Physics Reports vol.350

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MICROSCOPIC MODEL APPROACHES TO FRAGMENTATION OF NUCLEI AND PHASE TRANSITIONS IN NUCLEAR MATTER

J. RICHERT , P. WAGNER
Laboratoire de Physique The& orique, UMR 7085, CNRS/Universite& Louis Pasteur, 3, rue de l+Universite& , 67084 Strasbourg Cedex, France
Institut de Recherches Subatomiques, UMR 7500, IN2P3 } CNRS/Universite& Louis Pasteur, BP28, 67037 Strasbourg Cedex 2, France

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

Physics Reports 350 (2001) 1}92

Microscopic model approaches to fragmentation of nuclei and phase transitions in nuclear matter J. Richert *, P. Wagner
Laboratoire de Physique The& orique, UMR 7085, CNRS/Universite& Louis Pasteur, 3, rue de l 'Universite& , 67084 Strasbourg Cedex, France
Institut de Recherches Subatomiques, UMR 7500, IN2P3 } CNRS/UniversiteH Louis Pasteur, BP28, 67037 Strasbourg Cedex 2, France Received August 2000; editor: G.E. Brown Contents 1. Introduction 2. Properties of excited nuclei and nuclear matter: "rst experimental and theoretical attempts 2.1. Critical phenomena in nuclear matter: experimental facts and early interpretations 2.2. Microscopic interpretations 3. Thermodynamic equilibration and phase space descriptions of nuclear fragmentation 3.1. Thermodynamic equilibration in excited nuclear systems generated by nuclear collisions 3.2. Thermodynamic equilibrium: confrontation of experimental data with phase space models 4. Percolation models and fragment size distributions 4.1. Random-cluster models: general concepts and properties 4.2. Percolation models: de"nitions and general properties 4.3. Moments of the cluster size distribution and critical exponents 4.4. Finite size constraints on random-cluster systems 4.5. Models related to percolation and "rst confrontations with the experiment

4 5 6 8 14

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4.6. Relevant observables and percolation analysis of experimental data 4.7. Extensive tests on peripheral collision data: analysis of ALADIN experiments 4.8. Comparison with other fragmentation models 4.9. Final remarks 5. Lattice and cellular model approaches to nuclear fragmentation 5.1. Ensembles, thermodynamic stability, phase transitions in "nite and in"nite systems 5.2. Lattice models 5.3. Cellular models of nuclear fragmentation 6. Finite nuclei and phase transitions: present experimental and theoretical status 6.1. Caloric curves and phase transitions: experimental status 6.2. Signs for phase transitions in "nite systems: tests and applications to the nuclear case by means of lattice models 6.3. Experimental construction of the caloric curve and determination of the speci"c heat 7. Summary and conclusions Acknowledgements References

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* Corresponding author. Tel.: #00-33-388-35-8139; fax: 00-33-388-35-8144. E-mail addresses: [email protected] (J. Richert), [email protected] (P. Wagner). 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 1 2 0 - 4

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Abstract The properties of excited nuclear matter and the quest for a phase transition which is expected to exist in this system are the subject of intensive investigations. High-energy nuclear collisions between "nite nuclei which lead to matter fragmentation are used to investigate these properties. The present report covers e!ective work done on the subject over the two last decades. The analysis of experimental data is confronted with two major problems, the setting up of thermodynamic equilibrium in a time-dependent fragmentation process and the "nite size of nuclei. The present status concerning the "rst point is presented. Simple classical models of disordered systems are derived starting with the generic bond percolation approach. These lattice and cellular equilibrium models, like percolation approaches, describe successfully experimental fragment multiplicity distributions. They also show the properties of systems which undergo a thermodynamic phase transition. Physical observables which are devised to show the existence and to "x the order of critical behaviour are presented. Applications to the models are shown. Thermodynamic properties of "nite systems undergoing critical behaviour are advantageously described in the framework of the microcanonical ensemble. Applications to the designed models and to experimental data are presented and analysed. Perspectives of further developments of the "eld are suggested.  2001 Elsevier Science B.V. All rights reserved. PACS: 25.70.Pq; 64.60.Ak; 64.60.Fr; 05.70.Ce; 05.70.Jk; 05.50.#q Keywords: Nuclear fragmentation; Phase transitions; Lattice models; Cellular models

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1. Introduction Protons and neutrons are able to form bound nuclei with well-de"ned lifetimes. They are "nite and contain a relatively small number of nucleons which varies from one to several hundreds. The nucleons experience a strong short-range interaction which acts together with the long-range Coulomb interaction between the charged protons. The interactions between nucleons determine the ground-state properties and the low-energy excitation spectrum of nuclei which have been studied over several decades. Their description by means of very di!erent collective and microscopic, semi-classical and quantum models has led to a satisfactory understanding of their properties [1}3]. With increasing excitation energy the detailed information coded in the spectrum and the wave functions of the system gets less and less meaningful and stringent for a satisfactory description of the system. Statistical arguments get relevant. Random matrix theories, for instance, have proved to be a very e$cient tool in this regime [4,5]. Under certain circumstances experiments reveal that the behaviour of nuclei can be described in the framework of thermodynamics. Excited nuclei behave like equilibrated compounds which may decay through emission of gamma rays, nucleons or light clusters at energies corresponding to unbound states of the system [6]. The behaviour of nuclei in this regime and its description in terms of thermodynamic and statistical concepts leads to intriguing questions related to the general properties of thermodynamically equilibrated nuclear matter, either "nite like nuclei or (quasi)in"nite as it could be found in neutron stars, in the absence of the gravitational interaction. These questions are also triggered by the fact that nucleons are fermions and nuclear matter in its ground state, at zero temperature, shows the properties of a Fermi liquid of strongly interacting particles [7}10]. For high enough excitation energy it seems sensible to think that such a quantum liquid may go over to a quantum gas by undergoing a phase transition which would be observable in a in"nite system and for which one would "nd characteristic signals in "nite nuclei. This question has become a major source of interest over the last two decades in the nuclear physics community. The only way to get an experimental answer to it goes through energetic collisions between nuclei or nucleons and nuclei with the aim to generate excited "nite nuclear matter. The analysis of the experiments must then be able to reveal its properties and eventually deliver signs for the existence of one or several phase transitions in the in"nite system. Such ambitious objectives have led to intense experimental and theoretical investigations. Several reports have been written on the subject over the last ten years. In the present review we aim to discuss two aspects concerning excited nuclear matter. The "rst concerns the conditions under which it is experimentally generated. The second deals with speci"c microscopic models which we think allow a realistic study and description of experimental facts under the prerequisite that the physical conditions imposed by the experiment match those which are imposed by the theory. The content of the present review is the following. In Section 2 we describe the status of the subject at the beginning of the early 1980s. We present and discuss the attempts to interpret experimental results related to energetic heavy ion collisions and the theoretical models which were aimed to describe the outcome of such collisions, the fragmentation of the involved nuclei. We then turn over in Section 3 to the two major points which have to be taken in account in order to be able to compare experimental results with the type of models which will be introduced. These points

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concern the "niteness of the system and the problems related to its properties with respect to the concept of thermodynamic equilibrium. Section 4 is devoted to the outcome of very energetic nuclear collisions. Nuclei can break up into pieces. Fragments with di!erent sizes, mass and charge numbers, are generated. With the presently available detection instruments they can be recorded. Their charge and, to some extent, their mass can be measured. Charge and mass distributions as well as related quantities are a priori non-trivial observables which contain interesting information about the fragmentation process and the behaviour of excited nuclear matter. We present and discuss the concepts and models which have been introduced in order to describe the fragmentation characteristics of nuclei, as well as the possible reasons for which rather simple-minded models describe these observables with a remarkable success. In Section 5 we introduce microscopic statistical models which may behave as generic approaches for the study of the thermodynamic properties of excited systems composed of interacting nucleons. These models have either been borrowed from other "elds of physics or consist of adapted derivatives of these models. They describe systems which show phase transitions and whose outcome can be confronted with experimental data. Since they are "nite, tests which enable to characterize the phase transitions have been proposed. Some of these tests are introduced in Section 6 and their application both on theoretical models as well as experimental results is presented and discussed in the perspective of the existence of a phase transition in excited nuclear matter. In Section 7 we develop a summary, comment the present status of the "eld and suggest perspectives which may lead to further progress in the "eld. Nuclear fragmentation and the search of phase transitions is a very active "eld of research. This is best seen by looking at the number of preprints and publications which appear regularly. We tried to cover the newest results concerning the aspects of the subject with which the present report is concerned, extending to summer 2000. We also tried to be as exhaustive as possible in the presentation and quotation of the work which has been done. We apologize for the contributions which could have been forgotten because of lack of information about their existence.

2. Properties of excited nuclei and nuclear matter: 5rst experimental and theoretical attempts Finite nuclei in their ground state and low-lying excited states have been studied for over 50 years. They are still objects of interest, in particular the so called exotic nuclei which contain a larger number of protons or neutrons than those which lie in the so called valley of stability. These objects can be conceptually considered as "nite representatives of in"nite nuclear matter which may be realized in large objects of our universe like neutron stars where nucleons experience of course both the nuclear and gravitational interactions, the last one being absent from our present considerations. Extensive theoretical studies of the ground-state properties of in"nite nuclear matter have been pursued over several decades. They were essentially aimed to determine the non-relativistic nucleon}nucleon interaction which originates from meson exchanges between nucleons. Its precise expression can be "xed by means of nucleon}nucleon scattering amplitudes [11}15]. Di!erent expressions of this interaction have been introduced in order to construct e!ective interactions used in detailed spectroscopic studies of the ground state and low excited states of nuclei, see e.g. Refs. [16,17].

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At high enough excitation energy, nuclei which form liquid drops of fermions get unstable. It is tempting to trace a parallel with the behaviour of macroscopic liquids and to conjecture that the nuclear liquid may go over into vapour and that there may exist a transition from a liquid to a gas under precise thermodynamic conditions, in the limit of an in"nite system. But this is not necessarily the case, the evolution with increasing energy could well correspond to a smooth crossover from a bound to an unbound system of interacting particles. The question whether nuclear matter may exist in di!erent phases is one of the important questions raised by the nuclear community and a central point of the present review. The "rst serious attempts to answer this question started about two decades ago. 2.1. Critical phenomena in nuclear matter: experimental facts and early interpretations 2.1.1. Fragmentation of nuclei Experimental information about excited nuclear matter can only be gained through acceleratorinduced nuclear reactions by means of which excited nuclei can be generated and studied. Energetic beams of particles or heavy ions shot on nuclei lead to their fragmentation into pieces. The "nal multiplicity of fragments of di!erent sizes which are collected by the detectors is correlated with the degree of violence of the collision. To our knowledge, the "rst experiment which was concerned with the possible observation of signs related to a phase transition in nuclei was performed in 1982 by Minich et al. [18]. Protons with energies between 80 and 350 GeV were shot on xenon and krypton targets. The resulting experimentally detected fragments with A nucleons (124A431) and Z protons led to fragment yields >(A, Z) which could be parametrized in the form >(A, Z)JA\Of (A, Z, ¹) ,

(1)

where
is a positive exponent and f (A, Z, ¹) a Boltzmann factor which depends on a temperature ¹ and further constants which were "xed by a "t procedure to the experimental data. Expression (1) was in fact inspired by Fisher's droplet model [19] which for the purpose of the analysis was generalized to two types of particles, neutrons and protons. The value of the critical exponent extracted from the "t of the data was
"2.64 and 2.65 for xenon and krypton, respectively. It has to be compared to the value
"2.33 obtained in mean "eld theory [20]. Fisher's condensation theory predicts that the transition from a liquid to a gas proceeds via the formation of droplets whose size distribution follows a power law at the critical transition point. The authors do not claim that these results would be a proof for the existence of a critical behaviour but that they are consistent with it. Following the droplet model, the increase in excitation energy would generate instabilities in nuclear matter and would lead, at some critical temperature ¹ , to the disassembly  of the system, which is here "nite, into smaller pieces of all sizes which are experimentally detected. This pioneering experiment has been prolongated in further work which was pursued by Panagiotou et al. [21,22]. The authors introduced a systematic study of 12 di!erent high-energy reactions induced by protons and carbon ions on Ag, Kr, Xe and U targets. They measured fragment yields corresponding to nuclei with 34Z422. Using the same point of view as in Ref. [19] the yields of fragments of size A were parametrized in the explicit form P(A)JA\I exp[!a (¹)A!a (¹)A#(¹)A]/¹ Q T  "A\IX > ,

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where X"exp[!a (¹)/¹] Q and >"exp[!(a (¹)!(¹))/¹] . T Here ¹ is the temperature of the fragmenting system, a , a , are the surface, volume free energy per Q T particle and the chemical potential, respectively. ¹ was determined independently for the di!erent systems by di!erent means [21]. Under the assumption that there exists a critical temperature ¹ at  which a (¹) vanishes, the di!erent quantities were further parametrized as  a (¹)"18.4(1!¹/¹ ) , Q  a (¹)!(¹)"b(1!¹/¹ ) , T  where b is a coe$cient which has to be "xed. For ¹(¹ where the gas and liquid coexist  a (¹)!(¹)"0 , T hence >"1 at ¹"¹ . Furthermore  a (¹ )"0 , Q  hence X"1 . For ¹'¹ one assumes that a (¹)K0.  Q The expression P(A) was "tted to the experimental data in order to "x b and ¹ , for di!erent  values of k ("1.6}2.33) and in a temperature range ¹"¹$1 where ¹ is the supposed common temperature of all emitted fragments. The values of ¹ were chosen within an interval of 2 MeV in order to take the uncertainty related to this quantity in account. If one "xes the critical temperature to the lowest value of
(¹) in a parametrization P(A, ¹)JA\O2 where
is a so-called apparent exponent, then the "tted value of ¹ coincides for 1.74k41.8 with ¹ K12 MeV. Further   analysis of the same type [22] con"rmed these results, i.e. a critical temperature which lies in the range 9.5}13.5 MeV and an exponent k smaller than 2. 2.1.2. Thermodynamic interpretation of fragment yields: comments and discussions The model which is aimed to reproduce the outcome of the fragmentation reactions presented above presupposes that the process can be described in the framework of thermodynamics and the grand canonical ensemble. It assumes that the fragmenting system is in thermodynamic equilibrium. This assumption is strong since it implies that there exists a relaxation time which is not longer than the time interval which separates the beginning of the process and the time at which fragments are formed. We shall come back to this point at di!erent places below. The system undergoes a phase transition at a characteristic temperature ¹ . For ¹(¹ a gas   and a liquid coexist. The liquid phase is made of particles and composite droplets (fragments)

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whose energy is composed of a volume and a surface contribution. At ¹"¹ the surface  contribution disappears, the droplets disassemble into independent nucleons, hence form an homogeneous gas phase which is reached for ¹'¹ .  Taking the problem of thermal equilibrium apart, further thinking about this attractive picture leads to the following comments. First, the data analysis is restricted to a "nite range in the size of fragments. Light and heavy species are not taken in account. This may introduce a bias which is not under control. It has to be noticed that the exponents
and k which were obtained in Refs. [21,22] do not coincide with the exponent expected from the Fisher model which describes classical and in"nite systems. Finite size e!ects should play a role in the determination of critical exponents, here they are not taken in account. Finally fragments and particles are free, i.e. they do not interact with each other. Even if the nuclear interaction is weak at the fragmentation stage because the system has already expanded, the Coulomb energy is present and certainly non-negligible. Hence it should be taken into account. At this stage, even though there were encouraging signs for the existence of a critical behaviour of nuclear matter, it appeared that the observed phenomenon was not reducible to an ordinary liquid}gas phase transition as described by Fisher's droplet model. However, these "rst steps and the concepts they introduced paved the way to a huge amount of theoretical work which we shall brie#y describe below, as far as the existence of di!erent phases is concerned. 2.2. Microscopic interpretations Here we restrict ourselves to those approaches which have been introduced as stationary and non-stationary descriptions of "nite excited systems. They were aimed to describe the response and fate of nuclei when energy is imparted to them. 2.2.1. Temperature-dependent Hartree}Fock approaches The droplet model [19] is a classical and phenomenological description of an excited, inhomogeneous and unstable matter phase. The most common approximation introduced in the framework of the quantum many-body problem is Hartree}Fock theory in which nucleons move in a self-consistent mean "eld which can be described by an e!ective density-dependent interaction. This framework has been used up to the present time for the study of static nuclear properties, a detailed review of its applications can be found f.i. in [23] and references therein. The theory has been extended to the study of systems at "nite temperature in the framework of the canonical and grand canonical ensembles [24}29]. Using point contact and density-dependent e!ective interactions like Skyrme interactions it is easy to derive explicit expressions of di!erent equations of state such as the expression of the pressure P as a function of the density  for di!erent values of the temperature ¹. The typical behaviour of these quantities is presented in Fig. 1. One observes the existence of an area limited by an envelope, the spinodal line, where the pressure decreases with increasing density. This is the signal for the presence of an instability of matter which corresponds to the formation of an inhomogeneous medium. It is interpreted as a zone where liquid and gas coexist and which is not thermodynamically accessible. The transition from an homogeneous to the unstable regime corresponds to a "rst-order phase transition in the thermodynamic limit. The transition is of second order at the critical point corresponding to the critical temperature ¹ , at the  maximum of the spinodal line where dP/d"0. This behaviour establishes a connection with

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Fig. 1. Equation of state relating the pressure to the density in nuclear matter [25]. The curves represent isotherms. The spinodal region is indicated by dashed curves.

the droplet model physics. Condensation has been studied in this framework [20]. The calculations "x the critical density  and the critical temperatures. The presence and importance of the  Coulomb interaction was "rst taken into account by Levit and Bonche [30], who introduced a description in which the system is composed of a bound, excited nucleus surrounded by an external gas made of light particles in thermal equilibrium with the nucleus. The presence of the long-range Coulomb force produces a sizable qualitative e!ect. It results in the apparition of a so-called limiting temperature, ¹ , which restricts the instability zone to temperatures which are lower than ¹ and hence introduces an important modi"cation. Similar investigations were  performed in Refs. [31,32] with di!erent Skyrme interactions and re"nements concerning, in particular, the treatment of the vapour charge. The temperature-dependent Hartree}Fock approach with density-dependent e!ective interactions is a simple and elegant approximation to the many-body problem of "nite excited systems. A priori it is possible to "nd the correspondence with central ingredients of the droplet model like the concept of surface in the liquid phase. It is however di$cult to control the degree of realism of these microscopic quantum approaches. The fact that the process is described in the framework of the canonical or grand canonical ensemble when the fragmenting system is closed and has a "xed and small number of particles may lead to di$culties in the characterization of the transition.

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We shall come back to this point in the sequel. There are no means to obtain fragment yields which could be compared to the experiment since there are, only two types of species, a bound nucleus and light particles which constitute respectively, the liquid and the vapour phases. This is of course directly related to a major di$culty which comes from the mean "eld description. The phases are homogeneous. Many-body correlations induced by the neglected residual two-body interactions are absent, but their e!ect is very important at the considered excitation energies. This may also explain the high values of the temperature ¹ K15}20 MeV where the system becomes unstable.  Indeed, the interpretation of experiments indicates that these temperatures could be much lower, of the order of 4}6 MeV. Hence this kind of approach is at most indicative for the determination of the properties of "nite excited nuclear matter in thermodynamic equilibrium. It worked however as an incentive to develop more realistic approaches which take explicitly care of the many-body aspect of the problem. We describe some of them below.

2.2.2. Time-dependent descriptions of nuclear fragmentation The general microscopic framework describing energetic collisions leading to the decay of the system into species of all sizes is the quantum many-body scattering theory. In this fundamental approach the scattering S-matrix elements S measure the overlap between any arbitrary manyDG body initial scattering state i with any arbitrary "nal state f [33,34]. A full #edged microscopic approach of this type is however hopelessly out of scope because of its mathematical intricacy. Many pragmatic ways have been devised in order to circumvent this problem. They all rely on non-relativistic formalisms in which non-nucleonic degrees of freedom are parametrized by means of nucleon}nucleon potentials. All approaches which have been used in a time-dependent framework contain a minimum of classical ingredients. Indeed, all of them "x initial conditions at an initial time and follow the evolution of the system up to a "nal time in contradistinction with scattering theory which speci"es both the initial and the "nal state of the system. It is not necessarily clear that classical concepts and assumptions should work in the present physical context, except for the fact that high-energy processes are correlated with short wavelengths. One may however notice that processes like deep-inelastic or "ssion reactions which occur at lower collision energies can be reasonably and surprisingly well reproduced in terms of classical (or semi-classical) equations of motion [35,36]. An explicit time-dependent description generally called `dynamica description shows a priori a further advantage, since it allows to avoid the crucial point concerning thermodynamic equilibrium. Indeed, the dynamical description of the evolution of the system may lead or not to thermodynamic equilibrium which follows a transient period of time in a collision process. There exists essentially two classes of models of this type. The "rst one transcribes the many-body equations of motion into kinetic equations of motion for the one-particle phase-space distribution function f (r, p, t) at point r with momentum p. Each phase space cell [r, r#dr], [p, p#dp] evolves in time through the transport equation p Rf R< Rf Rf # ) # ) "I[ f ] , Rr Rp Rt m Rr

(2)

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where <(r) is a one-body potential and I[ f ], the so-called collision term, contains in principle all the information relative to 2-, 3-, n-body correlations which are generated by the two-body (possibly also 3-,2, n-body) potentials beyond the average "eld <. This quantity and I[ f ] are in principle correlated and "xed in the framework of the initial many-body equation of motion in such a way as to satisfy conservation laws. In practice however they are determined through phenomenological considerations which preserve more or less the quantal aspects of the initial problem. These approximations are particularly tricky for I[ f ] which has to be chosen in such a way that it can be expressed in terms of one-body distributions, in order to ensure the closed form of (2) [37,38]. The mean "eld potential < is generally "xed independently, from purely phenomenological formulations to self-consistent density-dependent expressions derived by means of static Hartree}Fock calculations. As a consequence, a large amount of models of the type of Eq. (2) have been proposed in the literature, see [37] and references therein. They have been used as descriptions of many-body systems, among which are high-energy nuclear collisions. It is not our aim to give a detailed report on the achievements of these descriptions. It has been done in di!erent publications and reports, see the references above. Here we want to restrict the subject to the discussion of some aspects of nuclear fragmentation which have been addressed in this framework. They are of interest for the understanding of the fragmentation process and problems related to thermodynamic equilibrium, the possible existence and (or) the coexistence of phases in nuclear matter. In the formulation of Eq. (2) the mean "eld < plays an essential role and its choice is central for a realistic description of the evolution of the system at high energy. As we have seen in Section 2.2.1, realistic mean "eld calculations in the framework of Hartree}Fock theory with e!ective densitydependent potentials < and the equation of state P() lead to a zone of instability and possibly of metastability whose separation from pure `liquida and `gasa phases corresponds to the existence of a "rst-order phase transition, see Fig. 1. Transport equations may be considered as time-dependent extensions of mean "eld approaches which furthermore avoid the problem of the existence or not of thermal equilibrium in the expanding and decaying system. Much work has been done in this framework in order to identify the existence and consequences of an instability zone where the initially homogeneous system gets inhomogeneous and appears as formed of clusters which can be identi"ed [39,40]. This has indeed been achieved in a pure mean "eld approximation, in the absence of the collision term, when the mean "eld is supposed to possess a #uctuating contribution [41}45], and (or) when the collision term induces #uctuations in the one-body distribution function f (r, p, t) [46,47]. These #uctuations develop collective modes like zero sound, some of which are unstable and break the system [48,49]. The expansion of the system a!ects also the development of the instabilities. The presence of two types of particles, protons and neutrons has been taken into account [50] and comparisons with experimental results have been made [51]. Quantum e!ects on the expansion dynamics have been investigated recently. It comes out that they are non-negligible, especially when the energy of the collective modes which develop in the system are larger than the temperature [52,53]. This approach which describes the fragmentation of nuclei as a mechanical process shows however limitations due to the already mentioned fact that Eq. (2) is an approximation to the many-body problem whose expression allows for many more or less controlled phenomenological ingredients, such as stochastic contributions in the mean "eld and (or) the collision term. These are supposed to simulate the missing many-body features. Unfortunately it is di$cult to control their degree of consistency, much physical information is hidden in numerics and may obscure the

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conclusions which can be drawn from comparisons with the experiment. On the other hand one can believe that transport models of the type given by Eq. (2) are able to describe the time evolution of global observables like inclusive reaction cross sections and collective properties like #ow [37]. The second approach concerns molecular dynamics. It is also an approximation to the timedependent many-body problem. Di!erent formulations starting from the most classical one, classical molecular dynamics (CMD) try to pick up quantal aspects at di!erent levels, going from `quantuma molecular dynamics (QMD) which tries to approximate the many-body wave function at the initial time by means of coherent (gaussian) states and takes care of the Pauli principle [54,55] to the more recent fermion molecular dynamics (FMD) [56}58] and antisymmetrized molecular dynamics (AMD) [59}62] which introduce the antisymmetrization of wave functions in a semi-classical way. In the simplest form (CMD) the time evolution of the system is described in terms of classical equations of motion for protons and neutrons which interact through a two-body potential representing the nuclear interaction and the Coulomb force which acts between protons. These interactions are implemented in such a way that the fundamental ground state properties of real nuclei are reproduced. The e!ects of the Pauli principle can be mimicked by means of a momentum-dependent potential [54]. Applications of this formalism have been developed into di!erent directions. We shall come back in the forthcoming section to the description of heavy ion collisions in which theoretically calculated and experimental determined observables are confronted in order to "x the possible relationship between fragmentation and phases in nuclear matter. Here we concentrate on applications in which molecular dynamics (MD) has been implemented in order to study the properties of a thermalized system for which the velocity distributions of particles are maxwellians. From there on the system evolves in time by expansion for di!erent initial values of the temperature and the density [63}65]. One follows the evolution of the temperature, the pressure and the fragment size distribution of the system with decreasing density. Fragments can be identi"ed by means of di!erent criteria which may be more or less realistic [64,66,67]. In some cases the system crosses the spinodal line and enters the instability zone. Di!erent observables show a behaviour which can be interpreted in terms of Fisher's droplet model, i.e. a fragment size distribution corresponding to a power law corrected for "nite size e!ects and a corresponding critical exponent
K 2.2}2.3, which goes over to other characteristic shapes for other choices of the temperature as expected in the droplet theory, see Fig. 2. These results along with the interpretation of other observables which will be de"ned below are interpreted as the outcome of a realistic simulation of the experimental situation and the sign for the existence of a critical behaviour of the system. The role and importance of the Coulomb interaction has been investigated [64]. Molecular dynamics has also been used in order to construct the equation of state [65], i.e. the thermodynamic properties which link density, pressure and temperature. It is again shown that there exists a critical point for "nite systems, and a power law distribution with
K2.23 which can be interpreted in the framework of Fisher's model. However these results and conclusions concerning criticality have been put in question by another study [68] which claims that the observed signs have not to be interpreted as being related to a critical behaviour but for the existence of an intermediate regime. This behaviour would correspond to a trajectory in the (¹, ) plane on which the evolving system does neither fall back to a drop nor develop into a gas. A power-law distribution can be extracted from the data. The power-law exponent depends sensitively on the initial temperature and the size of the system.

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Fig. 2. Mass distributions for an expanding system of A"100 particles for di!erent initial temperatures [64]. The dots are obtained by means of CMD calculations, the lines are "ts using Fisher's formula.

In Ref. [68] it is claimed that the power-law behaviour of mass yields expected for a system close to its critical point would be purely accidental. Time-dependent transport descriptions of nuclear fragmentation seem a priori to be the most realistic way to the description of high-energy nuclear collisions. They have been widely used in order to describe the possible out-of-equilibrium character of the fragmentation process which characterizes at least the "rst stage of the process [69]. In the present section we concentrated on aspects related to a time-dependent description of thermodynamically equilibrated systems which expand and cool down, exploring di!erent regions of the (P, ) and (¹, ) planes. Calculations show that the results can be interpreted as the entrance of the system into a thermodynamic unstable regime. It seems however not clear how this can e!ectively be related to the existence of a phase

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transition which would characterize an equilibrated system heated up in a "xed volume. In an interpretation in which the system gets critical it is claimed that it is possible to extract critical exponents, at least the power law exponent
[63]. However this quantity is obtained from "nite size systems and hence should be corrected for "nite size e!ects. The time coordinate which is introduced in these descriptions raises the question of time scales, i.e. the question concerning equilibration in energetic light particle reactions or heavy ion collisions. This is the subject of Section 3. Last but not necessarily least, all these time-dependent approaches need a detailed description of the dynamics in terms of more or less sophisticated mean "eld and two-body potentials. It is always di$cult to get a clear feeling for the sensivity of the outcome of numerical calculations and simulations to these details. Paradoxically, it is at criticality that the information on microscopic systems is the less sensitive to the details of the Hamiltonians which govern them. Finally, it is worthwhile to mention here that it has been shown recently [70] that even far-from-equilibrium, spatially chaotic systems can show equilibrium properties such as ergodicity (equivalence between time and ensemble averages), detailed balance in microscopic processes, partition functions and renormalization group #ow at coarse-grained lengths between microscopic and macroscopic scales. This suggests that it might be possible to describe the global behaviour of some far-from-equilibrium systems in the framework of equilibrium statistical mechanisms. Whether this result could be some clue to the actual question raised in the present framework is an open and highly interesting question.

3. Thermodynamic equilibration and phase space descriptions of nuclear fragmentation If an isolated system of particles is such that there is no unbalanced force acting between its constituents and if it does not experience internal structure changes of its constituents it is said to be in mechanical and chemical equilibrium. Thermal equilibrium is realized if the relevant degrees of freedom which characterize the system share equally its macroscopic energy. One can then de"ne a common temperature which characterizes the energy attributed to each degree of freedom. When all three conditions are realized the system is said to be in thermodynamic equilibrium [71] and can be described in terms of macroscopic coordinates, the thermodynamic coordinates like density, pressure and temperature which do not change with time and are linked together through equations of state. In the present section we aim to present and discuss the di$cult and for a large part unsettled problem of thermodynamic equilibrium in small systems like nuclei excited by means of energetic collisions between particles or nuclei called heavy ions if their mass is larger than 4. In Section 3.1 we discuss questions related to the interval of time over which equilibrium can be reached in terms of model estimates. A confrontation between experimental facts and phase space equilibrium models is presented in Section 3.2. 3.1. Thermodynamic equilibration in excited nuclear systems generated by nuclear collisions What is the scenario which governs energetic collisions? Data analysis and their interpretation show that several scenarii are at hand. Typically one expects that part of the bombarding energy is transferred to the internal degrees of freedom of the system, the system may fall into pieces,

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particles and fragments, and expand in time. The remnants are then collected at asymptotic distances in detectors which identify their charges and eventually their masses, along with their kinetic energies and distributions in space. The setting up of thermodynamic equilibrium over the whole or part of the interacting system of particles is a central but open question. Under favourable circumstances, one may "gure out that after some transient interval of time which starts at the beginning of the collision the system reaches at least partial equilibrium at some stage and then expands to in"nity. 3.1.1. Low-energy time scales At very low projectile energies the excitation of nuclei by means of light particles can lead to the formation of an excited complex whose behaviour can be interpreted within the framework of thermodynamics as a system in thermodynamic equilibrium and which can loose part of its excitation energy through the emission of particles, photons or through symmetric or asymmetric "ssion. The complex is the so-called compound nucleus whose formation and decay properties were already studied in the 1930s by N. Bohr [1,72], who conjectured that, in the framework of this scenario, the process can be explained as a two-step mechanism in which the decay of the equilibrated system would be totally uncorrelated from the way it was generated. From a theoretical point of view compound nucleus theory is a phenomenological approach. It has been largely investigated and worked out in terms of microscopic models, among others random matrix theory. There exists an extended litterature on this subject which covers several decades of intensive work [73]. Indeed, the consequences of the model have been abundantly veri"ed by means of physical observables like excitation functions and angular distributions of emitted particles [73,74]. The lifetime
of the compound system can in principle be read from the compound nucleus decay width E through E )
5 leading to
K10\}10\ s depending in particular on the size of the system. These times are substantially larger than
&l/3;10\ s corresponding to the time  necessary to light to cross a nucleus of diameter lfm. At higher energies, above the Coulomb barrier which corresponds to bombarding energies per particle E/AK1}2 MeV, reactions involving heavy ions lead to a new mechanism the so-called deep inelastic collision (DIC) process. Typical contact times during which the projectile and the target interact through the short range nuclear potential are much shorter than compound nucleus decay times, of the order of 10\ s, not much larger than the contact time in the so-called direct reactions in which the ions strike each other peripherally in typical time intervals of 10\ s. The interpretation of physical observables measured in these DICs shows that although the characteristic reaction times are quite short, the complex which is formed over the contact time of the two ions is su$cient to transfer sizable amounts of energy from the relative motion between ions into the excitation of the nucleonic degrees of freedom and the characteristic features of the system can be interpreted in terms of an equilibrated system whose equilibration is reached in as short as some units of 10\ s [35,75]. 3.1.2. Approaches to the equilibration problem at high energies How does this behaviour extrapolate to high-energy events which can lead to the fragmentation of the system into many pieces of di!erent sizes? The problem is intricate since it is di$cult to guess the scenario (or the di!erent scenarii) as already mentioned above. Hence one may proceed in steps, starting with simple-minded models. To our knowledge, the "rst attempt was made by Boal [76], at

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the time where the experiments, which led to the belief that signs for the existence of a phase transition had been seen, were performed [21,22]. In fact the question which was raised did not concern the equilibration but the fragment formation time. Former experiments [77] had shown that the ratio between inelastic proton reactions (p, p) and proton-neutron exchange reactions (p, n), (p, p)/(p, n), is of the order of 2 at 100 MeV incident energy after correction for the Z/N ratio (Z, N" proton and neutron number). This is evident from the fact that chemical equilibrium is not reached, hence strictly speaking thermodynamic equilibrium cannot be either, see above. The author of Ref. [76] postulates that a hot system generated at high-temperature cools down to a fraction of this temperature and, from thereon, experiences particle coalescence leading to the formation of clusters which come out as fragments. De"ning N (t) as the number of clusters of size G i which exist at time t normalized to the total number of particles A , the initial conditions are 2 supposed to be given by N (t"0)" ,  N (t"0)"0, 24i4A , G 2 where  is the matter density. The system starts at temperature ¹ from an homogeneous distribution of particles and its evolution is given by the postulated rate equations N (t)N (t) dN (t) H  I "
G !
N (t)N (t) , (3) GH G>HI G I GI 1# dt GH GH G where the "rst term on the r.h.s. of (3) corresponds to the coalescence of species of size i and j to species of size k and the second term to the decay of species of size k to species of size i. The coalescence and break-up rates are given by


 

 "4

  dv v(v)exp(!v/2¹) , 2¹

where  is the reduced mass of the colliding species, v their relative velocity and (v) the corresponding cross section. The solutions of the system of coupled equations (3) reaches the asymptotic regime after 4;10\ s. The results are shown in Fig. 3 by the histogram for the case of a reaction p#Kr at energies between 80 and 350 GeV. Their global trend agrees quite nicely with the experimental points. The asymptotic time is in agreement with the (p, p)/(p, n) ratio and the estimated rate of cooling of the excited source, &10\ s. One should mention here that the energy is very high and that thermal equilibrium is postulated since one de"nes a temperature ¹ which is furthermore kept constant over the duration time of the process. The expansion of the system in space is not taken into account. In a further work [78] Boal and Goodman abandoned the coalescence scenario of an initially homogeneous and expanded system. They proposed a break-up mechanism related to the entrance of the system into an instability region of the equation of state as already discussed in Section 2. The collision of energetic protons (E "300 MeV) with nuclei was simulated by means of a simpli"ed transport equation of the  Boltzmann}Uehling}Uhlenbeck (BUU) type. The authors emphasize the fact that thermal equilibrium, requested by a classical collision time between nucleons which is short compared to the expansion time of the nucleon gas, may not be established at a time where the system is already dilute, a regime which is reached after a very short characteristic time which is smaller than

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Fig. 3. Histogram of mass yields obtained by means of the rate equations (3) after t"4;10\ s [76]. The dots are experimental results of the reaction p#Kr at E"80}350 GeV.

