q!q) obeys Green's function equation (B.31) and F(q) has a vanishing mean L [Eq. (B.30)], d r (q)"F (q) . m dq

(B.35)

Hence we may make the shift and freely integrate by parts with no boundary contributions to evaluate the remaining functional integral and obtain






L@

 









nb L@ dq F(q) exp ! dq dq f (q!q)F(q) ) F(q) z [0] . (B.36) L L 2m   The "nal factor z [0] is a free particle path integral in the absence of any external force. This is just L a constant whose precise value is of no concern since the overall normalization will be trivially determined a posteriori by requiring that our result exhibit the correct free particle limit when the potential < vanishes. With these results in hand, we now see that the series (B.26) may be written as z [F]"(2p)J d i L

(dJr) 1r, nb"r, 02



(dJk ) k r  (!1)J L@  e  
 





(B.37)

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To illustrate the working of our results and to make contact with more familiar forms, we examine the two-point, charge density}charge density correlation function. This function is given by the double functional derivative of Eq. (B.37) with respect to






L@ k  eL@I dq e SO exp !nb f (q) . 1r, nb"r, 02 n (B.38) P(k, u)"e ($1)L> m L n  L In order to perform the sum and the Fourier transform, we recall Eqs. (B.19) and (B.33) to write

 

1r, nb"r, 02 exp !nb



"

   



 




(dJp) k p k q f (q) " exp !nb exp ! q 1! L (2p)J m 2m 2m nb



p q (dJp) exp !nb exp ! (k!2k ) p) , 2m 2m (2p)J

 (B.39)

where the second equality follows by making the translation pPp!qk/nb. Since the frequency u is a positive or negative integer multiple of 2p/b, we "nd that







p 1!exp +!(nb/2m) (k!2 p ) k), (dJp)  eL@I exp !nb P(k, u)"e ($1)L> 2m (2p)J (1/2m) (k!2 p ) k)!iu L (dJp) F ( p!k/2)!F ( p#k/2) ! ! "e , (B.40) (2p)J p ) k/m!iu



where

 

 

\ p F (p)" exp b !bk G1 ! 2m

(B.41)

are the free-particle Bose or Fermi distributions, and we have made a further translation pP!p#k/2. This is the familiar form for the density}density correlator in the `random phasea or single-ring approximation. Let us now restrict the discussion to the limit of classical, Maxwell}Boltzmann statistics where



P(k, u)"e





(dJp) 2 sinh (bp ) k/2m) b e@I exp ! (p#k/4) . (2p)J p ) k/m!iu 2m

 This later form is the result obtained by using operator methods to evaluate Tr e\L@NKe k  rOe\ k  r , where r(q)"r(0)!ipq/m is the operator free-particle motion in imaginary time.  See, for example, Eq. (30.9) and the discussion about it, in Fetter and Walecka [30].

(B.42)

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Taking the frequency to vanish and expanding in powers of the wave number gives



 



(B.43)

k jk en k "u "i , P(k, u)K C u C 2p(b u) m u

(B.44)

bk jk P(k, 0)Kebn 1! "i 1! . C 12m 24p

Here in the second form we have written i"ebn, which is the contribution to the squared C Debye wave number of a particle of generic charge e and density n, and j"2p b/m for the corresponding thermal wavelength. We have explicitly included the factor of  here to emphasize that this is a quantum correction. On the other hand, expanding in the wave number with the frequency non-zero gives

in which we have identi"ed the generic contribution to the squared plasma frequency u"en/m. C The plasma frequency is, of course, purely a classical quantity. However, the discrete frequencies that enter here are the quantum Matsubara frequencies that are integer multiples of 2p/ b (with b taken to have the units of inverse energy). The original form of the classical statistics limit is



P(k, u)"en

@







jk f (q) , e SO exp ! 2p

(B.45)

where we now write






"q" "q" f (q)"f (q)" 1! .  2b b

(B.46)

Since this is periodic in q with period b, it has the Fourier series representation  f (q)" f e\ SK O , (B.47) K K\ with u "2pm/b. Expanding Eq. (B.45) to order k and comparing with the results above, we K conclude that



!1/(2pm), mO0 ; f " K 1/12, m"0 .

(B.48)

These coe$cients are, of course, the same as those obtained directly from the Fourier transformation of f (q). We now return to the heavy mass limit, or equivalently the classical limit P0, in which the thermal wavelength becomes small, j"2p b/mP0. In this limit, the "nal exponential in Eq. (B.37) is set to one, and the resulting series may be trivially summed to reproduce the previous result (B.22). To obtain systematic corrections to this limit, it is worth noting that the sum which forms the integrand in Eq. (B.37) has the same structure as that of a classical grand canonical partition function for a system with pairwise interactions given by < ,!(n/m)k ) k f (q !q ). ?@ ? @L ? @ The imaginary-time integrals in Eq. (B.37) take the place of the spatial integrations that appear in a classical partition function, while the remaining factors in (B.37) may be interpreted as de"ning

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the single-particle measure. Consequently the usual `linked clustera theorem of classical statistical mechanics shows that the sum appearing in the integrand of Eq. (B.37) equals the exponential of the sum of `connected clustersa. In other words, if (dJk ) k r ? e ? 
(B.49)

denotes the `single-particlea measure, and





nb k ) k f (q !q ) !1 , g ,exp @ ?@ m ? @L ?

(B.50)

then the sum appearing in Eq. (B.37) may be expressed as the cluster expansion

 

 (!1)J l! J "exp

 

 

J J “ dk “ (1#g ) ? ?@ ? @? 1 1 dk # dk dk g # dk dk dk [3 g g #g g g ]#2 .  2!    3!        



(B.51) An expansion in powers of wave numbers is equivalent to an expansion in spatial gradients of the potential. Since g starts out proportional to k ) k , the jth term in the cluster expansion is of order ?@ ? @ wave number to the 2(j!1)th power. Thus the cluster representation (B.51) is a convenient vehicle for developing a systematic gradient expansion, in which each gradient will be accompanied by a factor of j"(2p b/m. Although the expansion is easily done for an arbitrary term n in the fugacity expansion, we shall need only the n"1 result corresponding to the classical limit of Maxwell}Boltzmann statistics. Hence we now restrict the discussion to n"1 and write f (q)"f (q)  as before. A little calculation, taking account of the remarks that we have just made, shows that to order j,



(dJr)e@I1r, b"r, 02



 

" (dJr)n exp !



 

@



j @ dq <(r, q)! dq dq f (q !q ) <(r, q ) <(r, q )     I  I  4p  

j  @ dq dq f (q !q ) <(r, q ) <(r, q )     I J  I J  4p  j  @ !2 dq dq dq f (q !q ) f (q !q ) <(r, q ) <(r, q ) <(r, q )        I J  I  J  4p 

#

#O(j) .

(B.52)

 See, for example Ref. [31, Section 10.1]. The connected nature of the expansion implies that, if the potentials
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To apply this result, so as to obtain the corresponding e!ective action interaction terms, we must replace, for each ion species a, !<(r, q)Pk (r, q)#ie (r, q). We are now allowing arbitrary ? ? space}time dependence in the chemical potential so that the resulting e!ective action will serve as a generating functional for frequency and wave-vector-dependent number density correlation functions. After summing over the various particle species, and inserting the Fourier series decompositions 

(r, q)" K(r) e\ SK O, K\ one "nds the action contribution



 k (r, q)" kK(r) e\ SK O , ? ? K\

(B.53)



S "! (dJr) n (r) exp ibe  ? ?  ? bj ! ? f [kK#ie K] [k\K#ie \K] K I ? ? I ? ? 4p K bj ? + [kK#ie K] [k\K#ie \K] # K I J ? ? I J ? ? 16p K bj # ? f f [kK>L#ie K>L] [k\K#ie \K] [k\L#ie \L]#O(j) . KL I J ? ? I ? ? J ? ? 8p KL (B.54)



Here + f , are the Fourier series coe$cients (B.48) for the function f (q), and + + , denote the K K Fourier series coe$cients for f (q), namely



!6/(2pm), mO0 ; + " K 1/120, m"0 .

(B.55)

In Eq. (B.54), the bare density n(r) is to be understood as containing the zero-frequency part of the ? chemical potential, so that n(r)"g j\J exp bk(r). ? ? ? ? In the "nal portion of Section 7 we need the contribution to the time-dependent number density operator generated by the action (B.54). This is the variational derivative of !S with respect to  bk (r, q). However, for the speci"c application of that section that involves the contribution to the ? long-distance tail of non-zero frequency correlators, speci"cally diagram (c) of Fig. 20 we may omit all terms involving  L, as such terms lead to spatial delta function contributions which cannot a!ect the leading long-distance behavior. The required correction to the number density operator arises only from the fourth line of Eq. (B.54) and, when Fourier transformed in time, becomes bje ? ? (f f !f f !f f ) \L L\K . (B.56) *n (r, u )"!n L K\L L \K K\L \K I J I J ? K ? 8p L Only terms involving exclusively non-zero frequency components of will be needed [as only these components have long-range 1/r correlations], which allows one to exclude the n"0 and n"m

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terms from the above sum and express the result as bje  ? ? [n(m!n)m]\ \L L\K . *n (r, u O0)"!n I J I J ? K ? (2p) L\ L$K

(B.57)

Appendix C. Required integrals C.1. Coulomb integrals The pure Coulomb potential for unit charges in l dimensions may be expressed as the Fourier transform



(dJk) e k  r < (r)" . J (2p)J k

(C.1)

To evaluate the potential explicitly, it is convenient to use the representation



 1 " (C.2) ds e\QI , k  interchange the s and k integrations, and perform the resulting Gaussian k integral. Writing s"1/t converts the result to the standard form of a C function, and yields




C(l/2!1) 1 J\ < (r)" . J 4pJ r

(C.3)

C.1.1. Powers of < The same procedure may be used to evaluate Fourier transforms of powers of the Coulomb potential,



CL(k), (dJr)e\ k  r < (r)L . J J

(C.4)

We insert the form (C.3) for the Coulomb potential, use


 

a  \ ds s?\e\QP , (C.5) 2  to represent the resulting power of r, interchange integrals and evaluate the Gaussian r integral. The variable change s"1/t once again produces the standard representation of the C function, yielding r\?"C




C(J !1)L C(n!J (n!1)) k JL\\L  . CL (k)"  J C(n(J !1)) (4p)L 4p 

(C.6)

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To obtain the lP3 limit of this result for various powers of the potential n, we make use of p C(z)C(1!z)" , sin pz

(C.7)

(from which follows C(1/2)"(p), use d t(z), ln C(z) , dz

(C.8)

with t(1)"!c, where c"0.577212 is Euler's constant, and Legendre's duplication formula C(2z)"2X\p\C(z)C(z#1/2) ,

(C.9)

(which shows that t(1/2)"!c!ln 4, a result that will also be needed). Using these ingredients, we "nd that CK(k) has a smooth limit as lP3, J







(!1)K> 1 K (p k K\ CK (k)" .  4 4p C(2m!1) 4p

(C.10)

In particular, we will need !1 (k . C (k)"  (16p)

(C.11)

For odd powers (greater than 1) there is a simple pole in 3!l arising from the last gamma function in (C.6), and one "nds that





1 K> k K>J\K> 1 CK>(k)"(!1)K J 4p 4p C(2m#3)



;



1 #(#m)[c#ln 4#t(#m)]#(1#m)t(1#m)#O(l!3) .   3!l (C.12)

In particular,


 
 



k J\ 1 1 #3!c#O(l!3) , C(k)" J 3!l 2(4p) 4p

(C.13)



k J\ 1 26 k ! ! #2c#O(l!3) . C(k)" J 3!l 3 4!(4p) 4p

(C.14)

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C.2. Debye integrals The Debye potential for a point charge in l spatial dimensions has the Fourier transform representation



(dJk) e k  r G (r)" . J (2p)J k#i

(C.15)

Writing the denominator as



 1 " ds e\I>GQ , k#i 

(C.16)

interchanging integrals, performing the resulting Gaussian integral in k, and scaling the resulting parameter integration variable by s"t(r/2i) expresses G (r) in terms of a standard representation J for a modi"ed Bessel function,










1 i J\1  ir 1 G (r)" dt t\J exp ! t# J (2p)J r 2 2 t  i J\ 1 K (ir) . " J\ (2p)J r

(C.17)

The power series development of the modi"ed Bessel function yields


 



1  (!1)K ir K G (r)" J 2(2p)J m! 2 K




#






2 J\ l C !m# !1 r 2

l i J\ C !m! #1 2 2

,

(C.18)

which displays the singular and regular terms for small r. C.2.1. Powers of G Let DL(k) denote the Fourier transform of the nth power of the Debye potential, J



DL(k), (dJr) e\ k  rG (r)L . J J

(C.19)

The density}density correlation function at l-loop order requires DL (k) for n up to l#1, and the J k"0 limits, DL (0) for n4l#1, are needed for the l-loop free energy. J D(k) is just the Fourier transformed Debye potential, J 1 , D(k)"GI (k)" J k#i

(C.20)

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while D (k) may be evaluated directly in three dimensions (and was already computed in J Section 2),



1  dk 1 k D (k)" " arctan .  k#k 4pk 4p 2i G It has the vanishing wave number limit

(C.21)

1 D (0)" .  8pi

(C.22)

For DL(k) with n53, one must work in l(3 dimensions and separate out the terms which J diverge as lP3, terms which arise from the small r region of the Fourier transform (C.19). Since the Coulomb potential in l dimensions, < (r), is the iP0 limit of G (r), the short-distance limit of the J J expansion (C.18) may be written as




l iJ\ C 1! [1#O((ir))] . G (r)"< (r) [1#O((ir))]# J J 2 (4p)J

(C.23)

To compute D(k), we note that as rP0, < (r)&(1/r)J\, and so [G (r)!< (r)] is less singular J J J J than 1/r when lP3. Hence the Fourier transform of this di!erence may be evaluated directly in l"3 dimensions, and we may write



D (k)" (dr)e\ k  r[G (r)!< (r)]#C(k)#O(l!3) , J   J

(C.24)

where C(k) is the Fourier transform of the cube of the Coulomb potential previously evaluated in J Eq. (C.13). To compute the integral of the di!erence of the cube of the Debye and Coulomb potentials, we represent



e\GP 1  e\IP G (r)" " , (C.25) dk(k!3i)  (4pr) (4p) 4pr G and use its iP0 limit to represent < (r). Placing an upper bound k"M on the these parametric  integrals, with the limit MPR reserved until the end of the computation, allows separate Fourier transforms to be taken, with the result, using Eq. (C.13), that


 





1 + k!3i + k D (k)" dk lim ! dk J (4p) k#k k#k  + G k J\1 1 1 # #3!c#O(l!3) . 2 3!l (4p) 4p



(C.26)

To keep the result in a dispersion relation or spectral form, we write k 1 1 k " ! k#k k k k#k

(C.27)

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in the "rst (Debye) integral and add the part






+ dk M "ln (C.28) k 3i G to the !ln(M/k) produced by the second (Coulomb) integral. The limit MPR can then be taken, and these two pieces reduce to  ln (k/9i). Since in the lP3 limit  k J\ 1 i J\ 1 i " #ln , (C.29) 4p 3!l 4p 3!l k










the ln (k/i) terms cancel, as they must, and there remains


 








1 9i J\ 1 1  k 1 D (k)" #3!c!2 dk #3i #O(l!3) . (C.30) J (4p) 4p 2 3!l k k#k G We have written an overall factor of (i)J\ so as to keep the dimensions correct when l!3O0 although this factor may be replaced by unity when it multiplies regular terms. It is a simple matter to evaluate the "nal integral and obtain the explicit result


 







9i J\1 1 6i k k 1 #3!c! arctan !ln 1# #O(l!3) , D (k)" J 2 3!l k 3i 9i (4p) 4p (C.31) whose kP0 limit is equal to


 



1 9i J\1 1 D (0) #1!c#O(l!3) . J (4p) 4p 2 3!l

(C.32)

The computation of D (k) may be performed in a similar fashion. Again referring to Eq. (C.23), J it is easy to check that




iJ\ l G (r)!< (r)!4< (r) C 1! J J J (4p)J 2

(C.33)

is less singular than 1/r when lP3. Hence,




iJ\ l D(k)"C(k)#4C(k) C 1! J J J (4p)J 2







1 e\GP 1 4i # (dr) e\ k  r ! # #O(l!3) . (4p) r r r

(C.34)

As before, we write the terms in the square brackets in the Fourier transform integral as parametric integrals over e\IP/r and interchange integrals to obtain





1 + (k!4i) 1 + k(k!8i) 1 lim ! D(k)" dk dk J 2 2 (4p) k#k k#k G  + iJ\ l # C(k)#4C(k) C 1! . J J (4p)J 2




 (C.35)

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With the aid of the results (C.11) and (C.13) for C(k) and C(k), it is a straightforward matter to J J compute D(k). Since we need only D(0), we shall simply state that J J 3 2i i J\ 1 #4! c!5 ln 2 . (C.36) D (0)"! J 3!l 2 (4p) 4p








C.2.2. Convolution integrals The Fourier transforms



DJKL(k)" (dJr)(dJr ) e\ k  rG (r!r )JG (r )KG (r)L , J  J  J  J

(C.37)

and



DIJKL(k)" (dJr)(dJr )(dJr ) e\ k  rG (r!r )IG (r !r )JG (r )KG (r)L ,   J  J   J  J J

(C.38)

were de"ned in the text in Eqs. (4.35) and (4.36). The two-loop correlators require the evaluation of D(k), D(k), and D(k), while the three-loop free energy involves D(0), D(0), and D(0). All of these quantities are well de"ned and may be evaluated directly in l"3 dimensions. The Fourier transform representation of D(k) reads



(dq) [q#i]\[(k!q)#i]\ . D(k)"  (2p)

(C.39)

This is just the derivative with respect to the (squared) Debye wave number of the Fourier transform of the square of the Debye Green's function, D(k),  1 1 1 dD(k)  " . (C.40) D (k)"!  8pk k#4i 2 di The other needed integrals are most easily evaluated using the spectral representation for the square of a Debye propagator in three dimensions,



 

 dk e\IP (dk)  dk e k  r " . (C.41) 4p 4pr (2p) 4p k#k G G Inserting this form into the de"nitions of D(k) and D(k), Fourier transforming, and   interchanging orders of integration produces G (r)" 

D(k)"D(k; i, i) , 

(C.42)

and R D(k)"! D(k; i, m)  Rm



,

(C.43)

m"i

in which

 

D(k; i, m),

 dk (dq) 1 1 1 . 4p (2p) (q#k) (q#m) (k!q)#i G

(C.44)

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The partial fraction decomposition



1 1 1 1 1 " ! q#k q#m k!m q#m q#k



(C.45)

yields two convolution integrals

 



(dq) 1 1 1 ! (2p) q#m q#k (k!q)#i

(C.46)

which just represent the Fourier transform of the di!erence of two products in coordinate space,







e\KP e\IP e\GP G>I du e\I P  ! " . 4pr 4p 4pr 4pr 4pr G>K

(C.47)

Hence





1  dk G>I dk  . D(k; i, m)" (4p) k!m k #k G G>K  Using



(C.48)



1 k#m 1 d "! ln , k!m k!m 2m dk

(C.49)

and integrating by parts gives



1 1 2i#m D(k; i, m)" ln (4p) 2m 2i!m





G



dk  dk k#m!i # ln k#k k#k k!m!i G>K G



.

(C.50)

This result yields

 











1 1 G dk  dk k D(k)" ln 3 # ln  (4p) 2i k#k k#k k!2i G G



,

(C.51)

and



1 1 ln 3  dk 1 4 G dk 1 D(k)" ! ! #  (4p) 4i k#4i 3i k#k k#k k k!2i G G 1 # D(k) . 2i 

 (C.52)

Simple integrations give the k"0 limits 1 1 D(0)"D(0)" ,   (4p) 6i

(C.53)

1 1 D(0)" ,  (4p) 18i

(C.54)

and

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where in Eq. (C.53) we have noted that at zero wave number D (0)"D (0). Again, we have   placed the results (C.51) and (C.52) in dispersion relation form. They may also be expressed in terms of elementary functions and Euler's dilogarithm



X dt ln(1#t) . (C.55) t  The dilogarithm contributions are exhibited by changing the dispersion relation integration variable to s"1/k, and then making partial fraction decompositions and further linear transformations on the s integration variable. The results are: Li (!z),! 











ik ik 1 1 i Li !2# !i Li !2! D(k)"    i i (4p) 4ki



ik k ik #i Li ! !i Li #2 ln 3 arctan ,   3i 3i 2i

(C.56)

and






 

1 8 1 8 k 4i k D(k)" ! arctan # # arctan  3 k#4i (4p) 8ki 3 2i 3i




2i k#2i k ! ln 1# k k#4i 9i

1 # D(k) . 2i 

(C.57)

The same techniques may be used to compute D(0) and D(0). It is easy to see that D(0)"¹(i, i) , 

(C.58)

and R D(0)"! ¹(i, m)  Rm



,

(C.59)

KG

where







 dk  dk (dq) 1   . (C.60) 4p 4p (2p) (q#k )(q#k )(q#m) G G   The three-dimensional q integral can be readily evaluated using spherical coordinates. Since the radial integral is even in q, it may be extended to run over !R(q(#R if it is multiplied by 1/2. The resulting integral over an in"nite range is trivially evaluated by contour integration. A little algebra puts the result in the form ¹(i, m),



 1 1 . dk dk ¹(i, m)"   (k #m)(k #m)(k #k ) (4p) G     The change of variables k "(m#2i)(y\!1)x#2i , 

k "(m#2i)(y\!1)(1!x)#2i , 

(C.61)

(C.62)

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converts the integration region to 0(x, y(1. The x integration is easily performed with the result that



 1 1 2 1 ln . (C.63) ¹(i, m)" dy (4p) (2i#m)#y(2i!m) 1#y y  A partial fraction decomposition, integration by parts, and a simple scale change for the integration variable in one of the terms gives the "nal form








2i!m 1 1 Li ! #Li (!1) . ¹(i, m)"   2i#m (4p) m

(C.64)

Hence, using Li (!1)"p/12, we have  1 p 1 1 Li ! # , D(0)"   3 12 (4p) i


 

(C.65)

and






1 1 p 4 3 1 Li ! # # ln . D(0)"   3 12 3 4 (4p) 2i

(C.66)

C.2.3. Even worse integrals The "nal integral needed for the three loop free energy is the `Mercedesa integral



D " (dr)(dr)(dr)G (r)G (r)G (r)G (r!r)G (r!r)G (r!r) ,       +

(C.67)

C , + . (4pi)

(C.68)

The pure number C may be shown [32] to be given by + dx 3 3#x x 4 x 3#x 1  ln #ln ! ln # ln C " + (2 4 2#x 4!x 2#x 2#x 3  (3!x 1 ? 1 "! [Cl (4a)!Cl (2a)] dh ln(2 sin(h/2))"  (2 ? (2  "0.02173762









(C.69)

(C.70)

Here Cl (h),Im Li (e F),  sin(nh)/n is the Clausen integral, and a,arcsin . The "rst    L integral form in (C.69) was derived by Rajantie [32]. The second integral form and the relation to Clausen integrals was `experimentallya reduced (and veri"ed to 1000 place precision) by Broadhurst [36]. This reduction to Clausen integrals has been proven analytically by Almkvist [37]. The "nal integral needed for the two-loop self-energy is



D (k)" (dr)(dr )(dr )e\ k  rG (r!r )G (r!r )G (r !r )G (r )G (r ) (              J(k/i) . , (4p) i

(C.71) (C.72)

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This integral is related to the discontinuity of the Mercedes integral if the screening length in one of the Debye potentials is analytically continued. It is not (so far as we know) expressible in terms of standard functions. However, Rajantie [32] has shown that it may be reduced to the one-dimensional form



 

 

z z  dx 2z z#(2#x) z\ arctan !arctan #ln J(z)" 2#x 2 (2#x) (z#3  (z#4!x 2#x



.

(C.73) For small z, 2z 1 #O(z) . J(z)" ! 36 243

(C.74)

Appendix D. Quantum Coulomb su(1, 1) symmetry exploited As discussed in the text, the ultraviolet divergences of classical two-loop order quantities are tamed by quantum #uctuations. The value of the "rst induced coupling which must be added to the classical theory can be inferred from the computation of the quantum-mechanical, two-particle, "nite-temperature correlation function. With the center-of-mass motion factored out as done in the text [Eq. (3.67)], the Fourier transform of the direct contribution to the relative motion correlation for particle species a, b reads



F (k)" (dr)e\ k  r 1r"e\@&"r2 . >

(D.1)

while the exchange contribution is



F (k)" (dr)e\ k  r1r"e\@&"!r2 . \

(D.2)

Here e e p # ? @, H" 4pr 2m ?@ is the Hamiltonian for the relative motion, with

(D.3)

1 1 1 " # (D.4) m m m ? @ ?@ the reduced mass of the two particles. To temporarily simplify the notation, we shall write m "m ?@ and e e /4p"e so that the Hamiltonian reads ? @ e p (D.5) H" # . r 2m

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Placing the factor of e\ k  r inside the matrix element in Eq. (D.1) and treating the coordinate r as an operator allows one to express the correlation function as a quantum-mechanical trace, F (k)"Tr e\@&e\ k  r . >

(D.6)

For the exchange contribution (D.2), we may write "!r2"P "r2, where P is the parity operator, so that F (k)"Tr Pe\@&e\ k  r . \

(D.7)

The evaluation of F (k) closely parallels that of F (k). To keep the presentation as simple as \ > possible we will focus on F (k), and then summarize the analogous results for the exchange > contribution F (k) at the end of this appendix. \ It proves convenient to write the correlation function F (k) as a contour integral involving > Green's function 1 e\ k  r ; H!E

(D.8)

dE e\@# G(k, E) , 2pi

(D.9)

G(k, E)"Tr namely



F (k)" >

!

where the contour C, shown in Fig. 21, wraps clockwise about the cut along the positive real E axis and also encircles all the bound-state poles which occur when e(0, corresponding to an attractive Coulomb potential. We shall "rst compute G(k, E) when the energy E is real and su$ciently negative so that E lies to the left of all singularities, and only later analytically continue to energies lying on the contour C. Thus at "rst we write c E"! , 2m

(D.10)

with c real and further restricted by c'"e" m when the potential is attractive. In view of the spherical symmetry of the problem, we may average over the orientations of k and use G(k, E)"Tr

1 sin kr . H!E kr

(D.11)

 This is slightly cavalier. Although the trace de"ning F (k) in (D.6) is well-de"ned, the corresponding trace in (D.8) > has a high-energy divergence in two or more dimensions. This divergence, which merely re#ects the growth of the density of states at high energy, is independent of the charge e. Therefore, we should really subtract the eP0 limit inside the trace de"ning G(k, E) and write F (k)"F (k)#*F (k), where F (k)"j\(2p)d(k) is the e"0 limit, so that the > > > > ?@ contour integral (D.9) becomes a representation just for the di!erence *F (k). But to keep the notation as simple as > possible, we will not indicate this subtraction explicitly.

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Fig. 21. Integration contour for F (k). Green's function G(k, E) has a cut along the positive real axis and, in the case of an > attractive potential, bound state poles at E "!me/(2n) for n"1, 2,2 . L

In view of the cyclic symmetry of the trace, this may be expressed as 1 1 sin kr . G(k, E)" Tr k (r(H!E)(r

(D.12)

D.1. Coulomb su(1, 1) symmetry This latter form permits a remarkably simple evaluation by group theory. To do this, we "rst de"ne the Hermitian operator c 1 J " (rp(r# r ,  2c 2

(D.13)

so that c (r (H!E)(r" J #e . m 

(D.14)

The (r transformation converts the energy eigenvalue problem to a coupling eigenvalue problem. To see this, we consider the Coulomb bound states "nlm2 which have the "xed energy !c/2m that corresponds to (mutually attractive) charges $e obeying the Bohr formula L e m c " L , (D.15) 2n 2m or c e"n . L m

(D.16)

 This su(1, 1) symmetry is a subgroup of a larger so(4, 2) `dynamicala symmetry of the hydrogen which was noted many years ago by Barut et al. [33]. The explicit construction used here of the generators in terms of canonical variables was, to our knowledge, "rst done by one of the authors (LSB) and G.J. Maclay and appears in the latter's Ph.D. dissertation [34]. Although we know of no other references, this construction may well appear elsewhere in the literature.

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Hence,





p e m c m 1 1 "nlm2" (r # "nlm2" (r L "nlm2"n "nlm2 , J  (r 2m 2m r c c (r

(D.17)

and so the eigenvalues j of J are the positive integers,   j "n , n"1, 2, 2 . (D.18)  For a "xed principal quantum number n, l ranges over 05l5n!1 and m in turn varies through !l5m5#l. Thus the degeneracy of the nth eigenvalue is L\ (2l#1)"n . (D.19) J To exploit the latent group properties, we introduce the Hermitian dilation operator which is conveniently labeled as J "(r ) p#p ) r) ,   and denote the commutator of J and J as J (times !i),    [J , J ]"!iJ .    Since

(D.20)

i[p, J ]"p , i[r, J ]"!r ,   J di!ers from J merely by a sign change,   1 c J " (rp(r! r .  2c 2

(D.22)

(D.21)

(D.23)

Moreover, a further commutation with J restores the original signs,  [J , J ]"!iJ . (D.24)    And a straightforward computation of the "nal commutator shows that the algebra closes, [J , J ]"iJ .    The three Pauli spin matrices p obey the su(2) Lie algebra I [p , p ]"2i e p . I J IJK K Thus, as far as the commutation relations go, we have the correspondences 1 i i J  p , J  p , J  p ,  2   2   2 

(D.25)

(D.26)

(D.27)

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which identi"es the commutators of the J with the Lie algebra su(1, 1). This, of course, corre? sponds to a non-compact group which has in"nite-dimensional irreducible representations. With these results in hand, we return to our computation. Since J !J "cr , (D.28)   the sine function in Eq. (D.12) may be written in terms of group generators. Using this and the expression (D.14) for the denominator in Eq. (D.12), we obtain 1 m Tr [e IJ \J A!e\ IJ \J A] . (D.29) G(k, E)" 2ick J #(me/c)  Representing the denominator in terms of the integral of an exponential now places the result in terms of the trace of the product of group elements:



m  (D.30) ds e\KCAQ Tr e\QJ [e IJ \J A!e\ IJ \J A] . G(k, E)" 2ick  The products of two group elements may be expressed as a third group element. Since the trace is invariant under similarity transformations, this third group element may be `rotateda into one involving only the generator J ,  J J J (D.31) Tr e\Q  e! I  \  A"Tr e\Q! J . The required parameters s will be determined momentarily. Evaluating the trace using the known ! eigenvalues j "n of J with multiplicity n yields   *  1  1 cosh s /2 ! . " (D.32) Tr e\Q! J " n e\Q! L" *s eQ! !1 4 sinh s /2 ! ! L Therefore










cosh s /2 cosh s /2 m  > ! \ . (D.33) G(k, E)" ds e\KCAQ sinh s /2 sinh s /2 8ick  > \ In view of the algebraic isomorphism between the group generators and the Pauli matrices, the parameters s may be found by replacing the generators in Eq. (D.31) by the equivalent 2;2 Pauli ! matrices. Hence,



ik 2 cosh s /2"tr exp+!s p ,"tr exp+!sp , exp $ (p !ip )    ! !  2c 







ik "tr exp+!sp , 1$ (p !ip )    2c 









ik ik "e\Q 1$ #e>Q 1G . 2c 2c

(D.34)

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139

We write







ik k 1# "e F 1# 2c 4c

(D.35)

so that Eq. (D.34) becomes



k cosh(sGih)/2 . cosh s /2" 1# ! 4c

(D.36)

A short calculation yields






k sinh s /2" 1# [sinh(sGih)/2#sin h/2] . ! 4c

(D.37)

For later use, note that h"2 arctan (k/2c)"k/c#O(k) .

(D.38)

Hence



1  m G(k, E)" ds e\KCAQ 8ick 1#(k/2c)  cosh(s!ih)/2 cosh(s#ih)/2 ! . ; [sinh(s!ih)/2#sin h/2] [sinh(s#ih)/2#sin h/2]





(D.39)

As noted in Footnote 75, all along we should have subtracted the e"0 contribution from G(k, E). In the integral representation above, this simply means replacing e\KCAQ by the subtracted exponential [e\KCAQ!1]. As anticipated, this subtraction removes what would otherwise be a singularity in the integral at s"0. D.2. Direct contribution To compute the integral (D.39) (with the e"0 piece removed), it is convenient to deform the path of integration into the contours shown in Fig. 22. For the "rst term in braces in the integrand, the contour is taken to run "rst over a portion of the imaginary axis, s"i , 0( (h, and then to continue along the line parallel to the real axis, sPs#ih, 0(s(R. The integration contour for the second term in the braces is the complex conjugate of the "rst. These contour deformations produce






1 me me me m J h, !sin h I h, G(k, E)" c c c 4ck 1#(k/2c)



,

(D.40)

where



J(h, z)"

F



d [cos z !1]

cos(h! )/2 , [sinh/2!sin(h! )/2]

(D.41)

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Fig. 22. Integration contours for G(k, E).

and





I(h, z)"

cosh s/2 . [sinh s/2#sin h/2]

(D.42)  Although this general result may be of interest in other contexts, here we are interested in the small k behavior, since this determines the induced couplings in the e!ective theory. In the "rst integral J(h, z), it is convenient to make the variable change "h(1!x) and write the integral as



J(h, z)"h

ds e\XQ



dx[cos zh(1!x)!1]

cos hx/2 . [sin h/2!sin hx/2]

(D.43)

 Recalling that hKk/c, we may expand the trigonometric functions in the integrand in Eq. (D.43) and keep only the leading terms to obtain



J(h, z)"!4z



dx

(1!x) #O(h) . (1#x)

(D.44)  Writing (1#x)\"!2(d/dx)(1#x)\ and integrating by parts produces an end-point contribution and an integral made trivial by the substitution x"sin s, and one "nds that J(h, z)"!z (8!2p)[1#O(h)] .

(D.45)

If I(h, z) is expanded in powers of z, the "rst three terms are singular as hP0, while all remaining terms have "nite hP0 limits. It is convenient to separate the singular terms by writing I(h, z)"I (h)!zI (h)#zI (h)#IM (h, z) ,     where



I (h), I



ds



sI cosh s/2 , [sinh s/2#sin h/2]

(D.46)

(D.47)

and



IM (h, z),





cosh s/2 ds [e\XQ!1#zs!zs] .  [sinh s/2#sin h/2]

(D.48)

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141

Since d(sinh s/2)"(ds/2) cosh s/2, the change of variable sinh s/2"sin(h/2) tan s makes the integral I (h) elementary,  2 8 2 p 2 I (h)" " # #O(h) . (D.49) ds cos s"  sin h/2 sin h/2 h 3  To evaluate I (h) we write   8s  cosh s/2 8 ds ds s I (h)" # ! . (D.50)  [s#h] [sinh s/2#sin h/2] [s#h]   The "rst integral, which is easy to evaluate, contains the piece which is singular as hP0, while the second integral is "nite as hP0 and may be evaluated directly at h"0. Therefore,


















 8 cosh s/2 8 I (h)" # ! #O(h) ds s  h sinh s/2 s  s cosh s/2 8  8 #O(h) #2 ! " ! sinh s/2 sinh s/2 s h  8 " !2#O(h) . h



(D.51)

A similar approach may be used for I (h) if one "rst splits the integral into the contributions from  s(1 and s'1,












 cosh s/2 8 8s # ds s ! [sinh s/2#sin h/2] [s#h] [s#h]    cosh s/2 ds s # [sinh s/2#sin h/2]  cosh s/2 8   cosh s/2 "!8(1#ln h/2)# ds s ! # #O(h) ds s sinh s/2 s sinh s/2   s cosh s/2  "!8(1#ln h/2)#lim 8 ln e# 8 ln sinh s/2! !4s #O(h) sinh s/2 sinh s/2 C C "4!8 ln h/2#O(h) . (D.52)

I (h)" 

ds









 

 

The "nal integral IM (h, z) is non-singular as hP0, and so we may simply set h equal to zero and then integrate-by-parts twice,

 

IM (0, z)"!

"!z





1 d ds [e\XQ!1#zs!zs]  ds sinh s/2





ds [e\XQ!1#zs]

1 sinh s/2

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"4z

ds [e\XQ!1#zs]





ds

d 1 ds eQ!1

e\XQ!1 . eQ!1

(D.53)  The denominator may be expanded in a geometric series and the resulting s integrals performed to give "4z

 1 . IM (0, z)"!4z l(l#z) J Using

(D.54)

1  , (D.55) t(z#1)#c"z l(l#z) J where t(z) is the logarithmic derivative of the gamma function and c is Euler's constant, yields the closed-form result IM (0, z)"!4z[t(z#1)#c] .

(D.56)

This form may be used to make contact with the literature on quantum Coulomb corrections [4,6]. A power series expansion in z is obtained if the denominator in the sum (D.54) is expanded in powers of z and the order of the resulting double sum interchanged. This process gives  IM (0, z)"4z (!z)Lf(n#1) , L where

(D.57)

 1 f(n)" lL J is the Riemann f function. Assembling the various pieces contributing to Green's function G(k, E) and inserting h"(k/c)!  (k/c)#O(k/c)  produces



(D.58)

(D.59)










2z p z 1 z z z z 2c G(k, E)"m ! # # ! !z ln ! ! IM (0, z) #O(k) , k 2 ck c 6 2 6 4 k

(D.60)

where z"me/c. This result is to be inserted into the contour integral (D.9) relating G(k, E) to the thermal correlator F (k) which, with the e"0 subtraction made explicit, reads > dE e\@#G(k, E) . (D.61) *F (k),F (k)!F (k)" > > > 2pi !



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143

Inserting the power series representation (D.57) for IM (0, z) and recalling that c"!2mE, the required contour integrals are easily performed using Hankel's formula



dt (!t)\?e\R , 2pi

(D.62)



dt ln(!t)(!t)\?e\R . 2pi

(D.63)

1 " C(a)

! and its derivative with respect to a, t(a) " C(a)

! The result, neglecting O(k) contributions, is


 

m  4pbe pbe *F (k)" ! # > 2pb k k




bk pbe 1 # ln #c!3# #f 2m 3 bme


  bme 2p

,

(D.64)

where 3 3(p  f(n#1) (!(py)L f (y),! ! . (D.65) 2 4y C((n#5)/2) L Returning to our rationalized units with ePe e /4p, replacing the mass parameter by the reduced ? @ mass, mPm , and writing the result in terms of the thermal wavelength corresponding to the ?@ reduced mass




2pb  j "

?@ m ?@

(D.66)

gives



be e (be e ) *F (k)"j\ ! ? @ # ? @ > ?@ k 16k


 







be e  p j k be e 8pj ? @ ? @ #O(k) . (D.67) ?@ #f ln ?@ #c!3# 4p 3 4p 4pj (be e ) ?@ ? @ Evaluating f (y) using the power series representation (D.65) is appropriate if be e /j is order ? @ ?@ one or smaller. But if be e /j is large, which corresponds to the formal mPR limit, one needs ? @ ?@ the asymptotic form of f (y) for large argument. The result di!ers depending on whether the Coulomb interactions are attractive or repulsive. Consider the repulsive case "rst, where z"me/(!2mE is positive on the negative real E axis. In this case IM (0, z) has no poles on the negative real E axis, which re#ects the absence of bound states for repulsive potentials. Thus, for repulsive interactions the contour integral (D.9) only wraps about the positive E axis, and IM (0, z) #

 See, for example, p. 245 of Whittaker and Watson [35].

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appears with "arg z"(p. Hence the large m limit may be obtained by using the large z asymptotic behavior of the t function,  B 1 L , ("arg z"(p) (D.68) t(z#1)&ln z# ! 2nzL 2z L where B are the Bernoulli numbers, to write the asymptotic form of Eq. (D.56) as L 1  B L . ("arg z"(p) (D.69) IM (0, z)&!4z ln z#c# ! 2nzL 2z L Using this form for IM (0, z) and re-evaluating the contour integral (D.61) yields the asymptotic expansion for large positive argument,






B 8 3  L C(n!) . f (y)&2 ln(2(py)#3c! ! (!1)L  2npLyL 3 2(p L Evaluating the "rst term in the sum with B "1/6 and C(!1/2)"!2(p yields  8 1 #O(1/y) . f (y)"2 ln(2(py)#3c! ! 3 4py

(D.70)

(D.71)

To obtain the corresponding limit in the attractive case, note that Eq. (D.56) gives  f(2m#2) 3 pKyK> . (D.72) f (y)!f (!y)"! #3p (m#2)! 2y K We insert the de"nition (D.58) of the f function and interchange the order of the summations. The sum over m now produces an exponential with its "rst two expansion coe$cients removed, and we obtain

  



py 3  py 3 n exp !1! . (D.73) f (y)!f (!y)"! # n n 2y py L As yP!R, the "rst term in the sum, which corresponds to the lowest bound state contribution, dominates, 3 exp+py, , f (y)& py

(D.74)

with exponentially small corrections. D.3. Exchange contribution The same approach may be used to evaluate the exchange contribution (D.75) F (k)"Tr Pe\@&e\ k  r . \ Since the parity operator P commutes with all the su(1, 1) group generators, all the previous formulas hold for this exchange term with the trivial change of an insertion of P in the trace

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145

de"ning Green's function. To evaluate the "nal trace Tr Pe\Q! J , we note that the "nlm2 basis which diagonalizes J as shown in Eq. (D.17) has the familiar parity  assignment of the hydrogen atom states, P"nlm2""nlm2(!1)J .

(D.76)

Hence, we have essentially the same evaluation of the trace as before except that the previous degeneracy factor (D.19) is replaced by L\ (!1)J(2l#1)"!(!1)Ln . J Thus we now have

(D.77)

 1 Tr Pe\Q! J "! (!1)Lne\Q! L" , 4 cosh s /2 ! L and Eq. (D.33) is replaced by





(D.78)



m  1 1 G (k, E)" ! . (D.79) ds e\KCAQ \ 8ick cosh s /2 cosh s /2  > \ We are interested in the ("nite) k"0 limit. Recalling Eq. (D.36) and that h&k/c, this is given by







1 m  1 1 G (0, E)" ! ds e\KCAQ lim \ 8c ih cosh(s!ih)/2 cosh(s#ih)/2  F 1 m  d "! . (D.80) ds e\KCAQ 4c ds cosh s/2  Expressing the hyperbolic cosine in terms of exponentials and performing two partial integrations produces






(D.81)



(D.82)

me m G (0, E)" h \ c c with

1 z  1 h(z)" ! #z ds e\XQ . 4 2 eQ#1  Expanding the denominator and performing the s integration gives





1 z z  1 1 h(z)" ! # ! . 4 2 l#(z!1)/2 l#z/2 2 J

(D.83)

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Adding and subtracting 1/l in the sum, combining denominators, and referring to the representation (D.55), identi"es the sum as the di!erence of two t functions, and yields the closed-form result





1 z z z#2 z#1 h(z)" ! # t !t 4 2 2 2 2

.

(D.84)

Expanding the denominators in Eq. (D.83) in powers of z and using  (!1)J> "[1!2\X]f(z) lX J gives the useful power series expansion

(D.85)

1 z  h(z)" ! #z (!z)L(1!2\L)f(n#1) . 4 2 L Here the n"0 member of the sum is to be understood as containing  (!1)J> lim [1!2\X]f(z)" "ln 2 . l X J Inserting this series into the contour integral



F (0)" \

(D.86)

(D.87)

dE e\@#G (0, E) , \ 2pi

(D.88)

! and again making use of Hankel's formula (D.62) produces









m  p (bme F (0)" be fI , \ 2p 2pb 3

(D.89)

where





3 f(n#2) 3 3 ln 2 3(p  1 fI (y)" (!(py)L 1! ! # ! . (D.90) 2 8py 2py 2y 2>L C((n#5)/2) L To obtain the behavior for the case of strong repulsion, that is, the large y"bme/2 limit, we return to the integral expression (D.82) for h(z). Writing 1 1 s 1 " ! tanh , 2 eQ#1 2 2

(D.91)

performing simple integrals, and rescaling the integration variable casts this integral representation in the form




 

u u 1  ! . (D.92) h(z)"! du e\S z tanh 2z 2 2  This result shows explicitly that h(z) is an even function of z"me/c. Writing E"p/2m sets c"(!2mE"ip, with no problem with the sign of i since only even functions of c appear. And, again because only even functions of p appear, we may replace the contour integral (D.88) in the

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147

E plane by a contour integral in the p plane having the exactly the same contour. Thus



F (0)"! \




dp 1 me e\@NKh . 2pi p ip

(D.93) ! Introducing the integral representation (D.92) into this contour integral, interchanging the integration order, and rescaling the contour integration variable by writing p"2mef/u yields









df 1 tan f 1  F (0)" exp+!2bmef/u, !1 . (D.94) du ue\S \ 2pi f f 4 !  The integrand of the contour integral has no pole at f"0 since the quantity in the square brackets vanishes there. Thus the only singularities of the integrand come from the factor in square brackets, which has a series of simple poles at odd integer multiples of p/2 with residue !1. Since these poles are encircled in a negative, clockwise sense, we obtain







2  1   exp+!bme(2n#1)p/2u, . (D.95) F (0)" du ue\S \ (2n#1)p 4  L The leading asymptotic behavior is obtained by evaluating the u integral, term-by-term, using the method of steepest descents. Only the n"0 term of the sum is relevant, since the remaining terms are exponentially smaller. Writing the result in terms of the function fI (y) de"ned in Eq. (D.89) with y"bme/2p gives



 3 fI (y)& du ue\S exp+!yp/u, , yp  whose steepest descent evaluation yields





3p 2(3 exp ! (2y) fI (y)" yp 2

(D.96)



1 17 1# #2 . 18p (2y)

(D.97)

Since the exchange term involves interactions of a single particle type, the reduced mass m appearing in the above formulae is m /2 for species a. Reverting to our rationalized units gives ? 3 be  be ? fI ? , F (0)"j\ (D.98) \ ?? p 4p 4pj ??





where j "(2j . Note that here the argument of fI (y) is always positive. ? ?? Appendix E. First quantum correction to classical one-component plasma Here we shall derive the leading, order , quantum correction to the N-particle canonical partition function of the classical, one-component plasma. This result appears in the literature [Eq. (24) of [5]], but we will give a self-contained pedagogical treatment. To do so, it is convenient "rst to examine the path integral representation of the single-particle partition function previously

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given in Eq. (B.21), namely





  





1 (dr) 1r, b"r, 02" [dr] exp !





@ m dr dr ) #<(r(q)) dq 2 dq dq

. (E.1)  We have explicitly displayed the factors of which appear when b and q have their conventional units of inverse energy and time, respectively. We state again that the functional integral is over all paths which are periodic with period b . It is therefore convenient to use a Fourier series representation for the path, r(q)"r#m(q), in which  2pinq m(q)" m exp ! (E.2) L

b L\ L$ contains the non-zero frequency #uctuations of the path about its mean position r. As we shall see, the size of the #uctuations m(q) are of order . Since the (imaginary) time average of these #uctuations vanish, the leading quantum-mechanical correction appears in














2pn  bm  m )m (dr)1r, b"r, 02" (dr) exp+!b<(r), [dm] exp ! L \L

b 2 L\ L$  ; 1!b mI mJ <(r)#O(m) . (E.3) L \L I J L\ L$ The path integral over the #uctuations de"nes a correlator which is just the inverse of the matrix de"ning the quadratic form in the exponential,








1 b  1mI mJ 2"d dIJ . L \LY LLY bm 2pn

(E.4)

Using 1  p "f(2)" , 2n 6 L\ L$ one immediately "nds



(E.5)







b  (dr)1r, b"r, 02"1r, b"r, 02 (dr) exp+!b<(r), 1!

<(r)#O( ) . 24m

(E.6)

In other words, to O( ), the e!ect of quantum #uctuations is equivalent to a shift in the potential energy <(r) appearing in the classical partition function by b  d<(r)"

<(r) . 24m

(E.7)

As a check on the result (E.6), we note that a partial integration of one of the gradients in the

 term, together with the identi"cation j"2p b/m and other minor notational changes, places this result in precisely the form of the "rst line of Eq. (B.54).

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149

This result is easily extended to the case of the canonical partition function for N particles. This case is represented by a path integral over the variables r (q), where a"1, 2,2, N, ? m dr dr 1 @ 1 ? ) ? # <(r (q)!r (q)) . (E.8) Z " “ [dr ] exp ! dq ? ? @ , 2

dq dq 2  ? ? ?$@ The leading quantum, order , corrections come from the quadratic #uctuations in the coordinates r (q) and r (q) in each of the potential terms. Expanding these terms out from the exponential ? @ is performed in a compact fashion if the N-particle number densities are introduced,



  



, n(r)" d(r!r ) , ? ? along with the canonical two-particle correlator



(E.9)



(E.10) K (r!r)" d(r!r )d(r!r ) "1n(r)n(r)!d(r!r)n(r)2 . ? @ , ?$@ With this notation, the change in the N-particle canonical partition function Z for a general , variation in the interparticle potential is given by



b d ln Z "! (dr)(dr) K (r!r)d<(r!r) . , , 2

(E.11)

Thus, in view of the previous one-particle result (E.7), but keeping in mind that both the coordinates in the potential undergo #uctuations, we see that the leading quantum correction is given by



b  b

<(r!r) . d ln Z "! (dr)(dr) K (r!r) , , 12m 2

(E.12)

Taking account of translational invariance, which gives an overall factor of the system volume V, and remembering the de"nition !bF"ln Z of the Helmholtz free energy F, we have , b  V (dr)K (r) <(r) . (E.13) dF" , 24m



This general result may be applied to a one-component plasma in the presence of a uniform neutralizing background charge density [since a strictly classical limit exists in this special case where the charge carriers all have a common sign of their charge]. However, one must be careful to properly handle the e!ect of the neutralizing background charge density before taking the thermodynamic limit. The easiest way to do so is to regard the interaction potential for the one-component plasma as the kP0 limit of the regularized potential




<(r)"e



e\IP 1 ! . 4pr Vk

(E.14)

The integral of this potential over the large volume V of the system vanishes, re#ecting that a proper subtraction of the uniform background charge density has been performed. [Equivalently, this amounts to using a regularized Coulomb potential e\IP/(4pr) with a total charge density of e[n(r)!n ], where n "N/V is the "xed average particle density.]

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Now using




! <(r)"e d(r)!k



e\IP , 4pr

(E.15)

Eq. (E.13) becomes







e\IP b e V !K (0)#lim k (dr) K (r) . dF" , , 4pr 24m I

(E.16)

Because of the singularity of the Coulomb potential when rP0, the two-particle correlation K (r) , vanishes as rP0, K (r)Jexp+!b<(r),P0 as rP0 , ,

(E.17)

and therefore the "rst term in Eq. (E.16) identically vanishes. On the other hand, at large separations, the number densities are not correlated, and so




N  "n  K (r)P , V

as rPR .

(E.18)

For in"nitesimal k, the integral in the second term of Eq. (E.16) is dominated by arbitrarily large distances, and hence one may simply replace K (r) in the integrand by its asymptotic value of n . , When multiplied by k, the resulting error one is making in the short distance part of the integral has no a!ect on the kP0 limit. Consequently,



e\IP b e b e Vn  lim k (dr) " Vn  . dF" 4pr 24m 24m I

(E.19)

Finally, the pressure is given by p"!RF/RV with N and b "xed. Thus, since Vn "N/V, we "nd that the "rst quantum correction to the pressure of classical one-component plasma is given by d




bp b  " ben . n 24m

(E.20)

Recalling the de"nitions i"ben, g"bei/4p, j "4pb /m, and g "be/(4pj ), we may CC CC CC rewrite this result as d










bp 1 b  4p  g " "g . n 24 4pg 24m be CC

(E.21)

This correction agrees with that which appears in Eq. (5.33) in the text, as well as with the discussion of Eq. (24) in Ref. [5]. Note that since the O( ) correction is proportional to g, it is entirely contained in the two-loop contribution of the equation of state; no O( ) corrections are contained in any higher-loop contributions.

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151

Appendix F. Some elements of quantum 5eld theory Here we shall brie#y derive and review some of the methods of quantum "eld theory that are used in the text. For simplicity we shall explicitly treat the case of the functional integral representation of the classical plasma without the induced interactions that provide the quantum corrections which make the theory "nite. These additional interactions entail no essential changes in the techniques that we are about to outline, and their e!ects are easily described by including the appropriate additional terms in the functional integrand. In the same vein, we shall also neglect the quantum-mechanical imaginary time dependence discussed in Section 3.6. Thus we shall examine the generating functional



Z[k]"Det[!b ] [d ] exp+!S[ ; k], ,

(F.1)

in which



b S[ ; k]" (dJr) (r)(! ) (r)#S [ ; k] ,  2

(F.2)

where



(F.3) S [ ; k]"! (dJr) n(r) e @C? (r . ?  ? With the generalized chemical potential functions k (r) taken to be constants, the generating ? functional Z[k] reduces to the grand canonical partition function. Functional derivatives with respect to the generalized chemical potentials, with these potentials then taken to be constants, yield number density correlation functions. F.1. Perturbation theory Perturbation theory developments of correlation functions can be done in essentially either of two ways: one can "rst perform the functional derivatives with respect to the generalized chemical potentials to bring down extra factors in the functional integrand that represent the particle densities and then set the chemical potentials to be constants and make a perturbative expansion. Or one can make a formal perturbative expansion of the functional integral with spatially varying, generalized chemical potentials k (r), and then expand the result in a spatial varying part of the ? chemical potentials to identify the correlation functions. Either case is subsumed in a slight generalization of the second way in which we write the functional integration "eld as

(r)" M (r)# (r), where M (r) is some suitable background "eld. We then take out and explicitly display the pieces of zeroth and second-order in the #uctuation "eld (r). Since the background "eld M appears as a `constanta translation in the (dummy) functional integration variable, [d ]"[d ], and so with this separation and with an operator or in"nite matrix notation, Z[k]"Det[!b ] exp+!S[ M ; k],



; [d ] exp+!(b/2) [! #V( M ; k)] ,F[ ] .

(F.4)

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Here V( M ; k; r)" be n(r) e @C? (M r , ? ? ?

(F.5)

and F[ ]"exp+!SI [ ; M , k],2 , (F.6)  where !SI [ ; M , k] contains the linear, cubic, and higher-order terms in  in the exponential.  The ellipsis 2 stands for possible insertions in the integrand of the factors of the form ne @C? (r ? that result in the "rst case above when functional derivative are "rst taken to construct correlation functions. We shall soon work out explicit examples that should make this perhaps somewhat abstract formulation clear. To obtain the perturbative development, we "rst note that by completing the square to obtain a Gaussian functional integral which produces an in"nite, Fredholm determinant, we have, using again an operator notation, the evaluation:



X[f]"Det[b(! )] [d ] exp+!(b/2) [! #V( M ; k)] #i f, "Det[b(! )] Det\[b(! #V)] exp+!(1/2b)fGf,



  



1 1 V exp ! (dJr)(dJr)f(r)G(r, r)f(r) , "Det\ 1# !  2b

(F.7)

where in the last line we have noted that the product of determinants is the determinant of the product of operators and used ordinary notation with G(r, r) Green's function de"ned by [! #V( M ; k; r)]G(r, r)"dJ(r!r) .

(F.8)

We next note that

   



1 (dJr)(dJr)f(r)G(r, r)f(r) exp ! 2b

 



1 d d (dJr)(dJr) G(r, r) exp i (dJr) (r)f(r) 2b d (r) d (r)

"exp

.

(F.9)

(

 The essence of the proof of this relation is obtained by replacing the functions by numbers, and by observing that



exp



d d g exp+ipx, dx dx



V

 


"exp+!ipx, exp




"exp

 





d d d d g exp+ipx,"exp e\ NV e NVge\ NV e NV dx dx dx dx

d d #ip g #ip dx dx

"exp+!pgp, .

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Hence the functional integral (F.7) de"ning X[f] may instead be replaced by an exponential functional derivative operation which, in the operator notation, reads



 







1 d 1 d V exp G exp+i f, X[f]"Det\ 1# !  2b d  d 



. (F.10) (Y Now the functional X[f] de"ned by the functional integral (F.7) has precisely the same form as the functional integral (F.4) de"ning the thermodynamic generating functional Z[k] except that F[ ] is replaced by the functional Fourier transform factor exp+i f,. Since this functional Fourier transform factor can be used to generate any functional, we conclude that 1 Z[k]"Det\ 1# V exp+!S[ M ; k], ! 

 

;exp



1 d d (dJr)(dJr) G(r, r) F[ ] 2b d (r) d (r)



. (F.11) (Y This is the functional form than lends itself to a perturbative development by expanding the exponential of the functional derivatives. Performing the functional derivatives produces the `Wick contractionsa that are familiar in quantum "eld theory and lead to the familiar graphical representation. The exact analytic form with the proper numerical factors associated with a given graph is easily obtained from the expansion of the exponential and the operation of the functional derivatives. F.2. Straightforward expansions We consider the ordinary partition function in which all the chemical potentials are constants. This will illustrate the use of this functional derivative formulation in a straightforward fashion. In this case we take the background "eld to vanish, M "0, and so the quadratic part of the action involves the lowest-order Debye (squared) wave number  i "b en , (F.12)  ? ? ? with VPi , and so the Green's function reduces to the Debye Green's function G (r!r) that  J was de"ned previously in Eq. (2.57). With constant chemical potentials and with M "0,



(F.13) !S[0; k]" n (dJr) , ? ? while the determinantal prefactor reduces to that evaluated previously in Eq. (2.77) of the text,







 

1 1 Det\ 1# i "exp !G (0) i (dJr) . J l  !  

(F.14)

 The functional derivatives may be viewed as `pacmena that eat up "elds sprouting from vertices with each pair of devoured "elds connected by a line that represents Green's function G(r, r).

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Therefore, the "rst two factors in the general perturbative formula (F.11) yield the partition function valid up to one-loop order,



Z "exp 

i n!G (0)  ? J l

 

(dJr) ,

(F.15)

the result (2.78) in the text, and hence

 



1 d d (dJr)(dJr) G (r!r) exp+!SI [ ; k], Z[k]"Z exp   2b d (r) J d (r)



.

(F.16)

(

Here



(F.17) !SI [ ; k]" (dJr) n+e @C? (r!1#[be (r)], . ?  ?  ? As a "rst application of this method, we derive the result (2.64) for 11 22 given in the text. To all orders

 



1 d d (dJr )(dJr ) G (r !r ) 11 (0)22"Z[k]\Z exp   J    2b d (r ) d (r )   ;+ (0) exp+!SI [ ; k],, " . (F.18)  ( To the desired one-loop order, with the linear coupling to counted as itself of one-loop order as explained in the discussion of Eq. (2.64) of the text,

         

 

1 d d (dJr )(dJr ) G (r !r )   d (r ) J   d (r ) 2b    ; (0) n (dJr) [(ibe (r))# (ibe (r))] ? ? r ? ( ? 1 d d  (dJr )(dJr ) G (r !r )

(0) n (dJr)(ibe (r)) "   d (r ) J   d (r ) ? ? 2b   ?  d d 1 1 (dJr )(dJr ) G (r !r ) #   d (r ) J   d (r ) 2 2b    ; (0) n (dJr)  (ibe (r)) r ? ? ?  "i (dJr)G (0!r) e n [1!beG (0)] . J ? ?  ? J ? In view of the Fourier representation (2.57) of the Debye Green's function, 11 (0)22"exp



1 (dJr)G (0!r)" , J i 







(F.19)

(F.20)

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and so i  11 22" e n [1!beG (0)] , (F.21) ? ?  ? J i  ? which is just the result (2.64) of the text. The perturbative expansions of the density and density}density correlations discussed in the text follow from

 

1 d d 11n (r)22"Z[k]\Z exp (dJr )(dJr ) G (r !r ) ?    J   2b d (r ) d (r )   r ? ;+n e @C (  exp+!SI [ ; k],, " , ?  (

 (F.22)

and

 

1 d d 11n (r)n (r)22"Z[k]\Z exp (dJr )(dJr ) G (r !r ) ? @    d (r ) J   d (r ) 2b   ;+n e @C? (rn e @C@ (rY exp+!SI [ ; k],, " . ? @  (

 (F.23)

F.3. Loop parameter It was emphasized in the text that the size of loop corrections is measured, in l dimensions, by the dimensionless parameter beG (0)&beiJ\, which reduces to bei in three dimensions: A perJ   turbative term corresponding to a graph containing l loops is of order [beiJ\]l, or in three  dimensions, [bei ]l. That is, the power of [bei ] counts the loop-order of the expression. It   should be noted that these loop graphs are connected and single-particle irreducible. In this counting, e denotes a generic, typical charge of any of the particle species, or, equivalently, one could write e "Z e, and e is the electron charge. Here we shall sketch the proof of this assertion. ? ? To do this, we examine the expansion of the perturbative formula (F.16) in powers of the unperturbed densities, which we write in the schematic form

 

1 1 d d (dJr)(dJr) G (r!r) Z [k]& exp , J 2b N! d (r) d (r)





 

; (dJr ) n(r)e @C(r  (dJr ) n(r)e @C(r 2 (dJr ) n(r)e @C(r,    ,



.

(F.24)

( This corresponds to a graph with N vertices. Functional derivatives with respect to k(r) may be taken to give number density correlators. We have omitted the subtraction of the unit and  terms which appear in the interaction part of the action (F.17), which we may do with the understanding that at least three functional derivatives are taken at each vertex or that each vertex emits at least three propagator lines. Let us "rst assume that functional derivatives have been taken so that each vertex is connected by a single propagator line. At this stage, we have a graph which is a polygon with N lines and N vertices. Note that since our counting applies to connected, single-particle irreducible graphs, any of these graphs must have such a perimeter polygon. To exhibit the parameters, we introduce

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dimensionless spatial coordinates by writing r"q/i . Then the propagator G (r!r) becomes  J iJ\ times a dimensionless function of the dimensionless variable (q!q). The functional deriva tive operations in Eq. (F.24) produce a propagator line times b\ with a factor of be at each end of the line. Thus for each propagator line and the vertex factors associated with both ends of the line we have an overall factor of beiJ\. Each vertex involves  (dJr)&i\J times n (a dimensionless   product), and so, all together for our skeleton polygon, we have N factors of beni\. But  ben&i , and so these are just N factors of 1. If we measure the size of this skeleton one-loop  graph in term of the unperturbed grand potential bX& (dJr)n, then one factor of beiJ\  remains to characterize the order of the one-loop graph. The remainder of the proof is now trivial. Each additional propagator line added to the skeleton polygon gives a factor beiJ\ and increases the number of loops by one.  Appendix G. Calculations using functional method We turn now to apply the functional methods using the alternative background "eld method mentioned in the preceding appendix. We choose the background "eld M (r) used there to be the solution (r) of the classical "eld equation of the total action which now contains a source: 



S [ ; p; k]"S[ ; k]!b (dJr) (r)p(r) 

 

" (dJr) Thus (r) is de"ned by 



b

(! ) ! n (r) exp+ibe ,!b p . ? ? 2 ?

(G.1)

!  (r)"i e n(r) exp+ibe (r),#p(r) . (G.2)  ? ? ? ? This choice is made because, since the action is stationary for variations about the solution of the classical "eld equation, with " # , there are no linear terms in the #uctuation "eld  and  the result (F.11) of the previous appendix takes the form





1 exp+=[p; k],"Det\ 1# V[ ; k] exp+!S [ ; p; k],    ! 

 

;exp



1 d d (dJr)(dJr) G(r, r) exp+!SI [ ; , k],   2b d (r) d (r)



.

(Y (G.3)

Here V( ; k; r)" be n(r) e @C? ( r ,  ? ? ?

(G.4)

+! #V( ; k; r),G(r, r)"d(r!r) , 

(G.5)

with

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and



!SI [ ; , k]" (dJr) n(r) e @C? ( r+e @C? (Yr!1!ibe (r)#[be (r)], .   ? ?  ? ?

(G.6)

G.1. Results through one loop We shall make use of this general result in the next section where the two-loop correction will be evaluated. Here we note that action of the exponential of functional derivatives on the exponential of SI produces only two- and higher-order loops since SI contains no linear terms in . Hence,   to the one-loop order with which we are concerned here, we have





1 V[ ; k] . =[p; k]"=[p; k]"!S [ ; p; k]! ln Det 1#     ! 

(G.7)

To obtain the e!ective action described in Appendix A, we "rst need the relation between d=[p; k] 1 (r)2N " @ dbp(r)

(G.8)

and (r). Since the action S [ ; p; k] is stationary for "eld variations about , the induced     variation in when the source p is varied does not contribute to dS and only the explicit source   variation contributes, giving dS [ ; p; k] !   " (r) .  dbp(r)

(G.9)

This is the classical or tree contribution. Using the formula d ln Det X"Tr X\dX, the one loop contribution is contained in



 



1 1 \ 1 d ln Det 1# V[ ; k] "Tr 1# V[ ; k] dV[ ; k]    !  !  !  "Tr[! #V[ ; k]]\dV[ ; k]  



" (dJr)G(r, r)ibg (r) d (r) ,  

(G.10)

where g (r)" be n(r) e @C? ( r .  ? ? ? The variation of Eq. (G.2) de"ning the classical solution yields

(G.11)

+! #V( ; k; r),d (r)"dp(r) ,   and so

(G.12)

d (r)  "G(r, r) . dp(r)

(G.13)

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Hence, to one-loop order, d=[p; k] 1 (r)2" " (r)#* (r) ,  dbp(r)

(G.14)

where



i * (r)"! (dJr)G(r, r)g (r)G(r, r) .  2

(G.15)

The correction * (r) is a one-loop contribution corresponding to a `tadpolea graph. This graph is just the same as the second graph in Fig. 11 except that now the vertex is given by g (r) and the  lines represent Green's function G(r, r). With the source vanishing, and with e n taken to be of ? ? one-loop order, "(i/i ) e n to this order. Thus, to one-loop order, can be taken to vanish   ? ?  in the explicitly one-loop term * , and the Green's functions there can be replaced by the Debye functions. With these remarks in mind, it is easy to check that general one-loop result (G.14) reduces to the previous result (F.21). The e!ective potential is taken to be a functional of the "eld expectation value which we relabel as M (r). The one-loop action (G.7) is a functional of the classical "eld (r) which di!ers from the  expectation value by the one-loop correction * (r). Since the classical action is stationary for variations about (r), replacing (r) in it by M (r) entails a correction involving * (r), which is of   two-loop order. Since the determinantal contribution is already of "rst order, replacing (r) in it  by M (r) also gives a two-loop correction. Thus, to one-loop order, we may replace (r) by M (r) in  the action functional (G.7). [The explicit form for * given in the previous paragraph is, of course, not needed to reach this conclusion. We made this explicit calculation because the result will be used in the next section on the two-loop e!ective action.] The e!ective action for the timeindependent "eld M is de"ned by simply restricting the Legendre transformation (A.21) to involve time-independent quantities so that the imaginary time integral is replaced by a factor of b. The source-"eld product in the Legendre transformation cancels the source term in the relation (G.1) between S [ ; pk] and S[ ; k], and we have to one-loop order  1 V[ M ; k] . (G.16) C[ M ; k]"S[ M ; k]# ln Det 1#  ! 





It proves convenient to rewrite the determinant in the form:





 

1 1 V "Det +! #i #(V!i ), Det 1#   !  ! 





1 "Det 1# i Det[1#G (V!i )] . J  !  

(G.17)

Here we have added and subtracted the (squared) Debye wave number i for the (lowest-order)  densities when the generalized chemical potentials reduce to their standard, spatially uniform form, k (r)Pk , and G is the Debye Green's function for this wave number. The "rst factor in Eq. (G.17) ? ? J is the one-loop correction to the standard partition function; in the limit in which the generalized

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159

chemical potentials become constant and M "0, V"0, and we see that since



S[ M ; k] " "! (dJr)n ,  ?

(G.18)

the grand partition function to one-loop order is given by ln Z "!C[ M ; k] "  





1 i , "nV! ln Det 1# ?  !  

(G.19)

in agreement with Eq. (2.54). To compute the number densities and number}density correlation functions to one-loop order, we "rst note that ln Det[1#G (V!i )] J  "Tr ln[1#G (V!i )] J  "Tr G (V!i )! Tr[G (V!i )]#2 J   J 

 

"G (0) (dJr )(V( M ; k; r )!i ) J    1 ! (dJr )(dJr )[G (r !r )](V( M ; k; r )!i )(V( M ; k; r )!i )   J       2 #2 .

(G.20)

Since (V( M ; k; r )!i )P0 when the generalized chemical potentials take on their constant values   and the "eld M vanishes, only the "rst term on the right-hand side of the last equality contributes to the number density which involves a single functional derivative before this limit is taken, only the "rst two terms contribute to the number density correlation function, and so forth for the higher correlators. To compute the density}chemical potential relation to one-loop order, we note that since V( M ; k; r ) is related to n(r) by Eq. (G.4) and  ? dn(r) @ "d d(r!r)n(r) , (G.21) ?@ ? dbk (r) ? it is easy to compute



dC[ M ; k] 1n 2" "n[1!beG (0)] , (G.22) ? @ ?  ? J dbk (r) ?  which is the result (2.66) in the text. As discussed in Appendix A, the density}density correlation function is determined by



d d C (r!r)"! C[ M ; k] . ?@ dbk (r) dbk (r)  ? @

(G.23)

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In the leading, tree approximation,



d d (G.24) S[ M ; k] "d n d(r!r) . C(r!r)"! ?@ ? ?@ dbk (r) dbk (r)  @ ? The correction arising from the "rst trace term in Eq. (G.20) is just to replace the lowest-order, chemical potential}density relation n here with the corrected functional form n. Of course, to ? ? whatever order we work, at the end we replace the chemical potential}density relation by the actual densities n . The contribution from the second trace in Eq. (G.20) is obtained with the same ? ingredients used in the number density evaluation, and we "nd that, through one-loop order, C(r!r)"d n d(r!r)# be n G (r!r) be n . @ @ ?@ ?@ ?  ? ? J This is precisely Eq. (2.104) of the text.

(G.25)

G.2. Two-loop ewective action We turn now to compute the e!ective action to two-loop order. Before obtaining the terms that contribute to two-loop order, it is instructive to examine some two-loop order terms that cancel among themselves. As was discussed in Appendix A, the e!ective action is single particle irreducible. We can now see explicitly how this works out to the two-loop order. In the preceding section, we noted that the replacement of by M " #* entailed two-loop corrections. First we note   that, since the action is stationary for variations about the classical solution, we have, to order * ,



dS [ ; p; k] 1  S [ M ; p; k]KS [ ; p; k]# (dJr)(dJr)    d (r)d (r) 2



* (r)* (r) . (G.26) ( Since * is already of one-loop order, we can replace by M in the second term here. Since the  second variation of the classical action brings in V( M ; k; r) and produces the inverse Green's function, to two-loop order,



b S [ ; p; k]"S [ M ; p; k]! (dJr)* (r)[! #V( M ; k; r)]* (r) .     2

(G.27)

Again to two-loop order, the correction to the determinant contribution is given, in view of Eq. (G.10), by









1 1  ln Det 1# V[ ; k] " ln Det 1# V[ M ; k]    !  ! 



1 ! (dJr)G(r, r)ibg (r) * (r) ,  2

(G.28)

We de"ne *C[ M ; k] by the sum of the two-loop corrections which appear above. Using the de"nitions (G.15) of * and (G.5) of G(r, r), and a little algebra, we "nd that



b *C[ M ; k]"! (dJr)(dJr)G(r, r)g (r)G(r, r)G(r, r)g (r) .   8

(G.29)

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This corresponds to the `dumbbella graph, (G.30) This graph is obviously single-particle reducible. Hence it must cancel the single-particle reducible piece of the remaining part *C>[ M ; k] of the e!ective action which we now turn to compute. The variational derivative expression (G.3) for exp+=[p; k], is a convenient tool to use to calculate this remaining part of the e!ective action. For the two-loop terms of interest, the exponential of the interaction terms (G.6) can be approximated by


 



b 1  (dJr )g (r ) (r ) (dJr )g (r ) (r ) exp+!SI [ ; , k],K!           2 3! b (dJr )g (r ) (r ) , #     4!

(G.31)

where we recall that (to our order) g (r)" be n(r) e @C? (M r  ? ? ? and de"ne

(G.32)

g (r)" be n(r) e @C? (M r . (G.33)  ? ? ? To our order of interest, the Legendre transform relation between = and C reduces to simply ="!C, the classical action and determinantal terms do not contribute, and Eq. (G.3) gives

  
 

!*C>[ M ; k]"exp

1 d d (dJr)(dJr) G(r, r) 2b d (r) d (r)





b 1  ; ! (dJr )g (r ) (r ) (dJr )g (r ) (r )         2 3!



b (dJr )g (r ) (r ) #     4!

. (G.34) (Y It is a straightforward matter to carry out the functional derivatives and verify that one set of terms precisely cancels the previous `dumbbella piece (G.29). Thus, we prove explicitly to two-loop order that the e!ective action functional has no single-particle reducible terms. The remaining terms give

  

b1 (dJr ) (dJr )g (r )G(r , r )g (r ) *C[ M ; k]"         2 3! b ! (dJr )g (r )G(r , r )#S [ M ; k] ,       4

(G.35)

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where the last term stands for the two-loop contribution of the induced interaction (3.32) that we have belatedly added. To our two-loop order, this additional term is given by the p"2 piece of Eq. (3.32) evaluated in the tree approximation which replaces the potential by its expectation value

M . Using the result (3.41) for the coupling constant gives






be e  p 1 ? @ kJ\ S [ M ; k]" g (k)! ?@  6 3!l 4p ?@



(dJr)n(r) exp+ibe M (r),n(r) exp+ibe M (r), ? ? @ @



(G.36) # h b (dJr) [ (k (r)#ie M (r))]n(r) e @C? (M r . ? ? ? ? ? To make use of the "rst part of this interaction to present an explicitly "nite result in the lP3 limit, we write the single particle irreducible two-loop e!ective action as a sum of two parts, *C[ M ; k]"*C[ M ; k]#*C[ M ; k] .   The "rst part, de"ned by

(G.37)

  

b1 *C[ M ; k]" (dJr ) (dJr )g (r )[G(r , r )!G (r !r )]g (r )             2 3! b ! (dJr )g (r )G(r , r ) ,      4

(G.38)

may be evaluated directly at l"3 since the subtraction of the cube of the three-dimensional Debye Green's function G (r !r ) in the "rst double integral renders it "nite while (with dimensional    continuation) the remaining single integral is well-behaved in the lP3 limit. Making a convenient rearrangement of the remaining part gives

  

b1 *C[ M ; k]" (dJr ) (dJr )g (r )G (r !r )[g (r )!g (r )]      J       2 3!



b1 (dJr )[g (r )] (dJr )G (r !r )#S [ M ; k] . #     J    2 3!

(G.39)

The lP3 limit may be taken in the "rst line on the right-hand side of this equation since a subtraction has been made that gives an integrable singularity when r "r . The result (C.32)   gives



1 1 (dJr )G (r !r )"  J   (4p) 2






i J\ 1  #1!c!2 ln 3 . 4p 3!l

(G.40)

Thus the pole terms on the second line on the right-hand side of Eq. (G.39) combine that give the well-de"ned lP3 limit


 
 




1 i J\ i \J i    . 1! P!ln 3!l 4p 4pk 4pk

(G.41)

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Thus, taking the lP3 limit and writing the "rst line in Eq. (G.39) in a symmetrical manner, we arrive at

  

  

b1 *C[ M ; k]"! (dJr ) (dJr )G (r !r )[g (r )!g (r )]    J       4 3!



i p be e  ? @  !1#c#2 ln 3 ln g (k)! ?@ 4pk 6 4p ?@ ; (dJr)n(r) exp+ibe M (r),n(r) exp+ibe M (r), ? ? @ @ # h b (dJr) [ (k (r)#ie M (r))]n(r) e @C? (M r . ? ? ? ? ?

(G.42)

The sum C[ M ; k]+C[ M ; k]#*C[ M ; k]#*C[ M ; k] (G.43)   is the generating functional for all connected, single-particle irreducible contributions through two-loop order. For example, the double functional derivative of this result with respect to the chemical potentials, with the chemical potentials then taken constant and M "0, produces the irreducible number density correlation function C through two-loop order, the result sum?@ marized in Eq. (4.38) of the text. The grand potential is given by the e!ective action with the generalized chemical potentials taking on constant values and with M "0, The two-loop contribution to the grand potential plus the previous lower-order terms give bX(b, k)"C[ M ; k] " #*C[ M ; k] " #*C[ M ; k] " . (G.44)      In this limit, the "rst piece of *C[ M ; k] in Eq. (G.38) vanishes while the second piece involves  G (0) which has the value !i /4p according to Eq. (2.60). Moreover, in this limit, the "rst line on   the right-hand side of the equation above for *C[ M ; k] also vanishes. With these remarks in  mind, it is a simple matter to verify that our e!ective action results agree with the result (4.7) of the text.

References [1] H. Georgi, E!ective Field Theory, Ann. Rev. Nucl. Part. Sci. 43 (1993) 209}252. [2] J. Polchinski, in: J. Harvey, J. Polchinski (Eds.), Proceedings of Recent Directions in Particle Theory: From Superstrings and Black Holes to the Standard Model (TASI-92), World Scienti"c, Singapore, 1993, pp. 235}276. [3] E. Braaten, A. Nieto, Phys. Rev. D 51 (1995) 6990; 53 (1996) 3421. [4] W. Ebeling, W.-D. Kraeft, D. Kremp, Theory of Bound States and Ionization Equilibrium in Plasmas and Solids, Akademie-Verlag, Berlin, 1976. [5] J.P. Hansen, Phys. Rev. A 8 (1973) 3096. [6] W.-D. Kraeft, D. Kremp, W. Ebeling, G. RoK pke, Quantum Statistics of Charged Particle Systems, Plenum Press, New York, 1986. [7] A. Alastuey, A. Perez, Phys. Rev. E 53 (1996) 5714. [8] J. Riemann, M. Schlanges, H.E. DeWitt, W.D. Kraeft, Physica A 219 (1995) 423. [9] H.E. DeWitt, M. Schlanges, A.Y. Sakakura, W.D. Kraeft, Phys. Lett. A 197 (1995) 326.

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[10] W.D. Kraeft, M. Schlanges, D. Kremp, J. Riemann, H.E. DeWitt, Z. Phys. Chem. 204 (1998) 199. [11] T. Kahlbaum, in: G.J. Kalman, J.M. Rommel, K. Blagoev (Eds.), Strongly Coupled Coulomb Systems, Plenum Press, New York, 1998. [12] J. Ortner, Phys. Rev. E 59 (1999) 6312. [13] H.E. DeWitt, J. Math. Phys. 3 (1962) 1216. [14] D. Brydges, A. Seiler, J. Stat. Phys. 42 (1986) 405. [15] A. Alastuey, Ph.A. Martin, Phys. Rev. A 40 (1989) 6485. [16] F. Cornu, Ph.A. Martin, Phys. Rev. A 44 (1991) 4893. [17] F. Cornu, Phys. Rev. Lett. 78 (1997) 1464; Phys. Rev. E 58 (1988) 5322. [18] D. Brydges, Ph.A. Martin, J. Stat. Phys. 96 (1999) 1163, cond-mat/9904122. [19] R.R. Netz, H. Orland, Euro. Phys. J. E1 (2000) 67, cond-mat/9902220. [20] M. Opher, R. Opher, Phys. Rev. Lett. 82 (1999) 4835. [21] J. Hubbard, Phys. Rev. Lett. 3 (1959) 77. [22] R.L. Stratonovitch, Dokl. Akad. Nauk SSSR 115 (1957) 1907. [23] L.S. Brown, Quantum Field Theory, Cambridge University Press, Cambridge, 1992. [24] S. Samuel, Phys. Rev. D 18 (1978) 1916. [25] M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995. [26] J. Negle, H. Orland, Quantum Many Particle Systems, Addison-Wesley, Reading, MA, 1988. [27] F.H. Stillinger, R. Lovett, J. Chem. Phys. 48 (1968) 3858; 49 (1968) 1991. [28] Ph.A. Martin, Rev. Mod. Phys. 60 (1988) 1075. [29] A. Alastuey, in: G. Chabrier, E. Schatzman (Eds.), The Equation of State in Astrophysics, Cambridge University Press, Cambridge, 1994. [30] A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. [31] K. Huang, Statistical Mechanics, 2nd Edition, Wiley, New York, 1987. [32] A. Rajantie, Nucl. Phys. B 480 (1996) 729; Nucl. Phys. B 513 (1998) 761(E). [33] C. Fronsdal, Phys. Rev. 171 (1968) 1811, and references therein. [34] G.J. Maclay, Ph.D. Dissertation, Yale University, 1972. [35] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1952. [36] D.J. Broadhurst, Eur. Phys. J. C 8 (1999) 363. [37] G. Almkvist, personal communication.

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EXPERIMENTAL TESTS OF THE STRONG INTERACTION AND ITS ENERGY DEPENDENCE IN ELECTRON+POSITRON ANNIHILATION O. BIEBEL III. Physikalisches Institut A, RWTH Aachen, Physikzentrum, 52056 Aachen, Germany

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 340 (2001) 165}289

Experimental tests of the strong interaction and its energy dependence in electron}positron annihilation O. Biebel III. Physikalisches Institut A, RWTH Aachen, Physikzentrum, 52056 Aachen, Germany Received April 2000; editor: J.V. Allaby

Contents 1. Introduction 2. QCD } a theory of the strong interaction 2.1. Group structure of QCD 2.2. Renormalization of QCD 3. The strong interaction in e>e\ annihilation 3.1. QCD and e>e\ annihilation 3.2. Phenomenology of QCD in e>e\ experiments 4. Studies of the energy dependence of QCD 4.1. Determination of the running of a 1 4.2. Quark mass e!ects 5. Tests of QCD treatments of hadronization 5.1. Inclusive fragmentation function and scaling violation

168 169 170 173 179 180 188 194 195 217 226 226

5.2. QCD at small x: MLLA and multiplicities 5.3. Power corrections 6. Studies related to the running of a 1 6.1. Higher-order corrections from energy dependence 6.2. Power corrections to the running 6.3. Additional coloured objects 6.4. a determinations from other hard 1 processes 6.5. A glance at asymptotic freedom 7. Summary and outlook Note added Acknowledgements References

236 251 265 265 268 270 273 274 276 279 279 280

Abstract e>e\ an nihilation into quark}antiquark pairs is a valuable platform for the investigation of the strong interaction. The level of sophistication reached by theory and experiment allows one to verify predictions with signi"cant precision for centre-of-mass energies ranging from the q lepton mass up to about 200 GeV. This report summarizes studies of the dependence of the strong interaction on the energy scale. Determinations of a1 from total cross-sections, hadronic branching fractions of the q lepton and of heavy quarkonia, jet rates, and event shape observables con"rm the energy dependence of the strong coupling constant. Tests of the #avour independence of the strong interaction and mass e!ects are reviewed. Perturbation calculations of

 Now at: Max-Planck-Institut fuK r Physik, FoK hringer Ring 6, 80805 MuK nchen, Germany. E-mail address: [email protected] (O. Biebel). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 7 2 - 7

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167

mass e!ects allow the determination of the bottom quark mass at high energies and, therefore, the scale dependence of quark masses predicted by QCD. Experimental studies of theoretical approaches to hadronization are presented. Besides fragmentation functions, scaling violations, and longitudinal cross-sections, successes of the modi"ed leading-logarithmic approximation and local parton}hadron duality are exempli"ed. Power suppressed corrections, which are expected to be related to hadronization, are discussed for mean values and distributions of event shape observables. From the energy dependence of the strong interaction missing higher-order terms of the perturbation series can be determined. The scrutiny of the scale dependence of a1 showed no evidence for power corrections, light gluinos, or anomalous strong couplings. The results on a1 from e>e\ annihilation are also very consistent with determinations of the strong coupling constant from other hard processes.  2001 Elsevier Science B.V. All rights reserved. PACS: 06.20.Tr; 12.38.!t Keywords: Fundamental constants; Quantum chromodynamics

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1. Introduction Over many decades nature has been probed by the scattering of elementary particles at higher and higher energies in order to reveal more and more the secrets of matter and forces. A fruitful connection of experimental and theoretical particle physics culminated in what is today called standard model of the electroweak and strong interactions. Based on quantum "eld theory, it is the foundation of the current understanding of all elementary particles, and it jointly describes all known forces but gravitation. Half-integer spin fermions act as the building blocks of matter with integer spin bosons acting as mediators of forces. Fermions appear in two species: leptons and quarks. These are, according to their respective quantum numbers, subject to the forces represented by the coupling and its strength. Each force has its bosons, namely the photon, the Z and the charged W! for the electroweak interaction, and the gluons for the strong interaction. All this is summarized in Table 1 which also lists the relevant quantum numbers. An enormous e!ort of both experimental and theoretical physics has brought knowledge about the constituents of matter and the forces. On the experimental side this has been achieved predominantly by scattering particles o! each other, observing the outcome, and analysing and understanding it in terms of basic and elementary processes between the particles. This would have been impossible without the theoretical advances. Calculations largely based on perturbation theory are the key to understanding the results of the measurements. More precise measurements required more precise calculations and vice versa, thus driving each other.

Table 1 Elementary fermions and bosons, that are known in the standard model of the electroweak and strong interactions, and the relevant quantum numbers assigned. Antifermions which are not listed have the signs of the charge, weak isospin, and colour charge quantum numbers inverted Fermions

Generations

Charge Q D

Weak isospin ¹ D

Colour charge

Spin

0 0

       

Colour charge

Spin

1st

2nd

3rd

Leptons

l  e\

l I k\

l O q\

0 !1

Quarks

u d

c s

t b

#  ! 

#  !  #  ! 

Bosons

Coupling

Charge Q D

Weak isospin ¹ D

Photon

Electromagn.

c

Weakons

Weak

W> Z W\

Gluons

Strong

G

r, g, or b r, g or b

0

0

0

1

#1 0 !1

#1 0 !1

0 0 0

1 1 1

0

0

1 colour#1 anticolour

1

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During the last 15 yr signi"cant contributions came from high-energy electron}positron scattering, in particular annihilation. Large particle accelerators such as PEP, SLC, PETRA, TRISTAN, LEP and others have been built to collide electrons and positrons at very high energy. Particle detectors have been installed to register and measure the outgoing particles of the scattering processes. Thus many valuable results on the standard model have been obtained for both the electroweak and the strong interaction. The LEP accelerator [1] which is still in operation has achieved a collision energy of electrons and positrons never reached before. Since the start of the PEP and PETRA colliders [2,3] which are no longer employed, the energy examined by LEP has almost increased twenty-fold. Results of very many investigations using the particle detectors installed at LEP have consolidated in particular the standard model of the electroweak interaction at an unprecedented level of precision [4}7]. This report will focus on results about the standard model of the strong interaction. After introducing the basic concepts in Section 2, Section 3 will explain how the strong interaction is scrutinized in electron}positron annihilation. This report will only touch on some topics of the inconceivably huge variety of investigations of the strong interaction done during the phase I operation of the LEP collider when it was tuned to produce Z bosons. More detailed reports on the LEP phase I results on the strong interaction can be found in [8}11] and also in textbooks [7,12]. The main topic of this report will be the combination of results from electron}positron collisions at various energies, thus taking advantage of the large energy range available. Naturally, the energy dependence of the strong interaction will be of particular interest. This will be addressed in Section 4 while Section 5 is dedicated to some recent theoretical developments which could allow even more precise tests of the standard model of the strong interaction. Before completing the report with a summary, concluding remarks, and a brief #ash on future examinations of the theory, Section 6 will show some investigations concerning extensions of and deviations from the standard model of the strong interaction.

2. QCD } a theory of the strong interaction The key to the understanding of strong interaction processes observed in the scattering of elementary particles is Quantum Chromodynamics (QCD) which was developed about 30 yr ago. It is a quantum "eld theory describing interactions between quarks and gluons based on the concept of a new charge, similar to but di!erent from electric charge. This new charge appears in three variations which are usually associated with colours. Quarks are, besides their one third integer electric charge, carriers of one unit of this colour charge. Thus quarks exist three times each with a di!erent colour charge. In QCD hadrons are composed of quarks and are colourless. The concept of colour charge is supported experimentally for example by the observation of hadrons like X\ and D>> which are made of three quarks with identical quantum numbers. Pauli's principle is recuperated due to the three di!erent colour charges of the constituent quarks. Other evidence for colour stems from the decay rate of nPcc. Without a colour factor of three the theoretical estimate for the decay rate would not agree with the experimentally measured value (for details see, e.g. Ref. [12]). Also in electron}positron scattering evidence has been found for the existence of colour. The virtual photon into which an electron and positron annihilate excites from the vacuum all electrically

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charged pairs of particle and antiparticle, in particular quark and antiquark. As quarks may carry one out of three colours the total excitement of quark}antiquark pairs is enhanced by a colour factor of three (see, e.g. Ref. [6]). In this section we will outline the structure of QCD. The focus will be on the origin of the energy dependence of the strong interaction. 2.1. Group structure of QCD QCD is a non-Abelian Yang-Mills type theory. The features of QCD are determined through the S;(3) group structure of colour. These features are re#ected in the Lagrangian density which describes the interaction of half-integer spin quarks of mass m and massless spin-1 gluons. O Suppressing a gauge and a ghost "eld term the Lagrangian density of QCD is given by [12}14]

1 LQCD "! G GIJ# q (icID !m ) q (1) ? I O ?@ @ 4 IJ    where the sum is over all quark #avours. Furthermore, repeated indices indicate a sum. In Eq. (1) D is the covariant derivative and cI are the four gamma matrices. The indices a, b of the quark I spinors q are the quark colour indices running from 1 to the number of colours, i.e. 3. Thus quarks appear as colour triplets. In addition, the quark spinors and also the gamma matrix cI have spinor indices that are suppressed for clarity. The "eld strength tensor G depends on the gluon "elds G. IJ I It is given by the relation G "R G!R G!g f  !G G! (2) IJ I J J I 1 I J where the indices A, B, C denote the eight elements of the gluon "eld octet of QCD. The last term in Eq. (2) allows for interactions between gluon "elds as will be demonstrated more clearly below. It contains the structure constants f  ! of S;(3) which determine the properties of QCD. Thus the last term is responsible for many peculiar features of QCD, in the "rst place that gluons, carrying two units of colour charge, may interact with each other. The constant g determines the strength of the 1 colour interaction. It is related to the strong coupling constant a "g/4n (3) 1 1 which is the analogue to the "ne structure constant a of Quantum Electrodynamics (QED). Thus  g can be regarded as the colour charge. 1 The covariant derivative D in Eq. (1) is given by the expression I (D ) "R d #ig (tG) . (4) I ?@ I ?@ 1 I ?@ It depends on the coupling strength g and on the generators t of S;(3). With three colour charges 1 forming the fundamental representation of S;(3), the generators are 3;3 matrices. The traditional

 A gauge has to be chosen to "x two of the four degrees of freedom for a massless gluon. This term explicitly breaks gauge invariance thus causing unphysical interaction terms which are compensated by introducing non-physical ghost terms. For details see, e.g. [12].

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choice for these matrices is the Gell}Mann matrices j, with A"1,2, 8. These are hermitian and traceless matrices that can be derived from the usual Pauli matrices. A full representation of the Gell}Mann matrices can be found in [12]. Using these matrices, the generators are de"ned as t"j . 

(5)

The relevant property that the generators inherit from the Gell}Mann matrices is the commutation relation which reads [t, t ]"if  !t!

(6)

thus de"ning the structure constants f  !. Conventionally the normalization of the generators is chosen such that Tr tt "¹ ) d $

with ¹ " . $ 

(7)

Two more colour factors can be derived using the generators and the structure constants. The sum over all colour indices A of the product of two generators de"nes the colour factor C of an S;(N) $ theory (t) (t) "C ) d ?@ @A $ ?A 

N!1 , 4 with C " " . $ 2N 3

(8)

Summing the product of two structure constants gives a de"ning relation for the C colour factor  of S;(N) f  !f  ""C ) d!" with C "N, " 3.   

(9)

These colour factors play the dominant role in the coupling between quarks and gluons. All features of QCD are related to the colour factors ¹ , C and C . This will become more obvious in $ $  a moment. The Lagrangian of Eq. (1), even though lacking gauge "xing and ghost terms, is rather involved. In order to make the physical implications of the Lagrangian visible, we plug in the covariant derivative Eq. (4). If, furthermore, colour indices of the quark "elds q and the sum over the quark #avours are suppressed one obtains LQCD "q (ic RI!m )q#ig (q cItq)G!G GIJ . I O 1 I  IJ 

(10)

Inserting the gluon "eld strength tensor G de"ned in Eq. (2) and recalling that q and G refer to IJ quark and gluon "elds, respectively, the Lagrangian can be written in a symbolic form which separates the di!erent interaction terms speci"c to QCD. Each of these terms can be associated to

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a Feynman graph characterizing the interacting particles and the kind of the interaction LQCD "**q q++# **G++# g ) t**q qG++# g ) f **G++# g ) f **G++ . 1 1 1

(11)

The "rst three terms are well known from their QED analogues. Terms one and two describe the propagation of a quark and a gluon, respectively. The third term contains the coupling between quarks and gluons. It can be interpreted as the decay of a gluon into a quark}antiquark pair (as shown above) and also as gluon radiation o! a quark (antiquark) by crossing the antiquark (quark) and gluon lines in the picture. The two processes have di!erent colour #ows which result in di!erent couplings as will be illustrated below. What distinguishes QCD from QED are the last two graphs which introduce gluon}gluon coupling in the form of a triple and a quartic gluon vertex, respectively. The symbolic representation of the QCD Lagrangian in Eq. (11) includes also the parameters that determine the strength of the couplings. In the case of the quark}gluon coupling, represented by the third term, the coupling depends on g and on the generators t. Also the details of the colour 1 #ow determined by the colour indices are involved. Considering the gluon splitting into a quark}antiquark pair the coupling is (12)

The additional factor n is due to the n di!erent quark #avours that the gluon may split into. For D D the radiation of a gluon from a quark one "nds a coupling of

(13)

In the triple gluon case the coupling is

(14)

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and for a quartic vertex it is

(15)

These are the elementary graphs of QCD. Every interaction between strongly interacting particles can be described perturbatively using these basic exchange graphs and the colour factors. Any experimental veri"cation of QCD as the theory of strong interaction, therefore, does not only require the discovery of colour, of quarks and gluons and their properties, but also requires the measurement of the colour factors and the observation of the consequences of the renormalization of QCD which we will address next. 2.2. Renormalization of QCD One of the bizarre features of quantum "eld theory is that it involves divergences when calculating, e.g. the self-interaction of a particle. Such divergences would render every calculation of quantum "eld theory unusable. Thus one has to eliminate these poles by renormalizing the terms containing the poles. A very complete description of renormalization can be found in Ref. [15]. In brief, renormalization is a prescription to introduce counterterms which absorb in"nities into physical quantities such as charge or mass of a particle. This concept seems to be rather arti"cial at "rst. However, an electron moving in a solid interacts with the atoms in the lattice such that its e!ective mass is di!erent from the nominal mass value. This is analogous to renormalization in quantum electrodynamics where the electron interacts with the vacuum. The di!erence is that one can remove the atoms of the solid to measure the mass of a free electron, but one cannot remove the vacuum. Following the lines of renormalization to absorb in"nities into physical mass or charge, a price has to be paid. It is that the renormalized theory acquires new properties. Only in 1971 was the renormalizability of non-abelian "eld theories such as QCD shown [16]. Today, dimensional regularization (see references in [15], particularly Ref. [17]) is frequently applied in QCD calculations. It uses the space}time dimension, d, as a regulator treating d"4!2e as a continuous variable. The renormalization prescription, in which counter-terms are pure 1/e poles at the physical value of d"4, is called the minimal subtraction renormalization scheme (MS) [18]. A modi"cation of this scheme is the MS scheme [19] which is used throughout this report. It di!ers from the MS scheme in that it also subtracts ln(4p)!c terms, responsible for large # coe$cients in the perturbation expansion, where c +0.5772 is Euler's constant. The MS and # especially the MS scheme are now widely used owing to their advantages. Both schemes automatically preserve many complicated symmetries except chiral symmetry. They have no problems with massless QCD theory. The calculations are convenient and the computation of the divergent part is not too di$cult. However, the MS and MS schemes are both unphysical because there is no physical reason for the introduction of the counter-terms. Another complication of minimal subtraction is that it is in e!ect a whole family of renormalization prescriptions with a single parameter k. This parameter represents an entirely arbitrary mass scale. It is introduced in the dimensional regularization process because from Eq. (2) it can be seen

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that the bare coupling g acquires a dimension if dO4. As a consequence every physical quantity 1 R depends not only on the coupling g and masses m but also on this scale k. In general, g and 1 1 m will also depend on k. Since the scale is entirely arbitrary, it cannot be related to any physical observable and, hence, physical observables should be invariant under the change of variables (k, g (k), m(k))P(k, g (k), m(k)). This invariance can be expressed by 1 1 k

dR "0 . dk

(16)

Replacing g by a with the help of Eq. (3) the total derivative with respect to k can be written as 1 1 k

d R Ra R Rm R "k #k 1 #k dk Rk Rk Ra Rk Rm 1 "k

R R R #b(k) !c (k)m K Rk Ra Rm 1

(17)

which de"nes two renormalization group coe$cients b and c which are usually called the K b function and the mass anomalous dimension, respectively. From Eq. (17) both coe$cients can be read o! respecting the usual sign convention for the mass anomalous dimension [20] b(k)"k

Ra 1, Rk

(18)

1 Rm . c (k)"!k K m Rk The solutions of these two di!erential equations reveal two fundamental properties of hence, the strong interaction } running coupling, a (Q), and running masses, m(Q). 1

(19) QCD

and,

2.2.1. The running coupling constant a (Q) 1 The value of the coupling a changes with the energy scale of the process under consideration. 1 This can easily be seen from the renormalization group equation (RGE) Q

Ra (Q) 1 "b(a ) 1 RQ

(20)

which is obtained from Eq. (18) after separation of variables, integration and derivation with respect to the squared energy scale of the process, Q. It still is a di!erential equation but we will obtain an explicit expression for a (Q) using the b function. It has been calculated by perturbation 1 expansion considering counter-terms to the divergent self-energy contributions involving Feynman diagrams with up to four loops. Joining the results of all calculations [21,22] leads to the expansion Ra b(a )"Q 1 "!b a!b a!b a!b a#O(a)  1  1  1  1 1 1 RQ

(21)

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with 1 (33!2n ) , b " D  12p 1 b " (153!19n ) ,  24p D 1 b + (22.320!4.3689n #0.09404n ) ,  p D D 1 b + (114.23!27.134n #1.5824n #0.00586n ) .  p D D D A very important property of QCD is to be noted from b . This coe$cient is positive as long as  n (17 causing a to decrease with increasing Q. Therefore, in contrast to QED, QCD obeys D 1 asymptotic freedom meaning that quarks and gluons behave at very short distances like free particles. Turning to very low-energy scales, the strong coupling constant grows, thus suggesting that coloured objects will eventually be con"ned in colourless compounds. These compounds are the usual mesons and baryons which one can "nd in the "nal state of scattering processes. An explicit expression for the running coupling a (Q) can be obtained by solving the renormal1 ization group equation (20) either numerically or by integration. In [23] the integration has been done using the 4-loop expression for the b function in the MS renormalization scheme. The result for the running coupling



1 b ln ¸ 1 b b  (ln ¸!ln ¸!1)#  a (Q)" !  # 1 b ¸ b ¸ b ¸ b b     









1 b 5 1 b b b  !ln ¸# ln ¸#2 ln ¸! !3   ln ¸#  # b ¸ b 2 2 b 2b     #O

1 ¸

 (22)

is expressed in inverse powers of ¸"ln(Q/KMS ) where KMS is the constant of integration which depends in general on the renormalization scheme. One notes that the explicit expression for a reveals a pole at Q"KMS . This is an unphysical Landau pole [24] indicating that the 1 perturbation expansion cannot be applied at very small scales. Non-perturbative e!ects become important at such scales. These will be discussed in Section 5. The running of the coupling a (Q) is 1 exempli"ed in Fig. 1 using the explicit expression Eq. (22) for various values of KMS .

 The coe$cients b are known exactly for i"0,2, 3 (see, e.g. [22]). Here we quote for simplicity only approximate G expressions for b and b .  

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Fig. 1. Running coupling constant a (Q) using the 4-loop expression Eq. (22) and di!erent values for KMS . 1

2.2.2. The running masses m(Q) The de"ning equation (19) for the mass anomalous dimension c describes the running of quark K masses. Solving this equation by separation of variables and integration one "nds an expression for the running mass:

   

m(Q)"m(k) exp !

/

dk c (k) k  K



I ?1 / da "m(k) exp ! (23) c (a ) 1 K 1 b(a ) ?1 I 1 when substituting with Eq. (18). One notes that the b function determines, together with c , also the K running of the masses. As for the b function, a perturbation expression for the mass anomalous dimension c has been K derived [25}27]:



1 Rm !c (a )" Q "!c a !c a!c a!c a#O(a) K 1 K 1 K 1 K 1 K 1 1 m RQ

(24)

with 1 c " , K p 1 c " (303!10n ) , K 72p D

 The coe$cients c are known exactly for i"0,2, 3 (see, e.g. [27]). For simplicity only approximate expressions for KG c and c are quoted. K K

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1 c + (19.516!2.2841n !0.027006n ) , K p D D 1 c + (98.943!19.1075n #0.27616n #0.005793n ) . K p D D D The expansions of the mass anomalous dimension c and of the beta function Eq. (21) can now be K inserted into the evolution equation for the quark masses Eq. (23). A 4-loop expansion for the running of the quark masses m(Q) in the MS scheme is given in [27]. For the charm (n "4) and D the bottom quark (n "5) these expansions read D






a (Q)  m (Q)"m (Q ) 1    a (Q ) 1  1#1.0141(a (Q)/p)#1.389(a (Q)/p)#1.09(a (Q)/p) 1 1 1 ; 1#1.0141(a (Q )/p)#1.389(a (Q )/p)#1.09(a (Q )/p) 1  1  1 




(25)



a (Q)  m (Q)"m (Q ) 1

 a (Q ) 1  1#1.1755(a (Q)/p)#1.501(a (Q)/p)#0.17(a (Q)/p) 1 1 1 ; 1#1.1755(a (Q )/p)#1.501(a (Q )/p)#0.17(a (Q )/p) 1  1  1 

(26)

where MS masses are denoted m and a reference scale Q is introduced. Fig. 2 shows the running of  the bottom quark mass in the MS renormalization scheme for n "5. The reference scale is chosen D such that m (Q "(4.25 GeV))"4.25 GeV. This mass value is the central value derived from the
 compilation of the Particle Data Group [28].

Fig. 2. Running of bottom quark mass with 4-loop precision in the

MS

renormalization scheme for 5 #avours.

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Recalling Eq. (20) and Eq. (21) it should be pointed out that the b function and consequently also a (Q) explicitly depend on the number of #avours n . This is of practical importance due to the 1 D very di!erent quark masses (see Ref. [28]). The dependence of a on the energy scale means that 1 one has to carefully consider how many quark #avours n actively contribute to the running of a . D 1 In particular, crossing the production threshold of a further quark #avour means a change of n and, hence, of a and m. This has to be taken into account when solving the renormalization D 1 group equations (21) and (24). For practical reasons, one usually evolves a using Eq. (22) for 1 n !1 #avours to the mass scale of the heavy quark #avour n . From a matching condition at this D D scale, aLD  is calculated from aLD \, and aLD  is evolved for n active #avours using Eq. (22) again. 1 1 1 D In [23,29] the matching conditions for a and for quark masses m have been calculated allowing an 1 arbitrary matching scale Q. The matching relation can be signi"cantly simpli"ed when choosing the MS mass for that scale, that is Q"m "m (m ), where a is matched for n !1 and n active 1 D D #avours. Then the ratio becomes [23]




aLD \(m ) (aLD (m )) 11 (aLD (m )) 1 "1# 1 # 1 (1.057!0.085n ) . D aLD (m ) p 72 p 1

(27)

It should be observed that the ratio is greater than unity at this matching point. Thus a (Q) is 1 reduced when increasing the number of active #avours. In summary it should be kept in mind that renormalizing QCD introduces rather peculiar properties } energy dependent coupling and mass. The coupling reveals that QCD is an asymptotic free theory. At very short distances or, equivalently, at very high energies quarks and gluons behave like free particles. On the other hand, we observe the con"nement of coloured objects in colourless hadrons, an experimental fact which still has to be proven as a result of the equations of QCD. 2.2.3. Choice of renormalization scale Up to now all details concerning renormalization have been neglected. In particular, the relation between the arbitrary renormalization scale k and the scale of a physical process Q has to be considered. Choosing Q&k would be natural. All calculations, however, will be done in some limited order of perturbation theory, in which Q&k is not conclusive. This can be readily seen from an explicit solution of the two-loop approximation of Eq. (20), yielding










b a (Q)"a (k) 1!a (k)X#a(k) X!X  #O(a) , 1 1 1 1 1 b 

(28)

with X"b ln(Q/k). This equation reveals the relation between the renormalized coupling  a (k) and the size of the coupling which is relevant for the physical process under consideration, 1 that is a (Q). 1 Each physical process must be independent of the renormalization scale k, as has been stated in Eq. (16). When calculating the physical observable R as a power series in a (k) 1 R(Q)"R #R a (k)#R a(k)#R a(k)#O(a) ,   1  1  1 1

(29)

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applying the derivative (17) and, for simplicity, neglecting mass terms, one "nds from this independence that dR(Q) RR RR "k #b(k) dk Rk Ra 1 RR RR RR RR  #a  !b R #a  !(2b R #b R ) #O(a) .  #a " 1 R ln k 1 R ln k   1 R ln k     1 R ln k

0"k



 



(30) In the calculation we have used a "a (k) and Eq. (21) for b(a ). In order to ensure the 1 1 1 independence from the renormalization scale order by order in a , the coe$cients R with n52 of 1 L the perturbation series of R must depend on k. For example, the next-to-leading order coe$cient R needs to be  Q Q "R (1)!b R ln . (31) R    k  k




Only the constant R and the leading order (LO) coe$cient, R , are independent of the scale.   Substituting Eq. (28) in Eq. (29) and comparing with Eq. (30), one "nds that the k scale dependence of the coe$cients exactly compensates the change in the coupling a (k), thus yielding 1 a k-independent result for the full series. If, however, the series is truncated, the k-independence breaks down. The scale dependence entails an uncertainty due to the choice of the scale. This needs to be considered in each perturbation calculation of observable physical quantities. The uncertainty due to a change of the scale is one order higher than the order of the perturbative calculation. Thus it is O(a) only for a next-to-next-to-leading order (NNLO) calculation, while for next-to-leading order 1 (NLO) it is O(a) and, hence, more pronounced. Such uncertainties will pop up in each experimental 1 determination of the strong coupling constant as will be seen in Section 4.

3. The strong interaction in e>e\ annihilation Strong interaction phenomena have been observed in the scattering of various particles, in particular protons, antiprotons, electrons and positrons. Among these, the collision and annihilation of electron and positron has several appealing features due to the well-de"ned initial state. Total centre-of-mass energy and momentum are precisely known as is the colour state. This rendered the electron}positron annihilation experiments at the colliders PEP, PETRA, TRISTAN, SLC, and LEP a valuable testing ground for the investigation of the strong interaction and for the scrutiny of QCD. This section addresses the question of how QCD can be examined in electron}positron annihilation processes. Broadly speaking, the process can in principle be subdivided into four steps as is illustrated in Fig. 3: E the actual annihilation possibly involving initial state photon radiation and the decay of the intermediate boson into a quark}antiquark pair,

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Fig. 3. Schematic representation of an e>e\ annihilation process into hadrons.

E the radiation of gluons and the gluon-splitting into quark}antiquark pairs leading to a parton shower, E the process of hadronization which summarizes the transition of quarks and gluons into hadrons, E and "nally the hadrons and their potential decays. The very "rst of these steps is well understood from electroweak theory. Details of this step can be found in many reports and textbooks, for example [4}7]. In this section the two middle steps will be considered in more detail. 3.1. QCD and e>e\ annihilation 3.1.1. e>e\ annihilation into quark}antiquark pairs In the collision of an electron and a positron an annihilation of the two into a virtual photon or a Z boson can occur. Both the virtual photon and the Z decay into pairs of a fermion and an antifermion, in particular a quark and an antiquark. This is depicted in the language of Feynman

 The word parton is used as a generic notation for quark, antiquark and gluon.

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Fig. 4. Lowest order Feynman graph for the annihilation of an electron}positron pair into a virtual photon cH or a Z boson and its decay into a quark}antiquark pair.

graphs in Fig. 4. Relevant for a study of the strong interaction is the fraction of annihilation events that generate a quark}antiquark "nal state and the amounts of the various quark #avours. Both quantities are related to the coupling of the virtual photon or the Z to the quark}antiquark pair. Instead of the absolute production cross-section for quark}antiquark pairs, its ratio to the lowest-order muon pair production cross-section is usually considered. At energies far below the mass of the Z this ratio is simply given by the electromagnetic coupling of the photon to the charge of the quarks Q [13] O p(e>e\Pqq ) "3 Q . (32) R " O O A p(e>e\Pk>k\) O The sum is over all quark #avours that can be created in the annihilation process. Additionally a factor of three enters the relation from the sum over quark colours. At energies close to the mass of the Z a pole occurs in the cross-section. It is due to the resonance production of the Z which dominates over the virtual photon exchange. In the vicinity of the pole the R ratio is expressed using the partial decay width of the Z into quark and muon pairs, respectively, [12] 3 (A#<) C(ZPqq ) O . " O O (33) R " O 8 C(ZPk>k\) A #< I I It depends on the axial, A, and vector, <, couplings of the fermions to the Z. These are related to the third component of the weak isospin ¹ and the charge Q of a fermion f, and also to the weak mixing angle h according to 5 < "¹ !2Q sin h , A "¹ . (34) D D D 5 D D The fermions known in the standard model of the strong and electroweak interactions are assigned the following values for the weak isospin and electric charge (see Table 1): Neutrinos and the up-type quarks (u, c, t) have ¹ "#1/2 while electron, muon, tau, and the down-type quarks D (d, s, b) have ¹ "!1/2. In units of the elementary charge the up-type quarks have electric charge D #2/3, the down-type quarks !1/3, neutrinos are neutral and electron, muon, tau have each !1. These values change sign when considering the respective antiparticles. Eqs. (32) and (33) are approximations for restricted energy regions. The complete R ratio is shown (dotted line) in Fig. 5(a) for a large region of centre-of-mass energies, (s, ranging from  Contributions due to W and Z pair production are not included.

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Fig. 5. (a) The ratio of the hadronic and the muonic cross-section R in e>e\ annihilation is shown with all but QCD corrections (dotted line). The dashed line represents RQCD which includes QCD corrections according to Eq. (37). (b) The branching ratio B of e>e\ annihilation into quarks is shown separately for up-type, down-type, and bottom quarks. Bottom quarks (dashed) deviate from the down-type behaviour due to their large mass and due to vertex corrections involving the top quark, and the W and Higgs bosons. The curves were produced by the ZFITTER program [30] considering initial state photon radiation. Pair production of W and Z bosons is not included.

close to the pair production threshold of bottom quarks at about (s"10 GeV to just below the threshold for top quark pairs at roughly 350 GeV. It has been calculated with the ZFITTER program [30]. At a centre-of-mass energy of (s"M the R ratio dominates while for energies much 8 8  Version 5.0 has been used.

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below the Z pole R is signi"cant. Above the Z pole the R ratio is given by a combination of A contributions from Z and c exchange. Due to the di!erent coupling of Z and c to the quarks, also the proportions of the individual quark #avours change with the centre-of-mass energy. This is shown in Fig. 5(b) for the "ve light quark #avours. At very low energies, up-type quarks are most frequently produced because of the dominance of the c exchange which yields a coupling proportional to the square of the quark charge (cf. Eq. (32)). Around the Z pole, down-type quarks dominate due to the Z boson exchange (cf. Eq. (33)), whereas at very high energies, the interference of c and Z entails again the domination of up-type quarks. Bottom quarks follow the trend of the down-type quarks but deviate a little due to mass e!ects and corrections which involve virtual top quarks, W and Higgs bosons. 3.1.2. Gluon radiation from quarks } full calculation The strong interaction comes into play with the radiation of gluons o! the quarks. It involves two contributions: the radiation of virtual and real gluons. These two are exempli"ed in Fig. 6. Only in the case of real gluon radiation is a gluon left over in the "nal state, together with the quark and the antiquark. The probability of real gluon emission as shown in Fig. 6 depends on the coupling strength. Applying Feynman rules (see for example [13]) the di!erential cross-section for this particular process, e>e\Pqq G, has been calculated. The result is [31] x#x a 1 dp O O "C 1 , $ 2p (1!x )(1!x  ) p dx dx  O O  O O

(35)

where x "2E /(s, i"q, q , are the centre-of-mass energy fractions of the massless "nal state G G quark and antiquark, respectively, and p is the total cross-section in absence of QCD radiation  which is usually denoted Born or tree level cross-section. The di!erential cross-section is directly proportional to the coupling strength a as is obvious from Eq. (35). 1 A complication for this di!erential cross-section is due to infrared and collinear divergences for real gluon emission. Such divergences occur if the gluon is either very low in energy or if it is emitted collinearly to the quark. In fact, the collinear divergence is due to neglecting masses in these calculations. These two fundamental divergences can be made evident in Eq. (35) using fourmomentum conservation for the exchange boson, p "(p #p  #p ) and s"p. One "nds for the % A A O O

Fig. 6. Lowest-order Feynman graphs for real (left) and virtual radiation (middle and right) of a gluon o! a quark or an antiquark. Analogous graphs exist for the radiation from the other "nal state quark.

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case of a gluon emitted by the antiquark (p  #p )"p!2p ) (p  #p )"p!2p ) p "s(1!x ) O % A O O % A O A O "2p  ) p "2E  E (1!cos h  ) O % O % O% 2E  E (1!cos h  ) O% N1!x " O % O s

(36)

where quarks and gluons are treated as massless which means p"0 for i"q, q , G. An infrared G gluon whose energy, E , is small with respect to the centre-of-mass energy, (s, leaves the % corresponding energy fractions x of the quark close to unity. Hence, the di!erential cross-section O is divergent (Eq. (35)). For the case of collinear gluon emission the angle between antiquark and gluon, h  , is small and, again, 1!x is close to zero such that the di!erential cross-section O O% diverges. In order to obtain the total cross-section, the di!erential cross-section Eq. (35) is to be integrated over the phase-space x , x  "021 and x "2!x !x  . In addition to this integration which % O O O O includes the regions of the divergences, the Feynman graphs involving virtual gluon lines shown in Fig. 6 have to be added. It can be shown that, for instance, in the total cross-section the divergences of the di!erential cross-section are cancelled by identical divergences with opposite sign from the virtual gluon graphs [12]. Thus, the total cross-section is "nite. Adding the contributions from real and virtual gluon radiation increases the total cross-section for the annihilation of electron and positron into a "nal state of partons. The increase can be expressed by a simple a dependent factor. This correction factor to the R ratio is known up to 1 third order in a [28]. In leading order the corrected R ratio is 1 a (37) RQCD "R 1# 1 #O(a) . 1 p






In Fig. 5(a) the RQCD ratio is shown as a dashed line versus the centre-of-mass energy (s. The size of the correction can be seen from the di!erence to the dotted line in this "gure which represents the R ratio without the QCD correction from Eq. (37). The a dependence entering the R ratio through 1 this correction allows a measurement of the strong coupling from total cross-sections alone. It is a fully inclusive measurement in the sense that nothing needs to be known about the details of the "nal state except that it has quarks. Experimentally the measurement is hampered because the correction is a rather small e!ect on a large R value, as can be seen from Fig. 5(a). This is in particular the case at higher energies where a assumes a small value. 1 To determine the coupling strength a , one can also investigate the details of the "nal state itself. 1 It is the di!erential cross-section from Eq. (35) which determines the "nal state of quark, antiquark and gluon depending on the coupling strength a . Common to all approaches is the measurement 1 of the probability of gluon radiation o! the quark or antiquark. A very direct method would be to

 Further corrections, in particular charm, bottom and top quark mass e!ects and the non-universality of the Z coupling to the quarks have to be taken into account. These modify the coe$cient but maintain the principle structure. See Section 4.1.1.

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detect quark, antiquark and gluon, to measure their energy fractions x , and to determine the G di!erential cross-section from counting rates. Other approaches consider the spatial distribution of quark, antiquark and gluon in the "nal state. At large centre-of-mass energy a quark}antiquark "nal state without a gluon is pencil-like in its rest frame, with a gluon it is planar, and with many partons the "nal state might look spherical. Such methods are measurements of the shape of the "nal state. The quantities, commonly called event shape observables, are calculated from the energies and momenta of the particles in the "nal state. Again a di!erential cross-section is given for each observable F. It results in leading order perturbation theory from the integration of the di!erential cross-section for real gluon emission Eq. (35) over the whole phase-space



x#x a O O dx dx  C 1 d(F!fF (x , x  , x ))#O(a) O O $ 2p (1!x )(1!x  ) O O % 1 O O a "A(F) ) 1 #O(a) 1 2p

1 dp " p dF 

(38)

where the function fF represents the value of the event shape observable F for the particular values of the energy fractions x of quark, antiquark and gluon. The result of the integral is given by the G function A(F) which can be obtained by either analytical or numerical integration. Measuring this di!erential cross-section for an event shape observable F thus allows a determination of the coupling constant a . 1 An example is the thrust observable ¹. It is determined from the fractional energies of quark, antiquark and gluon according to (39) ¹,f (x , x  , x )"max(x , x  , x ) . O O % 2 O O % Thrust acquires unity for "nal states without gluons whereas its value is 52/3 when gluon emission is present in the "nal state. Thus being sensitive to gluon radiation, a can be determined 1 from a measurement of ¹. Further explicit examples of event shape observables will be presented in Section 4.1.2. Experimentally it turns out, however, that all such determinations of the strong coupling from the "nal state of the partons are hampered since hadrons rather than quarks and gluons are observed by a particle detector. This problem will be addressed in more detail in Section 3.2. A theoretical complication is due to the aforementioned infrared and collinear divergences of the di!erential cross-section for the real gluon emission. In the case of the event shape observables the divergences are present in the function A(F) of Eq. (38). As a consequence the perturbative calculation of any di!erential cross-section is only applicable in a region of the phase-space not too close to the kinematical boundaries x , x  "1, or x #x  "1, that is x "0, where the di!erential O O % O O cross-section becomes divergent. Interpreted with the help of Eq. (36) this requirement translates into cut-o!s, namely that the gluon must be su$ciently energetic and not too close to either to the quark or antiquark. If a gluon in an event fails these requirements the corresponding three-parton con"guration is experimentally indistinguishable from the back-to-back parton con"guration in a two-parton "nal state. In other words, one has to require a resolvable gluon in the "nal state. In consequence, only within this phase-space region is a determination of the strong coupling possible using perturbative calculations as described above.

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One could improve on this situation. Since the physical cross-section is "nite, the theoretically calculated one has to be "nite too. This, however, would be the case for a perturbation series for the di!erential cross-section in Eq. (35) which includes all powers in a . Currently the di!erential 1 cross-section of three parton "nal states is known to second order O(a) [32,33] only. At this order 1 "nal states with up to four partons are possible, for example qq qq  and qq GG. The di!erential cross-section for these "nal states were calculated in leading order O(a) by several groups [32,34] 1 and also in next-to-leading order (O(a)) for some event shape observables [35]. Furthermore 1 there are computations of the "ve-parton "nal state in leading order, i.e. O(a) [36], and also the 1 six-parton "nal state was calculated in leading order [37]. 3.1.3. Logarithmic approximation of gluon radiation Although O(a) calculations constitute a signi"cant improvement over the O(a ) formulae, the 1 1 di!erential cross-sections obtained from the calculations still su!er divergences at the kinematical boundaries. The inclusion of even higher orders is progressing slowly. In particular the computation of virtual correction terms of such higher orders to the three-parton di!erential cross-section is very cumbersome. To obtain improved calculations di!erent methods have been developed which resum to all orders in a contributions to the di!erential cross-section that become large close to 1 the kinematic limit. The basic idea is to consider the probability of gluon emission in certain regions of phase-space. This probability becomes divergent close to the kinematical boundaries where the energy of the gluon becomes small, or the gluon is collinear with the parton emitting it. In such regions the simple perturbation expansion in powers of a is not reliable as the di!erential cross-section 1 Eq. (35) is logarithmically divergent. This can be seen if one considers a region of the phase-space where, for instance, the gluon is emitted by the antiquark and x is close to unity. Choosing the O nomenclature of Eq. (36), we have a kinematical invariant m"(p  #p )"s(1!x ). With this % O O invariant, one can change variables in Eq. (35) from x to m, yielding to the limit x +1 O O 1 1#z a 1 dp + 1 C , (40) p dz dm 2p !m $ 1!z  where z"x  /(x  #x ) is the fraction of energy of the antiquark. What has been achieved is, in fact, O O % a factorization of the di!erential cross-section into the product of the subprocesses, namely the mass of the antiquark}gluon system, m, and the energy fraction of the antiquark, z. Integrating over m or z while keeping the other variable "xed yields in both cases logarithms in the integration variable, for example











1#z dp dp a dm " &ln(m) 1 C , $ 1!z dz dm dz 2p

(41)

which become large as the integration approaches the kinematic limit at m"0 and z"1, respectively. These large logarithmic terms will dominate the cross-section in such regions of phase-space whereas contributions due to gluon emission at high energy and large angle are only important outside. In the framework of the leading-logarithmic approximation (LLA) only these logarithms in the perturbative expansion are kept and resummed to all orders in a . 1 Di!erent schemes have been devised such as the double leading-log approximation (DLLA) [38],

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the next-to-leading-log approximation (NLLA) [39], or the modixed leading-log approximation (MLLA) [40,41] in order to take into account some subleading corrections, which are suppressed by additional logarithmic factors. The leading-log approximation and its re"nements rely on simpli"cations in kinematic variables. Its predictive power for hard, wide angle parton emission remains limited. Such cases still have to be treated using the full calculation of the matrix element as in Eq. (35). What makes LLA a success is that it is possible to understand it together with the resummation in a probabilistic picture, which allows an implementation in Monte Carlo generators of multi-parton "nal states (see below). To reveal this picture, the ln(t)"ln(Q/KLLA ) derivative of Eq. (41), with m replaced by Q, is integrated over z yielding







a (Q) 1#z dP OO% & dz 1 C . $ 1!z 2p d ln(t)

(42)

This equation can be interpreted as the leading order probability P of a quark splitting, qPqG, into a quark of fractional energy z and a gluon. The term in square brackets of Eq. (42) determines this probability. Similar expressions, known as Altarelli-Parisi splitting functions [42], are derived for all basic branchings, yielding in leading order





1#z , P (z)"C OO% $ 1!z P (z)"2C %%% 

[1!z(1!z)] , z(1!z)

(z)"¹ [z#(1!z)] , (43) P $ %OO where as an additional requirement #avour and energy conservation have to be maintained. First-order corrections, O(a ), to these functions are also known [43,44]. 1 Cascading these branchings according to the probabilities P of Eq. (43), allows the creation of a multi-parton "nal state. An example is sketched in Fig. 3. A de"nite course of the cascading requires that t"Q/KLLA of Eq. (42) be regarded as a dimensionless evolution or ordering parameter. One of many possible choices for Q would be the square of the virtual mass, m, of the branching parton as was chosen for the derivation above. Repeated branchings which occur at subsequent steps in t of the evolution lead to a parton shower which, at the end, consists of many partons, mainly gluons due to the larger colour factor C in the gluon}gluon coupling. The  variable KLLA is related to the constant of integration of the renormalization group equation, see Eq. (22). It signals that perturbative QCD is applicable only for Q
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Predictions of the programs are compared with measurements quite extensively as will be indicated in the following sections. In summary, a parton shower calculation is complementary to an order-by-order calculation in the sense that the former may give a good description of the structure of partons which are close together. The parton shower approach, however, is not expected to cover the full information content available in the matrix-element expression, in particular when the partons are well separated. Nevertheless, patching up the parton shower Monte Carlo generators allows a reasonable description of hard gluon emission to be retrieved. 3.2. Phenomenology of QCD in e>e\ experiments In the preceding section "nal states were considered which consisted of partons only. What remains detectable from an annihilation into quarks and gluons, however, are hadrons built up from quarks and antiquarks. Fig. 7 shows two examples of e>e\ annihilation into a quark}antiquark pair developing a hadronic "nal state. It should be noticed that the plots in the "gure show two and three bundles of detected particles, respectively. Such bundles, usually called jets, re#ect the processes at the level of the partons. Therefore the right plot of Fig. 7 shows a jet of a well separated gluon high in energy in addition to the jets of quark and antiquark. So obvious as the connection between the observed jets and the underlying parton process might be, it is impossible to calculate the hadronization which is the transition of partons into hadrons in the framework of perturbative QCD. This is due to the very low energy scale Q involved in this transition which renders a too large for a useful perturbation expansion. The e!ects of hadroniz1 ation, however, which blur the view of the partons, have to be taken into account when deducing

Fig. 7. Two examples of hadronic "nal states in e>e\ annihilation recorded in a particle detector. The "nal state on the right results from an additional well separated gluon radiated by the quark}antiquark pair. The measured particles are shown in the plane perpendicular to the direction of e> and e\ which annihilated in the centre of each plot. Charged particles, bent by a solenoidal magnetic "eld, are shown as curves. The boxes represent the energy deposit of neutral and charged particles in outer components of the particle detector.

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information on the partons from the hadronic "nal state. Therefore many models have been devised which describe the hadronization of partons at the end of the parton shower development on a phenomenological basis. A detailed overview can be found in Refs. [45,46] and in Ref. [11] which also summarizes experimental results on hadronization e!ects. All existing models implement hadronization as a probabilistic and iterative procedure, usually named fragmentation, which applies one or more types of simple branchings: (IF) jetPhadron#remainder-jet, (SF) stringPhadron#remainder-string, (CF) clusterPhadron#hadron, or clusterPcluster#cluster. Probabilistic rules prescribe at each branching the production of new #avours and the sharing of energy and momentum between the fragments. In practice, these fragmentation rules depend on parameters which cannot be calculated from "rst principles but have to be adjusted to obtain a useful description of measured data. In the following a brief description of the main types of models and some relevant parameters is given. 3.2.1. Independent fragmentation The name of this hadronization model [47] suggests that each quark is hadronized independently. Fig. 8 illustrates the principle of the iterative fragmentation procedure. The quark jet q is split into a hadron consisting of qq and a remainder-jet q . The hadron takes a fraction z of the   available energy and momentum according to a probability function f (z), leaving 1!z for the remainder-jet. Usually, z is the light-cone energy-momentum fraction de"ned as (E#p ) ,   , (44) z" (E#p) O where p is the longitudinal momentum along the jet axis. The fragmentation function f (z) which is , used in the independent fragmentation model is assumed to be energy independent, thus being the same at each fragmentation step. The independent fragmentation model has a number of drawbacks. To start with it is not Lorentz invariant and does not exactly conserve energy, momentum and #avour which all have to be patched up at the end. On top of that independent fragmentation does not intrinsically contain

Fig. 8. Schematic representation of the independent fragmentation procedure (adapted from Ref. [45]).

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gluon interference e!ects which are expected from MLLA calculations [41,48] and were observed by the PETRA and the LEP experiments [49}53]. The independent fragmentation model is, therefore, now largely disfavoured. A Monte Carlo generator based on this model which is routinely used in comparisons with data is COJETS [54]. Irrespective of the principal de"ciencies of the independent fragmentation model, the concept is still commonly used when fragmenting the heavy charm and bottom quarks into hadrons containing these quarks. In particular, the Peterson et al. fragmentation function [55] is experimentally preferred (see [11]) because of its energy-momentum spectrum which is peaked at large values of z (see Fig. 9). The function is of the form






1 1 e \ f (z)& 1! ! / . z z 1!z

(45)

It is controlled by a single free parameter, e , which is expected to scale between #avours as / e &1/m . Fig. 9(a) shows this fragmentation function, respectively, for charm and bottom quarks, / / assuming e "0.031, e "0.0038 as given in Ref. [56]. 
3.2.2. String fragmentation The string fragmentation scheme, which was "rst proposed in Ref. [57] and later elaborated by the Lund group [58], considers the colour "eld between the partons. As a quark}antiquark pair of complementary colour moves apart the colour "eld between them collapses due to the gluon self-interaction into a uniform colour #ux tube, which is called a string. It has a transverse dimension of typical hadronic sizes (1 fm) and a constant tension i+1 GeV/fm. Energetic gluon emission can be regarded as energy-momentum carrying `kinksa of the string [59]. Hence, a complicated string moving in space}time is associated with a multiparton state. The fragmentation into hadrons occurs, if the potential energy stored in the string is su$cient to create a qq  pair from the vacuum, by breaking the string up into colour singlet systems as long as the invariant mass of the string pieces is larger than the on-shell mass of a hadron. Thus, at the end of the fragmentation each hadron corresponds to a small piece of string. This is illustrated in Fig. 10. The creation of the string breaking quark}antiquark pairs is governed by a quantum mechanical tunneling probability which depends on the hadron transverse mass m and the string tension i. ,

Fig. 9. (a) Peterson et al. [55] and (b) Lund symmetric [58] fragmentation function with parameters taken from Ref. [56].

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Fig. 10. Schematic representation of the string fragmentation procedure (adapted from Ref. [45]). A string stretched between q and q in two-dimensional space}time is repeatedly broken up by quark}antiquark pairs until on-mass-shell hadrons remain.

The probability is proportional to














pm pm pp exp ! , , exp ! , "exp ! i i i

(46)

where the transverse momentum p is locally compensated between quark and antiquark. As , a consequence the dependence on the hadron mass, m, results in a suppression of strange and, especially, charm and bottom quark production at this step of the fragmentation process. Finally, energy and momentum have to be shared between the string pieces such that the symmetry between the two ends of the string is maintained. The symmetry requirement restricts the choice of the fragmentation function which takes the simpli"ed form






bm 1 f (z)& (1!z)? exp ! , , z z

(47)

with two free parameters, a and b. These need to be adjusted so that the fragmentation is in accordance with measured data. The shape of this function is shown in Fig. 9(b) for a"0.11, b"0.52 GeV\, m"0.7 GeV, and p "0.4 GeV as given in Ref. [56]. , The concept of string fragmentation is implemented in several Monte Carlo generator programs, for example JETSET [60] and ARIADNE [61]. 3.2.3. Cluster fragmentation The cluster fragmentation scheme, which is implemented in the HERWIG [62] Monte Carlo generator, assumes a local compensation of colour based on the pre-conxnement property of

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Fig. 11. Schematic representation of the cluster fragmentation procedure (adapted from [45]).

perturbative QCD [63]. The whole process of cluster fragmentation is illustrated in Fig. 11. This scheme keeps track of the colour #ow during the parton shower evolution. To locally compensate colour at the end of the parton shower the remaining gluons are split into quark}antiquark pairs. A quark from such a splitting may form a colour singlet cluster with the antiquark from an adjacent splitting. Thus clusters are formed which have a typical mass of a couple of GeV. Finally, to obtain hadrons, a cluster is assumed to decay into two hadrons unless it is either too heavy, in which case it will decay into two clusters, or too light, in which case the cluster decays into a single hadron, requiring a rearrangement of energy and momentum with neighbouring clusters. For the decay into two hadrons, which is assumed to be isotropic in the rest frame of the cluster except if the primary quark is involved, a decay channel is chosen based on the phase-space probability only. It involves the density of states, in particular the spin degeneracy of the hadrons. Due to the phase-space dominance in the hadron formation, the cluster fragmentation has a compact description with few parameters. 3.2.4. Theoretical approaches to hadronization Apart from the many phenomenological fragmentation schemes that were proposed (see surveys in Refs. [45,46]), two approaches will be presented which are characterized by a simple but experimentally successful concept. Section 5 is devoted to the application of these two approaches to experimental data. One is the concept of local parton}hadron duality (LPHD) [41,64]. It assumes an immediate relation between the properties of the "nal state at the parton and at the hadron level based on the conjecture that the transition from partons to hadrons is local in phase space, blanching and hadronization of the coloured partons. The idea arose from the preconxnement property of

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193

perturbative QCD [63]. There is not an explicit formation of hadrons from the partons in the LPHD approach and, therefore, it does not involve dedicated fragmentation functions. The energy and momentum distributions at the hadron level are derived from the parton level distributions by employing a normalization factor. The other concept, which is now intensively investigated, considers non-perturbative corrections [65}74] to the perturbatively calculated standard cross-section. Non-perturbative corrections, which cannot be obtained from an expansion of the di!erential cross-section in powers of a , are 1 expected to become signi"cant because of hadronization. The e!ect of hadronization can be qualitatively estimated using the simple longitudinal phasespace or &tube' model [75] (see also [76]). In this model a parton produces a jet of light hadrons, each of them characterized by the rapidity y and the momentum p transverse to the direction of the R initial parton. The rapidity is de"ned as y"0.5 ln[(E#p )/(E!p )] where E is the energy and X X p is the momentum along the direction of the parton. The hadrons jointly occupy a tube in X (y, p )-space as is illustrated in Fig. 12. The transverse momentum of the hadrons in the tube is due R to the hadronization. Thus, from a hadron density o(p ) in the tube, one "nds its hadronization R scale j"dp o(p )p . Noting that cosh y"E /m and sinh y"p /m one can R R R         calculate the energy and momentum of a tube of rapidity length > for m +j, yielding   7 E" dy j cosh y"j sinh > ,  7 (48) P" dy j sinh y"j(cosh >!1) .  For ><1 one obtains approximately P+E!j. In order to "nd in this model the e!ects of hadronization on observables like thrust, a two-jet con"guration should be considered. The full energy of this system is Q. Thus each massless parton

 

Fig. 12. Schematic representation of the simple tube hadronization model in which hadrons occupy a tube in rapidity and transverse momentum. It can be deduced from this model that hadronization e!ects cause power corrections to observables.

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has E"Q/2 and, hence, x,E/(Q/2)"1 whereas for the tube x "2P/Q+2(E!j)/Q. Recall
 ing the de"nition of thrust from Eq. (39) one "nds for the tube ¹"max(x )+1!2j/Q . (49) 
 This exempli"es the impact of hadronization on thrust. Similar conclusions can be drawn for other observables such that hadronization e!ects are generally expected to be suppressed by one or more powers of 1/Q. 3.2.5. Treatment of hadronization in measurements Calculations of observables like thrust rely on perturbation theory, yielding power series in a as 1 shown in Eq. (38). When performed only up to a certain "nite order in a , the result of the 1 calculations is applicable to a "nal state with a small number of partons. However, observables are measured from the particles (hadrons) emerging from the hadronization of the partons which is still not calculable from basic QCD principles. E!ects due to hadronization are usually not negligible as one may deduce from the fact that at most four partons are described by an O(a) calculation 1 whereas about 40 charged and neutral particles are observed in the detector (see Fig. 7). The phenomenological models of hadronization, which are implemented in Monte Carlo programs and some of which have been described in this section, have to be invoked to bridge the gap between hadrons and partons. Broadly speaking, a measured distribution is numerically deconvolved for hadronization distortions using events simulated by Monte Carlo event generators. Given that the di!erential cross-section for an observable is measured as a histogram, two principal approaches are commonly pursued for the deconvolution: (i) bin-by-bin correction factors, and (ii) a correction matrix. In most studies correction factors are well-suited to account for hadronization so long as the di!erential cross-section varies only slightly with the value of the observable and if the width of the bins is su$ciently large. Otherwise hadronization may lead to signi"cant distortions due to bin migration e!ects which need to be corrected using a correction matrix. Such a matrix relates the values the observable assumes for a partonic "nal state with those of the associated hadronic "nal state. To apply a correction matrix to a measured distribution, it needs to be inverted. Doing this analytically usually yields unstable results due to statistical #uctuations. A variety of recipes are on the market like iterative inversion, singular value decomposition (SVD), Bayesian method and regularization, which will not be detailed in this report (for details see [77,78]). Since all these prescriptions rely heavily on the phenomenological hadronization models and their many non-calculable parameters, one naturally attempts to "nd observables which do not su!er signi"cant hadronization distortions. Moreover, the determination of the strong coupling constant, a , is usually restricted to regions with small hadronization e!ects. Examples of such Q observables will be described in the following section.

4. Studies of the energy dependence of QCD To obtain a "nite result for the calculation of cross-sections, renormalization of colour charge and quark mass has to be invoked. As a consequence both the renormalized charge and mass depend on the renormalization scale k. However, cross-sections of physical processes must not

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depend on it. This requirement leads to a scale dependence of the strong coupling constant (Eq. (22)) and the quark masses (Eq. (23)). The scale dependence due to renormalization is a fundamental prediction of QCD which has to be veri"ed by experiment. This section presents determinations of the strong coupling at various energy scales using di!erent methods, thus reducing potential biases of the energy dependence due to the method of determination. The review of results will be mostly limited to energy scales above the threshold for bottom quark pairs, that is +10 GeV, where perturbative calculations are assumed to be very reliable. At the end of the section the running of the quark masses will be considered. Experimental e!ects due to non-vanishing quark masses were looked for in studies of the #avour independence of a . Direct determinations of the scale dependence of quark masses are becoming available now 1 since the O(a) matrix elements including quark masses have been calculated. 1 4.1. Determination of the running of a 1 Many di!erent approaches to determine the strong coupling constant a in e>e\ annihilation 1 exist. Recalling the collinear and infrared singularities associated with the perturbative calculation of cross-sections, one may distinguish two di!erent classes of a determinations depending on the 1 inclusiveness of the measured quantity. Fully inclusive quantities have real and virtual contributions added such that infrared singularities cancel while remaining collinear divergences are factorizable and can be absorbed into process-independent parton distributions and fragmentation functions. Although inclusive quantities like event shape observables and jet rates consider the full "nal state, collinear and, in particular, infrared singularities a!ect the measurement through the sensitivity of the observables to the details of the "nal state. Another argument in favour of the fully inclusive and inclusive quantities concerns the renormalization scale dependence which has been explained in Section 2.2.3. Many fully inclusive quantities are calculated in a to orders higher than next-to-leading (NLO), mainly next-to-next-to-leading 1 (NNLO). While NLO predictions still signi"cantly depend on the choice of the renormalization scale k, the dependence at NNLO is greatly reduced because it is suppressed by a factor a, that is one 1 power more than for NLO. In general, the more terms of the perturbation series are added, the more stable is the prediction against arbitrary scale choices. Therefore, higher-order perturbation calculations are expected to yield results less sensitive to an arbitrary choice of the scale. 4.1.1. Fully inclusive quantities: cross-sections and hadronic branching fractions The total hadronic cross-section, p , is obviously a fully inclusive observable. In particular at  the Z pole high statistics measurements of cross-sections were done in order to precisely survey the Z resonance. With the use of dedicated subdetectors installed in each of the LEP experiments the luminosity, which is needed in the determination of cross-sections, is measured to an accuracy of better than 0.1% (see for example [5]). At such high precision, the strong coupling constant a can 1 be determined from the QCD corrections that have to be applied to the hadronic cross-section. Away from the Z pole, i.e. at higher energies at LEP II, at lower energies at TRISTAN, PETRA, PEP, and even near the bottom quark production threshold at about 10 GeV at CESR, measurements of the hadronic cross-section can be used to determine the strong coupling constant. Inclusive branching fractions of particle decays into hadrons are also fully inclusive quantities since they do not rely on any details of the "nal state. Precise measurements and perturbative

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predictions exist for the q lepton, the J/t(1S) and the B resonances. These have been used to determine the strong coupling constant at very low scales in the range of 1.8}10 GeV. Since a is 1 rather large at these scales, non-perturbative and mass e!ects are not negligible. Some of the corrections are even not calculable and, hence, have to be determined from the measured data. Nevertheless, determinations of a from these particles provide valuable input to test the running of 1 the strong coupling. 4.1.1.1. Hadronic cross-sections. The total hadronic cross-section, p , for e>e\ annihilation has  been calculated to third order in a (NNLO) [79], yielding 1 a a  a  (50) p "p 1# 1 #1.409 1 !12.808 1 #O(a) ,   1 p p p











where p is the Born level cross-section without QCD corrections. This formula receives further  corrections due to the "nite masses of the quarks. One also has to consider the di!erences between vector and axial contributions due to the Z weak coupling even for massless "nal state quarks. In Ref. [80] a parameterization applicable at the Z pole is obtained which includes these e!ects. It approximates the RQCD ratio which has been introduced in Section 3.1.1 at a precision of *R"0.0005. For a top quark mass of m "173.8 GeV [81] and a Higgs boson mass of  m "300 GeV the parameterization is [80] &  a  a  p (m ) a RQCD "   8 +R 1#1.060 1 #0.852 1 !15 1 . (51)  p p p (m ) p  8 Electroweak radiative corrections and the Higgs mass dependence are all absorbed into the factor R which assumes a value of 19.938 for the aforementioned top quark and Higgs boson masses.  The RQCD ratio has been inferred very precisely at the LEP collider from a measurement of the hadronic and leptonic decay widths, C and C , of the Z boson. A value of   RQCD "20.765$0.026 was determined in Ref. [4]. Solving Eq. (51) for a (m ) one obtains 1 8 a (m )"0.1217$0.0039 (exp.) $0.0040 (theor.) 1 8 "0.1217$0.0056 , (52)









where the "rst error is due to the uncertainty of RQCD . The second error is the quadratic sum of the contributions from the uncertainties of the electroweak calculations, from missing higher-order QCD corrections, and from the variation of the unknown Higgs boson mass between 60 and 1000 GeV. This result for the strong coupling, however, is based on the assumption that electroweak interactions (see Ref. [83]) are accurately described by the electroweak standard model, in particular C . One can reduce this sensitivity when in addition to R also the full decay width,  /!" C , and the hadronic pole cross-section of Z exchange are considered in a simultaneous "t of 8 m , a and m . Such a "t has been performed in Ref. [4]. Considering all available electroweak  1 & 

 Meanwhile a new parameterization based on the improved version of the ZFITTER program became available in Ref. [82], which slightly increases the central value of a (m ) in Eq. (52) by 0.0020. 1 8

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data, including direct measurements of m

5

and m



197

, the "t yielded

a (m )"0.119$0.003 (53) 1 8 with m "171.1$4.9 GeV and m "76> GeV. It should be noted that the lower value of  &  \ a from the simultaneous "t is due to the Higgs boson mass for which a value much lower than the 1 assumed value of 300 GeV, is preferred by this "t. The energy dependence of the strong coupling can be directly deduced from the cross-section measurements done by the experiments at the LEP collider above the Z pole at centre-of-mass energies of 130, 136, 161, 172, 183 and 189 GeV. The combination of these measurements [84}89], shown in Fig. 13 including measurements at lower centre-of-mass energies [90], is given in Table 2. Although the weighted averages of the measurements at each energy have a total relative error of only about 1.5%, this uncertainty is still too large for a signi"cant determination of a at each 1 energy. Therefore, the results at the six di!erent energies are combined in a "t in order to extract a value of the strong coupling at 161 GeV, which is just in the middle of the whole range of energies considered. Since both Z and c exchange contribute to the total hadronic cross-section above the pole, and due to the di!erences of the coupling of Z and c to the quark #avours one cannot simply apply Eq. (50) to determine a . Moreover, a substantial fraction of the total cross-section at these high 1 energies is due to very high energy initial state photon radiation, such that the e!ective centre-ofmass energy is close to the Z mass. In order to assess the value of the strong coupling the ZFITTER program [30] is employed to account for these details which are precisely known from perturbative

Fig. 13. The ratio of hadronic and muonic cross-section measured between 20 and 189 GeV is shown [84}90]. Measurements at identical energies are spread horizontally for clarity. For the same reason solely OPAL data are presented at the Z pole. The curve is calculated using the ZFITTER program [30]. It includes initial state photon radiation but not contributions from pair production of W and Z bosons.

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Table 2 Total hadronic cross-sections measured at LEP above the Z pole [84}89]. The R ratios are obtained using the /!" theoretical muon pair cross-sections calculated at the individual centre-of-mass energies by ZFITTER [30] (s (GeV)

hadr. cross-section (pb)

R /!"

130 136 161 172 183 189

334.7$5.0 275.1$4.6 152.8$2.3 124.4$2.2 106.3$1.2 98.2$1.1

15.15$0.23 14.56$0.24 13.52$0.20 12.96$0.23 12.79$0.14 12.72$0.14

calculations. ZFITTER takes a (m ) as input, evolving it internally to the appropriate energy scale by 1 8 means of the 3-loop b function. Applying the ZFITTER evolution, a s "t to the hadronic crosssection yielded for the coupling at 161 GeV a ((161 GeV))"0.128$0.033 (exp.)>  (top, Higgs) , (54) 1 \  where the "rst error is due to experimental uncertainties, whereas the second error stems from a variation of the top quark mass m "173.8$5.0 GeV [81] and of the Higgs mass between 60 and  1000 GeV. The large error of the result underlines that, even if all LEP II results are combined, the precision of the measured cross-section limits the sensitivity to the size of the strong coupling because it contributes a small correction to the total hadronic cross-section only (cf. Eq. (50)). Even worse, the size of this correction decreases with increasing energies thus rendering this determination barely usable for a veri"cation of the running of a at LEP II energies. 1 Since the expected energy dependence of the strong coupling is more pronounced towards lower scales, determinations of a from measurements of the total hadronic cross-section at lower 1 centre-of-mass energies might yield a signi"cant test of the running. In Ref. [91] measurements of the R ratio at centre-of-mass energies below approximately 60 GeV were used to determine a . A 1 A correction to the erroneous third order coe$cient of the theoretical calculation was applied to the original results in Ref. [9] which yielded a ((31.6 GeV))"0.163$0.022 P a (m )"0.133$0.015 . (55) 1 1 8 Eq. (22) has been applied to evolve to the m scale. Within the large errors, which are due to the 8 uncertainties of the cross-section measurements, this result agrees with the determinations (52) and (53) at the Z pole. At around (s"10 GeV many experiments precisely measured the R ratio for e>e\ annihiliation in the continuum [92,93]. A compilation can be found in Ref. [93]. The weighted average of these measurements is R((s+10 GeV)"3.53$0.05. To determine a from it one has to consider 1 the e!ects of quark masses and QED radiation. These were calculated in Ref. [94] to NNLO. The result is









a  a  a , R ((s"10 GeV)+R 1#1.0179 1 #1.9345 1 !10.7484 1  /!" p p p

(56)

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which di!ers only marginally from the result for massless quarks (cf. Eq. (50)). Since the measurements of R were performed below the B(4S) resonance, bbM production is kinematically forbidden. Thus only n "4 quark #avours, namely u, d, s, c, need to be considered in the calculation of D R according to Eq. (32), yielding R "10/3. Solving Eq. (56) for a one obtains   1 (57) aLD ((10 GeV))"0.169$0.040 . 1 With the help of Eq. (27) and using m "4.25$0.15 GeV [28] this result can be transformed to
a ((10 GeV))"0.173$0.042 for n "5 active quark #avours, where the error includes a negli1 D gible uncertainty ($0.0002) due to the error on the bottom quark mass m . Applying Eq. (22) the
value of the strong coupling becomes a (m )"0.116>  (58) 1 8 \  at the Z pole which is in agreement with the result obtained directly at the pole. There also exist many measurements of the R ratio towards even lower energies (see compilation in Ref. [28]). These, however, do not allow a precise determination of the strong coupling constant. Measurements of the R ratio at energies in the range of 2}5 GeV have been completed by the BES collaboration [95]. Given an uncertainty of the order of a few per mill, a precise determination of a will become possible, thus testing its scale dependence at very low energies. 1 4.1.1.2. Hadronic branching fraction of the q lepton. In the case of hadronic decays of the q lepton, hadrons are formed from the qq  pair which stems from the virtual W boson of the weak decay. The hadronic branching fraction of the q lepton, which is given by the ratio of the hadronic and electronic decay widths C and C , can be expressed as a power series up to third order in a [96] F  1 enhanced by additional correction terms [97] B 1!B !B C  I R " F "   " O C B B    a (m) a(m) a(m) "3.058 1# 1 O #5.2023 1 O #26.366 1 O #d #d #d .  K  p p p





(59)

The electroweak correction d "5a (m)/(12p)+0.001 is small [98]. The relative sizes of the   O corrections owing to "nite quark masses, d ,(m /m ), and owing to non-perturbative e!ects, d , K O O  are estimated in Refs. [96,99,100] to be !0.014$0.005 in total. The ratio R is derived in Ref. [28] from the measured branching fractions of the decays O into electrons, B , and muons, B , yielding R "3.642$0.024. Furthermore, a value R "  I O O 3.636$0.021 is determined from lifetimes and masses of q and k, from which B "(q /q )(m /m )  O I O I is calculated assuming lepton universality. Averaging these, a value of aLD (m)"0.35$0.03 can 1 O be derived for the strong coupling constant, where the error is dominated by the estimated theoretical error from missing a and higher-order terms in the power series, and from uncertainties 1 of the non-perturbative contributions. Instead of theoretically estimating the non-perturbative corrections they may be inferred from data by "tting to the invariant mass distribution of the hadronic q decay. This way some theoretical uncertainties can be avoided. One has to consider moments of the mass distribution, since the

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di!erential partial width dC /ds in the integral F 1 Q dC F R (s )" ds (60) O  C ds   is not yet directly accessible within existing theoretical skills. However, k, l-moments of the mass distribution, which are obtained by introducing a factor (1!(s/m))I(s/m)J into the integrand, O O were calculated in Ref. [97]. Aside from a mere determination of a at the q mass scale, that 1 is s "m, one may test the running of the coupling by restricting the moments to invariant mass  O squared of s (m. This allows the size of the strong coupling at various values of s to be  O  determined. Investigations of the invariant mass moments were performed by the ALEPH and OPAL collaborations at LEP, exploiting Z decays into q pairs, and also by the CLEO collaboration [101}103]. Fig. 14 exempli"es the results on a (s ) from "ts of the restricted moments. The band represents the 1  experimental error. Also shown is the expected running of a assuming 2 and 3 active #avours. 1 Despite large correlations between the a values determined at adjacent s , a satisfactory agree1  ment is found. Table 3 provides a compilation of the results on a obtained at s "m. These were 1  O found from the application of the "xed order perturbation theory to the values of R derived from O the separately measured vector and axial-vector contributions to dC /ds. Due to the large value F of a at the q mass scale the power series in Eq. (59) does not converge well. To improve the 1 convergence, attempts were made to obtain a resummation of some of the higher-order terms. Details can be found in Refs. [97,100,102]. In Ref. [102] variations in aLD (m) of up to $7%, 1 O that is $0.022, were found on average owing to modi"ed perturbative descriptions. The central result in Table 3 is obtained from a weighted average following the prescription of Ref. [104]. In brief, the value is calculated using the inverse squares of the total errors of the individual results as the weights. Ignoring tiny statistical correlations, but taking into account dominant systematic correlations, the error on the central result is determined by calculating a weighted average of the errors of the individual results using the same weights.



Fig. 14. The running of a (s ) obtained in Ref. [101] from "ts of an improved theoretical prediction to the invariant mass 1  moments restricted to s (m. A large correlation is present between the experimental values. The shaded band  O represents the experimental errors. Superimposed is the theoretical expectation of the scale dependence for 2 and 3 active #avours, respectively. Not shown is an overall uncertainty of about 7% due to the scatter from employing various improved theoretical predictions (see text and Refs. [101,102]).

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Table 3 Compilation of aLD (m) determined from R and from the measured vector and axial-vector decomposition of dC /ds 1 O O F [101}103]. The results were obtained using "xed-order perturbation theory. An overall error of $7% has to be added to the results to account for changes due to missing higher-order terms of the perturbative series. The "rst error denotes the experimental uncertainties which include errors propagated from external branching ratios. The second error is the theoretical uncertainty. A weighted average is calculated taking into account the correlations of the individual values Experiment

aLD (m) 1 O

ALEPH93

OPAL98

0.330$0.043$0.016 0.306$0.017$0.017 0.322$0.005$0.019 0.324$0.006$0.013

Average

0.321$0.009$0.015

CLEO95 ALEPH98

Experimental and theoretical error are estimated from the total errors using the respective error contributions quoted for R . O

To evolve a from the q mass scale to the Z pole Eq. (22) has to be applied. Since the number of 1 active #avours at the m scale is 3, the matching relations (27) have to be used at both the charm O (m (m )"(1.25$0.15) GeV [105]) and the bottom MS mass scale (m (m )"(4.25$0.15) GeV

  [28]). Thus evolved to the Z scale, aLD (m) assumes a value of 1 O a (m )"0.1191$0.0011 (exp.)$0.0019 (theor.) 1 8 $0.0029 (pert.)$0.0003 (evol.) , (61) where the "rst and second errors are experimental and theoretical, the third error is due to modi"ed perturbation series, and the fourth error propagates from uncertainties of the evolution and matching equations [29]. This result is in excellent agreement with the value obtained from the determination of a from R at the Z pole and also from the "t to the electroweak data. 1  4.1.1.3. Hadronic decays of B resonances. Although the J/t meson has a mass of about 3.1 GeV it is su$ciently light that relativistic and non-perturbative e!ects are very signi"cant. However, an a determination from hadronic decays of the B is possible. From its decay modes, two di!erent 1 R ratios can be de"ned for the B. Both are known in next-to-leading order NLO [106,107] C( BPGGGPhadrons) R " %%% C( BPk>k\)

 




 




k 10(p!9) a(k) 1 " 1# 9.1#6.3 ln a mB 9p  C( BPcGGPc#hadrons) R " A%% C( BPk>k\) k 8(p!9) a(k) 1 " 1# 3.7#4.2 ln a mB 9p 



a (k) 1 , p



a (k) 1 . p

(62)

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The "ne structure constant a in this equation is to be evaluated at the B scale, that is  mB +10 GeV. In Ref. [107] a value of a\(mB )+132.0 was used in order to determine the strong  coupling constant from measurements of the R ratio for the B and the J/t resonance states. At %%% the B scale the result is [108] (63) aLD (mB )"0.167>  Pa (m )"0.115>  , 1 \  1 8 \  where the value of the coupling at the Z pole is obtained from numerically solving Eq. (20). The ratio R has been measured to high accuracy by the CLEO experiment [109], yielding A%% aLD (mB )"0.163$0.014Pa (m )"0.112>  , (64) 1 1 8 \  where the error is dominated by theoretical uncertainties associated with the scale choice. In Ref. [110] an investigation of moments of the R(s) ratio for the "rst six B resonances was performed. From the perturbative series up to O(a) (NLO) for these moments, which are calculated 1 using measured masses and electronic decay widths of the B resonances, the MS bottom quark mass m "(4.13$0.06) GeV was obtained and the strong coupling constant was determined at this
scale to be (65) aLD ((4.13 GeV))"0.233>  Pa (m )"0.120>  . 1 \  1 8 \  The dominant contribution to the error comes from uncertainties due to the choice of the renormalization scales for the bottom quark mass and a . 1 4.1.1.4. Summary of a determinations from fully inclusive quantities. In Table 4 the various 1 a determinations presented in this section are listed. Motivated by the di!erent dominance of 1 theoretical, scale, and statistical uncertainties, the weighted average is calculated from the averages of each of the three groups of a determinations, viz. from R , from quarkonia and from R , 1 O /!" respectively. For the quarkonia the average a value and its error is determined using the total 1 Table 4 Listing of a determinations from fully inclusive quantities: the total hadronic cross-section and the hadronic branching 1 fraction of the q lepton and of the J/t and B mesons. The values at m were evolved from the values measured at (s by 8 numerically solving the 4-loop renormalization group equation (20) using m "1.25$0.15 GeV [105] and m " 
4.25$0.15 GeV [28]. The weighted average takes correlations into account Observable quantity

(s (GeV)

n D

a (s) 1

a (m ) 1 8

Theory

R O B [QCD moments] R (J/t,B) %%% R ( B) A%% R /!" R /!" R /!" R /!" Weighted average

1.777 4.13 10.0 10.0 10.0 31.6 91.2 161.0

3 4 4 4 4 5 5 5

0.321$0.017 0.233>  \  0.167>  \  0.163$0.014 0.169$0.040 0.163$0.022 0.122$0.006 0.130$0.035

0.119$0.004 0.120>  \  0.115>  \  0.112>  \  0.115>  \  0.133$0.015 0.122$0.006 0.141$0.044

NNLO

91.2

5

0.1195$0.0025

NLO NLO NLO NNLO NNLO NNLO NNLO

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errors for the weights, while when averaging the R results no correlation is assumed due to the /!" dominance of both the statistical uncertainties and the result at the Z pole. Also it has to be remembered that the a results obtained from the total hadronic cross-section are a!ected by tiny 1 uncertainties only due to the choice of the scale. The uncertainty of the overall average is, therefore, calculated taking the full systematic error from the R result as the sole common error for the O a determinations from R and the quarkonia decays. 1 O One notes that the value of the coupling falls o! signi"cantly when the energy scale increases. This can be seen in Fig. 15 which shows these results together with the expectation of QCD for a (m )"0.119. Comparing the a values evolved to the Z pole, one observes a good agreement 1 8 1 between the various results. From this agreement, one may deduce that the energy dependence as predicted by the renormalization group equation (20), which is used to evolve the a from their 1 respective to the m scale, is very consistent with the data. The total error of the individual 8 a determinations is still signi"cant at the level of several percent. 1 4.1.2. Inclusive quantities: jet rates and event shapes In order to determine a , one may directly investigate quarks and gluons in the "nal state of 1 a hadronic event. Gluon emission by the quarks is connected with the coupling constant. This becomes immediately obvious from the di!erential cross-section shown in Eq. (35). Thus, to lowest order, the probability for a parton to radiate a gluon is directly proportional to a . Exploiting this 1 property, one may pursue two approaches to assess the size of the coupling. One can either determine a by counting how often a gluon is emitted by a quark, that is by measuring the jet rates 1 which requires the reconstruction of quarks and gluons from the jets of particles measured by the detector. Alternatively, one may take advantage of the fact that gluon radiation changes the spatial

Fig. 15. The value of the strong coupling constant obtained from fully inclusive quantities at various centre-of-mass energies is shown. Superimposed on the data is the QCD expectation at 4-loop precision using a (m )"0.119. 1 8

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as well as the energy and momentum distributions of the particles in the detector, that is by performing a measurement of the shape of an event. Determinations of a based on both jet rates 1 and event shapes will be presented in this section, starting with a theoretical description of the observables in each case. 4.1.2.1. Perturbative predictions for jet rates and event shapes. Although jet rates and event shapes are formally di!erent in how they analyse an event, both are similarly described by perturbation theory. The general approach to the perturbative description of this kind of observable has already been presented in Eq. (38). Going beyond leading order (LO), analytical calculations become di$cult due to the complicated phase space for multiparton "nal state con"gurations. Moreover, a special treatment of infrared and collinear singularities in real and virtual contributions to the di!erential cross-section of an observable is necessary. Due to these di$culties full perturbation calculations of jet rates and event shapes are currently next-to-leading order (NLO) only. However, there exist improvements on these predictions using resummation of large logarithmic terms as will be detailed below. In NLO the distribution of an inclusive observable F is given by a series (cf. Eq. (38)) a (k) a(k) 1 dp " 1 A(F)# 1 (2pb A(F)ln x#B(F))#O(a) ,  I 1 2p 4p p dF 

(66)

where A and B are perturbatively calculable coe$cient functions of F which are tabulated in Ref. [111] for many event shape observables. The variable x"k/s is the renormalization scale I factor, which relates the physical scale (s to the renormalization scale k. In contrast to fully inclusive quantities one must not neglect the arbitrariness of the choice of the scale k at which a is 1 renormalized. Inclusive quantities are a!ected by infrared and collinear singularities since they are related to the di!erential cross-section for multiparton "nal states. As a consequence the perturbative prediction becomes unreliable for partons that are close together. This is in particular the case when approaching the 2-jet region with a back-to-back quark}antiquark "nal state, for which FP0. Due to large logarithmic contributions from ln(1/F) (see Section 3.1.3) perturbation calculations using a as expansion variable are not reliable in this region which occupies a signi"1 cant fraction of the whole phase space (see for instance [12]). To overcome this de"ciency of "xed-order perturbation theory, an expansion in a ¸ is envisaged, where ¸"ln(1/F). For this 1 purpose the normalized event shape cross-section is rewritten,



R(F)"

F




1 1 dp "C(a ) exp G a , ln dF 1 1 F p dF  



#D(a , F) , 1

(67)

using a coe$cient function C(a ), an exponential of a function G(a , ln(1/F)), and a remainder 1 1 function D(a , F), which vanishes for FP0 if F can be exponentiated this way. The functions 1

 Usually event shapes are normalized such that 0 corresponds to the 2-jet case.

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C and G can be expanded in a and ¸ 1  aL C(a )"1# C 1 1 L (2p)L L aL  L> 1 ¸K G(a , ¸)" G (68) LK (2p)L 1 L K  (a ¸)L  (a ¸)L 1 1 # G "¸ G LL (2p)L LL> (2p)L L L (a ¸)L a  1 #2 # 1 G LL\ (2p)L 2p L ,¸g (a ¸)#g (a ¸)#a g (a ¸)#2 .  1  1 1  1 The key point of the expansion of G is that some of the power series in a ¸ could be summed to all 1 orders for a number of event shape observables and jet rates [112}121], yielding speci"c functions g for i"1 and 2. The function g resums all leading logarithmic (LL) contributions, g all G   next-to-leading logarithms (NLL). Subleading logarithms are contained in g , etc. Obviously  g becomes important if ¸ is large, that is for F close to the 2-jet region. Provided g , D(a , F), etc.   1 behave reasonably in this region, one may exploit g and g to predict the distribution of F down   to a ¸:1, i.e. much lower than using a "xed-order perturbative expansion, which is applicable for 1 a ¸;1 only [12]. Thus the resummed expression complements the "xed order calculation 1 towards the 2-jet region. Before joining the two calculations, it has to be recalled that the logarithmic approximation (NLLA) presented above depends on the choice of the renormalization scale similarly to the "xed-order prediction (O(a)) in Eq. (66). While g is invariant under changes of the scale x"k/s, 1  I g assumes an explicit scale dependence according to [112]  dg (a (s)¸) . (69) g (a (k)¸)"g (a (s)¸)#(a (s)¸) b ln x  1  1  1 1  I d(a (s)¸) 1 Since the resummation accounts for large logarithms to all orders, one expects, however, the scale dependence of a as determined using the NLLA calculations to be reduced. This is, in fact, the case 1 as will be shown below. In order to take advantage of the improved prediction in determinations of the strong coupling constant from event shapes and jet rates, the resummed and the full second-order calculations [32,111] should be combined. Several prescriptions to match the two calculations have been proposed [77,112], which di!er in the treatment of subleading terms in third and higher-order of a ¸. In general one considers, for both the NLL approximation and the O(a) calculation, the 1 1 normalized cross-section R(F) given by Eq. (67) and by the integral FdF of Eq. (66), respectively.  To combine the two, one determines the remainder function D(a , F). This can be done by relating 1 the a expansions of the two R(F) formulae, yielding the so-called R-matching. Alternatively one 1 can do the same but for the logarithm, ln R(F), of the two formulae which is called ln R-matching. A di!erent treatment of higher-order terms which do not vanish for FP0, or imposing constraints

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at the kinematic limit FP1 leads, respectively, to modixed R- or modixed ln R-matching prescriptions (see for instance the appendix of Ref. [122]). The general formulae for these matching prescriptions can be found in Ref. [112]. Detailed formulae and tables of the coe$cients G and C for several event shapes are given in Refs. LK L [77,116,119,120], which include tables of the coe$cients G and C for several event shape LK L observables and jet rates. 4.1.2.2. Jet rates. The existence of jets of particles and the connection of a jet to a parton rely on the fact that the particles emerging from a parton receive during hadronization only a limited transverse momentum relative to the parton momentum. Thus the dominant direction of the particles is given by the parton. Although the existence of jets might be obvious from looking at event displays (see for example Fig. 7), in practice one has to apply an algorithm to build up a jet from the particles measured in the detector. A large variety of algorithms has been proposed. Algorithms that reconstruct a "xed number of jets, for example three jets, were popular in the early time of the PETRA experiments, when the gluon was discovered (see examples in Ref. [123]). A reconstruction prescription is widely used which was proposed by the JADE collaboration [124]. Starting with measured particles, the general principle is to determine a resolution parameter y for pairs of resolvable jets (particles) i and j. Jet pairs whose y exceeds a chosen threshold GH GH value y are combined into a single jet. A di$culty arises from jets acquiring mass due to the  recombination since the perturbative calculations are performed for massless partons. To resolve this potential problem, various algorithms were devised which di!er in the resolution parameter de"nition and the recombination prescription. Some of the frequently employed algorithms are listed in Table 5. Further algorithms and more detailed descriptions may be found in Ref. [125]. The CAMBRIDGE C-algorithm, which is a modi"cation of the D-algorithm, was proposed in Ref. [126]. Its aim is to reduce non-perturbative corrections and to provide a better resolution of the jet substructure. An implementation and results from an investigation using Monte Carlo generator events can be found in Ref. [127]. The algorithm applies a two-fold resolution criterion in order to &freeze' soft resolved jets. It begins by looking for the pair of objects which are closest in angle h , that is, the pair with the smallest v (see Table 5). The least energetic object is considered GH GH a jet if the resolution parameter y exceeds a given y , otherwise i and j are combined (see GH  Ref. [126]). To assess the size of the strong coupling constant from jets, one may recall that an event with three jets is due to the emission of a gluon carrying a signi"cant fraction of the centre-of-mass energy at large angle. Thus the ratio of the number of observed 3-jet to 2-jet events is to leading order proportional to a , and a measurement of the rate of 3-jet events allows a determination of 1 the coupling. The n-jet rate R (y), which depends on the choice for y"y , is de"ned in terms of L  the respective cross-sections for n52 jets





a (k) H p  R (y, (s), LU  " C (y, x) 1 LH I L 2p p  HL\ a (k) a (k)  p PR (y, (s), U  "C (y) 1 #2 # C (y, x) 1    I 2p 2p p 




(70)

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207

Table 5 De"nition of the resolution parameters and recombination prescriptions for various frequently used jet algorithms. Energy and three-momentum of jets are indicated as E and p, respectively, while upright boldface variables denote fourvectors. The centre-of-mass energy is (s, but often the total visible energy is used instead in experiments. For massless quarks the E0- and JADE-algorithms are identical in second-order perturbation theory. The C-algorithm has a two-stage resolution criterion which is described in the text Algorithm

Resolution parameter

Recombination

Remarks

Theory

E

(p #p ) H y " G GH s (p #p ) H y " G GH s

p "p #p I G H

Lorentz invariant

NLO

E "E #E I G H p #p H p "E ) G I I "p #p " G H E "E #E I G H p "p #p I G H p "p #p I G H E ""p " I I p "p #p I G H E ""p " I I

Conserves E,

NLO

E0

JADE

2E E (1!cos h ) GH y " G H GH s (p #p ) H y " G GH s

P

P0

(p #p ) H y " G GH s

Violates p Conserves E,

NLO

p Conserves p,

NLO

Violates E As p-scheme,

NLO

but E updated after each recombination

D,

G

k R

2min(E, E)(1!cos h ) G H GH y " GH s

E "E #E I G H p "p #p I G H

Conserves E,

NLO

p; avoids exp.

#

problems

NLLA NLO

8E E (1!cos h ) GH y " G H GH 9(E #E ) G H

p "p #p I G H

Conserves E,

v "2(1!cos h ) GH GH y "min(E, E)v GH G H GH

E "E #E I G H p "p #p I G H

Conserves E,

NLO

p; accounts for angular

#

ordering

NLLA

p; avoids exp. problems

C

such that  R "1. In this de"nition the renormalization scale factor x "k/s is introduced. L L I The coe$cients C of the perturbative expansion are known up to order O(a), which implies LH 1 j42. They were calculated by numeric integration of the full second-order matrix elements [32] and are tabulated in Ref. [111]. The jet rates R are thus predicted for up to n"4 jets. Recently the L 4-jet rate was calculated in next-to-leading (up to a terms) and the 5- and 6-jet rates in leading 1 order [35}37]. Considering the 3-jet rate obtained using the JADE jet algorithm at a "xed y "0.08, the scale  dependence of the coupling can be made apparent. In this speci"c case the coe$cients of the

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Table 6 Measured 3-jet rates and total errors using the Refs. [122,129}132]

JADE

jet algorithm at y "0.08. Data are compiled from 

(s (GeV)

DELPHI

L3

OPAL

130}136 161 172 183 189

18.2$2.4

19.4$3.3 19.0$2.8 14.0$4.7

18.9$2.6 14.5$2.8 14.9$3.8 15.5$2.8 16.5$1.6

Calculated from 1n 2 assuming a negligible 54-jet rate.



perturbative expansion (70) read [111] C "6.76$0.006 and C "163.5$0.3 , (71)   thus allowing a determination of a from the 3-jet rate. Measurements of this rate at and below the 1 Z pole are compiled in Ref. [128]. Comparable results for energies above the Z mass are summarized in Table 6. All the 3-jet rates are presented in Fig. 16 versus the centre-of-mass energy. Superimposed is the QCD expectation Eq. (70) with (71) for a (m )"0.121. 1 8 Another source of uncertainty is assumed negligible, namely the e!ects due to hadronization. The perturbation calculation, Eqs. (70) and (71), applies only to a "nal state with a small number of partons. The jets, however, are constructed from the many particles (hadrons) that emerged from the hadronization of the partons. In order to keep uncertainties low an observable like R should  have a small hadronization correction. Earlier investigations (see, e.g. [108]) demonstrated that the JADE jet algorithm exhibits small hadronization corrections over a large centre-of-mass energy range as is shown in Fig. 17. Also shown in the "gure is the DURHAM (D) jet algorithm [118] (see Table 5) which was devised after LEP came into operation to circumvent some de"ciencies of the JADE algorithm, for example that leading and next-to-leading logarithms (NLLA) cannot be resummed [133]. As can be seen from the "gure both the JADE and D-algorithm have fairly small hadronization corrections, those for D being smaller at higher centre-of-mass energies. For these reasons and owing to the resummation of leading and next-to-leading logarithms the DURHAM algorithm is now frequently used for determinations of a at LEP. 1 Not long ago the CAMBRIDGE or C-algorithm was proposed [126] (see Table 5) which takes into account coherence e!ects of the gluon radiation during the parton shower development implemented in the parton shower models as angular ordering of consecutive gluon emissions. Although very small hadronization corrections were found for the mean jet rate from the C-algorithm, detailed investigations of n-jet rates showed that the hadronization corrections are larger than for the D and JADE algorithms [127]. Exploiting the theoretically appealing features of the D algorithm in an investigation of 3- and 4-jet rates measured by the ALEPH collaboration at LEP I [53], the strong coupling was inferred in Ref. [134], where the R-matched NLO#NLLA calculations for 3- and 4-jet rates were used and certain soft logarithms were taken into account (K term). The resulting "ts, for which only statistical errors of the data were considered and the statistical correlation between the data points

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Fig. 16. The 3-jet rate (R ) at y "0.08 using the JADE jet "nder, for values of (s between 22 and 183 GeV. The O(a)   1 QCD prediction for a (m )"0.121 and x "1 is overlaid. 1 8 I

Fig. 17. Centre-of-mass energy dependence of the hadronization correction of the 3-jet rate for various jet algorithms. Figure adapted from Ref. [108].

was neglected, are presented in Fig. 18. In general the agreement between data and theoretical expectation gets better if more higher-order radiative corrections are included. Thus, a remarkably good description is achieved over a very large y range and, in particular, in the low y regime,   when parts of subleading terms (K) are included. Combining both 3- and 4-jet rates one arrives at a (m )"0.1175$0.0018 , 1 8

(72)

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Fig. 18. Leading order (LO), next-to-leading order (NLO), R-matched NLO#NLLA (next-to-leading-logarithmic approximation), and QCD prediction with subleading soft logarithms included, NLO#NLLA#K, for 3- (left) and 4-jet rates (right) are compared to ALEPH data showing statistical errors only. The lower part shows the relative deviation of the measured data from the NLO#NLLA prediction, where the band is the scale uncertainty from varying x between 0.5 and I 2. Figures adapted from Ref. [134].

where the error is dominated by the scale choice, whereas experimental uncertainties are negligible. Recently an analytic expression for the 4-jet rate has been calculated in which all the leading and next-to-leading kinematic logarithms have been resummed [135]. The n-jet rates for adjacent y values are strongly correlated, thus being favourable for studies  at a single y value only. Experimental investigations now widely consider the di!erential 2-jet  rate D (y) instead. It may be considered as the di!erential cross-section D (y )"1/p dp/dy of      y which is the lowest value of y in a 3-jet con"guration. The perturbative prediction for D is  GH  obtained from the derivative dR /dy of the 2-jet rate R , calculated as R "1!R !R      assuming negligible contributions from '4-jet rates. Fig. 19 shows some of the many measurements of the di!erential y cross-section based on the  DURHAM jet algorithm (references to the measurements next to the Z pole are listed in Ref. [9], measurements away from the Z mass can be found in Refs. [129,130,136}139]). The di!erential cross-section is shown for !ln y instead of y thus stretching the low y region. The "gure reveals    a change of the di!erential 2-jet rate D (!ln y ) with the energy scale. The distributions are shifted   towards higher values of !ln y as the centre-of-mass energy increases. This trend is a manifesta tion of the running of a , as it corresponds to a reduced amount of highly energetic and well 1 separated 3-jet events, thus indicating a decrease of a . 1 Moreover, the "gure shows results from the re-analysis of data measured at lower energies by experiments which already terminated long before the advent of the D algorithm. Several attempts have been undertaken [137}139] to re-analyse lower-energy data using the DURHAM algorithm, in order to accurately test the energy dependence of the strong coupling, to exploit the improved theoretical predictions for the D algorithm, and to exclude uncertainties due to di!erent

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Fig. 19. Di!erential 2-jet rate versus !ln y obtained at centre-of-mass energies between 22 and 183 GeV using the  DURHAM jet algorithm. The data, which are displaced vertically by multiples of 0.3, are corrected to hadron level. At each energy the result of a "t of the combined "xed second-order calculation (O(a)) and the next-to-leading logarithmic 1 approximation (NLLA) to the distributions is overlaid. The range of data used in the "t is indicated by a solid line. The extrapolations are shown as dashed and dotted curves. The upper "gure is taken from Ref. [136], the data of the lower "gure are compiled from Refs. [138,139].

observables. In combination with measurements obtained at the high energies of LEP II a signi"cant observation of the running of the strong coupling constant becomes possible. 4.1.2.3. Event shapes. In the spirit of the di!erential cross-section for y in the preceding section,  cross-sections of further observables commonly called event shapes can be used to survey the

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hadronic "nal state. As in the case of the di!erential jet rate these observables assign to each event a single number whose value indicates the presence of highly energetic, well separated gluon radiation. Besides the thrust observable already introduced in Section 3.1.2 as a measure of parton con"gurations, many other variables were proposed (see for instance Refs. [111,123,140]). In the following a few of the event shape observables will be brie#y introduced. A particular weight is given to those which are frequently used in investigations of hadronic "nal states, and for which leading and next-to-leading logarithms were resummed to all orders (NLLA) in addition to the next-to-leading order (NLO) perturbation calculation. All event shape observables are to be calculated from the three-momenta p of the particles measured in the detector. Although most G experiments have sophisticated detectors to identify particles, they are usually assumed to be either pions, if a charged particle is observed, or photons for neutral ones. The particles' energies E , when G required in the following, are calculated under this assumption. Thrust T is de"ned by the expression [141] ¹"max n

"p ) n" G G . "p " G G

(73)

The thrust axis n is the direction of the unit vector n which maximizes the ratio. Each event may 2 be divided into two hemispheres using the thrust axis, such that particle i belongs to hemisphere H  (H ) if p ) n '0 ((0). The vector n is one axis of an orthogonal coordinate system describing  G 2 2 the event. A second axis is the thrust major axis which is found as an axis perpendicular to n , 2 yielding the thrust major value ¹ , from the maximization in Eq. (73). The third axis is thrust

minor, given by the vector product of thrust and thrust major axis, from which the thrust minor value ¹ is obtained by calculating the ratio in Eq. (73).

 Heavy jet mass M or o,M /s is given by the larger value of the total invariant mass in each of & & the hemispheres H , k"1, 2, de"ned by the thrust axis, normalized by the centre-of-mass energy I (s, that is [111,142]




 



  E ! p . (74) G G I I GZ& GZ& Correspondingly, the light jet mass, M , is the lighter of the two hemispheres, M is the di!erence * " and the total jet mass, M , the sum of the heavy and light jet masses. 2 Jet broadening B is determined for each hemisphere, according to 1 M " max & (s I

"p ;n " 2 , (75) B " GZ&I G I 2 "p " G G from which the total, B "B #B , and the wide jet broadening observables, B "max(B , B ), 2   5   are derived [115]. C-parameter C is obtained from the three eigenvalues j , j , j of the linearized momentum    tensor [32]



p?)p@ h?@" G G "p " G "p " G G G

(76)

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according to C"3(j j #j j #j j ). It can also directly be obtained from the characteristic       equation of the tensor, yielding the expression [12] 3 "p ""p "!(p ) p )/("p ""p ") G H G H . C" GH G H (77) 2 ("p ""p ") GH G H All these event shapes except thrust acquire zero value in the extreme two jet region and adopt the maximum value for isotropic events. For consistency with the other event shapes 1!¹ is usually considered rather than ¹. As an example of event shapes, Fig. 20 presents some distributions of the C-parameter obtained from measurements at various centre-of-mass energies between 35 and 161 GeV [104,129,130,139]. A clear dependence of the di!erential cross-section on the centre-of-mass energy is visible. It can be considered as being due to a change of the coupling a . The curves overlaid on the distributions 1 resulted from "ts of NLLA and O(a) (NLO) calculations combined using the ln R matching scheme. 1

Fig. 20. Distributions of C-parameter as measured between 35 and 161 GeV. The "t results of ln R-matched NLLA#O(a) predictions are superimposed as dotted curves where the solid line indicates the "t range. Data are taken 1 from Refs. [104,129,130,139].

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Over the whole energy range an excellent description can be observed. Results of a "ts to the 1 C-parameter data of JADE at 35 and 44 GeV were published in Ref. [139], yielding a ((35 GeV))"0.1480$0.0017 (exp.)$0.0097 (had.)$0.0138 (scale) , 1 a ((44 GeV))"0.1470$0.0032 (exp.)$0.0073 (had.)$0.0133 (scale) . 1 where the experimental errors (exp.) are the statistical and experimental systematic uncertainties quadratically combined and where the hadronization error (had.) summarizes all MC modelling uncertainties. The last error stems from the choice of the renormalization scale. As has been mentioned in the beginning of this section, the uncertainties coming from the choice of the scale must not be neglected when applying NLO calculations. To estimate this uncertainty the renormalization scale factor x "k/(s is usually changed from its natural value, 1, to 0.5 and 2, respectiveI ly. The result for a at a varied scale, which is x m in this case, is evolved back to (s"m by 8 1 I 8 solving Eq. (20) numerically. Any deviation of a (m ) obtained for x O1 from the value obtained 1 8 I for x "1 is attributed to the uncertainty due to the choice of the renormalization scale. It yields I the dominating error contribution as can be seen from the result above, thus setting a theoretical limit on the precision of the a determination. 1 We performed similar "ts to the OPAL data [104,129,130], which yielded s/d.o.f. values of about 0.5 to 0.6 and a ((91.2 GeV))"0.1245$0.0013 (exp.)$0.0062 (had.)$0.0071 (scl.) , 1 a ((133 GeV))"0.1097$0.0099 (exp.)$0.0034 (had.)$0.0053 (scl.) , 1 a ((161 GeV))"0.1070$0.0060 (exp.)$0.0067 (had.)$0.0046 (scl.) . 1 The experimental and scale uncertainties are de"ned as for the JADE result. The hadronization uncertainty has been estimated by varying several parameters of the PYTHIA generator and also by employing HERWIG and ARIADNE to correct for hadronization e!ects choosing the same parameter as described in Refs. [104,129,130]. Exploiting the data taken above the Z pole, the ALEPH collaboration performed a simultaneous analysis of the distributions of thrust, heavy jet mass, wide jet broadening, C-parameter and !ln y [136]. The data taken at the Z pole were excluded because they would have dominated the  results owing to their large statistical weight. Although the statistical error is larger without the LEP I data, the systematic uncertainties were found to be essentially reduced because of a decreased impact of hadronization e!ects and the explicit enforcement of the energy scale dependence according to perturbative QCD. In all these results the uncertainty due to the choice of the renormalization scale x yields a large I contribution to the total error. Consequently, missing higher-orders, whose e!ect on the value of the coupling are assessed by varying x , are still important. The "tted value for a changes I 1 considerably for di!erent choices of the scale. This is demonstrated for the C-parameter in Fig. 21. Plot (a) shows the strong dependence of the "tted a (m ) on the renormalization scale factor 1 8 x (solid curve) for the ln R-matched "xed order and resummed next-to-leading logs. The s/d.o.f. I represented by the dotted curve has a #at minimum around x "1. Juxtaposing this result to the I scale dependence of a in (b), where only the "xed-order calculation was "tted, reveals that the 1

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215

Fig. 21. Renormalization scale factor (x ) dependence of the a "ts to C-parameter data as measured by the OPAL I 1 collaboration at the Z pole [104]. The solid curves show a (m ), the dotted curves show s/d.o.f. Plot (a) is the result 1 8 obtained from the ln R-matched NLLA#O(a) calculation, while in (b) only "xed-order calculations (O(a)) were used. 1 1

combination with the resummation results yields some improvement. A pronounced scale dependence is notable in (b) and, in particular, the s/d.o.f. is much worse so that one "nds a minimum at extremely low scales of about 0.02 close to the end of the region of stable "ts. Although the improvement from the inclusion of resummed leading and next-to-leading logs is appreciable, the calculation of higher orders is desperately needed to reduce the renormalization scale dependence of the determined a (m ) value. Until then, given the rather small experimental errors, this scale 1 8 uncertainty is a dominant contribution to the total error on a determinations from jet rates and 1 event shapes. Keeping in mind the large uncertainty from the scale choice, one "nds evidence for the energy scale dependence of the strong coupling from a comparison of the results on a (s) not only for the 1 C-parameter, but also for other event shapes. In general, the correspondence between the measured data and the perturbative calculations is very good when hadronization Monte Carlo models (MC) are invoked for the correction of hadronization e!ects. 4.1.2.4. Summary of a determinations from jet rates and event shapes. Although the variety of 1 event shape observables is large, only a fraction of them has been investigated at LEP II energies, in particular those for which the resummation of large leading and next-to-leading logarithms is available. These are the di!erential 2-jet rate D (y )"1/p dp/dy obtained using the DURHAM    jet "nder, thrust ¹, heavy jet mass M , total and wide jet broadening B and B , and the & 2 5 C-parameter. Table 7 lists the averaged a values at several centre-of-mass energies ranging from 22 1 to 183 GeV. The average value and its error are calculated as weighted means of the individual measurements and their errors, respectively, using the total error of each single measurement to determine the weights. Assuming a large correlation between the individual errors, it yields a conservative estimate of the total error. Using data from measurements at the Z pole, the L3 collaboration determined a at energy scales 1 far below m , down to 30 GeV [143]. They exploited photon bremsstrahlung o! electron and 8 positron before they annihilate (cf. Fig. 3), and also o! the quark and antiquark. Photon radiation in the initial state obviously lowers the centre-of-mass energy that is available for the annihilation and, therefore, for gluon radiation. When a photon is radiated from a quark at high energy and at a large angle to the remaining quark}antiquark "nal state, it can also be assumed to reduce the

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Table 7 Summary of a determinations based on matched "xed order and resummed perturbation calculations (O(a)#NLLA). 1 1 The values are averages of the measurements from the individual experiments (A: ALEPH, D: DELPHI, L: L3, O: OPAL, S: SLD, TPC: TPC/TWOGAMMA). The a determinations of these experiments were based on measurements of the following 1 observables: di!erential 2-jet rate D from the DURHAM jet "nder, thrust ¹, heavy jet mass M , total and wide jet  & broadening B and B , and C-parameter. Data are compiled from Refs. [87,88,104,129}132,136}139,143}145] 2 5 Exp.

Observables

a (s) 1

a (m ) 1 8

22

JADE

29

TPC

0.161>  \  0.160$0.012

0.124>  \  0.129$0.008

35

JADE

44

JADE

58

TOPAZ

0.1448>  \  0.1394>  \  0.1390$0.0080

0.1228>  \  0.1233>  \  0.1286$0.0068

41.2 55.3 65.4 75.7 82.3 85.1

L

D  D  D , ¹, m , B , B , C  & 2 5 D , ¹, M , B , B , C  & 2 5 D  ¹, M , B , B & 2 5

0.140$0.013 0.126$0.012 0.134$0.011 0.121$0.011 0.120$0.011 0.120$0.011

0.122$0.010 0.117$0.010 0.127$0.010 0.118$0.010 0.118$0.011 0.115$0.011

0.1211$0.0068 0.1132$0.0075 0.1070$0.0069 0.1012$0.0070 0.1084$0.0051 0.1076$0.0051

0.1211$0.0068 0.1197$0.0085 0.1160$0.0082 0.1102$0.0084 0.1200$0.0063 0.1196$0.0064

(s (GeV)

91.2 133 161 172 183 189 91.2

A, L, D, O, S A, L, D, O A, L, D, O A, L, D, O A, L, D, O L, O

Weighted average

D ,  D ,  D ,  D ,  D ,  D , 

¹, ¹, ¹, ¹, ¹, ¹,

M , & M , & M , & M , & M , & M , &

B , 2 B , 2 B , 2 B , 2 B , 2 B , 2

B 5 B , 5 B , 5 B , 5 B , 5 B , 5

C C C C C

0.1212$0.0079

centre-of-mass energy of this "nal state if the (time) scale involved is much less than the scale at which the parton shower develops from the quark}antiquark pair. This condition has to be guaranteed by the experimental selection cuts such that radiative events are characterized by a hadronic system recoiling against an isolated photon with energy E and large transverse A momentum k with respect to quark and antiquark. The reduced centre-of-mass energy (s is , given in terms of the nominal energy (s by s"s!2E (s. The coupling constant was deterA mined at this scale from the hadronic system using event shape observables. The averaged results of Table 7 are presented in Fig. 22. In addition to the total errors, which include the correlated uncertainties due to the hadronization correction and, in particular, the uncertainties from the choice of the scale, also the statistical and uncorrelated experimental uncertainties added in quadrature, are indicated in the "gure. The data agree nicely with the QCD prediction for a (m )"0.122. The same conclusion can be drawn from a direct comparison of the 1 8 individual a results in Table 7 after being evolved to the m scale. 1 8 The DELPHI collaboration determined a (m ) from 17 di!erential jet rate and event shape 1 8 quantities [146]. The study investigated the dependence of the di!erential distributions on the polar angle of the thrust axis. Good consistency of the single a values from "ts of the NLO 1 predictions to the data was achieved when allowing both a and the renormalization scale factor 1

O. Biebel / Physics Reports 340 (2001) 165}289

217

Fig. 22. Values of the strong coupling a obtained from jet rates and event shapes, their total errors (outer ticks), and the 1 quadratically combined statistical and experimental systematic uncertainties (inner ticks). The results from several determinations, which considered the ln R-matched O(a)# NLLA perturbation calculation for the di!erential 2-jet rate 1 D from the DURHAM jet "nder, thrust, heavy jet mass, total and wide jet broadening, and C-parameter, at various  centre-of-mass energies are shown. References for data are given in Table 7. Overlaid to the data is the QCD expectation at 4-loop precision using a (m )"0.121. 1 8

x to vary. This observation was also made in Refs. [104,147]. Fig. 23 is a compilation of the results I of a (m ) obtained for (a) "xed x "1 and (b) varying x . The optimized renormalization scale 1 8 I I factors range from x +0.057 for thrust up to +2.66 for the GENEVA jet "nder. To estimate the I uncertainty due to the choice of the renormalization scale the x factors were scaled by 1/(2 and I (2, respectively, and the "ts were repeated with the scale "xed to these values. In part (b) of the "gure, a weighted average, a (m )"0.1168$0.0026, is quoted which is in good agreement with 1 8 the other results although the error is smaller because of the smaller variation of x and since it has I been calculated using the `optimized correlationa method of Ref. [148] which takes into account an unknown correlation between the individual results. In brief, this method yields an average assuming a common correlation factor between the single measurements obtained from the requirement that s/d.o.f. must be unity. In conclusion, even not fully inclusive quantities yield an energy scale dependence in agreement with the QCD expectation. It has to be kept in mind, however, that the quoted errors are dominated by the uncertainty from the choice of the renormalization scale which would have limited any stringent conclusion if a deviation from the predicted energy dependence had been observed. 4.2. Quark mass ewects Most experimental investigations of the strong coupling constant which have been presented so far neglected e!ects due to "nite quark masses since massless quarks were assumed in many theoretical calculations. In the case of the jet "nders, where many "nal state particles are combined

218

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Fig. 23. Values of a (m ) obtained from "ts of the NLO prediction of QCD for oriented event shapes using (a) "xed x ,1 1 8 I and (b) allowing x to vary in the "t. The bars, which are the total errors from experimental, hadronization and scale I uncertainty added in quadrature, are subdivided in (b) to indicate by the dashed lines the contribution due to the choice of the renormalization scale. Figures taken from Ref. [146].

into jets, one has to take special measures in the recombination to ensure that the jets remain massless (for instance, rescaling of energy or momentum of the jets). The impact of quark masses is known for a long time in leading order (O(a )) [149] for c and Z exchange in e>e\ annihilation. Q A partial calculation of second-order terms in Ref. [150] was lately extended to full next-to-leading order (O(a)) by three independent groups [151}153] yielding consistent predictions [154]. 1 The e!ect of a "nite quark mass is twofold. Besides an obvious reduction of the phase space available for gluon emission, QCD radiation is also reduced. This follows from the di!erential cross-section of Eq. (35). After being complemented by further terms which explicitly depend on the quark mass m it reads in leading order [149] /






x #xM 1 a 4m 1 1 dp / / "C 1 ! / # $ 2p (1!x )(1!x M ) s 1!x 1!x M p dx dx M / / / /  / /









 

1 1  2m 1 4m 1 ! / # ! / # , s (1!x ) (1!x M ) s 1!x 1!x M / / / /

(78)

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219

where x for i"Q, QM , are the centre-of-mass energy fractions of the massive quark and antiquark, G respectively. The reduction of QCD radiation is immediately apparent in this equation since all terms that depend on the mass are subtracted. An important feature should be noted for the di!erential cross-section in Eq. (78). Collinear singularities do not a!ect the cross-section owing to the quark's "nite mass. When recalling Eq. (36), it can be seen that the mass of a Q}G system is restricted by the relation 1!x "m /s5m /s. Thus the limit x P1, which is the collinear singularity, does not occur. / /% / / The infrared singularity at x P0, however, is still present. % The cross-section in Eq. (78) reveals another remarkable feature if the gluon radiation is quite collinear with the heavy quark. In this region of the phase space close to the kinematical boundary, one "nds a dead cone for radiation which is due to the helicity conservation in the radiation of a spin-1 gluon o! a spin- massive quark. To derive an expression for the particular phase space  region, one considers a gluon of fractional energy x "2!x !x M ;1 which is close in angle to % / / the heavy quark, h+0. This yields a simpli"cation of Eq. (78) [12] 1 dp h a 1 +C 1 . $ p x (h#4m /s) p dx dh  % % /

(79)

Thus gluon radiation is suppressed if h:2m /(s. This is the dead cone the angular size of which / grows with the quark mass. In brief, according to QCD heavy quarks radiate fewer gluons than light quarks. Although the e!ect might be small at high energies, it has to be accounted for in precision determinations of the strong coupling constant. From the considerations above it is rather obvious that, in particular, the 3-jet rate is expected to be reduced for heavy quarks. In the energy range considered for this report the case of the bottom quark will be of special interest since it is the heaviest quark accessible at centre-of-mass energies between 10 and 190 GeV. Furthermore, at the Z pole all "ve quark #avours are produced at roughly the same frequency owing to their coupling to the Z (cf. Fig. 5(b)). While the impact of the massive bottom quark is small for investigations that consider all quark #avours inclusively, it will play a major role in studies of the properties of individual quark #avours. QCD, however, is a "eld theory in which the coupling a is independent of the quark 1 #avours. It therefore is a signi"cant test of QCD to verify the independence of the strong interaction of the #avours involved, and also to test the calculated quark mass e!ects. The latter procedure may be inverted. A measurement of quark mass e!ects can be exploited to determine the quark masses assuming #avour independence of the coupling. At this point it has to be recalled from Section 2.2.2 that within the MS renormalization scheme quark masses depend on the energy scale. Measuring di!erent quark masses at di!erent scales is therefore another stringent test of QCD and its renormalization. 4.2.1. Tagging the yavour of a quark Testing the dependence of the strong interaction on the #avour of a quark requires the identi"cation of the #avour from the hadronic remnants of quark, antiquark and gluons. In connection with the large data statistics and the high precision investigations in the bottom and charm sector at LEP I and SLC many sophisticated methods were established to tag these #avours

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which became possible after the installation of high precision silicon vertex detectors (see, e.g. Ref. [155]). For bottom quarks one uses the large decay multiplicity, the high mass, the semileptonic decays, and, in particular, the long lifetime of B hadrons (for details see, e.g. [156]) which allow for an event-by-event classi"cation. Although charm quarks can be tagged by the decay vertex of a D meson [157] one usually reconstructs the decay of a charged DH(2010)! meson into a D and a p! where the D is searched for in various decay channels [158] whose small branching ratios lead to marginal e$ciencies only. Since charm mesons also occur in the decay chain of B hadrons one has to "ght against bottom quarks spoiling the tagged charm events. To enrich a primary #avour the leading particle e!ect [159,160] is used, that is, one assumes that the highest energy hadron is likely to contain the primary quark. Exploiting this e!ect even the light uds quarks can be tagged, for instance the strange quark by looking for K or charged 1 K! [161]. Contributions from other quark #avours, however, can only be removed by statistical methods. More e$ciently and less biased by the demand for high energy particles, but without any distinction between primary u, d, and s quarks, these light quarks can be selected recalling the fact, that in such events no hadrons of long lifetime similar to bottom and charm hadrons can occur. Therefore, reversing the search for decay vertices yields a very pure light quark tag [156,158]. 4.2.2. Flavour independence of QCD All LEP and SLC collaborations performed tests of the #avour independence of the strong interaction at the Z pole employing various tagging methods to measure #avour dependent jet rates or event shapes [157,158,161}167]. In general, the studies are designed to determine the ratio of the coupling for a selected #avour f to the one obtained from the light (from all, or from the complementary, i.e. all but f ) quark #avours, that is aD/a (aD/a or aD/a  , respectively). 1 1 1 1 1 1 Such ratios have the advantage of reduced systematic uncertainties. For instance, the e!ects of the choice of the renormalization scale largely cancel in the ratio, thus becoming less signi"cant than experimental systematics. Instead of this, mass e!ects and the uncertainty of the mass of the bottom quark contribute to the total error. Fig. 24 shows results for the ratios a
/a and a /a which were obtained from 3-jet rates of 1 1 1 1 several jet "nders and from various event shapes, respectively. Considering di!erent #avours in numerator and denominator is advantageous in order to avoid correlations between them. Having included the mass e!ects, which are of the order of 5}7% for b quarks and 1% for c, the #avour dependent a ratios conform to unity for all 3-jet rates and event shapes considered, as expected for 1 a #avour independent strong coupling. Averaging the ratios in order to diminish the statistical and systematic #uctuations of the individual results, one arrives at the values listed in Table 8, which is a compilation of measurements of the LEP and SLC collaborations [157,158,162,164,166]. A weighted average of the determinations has been calculated deriving the weights from the total errors. For the calculation of the

 Unstable particles having lifetimes of '3;10\ s are usually regarded as stable owing to their tiny probability to decay within the detector.

O. Biebel / Physics Reports 340 (2001) 165}289

221

Fig. 24. Flavour dependent ratios of a for bottom and charm quarks to the light uds quarks, respectively. Shown are the 1 results obtained from 3-jet rates measured with di!erent jet algorithms [157], and from several event shapes [158]. All ratios were determined using NLO calculations with massive quarks [151,153].

total error of the average, the statistical uncertainties of the individual determinations are taken as uncorrelated. The systematic errors, which are considered as fully correlated, are averaged using the same weights as before. This prescription closely follows that of Ref. [104] which has already been outlined in Section 4.1.1. From the averages of Table 8 one can conclude that the #avour independence of the strong interaction is proven at the level of the systematic errors of 1}4%. The perturbation calculation to next-to-leading order (O(a)) for massive quarks, however, is inseparably connected to it. Even at 1 energies as high as at LEP I mass e!ects would imitate a #avour dependence at the level of several percent. 4.2.3. Running quark masses Assuming the #avour independence of the strong interaction one can turn the tables and use the perturbation calculations for massive quarks to determine the masses of the quarks. This constitutes another important test of QCD since one expects quark masses in the MS renormalization scheme to depend on the energy scale (see Section 2.2.2). In principle one would assume that quarks have a unique and constant mass value. Owing to con"nement, however, quarks can only be observed inside hadrons. In order to assess the value of the mass one needs a theoretical prescription that is based on perturbation calculations. Among several mass de"nitions (for an overview see the notes on quark masses in Ref. [28]), two are widely used in these calculations: the pole mass M and the MS mass m (k). The pole mass is related to   the pole of the heavy quark propagator in perturbation theory. This mass is, in fact, independent of

222

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Table 8 Compilation of ratios of a determined for bottom and charm quark events to the coupling determined for light uds 1 quarks, respectively [157,158,162,164,166] (A: ALEPH, D: DELPHI, L: L3, O: OPAL, S: SLD3). The "rst error is statistical, the second comprises systematic and theoretical uncertainties added in quadrature. The ALEPH and L3 values are derived from the measured a
/a by assuming a " cDaD, where cD"B(e>e\P+M ) is the standard model hadronic 1 1 1 D
1 branching fraction for Z decays into #avour f. For the determination of the averages and the total errors see the text Exp.

a
/a 1 1

A

1.002$0.009$0.022 1.007$0.005$0.009 1.07$0.05$0.06 0.998$0.005$0.012

D L O

a /a 1 1

1.002$0.017$0.027

S

1.004$0.005>  \ 

1.036$0.043$0.047

avg.