10\ s. This goes in the same direction as the non-equilibrium (p, p)/(p, n) which was experimentally found. Notice however that the energies involved here are very large, non-nucleonic degrees of freedom are not taken into account and the transport description is somewhat schematic. The di!erent models are used in order to explain the increase of entropy of the system when it crosses the spinodal line and enters the instability zone. This entropy agrees with the quantity extracted from the experiment. Some aspects of fragmentation related to time scales have been discussed by Gross and coll. [79]. The reaction process goes through a transient out-of-equilibrium phase which may ultimately lead to the formation of an equilibrated excited system. In order to estimate the transient time the authors use a BUU-type equation [80}83] which is a mean "eld description without collision term as described in Section 2. They argue that this description may be valid at the beginning of the collision process, up to and not further than the time when the system may break into pieces. At energies of 50 to 60 MeV A, this happens after 50}60 fm/c for a reaction Mo on Mo. Then the system expands and apparently reaches equilibrium at about 200 fm/c due to the dissipation of energy in the nucleonic degrees of freedom. How far the equilibrium is reached is however not "rmly established. Once the distance between the di!erent species gets larger than 3 fm, they do no longer interact by means of the nuclear potential. The system has then reached the so-called freeze-out stage in which it is in chemical equilibrium. A phase space description of fragments and particles which move under their mutual Coulomb interaction can then be introduced. The early time behaviour is con"rmed by experimental results [84].

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Recently relativistic transport equations [85] have been used in order to simulate the heavy ion reaction Au on Au which has been studied experimentally [86,87]. The calculations show that after the collision the excited system separates into a participant region of highly excited matter and a spectator which reaches thermal equilibrium, approximately after 2;10\ s. Indeed, a local temperature can be de"ned through the knowledge of fragment kinetic energies. It characterizes a system which is mechanically unstable and decays into fragments. The values of the temperatures obtained for di!erent beam energies compare well with those which are extracted from the experiments [88,89]. Other decay scenarii borrowed from low-energy reaction mechanisms have been introduced [90}95]. Fragments are formed sequentially in time, by means of the successive binary decay of the excited species which already exist at a given time step. The process starts from an equilibrated excited system. In Ref. [92] this mechanism is supplemented by a boost in time due to the early compression of the system at the beginning of the process. Most of these approaches rely on the Weisskopf formulation of the break-up rates [96] of an excited bound cluster of particles. In Ref. [94] the binary decay is interpreted in terms of "ssion barriers, using a formulation introduced by Swiatecki [97]. The use of the Weisskopf expressions raises problems which shed doubt about their physical relevance at high energies. First, they get unrealistically small at energies corresponding to temperatures above 4}5 MeV [98]. Second [79], each decay event in this formulation should be independent of the preceding one. This is only possible if the characteristic emission time exceeds the characteristic time over which the Coulomb emission barrier changes for the following emission. This time is of the order of 10\ s, much longer than the Weisskopf emission time and not compatible with characteristic fragment formation times as estimated from the experiments [99]. Hence, although confrontation of mass yields obtained in this framework with experiments may seem satisfactory it is di$cult to believe this type of description [100] for highly excited systems. It is of course sensible to believe that fragmentation of nuclei proceeds through a sequence of break-ups, but it is di$cult to guess a priori or to derive from some microscopic model the realistic expressions of the rates which govern them when the excitation energy is as high as several MeV/A. The evolution of the fragment emission time has been studied recently by means of two-fragment correlation measurements over a large range of excitation energies in reactions \ and p on Au [101]. The fragments are emitted from a unique source which is considered to be in thermal equilibrium. Confrontation of the data with results obtained by means of classical trajectory calculations allows to follow the emission time which decreases with increasing excitation energy. The result is interpreted as evidence for a cross-over from surface emission over long times as described by sequential decay and fast bulk emission which is characteristic of a break-up process due to mechanical instability. It has also been looked for further direct insight into the time scale problem. In Ref. [102] central-impact-parameter events from the reaction Ar#V with incident energies between 35 and 85 MeV A were analysed for the shape of the emitted particle and fragment events in momentum space. It is expected that fragments which are emitted quasi-simultaneously show a close to isotropic emission in the centre of mass system, hence are spherical in shape, whereas emission times which are long as it may be in the case of sequential emission would correspond to the existence of an elongated emission axis, hence an ellipsoidal shape. Experimental events were confronted with numerical simulations. It comes out that for energies below 35 MeV A the

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Fig. 4. Evolution of the average length of the interval of time between two successive break-ups in the fragmentation process with increasing excitation energy [104].

fragmentation process is slow. For higher energies, the experimental momentum distributions lie in between the predictions corresponding to a slow and a quasi simultaneous break-up process. Even though the result does not give any quantitative clue to the problem, it shows at least that the characteristic fragmentation time decreases with increasing energy. A fragmentation experiment induced by Ca#Ca at 35 MeV A [103] confronted with di!erent models related to quick and slow processes con"rms the fact that ordinary binary sequential decay may not be the appropriate fragmentation mechanism. Other characteristic time scales can be introduced such as the time for the occurrence of the "rst break-up of the system
, and the average length of the interval of time between successive  break-ups
[104]. These times can be estimated by means of relative angle correlation $$ measurements between breaking species. If the break-up time is slow, Coulomb repulsion will hinder the emission at small relative angle and low relative velocity. Simulations of the correlation functions indicate that
corresponds to prompt decay for excitation energies exceeding $$ 4 MeV/A. Estimates of
are shown in Fig. 4. A long time interval
has been observed for $$  excitation energies lower than 3 MeV/A in peripheral collision events. Even for high excitation energies of the order of 5 MeV/A
&100}150 fm/c. This leads to characteristic times of 300}400  fm/c for complete break-up which are rather large for heavy nuclei, close to the Fermi energy. Recently further emission times have been deduced from IMF}IMF correlation functions obtained from hadron-induced multifragmentation events [101]. One observes an evolution from typical emission times
K500 fm/c at excitation energies EH/A"2 MeV to
K20}50 fm/c for EH/A" 5 MeV and above. The observation of the transition is interpreted as a transition from surface to

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bulk-dominated emission. The characteristic times are somewhat shorter than those obtained in Refs. [104,105] for excitation energies lower than 6 MeV/A. Albeit the numerous and di!erent types of investigations which have been made up to now the arguments concerning a quick thermodynamic equilibration of the system and relying explicitly on time remain the subject of controversies since time is not directly under experimental control. These controversies are in particular raised by time-dependent descriptions of the collision process, in the framework of the quantum molecular dynamics (QMD) models [54,106,107]. These are classical molecular dynamics models which, to some extent, take care of the quantum nature of nucleons. In recent applications of this type of models to speci"c reactions which have been investigated experimentally in some details like Xe on Sn at 50 MeV A incident energy, the confrontation between calculations and experimental results shows good quantitative agreement, except for the mass yields corresponding to fragments with more than 30 nucleons and too small kinetic energies for fragments and particles [108]. It is argued that the system of interacting particles is far from equilibrium, there is no sign for a thermally equilibrated behaviour of the species which are generated during the early stages of the process. In a further study [109] QMD calculations are confronted with 50 MeV A Xe#Sn data which were also analysed in the framework of the statistical multifragmentation model (SMM, see below) [110]. The approach assumes thermodynamic equilibrium at the freeze-out. Experimental data related to the multiplicities of light charged particles (LCP), the average kinetic energy of LCPs and intermediate mass fragments (IMF) are very close to those obtained by means of QMD and SMM simulations. This result concerning the average transverse kinetic energy of IMFs and the total kinetic energy of LCPs contradicts two arguments which were supposed to help to distinguish between a fragmentation in an equilibrated and non-equilibrated system [111]. The preceding discussion shows once more that the experimental data related to fragmentation processes are ambiguous as far as the production mechanism is concerned. It raises the question why the outcome of di!erent descriptions should be so close to each other when the process is over, i.e. at the freeze-out. At the present stage where di!erent models lead to similar results which agree more or less satisfactorily with the experiment it is tempting to conclude that all of them contain part of the truth, at least to a su$cient degree as to agree with the measured quantities. Since there exists no trustful method to settle the point, one has to collect a maximum amount of informations, as exclusive as possible, and confront them directly with theoretical descriptions. In the next subsection we intend to show that in fact most of the data can be understood in terms of systems which reach thermodynamic equilibrium at the freeze-out. As a "nal remark, one may wonder why the occupation of phase space leading to equilibrium goes so fast, seemingly even faster at high than at low excitation energy. In a quantum mechanical microscopic description, the decay width of a state f into a state i at the energy E is given by the golden rule expression
(E)J2 <  (E) , DG D where < is the interaction matrix element between the initial and "nal state and  the density of DG D states at the energy of the "nal state. For increasing energy E,  (E) will increase exponentially and D if < does not decrease as fast as or faster than  which is certainly the case,
(E) increases with E. DG  If we associate a transition time
(E)J /
(E) to this process, one sees that the transition time

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between di!erent states of the system can decrease very fast and hence drastically accelerate the path to equilibration. 3.2. Thermodynamic equilibrium: confrontation of experimental data with phase space models Excited nuclei in thermodynamic equilibrium are described by means of the so-called statistical models. They generally describe a situation where the system is at the freeze-out, the stage at which particles and fragments present in the system are located at relative distances d from each other which are larger than 2}3 fm, so that the nuclear interaction between them is negligible. Before we show applications of this type of models to the analysis of experimental results we present and discuss some of their aspects. 3.2.1. The xrst phase space models To our knowledge, the "rst attempts to describe strongly excited systems of particles and bound clusters were made in the beginning of the 1980s. The concept of statistical multifragmentation was "rst introduced in Ref. [112]. In 1981, Randrup and Koonin proposed a thermodynamic phase space model [113]. They de"ned the partition function Z of a classical system in the grand canonical ensemble in which the number of particles A, N neutrons and Z protons, as well as the energy, is "xed in the average. If A is the average number of particles, one de"nes  "ln Z/A "
 ,  2 2 where ¹"1/2(N!Z). The multiplicity of fragments characterized by A and ¹ is given by





4 r   " 2 3 



2mA  z exp[!(< !A!¹)] . 2 2  

In the "rst term which is an e!ective volume,  is a parameter of order unity which can be "xed by comparison with experiment, r ("1.15 fm) the nuclear radius constant (R "r A). In the    second term which is related to the kinetic energy contribution m is the nucleon mass and  the inverse temperature. The third factor takes care of the internal energy of the fragments z "
gG exp(!G ) , 2 2 2 G where gG "2jG #1 is the degeneracy of the excited level i and G the corresponding energy. In 2 2 2 the fourth term < is the ground-state mass excess which is taken from a liquid drop formula 2 [114],  and  are Lagrange multipliers which are "xed in such a way that the total energy E ,  number of particles A and isotopic composition are "xed,  R E "! ln Z ,  R 1 R A " ln Z ,   R 1 R ln Z , ¹ "   R

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where Z"
exp[!(E !A !¹ )] D D D D and the sum over f extends over all "nal states which are characterized by the number of fragments, their mass numbers, isospin projections, internal excitations, positions and momenta. In this description, particles and fragments do not interact, hence their spatial location is not relevant. The "rst confrontation with experiment was made with an extension of this model in which unstable species decaying by particle emission were included [115]. The fragmentation was supposed to consist of a quick explosion followed by a slow evaporation from unstable excited fragments. The light particle generation ratios d/p, /p, He/p, t/p as well as the pion rates \/p and >/p were worked out and showed good agreement with experiment. Nuclei are "nite objects and closed systems, with a "xed number of particles and a "xed energy. An approximate microcanonical description was introduced later [116] along with an extension which allows for the existence of di!erent fragmentation sources generated in the collision process, which either `explodea for a large enough energy content or decay sequentially for low excitation energy. In this form the model has been used for explicit numerical simulations [117]. In a further step Koonin and Randrup [118] implemented the model with a more rigorous canonical and microcanonical description in which the interaction between fragments and particles is taken into account. The essential quantity in the microcanonical framework is the density of states for a system of A particles with energy E and located in a volume  $ $ (, A, E)"
(A !A) (E !E) , $ $ $ where F is the set of fragmentation con"gurations de"ned by a set of variables A , r , p ,   L L L L (n"1,2, N ) which speci"es the mass, charge, position, momentum and internal energy of the $ species labelled by n. The total energy is written as





p 1 $ L !B # #
< , E "
L L LLY $ 2mA 2 L LY$L L where B is the ground-state binding energy and L eZ Z L LY #< ( r !r ) , < " LSA L LY LLY r !r L LY the potentials which act between the fragments and the particles. Applications of the model which includes metastable states showed the e!ect of the "niteness of the system which is contained in a "xed volume and the importance of the interaction energy between the constituents. 3.2.2. The Berlin and Copenhagen phase space models The "rst statistical theory on fragmentation of "nite nuclei which included an exact treatment of the Coulomb energy between charged particles and fragments was developed in Ref. [119]. The most popular and successful models are the Berlin model called MMMC (microcanonical Metropolis Monte Carlo) [79,120}122] and the already quoted SMM (statistical multifragmentation

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model) [110,123}125]. A description closely related to MMMC in its spirit was developed in Beijing [126]. Except for technical though important points concerning Monte Carlo event samplings which are used in the practical implementation of these models, the main di!erence between MMMC and the Koonin}Randrup model concerns the fact that the system which is at the freeze-out density does not experience evaporation of light particles in its "nal stage. The volume of the system is kept "xed and spherical, the Coulomb interaction between the constituents is treated rigorously. The model has been extensively used in order to confront calculated and experimental observables such as fragment correlation functions at high-energy [127] and also low-energy data [128]. It has been tested for the possible existence of a "rst-order phase transition, see Section 5. Recently it has been extended to non-spherical shapes of the volume and the consequences of this generalization have been studied [129]. The Copenhagen approach has been initially worked out in the framework of the canonical ensemble [123,124] and later on a microcanonical formulation was developed [110]. The approach is rather close to MMMC, see Ref. [130]. The main di!erences concern the fact that particles can evaporate from excited fragments, the Coulomb interaction is treated in the Wigner}Seitz approximation and the volume in which the system is enclosed at freeze-out is not kept "xed. The problem of equilibration cannot be raised in the framework of the model. The question has been investigated by means of simple-minded transport equations [110]. The authors rely on the observation of collective #ow of matter which is generated and whose transverse energy carried perpendicularly to the incident beam axis by the particles and the fragments can be measured. In this framework the confrontation with experiment shows good agreement. The interpretation of the data indicates that fragments are generated very early in the process, over time intervals as small as 50 fm/c. This shows that fragment formation goes as a break-up and not a condensation process if the description is realistic. The instability growth is accelerated by the collective expansion of matter when the system enters the spinodal region. It is argued that the energy transferred to the system is converted into heat, generating a chaotic system which thermalizes somewhat before break-up, hence after a short interval of time. 3.2.3. Thermal equilibrium: confrontation of models with experimental facts Since there exists no direct way to measure the evolution and to "nd out experimentally whether thermodynamic equilibrium is reached or not, one has to rely on the confrontation between measured and model-dependent physical observables extracted from numerical simulations. This procedure has been followed a number of times in the recent past [131}137]. We give here some examples which show how far equilibration is at least compatible with the experimental outcome even if it is not possible to prove it rigorously. In an experiment Ar# Ni at di!erent energies (52, 74, 84 and 95 MeV A) Borderie et al. [138,139] analysed the outcome of fragmentation events in which 90% of the nucleons are identi"ed with the detector INDRA. They showed that these events correspond essentially to the vaporization of the system into light particles and clusters. The forward and backward spectra of particles are superimposable, which at low energy would be interpreted as a sign for the formation of a compound system. The energy distribution functions of the di!erent light species show experimental tails whose slopes are comparable to within 30%. The authors introduce simple thermodynamic models like the so called quantum statistical model (QSM) [140}142] which works

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in the framework of the grand canonical ensemble, in an attempt to reproduce the measured observables. A detailed comparison shows that all measured quantities are well reproduced, in particular light particle yields, their average kinetic energies and the variances of multiplicity distributions. The fragmentation events observed in the reaction Xe# Au at 30, 40, 50 and 60 MeV A projectile energy have been analysed in some detail [143]. The average transverse energy (E "
E sin  , where E sin  is the projection of the energy associated with species i in the R G G G G G direction perpendicular to the incident beam) measures the energy transferred from the relative motion of the ions to the excitation energy of the system. The analysis shows that the average number of light charged particles N and the average number of IMFs and their correspond*!. ing transverse energies increase monotonously with increasing E as shown in Fig. 5. This R contradicts the results of [144] where the transverse LCP energy saturates. This saturation can in fact be understood as being due to instrumental problems. The features observed in Fig. 5 can be interpreted and understood in the framework of a statistical decay mechanism [134]. The results can also be reproduced in the framework of phase space models like SMM [110]. Further analysis of the reaction Xe#Sn have been presented and discussed in the recent past [145}147]. In Ref. [146] correlation techniques were used in order to determine the multiplicities of H and He isotopes emitted by the excited fragments produced at the disassembly stage of an equilibrated hot source. The determination of LCP multiplicities and kinetic energies led to the fragment excitation energies. These energies per particle are the same for all IMFs, which is interpreted as a sign for the fact that thermodynamic equilibrium is established at the stage where the source decays. A very detailed analysis of the same reaction was presented and discussed in Ref. [147]. It is shown there that the experimental fragmentation data can be interpreted in terms of a unique source whose existence could be ascertained by means of kinematical arguments related to global observables like the collective #ow angle. For events which are selected under speci"c criteria the angular distributions of light particles and intermediate mass fragments show the characteristic behaviour of statistical evaporation, i.e. approximate symmetry with respect to 903, see Fig. 6, and excitation functions decrease exponentially as a function of energy. The experimental results were further confronted with simulations by means of the SMM model in its microcanonical version. This confrontation con"rmed that the assumption of thermodynamic equilibrium is compatible with information extracted from experimental data. The analysis relies on the so called backtracing method developed in Ref. [110] which solves the inverse problem by tracing back the issue of the process to the characteristics of its origin, i.e. the characteristics of an excited equilibrated source. This method has also been applied to experiments performed by the ALADIN collaboration at the GSI for reactions of Au on C, Al, Cu and Pb targets at 600 MeV A energy [148}150]. There exist some indications for equilibration from purely experimental origin [151], corresponding to so called spectator reactions, i.e. more or less peripheral heavy ion collisions, in particular Au on Au at high bombarding energies, 0.6 and 1 GeV A. In this type of reactions it is assumed that part of the nucleons, the spectators, form a highly excited system which experiences only disordered motion and from which any coherent collective motion is practically absent. Several arguments lead to the conclusion that these systems could be in thermodynamic equilibrium. Recently the relative velocity correlation between LCPs and fragments has been used in order to reconstruct the size and excitation energies of the so called primary fragments which are

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Fig. 5. Top panel: average number of IMFs (solid symbols) and LCPs (open symbols) as a function of the total transverse energy E . Bottom panel: corresponding average transverse energies as a function of E . The di!erent symbols correspond R R to di!erent bombarding energies indicated in the bottom panel [143]. See text. Fig. 6. Angular distribution of light particles (Z"1, 2) in the #ow angle 5603 corresponding to events generated by a unique source in the reaction Xe#Sn at 50 MeV A [147].

produced at the early stage of the collision of Xe on Sn at 32 and 50 MeV A [152]. The constancy of the excitation energy imparted to the system with increasing bombarding energy suggests that thermodynamic equilibrium may be reached at freeze-out. The fragment distributions are invariant with respect to the formation process, i.e. the entrance channel of the reaction. As we shall see in Section 5, there has been an attempt to "x a temperature to the spectator system by means of the measurement of the isotope ratios of light nuclei (He, Li) which are generated through fragmentation [153]. These ratios are invariant with respect to the bombarding energy. The result is however, at "rst sight, not compatible with other observables from which it is in principle possible to extract a temperature. This is so, in particular, for the slopes of the energy distributions of fragments which, if they are interpreted as Maxwellian distributions, lead to much higher temperatures. Several arguments which try to explain this fact have been proposed [151], among them the role played by the Fermi motion [154] of nucleons related to a fast fragmentation process. The equilibration problem has also been raised by the FOPI collaboration in the study of reactions involving more central heavy ion collisions [89]. In this type of events, one observes a collective #ow of particles. This fact complicates the experimental analysis and does not allow for a clear-cut conclusion about equilibration. Indeed, di!erent models, both time-independent and time-dependent seem to be able to reproduce the data [155]. Recently isospin arguments have been

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used in order to test equilibration [156]. They indicate that equilibration is not reached as far as isospin degrees of freedom are concerned.

4. Percolation models and fragment size distributions The detection and identi"cation of the charge and the mass of fragments generated through violent collisions is of fundamental importance in the study of excited nuclei. Many detectors like ALADIN, FOPI, INDRA, the EOS setup, the MINIBALL, LASSA, CHIMERA are able to detect and identify particles and fragments event by event and to determine their asymptotic kinematical properties which constitute the information which can be experimentally collected. As already seen above, the information contained in the outcome of processes like fragmentation are of extreme complexity and the formation mechanism largely unknown, certainly di!erent under di!erent physical conditions such as the energy range and the impact parameter. On the other hand, the use of the concepts of statistical mechanisms has shown that paradoxically the most complex systems can be described in terms of minimum information theories which bypass the detailed knowledge of the underlying microscopic systems. Here it may be tempting to follow this philosophy and to try to describe the outcome of nuclear fragmentation processes within the simplest possible physical framework. This has been done starting in the early 1980s by means of minimum information approaches like percolation models. These models ground on purely topological and statistical concepts, they avoid the explicit introduction of Hamiltonians. If they work, they raise of course the question why and how this is the case. In percolation theories, space contains empty regions and parts which are occupied by individual objects which can be considered as bound to form larger entities de"ned by means of di!erent criteria. The essential concepts are connectivity, localization and percolation which characterizes the properties of clusters (animals, fragments). Percolation itself is established when, in an in"nite space, at least one cluster of in"nite size is present. Percolation models have been used in many "elds which deal with geometry, disorder, statistics and physics such as aggregation, localization, di!usion, conductivity, etc. Before going over to its application in the "eld of nuclear fragmentation it may be worthwhile to recall general concepts, de"nitions and properties which appear in the framework of this type of descriptions. 4.1. Random-cluster models: general concepts and properties The "rst percolation model was introduced by Broadbent and Hammersley [157] in order to describe the spread of matter through a medium such as a liquid through a porous medium. It was then soon recognized that this type of model is acquainted to already known models like the Ising model [158] and other statistical spin models like those developed by Ashkin and Teller [159] and Potts [160]. All these descriptions can be studied in a general framework called random-cluster model. This has been worked out by Fortuin and Kasteleyn [161,162]. In the "rst paper [161] the authors showed the close connection between the connectivity problem of di!erent systems with uncorrelated bonds between sites which are occupied by single objects (particles) and the well-known

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Ising model. Since Ising systems exhibit phase transitions in 2d and 3d spaces it is not surprising that the random-cluster model shows itself a phase transition in the in"nite space limit. In later developments [162] the authors studied the general topological properties of the aforementioned models by means of graph theory, in particular those of the random-cluster model. They showed why the models quoted in Refs. [158}160] as well as linear resistance networks and percolation models can be studied in a unique framework by means of graphs de"ned in terms of vertices (sites), edges (bonds), connections between edges and vertices, and probabilities p for edges to exist or not. In practice they introduced a generalized partition function (cluster generation function) and `free energya which enables to work out physical observables like thermodynamic quantities in the case of the Ising, Ashkin}Teller and Potts models and correlation functions. Further analytical developments on percolation models were pursued by Kunz and Souillard [163,164] related to previous work by E.H. Lieb. In the case where sites do not interact, i.e. are not correlated, they introduced the free energy p!q.! e\FL "
e\F! , f (h)"
P N L n C ! L where p is the bond probability, q"1!p, P the probability that a given site belongs to a cluster L with exactly n sites, C are all "nite clusters containing the origin of space 0, C is the size of the clusters and R C the size of the boundary of clusters, i.e. their perimeters (surfaces). The study of the properties of f (h) shows that this function is analytic for h"0 and p(p , and develops N A a singularity for p5p if the system is in"nite, indicating a phase transition of second order at A p"p . For p5p the system percolates in the sense de"ned above, i.e. at least one cluster of in"nite A A size appears. 4.2. Percolation models: dexnitions and general properties Before discussing the applications of percolation concepts to nuclear fragmentation we present here some essential properties of percolation models we shall need in the sequel. Percolation models consist of a discrete number of sites which cover a "nite or in"nite part of a d-dimensional space [165}167]. Site percolation corresponds to the case where each site is either occupied by one `particlea or is empty. Bond percolation corresponds to the case where each site is occupied, neighbouring sites being bound or not to each other. The number of neighbouring sites is "xed by the connectivity which depends on the geometric structure of space occupation (f.i. a triangular lattice for d"2, a cubic lattice for d"3,2). The combination of site occupation and bond formation leads to hybrid site}bond models. Except for the results presented in Section 4.1 above [161}164], very little is known analytically about these models. Most information has been gained by means of numerical simulations. In the case of site percolation, one "xes a probability p3[0,1]. For each site, one draws a random number  taken from a uniform distribution in the interval [0,1]. If 4p the site is said to be occupied, if 'p it is empty. Hopping over all sites, one generates an ensemble of occupied and empty sites. If the number N of sites gets very large, one "nds Np occupied and N(1!p) empty sites. If neighbouring sites are occupied they belong to the same cluster. Space is "nally covered with clusters and empty space.

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A similar procedure works in the case of bond percolation. For a "xed bond probability p consider a pair of neighbouring sites and draw a random number 3[0,1]. If 4p the sites are linked by a bond, if 'p they are not. Testing this way all possible bonds between neighbouring sites generates linked clusters of bound sites, each cluster being made up of an ensemble of connected particles. The determination of the whole cluster network will be called event or realization. In each realization there are clusters of di!erent sizes. They appear with a given multiplicity which depends of course on p. Averaging over many realizations leads to a characteristic cluster size distribution in the in"nite system. Clusters with a "xed size can have di!erent shapes for d52. The average number of clusters of size r per lattice site taken over a set of realizations is given by n "
g pP(1!p)Q , P PQ Q where g is the number of clusters with size r and perimeter s, the perimeter being the number of PQ empty (non-connected) sites which surround the cluster. Generally g is not analytically accessible. PQ We shall come back to this point below. Individual clusters of size s can also be characterized by a `gyrationa radius R , de"ned by " Q r !r   , R "
G " s G where Q r r "
G  s G is the centre of mass of the cluster. Then the average-squared distance between two cluster sites is r !r  H . 2R"
G Q s G$H If g(r) is the probability that a site which is a distance r apart from an occupied site belongs to the same cluster, one can de"ne a correlation length  such that [167]
rg(r) , (4) " P
g(r) P 2R is the average distance between two sites of a cluster and sn the probability that a site belongs Q Q to a cluster of size s. Since such a site is connected to s other sites the correlation distance  can be rewritten as 2
Rsn " Q Q Q . (5)
sn Q Q As it has been mentioned above [161}164] there appears a percolation (second order) phase transition for d52 and some value p"p (1. For p'p one or several in"nite clusters are A A present. The system is said to percolate. The value of p depends on the dimensionality of the A

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system, the type of percolation (site or bond) and its connectivity (the number of maximum occupied sites in the next neighbourhood of a site or possible bonds of a site to the nearest neighbours) [167]. The central observable in percolation physics is of course the cluster size distribution. In practice, nothing is known analytically about it, except for conjectures which agree very precisely with numerical simulations. For p(p and clusters of large size (sPR) ln n J!s, i.e. one observes an A Q exponential tail when space occupation (site percolation) or the number of bonds ( bond percolation) is sparse. For p'p and sPR ln n J!s\B. As one can see the space dimension enters in A Q the case where space occupation or the number of bonds is large. This is not the case in the non-percolative regime. Finally for pKp and large clusters it has been conjectured [167] that A n (p)Ks\Of ((p!p )sN) , (6) Q A which is true for all percolation models,
and  being universal exponents and f a model-dependent function. For p"p , f (0)"1, leading to a power-law behaviour which intriguingly reminds A Fisher's expression (1). As stated above, these relations are very accurately veri"ed by numerical tests. 4.3. Moments of the cluster size distribution and critical exponents For in"nite systems it is possible to work out the expression of the cluster size distribution n (p) Q in the vicinity of the critical point p"p . A By de"nition the kth moment is m (p)"
sIn (p) . I Q Q Going over to the continuum limit



m (p)" I



ds sIn (p) Q

 and introducing the analytical expression (6) with f ((p!p )sN)"exp(! p!p Ns) A A for p in the vicinity of p&p leads to A m (p)& p!p II , I A where
!k!1  " . I 

(7)

(8)

For k'
!1 one sees that the moments diverge at p"p , a sign for the existence of a phase A transition. Hence, if
(3 which is the case for d"2,3 m (p)& p!p \A ,  A

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with "(3!
)/ .

(9)

For d"3 in a cubic lattice "1.74. Critical exponents characterize other quantities of interest. If in the in"nite system P(p) is the probability that a given site belongs to the in"nite cluster for p'p , then probability conservation A implies P(p)#(1!p)#
sn (p)"1 . Q Q The second term corresponds to the probability that the site is empty and the third that it belongs to a "nite cluster of size s. At p"p , P(p )"0 A A and
sn (p )"p . Q A A Q Hence in the vicinity of p , P can be written [167] as A P"
[n (p )!n (p)]s#O(p!p ) . (10) Q A Q A Q If one introduces the parametrization n (p)Js\O exp(!cs) with c"0 at p"p and goes over to Q A the continuum limit with (10) it comes PJ(p!p )@ , A where "(
!2)/ ,

(11)

which consistently tends to zero when
'2 at p"p . A Since percolation systems show a continuous transition at p"p one expects that the correlaA tion length  de"ned in (4) shows a singular behaviour at the threshold where an in"nitely large cluster merges. This is indeed the case J p!p \J (12) A and the positive exponent  can be determined numerically. Percolation clusters are generally fractal objects. The radius of gyration R is related to the " volume s through sJRBD , " where d is an e!ective, not necessarily integer space dimension. From expression (5) it follows that D  behaves like s>BD . The same type of derivation as the one used for the moments m (p) leads to I

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a power law expression of  whose numerator diverges with an exponent (3!
#2/d )/. Hence D  itself diverges like 2/d  and an identi"cation with (12) leads to D "1/d . (13) D At p"p R JsBD , hence the size of the largest cluster increases like ¸JsBD . If P is the strength A Q of the in"nite cluster as de"ned above, P¸B is the volume occupied by the largest cluster. If the linear extension of this cluster is of the order of the correlation length  P¸BJ¸BD , ¸JJ p!p \J . A Recalling the behaviour of P in the neighbourhood of p one gets A d "d!/ . (14) D In summary, general considerations lead to six critical exponents
, , , , d , and  which are D linked by four relations (9), (11), (13) and (14). Up to now we considered essentially the properties of in"nite systems and concentrated on their behaviour in the vicinity of the percolation threshold which allows to determine critical exponents. These exponents are very useful quantities since they "x the universality class to which a critical system belongs. If excited nuclei are the "nite counterparts of critical nuclear matter, the singular behaviour of observables at the phase transition is quantitatively characterized by critical exponents. We shall try to see and to understand below if, how and why percolation concepts could be related to nuclear fragmentation. But nuclei are "nite systems. Before we present and discuss the models which have been proposed, we present a study of the e!ects generated by the "niteness of percolation systems. 4.4. Finite size constraints on random-cluster systems Percolation deals with the critical behaviour of systems in the limit of in"nitely extended space. Microscopic and mesoscopic systems like nuclei, atoms, molecules, aggregates are "nite. As far as percolation concepts can be applied to their description and directly compared to experiment as we shall do below, it is necessary to get some insight into the behaviour of percolation models in "nite space. Finite size e!ects enter because of the presence of a surface which induces constraints on the system, especially when the space occupation is large. We concentrate here on site percolation. The aim of the study concerns the determination of the fragment size distribution of a system with < sites, with an occupation number equal to 0 or 1 for each site. The number of occupied sites A is "xed, the density is "A/<. Following percolation concepts one qualitatively expects a set of small clusters when  is small, hence a cluster size distribution which decays monotonously as a function of the cluster size and for large values of  one or several large clusters together with small clusters showing a kind of ;-shape. For intermediate values of  small and intermediate size clusters may coexist. These shapes are in fact observed in the experiment. The crucial point concerns the behaviour of the distribution for these intermediate values of . The behaviour of "nite percolation systems has been investigated in [168]. One considers a "nite volume < in a d-dimensional space. Space occupation by A particles is de"ned by O"(i ,2, i )  

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where i ,2, i label A occupied sites. There are (4),
(15)

Notice that, as generally admitted in random-cluster models, there exists no cluster}cluster interaction. This is due to the fact that the interaction is restricted to nearest-neighbour-occupied sites. The probability to "nd a cluster c of size a in a con"guration is given by P(c)"e\@&AZ( f, A!a, )/Z(<, A, ) .