1.004$0.013

1.009$0.035

Observables

Theory

¹, C, y (E0), y (D),   R (E0)  R (E0)  D (D), ¹, M , B , B , C  & 2 5 R (E),  R (E0),  R (P),  R (P0),  R (D), R (G)  

Partly massive Massive NLO Partly massive Massive NLO

Massive

NLO NLO

NLO

the renormalization scheme [25] and, therefore, independent of the energy scale. However, the pole mass is de"ned only within the context of perturbation theory because, owing to con"nement, the full quark propagator has no poles. Perturbatively, however, both M and m (k) are related in O(a) [168]. In O(a) the relation for   1 1 M is   a (k) 4 !2pc ) l M "m (k) 1# 1 K   3 p
















2 a(k) 8p 4 p # 1 K! c #p c ! b !pc l# c (b #c )l K K 3 K 3  p 3 K 2 K  # O(a) , 1

(80)

where b "(33!2n )/12p, c "1/p, and c "(303!10n )/72p are de"ned in Eqs. (21) and  D K K D (24), K+13.3 (12.4) for charm (bottom) quarks [169], and l"ln(m (k)/k). It must be pointed out that the convergence of the perturbative expansion for the expression of the MS mass via the pole mass is known to be worse [169] (see Ref. [23] for a review). This is mostly due to non-perturbative e!ects (NP) which are found to contribute an additional correction of [170] 2p (j d,."! 3 m

(81)

to the coe$cient of the term linear in a , where j is some regulator, acting like a small gluon mass, 1 to account for the pole at very small energy scales (see Section 5.3). Many estimates, in particular of the bottom quark mass from bottomonium and B hadron masses, were done (see compilation in Ref. [28]) which usually su!er large uncertainties due to the

O. Biebel / Physics Reports 340 (2001) 165}289

223

small scales involved and due to non-perturbative e!ects. All values of the MS bottom quark mass range between 4.1 and 4.4 GeV. A combination of some of the estimates in Ref. [171] yielded m (m )"(1.30$0.03) GeV and m (m )"(4.34$0.05) GeV. Next-to-next-to-leading order (NNLO) A A

corrections were considered in estimating the bottom quark mass, yielding m (m )"

(4.25$0.09) GeV [172] and m (m )"(4.19$0.06) GeV [173]. All these results are of the same

magnitude. Given the errors assigned to the masses, however, some results are in contradiction. In the following, the estimate of the Particle Data Group in Ref. [28] will therefore be adopted, that is m (m )"4.1}4.4 GeV.

In order to determine the quark masses at energies far above the region of the quarkonia, one has to employ perturbative calculations which consider massive quarks. Only recently such calculations became available for the gluon radiation from heavy quarks in O(a). The calculations 1 [153]. The consider both mass de"nitions, the MS mass m (k) [151,152] and the pole mass M   consistency of these calculations was veri"ed [154]. Using the large data statistics available from e>e\ annihilation at the Z pole and, furthermore, highly e$cient and very pure bottom quark tagging methods, the DELPHI [164,174] and SLD collaborations [156] investigated mass e!ects for bottom quarks. In Refs. [151,152] the ratio of the 3-jet rates for bottom and the light uds quarks, R
"R
/R, was proposed as being sensitive to    the value of the b mass. Fig. 25 shows this ratio versus y for the DURHAM and the CAMBRIDGE jet  "nders. Taking into account the large correlations between adjacent data points, a good agreement with the next-to-leading order (NLO) calculation for massive quarks is observed. From a "t of the theoretical expression to the 3-jet data of the DURHAM algorithm only, the DELPHI collaboration obtained a bottom quark mass at the Z scale of m (m )"(2.67$0.25(stat.)$0.34(hadr.)$0.27(theory)) GeV .
8

(82)

Fig. 25. The 3-jet rate ratio for bottom and light quarks is shown for the DURHAM and CAMBRIGDE jet algorithms [164,174]. The curves stem from leading (LO) and next-to-leading (NLO) calculations for the pole mass, M , or the MS
mass at the Z scale. Plots are taken from Ref. [174].

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The error on this result has contributions from statistics, the hadronization uncertainty (hadr.), which is dominated by the choice of the model (either STRING or CLUSTER), and from missing higher orders in the theoretical expression, which were assessed by varying the scale k by a factor of two around m , and by changing from the pole to MS running mass in the "tted expression. 8 Fixing y at a value speci"c for the jet "nder under consideration the dependence of the mass  value on the jet algorithm has been investigated in Ref. [156]. The theoretical expectation for the same ratio for six di!erent jet algorithms is shown versus m in Fig. 26, and the measured ratios
with their statistical uncertainties are overlaid as gray bands. It is remarkable at "rst sight to "nd some ratios increasing with the bottom quark mass. This can be understood from the resolution variables y of the jet "nders. If they are de"ned as a mass, as it is the case for the E, E0, P, P0 GH schemes, a "nite and large mass results in a shift of the 3-jet rate towards higher values of the resolution parameter y since y has to be at least as large as the quark mass. Since this is not the  GH case for massless quarks, one may "nd, particularly at rather low values of y , a higher 3-jet rate  for bottom than for light quarks.

Fig. 26. At "xed y the theoretically expected dependence of the 3-jet rate ratio, r ,R
/R, for bottom and light 
  quarks on the bottom quark mass is shown for various jet "nders as points with error bars and parameterized by a curve. The gray bands represent the experimentally measured values of the ratio and the respective statistical errors as obtained by the SLD collaboration. The "gure is taken from [156].

O. Biebel / Physics Reports 340 (2001) 165}289

225

Values for the mass were derived from the data by employing the theoretical expression and by combining the six results taking into account the correlations. This way an estimate of the MS bottom quark mass at the Z scale of m (m )"(2.56$0.27(stat.)>  (syst.)>  (theory)) GeV (83)
8 \  \  was obtained in Ref. [156] where the systematic error includes the hadronization uncertainties. The theory error comprises hadronization uncertainties and an additional uncorrelated 2% uncertainty on each ratio r ,R
/R, which is attributed due to missing higher-orders.
  Although these two determinations of the bottom quark mass at the Z scale agree within the signi"cant errors, the calculation of an average requires care because of the large spread of the results for di!erent jet "nders and the signi"cant but unknown correlations between the measurements. As already described in Section 4.2.2, a large correlation of systematic and theory errors is therefore assumed in the calculation of a weighted average of the two results. The weights are obtained from the symmetrized total errors. This yields m (m )"(2.65$0.18(stat.)$0.44(syst.)$0.30(theory)) GeV , (84)
8 where the systematic error is dominated by hadronization uncertainties. Fig. 27 compares the MS value of the bottom quark mass at the Z mass scale with that at the m scale. The solid curve in this "gure is the QCD prediction for the scale dependence of the running
mass, that is Eq. (26), where the bottom quark mass value of the particle data group PDG from Ref. [28] was used. The dashed and dotted bands indicate the uncertainties from the PDG bottom quark mass value and from a 0.006 error on a (m ). 1 8

Fig. 27. Values of the MS bottom quark mass at the Z and the m mass scales are shown. Overlaid is the theoretically
expected running of the mass assuming a (m )"0.119. The dashed lines indicate the uncertainty due to the error on 1 8 m (m  ), the dotted lines represent the error band when an additional 0.006 uncertainty on a (m ) is included.

1 8

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Although the uncertainty of the determined value for m (m ) is large, it supports the predicted
8 scale dependence of the MS quark masses. This becomes more obvious from the di!erence between the masses at the two scales m (m )!m (m )"(1.60$0.57) GeV , (85)


8 which deviates from zero by about 2.8 standard deviations. Further investigations of the e!ects on m from di!erent jet "nders are clearly necessary. But nevertheless, the agreement of the current
results with the expectation from the perturbative calculations in Fig. 27 is remarkable. 5. Tests of QCD treatments of hadronization Most of the investigations of strong interactions that have been described in the previous section rely on phenomenological models in order to account for hadronization e!ects. Although the models provided a well-suited representation of the e!ects, they contributed an inherent and hardly reducible uncertainty to each determination of the strong coupling constant. E!orts to open the hadronization phenomena to a proper treatment in the framework of perturbative QCD theory are required to diminish the in#uences of the various models. Such approaches necessarily consider techniques that go beyond a simple perturbative expansion in powers of a . 1 Two such methods will be presented in this section, viz. the modixed leading logarithm approximation (MLLA) in conjunction with the conjecture of local parton}hadron duality (LPHD), and the concept of renormalons which leads to power corrections to the perturbative calculations. As a start, light is shed on the perturbative picture of fragmentation functions and scaling violation. 5.1. Inclusive fragmentation function and scaling violation 5.1.1. Fragmentation function and evolution equation Inclusive hadron production in e>e\Pc, ZPhX, where h is either a given hadron species or a sum over all charged hadrons, is governed by the strong interaction and, therefore, by the strength of the coupling. The di!erential cross-section of this process in the case of unpolarized e> and e\ beams receives contributions from the polarization states of the exchange vector bosons c, Z, viz. transverse and longitudinal polarization, respectively. In addition, there is an asymmetric component due to the parity-violating interference. Each of these three contributes to the di!erential cross-section via its characteristic dependence on the polar angle h between the direction of the incoming e\ and the outgoing hadron h, according to the relation [175] 3 dpF 3 dpF 3 dpF dpF " (1#cos h) 2 # sin h * # cos h  , dx 4 dx 4 dx dx dcos h 8

(86)

where x,x "2E/Q, E being the particle's energy, is the fractional energy of the hadron and # Q"(s is the centre-of-mass energy. Recalling energy conservation, a sum rule for the inclusive cross-section summed over all outgoing hadrons h can be obtained



dpF 1 dx dcos h x "p "p #p ,  2 * dx dcos h 2 F

(87)

O. Biebel / Physics Reports 340 (2001) 165}289

227

where



1  dpF p , (88) dx x . (P"¹, ¸, A) . . 2 dx F  Thus the total inclusive cross-section is the sum of the transverse and the longitudinal crosssection. For unpolarized beams and high centre-of-mass energy, (s, the latter contribution is, depending on the fermion mass m , largely suppressed by a factor m /s due to the approximate D D helicity conservation at the cH/Z-+M vertex [176]. A signi"cant longitudinal cross-section is generated by gluon radiation o! quarks, rendering it proportional to a . 1 The three di!erential cross-sections dpF /dx, for P"¹, ¸, A, can be expressed by perturbative . QCD, based on the factorization property proven in Ref. [177], as a convolution [175]



dpF (s)  dz . (e>e\PhX)" C (z, a (k ), k /s) ) DF (x/z, k ) , 1 0 $ D $ dx z .D DOO % V

(89)

of coezcient functions C , being the cross-section for the inclusive production of f" quark q, .D antiquark q , or gluon G in the given process, and of the fragmentation functions DF (x, k ), which D $ represent the distributions of the energy fraction x of hadron h stemming from the fragmentation of parton f. The shape of the DF functions has to be obtained from experimental measurements, D because the production of a hadron is a non-perturbative process and the parton fragmentation functions are, therefore, not perturbatively calculable. The coe$cient functions C are known up to corrections of the order a [43,175,178]. In .D 1 leading order the transverse coe$cient is proportional to the Born level production cross-section of parton f, C "d(1!x)p (s). It is, for instance, zero for gluons. The longitudinal coe$cient 2D D C vanishes at this order. It solely receives corrections of the order a , thus allowing a determina*D 1 tion of the coupling from a measurement of the longitudinal cross-section. The full coe$cient functions up to O(a ) are too involved to be repeated here. They can be found in Ref. [175] for 1 instance. Although calculable, the coe$cient functions C contain collinear singularities whose renor.D malization renders both the coe$cient and the fragmentation functions dependent on an arbitrary factorization scale k which is analogous to the renormalization scale k introduced in $ 0 Section 2.2.3. However, physical cross-sections as in Eq. (89) must be independent of the k scale if $ they are determined from an all-order perturbation calculation. This fact can be used to derive the scale dependence of the parton fragmentation functions DF from the one of C (see Refs. D .D [12,175]). In the perturbative regime the scale dependence of DF is given by the DGLAP evolution D

 Here, the meaning of the term fragmentation function is slightly di!erent from that introduced in Section 3.2 in the context of hadronization models. The fragmentation function in such models determines the fraction z"(E#p ) /(E#p) of energy and momentum which is transferred from the parton to the hadron during ,     hadronization. Experimentally one measures x "2E/(s or x "2p/(s, that is the hadron's energy or momentum # N fraction of half of the centre-of-mass energy.  Dokshitzer, Gribov, Lipatov, Altarelli and Parisi.

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equation [42,179]



RDF(x, s)  dz a (k ) G 1 0 P (z, a (k ))DF(x/z, s) , " 1 0 H R ln s z 2p HG  HOO% V where i"q, q , G and a (k ) P (z, a (k ))"P(z)# 1 0 P(z)#O(a) HG 1 0 HG 1 2p HG

(90)

(91)

is the perturbative expansion of the Altarelli-Parisi splitting functions, whose leading order representation has already been introduced in Eq. (43) of Section 3.1.3. The full terms for P are HG lengthy and, therefore, not repeated here. They can be found for instance in Ref. [12]. The remarkable feature of the DGLAP evolution equation (90) is its dependence on a in the 1 convolution integral. Due to this the scaling of the fragmentation function D (x, s) with the energy D scale Q"(s is violated. A scale-independent x distribution is expected for instance in the independent fragmentation model. Historically it was this scaling violation which gave a "rst indication for the running nature of the strong coupling. 5.1.2. Determination of a from scaling violation 1 An experimental determination of a from the scaling violation was proposed in Ref. [180]. It 1 su!ers, however, from several complications. Firstly, in the comparison of fragmentation functions at very di!erent energies care must be devoted to the details of the #avour composition. In e>e\ annihilation the relative production rates of the various quark #avours change considerably from the very low energy region, which is dominated by photon exchange only, to the Z region, where the electroweak couplings dominate (see Section 3.1). Parton fragmentation functions may, in general, be di!erent for di!erent quark #avours, as can be seen from Eq. (89). Hence the fragmentation functions have to be measured separately for the di!erent quark #avours. This is possible with sophisticated #avour tagging methods, some of which have already been described in Section 4.2.1. Employing these techniques the fragmentation functions shown in Fig. 28 were obtained separately for bottom, charm and light uds quarks [53,140,181}183]. Another method to extract #avour dependent fragmentation functions is based on the momentum spectrum of a single particle species. Conjectures on the production of the particle species from a #avour f have to be applied in order to reduce the number of unknown fragmentation functions. This approach will not be described in this report, but details can be found in Ref. [178] (see also Ref. [160] for a di!erent approach). A particular property of these #avour-dependent fragmentation functions should be noted for b quarks. Although bottom quarks have been shown to transfer a large fraction of their energy and momentum to the b-#avoured hadron (see Section 3.2), the di!erential cross-section for hadrons of high scaled momentum x is signi"cantly less than for light quarks, u, d, s. This is due to the N cascade decays of the heavy b-#avoured hadrons via charmed into light hadrons. About 5.5 charged hadrons are found on average in the decay of one b-hadron [184]. The available energy is thus distributed over many particles, the decay products are low in energy, and the fragmentation function of b quarks is softer than for u, d, and s. As the #avour composition changes with the centre-of-mass energy, this mimics a scaling violation which has to be taken into account in the determination of the strong coupling constant.

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Fig. 28. Comparison of the inclusive and the #avour-dependent fragmentation functions obtained at (s"m by the 8 ALEPH (A), DELPHI (D), OPAL (D), and MARK II (M) collaborations. Statistical and systematic uncertainties are included in the error bars. The solid and dotted horizontal lines indicate the JETSET and HERWIG predictions, respectively. Figure taken from Ref. [181].

A second complication in the determination of a from fragmentation functions is due to the 1 gluon fragmentation function D (x, s) which is part of the convolution integral in Eq. (90). This % function cannot be measured directly in e>e\ annihilation since gluons enter the hadronic "nal state only through their radiation o! quarks, thus appearing always in conjunction with the quark fragmentation functions. One approach is to identify the gluon jet in a clear 3-jet "nal state by tagging both of the quark jets [185,186]. The results of this method are best at large x where the identi"cation of the gluon jet and the assignment of particles to the gluon jet is less ambiguous. Another approach to infer the gluon fragmentation function makes use of the lowest-order properties of the longitudinal coe$cient function C which, as has been mentioned above, is * proportional to a while the transverse coe$cient function for quarks is a d-function. For this 1 reason the longitudinal fragmentation function F (x)"(1/p )(dp /dx) can be regarded in leading *  * order as being composed of the transverse and the gluon fragmentation functions [175]

 


 

 dz z a F (z)#4 !1 D (z) #O(a) . F (x)" 1 C 2 % 1 * z x 2p $ V

(92)

Fig. 29 shows gluon fragmentation functions as obtained by the ALEPH, DELPHI, and OPAL collaborations. DELPHI's result is based on identifying the gluon jet in 3-jet con"gurations (Y and Mercedes topologies exempli"ed in Fig. 30; an example of the latter is shown in Fig. 7). In such

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Fig. 29. Gluon fragmentation function D at Q"m measured by ALEPH, DELPHI, and OPAL. The data points % 8 correspond to DELPHI's analysis of tagged gluon jets in very symmetric 3-jet events (Y or Mercedes topologies). The curves are from "ts of the parameterization described in the text to the gluon fragmentation function derived from the measured longitudinal and transverse fragmentation functions. Figure taken from Ref. [186].

Fig. 30. Y and Mercedes shaped 3-jet topologies used to investigate the properties of gluon jets. Figure taken from Ref. [52].

a con"guration the most energetic jet is very likely to originate from a quark. The second quark jet is tagged by "nding a displaced decay vertex of a bottom-#avoured heavy hadron (see Fig. 31 and Section 4.2.1). The remaining lower energetic jet is assumed to stem from a gluon. Biases due to the speci"c properties of b jets are avoided by statistical decomposition of gluon and quark jet contributions for which identical 3-jet con"gurations without tagging are considered. ALEPH and OPAL determined the gluon fragmentation function from the longitudinal and transverse fragmentation data by solving Eq. (92). The curves in Fig. 29 result from a purely phenomenological parameterization of the data [186}188]. Apart from some di!erences at very low x the agreement is good between the parameterized gluon fragmentation functions D (x) at the % Z scale. In every determination of a from scaling violation the problem of hadronization e!ects must 1 be considered which likewise a!ect the measured fragmentation functions and the event shapes.

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Fig. 31. Pictorial representation of the tagging of gluon jets by reconstruction of a displaced decay vertex from a bottom-#avoured heavy hadron.

On phenomenological and theoretical grounds such e!ects are expected to fall with reciprocal powers of the centre-of-mass energy (s (see Section 3.2.4), thus leading to power suppressed corrections. Approaches to account for hadronization e!ects involve either a shift or a rescaling of the scaled momentum fraction x by terms proportional to 1/(s as is discussed in Refs. [175,188]. The method to obtain a employs purely phenomenological parameterizations of the fragmenta1 tion functions for b, c, uds quarks, and gluons at (s"m based on the formula 8 D (x)"N x?D (1!x)@D exp(!c ln x) , (93) D D where except for c, which is assumed to be #avour independent by MLLA, the normalization N , and D the powers a and b depend on the #avour of the parton f. This function implies a strong D D correlation between the parameters N, a, b, and c such that no unique set of parameters may exist. The value of a (m ) can then be obtained from "ts to the fragmentation function for the inclusive 1 8 hadronic "nal state measured by many experiments over a wide range of centre-of-mass energies. The scale dependence is treated according to the evolution equation (90). Fig. 32 shows a compilation of some of the data available. The result of a "t is superimposed [188]. The scaling violation is immediately visible from the change of the slope with the energy scale, particularly at low values of x. The DELPHI and ALEPH collaborations performed such "ts, taking into account the variation of the #avour composition, and found consistent values [188}190] a (m )"0.126$0.007 (exp.) $0.006 (theory) (94) 1 8 DELPHI: a (m )"0.124>  (exp.)$0.009 (theory) (95) 1 8 \  average: a (m )"0.125$0.009 (96) 1 8 which are also in good agreement with the values obtained from completely inclusive quantities, and from jet rates and event shapes. Theoretical uncertainties, which dominate the total error, were estimated by the collaborations by varying both the renormalization scale k and the factorization 0 scale k in the range 0.5(s to 2(s. $ Until now the contributions of quarks and gluons have always been separated by the "t procedure. Exploiting the vast data statistics at LEP I and the high resolution vertex detectors of the experiments, jets can be classi"ed as to whether they originate from a quark or a gluon using the method described above. The DELPHI collaboration [191] tagged quark and gluon jets in their LEP I data. The scale relevant for the gluon jet under scrutiny was taken to be the maximum allowed ALEPH:

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Fig. 32. Inclusive fragmentation functions measured by several experiments at various centre-of-mass energies are shown vertically displaced. The errors on data points include statistical and systematic uncertainties. The full points were used in a "t which is shown as a curve. Figure taken from Ref. [188].

transverse momentum in a jet, that is h i"E sin  ,

 2

(97)

where h is the angle of that jet to the closest jet. The resulting fragmentation functions for quarks

 and gluons are shown versus the hardness scale i for "xed x intervals in Fig. 33. Parameterizing # the fragmentation functions at i"6.5 GeV for x between 0.15 and 0.90 using Eq. (93) and going # to other scales by means of the DGLAP evolution equations yields a good overall description of the data, except for very small values of x which were not considered in the parameterization of the # functions. Scaling violation is again clearly visible from the large di!erence between quark and gluon fragmentation functions regarding the i dependence and also from the slopes of the evolution. Since the DGLAP evolution equation (90) relates the logarithmic derivative of the fragmentation function to the colour factors via the Altarelli-Parisi splitting functions, the DELPHI collaboration

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233

Fig. 33. Scale dependence of (a) the quark and (b) the gluon fragmentation function. The functions are parameterized at i"5.5 GeV for x between 0.15 and 0.90 using Eq. (93). The evolution to di!erent scales as shown by the solid line is # done using the DGLAP evolution equations. Figures adapted from Ref. [191].

performed a measurement of the ratio C  "2.44$0.21 (stat.) C $ from a "t to the data in the range 8 GeV4i429 GeV. This ratio is in good agreement with the QCD expectation of 9/4. The "t also yielded a (m )"0.116>  to leading order consistent with 1 8 \  the higher-order results presented above within the errors. 5.1.3. Longitudinal and transverse cross-sections The sum rule in Eq. (87) yields a relation between the total, transverse, and longitudinal cross-sections. Recalling that the leading contributions to the longitudinal cross-section, p , are of * the order a , the coupling strength may be determined from a p measurement. The cross-section 1 * can be obtained experimentally from the integrals of the longitudinal fragmentation function



1  p (98) dx xF (x)& * , * 2 p   where p is the total hadronic cross-section de"ned in Eq. (50). Both the longitudinal and the  transverse fragmentation functions are shown in Fig. 34. It may be noticed from the "gure that F is signi"cantly below F which is due to the additional suppression of the former by a factor of * 2 a /p. From the measured fragmentation functions, the value of the above cross-section ratio was 1 determined by the OPAL and DELPHI collaborations to be [186,187] OPAL:

p * "0.057$0.005 , p 

(99)

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Fig. 34. Transverse and longitudinal fragmentation functions as measured by the (s"m . Figure taken from Ref. [190]. 8

OPAL

and

DELPHI

collaborations at

p * "0.051$0.007 , (100) p  p * "0.055$0.006 , average: (101) p  where a simple weighted average is formed from both individual values and their associated total errors which are dominated by systematic uncertainties. Using the perturbative prediction of the cross-section ratio [192] DELPHI:








a 11 s n s 37 a p * + 1 # 13.583! ln # D ln ! 1 (102) p 4 k 6 k 6 p p 0 0  for n "5 #avours one arrives at a value for the coupling strength at the Z mass of D a,*-(m )"0.128$0.011 (exp.)$0.009 (scale) , (103) 1 8 which is in agreement with other a determinations from cross-sections and also with those from jet 1 rates, event shapes and scaling violation. Corrections due to the production of heavy quarks at the Z scale were investigated in Ref. [193] and were found to be much smaller than the uncertainty due to the choice of the renormalization scale. It must be noted, however, that the longitudinal cross-section is known to be a!ected by signi"cant hadronization corrections, which were estimated under the assumptions of an infraredregular e!ective behaviour of a and of an ultraviolet dominance of higher-twist matrix elements 1 [74,194,195]. These theoretical considerations will be presented in more detail in Section 5.3. According to these estimates, the corrections, which are suppressed by reciprocal powers of the centre-of-mass energy (s, may be as large as d(p /p )"0.010$0.001, leading to a corrected *  value for the coupling of a,*->.-5(m )"0.118$0.014 , (104) 1 8 which is also in good agreement with the other a determinations mentioned before. 1 Besides the longitudinal and transverse cross-sections extracted from measurement of fragmentation functions in single hadron production, these cross-sections have also been investigated

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235

for the polar angle of the thrust axis, for which NLO calculations are available. The thrust axis is a good representation of the primary quark direction which is not accessible directly. Despite the same naming these cross-sections are di!erent from those measured from single hadron production. However, an expression completely analogous to that in Eq. (86) holds for h denoting the polar angle of the thrust axis instead of the hadron. Here, p , P"¹, ¸, refer to the corresponding . cross-sections, where the asymmetric term is absent since the de"nition of the thrust axis, Eq. (39), cannot distinguish the sense of the axis. The perturbative QCD prediction for the ratio of the longitudinal and the total hadronic cross-section is [196]









2 a a p 1 1#(l!2) 1 #O(a) , * "2C !3!8 ln (105) $ 1 3 2p 2p p  where l"0.7$0.2 governs the size of the next-to-leading term. and the di!erential The OPAL collaboration has measured both the cross-section ratio p /p *  longitudinal cross-section dp /d¹ for thrust [197]. Fig. 35 shows the di!erential cross-section with * the superimposed LO and NLO QCD expectation. A poor description of the distribution by the leading order prediction can be seen, indicating signi"cant higher-order corrections. The next-toleading order predictions were calculated in [197] using the program of Ref. [198]. Adding this contribution yields a much improved agreement although the statistical error on the calculation close to the kinematical boundary at ¹"1 is still large. The measurement of the cross-section ratio yielded, after correcting for detector and hadronization e!ects, p * "0.0127$0.0016 (stat.)$0.0013 (syst.) . p  This can be translated using Eq. (105) into a (m )"0.126$0.016 (stat.)$0.013 (syst.)$0.001 (theory) , (106) 1 8 where the theory error is due to the uncertainty of the l parameter. Despite the large error, this value of a (m ) is consistent with the results from other determinations. 1 8

Fig. 35. The di!erential longitudinal cross-section is shown for thrust measured at s"m and corrected for detector and 8 hadronization e!ects (parton level). The dashed curve is the leading-order prediction taking a (m )"0.119. The shaded 1 8 area represents the NLO prediction and its statistical error. Figure taken from Ref. [197].

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5.2.

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at small x: MLLA and multiplicities

Gluon radiation at large angle and high energy has a small probability. This can be seen from rewriting Eq. (35) to yield the di!erential spectrum of bremsstrahlung o! a quark (see e.g. Ref. [41])




k  dk dk a (k ) ,, duOO%"C 1 , 1# 1! $ 2p k k p ,

(107)

where p and k are the momenta of the quark and gluon, respectively, in the "nal state. The relevant scale for the coupling is set by k , the gluon transverse momentum with respect to the q direction. , When k &k&E, corresponding to gluon emission at large angle and at high energy, E"p for , massless partons, one rediscovers multijet topologies with u&a /p;1, which have been dis1 cussed in the previous section. The activity inside a jet is governed by quasi-collinear and soft partons which are characterized by k ;k;E. Their emission probability, according to the di!erential spectrum in Eq. (107), is , u&a ln E&1. In the formation of the parton shower, this `Double-Logarithmica (DL) qPqG 1 process has to be supplemented by the DL gluon radiation GPGG and the `Single-Logarithmica (SL) gluon splitting GPqq (see also Section 3.1.3). All these are not supposed to yield additional jets but to contribute to the jet initiated by the highly energetic parton. Furthermore, the contributions from such DL gluon radiation obey colour coherence which leads to a prominent e!ect: angular ordering. It means, in a classical picture, that for consecutive emissions of gluons the angle H between the emitter and the emitted gluon must decrease [41]. This e!ect can be explained by the transverse wavelength of the last emitted gluon which becomes too large to resolve the system formed by the emitter and the gluon emitted immediately before. Colour coherence and angular ordering entail a depletion of soft emission such that particles with intermediate energies are predominantly created in QCD cascades and, therefore, the particle spectrum has a hump-backed shape at small values of x. It follows that quasi-collinear and soft partons determine the details of the fragmentation function at small values of the scaled energy x. From the evolution equation (90) in this regime much can be learnt about the dynamics of low energy partons, in particular gluons. A widely used method to solve the evolution equation for D (x, Q) is that of the Mellin transformation, i.e. G considering moments with respect to x



DI (N, Q)" G





dx x,\D (x, Q) . G

(108)

In lowest order (leading-logarithmic approximation LLA) and using the explicit 1-loop expression for a the solution of Eq. (90) is of the form [12,199] 1



DI (N, Q)" DI (N, Q ) exp H  G HOO %






/ dq c (a (q), N) q HG 1 /



a (Q ) AHG @ , , " DI (N, Q ) 1  H  a (Q)  1 HOO%

(109)

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237

where c is the Mellin transform of a /2p ) P (z, a ), which are the Altarelli-Parisi splitting HG 1 HG 1 functions introduced in Eq. (91). This equation explicitly exhibits the scaling violation due to the scale dependence of the strong coupling. The power c is called anomalous dimension. HG Including the known next-to-leading corrections [43] Eq. (109) is supplemented by a coezcient function, C, and becomes





/ dq DI (N, Q)" C(a (Q), N)DI (N, Q ) exp c (a (q), N) , (110) G 1 H  q HG 1 / HOO % which, being of a similar form to Eq. (67), incorporates the exponentiation property of the elementary splitting processes. The coe$cient function C describes multijet contributions to the evolution of the system, given by large angle and high-energy parton emission. Small angle emission and, therefore, the evolution of a jet is determined by the anomalous dimension. Studying the fragmentation of quarks and gluons at small x involves the investigation of the "rst (N"1) moment of the anomalous dimension c and the coe$cient function C. This region is HG dominated by the DL gluon radiation processes for which c , evaluated to O(aL ), contains for %% 1 NP1 an infrared singularity of the form 1/(N!1)L\. Resumming the series to all orders of a one arrives at the expression [12,199] 1 1 a 1 , 3a 1, c (a , N)+ (N!1)#24 1 ! (N!1) P (111) %% 1 2p 4 4 p





which is in fact "nite at N"1. With Eqs. (110) and (111) the characteristics of the small-x regime of fragmentation, which concerns most of the produced particles, thus becomes accessible to perturbative predictions and their experimental testing. For instance, inserting the Taylor expansion of Eq. (111) at N!1 into Eq. (109) and performing the integration one "nds a Gaussian function of N. This yields, employing an inverse Mellin transformation, a Gaussian in the variable m ,ln(1/x ) for the small-x fragmentation function (for details see [12]) N N 1 (m !1m 2) . (112) xD(x, s)&exp ! N 2p N





Two omissions have to be recalled at this stage. First, the calculation of the anomalous dimension considered so far uses DL terms only, hence it is a double-logarithmic approximation (DLA) which also disregards energy conservation. It yielded c &(a . In the parton shower cascade, however, %% 1 the SL terms from GPqq contribute as well. The corrections due to these should be of the order of *c&a . One has to account for such next-to-leading terms, since they will cause essential 1 energy-dependent factors appearing in front of the exponential in Eq. (110). The approach of Refs. [41,64,200] adopted the shower picture, implying a dependence of the structure of the elementary parton decays (see Eq. (43)) on just the nearest forefathers of a considered parton. This yields an analytic prediction for the small-x fragmentation functions known as the modixed leading logarithmic approximation (MLLA) which obeys energy conservation. In Ref. [201] the next-to-leading corrections to the LLA prediction, which will be referred to as NLLA, were used to calculate higher moment corrections (skewness and kurtosis, see Ref. [199]) to the Gaussian in Eq. (112) of the small-x fragmentation function. Predictions of both approaches will be confronted with experimental data in the following section.

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Fig. 36. In standard hadronization models (left) the parton shower, being terminated at some virtuality scale cut-o! Q ,  is followed by a hadronization step. The MLLA plus LPHD concept assumes the hadronization phase to be represented on average by the parton cascade evolved down to Q close to the QCD scale K. Figure taken from Ref. [202]. 

Secondly, all the calculations quoted refer to partons and are, thus, not directly applicable to observables measured from the hadronic "nal state. Although phenomenological hadronization models have been shown in the previous section to be quite successful, a simple and fair assumption is that the hadronic spectra are directly related to and, in fact, are proportional to the partonic ones if the conversion from partons to hadrons occurs at a virtuality scale of the order of the hadron masses. This is depicted in Fig. 36. Thus, the formation of hadrons is independent of the primary hard process and its energy scale. This concept is known as the hypothesis of Local parton hadron duality (LPHD) [64] which originates from the preconxnement properties of QCD cascades [63]. It means, broadly speaking, that the long-range e!ects between partons in phase space are of secondary importance in the process of the blanching of coloured partons to form hadrons from them. Although intriguing, LPHD has a conceptual problem describing the production of massive hadrons such as baryons in the QCD jets [203]. It also assumes the spectrum of hadrons, whether they originate from decays of other hadrons or directly from the partons, to be reproduced by the calculated spectrum of the parton shower. Thus, "nding LPHD not to describe experimental data may open the view on the details of hadronization physics. 5.2.1. Small-x fragmentation function The spectra of the hadron's energy fraction x were measured by many experiments covering a vast energy range. In order to focus on the small-x region of the fragmentation function one usually considers the variable m ,ln(1/x ) where x is the momentum, rather than the energy N N N fraction of a particle. Fig. 37 shows (1/N)(dn/dm ), the di!erential distribution of the m variable for N N charged particles at centre-of-mass energies between 14 and 183 GeV [130,204}206]. Shown as curves are the NLLA and MLLA calculations and the dependence of the hump-backed shape of these spectra on the energy scale. The scale, which also controls the average multiplicity 1n (>)2 as will F be detailed below, is given by the variable >+ln(E /Q )"ln((s/2Q ), where the parton
   shower cut-o! Q equals K in the limiting spectrum. Both calculations, NLLA and MLLA,   consequently have only one free parameter, K , which is connected to the running of the strong  coupling constant. Since the leading and next-to-leading terms in the MLLA and NLLA calculations are scheme independent, the parameter K can be related to KMS only once the calculation has  been done in the MS renormalization scheme. In Fig. 37(a) K was obtained from a "t of the  distorted Gaussian to the data, whereas in Fig. 37(b), in order to be able to present the MLLA

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239

Fig. 37. Di!erential distribution of m ,ln(1/x ) for charged particles. In plot (a) Gaussian curves including skewness N N and kurtosis, which correspond to the NLLA calculation of Ref. [201] of the limiting shape, have been simultaneously "tted to the distributions, excluding the 91 GeV data. Plot (b) shows a comparison of the "tted distorted Gaussian (NLLA) together with the MLLA prediction of Ref. [41], which is a pure prediction with all parameters taken from a "t of the data at 91 GeV. The predictions of three MC hadronization models are also shown. Figures are taken from Refs. [130,206].

prediction of the hump-backed distribution, the respective value was taken from the "t of the m distribution at 91 GeV in Ref. [207]. N Although MLLA yields an analytic expression for the shape of the distribution it is not well-suited for numerical evaluation. In the limit of very high energies, however, the shape of the spectrum

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becomes insensitive to the cut-o! of very low energy gluon radiation. For this reason, one usually considers the limiting spectrum, i.e. Q "K , for which a more convenient, but still involved   expression was derived, for instance in Ref. [41]. At asymptotically high energies the distribution of partons is expected to be Gaussian in m (see Eq. (112) above). One can then represent the shape N even more conveniently by a distorted Gaussian [41,201]





1 1n (>)2 1 1 1 1 DM (m , >),x DM (x , >)+ F exp K! Sd! (2#K)d# Sd# Kd N N N 8 2 4 6 24 p(2p

(113)

with d"(m !1m 2)/p. The perturbative predictions for mean 1m 2, width p, skewness S, and N N N kurtosis K of this Gaussian for gluon initiated jets are mean: width: skewness: kurtosis:




1 o 1m 2" > 1# N 2 24





12 #O(>) , pb > 

p,1(m !1m 2)2 , N N 1(m !1m 2)2 N S, N , p 1(m !1m 2)2 N !3 , K, N p

(114)

where o is de"ned as o,11#2n /27 and b has already been given in Eq. (21). Some additional D  corrections need to be applied to account for jets initiated by quarks. These are known to leading order for all four parameters P of Eq. (114), and are of the order *P/P&O(0.1)(1#n /27)/> [201]. D Both perturbative calculations, MLLA and NLLA, yield good descriptions of the hump-backed shape of the small-x fragmentation function data over a large energy range. This becomes even more convincing when plotting the energy dependence of the peak position of the m spectra, m , as N  is done in Fig. 38 using the data from Refs. [87,88,129}131,204,206}214]. In the MLLA framework the energy scale dependence of the peak position is predicted to be






a (>) 1 a (>) m ,ln(1/x )"> #o 1 !o 1 #O(a) . 

 1 96p 2 96p

(115)

Fitting the single free parameter K to the data yields a very satisfactory description. Without  coherence in gluon radiation a decrease at very small x would be of purely kinematic origin due to the "nite particle masses, x&(s/m. Thus the dependence of the peak position on the energy scale would be expected to be m &>, which is twice the value including coherent radiation and not in  accordance with the data as can be seen in Fig. 38. Even though the e!ective K may not be related to KMS it is nevertheless instructive to "nd a (m)+0.118 assuming the crude but plausible 1 X approximation KMS &K . 

 Explicit expressions can be found, e.g. in Ref. [201].

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Fig. 38. Evolution of the peak position, m , of the m distribution with the centre-of-mass energy (s. The  N prediction [41] (solid) and the expectation without gluon coherence (dashed) were "tted to the data.

241

MLLA QCD

A further test of the MLLA calculation was conducted by the DELPHI collaboration [206] in order to examine the prediction m !1m 2+o/32+0.351. The m distributions shown in Fig. 37(a) were  N N re-"tted around the peak region considering 1m 2 as an additional free parameter since determinN ing it from the whole spectrum would ignore the fact that the distorted Gaussian is only valid close to the maximum. Averaging the results on m !1m 2 obtained from the data measured  N between 14 and 183 GeV, the DELPHI collaboration arrived at 0.35$0.14 which agrees with the MLLA expectation. The success of the MLLA plus LPHD approach was continued in an attempt to extend the perturbative predictions to the soft momentum domain [215}218]. Exploiting the representation of the analytic results for the spectrum D(m, >) in moments in m [219], an analysis over a large centre-of-mass energy range available in e>e\ annihilation was performed. The investigation of moments has a number of advantages over the analysis of the spectrum itself owing to simpli"ed theoretical calculations. In Refs. [215}218] the invariant hadronic density E(dn/dp) was studied in the limit of vanishing absolute hadron momentum p and compared with the prediction D (m, >) C dn % "2K $ , F C 4pN E(E!Q ) dp A  

(116)

where >"ln(E /Q ), E"(p#Q is the particle's energy and K is a normalization para
   F meter. The factor of 2 accounts for adding both hemispheres of an e>e\Pqq annihilation event. The function D (m, >) is the inclusive energy distribution of soft gluons originating from a primary % gluon for which an approximate solution of the MLLA evolution equation was derived in Refs. [216,217]. The appropriate leading order colour factor ratio C /C is introduced to translate this $  result to the case of a primary quark. Fig. 39 shows the momentum dependence of dn/dp for several centre-of-mass energies between 3 and 133 GeV in comparison with the extended MLLA prediction, Eq. (116), using K "0.45 and Q "270 MeV. Besides the overall good agreement F  a marginal excursion can be seen for the very high energy data at very small momenta of the order

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Fig. 39. Charged particle distribution dn/dp versus particle momentum p at several centre-of-mass energies [204,205,207,212,222]. The data are compared with the extended MLLA calculations which are detailed in the text. The "gure is taken from Ref. [217].

of 200 MeV which remains at even higher centre-of-mass energies (see Ref. [206]). In general, however, a remarkable agreement of the extended MLLA prediction [215}218] with the data can be observed and in particular the data in the "gure approach a common limit for pP0 as expected. Several further investigations of the analytical perturbative approach (APA,QCD#LPHD) have been conducted. An overview can be found in Refs. [220,221]. 5.2.2. Mean charged particle and mean jet multiplicities Perturbative calculations of the fragmentation function D(x, s) are primarily applicable in the neighbourhood of the maximum of the hump-backed m spectrum. Nevertheless, further quantities can be derived and confronted with experimental data. In the "rst place these are the Mellin moments of the fragmentation function in m or x



M(N, s)"



dx x,\D(x, s) .

(117)



For instance, the N"1 moment, being just the integral of the fragmentation function, corresponds to the mean parton multiplicity M(1, s),1N (s)2. Both the MLLA and the NLLA calculations arrive N

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at the same prediction for this multiplicity (see for instance Refs. [41,223]). In Refs. [224,225] the next-to-next-to-leading logarithmic approximation (NNLLA) and in Ref. [226] even the next-tonext-to-next-to-leading logarithmic approximation (3NLLA) was calculated for the mean parton multiplicity in jets initiated by a gluon. Rewriting the formula to explicitly show the dependence of the parton multiplicity on a one obtains 1 c a (Q) 1#6a 1 [1#O(a)] (118) 1N%(Q)2&a@ (Q) exp  1 N 1 p 4pb (a (Q)  1 where b is de"ned in Eq. (21) and  1 10 n D b" # 4 27 4pb  c"(96p








a +!0.502#0.0421n !0.00036n .  D D

(119)

5.2.2.1. Gluon to quark multiplicity ratio. Since the parton cascade in an e>e\ annihilation event is initiated by quarks rather than gluons, the above formulae have to be modi"ed to account for the di!erent colour factors of quarks and gluons which leads to a lower showering activity for quark jets compared with gluon jets. A namK ve estimate can be derived from the colour factors C and $ C since these determine the probability to radiate a gluon o! a quark or a gluon, respectively (see  Eqs. (13) and (14)). The ratio of multiplicities of gluon, 1N 2, and quark jets, 1N 2, is thus expected % $ to be asymptotically r"C /C "9/4 [227]. Including higher-order corrections and explicitly  $ exhibiting the dependence on a , this ratio becomes 1 C 1N 2 6a (Q) 6a (Q) % ,r(Q)"  1!r 1 1 !r #O(a) (120)   1 C 1N 2 p p $ $ where the coe$cients are (see [225] for exact expressions)




1 1 n , r " #  6 162 D



r +0.292#0.0457n !0.00041n .  D D



(121)

It should be noted that the above coe$cients were derived by means of generating functions and Taylor series in such a way that energy conservation is respected in three-parton vertices [224]. The respective values of r and r given in Refs. [228,229] are therefore slightly di!erent.   The multiplicity ratio was measured by the experiments at LEP I. The measurements applied the general technique of identifying the gluon jet by vertex tagging of the lower energetic bottom quark jet in very symmetric, Y-shaped 3-jet events and deriving multiplicities of pure gluon and quark jets by means of statistical decomposition using an identical sample of untagged symmetric 3-jet events

 There is a minor misprint in Eqs. (23) and (24) of Ref. [225]: the exponent of y should read !a /2B instead of a /2B.    The analytic expression for a can be found in Ref. [225]. 

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Fig. 40. (a) The scale dependence of the mean charged multiplicity in quark and gluon jets is shown. Part (b) presents the ratio of the multiplicities. The curves are results of "ts of the perturbative expression for the mean multiplicity allowing for a constant o!set. Figures adapted from Ref. [233].

from light quarks (cf. Section 5.1.2). All such investigations yielded a ratio r(m) of about 1.25 with X a total error of less than 3% [52,185,230,231]. Extending the study by including asymmetric 3-jet con"gurations, where one jet is less energetic than the others, the DELPHI collaboration [232] determined the energy scale dependence of the particle multiplicity in quark and gluon jets and of the ratio r(s). The relevant scale, i, of the jet was taken to be the transverse momentum of the jet with respect to the closest jet according to Eq. (97). Fig. 40(a) shows the average multiplicities found in quark and gluon jets, respectively, and Fig. 40(b) the ratio of these multiplicities. The data were "tted using the NLL approximation of the expressions in Eq. (118) and (120). In addition, a constant o!set was allowed which was determined to be approximately 2.6 for the quark and zero for the gluon multiplicities. The multiplicity ratio grows with an increasing scale i, seemingly approaching some asymptotic value, but still staying signi"cantly below the expectation of +1.7 from Eq. (120). The determinations of r discussed up to now are all based on the reconstruction of jets from the hadronic "nal state. The jets from highly energetic partons need not be well separated due to the collinear singularity. Thus the measurements might be biased by the use of a jet "nder. An investigation of the TOPAZ collaboration [234] avoids such biases using an approach which is based on the thrust observable rather than jets. They measured the multiplicity as a function of the thrust of an event and extrapolated to a thrust value of 2/3 which corresponds to the three-fold symmetric Mercedes topology of qq G events. The mean charged multiplicity of gluon jets follows by subtracting the qq contribution from the extrapolated multiplicity, 1N2 . For the subtraction 2 one can use qq events at the appropriate energy scale of (s"(s/3 (cf. Eq. (36)), that is the mass of the qq system recoiling against the gluon G. The ratio r can then be calculated by 1N2 !1N2  2 OO . r(s/3)" 1 1N2  OO 2

(122)

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Fig. 41. Topology of 3-jet events in which two quark jets are tagged by a secondary decay vertex in one hemisphere. The opposite hemisphere is considered as an inclusive gluon jet.

The TOPAZ collaboration calculated 1N2  at (s/3"57.8 GeV/(3 using the next-to-leadingOO logarithmic approximation of Eq. (118) which was "tted to mean charged multiplicity data measured from 12 up to 91 GeV. Inserting their extrapolated value for 1N2 , they "nally 2 obtained r((57.8 GeV)/3)"1.46>  [234]. This value is less than 1.7 which is expected from \  Eq. (120), but it agrees well with the scale dependence of the multiplicity ratio shown in Fig. 40(b). One has to be aware, however, that the calculation assumes a pair of either gluon or quark jets to emerge from a colour singlet point source whereas the gluon jets considered in the experimental investigations are radiated o! a quark and have to be reconstructed by employing some jet algorithm. In Ref. [235] a measurement was proposed yielding inclusive gluon jets that are similar to those used for analytic calculations. To this end, 3-jet events are selected, in which the identi"ed quark and antiquark jets appear in the same hemisphere. This situation is depicted in Fig. 41. All particles in the hemisphere opposite to the two tagged quark jets are inclusively considered as belonging to the gluon jet without application of a jet algorithm. Although gluon jets de"ned by this prescription still do not precisely match the jets considered in the theory calculations, they were found in Refs. [235,236] to be essentially identical to gluon jets in GG events from a colour singlet point source generated using a QCD Monte Carlo event generator. The OPAL collaboration performed a study in which inclusive gluon jets, after accounting for the di!erent mean jet energies, were compared with 2-jet events of light quarks [237,238]. Fig. 42 shows the multiplicity distributions for the inclusive gluon and light quark jets, respectively. The ratio of the mean multiplicities yields r(m /4)"1.471$0.024 (stat.)$0.043 (syst.) . (123) 8 This result is also shown in Fig. 40 to be consistent with the overall scale dependence of the multiplicity ratio. Although this value is still below the expectation from Eq. (120) the agreement is better. The r(s) ratio was calculated analytically in Ref. [239] by exact numerical integration of the complete MLLA evolution equations for parton multiplicities, derived from Eq. (90), with full account of energy conservation and the correct threshold behaviour, i.e. the mean parton multiplicity in a single jet approaches 1 for (sP0. In particular, this calculation refers to the multiplicity in the full hemisphere of a gluon jet emerging from a colour singlet state, and thus does not immediately apply to the symmetric >-shaped 3-jet events [239]. Choosing the parton shower cut-o! to be  Ref. [239] quotes the results for r at the scale m , but the energy scale of the experimental measurement is +m /4. 8 8 The Q value quoted in the reference is, therefore, scaled by one-half here. 

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Fig. 42. Corrected charged multiplicity distributions for (a) inclusive gluon and (b) light quark jets are shown with statistical (horizontal ticks) and total uncertainties. For comparison, the expectations from two phenomenological hadronization models, JETSET and HERWIG, implementing string and cluster fragmentation, respectively, are superimposed. Figures taken from Ref. [238].

Q +250 MeV, for which the mean particle multiplicities are well described as will be discussed  below, the numerical integration yielded r(m /4)+1.56. A similar calculation was done in the 8 context of the colour dipole cascade model in Ref. [240]. It employed the ARIADNE Monte Carlo generator [61] to "x the unknown parameters of the calculation, thus yielding r+1.5 consistent with the experimental measurement. In addition, the DELPHI collaboration [232] determined the colour factor ratio C /C from the  $ multiplicities in quark and gluon jets, using multiplicity data from e>e\ events at comparable scales obtained from previous measurements and also from events with hard photon radiation. Their result, C  "2.266$0.053 (stat.)$0.055 (syst.)$0.096 (theory) , C $ is, within the statistical, systematic and theoretical uncertainties, in good agreement with the QCD expectation of 9/4. It is also more precise than the corresponding determinations using 4-jet "nal states, that will be presented in Section 6.3.1.

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5.2.2.2. Mean charged particle multiplicity. From the combination of Eq. (118) with the multiplicity ratio for quark and gluon jets (120) and by relating parton and hadron multiplicities by a factor K according to the LPHD hypothesis, the mean multiplicity in e>e\Phadrons can be derived. F A low-energy constraint has to be considered in addition. In the limit of vanishing centre-of-mass energy the parton multiplicity should approach some non-zero value. This is most obvious for massive quarks for which the minimum parton multiplicity must be two since at a centre-of-mass energy of twice the quark mass there will be no phase space left for the quark}antiquark pair to radiate gluons [241]. The approximate expression for the mean hadron multiplicity, therefore, reads [239,241] 1N(s)2+2K F

1N%(s/4)2 N #c(s) , r(s/4)

(124)

where the factor 2 takes into account that both quark and antiquark will initiate a jet. It should be noted that for both the r ratio and the parton multiplicity the appropriate energy scale is s/4, i.e. that for a single jet. Moreover, at very high centre-of-mass energies the ratio, r(s), of gluon and quark multiplicities should approach the asymptotic value of C /C "9/4. The additive term c(s),  $ therefore, needs to vanish at asymptotic energies which could be achieved by a power suppressed behaviour like c(s)&const./1N%(s/4)2 or c(s)&const./(s. N ¹he upper plot of Fig. 43 shows the mean multiplicity of charged hadrons [53,87,88,129,130, 132,182,187,212,214,242}244] measured in e>e\ annihilation at centre-of-mass energies from about 10 up to 190 GeV. The predicted energy scale dependence of the mean multiplicity is superimposed on the data. The strong coupling constant is chosen to be a (m )"0.119 and the 1 8 two non-calculable parameters of Eq. (124) are adjusted appropriately. The description of the data by the NNLLA formula over this large energy range is satisfactory. The fact that the mean multiplicity exhibits a larger scaling violation than any other moment of the fragmentation function could render it one of the best quantities to determine KMS , the QCD parameter determining the scale dependence of the coupling a . It would, however, require 1 a calculation of Eq. (124) in the MS renormalization scheme [199,201]. 5.2.2.3. Moments of the multiplicity distribution. From a theoretical point of view, higher moments of the multiplicity distribution shown in Fig. 42, such as skewness and kurtosis, are more favourable since these scale with the mean multiplicity such that non-calculable factors like K cancel. One example of such moments is the second binomial moment R for which a NLLA F  prediction was obtained in Refs. [199,228]




5 20n 1N(N!1)2 11 R , " 1! ! D  2 1N2 8 891





a 1 #O(a ) . 1 6p

(125)

Some of the experiments also quoted values for R , but in most cases only the average multiplicity  1N2 and the ratio, 1N2/D of the mean to the dispersion D"(1N2!1N2, i.e. the width of the multiplicity distribution, are given. It is possible, however, to calculate R from these two numbers  according to R "1#(D/1N2)!1/1N2. This was done in Ref. [9] for the measurements at and  below the Z pole. In the lower part of Fig. 43 the data compiled in this reference, complemented by the values available or calculable from Refs. [87,88,129,130,132,212,244], are shown. It can be seen

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Fig. 43. In the upper plot, the mean multiplicity of charged particles measured in e>e\ annihiliation by various experiments is shown. The measurements include contributions from K and K decays. Overlaid is the NNLLA prediction 1 using a (m )"0.119. The lower plot shows the second binomial moment R of the multiplicity distribution. Superim1 8  posed as a solid line is the NLLA prediction for this moment. The dashed line shows the result when "tting for the unknown NNLLA coe$cient of the prediction. In both cases, the indicated errors comprise statistical and systematic errors added in quadrature. The data are compiled from Refs. [53,87,88,129,130,132,182,187,212,214,242}244].

from the "gure that the NLLA prediction using a (m )"0.119 overestimates the R moment. 1 8  Adding to Eq. (125) a term CNNLLA a /6p, a satisfactory description of the data with 1 a s/d.o.f."13.9/20 is obtained with the "tted value CNNLLA "!0.537$0.021, where the error is due to the experimental uncertainties only. The "tted NNLLA coe$cient is about 20% of the NLLA one in Eq. (125) which is approximately !2.38 for n "5. Calculated at the Z scale the value of D the NNLLA correction to Eq. (125) is only about 3.4;10\. The MLLA predictions for even higher moments of the multiplicity distribution show perfect agreement with the measurements. Exploiting the multiplicity distributions measured from the inclusive gluon hemisphere, as described above and shown in Fig. 42, the OPAL collaboration [238] determined the cumulant factorial moments K . In terms of the normalized factorial moments F of G G rank i, 1N(N!1)2(N!i#1)2 F, G 1N2G

(126)

these are de"ned as (i!1)! G\ K F . K ,F ! G G m!(i!1!m)! G\K K K

(127)

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Fig. 44. Cumulant factorial moments K of rank i"2}5 are shown as determined by the OPAL collaboration [238] G separately for (a) quark and (b) gluon jets. Overlaid are the results of NLLA [245] and MLLA [246] calculations.

Thus, R is identical with K . The K moments have several advantages over the F moments   G G because the former are more sensitive to the e!ects of higher-order QCD, largely independent of each other, and directly related to dispersion D, skewness S and kurtosis K (see [238]). Fig. 44 shows the measured cumulant factorial moments K of rank i"2}5 separately for quark and gluon jets. The G results of the NLLA [245] calculations and of the exact numerical solution of the MLLA DGLAP equation [246] are superimposed. In general, there is a remarkable qualitative and even quantitative agreement with the OPAL results for both gluon and quark jets, in particular for the exact numerical MLLA calculations. 5.2.2.4. Mean jet multiplicities. The MLLA calculations give a detailed insight into the low-x phenomena inside the jets, but they are also able to describe average hadron and jet multiplicities simultaneously over a wide range of centre-of-mass energies as is shown in Ref. [239]. In order to obtain predictions in the region of large values of the jet resolution parameter y (cf. Section 4.1.2),  i.e. for large angle 3-jet events, the lowest order contribution N of the evolution equation is O replaced by the explicit result for e>e\P3 partons in O(a ). The full contribution NU  follows 1 from numerical integration of Eq. (35) as in Eq. (38), but replacing the d function by the step function H(y !y ) which depends on y , the value of the jet resolution parameter at the #ip from    3 to 2 jets. In Ref. [239] this value is given for the DURHAM jet "nder in terms of the fractional energies of quark, antiquark and gluon by y "min(x  /x , x /x  ) (1!x ). The O(a ) corrected O 1  O % % O prediction is then applicable to both hadron multiplicities, choosing y ,(Q /Q), and jet multiA  plicities, i.e. y ,(Q /Q), and reads [239] A A > \ (128) N  (y )"2N (y )!2N(y )#NU (y ) ,  A O A O A A

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Fig. 45. The mean jet multiplicity at 91 GeV versus y ,y for the DURHAM jet "nder, and the mean charged particle A  multiplicity, the latter scaled by 3/2 to account for the unmeasured contribution from neutral particles, in the range of Q"1.6 to 91 GeV, versus y "(Q /Q) for Q +500 MeV are shown. Overlaid is the expectation from the exact A   solution of the evolution equation, using the same QCD parameter K . Figure taken from Ref. [239]. 

where N is obtained from the numerical solution of the evolution equation, Q is the usual cut-o! O  scale of the parton cascade adjusted to the typical mass of the hadrons produced, and Q "(sy . A  Fig. 45 shows mean jet multiplicities obtained with the DURHAM jet algorithm at (s"91 GeV. The jet multiplicity is used to "x the QCD parameter K which controls the scale dependence of the  coupling a . Moreover, the mean charged particle multiplicities are shown, as measured at 1 centre-of-mass energies from 1.6 to 91 GeV. These are scaled by 3/2 to account for the contributions from neutral particles. Using the value of K from the jet multiplicities, the values of the  parton shower cut-o! Q and the overall normalization factor K which relate parton and hadron  F multiplicities in Eq. (128) are adjusted to the data. A remarkably good description of the data is achieved for Q +500 MeV and K +1.  F 5.2.3. Summary of QCD at small-x Processes taking place at small values of the fractional energy x are, in general, not expected to be accessible to perturbative QCD calculations. This small-x regime should be governed by the physics of con"nement. It turns out, however, that perturbative QCD works down to scales as low as about 1 GeV. Moreover, when energy scales of a few hundred MeV are considered and the conjecture of local parton-hadron duality (LPHD) is invoked, perturbative calculations can even describe inclusive distributions of energy and multiplicity of hadrons originating from a hard process.

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Despite the support found for the hypothesis of LPHD adapted to the modi"ed leading-logarithmic approximation (MLLA) some questions still have to be addressed. Due to their larger colour charge and the gluon self-interaction, gluons play the dominant role in the development of the parton cascade, but the hadrons observed in the detector are built up of quarks and antiquarks. Although LPHD performs rather successfully it cannot explain the formation of hadrons from quarks and gluons emerging from the parton cascade. What the experimental veri"cation of LPHD can tell us is that hadronization and, hence, con"nement is likely to take place at very small energy scales as low as the mass of the lightest mesons despite the fact that many of these mesons arise from the decay of heavier hadrons. 5.3. Power corrections Precision measurements of the strong coupling constant a from hadronic event shapes require 1 a solid understanding of hadronization e!ects. During the past years the approach of trying to deduce as much information as possible about hadronization from perturbation theory has been intensively pursued [66}68,70,72}74,76,194,195,247,248]. When trying, however, to extend the standard perturbation expansion in powers of the strong coupling, one realizes that in high orders the coe$cients of the power series start to grow factorially and, hence, the series does not converge (see for instance Refs. [73,249,250]). At low scales, which one has to deal with in the context of hadronization, this is connected to the infrared renormalon divergence (see, e.g. Refs. [251,252]) that arises in the calculation of a propagating gluon with many successive insertions of virtual quark}antiquark loops as is depicted in Fig. 46. Thus, any perturbative treatment of hadronization e!ects has to somehow account for the renormalon divergence. By perturbative factorization it is possible to separate e!ects which are characterized by large energy scales from those of small scales up to inverse powers of the energy scale. This corresponds to the result, already obtained from the simple tube model described in Section 3.2.4, that hadronization e!ects induce corrections which are suppressed by reciprocal powers of the hard interaction energy scale Q,(s. In general, one expects for a perturbatively calculable (PT) observable, which is safe against collinear and infrared gluon radiation (see Section 3.1.2), deviations of the type [249] dpNP lnO Q & QN p

(129)

owing to non-perturbative (NP) e!ects from the physics of con"nement. What can be inferred from perturbation calculations are the powers p and q. In principle, one may regard hadronization as being governed by soft gluons as is suggested by the success of LPHD and MLLA. The contribution from these gluons is given by the size of the strong

Fig. 46. Graph related to an infrared renormalon. Figure taken from Ref. [247].

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Fig. 47. Strong coupling at low energy scales: The scale dependence of the perturbative expression for a which has a pole 1 at k "K is shown by the dashed line. The solid line represents a possible infrared-"nite behaviour of the strong , coupling at low scales. Figure adapted from Ref. [250].

coupling at a low energy scale, which is usually the transverse momentum k of the soft gluon with , respect to the radiating parton. At such a low energy scale the strong coupling is supposed to be large. From the explicit perturbative expression, Eq. (22), one may read o! that a pole is encountered when the energy scale approaches the QCD parameter K. It is known as the Landau pole [24]. The divergence of the coupling a close to k "K is shown by the dashed line in Fig. 47. 1 , On general grounds, the coupling is expected to be free of poles for k '0 if it is related to physical , observables. Several explicit models were proposed which modify the scale dependence of the coupling to cancel the Landau pole by the inclusion of non-perturbative contributions at low k [253}255]. , Instead of detailing these approaches in the following, a method yielding analyticity of the coupling will be brie#y introduced. It has been applied successfully to predict non-perturbative corrections to event shape observables (for further details see Ref. [71]). Its basic concept, in order to remove the Landau pole, is to introduce a gluon mass, which is not a physical mass, but is related to the virtual states of the gluon consisting of various parton con"gurations, like qq , GG, etc. These virtual states, appearing when renormalizing the gluon "eld, can be represented by a virtual mass m via the dispersion relation



a (Q)" 1

 Q dm a (m) , (m#Q)  

 A gluon mass would spoil the renormalizability of the theory.

(130)

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where a is an e!ective coupling constant whose de"nition is inspired by equivalent relations of an  Abelian theory like, for instance, QED. It should be regarded as a continuation of the physical perturbative coupling, a , into the non-perturbative domain. 1 Calculations based on the dispersion relation usually split the coupling constant into a perturbative (PT) and a non-perturbative part (NP), formally, a (k )"aPT(k )#aNP(k ) . (131) 1 , 1 , 1 , When substituting a (k ) according to this relation in the calculation of an observable, the PT part 1 , is connected to a perturbative expansion in powers of a (Q). The NP contribution is related to the 1 non-perturbative component of the e!ective coupling, da , which is to be calculated from  a using the dispersion relation Eq. (130). It is supposed to vanish in the regime of large  momentum transfer such that for some arbitrary "nite value k one has aNP(k )+0 for k 'k. 1 , , ' ' Hence, integrals of the type





 dk NP I' dk NP , a (k ) (k )N , a (k ) (k )N+ k 1 , , k 1 , ,  ,  , are convergent and will determine the 1/QN suppressed non-perturbative contribution to collinear and infrared safe observables. For instance, the p"1 correction obtained for the mean values of most event shapes is [71,249]

  

aF C  dm $ (m da (m) 1F2NP"  Q 2p m  aF 2C  $ " dk aNP(k ) , 1 , Q p  k aF 2C I' $ (132) " dk aNP(k )#O ' aNP , 1 , , Q 1 Q p  where aF is a calculable number depending on the observable. If aNP from Eq. (131) is used for 1 substitution, one can express the NP coupling by a and a.2, according to 1 1 I' I' I' (133) dk aNP(k )" dk a (k )! dk aPT(k ) . , 1 , , 1 , , 1 ,    The integral over a quanti"es the strength of the strong interaction in the region k (k. Its value 1 , ' is not perturbatively calculable at such low scales. One, therefore, introduces a non-perturbative quantity












p I' (k ), dk a (k )kN\ , (134) N\ ' , 1 , , kN '  which, in general, depends on the power p of the non-perturbative correction. An important property of the di!erence in Eq. (133) should be noted. Both, a and aPT have the same factorial 1 1 divergence in high orders which cancel in the di!erence such that the term on the left-hand side has, in fact, no renormalon embedded. a

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Table 9 Coe$cients of the perturbative prediction [32,115,198,256] and the coe$cients of the power correction of various event shapes [72,73]. In the case of the jet broadenings B and B the leading power correction is enhanced by additional 2 5 factors [249,257], see text. The coe$cient a  for the DURHAM 3-jet #ip parameter y is unknown, but the leading power W  correction is logarithmically enhanced [71] Observable F

AF

BF

aF

p

enhanced 1/QN

11!¹2 1C2 1M /s2 & 1B 2 2 1B 2 5 1y 2 

2.103 8.638 2.103 4.066 4.066 0.895

44.99 146.8 23.24 64.24 !9.53 12.68

2 3p 1 1 1/2

1 1 1 1 1 2

* * * O(1/(a ) 1 O(1/(a ) 1 ln Q

Performing the expansion of aPT in terms of a (Q), one obtains from Eq. (132) the prediction for 1 1 the power corrected mean of an event shape observable F




1F2"1F2PT#1F2NP" AF




a a  1 #(BF !2AF ) 1 #aF P , 2p 2p

(135)

where AF , BF and aF are observable-dependent constants. The term in square brackets is the general perturbative expression up to second order in a for the mean of an observable. The 1 coe$cients AF and BF can be derived from the O(a) perturbative calculations [32,115,198,256]. It 1 should be noted that the term !2AF accounts for the QCD corrections of the total hadronic cross-section as shown in Eq. (50) of Section 4.1.1. The aF coe$cient of the power correction P was determined in Refs. [72,73]. The numerical values of these coe$cients are shown in Table 9. With the exception of the DURHAM 3-jet #ip parameter y and the jet broadening measures,  which will be discussed below, the NP parameter P in Eq. (135) is given for the shapes as [74,249]












k K 4C M k ' a (k )! a (k )#2b a(k ) ln 0 #1# #2 P" $ 1 0  1 0 k 4pb p Q  ' '  where the factor K,C






5 67 p ! ! n  18 6 9 D

,

(136)

(137)

is due to the choice of the MS scheme. The expression in square brackets of Eq. (136) stems from the expansion of the aPT integral up to second order in a at some renormalization scale k . In addition, 1 1 0 there appears the Milan factor M. From a two-loop analysis of the 1/Q correction in Refs. [72,73], its value is determined as M+1#(2.437C !0.052n )/(4pb )+1.79, for n "3. Large  D  D  See: Note added at the end of this paper.  A value of 3 #avours is natural for the parton cascading considered here since the gluon splitting into pairs of heavy quark #avours is strongly suppressed.

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255

contributions from the next loop correction are expected such that a 20% uncertainty in the value cannot be excluded [249]. Universality of the power correction is an important issue since the concept of a "nite coupling at small scales, Eq. (134), should be independent of the type of event shape observable. The coe$cient aF contains the whole dependence on the observable F. In the above mentioned two-loop analysis the Milan factor is found to be universal for the event shapes considered in the following. Moreover, this analysis resolved the intrinsic ambiguity as to how the e!ects of "nite masses should be included in the de"nition of the event shape observables. The origin of this ambiguity is the inclusive treatment of gluon decays, where the actual contribution of "nal-state partons is replaced by that of a massive parent gluon. The analysis also takes into account that event shapes are not completely inclusive observables. In addition to the mean values of event shape observables, it was shown in Refs. [66,67,247] that di!erential distributions of the observables can also be described by power corrections using the same NP parameter aF P as in Eq. (135). In the region of small values of the observable F, but still large compared with the ratio of the QCD parameter K to Q, i.e. P
(F!DF ) ,

(138)

where the shift of the argument is DF ,aF P .

(139)

An extension for FP0 is possible but an additional phenomenological shape function has to be introduced [258]. The perturbative expression for the di!erential cross-section in Eq. (138) corresponds to the matched resummed NLLA and O(a) calculations, introduced in Section 4.1.2. 1 The non-perturbative prediction [73] for jet broadening had to be modi"ed because its comparison with experiment yielded results for the non-perturbative parameter a which did not  support the conjecture of universality when compared with those obtained for thrust and Cparameter [259]. In brief, a shift as in Eqs. (138) and (139) was found to be insu$cient to describe the data without an additional squeeze of the di!erential distribution. The squeeze is due to the interdependence of the perturbative and non-perturbative treatment. After being appropriately modi"ed in Refs. [249,257], the non-perturbative contributions to be added to the PT term 1B2PT for the mean values of the respective jet broadening observable are up to terms of O((a ) 1 p 3 2pb NP  #g # ! 1B 2 +a 2 P  2 3C 2(C a [1#Ka /(2p)] 4 $ $ 1 1 p 3 pb 1B 2NP+a 5 P # !  #g , (140) 5  2(2C a [1#Ka /(2p)] 4 3C$ $ 1 1





 

 Although in Refs. [73,120] the resummed perturbative expression was considered, the inclusion of the "xed-order calculations does not signi"cantly a!ect the shift. Therefore, for instance in Ref. [66], the ln R matched calculations were used. In fact, the power corrections are useful only in the regime where F is small, but still larger than P. It is this regime where the resummed calculations are more appropriate to represent the data.

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O. Biebel / Physics Reports 340 (2001) 165}289

where a is to be evaluated at the scale QM "Qe\, K is already given in Eq. (137), and 1 g +!0.6137 is a constant.  The shift that has to be applied to the argument of the perturbative expression for the di!erential broadening distributions is rather complicated and will, therefore, not be repeated here. It can be found in Ref. [257]. For most applications the following approximations of the shifts D can be used if the study is constrained to the region of B'0.05 and energy scales greater than about 35 GeV




 

1 #g  B 5 p 1 3 2pb  #g D (B )+a 2 P ln #g # # ! . (141) 2 2   B 4 3C 2(C a  [1#Ka  /(2p)] 2 $ $ 1 1 The more involved formula for the shift in the B case, although its "rst part is the same as that of 2 B , includes an additional contribution from the hemisphere where no signi"cant perturbative 5 gluon contribution is present. It is given by the term in braces and is identical with the correction to the mean value. In general, the ln(1/B) enhancement of the shift leads to the squeezing of the di!erential distribution. D (B )+a 5 5

5

P ln






5.3.1. Application of power corrections to mean values A huge collection of data on mean values of event shape observables covering a vast range of centre-of-mass energies is available [88,104,129}132,140,143,145,204,208,210}212,260}266], in particular for thrust and jet mass. Although the C-parameter has been known for a long time, measurements of this observable at energies below the Z pole became available only recently as is the case for the jet broadening and the DURHAM jet "nder which were proposed too late for the experiments prior to LEP [139]. All such measurements constitute a broad basis for scrutinizing the theoretical concept of power suppressed corrections to the mean values of event shape observables. Such investigations have been done rather extensively [74,131,136,139,145,212,257,267}270]. Considering the most prominent observables which were already used in the previous section, viz. thrust, heavy jet mass, C-parameter, total and wide jet broadening, and the 3P2-jet #ip parameter y obtained from the DURHAM jet "nder, one "nds a dependence on the centre-of-mass  energy as is shown in Fig. 48. The solid line shows the result of "ts with a (m ) and a as the only 1 8  free parameters. A very remarkable agreement of the theoretical prediction with the data is observed. From the di!erence between the dashed line which shows the perturbative contribution and the solid curve, one can infer the size of the power suppressed contribution. Since for the jet #ip parameter y the structure of the leading power corrections is only known to  include (ln Q)/Q and 1/Q terms [71], but the corresponding coe$cients are not yet calculated, several variations, 1/Q, (ln Q)/Q, 1/Q, (ln Q)/Q and omitting power correction terms, were tried in Refs. [259,268], introducing an unknown coe$cient a  as an additional "t parameter. In general, W none of these corrections is favoured by the s/d.o.f. of the "ts. When using 1/Q-type corrections the "ts yielded tiny values for a  of less than 10\ and values of a (m ) which exceed the world W 1 8 average by many standard deviations. Although "ts employing 1/Q-type corrections yielded a (m )+0.125, the coe$cients a  , which came out as about !0.2 to !0.5, have very large errors 1 8 W rendering them compatible with zero. In conclusion, there are still insu$cient data on 1y 2 to 

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257

Fig. 48. The energy dependence of the mean values of thrust 11!¹2, heavy jet mass 1M /s2, total 1B 2 and wide & 2 jet broadening 1B 2, C-parameter 1C2, and of the 3P2-jet #ip parameter 1y 2 is shown [88,104,129}132, 5  139,140,143,145,204,208,210}212,260}266,271]. The solid curve is the result of the "t using perturbative calculations plus power corrections, while the dashed line indicates the contribution from the perturbative prediction only.

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determine the details of the power correction for this observable. Recalling, however, that the expected leading correction is (ln Q)/Q, a very large coe$cient would be needed for a signi"cant power correction to 1y 2. The fact that the coe$cient is found to be small justi"es the neglect of  any power correction to the mean of the DURHAM 3P2-jet #ip parameter, yielding a very stable "t with s/d.o.f. of almost unity. The individual "t results for each of the six observables considered here are summarized in Table 10. It shows the dependence of the "t values on the choice of the renormalization scale, k , 0 varied between 0.5 and 2, and on the arbitrary matching scale, k , varied between 1 and 3 GeV, ' which enters the de"nition of the parameter a in Eq. (134) and which marks the boundary between  the non-perturbative and the perturbative regime. The dependence on unknown higher-order corrections of the Milan factor is also included. These are of the order a /p, where the coupling 1 strength is to be evaluated at some small scale resulting in an estimated uncertainty of about 20%

Table 10 Fitted values of (a) a (m ) and (b) a derived for k "2 GeV and x "1 using the O(a) calculations and two-loop power 1 8  ' I 1 corrections, including the Milan factor and the modi"ed predictions for the jet broadening variables [72,73,249,257]. Statistical and systematic uncertainties are also given. Signs indicate the direction in which a (m ) and a change with 1 8  respect to the standard analysis. The renormalization and infrared scale uncertainties are added asymmetrically to the errors of a (m ). a has error contributions from the choice of the renormalization scale and the Milan factor only 1 8  (a)

11!¹2

1M /s2 &

1B 2 2

1B 2 5

1C2

1y 2 

a (m ) 1 8 Q range (GeV) s/d.o.f. exp. x "0.5 I x "2.0 I M!20% M#20% k "1 GeV ' k "3 GeV ' Total error

0.1198 13}183

0.1141 14}183

0.1183 35}183

0.1190 35}183

0.1176 35}183

0.1215 22}183

52.2/39 $0.0013 !0.0049 #0.0061 #0.0011 !0.0011 #0.0025 !0.0019

22.0/33 $0.0010 !0.0026 #0.0037 #0.0013 !0.0001 #0.0013 !0.0011

22.1/25 $0.0016 !0.0038 #0.0048 #0.0008 !0.0007 #0.0017 !0.0014

18.8/26 $0.0020 #0.0017 #0.0003 #0.0005 !0.0005 #0.0011 !0.0009

18.8/16 $0.0013 !0.0043 #0.0053 #0.0009 !0.0009 #0.0020 !0.0016

13.6/13 $0.0014 !0.0040 #0.0054 * * * *

$0.0016 !0.0028 #0.0029 #0.0008 !0.0005 #0.0014 !0.0012

#0.0068 !0.0055

#0.0043 !0.0030

#0.0054 !0.0044

#0.0029 !0.0022

#0.0058 !0.0049

#0.0056 !0.0042

#0.0037 !0.0035

(b)

11!¹2

1M /s2 &

1B 2 2

1B 2 5

1C2

1y 2 

Average

a  exp. x "0.5 I x "2.0 I M!20% M#20%

0.509 $0.012 #0.003 !0.002 #0.058 !0.040

0.614 $0.018 #0.011 !0.005 #0.084 !0.064

0.442 $0.015 #0.020 !0.014 #0.046 !0.031

0.392 $0.028 #0.109 !0.042 #0.032 !0.022

0.451 $0.010 #0.005 !0.003 #0.050 !0.034

* * * * * *

0.473 $0.014 #0.018 !0.009 #0.053 !0.037

Total error

#0.059 !0.042

#0.087 !0.067

#0.052 !0.037

#0.117 !0.055

#0.051 !0.036

*

#0.058 !0.041

Average 0.1181

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259

for M, according to Refs. [73,249]. All "ts obtained s/d.o.f. close to unity. A large anticorrelation of over 90% is observed in all "ts. The individual results agree well within the correlated errors. To form a weighted average of the a (m ) results, taking into account the correlations, the procedure of 1 8 Ref. [104] which has already been detailed in Section 4.1.1 is applied again. The average a (m ) is in 1 8 very good agreement with other a determinations, quoted in this article, and also with the world 1 average a (m )"0.119$0.004 [272]. 1 8 Similarly, a fair agreement is found between the values of the non-perturbative parameter a ,  which is regarded as universal parameter in the theory calculations. Although the value from B is 5 a little low, and that from M /s somewhat high, all values agree within about 20%. In the table the & result of averaging these values is shown using the same procedure as for the strong coupling. The quoted error is determined from the combined experimental, renormalization scale, and Milan factor uncertainties only. It therefore does not consider the much larger spread of the individual a values. This could be accounted for by quadratically adding the r.m.s. of the "ve single  results, which is 0.076, to the quoted error. These results, in general, support the dispersion method to calculate power suppressed contributions to mean values of event shape observables. The conjecture of universality of the power correction, which was argued for in Refs. [66}68,70}73,195,247}249,257], is roughly con"rmed by the data. Moreover, there is also remarkable agreement of the measured values of a with the value  obtained from explicitly calculating the integral in Eq. (134) using an &analytically-improved' running coupling (see for instance Ref. [254]). The improvement concerns the cancellation of the Landau pole by adding to the explicit formula of a a &counter term' of the form K/(K!Q) 1 depending only on the QCD parameter K and the energy scale Q. No additional parameters are required. After adjusting K to reproduce the value of the strong coupling constant measured at the q mass scale, the integration in Eq. (134) yielded the prediction a (2 GeV)+0.46 [254] in good  agreement with the measured value. General properties of power suppressed corrections to mean values of event shapes were studied in Refs. [131,145,267], allowing also for di!erent integer and half-integer powers of the reciprocal energy scale Q. In particular, the power parameter p of a c/QN correction term was found in the case of the thrust variable to be p"0.92$0.19 from a simultaneous "t of p, a and the constant c that 1 determines the size of the non-perturbative correction [267]. No indication of higher-order power corrections was observed for thrust given the present precision of the data. In Ref. [131] the DELPHI collaboration investigated power corrections to di!erent observables shown in Fig. 49. Among them were the 3-jet rates, R , at "xed values of y "0.08 and 0.04 for the   JADE and DURHAM algorithms, respectively. Allowing for corrections of the order 1/Q and 1/Q satisfactory "ts were obtained for both quantities as shown in plot (a) of the "gure. While for the JADE case a signi"cant negative 1/Q correction is found, the DURHAM data yielded a 1/Q contribution consistent with zero within the errors. This agrees with the expectation mentioned above that the leading correction in the DURHAM case is of the order (ln Q)/Q. Observables are expected to lack signi"cant 1/Q corrections if the regions dominated by 2-jet events are excluded from the mean value. Such truncated mean values are shown in Fig. 49(b) for thrust (¹(0.8), heavy jet mass (M /s'0.1) and the energy}energy correlation EEC (see Ref. [140], & "cos t"(0.5). Fixing the value of a in order to obtain satisfactory "ts, the power correction to the 1 heavy jet mass was found to be predominantly of order 1/Q as predicted [70], while both thrust and EEC require considerable 1/Q corrections.

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Fig. 49. (a) Energy dependence of the 3-jet rates, R , from the JADE and the DURHAM jet "nders, and (b) of the truncated  mean values of thrust ((0.8), heavy jet mass ('0.1), and energy}energy correlation EEC, ("cos t"(0.5). Results of "ts of the perturbative calculations plus terms proportional to 1/Q and 1/Q, considering the coe$cients and a as parameters 1 of the "ts, are overlaid as solid curves. The bare perturbative contribution is represented by the dotted lines. Figure taken from Ref. [131].

Even higher moments of the event shape distributions have been invoked to determine the coupling strength. The non-perturbative contribution to the nth moment of an event shape distribution is found to be of the order 1/QL, which leads one to expect a signi"cant suppression of non-perturbative e!ects here. In fact, if one considers both the perturbative and non-perturbative contribution to the nth moment of an event shape, one "nds the leading power correction to be of the order a K/Q, i.e. suppressed by an additional factor of a (Q) only [249]. Considering 1 1 combinations of such moments, one may indeed construct observables in which the leading power corrections cancel. For instance such a cancellation was shown in Ref. [273] for the thrust variance p "1(1!¹)2!11!¹2+0.030a #0.037a . (142) \2 1 1 It has not yet been experimentally investigated. The measurement of the thrust variance will be di$cult because its size is tiny. Systematic uncertainties may, therefore, eventually render the measurement impossible if they are not well under control.

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261

The situation is more promising for the C-parameter whose variance should not have any leading power correction. The perturbative prediction is p "1C2!1C2+0.387a #0.0435a , (143) ! 1 1 which has a quite small second-order contribution. This prediction was derived from the results of a numerical integration of the full second-order matrix element for the moments of many event shape observables [274] using the program of Ref. [198]. Calculating the C-parameter variance from the high precision LEP I and SLC data [140,262,265] only and estimating the uncertainties using error propagation yields DELPHI: L3:

1C2!1C2"0.0331$0.0025 (stat.)$0.0106 (syst.) ,

1C2!1C2"0.0359$0.0016 (stat.)$0.0145 (syst.) ,

SLD:

1C2!1C2"0.0340$0.0046 (stat.)$0.0084 (syst.) ,

average: 1C2!1C2"0.0341$0.0013 (stat.)$0.0104 (syst.) , where the weighted average and its systematic uncertainty are calculated using the total errors for the weights while the statistical error is assumed to be uncorrelated. Using the weighted average and solving Eq. (143) for a (m ) one obtains 1 8 a (m )"0.087$0.003 (stat.)$0.026 (syst.)$0.007 (scale.) , 1 8 where the uncertainty due to the choice of the scale is estimated by changing x from unity to 0.5 I and 2. Although the total error is large, the central value is rather small compared with other determinations possibly indicating larger contributions from missing higher-order terms in a or 1 from power corrections. 5.3.2. Power corrections to diwerential distributions The investigation of power corrections to di!erential distributions of event shape observables has only just begun. A few such studies have been done up to now [136,257,259,267] which are based on the general concept presented in Ref. [66], extended to next-to-leading order accuracy and other event shape observables in Refs. [72,73]. The next-to-leading order treatment of the power corrections changed the shape dependent coe$cients aF and, "nally, yielded the values shown in Table 9. Fig. 50 shows results of the application of the power corrections to di!erential distributions of the thrust and the C-parameter observables. A good description of the distributions is found by applying the predicted shift to the matched "xed order and resummed calculation. Only at very small centre-of-mass energies is the agreement between data and expectation moderate. Since neither the perturbative nor the non-perturbative calculations applied in the "t accounts for "nite quark masses, one may expect mass e!ects to modify the distributions in the proximity of the bottom quark production threshold. The numerical values of a (m ) and a are summarized in Table 11. It has to be recalled that the 1 8  "t parameters a (m ) and a are strongly anti-correlated (60}84%), as is illustrated in Fig. 51. The 1 8  results shown were obtained by the ALEPH collaboration considering only data from the ALEPH experiment. Therefore, a detailed treatment of the correlation between the data at di!erent energies

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Fig. 50. (a) Thrust and (b) C-parameter distributions, measured at di!erent centre-of-mass energies (s,Q and corrected for detector e!ects, are shown vertically displaced. Overlaid as a solid curve is the result of a "t of each of the observables simultaneously for all energies using matched O(a) and resummed NLLA calculations complemented by 1 power corrections. The dashed line in (a) shows the result of a corresponding "t applying corrections from the JETSET Monte Carlo. The dotted line in (b) is the extrapolated "t result. Figures taken from Refs. [259,267].

was possible. The "gure shows the results obtained from both the di!erential thrust and Cparameter distributions. The impact of each systematic variation is indicated. Combining the values of Refs. [136,259] yields a (m )"0.1150$0.0036 , 1 8 a (2 GeV)"0.464$0.067 . (144)  These values are in good agreement with the world average of a "0.119$0.004 [272] and with 1 the average values obtained from the "ts to the mean values of the event shapes, respectively. Table 11 also lists the results obtained from similar "ts to the di!erential heavy jet mass and total jet broadening distributions, respectively. Further results on the broadening observables are available [136] but not listed because they were obtained using the older prediction for the power correction which neglected the interplay between perturbative and non-perturbative contributions (see Ref. [249]). Fig. 52 shows the result of the "t to the total jet broadening based on the new prediction at two distinct centre-of-mass energies. Numerical results of this "t are also listed in Table 11. Considering the results of the "ts to the di!erential distributions of single event shape observables, shown in Table 11, one "nds the overall concept of power corrections to be con"rmed. The results on a are all compatible with the world average, even though some of them tend to be small. 1 For the non-perturbative parameter a it has to be noted that the values agree very well with the 

O. Biebel / Physics Reports 340 (2001) 165}289

263

Table 11 Fitted values of a (m ) and a for x "1 using the ln R-matched O(a) and resummed NLLA calculations completed with 1 8  I 1 the 2-loop power corrections, including the Milan factor and the modi"cation of the prediction for the jet broadening variables [72,73,249,257] to describe the di!erential distributions. The errors correspond to the total uncertainties except for the values marked with an asterisk (H) where only the "t uncertainty is given in the references (1/p)(dp/d(1!¹)) Ref.

(s range

a (m ) 1 8

a (2 GeV) 

s/d.o.f.

[136]

91}183

0.1185

$0.0064

0.449

$0.082

140/42

[259]

35}183

0.1156

>  \ 

0.469

>  \ 

284/277

(1/p)(dp/dC) Ref.

(s range

a (m ) 1 8

a (2 GeV) 

s/d.o.f.

[136]

91}183

0.1145

$0.0043

0.443

$0.060

30/36

[259]

35}183

0.1137

>  \ 

0.437

>  \ 

163/170

(1/p)(dp/dM ) & Ref.

(s range

a (m ) 1 8

[136]

91}183

0.1157

a (2 GeV)  $0.0039

0.437

s/d.o.f. $0.003H

2.1

(1/p)(dp/dB ) 2 Ref.

(s range

a (m ) 1 8

a (2 GeV) 

s/d.o.f.

[257]

35}183

0.1140

$0.0007H

0.514

$0.007H

61/57

[259]

35}183

0.1125

>  \ 

0.562

>  \ 

161/171

results obtained from "ts to the mean values of event shape observables in the previous section. This gives further support for the conjecture of a being a universal non-perturbative  parameter. 5.3.3. Summary of power corrections In general, a remarkable ability of power corrections to describe non-perturbative hadronization e!ects is found. This holds for the mean values as well as for the di!erential distributions. Moreover, the power corrections are also applicable to other processes besides e>e\ annihilation. In the analysis of deep inelastic scattering of positrons o! protons at the HERA collider at DESY, for instance, power corrections were also found to be a good description of non-perturbative

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Fig. 51. The correlation contour of a (m ) and a obtained from combined "ts to thrust and C-parameter distributions is 1 8  shown at the 68% con"dence level. For its calculation statistical and systematic errors were considered. The points indicate the individual results from systematic variations of the analysis. Figure taken from Ref. [136].

Fig. 52. Total jet broadening distributions at 35 and 91 GeV are shown. The curves are from "ts of the ln R-matched perturbative calculations and the modi"ed prediction for the power correction to the di!erential B distribution. The 2 data sets considered for the "t ranged from 35 to 183 GeV in centre-of-mass energy. Figures taken from Ref. [257].

contributions to the mean values of event shapes (see summary in Refs. [83,275,276]), yielding results consistent with those presented in the previous sections (see Fig. 53). Although the success of power corrections might be surprising, their ability to describe nonperturbative e!ects could have been expected from the successes of the hypothesis of local parton}hadron duality (LPHD). In this respect, it is another indication that the strong interaction physics at very low energy scales is governed by soft gluons. As for the LPHD approach, power corrections cannot resolve the process of hadronization, but it increases the con"dence that one

O. Biebel / Physics Reports 340 (2001) 165}289

265

Fig. 53. Compilation of various results on a (m ) shown versus the corresponding a (2 GeV). The values have been 1 8  obtained from e>e\ annihilation using di!erent event shape observables and also from deep inelastic scattering of positrons o! protons [136,139,259,277].

may consider quarks and gluons for a perturbative treatment even at very small energy scales, close to the stage of con"nement [249,275]. 6. Studies related to the running of aS The results presented in the previous sections clearly show that the coupling strength of the strong interaction is dependent on the energy scale. Although the precision of the tests has improved considerably owing to improved and extended calculations and owing to new experiments at high energy colliders, details of the running are still unclear. Some aspects of possible deviations from and extensions to the standard S;(3) QCD structure will be brie#y addressed in this section. As a starting point, however, the exclusive validity of QCD will be assumed, thus allowing conclusions to be drawn concerning unknown higher-order corrections. 6.1. Higher-order corrections from energy dependence Higher-order corrections are a major source contributing to the overall uncertainty of every determination of the strong coupling constant a , in particular from quantities that are not 1 completely inclusive as was shown in Section 4. The e!ort of calculating the next-to-next-to-leading order corrections (NNLO) to such observables like jet rates and event shapes has just started (see Refs. [35,37]) and will very likely require many years until completion. And, in view of the renormalons as being connected with the factorial divergence of the coe$cients of a high-order Taylor expansion as discussed in Section 5.3, one might even argue that the next order of the series will not yield improved predictions.

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Other approaches might achieve useful results earlier, although they may be a!ected by intrinsic uncertainties. One such approach is related to the Pade& approximants [¸/M] (see Ref. [278] for an introduction) which expresses a Taylor series S(x) up to x, by the ratio of an ¸th and an Mth order polynomial. These two polynomials are chosen such that ¸#M"N and that S(x)"[¸/M] modulo terms of the order x,>. In detail this means that S(x)"c #c x#c x#2#c x,#O(x,>)    , is expressed by

(145)

a #a x#2#a x*  * #O(x,>) [¸/M],  1#b x#2#b x+  + "c #c x#c x#2#c x,#c x,>#2 . (146)    , ,> The PadeH approximant is formally a valid representation of S(x) at a given "nite order in x. Moreover, the PadeH approximants possess useful properties which are absent in the simple Taylor series of S(x). For instance, the coe$cient c of the Taylor expansion of [¸/M], which is usually ,> called Pade& Approximant Prediction (PAP), can provide a good estimate of the c coe$cient of the ,> Taylor series of S(x). The relative deviation of the estimate from c decreases exponentially with ,> N [278]. It seems obvious to apply the PAP to second-order perturbation series which are available for most observables. In Refs. [278,279] this was done for the 4-loop coe$cient, b , of the b-function in  Eq. (21), among other quantities. Once the exact result became available, it was realized that the quartic gluon vertices, which start to contribute at this order, were not predicted by simple PAP. A modi"cation was required to retrieve the exact b coe$cient with better than 1% accuracy. This  indicates a basic de"ciency of the approach due to its purely mathematical nature. Anyhow, it should be recalled, that no further gluon vertices are expected to contribute beyond that order, and, hence, PAP might indeed yield satisfactory results. Pade& approximant predictions were also employed for the determination of a . In Ref. [280] the 1 hadronic branching ratio of the q lepton, R , was investigated (see Section 4.1.1), to attempt to O estimate the next term of the slow converging perturbative series in Eq. (59) from PAP. The inclusion of the PadeH approximant term lowered the value of a(m) by about 10% [280]. 1 O In Ref. [281] the a determination from hadronic event shape observables was studied using the 1 Pade& approximant predictions. In total 15 observables were considered to determine the O(a) term 1 from the exact O(a) prediction. The extended series was then "tted to SLD data to determine a (m ), 1 8 1 "nding for a "xed renormalization scale factor x ,1 a reduced scatter of the individual results I about their common mean value. A similar study by the DELPHI collaboration [146] using 17 observables yielded a (m )"0.1168$0.0054 where the error includes statistical and systematic 1 8 uncertainties as well as uncertainties due to the scale choice. Although these results look promising, they can only be veri"ed upon completion of the full O(a) perturbative QCD calculation. 1 A completely distinct approach exploits the wealth of data available at very di!erent centre-ofmass energies [270]. It makes use of the fact that the basic structure of the perturbative expression for an event shape observable like thrust, appropriately normalized, is known to be






a a j a a R(Q)" 1 #r 1 #r 1 #2# 1#j 1 #2 ,  p  p p Q p

(147)

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which includes simpli"ed power corrections that have already been introduced in Section 5.3. Using the large range of energies available, one can avoid the renormalization scale uncertainty entirely by taking the derivative of R(Q) with respect to ln Q, which can be written as [270] dR "!2pR(b #pb R#b o R#2)#K R\@ @ e\p@ 0 ) (1#K R#2)       d ln Q ,!2pb o(R) 

(148)

where pb pb  !  r !r . o "r #    b  b   Here, b , b and b are the coe$cients from the renormalization group equation (21), o depends     on the NNLO coe$cient r , which is unknown, and K , in combination with KMS , is directly   connected with j. One may "t Eq. (148) to the observables R(Q) and dR/d ln Q, both of which can be determined from the data, in order to determine the unknowns, r and K . In fact, in Ref. [270] Eq. (148) was   integrated exploiting asymptotic freedom, that is R(Q)P0 as QPR, leaving, in principle, KMS as the constant of integration. Fig. 54 shows the results of three (o , j, KMS ) and one parameter (KMS )  "ts to the mean value of the thrust observable over a large range of centre-of-mass energies from 14 to 172 GeV. When "tting with o and j "xed to zero, KMS "266 MeV with s/d.o.f."82/32 was  found in Ref. [270] for 5 #avours. For the three parameter "t, however, the minimum was reached with s/d.o.f."40/30 for KMS "245> MeV, o "!16G11, and j"0.27>  GeV. This \  \  corresponds to a (m )"0.1194$0.0014 when using the 2-loop relation between KMS and a . The 1 1 8 value of o yields r "89$11 which comes out rather large compared with r +9.7, due to    the contribution r #cr in Eq. (148). Moreover, the value of j is rather small compared with the   results obtained in Section 5.3, which would suggest a value of about 1 GeV. This di!erence is due to the third-order contribution, o , which is determined from the "t, and the neglect of higher-order  terms when transforming K into a value of j [285]. Since o absorbs a signi"cant fraction of the   power corrections, there is a strong anticorrelation between o and j, which is the reason for the  large relative "t errors for o and j.  In general, exploiting the known structure of the unknown next order of a perturbative series together with the available data over a vast energy region is a promising approach to constrain the value of the higher-order coe$cients until the completion of the exact calculation. The approach should be applicable also to other observables, and maybe even to di!erential distributions. This method's very appealing feature is that the large uncertainties due to the choice of the renormalization scale, which are usually attributed to unknown higher-order corrections, no longer appear. In fact, a part of these higher-order corrections is calculable and resummable to all orders

 The value of r "(B !2A )/(2A )+9.7 can be determined from Table 9.  2 2 2

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Fig. 54. The energy dependence of the data on 11!¹2 [129}132,139,143,145,204,208,210,212,260,262}264,266,282}284] is shown. Overlaid are the "t results for a three parameter "t (see text) using KMS , o , and j as parameters (solid line), and for  a one parameter "t with o "0 and j"0 "xed (dashed). Figure taken from Ref. [270]. 

[286]. Once the resummation is done, this approach may lead to an improved precision for a determinations. 1 6.2. Power corrections to the running The data on a and the respective uncertainties shown in Figs. 15 and 22 in Sections 4.1.1 and 1 4.1.2, respectively, agree reasonably well with the expectation of QCD. Deviations from this expectation, however, cannot yet be ruled out. In fact, the Landau pole [24], appearing at low scales, induces 1/Q corrections at large Q [287]. Such corrections are also found from the investigation of the ultraviolet (UV) renormalons, which dominate the perturbation expansions in a (Q) at large orders n, because of factorially divergent coe$cients with alternating signs, in 1 contrast to the infrared (IR) renormalon discussed in Section 5.3, where the coe$cients have the same sign. The emergence of 1/Q contributions can easily be seen when one tries to remove the Landau pole of the coupling, as has been done by various approaches, to render the coupling "nite at low scales. For instance, in Ref. [254], a scale-dependent term is added to the explicit representation of a in order to cancel the pole, viz. 1 1 K a (Q)" # . (149) 1 b ln(Q/K) b (K!Q)  

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This &analytically-improved' coupling a no longer has a pole at Q"K and remains "nite for all 1 values of Q. Admittedly, it introduces a 1/Q contribution at large scales, Qe\ annihilation [71,253]. To overcome this potential problem, one could further extend the expression in Eq. (149) by adding K/(b Q), which  would remove the 1/Q contribution at large Q at the expense of bringing in a new pole, which, however, is now at the limit of the physical regime, i.e. at Q"0. Moreover, with this extension the improved coupling di!ers from the standard formula only by 1/Q terms. With the large quantity of available a determinations over a vast energy range in hand, one 1 could test the possibility of 1/Q and 1/Q contributions. Fig. 55 presents the data shown in Figs. 15 and 22. The 2-loop part of the explicit expression for a (Eq. (22)) is complemented by adding either 1 c /Q or c /Q and then, for illustrative purposes, "tting to the a data using experimental errors   1 only. Both "ts yield a value of a (m )"0.122 with a s/d.o.f."25/22. The values of the coe$cient 1 8 for the additional term are c "!0.66$0.60 and c "!0.18$0.19, respectively. Even though   the sign of the c parameters is just what is expected for the improved coupling, no indication of such contributions is seen from these data. The same observation has also been made for 1/QN corrections for other integer powers in the range of 0}7. One has to be aware, however, that the a values used in the "t were determined assuming 1 neither 1/Q nor 1/Q contributions. Thus, any in#uence of these could already have been absorbed into the value of a . A new determination of each a value, allowing for 1/Q or 1/Q corrections, 1 1 would be required to "nd the full contribution from such terms. Moreover, the a value determined 1 from the hadronic q lepton decays is of particular importance for the values of the c and  c parameters. If this a value comes out a little low (high) compared with the determinations at  1 high energies, the two parameters will have negative (positive) signs. Recalling the importance of

Fig. 55. Combined results on a from Figs. 15 and 22 are shown. The curves show the results of "ts of the explicit 2-loop 1 expression for a supplemented by either a 1/Q (solid) or a 1/Q (dashed) term. Only experimental errors are shown and 1 were considered in the "t.

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non-perturbative corrections in the determination of a from the q (see Section 4.1.1), no con1 clusions should be drawn about 1/Q or 1/Q corrections from the sign of the c , c parameters. In   fact, the "t is done for illustrative purposes only, indicating that, although the data look rather imprecise, there is no freedom to modify the running of the coupling by additional terms like 1/QN for any integer power p50. 6.3. Additional coloured objects In addition to gluons and quarks which are known to be coloured particles in the standard QCD theory, other coloured particles which participate in the strong interaction may exist. Two examples of candidate particles are: the gluino [288], which will be discussed in the following, and coloured Higgs particles [289]. Even though these particles might be heavy, they could contribute to loop corrections and, hence, a!ect the energy scale dependence of the strong coupling constant. The contribution of such particles would, in general, alter the number of active #avours, n , and D also the coe$cients of the b-function in Eq. (21). They might also introduce new and, therefore, anomalous strong couplings. 6.3.1. Light gluinos A particle whose existence is conjectured is the gluino, the supersymmetric spin- partner of the  gluon (for an introduction to supersymmetry (SUSY) see Ref. [28] and references therein and for the gluino see Ref. [288]). If it is su$ciently light, its appearance will increase the number of active #avours in lowest non-trivial order by up to 3 [270,290]. A high gluino mass will suppress its contribution to the loops by a kinematic factor such that the number of active #avours is raised by an amount less than 3, and which is not constrained to be an integer. In general, all cross-sections for strong interaction processes depend on the QCD colour factors C , C , and n ¹ . It is possible to infer the number of active #avours n from the measured values  $ D $ D of such cross-sections. For this purpose the dependence of the perturbative expansion on n has to D be made explicit for each cross-section. In Ref. [290] such an investigation was performed using, in addition to the 4-jet events which will be discussed below, the energy dependence of the 3-jet rate and the thrust distribution, and the R ratios. The result is n "6.3$1.1, where the error is D dominated by systematic uncertainties. With the bulk of LEP I data being available now, a more signi"cant study has become possible using 4-jet events. The number of active #avours appears in the determination of the ratios C /C  $ and ¹ /C from these events using 4-jet angular distributions which are sensitive to the colour $ $ factors [291]. Various angles were proposed [292] motivated by the notion of an intermediate gluon splitting into a pair of partons. Due to the spin structure at this vertex a quark}antiquark pair obeys a di!erent distribution than a gluon}gluon pair. Moreover, the coupling di!ers due to the colour factors involved (see Eqs. (12) and (14) of Section 2.1). This is the origin of the colour factor sensitivity of angular distributions when associating the 4 jets with the 2 partons from the splitting of the intermediate gluon plus quark and antiquark from the primary hard process. An overview of measurements of the QCD colour factors at LEP can be found in Ref. [293]. Fig. 56 shows a contour plot of the colour factor ratios obtained from angular distributions using LEP I data. Despite the large statistics available, most of the measurements were statistically limited. The combination of these results with those from jet rates and event shape observables, whose

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Fig. 56. Colour factor measurements based on 4-jet event angular distributions. The open plus sign shows the increased value of ¹ /C "3/5 due to the contribution of a massless gluino, without which it is ¹ /C "3/8. Figure taken from $ $ $ $ Ref. [293].

second order coe$cient of the perturbative prediction depends on the two colour factor ratios, might improve the precision. The ALEPH collaboration performed a simultaneous determination of a (m ), C /C , and n ¹ /C from the di!erential DURHAM 2-jet rate and the 4-jet angular 1 8  $ D $ $ distributions [294]. Accounting for mass e!ects, the colour factor ratios were determined to be C  "2.20$0.09 (stat.) $0.13 (syst.) , C $ ¹ $ "0.29$0.05 (stat.) $0.06 (syst.) , C $ in agreement with the expectations of C /C "9/4 and ¹ /C "3/8 for the S;(3) group structure  $ $ $ of QCD (see Section 2.1). One may now assume these to be the correct colour factors to determine a (m )"0.1162$0.0042 and the number of active #avours n "4.2$1.2, where the errors are 1 8 D statistical and systematic uncertainties added in quadrature [294]. The upper limit on an excess of the number of active #avours is thus *n (1.9 at a con"dence level of 95%. D Besides these measurements further investigations were done studying the running of a from the 1 R ratio and the hadronic cross-section at di!erent scales [295]. None of these found evidence for contributions from a light gluino (see also Ref. [7]). 6.3.2. Anomalous coupling New physics beyond the standard model may contribute to the strong interaction if the new particles involved are carriers of colour charge. Deviations of experimentally measured

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cross-sections from the QCD predictions are to be expected. These may, however, be small if the new particles are heavy. Without assuming a particular model for the extension of the standard model, the standard model Lagrangian was supplemented in Ref. [296] by non-standard operators of (energy) dimension six. Besides new couplings for left-handed quarks, these operators give rise to new qqG and qqGG vertices involving right-handed quarks and the Higgs "eld doublet. Recalling the experimental evidence for the #avour independence of the strong coupling presented in Section 4.2.2, the new couplings are assumed to be universal with respect to the quark #avour. The strength of the new anomalous couplings is expressed by a constant A and by a scale K which characterizes O%(  the new physics. Fig. 57 shows the net e!ect of the new anomalous coupling on some event shape distributions compared with the standard QCD expectation. A prominent di!erence in the shape of the distributions due to the anomalous couplings can be noted. By "tting the anomalous plus standard distributions to the data with KQCD and the renormalization scale "xed, the relative contributions of the anomalous parts were obtained. From these, bounds on the strength of the anomalous coupling and its characteristic scale were derived in Ref. [296] for each of the four shape distributions, yielding results of the order A /K (14 O%(  through 16 TeV\ at the 95% con"dence limit. If the unknown coupling strength A is chosen to O%(

Fig. 57. (a) Relative production of 3-jet events and di!erential distributions of (b) thrust, (c) spherocity, and (d) C-parameter are shown. The solid line corresponds to the standard QCD prediction. The contribution from the anomalous couplings described in the text is shown by the dashed line. The spherocity is de"ned as S"(4/p) min o ( "p ;n"/ "p ") [297]. Figures taken from Ref. [296]. L G G G G

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be of the order of unity, the scale of new physics associated with the anomalous coupling is above 270 GeV at least, and hence beyond the direct reach of LEP II. A comparable study considering the gluon energy was done by the SLD collaboration [298]. The gluons were identi"ed by tagging both bottom quark jets in a 3-jet con"guration by the reconstruction of a displaced decay vertex. The distribution of the gluon energy is potentially sensitive to an anomalous chromomagnetic moment of the bottom quark, but no evidence was found from the comparison of the distribution with perturbative QCD predictions [298]. Nevertheless, both results demonstrate the sensitivity of the qqG vertex to anomalous couplings due to new physics e!ects. 6.4. a determinations from other hard processes 1 The value of the strong coupling constant a can also be determined from hard processes other 1 than e>e\ annihilation. It is beyond the scope of this report to present all such determinations in detail. A compact description of the theoretical predictions used in these determinations can be found in Ref. [12], experimental details in the references to be mentioned in the following. Refs. [108,272,299] contain compilations of the most recent determinations of a from various 1 hard processes. Table 12, which is an excerpt of the results given in Refs. [12,108,272,299] for the processes other than e>e\ annihilation, complements the results from e>e\ annihilation listed in Table 4 of Section 4.1.1 and Table 7 of Section 4.1.2. In brief, a was obtained using: 1 E Deep inelastic scattering (DIS) of either leptons (e or k) or neutrinos o! nucleons from which the structure functions (polarized in the case of a polarized lepton beam) of proton, neutron, or deuteron were measured. The value of the strong coupling was extracted either directly from the Table 12 Listing of a determinations from processes other than e>e\ annihilation. Excerpt from Refs. [12,108,272,299]. 1 (DIS"deep inelastic scattering, Bj-SR"Bjorken sum rule, GLS-SR"Gross}Llewellyn}Smith sum rule, LGT"lattice gauge theory.) Ref.

Q (GeV)

[300] [301] [302] [303] [304] [305] [306] [307]

0.7}8 1.58 1.73 5.0 7.1 2}10 10}100 7}100

QQM states

[308]

0.7-8

pp PbbM X pp , ppPcX p(pp PW jets)

[309] [310] [311]

20 24.2 30}500

Process DIS DIS DIS DIS DIS DIS DIS DIS

[pol. strct. fctn.] [Bj-SR] [GLS-SR] [l; F , F ]   [k; F ]  [HERA; F ]  [HERA; jets] [HERA; ev.shps.]

a (Q) 1 0.375>  \  0.295>  \  0.215$0.016 0.280$0.014

0.145>  \  0.137>  \ 

a (m ) 1 8

Theory

0.120>  \  0.121>  \  0.114>  \  0.119$0.005 0.113$0.005 0.120$0.010 0.118$0.009 0.118>  \ 

NLO

0.120>  \ 

LGT

0.113$0.011 0.111>  \  0.121$0.009

NLO

NNLO NNLO NLO NLO NLO NLO NLO

NLO NLO

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structure functions, F , F , or g for polarized beams, or by exploiting higher-order corrections    to the Bjorken or Gross}Llewellyn}Smith sum rules for the structure functions. At su$ciently high momentum transfer q"!Q, available at the HERA electron}proton collider, a was also 1 extracted from jet rates and mean values of event shapes including power corrections as presented in Section 5.3. E Lattice gauge theory (LGT) calculations which, roughly speaking, discretize the four-dimensional space}time into hypercubes. The quark "elds reside on the corners of the cubes, the gauge "elds (gluons) are associated with the cubes' edges. One can explicitly calculate the action on such a lattice since the "nite lattice spacing serves as an ultraviolet cut-o! and, therefore, regulates the short distance divergence of the QCD Lagrangian (renormalization on the lattice, cf. Section 2.2). To obtain the value of the strong coupling constant from the lattice one usually calculates the B or charmonium spectrum and uses the true mass splitting from experimental measurements to set the scale for a . 1 E Hadron}hadron scattering by comparing heavy quark, direct photon and 2-jet cross-sections with next-to-leading order predictions. A more elaborate presentation of each topic can be found in Ref. [12]. The agreement of these results with the value of a obtained from the investigation of e>e\ 1 annihilation is remarkable. The values of the coupling at the Z mass scale all cluster closely around a value of approximately 0.119. Taking into account the unknown correlation between the individual results by the `optimized correlationa method of Ref. [148], a common average of all available data on a (m ) was calculated, yielding [272] 1 8 a (m )"0.119$0.004 , 1 8 where the overall correlation varies between about 50% and 80% depending on the subset of data chosen for the average. This average perfectly agrees with the value presented in this report. 6.5. A glance at asymptotic freedom Utilizing all results on a (Q) determined from e>e\ annihilation as well as other hard processes 1 at di!erent scales Q one obtains the behaviour depicted in Fig. 58. All individual values agree very well with the expectation of QCD over a vast energy range covering more than two orders of magnitude. The expectation of QCD shown in the "gure is the exact solution of the 4-loop renormalization group equation (21) using the world average of a (m )"0.119$0.004. These 1 8 data are presented di!erently in part (b) of the "gure, in order to make the property of asymptotic freedom visible. Fig. 58(b) shows, in addition, how the running of the strong coupling constant changes when the threshold to SUSY particle production, which is assumed for this plot to be at 1 TeV, is passed. A clear excursion from the straight path to asymptotic freedom can be seen which is due to new supersymmetric particles carrying colour charge and, therefore, participating in the strong interaction. Supersymmetry changes the b-function of Eq. (21) and, consequently, the energy dependence of the coupling constant a . Moreover the running of the quark masses is also a!ected, see for 1 instance Ref. [312]. The corresponding modi"cations of the 2-loop coe$cients of the b-function were taken from Ref. [313]. At the 2-loop level of the perturbation theory, one also has to consider

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Fig. 58. (a) Summary of a (Q). The QCD expectation for the energy scale dependence is shown using the world average 1 value a (m )"0.119$0.004 [272]. (b) shows the same data with the abscissa chosen as to emphasize the property of 1 8 asymptotic freedom. The dashed curve shows how the onset of supersymmetry at an assumed threshold of 1 TeV would change the energy dependence of the strong coupling.

the connection of the running of the strong coupling constant with that of the electromagnetic and weak coupling constants. The dashed curve in Fig. 58(b) is, in fact, the result of a numerical solution of a coupled system of three di!erential equations of the same type as Eq. (21) (see Ref. [313] for further details).

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The small size of the excursion indicates that a very precise determination of a is required to 1 notice the impact of SUSY. It is therefore more likely that SUSY particles will be discovered by a dedicated search rather than by an excursion in the running of a . Nevertheless, the "gure 1 demonstrates in principle the sensitivity of the strong interaction to new physics processes. Thus any signi"cant deviation of the strong coupling from the prediction could be evidence of yet unknown physics.

7. Summary and outlook Electron}positron annihilation experiments have reached a high level of sophistication in testing the predictions of QCD. This became possible only with the employment of colliders providing highly intense particle beams at very high energies. Below, at, and above the Z resonance an incredible amount of data statistics on annihilation into quark}antiquark pairs could be accumulated. By the juxtaposition of the results of the precise measurements with the predictions of perturbative QCD, the theory could be shown to be well-suited to describe the properties of the strong interaction. In this report QCD theory was "rst of all investigated using processes involving large energy scales. The property of asymptotic freedom, meaning that the strong coupling constant a dimin1 ishes as the energy scale goes towards in"nity, renders perturbative QCD predictions particularly reliable in these regions. The calculations were used to determine the strong coupling constant from (i) completely inclusive quantities as the hadronic cross-section and hadronic branching fractions of the q lepton and B mesons, and also (ii) from inclusive quantities as the production rate of n-jet "nal states and event shape observables which inclusively measure the distribution of the detected particles. Since QCD only operates on quarks and gluons, but experiments observe and measure hadrons, photons, and leptons, completely inclusive observables have a signi"cant advantage over inclusive quantities due to their insensitivity to the details of the "nal state. Precise determinations of the strong coupling constant are therefore possible from completely inclusive quantities, in particular, if next-to-next-to-leading order calculations are available. This is the case for the ratio R of the hadronic over the leptonic cross-section and for the hadronic q decays. However, the precision of these results is limited for the R ratio through its weak dependence on a contributing 1 only a small higher-order correction, and for the q decays by unknown higher-order terms which are important because of the large size of the coupling at such a low energy scale. Inclusive quantities as jet rates and event shapes are directly proportional to the coupling, thus promising an excellent sensitivity to the size of a . To perform any QCD test with them, however, 1 one needs to consider the e!ects from the transition of quarks and gluons into hadrons, that is, of hadronization. As the relevant energy scale at this stage is too low and the size of the coupling constant is too large, perturbation theory cannot be employed to describe this transition. The better sensitivity is traded for a considerable uncertainty coming from the necessity to use phenomenological models to account for hadronization e!ects. Perturbation calculations are available up to next-to-leading order only, but the large leading and next-to-leading logarithms could be resummed to all orders. Joining "xed order and resummed calculations allowed an extension of the range that can be used for the determination of a from a "t to the data 1 signi"cantly towards the 2-jet regime. The 2-jet region comprises the bulk of data characterized by

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hadronic "nal states where gluon emission at large angle and high energy is suppressed. Although improved, the contribution to the total error on a due to the arbitrariness of the choice of the 1 renormalization scale in next-to-leading order is still large. A missing ingredient of the perturbation theory calculation of inclusive quantities has "nally been added, viz. the exact treatment of "nite quark masses in jet rates and event shapes. Although mass e!ects may be expected to be small at high energy scales, their impact on #avour-dependent quantities has long been known. The high data statistics in conjunction with high resolution vertex detectors yielded very precise tests of the #avour dependence of the strong interaction. Deviations as large as 5% from the expected #avour independence of the strong coupling can now be fully explained by the e!ects of heavy bottom and charm quarks. Reversing the assumptions, a determination of the scale dependence of the bottom quark mass has become possible. It is found to be of the expected size of approximately 3 GeV at the Z pole, although the measurement is a!ected by large hadronization and renormalization scale uncertainties. Several approaches may be pursued to bypass, at least partly, the ignorance about hadronization, while still accounting for the details of the hadronic "nal state. The scaling violation, which yielded the "rst evidence for the energy dependence of the strong coupling in deep inelastic scattering, has been studied in inclusive fragmentation functions. In addition to the strong coupling, which was also obtained from the longitudinal cross-section, #avour-dependent fragmentation functions for quarks and gluons were determined. Stepping down the energy scale and looking inside the jets, the combination of the modi"ed leading-logarithmic approximation with the hypothesis of local parton}hadron duality is found to provide an excellent description of details connected with hadronization. The successful application of perturbative QCD in this framework at scales as low as a few hundred MeV leads to the conclusion that hadronization and con"nement of coloured partons into colour-neutral objects takes place at a scale which is of the order of the mass of the lightest mesons. A completely new approach to advance into the non-perturbative domain of hadronization and con"nement using the tools of perturbation theory is pursued by the investigation of power suppressed corrections to event shape observables. Although expected on general grounds from phenomenological hadronization models as well as from renormalons, their quantitative calculation became possible only with the assumption of a "nite coupling strength even at very small scales. The prediction could successfully be applied in the determination of a from both the mean 1 values and the di!erential distributions of event shape observables. Finally, the high precision of all the various determinations of a allowed further investigations 1 on the explicit energy dependence of the strong coupling. The known structure of the next order of the perturbation theory was used to determine its unknown expansion coe$cient from "ts to the data exploiting the energy dependence of a . Moreover, since no excursions from the QCD 1 expectation of energy dependence of the coupling constant were found, there is no evidence for either additional power suppressed contributions to the running of the coupling or additional coloured objects. After all the detailed investigations one might consider QCD as the theory of the strong interaction whose single unknown parameter is the strength of the coupling, a . Its size can be 1 obtained from joining the individual results from very many di!erent analyses of hadronic "nal states in e>e\ annihilation presented in this report and summarized in Table 13. The error of the a result from cross-sections and branching ratios is dominated by statistical uncertainties. It has, 1

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Table 13 Summary and weighted average of a (m ) determinations presented in this article 1 8 Quantity

a (m ) 1 8

Cross-sections, branching ratios Jet rates, event shapes Scaling violation Longitudinal cross-section Power corrections to mean values Power corrections to di!erential shapes Weighted average

0.1195$0.0025 0.1212$0.0079 0.125$0.009 0.118$0.014 0.1181>  \  0.1150$0.0036 0.1189$0.0025

therefore, been regarded as uncorrelated with the other a determinations which have been 1 averaged taking correlations due to theory uncertainties conservatively into account. Then the weighted average of all results yields a total relative error of about 2% and is in perfect agreement with the world average [272] which includes also a determinations from other hard scattering 1 processes. Recalling that new physics might slightly alter the running of the coupling, and because of the importance of knowing the precise value of the strong coupling, and also in view of future experiments at new e>e\ colliders (see Ref. [314]), the options to further improve on the precision need to be reviewed. Four major sources of error can be identi"ed, viz. (i) data statistics, (ii) "nite detector resolution and acceptance, (iii) hadronization, and (iv) the choice of the renormalization scale. Most of these can easily be reduced by improving experimental aspects of the determination: (i) data statistics, by improving the e$ciency of the event selection and by increasing the speci"c luminosity of the collider, and by combining the data of many experiments, (ii) detector e!ects, by better corrections or optimized detector designs, (iii) hadronization e!ects, by an ultimate tuning of the parameters of the phenomenological hadronization models using the vast amount of LEP I data statistics; by using higher centre-ofmass energies since the size of the corrections decreases like ln((s)/(s in the worst case; or by using power corrections, thus avoiding the e!ects completely. The uncertainty due to (iv) the arbitrariness of the choice of the renormalization scale can be solely reduced by a complete third-order calculation of jet rates and event shapes. The very "rst steps towards the completion of the calculation are complete. New computational tools for numerical integration of matrix elements might accelerate the pace of the progress [198,315]. And even the energy dependence of the strong interaction can be invoked to gain estimates of the size of the third-order terms. Moreover, further experimental tests can be performed at existing colliders using the hadronic "nal state of cc collisions. In particular the high centre-of-mass energy and huge luminosity at LEP II populates the region of cc events with high momentum transfer q,!Q, eventually allowing to study jet rates and event shapes over a signi"cant range of Q up to 40 000 GeV.

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279

Beyond the necessity to improve the precision on a , there are still more open questions to 1 be answered by QCD experiments. Besides topics like top quark physics, many questions concern the issue of hadronization and con"nement such as the existence of di-meson, di-baryon states, glue balls, quark}gluon plasma, just to mention a few. More generally, a proof of con"nement from the basic QCD Lagrangian would, besides asymptotic freedom, be another major supporting pillar of QCD as the theory of the strong interaction. Note added A re-evaluation of the Milan factor in Ref. [316] revealed an omission in the original derivation of M in Refs. [72,73]. The corrected formula is M+1#(1.575C !0.104n )/(4nb )+1.49 [317]  D  (cf. Section 5.3). Since this correction became available only after the investigations discussed in Sections 5.3.1 and 5.3.2 were completed, all results presented in these sections were obtained assuming M"1.79. The reduced size of M leads to increased values of a but has a negligible e!ect on the value  obtained for a (m ). First investigations of power corrections to the mean values of the event shape 1 8 observables presented in Section 5.3.1 (cf. Table 10) showed that a is increased by about 9%. The  value of a (m ) is found to be marginally enlarged by 0.6%. In general, the correction of the Milan  8 factor does not a!ect the observation that the non-perturbative parameter a has a universal  character at the level of 20%. Acknowledgements Many people were involved in the preparation of this manuscript, whether for giving the opportunity, for encouraging, teaching, discussing, helping, providing data, providing information, proof-reading, or many more -ings. In particular, I would like to thank: E S. Bethke for giving me the opportunity to write this report and also for his continuous support and encouragement; E A. Brandenburg for several discussions about the determination of running quark masses using his second-order calculations; E Yu. Dokshitzer for providing me information about the modi"ed power corrections for jet broadening observables; E Yu. Dokshitzer and B.R. Webber for answering my questions concerning power corrections; E T. Kawamoto for helping me to use the ZFITTER program; E E.W.N. Glover for providing additional information about his determination of higher-order corrections from the energy dependence of the mean thrust; E W. Bernreuther for interesting discussions about QCD and quark mass e!ects in QCD; E S. Kluth for providing the results from the numerical integration of the ERT matrix elements for several observables; E P.A. Movilla FernaH ndez for compiling di!erential event shape data from the Durham reaction database and from publications; E J. Letts, D.R. Ward, P. Pfeifenschneider, and M. ToK nnesmann for proof-reading this manuscript.

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WEAK GRAVITATIONAL LENSING

Matthias BARTELMANN, Peter SCHNEIDER Max-Planck-Institut f uK r Astrophysik, P.O. Box 1523, D-85740 Garching, Germany

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

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Weak gravitational lensing Matthias Bartelmann*, Peter Schneider Max-Planck-Institut fu( r Astrophysik, P.O. Box 1523, D-85740 Garching, Germany Received December 1999; editor: G.H.F. Diercksen Contents 1. Introduction 1.1. Gravitational light de#ection 1.2. Weak gravitational lensing 1.3. Applications of gravitational lensing 1.4. Structure of this review 2. Cosmological background 2.1. Friedmann}Lemam( tre cosmological models 2.2. Density perturbations 2.3. Relevant properties of lenses and sources 2.4. Correlation functions, power spectra, and their projections 3. Gravitational light de#ection 3.1. Gravitational lens theory 3.2. Light propagation in arbitrary spacetimes 4. Principles of weak gravitational lensing 4.1. Introduction 4.2. Galaxy shapes and sizes, and their transformation 4.3. Local determination of the distortion 4.4. Magni"cation e!ects 4.5. Minimum lens strength for its weak lensing detection 4.6. Practical consideration for measuring image shapes

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5. Weak lensing by galaxy clusters 5.1. Introduction 5.2. Cluster mass reconstruction from image distortions 5.3. Aperture mass and multipole measures 5.4. Application to observed clusters 5.5. Outlook 6. Weak cosmological lensing 6.1. Light propagation; choice of coordinates 6.2. Light de#ection 6.3. E!ective convergence 6.4. E!ective-convergence power spectrum 6.5. Magni"cation and shear 6.6. Second-order statistical measures 6.7. Higher-order statistical measures 6.8. Cosmic shear and biasing 6.9. Numerical approach to cosmic shear, cosmological parameter estimates, and observations 7. QSO magni"cation bias and large-scale structure 7.1. Introduction 7.2. Expected magni"cation bias from cosmological density perturbations 7.3. Theoretical expectations 7.4. Observational results

* Corresponding author. Tel.: #49-89-3299-3236; fax: #49-89-3299-3235. E-mail address: [email protected] (M. Bartelmann). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 8 2 - X

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M. Bartelmann, P. Schneider / Physics Reports 340 (2001) 291}472 7.5. Magni"cation bias of galaxies 7.6. Outlook 8. Galaxy}galaxy lensing 8.1. Introduction 8.2. The theory of galaxy}galaxy lensing 8.3. Results 8.4. Galaxy}galaxy lensing in galaxy clusters 9. The impact of weak gravitational light de#ection on the microwave background radiation 9.1. Introduction 9.2. Weak lensing of the CMB

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9.3. CMB temperature #uctuations 9.4. Auto-correlation function of the gravitationally lensed CMB 9.5. De#ection-angle variance 9.6. Change of CMB temperature #uctuations 9.7. Discussion 10. Summary and outlook Acknowledgements References

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Abstract We review theory and applications of weak gravitational lensing. After summarising Friedmann}Lemam( tre cosmological models, we present the formalism of gravitational lensing and light propagation in arbitrary space}times. We discuss how weak-lensing e!ects can be measured. The formalism is then applied to reconstructions of galaxy-cluster mass distributions, gravitational lensing by large-scale matter distributions, QSO}galaxy correlations induced by weak lensing, lensing of galaxies by galaxies, and weak lensing of the cosmic microwave background.  2001 Elsevier Science B.V. All rights reserved. PACS: 98.62.Sb; 95.30.Sf

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1. Introduction 1.1. Gravitational light deyection Light rays are de#ected when they propagate through an inhomogeneous gravitational "eld. Although several researchers had speculated about such an e!ect well before the advent of General Relativity (see Schneider et al., 1992 for a historical account), it was Einstein's theory which elevated the de#ection of light by masses from a hypothesis to a "rm prediction. Assuming light behaves like a stream of particles, its de#ection can be calculated within Newton's theory of gravitation, but General Relativity predicts that the e!ect is twice as large. A light ray grazing the surface of the Sun is de#ected by 1.75 arcsec compared to the 0.87 arcsec predicted by Newton's theory. The con"rmation of the larger value in 1919 was perhaps the most important step towards accepting General Relativity as the correct theory of gravity (Eddington, 1920). Cosmic bodies more distant, more massive, or more compact than the Sun can bend light rays from a single source su$ciently strongly so that multiple light rays can reach the observer. The observer sees an image in the direction of each ray arriving at their position, so that the source

Fig. 1. The gravitational lens system 2237#0305 consists of a nearby spiral galaxy at redshift z "0.039 and four  images of a background quasar with redshift z "1.69. It was discovered by Huchra et al. (1985). The image was taken by  the Hubble Space Telescope and shows only the innermost region of the lensing galaxy. The central compact source is the bright galaxy core, the other four compact sources are the quasar images. They di!er in brightness because they are magni"ed by di!erent amounts. The four images roughly fall on a circle concentric with the core of the lensing galaxy. The mass inside this circle can be determined with very high accuracy (Rix et al., 1992). The largest separation between the images is 1.8. Fig. 2. The radio source MG 1131#0456 was discovered by Hewitt et al. (1988) as the "rst example of a so-called Einstein ring. If a source and an axially symmetric lens are co-aligned with the observer, the symmetry of the system permits the formation of a ring-like image of the source centred on the lens. If the symmetry is broken (as expected for all realistic lensing matter distributions), the ring is deformed or broken up, typically into four images (see Fig. 1). However, if the source is su$ciently extended, ring-like images can be formed even if the symmetry is imperfect. The 6 cm radio map of MG 1131#0456 shows a closed ring, while the ring breaks up at higher frequencies where the source is smaller. The ring diameter is 2.1.

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appears multiply imaged. In the language of General Relativity, there may exist more than one null geodesic connecting the world-line of a source with the observation event. Although predicted long before, the "rst multiple-image system was discovered only in 1979 (Walsh et al., 1979). From then on, the "eld of gravitational lensing developed into one of the most active subjects of astrophysical research. Several dozens of multiply imaged sources have since been found. Their quantitative analysis provides accurate masses of, and in some cases detailed information on, the de#ectors. An example is shown in Fig. 1. Tidal gravitational "elds lead to di!erential de#ection of light bundles. The size and shape of their cross sections are therefore changed. Since photons are neither emitted nor absorbed in the process of gravitational light de#ection, the surface brightness of lensed sources remains unchanged. Changing the size of the cross section of a light bundle therefore changes the #ux observed from a source. The di!erent images in multiple-image systems generally have di!erent #uxes. The images of extended sources, i.e. sources which can observationally be resolved, are deformed by the gravitational tidal "eld. Since astronomical sources like galaxies are not intrinsically circular, this deformation is generally very di$cult to identify in individual images. In some cases, however, the distortion is strong enough to be readily recognised, most noticeably in the case of Einstein rings (see Fig. 2) and arcs in galaxy clusters (Fig. 3). If the light bundles from some sources are distorted so strongly that their images appear as giant luminous arcs, one may expect many more sources behind a cluster whose images are only weakly distorted. Although weak distortions in individual images can hardly be recognised, the net distortion averaged over an ensemble of images can still be detected. As we shall describe in Section 2.3, deep optical exposures reveal a dense population of faint galaxies on the sky. Most of these galaxies are at high redshift, thus distant, and their image shapes can be utilised to probe the tidal gravitational "eld of intervening mass concentrations. Indeed, the tidal gravitational "eld can be reconstructed from the coherent distortion apparent in images of the faint galaxy population, and from that the density pro"le of intervening clusters of galaxies can be inferred (see Section 4).

Fig. 3. The cluster Abell 2218 hosts one of the most impressive collections of arcs. This HST image of the cluster's central region shows a pattern of strongly distorted galaxy images tangentially aligned with respect to the cluster centre, which lies close to the bright galaxy in the upper part of this image. The frame measures about 80;160 (courtesy of J.-P. Kneib).

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1.2. Weak gravitational lensing This review deals with weak gravitational lensing. There is no generally applicable de"nition of weak lensing despite the fact that it constitutes a #ourishing area of research. The common aspect of all studies of weak gravitational lensing is that measurements of its e!ects are statistical in nature. While a single multiply imaged source provides information on the mass distribution of the de#ector, weak lensing e!ects show up only across ensembles of sources. One example was given above: The shape distribution of an ensemble of galaxy images is changed close to a massive galaxy cluster in the foreground, because the cluster's tidal "eld polarises the images. We shall see later that the size distribution of the background galaxy population is also locally changed in the neighbourhood of a massive intervening mass concentration. Magni"cation and distortion e!ects due to weak lensing can be used to probe the statistical properties of the matter distribution between us and an ensemble of distant sources, provided some assumptions on the source properties can be made. For example, if a standard candle at high redshift is identi"ed, its #ux can be used to estimate the magni"cation along its line-of-sight. It can be assumed that the orientation of faint distant galaxies is random. Then, any coherent alignment of images signals the presence of an intervening tidal gravitational "eld. As a third example, the positions on the sky of cosmic objects at vastly di!erent distances from us should be mutually independent. A statistical association of foreground objects with background sources can therefore indicate the magni"cation caused by the foreground objects on the background sources. All these e!ects are quite subtle, or weak, and many of the current challenges in the "eld are observational in nature. A coherent alignment of images of distant galaxies can be due to an intervening tidal gravitational "eld, but could also be due to propagation e!ects in the Earth's atmosphere or in the telescope. A variation in the number density of background sources around a foreground object can be due to a magni"cation e!ect, but could also be due to non-uniform photometry or obscuration e!ects. These potential systematic e!ects have to be controlled at a level well below the expected weak-lensing e!ects. We shall return to this essential point at various places in this review. 1.3. Applications of gravitational lensing Gravitational lensing has developed into a versatile tool for observational cosmology. There are two main reasons:

 The term standard candle is used for any class of astronomical objects whose intrinsic luminosity can be inferred independently of the observed #ux. In the simplest case, all members of the class have the same luminosity. More typically, the luminosity depends on some other known and observable parameters, such that the luminosity can be inferred from them. The luminosity distance to any standard candle can directly be inferred from the square root of the ratio of source luminosity and observed #ux. Since the luminosity distance depends on cosmological parameters, the geometry of the Universe can then directly be investigated. Probably, the best current candidates for standard candles are supernovae of Type Ia. They can be observed to quite high redshifts, and thus be utilised to estimate cosmological parameters (e.g. Riess et al., 1998).

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(1) The de#ection angle of a light ray is determined by the gravitational "eld of the matter distribution along its path. According to Einstein's theory of General Relativity, the gravitational "eld is in turn determined by the stress-energy tensor of the matter distribution. For the astrophysically most relevant case of non-relativistic matter, the latter is characterised by the density distribution alone. Hence, the gravitational "eld, and thus the de#ection angle, depend neither on the nature of the matter nor on its physical state. Light de#ection probes the total matter density, without distinguishing between ordinary (baryonic) matter or dark matter. In contrast to other dynamical methods for probing gravitational "elds, no assumption needs to be made on the dynamical state of the matter. For example, the interpretation of radial velocity measurements in terms of the gravitating mass requires the applicability of the virial theorem (i.e., the physical system is assumed to be in virial equilibrium), or knowledge of the orbits (such as the circular orbits in disk galaxies). However, as will be discussed in Section 3, lensing measures only the mass distribution projected along the line-of-sight, and is therefore insensitive to the extent of the mass distribution along the light rays, as long as this extent is small compared to the distances from the observer and the source to the de#ecting mass. Keeping this in mind, mass determinations by lensing do not depend on any symmetry assumptions. (2) Once the de#ection angle as a function of impact parameter is given, gravitational lensing reduces to simple geometry. Since most lens systems involve sources (and lenses) at moderate or high redshift, lensing can probe the geometry of the Universe. This was noted by Refsdal (1964), who pointed out that lensing can be used to determine the Hubble constant and the cosmic density parameter. Although this turned out later to be more di$cult than anticipated at the time, "rst measurements of the Hubble constant through lensing have been obtained with detailed models of the matter distribution in multiple-image lens systems and the di!erence in light-travel time along the di!erent light paths corresponding to di!erent images of the source (e.g. KundicH et al., 1997; Schechter et al., 1997; Biggs et al., 1999). Since the volume element per unit redshift interval and unit solid angle also depends on the geometry of space-time, so does the number of lenses therein. Hence, the lensing probability for distant sources depends on the cosmological parameters (e.g. Press and Gunn, 1973). Unfortunately, in order to derive constraints on the cosmological model with this method, one needs to know the evolution of the lens population with redshift. Nevertheless, in some cases, signi"cant constraints on the cosmological parameters (Kochanek, 1993; 1996; Maoz and Rix, 1993; Bartelmann et al., 1998; Falco et al., 1998), and on the evolution of the lens population (Mao and Kochanek, 1994) have been derived from multiple-image and arc statistics (see also the review by Chiba and Futamase, 1999). The possibility to directly investigate the dark-matter distribution led to substantial results over recent years. Constraints on the size of the dark-matter halos of spiral galaxies were derived (e.g. Brainerd et al., 1996), the presence of dark-matter halos in elliptical galaxies was demonstrated (e.g. Maoz and Rix, 1993; Gri$ths et al., 1996), and the projected total mass distribution in many cluster of galaxies was mapped (e.g. Kneib et al., 1996; Hoekstra et al., 1998; Kaiser et al., 1998). These results directly impact on our understanding of structure formation, supporting hierarchical structure formation in cold dark-matter (CDM) models. Constraints on the nature of dark matter were also obtained. Compact dark-matter objects, such as black holes or brown dwarfs, cannot be very abundant in the Universe, because otherwise they would lead to observable lensing e!ects

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(e.g. Schneider, 1993; Dalcanton et al., 1994). Galactic microlensing experiments constrained the density and typical mass scale of massive compact halo objects in our Galaxy (see PaczynH ski, 1996; Roulet and Mollerach, 1997; Mao, 2000 for reviews). We refer the reader to the reviews by Blandford and Narayan (1992), Schneider (1996a) and Narayan and Bartelmann (1991) for a detailed account of the cosmological applications of gravitational lensing. We shall concentrate almost entirely on weak gravitational lensing here. Hence, the #ourishing "elds of multiple-image systems and their interpretation, Galactic microlensing and its consequences for understanding the nature of dark matter in the halo of our galaxy, and the detailed investigations of the mass distribution in the inner parts of galaxy clusters through arcs, arclets, and multiply imaged background galaxies, will not be covered in this review. In addition to the references given above, we would like to point the reader to Refsdal and Surdej (1994), Fort and Mellier (1994), Wu (1996), and Hattori et al. (1999) for more recent reviews on various aspects of gravitational lensing, to Mellier (1999) for a very recent review on weak lensing, and to the monograph (Schneider et al., 1992) for a detailed account of the theory and applications of gravitational lensing. 1.4. Structure of this review Many aspects of weak gravitational lensing are intimately related to the cosmological model and to the theory of structure formation in the Universe. We therefore start the review by giving some cosmological background in Section 2. After summarising Friedmann}Lemam( tre}Robertson} Walker models, we sketch the theory of structure formation, introduce astrophysical objects like QSOs, galaxies, and galaxy clusters, and "nish the section with a general discussion of correlation functions, power spectra, and their projections. Gravitational light de#ection in general is the subject of Section 3, and the specialisation to weak lensing is described in Section 4. One of the main aspects there is how weak lensing e!ects can be quanti"ed and measured. The following two sections describe the theory of weak lensing by galaxy clusters (Section 5) and cosmological mass distributions (Section 6). Apparent correlations between background QSOs and foreground galaxies due to the magni"cation bias caused by large-scale matter distributions are the subject of Section 7. Weak lensing e!ects of foreground galaxies on background galaxies are reviewed in Section 8, and Section 9 "nally deals with weak lensing of the most distant and most extended source possible, i.e. the Cosmic microwave Background. We present a brief summary and an outlook in Section 10. We use standard astronomical units throughout: 1M "1 solar mass"2;10 g; > 1 Mpc"1 megaparsec"3.1;10 cm.

2. Cosmological background We review in this section those aspects of the standard cosmological model which are relevant for our further discussion of weak gravitational lensing. This standard model consists of a description for the cosmological background which is a homogeneous and isotropic solution of the "eld equations of General Relativity, and a theory for the formation of structure.

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The background model is described by the Robertson}Walker metric (Robertson, 1935; Walker, 1935), in which hypersurfaces of constant time are homogeneous and isotropic three spaces, either #at or curved, and change with time according to a scale factor which depends on time only. The dynamics of the scale factor is determined by two equations which follow from Einstein's "eld equations given the highly symmetric form of the metric. Current theories of structure formation assume that structure grows via gravitational instability from initial seed perturbations whose origin is yet unclear. Most common hypotheses lead to the prediction that the statistics of the seed #uctuations is Gaussian. Their amplitude is low for most of their evolution so that linear perturbation theory is su$cient to describe their growth until late stages. For general references on the cosmological model and on the theory of structure formation, cf. Weinberg (1972), Misner et al. (1973), Peebles (1980), BoK rner (1988), Padmanabhan (1993), Peebles (1993), and Peacock (1999). 2.1. Friedmann}Lemaı( tre cosmological models 2.1.1. Metric Two postulates are fundamental to the standard cosmological model, which are: (1) When averaged over suzciently large scales, there exists a mean motion of radiation and matter in the Universe with respect to which all averaged observable properties are isotropic. (2) All fundamental observers, i.e. imagined observers which follow this mean motion, experience the same history of the Universe, i.e. the same averaged observable properties, provided they set their clocks suitably. Such a Universe is called observer-homogeneous. General Relativity describes space}time as a four-dimensional manifold whose metric tensor g is considered as a dynamical "eld. The dynamics of the metric is governed by Einstein's "eld ?@ equations, which relate the Einstein tensor to the stress-energy tensor of the matter contained in space}time. Two events in space}time with coordinates di!ering by dx? are separated by ds, with ds"g dx? dx@. The eigentime (proper time) of an observer who travels by ds changes by c\ ds. ?@ Greek indices run over 0,2, 3 and Latin indices run over the spatial indices 1,2, 3 only. The two postulates stated above considerably constrain the admissible form of the metric tensor. Spatial coordinates which are constant for fundamental observers are called comoving coordinates. In these coordinates, the mean motion is described by dxG"0, and hence ds"g dt. If we  require that the eigentime of fundamental observers equal the cosmic time, this implies g "c.  Isotropy requires that clocks can be synchronised such that the space}time components of the metric tensor vanish, g "0. If this was impossible, the components of g identi"ed one particular G G direction in space}time, violating isotropy. The metric can therefore be written as ds"c dt#g dxG dxH , (2.1) GH where g is the metric of spatial hypersurfaces. In order not to violate isotropy, the spatial metric GH can only isotropically contract or expand with a scale function a(t) which must be a function of time only, because otherwise the expansion would be di!erent at di!erent places, violating homogeneity. Hence, the metric further simpli"es to ds"cdt!a(t) dl ,

(2.2)

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where dl is the line element of the homogeneous and isotropic three space. A special case of metric (2.2) is the Minkowski metric, for which dl is the Euclidean line element and a(t) is a constant. Homogeneity also implies that all quantities describing the matter content of the Universe, e.g. density and pressure, can be functions of time only. The spatial hypersurfaces whose geometry is described by dl can either be #at or curved. Isotropy only requires them to be spherically symmetric, i.e. spatial surfaces of constant distance from an arbitrary point need to be two spheres. Homogeneity permits us to choose an arbitrary point as coordinate origin. We can then introduce two angles h, which uniquely identify positions on the unit sphere around the origin, and a radial coordinate w. The most general admissible form for the spatial line element is then dl"dw#f  (w)(d #sinh dh),dw#f  (w) du . (2.3) ) ) Homogeneity requires that the radial function f (w) is either a trigonometric, linear, or hyperbolic ) function of w, depending on whether the curvature K is positive, zero, or negative. Speci"cally,



K\ sin(Kw)

(K'0) ,

(K"0) , f (w)" w ) (!K)\ sinh[(!K)w] (K(0) .

(2.4)

Note that f (w) and thus "K"\ have the dimension of a length. If we de"ne the radius r of the two ) spheres by f (w),r, the metric dl takes the alternative form ) dr dl" #r du . (2.5) 1!Kr 2.1.2. Redshift Due to the expansion of space, photons are redshifted while they propagate from the source to the observer. Consider a comoving source emitting a light signal at t which reaches a comoving  observer at the coordinate origin w"0 at time t . Since ds"0 for light, a backward-directed  radial light ray propagates according to "c dt""a dw, from the metric. The (comoving) coordinate distance between source and observer is constant by de"nition,







R R  c dt "constant (2.6) a  R and thus, in particular, the derivative of w with respect to t is zero. It then follows from   Eq. (2.6) w " 

dw"

dt a(t ) "  . (2.7) dt a(t )   Identifying the inverse time intervals (dt )\ with the emitted and observed light frequencies l ,   we can write dt l j " "  . dt l j   

(2.8)

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Since the redshift z is de"ned as the relative change in wavelength, or 1#z"j j\, we "nd   a(t ) 1#z"  . (2.9) a(t )  This shows that light is redshifted by the amount by which the Universe has expanded between emission and observation. 2.1.3. Expansion To complete the description of space}time, we need to know how the scale function a(t) depends on time, and how the curvature K depends on the matter which "lls space}time. That is, we ask for the dynamics of the space}time. Einstein's "eld equations relate the Einstein tensor G to the ?@ stress-energy tensor ¹ of the matter ?@ 8pG G " ¹ #Kg . (2.10) ?@ ?@ c ?@ The second term proportional to the metric tensor g is a generalisation introduced by Einstein to ?@ allow static cosmological solutions of the "eld equations. K is called the cosmological constant. For the highly symmetric form of the metric given by (2.2) and (2.3), Einstein's equations imply that ¹ has to have the form of the stress-energy tensor of a homogeneous perfect #uid, which is ?@ characterised by its density o(t) and its pressure p(t). Matter density and pressure can only depend on time because of homogeneity. The "eld equations then simplify to the two independent equations:




a  8pG Kc K " o! # a 3 a 3

(2.11)

and






aK 4 3p K "! pG o# # . a 3 c 3

(2.12)

The scale factor a(t) is determined once its value at one instant of time is "xed. We choose a"1 at the present epoch t . Eq. (2.11) is called Friedmann's equation (Friedmann, 1922, 1924). The two  Eqs. (2.11) and (2.12) can be combined to yield the adiabatic equation d da(t) [a(t)o(t)c]#p(t) "0 , dt dt

(2.13)

which has an intuitive interpretation. The "rst term ao is proportional to the energy contained in a "xed comoving volume, and hence the equation states that the change in &internal' energy equals the pressure times the change in proper volume. Hence, Eq. (2.13) is the "rst law of thermodynamics in the cosmological context. A metric of the form given by Eqs. (2.2)}(2.4) is called the Robertson}Walker metric. If its scale factor a(t) obeys Friedmann's equation (2.11) and the adiabatic equation (2.13), it is called the Friedmann}Lemam( tre}Robertson}Walker metric, or the Friedmann}Lemam( tre metric for short. Note that Eq. (2.12) can also be derived from Newtonian gravity except for the pressure term in

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(2.12) and the cosmological constant. Unlike in Newtonian theory, pressure acts as a source of gravity in General Relativity. 2.1.4. Parameters The relative expansion rate a a\,H is called the Hubble parameter, and its value at the present epoch t"t is the Hubble constant, H(t ),H . It has the dimension of an inverse time.    The value of H is still uncertain. Current measurements roughly fall into the range  H "(50}80) km s\ Mpc\ (see Freedman, 1996 for a review), and the uncertainty in H is   commonly expressed as H "100h km s\ Mpc\, with h"(0.5}0.8). Hence,  H +3.2;10\h s\+1.0;10\h yr\ . (2.14)  The time scale for the expansion of the Universe is the inverse Hubble constant, or H\+10h\ yr.  The combination 3H  ,o +1.9;10\h g cm\  8pG

(2.15)

is the critical density of the Universe, and the density o in units of o is the density parameter X ,    o X "  . (2.16)  o  If the matter density in the Universe is critical, o "o or X "1, and if the cosmological constant    vanishes, K"0, spatial hypersurfaces are #at, K"0, which follows from (2.11) and will become explicit in Eq. (2.30) below. We further de"ne K XK , . 3H  The deceleration parameter q is de"ned by  aK a q "!  a 

(2.17)

(2.18)

at t"t .  2.1.5. Matter models For a complete description of the expansion of the Universe, we need an equation of state p"p(o), relating the pressure to the energy density of the matter. Ordinary matter, which is frequently called dust in this context, has p;oc, while p"oc/3 for radiation or other forms of relativistic matter. Inserting these expressions into Eq. (2.13), we "nd o(t)"a\L(t)o 

(2.19)

with



n"

3 for dust, p"0 ,

4 for relativistic matter, p"oc/3 .

(2.20)

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The energy density of relativistic matter therefore drops more rapidly with time than that of ordinary matter. 2.1.6. Relativistic matter components There are two obvious candidates for relativistic matter today, photons and neutrinos. The energy density contained in photons today is determined by the temperature of the cosmic microwave background (CMB), ¹ "2.73 K (Fixsen et al., 1996). Since the CMB has an !+ excellent black-body spectrum, its energy density is given by the Stefan}Boltzmann law ) 1 p (k¹ !+ +4.5;10\ g cm\ . o " !+ c 15 ( c)

(2.21)

In terms of the cosmic density parameter X [Eq. (2.16)], the cosmic density contributed by the  photon background is X "2.4;10\h\ . (2.22) !+  Like photons, neutrinos were produced in thermal equilibrium in the hot early phase of the Universe. Interacting weakly, they decoupled from the cosmic plasma when the temperature of the Universe was k¹+1 MeV because later the time scale of their leptonic interactions became larger than the expansion time scale of the Universe, so that equilibrium could no longer be maintained. When the temperature of the Universe dropped to k¹+0.5 MeV, electron}positron pairs annihilated to produce c-rays. The annihilation heated up the photons but not the neutrinos which had decoupled earlier. Hence, the neutrino temperature is lower than the photon temperature by an amount determined by entropy conservation. The entropy S of the electron}positron pairs was  dumped completely into the entropy of the photon background S . Hence, A (S #S ) "(S ) , (2.23)  A
 A  where `beforea and `aftera refer to the annihilation time. Ignoring constant factors, the entropy per particle species is SJg¹, where g is the statistical weight of the species. For bosons g"1, and for "4 ) #2", while after fermions g" per spin state. Before annihilation, we thus have g
    the annihilation g"2 because only photons remain. From Eq. (2.23),






¹  11  " . (2.24) ¹ 4
 After the annihilation, the neutrino temperature is therefore lower than the photon temperature by the factor (). In particular, the neutrino temperature today is  4  ¹ "1.95 K . (2.25) ¹ " !+ J 11




Although neutrinos have long been out of thermal equilibrium, their distribution function remained unchanged since they decoupled, except that their temperature gradually dropped in the course of cosmic expansion. Their energy density can thus be computed from a Fermi}Dirac

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distribution with temperature ¹ , and be converted to the equivalent cosmic density parameter as J for the photons. The result is X "2.8;10\h\ (2.26) J per neutrino species. Assuming three relativistic neutrino species, the total density parameter in relativistic matter today is X "X #3;X "3.2;10\h\ . (2.27) 0 !+  J Since the energy density in relativistic matter is almost "ve orders of magnitude less than the energy density of ordinary matter today if X is of order unity, the expansion of the Universe today is  matter-dominated, or o"a\(t)o . The energy densities of ordinary and relativistic matter were  equal when the scale factor a(t) was X (2.28) a " 0 "3.2;10\X\h\   X  and the expansion was radiation-dominated at yet earlier times, o"a\o . The epoch of equality  of matter and radiation density will turn out to be important for the evolution of structure in the Universe discussed below. 2.1.7. Spatial curvature and expansion With the parameters de"ned previously, Friedmann's equation (2.11) can be written as





Kc H(t)"H a\(t)X #a\(t)X !a\(t) #XK .  0  H  Since H(t ),H , and X ;X , Eq. (2.29) implies   0  H  K"  (X #XK !1)  c




(2.29)

(2.30)

and Eq. (2.29) becomes (2.31) H(t)"H [a\(t)X #a\(t)X #a\(t)(1!X !XK )#XK ] .  0   The curvature of spatial hypersurfaces is therefore determined by the sum of the density contributions from matter, X , and from the cosmological constant, XK . If X #XK "1, space is #at, and   it is closed or hyperbolic if X #XK is larger or smaller than unity, respectively. The spatial  hypersurfaces of a low-density Universe are therefore hyperbolic, while those of a high-density Universe are closed [cf. Eq. (2.4)]. A Friedmann}Lemam( tre model universe is thus characterised by four parameters: the expansion rate at present (or Hubble constant) H , and the density parameters  in matter, radiation, and the cosmological constant. Dividing Eq. (2.12) by Eq. (2.11), using Eq. (2.30), and setting p"0, we obtain for the deceleration parameter q  X (2.32) q "  !XK .  2

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Fig. 4. Cosmic age t in units of H\ as a function of X , for XK "0 (solid curve) and XK "1!X (dashed curve).    

The age of the Universe can be determined from Eq. (2.31). Since dt"da a \"da(aH)\, we have, ignoring X , 0 1  t " (2.33) da [a\X #(1!X !XK )#aXK ]\ .  H     It was assumed in this equation that p"0 holds for all times t, while pressure is not negligible at early times. The corresponding error, however, is very small because the universe spends only a very short time in the radiation-dominated phase where p'0. Fig. 4 shows t in units of H\ as a function of X , for XK "0 (solid curve) and XK "1!X     (dashed curve). The model universe is older for lower X and higher XK because the deceleration  decreases with decreasing X and the acceleration increases with increasing XK .  In principle, XK can have either sign. We have restricted ourselves in Fig. 4 to non-negative XK because the cosmological constant is usually interpreted as the energy density of the vacuum, which is positive semi-de"nite. The time evolution (2.31) of the Hubble function H(t) allows one to determine the dependence of X and XK on the scale function a. For a matter-dominated Universe, we "nd



8pG X  X(a)" o a\" ,  3H(a) a#X (1!a)#XK (a!a)  K XK a " . (2.34) XK (a)" 3H(a) a#X (1!a)#XK (a!a)  These equations show that, whatever the values of X and XK are at the present epoch, X(a)P1  and XK P0 for aP0. This implies that for su$ciently early times, all matter-dominated Friedmann}Lemam( tre model universes can be described by Einstein}de Sitter models, for which K"0 and XK "0. For a;1, the right-hand side of the Friedmann equation (2.31) is therefore dominated

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by the matter and radiation terms because they contain the strongest dependences on a\. The Hubble function H(t) can then be approximated by H(t)"H [X a\(t)#X a\(t)] . (2.35)  0  Using the de"nition of a , a\X "a\X [cf. Eq. (2.28)], Eq. (2.35) can be written as   0    a . (2.36) H(t)"H Xa\ 1#    a






Hence,



aa\ H(t)"H X    a\

(a;a ) ,  (a ;a;1) .  Likewise, the expression for the cosmic time reduces to




2 a t(a)" X\ a 1!2   3H a 






(2.37)



a  1#  #2a  a

(2.38)

or



a\a (a;a ) , 1  t(a)" X\   (2.39) H  a (a ;a;1) .     Eq. (2.36) is called the Einstein}de Sitter limit of Friedmann's equation. Where not mentioned otherwise, we consider in the following only cosmic epochs at times much later than t , i.e.,  when a
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measurable quantities in Euclidean space. We de"ne here four di!erent distance scales, the proper distance, the comoving distance, the angular-diameter distance, and the luminosity distance. Distance measures relate an emission event and an observation event on two separate geodesic lines which fall on a common light cone, either the forward light cone of the source or the backward light cone of the observer. They are therefore characterised by the times t and t   of emission and observation, respectively, and by the structure of the light cone. These times can uniquely be expressed by the values a "a(t ) and a "a(t ) of the scale factor, or by the     redshifts z and z corresponding to a and a . We choose the latter parameterisation because     redshifts are directly observable. We also assume that the observer is at the origin of the coordinate system. The proper distance D (z , z ) is the distance measured by the travel time of a light ray which    propagates from a source at z to an observer at z (z . It is de"ned by dD "!c dt, hence     dD "!c daa \"!c da(aH)\. The minus sign arises because, due to the choice of coordi nates centred on the observer, distances increase away from the observer, while the time t and the scale factor a increase towards the observer. We get



c ?X  [a\X #(1!X !XK )#aXK ]\ da . (2.40) D (z , z )"      H  ?X  The comoving distance D (z , z ) is the distance on the spatial hyper-surface t"t between the     world-lines of a source and an observer comoving with the cosmic #ow. Due to the choice of coordinates, it is the coordinate distance between a source at z and an observer at z , dD "dw.    Since light rays propagate with ds"0, we have c dt"!a dw from the metric, and therefore dD "!a\c dt"!c da(aa )\"!c da(aH)\. Thus,  c ?X  [aX #a(1!X !XK )#aXK ]\ da D (z , z )"      H  ?X  "w(z , z ) . (2.41)   The angular-diameter distance D (z , z ) is de"ned in analogy to the relation in Euclidean space    between the physical cross section dA of an object at z and the solid angle du that it subtends for  an observer at z , duD "dA. Hence,   du dA " , (2.42) 4pa(z ) f  [w(z , z )] 4p  )   where a(z ) is the scale factor at emission time and f [w(z , z )] is the radial coordinate distance  )   between the observer and the source. It follows that



D




dA  "a(z ) f [w(z , z )] . (z , z )"  )      du

(2.43)

According to the de"nition of the comoving distance, the angular-diameter distance therefore is D

(z , z )"a(z ) f [D (z , z )] .     )   

(2.44)

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Fig. 5. Four distance measures are plotted as a function of source redshift for two cosmological models and an observer at redshift zero. These are the proper distance D (1, solid line), the comoving distance D (2, dotted line),   the angular-diameter distance D (3, short-dashed line), and the luminosity distance D (4, long-dashed   line).

The luminosity distance D (a , a ) is de"ned by the relation in Euclidean space between the    luminosity ¸ of an object at z and the #ux S received by an observer at z . It is related to the   angular-diameter distance through

 

a(z )  a(z )  D (z , z )"  f [D (z , z )] . D (z , z )"       a(z ) a(z ) )     

(2.45)

The "rst equality in (2.45), which is due to Etherington (1933), is valid in arbitrary space}times. It is physically intuitive because photons are redshifted by a(z )a(z )\, their arrival times are delayed   by another factor a(z )a(z )\, and the area of the observer's sphere on which the photons are   distributed grows between emission and absorption in proportion to [a(z )a(z )\]. This ac  counts for a total factor of [a(z )a(z )\] in the #ux, and hence for a factor of [a(z )a(z )\] in     the distance relative to the angular-diameter distance. We plot the four distances D , D , D , and D for z "0 as a function of z in      Fig. 5. The distances are larger for lower cosmic density and higher cosmological constant. Evidently, they di!er by a large amount at high redshift. For small redshifts, z;1, they all follow the Hubble law, cz distance" #O(z) . H 

(2.46)

2.1.10. The Einstein}de Sitter model In order to illustrate some of the results obtained above, let us now specialise to a model universe with a critical density of dust, X "1 and p"0, and with zero cosmological constant, XK "0. 

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Friedmann's equation then reduces to H(t)"H a\, and the age of the Universe becomes  t "2(3H )\. The distance measures are   2c D (z , z )" [(1#z )\!(1#z )\] ,      3H  2c D (z , z )" [(1#z )\!(1#z )\] ,      H  1 2c [(1#z )\!(1#z )\] , D (z , z )"      H 1#z   2c 1#z  [(1#z )\!(1#z )\] . D (z , z )" (2.47)      H (1#z )   2.2. Density perturbations The standard model for the formation of structure in the Universe assumes that there were small #uctuations at some very early initial time, which grew by gravitational instability. Although the origin of the seed #uctuations is yet unclear, they possibly originated from quantum #uctuations in the very early Universe, which were blown up during a later in#ationary phase. The #uctuations in this case are uncorrelated and the distribution of their amplitudes is Gaussian. Gravitational instability leads to a growth of the amplitudes of the relative density #uctuations. As long as the relative density contrast of the matter #uctuations is much smaller than unity, they can be considered as small perturbations of the otherwise homogeneous and isotropic background density, and linear perturbation theory su$ces for their description. The linear theory of density perturbations in an expanding universe is generally a complicated issue because it needs to be relativistic (e.g. Lifshitz, 1946; Bardeen, 1980). The reason is that perturbations on any length scale are comparable to or larger than the size of the horizon at su$ciently early times, and then Newtonian theory ceases to be applicable. In other words, since the horizon scale is comparable to the curvature radius of space}time, Newtonian theory fails for larger-scale perturbations due to non-zero space}time curvature. The main features can, nevertheless, be understood by fairly simple reasoning. We shall not present a rigorous mathematical treatment here, but only quote the results which are relevant for our later purposes. For a detailed qualitative and quantitative discussion, we refer the reader to the excellent discussion in Chapter 4 of the book by Padmanabhan (1993). 2.2.1. Horizon size The size of causally connected regions in the Universe is called the horizon size. It is given by the distance by which a photon can travel in the time t since the Big Bang. Since the appropriate time

 In this context, the size of the horizon is the distance ct by which light can travel in the time t since the Big Bang.

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scale is provided by the inverse Hubble parameter H\(a), the horizon size is d "cH\(a), and & the comoving horizon size is






\ c a c , (2.48) " X\a 1#  d " & aH(a) H  a  where we have inserted the Einstein}de Sitter limit (2.36) of Friedmann's equation. The length cH\"3h\ Gpc is called the Hubble radius. We shall see later that the horizon size at a plays   a very important ro( le for structure formation. Inserting a"a into Eq. (2.48), yields  c X\a+12(X h)\ Mpc , (2.49) d (a )"   &  (2H   where a from Eq. (2.28) has been inserted.  2.2.2. Linear growth of density perturbations We adopt the commonly held view that the density of the Universe is dominated by weakly interacting dark matter at the relatively late times which are relevant for weak gravitational lensing, a


a before a ,  (2.51) a after a  as long as the Einstein}de Sitter limit holds. For later times, a
g(a) ,d ag(a) , d(a)"d a   g(1)

(2.52)

where d is the density contrast linearly extrapolated to the present epoch, and the density dependent growth function g(a) is accurately "t by (Carroll et al., 1992)






5 X(a) g(a; X , XK )" X(a) X(a)!XK (a)# 1#  2 2




XK (a) 1# 70



\

.

(2.53)

The dependence of X and XK on the scale factor a is given in Eqs. (2.34). The growth function ag(a; X , XK ) is shown in Fig. 6 for a variety of parameters X and XK .   The cosmic microwave background reveals relative temperature #uctuations of order 10\ on large scales. By the Sachs}Wolfe e!ect (Sachs and Wolfe, 1967), these temperature #uctuations re#ect density #uctuations of the same order of magnitude. The cosmic microwave background originated at a+10\
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Fig. 6. The growth function ag(a),ag(a)/g(1) given in Eqs. (2.52) and (2.53) for X between 0.2 and 1.0 in steps of 0.2.  Top panel: XK "0; bottom panel: XK "1!X . The growth rate is constant for the Einstein}de Sitter model (X "1,   XK "0), while it is higher for a;1 and lower for a+1 for low-X models. Consequently, structure forms earlier in low than in high-X models. 

implies that the density #uctuations today, expected from the temperature #uctuations at a+10\, should only reach a level of 10\. Instead, structures (e.g. galaxies) with d<1 are observed. How can this discrepancy be resolved? The cosmic microwave background displays #uctuations in the baryonic matter component only. If there is an additional matter component that only couples through weak interactions, #uctuations in that component could grow as soon as it decoupled from the cosmic plasma, well before photons decoupled from baryons to set the cosmic microwave background free. Such #uctuations could therefore easily reach the amplitudes observed today, and thereby resolve the apparent mismatch between the amplitudes of the temperature #uctuations in the cosmic microwave background and the present cosmic structures. This is one of the strongest arguments for the existence of a dark-matter component in the Universe. 2.2.3. Suppression of growth It is convenient to decompose the density contrast d into Fourier modes. In linear perturbation theory, individual Fourier components evolve independently. A perturbation of (comoving) wavelength j is said to `enter the horizona when j"d (a). If j(d (a ), the perturbation enters & &  the horizon while radiation is still dominating the expansion. Until a , the expansion time scale,  t "H\, is determined by the radiation density o , which is shorter than the collapse time scale  0 of the dark matter, t : "+ t &(Go )\((Go )\&t . (2.54)  0 "+ "+

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Fig. 7. Sketch illustrating the suppression of structure growth during the radiation-dominated phase. The perturbation grows Ja before a , and Ja thereafter. If the perturbation is smaller than the horizon at a , it enters the horizon at   a (a while radiation is still dominating. The rapid radiation-driven expansion prevents the perturbation from   growing further. Hence, it stalls until a . By then, its amplitude is smaller by f "(a /a ) than it would be without     suppression.

In other words, the fast radiation-driven expansion prevents dark-matter perturbations from collapsing. Light can only cross regions that are smaller than the horizon size. The suppression of growth due to radiation is therefore restricted to scales smaller than the horizon, and larger-scale perturbations remain una!ected. This explains why the horizon size at a , d (a ), sets an  &  important scale for structure growth. Fig. 7 illustrates the growth of a perturbation with j(d (a ), that is small enough to enter the &  horizon at a (a . It can be read o! from the "gure that such perturbations are suppressed by   the factor




a  . f "   a 

(2.55)

It remains to be evaluated at what time a a density perturbation with comoving wavelength  j enters the horizon. The condition is c j"d (a )" . &  a H(a )  

(2.56)

Well in the Einstein}de Sitter regime, the Hubble parameter is given by Eq. (2.37). Inserting that expression into (2.56) yields



jJ

a  a 

(a

;a ) ,   (a ;a ;1) .  

(2.57)

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Let now k"j\ be the wave number of the perturbation, and k "d\(a ) the wave number  &  corresponding to the horizon size at a . The suppression factor (2.55) can then be written as 




k  . f "   k

(2.58)

From Eq. (2.49), k +0.083(X h) Mpc\+250(X h)(Hubble radii)\ .   

(2.59)

2.2.4. Density power spectrum The assumed Gaussian density #uctuations d(x) at the comoving position x can completely be characterised by their power spectrum P (k), which can be de"ned by (see Section 2.4) B 1dK (k)dK H(k)2"(2p)d (k!k)P (k) , " B

(2.60)

where dK (k) is the Fourier transform of d, and the asterisk denotes complex conjugation. Strictly speaking, the Fourier decomposition is valid only in #at space. However, at early times space is #at in any cosmological model, and at late times the interesting scales k\ of the density perturbations are much smaller than the curvature radius of the Universe. Hence, we can apply Fourier decomposition here. Consider now the primordial perturbation spectrum at some very early time, P (k)""dK (k)". Since the density contrast grows as dJaL\ [Eq. (2.51)], the spectrum grows as P (k)JaL\. At B a , the spectrum has therefore changed to  P (k)JaL\P (k)Jk\P (k) ,  

(2.61)

where Eq. (2.57) was used for k


k for k;k ,  P (k)J B k\ for k
(2.62)

An additional complication arises when the dark matter consists of particles moving with a velocity comparable to the speed of light. In order to keep them gravitationally bound, density perturbations then have to have a certain minimum mass, or equivalently a certain minimum size. All perturbations smaller than that size are damped away by free streaming of particles. Consequently, the density perturbation spectrum of such particles has an exponential cut-o! at large k. This clari"es the distinction between hot and cold dark matter: Hot dark matter (HDM) consists of fast particles that damp away small-scale perturbations, while cold dark-matter (CDM) particles are slow enough to cause no signi"cant damping.

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2.2.5. Normalisation of the power spectrum Apart from the shape of the power spectrum, its normalisation has to be "xed. Several methods are available which usually yield di!erent answers: (1) Normalisation by microwave-background anisotropies: The COBE satellite has measured #uctuations in the temperature of the microwave sky at the rms level of *¹/¹&1.3;10\ at an angular scale of &73 (Banday et al., 1997). Adopting a shape for the power spectrum, these #uctuations can be translated into an amplitude for P (k). Due to the large angular scale of the B measurement, this kind of amplitude determination speci"es the amplitude on large physical scales (small k) only. In addition, microwave-background #uctuations measure the amplitude of scalar and tensor perturbation modes, while the growth of density #uctuations is determined by the #uctuation amplitude of scalar modes only. (2) Normalisation by the local variance of galaxy counts, pioneered by Davis and Peebles (1983): Galaxies are supposed to be biased tracers of underlying dark-matter #uctuations (Kaiser, 1984; Bardeen et al., 1986; White et al., 1987). By measuring the local variance of galaxy counts within certain volumes, and assuming an expression for the bias, the amplitude of dark-matter #uctuations can be inferred. Conventionally, the variance of galaxy counts p is measured    within spheres of radius 8h\ Mpc, and the result is p +1. The problem of "nding the    corresponding variance p of matter-density #uctuations is that the exact bias mechanism of  galaxy formation is still under debate (e.g. Kau!mann et al., 1997). (3) Normalisation by the local abundance of galaxy clusters (White et al., 1993; Eke et al., 1996; Viana and Liddle, 1996): Galaxy clusters form by gravitational instability from dark-matterdensity perturbations. Their spatial number density re#ects the amplitude of appropriate dark-matter #uctuations in a very sensitive manner. It is therefore possible to determine the amplitude of the power spectrum by demanding that the local spatial number density of galaxy clusters be reproduced. Typical scales for dark-matter #uctuations collapsing to galaxy clusters are of order 10h\ Mpc, hence cluster normalisation determines the amplitude of the power spectrum on just that scale. Since gravitational lensing by large-scale structures is generally sensitive to scales comparable to k\&12(X h) Mpc, cluster normalisation appears to be the most appropriate normalisation   method for the present purposes. The solid curve in Fig. 8 shows the CDM power spectrum, linearly and non-linearly evolved to z"0 (or a"1) in an Einstein}de Sitter universe with h"0.5, normalised to the local cluster abundance. 2.2.6. Non-linear evolution At late stages of the evolution and on small scales, the growth of density #uctuations begins to depart from the linear behaviour of Eq. (2.52). Density #uctuations grow non-linearly, and #uctuations of di!erent size interact. Generally, the evolution of P(k) then becomes complicated and needs to be evaluated numerically. However, starting from the bold ansatz that the two-point correlation functions in the linear and non-linear regimes are related by a general scaling relation (Hamilton et al., 1991), which turns out to hold remarkably well, analytic formulae describing the non-linear behaviour of P(k) have been derived (Jain et al., 1995; Peacock and Dodds, 1996). It will turn out in subsequent chapters that the non-linear evolution of the density #uctuations is crucial for accurately calculating weak-lensing e!ects by large-scale structures. As an example, we show as

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Fig. 8. CDM power spectrum, normalised to the local abundance of galaxy clusters, for an Einstein-de Sitter universe with h"0.5. Two curves are displayed. The solid curve shows the linear, the dashed curve the non-linear power spectrum. While the linear power spectrum asymptotically falls o! Jk\, the non-linear power spectrum, according to Peacock and Dodds (1996), illustrates the increased power on small scales due to non-linear e!ects, at the expense of larger-scale structures.

the dashed curve in Fig. 8 the CDM power spectrum in an Einstein}de Sitter universe with h"0.5, normalised to the local cluster abundance, non-linearly evolved to z"0. The non-linear e!ects are immediately apparent: While the spectrum remains unchanged for large scales (k;k ),  the amplitude on small scales (k
U"4pGo , P

(2.63)

where o"(1#d)o is the total matter density, and U is the sum of the potentials of the smooth background UM and the potential of the perturbation U. The gradient operates with respect to the P physical, or proper, coordinates. Since Poisson's equation is linear, we can subtract the background contribution UM "4pGo. Introducing the gradient with respect to comoving coordinates P

"a , we can write Eq. (2.63) in the form V P

U"4pGaod . V

(2.64)

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In the matter-dominated epoch, o"a\o . With the critical density (2.15), Poisson's equation can  be re-written as 3H

U"  X d . V 2a 

(2.65)

2.3. Relevant properties of lenses and sources Individual reviews have been written on galaxies (e.g. Faber and Gallagher, 1979; Binggeli et al., 1988; Giovanelli and Haynes, 1991; Koo and Kron, 1992; Ellis, 1997), clusters of galaxies (e.g. Bahcall, 1977; Rood, 1981; Forman and Jones, 1982; Bahcall, 1988; Sarazin, 1986), and active galactic nuclei (e.g. Rees, 1984; Weedman, 1986; Blandford et al., 1990; Hartwick and Schade, 1990; Warren and Hewett, 1990; Antonucci, 1993; Peterson, 1997). A detailed presentation of these objects is not the purpose of this review. It su$ces here to summarise those properties of these objects that are relevant for understanding the following discussion. Properties and peculiarities of individual objects are not necessary to know; rather, we need to specify the objects statistically. This section will therefore focus on a statistical description, leaving subtleties aside. 2.3.1. Galaxies For the purposes of this review, we need to characterise the statistical properties of galaxies as a class. Galaxies can broadly be grouped into two populations, dubbed early- and late-type galaxies, or ellipticals and spirals, respectively. While spiral galaxies include disks structured by more or less pronounced spiral arms, and approximately spherical bulges centred on the disk centre, elliptical galaxies exhibit amorphous projected light distributions with roughly elliptical isophotes. There are, of course, more elaborate morphological classi"cation schemes (e.g. de Vaucouleurs et al., 1991; Buta et al., 1994; Naim et al., 1995a, 1995b), but the broad distinction between ellipticals and spirals su$ces for this review. Outside galaxy clusters, the galaxy population consists of about  spiral galaxies and  elliptical   galaxies, while the fraction of ellipticals increases towards cluster centres. Elliptical galaxies are typically more massive than spirals. They contain little gas, and their stellar population is older, and thus &redder', than in spiral galaxies. In spirals, there is a substantial amount of gas in the disk, providing the material for ongoing formation of new stars. Likewise, there is little dust in ellipticals, but possibly large amounts of dust are associated with the gas in spirals. Massive galaxies have of order 10 solar masses, or 2;10 g within their visible radius. Such galaxies have luminosities of order 10 times the solar luminosity. The kinematics of the stars, gas and molecular clouds in galaxies, as revealed by spectroscopy, indicate that there is a relation between the characteristic velocities inside galaxies and their luminosity (Faber and Jackson, 1976; Tully and Fisher, 1977); brighter galaxies tend to have larger masses. The di!erential luminosity distribution of galaxies can very well be described by the functional form U(¸)









¸ \J ¸ d¸ d¸ exp ! "U ,  ¸ ¸ ¸ ¸ H H H H

(2.66)

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proposed by Schechter (1976). The parameters have been measured to be l+1.1, ¸ +1.1;10¸ , U +1.5;10\h Mpc\ (2.67) H > H (e.g. Efstathiou et al., 1988; Marzke et al., 1994a,b). This distribution means that there is essentially a sharp cut-o! in the galaxy population above luminosities of &¸ , and the mean separation H between ¸ -galaxies is of order &U\+4h\ Mpc. H H The stars in elliptical galaxies have randomly oriented orbits, while by far the most stars in spirals have orbits roughly coplanar with the galactic disks. Stellar velocities are therefore characterised by a velocity dispersion p in ellipticals, and by an asymptotic circular velocity v in T  spirals. These characteristic velocities are related to galaxy luminosities by laws of the form




¸ ? v p T " "  , (2.68) ¸ v p H H TH where a ranges around 3}4. For spirals, Eq. (2.68) is called Tully}Fisher (1977) relation, for ellipticals Faber}Jackson (1976) relation. Both velocity scales p and v are of order 220 km s\. TH H Since v "(2p , ellipticals with the same luminosity are more massive than spirals. T  Most relevant for weak gravitational lensing is a population of faint galaxies emitting bluer light than local galaxies, the so-called faint blue galaxies (Tyson, 1988; see Ellis, 1997 for a review). There are of order 30}50 such galaxies per square arcminute on the sky which can be mapped with current ground-based optical telescopes, i.e. there are +20,000}40,000 such galaxies on the area of the full moon. The picture that the sky is covered with a &wall paper' of those faint and presumably distant blue galaxies is therefore justi"ed. It is this "ne-grained pattern on the sky that makes many weak-lensing studies possible in the "rst place, because it allows the detection of the coherent distortions imprinted by gravitational lensing on the images of the faint blue galaxy population. Due to their faintness, redshifts of the faint blue galaxies are hard to measure spectroscopically. The following picture, however, seems to be reasonably secure. It has emerged from increasingly deep and detailed observations (see e.g. Broadhurst et al., 1988; Colless et al., 1991, 1993; Lilly et al., 1991; Lilly, 1993; Crampton et al., 1995; and also the reviews by Koo and Kron, 1992; Ellis, 1997). The redshift distribution of faint galaxies has been found to agree fairly well with that expected for a non-evolving comoving number density. While the galaxy number counts in blue light are substantially above an extrapolation of the local counts down to increasingly faint magnitudes, those in the red spectral bands agree fairly well with extrapolations from local number densities. Further, while there is signi"cant evolution of the luminosity function in the blue, in that the luminosity scale ¸ of a Schechter-type "t increases with redshift, the luminosity function of the H galaxies in the red shows little sign of evolution. Highly resolved images of faint blue galaxies obtained with the Hubble Space Telescope are now becoming available. In red light, they reveal mostly ordinary spiral galaxies, while their substantial emission in blue light is more concentrated to either spiral arms or bulges. Spectra exhibit emission lines characteristic of star formation.

 The circular velocity of stars and gas in spiral galaxies turns out to be fairly independent of radius, except close to their centre. These #at rotations curves cannot be caused by the observable matter in these galaxies, but provide strong evidence for the presence of a dark halo, with density pro"le oJr\ at large radii.

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These "ndings support the view that the galaxy evolution towards higher redshifts apparent in blue light results from enhanced star-formation activity taking place in a population of galaxies which, apart from that, may remain unchanged even out to redshifts of z91. The redshift distribution of the faint blue galaxies is then su$ciently well described by




z @ b z exp ! dz . (2.69) p(z) dz" z z C(3/b)   This expression is normalised to 04z(R and provides a good "t to the observed redshift distribution (e.g. Smail et al., 1995b). The mean redshift 1z2 is proportional to z , and the  parameter b describes how steeply the distribution falls o! beyond z . For b"1.5, 1z2+1.5z .   The parameter z depends on the magnitude cuto! and the colour selection of the galaxy sample.  Background galaxies would be ideal tracers of distortions caused by gravitational lensing if they were intrinsically circular. Then, any measured ellipticity would directly re#ect the action of the gravitational tidal "eld of the lenses. Unfortunately, this is not the case. To "rst approximation, galaxies have intrinsically elliptical shapes, but the ellipses are randomly oriented. The intrinsic ellipticities introduce noise into the inference of the tidal "eld from observed ellipticities, and it is important for the quanti"cation of the noise to know the intrinsic ellipticity distribution. Let "e" be the ellipticity of a galaxy image, de"ned such that for an ellipse with axes a and b(a, a!b . "e", a#b

(2.70)

Ellipses have an orientation, hence the ellipticity has two components e , with "e""(e #e ). It    turns out empirically that a Gaussian is a good description for the ellipticity distribution, exp(!"e"/p) C p (e , e ) de de " de de (2.71) C     pp[1!exp(!1/p)]   C C with a characteristic width of p +0.2 (e.g. Miralda-Escude, 1991; Tyson and Seitzer, 1988; C Brainerd et al., 1996). We will later (Section 4.2) de"ne galaxy ellipticities for the general situation where the isophotes are not ellipses. This completes our summary of galaxy properties as required here. 2.3.2. Groups and clusters of galaxies Galaxies are not randomly distributed in the sky. Their positions are correlated, and there are areas in the sky where the galaxy density is noticeably higher or lower than average (cf. the galaxy count map in Fig. 9). There are groups consisting of a few galaxies, and there are clusters of galaxies in which some hundred up to a 1000 galaxies appear very close together. The most prominent galaxy cluster in the sky covers a huge area centred on the Virgo constellation. Its central region has a diameter of about 73, and its main body extends over roughly 153;403. It was already noted by Sir William Herschel in the 18th century that the entire Virgo cluster covers about th of the sky, while containing about rd of the galaxies observable at   that time. Zwicky (1933) noted that the galaxies in the Coma cluster and other rich clusters move so fast that the clusters required about ten to 100 times more mass to keep the galaxies bound than could

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319

Fig. 9. The Lick galaxy counts within 503 radius around the North Galactic pole (Seldner et al., 1977). The galaxy number density is highest at the black and lowest at the white regions on the map. The picture illustrates structure in the distribution of fairly nearby galaxies, viz., under-dense regions, long extended "laments, and clusters of galaxies.

be accounted for by the luminous galaxies themselves. This was the earliest indication that there is invisible mass, or dark matter, in at least some objects in the Universe. Several thousands of galaxy clusters are known today. The Abell (1958) cluster catalog lists 2712 clusters north of !203 declination and away from the Galactic plane. Employing a less restrictive de"nition of galaxy clusters, the catalog by Zwicky et al. (1968) identi"es 9134 clusters north of !33 declination. Cluster masses can exceed 10 g or 5;10M , and they have typical radii of > +5;10 cm or +1.5 Mpc. When X-ray telescopes became available after 1966, it was discovered that clusters are powerful X-ray emitters. Their X-ray luminosities fall within (10}10) erg s\, rendering galaxy clusters the most luminous X-ray sources in the sky. Improved X-ray telescopes revealed that the source of X-ray emission in clusters is extended rather than point-like, and that the X-ray spectra are best explained by thermal bremsstrahlung (free}free radiation) from a hot, dilute plasma with temperatures in the range (10}10) K and densities of &10\ particles per cm. Based on the assumption that this intra-cluster gas is in hydrostatic equilibrium with a spherically symmetric gravitational potential of the total cluster matter, the X-ray temperature and #ux can be used to estimate the cluster mass. Typical results approximately (i.e. up to a factor of &2) agree with the mass estimates from the kinematics of cluster galaxies employing the virial theorem. The mass of the intra-cluster gas amounts to about 10% of the total cluster mass. The X-ray emission thus independently con"rms the existence of dark matter in galaxy clusters. Sarazin (1986) reviews clusters of galaxies focusing on their X-ray emission. Later, luminous arc-like features were discovered in two galaxy clusters (Lynds and Petrosian, 1986; Soucail et al., 1987a,b; see Fig. 10). Their light is typically bluer than that from the cluster

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Fig. 10. The galaxy cluster Abell 370, in which the "rst gravitationally lensed arc was detected (Lynds and Petrosian, 1986; Soucail et al., 1987a,b). Most of the bright galaxies seen are cluster members at z"0.37, whereas the arc, i.e. the highly elongated feature, is the image of a galaxy at redshift z"0.724 (Soucail et al., 1988) (courtesy of J.-P. Kneib).

galaxies, and their length is comparable to the size of the central cluster region. PaczynH ski (1987) suggested that these arcs are images of galaxies in the background of the clusters which are strongly distorted by the gravitational tidal "eld close to the cluster centres. This explanation was generally accepted when spectroscopy revealed that the sources of the arcs are much more distant than the clusters in which they appear (Soucail et al., 1988). Large arcs require special alignment of the arc source with the lensing cluster. At larger distance from the cluster centre, images of background galaxies are only weakly deformed, and they are referred to as arclets (Fort et al., 1988; Tyson et al., 1990). The high number density of faint arclets allows one to measure the coherent distortion caused by the tidal gravitational "eld of the cluster out to fairly large radii. One of the main applications of weak gravitational lensing is to reconstruct the (projected) mass distribution of galaxy clusters from their measurable tidal "elds. Consequently, the corresponding theory constitutes one of the largest sections of this review. Such strong and weak gravitational lens e!ects o!er the possibility to detect and measure the entire cluster mass, dark and luminous, without referring to any equilibrium or symmetry assumptions like those required for the mass estimates from galactic kinematics or X-ray emission. For a review on arcs and arclets in galaxy clusters see Ford and Mellier (1994).

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Apart from being spectacular objects in their own right, clusters are also of particular interest for cosmology. Being the largest gravitationally bound entities in the cosmos, they represent the high-mass end of collapsed structures. Their number density, their individual properties, and their spatial distribution constrain the power spectrum of the density #uctuations from which the structure in the universe is believed to have originated (e.g. Viana and Liddle, 1996; Eke et al., 1996). Their formation history is sensitive to the parameters that determine the geometry of the universe as a whole. If the matter density in the universe is high, clusters tend to form later in cosmic history than if the matter density is low ("rst noted by Richstone et al., 1992). This is due to the behaviour of the growth factor shown in Fig. 6, combined with the Gaussian nature of the initial density #uctuations. Consequently, the compactness and the morphology of clusters re#ect the cosmic matter density, and this has various observable implications. One method to normalise the density-perturbation power spectrum "xes its overall amplitude such that the local spatial number density of galaxy clusters is reproduced. This method, called cluster normalisation and pioneered by White et al. (1993), will frequently be used in this review. In summary, clusters are not only regions of higher galaxy number density in the sky, but they are gravitationally bound bodies whose member galaxies contribute only a small fraction of their mass. About 80% of their mass is dark, and roughly 10% is in the form of the di!use, X-ray emitting gas spread throughout the cluster. Mass estimates inferred from galaxy kinematics, X-ray emission, and gravitational-lensing e!ects generally agree to within about a factor of two, typically arriving at masses of order 5;10 solar masses, or 10 g. Typical sizes of galaxy clusters are of order several megaparsecs, or 5;10 cm. In addition, there are smaller objects, called galaxy groups, which contain fewer galaxies and have typical masses of order 10 solar masses. 2.3.3. Active galactic nuclei The term &active galactic nuclei' (AGNs) is applied to galaxies which show signs of non-stellar radiation in their centres. Whereas the emission from &normal' galaxies like our own is completely dominated by radiation from stars and their remnants, the emission from AGNs is a combination of stellar light and non-thermal emission from their nuclei. In fact, the most prominent class of AGNs, the quasi-stellar radio sources, or quasars, have their names derived from the fact that their optical appearance is point-like. The nuclear emission almost completely outshines the extended stellar light of its host galaxy. AGNs do not form a homogeneous class of objects. Instead, they are grouped into several types. The main classes are: quasars, quasi-stellar objects (QSOs), Seyfert galaxies, BL Lacertae objects (BL Lacs), and radio galaxies. What uni"es them is the non-thermal emission from their nucleus, which manifests itself in various ways: (1) radio emission which, owing to its spectrum and polarisation, is interpreted as synchrotron radiation from a power-law distribution of relativistic electrons; (2) strong ultraviolet and optical emission lines from highly ionised species, which in some cases can be extremely broad, corresponding to Doppler velocities up to &20,000 km s\, thus indicating the presence of semi-relativistic velocities in the emission region; (3) a #at ultraviolet-to-optical continuum spectrum, often accompanied by polarisation of the optical light, which cannot naturally be explained by a superposition of stellar (Planck) spectra; (4) strong X-ray emission with a hard power-law spectrum, which can be interpreted as inverse Compton radiation by a population of relativistic electrons with a power-law energy distribution; (5) strong c-ray emission; (6) variability at all wavelengths, from the radio to the c-ray regime. Not all these

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phenomena occur at the same level in all the classes of AGNs. QSOs, for example, can roughly be grouped into radio-quiet QSOs and quasars, the latter emitting strongly at radio wavelengths. Since substantial variability cannot occur on time scales shorter than the light-travel time across the emitting region, the variability provides a rigorous constraint on the compactness of the region emitting the bulk of the nuclear radiation. In fact, this causality argument based on light-travel time can mildly be violated if relativistic velocities are present in the emitting region. Direct evidence for this comes from the observation of the so-called superluminal motion, where radiosource components exhibit apparent velocities in excess of c (e.g. Zensus and Pearson, 1987). This can be understood as a projection e!ect, combining velocities close to (but of course smaller than) the velocity of light with a velocity direction close to the line-of-sight to the observer. Observations of superluminal motion indicate that bulk velocities of the radio-emitting plasma components can have Lorentz factors of order 10, i.e., they move at &0.99c. The standard picture for the origin of this nuclear activity is that a supermassive black hole (or order 10M ), situated in the centre of the host galaxy, accretes gas from the host. In this process, > gravitational binding energy is released, part of which can be transformed into radiation. The appearance of an AGN then depends on the black-hole mass and angular momentum, the accretion rate, the e$ciency of the transformation of binding energy into radiation, and on the orientation relative to the line-of-sight. The understanding of the physical mechanisms in AGNs, and how they are related to their phenomenology, is still rather incomplete. We refer the reader to the books and articles by Begelman et al. (1984), Weedman (1986), Blandford et al. (1990), Peterson (1997), and Krolik (1999), and references therein, for an overview of the phenomena in AGNs, and of our current ideas on their interpretation. For the current review, we only make use of one particular property of AGNs: QSOs can be extremely luminous. Their optical luminosity can reach a factor of thousand or more times the luminosity of normal galaxies. Therefore, their nuclear activity completely outshines that of the host galaxy, and the nuclear sources appear point-like on optical images. Furthermore, the high luminosity implies that QSOs can be seen to very large distances, and in fact, until a few years ago QSOs held the redshift record. In addition, the comoving number density of QSOs evolves rapidly with redshift. It was larger than today by a factor of &100 at redshifts between 2 and 3. Taken together, these two facts imply that a #ux-limited sample of QSOs has a very broad redshift distribution, in particular, very distant objects are abundant in such a sample. However, it is quite di$cult to obtain a &complete' #ux-limited sample of QSOs. Of all point-like objects at optical wavelengths, QSOs constitute only a tiny fraction, most being stars. Hence, morphology alone does not su$ce to obtain a candidate QSO sample which can be veri"ed spectroscopically. However, QSOs are found to have very blue optical colours, by which they can e$ciently be selected. Colour selection typically yields equal numbers of white dwarfs and QSOs with redshifts below &2.3. For higher-redshift QSOs, the strong Lya emission line moves from the U-band "lter into the B-band, yielding redder U}B colours. For these higher-redshift QSOs, multi-colour or emission-line selection criteria must be used (cf. Fan et al., 1999). In contrast to optical selection, AGNs are quite e$ciently selected in radio surveys. The majority of sources selected at centimeter wavelengths are AGNs. A #ux-limited sample of radio-selected AGNs also has a very broad redshift distribution. The large fraction of distant objects in these samples make

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AGNs particularly promising sources for the gravitational lensing e!ect, as the probability of "nding an intervening mass concentration close to the line-of-sight increases with the source distance. In fact, most of the known multiple-image gravitational lens systems have AGN sources. In addition to their high redshifts, the number counts of AGNs are important for lensing. For bright QSOs with apparent B-band magnitudes B:19, the di!erential source counts can be approximated by a power law, n(S)JS\?>, where n(S) dS is the number density of QSOs per unit solid angle with #ux within dS of S, and a+2.6. At fainter magnitudes, the di!erential source counts can also be approximated by a power law in #ux, but with a much #atter index of a&0.5. The source counts at radio wavelengths are also quite steep for the highest #uxes, and #atten as the #ux decreases. The steepness of the source counts will be the decisive property of AGNs for the magni"cation bias, which will be discussed in Section 6. 2.4. Correlation functions, power spectra, and their projections 2.4.1. Dexnitions; homogeneous and isotropic random xelds In this subsection, we de"ne the correlation function and the power spectrum of a random "eld, which will be used extensively in later sections. One example already occurred above, namely the power spectrum P of the density #uctuation "eld d. B Consider a random "eld g(x) whose expectation value is zero everywhere. This means that an average over many realisations of the random "eld should vanish, 1g(x)2"0, for all x. This is not an important restriction, for if that was not the case, we could consider the "eld g(x)!1g(x)2 instead, which would have the desired property. Spatial positions x have n dimensions, and the "eld can be either real or complex. A random "eld g(x) is called homogeneous if it cannot statistically be distinguished from the "eld g(x#y), where y is an arbitrary translation vector. Similarly, a random "eld g(x) is called isotropic if it has the same statistical properties as the random "eld g(Rx), where R is an arbitrary rotation matrix in n dimensions. Restricting our attention to homogeneous and isotropic random "elds, we note that the two-point correlation function 1g(x)gH(y)2"C ("x!y") (2.72) EE can only depend on the absolute value of the di!erence vector between the two points x and y. Note that C is real, even if g is complex. This can be seen by taking the complex conEE jugate of (2.72), which is equivalent to interchanging x and y, leaving the right-hand side una!ected. We de"ne the Fourier-transform pair of g as



g( (k)"

1L



dLx g(x)e x  k; g(x)"

dLk g( (k)e\ x  k . (2p)L 1L

(2.73)

We now calculate the correlation function in Fourier space



1g( (k)g( H(k)2"

1L

dLx e x  k



1L

dLx e\ xY kY1g(x)gH(x)2 .

(2.74)

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Using (2.72) and substituting x"x#y, this becomes



1g( (k)g( H(k)2"

1L

dLx e x  k



1L

dLy e\ x>y kYC ("y") EE



"(2p)Ld (k!k) "

dLy e\ y  kC ("y") EE 1L

,(2p)Ld (k!k)P ("k") . (2.75) " E In the "nal step, we de"ned the power spectrum of the homogeneous and isotropic random "eld g,



P ("k")" E

dLy e\ y  kC ("y") , EE 1L

(2.76)

which is the Fourier transform of the two-point correlation function. Isotropy of the random "eld implies that P can only depend on the modulus of k. E Gaussian random xelds are characterised by the property that the probability distribution of any linear combination of the random "eld g(x) is Gaussian. More generally, the joint probability distribution of a number M of linear combinations of the random variable g(x ) is a multivariate G Gaussian. This is equivalent to requiring that the Fourier components g( (k) are mutually statistically independent, and that the probability densities for the g( (k) are Gaussian with dispersion P ("k"). Thus, a Gaussian random "eld is fully characterised by its power spectrum. E 2.4.2. Projections; Limber's equation We now derive a relation between the power spectrum (or the correlation function) of a homogeneous isotropic random "eld in three dimensions, and its projection onto two dimensions. Speci"cally, for the three-dimensional "eld, we consider the density contrast d[f (w)h, w], where h is ) a two-dimensional vector, which could be an angular position on the sky. Hence, f (w)h and w form ) a local comoving isotropic Cartesian coordinate system. We de"ne two di!erent projections of d along the backward-directed light cone of the observer at w"0, t"t , 



g (h)" dw q (w)d[f (w)h, w] G ) G

(2.77)

for i"1, 2. The q (w) are weight functions, and the integral extends from w"0 to the horizon G w"w . Since d is a homogeneous and isotropic random "eld, so is its projection. Consider now & the correlation function C "1g (h)g (h)2   





" dw q (w) dwq (w)1d[ f (w)h, w]d[f (w)h, w]2 .   ) )

(2.78)

We assume that there is no power in the density #uctuations on scales larger than a coherence scale ¸ . This is justi"ed because the power spectrum P declines Jk as kP0; see (2.62). This implies  B that the correlation function on the right-hand side of Eq. (2.78) vanishes for w <"w!w"9¸ . &  Although d evolves cosmologically, it can be considered constant over a time scale on which light

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travels across a comoving distance ¸ . We note that the second argument of d simultaneously  denotes the third local spatial dimension and the cosmological epoch, related through the light-cone condition "c dt""a dw. Furthermore, we assume that the weight functions q (w) do not G vary appreciably over a scale *w4¸ . Consequently, "w!w":¸ over the scale where C is   BB non-zero, and we can set f (w)+f (w) and q (w)"q (w) to obtain ) )  





C (h)" dw q (w)q (w) d(*w)C ((f  (w)h#(*w), w) .   BB ) 

(2.79)

The second argument of C now denotes the dependence of the correlation function on cosmic BB epoch. Eq. (2.79) is one form of the Limber (1953) equation, which relates the two-point correlation of the projected "eld to that of the three-dimensional "eld. Another very useful form of this equation relates the projected two-point correlation function to the power spectrum of the three-dimensional "eld. The easiest way to derive this relation is by replacing the d's in (2.78) by their Fourier transforms, where upon





 

dk dk C " dw q (w) dwq (w)    (2p) (2p) ;1dK (k, w)dK H(k, w)2e\ D) Uk,  he D) UYk,  hYe\ I Ue I UY .

(2.80)

k is the two-dimensional wave vector perpendicular to the line-of-sight. The correlator can be , replaced by the power spectrum P using (2.75). This introduces a Dirac delta function d (k!k), B " which allows us to carry out the k-integration. Under the same assumptions on the spatial variation of q (w) and f (w) as before, we "nd G ) dk C " dw q (w)q (w) P ("k", w) e\ D) Uk, h\hY e\ I U dw e I UY . (2.81)    (2p) B







The "nal integral yields 2pd (k ), indicating that only such modes contribute to the projected "  correlation function whose wave vectors lie in the plane of the sky (Blandford et al., 1991). Finally, carrying out the trivial k -integration yields  dk , P ("k ", w) e\ D) Uk,  h C (h)" dw q (w)q (w) (2.82)    (2p) B ,

 

 

kdk " dw q (w)q (w) P(k, w)J [f (w)hk] .    ) 2p

(2.83)

De"nition (2.73) of the Fourier transform, and relation (2.76) between power spectrum and correlation function allow us to write the (cross) power spectrum P (l) as 

  

P (l)" dh C (h)e l  h  



dk , P ("k ", w)(2p)d [l!f (w)k ] " dw q (w)q (w)   " I , (2p) B ,






l q (w)q (w)  P " dw  ,w , B f (w) f  (w) ) )

(2.84)

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which is Limber's equation in Fourier space (Kaiser, 1992,1998). We shall make extensive use of these relations in later sections.

3. Gravitational light de6ection In this section, we summarise the theoretical basis for the description of light de#ection by gravitational "elds. Granted the validity of Einstein's Theory of General Relativity, light propagates on the null geodesics of the space}time metric. However, most astrophysically relevant situations permit a much simpler approximate description of light rays, which is called gravitational lens theory; we "rst describe this theory in Section 3.1. It is su$cient for the treatment of lensing by galaxy clusters in Section 5, where the de#ecting mass is localised in a region small compared to the distance between source and de#ector, and between de#ector and observer. In contrast, mass distributions on a cosmic scale cause small light de#ections all along the path from the source to the observer. The magni"cation and shear e!ects resulting therefrom require a more general description, which we shall develop in Section 3.2. In particular, we outline how the gravitational lens approximation derives from this more general description. 3.1. Gravitational lens theory A typical situation considered in gravitational lensing is sketched in Fig. 11, where a mass concentration at redshift z (or angular diameter distance D ) de#ects the light rays from a source   at redshift z (or angular diameter distance D ). If there are no other de#ectors close to the   line-of-sight, and if the extent of the de#ecting mass along the line-of-sight is very much smaller than both D and the angular diameter distance D from the de#ector to the source, the actual   light rays which are smoothly curved in the neighbourhood of the de#ector can be replaced by two straight rays with a kink near the de#ector. The magnitude and direction of this kink is described by the deyection angle a( , which depends on the mass distribution of the de#ector and the impact vector of the light ray. 3.1.1. The deyection angle Consider "rst the de#ection by a point mass M. If the light ray does not propagate through the strong gravitational "eld close to the horizon, that is, if its impact parameter m is much larger than the Schwarzschild radius of the lens, m
(3.1)

This is just twice the value obtained in Newtonian gravity (see the historical remarks in Schneider et al., 1992). According to the condition m
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Fig. 11. Sketch of a typical gravitational lens system.

The "eld equations of General Relativity can be linearised if the gravitational "eld is weak. The de#ection angle of an ensemble of point masses is then the (vectorial) sum of the de#ections due to individual lenses. Consider now a three-dimensional mass distribution with volume density o(r). We can divide it into cells of size d< and mass dm"o(r) d<. Let a light ray pass this mass distribution, and describe its spatial trajectory by (m (j), m (j), r (j)), where the coordinates are    chosen such that the incoming light ray (i.e. far from the de#ecting mass distribution) propagates along r . The actual light ray is de#ected, but if the de#ection angle is small, it can be approximated  as a straight line in the neighbourhood of the de#ecting mass. This corresponds to the Born approximation in atomic and nuclear physics. Then, n(j),n, independent of the a$ne parameter j. Note that n"(m , m ) is a two-dimensional vector. The impact vector of the light ray relative to the   mass element dm at r"(m , m , r ) is then n!n, independent of r , and the total de#ection angle is     n!n 4G a( (n)" dm(m , m , r )    "n!n" c

 

4G n!n " dm dr o(m , m , r ) ,     "n!n" c

(3.2)

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which is also a two-dimensional vector. Since the last factor in Eq. (3.2) is independent of r , the  r -integration can be carried out by de"ning the surface mass density 



R(n), dr o(m , m , r ) ,    

(3.3)

which is the mass density projected onto a plane perpendicular to the incoming light ray. Then, the de#ection angle "nally becomes



n!n 4G dm R(n) . a( (n)" "n!n" c

(3.4)

This expression is valid as long as the deviation of the actual light ray from a straight (unde#ected) line within the mass distribution is small compared to the scale on which the mass distribution changes signi"cantly. This condition is satis"ed in virtually all astrophysically relevant situations (i.e. lensing by galaxies and clusters of galaxies), unless the de#ecting mass extends all the way from the source to the observer (a case which will be dealt with in Section 6). It should also be noted that in a lensing situation such as displayed in Fig. 11, the incoming light rays are not mutually parallel, but fall within a beam with opening angle approximately equal to the angle which the mass distribution subtends on the sky. This angle, however, is typically very small (in the case of cluster lensing, the relevant angular scales are of order 1 arcmin+2.9;10\). 3.1.2. The lens equation We now require an equation which relates the true position of the source to its observed position on the sky. As sketched in Fig. 11, the source and lens planes are de"ned as planes perpendicular to a straight line (the optical axis) from the observer to the lens at the distance of the source and of the lens, respectively. The exact de"nition of the optical axis does not matter because of the smallness of angles involved in a typical lens situation, and the distance to the lens is well de"ned for a geometrically thin matter distribution. Let g denote the two-dimensional position of the source on the source plane. Recalling the de"nition of the angular-diameter distance, we can read o! Fig. 11 D (3.5) g"  n!D a( (n) .  D  Introducing angular coordinates by g"D b and n"D h, we can transform Eq. (3.5) to   D b"h!  a( (D h),h!a(h) , (3.6)  D  where we de"ned the scaled de#ection angle a(h) in the last step. The interpretation of the lens equation (3.6) is that a source with true position b can be seen by an observer at angular positions h satisfying (3.6). If (3.6) has more than one solution for "xed b, a source at b has images at several positions on the sky, i.e. the lens produces multiple images. For this to happen, the lens must be &strong'. This can be quanti"ed by the dimensionless surface mass density R(D h)  i(h)" R 

D c  , with R "  4pG D D  

(3.7)

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where R is called the critical surface mass density (which depends on the redshifts of source and  lens). A mass distribution which has i51 somewhere, i.e. R5R , produces multiple images for  some source positions b (see Schneider et al., 1992, Section 5.4.3). Hence, R is a characteristic  value for the surface mass density which distinguishes between &weak' and &strong' lenses. Note that i51 is su$cient but not necessary for producing multiple images. In terms of i, the scaled de#ection angle reads 1 a(h)" p



1

dh i(h)

h!h . "h!h"

(3.8)

Eq. (3.8) implies that the de#ection angle can be written as the gradient of the deyection potential 1 t(h)" p



1

dh i(h) ln"h!h"

(3.9)

as a" t. The potential t(h) is the two-dimensional analogue of the Newtonian gravitational potential and satis"es the Poisson equation t(h)"2i(h). 3.1.3. Magnixcation and distortion The solutions h of the lens equation yield the angular positions of the images of a source at b. The shapes of the images will di!er from the shape of the source because light bundles are de#ected di!erentially. The most visible consequence of this distortion is the occurrence of giant luminous arcs in galaxy clusters. In general, the shape of the images must be determined by solving the lens equation for all points within an extended source. Liouville's theorem and the absence of emission and absorption of photons in gravitational light de#ection imply that lensing conserves surface brightness (or speci"c intensity). Hence, if IQ(b) is the surface-brightness distribution in the source plane, the observed surface-brightness distribution in the lens plane is I(h)"IQ[b(h)] .