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Then  "
P(c)(c) A for any observable . In particular, the multiplicity of clusters of size a is given by m(a)"
P(c) (a!a(s)) . A The essential constraint on m(a) is "xed by the "xed number of particles 
am(a)"A . (16) ? Monte Carlo simulations [168] reproduce the expected cluster size distributions which were predicted above, in particular the ;-shape at large density. This is true both in the case when the coupling constant e is zero or "nite except that the transition from the falling distribution for small  to the ;-shape is located at di!erent values of  for di!erent values of e. Hence in order to study the consequences of the constraint imposed by (16) one may restrict this study to the case e"0 which allows for the use of simple analytical arguments. Then





Z(<, A)"

< A

,





 (


s (a)J2#2(d!1)a ,

 in a d-dimensional hypercubic lattice. Of course s4
(18)

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Fig. 7. Plot of the (a, s) plane where the degeneracy g(a, s) is non-zero. For "xed A ("A or A ) only the hatched areas   contribute to the mass spectrum [168].

The overall behaviour of the surface as a function of the cluster size is shown in Fig. 7. If one de"nes an average surface (a)"
sP (s) , ? Q then for the small mass sector one may parametrize (a) in the following form: (a)"a# a;A , 42d!2   and for the large mass sector (a)"K(
   2(d!1)A #A "
(19)

A "(
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For a, <1 G(a)Ja\  exp( a) A with  "ln[(#1)H>/H] . A A similar expression for G(a) is obtained if P (s) is a Poisson distribution. As a consequence ? m(a)Ja\  exp( !)a A with "ln[1/(1!)H] .  is minimum for " at A  "1/(1#) . A Then m(a)Ja\  for " . One again "nds the manifestation of a power-law behaviour, which A is realized in the case of "nite systems. If < is large (a) lies close to 2(d!1)a#2, hence "2d!2 for cubic lattices. Then  "1/(2d!1)"A /< . A  In fact,  is determined by the crossing of the two topological limits "xed by the small and large A cluster sector. Hence in "nite systems the onset of a `criticala regime is correlated with a topological constraint which, in a "nite but large volume, corresponds to cluster surfaces with a maximum in the vicinity of  <"
 In summary, simple considerations lead to a di!erent behaviour of cluster size distributions in di!erent density sectors of "nite systems. This behaviour is essentially governed by the mass conservation constraint (16) imposed by the "nite volume in which the system is located. For small densities the size distribution decreases essentially as an exponential. There exists a density  for A which the distribution is a power law or close to it, with a "xed index irrespective of the dimensionality of space. For ' one observes both a decreasing light cluster sector and a large A size sector building up. The present results are general. As we shall show below they are in qualitative agreement with explicit simulation results and experimentally observed cluster size distributions.

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4.5. Models related to percolation and xrst confrontations with the experiment The "rst applications of percolation concepts to the description of nuclear fragmentation were proposed by Bauer and collaborators [170,171], Campi and Desbois [172]. The starting point of Bauer et al. was the observation of the power-law distribution of intermediate mass fragments [18,21,22] obtained by means of fragmentation processes A #A PA #X , N R D where A , A , A are the mass of the projectile, target and a speci"c fragment and X the rest of the N R D reaction outcome. In the absence of any precise information about the reaction mechanism the authors [170,171] suggested a minimal-information description, following the same spirit as the one which prevailed in Ref. [173]. They introduced a nucleus lattice model (NLM) in which a set of particles occupy lattice sites in a "nite space. The presence of bond clusters is tested by means of a bond percolation algorithm as described above applied to a 3d cubic lattice model in a "nite volume. The so-called standard percolation theory (SPT) deals with in"nite systems. The physical framework here is an A#1 particle system obtained by means of energetic collisions of a proton on a nucleus. The penetration of the proton generates a cylindrical tube of radius r with very excited matter ("reball), in an initial spherical nucleus at an impact parameter b. The nucleons of the "reball are kicked out, the remaining nucleons are sitting on lattice sites and experience a strong disturbance which re#ects in the de"nition of a bond-breaking probability p characterizing the link between the remaining spectator nucleons. In this description the exact number of particles is not "xed and there is no information about the energy imparted to the system. The bond-breaking probability p is supposed to be the larger, the larger the kinetic energy of the incoming projectile proton. A Monte Carlo percolation algorithm applied to the system for di!erent values of p3[0,1] leads to di!erent shapes of the fragment (cluster) size distribution, ranging from an ;-shape for small p to a typical exponential fall-o! with no heavy fragments available for large p, as predicted in Ref. [168]. For some pKp K0.74 the width of the mass distribution gets maximum and the mean A yield shows a power-law behaviour for light and intermediate mass fragments, with an exponent
K2.21. This value is somewhat lower than the one corresponding to the in"nite system. This bond-breaking model was confronted with experimental results concerning the reactions p#Ag, p#Ta and p#Au, for proton energies of several GeVs. If the bond probability p is parametrized in terms of the impact parameter as p(b)"p /[1#exp((b!R)/d)] ,  where R is the radius of the system, b the impact parameter and d a di!usion coe$cient, one "nds a very good agreement between calculated and experimental fragment size distributions, see Fig. 8, by sampling over the whole range of impact parameters, b"0 to R. In order to get a closer link between percolation calculations it is sensible to replace p which is not an observable by the average over realizations of the fragment multiplicities m obtained for a "xed value of p. This allows to relate
extracted from a power-law "t of the fragment size distribution to an average multiplicity at the `criticala point, m . A The strikingly good agreement between the application of simple percolation concepts and the outcome of inclusive observables extracted from complex collision processes raises several

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Fig. 8. Mass yields obtained with the parametrization of the bond probability p(b) given in the text for (a) p#Ag at 11.5 GeV, p "0.65; (b) p#Ta at 5.7 GeV, p "0.69; (c) p#Au at 11.5 GeV, p "0.70 [171].   

questions. The model does not care about mass and energy conservation, possible geometrical deformation e!ects when the fragmenting system is generated through heavy ion collisions. The bond probability is necessarily related to the microscopic dynamics. It may be asked why this unique parameter parametrizes so perfectly the underlying interaction between the nucleons in the excited nuclear medium. We shall come back to this point in Section 5.

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In a similar spirit, Campi and Desbois [172] developed a continuum site percolation approach. A particles occupy an ensemble of points r , p , i"1,2, A  in phase space chosen in a 2 G G 2 classical one-body Wigner distribution f (r, p) determined by a one-body Hamiltonian with 5 a Saxon}Woods potential. The collision process leaves pA nucleons of the original nucleus, 2 (1!p)A nucleons are kicked o! the original nucleus, p3[0,1] depends on the characteristics of 2 the reaction like impact parameter, energy, target size 2 Clusters are generated by means of phase space prescriptions which decide whether particles belong to fragments or not. A given number of particles forms a bound cluster if d " r !r ) p !p 42.5

GH G H G H for every pair of nearest neighbours belonging to the supposed cluster, where the maximum action on the r.h.s. of (4.5) corresponds to the value obtained for the deuteron. This condition is supplemented by a compactness constraint which imposes that the mean-square radius R and  mean-square momentum p veri"es  R 4(1#)r A ,   3  p , (21) p 4(1#) $  5




where p is the Fermi momentum and K0.12. If the constraints (21) are not ful"lled the longest $ link d is cut and the determination of clusters starts again. The "rst condition takes care of the GH balance between volume and surface energy of the cluster, the second eliminates nucleons with too large momenta. The model is constructed with percolation concepts which do however not correspond to the standard ones, essentially because the generation algorithm is not "xed by a single and simple bond probability parameter. The cluster multiplicity obtained in this way can be "tted by means of parametric analytical expressions [172]. The generated clusters are excited objects whose excitation energy EH can be estimated. Particle emission from fragments and their "ssion are taken into account. In order to enable the confrontation of the model with experiment, the quantity p which "xes the number of remaining nucleons after the collisions is related to the number of nucleon}nucleon collisions which are experienced in the system for a "xed impact parameter b and nucleon mean free path  which varies with the energy. This leads to a distribution for p, (p), for p larger than a value p corresponding to b"0. Depending on the physical conditions, p can reach or not

 the percolation threshold. One can then calculate explicitly inclusive mass yields for fragments of size A $  dp (p)m(p, A ) , (A )" $ $ N  where m(p, A ) is the multiplicity of fragments of size A , for a "xed value of p, and the total $ $ reaction cross section





 " 0



N 

dp (p) .

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The model reproduces very well the fragment size yields (A ) for low- and high-energy data $ corresponding to Ne#Ta at 400 MeV A, p#Ta at 5.7 GeV and p#Ta at 340 MeV. In the lower-energy data one observes the opening of the "ssion channel which is predominant along with particle evaporation at this energy. For high energy reactions, p#Xe at 80}350 GeV, both the slope and absolute value of (A ) are well reproduced. The trend of the apparent `criticala $ exponent (A )JA\O# where E is the bombarding energy shows a minimum
&2.4 at $ $ E"50 GeV. This is understood in the following way. When E is small p is large, p cannot be

 A reached, hence the system is essentially made up of one large cluster and a set of small particles. When E increases p decreases, p can be reached and one gets a power law distribution over the

 A whole range of fragments sizes, except for "nite size e!ects. For large value of E p continues to

 decrease, more and more smaller fragments are generated and the apparent slope of the distribution increases again. At very high energy the nucleon mean-free path (E) becomes constant, hence p and
saturate.

 4.6. Relevant observables and percolation analysis of experimental data The "rst extensive confrontation between standard percolation theory and nuclear fragmentation was initiated by Campi [174]. The phenomenon was interpreted as a bond-breaking process between bound nucleons due to the energy supplied to the system under violent collision conditions. As discussed in Section 4.3, the moments m of the fragment size distribution get singular at I p"p for k52, so do the normalized moments S "m /m . Hence one expects that they show A I I  a maximum in the vicinity of p&p . Since m Jexp(! ) with  "(
!1!k)/ (see (7) and (8)) A I I I at p , there should exist a linear relation between ln S and ln S at p . For k"2 and l"3 the slope A I J A can be written as ln S " ln S with "1#1/ and one can in principle read o! the value of the   power law exponent
. The introduction of m and S allows for a direct comparison between the moments obtained I I from bond percolation models (here 3d cubic) and those obtained from the experiment, mH , which I can be determined from the fragment size distributions of a set of events labelled by the index j by taking an average over all available events. Another interesting observable is given by [175] m m  "   ,  m  which is related to the variance  of the fragment size distribution by

(22)

 "1#m /m    and should also show a sizable enhancement at the `criticala threshold. Finally the last observable proposed in Ref. [174] concerns the behaviour of the largest cluster A in the system. Indeed, in the in"nite system, A should get in"nite at criticality, hence it is



 expected to get large when S gets large. Hence the correlation between A and S is expected to 

  contain information of interest. Bond percolation calculations in a cubic lattice in a "nite system were performed for di!erent sizes and S , S ,  and A were calculated. The average quantities were confronted with the   

 outcome of a few hundred events generated by the fragmentation of Au nuclei in emulsions [176]. Typical "ssion events involving two large fragments were excluded. Results are shown in

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Fig. 9. ln S as a function of ln S for single events. Part (a) corresponds to the break-up of gold nuclei, part (b) to a cubic   bond simulation with 216 sites and bond probabilities 0(p(1 [174].

Fig. 9 which relates S and S for both experimental events and percolation simulations. `Criticala   events correspond to large values of S and S . Violent (p"0) and smooth (p"1) collision events   correspond to small values of S and S . One observes that there is a striking agreement between   the two results. It should be noticed that in both cases one mixes events which correspond to di!erent values of p (as far as calculations are concerned) and multiplicities which are related to the degree of violence of the process (as far as the experiment is concerned). They both show the expected linear behaviour which can be related to the presence of `criticala events. It is also remarkable to notice that the slopes K2.22 are in both cases very close to the value "2.25 corresponding to bond percolation in the in"nite system. The same type of agreement can be observed when dealing with S vs. S [174]. In Fig. 10  is drawn as a function of the average    multiplicity m . The similarity of simulation results and experiment is again remarkable. One  observes an enhancement in the neighbourhood of m K0.25 which, when related to the bond  probability p, corresponds to p&p in 3d cubic systems. The fact that  '2 for m K0.25 A   indicates a power law distribution. Finally Fig. 11 compares the behaviour of A as a function of

 S in both cases. `Criticala events are concentrated in the right corner of the "gure. One observes  again a strong agreement between calculations and experiment. The possibility to extract values of  and
and the fact that these are close to standard percolation results may induce the temptation to conclude about the universality class of the apparent critical phenomenon unravelled by the present studies. One must however be careful, because di!erent types of transitions may lead to exponents which lie close to each other but correspond to di!erent classes. The determination of exponents needs some care. We shall come back to the problem concerning the determination of critical exponents in Section 5. 4.7. Extensive tests on peripheral collision data: analysis of ALADIN experiments Many experimental e!orts have been made over the last decade by the ALADIN collaboration of the GSI in order to collect a large amount of signi"cant information about particle and fragment

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Fig. 10. The quantity  de"ned in (22). The upper part corresponds to the fragmentation of gold and the lower part to  percolation simulations, [175].

Fig. 11. Size of the largest fragment for di!erent events as a function of S . (a) Experimental results obtained from 376  events. (b) Bond percolation results in a cubic volume with 216 sites, 4000 realizations [174].

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multiplicities generated in the fragmentation of the projectile by means of energetic peripheral heavy ion collisions. Such precise and complete information allowed for a careful comparison with theoretical approaches, in particular percolation and percolation-inspired models. The "rst measurements which were performed established correlations between the multiplicity of light particles M , the charge Z of the largest fragment in an event and the sum of the charges of the JN

 fragments with Z52, Z [148]. It was found that the correlation between the multiplicity of
 intermediate mass fragments (IMFs), i.e. fragments with 34Z430 and Z was independent of
 the target nucleus. This was interpreted as a sign for the equilibration of the projectile remnant (spectator particles) before decay. The "rst detailed investigation of the average multiplicity of IMFs, M , showed that with increasing energy transferred to the projectile in reactions of Au '+$ on C, Al and Cu targets M "rst increases to a maximum and then decays [149]. This could be '+$ interpreted in the following way. For small energies peripheral collisions lead essentially to one large projectile remnant and a few light particles. With increasing energy, the projectile breaks up into more and more IMFs. If the energy gets very large most fragments are very small, (Z42) and hence M decreases again. These results were then compared to theoretical investigations of '+$ M vs. Z [150] by means of the statistical models of Copenhagen [123}125] and Berlin '+$
 [122], and a sequential decay scenario [177]. It is seen that sequential evaporation cannot reproduce the data. An extensive analysis concerning the reactions with Au on C, Al, Cu and Pb at 600 MeV A was performed in Ref. [178]. The charge distribution was "tted by a power law and the exponent
showed a minimum somewhat below 2 when represented as a function of Z . The average
 largest fragment charge Z increases with increasing Z . Other correlation functions like


  (see above) and  A "[(Z !Z )#(Z !Z )#(Z !Z )]/(6 , 

   where Z "(Z #Z #Z )     and Z and Z are the second and third largest fragment charges were determined.   All results showed the striking feature that it is a simple-minded 3d site}bond percolation model with an occupation probability p for the site and a "xed bond probability p "0.45 obtained Q @ through a "t to the Z distribution which reproduces the best the available data. Masses in the

 percolation calculations were converted into charges by means of a semi-empirical mass formula [178]. The investigation of projectile fragmentation was "nally extended to other projectiles like Xe and U with Be, C, Al, Cu, In, Au and U targets and energies ranging from 400 to 1000 MeV A [179]. The striking feature about the results concerns the energy independence of M as a function of Z and its scaling properties, i.e. all curves get superposed on each '+$
 other if Z is rescaled to Z /Z , where Z is the charge of the projectile as it can be seen in

 N N Fig. 12. Statistical multifragmentation models reproduce qualitatively if not quantitatively the experimental results. Recent ALADIN data concerning fragment size multiplicities for Au on di!erent target reactions at 600 MeV A were analysed by means of a standard 3d cubic bond percolation model [180]. Depending on the value of the impact parameter, the number of spectator nucleons which remains in the projectile remnant is of course di!erent. In order to decide about this number,

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Fig. 12. Left: measured mean multiplicities M as a function of Z for U on Au (circles), Au on Au '+$
 (squares) and Xe on Au (triangles) for E"600 MeV A. Right: same quantities but rescaled by normalization with respect to the number of charges Z of the projectile [179]. N

a simulation by means of a BUU transport equation of the reaction process was performed. The spectator nucleons were arranged on a cubic lattice in a compact spherical arrangement and a bond percolation algorithm was introduced. Masses were converted into charges by means of the relation Z"A/(1.98#0.0155A) for A54. A "lter was applied to the results of the calculations in order to allow for a direct confrontation between the model calculations and the experimental data. A whole set of observables was analysed in this way. Fig. 13 shows the average value of A "(Z !Z )/(Z #Z ), 

 

  A "(Z !Z )/(Z #Z ) and A de"ned above as a function of Z . As one can see the      
 agreement with the experiment is nearly perfect and this is also the case for all other observables. A new analysis of the data [181] con"rms this statement. The authors also use a standard 3d bond percolation algorithm. The number of nucleons Z which is taken into account [182] is .1 parametrized in terms of Z . Simulations are done for "xed Z and observables are deter
 .1 mined for "xed Z . The analysis is again parameter free. The calculated quantities are the
 fragment charge distributions, the average size and #uctuations of the largest fragment, and the largest fragment distribution. The confrontation also shows that there appears a `criticala regime, characterized by a power law charge distribution with
K2.2. In the neighbourhood of the `criticala point the distribution of fragments with charge z can be parametrized as n(z)"n (z) f (z/z ) ,  where z "m /m , m being the moment of order k de"ned in Section 4.3.   I The authors consider two interpretations of these results. The "rst relies on the fact that a fragmenting nucleus is made of particles which interact essentially as nearest neighbours through a short-range two-body potential. If the relative kinetic energy between pairs of neighbouring

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Fig. 13. Average values of di!erent observables de"ned in the text as a function of Z




[180].

particles is larger than the potential energy they experience a bond is broken and hence independent fragments are generated if the excitation energy is large enough. The second interpretation requires the assumption of thermodynamic equilibrium. At di!erent "xed densities it is the temperature ¹ of the system which "xes its fragment content. For a given density  of the system, there exists a temperature ¹() where the distribution looks like a power law. We shall come back to explicit classical microscopic model illustrations of both interpretations in Section 5 below. It is clear that thermodynamic equilibrium is not a necessary prerequisite for an accurate description of the fragment content of energetic nuclear collisions. 4.8. Comparison with other fragmentation models The intriguing and impressive success of classical percolation models in the description of "nite nuclear fragmentation events and the existence of a continuous phase transition in in"nite percolation systems triggered many attempts to compare and possibly relate di!erent models with the percolation approach. Such attempts started with the Statistical Multifragmentation Model SMM [125]. The authors tried to establish a link between the percolation bond probability p and the excitation energy available in the fragmented system described by SMM [124] at freeze-out. They "xed p"< /(< #< ) where < is the volume of the initial compact system and < the   D  D

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volume at freeze-out. The investigations revealed that such a link is not clearly established and led to the conclusion that inclusive fragment size distributions may not contain enough information in order to distinguish the purely statistical features of the fragmentation process from those which concern physical information carried in phase space models. A detailed comparison of the fragment content obtained in the framework of the MMMC model [79] with percolation was carried out in Ref. [183] by means of an analysis of the relation between the charge of the heaviest fragment and the second moment of the charge distribution [174]. Close similarities between MMMC and percolation models were found, in particular the `criticala zone which was discussed in Section 4.6. The main qualitative di!erence between the two approaches is the manifestation of the presence of "ssion events (two heavy fragments) which are absent in the percolation simulations. This was interpreted as being due to the presence of the long range Coulomb interaction which is not taken into account by the percolation model. Three-body correlations between the three heaviest fragments were also in agreement with the percolation predictions. The so-called cracking events with more than two large fragments emphasized the role and importance of the Coulomb interaction. Moment correlations [173,174] have been used in order to test di!erent models and possibly discriminate between them [184]. The models which are introduced are an equal probability model for the generation of di!erent partitions [186,187], time-dependent sequential decay processes [94,185] and a standard bond percolation model on a "nite cubic lattice. All models agree qualitatively with percolation calculations but do not reach quantitative agreement. The peak corresponding to `criticalitya in percolation is always present, but it is shifted in the multiplicity and its height is not reproduced. As expected, the agreement between percolation calculations and experiment is excellent. A model analysis of data has also been performed by Kreutz et al. [178] with a statistical decay model [177], SMM and, as already discussed above, a site-bond percolation model. It appears again that all models lead to quantitatively di!erent results. Sequential decay fails to reproduce charge correlation measurements,  , charge distributions and IMF multiplicities come out  more or less closely to experiment in SMM, although the average largest fragment charge is too small. 4.9. Final remarks If one restricts the information concerning the fragmentation of highly excited nuclei to the description of the asymptotic fragment size distribution it appears clearly that parameter free standard percolation models are the only ones which are able to reproduce the experimental data quantitatively. It raises the up to now unanswered question why this is so. It shows that a minimum information model is su$cient in order to explain the a priori complex nuclear fragmentation process. The agreement is by essence independent of any detail concerning explicit two-body potentials. As it will be seen later, simple models governed by short-range interactions are able to reproduce percolation results. The long-range Coulomb interaction which induces "ssion-like fragmentation is out of reach. The fact that its absence does not a!ect the physics of the process may be understood if the excitation energy is large enough. The in#uence of the Coulomb interaction will be more extensively discussed in the framework of lattice models in Section 5. Taking a pessimistic attitude, it is tempting to conclude that the success of percolation concepts is

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the sign that the information content of the process is poor. But there remains the fact that the result is not trivial since the model is not. The success of a percolation description raises the problem of the presence of a phase transition in the in"nite nuclear system, its origin and its nature. The formal similarity with Fisher's model [19] is striking, since even the order of magnitude of the critical power law exponent
is reproduced. Fisher's model leads to a thermodynamic phase transition. Does there exist a connection of the second-order percolation phase transition and the thermodynamic properties of the system? The outcome of the analysis of "nite systems [168] should be kept in mind. Mass distributions of "nite systems are able to show a behaviour which corresponds or is close to a power-law behaviour, which may simply be induced by particle number conservation in a system con"ned in a "nite volume, suggesting that a power-law behaviour may not necessarily be the sign for the existence of a critical phenomenon. Experimentally, charge distributions are only a small part of the available information on nuclear fragmentation. This information has to be reproduced. It has to be related to the properties of observables related to the energy, in particular to the thermodynamic properties of the system if it is in equilibrium. This point has already been considered in Section 4.8. It will be raised again in Section 5. There we introduce generic microscopic cellular and lattice models which are close in spirit to percolation models but also hopefully realistic enough to be able to reproduce the thermodynamic properties of excited fragmenting nuclei.

5. Lattice and cellular model approaches to nuclear fragmentation Standard percolation provides a surprisingly good description of the asymptotic fragment multiplicity distributions generated in energetic nuclear collisions. By essence, percolation models cannot describe the thermodynamic properties of the decaying systems since they do not explicitly integrate the properties related to thermodynamic quantities like energy, volume, pressure. In the preceding section we presented models which try to link the purely geometrical aspects of percolation models to considerations related to phase space [125,170}173,183]. This link introduces phenomenologically reasonable but arbitrary parametrizations of the bond probability, generally in terms of excitation energy. The procedure may not be totally satisfactory. A hint to a more consistent approach can be found if one remembers the intimate relation between standard percolation models and spin models like the Ising model and its generalizations [162,163]. These last models introduce explicit Hamiltonians and are generic for many physical systems. The aforementioned formal relation induced the introduction of classical microscopic models, starting with Ising itself. These models allow for a thermodynamic description and the study of the fragment content of the system in a uni"ed framework. They can be considered as a simpli"ed though hopefully realistic description of a nucleonic system. In the sequel we aim to introduce and discuss di!erent types of cellular and lattice models which have been proposed and studied by di!erent groups in the recent past. In order to put this study in a comprehensive perspective we "rst recall some fundamental and useful concepts concerning statistical ensembles and thermodynamic phase transitions in "nite and in"nite systems at equilibrium.

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5.1. Ensembles, thermodynamic stability, phase transitions in xnite and inxnite systems We summarize here some general de"nitions and concepts which will be used for later developments and discussions presented below. They can be found in textbooks, see e.g. Refs. [188}190]. If one considers a large classical microscopic system which is in thermodynamic equilibrium, each particle is characterized by its phase space coordinates r , p , i"1,2, N. If r , p  is G G G G considered as a point in this space, there can exist a large number of points which correspond to the same macroscopic state. Each set r , p  de"nes a realization of the system. A collection of G G realizations which correspond to the same macroscopic thermodynamic state is called an ensemble with an arbitrary "xed number of physical properties such as a "xed number of particles, energy, temperature, angular momentum and possibly other conserved quantities. In statistical mechanics there are essentially three ensembles which are used in practical applications. The most common one is the canonical ensemble which corresponds to a closed isothermal system with a "xed number of N particles and which exchanges energy with an external ensemble generally called reservoir. If the system is made of N subsystems of n subsystems with R G energy E (i"1,2, N ) G R
n "N , G R G

(23)


n E "E. G G G The thermodynamic probability which characterizes a macroscopic state is

(24)



=(n )"N !  n ! G G R G and the entropy reads "k ln = ,

 where k is the Boltzmann constant and = is the maximum of = consistent with the constraints

 (23) and (24). The average entropy S"/N can be written as R S"!k
P ln P , G G G P "n /N "e\@#G /Z , G G R Z"
e\@#G , G P is the weight of the subsystem i and  can be identi"ed with the inverse thermodynamic G temperature ¹\. The knowledge of the entropy allows for the determination of all thermodynamic functions and quantities such as the free energy of a system in a volume < F(N, <, ¹)";!¹S"!k¹ ln Z , where ; is the energy.

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The concept of temperature has been extensively used in the canonical framework in low-energy nuclear physics, in particular in Hartree}Fock models [23] and compound nucleus theory [73]. Its use implicitly implies that the energy is not a rigorously "xed quantity. It raises also the question of the concept of temperature in very small systems like nuclei [191]. In the grand canonical ensemble both energy and number of particles are only "xed in the average by supplementing Eqs. (23) and (24) with
n N "N , G G G where N is the number of particles in subsystem i. Then G S"!k
P ln P , G G G where P "e\@#G >@ILG / , G "
e\@#G >@ILG G and  is the chemical potential which "xes the average number of particles of the system. One can again work out all thermodynamic functions, in particular the Gibbs free energy G"F#P< where P is the pressure in the system. If the system is closed, with "xed energy and "xed number of particles, it is described in the framework of the microcanonical ensemble. The temperature ¹ is then no longer a natural concept. Mathematically speaking the partition function with "xed temperature is the Laplace transform of the partition function with "xed energy, hence energy and temperature are conjugate through a Laplace transformation. The temperature can be introduced through the thermodynamic relation




¹\,

RS RE

,

4

where the entropy S"!k
P ln P G G G and P is the weight of subsystem i. The principle of equiprobability of occupation stipulates that G 1/ for E!E(E (E , G P" G 0 otherwise ,



where  is the partition sum of the system, the number of systems in the ensemble. Then S"k ln  .

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This shows that the central quantity to be known is the partition function which counts the number of states at energy E or the directly related entropy S. 5.1.1. Stability of thermodynamic systems Microscopic many-body systems can be macroscopically unstable. The Le Chatelier principle stipulates that any spontaneous change in the parameters of a system which is in stable equilibrium will give rise to a process which tends to restore the system to equilibrium [188]. This leads to the inequalities dQ C" '0 , d¹

(25)

d< K"! '0 dP

(26)

for the heat capacity C and the compressibility K which characterize the change in the quantity of heat Q when the temperature changes and the change of the volume < of the system if the pressure P varies. Eq. (25) concerns thermal equilibrium and (26) mechanical equilibrium. These inequalities show that the free energy F is a concave function of the temperature ¹ and a convex function of the volume whereas the Gibbs function G is a concave function of both ¹ and <. Indeed, in particular



 RF R¹

RG R¹

4





"!

RS R¹

RS "! R¹

4

1 "! C (0 , ¹ 4

1 "! C (0 ¹ .