(3.10)

If a source is much smaller than the angular scale on which the lens properties change, the lens mapping can locally be linearised. The distortion of images is then described by the Jacobian matrix









1!i!c !c Rt(h) Rb   " , A(h)" " d ! GH Rh Rh Rh !c 1!i#c G H   where we have introduced the components of the shear c,c #ic ""c"e P,   c "(t !t ), c "t       and i is related to t through Poisson's equation. Hence, if h is a point within an  corresponding to the point b "b(h ) within the source, we "nd from (3.10) using the   linearised lens equation I(h)"IQ[b #A(h ) ) (h!h )] .   

(3.11)

(3.12) image, locally (3.13)

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According to this equation, the images of a circular source are ellipses. The ratios of the semi-axes of such an ellipse to the radius of the source are given by the inverse of the eigenvalues of A(h ),  which are 1!i$"c", and the ratio of the solid angles subtended by an image and the unlensed source is the inverse of the determinant of A. The #uxes observed from the image and from the unlensed source are given as integrals over the brightness distributions I(h) and IQ(b), respectively, and their ratio is the magnixcation k(h ). From (3.13), we "nd  1 1 " . (3.14) k" det A (1!i)!"c" The images are thus distorted in shape and size. The shape distortion is due to the tidal gravitational "eld, described by the shear c, whereas the magni"cation is caused by both isotropic focusing caused by the local matter density i and anisotropic focusing caused by shear. Since the shear is de"ned by the trace-free part of the symmetric Jacobian matrix A, it has two independent components. There exists a one-to-one mapping from symmetric, trace-free 2;2 matrices onto complex numbers, and we shall extensively use complex notation. Note that the shear transforms as e P under rotations of the coordinate frame, and is therefore not a vector. Eqs. (3.9) and (3.12) imply that the complex shear can be written as 1 c(h)" p



1

dh D(h!h) i(h) ,

with !1 h !h !2ih h   " . D(h),  (h !ih ) "h"  

(3.15)

3.1.4. Critical curves and caustics Points in the lens plane where the Jacobian A is singular, i.e. where det A"0, form closed curves, the critical curves. Their image curves in the source plane are called caustics. Eq. (3.14) predicts that sources on caustics are in"nitely magni"ed; however, in"nite magni"cation does not occur in reality, for two reasons. First, each astrophysical source is extended, and its magni"cation (given by the surface brightness-weighted point-source magni"cation across its solid angle) remains "nite. Second, even point sources would be magni"ed by a "nite value since for them, the geometrical-optics approximation fails near critical curves, and a wave-optics description leads to a "nite magni"cation (e.g. Ohanian, 1983; Schneider et al., 1992, Chapter 7). For the purposes of this review, the "rst e!ect always dominates. Nevertheless, images near critical curves can be magni"ed and distorted substantially, as is demonstrated by the giant luminous arcs which are formed from source galaxies close to caustics. (Point) sources which move across a caustic have their number of images changed by $2, and the two additional images appear or disappear at the corresponding critical curve in the lens plane. Hence, only sources inside a caustic are multiply imaged. 3.1.5. An illustrative example: isothermal spheres The rotation curves of spiral galaxies are observed to be approximately #at out to the largest radii where they can be measured. If the mass distribution in a spiral galaxy followed the light

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distribution, the rotation curves would have to decrease at large radii in roughly Keplerian fashion. Flat rotation curves thus provide the clearest evidence for dark matter on galactic scales. They can be understood if galactic disks are embedded in a dark halo with density pro"le oJr\ for large r. The projected mass density then behaves like h\. Such density pro"les are obtained by assuming that the velocity dispersion of the dark-matter particles is spatially constant. They are therefore also called isothermal pro"les. We shall describe some simple properties of a gravitational lens with an isothermal mass pro"le, which shall later serve as a reference. The projected surface mass density of a singular isothermal sphere is p R(m)" T , 2Gm

(3.16)

where p is the line-of-sight velocity dispersion of the &particles' (e.g. stars in galaxies, or galaxies in T clusters of galaxies) in the gravitational potential of the mass distribution, assuming that they are in virial equilibrium. The corresponding dimensionless surface mass density is




p D h  (3.17) i(h)" # , where h "4p T # c D 2h  is called the Einstein deyection angle. As can easily be veri"ed from (3.8), the magnitude of the scaled de#ection angle is constant for this mass pro"le, "a""h , and the de#ection potential is t"h "h". # # From that, the shear is obtained using (3.12) h c(h)"! # e P 2"h"

(3.18)

and the magni"cation is "h" k(h)" . (3.19) "h"!h # This shows that "h""h de"nes a critical curve, which is called the Einstein circle. The correspond# ing caustic, obtained by mapping the Einstein circle back into the source plane under the lens equation, degenerates to a single point at b"0. Such degenerate caustics require highly symmetric lenses. Any perturbation of the mass distribution breaks the degeneracy and expands the singular caustic point into a caustic curve (see Chapter 6 in Schneider et al. (1992) for a detailed treatment of critical curves and caustics). Lens (3.17) produces two images with angular separation 2h for # a source with "b"(1, and one image otherwise. Mass distribution (3.17) has two unsatisfactory properties. The surface mass density diverges for "h"P0, and the total mass of the lens is in"nite. Clearly, both of these properties will not match real mass distributions. Despite this fact, the singular isothermal sphere "ts many of the observed lens

 For axially symmetric projected mass pro"les, the magnitude of the shear can be calculated from "c"(h)"i(h)!i(h), where i(h) is the mean surface mass density inside a circle of radius h from the lens centre. Accordingly, the magnitude of the de#ection angle is "a""hi(h).

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systems fairly well. In order to construct a somewhat more realistic lens model, one can cut o! the distribution at small and large distances, e.g. by h h # # i(h)" ! , (3.20) 2("h"#h 2("h"#h   which has a core radius h , and a truncation radius h . For h ;"h";h , this mass distribution     behaves like h\. This lens can produce three images, but only if h h (h #h )\(h /2. One of the     # three images occurs near the centre of the lens and is strongly de-magni"ed if h ;h . In most of  # the multiple-image QSO lens systems, there is no indication for a third central image, imposing strict upper bounds on h , whereas for some arc systems in clusters, a "nite core size is required  when a lens model like (3.20) is assumed. 3.2. Light propagation in arbitrary spacetimes We now turn to a more rigorous description of the propagation of light rays, based on the theory of geometrical optics in General Relativity. We then specialise the resulting propagation equations to the case of weak gravitational "elds and metric perturbations to the background of an expanding universe. These equations contain the gravitational lens equation discussed previously as a special case. We shall keep the discussion brief and follow closely the work of Schneider et al. (1992, Chapters 3 and 4), and Seitz et al. (1994), where further references can be found. 3.2.1. Propagation of light bundles In Section 3.1.2, we have derived the lens equation (3.5) in a heuristic way. A rigorous derivation in an arbitrary space}time must account for the fact that distance vectors between null geodesics are four vectors. Nevertheless, by choosing an appropriate coordinate system, the separation transverse to the line-of-sight between two neighbouring light rays can e!ectively be described by a two-dimensional vector n. We outline this operation in the following two paragraphs. We "rst consider the propagation of in"nitesimally thin light beams in an arbitrary space}time, characterised by the metric tensor g . The propagation of a "ducial ray c of the bundle is IJ  determined by the geodesic equation (e.g. Misner et al., 1973; Weinberg, 1972). We are interested here in the evolution of the shape of the bundle as a function of the a$ne parameter along the "ducial ray. Consider an observer O with four-velocity ;I, satisfying ;I; "1. The physical   I wave vector kI of a photon depends on the light frequency. We de"ne kI I,!c\u kI as  a past-directed dimensionless wave vector which is independent of the frequency u measured by  the observer. We choose an a$ne parameter j of the rays passing through O such that (1) j"0 at the observer, (2) j increases along the backward light cone of O, and (3) ;I kI "!1 at O. Then,  I with the de"nition of kI I, it follows that kI I"dxI/dj, and that j measures the proper distance along light rays for events close to O. Let cI(h, j) characterise the rays of a light beam with vertex at O, such that h is the angle between a ray and the "ducial ray with cI (j),cI(0, j). Further, let >I(h, j)"cI(h, j)!cI(0, j)"  [RcI(h, j)/Rh ]h denote the vector connecting the ray characterised by h with the "ducial ray at the I I same a$ne parameter j, where we assumed su$ciently small "h" so that >I can be linearised in h. We can then decompose >I as follows. At O, the vectors ;I and kI I de"ne a two-dimensional plane 

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perpendicular to both ;I and kI I. This plane is tangent to the sphere of directions seen by the  observer. Now choose orthonormal unit vectors E and E to span that plane. Hence, EI E "0,    I EIE "!1, EIkI "EI; "0, for k"1, 2. Transporting the four vectors kI I, ;I, EI , and I II I I I I   EI parallel along the "ducial ray de"nes a vierbein at each event along the "ducial ray. The  deviation vector can then be decomposed into >I(h, j)"!m (h, j) EI !m (h, j) EI !m (h, j) kI I . (3.21)      Thus, the two-dimensional vector n(h, j) with components m (h, j) describes the transverse  separation of two light rays at a$ne parameter j, whereas m allows for a deviation component  along the beam direction. Due to the linearisation introduced above, n depends linearly on h, and the choice of j assures that dn/dj(j"0)"h. Hence, we can write the linear propagation equation n(j)"D(j) h .

(3.22)

The 2;2 matrix D satis"es the Jacobi di!erential equation dD(j) "T(j) D(j) dj

(3.23)

with initial conditions D(0)"O and

dD (0)"I . dj

(3.24)

The optical tidal matrix T(j) is symmetric,




T(j)"



R(j)#R[F(j)]

I[F(j)]

I[F(j)]

R(j)!R[F(j)]

(3.25)

and its components depend on the curvature of the metric. R(z) and I(z) denote the real and imaginary parts of the complex number z. Speci"cally, R(j)"!R (j)kI I(j)kI J(j) , (3.26)  IJ where R (j) is the Ricci tensor at cI (j). The complex quantity F(j) is more complicated IJ  and depends on the Weyl curvature tensor at cI (j). The source of convergence R(j) leads to an  isotropic focusing of light bundles, in that a circular light beam continues to have a circular cross section. In contrast, a non-zero source of shear F(j) causes an anisotropic focusing, changing the shape of the light bundle. For a similar set of equations, see, e.g. Blandford et al. (1991) and Peebles (1993). To summarise this subsection, the transverse separation vector n of two in"nitesimally close light rays, enclosing an angle h at the observer, depends linearly on h. The matrix which describes this linear mapping is obtained from the Jacobi di!erential equation (3.23). The optical tidal matrix T can be calculated from the metric. This exact result from General Relativity is of course not easily applied to practical calculations in general space}times, as one "rst has to calculate the null geodesic cI (j), and from that the components of the tidal matrix have to be determined.  However, as we shall show next, the equations attain rather simple forms in the case of weak gravitational "elds.

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3.2.2. Specialisation to weak gravitational xelds We shall now specialise the transport equation (3.23) to the situation of a homogeneous and isotropic universe, and to weak gravitational "elds. In a metric of the Robertson}Walker type (2.2), the source of shear F must vanish identically because of isotropy; otherwise preferred directions would exist. Initially circular light bundles therefore remain circular. Hence, the optical tidal matrix T is proportional to the unit matrix, T(j)"R(j)I, and the solution of (3.23) must be of the form D(j)"D(j) I. According to (3.22), the function D(j) is the angular-diameter distance as a function of the a$ne parameter. As we shall demonstrate next, this function indeed agrees with the angular diameter distance as de"ned in (2.43). To do so, we "rst have to "nd R(j). The Ricci tensor deviates from the Einstein tensor by two terms proportional to the metric tensor g , one involving the Ricci scalar, the other conIJ taining the cosmological constant. These two terms do not contribute to (3.26), since kI I is a null vector. We can thus replace the Ricci tensor in (3.26) by the energy}momentum tensor according to Einstein's "eld equation. Since k"c\u"(1#z)c\u , we have kI "!(1#z), and the  spatial components of kI I are described by a direction and the constraint that kI I is a null vector. Then, using the energy-momentum tensor of a perfect #uid with density o and pressure p, (3.26) becomes






p 4nG o# (1#z) . R(j)"! c c

(3.27)

Specialising to a universe "lled with dust, i.e. p"0, we "nd from (2.16) and (2.19)




3 H   X (1#z) . R(j)"!  2 c

(3.28)

The transport equation (3.23) then transforms to




dD 3 H   X (1#z) D . "!  dj 2 c

(3.29)

In order to show that the solution of (3.29) with initial conditions D"0 and dD"dj at j"0 is equivalent to (2.43), we proceed as follows. First, we note that (2.43) for z "0 can be written as an  initial-value problem







d D D  "!K  dw a a

(3.30)

with D (0)"0 and dD "dw at w"0, because of the properties of the function f ; cf.   ) (2.4). Next, we need a relation between j and w. The null component of the photon geodesic is x"c(t !t). Then, from dxI"kI I dj, we obtain dj"!ac dt. Using dt"a \ da, we  "nd a da"! dj or ca

a dz" dj . ca

(3.31)

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Since c dt"!a dw for null rays, we have a \da"dt"!ac\ dw, which can be combined with (3.31) to yield dj"a dw .

(3.32)

We can now calculate the analogous expression of (3.30) for D,







d D d d D "a a dw a dj dj a

"a D!a a D ,

(3.33)

where a prime denotes di!erentiation with respect to j. From (3.31), a"!(ac)\a , and




1 d a  1 dH 1 d(a) " " a" 2c da a 2c da 2 da

(3.34)

with H given in (2.31). Substituting (3.29) into the "rst term on the right-hand side of (3.33), and (3.34) into the second term, we immediately see that D satis"es the di!erential equation (3.30). Since D has the same initial conditions as D , they indeed agree.  For computational convenience, we can also transform (3.29) into a di!erential equation for D(z). Using (3.31) and (2.31), one "nds






1 (1#z) (1#X z)!XK 1!  (1#z)



#






dD dz

7 X 2 X z#  #3!XK 3! 2  2 (1#z)



dD 3 # X D"0 . dz 2 

(3.35)

We next turn to the case of a weak isolated mass inhomogeneity with a spatial extent small compared to the Hubble distance cH\, like galaxies or clusters of galaxies. In that case, the metric  can locally be approximated by the post-Minkowskian line element











2U 2U c dt! 1! dx , ds" 1# c c

(3.36)

where dx is the line element of Euclidean three space, and U is the Newtonian gravitational potential which is assumed to be weak, U;c. Calculating the curvature tensor of metric (3.36), and using Poisson's equation for U, we "nd that for a light ray which propagates into the three direction, the sources of convergence and shear are 1 4nG o and F"! (U !U #2iU ) . R"!   c  c

(3.37)

Now, the question is raised as to how an isolated inhomogeneity can be combined with the background model of an expanding universe. There is no exact solution of Einstein's "eld equations which describes a universe with density #uctuations, with the exception of a few very special cases such as the Swiss}Cheese model (Einstein and Strauss, 1945). We therefore have to resort to approximation methods which start from identifying &small' parameters of the problem, and expanding the relevant quantities into a Taylor series in these parameters. If the length scales of density inhomogeneities are much smaller than the Hubble length cH\, the associated 

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Newtonian gravitational potential U;c (note that this does not imply that the relative density #uctuations are small!), and the peculiar velocities v;c, then an approximate metric is




ds"a(q)










2U 2U 1# c dq! 1! (dw#f  (w) du) , ) c c

(3.38)

where dq"a\ dt is the conformal time element, and U satis"es Poisson's equation with source *o, the density enhancement or reduction relative to the mean cosmic density (Futamase, 1989; Futamase and Sasaki, 1989; Jacobs et al., 1993). In the case of weak metric perturbations, the sources of convergence and shear of the background metric and the perturbations can be added. Recalling that both R and F are quadratic in kI IJ(1#z), so that the expressions in (3.37) have to be multiplied by (1#z), we "nd for the optical tidal matrix




(1#z) 3 H   X (1#z) d ! (2U #d U ) , T (j)"!  GH GH GH  GH c 2 c

(3.39)

where we have assumed that the local Cartesian coordinates are chosen such that the light ray propagates in the x -direction. The same result is obtained from metric (3.38).  The lens equation as discussed in Section 3.1 can now be derived from the previous relations. To do so, one has to assume a geometrically thin matter distribution, i.e. one approximates the density perturbation *o by a distribution which is in"nitely thin in the direction of photon propagation. It is then characterised by its surface mass density R(n). The corresponding Newtonian potential U can then be inserted into (3.39). The integration over U along the light ray vanishes, and (3.23)  can be employed to calculate the change of dD/dj across the thin matter sheet (the lens plane), whereas the components of D far from the lens plane are given by a linear combination of solutions of the transport equation (3.29). Continuity and the change of derivative at j , corresponding to the  lens redshift z , then uniquely "x the solution. If D(h, j ) denotes the solution at redshift z , then    D(h, j )"Rg/Rh in the notation of Section 3.1. Line integration of this relation then leads to the lens  equation (3.2). See Seitz et al. (1994) for details, and Pyne and Birkinshaw (1996) for an alternative derivation.

4. Principles of weak gravitational lensing 4.1. Introduction If the faint, and presumably distant, galaxy population is observed through the gravitational "eld of a de#ector, the appearance of the galaxies is changed. The tidal component of the gravitational "eld distorts the shapes of galaxy images, and the magni"cation associated with gravitational light de#ection changes their apparent brightness. If all galaxies were intrinsically circular, any galaxy image would immediately provide information on the local tidal gravitational "eld. With galaxies being intrinsically elliptical, the extraction of signi"cant information from individual images is impossible, except for giant luminous arcs (see Fig. 10, for an example) whose distortion is so extreme that it can easily be determined.

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However, assuming that the galaxies are intrinsically randomly oriented, the strength of the tidal gravitational "eld can be inferred from a sample of galaxy images, provided its net ellipticity surmounts the Poisson noise caused by the "nite number of galaxy images in the sample and by the intrinsic ellipticity distribution. Since lensing conserves surface brightness, magni"cation increases the size of galaxy images at a "xed surface-brightness level. The resulting #ux enhancement enables galaxies to be seen down to fainter intrinsic magnitudes, and consequently the local number density of galaxy images above a certain #ux threshold can be altered by lensing. In this section, we introduce the principles of weak gravitational lensing. In Section 4.2, we present the laws of the transformation between source and image ellipticities and sizes, and in particular we introduce a convenient de"nition of the ellipticity of irregularly shaped objects. Section 4.3 focuses on the determination of the local tidal gravitational "eld from an ensemble of galaxy images. We derive practical estimators for the shear and compare their relative merits. The e!ects of magni"cation on the observed galaxy images are discussed in Section 4.4. We derive an estimate for the detectability of a de#ector from its weak-lensing imprint on galaxy-image ellipticities in Section 4.5, and the "nal Section 4.6 is concerned with practical aspects of the measurement of galaxy ellipticities. 4.2. Galaxy shapes and sizes, and their transformation If a galaxy had elliptical isophotes, its shape and size could simply be de"ned in terms of axis ratio and area enclosed by a boundary isophote. However, the shapes of faint galaxies can be quite irregular and not well approximated by ellipses. In addition, observed galaxy images are given in terms of pixel brightness on CCDs. We therefore require a de"nition of size and shape which accounts for the irregularity of images, and which is well adapted to observational data. Let I(h) be the surface brightness of a galaxy image at angular position h. We "rst assume that the galaxy image is isolated, so that I can be measured to large angular separations from the centre hM of the image dh q [I(h)] h ' , (4.1) hM , dh q [I(h)] ' where q (I) is a suitably chosen weight function. For instance, if q (I)"H(I!I ) is the Heaviside ' '  step function, hM is the centre of the area enclosed by a limiting isophote I"I . Alternatively, if  q (I)"I, hM is the centre of light. As a third example, if q (I)"I H(I!I ), hM is the centre of light ' ' 

 This assumption is not seriously challenged. Whereas galaxies in a cluster may have non-random orientations relative to the cluster centre, or pairs of galaxies may be aligned due to mutual tidal interaction, the faint galaxies used for lensing studies are distributed over a large volume enclosed by a narrow cone with opening angle selected by the angular resolution of the mass reconstruction (see below) and length comparable to the Hubble radius, since the redshift distribution of faint galaxies is fairly broad. Thus, the faint galaxies typically have large spatial separations, which is also re#ected by their weak two-point angular auto-correlation (Brainerd et al., 1995; Villumsen et al., 1997).

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within the limiting isophote I"I . Having chosen q (I), we de"ne the tensor of second brightness  ' moments dh q [I(h)](h !hM ) (h !hM ) ' G G H H , i, j3+1, 2, Q " (4.2) GH dh q [I(h)] ' (e.g. Blandford et al., 1991). In writing (4.1) and (4.2), we implicitly assumed that q (I) is chosen such ' that the integrals converge. We can now de"ne the size of an image in terms of the two invariants of the symmetric tensor Q. For example, we can de"ne the size by u"(Q Q !Q ) , (4.3)    so that it is proportional to the solid angle enclosed by the limiting isophote if q(I) is a step function. We quantify the shape of the image by the complex ellipticity Q !Q #2iQ   . (4.4) s,  Q #Q   If the image has elliptical isophotes with axis ratio r41, then s"(1!r)(1#r)\ exp(2i0), where the phase of s is twice the position angle 0 of the major axis. This de"nition assures that the complex ellipticity is unchanged if the galaxy image is rotated by p, for this rotation leaves an ellipse unchanged. If we de"ne the centre of the source bM and the tensor of second brightness moments QQ of the GH source in complete analogy to that of the image, i.e. with I(h) replaced by IQ(b) in Eqs. (4.1) and (4.2), and employ the conservation of surface brightness (3.10) and the linearised lens equation (3.13), we "nd that the tensors of second brightness moments of source and image are related through QQ"A Q A2"A Q A ,

(4.5)

where A,A(hM ) is the Jacobian matrix of the lens equation at position hM . De"ning further the complex ellipticity of the source sQ in analogy to (4.4) in terms of QQ, ellipticities transform according to s!2g#gsH sQ" 1#"g"!2R(gsH)

(4.6)

(Schneider and Seitz, 1995; similar transformation formulae were previously derived by Kochanek, 1990; Miralda-Escude, 1991), where the asterisk denotes complex conjugation, and g is the reduced shear c(h) g(h), . 1!i(h)

(4.7)

The inverse transformation is obtained by interchanging s and sQ and replacing g by !g in (4.6). Eq. (4.6) shows that the transformation of image ellipticities depends only on the reduced shear, and not on the shear and the surface mass density individually. Hence, the reduced shear or

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functions thereof are the only quantities accessible through measurements of image ellipticities. This can also immediately be seen by writing A as






1!g !g   . (4.8) !g 1#g   The pre-factor (1!i) only a!ects the size, but not the shape of the images. From (4.5) and (4.3), we immediately see that the sizes of source and image are related through A"(1!i)

u"k(h)uQ .

(4.9)

We point out that the dimensionless surface mass density i, and therefore also the shear c, depend not only on the redshift of the lens, but also on the redshift of the sources, because the critical surface mass density (3.7) involves the source redshift. More precisely, for "xed lens redshift z , the  lens strength is proportional to the distance ratio D /D . This implies that transformation (4.6)   generally also depends on source redshift. We shall return to these redshift e!ects in Section 4.3, and assume for now that the lens redshift z is su$ciently small so that the ratio D /D is    approximately the same for all faint galaxy images. Instead of s, we can de"ne di!erent ellipticity parameters (see Bonnet and Mellier, 1995). One of these de"nitions turns out to be quite useful, namely Q !Q #2iQ    , (4.10) e, Q #Q #2(Q Q !Q )      which we shall also call complex ellipticity. (Since we shall use the notation s and e consistently throughout this article, there should be no confusion from using the same name for two di!erent quantities.) e has the same phase as s, and for elliptical isophotes with axis ratio r41, "e""(1!r)(1#r)\. e and s are related through 2e s , s" . e" 1#"e" 1#(1!"s") The transformation between source and image ellipticity in terms of e is given by e!g for "g"41 , 1!gHe eQ" 1!geH for "g"'1 eH!gH



(4.11)

(4.12)

(Seitz and Schneider, 1997), and the inverse transformation is obtained by interchanging e and eQ and replacing g by !g in (4.12). Although the transformation of e appears more complicated because of the case distinction, we shall see in the next subsection that it is often useful to work in terms of e rather than s; cf. Eq. (4.17) below. We note in passing that the possible polarisation of light of faint galaxies (Audit and Simmons, 1999) or faint radio sources (Surpi and Harari, 1999) may o!er a di!erent channel to detect shear. The orientation of the polarisation is unchanged in weak-"eld light de#ection (e.g., Schneider et al., 1992; Faraoni, 1993). Gravitational shear will turn the geometrical image, but not the polarisation of a galaxy. If the orientation of a galaxy is intrinsically strongly correlated with the direction of the

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polarisation of its light, then a mismatch of the observed directions provides information on the lensing distortion. However, the polarisation properties of faint galaxies are mostly unknown, and it is unclear whether such an intrinsic polarisation-orientation correlation exists. For the case of weak lensing, which we de"ne for the purpose of this section by i;1, "c";1, and thus "g";1, (4.12) becomes e+eQ#g, provided "e"+"eQ":1/2. Likewise, Eq. (4.6) simpli"es to s+sQ#2g in this case. 4.3. Local determination of the distortion As mentioned earlier, the observed ellipticity of a single galaxy image provides only little information about the local tidal gravitational "eld of the de#ector, for the intrinsic ellipticity of the source is unknown. However, based on the assumption that the sources are randomly oriented, information on the local tidal "eld can be inferred from a local ensemble of images. Consider for example galaxy images at positions h close enough to a "ducial point G h so that the local lens properties i and c do not change appreciably over the region encompassing these galaxies. The expectation value of their corresponding source ellipticities is assumed to vanish, E(sQ)"0"E(eQ) .

(4.13)

4.3.1. All sources at the same redshift We "rst consider the case that all sources are at the same redshift. Then, as mentioned following Eq. (3.13), the ellipticity of a circular source determines the ratio of the local eigenvalues of the Jacobian matrix A. This also holds for the net image ellipticity of an ensemble of sources with vanishing net ellipticity. From (3.11), we "nd for the ratio of the eigenvalues of A in terms of the reduced shear g 1G"g" . r" 1$"g"

(4.14)

Interestingly, if we replace g by 1/gH, r switches sign, but "r" and the phase of e remain unchanged. The sign of r cannot be determined observationally, and hence measurements cannot distinguish between g and 1/gH. This is called local degeneracy. Writing det A"(1!i)(1!"g"), we see that the degeneracy between g and 1/gH means that we cannot distinguish between observed images inside a critical curve (so that det A(0 and "g"'1) or outside. Therefore, only functions of g which are invariant under gP1/gH are accessible to (local) measurements, as for instance the complex distortion 2g . d, 1#"g"

(4.15)

Replacing the expectation value in (4.13) by the average over a local ensemble of image ellipticities, 1sQ2+E(sQ)"0, Schneider and Seitz (1995) showed that 1sQ2"0 is equivalent to s !d G "0 , u G 1!R(dsH) G G

(4.16)

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where the u are weight factors depending on "h !h" which can give larger weight to galaxies closer G G to the "ducial point. Additionally, the u can be chosen such as to account for measurement G uncertainties in the image ellipticities by giving less weight to images with larger measurement error. Eq. (4.16) has a unique solution d, so that the distortion can locally be determined. It is readily solved by a quickly converging iteration starting from d"1s2. The d obtained from (4.16) is an unbiased estimate of the distortion. Its dispersion about the true value depends on the dispersion p of the intrinsic ellipticity distribution, and on the number of Q galaxy images. A fairly accurate estimate of the rms error of d is p +p N\, where N is the B Q e!ective number of galaxies used for the local average, N"( u )( u)\. This overestimates the G G error for large values of "d" (Schneider and Seitz, 1995). It is important to note that the expectation value of s is not d, but di!ers from it by a factor which depends both on "d" and the intrinsic ellipticity distribution of the sources. In contrast to that, it follows from (4.13) and (4.12) that the expectation value of the complex ellipticity e of the images is the reduced shear or its inverse, E(e)"g if "g"(1 and E(e)"1/gH if "g"'1 (Schramm and Kayser, 1995; Seitz and Schneider, 1997). Hence, ue (4.17) 1e2" G G G u G G is an unbiased local estimate for g or 1/gH. The ellipticity parameter e is useful exactly because of this property. If one deals with sub-critical lenses (i.e. lenses which are not dense enough to have critical curves, so that det A(h)'0 everywhere), or with the region outside the critical curves in critical lenses, the degeneracy between g and 1/gH does not occur, and 1e2 is a convenient estimate for the local reduced shear. The rms error of this estimate is approximately p +p (1!"g") N\ (Schneider E C et al., 2000), where p is the dispersion of the intrinsic source ellipticity eQ. As we shall see in C a moment, e is the more convenient ellipticity parameter when the sources are distributed in redshift. The estimates for d and g discussed above can be derived without knowing the intrinsic ellipticity distribution. If, however, the intrinsic ellipticity distribution is known (e.g. from deep Hubble Space Telescope images), we can exploit this additional information and determine d (or g) through a maximum-likelihood method (Gould, 1995; Lombardi and Bertin, 1998a). Depending on the shape of the intrinsic ellipticity distribution, this approach can yield estimates of the distortion which have a smaller rms error than the estimates discussed above. However, if the intrinsic ellipticity distribution is approximately Gaussian, the rms errors of both methods are identical. It should be noted that the intrinsic ellipticity distribution is likely to depend on the apparent magnitude of the galaxies, possibly on their redshifts, and on the wavelength at which they are observed, so that this distribution is not easily determined observationally. Knowledge of the intrinsic ellipticity distribution can also be used to determine d from the orientation of the images (that is, the phase of s) only (Kochanek, 1990; Schneider and Seitz, 1995; Deiser, 1995, unpublished). This may provide a useful alternative to the method above since the orientation of images is much less a!ected by seeing than the modulus of s. We return to the practical estimate of the image ellipticities and the corresponding distortion in Section 4.5. In the case of weak lensing, de"ned by i;1 and "c";1, implying "g";1, we "nd from (4.11) to (4.16) that 1s2 d . c+g+ +1e2+ 2 2

(4.18)

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4.3.2. Sources distributed in redshift So far, we assumed that all source galaxies are at the same redshift, or more precisely, that the ratio D /D between the lens- and observer-source distances is the same for all sources.   This ratio enters into scaling (3.7) of the physical surface mass density R to the dimension-less convergence i. The de#ection angle, the de#ection potential, and the shear are all linear in i, so that the distance ratio D /D is su$cient to specify the lens strength as a function of source   redshift. Provided z :0.2, this ratio is fairly constant for sources with redshift z 90.8, so   that the approximation used so far applies to relatively low-redshift de#ectors. However, for higher-redshift lenses, the redshift distribution of the sources must explicitly be taken into account. For a "xed lens redshift z , the dimensionless surface mass density and the shear depend on the  source redshift. We de"ne lim R (z , z) Z(z), X   H(z!z )  R (z , z)   f [w(z , z)]  ") f [w(0, z)] )

f [w(0,R)] ) H(z!z ) ,  f [w(z ,R)] ) 

(4.19)

using the notation of Section 2.1. The Heaviside step function accounts for the fact that sources closer than the de#ector are not lensed. Then, i(h, z)"Z(z)i(h), and c(h, z)"Z(z)c(h) for a source at z, and i and c refer to a "ctitious source at redshift in"nity. The function Z(z) is readily evaluated for any cosmological model using (2.41) and (2.4). We plot Z(z) for various cosmologies and lens redshifts in Fig. 12.

Fig. 12. The function Z(z) de"ned in eq. (4.19) describes the relative lens strength as a function of source redshift z. We show Z(z) for three cosmological models as indicated in the "gure, and for three values for the lens redshift, z "0.2, 0.5, 0.8. By de"nition, Z(z)P0 as zPz , and Z(z)P1 as zPR. For sources close to the de#ector, Z(z) varies   strongly in a way depending relatively weakly on cosmology.

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The expectation value for the ellipticity of images with redshift z now becomes



Z(z) c 1!Z(z) i E[e(z)]"g(z)" 1!Z(z) i Z(z) cH

for k(z)50 , (4.20) for k(z)(0 ,

where k(z) is the magni"cation as a function of source redshift, k(z)"+[1!Z(z)i]!Z(z)"c",\ .

(4.21)

We refer to sub-critical lensing if k(z)'0 for all redshifts, which is equivalent to 1!i!"c"'0. Without redshift information, only the mean ellipticity averaged over all redshifts can be observed. We "rst consider this case, for which the source redshift distribution is assumed to be known. We de"ne the probability p (z) dz that a galaxy image (in the selected magnitude range) has X a redshift within dz of z. The image redshift distribution will, in general, be di!erent from the source redshift distribution since magni"ed sources can be seen to higher redshifts than unlensed ones. Therefore, the redshift distribution will depend on the local lens parameters i and c through magni"cation (4.21). If, however, the magni"cation is small, or if the redshift distribution depends only weakly on the #ux, the simpli"cation of identifying the two redshift distributions is justi"ed. We shall drop it later. Given p (z), the expectation value of the image ellipticity becomes the X weighted average



E(e)" dz p (z) E[e(z)]"c[X(i, c)#"c"\>(i, c)] X

(4.22)

with

 

X(i, c)"

IXY

>(i, c)"

Z(z) dz p (z) , X 1!Z(z)i dz p (z) X

1!Z(z)i Z(z)

(4.23) IX and the integration boundaries depend on the values of i and "c" through the magni"cation. If the lens is sub-critical, k(z)'0 for all z. Then >"0, and only the "rst term in (4.22) remains. Also, X no longer depends on c in this case, and E(e)"c X(i). An accurate approximation for X(i), valid for i:0.6, has been derived in Seitz and Schneider (1997),






E(e) 1Z2 c" 1! i , 1Z2 1Z2

(4.24)

where 1ZL2,dz p (z)ZL. X Specialising further to the weak-lensing regime, the expectation value of the image ellipticity is simply E(e)+1Z2c .

(4.25)

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Thus, in the weak-lensing case, a source redshift distribution can be collapsed on a single redshift z satisfying Z(z )"1Z2.   We now drop the simpli"cation introduced above and de"ne n (S, z)dS dz as the number of  galaxy images per unit solid angle with #ux within dS of S and redshift within dz of z in the absence of lensing. At a point h with surface mass density i and shear c, the number density can be changed by magni"cation. Images of a "xed set of sources are distributed over a larger solid angle, reducing the number density by a factor k\(z). On the other hand, the magni"cation allows the observation of fainter sources. In total, the expected number density becomes




S 1 n ,z n(S, z)"  k(z) k(z)

(4.26)

with k(z) given in (4.21). This yields the redshift distribution n [k\(z)S, z]  p(z; S, i, c)" , (4.27) k(z)dzk\(z)n [k\(z)S, z]  which depends on the #ux S and the local lens parameters i and c through the magni"cation. This function can now be substituted for p (z) in Eq. (4.22). X 4.3.3. Practical estimates of the shear We saw before that 1e2" u e / u is an unbiased estimate of the local reduced shear g if all G G G G G sources are at the same redshift. We now generalise this result for sources distributed in redshift. Then, the expectation value of e is no longer a simple function of i and c, and therefore estimates of c for an assumed value for i will be derived. We "rst assume that redshifts for individual galaxies are unavailable, but that only the normalised redshift distribution p (z) is known, or the distribution in Eq. (4.27). Replacing the X expectation value of the image ellipticity by the mean, Eq. (4.22) implies that the solution c of c"[X(i, c)#"c"\>(i, c)]\1e2

(4.28)

provides an unbiased estimator for the shear c. This is not a particularly explicit expression for the shear estimate, but it is still extremely useful, as we shall see in the next section. The shear estimate considerably simpli"es if we assume a sub-critical lens. Then,






1Z2 1e2 1! i , c"1e2X\(i)+ 1Z2 1Z2

(4.29)

where we used Eq. (4.24) in the second step. Specialising further to weak lensing, the shear estimate simpli"es to c "1e21Z2\ .

(4.30)

Next, we assume that the redshifts of all galaxy images are known. At "rst sight, this appears entirely unrealistic, because the galaxy images are so faint that a complete spectroscopic survey at the interesting magnitude limits seems to be out of reach. However, it has become clear in recent years that accurate redshift estimates, the so-called photometric redshifts, can be obtained from

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multi-colour photometry alone (see, e.g., Connolly et al., 1995). The accuracy of photometric redshifts depends on the number of wave bands for which photometry is available, the photometric accuracy, and the galaxy type; typical errors are *z&0.1 for faint, high-redshift galaxies. This uncertainty is small compared to the range over which the function Z(z) varies appreciably, so that photometric redshifts are (almost) as good as precise spectroscopic redshifts for our purposes. If the redshifts z of the galaxies are known, more precise shear estimates than before can be G derived. Consider the weighted sum F, u "e !E(e )", where the expectation value is given by G G G G Eq. (4.20), and Z"Z ,Z(z ). For an assumed value of i, an unbiased estimate of c is given by the G G c minimising F. Due to the case distinction in Eq. (4.20), this estimator is complicated to write down analytically, but can easily be calculated numerically. This case distinction is no longer necessary in the sub-critical case, for which the resulting estimator reads u Z e (1!Z i)\ G . c" G G G G u Z (1!Z i)\ G G G G In the case of weak lensing, this becomes

(4.31)

u Z e c " G G G G . (4.32) u Z G G G We now compare the accuracy of the shear estimates with and without redshift information of the individual galaxies. For simplicity, we assume sub-critical lensing and set all weight factors to unity, u "1. The dispersion of the estimate c"(N X)\ e for N galaxy images is G G G






p(c)"E("c")!"c""[NX(i)]\ E e eH !"c" . (4.33) G H GH The expectation value in the "nal expression can be estimated noting that the image ellipticity is to "rst order given by e "eQ#c, and that the intrinsic ellipticities are uncorrelated. If we further G G assume that the redshifts of any two galaxies are uncorrelated, we "nd





ZZ G H "c"#d p E(e eH)+ GH C G H (1!Z i)(1!Z i) G H "X(i)"c"#d (p "c"#p) , (4.34) GH 6 C where we used de"nition (4.23) of X(i), and de"ned p (i),1Z(1!Zi)\2!X. Angular 6 brackets denote averages over the redshift distribution p . Inserting (4.34) into (4.33) yields X p "c"#p C . (4.35) p(c)" 6 NX Likewise, the dispersion of the estimate c is Z Z (1!Z i)\(1!Z i)\E(e eH) G H G H !"c" p(c)" GH G H [ Z(1!Z i)\] G G G p p C C " + . Z(1!Z i)\ N[X(i)#p (i)] G G G 6

(4.36)

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Fig. 13. The fractional accuracy gain in the shear estimate due to the knowledge of the source redshifts is plotted, more precisely the deviation of the square root of (4.37) from unity in per cent. The four curves shown correspond to two di!erent values of the mean source redshift, and to the cases without lensing (i"0"c), and with lensing (i"0.3""c"), labelled NL and L, respectively. We assumed the redshift distribution (2.69) with b"3/2, and an Einstein}de Sitter cosmology. As expected, the higher the lens redshift z , the more substantially is the shear estimate improved by redshift  information, since for low values of z , the function Z(z) is nearly constant. Furthermore, the lower the mean redshift of  the source distribution, the more important the knowledge of individual redshifts becomes, for example to distinguish between foreground and background galaxies. Finally, redshift information is relatively more important for larger lens strength.

We used Eq. (4.34), but noted that Z is now no longer a statistical variable, so that we can put p "0 in (4.34). In the "nal step, we have replaced the denominator by its expectation value under 6 ensemble averaging. We then "nd the ratio of the dispersions









p p p(c) " 1#"c" 6 1# 6 . (4.37) p X p(c) C We thus see that the relative accuracy of these two estimates depends on the fractional width of the distribution of Z/(1!Zi), and on the ratio between the dispersion of this quantity and the ellipticity dispersion. Through its explicit dependence on "c", and through the dependence of p and X on i, the relative accuracy also depends on the lens parameters. Quantitative estimates of 6 (4.37) are given in Fig. 13. The "gure shows that the accuracy of the shear estimate is noticeably improved, in particular once the lens redshift becomes a fair fraction of the mean source redshift. The dependence of the lens strength on the de#ector redshift implies that the lens signal will become smaller for increasing de#ector redshift, so that the accuracy gained by redshift information becomes signi"cant. In addition, the assumptions used to derive (4.35) were quite optimistic, since we have assumed in (4.34) that the sample of galaxies over which the average is taken is a fair representation of the galaxy redshift distribution p (z). Given that these galaxies come from a small area (small enough to X assume that i and c are constant across this area), and that the redshift distribution of observed galaxies in pencil beams shows strong correlations (see, e.g., Broadhurst et al., 1990; Steidel et al.,

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1998; Cohen et al., 1999), this assumption is not very realistic. Indeed, the strong clustering of galaxy redshifts means that the e!ective p will be considerably larger than the analytical estimate 6 used above. The noise in the local determination of the shear due to the correlated galaxy redshifts does not decrease with the number N of galaxies used, and, therefore, its relative contribution becomes more important for larger number densities of source galaxies (Schneider and MoralesMerino, 2000). In any case, redshift information on the source galaxies will substantially improve the accuracy of weak lensing results. 4.4. Magnixcation ewects In addition to the distortion of image shapes, by which the (reduced) shear can be measured locally, gravitational light de#ection also magni"es the images, leaving the surface brightness invariant. The magni"cation changes the size, and therefore the #ux, of individual galaxy images. Moreover, for a "xed set of sources, the number density of images decreases by a factor k as the sky is locally stretched. Combining the latter e!ect with the #ux magni"cation, the lensed and unlensed source counts are changed according to (4.26). Two strategies to measure the magni"cation e!ect have been suggested in the literature, namely either through the change in the local source counts, perhaps combined with the associated change (4.27) in the redshift distribution (Broadhurst et al., 1995), or through the change of image sizes at "xed surface brightness (Bartelmann and Narayan, 1995). 4.4.1. Number density ewect Let n ('S, z) dz be the unlensed number density of galaxies with redshift within dz of z and with  #ux larger than S. Then, at an angular position h where the magni"cation is k(h, z), the number counts are changed according to (4.26),






S 1 n ' ,z . n('S, z)"  k(h, z) k(h, z)

(4.38)

Accordingly, magni"cation can either increase or decrease the local number counts, depending on the shape of the unlensed number-count function. This change of number counts is called magnixcation bias, and is a very important e!ect for gravitational lensing of QSOs (see Schneider et al., 1992 for references). Magni"cation allows the observation of fainter sources. Since the #ux from the sources is correlated with their redshift, the redshift distribution is changed accordingly, n ['k\(z)S, z]  , p(z;'S, i, c)" k(z) dz k\(z)n ['k\(z)S, z] 

(4.39)

 Bright QSOs have a very steep number-count function, and so the #ux enhancement of the sources outweighs the number reduction due to the stretching of the sky by a large margin. Whereas the lensing probability even for a high-redshift QSO is probably too small to a!ect the overall sources counts signi"cantly, the fraction of multiply imaged QSOs in #ux-limited samples is increased through the magni"cation bias by a substantial factor over the probability that any individual QSO is multiply imaged (see, e.g. Turner et al., 1984; Narayan and Wallington, 1993 and references therein).

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in analogy to the redshift distribution (4.27) at "xed #ux S. Since the objects of interest here are very faint, spectroscopic redshift information is in general di$cult to obtain, and so one can only observe the redshift-integrated counts



n('S)" dz

1 n ('k\(z)S, z) . k(z) 

(4.40)

The number counts of faint galaxies are observed to very closely follow a power law over a wide range of #uxes, and so we write the unlensed counts as n ('S, z)"aS\? p (z; S) , (4.41)   where the exponent a depends on the wave band of the observation (e.g. Smail et al., 1995a), and p (z; S) is the redshift probability distribution of galaxies with #ux 'S. Whereas this redshift  distribution is fairly well known for brighter galaxies which are accessible to current spectroscopy, little is known about the faint galaxies of interest here. The ratio of the lensed and unlensed source counts is then found by inserting (4.41) into (4.40),



n('S) " dz k?\(z)p (z; k\(z)S) . (4.42)  n ('S)  We should note that the lensed counts do not strictly follow a power law in S, for p depends on z.  Since the redshift distribution p (z, S) is currently unknown, the change of the number counts due  to the magni"cation cannot be predicted. For very faint #ux thresholds, however, the redshift distribution is likely to be dominated by galaxies at relatively high redshift. For lenses at fairly small redshift (say z :0.3), we can approximate the redshift-dependent magni"cation k(z) by the  magni"cation k of a "ducial source at in"nity, in which case n('S) "k?\ . (4.43) n ('S)  Thus, a local estimate of the magni"cation can be obtained through (4.43) and from a measurement of the local change of the number density of images. If the slope of the source counts is unity, a"1, there will be no magni"cation bias, while it will cause a decrease of the local number density for #atter slopes. Broadhurst et al. (1995) pointed out that one can immediately obtain (for sub-critical lensing, i.e. det A'0) an estimate for the local surface mass density from a measurement of the local magni"cation and the local reduced shear g, i"1![k(1!"g")]\. In the absence of shape information, (4.43) can be used in the weak-lensing limit [where i;1, "c";1, so that k+(1#2i)] to obtain an estimate of the surface mass density 1 n('S)!n ('S)  . i+ 2(a!1) n ('S) 

(4.44)

4.4.2. Size ewect Since lensing conserves surface brightness, the magni"cation can be obtained from the change in galaxy-image sizes at "xed surface brightness. Let I be some convenient measure of the surface brightness. For example, if u is the solid angle of an image, de"ned by the determinant of the tensor of second brightness moments as in (4.3), one can set I"S/u.

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Denoting by n(u, I, z) du the number density of images with surface brightness I, redshift z, and solid angle within du of u, the relation between the lensed and the unlensed number density can be written as




u 1 n(u, I, z)" n , I, z . k  k

(4.45)

For simplicity, we only consider the case of a moderately small lens redshift, so that the magni"cation can be assumed to be locally constant for all images, irrespective of galaxy redshift. We can then drop the variable z here. The mean image size 1u2(I) at "xed surface brightness I is then related to the mean image size 1u2 (I) in the absence of lensing through  1u2(I)"k1u2 (I) . (4.46)  If the mean image size in the absence of lensing can be measured (e.g. by deep HST exposures of blank "elds), the local value k of the magni"cation can therefore be determined by comparing the observed image sizes to those in the blank "elds. This method has been discussed in detail in Bartelmann and Narayan (1995). For instance, if we assume that the logarithm of the image size is distributed as a Gaussian with mean 1ln u2 (I) and dispersion p(I), we obtain an estimate for the  local magni"cation from a set of N galaxy images






1 \ , ln u !1ln u2 (I ) , G  G . ln k" p(I ) p(I ) G G G G A typical value for the dispersion is p(I)+0.5 (Bartelmann and Narayan, 1995).

(4.47)

4.4.3. Relative merits of shear and magnixcation ewect It is interesting to compare the prospects of measuring shear and magni"cation caused by a de#ector. We consider a small patch of the sky containing an expected number N of galaxy images (in the absence of lensing), which is su$ciently small so that the lens parameters i and c can be assumed to be constant. We also restrict the discussion to weak lensing case. The dispersion of a shear estimate from averaging over galaxy ellipticities is p/N, so that the C signal-to-noise ratio is




"c" (4.48) " (N . p   C According to (4.44), the expected change in galaxy number counts is "*N""2i"a!1"N. Assuming Poissonian noise, the signal-to-noise ratio in this case is S N


 S N

"2i"a!1"(N .  Finally, the signal-to-noise ratio for the magni"cation estimate (4.47) is


 S N

2i (N , " p(I)

  assuming all p(I) are equal.

(4.49)

(4.50)

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Comparing the three methods, we "nd 1 "c" (S/N) (S/N)  "2p(I)"a!1" .   " , i 2p "a!1" (S/N) (S/N) C   

(4.51)

If the lens situation is such that i+"c" as for isothermal spheres, the "rst of Eqs. (4.51) implies that the signal-to-noise of the shear measurement is considerably larger than that of the magni"cation. Even for number-count slopes as #at as a&0.5, this ratio is larger than "ve, with p &0.2. The C second of Eqs. (4.51) shows that the size e!ect yields a somewhat larger signal-to-noise ratio than the number-density e!ect. We therefore conclude from these considerations that shear measurements should yield more signi"cant results than magni"cation measurements. This, however, is not the end of the story. Several additional considerations come into play when these three methods of measuring lensing e!ects are compared. First, the shear measurement is the only one for which we know precisely what to expect in the absence of lensing, whereas the other two methods need to compare the measurements with calibration "elds void of lensing. These comparisons require very accurate photometry. Second, Eq. (4.49) overestimates the signal-tonoise ratio since we assumed Poissonian errors, while real galaxies are known to cluster even at very faint magnitudes (e.g., Villumsen et al., 1997), and so the error is substantially underestimated. A particularly bad example for this e!ect has been found by Athreya et al. 1999 where a cluster at z&0.9 seems to be behind the cluster (at z"0.3) they investigated with weak-lensing techniques, as identi"ed with photometric redshifts. Third, as we shall discuss in Section 4.6, observational e!ects such as atmospheric seeing a!ect the observable ellipticities and sizes of galaxy images, whereas the observed #ux of galaxies is much less a!ected. Hence, the shear and size measurements require better seeing conditions than the number-count method. Both the number counts and the size measurements (at "xed surface brightness) require accurate photometry, which is not very important for the shear measurements. As we shall see in the course of this article, most weak-lensing measurements have indeed been obtained from galaxy ellipticities. A more detailed study on the relative merits of shear and magni"cation methods has been performed by Schneider et al. (2000). Both methods were used to determine the parameters of mass pro"les of spherically symmetric clusters. The results of this study can be summarised as follows: The magni"cation in many cases yields tighter constraints on the slope of the mass pro"les, whereas the shear provides a more accurate determination of its amplitude (or lens strength). However, for the magni"cation methods to yield accurate results, the value of the unlensed number density n needs to be known fairly accurately. In particular, for measurements out to large  distances from the cluster centre (e.g., more than &10 Einstein radii), even an error of a few per cent on n destroys its relative advantage in the estimate of the shape relative to that of the shear.  But, as we shall see in the next section, the magni"cation e!ect is very important for breaking an invariance transformation in the lens reconstruction that is permitted by shear measurements alone. 4.5. Minimum lens strength for its weak lensing detection After our detailed discussion of shear estimates and signal-to-noise ratios for local lensing measurements, it is interesting to ask how strong a de#ecting mass distribution needs to be for

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a weak-lensing measurement to recognise it. Our simpli"ed consideration here su$ces to gain insight into the dependence on the lens mass of the signal-to-noise ratio for a lens detection, and on the redshifts of lens and sources. We model the de#ector as a singular isothermal sphere (see Section 3.1.5). Let there be N galaxy images with ellipticities e in an annulus centred on the lens and bounded by angular radii G h 4h 4h . For simplicity, we restrict ourselves to weak lensing, so that E(e)+c. For an axially  G  symmetric mass distribution, the shear is always tangentially oriented relative to the direction towards the mass centre, which is expressed by Eq. (3.18). We therefore consider the ellipticity component projected onto the tangential direction. It is formally de"ned by e ,!R(e e\ P),  where u is the polar angle of the galaxy position relative to the lens centre [see (3.18)]. We now de"ne an estimator for the lens strength by , X, a e . (4.52) G G G The factors a "a(h ) are arbitrary at this point, and will be chosen later such as to maximise the G G signal-to-noise ratio of estimator (4.52). Note that the expectation value of X is zero in the absence of lensing, so that a signi"cant non-zero value of X signi"es the presence of a lens. The expectation value for an isothermal sphere is E(X)"h a /(2h ), where we used (3.18), and G # G G , p , E(X)" a a E(e e )"[E(X)]# C a . (4.53) G H G H G 2 GH G We employed E(e e )"c (h )c (h )#d p/2 here, and the factor two is due to the fact that the G H  G  H GH C ellipticity dispersion only refers to one component of the ellipticity, while p is de"ned as the C dispersion of the two-component ellipticity. Therefore, the signal-to-noise ratio for a detection of the lens is h a h\ S G G G . " # (4.54) N (2p ( a C G G Di!erentiating (S/N) with respect to a , we "nd that (S/N) is maximised if the a are chosen Jh\. H G G Inserting this choice into (4.54) yields S/N"2\h p\( h\). We now replace the sum by # C G G its ensemble average over the annulus, 1 h\2"N1h\2"2nn ln(h /h ), where we used G G   N"nn(h !h ), with the number density of galaxy images n. Substituting this result into (4.54),   and using the de"nition of the Einstein radius (3.17), the signal-to-noise ratio becomes S h " # (pn(ln(h /h )   N p C n  p \ p  ln(h /h )  D C T    . "12.7 (4.55) 30 arcmin\ 0.2 600 km s\ ln 10 D  As expected, the signal-to-noise ratio is proportional the square root of the number density of galaxies and the inverse of the intrinsic ellipticity dispersion. Furthermore, it is proportional to the square of the velocity dispersion p . Assuming the "ducial values given in Eq. (4.55) and a typical T value of (D /D )&0.5, lenses with velocity dispersion in excess of &600 km s\ can be detected  











  

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with a signal-to-noise 96. This shows that galaxy clusters will yield a signi"cant weak lensing signal, and explains why clusters have been the main target for weak-lensing research up to now. Individual galaxies with p &200 km s\ cannot be detected with weak-lensing techniques. If one T is interested in the statistical properties of the mass distribution of galaxies, the lensing e!ects of N galaxies need to be statistically superposed, increasing (S/N) by a factor of (N . Thus, it is   necessary to superpose several hundred galaxies to obtain a signi"cant galaxy}galaxy lensing signal. We shall return to this topic in Section 7. We "nally note that (4.55) also demonstrates that the detection of lenses will become increasingly di$cult with increasing lens redshift, as the last factor is a sensitive function of z . Therefore, most  lenses so far investigated with weak-lensing techniques have redshifts below 0.5. High-redshift clusters have only recently become the target of detailed lensing studies. 4.6. Practical consideration for measuring image shapes 4.6.1. General discussion Real astronomical data used for weak lensing are supplied by CCD images. The steps from a CCD image to a set of galaxy images with measured ellipticities are highly non-trivial and cannot be explained in any detail in the frame of this review. Nevertheless, we want to mention some of the problems together with the solutions which were suggested and applied. The steps from CCD frames to image ellipticities can broadly be grouped into four categories; data reduction, image detection, shape determination, and corrections for the point-spread function. The data-reduction process is more or less standard, involving de-biasing, #at-"elding, and removal of cosmic rays and bad pixels. For the latter purpose, it is essential to have several frames of the same "eld, slightly shifted in position. This also allows the #at "eld to be determined from the images themselves (a nice description of these steps is given in Mould et al., 1994). To account for telescope and instrumental distortions, the individual frames have to be re-mapped before being combined into a "nal image. In order to do this, the geometric distortion has to be either known or stable. In the latter case, it can be determined by measuring the positions and shapes of stellar images (e.g., from a globular cluster). In Mould et al. (1994), the classical optical aberrations were determined and found to be in good agreement with the system's speci"cations obtained from ray-tracing analysis. With the individual frames stacked together in the combined image, the next step is to detect galaxies and to measure their shapes. This may appear simple, but is in fact not quite as straightforward, for several reasons. Galaxy images are not necessarily isolated on the image, but they can overlap, e.g. with other galaxies. Since weak-lensing observations require a large number density of galaxy images, such merged images are not rare. The question then arises whether a detected object is a single galaxy, or a merged pair, and depending on the choice made, the measured ellipticities will be much di!erent. Second, the image is noisy because of the "nite number of photons per pixel and the noise intrinsic to the CCD electronics. Thus, a local enhancement of counts needs to be classi"ed as a statistically signi"cant source detection, and a conservative signal-to-noise threshold reduces the number of galaxy images. Third, galaxy images have to be distinguished from stars. This is not a severe problem, in particular if the "eld studied is far from the Galactic plane where the number density of stars is small.

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Several data-analysis software packages exist, such as FOCAS (Jarvis and Tyson, 1981) and SExtractor (Bertin and Arnouts, 1996). They provide routines, based on algorithms developed from experience and simulated data, for objective selection of objects and measuring their centroids, their multipole moments, their magnitudes, and classify them as stars or extended objects. Kaiser et al. (1995) developed their own object detection algorithm. It is based on convolving the CCD image with two-dimensional Mexican hat-shaped "lter functions of variable width h . For each  value of h , the maxima of the smoothed intensity map are localised. Varying h , these maxima form   curves in the three-dimensional space spanned by h and h . Along each such curve, the signi"cance  of a source detection is calculated, and the maximum of the signi"cance is de"ned as the location h of an object with corresponding size h .  Once an object is found, the quadrupole moments can in principle be obtained from (4.2). In practice, however, this is not necessarily the most practical de"nition of the moment tensor. The function q (I) in (4.2) should be chosen such that it vanishes for surface brightnesses close to and ' smaller than the sky brightness; otherwise, one would sample too much noise. On the other hand, if q is cut o! at too bright values of I, the area within which the quadrupole moments are measured ' becomes too small, and the e!ects of seeing (see below) become overwhelming. Also, with a too conservative cut-o!, many galaxy images would be missed. Assume, for instance, that q (I)"IH(I!I ). One would then choose I such that it is close to, but a few p above the sky '     background, and the quadrupole moments would then be measured inside the resulting limiting isophote. Since this isophote is close to the sky background, its shape is a!ected by sky noise. This implies that the measured quadrupole moments will depend highly non-linearly on the brightness on the CCD; in particular, the e!ect of noise will enter the measured ellipticities in a non-linear fashion. A more robust measurement of the quadrupole moments is obtained by replacing the weight function q [I(h)] in (4.2) by I =(h), where =(h) explicitly depends on h. Kaiser et al. (1995) ' use a Gaussian of size h as their weight function =, i.e., the size of their = is the scale on which the  object was detected at highest signi"cance. It should be noted that the quadrupole moments obtained with a weight function =(h) do not obey the transformation law (4.5), and therefore, the expectation value of the ellipticity, E(e), will be di!erent from the reduced shear g. We return to this issue further below. Another severe di$culty for the determination of the local shear is atmospheric seeing. Due to atmospheric turbulence, a point-like source will be seen from the ground as an extended image; the source is smeared-out. Mathematically, this can be described as a convolution. If I(h) is the surface brightness before passing the Earth's atmosphere, the observed brightness distribution I
(h) is



I
(h)" d0 I(0) P(h!0) ,

(4.56)

where P(h) is the point-spread function (PSF) which describes the brightness distribution of a point source on the CCD. P(h) is normalised to unity and centred on 0. The characteristic width of the PSF is called the size of the seeing disc. The smaller it is, the less smeared the images are. A seeing well below 1 arcsec is required for weak-lensing observations, and there are only a handful of telescope sites where such seeing conditions are regularly met. The reason for this strong requirement on the data quality lies in the fact that weak-lensing studies require a high number density of galaxy images, i.e., the observations have to be extended to faint magnitudes. But the characteristic

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angular size of faint galaxies is below 1 arcsec. If the seeing is larger than that, the shape information is diluted or erased. The PSF includes not only the e!ects of the Earth's atmosphere, but also pointing errors of the telescope (e.g., caused by wind shake). Therefore, the PSF will, in general, be slightly anisotropic. Thus, seeing has two important e!ects on the observed image ellipticities: Small elliptical images become rounder, and the anisotropy of the PSF introduces a systematic, spurious image ellipticity. The PSF can be determined directly from the CCD once a number of isolated stellar images are identi"ed. The shape of the stars (which serve as point sources) re#ects the PSF. Note that the PSF is not necessarily constant across the CCD. If the number density of stellar images is su$ciently large, one can empirically describe the PSF variation across the "eld by a low-order polynomial. An additional potential di$culty is the chromaticity of the PSF, i.e. the dependence of the PSF on the spectral energy distribution of the radiation. The PSF as measured from stellar images is not necessarily the same as the PSF which applies to galaxies, due to their di!erent spectra. The di!erence of the PSFs is larger for broader "lters. However, it is assumed that the PSF measured from stellar images adequately represents the PSF for galaxies. In the idealised case, in which the quadrupole moments are de"ned with the weight function q (I)"I, the e!ect of the PSF on the observed image ellipticities can easily be described. If ' P denotes the quadrupole tensor of the PSF, de"ned in complete analogy to (4.2), then the GH observed quadrupole tensor Q
 is related to the true one by Q
"P #Q (see Valdes et al., GH GH GH GH 1983). The ellipticity s then transforms like s#¹s.1$ , s
" 1#¹

(4.57)

where P !P #2iP P #P   .  , s.1$"  (4.58) ¹"  P #P Q #Q     Thus, ¹ expresses the ratio of the PSF size to the image size before convolution, and s.1$ is the PSF ellipticity. It is evident from (4.57) that the smaller ¹, the less s
 deviates from s. In the limit of very large ¹, s
 approaches s.1$. In principle, relation (4.57) could be inverted to obtain s from s
. However, this inversion is unstable unless ¹ is su$ciently small, in the sense that noise a!ecting the measurement of s
 is ampli"ed by the inversion process. Unfortunately, these simple transformation laws only apply for the speci"c choice of the weight function. For weighting schemes that can be applied to real data, the resulting transformation becomes much more complicated. If a galaxy image features a bright compact core which emits a signi"cant fraction of the galaxy's light, this core will be smeared out by the PSF. In that case, s
 may be dominated by the core and thus contain little information about the galaxy ellipticity. This fact motivated Bonnet and Mellier (1995) to de"ne the quadrupole moments with a weight function =(h) which not only cuts o! at large angular separations, but which is also small near h"0. Hence, their weight function q is signi"cantly non-zero in an annulus with radius and width both being of the order of the size of the PSF. The di$culties mentioned above prohibit the determination of the local reduced shear by straight averaging over the directly measured image ellipticities. This average is a!ected by the use of an angle-dependent weight function = in the practical de"nition of the quadrupole moments, by

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the "nite size of the PSF and its anisotropy, and by noise. Bonnet and Mellier (1995) have performed detailed simulations of CCD frames which resemble real observations as close as possible, including an anisotropic PSF. With these simulations, the e$ciency of object detection, the accuracy of their centre positions, and the relation between true and measured image ellipticities can be investigated in detail, and so the relation between mean ellipticity and (reduced) shear can approximately be calibrated. Wilson et al. (1996) followed a very similar approach, except that the analysis of their simulated CCD frames was performed with FOCAS. Assuming an isotropic PSF, the mean image ellipticity is proportional to the reduced shear, g+f1e2, with a correction factor f depending on the limiting galaxy magnitude, the photometric depth of the image, and the size of the seeing disk. For a seeing of 0.8, Bonnet and Mellier obtained a correction factor f&6, whereas the correction factor in Wilson et al. for the same seeing is f&1.5. This large di!erence is not a discrepancy, but due to the di!erent de"nitions of the quadrupole tensor. Although the correction factor is much larger for the Bonnet and Mellier method, they show that their measured (and calibrated) shear estimate is more accurate than that obtained with FOCAS. Kaiser et al. (1995) used CCD frames taken with WFPC2 on board HST which are una!ected by atmospheric seeing, sheared them, and degraded the resulting images by a PSF typical for ground-based images and by adding noise. In this way, they calibrated their shear measurement and tested their removal of an anisotropic contribution of the PSF. However, calibrations relying on simulated images are not fully satisfactory since the results will depend on the assumptions underlying the simulations. Kaiser et al. (1995) and Luppino and Kaiser (1997) presented a perturbative approach for correcting the observed image ellipticities for PSF e!ects, with additional modi"cations made by Hoekstra et al. (1998) and Hudson et al. (1998). Since the measurement of ellipticities lies at the heart of weak-lensing studies, we shall present this approach in the next subsection, despite its being highly technical. 4.6.2. The KSB method Closely following the work by Kaiser et al. (1995), this subsection provides a relation between the observed image ellipticity and a source ellipticity known to be isotropically distributed. The relation corrects for PSF smearing and its anisotropy, and it also takes into account that transformation (4.5) no longer applies if the weight factor explicitly depends on h. We consider the quadrupole tensor



Q " dh(h !hM )(h !hM )I(h)=("h!hM "/p) , G G H H GH

(4.59)

where = contains a typical scale p, and hM is de"ned as in (4.1), but with the new weight function. Note that, in contrast to de"nition (4.2), this tensor is no longer normalised by the #ux, but this does not a!ect de"nition (4.4) of the complex ellipticity. The relation between the observed surface brightness I
(h) and the true surface brightness I is given by (4.56). We assume in the following that P is nearly isotropic, so that the anisotropic part of P is small. Then, we de"ne the isotropic part P  of P as the azimuthal average over P, and decompose P into an isotropic and an anisotropic part as



P(0)" du q(u)P (0!u) ,

(4.60)

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which de"nes q uniquely. In general, q(u) will be an almost singular function, but we shall show later that it has well-behaved moments. Both P  and q are normalised to unity and have vanishing "rst moments. With P , we de"ne the brightness pro"les

 

I (h)" du I(u)P (h!u) , I(h)" du I(u)P (h!u) .

(4.61)

The "rst of these would be observed if the true image was smeared only with an isotropic PSF, and the second is the unlensed source smeared with P . Both of these brightness pro"les are unobservable, but convenient for the following discussion. For each of them, we can de"ne a quadrupole tensor as in (4.59). From each quadrupole tensor, we de"ne the complex ellipticity s"s #is , in analogy to (4.4).   If we de"ne the centres of images including a spatial weight function, the property that the centre of the image is mapped onto the centre of the source through the lens equation is no longer strictly true. However, the deviations are expected to be very small, in general, and will be neglected in the following. Hence, we choose coordinates such that hM "0, and approximate the other centres to be at the origin as well. According to our fundamental assumption that the intrinsic ellipticities are randomly oriented, this property is shared by the ellipticities s de"ned in terms of I [see (4.61)], because it is una!ected by an isotropic PSF. Therefore, we can replace (4.13) by E(s)"0 in the determination of g. The task is then to relate the observed image ellipticity s
 to s. We break it into several steps. From s  to s
: We "rst look into the e!ect of an anisotropic PSF on the observed ellipticity. According to (4.60) and (4.61),



I
(h)" du q(h!u)I (u) .

(4.62)

Let f (h) be an arbitrary function, and consider



 



dh f (h)I
(h)" du I (u) d0 f (u#0)q(0)



1 Rf " du I (u)f (u)# q du I (u) #O(q) . 2 IJ Ru Ru I J We used the fact that q is normalised and has zero mean, and de"ned



q " du q(u)u u , q ,q !q , GH G H   

q ,2q .  

(4.63)

(4.64)

The tensor q is traceless, q "!q , following from (4.60). We consider in the following only GH   terms up to linear order in q. To that order, we can replace I  by I
 in the "nal term in (4.63), since the di!erence would yield a term JO(q). Hence,







1 Rf du I (u) f (u)+ dh f (h)I
(h)! q du I
(u) . 2 IJ Ru Ru I J

(4.65)

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Setting p "p ,p in the de"nition of the quadrupole tensors Q  and Q
, and choosing  
 f (h)"h h =("h"/p), yields G H (4.66) Q "Q
!Z q , GH GH  GHIJ IJ where the Einstein summation convention was adopted, and where








"u" R uu = Z " du I
(u) G H GHIJ p Ru Ru  I J This then yields

.

(4.67)

tr(Q )"tr(Q
)!x q , ? ? (Q  !Q  )"(Q
!Q
)!X q     ? ? and 2Q  "2Q
!X q ,   ? ? where the sums run over a"1, 2. Up to linear order in q , ? s "s
!P q ? ? ?@ @ with the de"nitions

(4.68)

(4.69)

1 P " (X !s
 x ) , ?@ tr Q
 ?@ ? @










= = =#2"u" d #g (u)g (u) , ?@ ? @ p p   = 2= x " du I
(u)g (u) #"u" , (4.70) ? ? p p   where d is the Kronecker symbol, and ?@ g (h)"h !h , g (h)"2h h . (4.71)       P was dubbed smear polarisability in Kaiser et al. (1995). It describes the (linear) response of the ?@ ellipticity to a PSF anisotropy. Note that P depends on the observed brightness pro"le. In ?@ particular, its size decreases for larger images, as expected: The ellipticities of larger images are less a!ected by a PSF anisotropy than those of smaller images. The determination of q : Eq. (4.69) provides a relation between the ellipticities of an observed ? image and a hypothetical image smeared by an isotropic PSF. In order to apply this relation, the anisotropy term q needs to be known. It can be determined from the shape of stellar ? images. X " du I
(u) ?@








 We use Greek instead of Latin indices a, b"1, 2 to denote that they are not tensor indices. In particular, the components of s do not transform like a vector, but like the traceless part of a symmetric tensor.

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Since stars are point-like and una!ected by lensing, their isotropically smeared images have zero ellipticity, sH "0. Hence, from (4.69), q "(PH )\sH
 . (4.72) ? ?@ @ In general, the PSF varies with the position of an image. If this variation is su$ciently smooth, q can be measured for a set of stars, and approximated by a low-order polynomial across the data "eld. As pointed out by Hoekstra et al. (1998), the scale size p in the measurement of q is best chosen to be the same as that of the galaxy image under consideration. Hence, for each value of p, such a polynomial "t is constructed. This approach works well and provides an estimate of q at the position of all galaxies, which can then be used in transformation (4.69). From s to s : We now relate s  to the ellipticity s of a hypothetical image obtained from isotropic smearing of the source. To do so, we use (4.61) and (3.10) in the form I(h)"IQ(Ah), and consider



I (h)" du I(Au)P (h!u)



1 " df I(f)P (h!A\f),IK (Ah) . det A

(4.73)

The second step is merely a transformation of the integration variable, and in the "nal step we de"ned the brightness moment



1 P (A\h) . IK (h)" du I(u)PK (h!u) with PK (h), det A

(4.74)

The function PK is normalised and has zero mean. It can be interpreted as a PSF relating IK to I. The presence of shear renders PK anisotropic. We next seek to "nd a relation between the ellipticities of I  and IK :






"b" QK " db b b IK (b) = G H GH p( 






"det A A A dh h h I (h) = GI HJ I J



"h"!d g (h) ? ? . p

(4.75)

The relation between the two "lter scales is given by p( "(1!i)(1#"g")p, and d is distortion (4.15). For small d, we can employ a "rst-order Taylor expansion of the weight function = in the previous equation. This results in the following relation between s( and s : s !s( "C g , ? ? ?@ @ where

(4.76)

2 2 C "2d !2s s # s ¸ ! B , ?@ ?@ ? @ tr (Q ) ? @ tr (Q ) ?@






"h" 1 g (h)g (h) B "! dh I (h)= @ ?@ p p ?

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"h" 1 ¸ "! dh "h"I (h)= g (h) . ? p p ?

359

(4.77)

C is the shear polarisability of Kaiser et al. (1995). Whereas C is de"ned in terms of I , owing to the assumed smallness of q, the di!erence of C calculated with I  and I
 would cause a second-order change in (4.76) and is neglected, so that we can calculate C directly from the observed brightness pro"le. In analogy to (4.60), we can decompose PK into an isotropic and an anisotropic part, the latter one being small due to the assumed smallness of the shear,



PK (h)" du PK (u) q( (h!u) .

(4.78)

De"ning the brightness pro"le which would be obtained from smearing the source with the isotropic PSF PK , IK (h)" du IQ(u) PK (h!u), one "nds



IK (h)" du IK (u)q( (h!u) .

(4.79)

Thus, the relation between IK and IK  is the same as that between I
 and I , and we can write s( "s( !P q( . (4.80) ? ? ?@ @ Note that P should in principle be calculated by using IK instead of I
 in (4.69). However, due to the assumed smallness of g and q, the di!erences between I
, I , and IK are small, namely of "rst order in g and q. Since q( is of order g [as is obvious from its de"nition, and will be shown explicitly in (4.82)], this di!erence in the calculation of P would be of second order in (4.80) and is neglected here. Eliminating s( from (4.76) and (4.80), we obtain s "s( #C g #P q( . (4.81) ? ? ?@ @ ?@ @ Now, for stellar objects, both s(  and s  vanish, which implies a relation between q( and g, q( "!(P H)\CH g , (4.82) ? ?@ @A A where the asterisk indicates that P and C are to be calculated from stellar images. Whereas the result should, in principle, not depend on the choice of the scale length in the weight function, it does so in practice. As argued in Hoekstra et al. (1998), one should use the same scale length in P H and CH as for the galaxy object for which the ellipticities are measured. De"ning now PE "C !P (P H)\CH ?@ ?@ ?A AB B@ and combining (4.69) and (4.81), we "nally obtain

(4.83)

s( "s
!P q !PE g . (4.84) ?@ @ ?@ @ This equation relates the observed ellipticity to that of the source smeared by an isotropic PSF, using the PSF anisotropy and the reduced shear g. Since the expectation value of s(  is zero, (4.84) yields an estimate of g. The two tensors P and PE can be calculated from the brightness pro"le of the images. Whereas the treatment has been con"ned to "rst order in the PSF anisotropy and the

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shear, the simulations in Kaiser et al. (1995) and Hoekstra et al. (1998) show that the resulting equations can be applied even for moderately large shear. A numerical implementation of these relations, the imcat software, is provided by N. Kaiser (see http://www.ifa.hawaii.edu/& kaiser). We also note that modi"cations of this scheme were recently suggested (Rhodes et al., 1999; Kaiser, 1999), as well as a completely di!erent approach to shear measurements (Kuijken, 1999). Kaiser et al. (1999b) provide a detailed description of the image analysis of weak-lensing data from a large CCD-array camera.

5. Weak lensing by galaxy clusters 5.1. Introduction So far, weak gravitational lensing has chie#y been applied to determine the mass distribution of medium-redshift galaxy clusters. The main reason for this can be seen from Eq. (4.55): Clusters are massive enough to be individually detected by weak lensing. More traditional methods to infer the matter distribution in clusters are (a) dynamical methods, in which the observed line-of-sight velocity distribution of cluster galaxies is used in conjunction with the virial theorem, and (b) the investigation of the di!use X-ray emission from the hot (&10 K) intra-cluster gas residing in the cluster potential well (see, e.g., Sarazin, 1986). Both of these methods are based on rather strong assumptions. For the dynamical method to be reliable, the cluster must be in or near virial equilibrium, which is not guaranteed because the typical dynamical time scale of a cluster is not much shorter than the Hubble time H\, and the  substructure abundantly observed in clusters indicates that an appreciable fraction of them is still in the process of formation. Projection e!ects and the anisotropy of galaxy orbits in clusters further a!ect the mass determination by dynamical methods. On the other hand, X-ray analyses rely on the assumption that the intra-cluster gas is in hydrostatic equilibrium. Owing to the "nite spatial and energy resolution of existing X-ray instruments, one often has to conjecture the temperature pro"le of the gas. Here, too, the in#uence of projection e!ects is di$cult to assess. Whereas these traditional methods have provided invaluable information on the physics of galaxy clusters, and will continue to do so, gravitational lensing o!ers a welcome alternative approach, for it determines the projected mass distribution of a cluster independent of the physical state and nature of the matter. In particular, it can be used to calibrate the other two methods, especially for clusters showing evidence of recent merger events, for which the equilibrium assumptions are likely to fail. Finally, as we shall show below, the determination of cluster mass pro"les by lensing is theoretically simple, and recent results show that the observational challenges can also be met with modern telescopes and instruments. Both shear and magni"cation e!ects have been observed in a number of galaxy clusters. In this section, we discuss the methods by which the projected mass distribution in clusters can be determined from the observed lensing e!ects, and show some results of mass reconstructions, together with a brief discussion of their astrophysical relevance. In principle, voids could also be measured using the same methods, but as shown in Amendola et al. (1999), their (negative) density contrast is too small for a detection under realistic assumptions. Section 5.2 presents the principles

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of cluster mass reconstruction from estimates of the (reduced) shear obtained from image ellipticities (also recently reviewed by Umetsu et al., 1999). In contrast to the two-dimensional mass maps generated by these reconstructions, the aperture mass methods discussed in Section 5.3 determine a single number to characterise the bulk properties of the cluster mass. Observational results are presented in Section 5.4. We outline further developments in the "nal section, including the combined analysis of shear and magni"cation e!ects, maximum-likelihood methods for the mass reconstruction, and a method for measuring local lens parameters from the extragalactic background noise. 5.2. Cluster mass reconstruction from image distortions We discussed in detail in Section 4 how the distortion of image shapes can be used to determine the local tidal gravitational "eld of a cluster. In this section, we describe how this information can be used to construct two-dimensional mass maps of clusters. Shortly after the discovery of giant luminous arcs (Soucail et al., 1987a; Lynds and Petrosian, 1989), Fort et al. (1988) detected a number of distorted galaxy images in the cluster A 370. They also interpreted these arclets as distorted background galaxy images, but on a weaker level than the giant luminous arc in the same cluster. The redshift determination of one arclet by Mellier et al. (1991) provided early support for this interpretation. Tyson et al. (1990) discovered a coherent distortion of faint galaxy images in the clusters A 1689 and Cl 1409#52, and constrained their (dark) mass pro"les from the observed &shear'. Kochanek (1990) and Miralda-Escude (1991) studied in detail how parameterised mass models for clusters can be constrained from such distortion measurements. The "eld began to #ourish after Kaiser and Squires (1993) found that the distortions can be used for parameter-free reconstructions of cluster surface mass densities. Their method, and several variants of it, will be described in this section. It has so far been applied to about 15 clusters, and this number is currently limited by the number of available dark nights with good observing conditions at the large telescopes which are required for observations of weak lensing. 5.2.1. Linear inversion of shear maps Eq. (3.15) shows that the shear c is a convolution of the surface mass density i with the kernel D. This relation is easily inverted in Fourier space to return the surface mass density in terms of a linear functional of the shear. Hence, if the shear can be observed from image distortions, the surface mass density can directly be obtained. Let the Fourier transform of i(h) be



i( (l)"

1

dh i(h) exp(ih ) l) .

(5.1)

The Fourier transform of the complex kernel D de"ned in (3.15) is (l !l #2il l )   . DK (l)"p  "l"

(5.2)

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Using the convolution theorem, Eq. (3.15) can be written c( (l)"p\DK (l)i( (l) for lO0. Multiplying both sides of this equation with DI H and using DI DI H"p gives i( (l)"p\c( (l)DK H(l) for lO0

(5.3)

and the convolution theorem leads to the "nal result 1 i(h)!i "  p 1 " p

 

1

1

dh DH(h!h) c(h) dh R[DH(h!h) c(h)]

(5.4)

(Kaiser and Squires, 1993). The constant i in (5.4) appears because a constant surface mass  density does not cause any shear and is thus unconstrained by c. The two expressions in (5.4) are equivalent because I(DK H c( ),0, as can be shown from the Fourier transforms of equations (3.12). In applications, the second form of (5.4) should be used to ensure that i is real. Relation (5.4) can either be applied to a case where all the sources are at the same redshift, in which case i and c are de"ned as in Eqs. (3.7) and (3.12), or where the sources are distributed in redshift, because i and c are interpreted as convergence and shear for a hypothetical source at in"nite redshift, as discussed in Section 4.3.2. In the case of a weak lens (i;1, "c";1), the shear map is directly obtained from observations, cf. (5.17). When inserted into (5.4), this map provides a parameter-free reconstruction of the surface mass density, apart from an overall additive constant. The importance of this result is obvious, as it provides us with a novel and simple method to infer the mass distribution in galaxy clusters. There are two basic ways to apply (5.4) to observational data. Either, one can derive a shear map from averaging over galaxy images by calculating the local shear on a grid in h-space, as described in Section 4.3; or, one can replace the integral in (5.4) by a sum over galaxy images at positions h , G 1 (5.5) i(h)" R[DH(h!h )e ] . G G np G Unfortunately, this estimate of i has in"nite noise (Kaiser and Squires, 1993) because of the noisy sampling of the shear at the discrete background galaxy positions. Smoothing is therefore necessary to obtain estimators of i with "nite noise. The form of Eq. (5.5) is preserved by smoothing, but the kernel D is modi"ed to another kernel DI . In particular, Gaussian smoothing with smoothing length h leads to  "h" "h" DI (h)" 1! 1# exp ! D(h) (5.6) h h   (Seitz and Schneider, 1995a). The rms error of the resulting i map is of order p N\, where N is C the number of galaxy images per smoothing window, N&nph. However, the errors will be  strongly spatially correlated. van Waerbeke (2000) showed that the covariance of a mass map obtained with the kernel (5.6) is














p "h!h" Cov(i(h)), i(h) " C exp ! . 4phn 2h  

(5.7)

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Thus, the correlation extends to scales of order the smoothing scale h (see also Lombardi and  Bertin, 1998b). Indeed, this result is surprising, as by reducing the smoothing scale, the correlation length of the noise can accordingly be reduced to small scales } although the surface mass density at each point depends on the galaxy ellipticities at all distances. It should be noted that the covariance in (5.7) is derived under the assumption of no lensing, c,0. In the presence of a shear } the interesting situation of course } an additional e!ect contributes to the noise, namely that the galaxy images are not uniformly, but randomly distributed. This e!ect contributes shot-noise to the covariance, quadratically in c (Schneider and Morales-Merino, 2000). Therefore, whereas the estimate (5.5) with D replaced by DI uses the observational data more directly than by "rst constructing a smoothed shear map and applying (5.4) to it, it turns out that the latter method yields a mass map which is less noisy than the estimate obtained from (5.5), because (5.5) contains the &shot noise' from the random angular position of the galaxy images (Seitz and Schneider, 1995a). A lower bound to the smoothing length h follows from the spatial number density of back ground galaxies, i.e. their mean separation. More realistically, a smoothing window needs to encompass several galaxies. In regions of strong shear signals, N&10 may su$ce, whereas mass maps in the outskirts of clusters where the shear is small may be dominated by noise unless N&100. These remarks illustrate that a single smoothing scale across a whole cluster may be a poor choice. We shall return to this issue in Section 5.5.1, where improvements will be discussed. Before applying the mass reconstruction formula (5.4) to real data, one should be aware of the following di$culties: (1) The integral in (5.4) extends over 1, while real data "elds are relatively small (most of the applications shown in Section 5.4 are based on CCDs with side lengths of about 7 arcmin). Since there is no information on the shear outside the data "eld, the integration has to be restricted to the "eld, which is equivalent to setting c"0 outside. This is done explicitly in (5.5). This cut-o! in the integration leads to boundary artefacts in the mass reconstruction. Depending on the strength of the lens, its angular size relative to that of the data "eld, and its location within the data "eld, these boundary artefacts can be more or less severe. They are less important if the cluster is weak, small compared to the data "eld, and centred on it. (2) The shear is an approximate observable only in the limit of weak lensing. The surface mass density obtained by (5.4) is biased low in the central region of the cluster where the weak-lensing assumption may not hold (and does not hold in those clusters which show giant arcs). Thus, if the inversion method is to be applied also to the inner parts of a cluster, the relation between c and the observable d has to be taken into account. (3) The surface mass density is determined by (5.4) only up to an additive constant. We demonstrate in the next subsection that there exists a slightly di!erent general invariance transformation which is present in all mass reconstructions based solely on image shapes. However, this invariance transformation can be broken by including the magni"cation e!ect. In the next three subsections, we shall consider points (1) and (2). In particular, we show that the "rst two problems can easily be cured. The magni"cation e!ects will be treated in Section 5.4.

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5.2.2. Non-linear generalisation of the inversion, and an invariance transformation In this section, we generalise the inversion equation (5.4) to also account for strong lensing, i.e. we shall drop the assumption i;1 and "c";1. In this case, the shear c is no longer a direct observable, but at best the reduced shear g, or in general the distortion d. In this case, the relation between i and the observable becomes non-linear. Furthermore, we shall assume here that all sources are at the same redshift, so that the reduced shear is well de"ned. Consider "rst the case that the cluster is sub-critical everywhere, i.e. det A'0 for all h, which implies "g(h)"(1. Then, the mean image ellipticity e is an unbiased estimate of the local reduced shear, so that c(h)"[1!i(h)]1e2(h) ,

(5.8)

where the "eld 1e2(h) is determined by the local averaging procedure described in Section 4.3.1. Inserting this into (5.4) leads to an integral equation for i(h), 1 i(h)!i "  p



1

dh [1!i(h)] R[DH(h!h) 1e2(h)]

(5.9)

(Seitz and Schneider, 1995a), which is readily solved by iteration. Starting from i,0, a "rst estimate of i(h) is obtained from (5.9), which after insertion into the right-hand side of (5.8) yields an update of c(h), etc. This iteration process converges quickly to the unique solution. The situation becomes only slightly more complicated if critical clusters are included. We only need to keep track of det A while iterating, because c must be derived from 1/1e2H rather than from 1e2 where det A(0. Hence, the local invariance between g and 1/gH is broken due to non-local e!ects: A local jump from g to 1/gH cannot be generated by any smooth surface mass density. After a minor modi"cation, this iteration process converges quickly. See Seitz and Schneider (1995a) for more details on this method and for numerical tests done with a cluster mass distribution produced by a cosmological N-body simulation. It should have become clear that the non-linear inversion process poses hardly any additional problem to the mass reconstruction compared to the linear inversion (5.4). This non-linear inversion still contains the constant i , and so the result will depend on this  unconstrained constant. However, in contrast to the linear (weak lensing) case, this constant does not correspond to adding a sheet of constant surface mass density. In fact, as can be seen from (5.9), the transformation i(h)Pi(h)"ji(h)#(1!j) or [1!i(h)]"j[1!i(h)]

(5.10)

 At points where i"1, 1/gH"0 and E(e)"0, while c remains "nite. During the iteration, there will be points h where the "eld i is very close to unity, but where 1e2 is not necessarily small. This leads to large values of c, which render the iteration unstable. However, this instability can easily be removed if a damping factor like (1#"c(h)") exp(!"c(h)") is included in (5.4). This modi"cation leads to fast convergence and a!ects the result of the iteration only very slightly.

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leads to another solution of the inverse problem for any value of jO0. Another and more general way to see this is that the transformation iPi changes c to c(h)"jc(h) (cf. (3.15)). Hence, the reduced shear g"c(1!i)\ is invariant under transformation (5.10), so that the relation between intrinsic and observed ellipticity is unchanged under the invariance transformation (5.10). This is the mass-sheet degeneracy pointed out by Falco et al. (1985) in a di!erent context. We thus conclude that the degeneracy due to the invariance transformation (5.10) cannot be lifted if only image shapes are used. However, the magni"cation transforms like k(h)"j\k(h) ,

(5.11)

so that the degeneracy can be lifted if magni"cation e!ects are taken into account (see Section 4.4). The invariance transformation leaves the critical curves of the lens mapping invariant. Therefore, even the location of giant luminous arcs which roughly trace the critical curves does not determine the scaling constant j. In addition, the curve i"1 is invariant under (5.10). However, there are at least two ways to constrain j. First, it is reasonable to expect that on the whole the surface mass density in clusters decreases with increasing separation from the cluster ¢re', so that j'0. Second, since the surface mass density i is non-negative, upper limits on j are obtained by enforcing this condition. 5.2.3. Finite-xeld inversion techniques We shall now turn to the problem that inversion (5.4) in principle requires data on the whole sky, whereas the available data "eld is "nite. A simple solution of this problem has been attempted by Seitz and Schneider (1995a). They extrapolated the measured shear "eld on the "nite region U outside the data "eld, using a parameterised form for the radial decrease of the shear. From a sample of numerically generated cluster mass pro"les, Bartelmann (1995a) showed that this extrapolation yields fairly accurate mass distributions. However, in these studies the cluster was always assumed to be isolated and placed close to the centre of the data "eld. If these two conditions are not met, the extrapolation can produce results which are signi"cantly o!. In order to remove the boundary artefacts inherent in applying (5.4) to a "nite "eld, one should therefore aim at constructing an unbiased "nite-"eld inversion method. The basis of most "nite-"eld inversions is a result "rst derived by Kaiser (1995). Eq. (3.12) shows that shear and surface mass density are both given as second partial derivatives of the de#ection potential t. After partially di!erentiating (3.12) and combining suitable terms we "nd






c #c  ,u (h) . (5.12)

i"  A c !c   The gradient of the surface mass density can thus be expressed by the "rst derivatives of the shear, hence i(h) can be determined, up to an additive constant, by integrating (5.12) along appropriately selected curves. This can be done in the weak-lensing case where the observed smoothed ellipticity "eld 1e2(h) can be identi"ed with c, and u (h) can be constructed by "nite di!erencing. If we insert A c"(1!i) g into (5.12), we "nd after some manipulations




1!g !1 

K(h)" 1!g !g !g    ,u (h) , E

!g  1#g 




g #g   g !g  

 (5.13)

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where K(h),ln[1!i(h)] .

(5.14)

Hence, using the smoothed ellipticity "eld 1e2(h) as an unbiased estimator for g(h), and assuming a sub-critical cluster, one can obtain the vector "eld u (h) by "nite di!erencing, and thus determine E K(h) up to an additive constant from line integration, or, equivalently, 1!i(h) up to an overall multiplicative constant. This is again the invariance transformation (5.10). In principle, it is now straightforward to obtain i(h) from the vector "eld u (h), or K(h) from A u (h), simply by a line integration of the type E



i(h, h )"i(h )#  

h

h



dl ) u (l) , A

(5.15)

where l is a smooth curve connecting h with h . If u is a gradient "eld, as it ideally is, the resulting  A surface mass density is independent of the choice of the curves l. However, since u is obtained A from noisy data (at least the noise resulting from the intrinsic ellipticity distribution), it will in general not be a gradient "eld, so that (5.12) has no solution. Therefore, the various line integration schemes proposed (Schneider, 1995; Kaiser et al., 1994; Bartelmann, 1995a) yield di!erent results. Realising that Eq. (5.12) has no exact solution for an observed "eld u , we wish to "nd a A mass distribution i(h) which satis"es (5.12) &best'. In general, u can be split into a gradient A "eld and a curl component, but this decomposition is not unique. However, as pointed out in Seitz and Schneider (1996), since the curl component is due to noise, its mean over the data "eld is expected to vanish. Imposing this condition, which determines the decomposition uniquely, they showed that



i(h)!i"

U

dh H(h, h) ) u (h) , A

(5.16)

where i is the average of i(h) over the data "eld U, and the kernel H is the gradient of a scalar function which is determined through a von Neumann boundary value problem, with singular source term. This problem can be solved analytically for circular and rectangular data "elds, as detailed in the appendix of Seitz and Schneider (1996). If the data "eld has a more complicated geometry, an analytic solution is no longer possible, and the boundary value problem with a singular source term cannot be solved numerically. An alternative method starts with taking the divergence of (5.12) and leads to the new boundary value problem

i" ) u

with n ) i"n ) u on RU , (5.17) A A where n is the outward-directed normal on the boundary of U. As shown in Seitz and Schneider (1998), Eqs. (5.16) and (5.17) are equivalent. An alternative and very elegant way to derive (5.17) has been found by Lombardi and Bertin (1998b). They noticed that the &best' approximation to a solution of (5.12) minimises the &action'



U

dh " i(h)!u (h)" . A

(5.18)

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Euler's equations of the variational principle immediately reproduce (5.17). This von Neumann boundary problem is readily solved numerically, using standard numerical techniques (see Section 19.5 of Press et al., 1986). Lombardi and Bertin (1999a) proposed a direct method for solving the variational principle (5.18) which, for rectangular "elds, is equivalent to Fourier methods for the solution of the Neumann problem (5.17). A comparison between these di!erent "nite-"eld inversion equations was performed in Seitz and Schneider (1996) and in Squires and Kaiser (1996) by numerical simulations. Of all the inversions tested, inversion (5.17) performs best on all scales (Seitz and Schneider, 1996; Fig. 6 of Squires and Kaiser, 1996). Indeed, Lombardi and Bertin (1998b) showed analytically that the solution of Eq. (5.17) provides the best unbiased estimate of the surface mass density. Relations (5.15)}(5.18) can be generalised to the non-weak case by replacing i with K and u with u . A E 5.2.4. Accounting for a redshift distribution of the sources We now describe how the preceding mass reconstructions must be modi"ed if the sources have a broad redshift distribution. In fact, only minor modi"cations are needed. The relation 1e2"g for a single source redshift is replaced by Eq. (4.28), which gives an estimate for the shear in terms of the mean image ellipticities and the surface mass density. This relation can be applied iteratively: Begin with i"0; then, Eq. (4.28) yields a "rst guess for the shear c(h) by setting c"0 on the right-hand side. From (5.16), or equivalently by solving (5.17), the corresponding surface mass density i(h) is obtained. Inserting i and c on the right-hand side of Eq. (4.28), a new estimate c(h) for the shear is obtained, and so forth. This iteration process quickly converges. Indeed, the di$culty mentioned in footnote 9 no longer occurs since the critical curves and the curve(s) i"1 are e!ectively smeared out by the redshift distribution, and so the iteration converges even faster than in the case of a single-source redshift. Since iL is determined only up to an additive constant for any cL, the solution of the iteration depends on the choice of this constant. Hence, one can obtain a one-parameter family of mass reconstructions, like in (5.10). However, the resulting mass-sheet degeneracy can no longer be expressed analytically due to the complex dependence of (4.28) on i and c. In the case of weak lensing, it corresponds to adding a constant, as before. An approximate invariance transformation can also be obtained explicitly for mildly non-linear clusters with i:0.7 and det A'0 everywhere. In that case, Eq. (4.29) holds approximately, and can be used to show (Seitz and Schneider, 1997) that the invariance transformation takes the form (1!j)1Z2 . i(h)Pi(h)"j i(h)# 1Z2

(5.19)

In case of a single redshift z , such that Z(z )"1Z2, this transformation reduces to (5.10) for 1Z2i.   We point out that the invariance transformation (5.19) in the case of a redshift distribution of sources is of di!erent nature than that for a single-source redshift. In the latter case, the reduced shear g(h) is invariant under transformation (5.10). Therefore, the probability distribution of the observed galaxy ellipticities is invariant, since it involves only the intrinsic ellipticity distribution and g. For a redshift distribution, the invariance transformation keeps the mean image ellipticities invariant, but the probability distributions are changed. Several strategies were explored in Seitz

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and Schneider (1997) to utilise this fact for breaking the invariance transformation (see also Lombardi and Bertin, 1999b; Gautret et al., 2000). While possible, in principle, the corresponding e!ect on the observed ellipticity distribution is too small for this approach to be feasible with existing data. 5.2.5. Breaking the mass-sheet degeneracy Eq. (5.11) shows that the invariance transformation (5.10) a!ects the magni"cation. Hence, the degeneracy can be lifted with magni"cation information. As discussed in Section 4.4, two methods to obtain magni"cation information have been proposed. Detections of the number-density e!ect have so far been reported for two clusters (Cl 0024#16, Fort et al., 1997; Abell, 1689; Taylor et al., 1998). Whereas the information provided by the number density e!ect is less e$cient than shear measurements (see Section 4.4.3), these two clusters appear to be massive enough to allow a signi"cant detection. In fact, Taylor et al. (1998) obtained a two-dimensional mass reconstruction of the cluster A 1689 from magni"cation data. In the case of weak lensing, and thus small magni"cations, the magni"cation can locally be translated into a surface mass density } see (4.44). In general, the relation between k and i is non-local, since k also depends on the shear. Various attempts to account for this non-locality have been published (van Kampen, 1998; Dye and Taylor, 1998). However, it must be noted that the surface mass density cannot be obtained from magni"cation alone since the magni"cation also depends on the shear caused by matter outside the data "eld. In practice, if the data "eld is su$ciently large and no mass concentration lies close to but outside the data "eld, the mass reconstruction obtained from magni"cation can be quite accurate. In order to break the mass-sheet degeneracy, it su$ces in principle to measure one value of the magni"cation: Either the magni"cation at one location in the cluster, or the average magni"cation over a region. We shall see later in Section 5.5.1 how local magni"cation information can be combined with shear measurements. Doing it the naive way, expressing i in terms of k and c, is a big waste of information: Since there is only one independent scalar "eld (namely the de#ection potential t) describing the lens, one can make much better use of the measurements of c and k than just combining them locally; the relation between them should be used to reduce the error on i. 5.2.6. Accuracy of cluster mass determinations The mass-sheet degeneracy fundamentally limits the accuracy with which cluster masses can be determined from shear measurements if no additional assumptions are introduced. Furthermore, cosmologists are traditionally interested in the masses of clusters inside spherical volumes (e.g., inside the virial radius), whereas lensing measures the mass in cylinders, i.e., the projected mass. On the other hand, cosmological simulations show that cluster mass pro"les are quite similar in shape (e.g., Navarro et al., 1996b). Assuming such a universal density pro"le, both of these e!ects can approximately be accounted for. The relation between projected mass within the virial radius and that inside a sphere with virial radius has been investigated by Reblinsky and Bartelmann (1999a) and Metzler et al. (1999), using numerical cluster simulations. The ratio of these two masses is by de"nition 51, but as these authors show, this ratio can be larger than unity by several tens of a per cent, due to projection of additional mass in front of or behind the cluster proper. As clusters are preferentially located inside "laments, the largest deviations occur when the "lament is oriented along the line-of-sight to the

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cluster. The amplitude of this e!ect decreases with higher cluster masses. The projection bias is of interest only when comparing lensing masses with cosmological predictions of spherical masses. However, at least when cosmological predictions are derived from numerical simulations, one can equally well extract the projected masses; the projection bias therefore does not a!ect the use of cluster mass estimates from lensing for cosmology. Using cluster mass models obtained from N-body simulations, Brainerd et al. (1999) showed that, when the observed shear signal is related to the mass using the isothermal relation (cf. Section 3.1.5) M ((r)"4rc(r)p for the mass inside spheres of radius r, or M ((r)"2prc(r)R     for the projected mass inside r"D h, fairly accurate masses of clusters can be derived from weak  lensing. In particular, the virial masses of clusters can be determined with high accuracy, provided the shear measurements extend to such large distances. Whereas most of the previous weak-lensing cluster studies, as described in Section 5.4, do not cover such large an area, the upcoming wide-"eld imaging cameras will allow one to do this in the near future. Nevertheless, the projection bias needs to be kept in mind when masses of clusters are quoted from weak-lensing analyses using relatively small angular "elds. 5.3. Aperture mass and multipole measures Having reconstructed the mass distribution, we can estimate the local dispersion of i (e.g. Lombardi and Bertin, 1998b; van Waerbeke, 2000); cf. Eq. (5.7). However, the errors at di!erent points are strongly correlated, and so it makes little sense to attach an error bar to each point of the mass map. Although mass maps contain valuable information, it is sometimes preferable to reduce them to a small set of numbers such as the mass-to-light ratio, or the correlation coe$cient between the mass map and the light distribution. One of the quantities of interest is the total mass inside a given region. As became clear in the last section, this quantity by itself cannot be determined from observed image ellipticities due to the invariance transformation. But a quantity related to it, (5.20) f(h; 0 , 0 ),i(h; 0 )!i(h; 0 , 0 ) ,      the di!erence between the mean surface mass densities in a circle of radius 0 around h and in an  annulus of inner and outer radii 0 and 0 , respectively, can be determined in the weak-lensing   case, since then the invariance transformation corresponds to an additive constant in i which drops out of (5.20). We show in this section that quantities like (5.20) can directly be obtained from the image ellipticities without the need for a two-dimensional mass map. In Section 5.3.1, we derive a generalised version of (5.20), whereas we consider the determination of mass multipoles in Section 5.3.2. The prime advantage of all these aperture measures is that the error analysis is relatively straightforward. 5.3.1. Aperture mass measures Generally, aperture mass measures are weighted integrals of the local surface mass density



M (h )" dh i(h) ;(h!h )   

(5.21)

with weight function ;(h). Assume now that the weight function is constant on self-similar concentric curves. For example, the f-statistics (5.20), introduced by Kaiser (1995), is of the form

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(5.21), with a weight function that is constant on circles, ;(0)"(p0 )\ for 04040 ,   ;(0)"[p(0 !0 ]\ for 0 (040 , and zero otherwise.     Let the shape of the aperture be described by a closed curve c(j), j3I, where I is a "nite interval, such that c;c ,c c !c c '0 for all j3I. We can then uniquely de"ne a new coordinate     system (b, j) by choosing a centre h and de"ning h"h #bc(j). The weight function should be   constant on the curves c(j) so that it depends only on b. In the new coordinate system, (5.21) reads







(5.22) db b ;(b) dj c;c i[h #bc(j)] ,  '  where the factor b c;c is the Jacobian determinant of the coordinate transformation. Eq. (5.22) can now be transformed in three steps; (1) by a partial integration with respect to b; (2) by replacing partial derivatives of i with partial derivatives of c using Eq. (5.12); and (3) by removing partial derivatives of c in another partial integration. In carrying out these steps, we assume that the weight function is compensated, M (h )"  



db b ;(b)"0 .

(5.23)

Introducing



2 @ Q(b), db b ;(b)!;(b) (5.24) b  and writing the curve c in complex notation, C(j)"c (j)#i c (j), leads to the "nal result   (Schneider and Bartelmann, 1997)



I[c(h)CHCQ H] , M (h )" dh Q[b(h)]   I[CHCQ ]

(5.25)

where the argument j of C is to be evaluated at position h"h #bc(j). The numerator in the  "nal term of (5.25) projects out a particular component of the shear, whereas the denominator is part of the Jacobian of the coordinate transformation. Constraint (5.23) assures that an additive constant in i does not a!ect M . Expression (5.25) has several nice properties which render it  useful: (1) If the function ;(b) is chosen such that it vanishes for b'b , then from (5.23) and (5.24),  Q(b)"0 for b'b . Thus, the aperture mass can be derived from the shear in a "nite region.  (2) If ;(b)"const for 04b4b , then Q(b)"0 in that interval. This means that the aperture mass  can be determined solely from the shear in an annulus b (b(b . This has two advantages   which are relevant in practice. First, if the aperture is centred on a cluster, the bright central cluster galaxies may prevent the detection of a large number of faint background galaxies there, so that the shear in the central part of the cluster may be di$cult to measure. In that case it is

 There are of course other ways to derive (5.25), e.g. by inserting (5.4) into (5.21). See Squires and Kaiser (1996) for a di!erent approach using Gauss's law.

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still possible to determine the total mass inside the cluster core using (5.25) with an appropriately chosen weight function ;. Second, although in general the shear cannot be determined directly from the image ellipticities [but only the reduced shear c(1!i)\], we can choose the size b of the inner boundary of the annulus su$ciently large that i;1 in the annulus, and  then c+g is an accurate approximation. Hence, in that case the mean image ellipticity directly yields an estimate of the shear. Then, integral (5.25) can be transformed into a sum over galaxy images lying in the annulus, yielding M directly in terms of the observables. This in turn has  the great advantage that an error analysis of M is fairly simple.  We consider circular apertures as an example, for which (b, j)"(h, u) and C(u)"exp(iu). Then, I(CHCQ )"1, and I(cCHCQ H)"c (h; h ) " : ![c cos(2u)#c sin(2u)]"!R[c(h#h )e\ P] , (5.26)      where we have de"ned the tangential component c of the shear relative to the point h . Hence, for   circular apertures (5.25) becomes



M (h )" dh Q("h") c (h; h )    

(5.27)

(Kaiser et al., 1994; Schneider, 1996b). The f-statistics (5.20) is obtained from (5.27) by setting Q(h)"0 h\[p(0 !0 )]\ for 0 4h40 and Q(h)"0 otherwise, so that      c (h; h ) 0  ,  dh  (5.28) f(h ; 0 , 0 )"    "h" [p(0 !0 )]   where the integral is taken over the annulus 0 4h40 .   For practical purposes, the integral in (5.27) is transformed into a sum over galaxy images. Recalling that e is an estimator for c in the weak-lensing case, and that the weight function can be chosen to avoid the strong-lensing regime, we can write



1 (5.29) M (h )" Q("h !h ") e (h ) , G  G    n G where we have de"ned, in analogy to c , the tangential component e of the ellipticity of an image at  G h relative to the point h by G  e "!R(e e\ P) , (5.30) G u is the polar angle of h!h , and n is the number density of galaxy images. The rms dispersion  p(M ) of M in the case of no lensing is found from the (two dimensional) dispersion p of the   C intrinsic ellipticity of galaxies,





 p . (5.31) p(M )" C Q("h !h ") G   2n G The rms dispersion in the presence of lensing will deviate only weakly from p(M ) as long as  the assumption of weak lensing in the annulus is satis"ed. Hence, p(M ) can be used as an  error estimate for the aperture mass and as an estimate for the signal-to-noise ratio of a mass measurement.

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This opens the interesting possibility to search for (dark) mass concentrations using the aperture mass (Schneider, 1996b). Consider a weight function ; with the shape of a Mexican hat, and a data "eld U on which apertures of angular size h can be placed. For each aperture position, one can calculate M and the dispersion. The dispersion can be obtained either from the analytical formula  (5.31), or it can be obtained directly from the data, by randomising the position angles of all galaxy images within the aperture. The dispersion can be obtained from many realisations of this randomisation process. Large values of M will be obtained for mass concentrations whose  characteristic size and shape is close to that of the chosen "lter function ;. Thus, by varying the size h of the "lter, di!erent mass concentrations will preferentially be selected. The aperture mass is insensitive to mass concentrations of much smaller and much larger angular scales than the "lter size. We have considered in Section 4.5 the signal-to-noise ratio for the detection of a singular isothermal sphere from its weak-lensing e!ect. Estimate (4.54) was obtained by an optimal weighting scheme for this particular mass distribution. Since real mass concentrations will deviate from this pro"le, and also from the assumed symmetry, the "lter function ; should have a more generic shape. In that case, the S/N will have the same functional behaviour as in (4.54), but the prefactor depends on the exact shape of ;. For the "lter function used in Schneider (1996b), S/N is about 25% smaller than in (4.54). Nevertheless, one expects that the aperture-mass method will be sensitive to search for intermediate-redshift halos with characteristic velocity dispersions above &600 km s\. This expectation has been veri"ed by numerical simulations, which also contained larger and smaller-scale mass perturbations. In addition, a detailed strong-lensing investigation of the cluster MS 1512#62 has shown that its velocity dispersion is very close to &600 km s\, and it can be seen from the weak-lensing image distortion alone with very high signi"cance (Seitz et al., 1998b), supporting the foregoing quantitative prediction. Thus, this method appears to be a very promising way to obtain a mass-selected sample of halos which would be of great cosmological interest (cf. Reblinsky and Bartelmann, 1999b). We shall return to this issue in Section 6.7.2. 5.3.2. Aperture multipole moments Since it is possible to express the weighted mass within an aperture as an integral over the shear, with the advantage that in the weak-lensing regime this integral can be replaced by a sum over galaxy ellipticities, it is natural to ask whether a similar result holds for multipole moments of the mass. As shown in Schneider and Bartelmann (1997), this is indeed possible, and we shall brie#y outline the method and the result. Consider a circular aperture centred on a point h . Let ;("h") be a radial weight function, and  de"ne the nth multipole moment by



QL,







dh hL> ;(h)

p



du eL Pi(h #h) . 

(5.32)

 The method is not restricted to circular apertures, but this case will be most relevant for measuring multipole moments.

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This can be replaced by an integral over c in two ways: (5.32) can be integrated by parts with respect to u (for nO0), or with respect to h, again utilising (5.12). The resulting expressions are assumed to contain no boundary terms, which restricts the choice for the weight function ;(h). The remaining integrals then contain partial derivatives of i with respect to u and h, respectively. Writing (5.12) in polar coordinates, these partial derivatives can be replaced by partial derivatives of the shear components with respect to u and h. Integrating those by parts with respect to the appropriate coordinate, and enforcing vanishing boundary terms, we "nd two di!erent expressions for the QL:



QL " dh qL (h)c(h #h) . PF  PF

(5.33)

The two expressions for qL are formally very di!erent, although it can be shown that the resulting two expressions for QL are equivalent. The two very di!erent equations for the same result are due to the fact that the two components of the shear c are not mutually independent, which was not used in the derivation of (5.33). We now have substantial freedom to choose the weight function and to select one of the two expressions for QL, or even to take a linear combination of them. We note the following interesting examples: (1) The weight function ;(h) can be chosen to vanish outside an annulus, to be piece-wise di!erentiable, and to be zero on the inner and outer boundary of the annulus. The QL for nO0 can then be expressed as integrals of the shear over the annulus, with no further restrictions on ;. In particular, ;(h) does not need to be a compensated weight function. (2) ;(h) can be a piece-wise di!erentiable weight function which is constant for h4h , and  decreases smoothly to zero at h"h 'h . Again, QL for nO0 can be expressed as an integral   of the shear in the annulus h 4h4h . Hence, as for the aperture mass, multipole moments in   the inner circle can be probed with the shear in the surrounding annulus. (3) One can choose, for n'2, a piece-wise di!erentiable weight function ;(h) which behaves like h\L for h'h and decreases to zero at h"h (h . In that case, the multipole moments of the    matter outside an annulus can be probed with data inside the annulus. For practical applications, the integral in (5.33) is replaced by a sum over galaxy ellipticities. The dispersion of this sum is easily obtained in the absence of lensing, with an expression analogous to (5.31). Therefore, the signal-to-noise ratio for the multipole moments is easily de"ned, and thus also the signi"cance of a multipole-moment detection. 5.4. Application to observed clusters Soon after the parameter-free two-dimensional mass reconstruction was suggested by Kaiser and Squires (1993), their method was applied to the cluster MS 1224 (Fahlman et al., 1994). Since then, several groups have used it to infer the mass pro"les of clusters. In parallel to this, several methods have been developed to measure the shear from CCD data, accounting for PSF smearing and anisotropy, image distortion by the telescope, noise, blending etc. } see the discussion in Section 4.6. We will now summarise and discuss several of these observational results. Tyson et al. (1990) made the "rst attempt to constrain the mass distribution of a cluster from a weak-lensing analysis. They discovered a statistically signi"cant tangential alignment of faint

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galaxy images relative to the centre of the clusters A 1689 and Cl 1409#52. Their `lens distortion mapa obtained from the image ellipticities yields an estimate of the mass distribution in these clusters. A detailed analysis of their method is given in Kaiser and Squires (1993). From a comparison with numerical simulations, Tyson et al. showed that the best isothermal sphere model for the clusters has a typical velocity dispersion of p &1300$200 km s\ for both clusters. In particular, T their analysis showed that di!use dark matter in the cluster centres is needed to account for the observed image distortions. The inversion method developed by Kaiser and Squires (1993) provided a systematic approach to reconstruct the mass distribution in clusters. It was "rst applied to the cluster MS 1224#20 (Fahlman et al., 1994) at redshift z "0.33, which had been selected for its high X-ray luminosity.  Their square data "eld with side length &14 was composed of several exposures, most of them with excellent seeing. They estimated the shear from image ellipticities, corrected for the PSF anisotropy, and applied a correction factor f as de"ned in Section 4.6.1. They found f&1.5 in simulations, in very good agreement with Wilson et al. (1996). The resulting shear pattern, obtained from 2147 galaxy images, clearly shows a circular pattern around the cluster centre as de"ned by the centroid of the optical and X-ray light. Using the Kaiser and Squires reconstruction method (5.8), Fahlman et al. produced maps of the dimensionless surface mass density i(h), both by taking all galaxy images into account, and after splitting the galaxy sample into a &brighter' and &fainter' sample of roughly equal size. Although di!ering in detail, the resulting mass show an overall similarity. In particular, the position of the mass centre is very similar in all maps. Fahlman et al. applied the aperture mass method to determine the cluster mass } see (5.21) and (5.29) } in an annulus centred on the cluster centre with inner radius 0 "2.76 and an outer radius  such that the annulus nearly "ts into their data "eld. The lower limit to the mean surface mass density in the annulus is i(2.76)5f"0.06$0.013. To convert this into an estimate of the physical surface mass density and the total mass inside the aperture, the mean distance ratio D /D   for the galaxy population has to be estimated, or equivalently the mean value of Z as de"ned after (5.19). While the redshift distribution is known statistically for the brighter sub-sample from redshift surveys, the use of the fainter galaxies requires an extrapolation of the galaxy redshifts. From that, Fahlman et al. estimated the mass within a cylinder of radius 0 "2.76, corresponding to  0.48h\ Mpc for an Einstein}de Sitter cosmology, to be &3.5;10h\ M . This corresponds to > a mass-to-light ratio (in solar units) of M/¸&800h. Carlberg et al. (1994) obtained 75 redshifts of galaxies in the cluster "eld, of which 30 are cluster members. From their line-of-sight velocity dispersion, the cluster mass can be estimated by a virial analysis. The resulting mass is lower by a factor &3 than the weak-lensing estimate. The mass-to-light ratio from the virial analysis is much closer to typical values in lower-redshift clusters like Coma, which has M/¸+270h\. The high mass estimate of this cluster was recently con"rmed in a completely independent study by Fischer (1999). The origin of this large apparent discrepancy is not well understood yet, and several possibilities are discussed in Kaiser et al. (1994). It should be pointed out that lensing measures the total mass inside a cone, weighted by the redshift-dependent factor D D /D , and hence the lensing mass    estimate possibly includes substantial foreground and background material. While this may cause an overestimate of the mass, it is quite unlikely to cause an overestimate of the mass-to-light ratio of the total material inside the cone. Foreground material will contribute much more strongly to

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the light than to the measured mass, and additional matter behind the cluster will not be very e$cient as a lens. The uncertainty in the redshift distribution of the faint galaxies translates into an uncertainty in the mass. However, all background galaxies would have to be put at a redshift &4 to explain the mass discrepancy, while redshift surveys show that the brighter sub-sample of Fahlman et al. has a mean redshift below unity. The mass estimate is only weakly dependent on the assumed cosmological model. On the other hand, the light distribution of the cluster MS 1224 is not circular, and it cannot be excluded that this cluster is not in virial equilibrium. Two images of the cluster Cl 0024#17 were analysed by Bonnet et al. (1994). One was centred on the cluster itself and yielded the shear in the inner part of the cluster. The second image was o!-centred by several arcmin and allowed, for the "rst time, a shear measurement out to large radial distances. They detected a clear shear signal out to distances 91h\ Mpc. In addition, they found an apparent distortion of the nearly circular shear pattern from the cluster which is most directly interpreted as a mass concentration. However, it does not show an obvious concentration of galaxies. In fact, an X-ray observation of this cluster reveals a weak X-ray source close to the position where the mass concentration was seen in the shear map (Soucail et al., 2000), although with marginal signi"cance. This cluster (at z"0.39) hosts a giant arc system and has an Einstein radius of &30; together with the redshift of z"1.675 of the arc (Broadhurst et al., 1999), this indicates that the cluster is indeed very massive. Despite that, the cluster is a comparatively faint and cool X-ray source, indicating a clear and interesting discrepancy between mass estimates from the X-rays and both strong and weak lensing. Squires et al. (1996a) compared the mass pro"les derived from weak lensing data and the X-ray emission of the cluster A 2218. Under the assumption that the hot X-ray-emitting intra-cluster gas is in hydrostatic equilibrium between gravity and thermal pressure support, the mass pro"le of the cluster can be constrained. The reconstructed mass map qualitatively agrees with the optical and X-ray light distributions. Using the aperture mass estimate, a mass-to-light ratio of M/¸"(440$80)h in solar units is found. The radial mass pro"le appears to be #atter than isothermal. Within the error bars, it agrees with the mass pro"le obtained from the X-ray analysis, with a slight indication that at large radii the lensing mass is larger than the mass inferred from X-rays. Abell 2218 also contains a large number of arcs and multiply imaged galaxies which have been used by Kneib et al. (1996) to construct a detailed mass model of the cluster's central region. In addition to the main mass concentration, there is a secondary clump of cluster galaxies whose e!ects on the arcs is clearly visible. The separation of these two mass centres is 67. Whereas the resolution of the weak lensing mass map as obtained by Squires et al. is not su$cient to reveal a distinct secondary peak, the elongation of the central density contours extend towards the secondary galaxy clump. General agreement between the reconstructed mass map and the distribution of cluster galaxies and X-ray emission has also been found for the two clusters Cl 1455#22 (z"0.26) and Cl 0016#16 (z"0.55) by Smail et al. (1995a). Both are highly X-ray luminous clusters in the Einstein Extended Medium Sensitivity Survey (EMSS; Stocke et al., 1991). The orientation and ellipticity of the central mass peak is in striking agreement with those of the galaxy distribution and the X-ray map. However, the authors "nd some indication that the mass is more centrally condensed than the other two distributions. In addition, given the "nite angular resolution of the mass map, the core size derived from weak lensing is most likely only an upper bound to the true value, and in both

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Fig. 14. Left panel: WFPC2 image of the cluster Cl0939#4713 (A 851); North is at the bottom, East to the right. The cluster centre is located at about the upper left corner of the left CCD, a secondary maximum of the bright (cluster) galaxies is seen close to the interface of the two lower CCDs, and a minimum in the cluster light is at the interface between the two right CCDs. In the lensing analysis, the data from the small CCD (the Planetary Camera) were not used. Right panel: The reconstructed mass distribution of A 851, assuming a mean redshift of the N"295 galaxies with 244R425.5 of 1z2"1 (from Seitz et al., 1996).

clusters the derived core size is signi"cantly larger than found in clusters with giant luminous arcs (see, e.g., Fort and Mellier, 1994). The mass-to-light ratios for the two clusters are &1000h and &740h, respectively. However, at least for Cl 0016#16, the mass scale is fairly uncertain, owing to the high cluster redshift and the unknown redshift distribution of the faint galaxies. The mean value of D /D must be estimated   from an assumed distribution p(z). The unprecedented imaging quality of the refurbished Hubble Space Telescope (HST) can be used pro"tably for weak lensing analyses. Images taken with the Wide Field Planetary Camera 2 (WFPC2) have an angular resolution of order 0.1, limited by the pixel size. Because of this superb resolution and the lower sky background, the number density of galaxy images for which a shape can reliably be measured is considerably larger than from the ground, so that higher-resolution mass maps can be determined. The drawback is the small "eld covered by the WFPC2, which consists of 3 CCD chips with 80 side-length each. Using the "rst publicly available deep image of a cluster obtained with the WFPC2, Seitz et al. (1996) have constructed a mass map of the cluster Cl 0939#47 (z"0.41). Fig. 14 clearly shows a mass peak near the left boundary of the frame shown. This maximum coincides with the cluster centre as determined from the cluster galaxies (Dressler and Gunn, 1992). Furthermore, a secondary maximum is clearly visible in the mass map, as well as a pronounced minimum. When compared to the optical image, a clear correlation with the bright (cluster) galaxies is obvious. In particular, the secondary maximum and the minimum correspond to the same features in the bright galaxy distribution. A formal correlation test

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con"rms this similarity. Applying the maximum-likelihood mass reconstruction technique (Seitz et al., 1998c; see Section 5.4) to the same HST image, Geiger and Schneider (1999) constructed a higher-resolution map of this cluster. The angular resolution achieved is much higher in the cluster centre, predicting a region in which strong-lensing e!ects may occur. Indeed, Trager et al. (1997) reported on a highly elongated arc and a triple image, with both source galaxies having a redshift z+3.97. The X-ray map of this cluster (Schindler and Wambsganss, 1997) shows that the two mass peaks are also close to two X-ray components. The determination of the total mass inside the WFPC2 frame is di$cult, for two reasons: First, the high redshift of the cluster implies that the mean value of D /D depends quite sensitively on the assumed redshift distribution of the background   galaxies. Second, the small "eld of the WFPC2 precludes the measurement of the surface mass density at large distance where i tends to zero, and thus the mass-sheet degeneracy implies a considerable uncertainty in the mass scale. Attempting to lift the mass sheet degeneracy with the number-density e!ect } see Section 4.4.1 } a mass-to-light ratio of &250h was derived within the WFPC2 aperture. This value is also a!ected by the unknown fraction of cluster members in the catalog of faint galaxies. Seitz et al. (1996) assumed that the spatial distribution of faint cluster galaxies follows that of brighter cluster galaxies. The striking di!erence between the M/¸ ratios for this and the other clusters described above may be related to the fact that Cl 0939#47 is the highest-redshift cluster in the Abell catalog (A 851). Hence, it was selected by its high optical luminosity, whereas the previously mentioned clusters are all X-ray selected. The X-ray luminosity of Cl 0939#47 is fairly small for such a rich cluster (Schindler and Wambsganss, 1996). Since X-ray luminosity and cluster mass are generally well correlated, the small M/¸-ratio found from the weak lensing analysis is in agreement with the expectations based on the high optical #ux and the low X-ray #ux. Note that the large spread of mass-to-light ratios as found by the existing cluster mass reconstructions is unexpected in the frame of hierarchical models of structure formation and thus poses an interesting astrophysical problem. Hoekstra et al. (1998) reconstructed the mass distribution in the cluster MS 1358#62 from a mosaic of HST images, so that their data "eld in substantially larger than for a single HST pointing (about 8;8). This work uses the correction method presented in Section 4.6.2, thus accounting for the relatively strong PSF anisotropy at the edges of each WFPC2 chip. A weaklensing signal out to 1.5 Mpc is found. The X-ray mass is found to be slightly lower than the dynamical mass estimate, but seems to agree well with the lensing mass determination. Luppino and Kaiser (1997) found a surprisingly strong weak-lensing signal in the "eld of the high-redshift cluster MS 1054!03 (z"0.83). This implies that the sheared galaxies must have an appreciably higher redshift than the cluster, thus strongly constraining their redshift distribution. In fact, unless the characteristic redshift of these faint background galaxies is 91.5, this cluster would have an unrealistically large mass. It was also found that the lensing signal from the bluer galaxies is stronger than from the redder ones, indicating that the characteristic redshift of the bluer sample is higher. In fact, the mass estimated assuming 1z 2"1.5 agrees well  with results from analyses of the X-ray emission (Donahue et al., 1998) and galaxy kinematics (Tran et al., 1999). Using an HST mosaic image in two "lters, Hoekstra et al. (2000) also studied MS 1054. They found a tangential distortion which is smaller than that obtained by Luppino and Kaiser (1997) by about a factor of 1.5, but fairly well in agreement with that obtained by Clowe et al. (2000) from

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Keck imaging. They estimated the redshift distribution of the background galaxies from the photometric redshifts obtained in the Hubble Deep Fields, both as a function of magnitude and of colour. This enabled them to study the relative lensing strength 1Z2 as a function of these two observables, "nding, as expected, the lensing strength increasing towards fainter magnitudes and, in agreement with Luppino and Kaiser (1997) and Clowe et al. (2000), with bluer colour. The estimated mass is in very good agreement with that obtained from the X-ray temperature of this cluster. The mass map shows three distinct peaks which are in good correspondence with the observed distribution of cluster galaxies. Clowe et al. (1998) derived weak-lensing maps for two additional clusters at z&0.8, namely MS 1137#66 at z"0.783 and RXJ 1716#67 at z"0.813. The large-format CCD cameras allow weak-lensing studies of low-redshift clusters which subtend a larger solid angle on the sky. As a "rst example, Jo!re et al. (1999) obtained the mass map for the cluster Abell 3667 (z"0.055). Investigations of low-redshift clusters are particularly useful since for them more detailed X-ray and optical information is available than for higher-redshift ones. The mass distribution in the supercluster MS 0302#17 at z"0.42 was reconstructed by Kaiser et al. (1998) in a wide-"eld image of size &30. The supercluster consists of three clusters which are very close together on the sky and in redshift. The image contains about 30,000 galaxies from which a shear can be measured. This shear was found to correlate strongly with the distribution of the early-type (foreground) galaxies in the "eld, provided that the overall mass-to-light ratio is about 250h. Each of the three clusters, which are also seen in X-rays, is recovered in the mass map. The ratios between mass and light or X-ray emission di!er slightly across the three clusters, but the di!erences are not highly signi"cant. A magni"cation e!ect was detected from the depletion of the number counts (see Section 4.4.1) in two clusters. Fort et al. (1997) discovered that the number density of very faint galaxies drops dramatically near the critical curve of the cluster Cl 0024#16, and remains considerably lower than the mean number density out to about twice the Einstein radius. This is seen in photometric data with two "lters. Fort et al. (1997) interpret this broad depletion curve in terms of a broad redshift distribution of the background galaxies, so that the location of the critical curve of the cluster varies over a large angular scale. A spatially dependent number depletion was detected in the cluster A 1689 by Taylor et al. (1998). These examples should su$ce to illustrate the current status of weak-lensing cluster mass reconstructions. For additional results, see Squires et al. (1996b), Squires et al. (1997), Fischer et al. (1997), Fischer and Tyson (1997), and Athreya et al. (1999). Many of the di$culties have been overcome; e.g., the method presented in Section 4.6.2 appears to provide an accurate correction method for PSF e!ects. The quantitative results, for example for the M/¸-ratios, are somewhat uncertain due to the lack of su$cient knowledge on the source redshift distribution, which applies in particular to the high-redshift clusters. Further large-format HST mosaic images either are already or will soon become available, e.g. for the clusters A 2218, A 1689 and MS 1054!03. Their analysis will substantially increase the accuracy of cluster mass determinations from weak lensing compared to ground-based imaging. 5.5. Outlook We have seen in the preceding subsection that "rst results on the mass distribution in clusters were derived with the methods described earlier. Because weak lensing is now widely regarded as

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the most reliable method to determine the mass distribution of clusters, since it does not rely on assumptions on the physical state and symmetries of the matter distribution, further attempts at improving the method are in progress, and some of them will brie#y be outlined below. In particular, we describe a method which simultaneously accounts for shear and magni"cation information, and which can incorporate constraints from strong-lensing features (such as arcs and multiple images of background sources). A method for the determination of the local shear is described next which does not rely on the detection and the quadrupole measurement of individual galaxies, and instead makes use of the light from very faint galaxies which need not be individually detected. We will "nally consider the potential of weak lensing for determining the redshift distribution of galaxies which are too faint to be investigated spectroscopically, and report on "rst results. 5.5.1. Maximum-likelihood cluster reconstructions The mass reconstruction method described above is a direct method: the locally averaged observed image ellipticities 1e2 are inserted into an inversion equation such as (5.10) to "nd the mass map i(h). The beauty of this method is its simplicity and computational speed. Mass reconstructions from the observed image ellipticities are performed in a few CPU seconds. The drawback of this method is its lack of #exibility. No additional information can be incorporated into the inversion process. For example, if strong-lensing features like giant arcs or multiple galaxy images are observed, they should be included in the mass reconstruction. Since such strong-lensing features typically occur in the innermost parts of the clusters (at :30 from cluster centres), they strongly constrain the mass distribution in cluster cores which can hardly be probed by weak lensing alone due to its "nite angular resolution. A further example is the incorporation of magni"cation information, as described in Section 4.4, which can in principle not only be used to lift the mass-sheet degeneracy, but also provides local information on the shape of the mass distribution. An additional problem of direct inversion techniques is the choice of the smoothing scale which enters the weight factors u in (5.16). We have not given a guideline on how this scale should be G chosen. Ideally, it should be adapted to the data. In regions of strong shear, the signal-to-noise ratio of a shear measurement for a "xed number of galaxy images is larger than in regions of weak shear, and so the smoothing scale can be smaller there. Recently, these problems have been attacked with inverse methods. Suppose the mass distribution of a cluster is parameterised by a set of model parameters p . These model parameters could I then be varied until the best-"tting model for the observables is found. Considering for example the observed image ellipticities e and assuming a non-critical cluster, the expectation value of e is the G G reduced shear g at the image position, and the dispersion is determined (mainly) by the intrinsic dispersion of galaxy ellipticities p . Hence, one can de"ne a s-function C , "e !g(h )" G (5.34) s" G p C G and minimise it with respect to the p . A satisfactory model is obtained if s is of order N at its I  minimum, as long as the number of parameters is much smaller than N . If the chosen para meterisation does not achieve this minimum value, another one must be tried. However, the

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resulting mass model will depend on the parameterisation which is a serious drawback relative to the parameter-free inversion methods discussed before. This problem can be avoided with &generic' mass models. For instance, the de#ection potential t(h) can be composed of a "nite sum of Fourier modes (Squires and Kaiser, 1996), whose amplitudes are the parameters p . The number of Fourier modes can be chosen such that the I resulting s per degree of freedom is approximately unity. Additional modes would then start to "t the noise in the data. Alternatively, the values of the de#ection potential t on a (regular) grid can be used as the p . I Bartelmann et al. (1996) employed the locally averaged image ellipticities and the size ratios 1u2/1u2 } see (4.47) } on a grid. The corresponding expectation values of these quantities, the  reduced shear g and the magni"cation k, were calculated by "nite di!erencing of the discretised de#ection potential t. Since both c and i, and thus k, are unchanged under the transformation t(h)Pt(h)#t #a ) h, the de#ection potential has to be kept "xed at three grid points. If no  magni"cation information is used, the mass-sheet degeneracy allows a further transformation of t which leaves the expected image ellipticities invariant, and the potential has to be kept "xed at four grid points. A s-function was de"ned using the local dispersion of the image ellipticities and image sizes relative to unlensed sizes of galaxies with the same surface brightness, and it was minimised with respect to the values of t on the grid points. The grid spacing was chosen such that the resulting minimum s has approximately the correct value. Tests with synthetic data sets, using a numerically generated cluster mass distribution, showed that this method reconstructs very satisfactory mass maps, and the total mass of the cluster was accurately reproduced. If a "ner grid is used, the model for the de#ection potential will reproduce noise features in the data. On the other hand, the choice of a relatively coarse grid which yields a satisfactory s implies that the resolution of the mass map is constant over the data "eld. Given that the signal increases towards the centre of the cluster, one would like to use a "ner grid there. To avoid over-"tting of noise, the maximum-likelihood method can be complemented by a regularisation term (see Press et al., 1986, Chapter 18). As shown by Seitz et al. (1998c), a maximum-entropy regularisation (Narayan and Nityananda, 1986) is well suited for the problem at hand. As in maximum-entropy image restoration (e.g., Lucy, 1994), a prior is used in the entropy term which is a smoothed version of the current density "eld, and thus is being adapted during the minimisation. The relative weight of the entropy term is adjusted such that the resulting minimum s is of order unity per degree of freedom. In this scheme, the expectation values and dispersions of the individual image ellipticities and sizes are found by bi-linear interpolation of i and c on the grid which themselves are obtained by "nite di!erencing of the potential. When tested on synthetic data sets, this re"ned

 It is important to note that the de#ection potential t rather than the surface mass density i (as in Squires and Kaiser, 1996) should be parameterised, because shear and surface mass density depend on the local behaviour of t, while the shear cannot be obtained from the local i, and not even from i on a "nite "eld. In addition, the local dependence of i and c on t is computationally much more e$cient than calculating c by integrating over i as in Bridle et al. (1998).

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maximum-likelihood method produces mass maps with considerably higher resolution near the cluster centre without over-"tting the noise at larger cluster-centric distances. The practical implementation of this method is somewhat complicated. In particular, if critical clusters are studied, some modi"cations have to be included to allow the minimisation algorithm to move critical curves across galaxy images in the lens plane. However, the quality of the reconstruction justi"es the additional e!ort, especially if high-quality data from HST images are available. A "rst application of this method is presented by Geiger and Schneider (1999). Inverse methods such as the ones described here are likely to become the standard tool for cluster mass pro"le reconstruction, owing to their #exibility. As mentioned before, additional constraints from strong-lensing signatures such as arcs and multiply imaged sources, can straightforwardly be incorporated into these methods. The additional numerical e!ort is negligible compared to the e!orts needed to gain the observational data. Direct inversion methods will certainly retain an important role in this "eld, to obtain quick mass maps during the galaxy image-selection process (e.g., cuts in colour and brightness can be applied). Also, a mass map obtained by a direct method as a starting model in the inverse methods reduces the computational e!ort. 5.5.2. The auto-correlation function of the extragalactic background light So far, we described how shear can be determined from ellipticities of individual galaxy images on a CCD. In that context, a galaxy image is a statistically signi"cant #ux enhancement on the CCD covering several contiguous pixels and being more extended than the PSF as determined from stars. Reducing the threshold for the signal-to-noise per object, the number density of detected galaxies increases, but so does the fraction of misidenti"cations. Furthermore, the measured ellipticity of faint galaxies has larger errors than that of brighter and larger images. The detection threshold therefore is a compromise between high number density of images and signi"cance per individual object. Even the faintest galaxy images whose ellipticity cannot be measured reliably still contain information on the lens distortion. It is therefore plausible to use this information, by &adding up' the faintest galaxies statistically. For instance, one could co-add their brightness pro"les and measure the shear of the combined pro"led. This procedure, however, is a!ected by the uncertainties in de"ning the centres of the faint galaxies. Any error in the position of the centre, as de"ned in (5.1), will a!ect the resulting ellipticity. To avoid this di$culty, and also the problem of faint object de"nition at all, van Waerbeke et al. (1997) have suggested considering the auto-correlation function (ACF) of the &background' light. Most of the sky brightness is due to atmospheric scattering, but this contribution is uniform. Fluctuations of the brightness on the scale of arcseconds is supposedly mainly due to very faint galaxies. Therefore, these #uctuations should intrinsically be isotropic. If the light from the faint galaxies propagates through a tidal gravitational "eld, the isotropy will be perturbed, and this provides a possibility to measure this tidal "eld. Speci"cally, if I(h) denotes the brightness distribution as measured on a CCD, and IM is the brightness averaged over the CCD (or a part of it, see below), the auto-correlation function m(h) of the brightness is de"ned as m(h)"1(I(0)!IM )(I(0#h)!IM )20 ,

(5.35)

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where the average is performed over all pairs of pixels with separation h. From the invariance of surface brightness (3.10) and the locally linearised lens mapping, I(h)"I(Ah), one "nds that the observed ACF is related to the intrinsic ACF m, de"ned in complete analogy to (5.35), by m(h)"m(Ah) .

(5.36)

Thus, the transformation from intrinsic to observed ACF has the same functional form as the transformation of surface brightness. In analogy to the de"nition of the quadrupole tensor Q for galaxy images } see (5.2) } the tensor of second moments of the ACF is de"ned as dh m(h)h h G H . M " GH dh m(h)

(5.37)

The transformation between the observed quadrupole tensor M and the intrinsic one, M, is the same as for the moment tensor of image ellipticities, (5.5), M"AMA. As shown by van Waerbeke et al. (1997), the tensor M directly determines the distortion d, M !M #2iM   . (5.38) d"  M #M   Hence, d is related to M in the same way as the complex ellipticity s is related to Q. In some sense, the ACF plays the role of a single &equivalent' image from which the distortion can be determined, instead of an ensemble average over individual galaxy ellipticities. Working with the ACF has several advantages. First, centres of galaxy images do not need to be determined, which avoids a potential source of error. Second, the ACF can be used with substantial #exibility. For instance, one can use all galaxy images which are detected with high signi"cance, determine their ellipticity, and obtain an estimate of d from them. Su$ciently large circles containing these galaxies can be cut out of the data frame, so that the remaining frame is reminiscent of a Swiss cheese. The ACF on this frame provides another estimate of d, which is independent information and can statistically be combined with the estimate from galaxy ellipticities. Or one can use the ACF only on galaxy images detected within a certain magnitude range, still avoiding the need to determine centres. Third, on su$ciently deep images with the brighter objects cut out as just described, one might assume that the intrinsic ACF is due to a very large number of faint galaxies, so that the intrinsic ACF becomes a universal function. This function can in principle be determined from deep HST images. In that case, one also knows the width of the intrinsic ACF, as measured by the trace or determinant of M, and can determine the magni"cation from the width of the observed ACF, very similar to the method discussed in Section 4.4.2, but with the advantage of dealing with a single &universal source'. If this universal intrinsic ACF does exist, corrections of the measured M for a PSF considerably simplify compared to the case of individual image ellipticities, as shown by van Waerbeke et al. (1997). They performed several tests on synthetic data to demonstrate the potential of the ACF method for the recovery of the shear applied to the simulated images. van Waerbeke et al. determined shear "elds of two clusters, with several magnitude thresholds for the images which were punched out. A comparison of these shear "elds with those obtained from the standard method using galaxy ellipticities clearly shows that the ACF method is at least competitive, but

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since it provides additional information from those parts of the CCD which are unused by the standard method, it should in be employed any case. The optimal combination of standard method and ACF still needs to be investigated, but detailed numerical experiments indicate that the ACF may be the best method for measuring very weak shear amplitudes (L. van Waerbeke, Y. Mellier, private communication). 5.5.3. The redshift distribution of very faint galaxies Galaxy redshifts are usually determined spectroscopically. A successful redshift measurement depends on the magnitude of the galaxy, the exposure time, and the spectral type of the galaxy. If it shows strong emission or absorption lines, as star-forming galaxies do, a redshift can much easier be determined than in absence of strong spectral features. The recently completed Canadian} French Redshift Survey (CFRS) selected 730 galaxies in the magnitude interval 17.54I422.5 (see Lilly et al., 1995 and references therein). For 591 of them (81%), redshifts were secured with multi-slit spectroscopy on a 3.6 m telescope (CFHT) with a typical exposure time of &8 h. Whereas the upcoming 10 m-class telescopes will be able to perform redshift surveys to somewhat fainter magnitude limits, it will be di$cult to secure fairly complete redshift information of a #ux-limited galaxy sample fainter than I&24. In addition, it can be expected that many galaxies in a #ux-limited sample with fainter threshold will have redshifts between &1.2 and &2.2, where the cleanest spectral features, the OII emission line at j"372.7 nm and 400 nm break are shifted beyond the region where spectroscopy can easily be done from the ground. As we have seen, the calibration of cluster mass distributions depends on the assumed redshift distribution of the background galaxies. Most of the galaxies used for the reconstruction are considerably fainter than those magnitude limits for which complete redshift samples are available, so that this mass calibration requires an extrapolation of the redshift distribution from brighter galaxy samples. The fact that lensing is sensitive to the redshift distribution is not only a source of uncertainty, but also o!ers the opportunity to investigate the redshift distribution of galaxies too faint to be investigated spectroscopically. Several approaches towards a redshift estimate of faint galaxies by lensing have been suggested, and some of them have already shown spectacular success, as will be discussed next. First of all, a strongly lensed galaxy (e.g. a giant luminous arc) is highly magni"ed, and so the gravitational lens e!ect allows to obtain spectra of objects which would be too faint for a spectroscopic investigation without lensing. It was possible in this way to measure the redshifts of several arcs, e.g., the giant arc in A 370 at z"0.724 (Soucail et al., 1988), the arclet A 5 in A 370 at z"1.305 (Mellier et al., 1991), the giant arc in Cl 2244!02 at z"2.237 (Mellier et al., 1991), and the &straight arc' in A 2390 at z"0.913 (Pello et al., 1991). For a more complete list of arc redshifts, see Fort and Mellier (1994). A fair fraction of galaxies with redshift z94 have been found behind clusters, for example two arclet sources at z+4.05 behind A 2390 (Frye and Broadhurst, 1998; PelloH et al., 1999b), two sources at z+3.97 behind Cl 0939#4713 (Trager et al., 1997), and two sources behind MS 1358#62, which for a few months held the redshift record of z"4.92 (Franx et al., 1997). If the cluster contains several strong-lensing features, the mass model can be su$ciently well constrained to determine the arc magni"cations (if they are resolved in width, which has become possible only from imaging with the refurbished HST), and thus to determine the unlensed magnitude of the source galaxies, some of which are fainter that B&25.

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Some clusters, such as A 370 and A 2218, were observed in great detail both from the ground and with HST, and show a large number of strongly lensed images. They can be used to construct very detailed mass models of the cluster centre (e.g., Kneib et al., 1993, 1996). An example is A 2218, in which at least "ve multiply imaged systems were detected (Kneib et al., 1996), and several giant arcs were clearly seen. Re"ning the mass model for A 2218 constructed from ground-based data (Kneib et al., 1995) with the newly discovered or con"rmed strong-lensing features on the WFPC2 image, a strongly constrained mass model for the cluster can be computed and calibrated by two arc redshifts (a "ve-image system at z"0.702 and 1.034). Up to now, the deepest HST image taken in the direction of a cluster was on A 1689, perhaps the strongest lensing cluster yet detected (HST proposal number 6004, PI J. A. Tyson). This impressive image provides a wealth of strong-lensing features which should allow the construction of a very detailed mass model for its central region. In addition, a large-scale, though fairly shallow, image mosaic has been obtained with HST (HST proposal number 5993, PI N. Kaiser). These two data sets will yield the most detailed mass pro"le currently obtainable. Visual inspection of the WFPC2 image immediately shows a large number of arclets in A 2218, which surround the cluster centre in a nearly perfect circular pattern. These arclets have very small axis ratios, and most of them are therefore highly distorted. The strength of the distortion depends on the redshift of the corresponding galaxy. Assuming that the sources have a considerably smaller ellipticity than the observed images, one can then estimate a redshift range of the galaxy. To be more speci"c, let p(e) be the probability density of the intrinsic source ellipticity, assumed for simplicity to be independent of redshift. The corresponding probability distribution for the image ellipticity is then




p(e)"p(e(e)) det

Re , Re

(5.39)

where the transformation e(e) is given by Eq. (4.12), and the "nal term is the Jacobian of this transformation. For each arclet near the cluster centre where the mass pro"le is well constrained, the value of the reduced shear g is determined up to the unknown redshift of the source } see Eq. (4.20). One can now try to maximise p(e) with respect to the source redshift, and in that way "nd the most likely redshift for the arc. Depending on the ellipticity of the arclet and the local values of shear and surface mass density, three cases have to be distinguished: (1) the arclet has the &wrong' orientation relative to the local shear, i.e., if the source lies behind the cluster, it must be even more elliptical than the observed arclet. For the arclets in A 2218, this case is very rare. (2) The most probable redshift is &at in"nity', i.e., even if the source is placed at very high redshift, the maximum of p(e) is not reached. (3) p(e) attains a maximum at a "nite redshift. This is by far the most common case in A 2218. This method, "rst applied to A 370 (Kneib et al., 1994), was used to estimate the redshifts of &80 arclets in A 2218 brighter than R&25. Their typical redshifts are estimated to be of order unity,

 This simpli"ed treatment neglects the magni"cation bias, i.e. the fact that at locations of high magni"cation the redshift probability distribution is changed } see Section 4.3.2.

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with the fainter sub-sample 244R425 extending to somewhat higher redshifts. For one of them, a redshift range 2.6:z:3.3 was estimated, and a spectroscopic redshift of z"2.515 was later measured (Ebbels et al., 1996), providing spectacular support for this method. Additional spectroscopic observations of arclets in A 2218 were conducted and further con"rmed the reliability of the method for the redshift estimates of individual arclets (Ebbels et al., 1998). Another success of this arclet redshift estimate was recently achieved in the cluster A 2390, which can also be modelled in great detail from HST data. There, two arclets with very strong elongation did not "t into the cluster mass model unless they are at very high redshift. Spectroscopic redshifts of z&4.05 were recently measured for these two arclets (Frye and Broadhurst, 1998; PelloH et al., 1999a). However, several issues should be kept in mind. First, the arclets for which a reliable estimate of the redshift can be obtained are clearly magni"ed, and thus the sample is magni"cation biased. Since it is well known that the galaxy number counts are considerable steeper in the blue than in the red (see, e.g., Smail et al., 1995a), blue galaxies are preferentially selected as arclets } see Eq. (4.42). This might also provide the explanation why most of the giant arcs are blue. Therefore, the arclets represent probably a biased sample of faint galaxies. Second, the redshift dependence of p(e) enters through the ratio D /D . For a cluster at relatively low redshift, such as A 2218   (z "0.175), this ratio does not vary strongly with redshift once the source redshift is larger than  &1. Hence, to gain more accurate redshift estimates for high-redshift galaxies, a moderately high-redshift cluster should be used. The method just described is not a real &weak lensing' application, but lies on the borderline between strong and weak lensing. With weak lensing, the redshifts of individual galaxy images cannot be determined, but some statistical redshift estimates can be obtained. Suppose the mass pro"le of a cluster has been reconstructed using the methods described in Section 5.2 or Section 5.5.1, for which galaxy images in a certain magnitude range were used. If the cluster contains strong-lensing features with spectroscopic information (such as a giant luminous arc with measured redshift), then the overall mass calibration can be determined, i.e., the factor 1Z2 } see Section 4.3.2 } can be estimated, which provides a "rst integral constraint on the redshift distribution. Repeating this analysis with several such clusters at di!erent redshifts, further estimates of 1Z2 with di!erent D are obtained, and thus additional constraints on the redshift distribution. In  addition, one can group the faint galaxy images into sub-samples, e.g., according to their apparent magnitude. Ignoring for simplicity the magni"cation bias (which can safely be done in the outer parts of clusters), one can determine 1Z2 for each magnitude bin. Restricting our treatment to the regions of weak lensing only, such that "c";1, i;1, the expectation value of the ellipticity e of G a galaxy at position h is 1Z2c(h ), and so an estimate of 1Z2 for the galaxy sub-sample under G G consideration is R(c(h )eH) G G . 1Z2" "c(h )" G

(5.40)

In complete analogy, Bartelmann and Narayan (1995) suggested the &lens parallax method', an algorithm for determining mean redshifts for galaxy sub-samples at "xed surface brightness, using the magni"cation e!ect as described in Section 4.4.2. Since the surface brightness I is most likely

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much more strongly correlated with galaxy redshift than the apparent magnitude (due to the (1#z)\ decrease of bolometric surface brightness with redshift), a narrow bin in I will probably correspond to a fairly narrow distribution in redshift, allowing to relate 1Z2 of a surface brightness bin fairly directly to a mean redshift in that bin, while 1Z2 in magnitude bins can only be translated into redshift information with a parameterised model of the redshift distribution. On the other hand, apparent magnitudes are easier to measure than surface brightness and are much less a!ected by seeing. Even if a cluster without strong lensing features is considered, the two methods just described can be applied. The mass reconstruction then gives the mass distribution up to an overall multiplicative constant. We assume here that the mass-sheet degeneracy can be lifted, either using the magni"cation e!ect as described in Section 5.4, or by extending the observations so su$ciently large distances so that i+0 near the boundary of the data "eld. The mass scale can then be "xed by considering the brightest sub-sample of galaxy images for which a shear signal is detected if they are su$ciently bright for their redshift probability distribution to be known from spectroscopic redshift surveys (Bartelmann and Narayan, 1995). Whereas these methods have not yet rigorously been applied, there is one observational result which indicates that the faint galaxy population has a relatively high median redshift. In a sequence of clusters with increasing redshift, more and more of the faint galaxies will lie in the foreground or very close behind the cluster and therefore be unlensed. The dependence of the observed lensing strength of clusters on their redshift can thus be used as a rough indication of the median redshift of the faint galaxies. This idea was put forward by Smail et al. (1994), who observed three clusters with redshifts z"0.26, 0.55 and 0.89. In the two lower-redshift clusters, a signi"cant weak lensing signal was detected, but no signi"cant signal in the high-redshift cluster. From the detection, models for the redshift distribution of faint I425 can be ruled out which predict a large fraction to be dwarf galaxies at low redshift. The non-detection in the high-redshift cluster cannot easily be interpreted since little information (e.g., from X-ray maps) is available for this cluster, and thus the absence of a lensing signal may be due to the cluster being not massive enough. However, the detection of a strong shear signal in the cluster MS 1054!03 at z"0.83 (Luppino and Kaiser, 1997) implies that a large fraction of galaxies with I425.5 must lie at redshifts larger than z&1.5. They split their galaxy sample into red and blue sub-samples, as well as into brighter and fainter sub-samples, and found that the shear signal is mainly due to the fainter and the blue galaxies. If all the faint blue galaxies have a redshift z "1.5, the mass-to-light ratio of this cluster is  estimated to be M/¸&580h, and if they all lie at redshift z "1, M/¸ exceeds &1000h. This  observational result, which is complemented by several additional shear detections in high-redshift clusters, one of them at z"0.82 (G. Luppino, private communication), provides the strongest evidence for the high-redshift population of faint galaxies. In addition, it strongly constrains cosmological models; an X "1 cosmological model predicts the formation of massive clusters  only at relatively low redshifts (e.g., Richstone et al., 1992; Bartelmann et al., 1993) and has di$culties to explain the presence of strong lensing clusters at redshift z&0.8. Recently, Lombardi and Bertin (1999c) and Gautret et al. (2000) suggested that weak lensing by galaxy clusters can be used to constrain the cosmological parameters X and XK . Both of these two  di!erent methods assume that the redshift of background galaxies can be estimated, e.g. with su$ciently precise photometric-redshift techniques. Owing to the dependence of the lensing strength on the angular-diameter distance ratio D /D , su$ciently detailed knowledge of the mass  

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distribution in the lens and of the source redshifts can be employed to constrain these cosmological parameters. Such a determination through purely geometrical methods would be very valuable, although the observational requirements for applying these methods appear fairly demanding at present.

6. Weak cosmological lensing In this section, we review how weak density perturbations in otherwise homogeneous and isotropic Friedmann}Lemam( tre model universes a!ect the propagation of light. We "rst describe how light propagates in the homogeneous and isotropic background models, and then discuss how local density inhomogeneities can be taken into account. The result is a propagation equation for the transverse separation between the light rays of a thin light bundle. The solution of this equation leads to the de#ection angle a of weakly de#ected light rays. In close analogy to the thin-lens situation, half the divergence of the de#ection angle can be identi"ed with an e!ective surface-mass density i . The power spectrum of i is closely related to the   power spectrum of the matter #uctuations, and it forms the central physical object of the further discussion. Any two-point statistics of cosmic magni"cation and cosmic shear can then be expressed in a fairly simple manner in terms of the e!ective-convergence power spectrum. We discuss several applications, among which are the uncertainty in brightness determinations of cosmologically distant objects due to cosmic magni"cation, and several measures for cosmic shear, one of which is particularly suited for determining the e!ective-convergence power spectrum. At the end of this chapter, we turn to higher-order statistical measures of cosmic lensing e!ects, which re#ect the non-Gaussian nature of the non-linearly evolved density perturbations. When we give numerical examples, we generally employ four di!erent model universes. All have the CDM power spectrum for density #uctuations, but di!erent values for the cosmological parameters. They are summarised in Table 1. We choose two Einstein}de Sitter models, SCDM and pCDM, normalised either to the local abundance of rich clusters or to p "1, respectively, and  two low-density models, OCDM and KCDM, which are cluster normalised and either open or spatially #at, respectively. Light propagation in inhomogeneous model universes has been the subject of numerous studies. Among them are Zel'dovich (1964), Dashevskii and Zel'dovich (1965), Kristian and Sachs (1966), Gunn (1967), Jaroszynski et al. (1990), Babul and Lee (1991), Bartelmann and Schneider (1991),

Table 1 Cosmological models and their parameters used for numerical examples Model

X 

XK

h

Normalisation

p 

SCDM pCDM OCDM KCDM

1.0 1.0 0.3 0.3

0.0 0.0 0.0 0.7

0.5 0.5 0.7 0.7

Cluster p  Cluster Cluster

0.5 1.0 0.85 0.9

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Blandford et al. (1991), Miralda-EscudeH (1991), and Kaiser (1992). Non-linear e!ects were included analytically by Jain and Seljak (1997), who also considered statistical e!ects of higher than second order, as did Bernardeau et al. (1997). A particularly suitable measure for cosmic shear was introduced by Schneider et al. (1998a). 6.1. Light propagation; choice of coordinates As outlined in Section 3.2.1, the governing equation for the propagation of thin light bundles through arbitrary space times is the equation of geodesic deviation (e.g. Misner et al., 1973, Section 11; Schneider et al., 1992, Section 3.5), or Jacobi equation (3.23). This equation implies that the transverse physical separation n between neighbouring rays in a thin light bundle is described by the second-order di!erential equation dn "Tn , dj

(6.1)

where T is the optical tidal matrix (3.25) which describes the in#uence of space}time curvature on the propagation of light. The a$ne parameter j has to be chosen such that it locally reproduces the proper distance and increases with decreasing time, hence dj"!ca dt. The elements of the matrix T then have the dimension [length]\. We already discussed in Section 3.2.1 that the optical tidal matrix is proportional to the unit matrix in a Friedmann}Lemam( tre universe, T"R I ,

(6.2)

where the factor R is determined by the Ricci tensor as in Eq. (3.26). For a model universe "lled with a perfect pressure-less #uid, R can be written in the form (3.28). It will prove convenient for the following discussion to replace the a$ne parameter j in Eq. (6.1) by the comoving distance w, which was de"ned in Eq. (2.3) before. This can be achieved using Eqs. (3.31) and (3.32) together with the de"nition of Hubble's parameter, H(a)"a a\. Additionally, we introduce the comoving separation vector x"a\n. These substitutions leave the propagation equation (6.1) in the exceptionally simple form dx #K x"0 , dw

(6.3)

where K is the spatial curvature given in Eq. (2.30). Eq. (6.3) has the form of an oscillator equation, hence its solutions are trigonometric or hyperbolic functions, depending on whether K is positive or negative. In the special case of spatial #atness, K"0, the comoving separation between light rays is a linear function of distance. 6.2. Light deyection We now proceed by introducing density perturbations into the propagation equation (6.3). We assume throughout that the Newtonian potential U of these inhomogeneities is small, "U";c, that they move with velocities much smaller than the speed of light, and that they are localised, i.e. that

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the typical scales over which U changes appreciably are much smaller than the curvature scale of the background Friedmann}Lemam( tre model. Then, there exists a local neighbourhood around each density perturbation which is large enough to contain the perturbation completely and still small enough to be considered #at. Under these circumstances, the metric is well approximated by the "rst post-Newtonian order of the Minkowski metric (3.36). It then follows from Eq. (3.36) that the e!ective local index of refraction in the neighbourhood of the perturbation is dl 2U "n"1! . dt c

(6.4)

Fermat's principle (e.g. Blandford and Narayan, 1986; Schneider, 1985) demands that the light travel time along actual light paths is stationary, hence the variation of n dl must vanish. This condition implies that light rays are de#ected locally according to 2 dx "! U . c , dw

(6.5)

In weakly perturbed Minkowski space, this equation describes how an actual light ray is curved away from a straight line in unperturbed Minkowski space. It is therefore appropriate for describing light propagation through, e.g. the Solar system and other well-localised mass inhomogeneities. This interpretation needs to be generalised for large-scale mass inhomogeneities embedded in an expanding cosmological background, since the meaning of a `straighta "ducial ray is then no longer obvious. In general, any physical "ducial ray will also be de#ected by potential gradients along its way. We can, however, interpret x as the comoving separation vector between an arbitrarily chosen "ducial light ray and a closely neighbouring light ray. The right-hand side of Eq. (6.5) must then contain the diwerence *( U) of the perpendicular potential gradients between , the two rays to account for the relative de#ection of the two rays. Let us therefore imagine a "ducial ray starting at the observer (w"0) into direction h"0, and a neighbouring ray starting at the same point but in direction hO0. Let further x(h, w) describe the comoving separation between these two light rays at comoving distance w. Combining the cosmological contribution given in Eq. (6.3) with the modi"ed local contribution (6.5) leads to the propagation equation 2 dx #K x"! D+ U[x(h, w), w], . , c dw

(6.6)

The notation on the right-hand side indicates that the di!erence of the perpendicular potential gradients has to be evaluated between the two light rays which have comoving separation x(h, w) at comoving distance w from the observer. Linearising the right-hand side of Eq. (6.6) in x immediately returns the geodesic deviation equation (6.1) with the full optical tidal matrix, which combines the homogeneous cosmological contribution (3.28) with the contributions of local perturbations (3.37). Strictly speaking, the comoving distance w, or the a$ne parameter j, are changed in the presence of density perturbations. Here, we assume that the global properties of the weakly perturbed

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Friedmann}Lemam( tre models remain the same as in the homogeneous and isotropic case, and under this assumption the comoving distance w remains the same as in the unperturbed model. To solve Eq. (6.6), we "rst construct a Green's function G(w, w), which has to be a suitable linear combination of either trigonometric or hyperbolic functions since the homogeneous equation (6.6) is an oscillator equation. We further have to specify two boundary conditions. According to the situation we have in mind, these boundary conditions read x"0,

dx "h dw

(6.7)

at w"0. The "rst condition states that the two light rays start from the same point, so that their initial separation is zero, and the second condition indicates that they set out into directions which di!er by h. The Green's function is then uniquely determined by



f (w!w) for w'w, G(w, w)" ) 0 otherwise

(6.8)

with f (w) given in Eq. (2.4). As a function of distance w, the comoving separation between the two ) light rays is thus



2 U dw f (w!w) *+ U[x(h, w), w], . (6.9) x(h, w)"f (w)h! ) , ) c  The perpendicular gradients of the Newtonian potential are to be evaluated along the true paths of the two light rays. In its exact form, Eq. (6.9) is therefore quite involved. Assuming that the change of the comoving separation vector x between the two actual rays due to light de#ection is small compared to the comoving separation of unperturbed rays, "x(h, w)!f (w)h" ) ;1 , (6.10) " f (w)h" ) we can replace x(h, w) by f (w)h in the integrand to arrive at a much simpler expression which ) corresponds to the Born approximation of small-angle scattering. The Born approximation allows us to replace the di!erence of the perpendicular potential gradients with the perpendicular gradient of the potential di!erence. Taking the potential di!erence then amounts to adding a term to the potential which depends on the comoving distance w from the observer only. For notational simplicity, we can therefore rename the potential di!erence *U between the two rays to U. It is an important consequence of the Born approximation that the Jacobian matrix of the lens mapping (3.11) remains symmetric even in the case of cosmological weak lensing. In a general multiple lens-plane situation, this is not the case (Schneider et al., 1992, Chapter 9). If the two light rays propagated through unperturbed space}time, their comoving separation at distance w would simply be x(h, w)"f (w)h, which is the "rst term on the right-hand side of ) Eq. (6.9). The net de#ection angle at distance w between the two rays is the di!erence between x and x, divided by the angular diameter distance to w, hence



f (w!w) 2 U f (w)h!x(h, w) dw ) "

U[ f (w)h, w] . a(h, w)" ) , ) f (w) c f (w)  ) )

(6.11)

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Again, this is the de#ection angle of a light ray that starts out at the observer into direction h relative to a nearby "ducial ray. Absolute de#ection angles cannot be measured. All measurable e!ects of light de#ection therefore only depend on derivatives of the de#ection angle (6.11), so that the choice of the "ducial ray is irrelevant for practical purposes. For simplicity, we call a(h, w) the de#ection angle at distance w of a light ray starting into direction h on the observer's sky, bearing in mind that it is the de#ection angle relative to an arbitrarily chosen "ducial ray, so that a(h, w) is far from unique. In an Einstein}de Sitter universe, f (w)"w. De"ning y"w/w, Eq. (6.11) simpli"es to ) 2w  a(h, w)" d(1!y) U(wyh, wy) . (6.12) , c  Clearly, the de#ection angle a depends on the direction h on the sky into which the light rays start to propagate, and on the comoving distance w to the sources. Recall the various approximations adopted in the derivation of Eq. (6.11): (i) The density perturbations are well localised in an otherwise homogeneous and isotropic background, i.e. each perturbation can be surrounded by a spatially #at neighbourhood which can be chosen small compared to the curvature radius of the background model, and yet large enough to encompass the entire perturbation. In other words, the largest scale on which the density #uctuation spectrum P (k) has appreciable power must be much smaller than the Hubble radius c/H . (ii) The B  Newtonian potential of the perturbations is small, U;c, and typical velocities are much smaller than the speed of light. (iii) Relative de#ection angles between neighbouring light rays are small enough so that the di!erence of the transverse potential gradient can be evaluated at the unperturbed path separation f (w)h rather than the actual one. Reassuringly, these approxi) mations are very comfortably satis"ed even under fairly extreme conditions. The curvature radius of the Universe is of order cH\"3000h\ Mpc and therefore much larger than perturbations of  even several tens of Mpc's in size. Typical velocities in galaxy clusters are of order 10 km s\, much smaller than the speed of light, and typical Newtonian potentials are of order U:10\ c.



6.3. Ewective convergence 6.3.1. Dexnition and derivation In the thin-lens approximation, convergence i and de#ection angle a are related by 1 Ra (h) 1 G , (6.13) i(h)" ) a(h)" 2 Rh 2 F G where summation over i is implied. In exact analogy, an e!ective convergence i (w) can be de"ned  for cosmological weak lensing, 1 i (h, w)" ) a(h, w)  2 F



f (w!w) f (w) R 1 U ) dw ) U[ f (w)h, w] . " ) f (w) Rx Rx c  ) G G

(6.14)

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Had we not replaced x(h, w) by f (w)h following Eq. (6.9), Eq. (6.14) would have contained second ) and higher-order terms in the potential derivatives. Since Eq. (6.9) is a Volterra integral equation of the second kind, its solution (and derivatives thereof) can be expanded in a series, of which the foregoing expression for i is the "rst term. Eq. (6.16) below shows that this term is of the order of  the line-of-sight average of the density contrast d. The next higher-order term, explicitly written down in the appendix of Schneider et al. (1998a), is determined by the product d(w) d(w), averaged along the line-of-sight over w(w. Analogous estimates apply to higher-order terms. Whereas the density contrast may be large for individual density perturbations passed by a light ray, the average of d is small compared to unity for most rays, hence i ;1, and higher-order terms  are accordingly negligible. The e!ective convergence i in Eq. (6.14) involves the two-dimensional Laplacian of the  potential. We can augment it by (RU/Rx ) which involves only derivatives along the light  path, because these average to zero in the limit to which we are working; the validity of this approximation has been veri"ed with numerical simulations by White and Hu (1999). The threedimensional Laplacian of the potential can then be replaced by the density contrast via Poisson's equation (2.65), 3H X *U"   d . 2a

(6.15)

Hence, we "nd for the e!ective convergence, 3H X i (h, w)"    2c



f (w) f (w!w) d[ f (w)h, w] ) ) . (6.16) dw ) a(w) f (w)  ) The e!ective convergence along a light ray is therefore an integral over the density contrast along the (unperturbed) light path, weighted by a combination of comoving angular-diameter distance factors, and the scale factor a. The amplitude of i is proportional to the cosmic density  parameter X .  Expression (6.16) gives the e!ective convergence for a "xed source redshift corresponding to the comoving source distance w. When the sources are distributed in comoving distance, i (h, w)  needs to be averaged over the (normalised) source-distance distribution G(w),



i (h)" 

U

U&

dw G(w) i (h, w) , (6.17)   where G(w) dw"p (z) dz. Suitably re-arranging the integration limits, we can then write the X source-distance weighted e!ective convergence as 3H X i (h)"    2c



U&

dw = M (w) f (w) )

d[ f (w)h, w] ) , a(w)

(6.18)

 where the weighting function = M (w) is now



= M (w),

U&

f (w!w) dw G(w) ) . f (w) U )

(6.19)

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The upper integration boundary w is the horizon distance, de"ned as the comoving dis& tance obtained for in"nite redshift. In fact, it is easily shown that the e!ective convergence can be written as



4pG D D dD    (o!o) , (6.20) i " dz  dz c D  and the weighting function = M is the distance ratio 1D /D 2, averaged over the source distances at   "xed lens distance. Naively generalising the de"nition of the dimensionless surface-mass density (3.7), to a three-dimensional matter distribution would therefore directly have led to the cosmologically correct expression for the e!ective convergence. 6.4. Ewective-convergence power spectrum 6.4.1. The power spectrum from Limber's equation Here, we are interested in the statistical properties of the e!ective convergence i , especially its  power spectrum P (l). We refer the reader to Section 2.4 for the de"nition of the power spectrum. G We also note that the expression for i (h) is of the form (2.77), and so the power spectrum P (l) is  G given in terms of P (k) by Eq. (2.84), if one sets B f (w) 3 H X = M (w) ) . (6.21) q (w)"q (w)"   a(w) 2 c  We therefore obtain 9H X P (l)"   G 4c



U&






l = M (w) P ,w (6.22) a(w) B f (w)  ) with the weighting function = M given in Eq. (6.19). This power spectrum is the central quantity for the discussion in the remainder of this chapter. Fig. 15 shows P (l) for "ve di!erent realisations of the CDM cosmogony. These are the four G models whose parameters are detailed in Table 1, all with non-linearly evolving density power spectrum P , using the prescription of Peacock and Dodds (1996), plus the SCDM model with B linearly evolving P . Sources are assumed to be at redshift z "1. Curves 1 and 2 (solid and dotted; B  SCDM with linear and non-linear evolution, respectively) illustrate the impact of non-linear density evolution in an Einstein}de Sitter universe with cluster-normalised density #uctuations. Non-linear e!ects set in on angular scales below a few times 10, and increase the amplitude of P (l) G by more than an order of magnitude on scales of +1. Curve 3 (short-dashed; pCDM), obtained for CDM normalised to p "1 rather than the cluster abundance, demonstrates the potential  in#uence of di!erent choices for the power-spectrum normalisation. Curves 4 and 5 (dashed}dotted and long-dashed; OCDM and KCDM, respectively) show P (l) for cluster-normalised CDM in an G open universe (X "0.3, XK "0) and in a spatially #at, low-density universe (X "0.3, XK "0.7).   It is a consequence of the normalisation to the local cluster abundance that the various P (l) are G very similar for the di!erent cosmologies on angular scales of a few arc minutes. For the low-density universes, the di!erence between the cluster- and the p normalisation is substantially  smaller than for the Einstein}de Sitter model. dw

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Fig. 15. Five e!ective-convergence power spectra P (l) are shown as functions of the angular scale 2pl\, expressed in arc G minutes. All sources were assumed to lie at z "1. The "ve curves represent the four realisations of the CDM cosmogony  listed in Table 1, all with non-linearly evolving density-perturbation power spectra P , plus the SCDM model with B linearly evolving P . Solid curve (1) linearly evolving SCDM model; dotted curve, (2) non-linearly evolving SCDM; B short-dashed curve, (3) non-linearly evolving pCDM; dashed}dotted and long-dashed curves (4) and (5) non-linearly evolving OCDM and KCDM, respectively. Fig. 16. Di!erent representation of the curves in Fig. 15. We plot here l P (l), representing the total power in the e!ective G convergence per logarithmic l interval. See the caption of Fig. 15 for the meaning of the di!erent line types. The "gure demonstrates that the total power increases monotonically towards small angular scales when non-linear evolution is taken into account (i.e. with the exception of the solid curve). On angular scales still smaller than +1, the curves level o! nd decrease very slowly. This shows that weak lensing by cosmological mass distributions is mostly sensitive to structures smaller than +10.

Fig. 16 gives another representation of the curves in Fig. 15. There, we plot l P (l), i.e. the total G power in the e!ective convergence per logarithmic l interval. This representation demonstrates that density #uctuations on angular scales smaller than +10 contribute most strongly to weak gravitational lensing by large-scale structures. On angular scales smaller than +1, the curves level o! and then decrease very gradually. The solid curve in Fig. 16 shows that, when linear density evolution is assumed, most power is contributed by structures on scales above 10, emphasising that it is crucial to take non-linear evolution into account to avoid misleading conclusions. 6.4.2. Special cases In the approximation of linear density evolution, applicable on large angular scales 930, the density contrast grows in proportion with ag(a), as described following Eq. (2.52). The power

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spectrum of the density contrast then evolves Ja g(a). Inserting this into Eq. (6.22), the squared scale factor a(w) cancels, and we "nd 9H X P (l)"   G 4c






U&

l dw g[a(w)] = M (w) P . (6.23) B f (w)  ) Here, P(k) is the density-contrast power spectrum linearly extrapolated to the present epoch. B In an Einstein}de Sitter universe, the growth function g(a) is unity since P grows like the B squared scale factor. In that special case, the expression for the power spectrum of i further  reduces to 9H P (l)"  G 4c






U&

l dw = M (w) P B w

(6.24)

 and the weight function = M simpli"es to



= M (w)"

U&






w dw G(w) 1! . w

(6.25) U In some situations, the distance distribution of the sources can be approximated by a delta peak at some distance w , G(w)"d (w!w ). A typical example is weak lensing of the Cosmic Microwave  "  Background, where the source is the surface of last scattering at redshift z +1000. Under such  circumstances,






w H(w !w) , (6.26) = M (w)" 1!  w  where the Heaviside step function H(x) expresses the fact that sources at w are only lensed by mass  distributions at smaller distance w. For this speci"c case, the e!ective-convergence power spectrum reads






 l 9H , (6.27) P (l)"  w dy (1!y) P G B wy 4c    where y"w/w is the distance ratio between lenses and sources. This equation illustrates that all  density-perturbation modes whose wave numbers are larger than k "w\l contribute to P (l),

  G or whose wavelengths are smaller than j "w h. For example, the power spectrum of weak

  lensing on angular scales of h+10 on sources at redshifts z +2 originates from all density  perturbations smaller than +7h\ Mpc. This result immediately illustrates the limitations of the foregoing approximations. Density perturbations on scales smaller than a few Mpc become non-linear even at moderate redshifts, and the assumption of linear evolution breaks down. 6.5. Magnixcation and shear In analogy to the Jacobian matrix A of the conventional lens equation (3.11), we now form the matrix 1 Rx(h, w) Ra(h, w) " . A(h, w)"I! f (w) Rh Rh )

(6.28)

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The magni"cation is the inverse of the determinant of A (see Eq. (3.14)). To "rst order in the perturbations, we obtain for the magni"cation of a source at distance w seen in direction h 1 +1# ) a(h, w)"1#2i (h, w) k(h, w)" F  detA(h, w) ,1#dk(h, w) .

(6.29)

In the weak-lensing approximation, the magni"cation #uctuation dk is simply twice the e!ective convergence i , just as in the thin-lens approximation.  We emphasise again that the approximations made imply that the matrix A is symmetric. In general, when higher-order terms in the Newtonian potential are considered, A attains an asymmetric contribution. Jain et al. (2000) used ray-tracing simulations through the density distribution of the Universe computed in very high resolution N-body simulations to show that the symmetry of A is satis"ed to very high accuracy. Only for those light rays which happen to propagate close to more than one strong de#ector can the deviation from symmetry be appreciable. Further estimates of the validity of the various approximations have been carried out analytically by Bernardeau et al. (1997) and Schneider et al. (1998a). Therefore, as in the single lens-plane situation, the anisotropic deformation, or shear, of a light bundle is determined by the trace-free part of the matrix A (cf. Eq. (3.11)). As explained there, the shear makes elliptical images from circular sources. Let a and b be the major and minor axes of the image ellipse of a circular source, respectively, then the ellipticity is a!b +2"c" , "s"" a#b

(6.30)

where the latter approximation is valid for weak lensing, "c";1; cf. Eq. (4.18). The quantity 2c was sometimes called polarisation in the literature (Blandford et al., 1991; Miralda-EscudeH , 1991; Kaiser, 1992). In the limit of weak lensing which is relevant here, the two-point statistical properties of dk and of 2c are identical (e.g. Blandford et al., 1991). To see this, we "rst note that the "rst derivatives of the de#ection angle occurring in Eqs. (6.29) can be written as second derivatives of an e!ective de#ection potential t which is de"ned in terms of the e!ective surface mass density i in the same  way as in the single lens-plane case; see (3.9). We then imagine that dk and c are Fourier transformed, whereupon the derivatives with respect to h are replaced by multiplications with G components of the wave vector l conjugate to h. In Fourier space, the expressions for the averaged quantities 1dk2 and 4 1"c"2 di!er only by the combinations of l and l which appear under the   average. We have (l #l )""l" for 1dk2 ,   (6.31) (l !l )#4 l l ""l" for 4 1"c"2"4 1c #c 2      and hence the two-point statistical properties of dk and 2 c agree identically. Therefore, the power spectra of e!ective convergence and shear agree, 1i( (l )i( H (l)2"1c( (l )c( H(l)2NP (l )"P (l ) .   G A

(6.32)

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Thus, we can concentrate on the statistics of either the magni"cation #uctuations or the shear only. Since dk"2i , the magni"cation power spectrum P is 4P , and we can immediately employ the  I G convergence power spectrum P . G 6.6. Second-order statistical measures We aim at the statistical properties of the magni"cation #uctuation and the shear. In particular, we are interested in the amplitude of these quantities and their angular coherence. Both can be described by their angular auto-correlation functions, or other second-order statistical measures that will turn out to be more practical later. As long as the density #uctuation "eld d remains Gaussian, the probability distributions of dk and c are also Gaussians with mean zero, and two-point statistical measures are su$cient for their complete statistical description. When nonlinear evolution of the density contrast sets in, non-Gaussianity develops, and higher-order statistical measures become important. 6.6.1. Angular auto-correlation function The angular auto-correlation function m ( ) of some isotropic quantity q(h) is the Fourier O transform of the power spectrum P (l) of q(h). In particular, the auto-correlation function of the O magni"cation #uctuation, m ( ), is related to the e!ective-convergence power spectrum P (l) I G through m ( )"1dk(h)dk(h# )2"4 1i (h)i (h# )2 I   dl ldl P (l) exp(!i l ) )"4 P (l) J (l ) , (6.33) "4 1c(h)cH(h# )2"4  (2p) G 2p G  where is a vector with norm . The factor four in front of the integral accounts for the fact that dk"2i in the weak-lensing approximation. For the last equality in (6.33), we integrated over the  angle enclosed by l and , leading to the zeroth-order Bessel function of the "rst kind, J (x).  Eq. (6.33) shows that the magni"cation (or shear) auto-correlation function is an integral over the power spectrum of the e!ective convergence i , "ltered by the Bessel function J (x). Since the   latter is a broad-band "lter, the magni"cation auto-correlation function is not well suited for extracting information on P . It would be desirable to replace m ( ) by another measurable G I quantity which involves a narrow-band "lter. Nonetheless, inserting Eq. (6.22) into Eq. (6.33), we obtain the expression for the magni"cation auto-correlation function,



9H X m ( )"   I c



U&





 kdk P (k, w) J [ f (w)k ] . (6.34)  ) 2p B   The magni"cation autocorrelation function therefore turns out to be an integral over the density#uctuation power spectrum weighted by a k-space window function which selects the contributing density perturbation modes. The correlation function of the image ellipticity (or the shear) is then 1eeH2( )"m ( )"m ( )/4. A I Since the ellipticity has two components, one can de"ne and calculate the corresponding correlations functions as well: Any pair of galaxy images de"nes the direction u of their separation vector. dw f (w) = M (w, w) a\(w); )

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With respect to this direction, one can de"ne in complete analogy to (5.31) the tangential and cross-components of the ellipticities, e and e "!I(e exp(!2iu)), respectively. One then "nds  " (Kaiser, 1992) that




 






1e e 2( ) J (l )#J (l ) 1 l dl    " P (l)  (6.35) G 2p 2 1e e 2( ) J (l )!J (l )  " "   and 1e e 2( )"0. The latter expression can be used to estimate systematic errors on a given data  " set from which the correlation functions are calculated. 6.6.2. Special cases and qualitative expectations In order to gain some insight into the expected behaviour of the magni"cation auto-correlation function m ( ), we now make a number of simplifying assumptions. Let us "rst specialise to linear I density evolution in an Einstein}de Sitter universe, and assume sources are at a single distance w .  Eq. (6.34) then immediately simpli"es to





  kdk 9H P(k)J (wyk ) (6.36) m ( )"  w dy y(1!y)   I 2p B c   with y,w\w.  We now introduce two model spectra P(k), one of which has an exponential cut-o! above some B wave number k , while the other falls o! like k\ for k'k . For small k, both spectra increase   like k. They approximately describe two extreme cases of popular cosmogonies, the HDM and the CDM model. We choose the functional forms




k 9k  , P "Ak , (6.37) P "Ak exp ! B!"+ B&"+ k (k#3k )   where A is the normalising amplitude of the power spectra. The numerical coe$cients in the CDM model spectrum are chosen such that both spectra peak at the same wave number k"k . Inserting  these model spectra into Eq. (6.36), performing the k integration, and expanding the result in a power series in , we obtain (Bartelmann, 1995b) m m

9A 3A (w k )! (w k ) #O( ) , ( )" I&"+ 35p   10p   I!"+

9(3A 27A ( )" (w k )! (w k ) #O( ) ,   80 40p  

(6.38)

where A"(H c\)A. We see from Eq. (6.38) that the magni"cation correlation function for the  HDM spectrum is #at to "rst order in , while it decreases linearly with for the CDM spectrum. This demonstrates that the shape of the magni"cation auto-correlation function m ( ) re#ects the I shape of the dark-matter power spectrum. Motivated by the result of a large number of cosmological studies showing that HDM models have the severe problem of structure on small scales forming at times much later than observed (see e.g. Peacock, 1999), we now neglect the HDM model and focus on the CDM power spectrum only.

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We can then expect m ( ) to increase linearly with as goes to zero. Although we assumed I linear evolution of the power spectrum to achieve this result, this qualitative behaviour remains valid when non-linear evolution is assumed, because for large wave numbers k, the non-linear CDM power spectra also asymptotically fall o! Jk\ for large k. Although the model spectra (6.37) are of limited validity, we can extract some useful information from the small-angle approximations given in Eq. (6.38). First, the correlation amplitude m (0) scales with the comoving distance to the sources w as w. In the Einstein}de Sitter case, I   for which Eq. (6.38) was derived, w "(2c/H ) [1!(1#z )\]. For low source redshifts, z ;1,     w +(c/H ) z , so that m (0)Jz. For z <1, w P(2c/H ), and m (0) becomes independent of    I     I source redshift. For intermediate source redshifts, progress can be made by de"ning f ,ln(z ) and   expanding ln w[exp(f )] in a power series in f . The result is an approximate power-law expression,   w(z )JzC , valid in the vicinity of the zero point of the expansion. The exponent e changes from   +0.6 at z +1 to +0.38 at z +3.   Second, typical source distances are of order 2 Gpc. Since k is the wave number corresponding  to the horizon size when relativistic and non-relativistic matter had equal densities, k\"d (a )+12 (X h)\ Mpc. Therefore, w k +150. Typically, the spectral amplitude A  &     ranges between 10\ and 10\. A rough estimate for the correlation amplitude m (0) thus ranges I between 10\ and 10\ for &typical' source redshifts z 91.  Third, an estimate for the angular scale of the magni"cation correlation is obtained by  determining the angle where m ( ) has dropped to half its maximum. From the small-angle I approximation (6.38), we "nd "p(3(12 w k )\. Inserting as before w k +150, we obtain     

+10, decreasing with increasing source redshift.  Summarising, we expect m ( ) in a CDM Universe to I (1) start at 10\}10\ at "0 for source redshifts z &1;  (2) decrease linearly for small on an angular scale of +10; and  (3) increase with source redshift roughly as Jz  around z "1.   6.6.3. Realistic cases After this digression, we now return to realistic CDM power spectra normalised to "t observational constraints. Some representative results are shown in Fig. 17 for the model parameter sets listed in Table 1. The "gure shows that typical values for m( ) in cluster-normalised CDM models with I non-linear density evolution are +6% at +1, quite independent of the cosmological model. The e!ects of non-linear evolution are considerable. Non-linear evolution increases the m by I factors of three to four. The uncertainty in the normalisation is illustrated by the two curves for the Einstein}de Sitter model, one of which was calculated with the cluster-, the other one with the p "1 normalisation, which yields about a factor of two larger results for m. For the other  I cosmological models (OCDM and KCDM), the e!ects of di!erent normalisations (cluster vs. COBE) are substantially smaller. 6.6.4. Application: magnixcation yuctuations At zero lag, the magni"cation auto-correlation function reads m (0)"1[k(h)!1]2,1dk2 , I

(6.39)

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Fig. 17. Four pairs of magni"cation auto-correlation functions are shown for the cosmological model parameter sets listed in Table 1, and for an assumed source redshift z "1. For each pair, plotted with the same line type, the curve with  lower amplitude at small angular scale was calculated assuming linear, and the other one non-linear density evolution. Solid curves: SCDM; dotted curves: pCDM; short-dashed curve: OCDM; and long-dashed curve: KCDM. Non-linear evolution increases the amplitude of m( ) on small angular scales by factors of three to four. The results for I the cluster-normalised models di!er fairly little. At +1, m( )+6% for non-linear density evolution. For the I Einstein}de Sitter models, the di!erence between cluster- and p "1 normalisation amounts to about a factor of two in  m( ). I Fig. 18. The rms magni"cation #uctuation dk is shown as a function of source redshift z for non-linearly evolving    density #uctuations in the four di!erent realisations of the CDM cosmogony detailed in Table 1. Solid curve: SCDM; dotted curve: pCDM; short-dashed curve: OCDM; and long-dashed curve: KCDM. Except for the pCDM model, typical rms magni"cation #uctuations are of order 20% at z "2, and 25% for z "3.  

which is the variance of the magni"cation #uctuation dk. Consequently, the rms magni"cation #uctuation is dk "1dk2"m(0) . (6.40)   I Fig. 18 shows dk as a function of source redshift for four di!erent realisations of the CDM   cosmogony. For cluster-normalised CDM models, the rms magni"cation #uctuation is of order dk +20% for sources at z +2, and increases to dk +25% for z +3. The strongest e!ect       occurs for open CDM (OCDM) because there non-linear evolution sets in at the highest redshifts. The results shown in Fig. 18 indicate that for any cosmological source, gravitational lensing causes a statistical uncertainty of its brightness. In magnitudes, a typical e!ect at z +2 is  dm+2.5;log(1.2)+0.2. This can be important for, e.g. high-redshift supernovae of Type Ia, which are used as cosmological standard candles. Their intrinsic magnitude scatter is of order dm+0.1}0.2 magnitudes (e.g. Phillips, 1993; Riess et al., 1995, 1996; Hamuy et al., 1996). Therefore, the lensing-induced brightness #uctuation is comparable to the intrinsic uncertainty at redshifts z 92 (Frieman, 1996; Wambsganss et al., 1997; Holz, 1998; Metcalf and Silk, 1999).  Since the magni"cation probability can be highly skewed, the most probable observed #ux of a high-redshift supernova can deviate from the mean #ux at given redshift, even if the intrinsic

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luminosity distribution is symmetric. This means that particular care needs to be taken in the analysis of future large SN surveys. However, if SNe Ia are quasi-standard candles also at high redshifts, with an intrinsic scatter of *¸"4pD (z)*S(z) around the mean luminosity  ¸ "4pD (z)S (z), then it is possible to obtain volume-limited samples (in contrast to #ux-limited    samples) of them. If, for a given redshift, the sensitivity limit is chosen to be S :k (S !3*S), one can be sure



  to "nd all SNe Ia at the redshift considered. Here, k is the minimum magni"cation of a source at

 the considered redshift. Since no source can be more de-magni"ed than one that is placed behind a hypothetical empty cone (see Dyer and Roder (1973) the discussion in Section 4.5 of Schneider et al., 1992), k is not much smaller than unity. Flux conservation (e.g. Weinberg, 1976) implies

 that the mean magni"cation of all sources at given redshift is unity, 1k(z)2"1, and so the expectation value of the observed #ux at given redshift is the unlensed #ux, 1S(z)2"S (z). It  should be pointed out here that a similar relation for the magnitudes does not hold, since magnitude is a logarithmic measure of the #ux, and so 1m(z)2Om (z). This led to some confusing  conclusions in the literature claiming that lensing introduces a bias in cosmological parameter estimates from lensing, but this is not true: One just has to work in terms of #uxes rather than magnitudes. However, a broad magni"cation probability distribution increases the con"dence contours for X and XK (e.g. Holz, 1998). If the probability distribution was known, more sensitive estimators  of the cosmological model than the mean #ux at given redshift could be constructed. Furthermore, if the intrinsic luminosity distribution of the SNe was known, the normalisation of the power spectrum as a function of X and XK could be inferred from the broadened observed  #ux distribution (Metcalf, 1999). If part of the dark matter is in the form of compact objects with mass 910\M , these objects can individually magnify a SN (Schneider and Wagoner, > 1987), additionally broadening the magni"cation probability distribution and thus enabling the nature of dark matter to be tested through SN observations (Metcalf and Silk, 1999; Seljak and Holz, 1999). 6.6.5. Shear in apertures We mentioned below Eq. (6.33) that measures of cosmic magni"cation or shear other than the angular auto-correlation function which "lter the e!ective-convergence power spectrum P with G a function narrower than the Bessel function J (x) would be desirable. In practice, a convenient  measure would be the variance of the e!ective convergence within a circular aperture of radius h. Within such an aperture, the averaged e!ective convergence and shear are



i (h)" 



(6.41)

Fd Fd  1i ( )i ( )2"1"c "2(h) .    ph ph  

(6.42)

Fd

Fd

i ( ), c (h)" c( )   ph ph  

and their variance is

 

1i 2(h)" 

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Fig. 19. The rms shear c (h) in circular apertures of radius h is plotted as a function of h for the four di!erent   realisations of the CDM cosmogony detailed in Table 1, where all sources are assumed to be at redshift z "1. A pair  of curves is plotted for each realisation, where for each pair the curve with lower amplitude at small h is for linearly, the other one for non-linearly evolving density #uctuations. Solid curves: SCDM; dotted curves: pCDM; shortdashed curves: OCDM; and long-dashed curves: KCDM. For the cluster-normalised models, typical rms shear values are +3% for h+1. Non-linear evolution increases the amplitude by about a factor of two at h+1 over linear evolution.

The remaining average is the e!ective-convergence auto-correlation function m (" ! "), which G can be expressed in terms of the power spectrum P . The "nal equality follows from m "m . G G A Inserting (6.42) and performing the angular integrals yields



 

 J (lh)  l dl P (l)  ""c "(h) , (6.43) G  plh  where J (x) is the "rst-order Bessel function of the "rst kind. Results for the rms shear in  apertures of varying size are shown in Fig. 19 (cf. Blandford et al., 1991; Kaiser, 1992; Jain and Seljak, 1997). 1i 2(h)"2p 

6.6.6. Aperture mass Another measure for the e!ects of weak lensing, the aperture mass M (h) (cf. Section 5.3.1), was  introduced for cosmic shear by Schneider et al. (1998a) as



F

d ;( ) i ( ) ,   where the weight function ;( ) satis"es the criterion M (h)" 



F



d ;( )"0 .

(6.44)

(6.45)

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In other words, ;( ) is taken to be a compensated radial weight function across the aperture. For such weight functions, the aperture mass can be expressed in terms of the tangential component of the observable shear relative to the aperture centre,



M (h)" 

F

d Q( )c ( ) , 

(6.46)

 where Q( ) is related to ;( ) by (5.24). M is a scalar quantity directly measurable in terms of the  shear. The variance of M reads   F  (6.47) l dl P (l)

d ;( )J (l ) . 1M 2(h)"2p G     Eqs. (6.43) and (6.47) provide alternative observable quantities which are related to the e!ectiveconvergence power spectrum P through narrower "lters than the auto-correlation function m . G G The M statistic in particular permits one to tune the "lter function through di!erent choices of  ;( ) within constraint (6.45). It is important that M can also be expressed in terms of the shear  [see Eq. (5.27)], so that M can directly be obtained from the observed galaxy ellipticities.  Schneider et al. (1998a) suggested a family of radial "lter functions ;( ), the simplest of which is










1 9 (1!x) !x , ;( )" 3 ph



6 Q( )" x(1!x) , ph

where xh" . With this choice, the variance 1M 2(h) becomes   1M 2(h)"2p l dl P (l) J(lh)  G  with the "lter function

(6.48)



(6.49)

12 J (g) , J(g)" pg 

(6.50)

where J (g) is the fourth-order Bessel function of the "rst kind. Examples for the rms aperture  mass, M (h)"1M 2(h), are shown in Fig. 20.    The curves look substantially di!erent from those shown in Figs. 17 and 19. Unlike there, the aperture mass does not increase monotonically as hP0, but reaches a maximum at "nite h and drops for smaller angles. When non-linear evolution of the density #uctuations is assumed, the maximum occurs at much smaller h than for linear evolution: Linear evolution predicts the peak at angles of order 13, non-linear evolution around 1! The amplitude of M (h) reaches +1%   for cluster-normalised models, quite independent of the cosmological parameters. Some insight into the expected amplitude and shape of 1M 2(h) can be gained by noting that J(g) is well  approximated by a Gaussian



(g!g )  J(g)+A exp ! 2p



(6.51)

with mean g +4.11, amplitude A+4.52;10\, and width p+1.24. At aperture radii of h+1,  the peak g +4.11 corresponds to angular scales of 2pl\+1.6, where the total power lP (l) in  G

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Fig. 20. The rms aperture mass, M (h), is shown in dependence of aperture radius h for the four di!erent realisations   of the CDM cosmogony detailed in Table 1 where all sources are assumed to be at redshift z "1. For each realisation,  a pair of curves is plotted; one curve with lower amplitude for linear, and the second curve for non-linear density evolution. Solid curves: SCDM; dotted curves: pCDM; short-dashed curves: OCDM; and long-dashed curves: KCDM. Non-linear evolution has a pronounced e!ect: The amplitude is approximately doubled, and the peak shifts from degreeto arcmin scales. Fig. 21. The rms aperture mass M (h) is shown together with the approximation M I (h) of Eq. (6.53). The three     curves correspond to the three cluster-normalised cosmological models (SCDM, OCDM and KCDM) introduced in Table 1 for non-linearly evolving matter perturbations. All sources were assumed to be at redshift z "1. Clearly, the rms  aperture mass is very accurately approximated by M I on angular scales h910, and even for smaller aperture sizes of   order &1 the deviation between the curves is smaller than +5%. The observable rms aperture mass therefore provides a very direct measure for the e!ective-convergence power spectrum P (l). G

the e!ective convergence is close to its broad maximum (cf. Fig. 16). The "lter function J(g) is therefore fairly narrow. Its relative width corresponds to an l range of dl/l+p/g &0.3. Thus, the  contributing range of modes l in integral (6.49) is very small. Crudely approximating the Gaussian by a delta distribution J(g)+A(2ppd (g!g ) , "  we are led to

(6.52)





g (2p)Ap g   P  +2.15;10\ l P (l ) (6.53) 1M 2+1M I  2, G h  G    h g  with l ,g h\. Hence, the mean-square aperture mass is expected to directly yield the total   power in the e!ective-convergence power spectrum, scaled down by a factor of +2.15;10\. We saw in Fig. 16 that lP (l)+3;10\ for 2pl\+1 in cluster-normalised CDM models, so that G 1M 2+0.8% at h+1 (6.54)  for sources at redshift unity. We compare M (h) and the approximation M I (h) in Fig. 21.     Obviously, the approximation is excellent for h910, but even for smaller aperture radii of &1 the

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Fig. 22. The three "lter functions F(g) de"ned in Eq. (6.56) are shown as functions of g"lh. They occur in the expressions for the magni"cation auto-correlation function, m (solid curve), the mean-square shear in apertures, 1c2 (dotted curve), I and the mean-square aperture mass, 1M 2 (dashed curve). 

relative deviation is less than +5%. At this point, the prime virtue of the narrow "lter function J(g) shows up most prominently. Up to relatively small errors of a few per cent, the rms aperture mass (h) very accurately re#ects the e!ective-convergence power spectrum P (l). Observations of M G   are therefore most suitable to obtain information on the matter power spectrum (cf. Bartelmann and Schneider, 1999). 6.6.7. Power spectrum and xlter functions The three statistical measures discussed above, the magni"cation (or, equivalently, the shear) auto-correlation function m , the mean-square shear in apertures 1c2, and the mean-square I aperture mass 1M 2, are related to the e!ective-convergence power spectrum P in very similar  G ways. According to Eqs. (6.33), (6.44) and (6.49), they can all be written in the form



Q(h)"2p



l dl P (l) F(lh) , G

(6.55)

 where the "lter functions F(g) are given by



J (g)  p J (g)   F(g)" pg 12J (g)   pg

   

for Q"m , I for Q"1c 2 , 

(6.56)

for Q"1M 2 . 

Fig. 22 shows these three "lter functions as functions of g"lh. Firstly, the curves illustrate that the amplitude of m is largest (owing to the factor of four relative to the de"nition of m ), and that of I A 1M 2 is smallest because the amplitudes of the "lter functions themselves decrease. Secondly, it 

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becomes evident that, for given h, the range of l modes of the e!ective-convergence power spectrum P (l) convolved into the weak-lensing estimator is largest for m and smallest for 1M 2. Thirdly, G I  the envelope of the "lter functions for large g decreases most slowly for m and most rapidly for I 1M 2. Although the aperture mass has the smallest signal amplitude, it is a much better probe for  the e!ective-convergence power spectrum P (l) than the other measures because it picks up the G smallest range of l modes and most strongly suppresses the l modes smaller or larger than its peak location. We can therefore conclude that, while the strongest weak-lensing signal is picked up by the magni"cation auto-correlation function m , the aperture mass is the weak-lensing estimator most I suitable for extracting information on the e!ective-convergence power spectrum. 6.6.8. Signal-to-noise estimate of aperture-mass measurements The question then arises whether the aperture mass can be measured with su$cient signi"cance in upcoming wide-"eld imaging surveys. In practice, M is derived from observations of image  distortions of faint background galaxies, using Eq. (5.27) and replacing the integral by a sum over galaxy ellipticities. If we consider N independent apertures with N galaxies in the ith aperture, an  G unbiased estimator of 1M 2 is  1 ,G (ph) ,  Q Q e e , (6.57) M" GH GI GH GI N (N !1) N  G G G H$I where Q is the value of the weight function at the position of the jth galaxy in the ith aperture, and GH e is de"ned accordingly. GH The noise properties of this estimator were investigated in Schneider et al. (1998a). One source of noise comes from the fact that galaxies are not intrinsically circular, but rather have an intrinsic ellipticity distribution. A second contribution to the noise is due to the random galaxy positions, and a third one to cosmic (or sampling) variance. Under the assumptions that the number of galaxies N in the apertures is large, N <1, it turns out that the second of these contributions can G G be neglected compared to the other two. For this case, and assuming for simplicity that all N are G equal, N ,N, the signal-to-noise of the estimator M becomes G  \ S 1M 2 6p  "N k # (2# C , (6.58) ,   N p(M) 5(2N1M 2  where p +0.2 (e.g. Hudson et al., 1998) is the dispersion of the intrinsic galaxy ellipticities, and C k "1M 2/1M 2!3 is the curtosis of M , which vanishes for a Gaussian distribution. The     two terms of (6.58) in parentheses represent the noise contributions from Gaussian sampling variance and the intrinsic ellipticity distribution, respectively, and k accounts for sampling  variance in excess of that for a Gaussian distribution. On angular scales of a few arcmin and smaller, the intrinsic ellipticities dominate the noise, while the cosmic variance dominates on larger scales. Another convenient and useful property of the aperture mass M follows from its "lter function  being narrow, namely that M is a well localised measure of cosmic weak lensing. This implies that  M measurements in neighbouring apertures are almost uncorrelated even if the aperture centres  are very close (Schneider et al., 1998a). It is therefore possible to gain a large number of (almost)






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Fig. 23. The signal-to-noise ratio S/N(h) of measurements of mean-square aperture masses 1M 2 is plotted as a function  of aperture radius h for an experimental setup as described in the text. The curtosis was set to zero here. The four curves are for the four di!erent realisations of the CDM cosmogony listed in Table 1. Solid curve: SCDM; dotted curve: pCDM; short-dashed curve: OCDM; and long-dashed curve: KCDM. Quite independently of the cosmological parameters, the signal-to-noise ratio S/N reaches values of '10 on scales of +1}2.

independent M measurements from a single large data "eld by covering the "eld densely with  apertures. This is a signi"cant advantage over the other two measures for weak lensing discussed above, whose broad "lter functions introduce considerable correlation between neighbouring measurements, implying that for their measurement imaging data on widely separated "elds are needed to ensure statistical independence. Therefore, a meaningful strategy to measure cosmic shear consists in taking a large data "eld, covering it densely with apertures of varying radius h, and determining 1M 2 in them via the ellipticities of galaxy images. Fig. 23 shows an example for the  signal-to-noise ratio of such a measurement that can be expected as a function of aperture radius h. Computing the curves in Fig. 23, we assumed that a data "eld of size 53;53 is available which is densely covered by apertures of radius h, hence the number of (almost) independent apertures is N "(300/2h). The number density of galaxies was taken as 30 arcmin\, and the intrinsic  ellipticity dispersion was assumed to be p "0.2. Evidently, high signal-to-noise ratios of '10 are C reached on angular scales of +1 in cluster-normalised universes quite independent of the cosmological parameters. The decline of S/N for large h is due to the decreasing number of independent apertures on the data "eld, whereas the decline for small h is due to the decrease of the signal 1M 2, as seen in Fig. 20. We also note that for calculating the curves in Fig. 23, we have put  k "0. This is likely to be an overly optimistic assumption for small angular scales where the  density "eld is highly non-linear. Unfortunately, k cannot easily be estimated analytically. It was  numerically derived from ray-tracing through N-body simulations of large-scale matter distributions by Reblinsky et al. (1999). The curtosis exceeds unity even on scales as large as 10, demonstrating the highly non-Gaussian nature of the non-linearly developed density perturbations. Although the aperture mass is a very convenient measure of cosmic shear and provides a localised estimate of the projected power spectrum P (l) [see (6.53)], it is by no means clear that it G

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is an optimal measure for the projected power spectrum. Kaiser (1998) considered the case of a square-shaped data "eld and employed the Fourier-transformed Kaiser and Squires inversion formula, Eq. (5.3). The Fourier transform of the shear is then replaced by a sum over galaxy ellipticities e , so that i( (l) is expressed directly in terms of the e . The square "i( (l)" yields an G  G  estimate for the power spectrum which allows a simple determination of the noise coming from the intrinsic ellipticity distribution. As Kaiser (1998) pointed out that, while this noise is very small for angular scales much smaller than the size of the data "eld, the sampling variance is much larger, so that di!erent sampling strategies should be explored. For example, he suggests to use a sparse sampling strategy. Seljak (1998) developed an estimator for the power spectrum which achieves minimum variance in the case of a Gaussian "eld. Since the power spectrum P (l) deviates G signi"cantly from its linear prediction on angular scales below 13, one expects that the "eld attains signi"cant non-Gaussian features on smaller angular scales, so that this estimator does no longer need to have minimum variance. 6.7. Higher-order statistical measures 6.7.1. The skewness As the density perturbation "eld d grows with time, it develops non-Gaussian features. In particular, d is bounded by !1 from below and unbounded from above, and therefore the distribution of d is progressively skewed while evolution proceeds. The same then applies to quantities like the e!ective convergence i derived from d (cf. Jain and Seljak, 1997; Bernardeau  et al., 1997; Schneider et al., 1998a). Skewness of the e!ective convergence can be quanti"ed by means of the three-point correlator of i . In order to compute that, we use expression (6.18),  Fourier transform it, and also express the density contrast d in terms of its Fourier transform. Additionally, we employ the same approximation used in deriving Limber's equation in Fourier space, namely that correlations of the density contrast along the line-of-sight are negligibly small. After carrying out this lengthy but straightforward procedure, the three-point correlator of the Fourier transform of i reads (suppressing the subscript &e! ' for brevity)  27H X   1i( (l )i( (l )i( (l )2"    8c




; dK



U&



dw

 


 dk = M (w)  exp(ik w)  2p a(w) f  (w) \ )






l l l  , k dK  , 0 dK  ,0 f (w)  f (w) f (w) ) ) )

.