. . for "xed volume < and pressure P, respectively. If (25) and (26) are not realized the considered system is unstable and cannot be described in the framework of equilibrium thermodynamics. 5.1.2. Some essential reminders concerning phase transitions Phase transitions can occur in many types of systems. In the traditional acceptance of the word they are observed in in"nite systems. Only there can the correlation length between di!erent points in the system or thermodynamic quantities like the heat capacity get in"nitely large. In practice the phenomenon can be observed and mathematically extrapolated to in"nite systems in macroscopic large systems like those which are at hand in condensed matter physics where the phenomenon has been studied for all kinds of materials. There exist di!erent types of phase transitions in nature. They have been classi"ed in universality classes. Thermodynamic transitions are said to be of "rst order if there exists a discontinuity in one or more "rst derivatives of the appropriate thermodynamic potential. If, for instance, the entropy is not continuous as a function of temperature then the transition is associated with latent heat generation. If the "rst derivatives are continuous but second or higher ones are discontinuous the transition is said to be of second order or continuous. There appears then divergences in quantities like the susceptibility, correlations get in"nite and decay as power laws. The order of a transition is

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a universal characteristic of the phenomenon. Di!erent though closely related systems can experience di!erent types of transitions. Such is for instance the case of the Ising model which shows a second-order transition at some "xed value ¹ of the temperature and a "rst-order  transition for ¹(¹ for B"0 if the system experiences a magnetic "eld B. Then the magnetiz ation which is the derivative of the free energy with respect to B shows a jump. It is continuous at ¹"¹ but shows an in"nite slope there.  Phase transitions can be characterized by order parameters. These are useful physical quantities which are a help in order to distinguish between di!erent phases. They can be chosen arbitrarily among the extensive variables. In spin systems the magnetization works as an order parameter. For each phase there exists a characteristic value of the order parameter, the magnetization is zero above ¹ and "nite for ¹(¹ . Order parameters behave either continuously or jump at   a transition crossing point. This is the reason for their usefulness when trying to "nd the order of a transition. 5.1.3. Critical points, singularities, universality and critical exponents Phase transitions occur at speci"c values or intervals of values of the thermodynamic variables such as the temperature. As already seen in Section 4 these values which correspond to the so-called critical points, lines, surfaces, are marked by divergences in physical observables. In the case of a thermodynamic transition one expects that in the vicinity of a critical point ¹ observables may  behave like lim F(t)"A t H(1#ptH #2) R with  '0, as a function of the dimensionless parameter t"(¹!¹ )/¹ .    For instance for a #uid system one observes that the speci"c heat at constant volume C behaves 4 like C & t \?, the liquid}vapor density di!erence ( ! )&(!t)@, the isothermal compressibil4 J E ity K & t \A and the correlation length & t \J, where all the indices are positive. The interest in 2 the knowledge of critical exponents lies in the fact that they possess a universal character. While the location of the critical exponent itself may depend on the details of the interparticle interaction, the exponents are universal in the sense that they depend only on quantities like the dimensionality d of space and the symmetry properties of the order parameter. Evidence for this has been veri"ed number of times, in particular for the "rst time by Guggenheim [192] in the phase separation line of the liquid}gas transition for di!erent species, see Fig. 14, for which it can be seen that the exponent "1/3. The very same value for  comes in the three-dimensional Ising model, the magnetization M at ¹"¹ behaves like K(!t). These properties lead to de"ne universality  classes for systems showing the same behaviour at phase transitions. They allow to classify a priori very di!erent physical systems by means of their critical exponents which characterize the behaviour of their order parameters at a critical point. Scaling relations which govern the behaviour of physical quantities when one looks at systems at di!erent scales allow to establish inequality or equality relations between the exponents. This has already been shown in Section 4 in the framework of percolation, see also below. The importance of critical exponents raises the question of their explicit determination. Generally this can only be done numerically. Since critical exponents only make sense in in"nite systems

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Fig. 14. Phase diagram showing a "rst-order transition for eight #uids indicated in the "gure [192]. The collapse of the coexistence curve shows the universality character of the phenomenon.

it is necessary to "nd methods which lead to an extrapolation of the behaviour of the critical observables to the in"nite limit in order to determine them. There exists di!erent, more or less sophisticated methods and numerical techniques which allow for very precise determinations of exponents, see e.g. Refs. [193,194]. A whole class of methods relies on the so-called "nite size scaling assumption (FSS). The scaling hypothesis implies that in the neighbourhood of a critical point a system shows self-similarity properties when it is examined at di!erent scales. These properties are re#ected in the mathematical behaviour of thermodynamic functions and physical observables which show the so-called homogeneity properties, i.e. f (ab)"f (a) f (b) ,   where f (a)"aN, p is a real number, and for an arbitrary number of variables  f (x , x ,2x )"Nf (x , x ,2x ) .   L   L If, for the case of two variables, one chooses "x\ then  f (x , x )"xN f (1, x /x )"xN g(x /x ) ,         which shows that in practice F does not depend on the two independent variables x and x , but   on x and the ratio x /x whatever x . For "xed x , g is a scale independent function of x /x ,       

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g(x /x )"g(x /x ). The FSS assumption allows to relate the behaviour of an observable at     some scale to its behaviour at another scale. As an example of application consider an observable O (t) which depends on a variable * t"(t !t)/t where t"t is a critical point of the in"nite system and describes a "nite system of    linear dimension ¸. If one de"nes x(¸, t)" (t)/¸ where  (t) is the correlation length in the "nite * * system for a "xed value of t in the neighbourhood of t , FSS stipulates that [195]  O (t)"O(t)Q (x(¸, t)) , (27) *  where O(t) corresponds to the value of the observable in the corresponding in"nite system and Q (x(¸, t)) is a scaling function which is universal in the sense that it does not depend on t but only  on the ratio x in the neighbourhood of t , i.e. all points corresponding to di!erent t's lie on the same  curve when Q is plotted as a function of x. If O (t) converges to a "nite O(t) with increasing ¸,  * Q (x) can be determined from (27) and close to t one expects that   O(t)"O (t)/Q (x)&(t !t)\? (28) *   if O shows a critical behaviour, and  is a critical exponent. If Q (x) can be obtained numerically it  is generally "tted to a polynomial in x Q (x)"1#b x#b x#2    with Q (0)"1 if  (t)/¸ goes to zero when ¸ goes to in"nity. This type of methods has been applied  * in di!erent places, in particular to percolation (t,p) and spin systems [195}197]. Their use leads generally to reliable and accurate values of t and critical exponents. Their practical implementa tion shows however that the numerical determination of these quantities can be a cumbersome task. Other methods have been proposed by di!erent authors [55,198], in particular in order to "x the exponents which describe experimental fragment size distributions in the vicinity of the expected critical point where the distribution should follow a power law in the corresponding in"nite system. However these methods rely on expressions which are valid for large systems and do not extrapolate to the thermodynamic limit. Even though the information on criticality extracted this way looks reasonable it remains suspicious to use it in order to conclude about the determination of the universality class of the phase transition which is the ultimate goal of such an analysis. On the other hand, it is di$cult to believe that the use of scaling assumptions implemented by means of FSS can be used in practice in the framework of the analysis of experimental events since, among other severe problems, the number of points necessary to construct scaling functions would be rather restricted because of the small interval of sizes which can be reached with nuclei. Critical exponents are rather re"ned quantities. There remains hope to be able to detect the existence and eventually the order of phase transitions in nuclear matter by other means. Considering thermodynamic aspects of fragmentation, this can possibly be reached restricting ones considerations to "nite systems. The question concerning the order of a phase transition has been raised recently on theoretical grounds [199,200] in the framework of the Lee and Yang theory [201] and the approach of Grossmann and collaborators [202]. The criterion which allows to distinguish between di!erent orders concerns the location of the zeroes of the grand canonical and the canonical partition function in the complex temperature plane. But these considerations are theoretical, possibly di$cult to implement in practice for interacting systems and the connection

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with experiment is lacking. In Section 6 we shall show that there may exist simple operational means to overcome the problem. Finite size e!ects can hide or even delude the truth with respect to the order of a phase transition. Di!erent ensembles may lead to a di!erent behaviour of observables when applied to "nite systems. We shall indeed see below that characteristic features of observables when calculated in the framework of a speci"c ensemble may be of some help in order to decide about the nature of the transition. The assertion has however again to be taken with a grain of salt since there does not exist any guarantee that these characteristic features will maintain their validity in the thermodynamic limit. 5.2. Lattice models We present and discuss here di!erent lattice models which have been introduced and studied by di!erent groups. The essential point of interest concerns the existence of signs for the existence of one or several phase transition(s) in nuclear matter. 5.2.1. The Ising model as a random cluster model The success of percolation raised the question for the reasons of its achievements. As we saw in Section 4, classical lattice systems described by the Ising, Ashkin}Teller and Potts models are generally close to percolation models [161,162]. They describe Hamiltonian systems, hence they introduce an explicit, physical description of an N-body system. This was the main motivation for the introduction of the Ising model in order to interpret the cluster size distributions of an excited, fragmented system [169,203]. In order to be able to get a clearer understanding of the behaviour of observables the model was "rst studied in d"2 which allows the use of analytical techniques [169]. The Hamiltonian reads , + , + H"!J

  !J

  ,  LK L>K  LK LK> L K L K where J and J are positive constants and  is a classical spin ("$1) located at the position   LK (n, m) on a 2d lattice with N lines and M columns. An ensemble of sites are said to form a cluster if the corresponding spins show the same value and each site of the cluster has at least one neighbour site with a spin of the same value (connexity). The perimeter of a cluster is the ensemble of the nearest-neighbour sites whose spin takes the value which is opposite to the spins in the cluster. With this de"nition and after a mapping of the classical system into a quantum system of independent fermions [204] it is convenient to introduce projection operators along lines (l ) and columns (c ) ), PJ "(1#V V LK LK> >  PA"(1#V V ) >  LK L>K and PJ "(1!V V ), \  LK LK> PA"(1!V V ), \  LK L>K

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where V is the standard Pauli matrix. The correlation operators P and anticorrelation operators > P have a classical interpretation, PJA"1 and PJA"0 for aligned spins, and PJA"0 and \ > \ > PJA"1 for anti-aligned spins. \ These operators allow to de"ne the probability of presence of a cluster of a given size and geometry. (29) G"P*G> P*G\ , > \ where the index i characterizes the linked cluster which is considered, ¸G is the number of bonds > between cluster sites and ¸G the number of bonds between the surface of the cluster and the \ perimeter sites. For a given cluster size N , one "nds 1 2¸G #¸G "4N . > \ 1 The brackets in (29) indicate a trace over the many-body quantum states [169]. The multiplicity corresponding to clusters of a "xed number of sites N and a "xed topology (i) is 1 given by MG(N )"DG(N , N )G , 1 1 2 where DG(N , N ) is the degeneracy of cluster of size N , and N is the total number of sites. The 1 2 1 2 total multiplicity of clusters of size N reads 1 MM (N )"
MG(N ) . 1 1 G Since the number of spins with a given sign is not "xed in the model (see discussion below) one introduces a normalization factor N and 1 M M (N )" MM (N ) , 1 1 N such that ,2 N "N
N MM (N ) . 2 1 1 ,1  The interest of this approach lies in the fact that it is easy to work out the evolution of quantities like the average size of clusters as a function of the temperature



,2 ,2 N (¹) "
N MM (N , N )
M M (N , N ) , 1 1 1 2 1 2 ,1  ,1  the corresponding variance (¹)"N(¹) !N (¹)  1 1 and the fragment size distributions. The expected qualitative features obtained in a "nite system when the temperature of the system increases is observed. The results can be interpreted in terms of site percolation concepts. `Occupieda sites in the percolation framework correspond to cluster sites in the present case and `emptya sites to perimeter sites. The relative amount of `occupieda sites is governed by the temperature. The number of perimeter sites increases with increasing temperature

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Fig. 15. Multiplicities of clusters made of parallel spins in the Ising model for (1) short-range and (2) long-range interactions and di!erent temperatures [203].

and the relative amount of large clusters diminishes. In the percolation picture the number of empty sites increases with decreasing occupation probability p, leading to the same trend. The model which deals with short-range nearest-neighbour interactions has been extended to include long-range interactions [203]. Numerical studies show that the range of the interaction in "nite systems does not change the qualitative behaviour of multiplicity distributions in "nite systems, except for the fact that a long-range attractive interaction favours the survival of large clusters over a large temperature interval. At high temperature both distributions collapse onto each other, as it can be seen in Fig. 15. The presence of `criticala events is seen in the behaviour of the second moment of the cluster size distribution. Similar results are obtained with 3d systems. The a priori general di$culty to relate lattice models with nuclear fragmentation will be discussed later. The use of the Ising model presents a further weakness which stems from the fact that the identi"cation of clusters in terms of spins which are aligned in a given direction breaks the Z ($1) 

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symmetry which is present by construction and leads to non conservation of the number of `particlesa. The remedy to this disease consists of the introduction of a term governed by a Lagrange multiplier. We shall come to this point below. The present models were thought to build up a bridge between the purely geometrical concepts introduced by percolation models and sophisticated liquid}gas descriptions. They triggered the construction of more re"ned models aimed to describe and understand the physics of fragmentation and critical behaviour in "nite nuclei. 5.2.2. The grand canonical lattice gas model (LGM) The preceeding study was aimed to look for the connection between the geometrical and probabilistic concepts which de"ne percolation models and the thermodynamic properties of spin models which are de"ned in terms of classical Hamiltonians. It did however not explicitly de"ne the link between bond probabilities, energy and temperature. This link was established by Jicai Pan and Das Gupta [205}207] and Campi, Krivine and collaborators [208}210]. In Ref. [208] the authors consider a classical lattice gas model de"ned by the Hamiltonian , p H"
n G !
n n , G H G 2m 6GH7 G where n "0,1 is the occupation number of the lattice site i, N the total number of lattice sites and G  the energy between nearest neighbours i, j . The number of particles A"
n is conserved in G G the average by means of a constraint induced by a Lagrange multiplier procedure. The model is treated by means of Metropolis Monte Carlo simulations, in the framework of the grand canonical ensemble which can be formally related to the canonical description of the Ising model with magnetic "eld [211]. The exploration of the properties of the system in the density}temperature plane (, ¹) leads to the typical phase diagram shown in Fig. 16. One observes the presence of a full line which separates two phases above the line, a gas at low  and a liquid at large . Below the separation line gas and liquid coexist in an inhomogeneous phase forming domains (clusters) of di!erent sizes. In the thermodynamic limit the transition line corresponds to a "rstorder transition. At "1/2 in units of the normal density  there appears a critical point at  ¹"¹ which corresponds to a continuous transition characterized by critical exponents which  are universal, i.e. do not depend on the detail of the interaction between particles. The main point of interest in the study of this model lies in the analysis of the cluster content of the system for di!erent values of  and ¹. The identi"cation of clusters raises the question of their de"nition. One gets confronted with two problems [208]. The most disturbing one concerns the fact that the thermodynamic critical point does not coincide with the percolation critical point at which the cluster size distribution shows a power-law behaviour if clusters are de"ned as an ensemble of particles occupying connex (nearest neighbour) sites [212]. If clusters are de"ned in a purely geometric way as made of connected occupied sites percolation events generate a whole line of points in the phase diagram, an in"nite cluster is obtained in the in"nite limit on the right-hand side of the dot}dashed line shown in Fig. 16. This is of course unphysical, since at high temperature one expects essentially small clusters and isolated particles. A remedy to this problem was proposed by Coniglio and Klein [213], who introduced the energy concept in order to de"ne

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Fig. 16. Typical phase diagram corresponding to a lattice gas model [208]. See discussion in the text.

clusters as made of connected ensembles of nearest-neighbour particles with a bond probability smaller or equal than p (/¹)"1!exp(!/2¹). (30) !) If one uses this de"nition the thermodynamic and percolation critical point coincide and there exists a whole continuous line of points in the (, ¹) plane called the KerteH sz line [214], along which the cluster size distribution shows a power-law behaviour. The important point concerns the fact that de"nition (30) is shown to be very close to the bond probability prescription between neighbour particles given by







 "Prob p @ ¹

 



4  p ! (0 "1! du ue\S , (31) 2m ( ( C2 which stipulates that a pair of particles is bound if (p/2m!)(0 where the momentum p is drawn from a Maxwell distribution [206,208]. It is shown [208] that if (31) is chosen as a bond criterion then the separation energy S(A) of one particle from a bound cluster of A particles satis"es approximately the inequality S(A)"B(A!1)!B(A)(0 , where B(i) is the binding energy of a system of i bound particles, and the `criticala percolation line coincides approximately with the KerseH tz line de"ned above. Following this prescription one obtains cluster size distributions in di!erent part of the ( , ¹) plane, where  is the average density in the grand canonical ensemble. These distributions show the same properties as those obtained in the framework of percolation models, i.e. a power-law behaviour corrected for "nite size e!ects along the KerseH tz line, with a power-law exponent

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K2.2}2.3, an exponential decay at low density and high temperature, the characteristic large cluster peak at high density. One may notice also that contrary to the case of multifragmentation models [79,110] in which clusters are by construction spherical the cluster along the KerseH tz line have surfaces with fractal properties which allows for a freeze-out density larger than the `criticala density. Furthermore one observes that the excitation energy per particle above the ground state keeps approximately constant along the critical line [210]. Further work relying on lattice models in the framework of nuclear fragmentation was developed in the recent past. It concerns the study of the two types of phase transitions (thermodynamic and percolation) mentioned above which may be of di!erent orders ("rst and second), the importance of the Coulomb interaction, problems related to the "nite nature of the systems. We examine di!erent approaches and their contribution to the understanding and explanation of these questions in the next subsections. 5.2.3. Lattice models and their application in the framework of nuclear fragmentation The generic character of lattice gas models (LGM), their kinship with percolation models and classical systems of particles with short range interactions lead to a large amount of theoretical investigations. Das Gupta and Jicai Pan proposed di!erent variants of this type of models [205,206] and applied them for direct comparison with experimental data [207]. They worked out the canonical partition function Z on the lattice in an analytic form by using the Bragg}Williams approximation for the calculation of the potential contribution to the free energy. In this approximation the number of sites which are occupied in the neighbourhood of a "xed occupied site is "xed in the average as zn/N, where z is the number of neighbours (coordinance) and n/N the average number of occupied sites (n) in a lattice with N sites. The approximate analytical expression of Z allows to determine di!erent equations of state, in particular the relation between the pressure and the volume which comes out as a van der Waals-type equation [205]. In a further development [206] the authors introduced a grand canonical description, the mean "eld Bragg}Williams approximation was replaced by the Bethe}Peierls approximation which corresponds to the introduction of blocks of lattice sites inside which occupied nearest neighbour sites interact. An interaction  between blocks is taken into account and "xed along with the chemical potential of the particles by means of a self-consistent procedure. This further improvement brings in fact rather small changes when compared with the Bragg}Williams approximation and is rather close to the outcome of a mean "eld approach using a realistic Skyrme interaction. In all cases the descriptions show signs for the existence of a "rst order phase transition for all values of , a second-order transition at  "0.5 where  is the normal nuclear matter density.    Emphasis was put on the study of the cluster content of "nite lattice gas systems. It was shown by means of explicit calculations that the second moment of the fragment size distribution shows a maximum at di!erent values of the temperature for di!erent values of the density [205]. In an interpretation in terms of percolation concepts this is the sign for the existence of a KerteH sz line. If one introduces p/2!40 P or'0

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(where p is the relative momentum between neighbouring particles,  the reduced mass and  the P strength of the interaction) as a criterion to decide whether the considered particles are bound or not the bond probability is the same as the one given by (31) in Section 5.2.2 [206]. E!ective exponents
extracted from the experiment in the spirit of Refs. [21,22] (see Section 2.1) were compared with lattice calculations in an attempt to obtain informations about characteristic freeze-out densities of systems formed at di!erent beam energies. The freeze-out density determined in this way came out to be K0.4  .  One very important point in nuclear fragmentation concerns the role and importance of the Coulomb interaction. This point has been emphasized in the framework of multifragmentation models [79]. A calculation in the framework of an LGM for a large system obtained through the collision of Au on Au shows that the Coulomb interaction may have a sizable e!ect and may lead to large discrepancies between LGM and the experiment [207]. One may introduce two types of binding energies between nearest neighbours which correspond to the interaction between identical and di!erent types of particles (protons and neutrons) [215]. This has been done by means of Monte Carlo simulations and the results were compared to the outcome of molecular dynamics (MD) calculations which coincide with the lattice gas results in the absence of the Coulomb interaction. It is no longer the case when this interaction is taken into account. The LGM results deviate from the MD results in large systems with a sizable amount of protons. We shall come back to the Coulomb problem below. Further comparisons with di!erent models have been performed [216] such as percolation and multifragmentation models. The analysis of di!erent observables shows that these models share common features, in particular the statistical multifragmentation and lattice gas approaches, except for quantities like the speci"c heat C at low temperature. The multifragmentation 4 approach possesses quantum mechanical properties and C increases linearly with ¹ while in the 4 LGM C stays at the constant value 3/2. It should be mentioned that the algorithm used in the 4 simulations [205}207] does not sample the events in a rigorous fashion [217]. This should however not qualitatively a!ect the results and change the conclusions. 5.2.4. Extensions of the LGM in the framework of the canonical ensemble Lattice gas models were further developed and analysed in the framework of the canonical ensemble by Gulminelli and Chomaz [218}221], Borg et al. [222] and Carmona et al. [197]. Gulminelli and Chomaz [218,220] constructed the phase diagram of a "nite system by working out the compressibility coe$cient. This quantity shows an anomalous behaviour, it becomes negative at some temperature ¹ over some density interval, the sign for the existence of a "rst order phase transition in the in"nite system. Fig. 17 shows the behaviour of the chemical potential represented as a function of pressure exempli"es this phenomenon. It has also been observed that the fragment size distribution possesses a separation line (KerteH sz line) which crosses the phase separation line and enters the coexistence zone at some low value of the density. This is a priori somewhat surprising since it means that in the thermodynamic limit one would observe the existence of a second-order transition in the coexistence zone. However, it comes out that for large systems the separation line e!ectively stops at the thermodynamic critical point located at / "0.5, hence the authors interpret the e!ect as being due to the "nite size of the system. They  comfort this point of view by showing that in this region of the phase diagram the fragment size distribution does not go through a power law distribution when one crosses the separation line in

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Fig. 17. Exact (full lines) and mean "eld (dotted lines) chemical potentials as a function of pressure for di!erent temperatures [223].

the coexistence zone. If this is so, it means that the system never shows the signs for a critical behaviour at subcritical densities. Lattice gas models were also recently extended to the case where two types of particles, protons (p) and neutrons (n) coexist. The re"nement is implemented by the choice of di!erent nearest neighbour interaction strengths,  O " . These constant quantities were chosen such as to LN LL NN reproduce the nuclear binding energies of the corresponding quantum systems. One may ask for the consequences of the introduction of the isospin degree of freedom [219]. In fact, the extension of the model has only a small e!ect on the phase diagram, so that in practice there exists only one relevant order parameter,  ! , where  and  are the liquid and gas density, respectively. * % * % Critical exponents do not depend on the isospin degree of freedom either, and the critical temperature shows a quadratic dependence on asymmetry. The sign for isospin e!ects may perhaps be found in the fact that in the coexistence region the vapour mode of small fragments is more isospin asymmetric than the liquid mode of larger fragments. This can e!ectively be observed in the experiment [268,269]. A similar model with protons and neutrons has been studied in Ref. [222]. The thermodynamic properties of the system are determined for open boundary conditions and the caloric curve is constructed at constant density and constant pressure. For "xed pressure one observes clearly the appearance of a plateau which is the sign for a "rst-order transition. For "xed density, the caloric curve is monotonously increasing with energy. Fragment size distributions are analysed along isobars and isochores. They show the usual trend, with increasing temperature the large fragment yield shrinks and the small fragment and light particle contribution increases continuously. Recently the e!ect of the shape of the surface of the fragmenting system on the fragment size distribution has also been investigated [221]. Cubic shapes with and without periodic boundary conditions as well as spherical shapes with sharp and di!use surfaces have been considered. These shapes have no real in#uence on the thermodynamic properties and fragment size distributions of small systems with typically 100 particles at the critical density. Similar tests have been performed

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on other models, see below [224]. It raises the question whether the spatial shape of fragmentation events may be reconstructed in experimental data. 5.3. Cellular models of nuclear fragmentation As already mentioned the success of percolation concepts for the reproduction of experimental fragment size distributions raised the question of the correlations between theses quantities and the actual dynamics of highly excited nuclei [169,203]. Peripheral collisions [225] indicate that the spectator remnant behaves essentially as an excited system of particles which is thermalized. It leads to the idea that fragmenting nuclei can essentially be described as completely disordered systems of interacting particles, at least under the experimental conditions described above. Furthermore, it is a non-trivial but established fact that disorder may be connected with complexity and universality [226]. This property re#ects the fact that disordered systems are not sensitive to the details of their dynamical characteristics but show features which are common to many of them such as the scaling properties discussed above. In the present section we introduce and discuss the so-called cellular models which are aimed to relate the phase space properties of a classical system of particles to bond probabilities and hence to the generation of bound clusters. They can be considered as extensions of lattice models. 5.3.1. Disordered systems and cluster identixcation A fragmenting nucleus is considered as an assembly of interacting particles contained in a "nite three-dimensional volume < divided into A cubic cells, <"Ad where d is the linear size of each cell. Each cell contains one particle [224,227]. Each particle inside its cell is characterized by its position r (i"1,2, A) and linear momentum p , both phase space coordinates being taken at G G random and obeying r !r p !p 5 which has to be veri"ed for particles located in G H G H neighbouring cells. The interaction is chosen as a short-range two-body potential adapted to a classical description of the nucleus [228,229]. It acts between nearest neighbours only. The total energy of two such particles can be split into a centre of mass and a relative contribution E"P/2M#p/2#<(r)"P/2M# with P"p #p , p"p !p , r" r !r , M"2m and "m/2, where m is the mass of the G H G H G H nucleon. Each coordinate r is de"ned as x "x $d/2 (same for y and z ), where x locates the G G G G G G centre of the cell and  is drawn uniformly in the interval [0,1]. The momenta  pi  are taken uniformly in a sphere of radius p . Two neighbouring particles are said to be bound to each other

 if (0 and unbound if '0. By means of numerical simulations the criterion allows to construct a bond probability p(E)"n(E)/N(E) where N(E) is the total number of draws in an energy interval [E, E# E] and n(E) the total number of events out of N(E) for which the two-particle system is bound at energy E. The violation of the inequality relating relative distances and momenta de"ned above leads to the rejection of the corresponding event. Three di!erent procedures have been introduced in order to identify bound clusters. In the "rst called A2, the energy E of each pair is drawn from the distribution N(E) and the corresponding p(E) is compared to a random number  belonging to [0,1]. If p(E)' the pair is considered as bound and unbound in the reverse case. This procedure is close to a percolation algorithm but p(E)

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is not chosen in a uniform distribution. It neglects also the correlations which exist between pairs sharing common particles. The second procedure (A1) "xes the energy of a pair E"E(r , p ; r , p ) of neighbouring particles G G H H i and j by means of their phase space coordinates which are "xed as described above. This "xes also p(E) and the bond criterion is the same as for A2. Finally, correlations are fully taken into account by means of algorithm A0 for which the generated phase space r , p , i"1,2, A is used to G G calculate explicitly  for each pair. It corresponds to a deterministic determination of the clusters for each generated event. It is important to notice that none of these descriptions implies that the considered systems are necessarily in any kind of equilibrium. 5.3.2. Physical observables. Confrontation with bond percolation The fragment content can be characterized by the mass distribution, its moments and associated observables like  which has already been de"ned above. In order to compare directly with the  experiment these observables are determined as a function of the multiplicity m which can be measured and is univoquely related to the bond probability p. For each event k

or

mI"
iPI(m)  G G

SI(m)"mI(m)/mI(m) ,    where PI(m) is the number of clusters of size i and the prime index indicates that the heaviest G fragment is omitted. Similarly I"mI(m)mI(m)/mI(m)     "1#I(m)mI(m)/mI(m)   where I,(mI(m)mI(m)!mI(m))/mI(m) .     Average quantities are de"ned as , m (m) "N\
mI(m) ,    I ,  (m) "N\
I(m) .    I Numerical investigations have "rst been performed with the algorithm A2. The comparison with bond percolation is shown in Fig. 18. One observes an amazing agreement between the two descriptions although bond probabilities are taken from di!erent distributions and this agreement is very robust with respect to the strength of the two-body potential. In fact, if () is the probability distribution of "p(E ) for a given E the equivalent probability in bond percolation is   E "1! d() . 



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Fig. 18. Comparison of percolation results (upper part) with A2 algorithm (lower part) [224].

Through this relation m and  are related to each other in the same way as m and p in bond percolation. In particular, one observes a critical value  for which an in"nite cluster appears in  the in"nite system. The agreement survives also if protons and neutrons are di!erentiated by means of di!erent bond probabilities p (E)"p (E)Op (E) [224]. Qualitatively similar but quantitatively di!erent NN LL NL results are obtained with algorithm A1, in particular the location of the maxima of m and    are shifted to higher values of m. The deviations are indicated by the correlations which are built in A1 and absent from A2. One expects that the e!ect is due to the hybrid nature of the algorithm which mixes probabilistic and deterministic properties. Algorithm A0 provides a priori the most realistic description since it takes all correlations consistently into account. It allows also for an easy treatment of the long-range Coulomb interaction. For a "xed value of the length d of the cubic cells the moments m and   "m ) m /m  can di!er sensibly from the percolation results. If, however, one averages     over events corresponding to values of d which are uniformly distributed over a given interval of densities , say from 5;10\ to 0.17 fm\, one obtains mass distributions, m and  which are   remarkably close to those obtained with A2 and robust with respect to modi"cations of the

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strength of the short-range potential as it is the case with A2. The agreement between the results obtained with A0 and A2 is not easy to understand. It seems to indicate that averaging over densities acts as a mean to wash out correlation e!ects. 5.3.3. Confrontation with experimental data Multifragmentation data available from experiments made by the ALADIN collaboration [178,230,231] have been confronted with the outcome of the A2 algorithm in the same way as it was done for the percolation approach in Section 4.7 [224,227]. The charge Z to mass A relation used is the same as in this section. For fragments of mass A"3 the isobaric ratio H/He is "xed to 2.2. The mass of the spectator part of the fragmented system depends on the impact parameter b, the number of particles involved is divided into 10 di!erent sets corresponding to 10 intervals for the impact parameter. Each system with A (i"1,2,10) particles is put in a cube of 7 cells in as G compact and symmetric a geometry as possible. Each calculated observable O is averaged over the whole impact parameter range



  O "
b b O(b )
b b . G G G G G G G Results of this analysis are shown in Fig. 19 for di!erent observables, the average value of the charge of the largest fragment Z , the IMF (34Z430) multiplicity, the He isotope multiplicity

 which are represented as a function of Z and the average number of fragments with charge
 64Z430. As it can be seen, the agreement between experiment and simulations is excellent. This is not really surprising since bond percolation and A2 lead to very close results and, as it has been seen in Section 4, percolation reproduces experimental results. 5.3.4. Geometric and collective ewects The use of simple cubic shapes for the description of the fragmenting system may appear as too restrictive. This point has been investigated by introducing other shapes, spheres, parallelepipeds, streaked, crenellated and crumpled surfaces which were generated by means of a random growing process leading to fractal surface structures which may introduce empty cells in the bulk of the system [224]. The moments of the mass distributions show deviations from the cubic case, but these deviations are always surprisingly small, they never lead to qualitative changes in any observable related to fragment size distributions. Cellular models describe disordered systems which show a power-law behaviour with deviations due to "nite size e!ects at a value of the fragment multiplicity corresponding to criticality in the in"nite system. Experimentally this situation prevails for peripheral collisions [178,230,231]. In the case of more central collisions fragment size distributions show rather an exponential fall o! as a function of the mass number [232]. These collisions are also characterized by a sizable collective radial #ow [155,233]. It is out of scope to reproduce this dynamic e!ect with the present simple-minded approach. It is however possible to mimic it. This was done with algorithm A0, mixing events with linear cell size 1.84d46 fm for A"216 particles. The linear momenta strengths of the particles are chosen randomly but their directions are "xed along a line which joins the centre of the system to their actual position, pointing out of the system. The typical behaviour of the mass distribution in this case is shown in Fig. 20, along with the case where collective e!ects

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Fig. 19. Di!erent observables represented as a function of Z . The experimental average largest cluster Z , IMF


 multiplicities M and He multiplicities M represented by dots are compared to calculations performed with the '+$ & A0 algorithm, see text [224].

are absent. The shape which is power-law like in the case where the directions of the momenta are random gets very close to an exponential for sizes between 20 and 90 units of mass. Deviations are observed for light and heavy fragments. Such a behaviour has been con"rmed in Ref. [155] where a more detailed analysis on a more sophisticated approach has been performed. This result comforts the idea that peripheral collisions induce the generation of full disorder and are very weakly a!ected by collective e!ects whereas central collisions show sizable collective e!ects. 5.3.5. Thermodynamic properties of systems described by cellular models: an analytical model The encouraging results concerning fragment multiplicities obtained in this type of simple models raises the question whether the present description supports the existence of a thermodynamic phase transition as it is the case for the lattice gas models described above. The problem which comes up is the fact that the generated events do not need to possess the property of being in

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Fig. 20. Mass distributions for a 6 system. The upper curve corresponds to a calculation using the A0 algorithm, the lower corresponds to the simulation of collective e!ects, see text [224].

thermodynamic equilibrium since the algorithm which is used does not guarantee it. Indeed, a necessary condition for thermodynamic equilibrium in the canonical ensemble is the determination of an absolute minimum in the energy for a "xed temperature through the application of detailed balance. This has not been implemented. However there are reasons to believe that the generated systems are close to if not at equilibrium. This led to the analysis of their thermodynamic properties in the framework of the microcanonical ensemble [234]. The entropy S(E) has been determined as a function of the energy E and the caloric curve constructed by de"ning the temperature as ¹\"(RS/RE) where < is the volume of the system. The outcome of the study is 4 the fact that ¹ increases monotonously with E and there is no sign for the presence of a transition. An analytical investigation which describes the system in a way which is close to the one which is obtained through numerical simulations con"rms this result [234].

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The absence of a transition is in fact qualitatively understandable if one goes back to the lattice gas models of Section 5.2. Indeed, the discussed cellular models correspond to the extreme case where the system is homogeneously "lled with particles, hence / "1 in Fig. 16 where the system  is in a pure liquid phase. In order to study the behaviour of a system for every density one may introduce a simple three-dimensional cubic lattice whose volume < is divided into N cells of linear size d, <"(Nd). This geometry can be straightforwardly extended to a parallelepiped <"N N N d [235]. The    restriction to attractive nearest-neighbour interactions leads to a Hamiltonian  p (32) H "
G !<
s s ,  G H  2m 6L L7 G where A is the total number of particles, n.n stands for the `kth nearest-neighboura sites(i, j), I < is the strength of the interaction and s "0 (resp. 1) corresponds to a cell which is empty (resp.  I occupied by a particle). The model which describes an inhomogeneous system can be trivially mapped into a spin-model by means of the change of variables  "2s !1 . I I Particle number conservation requires 1 ,
(1# )"A . I 2 I In terms of the new variables the Hamiltonian reads





< ,  p H "
I ! 
  #6
 #3N , I J I  2m 4 6LL7 I I which corresponds to an Ising model with "xed magnetic "eld h"3< /2. In order to work out its  thermodynamic properties one goes over to the continuum limit by replacing  "$1 by the I continuous variables u in [!R,#R]. This leads to a description in the framework of the I spherical model which is exactly solvable in any number of dimensions for di!erent types of interactions and even in the presence of disorder [236}240]. Constraining the continuum variables u  by means of I ,
u"N (33) I I and introducing the Lagrange multipliers  and  leads to the Hamiltonian






 , ,
u #N H"H #
u# I 2 I  I I and a canonical partition function



Z(, A, N)"

>



>  dpG du exp[!H(u )] , I I I \ IG \

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where pG corresponds to the ith component of the moments p of the particle k. The parameters I I  and  are "xed by !\R ln Z/R"N ,

(34)

!\R ln Z/R"A .

(35)

The physical justi"cation for the introduction of the continuum limit variable u  is the fact that I the interaction in H can now be interpreted as taking a continuum of values, not only $< /4,   which is sensible in the case of cellular models, since the location of the particles in cells is not restricted to the centre of the cells. Because of the constraints imposed by the volume constraint (33) the important contributions to <  remain "nite. The procedure does not take care of the fact 6LL7 that positions of a set of interacting particles are correlated which induces correlations on the two-body contributions. These correlations are not taken into account here, like in the use of algorithm A2 in numerical simulations. The partition function can be worked out explicitly. It can be cast in the form Z(, A, N)Jexp[!Nf (, A, N)] with the free energy per cell ,\ f (, A, N)"[(!3< )/2]/[4(3< /2!]#(1/2N)
ln [(!<  /2)] ,  G   G  where  ,    "
cos  I G G GGG I  I "2i /N (i "0, 1,2, N!1) . G I I The constraints (34) and (35) lead to the relation ,\
[!<  /2]\"2N(1!M) ,  G G which "xes . Then "4(3< /2!)M#3< ,   where

(36)

(37)

M"![1#2(N)\R ln Z/R] "!(!3< )/[4(!3< /2)]   "!1#2A/N "2/ !1 ,  which shows that M3[!1,#1]. The dispersion relation (36) has N poles. There exists one solution which corresponds to '<  /2 with  "3 and guarantees that the corresponding value of  leads to an  

 expression of Z which makes physical sense.

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Fig. 21. Phase diagrams for di!erent types of interactions, see explanations in the text [235].

It can be shown [239] that for M"0 (/ )"0.5) the free energy per unit volume gets singular  for a given value of the temperature ¹"¹ (0)"< /2= (0) where = (0)"0.2527 [235,239]. The     same behaviour of the free energy is observed for all other values of M leading to critical temperatures ¹ (M). The speci"c heat is given by  C "! d/d , 4 where "E/A is the energy per particle. C shows a maximum for every value of M at ¹ (M) 4  which signals a continuous phase transition between two phases. The corresponding phase diagram is shown in Fig. 21. The analytic expression of Z allows for the determination of all thermodynamic quantities and the construction of the caloric curve ¹"f (E) which is a monotonously increasing function. This is due to the fact that C remains "nite at the transition point for 4 any value of / [235]. Only the derivative of C shows a discontinuity at the transition points.  4 The model can be extended to the case where particles interact through short or/and long-range interactions



H"K! <




s s #<
s s #2#<
ss G H  G H K G H 6LL7 6LL7 6LL7K



#
s #
(2s !1) , G G G G where K is the kinetic energy and nn has been de"ned below (32). A sensible parametrization in I the case of the nucleus would correspond to <  <  < K

" < #<  , Q J " <  J  " < K , J

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where < '0 stands for the short-range nuclear potential and <I(0 for the long-range repulsive Q J Coulomb interaction parametrized in the following form <I"(Z /A)e/kd , J N where Z is the total number of protons. N The partition function can be worked out in the framework of the spherical model in the same way as in the former case [235]. Fig. 21 shows typical phase diagrams for di!erent cases involving long-range interactions. As one can see, they lead to qualitatively similar results. In the usual spherical model in which the constraint (35) is not implemented  plays the role of a "xed magnetic "eld. Then, for O0, the magnetization can jump from !M to #M which corresponds to a "rst-order transition. For "0 the transition is continuous. This is not the case here since the magnetization M (density ) is "xed, and not  which is determined through (35). It explains why the transition is continuous in the present case for any value of M or (/ ).  5.3.6. Thermodynamic properties of systems described by cellular models: numerical simulations The former approach is a simpli"ed even though quite realistic thermodynamic description of a disordered system. However, it does not allow to work out the fragment content of the system, which is accessible by means of numerical simulations [241]. This has been performed in the framework of the canonical ensemble for particles which experience short range nearest-neighbour interactions. The Hamiltonian is written as  p H"
G s #
v (r )s s ,  GH G H 2m G 6GH7 G where v is the potential which acts between particles located in neighbour cells [228] at  a distance r from each other, A is the number of particles, "A/¸ the density of particles in GH a volume <"¸ and the s 's are occupation numbers de"ned above. Periodic boundary condiG tions are imposed in the numerical simulations. A Metropolis Monte Carlo procedure is used in order to generate events corresponding to di!erent densities  and temperatures ¹. For each (, ¹) the total number of events is 6;10 and for each event about 5000 Metropolis iterations have been realized. Each iteration consists of either the move of a particle from its original cell to an empty cell, or its displacement inside a cell, or the determination of a new momentum. Each operation is realized with a probability 1/3. The latent heat is de"ned as 1 C " (; !; ) , 4 A where the average is an ensemble average and ; the potential energy of the system. The calculations have also been extended to the microcanonical ensemble with "xed A and energy E, by means of an algorithm described in Ref. [242] which is well adapted to the present type of models. The probability for the existence of an event which has a con"guration s characterizing the positions and momenta of the particle is given by = (s)J(E!;(s))4\(E!;(s)) , #

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where d is the space dimension, < the volume and  the Heaviside function. Events are selected by means of a Metropolis procedure which accepts a con"guration s starting from a con"guration s with a probability






= (s) . = "min 1, # QQY = (s) # The temperature is obtained from ¹"2K /
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Fig. 22. Caloric curve and speci"c heat for a "nite size system, ¸"8, and density / "0.5. Canonical and  microcanonical simulations coincide [241].

6. Finite nuclei and phase transitions: present experimental and theoretical status The experimental quest for a thermodynamic phase transition in nuclear matter received a new impetus in 1995 with the work of Pochodzalla et al., who tried to construct the so-called caloric curve [243]. If nuclear matter would behave like a Fermi gas one would expect an increase of the temperature ¹ with excitation energy EH such that ¹JEH. Deviations could indicate the existence of a phase transition which, following classical arguments, is expected to be of the liquid}gas type.

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Fig. 23. The phase diagram obtained with the cellular model described in the text [241]. See discussion in the text.

However, nuclei are "nite and the asymptotic limit of in"nite matter is experimentally out of reach. The situation which is common to other "elds like physics of aggregates and mesoscopic systems in condensed matter physics needs the development of new tools which may allow to detect the existence of the transition through the study of the behaviour of physical observables which characterize "nite systems. In the present section we try to give an overview of the recent experimental e!orts which have been made and the theoretical investigations which have been initiated, speci"cally with the help of lattice-type models in order to detect a possible phase transition and unravel its nature in excited nuclear matter under the assumption that experimental measures are made on thermodynamically equilibrated systems. 6.1. Caloric curves and phase transitions: experimental status The caloric curve which relates the internal energy of an excited system at thermodynamic equilibrium to its temperature is a priori the simplest experimental tool to look for the existence of a phase transition. The "rst determination attempted by the ALADIN collaboration [243] collected the outcome of Au on Au collisions at 600 MeV A with data obtained by means of less energetic collisions. The excitation energy was determined by following the procedure prescribed in Ref. [182] and the corresponding temperature "xed by means of arguments relating this quantity to the so-called double ratio procedure [244], in the present case the ratios of He/He and Li/Li isotopes. The corresponding curve showed the features of a rather strong "rst-order

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transition, with a characteristic close to constant temperature ¹ over a large energy interval lying between 3 and 10 MeV excitation energy per nucleon, which may be interpreted as a sign for the generation of latent heat and followed by a strong increase of ¹ with excitation energy above 10 MeV. A critical discussion followed this observation, in which the hypotheses and simpli"cations underlying the de"nition of the temperature, the freeze-out density and the increase of ¹ at high excitation energy were examined [245]. A similar analysis by means of 1 GeV A collisions of C on Au was initiated soon after this "rst attempt [246]. The reaction was interpreted as a two-step process, a prompt stage with the emission of light particles and a second step corresponding to the decay of an equilibrated system undergoing fragmentation. The outcome of the analysis leads to a smooth caloric curve from which any clear sign corresponding to a "rst-order transition was seemingly absent, showing at most the possible existence of a continuous transition. In this experiment the temperature was determined in terms of two di!erent isotope ratios, (He/He)/(Li/Li) and (He/He)/(d/t). The results were essentially con"rmed later in a further experiment [247]. The INDRA collaboration considered Ar on Ni collisions at 52 and 92 MeV A in order to investigate this point [248]. Several double isotopic yield ratios were used in order to de"ne the temperature. They led to di!erent slopes in the temperature increase as a function of the excitation energy. The issue of the experiment and its interpretation have been performed by means of statistical models [138,139,142]. A criticism has however been raised [249] which claims that the observation of the transition may have been missed because of the choice of too large steps in energy binnings. At this stage the ALADIN collaboration considered the problem of the determination of the temperature by means of an analysis of Au on Au collisions at 1000 MeV A [225] which was determined through di!erent isotopic yield ratios and corrected for sequential evaporation from primary fragments. In this experiment, temperatures were also de"ned by means of excited state populations [250,251]. Both types of temperatures are shown in Fig. 24. Those obtained from population yields are remarkably constant as a function of the excitation energy per particle, in contrast to the temperatures obtained by means of isotopic yield ratios. The interpretation of this apparent inconsistency leads to the conclusion that these temperatures may correspond to the "nal stage of fragment emission. The corrected new caloric curve emerging from the use of isotopic yields does no longer show a clear plateau-like behaviour. At the highest value of the excitation energy there remains however a sizable rise in the temperature which seems to indicate the entrance into a new phase. A further attempt has been performed by bombarding Ag and Au with 4.8 GeV He ions [252]. The temperature was "xed by means of the double ratio yield (H/H)/(He/He). A slope discontinuity is observed for EH/AK2}3 MeV which is followed, with increasing excitation energy, by a monotonic rise up to the measured energy EH/AK10 MeV. The curve is in qualitative agreement with the theoretical predictions of the Expanding-Evaporating Source model (EES) [92,253] and the statistical multifragmentation model (SMM) [110,254]. More recently, a further result has been obtained on excited systems with AK110 particles produced by means of di!erent projectiles and targets with a bombarding energy of 47 MeV A [255]. The temperature was obtained from the double ratios (d/t)/(He/He). An analysis based on coalescence models [140,256}259] was used in order to follow the evolution of temperature and densities. Calorimetry measurements "x the excitation energies. The experimental points combined

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Fig. 24. Isotope temperature ¹ (full triangles) and temperatures obtained by means of excited state population rate &* measurements (open symbols) as a function of the excitation energy per nucleon for Au on Au at E"1000 MeV A [225].

with former results [260] show a steep increase of ¹ up to EH/AK3.5 MeV. The temperature stabilizes for 3.54EH/A47 MeV around ¹&7 MeV. The analysis and comparison of results obtained by di!erent groups show hardly more than qualitative similarities and are rather non conclusive as far as the existence and possible order of a phase transition are concerned. They are plagued by di!erent problems which are not only related to "nite size e!ects or the equilibration problem, but also by the fact that in the experiment the system undergoes expansion, sequential decay of primary excited fragments, apparently temperature determinations can lead to inconsistent results. These problems have been investigated by several groups [261}267]. Further new methods for the determination of the temperature have been proposed [268]. As the matter stands, there are signs which point towards the existence of a transition. The caloric curve may however not be the most adequate tool to con"rm it. We shall discuss this point in the next section. 6.2. Signs for phase transitions in xnite systems: tests and applications to the nuclear case by means of lattice models Nuclei are "nite systems, hence experimental observations refer to small systems, far from the thermodynamic limit. Contrary to other aggregates or quantum dots whose size can be experimentally varied and controlled, there is no such #exibility in "xing the number of constituents over large scales here. It shows that even if the experimental determination of physical observables could be fully tested, there would remain the problem of the small size limit. This point raises the question of the detection of the existence and nature of a phase transition through the measurements on

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"nite systems. The signs for the existence of transitions clearly exist, they are seen in "nite systems since in"nite systems do not exist, and one may speak about phase transitions in "nite systems [270]. The determination of their order necessitates good tests and a great deal of caution. This point has been raised in several "elds of physics in the recent past. We review some of the methods and tests which have been proposed in the next section before applying them to lattice models and the analysis of experimental data. 6.2.1. Tests for the existence of phase transitions from xnite system analysis There exist several characteristic properties which may signal that a "nite system can show a critical behaviour related to a phase transition. Close to a critical point observables follow scaling laws which relate their behaviour at di!erent scales, i.e. for di!erent sizes of the system. They allow for the determination of critical exponents which characterize the singular behaviour at the critical point in the limit of an in"nite system. This property has already been presented and discussed in Section 4.3 [197]. We shall come back to it later. A second characteristic of "nite systems which are close to a phase transition is the enhancement and universal behaviour of #uctuations in observables and order parameters. This type of properties is commonly studied [271] and has been used [272] as will be shown later. Recently the properties of order parameters of "nite systems close to a second-order transition have been studied in the framework of a "nite size scaling approach [273]. It comes out that order parameters can be characterized by probability densities which possess speci"c properties depending on their location with respect to the critical point. In particular, at the critical point, they show universal scaling and deviations from a gaussian behaviour in the tails. This behaviour is valid for both equilibrium as well as o!-equilibrium processes in systems which undergo second-order transitions. The method could work as an alternative to the determination of critical exponents. Applications will certainly be developed in the near future. Informations about the thermodynamic properties of small extensive and non-extensive systems (for which the sum of the energies and entropies is not the sum of energies and entropies of their parts, e.g. if long-range interactions are present) can be very e$ciently analysed in the framework of the microcanonical ensemble. This aspect has been extensively discussed, see e.g. [270] and references therein. The important point of practical importance concerns the topology of the entropy S(E, A) of a system with A particles and total energy E [270]. In the presence of a "rst-order transition, the entropy shows convexity (negative curvature) in certain ranges of energy. This behaviour is speci"c to the microcanonical ensemble which is able to give a faithful description of inhomogeneous systems. In the caloric curve the temperature is a multivalued function of the energy, showing a bend (`Sa curve) and, as a consequence, the speci"c heat can get negative over some intervals of energy. The physical explanation for this phenomenon is the existence of a surface tension which is generated by the coexistence of two phases. Indeed, the creation of intra-phase surfaces generates an entropy loss S which can be obtained from 1 the relation [274]   , S " 1 ¹ A 

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where  is the surface tension, ¹ the transition temperature,  the surface area of all species with  mass larger than 1, and A the number of particles. This remarkable property shown by "nite systems will be used below as a main tool in order to detect and characterize the behaviour of fragmenting nuclear systems. The present collection of tests is certainly not exhaustive, there may exist other means to detect signs for phase transitions. One of them which grounds on the Yang and Lee theory has already been mentioned in Section 5.1.3 [199]. Recently Ma [275,276] proposed to use the enhancement of entropy production at the pseudo-transition point and claimed that Zipf's law [277] is veri"ed at the transition point. This has been tested numerically by means of calculations involving an LGM model and molecular dynamics simulations. Ref. [278] discusses the correlation between an `Sa curve in the caloric curve in the microcanonical framework, the thermodynamic temperature ¹"(RE/RS) and the temperature determined from the kinetic energies of the constituents of the 4 system. It is shown that the caloric curve obtained by means of the thermodynamic temperature shows a bend if this is the case for the kinetic temperature, but that the reverse is not true. 6.2.2. Applications of the tests to models of nuclear fragmentation. Finite size scaling and microcanonical description The "nite size scaling test for the determination of the order of a phase transition by means of the determination of critical exponents has been recently applied to a lattice model, the Ising Model with Fixed Magnetization (IMFM) [197]. Its canonical partition function reads






Z "
exp <
  K ,  G H ++ 4 6LL7 N

where < is a negative potential strength, M"2/ !1 the `magnetizationa related to the   relative density / , M K a "xed value of M,  "$1 related to the lattice site occupations  G s "0, 1 in the notation of subsection 5.3.5. G The model is equivalent to an LGM except for the fact that the number of particles, A"
s , is G G strictly "xed and, as a consequence, / is "xed if the volume < is "xed. It is a non trivial model  whose properties have been investigated recently [279]. In the canonical ensemble the kinetic energy plays a trivial role, therefore there is no need to introduce it if one looks for critical properties of the system in this framework. The system governed by the IMFM shows a phase transition with a phase diagram (, ¹) which has the same topology as LGM and related models, see e.g. Fig. 16. However in the ordinary Ising model with zero magnetic "eld contribution, the magnetization jumps when the system enters the coexistence phase at ¹(¹ . Since the present description keeps the magnetization "xed, the order  of the transition is a priori not clear. The problem has been investigated numerically, "rst in the canonical framework for a system with "xed volume. The sensible quantity which has been worked out is the speci"c heat C "3<(E !E ) , 4 where < is the volume and E the energy of the system. At the separation line, one looks for the asymptotic behaviour of C in the form 4 C J(!2)\? 4 

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Fig. 25. Fit of C  as a function of the linear size ¸ of the system for di!erent densities. Crosses indicate di!erent sizes. 4 The largest size is ¸"48 [197].

2"¹\, 2"(¹2)\, ¹2 being the thermodynamic transition temperature of the in"nite    system for "xed density  and  a real positive exponent which may help to characterize the universality class to which the transition belongs. Using "nite size scaling arguments and writing C (¸)"A#B¸?J , 4  (¸)! (R)"AM ¸\J , (38)   where A, B, AM are real constants and ¸ is the linear size of the system, it is possible to obtain  and /, hence , corresponding to the thermodynamic limit [197]. Typical results for the evolution of C with the size of the system by identi"cation of the maximum of this quantity for "xed  are 4 shown in Fig. 25 for two values of / "(MK #1)/2. In this "gure / is read from the slope of the  curves which should behave like rigorously straight lines if the scaling hypothesis (38) is working. The straight line behaviour is observed for ¸ large enough (¸528), although there are deviations at close inspection. The values of  and  for / "0.3 and 0.5 could be compatible with the  universality class of the Ising model although this is not absolutely established since the behaviour for ¸"48 could still be transitory. This value of ¸ in 3d space requires already a large numerical e!ort. The non-completely conclusive situation led to further investigations [280]. First, the topology of the system on the critical line is very similar for di!erent values of the density, showing a quite homogeneous pattern. Second, the evolution of the caloric curve shows a change of slope which

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Fig. 26. Caloric curve calculated for a 3D IMFM for a system of linear size ¸"10, / "0.3 (upper part), / "0.5   (lower part). Canonical and microcanonical calculations are indistinguishable. See comments in the text [280].

gets more accentuated with increasing size of the system, with the presence of an in#exion point which is characteristic of a continuous transition. A sharper test consists of the use of the microcanonical ensemble. Here the kinetic energy part is taken into account and the procedure is described in Section 5.3.6. As discussed above, a "rst-order transition should manifest itself by a temperature which is a multivaluated function of energy, and this is not the case if the transition is continuous. This phenomenon was numerically checked on the Potts model which can e!ectively be of "rst or second order for di!erent numbers of spin directions on a given site [160,281]. The application of the microcanonical algorithm to the IMFM does not show the expected `Sa curve as it would be the case for a "rst-order transition, see Fig. 26. This fact does however not lead to a de"nite answer because it is expected that for a "xed volume the caloric curve in the in"nite system never shows a plateau in the presence of a "rst-order

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liquid}gas-type phase transition but only a more or less steep change of slope, the plateau arises for "xed pressure as recalled in Ref. [282]. Furthermore, as already mentioned, the "nite system announces a transition which may not necessarily be the one which is really observed in the thermodynamic limit. It has been shown recently that the multivaluedness of the energy on the caloric curve appears when the system gets large enough [292]. Two conclusions can be drawn from this study. First, the present model deals with two constraints, "xed volume and "xed number of particles. These constraints may play a critical role in "nite systems. This can be seen on the Potts model. For q55 where q is the number of spin directions on a site, the caloric curve should show an `Sa curve. When constraints are applied on the spin population of di!erent spin directions at the di!erent sites of the system the bend disappears [280]. The importance of constraints has of course also consequences for the interpretation of experimental results which depend of the physical conditions under which data are collected. Fixed volume is an illustration of the e!ect of constraints. 6.2.3. Fluctuations and criticality As already seen above [273], critical phenomena are strongly correlated with large #uctuations in the thermodynamic variables. This property has been exploited recently by Chomaz and Gulminelli [272,283,284] in the framework of the microcanonical ensemble. These authors have explicitly shown on a simple model how the speci"c heat which can be de"ned in terms of energy #uctuations can reach negative values. A closed system with energy E is divided into two subsystems, 1 and 2, with energy E and E , R   E "E #E . Then the energy distribution in subsystem 1 can be written as R   = (E )= (E !E )  P#R (E )"    R   =(E ) R "N exp[S (E )#S (E !E )!S(E )] , (39)    R  R where N is a normalization factor, = (E ) is the density of states of the system i at the energy G G E and S (E ) its entropy. =(E ) is a convolution of partial densities G G G R #2 =(E )" dE exp[S (E )#S (E !E )] . R     R   At equilibrium =(E ) is maximum for R RS RS  "  , (40) RE M  RE R M   # \# # where EM corresponds to the stationary point where the equality is realized. Relation (40) shows  that the temperatures ¹ and ¹ are the same in both subsystems, ¹ "¹ "¹M as expected for     a system in thermodynamic equilibrium. Close to the maximum of =(E ) it is reasonable to parametrize P#R by a gaussian  R R (41) P# (E )"(2 )\ exp[!(E !EM )/2 ] .      







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Then



RP#R (E )   "!(2)\\ .  RE M  #  From (39), for N"(2 )\  RP#R 1 C #C  "!   , RE M  ¹M  C C  #   where








(42)

(43)




1 RS \ dEM G C "! " G G ¹ RE dE G G R is the speci"c heat of the system i. Equating (42) and (43) and introducing CM "C #C   leads to

(44)

C  . CM " C ! /¹M    If now subsystem 1 corresponds to the kinetic energy part of the system with energy E and speci"c I heat C and subsystem 2 to the potential contribution E then I N C I , (45) CM " C !/¹M  I I where C "A and A is the number of particles. The knowledge of the temperature ¹M of the system I  and the kinetic energy #uctuations characterized by  allow then to determine the heat capacity of I the system in the framework of the microcanonical ensemble. It is clear that if  gets larger than I C ¹M , CM gets negative which corresponds to an `Sa bend in the caloric curve. The analysis requires I however some care. There is a priori no justi"cation for the distribution (41) to be gaussian nor for (44) to be strictly true in the vicinity of a critical point. The lack of concavity of the entropy is the sign for it. There are however signs that, even if (44) is not strictly correct, the variance of the real distribution can be assimilated to the variance of a gaussian distribution if the number of particles is not too small [285]. The correction to the heat capacity due to higher moment contributions to (42) has been discussed in Ref. [272]. The kinetic energy distribution is in principle experimentally accessible. Relation (44) has been applied to lattice models of the same type as the one whose Hamiltonian is used in Section 5.2.2 [284]. The partition function reads Z (E)"
= (E)e\H4 H 4 4 where = (E) is the density of states at energy E. The volume of the system is not "xed here but 4 #uctuates. This is described by means of the introduction of a Lagrange multiplier  which is interpreted as a quantity proportional to a pressure P"¹ where ¹ "R ln Z /RE. H H H

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Fig. 27. Caloric curve, speci"c heat and kinetic energy #uctuations at constant pressure (left part) and constant volume (right part) calculated in the framework of the microcanonical ensemble [284]. See comments in the text.

The equation of state for this model has been studied numerically in the three-dimensional space (¹, E, ). One observes a backbending e!ect if one represents P"¹ as a function of the average volume < which is interpreted here as being the consequence of mass and energy conservation constraints. An `Sa bend appears also in the caloric curve on the (E, ¹) surface. For strictly "xed volume the caloric curve does not show any backbending. This is illustrated in Fig. 27 in which the speci"c heat is represented as a function of energy using (45). With this interpretation the puzzle concerning the order of the transition which was discussed in subsection 6.2.2 [280] is apparently explained, the shape of the caloric curve depends on the thermodynamic variables which are considered [282]. This message is of some importance in the framework of experimental analyses. One should however notice that the argument is at variance with the recent results of Ref. [292]. 6.3. Experimental construction of the caloric curve and determination of the specixc heat Di!erent attempts have been made since the pioneering work already mentioned above [ALADIN, INDRA, EOS, . . .] in order to construct the caloric curve starting from experimental data. The reaction Si on Mo leading to an incomplete fused system has been used in order to relate the excitation energy per particle eH of the system to a so called apparent temperature ¹ which is obtained from the slopes of the energy distributions of emitted light particles [286]. 





(E !B) d J(E !B) exp !   ,   ¹ dE    where E is the kinetic energy and B the Coulomb barrier. ¹ which is not the thermodynamic    temperature ¹ is shown to be a sensible quantity, both ¹ and ¹ show the same trend as   

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a function of the energy. The outcome of the experience is in reasonable agreement with calculations performed in the framework of the MMMC model. Since this model indicates the existence of a phase transition, it is tempting to conclude that it is also experimentally true since model calculations and experimental data have been worked out under the same physical conditions. It should nevertheless be mentioned that the results do not constitute a full proof because there remain some di!erences between the calculations and the data analysis. Former data from the Texas A & M-group [260] are in agreement with those discussed here. The problem concerning the determination of the temperature has also been raised in the framework of a sequential decay model in which fragmentation evolves in time by means of successive binary break-ups of excited fragments [287] in a way which is similar to the approach of Ref. [94]. The analysis [142,288] of di!erent de"nitions of the temperature leads to di!erent results, due to the time-dependence of the process. It comes out that the experimental caloric curve [243] can be reproduced over a large range of excitation energies. Very recently an attempt to detect the signal of a phase transition through the determination of the speci"c heat has been made by the Multics}Miniball collaboration at the cyclotron of the Michigan State University [289,290]. The excited system has been generated by means of Au on Au collisions at 35 MeV A. The angular distributions of fragments show isotropy which is a sign for possible equilibration [289]. The conditions are such that the number of involved particles and the total energy are "xed, so that the arguments which characterize microcanonical ensembles can be applied. The correlation between the excitation energy EH and the temperature is "xed through the balance in energy at the freeze out + 3 m #EH"
(m #E )#E # (M!1)¹ ,  G G !
2 G where m is the initial mass excess, m the mass excesses of the fragments, EH the excitation energy  G of the source, E  the internal excitation energy of fragment i, E the Coulomb repulsion energy, G !
M the multiplicity and ¹ the temperature which is "xed by the balance equation. No collective energy is taken into account. A backtracing procedure is used in order to correlate the asymptotically observed events to the freeze-out con"gurations. The comparison of EH(¹) with SMM predictions are in good though not perfect agreement. The heat capacity shows a sizable enhancement for temperatures lying between 4 and 6 MeV. In a further step the experiment has been analysed following the microcanonical prescriptions of Chomaz and Gulminelli, see Section 6.2.3 [283,284]. The total energy E has been decomposed into R a kinetic and a potential contribution E "m #EH"E #E , R    where + E "
m #E  G !
G and E is the Coulomb energy acting between the fragments at freeze-out. This energy has been !
evaluated by means of simulations in which primary fragments are randomly distributed in a freeze-out volume equal to three times the normal volume. Two extreme hypotheses have been

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Fig. 28. Experimental heat capacities for the two extreme hypotheses which were made in order to evaluate E (see !
text). The dotted curve corresponds to the Fermi-gas calculation [290].

used. Either (1) all charged species including all light particles were considered as primary or (2) the totality of neutrons and light particles was distributed over the "nal fragments. These choices correspond to the procedures used in the framework of the SMM and MMMC models. In this way, one "xes the ensemble averaged quantities over sets of events E , E ,  and    C "(M!1) where  

 





+ 3
a ¹# (M!1) ¹ G 2 G and a is the level density of fragments i. G The results corresponding to cases (1) and (2) are shown in Fig. 28. In both cases one observes the signs for a divergent behaviour of the heat capacity at two values of the energy and the expected negative contributions to the speci"c heat in the microcanonical framework. The stability of this result with respect to variations in the size of excitation energy bins, the assumptions on EH, E and R the choice of the a 's have been checked. G The same type of analysis has been performed with the INDRA detector on the outcome of the reaction Xe on Sn at 32 and 50 MeV A central collisions [147,291]. In order to extract the kinetic energy #uctuations which enter the determination of the speci"c heat, it has been necessary to substract the collective e!ects which are present and to take care of the particle emission which takes place during the expansion phase of the process. One of the two divergences which develop in the heat capacity is observed as well as the onset of negative values for this quantity. The second singularity is not seen because the excitation range does not cover the whole interval of energy corresponding to the coexistence of the phases. E " 

7. Summary and conclusions In the present report we tried to summarize the present knowledge about the properties of excited nuclear matter which can be gained through the confrontation of simple classical models

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describing "nite systems and experimental facts obtained from energetic collisions between "nite nuclei. As an introduction we developed in Section 2 some historical facts concerning the beginning of the activities in the "eld of nuclear fragmentation, both on the side of the experiment with the observation of power laws in the fragment size distributions, the introduction of the droplet model and the theoretical developments based on mean "eld approaches to nuclear matter and microscopic transport descriptions in the mean "eld approximation and beyond. We recalled that there are two main di$culties in the interpretation of experimental data obtained by means of energetic nuclear collisions. The "rst concerns the "nite size of nuclei and the second the fact that the colliding objects evolve with time, from an initial compact system to an expanded system at the nuclear freeze-out where the nuclear interaction between fragments does no longer act, and "nally to the asymptotic stage where the remnants of the reaction are identi"ed by detectors. The second point leads to the question of the setting up of chemical and thermodynamic equilibrium. At lower energies there are signs which show that nuclear systems are able to absorb and spread sizable amounts of energy more or less uniformly over phase space. Thermodynamic equilibration is discussed in Section 3 in which some of the e!orts which have been made in order to get information about this point are developed. For the moment, equilibration seems to be a sensible concept since experimental results can be consistently interpreted in this framework, even though no strong and clear-cut proof for it exists up till now. One of the "rst and more spectacular successes in the "eld concerns the multiplicities of fragments of di!erent sizes which are amazingly well reproduced by means of simple minded percolation models. This point is presented and discussed at length in Section 4. Universality properties are clearly identi"ed and the intriguing power law behaviour which was observed and was of central interest in the early days appears again in a new context. Since percolation concepts are minimal information concepts it may appear that there is not much information to be gained from observables like fragment size distributions. The reasons for this may be related to the fact that highly excited nuclei behave like disordered systems, and as it is the case for many other systems of this type in other "elds of physics, the details of their dynamical properties are of very restricted importance. Universality properties of physical systems are remarkable because they work as a link between objects and phenomena which are seemingly very di!erent from each other and at the same time are insensitive to the detailed information about the system. The success of percolation drew the attention towards other simple classical models which have a strong kinship with percolation but are de"ned in terms of Hamiltonians. Some of them, the so-called lattice and cellular models, are described and analysed in Section 5. Their simplicity and at the same time their universal character leads to the observation that they work as good tools for the investigation of the properties of fragmenting nuclei. Most of them are able to reproduce the properties related to fragment multiplicities. They show very useful for the investigations of e!ects related to "niteness in search for and analysis of phase transitions. They predict the possible existence of two types of transitions which are not necessarily correlated in an evident way, a thermodynamic transition and a continuous percolation type transition which for systems with a density between nuclear density and half nuclear density occurs at temperatures which are higher than those which correspond to the thermodynamic transition. This point is not yet clearly understood, although it does not lead, a priori, to any contradiction. It opens up the old question

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concerning the link with Fisher's droplet model. Numerical simulations on di!erent models lead to very similar thermodynamic properties. The "nal part, Section 6, deals more speci"cally with the problem of the existence and characterization of phase transitions and signs which show their existence in "nite systems. For such types of analyses speci"c tools are needed. Some of them have been proposed very recently, at a time where similar considerations are developed in other "elds of physics dealing with small systems. Some of these tools are explicitly described. Theoretical methods relying on a microcanonical description and aimed to detect transitions which have been developed in the framework of lattice models have been presented. Their application in the analysis of experimental data is described at the end of the section. A great wealth of information has been accumulated about the characteristics and behaviour of highly excited nuclei, their fragmentation properties and the existence of a genuine phase transition of a type which may be similar to the classical liquid}gas transition. The last point remains however to be con"rmed. Many pieces of the puzzle have been brought together, however the whole picture is not yet totally clear. Experimental con"rmation of the transition has to be brought, scaling properties further investigated, the question of the possible coexistence of two types of transitions elucidated. Quantum mechanical aspects which have been quoted and the importance of quantum e!ects have to be further worked out and analysed. Last, there remains the di$cult problem of the possible o!-equilibrium character of the fragmentation process of a "nite critical system. The study of critical o!-equilibrium dynamics is a relatively recent "eld of research, still in its infancy. It is a "eld to which the nuclear physics community interested in the behaviour of highly excited and decaying systems of particles could certainly bring its contribution.

Acknowledgements The present report would not have been written without informative and clarifying discussions with many colleagues who provided advice and insight. We would like to thank A. Bonasera, M. Ploszajczak, J. Perdang for interesting exchanges of ideas and informations. We are grateful to C. Barbagallo, T. BiroH , J.M. Carmona, B. Elattari, M. Henkel, J. Knoll, A. Lejeune, S.K. Samaddar, A. TarancoH n, Y.M. Zheng for their active collaboration in those parts which concern our own contribution. Special thanks go to X. Campi, Ph. Chomaz, D.H.E. Gross, F. Gulminelli, H. Krivine and W. Trautmann for the direct and indirect supply of good ideas and e$cient information, illuminating explanations, critical and constructive remarks. We acknowledge clarifying remarks made by R. Bougault and L. Phair about the interpretation of experimental "gures. References [1] [2] [3] [4] [5] [6]

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PARTON-BASED GRIBOV+REGGE THEORY

H.J. DRESCHERa, M. HLADIKa, S. OSTAPCHENKOa,b, T. PIEROGa, K. WERNERa SUBATECH, UniversiteH de Nantes } IN2P3/CNRS } EMN, Nantes, France
Moscow State University, Institute of Nuclear Physics, Moscow, Russia now at SAP AG, Berlin, Germany

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

Physics Reports 350 (2001) 93}289

Parton-based Gribov}Regge theory H.J. Drescher , M. Hladik , S. Ostapchenko 
, T. Pierog , K. Werner * SUBATECH, Universite& de Nantes } IN2P3/CNRS } EMN, Nantes, 44072 Nantes, France
Moscow State University, Institute of Nuclear Physics, Moscow, Russia Received October 2000; editor: G.E. Brown Contents 1. Introduction 1.1. Present status 1.2. Parton-based Gribov}Regge theory 2. Formalism 2.1. Pro"le functions 2.2. Parton}parton scattering 2.3. Hadron}hadron scattering 2.4. Nucleus}nucleus scattering 2.5. Di!ractive scattering 2.6. AGK cancellations in hadron}hadron scattering 2.7. AGK cancellations in nucleus}nucleus scattering 2.8. Outlook 3. Virtual emissions 3.1. Parameterizing the elementary interaction 3.2. Calculating
for proton}proton collisions 3.3. Unitarity problems 3.4. A phenomenological solution: unitarization of
3.5. Properties of the unitarized theory 3.6. Comparison with the conventional approach 3.7. Unitarization for nucleus}nucleus scattering

97 98 100 101 101 103 108 112 116

4.

116 118 119 120

5.

120 6. 122 124 126 129 133 134

7.