(6.59)

Hats on symbols denote Fourier transforms. Note the fairly close analogy between (6.59) and (6.22): The three-point correlator of i( is a distance-weighted integral over the three-point correlator of the Fourier-transformed density contrast dK . The fact that the three-component k of the wave vector  k appears only in the "rst factor dK re#ects the approximation mentioned above, i.e. that correlations of d along the line-of-sight are negligible. Suppose now that the density contrast d is expanded in a perturbation series, d" dG such that dG"O([d]G), and truncated after the second order. The three-point correlator of dK  vanishes because d remains Gaussian to "rst perturbation order. The lowest-order, non-vanishing threepoint correlator of d can therefore symbolically be written 1dK dK dK 2, plus two permutations

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of that expression. The second-order density perturbation is related to the "rst order through (Fry, 1984; Goro! et al., 1986; Bouchet et al., 1992)



dk dK (k)dK (k!k) F(k, k!k) , dK (k, w)"D (w)  > (2p) 

(6.60)

where d is the "rst-order density perturbation linearly extrapolated to the present epoch, and  D (w) is the linear growth factor, D (w)"a(w) g[a(w)] with g(a) de"ned in Eq. (2.52). The function > > F(x, y) is given by






5 1 1 1 2 (x ) y) F(x, y)" # # x ) y# . 7 2 "x" "y" 7 "x""y"

(6.61)

Relation (6.60) implies that the lowest-order three-point correlator 1dK dK dK 2 involves four-point correlators of dK . For Gaussian "elds like d, four-point correlators can be decomposed into sums of products of two-point correlators, which can be expressed in terms of the linearly extrapolated density power spectrum P. This leads to B 1dK (k )dK (k )dK (k )2"2 (2p) D (w) P(k )P(k )d (k #k #k ) F(k , k ) . (6.62)    > B  B  "      The complete lowest-order three-point correlator of dK is a sum of three terms, namely the left-hand side of (6.62) and two permutations thereof. Each permutation yields the same result, so that the complete correlator is three times the right-hand side of (6.62). We can now work our way back, inserting the three-point density correlator into Eq. (6.59) and Fourier-transforming the result with respect to l . The three-point correlator of the e!ective convergence so obtained can then in  a "nal step be used to compute the third moment of the aperture mass. The result is (Schneider et al., 1998a)








81H X U& l = M (w)D (w)    > 1M (h)2" dl P J(l h) dw   B f (w)  8pc a(w) f (w)  ) ) l  ; dl P J(l h) J("l #l "h) F(l , l ) (6.63)  B f (w)      ) with the "lter function J(g) de"ned in Eq. (6.49). Commonly, third-order moments are expressed in terms of the skewness






1M (h)2  S(h), , (6.64) 1M (h)2  where 1M (h)2 is calculated with the linearly evolved power spectrum. As seen earlier in Eq. (6.49),  1M 2 scales with the amplitude of the power spectrum, while 1M 2 scales with the square of it. In   this approximation, the skewness S(h) is therefore independent of the normalisation of the power spectrum, removing that major uncertainty and leaving cosmological parameters as primary degrees of freedom. For instance, the skewness S(h) is expected to scale approximately with X\.  Fig. 24 shows three examples. As expected, lower values of X yield larger skewness, and the skewness is reduced when XK is  increased keeping X "xed. Despite the sensitivity of S(h) to the cosmological parameters, it 

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Fig. 24. The skewness S(h) of the aperture mass M (h) is shown as a function of aperture radius h for three of the  realisations of the cluster-normalised CDM cosmogony listed in Table 1: SCDM (solid curve); OCDM (dotted curve); and KCDM (dashed curve). The source redshift was assumed to be z "1. 

should be noted that the source redshift distribution [entering through = M (w)] needs to be known su$ciently well before attempts can be made at constraining cosmological parameters through measurements of the aperture-mass skewness. However, photometric redshift estimates are expected to produce su$ciently well-constrained redshift distributions in the near future (Connolly et al., 1995; Gwyn and Hartwick, 1996; Hogg et al., 1998). We have con"ned the discussion of the skewness to the aperture mass since M is a scalar  measure of the cosmic shear which can directly be expressed in terms of the observed image ellipticities. One can, of course, also consider the skewness directly in terms of i, since i can be obtained from the observed image ellipticities through a mass reconstruction algorithm as described in Section 5. Analytical and numerical results for this skewness have been presented in, e.g., Bernardeau et al. (1997), van Waerbeke et al. (1999b), Jain et al. (2000) and Reblinsky et al. (1999). We shall discuss some of their results in Section 6.9.1. As pointed out by Bernardeau (1998), the fact that the source galaxies are clustered in threedimensional space, and therefore also in redshift space, generates an additional contribution to the skewness. This e!ect is more important than the contributions by the approximations made in the light propagation equations; in fact, Bernardeau (1998) estimated that the skewness can change by &25% due to source clustering. Whereas the expectation values of second-order statistics of cosmic shear is una!ected by this clustering, the dispersion of any estimator increases. Of course, if the redshifts of the source galaxies are known, these e!ects can be avoided by suitably de"ning estimators for the quantities under consideration. In the regime of small angular scales, where the relevant density contrast is highly non-linear, di!erent approximations apply for calculating higher-order statistical quantities. One of them is based on the so-called stable-clustering ansatz, which predicts a scaling relation for the n-point correlation function of the density contrast (Peebles, 1980). Based on this assumption, and variants thereof, higher-order moments of cosmic-shear measures can be derived (e.g., Hui, 1999a; Munshi

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and Coles, 2000; Munshi and Jain, 1999a), as well as approximations to the probability distribution for i itself and "ltered (smoothed) versions thereof (Valageas, 2000; Munshi and Jain, 1999b;  Valageas, 1999). The resulting expressions, when compared to numerical simulations of light propagation through large-scale structures, are surprisingly accurate. 6.7.2. Number density of (dark) halos In Section 5.3.1, we discussed the possibility to detect mass concentrations by their weak lensing e!ects on background galaxies by means of the aperture mass. The number density of mass concentrations that can be detected at a given threshold of M depends on the cosmological  model. Fixing the normalisation of the power spectrum so that the local abundance of massive clusters is reproduced, the evolution of the density "eld proceeds di!erently in di!erent cosmologies, and so the abundances will di!er at redshifts z&0.3 where the aperture-mass method is most sensitive. The number density of halos above a given threshold of M (h) can be estimated analytically,  using two ingredients. First, the spatial number density of halos at redshift z with mass M can be described by the Press}Schechter theory (Press and Schechter, 1974), which numerical simulations (Lacey and Cole, 1993, 1994); have shown to be a fairly accurate approximation. Second, in a series of very large N-body simulations, Navarro et al. (1996a, 1997) found that dark matter halos have a universal density pro"le which can be described by two parameters, the halo mass and a characteristic scale length, which depends on the cosmological model and the redshift. Combining these two results from cosmology, Kruse and Schneider (1999b) calculated the number density of halos exceeding M . Using the signal-to-noise estimate (6.58), a threshold  value of M can be directly translated into a signal-to-noise threshold S . For an assumed   number density of n"30 arcmin\ and an ellipticity dispersion p "0.2, one "nds S +(h/1 arC  cmin)(M (h)/0.016).  For the redshift distribution (2.69) with b"3/2 and z "1, the number density of halos with  S 55 exceeds 10 per square degree for cluster-normalised cosmologies, across angular scales  1:h:10, and these halos have a broad redshift distribution which peaks at z &0.3. This implies  that a wide-"eld imaging survey should be able to detect a statistically interesting sample of medium redshift halos, thus allowing the de"nition of a mass-selected sample of halos. Such a sample will be of utmost interest for cosmology, since the halo abundance is considered to be one of the most sensitive cosmological probes (e.g., Eke et al., 1996; Bahcall and Fan, 1998). Current attempts to apply this tool are hampered by the fact that halos are selected either by the X-ray properties or by their galaxy content. These properties are much more di$cult to predict than the dark-matter distribution of halos which can directly be determined from cosmological N-body simulations. Thus, these mass-selected halos will provide a much closer link to cosmological predictions than currently possible. Kruse and Schneider (1999b) estimated that an imaging survey of several square degrees will allow one to distinguish between the cosmological models given in Table 1, owing to the di!erent number density of halos that they predict. Using the aperture-mass statistics, Erben et al. (2000) recently detected a highly signi"cant matter concentration on two independent wide-"eld images centred on the galaxy cluster A 1942. This matter concentration 7 South of A 1942 is not associated with an overdensity of bright foreground galaxies, which sets strong lower limits on the mass-to-light ratio of this putative cluster.

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6.8. Cosmic shear and biasing Up to now, we have only considered the mass properties of the large-scale structure and tried to measure them with weak lensing techniques. An interesting question arises when the luminous constituents of the Universe are taken into account. Most importantly, the galaxies are supposed to be strongly tied to the distribution of dark matter. In fact, this assumption underlies all attempts to determine the power spectrum of cosmic density #uctuations from the observed distribution of galaxies. The relation between the galaxy and dark-matter distributions is parameterised by the so-called biasing factor b (Kaiser, 1984), which is de"ned such that the relative #uctuations in the spatial number density of galaxies are b times the relative density #uctuations d, n(x)!1n2 "bd(x) , 1n2

(6.65)

where 1n2 denotes the mean spatial number density of galaxies at the given redshift. The bias factor b is not really a single number, but generally depends on redshift, on the spatial scale, and on the galaxy type (see, e.g., Efstathiou, 1996; Peacock, 1997; Kau!mann et al., 1997; Coles et al., 1998). Typical values for the bias factor are assumed to be b&1}2 at the current epoch, but can increase towards higher redshifts. The clustering properties of UV dropout galaxies (Steidel et al., 1998) indicate that b can be as large as 5 at redshifts z&3, depending on the cosmology. The projected surface mass density i (h) should therefore be correlated with the number density  of (foreground) galaxies in that direction. Let G (w) be the distribution function of a suitably % chosen population of galaxies in comoving distance (which can be readily converted to a redshift probability distribution). Then, assuming that b is independent of scale and redshift, the number density of the galaxies is

 



n (h)"1n 2 1#b dw G (w) d( f (w)h, w) , % ) % %

(6.66)

where 1n 2 is the mean number density of the galaxy population. The distribution function G (w) % % depends on the selection of galaxies. For example, for a #ux-limited sample it may be of the form (2.69). Narrower distribution functions can be achieved by selecting galaxies in multi-colour space using photometric redshift techniques. The correlation function between n (h) and i (h) can %  directly be obtained from Eq. (2.83) by identifying q (w)"3H X = M (w)f (w)/[2ca(w)] [see    ) Eq. (6.18)], and q (w)"1n 2bG (w). It reads  % % = M (w)f (w) 3H X ) G (w) m (h),1n i 2(h)"   b1n 2 dw % % %G %  a(w) 2c



dk k P (k, w) J ( f (w)hk) .  ) 2p B



(6.67)

Similar equations were derived by, e.g., Kaiser (1992), Bartelmann (1995b), Dolag and Bartelmann (1997), Sanz et al. (1997).

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One way to study the correlation between foreground galaxies and the projected density "eld consists in correlating the aperture mass M (h) with a similarly "ltered galaxy number density,  de"ned as



N(h)" d0 ;("0") n (0) %

(6.68)

with the same "lter function ; as in M . The correlation between M (h) and N(h) then becomes  





m(h),1M (h)N(h)2" d0 ;("0") d0 ;("0")m ("0!0")  %G











H  l = M (w)G (w)  X b1n 2 dw % dl lP , w J(lh) , (6.69)  % B c f (w) a(w) f (w) ) ) where we used Eq. (2.83) for the correlation function m in the "nal step. The "lter function J is %G de"ned in Eq. (6.50). Note that this correlation function "lters out the power spectrum P at B redshifts where the foreground galaxies are situated. Thus, by selecting galaxy populations with narrow redshift distribution, one can study the cosmological evolution of the power spectrum or, more accurately, the product of the power spectrum and the bias factor. The convenient property of this correlation function is that one can de"ne an unbiased estimator for m in terms of observables. If N galaxies are found in an aperture of radius h at positions 0 with G
tangential ellipticity e , and N foreground galaxies at positions u , then G G  ph ,
, mI (h)" Q("0 ") e ;("u ") (6.70) G G I N
G I is an unbiased estimator for m(h). Schneider (1998) calculated the noise properties of this estimator, concentrating on an Einstein}de Sitter model and a linearly evolving power spectrum which can locally be approximated by a power law in k. A more general and thorough treatment is given in van Waerbeke (1998), where various cosmological models and the non-linear power spectrum are considered. van Waerbeke (1998) assumed a broad redshift distribution for the background galaxies, but a relatively narrow redshift distribution for the foreground galaxies, with dz /z &0.3.   For an open model with X "0.3, m(h) declines much faster with h than for #at models, implying  that open models have relatively more power on small scales at intermediate redshift. This is a consequence of the behaviour of the growth factor D (w); see Fig. 6. For foreground redshifts > z 90.2, the signal-to-noise ratio of the estimator (6.70) for a single aperture is roughly constant for  h95, and relatively independent of the exact value of z over a broad redshift interval, with  a characteristic value of &0.4. van Waerbeke (1998) also considered the ratio "3p

m(h) R, 1N(h)2

(6.71)

and found that it is nearly independent of h. This result was shown in Schneider (1998) to hold for linearly evolving power spectra with power-law shape, but surprisingly it also holds for the fully non-linear power spectrum. Indeed, varying h between 1 and 100, R varies by less than 2% for the models considered in van van Waerbeke (1998). This is an extremely important result, in that any

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observed variation of R with angular scale indicates a corresponding scale dependence of the bias factor b. A direct observation of this variation would provide valuable constraints on the models for the formation and evolution of galaxies. We point out that the ratio R depends, in the linear regime, on the combination X /b,  independent of the normalisation of the power spectrum. This is to be compared with the combination X /b determined by peculiar motions of galaxies (e.g., Strauss and Willick, 1995 and  references therein). Since these combinations of the two parameters di!er, one might hope that they can be derived separately by combining them. 6.9. Numerical approach to cosmic shear, cosmological parameter estimates, and observations 6.9.1. Cosmic shear predictions from cosmological simulations So far, we have treated the lensing e!ect of the large-scale structure with analytic means. This was possible because of two assumptions. First, we considered only the lowest-order lensing e!ect, by employing the Born approximation and neglecting lens-lens coupling in going from Eq. (6.9) to Eq. (6.11). Second, we used the prescription for the non-linear power spectrum as given by Peacock and Dodds (1996), assuming that it is a su$ciently accurate approximation. Both of these approximations may become less accurate on small angular scales. Providing a two-point quantity, the analytic approximation of P is applicable only for two-point statistical measures of G cosmic shear. In addition, the error introduced with these approximations cannot be controlled, i.e., we cannot attach &error bars' to the analytic results. A practical way to avoid these approximations is to study the propagation of light in a model Universe which is generated by cosmological structure-formation simulations. They typically provide the three-dimensional mass distribution at di!erent redshifts in a cube whose sidelength is much smaller than the Hubble radius. The mass distribution along a line-of-sight can be generated by combining adjacent cubes from a sequence of redshifts. The cubes at di!erent redshifts should either be taken from di!erent realisations of the initial conditions, or, if this requires too much computing time, they should be translated and rotated such as to avoid periodicity along the line-of-sight. The mass distribution in each cube can then be projected along the line-of-sight, yielding a surface mass density distribution at that redshift. Finally, by employing the multiple lens-plane equations, which are a discretisation of the propagation equation ((6.9); Seitz et al., 1994), shear and magni"cation can be calculated along light rays within a cone whose size is determined by the sidelength of the numerical cube. This approach was followed by many authors (e.g., Jaroszynski et al., 1990; Jaroszynski, 1991; Bartelmann and Schneider, 1991; Blandford et al., 1991; Waxman and Miralda-EscudeH , 1995), but the rapid development of N-body simulations of the cosmological dark-matter distribution render the more recent studies particularly useful (Wambsganss et al., 1998; van Waerbeke et al., 1999b; Jain et al., 2000). As mentioned below Eq. (6.30), the Jacobian matrix A is generally asymmetric when the propagation equation is not simpli"ed to (6.11). Therefore, the degree of asymmetry of A provides one test for the accuracy of this approximation. Jain et al. (2000) found that the power spectrum of the asymmetric component is at least three orders of magnitude smaller than that of i . For  a second test, we have seen that the power spectrum of i should equal that of the shear in the  frame of our approximations. This analytic prediction is very accurately satis"ed in the numerical simulations.

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Jain et al. (2000) and Reblinsky et al. (1999) found that analytic predictions of the dispersions of i and M , respectively, are very accurate when compared to numerical results. For both  cosmic shear measures, however, the analytic predictions of the skewness are not satisfactory on angular scales below &10. This discrepancy re#ects the limited accuracy of the second-order Eulerian perturbation theory employed in deriving the analytic results. Hui (1999b) showed that the accuracy of the analytic predictions can be much increased by using a prescription for the highly nonlinear three-point correlation function of the cosmic density contrast, as developed by Scoccimarro and Frieman (1999). On larger angular scales, the predictions from perturbation theory as described in Section 6.7.1 are accurate, as shown by Gaztanaga and Bernardeau (1998). The signal-to-noise ratio of the dispersion of the cosmic shear, given explicitly for M in  Eq. (6.58), is determined by the intrinsic ellipticity dispersion of galaxies and the sampling variance, expressed in terms of the curtosis. As shown by van Waerbeke et al. (1999b), Reblinsky et al. (1999), and White and Hu (1999), this curtosis is remarkably large. For instance, the curtosis of the aperture mass exceeds unity even on scales larger than 10, revealing non-Gaussianity on such large scales. Unfortunately, this large sampling variance implies not only that the area over which cosmic shear needs to be measured to achieve a given accuracy for its dispersion must be considerably larger than estimated for a Gaussian density "eld, but also that numerical estimates of cosmic shear quantities need to cover large solid angles for an accurate numerical determination of the relevant quantities. From such numerical simulations, one can not only determine moments of the shear distribution, but also consider its full probability distribution. For example, the predictions for the number density of dark matter halos that can be detected through highly signi"cant peaks of M } see  Section 6.7.2 } have been found by Reblinsky et al. (1999) to be fairly accurate, perhaps surprisingly so, given the assumptions entering the analytic results. Similarly, the extreme tail (say more than "ve standard deviations from the mean) of the probability distribution for M , calculated  analytically in Kruse and Schneider (1999a), does agree with the numerical results; it decreases exponentially.

6.9.2. Cosmological parameter estimates Since the cosmic shear described in this section directly probes the total matter content of the Universe, i.e., without any reference to the relation between mass and luminosity, it provides an ideal tool to investigate the large-scale structure of the cosmological density "eld. Assuming the dominance of cold dark matter, the statistical properties of the cosmic mass distribution are determined by a few parameters, the most important of which are X , XK ,  the shape parameter of the power spectrum, C, and the normalisation of the power spectrum expressed in terms of p . For each set of these parameters, the corresponding cosmic shear  signals can be predicted, and a comparison with observations then constrains the cosmological parameters. Furthermore, since weak lensing probes the shape of the projected power spectrum, modi"cations of the CDM power spectrum by a contribution from hot dark matter (such as massive neutrinos) may be measurable; e.g. Cooray (1999a) estimated that a deep weak-lensing survey of 100 square degrees may yield a lower limit on the neutrino mass of 3.5 eV.

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Several approaches to this parameter estimation have been discussed in the literature. For example, van Waerbeke et al. (1999b) used numerical simulations to generate synthetic cosmic shear data, "xing the normalisation of the density #uctuations to p X "0.6, which is essentially   the normalisation by cluster abundance. A moderately wide and deep weak-lensing survey, covering 25 square degrees and reaching a number density of 30 galaxies per arcmin with characteristic redshift z &1, will enable the distinction between an Einstein}de Sitter model and  an open universe with X "0.3 at the 6-p level, though each of these models is degenerate in the  X vs. XK plane. For this conclusion, only the skewness of the reconstructed e!ective surface mass  density or the aperture mass was used. Kruse and Schneider (1999a) instead considered the highly non-Gaussian tail of the aperture mass statistics to constrain cosmological parameters, whereas Kruse and Schneider (1999b) considered the abundance of highly signi"cant peaks of M as  a probe of the cosmological models. The peak statistics of reconstructed surface density maps (Jain and van Waerbeke, 2000) also provides a valuable means to distinguish between various cosmological models. Future work will also involve additional information on the redshifts of the background galaxies. Hu (1999) pointed out that splitting up the galaxy sample into several redshift bins substantially increases the ability to constrain cosmological parameters. He considered the power spectrum of the projected density and found that the accuracy of the corresponding cosmological parameters improves by a factor of &7 for XK , and by a factor of &3 for X , estimated for a median redshift  of unity. All of the quoted work concentrated mainly on one particular measure of cosmic shear. One goal of future theoretical investigations will certainly be the construction of a method which combines the various measures into a &global' statistics, designed to minimise the volume of parameter space allowed by the data of future observational weak lensing surveys. Future, larger-scale numerical simulations will guide the search for such a statistics and allow one to make accurate predictions. In addition to a pure cosmic shear investigation, cosmic shear constraints can be used in conjunction with other measures of cosmological parameters. One impressive example has been given by Hu and Tegmark (1999), who showed that even a relatively small weak lensing survey could dramatically improve the accuracy of cosmological parameters measured by future Cosmic Microwave Background missions. 6.9.3. Observations One of the "rst attempts to measure cosmic shear was reported in Mould et al. (1994), where the mean shear was investigated across a "eld of 9.6;9.6, observed with the Hale 5-m Telescope. The image is very deep and has good quality (i.e., a seeing of 0.87 FWHM). It is the same data as used by Brainerd et al. (1996) for the "rst detection of galaxy}galaxy lensing (see Section 8). The mean ellipticity of the 4363 galaxies within a circle of 4.8 radius with magnitudes 234r426 was found to be (0.5$0.5)%. A later, less conservative reanalysis of these data by Villumsen (unpublished), where an attempt was made to account for the seeing e!ects, yielded a 3-p detection of a nonvanishing mean ellipticity. Following the suggestion that the observed large-angle QSO-galaxy associations are due to weak lensing by the large-scale structure in which the foreground galaxies are embedded (see Section 7), Fort et al. (1996) searched for shear around "ve luminous radio quasars. In one of the

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"elds, the number density of stars was so high that no reasonable shear measurement on faint background galaxies could be performed. In the remaining four QSO "elds, they found a shear signal on a scale of &1 for three of the QSOs (those which were observed with SUSI, which has a "eld-of-view of &2.2), and on a somewhat larger angular scale for the fourth QSO. Taken at face value, these observations support the suggestion of magni"cation bias caused by the largescale structure. A reanalysis of the three SUSI "elds by Schneider et al. (1998b), considering the rms shear over the "elds, produced a positive value for 1"c"2 at the 99% signi"cance level, as determined by numerous simulations randomising the orientation angles of the galaxy ellipticities. The amplitude of the rms shear, when corrected for the dilution by seeing, is of the same magnitude as expected from cluster-normalised models. However, if the magni"cation bias hypothesis is true, these three lines-of-sight are not randomly selected, and therefore this measurement is of no cosmological use. Of course, one or a few narrow-angle "elds cannot be useful for a measurement of cosmic shear, owing to cosmic variance. Therefore, a meaningful measurement of cosmic shear must either include many small "elds, or must be obtained from a wide-"eld survey. Using the "rst strategy, several projects are under way: The Hubble Space Telescope has been carrying out the so-called parallel surveys, where one or more of the instruments not used for primary observations are switched on to obtain data of a "eld located a few arcmin away from the primary pointing. Over the past few years, a considerable database of such parallel data sets has accumulated. Two teams are currently analysing parallel data sets taken with WFPC2 and STIS, respectively (see Seitz et al., 1998a; Rhodes et al., 1999). In addition, a cosmic-shear survey is currently under way, in which randomly selected areas of the sky are mapped with the FORS instrument (&6.7;6.7 ) on the VLT. Some of these areas include the "elds from the STIS parallel survey. The alternative approach is to map big areas and measure the cosmic shear on a wide range of scales. The wide-"eld cameras currently being developed and installed are ideally suited for this purpose, and several groups are actively engaged in this work (see the proceedings of the Boston lens conference, July 1999). Very recently, four groups have independently and almost simultaneously reported statistically signi"cant detections of cosmic shear. In alphabetic order: Bacon et al. (2000) used 14 independent "elds of size 8;16 obtained with the WHT to measure the rms shear in squares of 8;8. Kaiser et al. (2000) used six independent images taken with the UH8K camera on CFHT, each 30;30 in size, to measure the cosmic shear on scales between 2 and 30. van Waerbeke et al. (2000) observed eight independent "elds with the UH8K and UH12K (30;45) cameras at CFHT and measured the rms shear on scales below 3.5 since they avoided measurements in apertures crossing chip edges. Finally, Wittman et al. (2000) took three independent "elds of size 43;43 with the BTC at CTIO to measure the two-point correlation function of galaxy ellipticities on scales between 2 and 30. All four groups discuss their statistical and systematic uncertainties in detail and employ various tests to convincingly demonstrate the physical reality of the signal. In particular, they show that remaining systematics most probably contribute to the shear signal at a level below 1%, i.e. much less than the measured signal on scales :10}15. The results of these groups are presented in Fig. 25. The yet unpublished result by Maoli et al. is not included. It was obtained from 45 images

 This "eld was subsequently used to demonstrate the superb image quality of the SUSI instrument on the ESO NTT.

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Fig. 25. Compilation of the results of four di!erent measurements of the cosmic-shear dispersion (with the two-point shear correlation function of Wittman et al. (2000) transformed into an equivalent dispersion for comparison). Open triangle: Bacon et al. (2000); "lled squares: Kaiser et al. (2000); open squares: van Waerbeke et al. (2000); crosses: Wittman et al. (2000). The error bars include both statistical errors and cosmic variance. Points from the same group at di!erent angular scales are not statistically independent. The dotted curves are predictions for a cluster-normalised KCDM model with e!ective source redshifts of z "1 (lower curve) and z "2 (upper curve), taken from Jain and Seljak (1997) (adapted   from Kaiser et al., 2000).

taken with the FORS1 instrument (6.7;6.7) on UT1 of VLT. Evidently, the results of the various groups are in excellent agreement despite the data being taken with di!erent optical "lters, di!erent cameras, di!erent telescopes, and reduced with di!erent data analysis techniques. This provides additional evidence for the reality of the cosmic-shear signal. The signi"cance of the results extends up to 5.5p, dependent of course on the total size of the "elds used for the respective analyses. Since, except for the VLT data, the number of independent "elds used for these studies is small, the error is entirely dominated by cosmic variance. These impressive results prove the power of cosmic-shear measurements as a novel tool for probing the statistical properties of large-scale structures on small scales and at late times in the universe. In the near future, such measurements will become comparably important, and will provide complementary cosmological information to that obtained from CMB experiments. There is nothing special about weak lensing being carried out predominantly in the optical wavelength regime, except that the optical sky is full of faint extended sources, whereas the radio sky is relatively empty. The FIRST radio survey covers at present about 4200 square degrees and contains 4;10 sources, i.e., the number density is smaller by about a factor &1000 than in deep optical images. However, this radio survey covers a much larger solid angle than current or foreseeable deep optical surveys. As discussed in Refregier et al. (1998), this survey may yield a signi"cant measurement of the two-point correlation function of image ellipticities on angular scales 910. On smaller angular scales, sources with intrinsic double-lobe structure cannot be separated from individual independent sources. The Square Kilometer Array (van Haarlem and van der Hulst, 1999) currently being discussed will yield such a tremendous increase in sensitivity for cm-wavelength radio astronomy that the radio sky will then be as crowded as the current optical sky. Finally, the recently commissioned Sloan telescope will map a quarter of the sky

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in "ve colours. Although the imaging survey will be much shallower than current weak-lensing imaging, the huge area surveyed can compensate for the reduced galaxy number density and their smaller mean redshift Stebbins et al. (1996). Indeed, "rst weak-lensing results were already reported at the Boston lensing conference (July 1999) from commissioning data of the telescope (see also Fischer et al., 1999).

7. QSO magni5cation bias and large-scale structure 7.1. Introduction Magni"cation by gravitational lenses is a purely geometrical phenomenon. The solid angle spanned by the source is enlarged, or equivalently, gravitational focusing directs a larger fraction of the energy radiated by the source to the observer. Sources that would have been too faint without magni"cation can therefore be seen in a #ux-limited sample. However, these sources are now distributed over a larger patch of the sky because the solid angle is stretched by the lens, so that the number density of the sources on the sky is reduced. The net e!ect on the number density depends on how many sources are added to the sample because they appear brighter. If the number density of sources increases steeply with decreasing #ux, many more sources appear due to a given magni"cation, and the simultaneous dilution can be compensated or outweighed. This magni"cation bias was described in Section 4.4.1 and quanti"ed in Eq. (4.38). As introduced there, let k(h) denote the magni"cation into direction h on the sky, and n ('S) the intrinsic counts  of sources with observed #ux exceeding S. In the limit of weak lensing, k(h)91, and the #ux will not change by a large factor, so that it is su$cient to know the behaviour of n ('S) in a small  neighbourhood of S. Without loss of generality, we can assume the number-count function to be a power law in that neighbourhood, n ('S)JS\?. We can safely ignore any redshift dependence  of the intrinsic source counts here because we aim at lensing e!ects of moderate-redshift mass distributions on high-redshift sources. Eq. (4.43) then applies, which relates the cumulative source counts n('S, h) observed in direction h to the intrinsic source counts n('S, h)"k?\(h)n ('S) . (7.1)  Hence, if a'1, the observed number density of objects is increased by lensing, and reduced if a(1. This e!ect is called magnixcation bias or magnixcation anti-bias (e.g. Schneider et al., 1992). The intrinsic number-count function of QSOs is well "t by a broken power law with a slope of a&0.64 for QSOs fainter than &19th blue magnitude, and a steeper slope of a&2.52 for brighter QSOs (Boyle et al., 1988; Hartwick and Schade, 1990; Pei, 1995). Faint QSOs are therefore anti-biased by lensing, and bright QSOs are biased. In the neighbourhood of gravitational lenses, the number density of bright QSOs is thus expected to be higher than average, in other words, more bright QSOs should be observed close to foreground lenses than expected without lensing. According to Eq. (7.1), the overdensity factor is n('S, h) "k?\(h) . q(h)" n ('S) 

(7.2)

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If the lenses are individual galaxies, the magni"cation k(h) drops rapidly with increasing distance from the lens. The natural scale for the angular separation is the Einstein radius, which is of order an arcsec for galaxies. Therefore, individual galaxies are expected to increase the number density of bright QSOs only in a region of radius a few arcsec around them. Fugmann (1990) reported an observation which apparently contradicts this expectation. He correlated bright, radio-loud QSOs at moderate and high redshifts with galaxies from the Lick catalogue (Seldner et al., 1977) and found that there is a signi"cant overdensity of galaxies around the QSOs of some of his sub-samples. This is intriguing because the Lick catalogue contains the counts of galaxies brighter than &19th magnitude in square-shaped cells with 10 side length. Galaxies of :19th magnitude are typically at much lower redshifts than the QSOs, z:0.1}0.2, so that the QSOs with redshifts z90.5}1 are in the distant background of the galaxies, with the two samples separated by hundreds of megaparsecs. Physical correlations between the QSOs and the galaxies are clearly ruled out. Can the observed overdensity be expected from gravitational lensing? By construction, the angular resolution of the Lick catalogue is of order 10, exceeding the Einstein radii of individual galaxies by more than two orders of magnitude. The result that Lick galaxies are correlated with bright QSOs can thus neither be explained by physical correlations nor by gravitational lensing due to individual galaxies. On the other hand, the angular scale of &10 is on the right order of magnitude for lensing by large-scale structures. The question therefore arises whether the magni"cation due to lensing by large-scale structures is su$cient to cause a magni"cation bias in #ux-limited QSO samples which is large enough to explain the observed QSO}galaxy correlation. The idea is that QSOs are then expected to appear more abundantly behind matter overdensities. More galaxies are expected where the matter density is higher than on average, and so the galaxies would act as tracers for the dark material responsible for the lensing magni"cation. This could then cause foreground galaxies to be overdense around background QSOs. This exciting possibility clearly deserves detailed investigation. Even earlier than Fugmann, Tyson (1986) had inferred that galaxies apparently underwent strong luminosity evolution from a detection of signi"cant galaxy overdensities on scales of 30 around 42 QSOs with redshifts 14z41.5, assuming that the excess galaxies were at the QSO redshifts. In the light of later observations and theoretical studies, he probably was the "rst to detect weak-lensing-induced associations of distant sources with foreground galaxies. 7.2. Expected magnixcation bias from cosmological density perturbations To estimate the magnitude of the e!ect, we now calculate the angular cross-correlation function m ( ) between background QSOs and foreground galaxies expected from weak lensing due to /% large-scale structures (Bartelmann, 1995b; Dolag and Bartelmann, 1997; Sanz et al., 1997). We employ a simple picture for the relation between the number density of galaxies and the density contrast of dark matter, the linear biasing scheme (e.g. Kaiser, 1984; Bardeen et al., 1986; White et al., 1987). Within this picture, and assuming weak lensing, we shall immediately see that the desired correlation function m is proportional to the cross-correlation function m between /% IB magni"cation k and density contrast d. The latter correlation can straightforwardly be computed with the techniques developed previously.

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7.2.1. QSO}galaxy correlation function The angular cross-correlation function m ( ) between galaxies and QSOs is de"ned by /% m

%/

1 ( )" 1[n (h)!1n 2][n (h# )!1n 2]2 , / / % % 1n 21n 2 / %

(7.3)

where 1n 2 are the mean number densities of QSOs and galaxies averaged over the whole sky. /% Assuming isotropy, m ( ) does not depend on the direction of the lag angle . All number /% densities depend on #ux (or galaxy magnitude), but we leave out the corresponding arguments for brevity. We saw in Eq. (7.1) in the introduction that n (h)"k?\(h)1n 2. Since the magni"cation / / expected from large-scale structures is small, k"1#dk with "dk";1, we can expand k?\+1#(a!1)dk. Hence, we can approximate n (h)!1n 2 / / +(a!1)dk(h) , 1n 2 /

(7.4)

so that the relative #uctuation of the QSO number density is proportional to the magni"cation #uctuation, and the factor of proportionality quanti"es the magni"cation bias. Again, for a"1, lensing has no e!ect on the number density. The linear biasing model for the #uctuations in the galaxy density asserts that the relative #uctuations in the galaxy number counts are proportional to the density contrast d n (h)!1n 2 % % "bdM (h) , 1n 2 %

(7.5)

where dM (h) is the line-of-sight integrated density contrast, weighted by the galaxy redshift distribution, i.e. the w-integral in Eq. (6.66). The proportionality factor b is the e!ective biasing factor appropriately averaged over the line-of-sight. Typical values for the biasing factor are assumed to be b91}2. Both the relative #uctuations in the galaxy number density and the density contrast are bounded by !1 from below, so that the right-hand side should be replaced by max[bdM (h),!1] in places where dM (h)(!b\. For simplicity we use (7.5), keeping this limitation in mind. Using Eqs. (7.4) and (7.5), the QSO}galaxy cross-correlation function (7.3) becomes m

/%

( )"(a!1)b1dk(h)dM (h# )2 .

(7.6)

Hence, it is proportional to the cross-correlation function m between magni"cation and IB density contrast, and the proportionality factor is given by the steepness of the intrinsic QSO number counts and the bias factor (Bartelmann, 1995b). As expected from the discussion of the magni"cation bias, the magni"cation bias is ine!ective for a"1, and QSOs and galaxies are anti-correlated for a(1. Furthermore, if the number density of galaxies does not re#ect the dark-matter #uctuations, b would vanish, and the correlation would disappear. In order to "nd the QSO}galaxy cross-correlation function, we therefore have to evaluate the angular crosscorrelation function between magni"cation and density contrast.

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7.2.2. Magnixcation-density correlation function We have seen in Section 6 that the magni"cation #uctuation is twice the e!ective convergence dk(h)"2i (h) in the limit of weak lensing, see Eq. (6.29). The latter is given by Eq. (6.19), in which  the average over the source-distance distribution has already been performed. Therefore, we can immediately write down the source-distance averaged magni"cation #uctuation as



3H X U& d[ f (w)h, w] ) . (7.7) dk(h)"   dw = M (w)f (w) / ) c a(w)  Here, = M (w) is the modi"ed QSO weight function / U& f (w!w) = M (w), (7.8) dw G (w) ) / / f (w) U ) and G (w) is the normalised QSO distance distribution. / Both the average density contrast dM and the average magni"cation #uctuation dk are weighted projections of the density #uctuations along the line-of-sight, which is assumed to be a homogeneous and isotropic random "eld. As in the derivation of the e!ective-convergence power spectrum in Section 6, we can once more employ Limber's equation in Fourier space to "nd the cross power spectrum P (l) for projected magni"cation and density contrast, IB 3H X U& = M (w)G (w) l % P P (l)"   dw / . (7.9) IB B f (w) c a(w)f (w)  ) ) The cross-correlation function between magni"cation and density contrast is obtained from Eq. (7.9) via Fourier transformation, which can be carried out and simpli"ed to yield












k dk 3H X U& dw f (w)= M (w)G (w)a\(w) m ( )"   P (k, w)J [f (w)k ] . (7.10) ) / % IB  ) 2p B c   Quite obviously, there is a strong similarity between this equation and that for the magni"cation auto-correlation function, Eq. (6.34). We note that Eq. (7.10) automatically accounts for galaxy auto-correlations through the matter power spectrum P (k). B We point out that the dependence of the QSO}galaxy correlation function scales like m JbX P (k ), where k is the comoving wave number determined by the peak of the redshift /%  B   distribution of the foreground galaxies and the angular separation considered. On the other hand, the auto-correlation function of the foreground galaxies behaves like m JbP (k ), which %% B  implies that the ratio m /m JX/b, the same dependence as already stressed earlier (Section 6.8). /% %% Again, this ratio is nearly independent of the normalisation of the power spectrum, and therefore a convenient measure of the ratio X/b (BenmH tez and Sanz, 1999). 7.2.3. Distance distributions and weight functions The QSO and galaxy weight functions G (w) are normalised representations of their respective /% redshift distributions, where the redshift needs to be transformed to comoving distance w. The redshift distribution of QSOs has frequently been measured and parameterised. Using the functional form and the parameters determined by Pei (1995), the modi"ed QSO weight function

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Fig. 26. QSO and galaxy weight functions, = M (w) and G (w), respectively. Top panel: = M (w) for "ve di!erent choices of / % / the lower cut-o! redshift z imposed on the QSO sample; z increases from 0.0 (solid curve) to 2.0 in steps of 0.5. The peak   in = M (w) shifts to larger distances for increasing z . Bottom panel: G (w) for "ve di!erent galaxy magnitude limits m , /  %  increasing from 18.5 to 22.5 (solid curve) in steps of one magnitude. The peak in the galaxy distance distribution shifts towards larger distances with increasing m , i.e. with decreasing brightness of the galaxy sample. 

= M (w) has the shape illustrated in the top panel of Fig. 26. It is necessary for our present purposes / to be able to impose a lower redshift limit on the QSO sample. Since we want to study lensing-induced correlations between background QSOs and foreground galaxies, there must be a way to exclude QSOs physically associated with galaxy overdensities. This is observationally achieved by choosing a lower QSO redshift cut-o! high enough to suppress any redshift overlap between the QSO and galaxy samples. This procedure must be reproduced in theoretical calculations of the QSO}galaxy cross-correlation function. This can be achieved by cutting o! the observed redshift distribution G below some redshift z , re-normalising it, and putting the result /  into Eq. (7.8) to "nd = M . The "ve curves shown in the top panel of Fig. 26 are for cut-o! redshifts / z increasing from 0.0 (solid curve) to 2.0 in steps of 0.5. Obviously, the peak in = M shifts to larger  / w for increasing z .  Galaxy redshift distributions G can be obtained by extrapolating local galaxy samples to higher % redshifts, adopting a constant comoving number density and a Schechter-type luminosity function. For the present purposes, this is a safe procedure because the galaxies to be correlated with the QSOs should be at su$ciently lower redshifts than the QSOs to avoid overlap between the samples. Thus, the extrapolation from the local galaxy population is well justi"ed. In order to convert galaxy luminosities to observed magnitudes, k-corrections need to be taken into account. Conveniently, the resulting weight functions should be parameterised by the brightness cut-o! of the galaxy sample, in practice by the maximum galaxy magnitude m (i.e. the minimum luminosity) required  for a galaxy to enter the sample. The "ve representative curves for G (w) in the lower panel of %

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Fig. 26 are for m increasing from 18.5 to 22.5 (solid curve) in steps of one magnitude. R-band  magnitudes are assumed. For increasing cut-o! magnitude m , i.e. for fainter galaxy samples, the  distributions broaden, as expected. The correlation amplitude as a function of m peaks if m is   chosen such that the median distance to the galaxies is roughly half the distance to the bulk of the QSO population considered. 7.2.4. Simplixcations It turns out in practice that the exact shapes of the QSO and galaxy weight functions = M (w) and / G (w) are of minor importance for the results. Allowing inaccuracies of order 10%, we can replace % the functions G (w) by delta distributions centred on typical QSO and galaxy distances w and /% / w (w . Then, from Eq. (7.8), % / f (w !w) H(w !w) , (7.11) = M (w)" ) / / / f (w ) ) / where H(x) is the Heaviside step function, and the line-of-sight integration in Eq. (7.7) becomes trivial. It is obvious that matter #uctuations at redshifts higher than the QSO redshift do not contribute to the cross-correlation function m ( ): Inserting (7.11) together with G "d(w!w ) IB % % into Eq. (7.10), we "nd m ( )"0 if w 'w , as it should be. IB % / The expression for the magni"cation-density cross-correlation function further simpli"es if we specialise to a model universe with zero spatial curvature, K"0, such that f (w)"w. ) Then,






w H(w !w) = M (w)" 1! / / w / and the cross-correlation function m ( ) reduces to IB k dk w 3H X w P (k, w )J (w k ) m ( )"   % 1! % %  % IB 2p B w c a(w ) % /  for w 'w , and m ( )"0 otherwise. / % IB






(7.12)

(7.13)

7.3. Theoretical expectations 7.3.1. Qualitative behaviour Before we evaluate the magni"cation-density cross-correlation function fully numerically, we can gain some insight into its expected behaviour by inserting the CDM and HDM model spectra de"ned in Eq. (6.37) into Eq. (7.10) and expanding the result into a power series in (Bartelmann, 1995b). As in the case of the magni"cation auto-correlation function before, the two model spectra produce qualitatively di!erent results. To "rst order in , m ( ) decreases linearly with increasing IB

for CDM, while it is #at for HDM. The reason for this di!erent appearance is the lack of small-scale power in HDM, and the abundance thereof in CDM. The two curves shown in Fig. 27 illustrate this for an Einstein}de Sitter universe with Hubble constant h"0.5. The underlying density-perturbation power spectra were normalised by the local abundance of rich clusters, and linear density evolution was assumed.

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Fig. 27. Cross-correlation functions between magni"cation and density contrast, m ( ), are shown for an Einstein}de IB Sitter universe with h"0.5, adopting CDM (solid curve) and HDM (dotted curve) density #uctuation spectra. Both spectra are normalised to the local cluster abundance, and linear density evolution is assumed. The lower cut-o! redshift of the QSOs is z "0.3, the galaxy magnitude limit is m "20.5. In agreement with the expectation derived from the   CDM and HDM model spectra (6.37), the CDM cross-correlation function decreases linearly with increasing for small

, while it is #at to "rst order in for HDM. The small-scale matter #uctuations in CDM compared to HDM cause m ( ) to increase more steeply as P0. IB Fig. 28. Angular magni"cation-density cross-correlation functions m ( ) are shown for the four cosmological models IB speci"ed in Table 1. Two curves are shown for each cosmological model; those with the higher (lower) amplitude at "0 were calculated with the non-linearly (linearly) evolving density-perturbation power spectra, respectively. The models are: SCDM (solid curves), pCDM (dotted curves), OCDM (short-dashed curves), and KCDM (long-dashed curves). Obviously, non-linear evolution has a substantial e!ect. It increases the correlation amplitude by about an order of magnitude. The Einstein}de Sitter model normalised to p "1 has a signi"cantly larger cross-correlation amplitude than  the cluster-normalised Einstein}de Sitter model. For the low-density models, the di!erence is much smaller. The curves for the cluster-normalised models are very similar, quite independent of cosmological parameters.

The linear correlation amplitude, m (0), for CDM is of order 3;10\, and about a factor of "ve IB smaller for HDM. The magni"cation-density cross-correlation function for CDM drops to half its peak value within a few times 10 arcmin. This, and the monotonic increase of m towards small , IB indicate that density perturbations on angular scales below 10 contribute predominantly to m . At IB typical lens redshifts, such angular scales correspond to physical scales up to a few Mpc. Evidently therefore, the non-linear evolution of the density perturbations needs to be taken into account, and its e!ect is expected to be substantial. 7.3.2. Results Fig. 28 con"rms this expectation; it shows magni"cation-density cross-correlation functions for the four cosmological models detailed in Table 1. Two curves are shown for each model, one for

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linear and the other for non-linear density evolution. The two curves of each pair are easily distinguished because non-linear evolution increases the cross-correlation amplitude at small by about an order of magnitude above linear evolution, quite independent of the cosmological model. At the same time, the angular cross-correlation scale is reduced to a few arcmin. At angular scales :30, the non-linear cross-correlation functions are above the linear results, falling below at larger scales. The correlation functions for the three cluster-normalised models (SCDM, OCDM and KCDM; see Table 1) are very similar in shape and amplitude. The curve for the pCDM model lies above the other curves by a factor of about "ve, but for low-density universes, the in#uence of di!erent power-spectrum normalisations are much less prominent. The main results to be extracted from Fig. 28 are that the amplitude of the magni"cation-density cross-correlation function, m (0), reaches approximately 5;10\, and that m drops by an order IB IB of magnitude within about 20. This behaviour is quite independent of the cosmological parameters if the density-#uctuation power spectrum is normalised by the local abundance of rich galaxy clusters. More detailed results can be found in Dolag and Bartelmann (1997) and Sanz et al. (1997). 7.3.3. Signal-to-noise estimate The QSO}galaxy correlation function m ( ) is larger than m ( ) by the factor (a!1)b. The /% IB value of the bias factor b is yet unclear, but it appears reasonable to assume that it is between 1 and 2. For optically selected QSOs, a+2.5, so that (a!1)b+2}3. Combining this with the correlation amplitude for CDM read o! from Fig. 28, we can expect m (0):0.1. /% Given the meaning of m ( ), the probability to "nd a foreground galaxy close to a background /% QSO is increased by a factor of [1#m ( )]:1.1 above random. In a small solid angle du /% around a randomly selected background QSO, we thus expect to "nd N +[1#m (0)]1n 2 du,[1#m (0)]1N 2 (7.14) % /% % /% % galaxies, where 1N 2 is the average number of galaxies within a solid angle of du. In a sample of % N "elds around randomly selected QSOs, the signal-to-noise ratio for the detection of a galaxy / overdensity is then N (N !1N 2) S % "(N 1N 2)m (0) . + / % (7.15) / % /% (N 1N 2) N / % Typical surface number densities of reasonably bright galaxies are of order n &10 per square % arcmin. Therefore, there should be of order 1N 2&30 galaxies within a randomly selected disk of % 1 arcmin radius, in which the QSO}galaxy cross correlation is su$ciently strong. If we require a certain minimum signal-to-noise ratio such that S/N5(S/N) , the number of QSO "elds to be  observed in order to meet this criterion is



 

S  N 5 m\(0)1N 2\ / /% % N  S  [(a!1)b]\m\(0)1N 2\ " IB % N  (S/N)  (a!1)b \ m (0) \ 30  IB , "20 5 4 0.05 1N 2 %









(7.16)

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where we have inserted typical numbers in the last step. This estimate demonstrates that gravitational lensing by non-linearly evolving large-scale structures in cluster-normalised CDM can produce correlations between background QSOs and foreground galaxies at the 5p level on arcmin scales in samples of 920 QSOs. The angular scale of the correlations is expected to be of order 1}10 arcmin. Eq. (7.16) makes it explicit that more QSO "elds need to be observed in order to establish the signi"cance of the QSO}galaxy correlations if (i) the QSO number count function is shallow (a close to unity), and (ii) the galaxy bias factor b is small. In particular, no correlations are expected if a"1, because then the dilution of the sources and the increase in QSO number exactly cancel. Numerical simulations (Bartelmann, 1995b) con"rm estimate (7.16). The Fugmann (1990) observation was also tested in a numerical model universe based on the adhesion approximation to structure formation (Bartelmann and Schneider, 1992). This model universe was populated with QSOs and galaxies, and QSO}galaxy correlations on angular scales on the order of &10 were investigated using Spearman's rank-order correlation test (Bartelmann and Schneider, 1993a). Light propagation in the model universe was described with the multiple lens-plane approximation of gravitational lensing. In agreement with the analytical estimate presented above, it was found that lensing by large-scale structures can indeed account for the observed correlations between high-redshift QSOs and low-redshift galaxies, provided the QSO number-count function is steep. Lensing by individual galaxies was con"rmed to be entirely negligible. 7.3.4. Multiple-waveband magnixcation bias The magni"cation bias quanti"ed by the number-count slope a can be substantially increased if QSOs are selected in two or more mutually uncorrelated wavebands rather than one (Borgeest et al., 1991). To see why, suppose that optically bright and radio-loud QSOs were selected, and that their #uxes in the two wavebands are uncorrelated. Let S be the #ux thresholds in the optical  and in the radio regimes, respectively, and n the corresponding number densities of either  optically bright or radio-loud QSOs on the sky. As in the introduction, we assume that n can be  written as power laws in S , with exponents a .   In a small solid angle du, the probability to "nd an optically bright or radio-loud QSO is then p (S )"n (S ) du, and the joint probability to "nd an optically bright and radio-loud QSO is the G G G G product of the individual probabilities, or p(S , S )"p (S )p (S )"[n (S )n (S )] duJS\? S\? du ,            

(7.17)

provided there is no correlation between the #uxes S so that the two probabilities are indepen dent. Suppose now that lensing produces a magni"cation factor k across du. The joint probability is then changed to





k ? k ? du p(S , S )J "k? >? \p(S , S ) .     S S k  

(7.18)

Therefore, the magni"cation bias in the optically bright and radio-loud QSO sample is as e$cient as if the number-count function had a slope of a"a #a .  

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More generally, the e!ective number-count slope for the magni"cation bias in a QSO sample that is #ux limited in m mutually uncorrelated wave bands is K a" a , (7.19) G G where a are the number-count slopes in the individual wavebands. Then, the QSO}galaxy G cross-correlation function is






K (7.20) ( )" a !1 bm ( ) G IB G and can therefore be noticeably larger than for a QSO sample which is #ux limited in one waveband only. m

/%

7.4. Observational results After this theoretical investigation, we turn to observations of QSO}galaxy cross-correlations on large angular scales. The existence of QSO}galaxy correlations was tested and veri"ed in several studies using some very di!erent QSO- and galaxy samples. Bartelmann and Schneider (1993b) repeated Fugmann's analysis with a well-de"ned sample of background QSOs, namely the optically identi"ed QSOs from the 1-Jansky catalogue (KuK hr et al., 1981; Stickel et al., 1993; Stickel and KuK hr, 1993). Optically identi"ed QSOs with measured redshifts need to be bright enough for detection and spectroscopy, hence the chosen sample is implicitly also constrained by an optical #ux limit. Optical and radio QSO #uxes are generally not strongly correlated, so that the sample is a!ected by a double-waveband magni"cation bias, which can further be strengthened by explicitly imposing an optical #ux (or magnitude) limit. Although detailed results di!er from Fugmann's, the presence of the correlation is con"rmed at the 98% con"dence level for QSOs with redshifts 50.75 and brighter than 18th magnitude. The number of QSOs matching these criteria is 56. The correlation signi"cance decreases both for lower- and higher-redshift QSO samples, and also for optically fainter ones. This is in accordance with an explanation in terms of a (double-waveband) magni"cation bias due to gravitational lensing. For low-redshift QSOs, lensing is not e$cient enough to produce the correlations. For high-redshift QSOs, the most e$cient lenses are at higher redshifts than the galaxies, so that the observed galaxies are uncorrelated with the structures which magnify the QSOs. Hence, the correlation is expected to disappear for increasing QSO redshifts. For an optically unconstrained QSO sample, the e!ective slope of the number-count function is smaller, reducing the strength of the magni"cation bias and therefore also the signi"cance of the correlation. With a similar correlation technique, correlations between the 1-Jansky QSO sample and IRAS galaxies (Bartelmann and Schneider, 1994) and di!use X-ray emission (Bartelmann et al., 1994; see also Cooray, 1999b) were investigated, leading to qualitatively similar results. IRAS galaxies are correlated with optically bright, high-redshift z51.5 1-Jansky sources at the 99.8% con"dence level. The higher QSO redshift for which the correlation becomes signi"cant can be understood if the IRAS galaxy sample is deeper than the Lick galaxy sample, so that the structures responsible for the lensing can be traced to higher redshift.

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Bartsch et al. (1997) re-analysed the correlation between IRAS galaxies and 1-Jansky QSOs using a more advanced statistical technique which can be optimised to the correlation function expected from lensing by large-scale structures. In agreement with Bartelmann and Schneider (1994), they found signi"cant correlations between the QSOs and the IRAS galaxies on angular scales of &5, but the correlation amplitude is higher than expected from large-scale structure lensing, assuming linear evolution of the density-perturbation power spectrum. Including nonlinear evolution, however, the results by Bartsch et al. (1997) can well be reproduced (Dolag and Bartelmann, 1997). X-ray photons from the ROSAT All-Sky Survey (e.g. Voges, 1992) are correlated with optically bright 1-Jansky sources both at low (0.54z41.0) and at high redshifts (1.54z42.0), but there is no signi"cant correlation with QSOs in the intermediate redshift regime. A plausible explanation for this is that the correlation of X-ray photons with low-redshift 1-Jansky QSOs is due to hot gas which is physically associated with the QSOs, e.g. which resides in the host clusters of these QSOs. Increasing the source redshift, the #ux from these clusters falls below the detection threshold of the All-Sky Survey, hence the correlation disappears. Upon further increasing the QSO redshift, lensing by large-scale structures becomes e$cient, and the X-ray photons trace hot gas in the lenses. Rodrigues-Williams and Hogan (1994) found a highly signi"cant correlation between optically selected, high-redshift QSOs and Zwicky clusters. Their cluster sample was fairly bright, which indicates that the clusters are in the foreground of the QSOs. This rules out that the clusters are physically associated with the QSOs and thus exert environmental e!ects on them which might lead to the observed association. Rodrigues-Williams and Hogan discussed lensing as the most probable reason for the correlations, although simple mass models for the clusters yield lower magni"cations than required to explain the signi"cance of the e!ect. Seitz and Schneider (1995b) repeated their analysis with the 1-Jansky sample of QSOs. They found agreement with RodriguesWilliams and Hogan's result for intermediate-redshift (z&1) QSOs, but failed to detect signi"cant correlations for higher-redshift sources. In addition, a signi"cant under-density of low-redshift QSOs close to Zwicky clusters was found, for which environmental e!ects like dust absorption are the most likely explanation. A variability-selected QSO sample was correlated with Zwicky clusters by Rodrigues-Williams and Hawkins (1995). They detected a signi"cant correlation between QSOs with 0.44z42.2 with foreground Zwicky clusters (with 1z2&0.15) and interpreted it in terms of gravitational lensing. Again, the implied average QSO magni"cation is substantially larger than that inferred from simple lens models for clusters with velocity dispersions of &10 km s\. Wu and Han (1995) searched for associations between distant 1- and 2-Jansky QSOs and foreground Abell clusters. They found no correlations with the 1-Jansky sources, and a marginally signi"cant correlation with 2-Jansky sources. They argue that lensing by individual clusters is insu$cient if cluster velocity dispersions are of order 10 km s\, and that lensing by large-scale structures provides a viable explanation. BenmH tez and MartmH nez-GonzaH lez (1995) found an excess of red galaxies from the APM catalog with moderate-redshift (z&1) 1-Jansky QSOs on angular scales (5 at the 99.1% signi"cance level. Their colour selection ensures that the galaxies are most likely at redshifts 0.24z40.4, well in the foreground of the QSOs. The amplitude and angular scale of the excess is compatible with its originating from lensing by large-scale structures. The measurements by BenmH tez and MartmH nezGonzaH lez (1995) are plotted together with various theoretical QSO}galaxy cross-correlation

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Fig. 29. QSO}galaxy cross-correlation measurements are plotted together with theoretical cross-correlation functions m ( ) for various cosmological models as indicated by line type. The CDM density-perturbation power spectrum was /% cluster-normalised, and non-linear evolution was taken into account. The "gure shows that the measurements fall above the theoretical predictions at small angular scales, :2. This excess can be attributed to gravitational lensing by individual galaxy clusters (see the text for more detail). The theoretical curves depend on the Hubble constant h through the shape parameter C"X h, which determines the peak location of the power spectrum. 

functions in Fig. 29, which clearly shows that the QSO}galaxy cross-correlation measurements agree quite well with the cross-correlation functions m ( ), but they fall above the range of /% theoretical predictions at small angular scales, :2. This can be attributed to the magni"cation bias due to gravitational lensing by individual clusters. Being based on the weak-lensing approximation, our approach breaks down when the magni"cation becomes comparable to unity, k91.5, say. This amount of magni"cation occurs for QSOs closer than &3 Einstein radii to cluster cores. Depending on cosmological parameters, QSO and galaxy redshifts, &3 Einstein radii correspond to &1}2. Hence, we expect the theoretical expectations from lensing by large-scale structures alone to fall below the observations on angular scales :1}2. Norman and Impey (1999) took wide-"eld R-band images centred on a subsample of 1-Jansky QSOs with redshifts between 1 and 2. They searched for an excess of galaxies in the magnitude range 19.5(R(21 on angular scales of 910 around these QSOs and found a correlation at the 99% signi"cance level. The redshift distribution of the galaxies is likely to peak around z&0.2. The angular cross-correlation function between the QSOs and the galaxies agrees well with the theoretical expectations, although the error bars are fairly large. All these results indicate that there are correlations between background QSOs and foreground &light', with light either in the optical, the infrared, or the (soft) X-ray wavebands. The angular scale of the correlations is compatible with that expected from lensing by large-scale structures, and the amplitude is either consistent with that explanation or somewhat larger. Wu and Fang (1996)

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discussed whether the auto-correlation of clusters modelled as singular isothermal spheres can produce su$cient magni"cation to explain this result. They found that this is not the case, and argued that large-scale structures must contribute substantially. If lensing is indeed responsible for the correlations detected, other signatures of lensing should be found in the vicinity of distant QSOs. Indeed, Fort et al. (1996) searched for the shear induced by weak lensing in the "elds of "ve luminous QSOs with z+1 and found coherent shear signals in four of them (see also Schneider et al., 1998b). In addition, they detected galaxy groups in three of their "elds. Earlier, Bonnet et al. (1993) had found evidence for coherent weak shear in the "eld of the potentially multiply-imaged QSO 2345#007, which was later identi"ed with a distant cluster (Mellier et al., 1994; Fischer et al., 1994). Bower and Smail (1997) searched for weak-lensing signals in "elds around eight luminous radio sources at redshifts &1. They con"rmed the coherent shear detected earlier by Fort et al. (1996) around one of the sources (3C336 at z"0.927), but failed to "nd signatures of weak lensing in the combined remaining seven "elds. A cautionary note was recently added to this discussion by Williams and Irwin (1998) and Norman and Williams (1999). Cross-correlating LBQS and 1-Jansky quasars with APM galaxies, they claimed signi"cant galaxy overdensities around QSOs on angular scales of order one degree. As discussed above, lensing by currently favoured models of large-scale structures is not able to explain such large correlation scales. Thus, if these results hold up, they would provide evidence that there is a fundamental di$culty with the current models of large-scale structure formation. 7.5. Magnixcation bias of galaxies The investigation of the angular correlation between QSOs and foreground galaxies was motivated by observational evidence of this e!ect, as described in the previous subsection. However, the magni"cation bias generates a similar correlation function between foreground galaxies and di!erent classes of background sources, provided the latter have a slope of the cumulative sources counts di!erent from unity. QSOs are particularly convenient due to their steep number counts and their high redshift. Moessner et al. (1998) and Moessner and Jain (1998) studied the angular correlation between two di!erent populations of galaxies. If, for example, the two populations of galaxies were selected by their apparent magnitude, the fainter one will on average be more distant than the brighter one; therefore, matter traced by the brighter galaxies magni"es the fainter population of galaxies. Unfortunately, owing to the broad redshift distribution of galaxies at "xed apparent magnitude, there will be a signi"cant overlap in redshift between these two populations. Since galaxies are auto-correlated, this intrinsic clustering contribution is likely to swamp any lensing-induced correlation. Note that, owing to the high-redshift cut used for the QSO samples considered in the previous subsection, this intrinsic correlation is of little or no importance there. However, if the foreground and background populations can be better separated, the lensing e!ect may be stronger than the intrinsic correlation. For example, by using photometric redshift estimates, the two galaxy populations may be nicely separated in their redshift distribution. In that case, the cross-correlation function will take the form m ( )"(a !1)b m ( )#m( ) ,    IB

(7.21)

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where the "rst term is the contribution due to the magni"cation and has the same form as that derived for the QSO}galaxy correlation in the the previous sub-section, and m is the intrinsic cross-correlation function coming from imperfect redshift separation of the two galaxy populations. Note that a is the number-count slope of the background galaxies, and b the bias factor of   the foreground population. If m and m have di!erent functional forms with respect to , these two IB contributions to the cross-correlation function may be separable. From early commissioning data of the Sloan Digital Sky Survey, covering 100 square degrees in "ve passbands, Jain et al. (1999) attempted to detect this magni"cation bias-induced crosscorrelation between two galaxy populations. From their photometric redshift estimates for the galaxies, they de"ne the foreground and background galaxy samples by 04z 40.15 and  0.354z 40.45, together with a magnitude cut at r420.5. The large gap between the two redshift  ranges accounts for the fact that photometric redshifts have an uncertainty of slightly less that *z"0.1, so that this conservative cut should minimise the overlap between the two populations. At the magnitude cut, the so-de"ned background sample exhibits an e!ective slope of a&0.5, so that lensing should produce an anti-correlation. In fact, Jain et al. (1999) found that m is negative  for 91, but slightly positive for smaller angular separations. Note that this behaviour is expected from Eq. (7.21), since the positive correlation at small angles is due to the prevalence of the intrinsic cross-correlation owing to the redshift overlap of the two samples. In order to strengthen their interpretation of this result, Jain et al. (1999) split their background sample into a red and a blue half. The number-count slope of these two sub-samples of background galaxies at the magnitude cut is a&0 and 1, respectively. Correspondingly, they "nd that m calculated with the blue  subsample shows no sign of an anti-correlation at any angular separation, whereas the red subsample shows a stronger anti-correlation than for the total sample of background galaxies. Hence, it seems that the magni"cation bias of galaxies has been measured; given that the data on which this result is based constitutes only &1% of the total imaging data the Sloan Survey will accumulate, it is clear that the correlation function m will be measurable with high precision out  to large angular separations, providing a very convenient handle on X/b, and the scale dependence of b at redshifts z&0.1. 7.6. Outlook Cross-correlations between distant QSOs and foreground galaxies on angular scales of about 10 arcmin have been observed, and they can be attributed to the magni"cation bias due to gravitational lensing by large-scale structures. Coherent shear patterns have been detected around QSOs which are signi"cantly correlated with galaxies. The observations so far are in reasonable agreement with theoretical expectations, except for the higher observed signal in the innermost few arcmin, and the claimed correlation signal on degree scales. While the excess cross-correlation on small scales can be understood by the lensing e!ects of individual galaxy clusters, correlations on degree scales pose a severe problem for the lensing explanation if they persist, because the lensing-induced cross-correlation quickly dies o! beyond scales of approximately 10. QSO}galaxy cross-correlations have the substantial advantage over other diagnostics of weak lensing by large-scale structures that they do not pose any severe observational problems. In particular, it is not necessary to measure either shapes or sizes of faint background galaxies accurately, because it is su$cient to detect and count comparatively bright foreground galaxies

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near QSOs. However, such counting requires homogeneous photometry, which is di$cult to achieve in particular on photographic plates, and requires careful calibration. Since the QSO}galaxy cross-correlation function involves "ltering the density-perturbation power spectrum with a fairly broad function, the zeroth-order Bessel function J (x) [cf. Eq. (7.10)],  these correlations are not well suited for constraining the power spectrum. If the cluster normalisation is close to the correct one, the QSO}galaxy cross-correlation function is also fairly insensitive to cosmological parameters. Rather, QSO}galaxy cross correlations are primarily important for measuring the bias parameter b. The rationale of future observations of QSO}galaxy correlations should therefore be to accurately measure the correlation amplitude on scales between a few and 10 arcmin. On smaller scales, the in#uence of individual galaxy clusters sets in, and on larger scales, the correlation signal is expected to be weak. Once it becomes possible to reliably constrain the density-#uctuation power spectrum, such observations can then be used to quantify the bias parameter, and thereby provide most valuable information for theories of galaxy formation. A possible dependence of the bias parameter on scale and redshift can also be extracted. Su$ciently large data "elds for this purpose will soon become available, in particular through wide-"eld surveys like the 2dF Survey (Colless, 1998) and the Sloan Digital Sky Survey (Gunn and Knapp, 1993; Loveday and Pier, 1998). It therefore appears feasible that within a few years weak lensing by large-scale structures will be able to quantify the relation between the distributions of galaxies and the dark matter.

8. Galaxy}galaxy lensing 8.1. Introduction Whereas the weak lensing techniques described in Section 5 are adequate to map the projected matter distribution of galaxy clusters, individual galaxies are not su$ciently massive to show up in the distortion of the images of background galaxies. From the signal-to-noise ratio (4.55) we see that individual isothermal halos with a velocity dispersion in excess of &600 km s\ can be detected at a high signi"cance level with the currently achievable number densities of faint galaxy images. Galaxies have halos of much lower velocity dispersion: The velocity dispersion of an ¸ elliptical galaxy is &220 km s\, that of an ¸ spiral &145 km s\. H H However, if one is not interested in the mass properties of individual galaxies, but instead in the statistical properties of massive halos of a population of galaxies, the weak lensing e!ects of several such galaxies can statistically be superposed. For example, if one considers N identical foreground  galaxies, the signal-to-noise ratio of the combined weak lensing e!ect increases as N, so that for  a typical velocity dispersion for spiral galaxies of p &160 km s\, a few hundred foreground T galaxies are su$cient to detect the distortion they induce on the background galaxy images. Of course, detection alone does not yield new insight into the mass properties of galaxy halos. A quantitative analysis of the lensing signal must account for the fact that &identical' foreground galaxies cannot be observed. Therefore, the mass properties of galaxies have to be parameterised in order to allow the joint analysis of the foreground galaxy population. In particular, one is interested in the velocity dispersion of a typical (¸ , say) galaxy. Furthermore, the rotation curves H

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of (spiral) galaxies which have been observed out to &30h\ kpc show no hint of a truncation of the dark halo out to this distance. Owing to the lack of dynamical tracers, with the exception of satellite galaxies (Zaritsky and White, 1994), a direct observation of the extent of the dark halo towards large radii is not feasible with conventional methods. The method described in this section uses the light bundles of background galaxies as dynamical tracers, which are available at all distances from the galaxies' centres, and are therefore able, at least in principle, to probe the size (or the truncation radius) of the halos. Methods for a quantitative analysis of galaxy halos will be described in Section 8.2. The "rst attempt at detecting this galaxy}galaxy lensing e!ect was reported by Tyson et al. (1984), but the use of photographic plates and the relatively poor seeing prevented them from observing a galaxy}galaxy lensing signal. The "rst detection was reported by Brainerd et al. (1996), and as will be described in Section 8.3, several further observational results have been derived. Gravitational light de#ection can also be used to study the dark matter halos of galaxies in clusters. The potential in#uence of the environment on the halo properties of galaxies can provide a strong hint on the formation and lifetimes of clusters. One might expect that galaxy halos are tidally stripped in clusters and therefore physically smaller than those of "eld galaxies. In Section 8.4, we consider galaxy}galaxy lensing in clusters, and report on some "rst results. Intermediate in mass between clusters and galaxies are groups of galaxies. With a characteristic velocity dispersion of &300 km s\, they are also not massive enough to be detected individually with weak lensing techniques. For them, the foregoing remarks also apply: as galaxies, groups can be statistically superposed to investigate the statistical properties of their mass pro"le. Hoekstra et al. (1999) describe a "rst application of this technique, "nding a highly signi"cant shear signal in a sample of 59 groups detected by spectroscopic methods, which yields an average velocity dispersion of &320 km s\ and a mass-to-light ratio of &250h\. 8.2. The theory of galaxy}galaxy lensing A light bundle from a distant galaxy is a!ected by the tidal "eld of many foreground galaxies. Therefore, in order to describe the image distortion, the whole population of foreground galaxies has to be taken into account. But "rst we shall consider the simple case that the image shape is a!ected (mainly) by a single foreground galaxy. Throughout this section we assume that the shear is weak, so that we can replace (4.12) by e"e!c .

(8.1)

Consider an axi-symmetric mass distribution for the foreground galaxy, and background images at separation h from its centre. The expectation value of the image ellipticity then is the shear at h, which is oriented tangentially. If p(e) and p(e) denote the probability distributions of the image and source ellipticities, then according to (8.1), p(e)"p(e!c)"p(e)!c

R p(e) , ? Re ?

(8.2)

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where the second equality applies for "c";1. If u is the angle between the major axis of the image ellipse and the line connecting source and lens centre, one "nds the probability distribution of u by integrating (8.2) over the modulus of e,





1 1 d"e" p(e) , p(u)" d"e" "e" p(e)" !c cos(2u)  2p 2p

(8.3)

where u ranges within [0, 2p]. Owing to the symmetry of the problem, we can restrict u to within 0 and p/2, so that the probability distribution becomes



 



1 2 cos(2u) , p(u)" 1!c  e p

(8.4)

i.e., the probability distribution is skewed towards values larger than p/4, showing preferentially a tangential alignment. Lensing by additional foreground galaxies close to the line-of-sight to the background galaxy does not substantially change the probability distribution (8.4). First of all, since we assume weak lensing throughout, the e!ective shear acting on a light bundle can well be approximated by the sum of the shear contributions from the individual foreground galaxies. This follows either from the linearity of the propagation equation in the mass distribution, or from the lowest-order approximation of multiple-de#ection gravitational lensing (e.g., Blandford and Narayan, 1986; Seitz and Schneider, 1992). Second, the additional lensing galaxies are placed at random angles around the line-of-sight, so that the expectation value of their combined shear averages to zero. Whereas they slightly increase the dispersion of the observed image ellipticities, this increase is negligible since the dispersion of the intrinsic ellipticity distribution is by far the dominant e!ect. However, if the lens galaxy under consideration is part of a galaxy concentration, such as a cluster, the surrounding galaxies are not isotropically distributed, and the foregoing argument is invalid. We shall consider galaxy}galaxy lensing in clusters in Section 8.4, and assume here that the galaxies are generally isolated. For an ensemble of foreground}background pairs of galaxies, the probability distribution for the angle u simply reads



 



1 2 cos(2u) , p(u)" 1!1c 2  e p

(8.5)

where 1c 2 is the mean tangential shear of all pairs considered. The function p(u) is an observable.  A signi"cant deviation from a uniform distribution signals the presence of galaxy}galaxy lensing. To obtain quantitative information on the galaxy halos from the amplitude of the cosine term, one needs to know 11/e2. It can directly be derived from observations because the weak shear assumed here does not signi"cantly change this average between source and image ellipticities, from a parameterised relation between observable galaxy properties, and from the mean shear 1c 2. Although, in principle, "ne binning in galaxy properties (like colour, redshift, luminosity,  morphology) and angular separation of foreground}background pairs is possible in order to probe the shear as a function of angular distance from a well-de"ned set of foreground galaxies and thus to obtain its radial mass pro"le without any parameterisation, this approach is currently unfeasible owing to the relatively small "elds across which observations of su$cient image quality are available.

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A convenient parameterisation of the mass pro"le is the truncated isothermal sphere with surface mass density






m p , R(m)" T 1! 2Gm (s#m

(8.6)

where s is the truncation radius. This is a special case of the mass distribution (3.20). Brainerd et al. (1996) showed that this mass pro"le corresponds to a physically realisable dark-matter particle distribution. The velocity dispersion is assumed to scale with luminosity according to (2.68), which is supported by observations. A similar scaling of s with luminosity ¸ or velocity dispersion p is also assumed, T p  ¸ ? T "s , (8.7) s"s H ¸ H p TH H where the choice of the exponent is largely arbitrary. The scaling in (8.7) is such that the ratio of truncation radius and Einstein radius at "xed redshift is independent of ¸. If, in addition, a"4, the total mass-to-light ratio is identical for all galaxies. The "ducial luminosity ¸ may depend on H redshift. For instance, if the galaxies evolve passively, their mass properties are una!ected, but aging of the stellar population cause them to become fainter with decreasing redshift. This e!ect may be important for very deep observations, such as the Hubble Deep Field (Hudson et al., 1998), in which the distribution of lens galaxies extends to high redshifts. The luminosity ¸ of a lens galaxy can be inferred from the observed #ux and an assumed redshift. Since the scaling relation (2.68) applies to the luminosity measured in a particular waveband, the calculation of the luminosity from the apparent magnitude in a speci"ed "lter needs to account for the k-correction. If data are available in a single waveband only, an approximate average k-correction relation has to be chosen. For multi-colour data, the k-correction can be estimated for individual galaxies more reliably. In any case, one assumes a relation between luminosity, apparent magnitude, and redshift,




¸"¸(m, z) .




(8.8)

The "nal aspect to be discussed here is the redshift of the galaxies. Given that a galaxy}galaxy analysis involves at least several hundred foreground galaxies, and even more background galaxies, one cannot expect that all of them have spectroscopically determined redshifts. In a more favourable situation, multi-colour data are given, from which a redshift estimate can be obtained, using the photometric redshift method (e.g., Connolly et al., 1995; Gwyn and Hartwick, 1996; Hogg et al., 1998). These redshift estimates are characteristically accurate to *z&0.1, depending on the photometric accuracy and the number of "lter bands in which photometric data are measured. For a single waveband only, one can still obtain a redshift estimate, but a quite unprecise one. One then has to use the redshift distribution of galaxies at that particular magnitude, obtained from

 It is physically realisable in the sense that there exists an isotropic, non-negative particle distribution function which gives rise to a spherical density distribution corresponding to (8.6).

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spectroscopic or multi-colour redshift surveys in other "elds. Hence, one assumes that the redshift probability distribution p (z; m) as a function of magnitudes is known su$ciently accurately. X Suppose for a moment that all galaxy redshifts were known. Then, one can predict the e!ective shear for each galaxy, caused by all the other galaxies around it, c " c (h !h , z , z , m ) , (8.9) G GH G H G H H H where c is the shear produced by the jth galaxy on the ith galaxy image, which depends on the GH angular separation and the mass properties of the jth galaxy. From its magnitude and redshift, the luminosity can be inferred from (8.8), which "xes p and the halo size s through the scaling relations T (2.68) and (8.7). Of course, for z 4z , c "0. Although the sum in (8.9) should, in principle, extend G H GH over the whole sky, the lensing e!ect of all foreground galaxies with angular separation larger than some h will average to zero. Therefore, the sum can be restricted to separations 4h . We shall



 discuss the value of h further below.

 In the realistic case of unknown redshifts, but known probability distribution p (z; m), the shear X c cannot be determined. However, by averaging (8.9) over p (z; m), the mean and dispersion, 1c 2 G X G and p , of the shear for the ith galaxy can be calculated. Instead of performing the highAG dimensional integration explicitly, this averaging can conveniently be done by a Monte-Carlo integration. One can generate multiple realisations of the redshift distribution by randomly drawing redshifts from the probability density p (z; m). For each realisation, the c can be calculated X G from (8.9). By averaging over the realisations, the mean 1c 2 and dispersion p of c can be G AG G estimated. 8.3. Results The "rst attempt at detecting galaxy}galaxy lensing was made by Tyson et al. (1984). They analysed a deep photographic survey consisting of 35 prime-focus plates with the 4-m Mayall Telescope at Kitt Peak. An area of 36 arcmin on each plate was digitised. After object detection, &12,000 &foreground' and &47,000 &background' galaxies were selected by their magnitudes, such that the faintest object in the &foreground' class was one magnitude brighter than the brightest &background' galaxy. This approach assumes that the apparent magnitude of an object provides a good indication for its redshift, which seems to be valid, although the redshift distributions of &foreground' and &background' galaxies will substantially overlap. There were &28,000 foreground-background pairs with *h463 in their sample, but no signi"cant tangential alignment could be measured. By comparing their observational results with Monte-Carlo simulations, Tyson et al. concluded that the characteristic velocity dispersion of a foreground galaxy in their sample must be smaller than about 120 km s\. This limit was later revised upwards to &230 km s\ by Kovner and Milgrom (1987) who noted that the assumption made in the Tyson et al. analysis that all background galaxies are at in"nite distance (i.e., D /D "1) was critical. This   upper limit is fully compatible with our knowledge of galaxy masses. This null-detection of galaxy}galaxy lensing in a very large sample of objects apparently discouraged other attempts for about a decade. After the "rst weak-lensing results on clusters became available, it was obvious that this method requires deep data with superb image quality. In particular, the non-linearity of photographic plates and mediocre seeing conditions are probably

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Fig. 30. The probability distribution p(u) for the 3202 foreground-background pairs (204r423 and 234r424, respectively) with 54*h434 in the sample used by Brainerd et al. (1996), together with the best "t according to (8.5). The observed distribution is incompatible with a #at distribution (dotted line) at a high con"dence level of 99.9% (from Brainerd et al.).

fatal to the detection of this e!ect, owing to its smallness. The shear at 5 from an ¸ galaxy with H p "160 km s\ is less than 5%, and pairs with smaller separations are very di$cult to investigate T as the bright galaxy will a!ect the ellipticity measurement of its close neighbour on ground-based images. Using a single 9.6;9.6 blank "eld, with a total exposure time of nearly seven hours on the 5-m Hale Telescope on Mount Palomar, Brainerd et al. (1996) reported the "rst detection of galaxy} galaxy lensing. Their co-added image had a seeing of 0.87 at FWHM, and the 97% completeness limit was r"26. They considered &foreground' galaxies in the magnitude range 204r423, and several fainter bins for de"ning the &background' population, and investigated the distribution function p(u) for pairs with separation 54*h434. The most signi"cant deviation of p(u) from a #at distribution occurs for &background' galaxies in the range 234r424. For fainter (and thus smaller) galaxies, the accuracy of the shape determination deteriorates, as Brainered et al. explicitly show. The number of &foreground' galaxies, &background' galaxies, and pairs, is N "439, N "506 and N "3202. The binned distribution for this &background' sample is 
  shown in Fig. 30, together with a "t according to (8.5). A Kolmogorov}Smirnov test rejects a uniform distribution of p(u) at the 99.9% level, thus providing the "rst detection of galaxy}galaxy lensing. Brainerd et al. performed a large number of tests to check for possible systematic errors, including null tests (e.g., replacing the positions of &foreground' galaxies by random points, or stars), splitting the whole sample into various subsamples (e.g., inner part vs. outer part of the image, upper half vs. lower half, etc.), and these tests were passed satisfactorily. Also a slight PSF anisotropy in the data, or contamination of the ellipticity measurement of faint galaxies by brighter neighbouring galaxies, cannot explain the observed relative alignment, as tested with extensive simulations, so that the detection must be considered real.

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Fig. 31. Contours of constant s in the < }h s parameter plane, where < "(2p , obtained from a comparison of H H H TH the observed tangential alignment 1c 2 with the distribution found in Monte-Carlo simulations. The solid contours range  from 0.8 (innermost) to 8 per degree of freedom; the dotted curve displays s"1 per degree of freedom. (from Brainerd et al., 1996).

Brainerd et al. then quantitatively analysed their observed alignment, using the model outlined in Section 8.2, with a"4. The predictions of the model were inferred from Monte-Carlo simulations, in which galaxies were randomly distributed with the observed number density, and redshifts were assigned according to a probability distribution p (z;m), for which they used a slight X extrapolation from existing redshift surveys, together with a simple prescription for the k-correction in (8.8) to assign luminosities to the galaxies. The ellipticity for each background galaxy image was then obtained by randomly drawing an intrinsic ellipticity, adding shear according to (8.9). The simulated probability distribution p(u) was discretised into several bins in angular separation *h, and compared to the observed orientation distribution, using s-minimisation with respect to the model parameters p and s . The result of this analysis is shown in Fig. 31. The shape of the TH H s-contours is characteristic in that they form a valley which is relatively narrow in the p TH direction, but extends very far out into the s -direction. Thus, the velocity dispersion p can H TH signi"cantly be constrained with these observations, while only a lower limit on s can be derived. H Formal 90% con"dence limits on p are &100 and &210 km s\, with a best-"tting value TH of about 160 km s\, whereas the 1- and 2-p lower limits on s are 25 h\ and &10 h\ kpc, H respectively. Finally, Brainerd et al. studied the dependence of the lensing signal 1c 2 on the colour of their  &background' sample, by splitting it into a red and a blue half. The lensing signal of the former is compatible with zero on all scales, while the blue sample reveals a strong signal which decreases with angular separation as expected. This result is in accordance with that discussed in Section 5.5.3, where the blue galaxies showed a stronger lensing signal as well, indicating that their redshift distribution extends to larger distances.

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We have discussed the work of Brainerd et al. (1996) in some detail since it provided the "rst detection of galaxy}galaxy lensing, and since it was the only one obtained from the ground until recently. Also, their careful analysis exempli"es the di$culties in deriving a convincing result. Gri$ths et al. (1996) analysed the images from the Hubble Space Telescope Medium Deep Survey (MDS) in terms of galaxy}galaxy lensing. The MDS is an imaging survey, using parallel data obtained with the WFPC2 camera on-board HST. They identi"ed 1600 &foreground' (15(I(22) and 14000 &background' (22(I(26) galaxies. Owing to the spatial resolution of the HST, a morphological classi"cation of the foreground galaxies could be performed, and spiral and elliptical galaxies could separately be analysed. They considered the mean orientation angle 1u2"p/4#p\1c 2/"e"2 as a statistical variable, and scaled the truncation radius in their mass  models in proportion to the half-light radius. They found that p "220 and 160 km s\ are TH compatible with their shear data for elliptical and spiral galaxies, respectively. For their sample of elliptical foreground galaxies, they claim that the truncation radius must be more than 10 times the half-light radius to "t their data, and that a de Vaucouleurs mass pro"le is excluded. Unfortunately, no signi"cance levels are quoted. A variant of the method for a quantitative analysis of galaxy}galaxy lensing was developed by Schneider and Rix (1997). Instead of a s-analysis of 1c 2 in angular separation bins, they suggested  a maximum-likelihood analysis, using the individual galaxy images. In their Monte-Carlo approach, the galaxy positions (and magnitudes) are kept "xed, and only the redshifts of the galaxies are drawn from their respective probability distribution p (z; m), as described at the end X of Section 8.2. The resulting log-likelihood function "e !1c 2" G ! ln[p(o#p )] , l"! G AG o#p AG G G

(8.10)

where o is the dispersion of intrinsic ellipticity distribution, here assumed to be a Gaussian, can then be maximised with respect to the model parameters, e.g., p and s . Extensive simulations TH H demonstrated that this approach, which utilises all of the information provided by observations, yields an unbiased estimate of these model parameters. Later, Erben (1997) showed that this remains valid even if the lens galaxies have elliptical projected mass pro"les. This method was applied to the deep multi-colour imaging data of the Hubble Deep Field (HDF; Williams et al., 1996) by Hudson et al. (1998), after Dell'Antonio and Tyson (1996) detected a galaxy}galaxy lensing signal in the HDF on an angular scale of :5. The availability of data in four wavebands allows an estimate of photometric redshifts, a method demonstrated to be quite reliable by spectroscopy of HDF galaxies (e.g., Hogg et al., 1998). The accurate redshift estimates, and the depth of the HDF, compensates for the small "eld-of-view of &5 arcmin. A similar study of the HDF data was carried out by the Caltech group (see Blandford et al., 1998). In order to avoid k-corrections, using the multi-colour photometric data to relate all magnitudes to the rest-frame B-band, Hudson et al. considered lens galaxies with redshift z:0.85 only, leaving 208 galaxies. Only such source-lens pairs for which the estimated redshifts di!er by at least 0.5 were included in the analysis, giving about 10 foreground}background pairs. They adopted the same parameterisation for the lens population as described in Section 8.2, except that the depth of the HDF suggests that the "ducial luminosity ¸ should be allowed to depend on redshift, H ¸ J(1#z)D. Assuming no evolution, f"0, and a Tully}Fisher index of 1/a"0.35, they found H

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p "(160$30) km s\. Various control tests were performed to demonstrate the robustness of TH this result, and potential systematic e!ects were shown to be negligible. As in the previous studies, halo sizes could not be signi"cantly constrained. The lensing signal is dominated by spiral galaxies at a redshift of z&0.6. Comparing the Tully}Fisher relation at this redshift to the local relation, the lensing results indicate that intermediate-redshift galaxies are fainter than local spirals by 1$0.6 magnitudes in the B-band, at "xed circular velocity. Hence, all results reported so far yield compatible values of p , but do not allow upper bounds TH on the halo size to be set. The #atness of the likelihood surface in the s -direction shows that H a measurement of s requires much larger samples than used before. We can understand the H insensitivity to s in the published analyses at least qualitatively. The shear caused by a galaxy at H a distance of, say, 100 kpc is very small, of order 1%. This implies that the di!erence in shear caused by galaxies with truncation radius of 20 kpc and s"100 kpc is very small indeed. In addition, there are typically other galaxies closer to the line-of-sight to background galaxies which produce a larger shear, making it more di$cult to probe the shear of widely separated foreground galaxies. Hence, to probe the halo size, many more foreground}background pairs must be considered. In addition, the angular scale h within which pairs are considered needs to be larger than the

 angular scale of the truncation radius at typical redshifts of the galaxies, and on the other hand, h should be much smaller than the size of the data "eld available. Hence, to probe large scales of

 the halo, wide-"eld imaging data are needed. There is a related problem which needs to be understood in greater detail. Since galaxies are clustered, and probably (biased) tracers of an underlying dark-matter distribution (e.g., most galaxies may live in groups), it is not evident whether the shear caused by a galaxy at a spatial separation of, say, 100 kpc is caused mainly by the dark-matter halo of the galaxy itself, or rather by the dark-matter halo associated with the group. Here, numerical simulations of the dark matter may indicate to which degree these two e!ects can be separated, and observational strategies for this need to be developed. In fact, the two points just mentioned were impressively illustrated by a galaxy}galaxy lensing analysis of early commissioning imaging data from the Sloan Digital Sky Survey (Fischer et al., 1999), covering 225 square degrees. The separation between foreground and background galaxies was based on apparent magnitude, with an estimated mean redshift of the foreground sample of 1z 2+0.17. Fischer et al. (1999) used data in three optical "lters for their analysis; the number of  foreground (background) galaxies in each "lter is &28,000 (1.4;10). The galaxy}galaxy lensing signal is seen out to &10 in all three "lters, and the mean tangential shear in the annulus 104h410 is +6;10\. With an assumed redshift distribution of foreground and background galaxies, the characteristic velocity dispersion could be estimated to be p "170$20 km s\ at H 95% con"dence. Even at the large angular separation probed by this data set, no sign of a cut-o! radius of the galaxy halos is seen, and a lower limit of s 5275h\ kpc can be derived. At such H scales, the shear is probably no longer dominated by the foreground galaxy used as the origin for the de"nition of tangential shear, but by neighbouring galaxies and/or dark matter correlated with the galaxy. Therefore, the results of such a study may best be interpreted as a galaxymass correlation function (Kaiser, 1992), which brings us back to the issue of biasing discussed in Section 6.8. A preliminary analysis presented in Fischer et al. (1999) yields X /b&0.3, if a linear  biasing factor b is assumed. At least as important as the quantitative results from the Sloan Survey is the fact that they demonstrate the enormous potential of this method } this analysis used about

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2% of the imaging data the full Sloan Survey will provide, and did not yet utilise photometric redshift information which, as mentioned before, will increase the accuracy of the physical parameters derived. 8.4. Galaxy}galaxy lensing in galaxy clusters An interesting extension of the work described above aims at the investigation of the dark-matter halo properties of galaxies within galaxy clusters. In the hierarchical model for structure formation, clusters grow by mergers of less massive halos, which by themselves formed by merging of even smaller substructures. Tidal forces in clusters, possible ram-pressure stripping by the intra-cluster medium, and close encounters during the formation process, may a!ect the halos of galaxies, most of which presumably formed at an early epoch. Therefore, it is unclear at present whether the halo properties of galaxies in clusters are similar to those of "eld galaxies. Galaxy}galaxy lensing o!ers an exciting opportunity to probe the dark galaxy halos in clusters. There are several di!erences between the investigation of "eld and of cluster galaxies. First, the number of massive galaxies in a cluster is fairly small, so the statistics for a single cluster will be limited. This can be compensated by investigating several clusters simultaneously. Second, the image distortion is determined by the reduced shear, g"c/(1!i). For "eld galaxies, where the shear and the surface mass density is small, one can set g+c, but this approximation no longer holds for galaxies in clusters, where the cluster provides i substantially above zero. This implies that one needs to know the mass distribution of the cluster before the statistical properties of the massive galaxy halos can be investigated. On the other hand, it magni"es the lensing signal from the galaxies, so that fewer cluster galaxies are needed to derive signi"cant lensing results compared to "eld galaxies of similar mass. Third, most cluster galaxies are of early type, and thus their p } and consequently, their lensing e!ect } is expected to be larger than for typical "eld galaxies. TH In fact, the lensing e!ect of individual cluster galaxies can even be seen from strong lensing. Modelling clusters with many strong-lensing constraints (e.g., several arcs, multiple images of background galaxies), the incorporation of individual cluster galaxies turns out to be necessary (e.g., Kassiola et al., 1992; Wallington et al., 1995; Kneib et al., 1996). However, the resulting constraints are relevant only for a few cluster galaxies which happen to be close to the stronglensing features, and mainly concern the mass of these galaxies within &10h\ kpc. The theory of galaxy}galaxy lensing in clusters was developed in Natarajan and Kneib (1997) and Geiger and Schneider (1998), using several di!erent approaches. The simplest possibility is related to the aperture mass method discussed in Section 5.3.1. Measuring the tangential shear within an annulus around each cluster galaxy, perhaps including a weight function, permits a measurement of the aperture mass, and thus to constrain the parameters of a mass model for the galaxies. Provided the scale of the aperture is su$ciently small, the tidal "eld of the cluster averages out to "rst order, and the local in#uence of the cluster occurs through the local surface mass density i. In particular, the scale of the aperture should be small enough in order to exclude neighbouring cluster galaxies. A more sophisticated analysis starts from a mass model of the cluster, as obtained by one of the reconstruction techniques discussed in Section 5, or by a parameterised mass model constructed from strong-lensing constraints. Then, parameterised galaxy models are added, again with a prescription similar to that of Section 8.2, and simultaneously the mass model of the cluster is

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multiplied by the relative mass fraction in the smoothly distributed cluster mass (compared to the total mass). In other words, the mass added by inserting galaxies into the cluster is subtracted from the smooth density pro"le. From the observed galaxy ellipticities, a likelihood function can be de"ned and maximised with respect to the parameters (p , s ) of the galaxy model. TH H Natarajan et al. (1998) applied this method to WFPC2 images of the cluster AC 114 (z "0.31).  They concluded that most of the mass of a "ducial ¸ cluster galaxy is contained in a radius of H &15 kpc, indicating that the halo size of galaxies in this cluster is smaller than that of "eld galaxies. Using their HST mosaic image, Hoekstra et al. (2000) also detected galaxy}galaxy lensing in the high-redshift cluster MS1054!03 at z"0.83. Avoiding the densest part of the cluster in selecting their foreground galaxies, they investigated the average tangential shear around them, after subtracting the shear from the cluster as determined from the mass reconstruction (see Section 5.3), also using scaling (2.68). The galaxy}galaxy lensing signal is seen at the 99.8% con"dence level. Using the redshift distribution of background galaxies as determined from photometric redshift estimates in the Hubble Deep Fields, their lensing signal yields p +200$35 km s\ for the cluster galaxies. Not H unexpectedly, this value is larger than those obtained from "eld galaxies, since the cluster preferentially hosts early-type galaxies for which p is known to be larger than for spirals. It is indeed H encouraging that this method is able to measure the mass of high-redshift galaxies. Once the mass contained in the cluster galaxies is a signi"cant fraction of the total mass of the cluster, this method was found to break down, or give strongly biased results. Geiger and Schneider (1999) modi"ed this approach by performing a maximum-likelihood cluster mass reconstruction for each parameter set of the cluster galaxies, allowing the determination of the best representation of the global underlying cluster component that is consistent with the presence of the cluster galaxies and the observed image ellipticities of background galaxies. This method was then applied to the WFPC-2 image of the cluster Cl0939#4713, already described in Section 5.4. The entropy-regularised maximum-likelihood mass reconstruction of the cluster is very similar to the one shown in Fig. 14, except that the cluster centre is much better resolved, with a peak very close to the observed strong lensing features (Trager et al., 1997). Cluster galaxies were selected according to their magnitudes, and divided by morphology into two subsamples, viz., early-type galaxies and spirals. In Fig. 32 we show the likelihood contours in the s }p plane, for both subsets of cluster galaxies. Whereas there is no statistically signi"cant H TH detection of lensing by spiral galaxies, the lensing e!ect of early-type galaxies is clearly detected. Although no "rm upper limit of the halo size s can be derived from this analysis owing to the small H angular "eld of the image (the maximum of the likelihood function occurs at 8h\ kpc, and a 1-p upper limit would be &50h\ kpc), the contours &close' at smaller values of s compared to the H results obtained from "eld galaxies. By statistically combining several cluster images, a signi"cant upper limit on the halo size can be expected. The maximum-likelihood estimate of p for the early-type galaxies is &200 km s\, in agreeH ment with that found by Hoekstra et al. (2000). It should be noted that the results presented above still contain some uncertainties, most notably the unknown redshift distribution of the background galaxies and the mass-sheet degeneracy, which becomes particularly severe owing to the small "eld-of-view of WFPC2. Changing the assumed redshift distribution and the scaling parameter j in (5.10) shifts the likelihood contours in Fig. 32 up or down, i.e., the determination of p is a!ected. As for galaxy}galaxy lensing of "eld TH galaxies, the accuracy can be increased by using photometric redshift estimates. Similarly, the

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Fig. 32. Results of applying the entropy-regularised maximum-likelihood method for galaxy}galaxy lensing to the WFPC2 image of the cluster Cl0939#4713. The upper and lower panels correspond to early-type and spiral galaxies, respectively. The solid lines are con"dence contours at 68.3%, 95.4% and 99.7%, and the cross marks the maximum of the likelihood function. Dashed lines correspond to galaxy models with equal aperture mass M ((8h\ kpc)" H (0.1, 0.5, 1.0);10h\M . Similarly, the dotted lines connect models of constant total mass for an ¸ -galaxy, > H of M "(0.1, 0.5, 1.0, 5.0, 10);10h\M , which corresponds to a mass fraction contained in galaxies of H > (0.15, 0.75, 1.5, 7.5, 15)%, respectively (from Geiger and Schneider, 1999).

allowed range of the mass-sheet transformation can be constrained by combining these small-scale images with larger-scale ground-based images, or, if possible, by using magni"cation information to break the degeneracy. Certainly, these improvements of the method will be a "eld of active research in the immediate future. 9. The impact of weak gravitational light de6ection on the microwave background radiation 9.1. Introduction The Cosmic Microwave Background originated in the hot phase after the Big Bang, when photons were created in thermal equilibrium with electromagnetically interacting particles. While the Universe expanded and cooled, the photons remained in thermal equilibrium until the temperature was su$ciently low for electrons to combine with the newly formed nuclei of mainly hydrogen and helium. While the formation of atoms proceeded, the photons decoupled from the matter due to the rapidly decreasing abundance of charged matter. Approximately 300,000 years after the Big Bang, corresponding to a redshift of z+1000, the universe became transparent for the radiation, which retained the Planck spectrum it had acquired while it was in thermal equilibrium, and the temperature decreased in proportion with the scale factor as the Universe expanded. This relic radiation, cooled to ¹"2.73 K, forms the Cosmic Microwave Background (hereafter CMB).

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Penzias and Wilson (1965) detected it as an `excess antenna temperaturea, and Fixsen et al. (1996) used the COBE-FIRAS instrument to prove its perfect black-body spectrum. Had the Universe been ideally homogeneous and isotropic, the CMB would have the intensity of black-body radiation at 2.73 K in all directions on the sky, and would thus be featureless. Density perturbations in the early Universe, however, imprinted their signature on the CMB through various mechanisms, which are thoroughly summarised and discussed in Hu (1995). Photons in potential wells at the time of decoupling had to climb out, thus losing energy and becoming slightly cooler than the average CMB. This e!ect, now called the Sachs}Wolfe ewect was originally studied by Sachs and Wolfe (1967), who found that the temperature anisotropies in the CMB trace the potential #uctuations on the &surface' of decoupling. CMB #uctuations were "rst detected by the COBE-DMR experiment (Smoot et al., 1992) and subsequently con"rmed by numerous groundbased and balloon-borne experiments (see Smoot, 1997 for a review). The interplay between gravity and radiation pressure in perturbations of the cosmic &#uid' before recombination gave rise to another important e!ect. Radiation pressure is only e!ective in perturbations smaller than the horizon. Upon entering the horizon, radiation pressure provides a restoring force against gravity, leading to acoustic oscillations in the tightly coupled #uid of photons and charged particles, which cease only when radiation pressure drops while radiation decouples. Therefore, for each physical perturbation scale, the acoustic oscillations set in at the same time, i.e. when the horizon size becomes equal the perturbation size, and they end at the same time, i.e. when radiation decouples. At "xed physical scale, these oscillations are therefore coherent, and they show up as distinct peaks (the so-called Doppler peaks) and troughs in the power spectrum of the CMB #uctuations. Perturbations large enough to enter the horizon after decoupling never experience these oscillations. Going through the CMB power spectrum from large to small scales, there should therefore be a &"rst' Doppler peak at a location determined by the horizon scale at the time of decoupling. A third important e!ect sets in on the smallest scales. If a density perturbation is small enough, radiation pressure can blow it apart because its self-gravity is too weak. This e!ect is comparable to the Jeans' criterion for the minimal mass required for a pressurised perturbation to collapse. It amounts to a suppression of small-scale #uctuations and is called Silk damping, leading to an exponential decline at the small-scale end of the CMB #uctuation power spectrum. Other e!ects arise between the &surface' of decoupling and the observer. Rees and Sciama (1968) pointed out that large non-linear density perturbations between the last-scattering surface and us can lead to a distinct e!ect if those #uctuations change while the photons traverse them. Falling into the potential wells, they experience a stronger blue-shift than climbing out of them because expansion makes the wells shallower in the meantime, thus giving rise to a net blue-shift of photons. Later, this e!ect was re-examined in the framework of the &Swiss-Cheese' (Dyer, 1976) and &vacuole' (Nottale, 1984) models of density perturbations in an expanding background space}time. The masses of such perturbations have to be very large for this e!ect to become larger than the Sunyaev}Zel'dovich e!ect due to the hot gas contained in them; Dyer (1976) estimated that

 The (thermal) Sunyaev}Zel'dovich e!ect is due to Compton-upscattering of CMB photons by thermal electrons in the hot plasma in galaxy clusters. Since the temperature of the electrons is much higher than that of the photons, CMB photons are e!ectively re-distributed towards higher energies. At frequencies lower than +272 GHz, the CMB intensity is thus decreased towards galaxy clusters; in e!ect, they cast shadows on the surface of the CMB.