3.8. Pro"le functions in nucleus}nucleus scattering 3.9. Inclusive cross sections in nucleus}nucleus scattering Markov chain techniques 4.1. Probability distributions for con"gurations 4.2. Interaction matrix 4.3. Markov chain method 4.4. Convergence 4.5. Some tests for proton}proton scattering Enhanced Pomeron diagrams 5.1. Calculating lowest-order enhanced diagrams 5.2. Cutting enhanced diagrams 5.3. Important features of enhanced diagrams Parton con"gurations 6.1. General procedure of parton generation 6.2. Generating the parton ladder 6.3. Time-like parton cascade Hadronization 7.1. Hadronic structure of cut Pomerons 7.2. Lagrange formalism for strings 7.3. Identifying partons and kinks 7.4. Momentum bands in parameter space 7.5. Area law

* Corresponding author. E-mail address: [email protected] (K. Werner).  Now at SAP AG, Berlin, Germany. 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 1 2 2 - 8

136 137 138 138 139 141 143 143 146 146 151 155 158 158 161 163 165 166 170 172 173 175

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289 7.6. Generating break points 7.7. From string fragments to hadrons 7.8. Transverse momentum 7.9. Fragmentation algorithm 8. Parameters 8.1. Hadronization 8.2. Time-like cascade 8.3. Parton}parton scattering 8.4. Hadron}hadron scattering 9. Testing time-like cascade and hadronization: electron}positron annihilation 9.1. Basic diagram 9.2. Time-like parton cascade and string formation 9.3. Event shape variables 9.4. Charged particle distributions 9.5. Identi"ed particles 9.6. Jet rates 10. Testing the semi-hard Pomeron: photon}proton scattering 10.1. Kinematics 10.2. Cross sections 10.3. Parton momentum distributions 10.4. Structure function F  10.5. Parton con"gurations: basic formulas 10.6. Generating initial conditions for the perturbative evolution 10.7. Generating the ladder partons 10.8. Hadron production 10.9. Results

177 180 183 185 185 185 186 187 187 190 190 191 192 198 201 202 203 210 213 215 221 223 226 229 232 233

11. Results for proton}proton scattering 11.1. Energy dependence 11.2. Charged particle and pion spectra 11.3. Proton spectra 11.4. Strange particle spectra 12. Results for collisions involving nuclei 12.1. Proton}nucleus scattering 12.2. Nucleus}nucleus scattering 13. Summary Acknowledgements Appendix A. Kinematics of two-body collisions A.1. Conventions A.2. Proof of the impossibility of longitudinal excitations Appendix B. Partonic interaction amplitudes B.1. Semi-hard parton}parton scattering B.2. Parton evolution B.3. Time-like parton splitting Appendix C. Hadron}hadron amplitudes C.1. Neglecting valence quark scatterings C.2. Including valence quark hard scatterings C.3. Enhanced diagrams C.4. Parton generation for triple-Pomeron diagrams Appendix D. Calculation for
and H D.1. Calculation of
NN D.2. Calculation of H D.3. Calculation of
 D.4. Exponentiation of
 References

95 238 240 240 240 241 241 241 245 253 255 256 256 257 258 258 262 263 265 265 269 270 274 275 275 278 281 283 284

Abstract We present a new parton model approach for hadron}hadron interactions and, in particular, for the initial stage of nuclear collisions at very high energies (RHIC, LHC and beyond). The most important aspect of our approach is a self-consistent treatment, using the same formalism for calculating cross sections and particle production, based on an e!ective, QCD-inspired "eld theory, where many of the inconsistencies of presently used models will be avoided. In addition, we provide a uni"ed treatment of soft and hard scattering, such that there is no fundamental cuto! parameter any more de"ning an arti"cial border between soft and hard scattering. Our approach cures some of the main de"ciencies of two of the standard procedures currently used: the Gribov}Regge theory and the eikonalized parton model. There, cross section calculations and particle production cannot be treated in a consistent way using a common formalism. In particular, energy conservation is taken care of in case of particle production, but not concerning cross section calculations. In addition, hard contributions depend crucially on some cuto!, being divergent for the cuto! being zero. Finally, in case of several elementary scatterings, they are not treated on the same level: the "rst collision is always treated di!erently from the subsequent ones. All these problems are solved in our new approach. For testing purposes, we make very detailed studies of electron}positron annihilation and lepton}nucleon scattering before applying our approach to proton}proton and nucleus}nucleus collisions.

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In order to keep a clean picture, we do not consider secondary interactions. We provide a very transparent extrapolation of the physics of more elementary interactions towards nucleus}nucleus scattering, without considering any nuclear e!ects due to "nal state interactions. In this sense we consider our model a realistic and consistent approach to describe the initial stage of nuclear collisions.  2001 Elsevier Science B.V. All rights reserved. PACS: 13.85.!t Keywords: Gribov}Regge theory; Ultrarelativistic nucleus}nucleus scattering; Pomeron

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97

1. Introduction The purpose of this paper is to provide the theoretical framework to treat hadron}hadron scattering and the initial stage of nucleus}nucleus collisions at ultra-relativistic energies, in particular with view to RHIC and LHC. The knowledge of these initial collisions is crucial for any theoretical treatment of a possible parton}hadron phase transition, the detection of which being the ultimate aim of all the e!orts of colliding heavy ions at very high energies. It is quite clear that coherence is crucial for the very early stage of nuclear collisions, so a real quantum mechanical treatment is necessary and any attempt to use a transport theoretical parton approach with incoherent quasi-classical partons should not be considered at this point. Also semi-classical hadronic cascades cannot be stretched to account for the very "rst interactions, even when this is considered to amount to a string excitation, since it is well known [1] that such longitudinal excitation is simply kinematically impossible (see Section A.2). There is also the very unpleasant feature of having to treat formation times of leading and non-leading particles very di!erent. Otherwise, due to a large gamma factor, it would be impossible for a leading particle to undergo multiple collisions. So what are the currently used fully quantum mechanical approaches? There are presently considerable e!orts to describe nuclear collisions via solving classical Yang}Mills equations, which allows to calculate inclusive parton distributions [2}19]. This approach is to some extent orthogonal to ours: here, screening is due to perturbative processes, whereas we claim to have good reasons to consider soft processes to be at the origin of screening corrections. Provided factorization works for nuclear collisions, on may employ the parton model, which allows to calculate inclusive cross sections as a convolution of an elementary cross section with parton distribution functions, with these distribution function taken from deep inelastic scattering. In order to get exclusive parton level cross sections, some additional assumptions are needed, as discussed later. Another approach is the so-called Gribov}Regge theory. This is an e!ective "eld theory, which allows multiple interactions to happen `in parallela, with the phenomenological object called `Pomerona representing an elementary interaction. Using the general rules of "eld theory, on may express cross sections in terms of a couple of parameters characterizing the Pomeron. Interference terms are crucial, they assure the unitarity of the theory. A disadvantage is the fact that cross sections and particle production are not calculated consistently: the fact that energy needs to be shared between many Pomerons in case of multiple scattering is well taken into account when calculating particle production (in particular in Monte Carlo applications), but energy conservation is not taken care of for cross section calculations. This is a serious problem and makes the whole approach inconsistent. Also a problem is the question of how to include in a consistent way hard processes, which are usually treated in the parton model approach. Another unpleasant feature is the fact that di!erent elementary interactions in case of multiple scattering are usually not treated equally, so the "rst interaction is usually considered to be quite di!erent compared to the subsequent ones. Here, we present a new approach which we call Parton-based Gribov}Regge Theory, where we solve some of the above-mentioned problems: we have a consistent treatment for calculating cross sections and particle production considering energy conservation in both cases; we introduce hard processes in a natural way, and, compared to the parton model, we can deal with exclusive cross

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sections without arbitrary assumptions. A single set of parameters is su$cient to "t many basic spectra in proton}proton and lepton}nucleon scattering, as well as for electron}positron annihilation (with the exception of one parameter which needs to be changed in order to optimize electron}positron transverse momentum spectra). The basic guideline of our approach is theoretical consistency. We cannot derive everything from "rst principles, but we use rigorously the language of "eld theory to make sure not to violate basic laws of physics, which is easily done in more phenomenological treatments (see discussion above). There are still problems and open questions: there is clearly a problem with unitarity at very high energies, which should be cured by considering screening corrections due to so-called triplePomeron interactions, which we do not treat rigorously at present but which is our next project. 1.1. Present status Before presenting new theoretical ideas, we want to discuss a little bit more in detail the present status and, in particular, the open problems in the parton model approach and in Gribov}Regge theory. 1.1.1. Gribov}Regge theory Gribov}Regge theory [20}57] is by construction a multiple scattering theory. The elementary interactions are realized by complex objects called `Pomeronsa, whose precise nature is not known, and which are therefore simply parameterized: the elastic amplitude ¹ corresponding to a single Pomeron exchange is given as ¹(s, t)&is? >?YR

(1.1)

with a couple of parameters to be determined by experiment. Even in hadron}hadron scattering, several of these Pomerons are exchanged in parallel, see Fig. 1.1. Using general rules of "eld theory (cutting rules), one obtains an expression for the inelastic cross section,




F F " db1!exp(!G(s, b)) , 

(1.2)

where the so-called eikonal G(s, b) (proportional to the Fourier transform of ¹(s, t)) represents one elementary interaction (a thick line in Fig. 1.1). One can generalize to nucleus}nucleus collisions, where corresponding formulas for cross sections may be derived.

Fig. 1.1. Hadron}hadron scattering in GRT. The thick lines between the hadrons (incoming lines) represent a Pomeron each. The di!erent Pomeron exchanges occur in parallel.

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GRT has been applied extensively to calculate successfully many characteristics of proton}proton, proton}nucleus, and nucleus}nucleus scattering, using the Dual Parton Model [58}73], the quark gluon string model [74}86], and VENUS [87}101]. In order to calculate exclusive particle production, one needs to know how to share the energy between the individual elementary interactions in case of multiple scattering. We do not want to discuss the di!erent recipes used to do the energy sharing (in particular in Monte Carlo applications). The point is, whatever procedure is used, this is not taken into account in calculation of cross sections discussed above. So, actually, one is using two di!erent models for cross section calculations and for treating particle production. Taking energy conservation into account in exactly the same way will modify the cross section results considerably, as we are going to demonstrate later. This problem has "rst been discussed in [102,103]. The authors claim that following from the non-planar structure of the corresponding diagrams, conserving energy and momentum in a consistent way is crucial, and therefore the incident energy has to be shared between the di!erent elementary interactions, both real and virtual ones. Another very unpleasant and unsatisfactory feature of most `recipesa for particle production is the fact, that the second Pomeron and the subsequent ones are treated di!erently than the "rst one, although in the above-mentioned formula for the cross section all Pomerons are considered to be identical. 1.1.2. The parton model The standard parton model approach to hadron}hadron or also nucleus}nucleus scattering amounts to presenting the partons of projectile and target by momentum distribution functions, f  and f  , and calculating inclusive cross sections for the production of parton jets with the F F squared transverse momentum p larger than some cuto! Q as , 

  

d
(
F F "
dp dx> dx\f G (x>, p ) f H (x\, p ) GH (x>x\s)(p !Q ) , , F , F , dp ,   , GH

(1.3)

where d
( /dp is the elementary parton}parton cross section and i, j represent parton #avors. GH , Many Monte Carlo applications are based on the above formula: ISAJET [104], PYTHIA [105], HERWIG [106}108], for more details see the review [109]. The simple factorization formula is the result of cancellations of complicated diagrams (AGK cancellations) and hides therefore the complicated multiple scattering structure of the reaction. The most obvious manifestation of such a structure is the fact that at high energies ((s<10 GeV) the inclusive cross section in proton}(anti)proton scattering exceeds the total one, so the average number NM NN of elementary interactions must be bigger than one:  NM F F "
F F /
F F '1 .   

(1.4)

The usual solution is the so-called eikonalization [110}119], which amounts to re-introducing multiple scattering, based on the above formula for the inclusive cross section:




F F (s)" db1!exp(!A(b)
F F (s))"

F F (s) ,   K

(1.5)

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with




F F (s)" db K

(A(b)
F F (s))K  exp(!A(b)
F F (s))  m!

(1.6)

representing the cross section for m scatterings; A(b) being the proton}proton overlap function (the convolution of the two proton pro"les). In this way the multiple scattering is `recovereda. The disadvantage is that this method does not provide any clue how to proceed for nucleus}nucleus (AB) collisions. One usually assumes the proton}proton cross section for each individual nucleon}nucleon pair of a AB system. We are going to demonstrate that this assumption is incorrect. Another problem, in fact the same one as discussed earlier for the GRT, arises in case of exclusive calculations (event generation), since the above formulas do not provide any information on how to share the energy between the many elementary interactions. The Pythia method [116] amounts to generating the "rst elementary interaction according to the inclusive di!erential cross section, then taking the remaining energy for the second one and so on. In this way, the event generation will reproduce the theoretical inclusive spectrum for hadron}hadron interaction (by construction), as it should be. The method is, however, very arbitrary, and is certainly not a convincing procedure for the multiple scattering aspects of the collisions. The factorization formula Eq. (1.3) is also used in order to introduce hard processes in the GRT approach [71,120], which we refer to as GRT#minijets. The eikonalized Parton model and GRT#minijets is essentially the same thing, just the original starting point is di!erent. 1.2. Parton-based Gribov}Regge theory In this paper, we present a new approach for hadronic interactions and for the initial stage of nuclear collisions, which is able to solve several of the above-mentioned problems [121}125]. We provide a rigorous treatment of the multiple scattering aspect, such that questions as energy conservation are clearly determined by the rules of the "eld theory, both for cross section and particle production calculations. In both (!) cases, energy is properly shared between the di!erent interactions happening in parallel, see Fig. 1.2 for proton}proton and Fig. 1.3 for proton}nucleus collisions (generalization to nucleus}nucleus is obvious). This is the most important and new aspect of our approach, which we consider to be a "rst necessary step to construct a consistent model for high energy nuclear scattering. The elementary interactions, shown as the thick lines in the above "gures, are in fact a sum of a soft, a hard, and a semi-hard contribution, providing a consistent treatment of soft and hard scattering. To some extent, our approach provides a link between the Gribov}Regge approach and the parton model, we call it `Parton-based Gribov}Regge theorya. There are still many problems to be solved: as we are going to show later, a rigorous treatment of energy conservation will lead to unitarity problems (increasingly severe with increasing energy), which is nothing but a manifestation of the fact that screening corrections will be increasingly important. In this paper we will employ a `unitarization procedurea to solve this problem, but this is certainly not the "nal answer. The next step should be a consistent approach, taking into account both energy conservation and screening corrections due to multi-Pomeron interactions.

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Fig. 1.2. Graphical representation of a contribution to the elastic amplitude of proton}proton scattering. Here, energy conservation is taken into account: the energy of the incoming protons is shared among several `constituentsa (shown by splitting the nucleon lines into several constituent lines), and so each Pomeron disposed only a fraction of the total energy, such that the total energy is conserved. Fig. 1.3. Graphical representation of a contribution to the elastic amplitude of proton}nucleus scattering, or more precisely a proton interacting with (for simplicity) two target nucleons, taking into account energy conservation. Here, the energy of the incoming proton is shared between all the constituents, which now provide the energy for interacting with two target nucleons.

Our approach is realized as the Monte Carlo code NEXUS, which is nothing but the direct implementation of the formalism described in this paper. All our numerical results are calculated with NEXUS version 2.00.

2. Formalism We want to calculate cross sections and particle production in a consistent way, in both cases based on the same formalism, with energy conservation being ensured. The formalism operates with Feynman diagrams of the QCD-inspired e!ective "eld theory, such that calculations follow the general rules of the "eld theory. A graphical representation of a contribution to the elastic amplitude of nucleus}nucleus scattering (related to particle production via the optical theorem) is shown in Fig. 2.1: here the nucleons are split into several `constituentsa, each one carrying a fraction of the incident momentum, such that the sum of the momentum fractions is one (momentum conservation). Per nucleon there are one or several `participantsa and exactly one `spectatora or `remnanta, where the former ones interact with constituents from the other side via some `elementary interactiona (vertical lines in Fig. 2.1). The remnant is just all the rest, i.e. the nucleon minus the participants. After a technical remark concerning pro"le functions, we are going to discuss parton}parton scattering, before we develop the multiple scattering theory for hadron}hadron and nucleus} nucleus scattering. 2.1. Proxle functions Pro"le functions play a fundamental role in our formalism, so we brie#y sketch their de"nition and physical meaning.

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Fig. 2.1. A contribution to the elastic amplitude of a nucleus}nucleus collision, or more precisely two nucleons from projectile A interacting with two nucleons from target B, taking into account energy conservation. The energy of the incoming nucleons is shared between all the constituents.

Let ¹ be the elastic scattering amplitude ¹ for the two-body scattering depicted in Fig. 2.2. The 4-momenta p and p are the ones for the incoming particles, p "p#q and p "p!q the ones for the outgoing particles, and q the 4-momentum transfer in the process. We de"ne as usual the Mandelstam variables s and t (see Section A.1). Using the optical theorem, we may write the total cross section as 1
(s)" 2Im ¹(s, t"0) .  2s

(2.1)

We de"ne the Fourier transform ¹I of ¹ as



1 ¹I (s, b)" dq e\ Oo , @o ¹(s, t) , , 4 using t"!q (see Section A.2), and a so-called `pro"le functiona G(s, b) as , 1 G(s, b)" 2Im ¹I (s, b) . 2s

(2.2)

(2.3)

One can easily verify that




(s)" db G(s, b) , 

(2.4)

which allows an interpretation of G(s, b) to be the probability of an interaction at impact parameter b. In the following, we are working with partonic, hadronic, and even nuclear pro"le functions. The central result to be derived in the following sections is the fact that hadronic and also nuclear pro"le functions can be expressed in terms of partonic ones, allowing a clean formulation of a multiple scattering theory.

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Fig. 2.2. The general elastic scattering amplitude ¹. Fig. 2.3. The soft contribution.

2.2. Parton}parton scattering We distinguish three types of elementary parton}parton scatterings, referred to as `softa, `harda and `semi-harda, which we are going to discuss brie#y in the following. The detailed derivations can be found in Section B.1. 2.2.1. Soft contribution Let us "rst consider the pure non-perturbative contribution to the process of Fig. 2.2, where all virtual partons appearing in the internal structure of the diagram have restricted virtualities Q(Q , where Q &1 GeV is a reasonable cuto! for perturbative QCD being applicable. Such   soft non-perturbative dynamics is known to dominate hadron}hadron interactions at not too high energies. Lacking methods to calculate this contribution from "rst principles, it is simply parameterized and graphically represented as a `bloba, see Fig. 2.3. It is traditionally assumed to correspond to multi-peripheral production of partons (and "nal hadrons) [126] and is described by the phenomenological soft Pomeron exchange contribution [25]:




s( ?  exp(  (s( /s )t) , ¹ (s( , t)"8s (t)       s 

(2.5)

with L (z)"nR #  ln z , (2.6)     where s( "(p#p). The parameters (0),  are the intercept and the slope of the Pomeron   trajectory,  and R are the vertex value and the slope for the Pomeron}parton coupling, and     s K1 GeV is the characteristic hadronic mass scale. The so-called signature factor  is given as   (t) (t)"i!cot . Ki . (2.7) 2 Cutting the diagram of Fig. 2.3 corresponds to the summation over multi-peripheral intermediate hadronic states, connected via unitarity to the imaginary part of the amplitude (2.5), 1 1 disc ( ¹ (s( , t)" [¹ (s( #i0, t)!¹ (s( !i0, t)] Q   i i 

(2.8)

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"2Im ¹ (s( , t) 

(2.9)



d ¹ ¹H , (2.10) "
L NNY6L N N Y6L 2 L  is the amplitude for the transition of the initial partons p, p into the n-particle state where ¹ NNY6L X , d is the invariant phase space volume for the n-particle state X and the summation is done L L L over the number of particles n and over their spins and species, the averaging over initial parton colors and spins is assumed; disc ( ¹ (s( , t) denotes the discontinuity of the amplitude ¹ (s( , t) on  Q  the right-hand cut in the variable s( . For t"0 one obtains via the optical theorem the contribution
of the soft Pomeron  exchange to the total parton interaction cross section,




s( ? \ 1 , (2.11)
(s( )" 2Im ¹ (s( , 0)"8     s 2s(  where 2s( de"nes the initial parton #ux. The corresponding pro"le function for parton}parton interaction is expressed via the Fourier transform ¹I of ¹ divided by the #ux 2s, 1 D (s( , b)" 2Im ¹I (s( , b) ,   2s(

(2.12)

which gives



1 D (s( , b)" dq exp(!iqo bo ) 2Im ¹ (s( ,!q )  , ,  , 8s(







(2.13)



s( ? \ b 2   exp ! . (2.14) " 4  (s( /s )  (s( /s ) s      The external legs of the diagram of Fig. 2.3 are `partonic constituentsa, which are assumed to be quark}antiquark pairs. 2.2.2. Hard contribution Let us now consider the other extreme, when all the processes in the `boxa of Fig. 2.2 are perturbative, i.e. all internal intermediate partons are characterized by large virtualities Q'Q .  In that case, the corresponding hard parton}parton scattering amplitude ¹HI (s( , t) ( j, k denote the   types (#avors) of the initial partons) can be calculated using the perturbative QCD techniques [127,128], and the intermediate states contributing to the absorptive part of the amplitude of Fig. 2.2 can be de"ned in the parton basis. In the leading logarithmic approximation of QCD, summing up terms where each (small) running QCD coupling constant (Q) appears together Q with a large logarithm ln(Q/  ) (with being the infrared QCD scale), and making use of the /!" /!" factorization hypothesis, one obtains the contribution of the corresponding cut diagram for t"q"0 as the cut parton ladder cross section
HI (s( , Q ), which will correspond to the cut    diagram of Fig. 2.4, where all horizontal rungs are the "nal (on-shell) partons and the virtualities of  Strictly speaking, one obtains the ladder representation for the process only using axial gauge.

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Fig. 2.4. The hard (or val}val) contribution.

the virtual t-channel partons increase from the ends of the ladder towards the largest momentum transfer parton}parton process (indicated symbolically by the `bloba in the middle of the ladder): 1
HI (s( , Q )" 2Im ¹HI (s( , t"0)      2s(



d
KJ  (x>x\s( , p ) "K
dx> dx\ dp , dp , , KJ ;EHK (x>, Q , M ) EIJ (x\, Q , M )(M !Q ) . (2.15) /!"  $ /!"  $ $  Here d
KJ /dp is the di!erential 2P2 parton scattering cross section, p is the parton transverse  , , momentum in the hard process, m, l and x! are correspondingly the types and the shares of the light cone momenta of the partons participating in the hard process, and M is the factorization $ scale for the process (we use M "p /4). The `evolution functiona EHK (Q , M , z) represents the $ , /!"  $ evolution of a parton cascade from the scale Q to M , i.e. it gives the number density of partons of  $ type m with the momentum share z at the virtuality scale M , resulted from the evolution of the $ initial parton j, taken at the virtuality scale Q . The evolution function satis"es the usual DGLAP  equation [129}132] with the initial condition EHK (Q , Q , z)" H (1!z), as discussed in detail in /!"   K Section B.3. The factor KK1.5 takes e!ectively into account higher-order QCD corrections. In the following we shall need to know the contribution of the uncut parton ladder ¹HI (s( , t)   with some momentum transfer q along the ladder (with t"q). The behavior of the corresponding amplitudes was studied in [133] in the leading logarithmic(1/x) approximation of QCD. The precise form of the corresponding amplitude is not important for our application; we just use some of the results of [133], namely that one can neglect the real part of this amplitude and that it is nearly independent of t, i.e. that the slope of the hard interaction R is negligibly small, i.e.   compared to the soft Pomeron slope one has R K0. So we parameterize ¹HI (s( , t) in the region     of small t as [134] ¹HI (s( , t)"is(
HI (s( , Q ) exp(R t) . (2.16)        The corresponding pro"le function is obtained by calculating the Fourier transform ¹I of   ¹ and dividing by the initial parton #ux 2s( ,   1 (2.17) DHI (s( , b)" 2Im ¹I HI (s( , b) ,     2s(

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which gives



1 dq exp(!iqo bo ) 2Im ¹HI (s( ,!q ) DHI (s( , b)" , ,   ,   8s(






b 1 exp ! , (2.18) "
HI (s( , Q )    4R 4R     In fact, the above considerations are only correct for valence quarks, as discussed in detail in the next section. Therefore, we also talk about `valence}valencea contribution and we use D }   instead of D :   DHI } (s( , b),DHI (s( , b) , (2.19)     so these are two names for one and the same object. 2.2.3. Semi-hard contribution The discussion of the preceding section is not valid in case of sea quarks and gluons, since here the momentum share x of the `"rsta parton is typically very small, leading to an object with  a large mass of the order Q /x between the parton and the proton [135]. Microscopically, such   `slowa partons with x ;1 appear as a result of a long non-perturbative parton cascade, where  each individual parton branching is characterized by a small momentum transfer squared Q(Q  and nearly equal partition of the parent parton light cone momentum [44,126]. When calculating proton structure functions or high-p jet production cross sections that non-perturbative contribuR tion is usually included into parameterized initial parton momentum distributions at Q"Q .  However, the description of inelastic hadronic interactions requires to treat it explicitly in order to account for secondary particles produced during such non-perturbative parton pre-evolution, and to describe correctly energy-momentum sharing between multiple elementary scatterings. As the underlying dynamics appears to be identical to the one of soft parton}parton scattering considered above, we treat this soft pre-evolution as the usual soft Pomeron emission, as discussed in detail in Section B.1. So for sea quarks and gluons, we consider so-called semi-hard interactions between parton constituents of initial hadrons, represented by a parton ladder with `soft endsa, see Fig. 2.5. As in case of soft scattering, the external legs are quark}antiquark pairs, connected to soft Pomerons. The outer partons of the ladder are on both sides sea quarks or gluons (therefore the index `sea}seaa). The central part is exactly the hard scattering considered in the preceding section. As discussed in length in Section B.1, the mathematical expression for the corresponding amplitude is given as









 dz> dz\ s s  , t Im ¹I  , t i¹HI (z>z\s( , t) , Im ¹H (2.20) i¹ } (s( , t)"
      z> z\ z> z\ HI  with z! being the momentum fraction of the external leg-partons of the parton ladder relative to the momenta of the initial (constituent) partons. The indices j and k refer to the #avor of these external ladder partons. The amplitudes ¹H are the soft Pomeron amplitudes discussed earlier, but with  modi"ed couplings, since the Pomerons are now connected to the ladder on one side. The arguments s /z! are the squared masses of the two soft Pomerons, z>z\s( is the squared mass of the hard piece. 

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Fig. 2.5. The semi-hard `sea}seaa contribution: parton ladder plus `soft endsa. Fig. 2.6. Two `mixeda contributions.

Performing as usual the Fourier transform to the impact parameter representation and dividing by 2s( , we obtain the pro"le function 1 D } (s( , b)" 2Im ¹I } (s( , b) ,     2s(

(2.21)

which may be written as



 dz> dz\EH (z>) EI (z\)
HI (z>z\s( , Q ) D } (s( , b)"
       HI  b 1 exp ! (2.22) ; 4  (1/(z>z\)) 4  (1/(z>z\))   with the soft Pomeron slope  and the cross section
HI being de"ned earlier. The functions    EH (z!) representing the `soft endsa are de"ned as  s  , t"0 , (2.23) EH (z!)"Im ¹H   z>











or explicitly EE (z)"8s   z\?  (1!z)@E ,     E  z , EO (z)" d PO ( ) EE  OE E  X

(2.24)

  "w  , OE E   E

(2.26)






(2.25)

with  "(1!w ) , E   E

and (3! (0)# ) 1  E  " E 8s  (2! (0))(1# )  E   

(2.27)

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(see Section B.1). We neglected the small hard scattering slope R compared to the Pomeron   slope  . We call E also the `soft evolutiona, to indicate that we consider this as simply   a continuation of the QCD evolution, however, in a region where perturbative techniques do not apply any more. As discussed in Section B.1, EH (z) has the meaning of the momentum distribution  of parton j in the soft Pomeron. Consistency requires to also consider the mixed semi-hard contributions with a valence quark on one side and a non-valence participant (quark}antiquark pair) on the other one, see Fig. 2.6. We have



i¹H } (s( , t)"  




 dz\ s  , t i¹HI (z\s( , t)
Im ¹I    z\ z\  I

(2.28)

and



 dz\ EI (z\)
HI (z\s( , Q ) DH } (s( , b)"
      I  1 b ; exp ! (2.29) 4  (1/z\) 4  (1/z\)   where j is the #avor of the valence quark at the upper end of the ladder and k is the type of the parton on the lower ladder end. Again, we neglected the hard scattering slope R compared to   the soft Pomeron slope. A contribution DH } (s( , b), corresponding to a valence quark participant   from the target hadron, is given by the same expression,






(2.30) DH } (s( , b)"DH } (s( , b) ,     since Eq. (2.29) stays unchanged under replacement z\Pz> and only depends on the total c.m. energy squared s( for the parton}parton system. 2.2.4. Pomeron The name Pomeron refers to the amplitudes of elementary parton}parton scatterings, the corresponding pro"le functions being the above-mentioned functions D , D } , D } ,      and D } . Correspondingly, we have Pomerons of di!erent types, namely `softa, D }     `val}vala, `sea}seaa, `val}seaa, and `sea}vala. We also refer to the ensemble of all Pomerons with the exception of the soft one as semi-hard Pomerons. Our semi-hard Pomeron should not be confused with the BFKL Pomeron [133,136}147], which is sometimes also referred to as `harda Pomeron. In our case, the QCD part is strictly based on the standard DGLAP evolution [127] with ordered parton virtualities in the ladder. For completeness, we mention also the fact that a so-called non-perturbative Pomeron has been introduced [148}151], based on non-perturbative, "nite-range gluon propagators. 2.3. Hadron}hadron scattering Let us now consider hadron}hadron interactions (a more detailed treatment can be found in Appendix C). We ignore "rst contributions involving valence quark scatterings. In the general case, the expression for the hadron}hadron scattering amplitude includes contributions from multiple

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109

scattering between di!erent parton constituents of the initial hadrons, as shown in Fig. 2.7; and can be written according to the standard rules [25,44] as

 



 1 L dk dk dq J J J NL(p, k ,2, k , q ,2, q )  i¹   (s, t)"
FF  L  L n! (2) (2) (2) F J L L L ;  [i¹ / (s( , q)]NL (p, k ,2, k ,!q ,2,!q )(2) 
q !q , (2.31) F  L  L  J J G J I with t"q, s"(p#p)Kp>p\, with p, p being the 4-momenta of the initial hadrons, and with s( "(k #k )Kk>k \. ¹ / is the sum of partonic one-Pomeron-exchange scattering amplitudes, J J J J J  discussed in the preceding section, ¹ / "¹ #¹ } . The momenta k , k and q denote J J J     correspondingly the 4-momenta of the initial partonic constituents (quark}antiquark pairs) for the lth scattering and the 4-momentum transfer in that partial process. The factor 1/n! takes into account the identical nature of the n re-scattering contributions. NL(p, k ,2, k , q ,2, q ) denotes F  L  L the contribution of the vertex for n-parton coupling to the hadron h. As discussed in Appendix C, the hadron}hadron amplitude (2.31) may be written as






 





L 1  1  L  dx> dx\  i¹   (s, t)"8s
dq , i¹F/F (x>, x\, s,!q, ) J J  FF J J J J n! 8s(  J J J L L L L ;FF 1!
x> FF 1!
x\ 
qo , !qo . (2.32) I H H ,     I H H (see Eq. (C.22)), where the partonic amplitudes are de"ned as ¹F/F "¹F F #¹F F} , with     (x>x\s,!q )FF (x>)FF (x\) ¹F F } (x>, x\, s,!q )"¹   ,  } ,     (2.33) ;exp(![R #R ]q ) F , F representing the contributions of `elementary interactions plus external legsa; the functions FF , FF are de"ned in (C.18)}(C.21) as     (2.34) FF (x)" x\?  ,   F (2.35) FF (x)"x?F  .   Formula (2.32) is also correct if one includes valence quarks (see Appendix C), if one de"nes










¹F/F "¹F F #¹F F} #¹F F} #¹F F} #¹F F} ,           with the hard contribution







(2.36)

V> x> V\ x\ dx> dx\
¹HI (x>x\s,!q ) T x> T x\   T T ,  T  T HI ;FM F H (x>, x>!x>)FM F I (x\, x\!x\) exp(![R #R ]q ) ,   T T   T T F F , (2.37)

¹F F} (x>, x\, s,!q )"   ,

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with the mixed semi-hard `val}seaa contribution



V> x> dx>
¹H } (x>x\s,!q ) ,   T T x>  T H ;FM F H (x>, x>!x>)FF (x\) exp(![R #R ]q ) ,   T T   F , F and with the contribution `sea}vala obtained from `val}seaa by exchanging variables, ¹F F} (x>, x\, s,!q )"   ,

(2.38)

¹F F} (x>, x\, s,!q )"¹F F} (x\, x>, s,!q ) .   ,   , Here, we allow formally any number of valence type interactions (based on the fact that multiple valence type processes give negligible contribution). In the valence contributions, we have convolutions of hard parton scattering amplitudes ¹HI and valence quark distributions FM H over the     valence quark momentum share x! rather than a simple product, since only the valence quarks are T involved in the interactions, with the anti-quarks staying idle (the external legs carrying momenta x> and x\ are always quark}antiquark pairs). The functions FM are given as FM FG (x , x  )"N\qG (x , Q )(1!x )?1 \\?  (x  )\?1 ,  T  T O   T O with the normalization factor

(2.39)

(1#

)(1! 1 )   , (2.40) N" (2#

! 1 )   where qG is a usual valence quark distribution function. The Fourier transform ¹I   of amplitude  FF (2.32) is given as



L i  1  L i  dx> dx\  ¹I (s, b)"
¹I F/F (x>, x\, s, b) J J J J 2s(  n! 2s F F  J J J L L L 1!
x> FF 1!
x\ , (2.41) ;FF     H H H H with ¹I F/F being the Fourier transform of ¹F/F . The pro"le function  is as usual de"ned as   1    (s, b)" 2Im ¹I   (s, b) , (2.42) FF FF 2s









which may be evaluated using the AGK cutting rules [30],

 


 1    (s, b)"
FF m! K  1 ;
l! J

 K K 1 2Im ¹I F/F (x>, x\, s, b)  dx> dx\   I I I I 2s(  I I I  J J 1  dx > dx \  (!2) Im ¹I F/F (x >, x \, s, b)  H H H H 2s(  H H H K J K J ;FF 1!
x>!
x > FF 1!
x\!
x \ , (2.43)     H I H I H I H I where 2Im ¹I F F represents a cut elementary diagram and !2Im ¹I F F an uncut one (taking into account that the uncut contribution may appear on either side from the cut plane). It is therefore






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Fig. 2.7. Hadron}hadron interaction amplitude. Fig. 2.8. The hadronic pro"le function  expressed in terms of partonic pro"le functions G,GF/F . 

useful to de"ne a partonic pro"le function G via 1 2Im ¹I F/F (x>, x\, s, b) , (2.44) GF/F (x>, x\, s, b)"   H H H H 2x>x\s H H which allows to write the integrand of the right-hand side of Eq. (2.43) as a product of G and (!G) terms:

 


 1    (s, b)"
FF m! K  1 ;
l! J

 K K  dx> dx\  GF/F (x>, x\, s, b)  I I I I  I I  J J  dx > dx \  !GF/F (x >, x \, s, b)  H H H H  H H






;F x !
x > F x !
x \ ,   H   H H H

(2.45)

see Fig. 2.8, with x  "1!
x! I being the momentum fraction of the projectile/target remnant. This is a very important result, allowing to express the total pro"le function    via the elementary pro"le functions GF/F .  FF Based on the above de"nitions, we may write the pro"le function GF/F as  (2.46) GF/F "GF F #GF F} #GF F} #GF F} #GF F} ,           with GF F } (x>, x\, s, b)"DF F } (x>x\s, b)FF (x>)FF (x\)         for the soft and semi-hard `sea}seaa contribution, with

(2.47)



GF F} (x>, x\, s, b)" dx>T dx\T
DF F} GH (x>T x\T s, b)   O O   O O GH ;FM F G (x>T , x>!x>T )FM F H (x\T , x\!x\T )   O O   O O

(2.48)

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for the hard `val}vala contribution, and with



(2.49)



(2.50)

GF F} (x>, x\, s, b)" dx>T
DF F} G (x>T x\s, b)FM F G (x>T , x>!x>T )FF (x\)   O   O   O O   G and GF F} (x>, x\, s, b)" dx\T
DF F} G (x>x\T s, b)FM F G (x\T , x\!x\T )FF (x>) , O     O O     O G

for the mixed semi-hard `val}seaa and `sea}vala contributions. For the soft and `sea}seaa contributions, the D-functions are given as







1 b (s( , bo !bo ) exp ! , DF F } (s( , b)" dbD }     4(R #R ) 4(R #R ) F F F F

(2.51)

has the same functional form as D , with  ( ) being which means that DF F }    }  replaced by F F ( )"  ( )#R #R F   F KR #R #  ln , F  F

(2.52) (2.53)

where we neglected the parton slope R compared to the hadron slope R.   F For the hard contribution, we have correspondingly DF F} GH (s( , b) being given in Eq. (2.18) with   R being replaced by R #R (neglecting the hard scattering slope R as compared to the   F F   hadron Regge slopes R , R ). In case of mixed Pomerons, we have DF F} G (s( , b) given in Eq. (2.29) F   F with R being replaced by R #R .   F F 2.4. Nucleus}nucleus scattering We generalize the discussion of the last section in order to treat nucleus}nucleus scattering. In the Glauber}Gribov approach [28,152], the nucleus}nucleus scattering amplitude is de"ned by the sum of contributions of diagrams, shown in Fig. 2.1, corresponding to multiple scattering processes between parton constituents of projectile and target nucleons. Nuclear form factors are supposed to be de"ned by the nuclear ground state wave functions. Assuming the nucleons to be uncorrelated, one can make the Fourier transform to obtain the amplitude in the impact parameter representation. Then, for given impact parameter bo between the nuclei, the only formal di!erences  from the hadron}hadron case will be the possibility for a given nucleon to interact with a number of nucleons from the partner nucleus as well as the averaging over nuclear ground states, which amounts to an integration over transverse nucleon coordinates bo  and bo in the projectile and in G H the target correspondingly. We write this integration symbolically as



d¹





  db ¹ (b)  db ¹ (b ) , " : G  G H H G H

(2.54)

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with A, B being the nuclear mass numbers and with the so-called nuclear thickness function ¹ (b)  being de"ned as the integral over the nuclear density  ,  



dz  ((b#z) . ¹ (b) " :  

(2.55)

For the nuclear densities, we use a parameterization of experimental data of [153],

 

if A"2 ,

exp!r/1.72

 (r)J exp!r/(0.9A)  r!0.7A  1 1#exp 0.545




if 2(A(10 ,



(2.56)

if A510 .

It is convenient to use the transverse distance b between the two nucleons from the kth I nucleon}nucleon pair, i.e. b "bo #bo  !bo  , (2.57) I  LI OI where the functions (k) and (k) refer to the projectile and the target nucleons participating in the kth interaction (pair k). In order to simplify the notation, we de"ne a vector b whose components are the overall impact parameter b as well as the transverse distances b ,2, b of the nucleon    pairs, b"b , b ,2, b  . (2.58)    Then the nucleus}nucleus interaction cross section can be obtained applying the cutting procedure to elastic scattering diagrams of Fig. 2.1 and written in the form

 


 (s)" db d¹  (s, b) ,    

(2.59)

where the so-called nuclear pro"le function  represents an interaction for given transverse  coordinates of the nucleons. The calculation of the pro"le function  as the sum over all cut diagrams of the type shown in  Fig. 2.9 does not di!er from the hadron}hadron case and follows the rules formulated in the preceding section: E For a remnant carrying the light cone momentum fraction x (x> in case of projectile, or x\ in case of target), one has a factor F (x), de"ned in Eq. (2.35).   E For each cut elementary diagram (real elementary interaction"dashed vertical line) attached to two participants with light cone momentum fractions x> and x\, one has a factor G(x>, x\, s, b), given by Eqs. (2.46)}(2.50). Apart from x> and x\, G is also a function of the total squared energy s and of the relative transverse distance b between the two corresponding nucleons (we use G as for nucleon}nucleon scattering). an abbreviation for G,, / E For each uncut elementary diagram (virtual emissions"full vertical line) attached to two participants with light cone momentum fractions x> and x\, one has a factor !G(x>, x\, s, b), with the same G as used for the cut diagrams.

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Fig. 2.9. Example for a cut multiple scattering diagram, with cut (dashed lines) and uncut (full lines) elementary diagrams. This diagram can be translated directly into a formula for the inelastic cross section (see text).

E Finally one sums over all possible numbers of cut and uncut Pomerons and integrates over the light cone momentum fractions. So we "nd

 

 KI JI  (s, b)"
2
(1!
I )   dx> dx\  dx > dx \  K II II IH IH  I I K J H K J 1 1 KI  JI ;  G(x> , x\ , s, b )  !G(x > , x \ , s, b ) II II I IH IH I m !l ! I I I I H  ; F x !
x >  F x !
x \   H   G IH IH H G LIG OIH











 

 (2.60)

with x "1!
x> , (2.61) G II LIG x "1!
x\ . (2.62) II H OIH The summation indices m refer to the number of cut elementary diagrams and l to the number of I I uncut elementary diagrams, related to nucleon pair k. For each possible pair k (we have altogether AB pairs), we allow for any number of cut and uncut diagrams. The integration variables x! refer II to the th elementary interaction of the kth pair for the cut elementary diagrams, the variables x ! refer to the corresponding uncut elementary diagrams. The arguments of the remnant IH functions F are the remnant light cone momentum fractions, i.e. unity minus the momentum   fractions of all the corresponding elementary contributions (cut and uncut ones). We also introduce the variables x and x , de"ned as unity minus the momentum fractions of all the correspondG H ing cut contributions, in order to integrate over the uncut ones (see below). The expression for  (2) sums up all possible numbers of cut Pomerons m with one exception  I due to the factor (1!
I ): one does not consider the case of all m 's being zero, which I  K

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corresponds to `no interactiona and therefore does not contribute to the inelastic cross section. We may therefore de"ne a quantity  (2), representing `no interactiona, by taking the expression for   (2) with (1!
I ) replaced by (
I ):  K   K  JI  1 JI  (s, b)"
2
  dx > dx \   !G(x > , x \ , s, b )  IH IH IH IH I l ! J J I H I I H  ;  F> 1!
x >  F\ 1!
x \ . (2.63) IH IH H G LIG OIH One now may consider the sum of `interactiona and `no interactiona, and one obtains easily

 


  






 (s, b)# (s, b)"1 . (2.64)   Based on this important result, we consider  to be the probability to have an interaction and  correspondingly  to be the probability of no interaction, for "xed energy, impact parameter and  nuclear con"guration, speci"ed by the transverse distances b between nucleons, and we refer to I Eq. (2.64) as `unitarity relationa. But we want to go even further and use an expansion of  in  order to obtain probability distributions for individual processes, which then serves as a basis for the calculations of exclusive quantities. The expansion of  in terms of cut and uncut Pomerons as given above represents a sum of  a large number of positive and negative terms, including all kinds of interferences, which excludes any probabilistic interpretation. We have therefore to perform summations of interference contributions and sum over any number of virtual elementary scatterings (uncut Pomerons), for given non-interfering classes of diagrams with given numbers of real scatterings (cut Pomerons) [30]. Let us write the formulas explicitly. We have

 



 KI  (s, b)"
2
(1!
I )   dx> dx\  K II II  K I I K 1 KI  ;  G(x> , x\ , s, b )
(x , x , s, b) , (2.65) II II I  m ! I I I where the function
representing the sum over virtual emissions (uncut Pomerons) is given by the following expression:



 




  




 JI  1 JI  !G(x > , x \ , s, b )
(x , x , s, b)"
2
  dx > dx \  IH IH IH IH I  l ! I I H J I H J  ; F x !
x >  F x !
x \ . (2.66)   H   G IH IH H G LIG OIH This summation has to be carried out, before we may use the expansion of  to obtain probability  distributions. This is far from trivial, as we are going to discuss in the next section, but let us assume for the moment that it can be done. To make the notation more compact, we de"ne matrices X> and X\, as well as a vector m, via X>"x>  , II



(2.67)

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X\"x\  , II m"m  , I which leads to

(2.68) (2.69)



 (s, b)"
(1! ) dX> dX\Q@(m, X>, X\) ,  K  K  (s, b)"Q@(0, 0, 0) ,  

(2.70) (2.71)

with





1 KI  Q@(m, X>, X\)"   G(x> , x\ , s, b )
(x , x , s, b) .  II II I  m ! I I I This allows to rewrite the unitarity relation Eq. (2.64) in the following form:

(2.72)



(2.73)
dX> dX\Q@(m, X>, X\)"1 .  K This equation is of fundamental importance, because it allows us to interpret Q@(m, X>, X\) as probability density of having an interaction con"guration characterized by m, with the light cone momentum fractions of the Pomerons being given by X> and X\. 2.5. Diwractive scattering We do not have a consistent treatment of di!ractive scattering at the moment, this is left to a future project in connection with a complete treatment of enhanced diagrams. For the moment, we introduce di!raction `by handa: in case of no interaction in pp or pA scattering, we consider the projectile to be di!ractively excited with probability w  

(1!(
(1, x , s, b)) ,
(1, x , s, b)

(2.74)

with a "t parameter w . Nucleus}nucleus scattering is here (but only here!) considered as   composed of pA or Ap collisions. 2.6. AGK cancellations in hadron}hadron scattering As a "rst application, we are going to prove that AGK cancellations apply perfectly in our model. As we showed above, the description of high energy hadronic interaction requires to consider explicitly a great number of contributions, corresponding to multiple scattering process, with  We speak here about the contribution of elementary interactions (Pomeron exchanges) to the secondary particle production; the AGK cancellations do not hold for the contribution of remnant states (spectator partons) hadronization [102].

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a number of elementary parton}parton interactions happening in parallel. However, when calculating inclusive spectra of secondary particles, it is enough to consider the simplest hadron}hadron (nucleus}nucleus) scattering diagrams containing a single elementary interaction, as the contributions of multiple scattering diagrams with more than one elementary interaction exactly cancel each other. This so-called AGK cancellation is a consequence of the general Abramovskii} Gribov}Kancheli cutting rules [30]. Let us consider the most fundamental inclusive distribution, where all other inclusive spectra may be derived from: the distribution dnF F /dx> dx\, with dnF F being the number of Pomerons . . with light cone momentum fractions between x> and x>#dx>and between x\ and x\#dx\ respectively, at a given value of b and s. If AGK cancellations apply, the result for dnF F /dx> dx\ . should coincide with the contribution coming from exactly one elementary interaction (see Eq. (2.45)): dnF F . (x>, x\, s, b)"GF/F (x>, x\, s, b)FF (1!x>)FF (1!x\) ,      dx> dx\

(2.75)

and the contributions from multiple scattering should exactly cancel. We have per de"nition



K K>J   dnF F . (x>, x\, s, b)"

 dx> dx\  dx> dx\ I I H H dx> dx\ HK> K J I 1 1 K K>J ;  GF/F (x>, x\, s, b)  !GF/F (x>, x\, s, b)   I I H H m! l! I HK> K>J K>J 1!
x> FF 1!
x\ ;FF J J     J J K ;
(x>!x>) (x\!x\) . (2.76) IY IY IY Due to the symmetry of the integrand in the r.h.s. of Eq. (2.76) in the variables x!,2, x!, the sum  K of delta functions produces a factor mG(x>, x\, s, b), and removes one dx> dx\ integration. Using I I  1 K\  1 KY m
2  2"
2  2, (2.77) m! m! K I KY I with m"m!1, we get











1  L dnF F . (x>, x\, s, b)"GF/F (x>, x\, s, b)
 dx> dx\  J J n! dx> dx\ L J L n K L  GF/F (x>, x\, s, b)  !GF/F (x>, x\, s, b) ;
I I H H   m K I HK> L L ;FF 1!x>!
x> FF 1!x\!
x\ .     J J J J












(2.78)

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The term in curly brackets 2 is 1 for n"0 and zero otherwise, so we get the important "nal result dnF F dnF F . (x>, x\, s, b)" . (x>, x\, s, b) , dx> dx\ dx> dx\

(2.79)

which corresponds to one single elementary interaction; the multiple scattering aspects completely disappeared, so AGK cancellations indeed apply in our approach. AGK cancellations are closely related to the factorization formula for jet production cross section, since as a consequence of Eq. (2.79), we may obtain the inclusive jet cross section in a factorized form as

  

d
HI  (z>z\s, p ) ,
F F "
dp dz> dz\ f F (z>, M ) f F (z\, M ) , ,

 H $ I $ dp , HI with f F and f F representing the parton distributions of the two hadrons. H I

(2.80)

2.7. AGK cancellations in nucleus}nucleus scattering We have shown in the previous section that for hadron}hadron scattering AGK cancellations apply, which means that inclusive spectra coincide with the contributions coming from exactly one elementary interaction. For multiple Pomeron exchanges we have a complete destructive interference, they do not contribute at all. Here, we are going to show that AGK cancellations also apply for nucleus}nucleus scattering, which means that the inclusive cross section for A#B scattering is AB times the corresponding inclusive cross section for proton}proton interaction. The inclusive cross section for forming a Pomeron with light cone momentum fractions x> and x\ in nucleus}nucleus scattering is given as

 

d
 . (x>, x\, s)" db d¹   dx>dx\

 



 KI >JI  dx> dx\ ;
2
(1!
I )   K II II I I K J K J 1 KI  KI >JI ;  G(x> , x\ , s, b )  !G(x> , x\ , s, b ) II II I IH IH I m !l ! I I I I HKI >  ; F 1!
x>  F 1!
x\     IH IH H G LIG OIH KI  ;

(x>!x> ) (x\!x\ ) . (2.81) IYIY IYIY IY IY The factor (1!
I ) makes sure that at least one of the indices m is bigger than zero. Integrating  K I over the variables appearing in the delta functions, we obtain a factor
G(x>, x\, s, b )m which IY I IY may be written in front of
  2. In the following expression one may rename the integration KJ variables such that the variables x> IY and x\ IY disappear. This means for the arguments of the IYK IYK functions F that for i"(k) and j"(k) one replaces 1 by 1!x> and 1!x\ respectively.  















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Then one uses the factor m mentioned above to replace m ! by (m !1)!. One "nally renames IY IY IY (m !1) by m , as a consequence of which one may drop the factor (1!
I ). This is crucial, IY IY  K since now we have factors of the form   KI >JI 1 KI

2  G(x> , x\ , s, b)  !G(x> , x\ , s, b)2. (2.82) II II IH IH m !l ! I I I KI  JI  HKI > In this sum only the term for m "0 and l "0 is di!erent from zero, namely 1, and so we get I I d
 . (x>, x\, s, b)" db d¹
G(x>, x\, s, b )F (1!x>)F (1!x\) . (2.83)   IY     dx>dx\ IY Using the de"nition of d¹ , writing b explicitly as bo #bo  !bo , we obtain  IY  LI OI d
 . (x>, x\, s)"AB db db ¹(b ) db ¹(b )G(x>, x\, s, bo #bo !bo )  > > \ \  > \ dx>dx\

 

 

;F



(1!x>)F (1!x\) .     Changing the order of the integrations, we obtain "nally d
NN d
 . (x>, x\, s) . (x>, x\, s)"AB dx>dx\ dx>dx\

(2.84)

(2.85)

with



d
NN . (x>, x\, s)" db G(x>, x\, s, b)F (1!x>)F (1!x\) .     dx> dx\

(2.86)

Since any other inclusive cross section d
/dq may be obtained from the inclusive Pomeron  distribution via convolution, we obtain the very general result d
NN d
  (q, s, b)"AB  (q, s, b) , dq dq

(2.87)

so nucleus}nucleus inclusive cross sections are just AB times the proton}proton ones. So, indeed, AGK cancellations apply perfectly in our approach. 2.8. Outlook What did we achieve so far? We have a well de"ned model, introduced by using the language of "eld theory (Feynman diagrams). We were able to prove some elementary properties (AGK cancellations in case of proton}proton and nucleus}nucleus scattering). To proceed further, we have to solve (at least) two fundamental problems: E the sum over virtual emissions has to be performed, E tools have to be developed to deal with the multi-dimensional probability distribution Q@(m, X>, X\),  both being very di$cult tasks.

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Calculating the sum over virtual emissions (
) is not only technically di$cult, there are also  conceptual problems. By studying the properties of
, we "nd that at very high energies the  theory is no longer unitary without taking into account additional screening corrections. In this sense, we consider our work as a "rst step to construct a consistent model for high energy nuclear scattering, but there is still work to be done. Concerning the multi-dimensional probability distribution Q@(m, X>, X\), we are going to  develop methods well known in statistical physics (Markov chain techniques), which we also are going to discuss in detail later. So "nally, we are able to calculate the probability distribution Q@(m, X>, X\), and are able to generate (in a Monte Carlo fashion) con"gurations (m, X>, X\)  according to this probability distribution. The two above-mentioned problems will be discussed in detail in the following sections.

3. Virtual emissions In order to proceed, we need to calculate the sum over virtual emissions, represented by the function
. Understanding the behavior of
is crucial, since this function is related to  and    plays therefore a crucial role in connection with unitarity, the unitarity equation being given as  # "1. By studying the properties of
, we "nd inconsistencies in the limit of high    energies, in the sense that the individual terms appearing in the unitarity equation are not necessarily positive. Attempting to understand this unphysical behavior, we "nd that any model where AGK cancellations apply (so most of the models used presently) has to run asymptotically into this problem. So eventually one needs to construct models, where AGK cancellations are violated, which is going to be expected when contributions of Pomeron}Pomeron interactions are taken into consideration. As a "rst phenomenological solution of the unitarity problem, we are going to `unitarizea the `bare theorya introduced in the preceding section `by handa, such that the theory is changed as little as possible, but the asymptotic problems disappear. The next step should of course be a consistent treatment including contributions of enhanced Pomeron diagrams. In the following, we are going to present the calculation of
, we discuss the unitarity  problems and the phenomenological solution, as well as properties of the `unitarized theorya. 3.1. Parameterizing the elementary interaction , which is the pro"le function The basis for all the calculations which follow is the function G,, / representing a single elementary nucleon}nucleon (NN) interaction. For simplicity, we write simply . This function G is a sum of several terms, representing soft, semi-hard, valence, and G,G,, / screening contributions. In case of soft and semi-hard, one has G"(x>x\)\?  D, where D represents the Pomeron exchange and the factor in front of D the `external legsa, the nucleon participants. For the other contributions the functional dependence on x>, x\ is somewhat more complicated, but nevertheless it is convenient to de"ne a function G(x>, x\, s, b) . D(x>, x\, s, b)" (x>x\)\? 

(3.1)

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We obtain G and therefore D as the result of a quite involved numerical calculation, which means that these functions are given in a discretized fashion. Since this is not very convenient and since the dependence of x>, x\, and b are quite simple, we are going to parameterize our numerical results and use this analytical expression for further calculations. This makes the following discussions much easier and more transparent. We "rst consider the case of zero impact parameter (b"0). In Fig. 3.1, we plot the function D together with the individual contributions as functions of x"x>x\, for b"0 and for di!erent values of s. To "t the function D, we make the following ansatz: , D(x>, x\, s, b"0)"
 G # HG (x>x\)@H"G (x>x\s)@"G , " " G GFFFFFHFFFFFI "G

(3.2)

where the parameters may depend on s, and the parameters marked with a star are non-zero for a given i only if the corresponding G is zero. This parameterization works very well, as shown in " Fig. 3.2, where we compare the original D function with the "t according to Eq. (3.2). Let us now consider the b-dependence for "xed x> and x\. Since we observe almost a Gaussian shape with a weak (logarithmic) dependence of the width on x"x>x\, we could make the following ansatz:






!b , . D(x>, x\, s, b)"
 G # HG (x>x\)@H"G (x>x\s)@"G exp " " G # G ln(x>x\s) " " G

(3.3)

Fig. 3.1. The function D (solid line) as well as the di!erent contributions to D as a function of x"x>x\, for b"0, at di!erent energies E"(s. We show D (dotted), D (dashed), and D (dashed}dotted).    

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Fig. 3.2. The exact (full) and the parameterized (dashed) function D as a function of x"x>x\, for b"0, at di!erent energies E"(s.

However, we can still simplify the parameterization. We have




exp










!b !b  + exp 1! ln(x>x\s) # ln x "exp

(3.4)

!b (x>x\s)A@ .

(3.5)

So we make "nally the ansatz , (3.6) D(x>, x\, s, b)"
 G # HG (x>x\)@H"G (x>x\s)@"G >A"G @e\@B"G , " " G which provides a very good analytical representation of the numerically obtained function D, as shown in Fig. 3.3. 3.2. Calculating
for proton}proton collisions We "rst consider proton}proton collisions. To be more precise, we are going to derive an expression for
which can be evaluated easily numerically and which will serve as the basis to NN investigate the properties of
. We have NN  1 J
(x>, x\, s, b)"
dx> dx\2dx> dx\  !G(x>, x\, s, b) NN H H   J J l! J H ;F (x>!
x>)F (x\!
x\) (3.7)   H   H







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Fig. 3.3. The b-dependence of b ) D(x>, x\, s, b) for a "xed value of x"x>x\, with x>"x\, at di!erent energies E"(s. Exact results are represented as solid lines, the parameterized ones by dashed lines.

with F (x)"x?  (x)(1!x) ,   where (x) is the Heavyside function, and

(3.8)

G(x>, x\, s, b)"(x>x\)\?  D(x>, x\, s, b) . H H H H H H Using Eq. (3.6), we have

(3.9)

with

, G(x>, x\, s, b)"
(x>x\)@G H H G H H G GFHFI %GH

(3.10)

"( G # HG )s@"G >A"G @ e\@B"G , (3.11) G " "  " G #HG # G b!

, (3.12) G " " "   with HGO0 and HGO0 only if G"0. Using Eq. (3.10), we obtain from Eq. (3.7) the following " " " expression:



 1 P >2>P,  1
(x>, x\s, b)"
2
2  dx> dx\ NN H H r ! r ! , P  H P,   2 P > >P, P  ;  !G  2 !G , M ,M 2 M  M, P > >P,\ > ;F

 









x>!
x> F x\!
x\ . H   H H H

(3.13)

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Using the fact that the functions G are separable, GH G " (x>)@G (x\)@G , H GH G H one "nds "nally (see Section D.1)



(3.14)



 (1#

)    
(x>, x\, s, b)"x? 
2
NN (1#

#r I #2#r I )     , , P  P,  I (! x@ (I ))P (! x@I , (I ))R  ,  2 , ; (3.15) r ! r !  , with x"x>x\ and I " #1. Since the sums converge very fast, this expression can be easily G G evaluated numerically. 3.3. Unitarity problems In this section, we are going to present numerical results for
, based on Eq. (3.15). We will NN observe an unphysical behavior in certain regions of phase space, which amounts to a violation of unitarity. Trying to understand its physical origin, we "nd that AGK cancellations, which apply in our model, automatically lead to unitarity violations. This means on the other hand that a fully consistent approach requires explicit violation of AGK cancellations, which should occur in case of considering contributions of enhanced Pomeron diagrams. In which way is
related to unitarity? We have shown in the preceding section that the NN inelastic non-di!ractive cross section
(s) may be written as 




(s)" db  (s, b) , NN 

(3.16)

with the pro"le function  (s, b) representing all diagrams with at least one cut Pomeron. We NN de"ned as well the corresponding quantity  (s, b) representing all diagrams with zero cut NN Pomerons. We demonstrated that the sum of these two quantities is one (2.64):  (s, b)# (s, b)"1 , (3.17) NN NN which represents a unitarity relation. The function
enters "nally, since we have the relation NN  (s, b)"
(1, 1, s, b) . (3.18) NN NN Based on these formulas, we interpret  (s, b)"
(1, 1, s, b) as the probability of having no NN NN interaction, whereas  (s, b)"1!
(1, 1, s, b) represents the probability to have an interaction, NN NN at given impact parameter and energy. Such an interpretation of course only makes sense as long as any of the 's is positive, otherwise unitarity is said to be violated, even if Eq. (3.17) still holds. In Fig. 3.4, we plot
as a function of x"x>x\ for (s"200 GeV for two di!erent values of NN b. The curve for b"1.5 fm (solid curve) is close to one with a minimum of about 0.8 at x"1. The x-dependence for b"0 fm (dashed curve) is much more dramatic: the curve deviates from 1 already at relatively small values of x and drops "nally to negative values at x"1. The values for x"1 are of particular interest, since 1!
(1, 1, s, b)" (s, b) represents the pro"le function in NN NN the sense that the integration over b provides the inelastic non-di!ractive cross section. Therefore,

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Fig. 3.4. The expression
(x>, x\, s, b)/x?  as a function of x"x>x\ for b"0 (dashed) and for b"1.5 fm NN (solid curve). Fig. 3.5. The pro"le function 1!
(1, 1, s, b) as a function of impact parameter b. This function should represent the NN probability to have an interaction at a given impact parameter.

in Fig. 3.5, we plot the b-dependence of 1!
(1, 1, s, b), which increases beyond 1 for small values NN of b, since
is negative in this region, as discussed above for the case of b"0 fm. On the other NN hand, an upper limit of 1 is really necessary in order to assure unitarity. So the fact that
grows NN to values bigger than one is a manifestation of unitarity violation. In the following, we try to understand the physical reason for this unitarity problem. We are going to show, that it is intimately related to the fact that in our approach AGK cancellations are ful"lled, as shown earlier, which means that any approach where AGK cancellations apply will have exactly the same problem. We are going to demonstrate in the following that AGK cancellations imply automatically unitarity violation. The average light cone momentum taken by a Pomeron may be calculated from the Pomeron inclusive spectrum dn /dx> dx\ as . 1 dn . (x>, x\, s, b) . db dx\ (3.19) x>" dx>x>
(s) dx> dx\  If AGK cancellations apply, we have





 



dn /dx> dx\"dn /dx> dx\"G(x>, x\, s, b)F (x>)F (x\) , . .     and therefore dn . (x>, x\, s, b)"(s) (x>x\s)Q e\@HQf (x>) f (x\) , dx> dx\

(3.20)

(3.21)

where (s) is greater than zero and increases with energy and (s) and (s) depend weakly (logarithmically) on s, whereas f is an energy independent function. We obtain (s)sQ x>" ,
(s) 

(3.22)

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where (s) depends only logarithmically on s. Since x> must be less than or equal to one, we "nd (3.23)
(s)5(s)sQ ,  which violates the Froissard bound and therefore unitarity. This problem is related to the old problem of unitarity violation in case of single Pomeron exchange. The solution appeared to be the observation that one needs to consider multiple scattering such that virtual multiple emissions provide su$cient screening to avoid the unreasonably fast increase of the cross section. If AGK cancellations apply (as in our model), the problem comes back by considering inclusive spectra, since these are determined by single scattering. Thus, we have shown that a consistent application of the eikonal Pomeron scheme both to interaction cross sections and to particle production calculations unavoidably leads to the violation of the unitarity. This problem is not observed in many models currently used, since there simply is no consistent treatment provided, and the problem is therefore hidden. The solution of the unitarity problem requires to employ the full Pomeron scheme, which includes also so-called enhanced Pomeron diagrams, to be discussed later. The simplest diagram of that kind } so-called >-diagram, for example, contributes a negative factor to all inclusive particle distributions in the particle rapidity region y (y(>, where > is the total rapidity range for the interaction and y corre  sponds to the rapidity position of the Pomeron self-interaction vertex. Thus, one speaks about breaking of the AGK cancellations in the sense that one gets corrections to all inclusive quantities calculated from just one Pomeron exchange graph [154,155]. In particular, presenting the inclusive Pomeron distribution d
(x>, x\)/dx> dx\ by the formula (2.75) implies that the functions . F(1!x!) acquire a dependence on the energy of the interaction s. It is this dependence which is expected to slow down the increase of the Pomeron number, to modify the inclusive hadron spectra, and thus to cure the unitarity problem. So we think it is mandatory to proceed in the following way: "rst one needs to provide a consistent treatment of cross sections and particle production, which will certainly lead to unitarity problems, and second one has to re"ne the theory to solve the unitarity problem in a consistent way, via screening corrections. The "rst part of this program is provided in this paper, the second one will be treated in some approximate fashion later, but a rigorous, self-consistent treatment of this second has still to be done. 3.4. A phenomenological solution: unitarization of
As we have seen in the preceding sections, unitarity violation manifests itself by the fact that the virtual emission function
appears to be negative at high energies and small impact parameter NN for large values of x> and x\, particularly for x>"x\"1. What is the mathematical origin of these negative values? In Eq. (3.15), the sums over r contains terms of the form (2)PG /r ! and an G G additional factor of the form





(1#

)   . (3.24) (1#

#r I #2#r I )     , , It is this factor which causes the problem, as it strongly suppresses contributions of terms with large r , which are important when the interaction energy s increases. Physically it is connected to G

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the reduced phase space in case of too many virtual Pomerons emitted. By dropping this factor one would obtain a simple exponential function which is de"nitely positive. Our strategy is to modify the scheme such that
stays essentially unchanged for values of NN s, b, x>, and x\, where
is positive and that
is `correcteda to assure positive values in NN NN regions where it is negative. We call this procedure `unitarizationa, which should not be considered as an approximation, since one is really changing the physical content of the theory. This is certainly only a phenomenological solution of the problem, the correct procedure should amount to taking into account the mentioned screening corrections due to enhanced Pomeron diagrams, which should provide a `natural unitarizationa. Nevertheless we consider our approach as a necessary "rst step towards a consistent formulation of multiple scattering theory in nuclear (including hadron}hadron) collisions at very high energies. Let us explain our `unitarizationa in the following. We de"ne (1#

)   g(z)" (1#

#z)   such that g(r I #2#r I )   , , is the factor causing unitarity problems. This expression should be of the form

(3.25)

(3.26)

(2)P 2(2)P, , which would make
a well-behaved exponential function. In order to achieve this, the function  g should be an exponential. So we replace g(z) by g (z), where the latter function is de"ned as  g (z)"e\C X , (3.27)  where the parameter  should be chosen such that g(z) is well approximated for values of z between  (say) 0 and 0.5 (see Fig. 3.6). The index `ea refers to `exponentiationa. So, we replace the factor g(r I #2#r I ) by   , , (3.28) g (r I #2#r I )"e\C P @I  >2>P, @I ,     , , "(e\C @I  )P 2(e\C @I , )P, , (3.29)

Fig. 3.6. Function g(z) (full line) and unitarized function g (z) (dashed). 

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and obtain correspondingly instead of


NN  
NN (x>, x\, s, b)"x? 
2
(e\C @I  )P 2(e\C @I , )P,  P  P,  (! x@I  (I ))P (! x@I , (I ))R,  2 ,  , ; . r ! r !  , Now the sums can be performed and we get ,
NN (x>, x\, s, b)"x?   exp! x@I G (I ) e\C @I G  G G  G which may be written as

(3.30)

(3.31)


NN (x>, x\, s, b)"(x>x\)?  exp!GI (x>x\, s, b) 

(3.32)

, GI (x, s, b)"
 x@I G G G

(3.33)

with

with (3.34)

 " (I ) e\C @I G , G G G I " #1 , (3.35) G G where and  are given as G G (3.36)

"( G # HG )s@"G >A"G @ e\@B"G , " G " , (3.37)  " G #HG # G b!