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masses beyond 10M would be necessary, a value four to "ve orders of magnitude larger than > that of typical galaxy clusters. The gravitational lens e!ect of galaxy clusters moving transverse to the line-of-sight was investigated by Birkinshaw and Gull (1983) who found that a cluster with &10M and > a transverse velocity of &6000 km s\ should change the CMB temperature by &10\ K. Later, Gurvits and Mitrofanov (1986) re-investigated this e!ect and found it to be about an order of magnitude smaller. Cosmic strings as another class of rapidly moving gravitational lenses were studied by Kaiser and Stebbins (1984) who discussed that they would give rise to step-like features in the CMB temperature pattern. 9.2. Weak lensing of the CMB The introduction shows that the CMB is expected to display distinct features in a hierarchical model of structure formation. The CMB power spectrum should be featureless on large scales, then exhibit pronounced Doppler peaks at scales smaller than the horizon at the time of decoupling, and an exponential decrease due to Silk damping at the small-scale end. We now turn to investigate whether and how gravitational lensing by large-scale structures can alter these features. The literature on the subject is rich (see Blanchard and Schneider, 1987; CayoH n et al., 1993b; CayoH n et al., 1993a; Cole and Efstathiou, 1989; Fukugita et al., 1992; Kashlinsky, 1988; Linder, 1988; Linder, 1990a,b; MartmH nez-GonzaH lez et al., 1990; Sasaki, 1989; Tomita, 1989; Watanabe and Tomita, 1991), but di!erent authors have sometimes arrived at contradicting conclusions. Perhaps, the most elegant way of studying weak lensing of the CMB is the power-spectrum approach, which was most recently advocated by Seljak (1994, 1996). We should like to start our discussion by clearly stating two facts concerning the e!ect of lensing on #uctuations in the Cosmic Microwave Background which clarify and resolve several apparently contradictory discussions and results in the literature: (1) If the CMB was completely isotropic, gravitational lensing would have no ewect whatsoever because it conserves surface brightness. In this case, lensing would only magnify certain patches in the sky and de-magnify others, but since it would not alter the surface brightness in the magni"ed or de-magni"ed patches, the temperature remained una!ected. An analogy would be observers facing an in"nitely extended homogeneously coloured wall, seeing some parts of it enlarged and others shrunk. Regardless of the magni"cation, they would see the same colour everywhere, and so they would notice nothing despite the magni"cation. (2) It is not the absolute value of the light deyection due to lensing which matters, but the relative deyection of neighbouring light rays. Imagine a model universe in which all light rays are isotropically de#ected by the same arbitrary amount. The pattern of CMB anisotropies seen by an observer would then be coherently shifted relative to the intrinsic pattern, but remain unchanged otherwise. It is thus merely the dispersion of de#ection angles what is relevant for the impact of lensing on the observed CMB #uctuation pattern.

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9.3. CMB temperature yuctuations In the absence of any lensing e!ects, we observe at the sky position h the intrinsic CMB temperature ¹(h). There are #uctuations *¹(h) in the CMB temperature about its average value 1¹2"2.73 K. We abbreviate the relative temperature #uctuations by *¹(h) ,q(h) 1¹2

(9.1)

in the following. They can statistically be described by their angular auto-correlation function m ( )"1q(h) q(h# )2 (9.2) 2 with the average extending over all positions h. Due to statistical isotropy, m ( ) depends neither on 2 the position h nor on the direction of , but only on the absolute separation of the correlated points. Commonly, CMB temperature #uctuations are also described in terms of the coe$cients a of JK an expansion into spherical harmonics  J q(h, )" a >K(h, ) (9.3) JK J J K\J and the averaged expansion coe$cients constitute the angular power spectrum C of the CMB J #uctuations C "1"a "2 . (9.4) J JK It can then be shown that the correlation function m ( ) is related to the power-spectrum 2 coe$cients C through J p (9.5) C " d sin( )P (cos )m ( ) J 2 J  with the Legendre functions P (cos ). J



9.4. Auto-correlation function of the gravitationally lensed CMB 9.4.1. Dexnitions If there are any density inhomogeneities along the line-of-sight towards the last-scattering surface at z+1000 (the &source plane' of the CMB), a light ray starting into direction h at the observer will intercept the last-scattering surface at the de#ected position b"h!a(h) ,

(9.6)

where a(h) is the (position-dependent) de#ection angle experienced by the light ray. We will therefore observe, at position h, the temperature of the CMB at position b, or ¹(b),¹(h)"¹[h!a(h)] .

(9.7)

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The intrinsic temperature auto-correlation function is thus changed by lensing to m ( )"1q[h!a(h)] q[(h# )!a(h# )]2 . (9.8) 2 For simplicity of notation, we further abbreviate a(h),a and a(h# ),a in the following. 9.4.2. Evaluation In this section we evaluate the modi"ed correlation function (9.8) and quantify the lensing e!ects. For this purpose, it is convenient to decompose the relative temperature #uctuation q(h) into Fourier modes



q(h)"

dl q( (l) exp(i l h) . (2p)

(9.9)

The expansion of q(h) into Fourier modes rather than into spherical harmonics is permissible because we do not expect any weak-lensing e!ects on large angular scales, so that we can consider ¹(h) on a plane locally tangential to the sky rather than on a sphere. We insert the Fourier decomposition (9.9) into the expression for the correlation function (9.8) and perform the average. We need to average over ensembles and over the random angle between the wave vector l of the temperature modes and the angular separation of the correlated points. The ensemble average corresponds to averaging over realisations of the CMB temperature #uctuations in a sample of universes or, since we focus on small scales, over a large number of disconnected regions on the sky. This average introduces the CMB #uctuation spectrum P (l), 2 which is de"ned by 1q( (l) q( H(l)2,(2p) d(l!l) P (l) . (9.10) 2 Averaging over the angle between l and the position angle gives rise to the zeroth-order Bessel function of the "rst kind, J (x). These manipulations leave Eq. (9.8) in the form   l dl P (l) 1exp[i l(a!a)]2 J (l ) . (9.11) m ( )"  2 2p 2  The average over the exponential in Eq. (9.11) remains to be performed. To do so, we "rst expand the exponential into a power series



 1(i l da)H2 , (9.12) 1exp(i l da)2" j! H where da,a!a is the de#ection-angle di!erence between neighbouring light rays with initial angular separation . We now assume that the de#ection angles are Gaussian random "elds. This is reasonable because (i) de#ection angles are due to Gaussian random #uctuations in the density-contrast "eld as long as the #uctuations evolve linearly, and (ii) the assumption of linear evolution holds well for redshifts where most of the de#ection towards the last-scattering surface occurs. Of course, this makes use of the commonly held view that the initial density #uctuations are of Gaussian nature. Under this condition, the odd moments in Eq. (9.12) all vanish. It can then be shown that 1exp(i l da)2"exp(!lp( )) 

(9.13)

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holds exactly, where p( ) is the dispersion of one component of the de#ection angle, p( ), 1(a!a)2 . (9.14)  Even if the assumption that da is a Gaussian random "eld fails, Eq. (9.13) still holds approximately. To see this, we note that the CMB power spectrum falls sharply on scales l9l +(10 X)\. The   scale l is set by the width of the last-scattering surface at redshift z&1000. Smaller-scale  #uctuations are e$ciently damped by acoustic oscillations of the coupled photon}baryon #uid. Typical angular scales l\ in the CMB #uctuations are therefore considerably larger than the di!erence between gravitational de#ection angles of neighbouring rays, da, so that l(a!a) is a small number. Hence, ignoring fourth-order terms in lda, the remaining exponential in (9.11) can be approximated by (9.15) 1exp(i l da)2+1!lp( )+exp[!lp( )] .   Therefore, the temperature auto-correlation function modi"ed by gravitational lensing can safely be written







 l dl 1 P (l) exp ! lp( ) J (l ) . (9.16) 2  2p 2  This equation shows that the intrinsic temperature-#uctuation power spectrum is convolved with a Gaussian function in wave number l with dispersion p\( ). The e!ect of lensing on the CMB temperature #uctuations is thus to smooth #uctuations on angular scales of order or smaller than p( ). m ( )" 2

9.4.3. Alternative representations Eq. (9.16) relates the unlensed CMB power spectrum to the lensed temperature auto-correlation function. Noting that P (l) is the Fourier transform of m ( ), 2 2





P (l)" d m ( ) exp(!i l )"2p d m ( ) J (l ) , 2 2  2

(9.17)

we can substitute one for the other. Isotropy permitted us to perform the integration over the (random) angle between l and in the last step of (9.17). Inserting (9.17) into (9.16) leads to



m ( )"  d  m ( ) K( , ) . 2 2

(9.18)

The kernel K( , ) is given by



K( , ),



    

1 l dl J (l ) J (l ) exp ! l p( )   2



 1

#  " exp ! I ,  p( ) p( ) 2p( )



(9.19)

where I (x) is the modi"ed zeroth-order Bessel function. Eq. (6.66) (3.2) of Gradshteyn and Ryzhik  (1994) was used in the last step. As will be shown below, p( ); , so that the argument of I is 

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generally a very large number. Noting that I (x)+(2px)\ exp(x) for xPR, we can write  Eq. (9.18) in the form







( ! ) 1 d   m ( ) exp ! . m ( )+ 2 2 2p( ) (2p ) p( )

(9.20)

Like Eq. (9.16), this expression shows that lensing smoothes the intrinsic temperature autocorrelation function m ( ) over angular scales of p( ). Note in particular that, if p( ), the 2 exponential in (9.20) tends towards a Dirac delta distribution





( ! ) 1 exp ! "d( ! ) , (9.21) lim 2p( ) N( (2p p( ) so that the lensed and unlensed temperature auto-correlation functions agree, m ( )"m ( ). 2 2 Likewise, one can Fourier back-transform Eq. (9.16) to obtain a relation between the lensed and the un-lensed CMB power spectra. To evaluate the resulting integral, it is convenient to assume p( )"e , with e being either a constant or a slowly varying function of . This assumption will be justi"ed below. One then "nds









(9.22)



(9.23)

 dl l l#l P (l) exp ! I .  el el 2 2el  For e;1, this expression can be simpli"ed to P (l)" 2



P (l)" 2

 dl





(l!l) P (l) exp ! . 2 2el (2pel

9.5. Deyection-angle variance 9.5.1. Auto-correlation function of deyection angles We proceed by evaluating the dispersion p( ) of the de#ection angles. This is conveniently derived from the de#ection-angle auto-correlation function ma ( ),1a a2 .

(9.24)

Note that the correlation function of a is the sum of the correlation functions of the components of a, ma "1a a2"1a a 2#1a a 2"m  #m  .     ? ? In terms of the auto-correlation function, the dispersion p( ) can be written as

(9.25)

p( )" 1[a!a]2"ma (0)!ma ( ) . (9.26)  The de#ection angle is given by Eq. (6.11) in terms of the Newtonian potential U of the density #uctuations d along the line-of-sight. For lensing of the CMB, the line-of-sight integration extends along the (unperturbed) light ray from the observer at w"0 to the last-scattering surface at w (z+1000); see the derivation in Section 6.2 leading to Eq. (6.11).

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Fig. 33. The "lter function F(l, ) as de"ned in Eq. (9.30), divided by , is shown as a function of l . Compare Fig. 22. For "xed , the "lter function emphasises large-scale projected density perturbations (i.e. structures with small l).

We introduced the e!ective convergence in (6.14) as half the divergence of the de#ection angle. In Fourier space, this equation can be inverted to yield the Fourier transform of the de#ection angle 2i i( (l) a( (l)"!  l . "l"

(9.27)

The de#ection-angle power spectrum can therefore be written as 4 Pa ( )" P (l) . l G

(9.28)

The de#ection-angle auto-correlation function is obtained from Eq. (9.28) via Fourier transformation. The result is





dl  J (l ) l dl P (l)  Pa (l) exp(!i l )"2p , (9.29) G (2p) (pl)  similar to the form (6.59), but here the "lter function is no longer a function of the product l only, but of l and separately, ma ( )"

J (l ) J (l ) "   . F(l, )"  (pl ) (pl)

(9.30)

We plot \F(l, ) in Fig. 33. For "xed , the "lter function suppresses small-scale #uctuations, and it tends towards F(l, )P(pl)\ for lP0. Inserting P (l) into (9.29), we "nd the explicit expression for the de#ection-angle auto-correlation G function 9H X ma ( )"   c





U  dk dw =(w, w) a\(w) P (k, w) J [ f (w)k ] .  ) 2pk B  

(9.31)

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Despite the obvious similarity between this result and the magni"cation auto-correlation function (6.34), it is worth noting two important di!erences. First, the weighting of the integrand along the line-of-sight di!ers by a factor of f  (w) because we integrate de#ection-angle components rather ) than the convergence, i.e. "rst rather than second-order derivatives of the potential U. Consequently, structures near the observer are weighted more strongly than for magni"cation or shear e!ects. Secondly, the wave-number integral is weighted by k\ rather than k, giving most weight to the largest-scale structures. Since their evolution remains linear up to the present, it is expected that non-linear density evolution is much less important for lensing of the CMB than it is for cosmic magni"cation or shear. 9.5.2. Typical angular scale A typical angular scale for the coherence of gravitational light de#ection can be ob tained as




1

,  ma (0)

Rma ( ) R 

 

\

.

(9.32)

(

As Eq. (9.31) shows, the de#ection-angle auto-correlation function depends on only through the argument of the Bessel function J (x). For small arguments x, the second-order derivative of the  J (x) is approximately J (x)+!J (x)/2. Di!erentiating ma ( ) twice with respect to , and    comparing the result to the expression for the magni"cation auto-correlation function m ( ) in I Eq. (6.34), we "nd 1 Rma ( ) +! m ( ) R  2 I

(9.33)

and thus ma (0)

+2 .  m (0) I

(9.34)

We shall estimate later after giving a simple expression for ma ( ). The angle gives an estimate   of the scale over which gravitational light de#ection is coherent. 9.5.3. Special cases and qualitative expectations We mentioned before that it is less critical here to assume linear density evolution because large-scale density perturbations dominate in the expression for ma ( ). Specialising further to an Einstein}de Sitter universe so that w+2c/H , Eq. (9.31) simpli"es to 





9H   dk P(k)J (wyk ) ma ( )"  w dy(1!y)  c 2pk B   with wy,w.

(9.35)

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Adopting the model spectra for HDM and CDM speci"ed in Eq. (6.37) and expanding ma ( ) in a power series in , we "nd, to second order in ,







 3 1! (wk )  20 2p

for HDM ,

ma ( )"Awk  3(3 3  1! (wk )  40 8





(9.36) for CDM .

Combining these expressions with Eqs. (9.34) and (6.38), we "nd for the de#ection-angle coherence scale



+3(wk )\ . (9.37)   It is intuitively clear that should be determined by (wk )\. Since k\ is the typical length scale    of light-de#ecting density perturbations, it subtends an angle (wk )\ at distance w. Thus, the  coherence angle of light de#ection is given by the angle under which the de#ecting density perturbation typically appears. The source distance w in the case of the CMB is the comoving distance to z"1000. In the Einstein}de Sitter case, w"2 in units of the Hubble length. Hence, with k\+12(X h) Mpc [cf. Eq. (2.49)], we have wk +500. Therefore, the angular scale of the    de#ection-angle auto-correlation is of order

+6;10\+20 .  To lowest order in , the de#ection-angle dispersion (9.26) reads

(9.38)

p( )J(wk )  . (9.39)  The dispersion p( ) is plotted in Fig. 34 for the four cosmological models speci"ed in Table 1 for linear and non-linear evolution of the density #uctuations. The behaviour of p( ) expressed in Eq. (9.39) can qualitatively be understood describing the change in the transverse separation between light paths as a random walk. Consider two light paths separated by an angle such that their comoving transverse separation at distance w is w . Let k\ be the typical scale of a potential #uctuation U. We can then distinguish two di!erent cases depending on whether w is larger or smaller than k\. If w 'k\, the transverse separation between the light paths is much larger than the typical potential #uctuations, and their de#ection will be incoherent. It will be coherent in the opposite case, i.e. if w (k\. When the light paths are coherently scattered passing a potential #uctuation, their angular separation changes by d +w (2k\ U/c), which is the change in the de#ection angle  , , across w . If we replace the gradients by the inverse of the typical scale, k, we have d +2 w kU/c. Along a distance w, there are N+kw such potential #uctuations, so that the  total change in angular separation is expected to be d +Nd .  In case of incoherent scattering, the total de#ection of each light path is expected to be d +N (2k\ U/c)+N 2U/c, independent of . Therefore, , Nd  +(2U/c) (wk)  for ((wk)\,  (9.40) p( )+ N (2U/c)+(2U/c) (wk) for '(wk)\.



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Fig. 34. The de#ection-angle variance p( ) is shown for the four cosmological models speci"ed in Table 1. Two curves are shown for each model, one for linear and one for non-linear evolution of the density #uctuations. Solid curves: SCDM; dotted curve: pCDM; short-dashed curves: OCDM; and long-dashed curves: KCDM. The somewhat steeper curves are for linear density evolution. Generally, the de#ection-angle variance increases linearly with for small , and #attens gradually for 920. At +10, p( ) reaches +0.1, or +0.01 , for the cluster-normalised model universes (all except pCDM; dotted curves). As expected, the e!ect of non-linear density evolution is fairly moderate, and most pronounced on small angular scales, :10.

This illustrates that the dependence of p( ) on (wk)  for small is merely a consequence of the random coherent scattering of neighbouring light rays at potential #uctuations. For large , p( ) becomes constant, and so p( ) \P0. As Fig. 34 shows, the dispersion p( ) increases linearly with

for small and #attens gradually for ' +(10}20) as expected, because divides coherent   from incoherent scattering. 9.5.4. Numerical results The previous results were obtained by specialising to linear evolution of the density contrast in an Einstein}de Sitter universe. For arbitrary cosmological parameters, the de#ection-angle dispersion has to be computed numerically. We show in Fig. 34 examples for p( ) numerically calculated for the four cosmological models detailed in Table 1. Two curves are plotted for each model. The somewhat steeper curves were obtained for linear, the others for non-linear density evolution. Fig. 34 shows that typical values for the de#ection-angle variance in cluster-normalised model universes are of order p( )+(0.03}0.1) on angular scales between +(1}10). While the results for di!erent cosmological parameters are fairly close for cluster-normalised CDM, p( ) is larger by about a factor of two for CDM in an Einstein}de Sitter model normalised to p "1. For the other  cosmological models, the di!erences between di!erent choices for the normalisation are less pronounced. The curves shown in Fig. 34 con"rm the qualitative behaviour estimated in the previous section: The variance p( ) increases approximately linearly with as long as is small, and it gradually #attens o! at angular scales 9 +20.  In earlier chapters, we saw that non-linear density evolution has a large impact on weak gravitational lensing e!ects, e.g. on the magni"cation auto-correlation function m ( ). As menI tioned before, this is not the case for the de#ection-angle auto-correlation function ma ( ) and the

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variance p( ) derived from it, because the "lter function F(l, ) relevant here suppresses small-scale density #uctuations for which the e!ect of non-linear evolution are strongest. Therefore, non-linear evolution is expected to have less impact here. Only on small angular scales , the "lter function extends into the su$ciently non-linear regime. The curves in Fig. 34 con"rm and quantify this expectation. Only on scales of :10, the non-linear evolution does have some e!ect. Obviously, non-linear evolution increases the de#ection-angle variance in a manner quite independent of cosmology. At angular scales +1, the increase amounts to roughly a factor of two above the linear results. 9.6. Change of CMB temperature yuctuations 9.6.1. Summary of previous results We are now ready to justify assumptions and approximations made earlier, and to quantify the impact of weak gravitational lensing on the Cosmic Microwave Background. The main assumptions were that (i) the de#ection-angle variance p( ) is small, and (ii) p( )+e , with e a (small) constant or a function slowly varying with . The results obtained in the previous section show that p( ) is typically about two orders of magnitude smaller than , con"rming e;1. Likewise, Fig. 34 shows that the assumption p( )J is valid on angular scales smaller than the coherence scale for the de#ection, : +20. As we have seen, this proportionality is a mere consequence of random  coherent scattering of neighbouring light rays in the #uctuating potential "eld. For angles larger than , p( ) gradually levels o! to become constant, so that the ratio between p( ) and tends to  zero while increases further beyond . We can thus broadly summarise the numerical results on  the de#ection-angle variance by



p( )+

0.01 for : 20, 0.7

for <20

(9.41)

which is valid for cluster-normalised CDM quite independent of the cosmological model; in particular, p( )(1+3;10\ radians for all . 9.6.2. Simplixcations Accordingly, the argument of the exponential in Eq. (9.16) is a truly small number. Even for large l+10, lp( );1. We can thus safely expand the exponential into a power series, keeping only the lowest-order terms. Then, Eq. (9.16) simpli"es to



m ( )"m ( )!p( ) 2 2

 l dl P (l)J (l ) , 4p 2  

(9.42)

where we have used that the auto-correlation function m ( ) is the Fourier transform of the power 2 spectrum P (l). Employing again the approximate relation J (x)+!J (x)/2 which holds for 2   small x, we notice that



 l dl Rm ( ) P (l)J (l )+! 2 . 4p 2  R  

(9.43)

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We can introduce a typical angular scale for the CMB temperature #uctuations in the same  manner as for light de#ection in Eq. (9.32). We de"ne by  1 Rm ( ) 2

\,! , (9.44)  m (0) R  ( 2 so that, up to second order in , Eq. (9.42) can be approximated as



p( ) m ( )+m ( )! m (0) . (9.45) 2 2

 2  We saw earlier that p( )+e for : . Eq. (9.45) can then further be simpli"ed to read 

 m ( )+m ( )!em (0) . (9.46) 2 2 2   In analogy to Eq. (9.26), we can write the mean-square temperature #uctuations of the CMB between two beams separated by an angle as p ( )"1[q(h)!q(h# )]2"2[m (0)!m ( )] . 2 2 2 Weak gravitational lensing changes this relative variance to

(9.47)

p "2[m (0)!m ( )] . 2 2 2 Using Eq. (9.46), we see that the relative variance is increased by the amount

(9.48)

 *p ( )"p ( )!p ( )+em (0) . (9.49) 2 2 2 2   Now, the auto-correlation function at zero lag, m (0), is the temperature-#uctuation variance, p . 2 2 Hence, we have for the rms change in the temperature variation

[*p ( )]"ep . (9.50) 2 2

 Weak gravitational lensing thus changes the CMB temperature #uctuations only by a very small amount, of order e+10\ for + .  9.6.3. The lensed CMB power spectrum However, we saw in Eq. (9.23) that the gravitationally lensed CMB power spectrum is smoothed compared to the intrinsic power spectrum. Modes on an angular scale are mixed with modes on angular scales $p( ), i.e. the relative broadening d / is of order 2p( )/ . For : +20, this  relative broadening is of order 2e+2;10\, while it becomes negligible for substantially larger scales because p( ) becomes constant. This e!ect is illustrated in Fig. 35, where we show the unlensed and lensed CMB power spectra for CDM in an Einstein}de Sitter universe. The "gure clearly shows that lensing smoothes the CMB power spectrum on small angular scales (large l), while it leaves large angular scales una!ected. Lensing e!ects become visible at l 9 500, corresponding to an angular scale of : (p/500) rad+20, corresponding to the scale where coherent gravitational light de#ection sets in. An important e!ect of lensing is seen at the high-l tail

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Fig. 35. The CMB power spectrum coe$cients l(l#1)C are shown as a function of l. The solid line displays the intrinsic J power spectrum, the dotted line the lensed power spectrum for an Einstein}de Sitter universe "lled with cold dark matter. Evidently, lensing smoothes the spectrum at small angular scales (large l), while it has no visible e!ect on larger scales. The curves were produced with the CMBfast code, see Zaldarriaga and Seljak (1998).

of the power spectra, where the lensed power spectrum falls systematically above the unlensed one (Metcalf and Silk, 1997). This happens because the Gaussian convolution kernel in Eq. (9.23) becomes very broad for very large l, so that the lensed power spectrum at l can be substantially increased by intrinsic power from signi"cantly smaller l. In other words, lensing mixes power from larger angular scales into the otherwise featureless damping tail of P (l). 2 9.7. Discussion Several di!erent approximations entered the preceding derivations. Firstly, the de#ection-angle variance p( ) was generally assumed to be small, and for some expressions to be proportional to

with a small constant of proportionality e. The numerical results showed that the "rst assumption is very well satis"ed, and the second assumption is valid for : , the latter being the  coherence scale of gravitational light de#ection. We further assumed the de#ection-angle "eld to be a Gaussian random "eld, the justi"cation being that the de#ecting matter distribution is also a Gaussian random "eld. While this fails to be exactly true at late stages of the cosmic evolution, we have seen that the resulting expression can also be obtained when p( ) is small and a is not a Gaussian random "eld; hence, in practice this assumption is not a limitation of validity. A "nal approximation consists in the Born approximation. This should also be a reasonable assumption at least in the case considered here, where we focus on statistical properties of light propagation. Even if the light rays would be bent considerably, the statistical properties of the potential gradient along their true trajectories are the same as along the approximated unperturbed rays. Having found all the assumptions made well justi"able, we can conclude that the random walk of light rays towards the surface of recombination leads to smoothing of small-scale features in the

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CMB, while large-scale features remain una!ected. The border line between small and large angular scales is determined by the angular coherence scale of gravitational light de#ection by large-scale matter distributions, which we found to be of order +20, corresponding to  l "2p \+1000. For the smallest angular scales, well into the damping tail of the intrinsic CMB   power spectrum, this smoothing leads to a substantial re-distribution of power, which causes the lensed CMB power spectrum to fall systematically above the unlensed one at l92000, or

:2pl\+10. Future space-bound CMB observations, e.g. by the Planck Surveyor satellite, will achieve angular resolutions of order 95, so that the lensed regime of the CMB power spectrum will be well accessible. Highly accurate analyses of the data of such missions will therefore need to take lensing e!ects by large-scale structures into account. One of the foremost goals of CMB observations is to derive cosmological parameters from the angular CMB power spectrum C . Unfortunately, there exists a parameter degeneracy in the sense J that for any given set of cosmological parameters "tting a given CMB spectrum, a whole family of cosmological models can be found that will "t the spectrum (almost) equally well (Zaldarriaga et al., 1997). Metcalf and Silk (1998) and Stompor and Efstathiou (1999) showed that the rise in the damping-tail amplitude due to gravitational lensing of the CMB can be used to break this degeneracy once CMB observations with su$ciently high angular resolution become available. We discussed in Section 4.2 how shapes of galaxy images can be quanti"ed with the tensor Q of GH second surface-brightness moments. Techniques for the reconstruction of the intervening projected matter distribution are then based on (complex) ellipticities constructed from Q , e.g. the quantity GH s de"ned in (4.4). Similar reconstruction techniques can be developed by constructing quantities comparable to s from the CMB temperature #uctuations q(h). Two such quantities were suggested in the literature, namely q !q #2i q q    

(9.51)

(Zaldarriaga and Seljak, 1999) and q !q #2 i q   

(9.52)

(Bernardeau, 1997). As usual, comma-preceded indices i denote di!erentiation with respect to h . G The transformation of the tensor q q between the lensed and unlensed CMB anisotropy G H distribution is mediated by the e!ective surface mass distribution i (h), de"ned as in (6.16) with  w set to the comoving distance to the last-scattering surface at z&1000. As shown by Zaldarriaga and Seljak (1999) and Seljak and Zaldarriaga (1999), one can reconstruct the power spectrum of the projected surface density from the observed statistical properties of q q ; in fact, this power G H spectrum can be obtained either from the trace-part of this tensor, corresponding to i itself, or  from the trace-free part, corresponding to the power spectrum of the shear which, as was shown earlier, is the same as that of i . In contrast to similar studies based on the distortion of faint  galaxies, the power-spectrum estimate from the CMB has the advantage that the redshift of the source is known. Furthermore, the power spectrum of the projected matter distribution can be obtained over a wide range of angular scales, corresponding to a wide range of spatial scales. Even if the temperature anisotropies are intrinsically Gaussian, lensing will induce non-Gaussian features of the measured temperature map (e.g. Winitzki, 1998). Hence, measurements of nonGaussian temperature #uctuations must be interpreted with care. However, the lensing-induced

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non-Gaussian features on small angular scales are correlated with large-scale temperature gradients (Zaldarriaga, 1999), thus providing a signature of the presence of lensing e!ects in the maps. Lensing of the CMB can also be correlated with lensing e!ects of faint galaxies at lower redshift. The shear acting on these galaxies is part of the shear acting on the CMB, the di!erence being due to the di!erent redshift of galaxies and the last scattering surface. Hence, one expects a correlation between these two shears (van Waerbeke et al., 1999a), as can be measured by correlating galaxy ellipticities with either of quantities (9.51) or (9.52). Finally, it is worth noting that gravitational lensing mixes di!erent types of CMB polarisation (the `electrica and `magnetica, or E and B modes, respectively) and can thus create B-type polarisation even when only E-type polarisation is intrinsically present (Zaldarriaga and Seljak, 1998). This e!ect, however, is fairly small in typical cosmological models and will only marginally a!ect future CMB polarisation measurements.

10. Summary and outlook We have summarised the basic ideas, theoretical developments, and "rst applications of weak gravitational lensing. In particular, we showed how the projected mass distribution of clusters can be reconstructed from the image distortion of background galaxies, using parameter-free methods, how the statistical mass distribution of galaxies can be obtained from galaxy}galaxy lensing, and how the larger-scale mass distribution in the Universe a!ects observations of galaxy shapes and #uxes of background sources, as well as the statistical properties of the CMB. Furthermore, weak lensing can be used to construct a mass-selected sample of clusters of galaxies, making use only of their tidal gravitational "eld which leaves an imprint on the image shapes of background galaxies. We have also discussed how the redshift distribution of these faint and distant galaxies can be derived from lensing itself, well beyond the magnitude limit which is currently available through spectroscopy. Given that the "rst coherent image alignment of faint galaxies around foreground clusters was discovered only a decade ago (Fort et al., 1988; Tyson et al., 1990), the "eld of weak lensing has undergone a rapid evolution in the last few years, for three main reasons: (i) Theoreticians have recognised the potential power of this new tool for observational cosmology, and have developed speci"c statistical methods for extracting astrophysically and cosmologically relevant information from astronomical images. (ii) Parallel to that e!ort, observers have developed new observing strategies and image analysis software in order to minimise the in#uence of instrumental artefacts on the measured properties of faint images, and to control as much as possible the point-spread function of the resulting image. It is interesting to note that several image analysis methods, particularly aimed at shape measurements of very faint galaxies for weak gravitational lensing, have been developed by a coherent e!ort of theoreticians and observers (Bonnet and Mellier, 1995; Kaiser et al., 1995; Luppino and Kaiser, 1997; van Waerbeke et al., 1997; Kaiser, 1999; Rhodes et al., 1999; Kuijken, 1999), indicating the need for a close interaction between these two groups which is imposed by the research subject. (iii) The third and perhaps major reason for the rapid evolution is the instrumental development that we are witnessing. Most spectacular was the refurbishment of the Hubble Space Telescope

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(HST) in December 1993, after which this telescope produced astronomical images of angular resolution unprecedented in optical astronomy. These images have not only been of extreme importance for studying multiple images of galaxy-scale lens systems (where the angular separation is of order 1 arcsecond) and for detailed investigations of giant arcs and multiple galaxy images in clusters of galaxies, but also for several of the most interesting results of weak lensing. Owing to the lack of atmospheric smearing and the reduced sky background from space, the shape of fainter and smaller galaxy images can be measured on HST images, increasing the useful number density of background galaxies, and thus reducing the noise due to the intrinsic ellipticity distribution. Two of the most detailed mass maps of clusters have been derived from HST data (Seitz et al., 1996; Hoekstra et al., 1998), and all but one published results on galaxy}galaxy lensing are based on data taken with the HST. In parallel to this, the development of astronomical detectors has progressed quickly. The "rst weak-lensing observations were carried out with CCD detectors of &1000 pixels, covering a fairly small "eld-of-view. A few years ago, the "rst (8 K) camera was used for astronomical imaging. Its 30;30 "eld can be used to map the mass distribution of clusters at large cluster-centric radii, to investigate the potential presence of "laments between neighbouring clusters (Kaiser et al., 1998), or simply to obtain high-quality data on a large area. Such data will be useful for galaxy}galaxy lensing, the search for halos using their lensing properties only, for the investigation of cosmic shear, and for homogeneous galaxy number counts on large "elds, needed to obtain a better quanti"cation of the statistical association of AGNs with foreground galaxies. It is easy to foresee that the instrumental developments will remain the driving force for this research "eld. By now, several large-format CCD cameras are either being built or already installed, including three cameras with a one square degree "eld-of-view and adequate sampling of the PSF (MEGAPRIME at CFHT, MEGACAM at the refurbished MMT, and OMEGACAM at the newly built VLT Support Telescope at Paranal; see the recent account of wide-"eld imaging instruments in Arnaboldi et al., 1998). Within a few years, more than a dozen 8- to 10-m telescopes will be operating, and many of them will be extremely useful for obtaining high-quality astronomical images, due to their sensitivity, their imaging properties and the high quality of the astronomical site. In fact, at least one of them (SUBARU on Mauna Kea) will be equipped with a largeformat CCD camera. One might hypothesise that weak gravitational lensing is one of the main science drivers to shift the emphasis of optical astronomers more towards imaging, in contrast to spectroscopy. For example, the VLT Support Telescope will be fully dedicated to imaging, and the fraction of time for wide-"eld imaging on several other major telescopes will be substantial. The Advanced Camera for Surveys (ACS) is planned to be installed on the HST in 2001. Its larger "eld-of-view, better sampling, and higher quantum e$ciency } compared to the current imaging camera WFPC2 } promises to be particularly useful for weak lensing observations. Even more ambitious ground-based imaging projects are currently under discussion. Funding has been secured for the VISTA project of a 4 m telescope in Chile with a "eld-of-view of at least one square degree. Another 4 m Dark Matter Telescope with a substantially larger "eld-of-view (nine square degrees) is being discussed speci"cally for weak lensing. Kaiser et al. (1999a) proposed a new strategy for deep, wide-"eld optical imaging at high angular resolution, based on an array of relatively small (D&1.5 m) telescopes with fast guiding capacity and a `rubbera focal plane.  See http://www-star.qmw.ac.uk/jpe/vista/

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Associated with this instrumental progress is the evolution of data-analysis capabilities. Whereas a small-format CCD image can be reduced and analysed &by hand', this is no longer true for the large-format CCD images. Semi-automatic data-reduction pipelines will become necessary to keep up with the data #ow. These pipelines, once properly developed and tested, can lead to a more &objective' data analysis. In addition, specialised software, such as for the measurement of shapes of faint galaxies, can be implemented, together with tools which allow a correction for PSF anisotropies and smearing. Staying with instrumental developments for one more moment, the two planned CMB satellite missions (MAP and Planck Surveyor) will provide maps of the CMB at an angular resolution and a signal-to-noise ratio which will most likely lead to the detection of lensing by the large-scale structure on the CMB, as described in Section 9. Last but not least, the currently planned Next Generation Space Telescope (NGST, Kaldeich, 1999), with a projected launch date of 2008, will provide a giant step in many "elds of observational astronomy, not the least for weak lensing. It combines a large aperture (of order 8 m) with a position far from Earth to reduce sky background and with large-format imaging cameras. Even a relatively short exposure with the NGST, which will be optimised for observations in the near infrared, will return images with a number density of several hundred background galaxies per square arcminute, for which a shape can be reliably measured; more accurate estimates are presently not feasible due to the large extrapolation into unknown territory. Comparing this number with the currently achievable number density in ground-based observations of about 30 per square arcminute, NGST will revolutionise this "eld. In addition, the corresponding galaxies will be at much higher mean redshift than currently observable galaxy samples. Taken together, these two facts imply that one can detect massive halos at medium redshifts with only half the velocity dispersion currently necessary to detect them with ground-based data, or that the investigations of the mass distribution of halos can be extended to much higher redshifts than currently possible (see Schneider and Kneib, 1998). The ACS on board HST will provide an encouraging hint of the increase in capabilities that NGST has to o!er. Progress may also come from somewhat unexpected directions. Whereas the Sloan Digital Sky Survey (SDSS; e.g. Szalay, 1998) will be very shallow compared to more standard weak-lensing observations, its huge angular coverage may compensate for it (Stebbins, 1996). The VLA-FIRST survey of radio sources (White et al., 1997) su!ers from the sparsely populated radio sky, but this is also compensated by the huge sky coverage (Refregier et al., 1998). The use of both surveys for weak lensing will depend critically on the level down to which the systematics of the instrumental image distortion can be understood and compensated for. Gravitational lensing has developed from a stand-alone research "eld into a versatile tool for observational cosmology, and this also applies to weak lensing. But, whereas the usefulness of strong lensing is widely accepted by the astronomical community, weak lensing is only beginning

 Whereas with the 8- to 10-m class ground-based telescopes deeper images can be obtained, this does not drastically a!ect the &useful' number density of faint galaxy images. Since fainter galaxies also tend to become smaller, and since a reliable shape estimate of a galaxy is feasible only if its size is not much smaller than the size of the seeing disk, very much deeper images from the ground will not yield much larger number densities of galaxy images which can be used for weak lensing.

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to reach that level of wide appreciation. Part of this di!erence in attitude may be due to the fact that strong-lensing e!ects, such as multiple images and giant arcs, can easily be seen on CCD images, and their interpretation can readily be explained also to the non-expert. In contrast, weak lensing e!ects are revealed only through thorough statistical analysis of the data. Furthermore, the number of people working on weak lensing on the level of data analysis is still quite small, and the methods used to extract shear from CCD data are rather intricate. However, the analysis of CMB data is certainly more complicated than weak-lensing analyses, but there are more people in the latter "eld, who checked and cross-checked their results; also, more people implies that much more development has gone into this "eld. Therefore, what is needed in weak lensing is a detailed comparison of methods, preferably by several independent groups, analysing the same data sets, together with extensive work on simulated data to investigate down to which level a very weak shear can be extracted from them. Up to now, no show stopper has been identi"ed which prohibits the detection of shear at the sub-percent level. Weak-lensing results and techniques will increasingly be combined with other methods. A few examples may su$ce to illustrate this point. The analysis of galaxy clusters with (weak) lensing will be combined with results from X-ray measurements of the clusters and their Sunyaev}Zel'dovich decrement. Once these methods are better understood, in particular in terms of their systematics, the question will no longer be, `Are the masses derived with these methods in agreement?a, but rather, `What can we learn from their comparisona? For instance, while lensing is insensitive to the distribution of matter along the line-of-sight, the X-ray emission is, and thus their combination provides information on the depth of the cluster (see, e.g., Zaroubi et al., 1998). Joint analyses of weak-lensing, X-ray and Sunyaev}Zel'dovich data on galaxy clusters promise to substantially improve determinations of the baryonic-matter fraction in clusters and of the structure and distribution of cluster-galaxy orbits. One might expect that clusters will continue for some time to be main targets for weak-lensing studies. In addition to clusters selected by their emission, mass concentrations selected only by their weak-lensing properties shall be investigated in great detail, both with deeper images to obtain a more accurate measurement of the shear, and by X-ray, IR, sub-mm, and optical/IR multi-colour techniques. It would be spectacular, and of great cosmological signi"cance, to "nd mass concentrations of exceedingly high mass-to-light ratio (well in excess of 1000 in solar units), and it is important to understand the distribution of M/L for clusters. A "rst example may have been found by Erben et al. (2000). As mentioned before, weak lensing is able to constrain the redshift distribution of very faint objects which do not allow spectroscopic investigation. Thus, lensing can constrain extrapolations of the z-distribution, and the models for the redshift estimates obtained from multi-colour photometry (&photometric redshifts'). On the other hand, photometric redshifts will play an increasingly important role for weak lensing, as they will allow to increase the signal-to-noise ratio of local shear measurements. Furthermore, if source galaxies at increasingly higher redshifts are considered (as will be the case with the upcoming giant telescopes, cf. Clowe et al., 1998), the probability increases that more than one de#ector lies between us and this distant screen of sources. To disentangle the corresponding projection e!ects, the dependence of the lensing strength on the lens and source redshift can be employed. Lenses at di!erent redshifts cause di!erent sourceredshift dependences of the measured shear. Hence, photometric redshifts will play an increasingly important role for weak lensing. Whereas a fully three-dimensional mass distribution will probably

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be di$cult to obtain using this relatively weak redshift dependence, a separation of the mass distribution into a small number of lens planes appears feasible. Combining results from cosmic-shear measurements with the power spectrum of the cosmic density #uctuations as measured from the CMB will allow a sensitive test of the gravitational instability picture for structure formation. As was pointed out by Hu and Tegmark (1999), cosmic-shear measurements can substantially improve the accuracy of the determination of cosmological parameters from CMB experiments, in particular by breaking the degeneracies inherent in the latter (see also Metcalf and Silk, 1998). The comparison between observed cosmic shear and theory will at least partly involve the increasingly detailed numerical simulations of cosmic structure evolution, from which predictions for lensing observations can directly be obtained. For example, if the dark-matter halos in the numerical simulations are populated with galaxies, e.g., by using semi-empirical theories of galaxy evolution (Kau!mann et al., 1997), detailed prediction for galaxy}galaxy lensing can be derived and compared with observations, thus constraining these theories. The same numerical results will predict the relation between the measured shear and the galaxy distribution on larger scales, which can be compared with the observable correlation between these quantities to investigate the scale- and redshift dependence of the bias factor. The range of applications of weak lensing will grow in parallel to the new instrumental developments. Keeping in mind that many discoveries in gravitational lensing were not really expected (like the existence of Einstein rings, or giant luminous arcs), it seems likely that the introduction and extensive use of wide-"eld cameras and giant telescopes will give rise to real surprises. Acknowledgements We are deeply indebted to Lindsay King and Shude Mao for their very careful reading of the manuscript and their numerous constructive remarks. Helpful and constructive comments were provided by Nick Kaiser, Guido Kruse, Yannick Mellier and Martin White. We wish to thank Tereasa Brainerd, Klaus Dolag, Jean-Paul Kneib, Yannick Mellier and Robert Schmidt for allowing us to use their original illustrations and "gures in this review. Finally, and most importantly, we express our gratitude towards our past and present collaborators in the "eld of weak gravitational lensing. Whereas they cannot all be mentioned here, we particularly mention Thomas Erben, Bhuvnesh Jain, Yannick Mellier, Ramesh Narayan, Carolin Seitz, Stella Seitz and Ludovic van Waerbeke for long and fruitful collaborations which have made this review possible. This work was supported in part by TMR Network `Gravitational Lensing: New Constraints on Cosmology and the Distribution of Dark Mattera of the EC under contract No. ERBFMRXCT97-0172, and by the `Sonderforschungsbereich 375a on Astro-Particle Physics by the Deutsche Forschungsgemeinschaft. References Abell, G.O., 1958. Astrophys. J. Suppl. Ser. 3, 211. Amendola, L., Frieman, J.A., Waga, I., 1999. Mon. Not. R. Astron. Soc. 309, 465.

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CHIRAL MULTI-ELECTRON EMISSION

Jamal BERAKDARa, Hubert KLARb Max-Planck-Institut fuK r Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
Theoretical Quantum Dynamics, FakultaK t fuK r Physik, Hermann-Herder-Stra}e 3, 79104 Freiburg, Germany

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 340 (2001) 473}520

Chiral multi-electron emission Jamal Berakdar *, Hubert Klar
Max-Planck-Institut fu( r Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
Theoretical Quantum Dynamics, Fakulta( t fu( r Physik, Hermann-Herder-Stra}e 3, 79104 Freiburg, Germany Received April 2000; editor: J. Eichler

Contents 1. Introduction 2. One-electron photoemission 3. Production of chiral electron pairs by onephoton absorption 3.1. General analytical properties of the circular dichroism in one-photon twoelectron transitions 3.2. Propensity rules for the circular dichroism 3.3. Alternative routes to the circular dichroism 3.4. The circular dichroism and the role of the Pauli principle 3.5. Consistency conditions 3.6. Closed analytical expressions for the circular dichroism 3.7. Dependence of the circular dichroism on the strength of the residual ion's "eld 3.8. The photon-frequency dependence 3.9. Calculational schemes of the polarised onephoton two-electron transitions

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3.10. The convergent close coupling technique 3.11. Circular dichroism in the Auger-electron photoelectron spectroscopy 4. Chiral electron pair emission from laserpumped atoms 4.1. Formal development 4.2. Symmetry relations of the orientational dichroism 4.3. Analytical results 4.4. Calculational schemes and experimental "ndings 5. Conclusions and outlook Acknowledgements Appendix A A.1. The analytical form of the circular dichroism A.2. Analytical expressions for the one-photon double-ionisation cross-section References

* Corresponding author. Tel: #49-345-5582666; fax: #49-345-5511223. E-mail address: [email protected] (J. Berakdar). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 8 3 - 1

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Abstract In this report we review recent progress in the understanding of the role of chirality in the multi-electron emission. A brief account of the chiral single-electron photoemission is given. In this case the chirality of the experimental set-up is brought about by an initial orientation of the target or/and by specifying a certain projection of the photoelectron spin. The dependence of the photoelectron spectrum on the chirality of the experiment is probed by changing the initial orientation of the target or by inverting the photoelectron spin projection. In a further section we envisage the direct transition of chiral electron pairs from an isotropic bound initial state into a double-continuum state following the absorption of a circularly polarised photon. We work out the necessary conditions under which the spectrum of the correlated photoelectron pair shows a chiral character, i.e. a dependence on the chirality of the exciting photon. The magnitude and the general behaviour of the chiral e!ects are estimated from simple analytical models and more elaborate numerical methods are presented for a more quantitative predictions. As a further example for the chiral multi-electron emission we study the photoelectron Auger-electron coincidence spectrum. The Auger hole is created by ionising a randomly oriented target by a circular polarised photon. We investigate how the helicity the photon is transferred to the emitted photoelectron pair. The theoretical "ndings are analysed and interpreted in light of recent experiments. In a "nal section we focus on the emission of correlated electrons where the initial state is already oriented, e.g. via optical pumping by circularly polarised light. The initial orientation of the atom is transferred to the continuum states following the ionisation of the target by low-energy electrons. We formulate and analyse the theoretical concepts for the transition of the screw sense of the initially bound atomic electron to the continuum electron pair. Numerical methods for the calculations of the cross-sections for the electron-impact ionisation of oriented atoms are presented and their results are contrasted against recent experimental data.  2001 Elsevier Science B.V. All rights reserved. PACS: 33.55.!b; 32.80.!t Keywords: Circular dichroism; Chirality; Photoionisation; One-photon double ionisation; Chirality in scattering reactions; Ionisation of oriented targets; Few particle scattering dynamics

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1. Introduction Since the early days of chemistry it has been observed that some compounds exist in two states that have similar physical and chemical properties. Their optical properties, however, are distinctively di!erent: The plane of polarisation of the light passing through these chemicals is rotated in a di!erent way. As it turned out the reason for this `optical activitya is a symmetry break caused by the special inner construction of the optically active molecules [1]. These compounds possess two di!erent molecular structures that are mirror images of each other just like the left and the right hand. This special case of spatial symmetry has been termed `stereoisomerya. Stereoisomery occurs only when the molecules are spatially asymmetric or `chirala, as termed by Lord Keliv, i.e. if they do not possess a symmetry plane. The reduction of symmetry, regardless of the way this reduction has been brought about, is in fact the underlying reason for the optical activity. As demonstrated experimentally [2] the optical activity can indeed be increased by sculpturing thin "lms that reveals appropriate chirality. Another well established example of the symmetry reduction leading to optical e!ects is observed in magnetic materials. In this case the presence of a time reversal symmetry-breaking magnetisation of the medium results in a non-reciprocal optical e!ects, such as the polarisation rotation. For metallic, re#ecting media, this phenomenon is known as the magneto-optical Kerr e!ect whereas for transparent media it is called the Faraday e!ect [3]. Chirality can also be introduced in the reaction via a suitable choice of the experimental set-up [4,5]. For example, in Ref. [6] two input laser beams (a control and a probe beam) have been used to study second harmonic generation from achiral thin "lms with in-plane isotropy, i.e. the sample as such has no chirality. The linear polarisation of the control beam breaks the re#ection symmetry of the set-up leading thus to a di!erent second harmonic e$ciency for a left and a right-handed circular polarisation of the probe beam. A further example of an experiment-induced chirality is the photoelectron emission from oriented linear molecules. As a matter of de"nition, a chiral linear molecule does not exist, for an arbitrary plane containing the molecular bond can always be chosen as a plane of symmetry. However, for linear molecules either adsorbed on surfaces [7] or "xed in space [8] chirality can be triggered by angular-resolved photoemission experiments. In this case, the molecular axis, the direction of the incident radiation and the momentum vector of the photoelectron form a right-handed frame whose left-handed re#ection is being well de"ned and experimentally controllable, e.g. by #ipping the helicity of the impinging photon [9}14]. For atoms the situation is slightly di!erent due to the absence of the molecular bond axis. In fact, single photoionisation of unpolarised atoms is insensitive to the helicity of the photon and thus the photoelectron spectra show no chiral e!ects [15,16]. Nonetheless, it is possible to observe chiral e!ects in atomic systems by constructing a chiral experimental set-up. This can be achieved, e.g., by performing the photoionisation experiment with the spin of the photoelectron being resolved [17}21]. In this case a co-ordinate system with well-de"ned orientation can be formed by the wave vector of the photon, the spin projection of the emitted photoelectron and the photoelectron's vector momentum. Alternatively, one might dismiss the laborious spin detection of the photoelectrons but use instead optical pumping to polarise the bound initial state [22}31]. A chiral e!ect appears then because the orientation of the target, the photon's wave vector and the vector momentum of the ionised electron build a right or a left handed co-ordinate frame depending, for example, on the orientation of the initial state. For naturally polarised targets, such as magnetised

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samples, this dichroic e!ect serves as a standard method for the investigation of element selective magnetic properties [32}35]. In recent years, it became clear that chirality can also be observed in one photon two-electron emission from completely isotropic targets and without resolving the spins of the photoelectrons [36}54]. In this case, which is the subject of the present article, the co-ordinate frame is spanned by the vector momenta of the two photoelectrons (k and k ) and the wave vector of the photon k. Since ? @ the photon is absorbed by the two photoelectrons and since the target initial state and the residual ion state are achiral, one can argue that the chirality of the photon is transferred to the two electron pair. That is, the pair attains a chirality as an internal degree of freedom that can be probed by varying the helicity of the photon. At "rst sight the occurrence of such a chirality may seem surprising as the initial target has no internal sense of rotation and the "nal state attains its chiral character by resolving the wave vectors of the two photoelectrons. On the other hand, due to the Pauli principle, the experimental out-come has to be invariant under the exchange of the two photoelectrons. However, as illustrated in Section 3, the orientation of the co-ordinate system (spanned by k , k and ? @ k) is inverted only when kK and kK or k and k are inverted but not when k is replaced by k . This ? @ ? @ ? @ subtle feature of `chiral two-electrona photoionisation is unravelled by a formal mathematical analysis presented in Section 3 which gives insight into the geometrical structure of the chiral e!ect. Furthermore, we discuss in Section 3 the interplay between geometry and dynamics and point out ways and models to quantify and present the phenomena of chirality in two-electron photoemission. As mentioned above, the orientation of an atomic target, e.g. via optical pumping, results in photoelectron spectra that are sensitive to the helicity of the photon. This dichroic e!ect has also been observed in the scattering of charged particle from oriented targets, in particular in the capture channel [55}57]. In this case the appearance of the dichroism (with respect to inversion of the target orientation) is to be expected since for high energies and small momentum transfer the charged-particle scattering is intimately related to photoabsorption process via the dipolar limit. Very recent advances in coincidence detection techniques have rendered possible the investigation of the chiral two-electron continuum following the electron-impact ionisation of laser-oriented targets [58,59]. This is particularly interesting, for such studies o!er an opportunity to zoom in the chirality transfer from a bound system onto a correlated electron pair and to compare with the case where the chirality of the electron pair had been put into the system by an external perturbation, as is the case for double-electron emission upon the absorption of a circularly polarised photon. Thus in Section 4, we envisage theoretically, using a tensorial analysis, the physical and mathematical structure of the chiral electron-pair continuum states achieved upon electron-impact ionisation of oriented atoms. Emphasis is put on the symmetry analysis and on the role of the orientation transfer from the initial target to the two correlated electrons. The formal development is complemented with numerical studies and the theoretical results are contrasted against recent experimental "nding. Finally, we conclude this article by a summary and a brief outlook of future directions. Unless otherwise stated we use atomic units throughout.

2. One-electron photoemission In this section we present compactly the main feature of the photoelectron spectra in single photoemission and point out the reason for the appearance of dichroism, i.e. a di!erence in the

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spectrum when the helicity of the photon is inverted. At a "rst glance one might reject a dichroism in consequence of Yang's theorem [15]. According to this theorem the angular distribution of the photoelectrons, i.e. the cross-section for the emission of a photoelectron under a solid angle X, is described by dp p "  [1#bP (cos h)] ,  dX 4p

(2.1)

where p is the total cross section, b is the asymmetry parameter, P is the second Legendre   polynomial and h is the angle between the photoelectron momentum and the light propagation direction (for circular polarisation) or between the photoelectron momentum and the electric "eld (for linear polarisation). In particular, we see that the helicity of the light does not enter into Eq. (2.1), i.e. there is no circular dichroism. Yang's formula however holds only for unpolarised target atoms and provided the spin of the photoelectron is not being observed. As stated in the introduction, the situation changes when the spin of the electron is resolved or when the target atoms are polarised, e.g., if the atomic target is polarised with a( being a unit vector along its quantisation axis, the angular distribution of the photoelectrons has the general form Ref. [23] dp "4paa u(!1)>O 11q1!q " >02o B(¸, K, >)Y*)( p( , a( ) . (2.2)  ) 7 dX *)7 Here a is the "ne-structure constant, a is the Bohr radius, u is the frequency of the light and p( is  the emission direction of the photoelectron. q quanti"es the polarisation state of the light, i.e. q"$1 indicates right/left circular polarisation whereas q"0 means linear polarisation. In Eq. (2.2) we assumed the hyper"ne structure of the target to be resolved and to be labelled by the quantum numbers F . The population of hyper"ne states of the target is conveniently described by  the density matrix o     (cf. Ref. [60]). In Eq. (2.2) the density matrix has been expressed $+$+ through its state multipoles o (see Ref. [60] for details) via the relation ) o     " (!1))\$ \+ $+$+ )



4p 1F !M F M "KM !M 2o >   (a( ) ,      ) )+ \+ 2K#1  (2.3)

where > is a standard spherical harmonics. The angular function Y*)( p( , a( ) in Eq. (2.2) is the )/ 7 result of coupling two spherical harmonics associated with the directions p( and a( and will be discussed in more details in the next sections. In Eq. (2.2) the generalised asymmetry parameters B appear. Those are given by



K J  B(¸, K, >)"(2F #1)  I F 

J

 F 



(!1)( >'>$ >( >(\ ([(2l#1)(2l#1)(2j#1)(2j#1) JH(JYHY(Y

J. Berakdar, H. Klar / Physics Reports 340 (2001) 473}520



(2J#1)(2J#1)1l0l0"¸02



¸

J

K J

>

1

J 

J  1






¸

J J

J 

j

1 J l ; J""r""J  2 

j



¸ j j  




l

l

479

 

H 1 . J l j; J""r""J  2 

(2.4)

Here the familiar de"nition of angular momentum quantum numbers is used, i.e. ¸ is the total orbital angular momentum, J is the total electronic angular momentum with coupling scheme (¸S)J, I is the nuclear spin, and F is the overall angular momentum with coupling scheme (JI)F. From the symmetry properties of the Clebsch}Gordan coe$cients [cf. Eq. (2.2)] it follows that the term >"1 describes a circular dichroism provided B and Y are non-vanishing. In the simplest case (K"¸"1) the corresponding angular function is given by Y ( p( , a( )Jp( ;a( . This means that  the circular dichroism can be observed provided the vectors p( and a( are linearly independent (i.e. not (anti)parallel). The dichroism should be largest if the photoelectron is observed perpendicular to the quantisation axis a( of the target. A "nite value of B requires J 5. The total cross-section   does not reveal any dichroism since the angular integration over the emission direction of the photoelectron selects ¸"0. The physical origin of the circular dichroism in this case is the existence of an initial target orientation (described by K"1 and realised for instance by optical pumping). In course of the photoionisation process this orientation is transferred to the photoelectron continuum. An absence of the orientation of the target will destroy the dichroic e!ect, as is the case for isotropic target. However, as we will see in the next section the situation is di!erent when two photoelectrons are emitted and simultaneously detected.

3. Production of chiral electron pairs by one-photon absorption In this section we focus on the direct emission of correlated electron pairs with chirality as an internal degree of freedom. By means of a general mathematical analysis we discuss how the chirality of the photon (absorbed by the electron pair) is transferred to the two coupled electrons and in which way the electronic correlation is interfering with this new feature of the electrons. A numerical analysis with various levels of sophistication allows some insights in the actual values of this chiral e!ect and in its dependence on the energy of the photon. The di!erent theoretical approaches are contrasted against the experimental "ndings and the exact analytical results. The case of double ionisation with one linear polarised photon will not be discussed here unless it is of direct relevance to the chiral e!ects. The interested reader is referred to the excellent recent review article [54] and to the references therein. 3.1. General analytical properties of the circular dichroism in one-photon two-electron transitions At a "xed light frequency u, the energy- and angle-resolved cross-section = for the one-photon two-electron transition can be written in the form (see for instance [61,62]) 1 "1W\ (3.1) =(X , X , E )"C k k "e( ) D"U 2" . ? @ ? @ ? 2J #1 + +

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Here the momenta of the two escaping electrons are labelled by k and k whereas the E is the ? @ A incident radiation energy and C"4paa E k k . The many-body dipole operator D" d L L  A ? @ (in the length form) can be expressed in terms of the single particle dipole operator d . The light G polarisation vector is referred to by e( . In Eq. (2) we average over the initial magnetic sublevels M , and sum over the magnetic sublevels M of the photoion. If we de"ne E and E as the energies of  ? @ the escaping electrons and E"E #E as their total kinetic energy, Eq. (3.1) can be rewritten in ? @ the form =(k , k )"C 1W\ (3.2) k k "DSDR"W\ k k 2 , ? @ ? @ ? @ + where C"4paa E and D"e( ) D is the two-particle photon dipole operator. The operator S is  A given by 1 "U 21U "d(E #e !E) . (3.3) S" A 2J #1 + Here e is the (negative) binding energy of the two electrons in the initial state. Eq. (3.3) quanti"es the density of occupied states. In the context of many-particle theory S is usually expressed in terms of the imaginary part of the Green function G of the occupied states (this is valid for an in"nite two-particle life time) 1 ! Im G(E)" "U 2d(E!e )1U " . p + Using Eq. (3.4) the cross-section (3.2) then reads

(3.4)

C =(k , k )"! 1W\ (3.5) k k "D Im G(E!E )DR"W\ k k 2 . ? @ ? @ ? @ A p + Most of modern theories of single photoemission from extended systems relies on the so-called one-step photoelectron current formula that has been derived by Caroli et al. [63]. This formula rests on the non-equilibrium Green function formalism. The Caroli formula states that upon the absorption of a one low-energy photon by an electronic system, the photoelectron current J(k) can be calculated as J(k)"!(1/p)1tk "D Im G(E !E )DR"tk 2 where "tk 2 is the excited ("nal) state of I A the electron with a wave vector k and energy Ek . Thus, relation (3.5) is nothing but the exact analogue of the Caroli formula for the two-particle current [64]. Hence, the conclusions drawn in the subsequent sections are readily generalised to extended systems, such as solids and surfaces [67]. A thorough discussion of the many-particle currents is beyond the scope of the present work, some details can be found in Refs. [64}66]. For the production of chiral electrons it is decisive that the target (prior to the absorption process) is randomly oriented, i.e. the chirality of the electron pair is imparted by the photon "eld. This isotropy of the target state ensures that the projection operator S (as given by Eq. (3.3)) is a scalar. Therefore the operator D"SD is a polar vector operator with respect to overall rotations. It should be emphasised, however, that this conclusion would not be true if the initial state were oriented, as is the case in the previous and the next section.

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Our strategy to extract formal analytical expressions for the circular dichroism is based on a decoupling of geometrical from dynamical e!ects. This is achieved by expressing the vectors e( , e( H, D and D as spherical tensors of rank one and employing a tensorial re-coupling scheme that formally resembles the one used by Fano and Macek [68] in their polarisation analysis of the dipole radiation emitted from oriented and aligned targets. The product (e( ) D)(e( H ) D) can be expanded symbolically in the outer and inner products as (3.6) (e( ) D)(e( H ) D)"(D ) D)#(e( ;e( H) ) (D;D)#¹ (e( , e( H) ) ¹ (D, D) ,     where ¹ (x, y) stands for a tensor of rank 2 formed from the two spherical tensors of rank one  (the components of ¹ (x, y) are given by the formula ¹ (x, y)" 11p1!q"2Q2x y where  / NO N O 12"22 denotes the Clebsch}Gordon coe$cients). A change of the light helicity corresponds to a replacement of e( by its complex conjugate e( H. The vector product term in relation (3.6) is the only term being odd with respect to this replacement. This term is therefore the quantity sensitive to inversion of the helicity of the photon. In what follows we make the convention e( (e( H) to describe left (right) circularly polarised light. Furthermore, we denote the di!erence of the cross-sections for left and right circularly polarised light as the circular dichroism (hereafter referred to by CD), i.e. CD"=(p>)!=(p\) where p> (p\) stands for the helicity of the left (right) hand circularly polarised photon. Further simpli"cation is achieved by assuming the z-axis to be the direction k of the incident light. With e( "( (1, i, 0) and e( ;e( H"!ikK we conclude for the dichroism the relation  2, (3.7) CD(p>, k , k )"!iC 1W\ k k "[D;D] "W\ ? @  k? k@ ? @ + where the index zero refers to the z-component, i.e. [D;D] "[D;D] ) kK . This identi"es the CD  as an expectation value of the pseudovector operator D;D. We note in this context that, for a general N electron system, the dichroism CD cannot be regarded as an orientation in the sense of Fano and Macek [68]. This is because, in contrast to the cases treated in Ref. [68], the present "nal state is generally not an eigenstate of total angular momentum (except for a S initial state). So far this formal development applies as it stands for single photoionisation. However, as we mentioned earlier, according to Yang's formula [15] the CD should vanish in case of single photoemission. Thus, it is instructive to show (a) that our mathematical analysis is consistent with Yang's formula and (b) that the CD is generally "nite for many-electron emission. To this end we de"ne the circular dichroism d in the case of single photoemission as the correspondence to Eq. (3.7), namely (3.8) dJ1tk "[D;D] "tk 2 .  As we will show d vanishes due to conservation of parity. This is deduced from the partial wave expansion "tk 2" "t 2C (kK ) (3.9) JK JK JK where C (kK ) is a spherical harmonic in the notation of Ref. [69]. Upon substituting (3.9) into JK the expression for d, applying the Wigner}Eckart theorem, and performing the sum over m,

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we conclude






l l 1 C (kK )1t ""D;D""W\2 . dJ (!)JY JY J 0 0 0  JYJ

(3.10)

The symbol




j  m 

j  m 

j

 m 



is a 3!j symbol. The point is now that only even or only odd values of l and l contribute because of parity conservation. We assume here that the initial state is a parity eigenstate which is usually the case. The 3!j symbol in (9) is then equal to zero because l#l#1"odd. Moreover l"l because (l, l, 1) satisfy a triangular relation. In contrast to single photoemission, for one-photon double ionisation (PDI) parity conservation does not imply the absence of dichroism. To show this we proceed as in the case of single photoionisation and expand the "nal state into partial waves, 2BJ? J@ (kK , kK ) . "W\ k k 2" "W\ ? @ J? J@ JK JK ? @ J? J@ JK

(3.11)

The bipolar spherical harmonics BJ? J@ (kK , kK ) [69] are basically the tensor products of two spherical JK ? @ harmonics (with appropriate normalisation) and are given by BJ J (x( , y( )" 1l m l m " lm2C   (x( )C   (y( ) .    JK JK JK K K Parity conservation in (10) implies that l #l is either even or odd. We substitute the partial wave ? @ expansion (3.11) into (3.7) and apply the Clebsch}Gordan series for bipolar harmonics [70] BJ? J@ (a( , bK )BJY? JY@ (a( , bK )H"(!)J? >J@ >JY>KY((2l#1)(2l#1) JK JYKY




l

(2¸ #1)(2¸ #1) ? ? @ 0 *? *@ )/ 1lml!m " KQ2B*? *@ (a( , bK ) . )/

l ? 0

¸ 0

?




l

@ 0

l @ 0

¸ @ 0





l l

l

? @

l

K

l ? l @

¸

?

¸

@

 (3.12)

In the "nal state we couple the angular momentum J of the ion with the angular momentum l of  the electron pair to the resultant J. Applying the Wigner}Eckart theorem, the summation over all magnetic quantum numbers can then be performed, and we "nd for the dichroism CD(p>, k , k )"!i c ? @ B*? *@ (kK , kK ) * *  ? @ ? @ *? *@

(3.13)

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483

with c ? @ "C **

(!)J? >J@ >JY>(Y>( >(2J#1)(2¸ #1)(2¸ #1) ? @   J? J@ JJ? J@ JY((Y l l 1 (2l#1)(2l#1) l? l? ¸? l@ l@ ¸@ l J J l l ¸ ? ? ? 3 0 0 0 0 0 0 J 1 l l l ¸ @ @ @ 1J (l l )l; J""D;D""J (l l )l; J2 . (3.14)  ?@  ?@ To simplify further Eqs. (3.13) and (3.14) we remark that, usually, in a photoionisation experiment the initial target as well as the "nal ion are in parity eigenstates. The pair of escaping electrons in one-photon double ionisation will therefore be in a parity eigenstate with parity p"(!)J? >J@ where l and l are the orbital angular momenta of the electrons. Since the two electrons are ? @ exchanging energy and momentum the one-electron angular momentum states are not useful to characterise the pair's states. In fact, many pairs of angular momenta (l , l ) will contribute to the ? @ two-electron continuum state such that l #l is either even or odd. For example, double ? @ photoionisation of He or H\ in the ground state (S) leads to the P symmetry with con"gurations (l , l )"(s, p), (p, d), (d, f ),2 . ? @ From (3.14) it follows that only pairs of (¸ , ¸ ) with ¸ "¸ contribute to the circular ? @ ? @ dichroism (3.13). Further inspection of the 3!j symbols in (3.14) reveals that "nite coe$cients require the following relation be ful"lled:

















l #l #¸ "even , ? ? ? l #l #¸ "even . @ @ @ We add these two equations and conclude ¸ #¸ "even because l #l and l #l are both ? @ ? @ ? @ either even or odd. Then we see from the 9!j symbol that the three numbers (1, ¸ , ¸ ) satisfy ? @ a triangular relation. Since the case ¸ "¸ $1 leads to odd values of ¸ #¸ we arrive at ? @ ? @ ¸ "¸ . Therefore, Eq. (3.13) can be simpli"ed as ? @ CD(p>, k , k )"!i c B** (kK , kK ) . (3.15) ? @ **  ? @ * Since only the diagonal elements c contribute to Eq. (3.15) one might expect that the dichroism is ** less sensitive a quantity to the description of the scattering dynamics than the cross-sections. As discussed below, the CD carries particular information related to phase di!erences of the optical transition amplitudes. A measurement of the CD alone is not su$cient for a complete description of the photodouble-ionisation (PDI) process. 3.2. Propensity rules for the circular dichroism As any physical observable, the CD must be invariant under an exchange of the two electrons, i.e. CD(p>, k , k )"CD(p>, k , k ) . ? @ @ ?

(3.16)

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In addition as clear from the de"nition of the dichroism the CD is odd with respect to inversion of the helicity of the photon, i.e. CD(p>, k , k )"!CD(p\, k , k ) . ? @ ? @ From Eqs. (3.16) and (3.17) we conclude

(3.17)

CD(p>, k , k )"!CD(p\, k , k ) (3.18) ? @ @ ? this means a grand re#ection of the experimental set-up results in a sign change of the CD (the CD is a pseudoscalar in the laboratory coordinate frame). To analyse the behaviour of the CD when the electrons' energies and the solid angles X , X are the relevant parameters we have to discuss the ? @ angular functions B** (kK , kK ). These functions are decisive for the angular behaviour of the CD as  ? @ seen from Eq. (3.15). They are explicitly given by B** (kK , kK )" 1¸M¸!M " 102C (kK )C (kK ) .  ? @ *+ ? *\+ @ + From this equation we deduce the following properties:

(3.19)

1. B** (kK , kK ) are purely imaginary. This follows from the relation C (x( )H"(!)+C (x( ) for  ? @ *+ *\+ spherical harmonics and the symmetry formula 1¸!M¸M " 102"!1¸M¸!M " 102 for Clebsch}Gordan coe$cients. The dichroism CD is a di!erence of cross-sections and as such must be real. Therefore, we conclude that the coe$cients c in (14) are real as well. ** 2. B** (kK , kK ) are parity-even in the solid angles associated with the momenta k , k of the two  ? @ ? @ photoelectrons, i.e. B** (!kK ,!kK )"B** (kK , kK ) which follows from the parity of spherical  ? @  ? @ harmonics given by C (!x( )"(!)*C (x( ). *+ *+ 3. From the symmetry of Clebsch}Gordan coe$cients, we deduce furthermore that Eq. (3.19) is odd with respect to exchange of the electrons, i.e. B** (kK , kK )"!B** (kK , kK ). This relation  @ ?  ? @ implies that: 4. B** (kK , kK ) vanishes when the two electrons escape in the same direction and, due to relation  ? @ (3.16), the functions c has to satisfy the condition ** c (k , k )"!c (k , k ) . (3.20) ** ? @ ** @ ? This leads to a vanishing dichroism for emission of two electrons with equal energies. 5. B** vanishes when the electrons recede in a back-to-back con"guration (kK "" !kK ). This is  @ ? concluded by considering the quantity B** (x( , x( )" 1¸M¸!M " 102C (x( )C (x( )  *+ *\+ + and substituting the expansion

(3.21)

C (x( )C (x( )" 1¸M¸!M " K021¸0¸0 " K02C (x( ) . (3.22) *+ *\+ ) ) The orthogonality of Clebsch}Gordan coe$cients selects then the only value K"1 for which, however, 1¸0¸0 " 102"0. For kK "!kK we use C (!x( )"(!)*C (x( ) and repeat the @ ? *+ *+ arguments above.

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6. The CD vanishes if the direction of the incident light kK and the electrons' vector momenta k , k ? @ are linearly dependent. The above consideration assumes a co-ordinate frame with the z-axis being along kK . Now let us select an arbitrary direction of kK described by the polar angles h , u . ? ? ? If the three vectors kK and k , k are linearly dependent the spherical position of kK is determined ? @ @ by h , u "u #Np with N"integer. The product of phase factors of the spherical harmonics
@ ? is then real. For this reason also the bipolar harmonics (3.19) are real which contradicts the prediction that they are purely imaginary, except for the case when they are equal to zero. 7. The CD vanishes in a non-coincidence experiment, i.e. if we integrate over one of the directions kK or kK . This follows directly from the orthogonality of spherical harmonics and ¸51. Thus, ? @ one can interpret the CD as a feature which is shared by the correlated chiral electron pair, exclusively. If one of the electrons of the chiral pair is not detected the CD is destroyed. This fact together with the exact geometrical propensity rules exposed above makes the CD a predestine candidate for the study of entanglement in quantum systems. 3.3. Alternative routes to the circular dichroism As shown by an elegant mathematical analysis [40,48,49], the propensity rules of CD, as listed above, appear naturally from the parameterisation of the cross-section (3.1) as =(X , X , E )"p #mp kK ) (kK ;kK )#p +3Re[(e( ) kK )(e( H ) kK )]!kK ) kK , ? @ ?   ? @  ? @ ? @ #p(3"e( ) kK "!1)#p(3"e( ) kK "!1) . (3.23)  ?  @ Here m is the degree of circular polarisation and the "ve dependent parameters p , p , p , p     depend on the energies of the ejected electrons and on the mutual angle between the emission directions of the two electrons. It is the term mp kK ) (kK ;kK ) in Eq. (3.23) that describes the CD. The  ? @ triple product in this term encompasses, basically all the geometrical properties of the CD. The dynamical features of the CD are contained in the function p . The numerical evaluation of this  function requires a dynamical model for the motion of the electron pair in the presence of the "eld of the atom. Such models will be presented and discussed in the forthcoming sections. In Ref. [40] Eq. (3.23) has been derived using a reduction scheme for bipolar harmonics. Another approach that proved very useful in analysing the PDI process relies on the Wannier}Peterkop}Rau theory [71}73], as formulated in Ref. [74]. To illustrate the use of this theory we consider an initial state of the electron pair with a S symmetry. Thus, the two coupled photoelectrons go over into a P symmetry with magnetic quantum numbers M"$1 upon the absorption of a circularly polarised photon. According to Table 3 of Ref. [74] the general form of the "nal state wave function reads (3.24) W "$F+sin 0 e! P? #sin 0 e! P@ ,$F+!sin 0 e! P? #sin 0 e! P@ , , ? @ ? @ +! where the functions F and F depend on the rotationally invariant positions r , r of the two ? @ electrons and the inter-electronic angle 0 "cos\(r( ) r( ). In Eq. (3.24) the angles 0 , 0 and u , u ?@ ? @ ? @ ? @ are the polar and azimuthal angles of r and r in a frame with the z-axis along the direction of the ? @ photon beam. The functions F and F are, respectively, symmetric and antisymmetric under an interchange r  r . To arrive at the angular structure of the PDI cross-section one employs the asymptotic ? @

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form of (3.24) which yields =(X , X , E )" ""a +sin h #sin h e! P,#a +!sin h #sin h e! P," . (3.25) ? @ ? +!  ?
 ?
The complex functions a (E , E , h ) and a (E , E , h ) are given by the asymptotic behavior of  ? @ ?@  ? @ ?@ quantities F and F in case the azimuthal and polar angles and h are now being associated ?@ ?@ with the directions of the momenta k and k as measured with respect to a z direction pointing into ? @ the photon beam direction, and u is given by u" ! . In the particular case of symmetric @ ? energy sharing (E "E ) the asymptotic form of the "nal state is symmetric in r  r , and this ? @ ? @ causes the second term in (3.25) to vanish, i.e. a "0.  The CD is derived from Eq. (3.25) to be CD"8 Im(a aH) sin h sin h sin u . (3.26)   ?
Again we verify from this relation the propensity rules for the CD, e.g., there is no dichroism if k and k are linearly dependent, i.e. if (u"0 or p). The CD vanishes in symmetric energy sharing ? @ (E "E ) because in this case a "0. Eq. (3.26) suggests "rst rough estimate for the value of the ? @  CD, namely the CD is largest for h "h "u"p/2 (note, however, that the gerade and the ?
ungerade amplitudes depend on the inter-electronic angle). The angular dependence of CD as given in (3.26) is consistent with the structure of the CD as given by Eq. (3.15) in terms of the bipolar harmonics. In fact, it is straightforward to show that the angular part sin h sin h sin u of Eq. (3.26) can be written as ?
i kK ) (kK ;kK ) . (3.27) sin h sin h sin u"!2iB (kK , kK )"(!2i) ? @ ?
 ? @ (2 Furthermore, one can verify that the portion B is contained in all angular functions B** . In other   words, B** can be written as B** "B BI ** "(i/(2)kK ) (kK ;kK )BI ** . This relation allows to rewrite ? @      Eq. (3.15) in the form i CD(p>, k , k )" c BI ** kK ) (kK ;kK ) . ? @ ? @ (2 * ** 

(3.28)

Hence, all three forms of the CD given by Eqs. (3.15), (3.23) and (3.26) have the same angular structure. In fact, the general parameterisation (3.23) can be expressed in terms of the gerade and the ungerade amplitudes a and a as [48,49]   p "p#p#p cos h ,     ?@ p "!i[a aH!aHa ] ,      p"+"a "#(a aH#aHa )#"a ", ,         p"+"a "!(a aH#aHa )#"a ", ,         p "+"a "!"a ", ,     p "p#p#p cos h . (3.29)     ?@

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3.4. The circular dichroism and the role of the Pauli principle The angular dependence of the CD in the laboratory frame, as given by Eqs. (3.28), (3.23) and (3.26), allows the interpretation that the three vectors kK , kK and kK span a (generally non-ortho? @ gonal) co-ordinate system S in space. An inversion of the helicity of the photon corresponds to an inversion of the orientation of S. Such an inversion of the orientation of S (and hence an inversion of the helicity) can also be achieved by exchanging the momentum directions kK , kK . On the other ? @ hand, the Pauli principle imposes the condition that for a given helicity state of the photon, the coincidence rate should be invariant under an exchange of the roles of the two emerging electrons, as stated in Eq. (3.16). Thus, it is useful to look into the role of the Pauli principle and its e!ect on the CD. For this we choose a situation where the photon beam is perpendicular to the plane spanned by kK , kK . The coincidence rate depends only on the electrons' energies and the inter? @ electronic angle h . Therefore, we choose, without any loss of generality, the x-axis as the bisector ?@ of h (cf. Fig. 1). In Fig. 1(a) and Fig. 1(b) we consider the same experimental set-up with, ?@ respectively, positive and negative helicity of the photon. Thus the CD is the di!erent in the coincidence rate between the situation in Fig. 1(a) and Fig. 1(b). It is clear that Fig. 1(a) cannot be obtained from Fig. 1(b) by any rotation operation. The Pauli-principle requires that the outcome of the experiment should be the same for cases of Fig. 1(a) and Fig. 1(d). Equivalently, the set-up of Figs. 1(b) and (e) should yield the same results. It is clear from the diagrams that the fast electron (indicated by the longer arrow) is situated always to the right of the x-axis and this situation

Fig. 1. An illustration of the role of the Pauli principle in the chiral electron-pair emission following the absorption of a single photon. The wave vector of the photon is perpendicular to the plane of the drawing and its helicity is indicated by c . The x-axis is the bisector of the inter-electron emission angle. The arrows labelled by e and e indicate the emission N! ? @ direction of the photoelectrons and their lengths correspond to their relative energies.

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remains unchanged upon inclusion of the Pauli principle. As shown in Fig. 1(b) and Fig. 1(c) as well as in Fig. 1(e) and Fig. 1(f ), the helicity state can be #ipped from p\ to p> by a 1803 rotation at the x-axis. However, in this case the faster electron will emerge left to the x-axis. From these diagrams (cf. in particular Fig. 1(b) and Fig. 1(c)) it is obvious that the CD disappears when the two electrons escape with equal energies and/or they emerge parallel or anti-parallel to each others. In this sense the CD can be regarded as a left}right asymmetry e!ect. The appearance of this asymmetry is a manifestation of the parity conservation: The system as a whole (the photon and the atom) has a certain parity, carried in the initial state by the photon. After the photon had been absorbed, this parity state is given over to the two electrons that become a chiral electron pair. 3.5. Consistency conditions Having established the general features of the CD we turn now to the actual calculations of its value for various kinematics and targets. Before introducing sophisticated dynamical models it is instructive to develop consistency checks that are independent of the speci"c theoretical or experimental approach. To this end we construct the circular polarised state of the photons from two independent linear polarised states. Using a co-ordinate system where the z-axis is aligned along the wave vector of the circular photon, the optical transition amplitude with left (right)-hand circularly polarised light, labelled as ¹ > (¹ \ ), can thus be written as N N (3.30) ¹ ! "c(¹ $i¹ ) . V W N Here, ¹ (¹ ) is the transition amplitude for the DPE with linear polarised light where the electric V W "eld vector is aligned along the x (y) direction, and c"1/(2. The photoelectrons are emitted with momenta k and k determined in the co-ordinate system x, y, z. For the following it is instructive ? @ to write Eq. (3.30) in the form (3.31) ¹ ! "c["¹ "exp(i )#"¹ "exp(i $ip/2)] , V V W W N where ( ) is the phase of ¹ (¹ ). Thus, the quantities "¹ ! " that determine the cross-section V W V W N (3.1) attain the form "¹ ! ""["¹ "#"¹ "$2"¹ ""¹ " sin( ! )] . N  V W V W W V For the following, it is useful to de"ne a normalised CD as CD such that L CD CD " : . L =(p>, k , k )#=(p\, k , k ) ? @ ? @ From Eq. (3.32) we obtain 2"¹ ""¹ " V W sin( ! ) . CD " W V L "¹ "#"¹ " V W Equivalently, one can show that 2 CD "! Im(¹ ¹H) . L W V "¹ "#"¹ " V W

(3.32)

(3.33)

(3.34)

(3.35)

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In line with the preceding sections we conclude therefore that by changing the polarisation state of the photon information on the phase diwerences of the optical transition amplitudes can be obtained. A quantity that is independent of the polarisation state of the photon can be deduced from "¹ " and "¹ " to be V W (3.36) R" : "¹ > "#"¹ \ """¹ "#"¹ " . N V W N In addition to this polarisation independent quantity, we can characterise the dependence of the PDI process with linear polarised light on the direction of the polarisation axis by de"ning a normalised linear dichroism (LD ) as L "¹ "!"¹ " W . (3.37) LD " V L R We remark however, that "¹ " and "¹ " di!er only in the orientation of the reference axis de"ned V W by the oscillating electronic "eld vector of the photon. Therefore, the LD does not provide further L dynamical details (such as the phase information) other than those already contained in "¹ " and V "¹ ". W The polarisation-independent relation (3.36) does not rely on a speci"c theoretical model for the absorption dynamics (except for the dipole approximation for the radiation "eld). Thus, Eq. (3.36) is in sofar useful as it can be used to check the consistency of the cross-sections "¹ ! " with "¹ " VW N (calculated or measured), as done in Ref. [45]. In addition, we can use Eq. (3.36) to test the consistency of PDI cross-sections with linear polarised light, i.e. the internal consistency of "¹ " and V "¹ ". To see this let us consider the situation where both electrons are detected in the x}y plane. In W this case one can show that ¹ ! (u !u )"¹ ! (u !u ), +∀u , u , u , u 3[0, 2p] " u !u "u !u , . @ N ? @ ? @ ? @ ? @ ? @ N ? This means, ¹ ! depend on the inter-electronic relative angle only. Therefore, according to N Eq. (3.36), the following equality applies: R(u !u )"R(u !u )""¹ (u !u )"#"¹ (u !u )" ? @ ? @ V ? @ W ? @ ""¹ (u !u )"#"¹ (u !u )" . V ? @ W ? @ We note on the other hand that

(3.38)

¹ (u !u )O¹ (u !u ) for u Ou . VW ? @ VW ? @ ?@ ?@ Hence, the inter-relation Eq. (3.38) for the PDI measurement with linear polarised light must be given. From Eq. (3.34) it is readily deduced that (a) the CD vanishes for ! "np and n is an integer, this is for example the case where ¹ and W V W ¹ are both pure imaginary or pure real. We remark here, that the phases depend on the V frequency in a dynamical way. Thus, it might well be that at a certain frequency the phase relation is such that the CD vanishes (see the calculations below). L (b) The CD vanishes when ¹ and/or ¹ vanishes. This observation is most valuable for L V W interpreting the structure of the measured cross-sections with polarised light, as done below.

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(c) The CD diminishes as "¹ "/"¹ "; j"x, y; l"y, x for "¹ "<"¹ ". This condition can be used to L J H H J reveal the high- and low-frequency behaviour of the PDI cross-section [45]. 3.6. Closed analytical expressions for the circular dichroism The previous sections revealed a wealth of information about the properties of the circular dichroism and the information contained within. However, we were basically able to make some predictions concerning the geometry where the dichroism vanishes. It is clear that if the CD is to be "nite there must be kind of a peak or probably structured peak between the various situations where the dichroism diminishes. For an accurate prediction of the size and shape of the CD one has to resort to some modelling of the many-body states (at least two electrons in the "eld of a positive point charge). Before presenting sophisticated theories that cannot give a closed analytical formula for the CD we investigate simpler cases in which the rough behaviour and size of the CD can be deduced analytically. As it is clear from Eq. (3.1), expressions for the many-body wave functions in the initial ("U 2) and the "nal state ("W\2) are needed. While, the initial bound state can be described theoretically with standard methods to a very good accuracy, the correlated dynamics in the "nal state is still a challenge for theorists. Here we have to resort to simple expressions for "U 2 and "W\2 to arrive at analytical results for the CD. For an initial two-electron state with S symmetry we employ the expression U "N exp[!Z (r #r )] , (3.39) Q  Q ? @ where r and r are the positions of the two electrons with respect to the nucleus. The parameter ? @ Z is variationally determined by minimising the binding energy and N is a normalisation factor. Q Q A Ritz variational procedure [81] yields Z "Z!5/16, N "Z/p, where Z is the nuclear charge. Q Q Q This means by using (3.39) we account for the electron}electron interaction as a mere e!ective (angular and radially independent) screening of the nuclear interaction. According to (3.39), in the limit of Z<1 we obtain Z +Z, i.e. the electron}electron interaction can be neglected altogether Q in favour of the nuclear one. In the "nal state we assume the two electrons to move in an e!ective "eld of the nucleus. The electron}electron interaction is subsumed into a dynamical screening of the strength of interaction of the electrons with the residual ion. Mathematically formulated this yields for "W\2 the expression [81] " : (2p)\N N e k?  r? > k@  r@ F [ib , 1,!i(k r #k ) r )] W\ k k (r , r )+W ? @ ? @   ? ? ? ? ? A ? @ F [ib , 1,!i(k r #k ) r )] . (3.40)   @ @ @ @ @ Here, the functions F (a, b, z) are the con#uent hypergeometric functions [75] and the Sommer  feld parameters, that characterise the strength of the two-particle Coulomb interaction, are given by b " : !Z /k . There are various models for deriving reasonable values of the e!ective ?@ ?@ ?@ charges ([76] and references therein). It is not our purpose here to go into the details of this subject. For the sake of simplicity and clarity we make the replacement Z "Z"Z , i.e. we neglect ? @ screening e!ects. The same mathematical steps involved in the derivation of the CD can be repeated when Z depend on the momenta of the two electrons. The normalisation constants in ?@ Eq. (3.40) N , j"a, b are given by N "exp(!pb /2)C(1!ib ), j"a, b. Upon the replacement H H H H

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b ,0,b in Eq. (3.40) we arrive at the plane wave approximation for the photoelectrons and the ? @ optical transition amplitude in velocity form derives from the expression



1k , k "e( ) ( # )"U 2" dp dq1k , k "e( ) ( # )" p, q21 p, q"U 2 ? @ ? @ ? @ ? @ "ie( ) (k #k )UI (k , k ) , (3.41) ? @ ? @ where " p, q2 is a complete set of plane waves and UI (k , k ) is the (six-dimensional) Fourier ? @ transform of the initial state. Since the initial state is randomly oriented the Fourier transform is real and the expression (3.41) is imaginary for any real polarisation vector. Thus, according to Eq. (3.35), the CD vanishes in the plane-wave approximation. Eq. (3.41) makes clear that the phases

and are primarily related to the functional dependence of the phases of the "nal state, for V W Eq. (3.41) is valid for the exact "nal-state wave function if rewritten in the form



I k\H I ( p, q) . 1W\ k k "e( ) ( # )"U 2"ie( ) dp dq(p#q)W k ( p, q)U ? @ ? @ ? @

(3.42)

Again, the random initial state does not contribute to the phase accumulation (as described by the integral) to result in the dynamical phases and . In Eq. (3.42) WI k\? k@ is the Fourier transform of  V W the two-electron "nal state whose phase is the decisive quantity for the CD. This makes comprehensible why the s con"guration in the "nal state cannot contribute to the dichroism whereas the simplest approximation of the "nal state as being (s, p) P leads to a "nite dichroism [37]. Using expressions (3.40) and (3.39), the dichroism, CD(k , k ), is calculated in closed form ? @ (see Appendix A): CD"!ZF(k !k )(kK ;kK ) ) kK . ? @ ? @ The function F reads

(3.43)

F(k , k )"2C (Z !Z)(2Z !Z)(2f f )(k#Z)\(k#Z)\ , (3.44) ? @ ? Q Q ? @ ? Q @ Q where C and f , j"a, b are given by Eqs. (A.4) and (A.7), respectively. Again we recover, the ? H geometric and energetic symmetry properties of CD, as respectively given by the triple vectorial product and the factor (k !k ). The function F contains further dynamical information. Since ? @ (k !k ) and (kK ;kK ) ) kK are both antisymmetric with respect to exchange of the two electrons, we ? @ ? @ deduce F(k , k )"F(k , k ), for Eq. (3.35) must be satis"ed. Furthermore, upon inspection of ? @ @ ? F(k , k ) (Eq. (3.44)) we verify that F is positive de"nite for all combinations of k and k . This ? @ ? @ means that, within the approximations (3.39) and (3.40), CD does not vanish except for the zero points of (k !k )(kK ;kK ) ) kK . This conclusion is also valid for the normalised dichroism CD , as ? @ ? @ L de"ned by Eq. (3.33), since F/R (R " : =(p>)#=(p\)) is also positive de"nite (R'0). These statements concerning the CD apply for all wave functions that contain the inter-electronic L interaction via a co-ordinate-independent multiplicative factor [44], since this factor will cancel out when taking the ratio CD/R. Therefore, we conclude, that a vanishing CD at situations other than those dictated by (k !k )(kK ;kK ) ) kK are traced back to e!ects of the electron}electron interaction other than the ? @ ? @ static screening of the residual-ion "eld.

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3.7. Dependence of the circular dichroism on the strength of the residual ion's xeld As remarked previously the approximate forms (3.40) and (3.39) become more appropriate with increasing nuclear charge Z. Therefore, it is worthwhile to study the CD using Eqs. (3.40) and (3.39) for moderately large Z (we still neglect however the L ) S interaction, i.e. Z should not be too large for the L ) S to become sizeable). Generally, with increasing Z the photodouble-ionisation crosssection decreases rapidly; in contrast the CD remains "nite. This follows from the expression for L CD , L Z (3.45) CD "! (k !k )(kK ;kK ) ) kK , @ ? @ L F ? where F"R/F is given by Eq. (A.18). As R and F are positive-de"nite functions of k and k , we ? @ conclude F(k , k )'0. For the case that the photon beam is perpendicular to the plane spanned ? @ by kK and kK , the function F simpli"es to ? @ k#Z k#Z Q (k#Z)#2(k k #Z) cos h . Q (k#Z)# ? (3.46) F" @ ? @ ?@ k#Z @ k#Z ? @ Q ? Q The behaviour of CD as function of Z is pretty much dependent on the experimental situation. L For example, if we consider the CD as function of h for energies of the electrons such that L ?@ k (S), Be>(S) and B>(S) as targets. From Fig. 2 it is also clear that the CD is less sensitive to changes in the ionisation dynamics than the cross-sections. UnfortuL nately, there is as yet no experimental data for the CD for a varying residual ion charge. L 3.8. The photon-frequency dependence The photon-frequency dependence of the CD close to the double ionisation threshold L can be investigated within the Wannier}Peterkop}Rau (WPR) theory that has been discussed in Section (3.3). As well known the gerade and the ungerade amplitudes a and a obey di!erent   threshold laws, namely "a "JEL\ ,  "a "JEL\ ,  where E"E #E is the total excess energy and n is the Wannier exponent [71] (its numerical ? @ value is, e.g. n"1.127 for H\ and n"1.056 for He). For small E, the gerade amplitude a dominates over the ungerade amplitude a within a small energy range of unknown extension.   The dichroism being proportional to the interference between a and a possesses the energy   dependence CDJEL\ sin h sin h sin u . ? @

(3.47)

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Fig. 2. The one-photon double ionisation of Li>(S), Be>(S), B>(S). As sketched, the two electrons escape in the plane of the drawing that is perpendicular to the photon beam. For a "xed relative angle between the electrons h , the ?@ energy of one the electrons, say E , is varied while the the excess energy E"E #E is "xed to be 14.5 eV @ ? @ [E "14.5!E (eV)]. The photon energy is varied as to compensate for the di!erent double-ionisation potentials of the ? @ di!erent targets. The "gures show: =(Li>, p>): thick solid curve, =(Li>, p\): thick dashed curve, =(Be>, p>): solid thin curve, =(Be>, p\): dotted curve, =(B>, p>): dashed thin curve, and =(B>, p\): dash-dotted curve. Also shown is the dichroism CD for the case of Li> (solid curve), Be> (dotted curve), and B> (dash-dotted curve). The cross-section for L Be> (B>) has been multiplied by a factor of 2 (3). For the calculations we employed the analytical formulas, as given by Eqs. (3.40) and (3.39).

A "nite dichroism close to threshold is not forbidden by this consideration but approaching threshold we expect a decreasing dichroism as predicted by (3.47). The range of validity of this law is closely related to that of the WPR theory. When compared with the measured total photodouble-ionisation cross-sections [82] the WPR theory provides reliable predictions within 2 eV above threshold. It should be stressed however, that the CD is essentially a di!erential quantity (it vanishes when integrated over one of the electrons) whereas the WPR is designed for integrated cross-sections. The predictions of the WPR theory for the frequency dependence of CD and of CD L (near threshold the behaviour of R(E) is determined by "a ") can now be compared with the 

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analytical results of Section 3.6. To do this we note that the photon-frequency dependence of R and CD , as described by Eqs. (A.16) and (3.45), is more transparent in the parameterisL ation k "(2E sin a, k "(2E cos a (E is the excess energy). Eq. (3.45) reads then, CD " @ L ? Z(2E(cos a!sin a)(kK ;kK ) ) kK /F, where ? @ 2E cos a#Z 2E sin a#Z Q (2E sin a#Z)# Q (2E cos a#Z) F" 2E sin a#Z 2E cos a#Z Q Q #2E(sin 2a#Z) cos h . ?@ Since F"F/R and the function F(E), as given by Eq. (A.12), is positive de"nite it follows that F(E) is positive de"nite as well. At threshold (EP0) the CD decreases as (E. For high photon L frequency (EPR) the CD is proportional to 1/(E. Except for these two limits the CD possesses L L no additional zero points as function of E (this is due to F(E)'0, ∀E and CD J(E/F). These L conclusions are applicable when employing the approximate wave functions (3.40) and (3.39). Any additional zero points in the CD as function of E has to be assigned to the electron}electron L interaction in the initial and/or "nal state, as discussed later on. 3.9. Calculational schemes of the polarised one-photon two-electron transitions In Section 3.6, we were able to derive analytical results for the dichroism and the cross-sections. From the structure of the wave functions (3.40) it is clear however that the double-ionisation dynamics is probably oversimpli"ed, at least for cases with strong electronic correlation. More accurate results are provided by including the electron}electron interaction either analytically, as in the three-body Coulomb wave model (3C), or full numerically, as in the convergent close coupling method (CCC). It should be noted here that there is a considerable body of theoretical treatment of the one-photon double ionisation by linear polarised photon which have been summarised recently in Ref. [54]. Here we focus on those models that have been employed for the evaluation of the dichroic e!ects in double photoionisation. In the 3C treatment Eq. (3.40) is extended to include the "nal-state electron}electron interaction in the form (cf. Refs. [76}79]) W "N W F [ib , 1,!i(k r #k ) r )] , (3.48) ! ?@ A   ?@ ?@ ?@ ?@ ?@ where r " : r !r and k is its conjugate momentum, b " : 1/2k and N " ?@ ? @ ?@ ?@ ?@ ?@ exp(!pb /2)C(1!ib ). In what follows we improve on the description of the initial state by ?@ ?@ employing a correlated Hylleraas initial state U "N [exp(!a r !a r )#exp(!a r !a r )]exp(br ) . (3.49) F F F ? F @ F @ F ? ?@ The parameters a , b are variationally determined by minimising the binding energy. FF It is out of the scope of this work to go into the very details of the approximate wave functions (3.48) and (3.49), the interested reader can "nd comprehensive details in Refs. [76}79]. As shown in Ref. [76], when expressed in appropriate co-ordinates, the three-body Hamiltonian (two continuum electrons in the "eld of a residual ion) can be written as a sum of three two-body Hamiltonians that commute with each other. This means that the three-body system is considered

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as a sum of three two-body systems. Thus, in the appropriate non-orthogonal six-dimensional space, the coupling between these individual subsystems is not accounted for by (3.48). This coupling can be incorporated approximately into the theory, but on the considerable expense of introducing a radial dependence of the two-body interaction strength Ref. [76]. This makes the numerical implementation quite involved. In addition, it has not yet been possible to "nd a method to normalise the coupled three-body wave function in a mathematically sound way. Nevertheless, there are already some numerical implementations of this radially coupled three-body wave function for the double photoionisation, but in view of the above statements concerning the normalisation, these numerical methods and their results are not conceivably conclusive. Nonetheless, we stress here that analytical methods can only be approximate, for the many-problem is not separable. Their role is to serve as a tool to understand the underlying physics, rather than to compete in accuracy with full numerical methods, such as those outlined below. To get an insight into the e!ect of the electronic correlation on the chiral electron-pair emission we consider in Fig. 3 the same case studied in Fig. 2 using the wave functions (3.48) and (3.49). Comparing Figs. 2 and 3 we observe that the e!ect of the electron}electron interaction is less pronounced when the two electrons escape in almost opposite directions (h "1503). This is ?@

Fig. 3. The same geometry as in Fig. 2 (schematically shown by the drawing), however, the wave function (3.48) has been employed for the "nal state. The theoretical cross-sections at h "1503 (853) have been scaled down by a factor of 2 (3). The ?@ target is a helium atom in its ground state. The total energy of the pair is 14.5 eV. Experimental data are due to Ref. [42].

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understandable from the properties of the electron}electron repulsion which is strongest when the two electrons escape close to each other. When h becomes smaller the in#uence of electronic ?@ correlation becomes prevalent. Probably the most notable feature in Fig. 3, as compared to Fig. 2 and to our "nding of Section 3.1, is that the CD becomes quite small at h "1253 and changes sign when varying the L ?@ mutual angle further to h "853. This behaviour, which is con"rmed by the experimental "nding, ?@ is in so far surprising as it cannot be explained by the analytical results deduced from the approximation (3.40) and (3.39) nor by the general tensorial analysis of Section 3.1. Since this behaviour of the CD occurs only when correlated states are used, we assign it to the elecL tron}electron coupling. However, the underlying physics for this new feature is not clear, i.e. it is still an open question why the phase of the correlated "nal state wave function behaves such that the phase di!erence between ¹ and ¹ vanishes. V W For the explanation of the evolution of the size of the CD as revealed by Fig. 3 we refer to L Ref. [44]. In Figs. 2 and 3 we analysed the energy-sharing behaviour of the CD . The angular correlation of L the CD is investigated in Fig. 4. Since the behaviour of the double ionisation cross-section by L linearly