" "   G " with HG O0 and HG O0 only if G O0. We are not yet done. We modi"ed
such that the new " " NN " function
NN is surely positive. But what happened to our unitarity equation? If we replace
by NN 
NN , we obtain    # "
dX> dX\ Q@ (m, X>, X\) , (3.38) NN NN NN K with







1 K  G(s, x>, x\, b)
NN (x>, x\, s, b) , (m, X>, X\)" (3.39) Q@ I I  NN m! I where x> and x\ refer to the remnant light cone momenta. Since
NN is always bigger than
for  NN small values of b, the sum  # is bigger than one, so the unitarity equation does not hold any NN NN more. This is quite natural, since we modi"ed the virtual emissions without caring about the real ones. In order to account for this, we de"ne







1 K   G(s, x>, x\, b)
NN (x>, x\, s, b) , Z(s, b)"
dX> dX\ I I  m! I K

(3.40)

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129

which is equal to one in the exact case, but which is di!erent from one if we use
NN instead of
.  NN In order to recover the unitarity equation, we have to `renormalizea
NN , and we de"ne therefore  the `unitarizeda virtual emission function
NN via 
(x>, x\, s, b)
NN (x>, x\, s, b)" NN . (3.41)  Z(s, b) Now, the unitarity equation holds,



  # "
dX> dX\ Q@ (m, X>, X\)"1 , NN NN NN K

(3.42)

with





1 K  G(s, x>, x\, b)
NN (x>, x\, s, b) (m, X>, X\)" Q@ NN I I   m! I being strictly positive, which allows "nally the probability interpretation.

(3.43)

3.5. Properties of the unitarized theory We are now going to investigate the consequences of our unitarization, in other words, how the results are a!ected by this modi"cation. In Fig. 3.7 we compare the exact and the exponentiated version of the virtual emission function (
and
NN ) for a large value of the impact parameter NN  (b"1.5 fm). The exponentiated result (dashed) is somewhat below the exact one (solid curve), but the di!erence is quite small. The situation is somewhat di!erent in case of zero impact parameter b as shown in Fig. 3.8. For small values of x the two curves coincide more or less, however for x"1 the exponentiated result (dashed) is well above the exact one (solid curve). In particular, and this is most important, the dashed curve rests positive and in this sense corrects for the unphysical behavior (negative values) for the exact curve. The behavior for x"1 for di!erent values of b is summarized in Fig. 3.9, where we plot 1!
(1, 1, s, b) as a function of b. We clearly observe that NN

Fig. 3.7. Behavior of
(x>, x\, s, b)/(x>x\)?  as a function of x"x>x\, for the exact function (solid) and the NN exponentiated one (dashed curve) for impact parameter b"1.5 fm. Fig. 3.8. Behavior of
(x>, x\, s, b)/(x>x\)?  as a function of x"x>x\, for the exact function (solid) and the NN exponentiated one (dashed curve) for impact parameter b"0 fm.

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Fig. 3.9. The function 1!
(1, 1, s, b) as a function of impact parameter b. We show the exact result (solid) as well as the NN exponentiated one (dashed curve). Fig. 3.10. The normalization function Z(s, b) as a function of the impact parameter b.

for large b exact (solid) and exponentiated (dashed curve) result agree approximately, whereas for small values of b they di!er substantially, with the exponentiated version always staying below 1, as it should be. So the e!ect of our exponentiation is essentially to push the function below 1. Next we calculate explicitly the normalization function Z(s, b). We have



K 1 K   dx> dx\  G(x>, x\, s, b) Z(s, b)"
I I m! I I I K I K K ;
NN 1!
x>, 1!
x\ ,  J J J J which may be written as






(3.44)



Z(s, b)"
NN (1, 1, s, b)# dz> dz\ H(z>, z\)
NN (z>, z\, s, b) ,  

(3.45)

with




K 1 K  dx> dx\  G(x>, x\, s, b) I I m! I I I I K K ; 1!z>!
x> 1!z\!
x\ . I I I I Using the analytical form of G, we obtain  H(z>, z\)"
K






(3.46)

  [(1!z>)(1!z\)]P @I  >2>P, @I , \ H(z>, z\)"
2
(r I #2#r I )     , , PGFHFI P,  P >2>P) $

( (I ))P ( (I ))P, ,  2 , ;  r ! r !  ,

(3.47)

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131

(see Section D.2). This can be calculated, and after numerically doing the integration over z>, z\, we obtain the normalization function Z(s, b), as shown in Fig. 3.10. We observe, as expected, a value close to unity at large values of b, whereas for small impact parameter Z(s, b) is bigger than one, since only at small values of b the virtual emission function
has been changed substantially NN towards bigger values. Knowing
NN and Z, we are ready to calculate the unitarized emission functions
NN which   "nally replaces
in all formulas for cross section calculations. The results are shown in Fig. 3.11, NN where we plot 1!
NN together with 1!
NN and 1!
for both x> and x\ being one. We  NN  observe that compared to 1!
NN the function 1!
NN is somewhat increased at small values of   b due to the fact that here Z(s, b) is greater than one, whereas for large impact parameters there is no di!erence. Since the unitarity equation holds, we may integrate 1!
NN (1, 1, s, b) over impact parameter, to  obtain the inelastic non-di!ractive cross section,




(s)" db1!
NN (1, 1, s, b) ,  

(3.48)

the result being shown in Fig. 3.12. Here the exact and the unitarized result (using
and NN
NN respectively) are quite close due to the fact that one has a two-dimensional b-integration, and  therefore the small values of b, where we observe the largest di!erences, do not contribute much to the integral. We now turn to inclusive spectra. We consider the inclusive x-spectrum of Pomerons, dn /dx, . where x"x>x\ is the squared mass of the Pomeron divided by s. In the exact theory, we may take advantage of the AGK cancellations, and obtain





\ (V dn dn . (x>, x\, s, b) . (x, s, b)" dy , (3.49) dx> dx\ dx ( >  V V>(VW V\(V\W where dn /dx> dx\ is the corresponding inclusive distribution for one single elementary interac. tion, which is given in Eq. (2.75). The y-integration can be easily performed numerically, and we

Fig. 3.11. The function 1!
(1, 1, s, b) as a function of the impact parameter b. We show the exact result 1!
(solid NN NN line) as well as the unitarized one 1!
NN (dashed) and the exponentiated one 1!
NN (dotted).   Fig. 3.12. The inelastic non-di!ractive cross section as a function of the energy, using the exact (solid) and the unitarized (dashed)
-function.

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Fig. 3.13. Example of an inclusive spectrum: x-distribution of Pomerons. The variable x is de"ned as x"x>x\ and is therefore the squared mass of the Pomeron divided by s. We show the exact (solid) and the unitarized results (dashed) for b"0 fm (upper curves) and b"1.5 fm (lower curves). In fact, for b"1.5 fm the two curves coincide. Fig. 3.14. The x-distribution of Pomerons, averaged over impact parameter. We show the exact (solid) and the unitarized results (dashed).

obtain the results shown in Fig. 3.13 as solid curves, the upper one for b"0 fm and the lower one for b"1.5 fm. The calculation of the unitarized result is more involved, since now we cannot use the AGK cancellations any more. We have



dn  . (x>, x\, s, b)"
dx> dx\ K 1 ; m!

K  dx> dx\ I I I K K K  G(x>, x\, s, b)
NN 1!
x>, 1!
x\, s, b I I  I I I I I




K ;
(x>!x>) (x\!x\) , IY IY IY where we used the unitarized version of
. We "nd NN dn . (x>, x\, s, b)"G(x>, x\, s, b) dx> dx\

 

;


NN

 (3.50)

(1!x>, 1!x\, s, b)



# dz> dz\H(z>#x>, z\#x\)
NN (z>, z\, s, b) , 

(3.51)

where H is de"ned in Eq. (3.46), with the "nal result given in Eq. (3.47). The integration over z>, z\ can now be done numerically. Expressing x> and x\ via x and y and integrating over y, we "nally obtain dn /dx, as shown in Fig. 3.13 (dashed curves). In Fig. 3.14 we show the b-averaged . inclusive spectra, which are given as 1



db

dn . (x, s, b) dx


 for both, the exact and the unitarized version.

(3.52)

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133

3.6. Comparison with the conventional approach At this point it is noteworthy to compare our approach with the conventional one [73,74]. There one neglects the energy conservation e!ects in the cross section calculation and sums up virtual Pomeron emissions, each one taken with the initial energy of the interaction s. We can recover the conventional approach by simply considering independent (planar) emission of all the Pomerons, neglecting energy-momentum sharing between them. In the cross section formulas (2.65)}(2.66) this amounts to performing formally the convolutions of the Pomeron eikonals G(x>, x\, s, b) with the remnant functions F (x!) for all the Pomerons. In case of proton}proton scattering, we then get   (3.53)
NN (x>, x\, s, b)"e\QQ@ ,  with



(s, b)" dx> dx\G(x>, x\, s, b)F (1!x>)F (1!x\) ,    

(3.54)

where
does not depend on x> and x\ anymore. We obtain a unitarity relation of the form 
 "1 , K K

(3.55)

((s, b))K  " e\QQ@ K m!

(3.56)

with

representing the probability of having m cut Pomerons (Pomeron multiplicity distribution). So in the traditional case, the Pomeron multiplicity distribution is a Poissonian with the mean value given by (s, b). As already mentioned above, that approach is not self-consistent as the AGK rules are assumed to hold when calculating interaction cross sections but are violated at the particle production generation. This inconsistency was already mentioned in [102,103], where the necessity to develop the correct, Feynman-diagram-based scheme, was "rst argued. The exact procedure is based on the summation over virtual emissions with the energymomentum conservation taken into account. This results in formula (3.7) or, using our parameterization, (3.15) for
(x>, x\, s, b), explicitly dependent on the momentum, left after cut NN Pomerons emission, and in the formula




(s)" db1!
(1, 1, s, b) NN 

(3.57)

for the inelastic cross section; AGK rules are exactly ful"lled both for the cross sections and for the particle production. But with the interaction energy increasing the approach starts to violate the unitarity and is no longer self-consistent. The `unitarizeda procedure, which amounts to replacing
by
NN , allows to avoid the NN  unitarity problems. The expressions for cross sections and for inclusive spectra are consistent with each other and with the particle generation procedure. The latter one assures the AGK cancellations validity in the region, where unitarity problems do not appear yet (not too high energies or large impact parameters).

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In order to see the e!ect of energy conservation we calculate  as given in Eq. (3.54) with the same parameters as we use in our approach for di!erent values of b, and we show the corresponding Pomeron multiplicity distribution in Fig. 3.15 as dashed lines. We compare this traditional approach with our full simulation, where energy conservation is treated properly (solid lines in the "gures). One observes a huge di!erence between the two approaches. So energy conservation makes the Pomeron multiplicity distributions much narrower, in other words, the mean number of Pomerons is substantially reduced. The reason is that due to energy conservation the phase space of light cone momenta of the Pomeron ends is considerably reduced. Of course, in the traditional approach one chooses di!erent parameters in order to obtain reasonable values for the Pomeron numbers in order to reproduce the experimental cross sections. But this only `simulatesa in some sense the phase space reduction due to energy conservation in an uncontrolled way. We conclude that considering energy conservation properly in the cross section formulas has an enormous e!ect and cannot be neglected. 3.7. Unitarization for nucleus}nucleus scattering In this section, we discuss the unitarization scheme for nucleus}nucleus scattering. The sum over virtual emissions is de"ned as  JI  1 JI  dx > dx \   !G(x > , x \ , s, b)
(X>, X\, s, b)"
2
 IH IH IH IH  l ! J I H I I H J  ; F x>!
x >  F x\!
x \ (3.58)   H   G IH IH H G LIG OIH

 




 








Fig. 3.15. Distribution of the number m of Pomerons for di!erent impact parameters. We show the results of a full simulation (solid lines) as well as the Poissonian distribution obtained by ignoring energy conservation (dashed line).

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135

where X>"x>2x>, X\"x\2x\ and (k) and (k) represent the projectile or target    nucleon linked to pair k. This calculation is very close to the calculation for proton}proton scattering. Using expression Eq. (3.10) of G(x > , x \ , s, b), de"nition Eq. (3.8) of F (x), one "nally IH IH   "nds
(X>, X\, s, b)  "

 (! )PI (! )P,I ,  2 
2
r ! r ! 2 2  ,  , I ,I P I P P P





 ;  (x>)?   ((I )(x>)@I  )PI 2((I )(x>)@I , )P,I g
r I #2#r I G  G , G I  ,I , G LIG LIG ;  (x\)?   ((I )(x\)@I  )PI 2((I )(x\)@I , )P,I g
r I #2#r I H  H , H I  ,I , H OIH OIH

 

(3.59) (see Section D.3), where the function g(z) is de"ned in Eq. (3.25), and the parameters and I are G G the same ones as for proton}proton scattering. In case of nucleus}nucleus scattering, we use the same unitarization prescription as already applied to proton}proton scattering. The "rst step amounts to replace the function g(z), which appears in the "nal expression of
, by the exponential form g (z). This allows to perform the   sums in Eq. (D.66), and we obtain  
 (X>, X\, s, b)"  (x>)?   (x\)?   e\%I V>LI V\OI  G  H G H I

(3.60)

(see Section D.3), where GI (x) is de"ned in Eq. (3.33). Having modi"ed
, the unitarity equation
K



dX> dX\ Q@ (m, X>, X\)"1 

(3.61)

does not hold any more, since  depends on
, and only the exact
assures a correct unitarity   relation. So as in proton}proton scattering, we need a second step, which amounts to renormalizing
. So we introduce a normalization factor  Z (s, b)"
 K



dX> dX\ Q@ (m, X>, X\) , 

(3.62)

with   de"ned in the same way as  but with
replaced by
 , which allows to de"ne the    unitarized
 function as 
(x>, x\, s, b)
 (x>, x\, s, b)"  .  Z (s, b) 

(3.63)

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In this way we recover the unitarity relation,




dX> dX\ Q@ (m, X>, X\)"1 ,  K

(3.64)

with





 1 KI (m, X>, X\)"   G(s, x> , x\ , b)
 (x>, x\, s, b) , Q@ II II   m ! I I I

(3.65)

and Q@(m, X>, X>) may be interpreted as probability distribution for con"gurations (m, X>, X>).  3.8. Proxle functions in nucleus}nucleus scattering In case of nucleus}nucleus scattering, the conventional approach [73,74] represents a `Glaubertype modela, where nucleus}nucleus scattering may be considered as a sequence of nucleon} nucleon scatterings with constant cross sections; the nucleons move through the other nucleus along straight line trajectories. In order to test this picture, we consider all pairs of nucleons, which due to their distributions inside the nuclei provide a more or less #at b-distribution. We then simply count, for a given b-bin, the number of interacting pairs and then divide by the number of pairs in the corresponding bin. The resulting distribution, which we call nucleon}nucleon pro"le function for nucleus}nucleus scattering, represents the probability density of an interaction of a pair of nucleons at given impact parameter. This may be compared with the proton}proton pro"le function 1!
NN (1, 1, s, b). In the Glauber model, these two distributions coincide. As demon strated in Fig. 3.16 for S#S scattering this is absolutely not the case. The pro"le function in case of S#S scattering is considerably reduced as compared to the proton}proton one. Since integrating the proton}proton pro"le function represents the inelastic cross section, one may also de"ne the corresponding integral in nucleus}nucleus scattering as `individual nucleon}nucleon cross sectiona. So we conclude that this cross section is smaller than the proton}proton cross section. This is due to the energy conservation, which reduces the number of Pomerons connected to any nucleon from the projectile and the target and "nally a!ects also the `individual cross sectiona.

Fig. 3.16. The numerically determined pro"le function in S#S scattering (points) compared to the proton}proton pro"le function (solid curve).

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3.9. Inclusive cross sections in nucleus}nucleus scattering We have shown in the preceding section that in the `barea theory AGK cancellations apply perfectly, which means that nucleus}nucleus inclusive cross sections are just AB times the proton}proton ones, d
 d
NN  (q, s, b)"AB  (q, s, b) . dq dq

(3.66)

In the unitarized theory, the results is somewhat di!erent. Unfortunately, we cannot calculate cross sections analytically any more, so we perform a numerical calculation using the Markov chain techniques explained later. In order to investigate the deviation from exact AGK cancellations, we calculate the inclusive nucleus}nucleus cross section for Pomeron production (being the basic inclusive cross section), divided by AB, 1 d
 . (x, s, b) , AB dx

(3.67)

and compare the result with the corresponding proton}proton cross section, see Figs. 3.17 and 3.18. For large and for small values of x, we still observe AGK cancellations (the two curves agree), but for intermediate values of x, the AGK cancellations are violated, the nucleus}nucleus cross section is smaller than AB times the nucleon}nucleon one. The e!ect is, however, relatively moderate. If one writes the proton}nucleus cross section as d
NN d
N . (x, s, b)"A?V . , dx dx

(3.68)

we obtain for (x) values between 0.85 and 1. So one may summarize that `AGK cancellations are violated, but not too stronglya.

Fig. 3.17. Inclusive cross section of Pomeron production for S#S scattering, divided by AB"32 (points), compared to the corresponding proton}proton cross section (solid line). Fig. 3.18. Inclusive cross section of Pomeron production for p#Au scattering, divided by AB"197 (points), compared to the corresponding proton}proton cross section (solid line).

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4. Markov chain techniques In this section we discuss how to deal with the multi-dimensional probability distribution  (K) with K"m, X>, X\, where the vector m characterizes the type of interaction of each pair  of nucleons (the number of elementary interactions per pair), and the matrices X>, X\ contain the light cone momenta of all Pomerons (energy sharing between the Pomerons). 4.1. Probability distributions for conxgurations In this section we essentially repeat the basic formulas of the preceding sections which allowed us to derive probability distributions for interaction con"gurations in a consistent way within an e!ective theory based on Feynman diagrams. Our basic formula for the inelastic cross section for a nucleus}nucleus collision (which includes also as a particular case proton}proton scattering) could be written in the following form:

 


(s)" db d¹  (s, b , b 2b ) ,       

(4.1)

where d¹ represents the integration over the transverse coordinates b and b of projectile and  G H target nucleons, b is the impact parameter between the two nuclei, and b "bo #bo  !bo  is  I LI OI the transverse distance between the nucleons of kth pair. Using the compact notation b"b , m"m , X>"x> , X\"x\  , I I II II the function  is given as 

(4.2)



(m, X>, X\) , (4.3)  (s, b)"
(1! ) dX> dX\ Q@   K K which represents all diagrams with at least one cut Pomeron. One may de"ne a corresponding quantity  , which represents the con"guration with exactly zero cut Pomerons. The latter one  can be obtained from (4.3) by exchanging 1! by , which leads to K K  (s, b)"Q@ (0, 0, 0) . (4.4)   The expression for  is given as





1 KI  Q@ (m, X>, X\)"   G(x> , x\ , s, b )
 (x>, x\, s, b) ,   II II I  m ! I I I

(4.5)

with 1 
 (x>, x\, s, b)"  exp(!GI (x> x\ , s, b ))  LI OI I Z  I  ;  (x>)?  (x>) (1!x>)  (x\)?  (x\) (1!x\) . G G G H H H G H

(4.6)

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139

The arguments of
 are the momentum fractions of projectile and target remnants,  x>"1!
x> , x\"1!
x\ , (4.7) G II H II LIG OIH where (k) and (k) point to the remnants linked to the kth interaction. In the following, we perform the analysis for given s and b"(b , b ,2, b ), so we do not write    these variables explicitly. In addition, we always refer to the unitarized functions, so we will also suppress the subscript `ua. Furthermore, we suppress the index AB. Crucial for our applications is the probability conservation constraint #"1 ,

(4.8)

which may be written more explicitly as



(4.9)
dX> dX\ (m, X>, X\)"1 . K This allows us to interpret (m, X>, X\) as the probability distribution for a con"guration (m, X>, X\). For any given con"guration the function  can be easily calculated using the techniques developed in the Section 2. The di$culty with the Monte Carlo generation of interaction con"gurations arises from the fact that the con"guration space is huge and rather non-trivial in the sense that it cannot be written as a product of single Pomeron contributions. We are going to explain in the next sections, how we deal with this problem. 4.2. Interaction matrix Since (m, X>, X\) is a high-dimensional and nontrivial probability distribution, the only way to proceed amounts to employing dynamical Monte Carlo methods, well known in statistical and solid state physics. We "rst need to choose the appropriate framework for our analysis. So we translate our problem into the language of spin systems [156]: we number all nucleon pairs as 1, 2,2, AB and for each nucleon pair k the possible elementary interactions as 1, 2,2, m Let m be the maximum I

 number of elementary interactions per nucleon pair one may imagine. We now consider a two dimensional lattice with AB lines and m columns, see Fig. 4.1. Lattice sites are occupied ("1) or

 empty ("0), representing an elementary interaction (1) or the case of no interaction (0), for the kth pair. In order to represent m elementary interactions for the pair k, we need m occupied cells (1's) I I in the kth line. A line containing only empty cells (0's) represents a pair without interaction. Any possible interaction may be represented by this `interaction matrixa M with elements m 30, 1 . (4.10) II Such an `interaction con"gurationa is exactly equivalent to a spin con"guration of the Ising model. Unfortunately the situation is somewhat more complicated in case of nuclear collisions: we need to consider the energy available for each elementary interaction, represented via the momentum fractions x> and x\ . So we have a `generalizeda matrix K, II II K"(M, X>, X\) , (4.11)

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Fig. 4.1. The interaction lattice.

representing an interaction con"guration, with elements K "(m x> , x\ ) . (4.12) II II II II It is important to note that a number of matrices M represents one and the same vector m. In fact, m is represented by all the matrices M with K 
m "m , (4.13) II I I for each k. Since all the corresponding con"gurations (M, X>, X\) should have the same weight, and since there are m ! 

 (4.14) c"  m !(m !m )! I I I  con"gurations (M, X>, X\) representing the same con"guration (m, X>, X\), the weight for the former is c\ times the weight for the latter, so we obtain the following probability distribution for K"(M, X>, X\):





(m !m )! KI 

 I  G(s, x> , x\ , b)
(x>, x\, s, b) , (K)"  II II  m !

 I I or, using the expression for
 ,  (m !m )! KI 1 

 I  G(s, x> , x\ , b ) exp(!GI (x> x\ , s, b ))  (K)" II II I LI OI I m ! Z

  I I  ;  (x>)?  (x>)(1!x>)  (x\)?  (x\)(1!x\) . G G G H H H G H The probability conservation now reads





(4.15)

 (4.16)


(K)"
dX>dX\ (M, X>, X\)"1 . (4.17) ) + In the following, we shall deal with the `interaction matrixa K, and the probability distribution (K).

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141

4.3. Markov chain method In order to generate K according to the given distribution (K), de"ned earlier, we construct a Markov chain K, K, K,2, KR  

(4.18)

such that the "nal con"gurations KR   re distributed according to the probability distribution (K), if possible for a t not too large!

 Let us discuss how to obtain a new con"guration KR>"¸ from a given con"guration KR"K. We use Metropolis' Ansatz for the transition probability p(K, ¸)"prob(KR>"¸  KR"K)

(4.19)

as a product of a proposition matrix w(K, ¸) and an acceptance matrix u(K, ¸):



w(K, ¸)u(K, ¸)

p(K, ¸)"

w(K, K)#


*$)

if ¸OK , w(K, ¸)1!u(K, ¸) if ¸"K ,

(4.20)

where we use




u(K, ¸)"min



(¸) w(¸, K) ,1 , (K) w(K, ¸)

(4.21)

in order to assure detailed balance. We are free to choose w(K, ¸), but of course, for practical reasons, we want to minimize the autocorrelation time, which requires a careful de"nition of w. An e$cient procedure requires u(K, ¸) to be not too small (to avoid too many rejections), so an ideal choice would be w(K, ¸)"(¸). This is of course not possible, but we choose w(K, ¸) to be a `reasonablea approximation to (¸) if K and ¸ are reasonably close, otherwise w should be zero. So we de"ne



w(K, ¸)"

 (¸) if d(K, ¸)41 ,  0 otherwise ,

(4.22)

where d(K, ¸) is the number of lattice sites being di!erent in ¸ compared to K, and where  is de"ned by the same formulas as  with one exception:
 is replaced by 1. So we get  KI  (4.23)  (¸)&  (m !m )!  G(x> , x\ , s, b ) .

 I II II I  I I The above de"nition of w(K, ¸) may be realized by the following algorithm:





E choose randomly a lattice site (k, ), E propose a new matrix element (m , x> , x\ ) according to the probability distribution II II II (m , x> , x\ ), II II II where we are going to derive the form of  in the following. From Eq. (4.23), we know that  should be of the form



G(x>, x\, s, b) if m"1 , (m, x>, x\)&m !  1 if m"0 ,

(4.24)

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where m "m !m is the number of zeros in the row k. Let us de"ne m as the number of zeros 

  (empty cells) in the row k not counting the current site (k, ). Then the factor m ! is given as m ! in   case of mO0 and as m !(m #1) in case of m"0, and we obtain  (m, x>, x\)&(m #1) #G(x>, x\, s, b) . (4.25)  K K Properly normalized, we obtain (m, x>, x\)"p #(1!p )  K 

G(x>, x\, s, b) , K 

(4.26)

where the probability p of proposing no interaction is given as  m #1  , p "  m #1#(s, b)  with  being obtained by integrating G over x> and x\,



(s, b)"

(4.27)



dx>dx\ G(x>, x\, s, b) . (4.28)  Having proposed a new con"guration ¸, which amounts to generating the values m , x> , x\ for II II II a randomly chosen lattice site as described above, we accept this proposal with the probability u(K, ¸)"min(z z , 1) ,  

(4.29)

w(¸, K) (¸) , z " . z "  w(K, ¸)  (K)

(4.30)

with

Since K and ¸ di!er in at most one lattice site, say (k, ), we do not need to evaluate the full formula for the distribution  to calculate z , we rather calculate  II(¸) , (4.31) z "  II(K) with






GI (x> x\ , s, b ) II(K)"(m , x> , x\ ) exp !
LJ OJ J II II II J   I ;(x> )?  (x> )(1!x> ) (x\ )?  (x\ )(1!x\ ) , (4.32) LI LI LI OI OI OI which is technically quite easy. Our "nal task is the calculation of the asymmetry z . In many  applications of the Markov chain method one uses symmetric proposal matrices, in which case this factor is simply one. This is not the case here. We have  (K) II(K) z "  "  ,   (¸) II(¸)  

(4.33)

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with II(K)"(m , x> , x\ ) , (4.34)  II II II which is also easily calculated. So we accept the proposal ¸ with the probability min(z z , 1), in   which case we have KR>"¸, otherwise we keep the old con"guration K, which means KR>"K. 4.4. Convergence A crucial item is the question of how to determine the number of iterations, which are su$cient to reach the stationary region. In principle one could calculate the autocorrelation time, or better one could estimate it based on an actual iteration. One could then multiply it with some `reasonable numbera, between 10 and 20, in order to obtain the number of iterations. Since this `reasonable numbera is not known anyway, we proceed di!erently. We consider a number of quantities like the number of binary interactions, the number of Pomerons, and other observables, and we monitor their values during the iterations. Simply by inspecting the results for many events, one can quite easily convince oneself if the numbers of iterations are su$ciently large. As a "nal check one makes sure that the distributions of some relevant observables do not change by doubling the number of iterations. In Fig. 4.2, we show the number of collisions (left) and the number of Pomerons (right) as a function of the iteration step t for a S#S collision, where the number of iterations t has been determined according to some empirical procedure described

 below. We observe that these two quantities approach very quickly the stationary region. In order to determine the number t of iterations for a given reaction A#B, we "rst calculate the upper

 limit for the number of possibly interacting nucleon pairs as the number of pairs k with

 a transverse distance smaller than some value b being de"ned as

 (4.35) 1!e\QQ@  "0.001 , and we then de"ne t "100 ) k . (4.36)

   Actually, in the real calculations, we never consider sums of nucleon pairs from 1 to AB, but only from 1 to k , because for the other ones the chance to be involved in an interaction is so small that

 one can safely ignore it. 4.5. Some tests for proton}proton scattering As a "rst test, we check whether the Monte Carlo procedure reproduces the theoretical pro"le function  . So we make a large number of simulations of proton}proton collisions at a given  energy (s, where the impact parameters are chosen randomly between zero and the earlier de"ned maximum impact parameter b . We then count simply the number n (b) of inelastic interac   tions in a given impact parameter bin [b!b/2, b#b/2], and divide this by the number n (b)  of simulations in this impact parameter interval. Since the total number n (b) of simulated  con"gurations for the given b-bin splits into the number n (b) of `interactionsa and the number 

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Fig. 4.2. Number of collisions (left) and number of Pomerons (right) as a function of the iteration step t (in percent relative to the maximum number of iterations) for three di!erent S#S events.

n (b) of `non-interactionsa, with n (b)"n (b)#n (b), the result       n (b) (4.37) P(b)"  n (b)  represents the probability to have an interaction at a given impact parameter b, which should coincide with the pro"le function (4.38)  (s, b)"1!
 (1, 1, s, b)   for the corresponding energy. In Fig. 4.3, we compare the two quantities for a proton}proton collision at (s"200 GeV and we "nd an excellent agreement, as it should be. Another elementary quantity is the inclusive momentum spectrum of Pomerons. Pomerons, representing elementary interactions, are characterized by their light cone momentum fractions

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

145

Fig. 4.3. The simulated probability P(b) (points) of an inelastic interaction for a proton}proton collision at (s"200 GeV compared to the pro"le function  (s, b) (solid line). 

x> and x\, so one might study two-dimensional distributions, or for example a distribution in x"x>x\, where the second variable y"0.5 log(x>/x\) is integrated over. So again we simulate many proton}proton events at a given energy (s and we count the number of Pomerons N . within a certain interval [x!x/2, x#x/2] and we calculate dn+! 1 N . " . , (4.39) dx x N  representing the Monte Carlo Pomeron x-distribution, which may be compared with the analytical result calculated earlier, as shown in Fig. 4.4. The analytical results of course refer to the unitarized theory. Again we "nd perfect agreement between Monte Carlo simulations and analytical curves, as it should be. In Fig. 4.5, we compare inclusive Pomeron cross sections (integrated over impact parameter). Here, the impact parameters are generated randomly between 1 and some b , one

 counts the number of Pomerons N within a certain interval of size x, and one calculates . r N d
+! . , . "  (4.40) x N dx  which is compared with the analytical result



d
dn . (x, s)" db . (x, s, b) , dx dx

(4.41)

which again show an excellent agreement. These Pomeron distributions are of particular interest, because they are elementary distributions based on which other inclusive spectra like a transverse momentum distribution of pions may be obtained via convolution. The two examples of this section provide on one hand a check that the numerical procedures work properly, on the other hand they demonstrate nicely that our Monte Carlo procedure is a very well de"ned numerical method to solve a particular mathematical problem. In simple cases where analytical results exist, they may be compared with the Monte Carlo results, and they must absolutely agree.

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Fig. 4.4. Inclusive x-distribution of Pomerons. The variable x is de"ned as x"x>x\ and is therefore the squared mass of the Pomeron divided by s. We show unitarized analytical results (solid lines) for b"0 fm (upper curves) and b"1.5 fm (lower curves) and the corresponding simulation results (points). Fig. 4.5. Inclusive cross section of Pomerons versus x. The variable x is de"ned as x"x>x\ and is therefore the squared mass of the Pomeron divided by s. We show unitarized analytical results (solid line) and the corresponding simulation results (points).

5. Enhanced Pomeron diagrams The eikonal type diagrams shown in Fig. 5.1, considered in the previous sections, correspond to pair-like scatterings between hadron constituents and form the basis for the description of hadronic interactions at not too high energies. However, when the interaction energy increases, the contribution of so-called enhanced Pomeron diagrams as, for example, the diagrams shown in Fig. 5.2, become more and more important. The latter ones take into account interactions of Pomerons with each other. The corresponding amplitudes increase asymptotically much faster than the usual eikonal type contributions considered so far. The problem of consistent treatment of Pomeron} Pomeron interactions was addressed in [44,46,52,55,154,157}166]. The number of diagrams, which contribute essentially to the interaction characteristics, increases fast with the energy. In this paper, we restrict ourselves to the lowest order diagrams (>-diagrams and inverted >-diagrams). In the following sections, we discuss the amplitudes corresponding to the lowest-order enhanced diagrams and the modi"cation of the hadronic pro"le function in the presence of these diagrams, before we discuss their most important features. 5.1. Calculating lowest-order enhanced diagrams To introduce enhanced type diagrams let us come back to the process of double soft Pomeron exchange, which is a particular case of the diagram of Fig. 2.7. The corresponding contribution to the elastic scattering amplitude is given in Eqs. (2.31), (C.11) with n"2 and with ¹ / being  replaced by ¹ :  1 dk dk dk dk dq      (s>)(s\)(s>)(s\)(s> )(s\ ) (s, t)" i¹     O O F F 2 (2) (2) (2) (2) (2)



(p, k , k , q , q!q ) ;disc > > >O N     Q Q Q F  ;  [i¹ (s( , q)]disc \ \ \O N (p, k , k ,!q ,!q#q )  J J Q Q Q F     J

(5.1)

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147

Fig. 5.1. Eikonal type diagrams.

Fig. 5.2. Enhanced diagrams.

see Fig. 5.3. We are now interested in the contribution with some of the invariants s>"(p!k )K!p>k\ , (5.2)    s>"(p!k !k )K!p>(k\#k\) , (5.3)      (5.4) s> "(p#q )Kp>q\ ,   O being large, implying k\, q\ to be not too small. As shown in Section C.3, in that case the above G  amplitude may be written as



i¹/\ (s, t)" F F

 dx> dx\ FF (1!x>)FF (1!x\) i¹F/F (x>, x\, s, t)    \ x> x\   

(5.5)

with







V> dx> 1  Im ¹F (x>, s>,!q ) , 2s> \ x> Q V  V\ 1  ; dz> dq , dq , i¹F (x\, s( ,!q, ) dx\ dx\    J J J   8s(  J J ; (x\!x\!x\) (qo !qo , !qo , ) ,  ,    r

/ i¹F F (x>, x\, s, t)"8x>x\s   \ 2 / 

 





 (5.6)

with ¹F(x, s,!q )"¹F (x, s,!q )"¹ (s,!q )FF (x) exp(!Rq ) ,  ,  ,   F ,

(5.7)

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Fig. 5.3. Double Pomeron exchange.

and s( "x> z>x\s, s( "x> (1!z>)x\s ,       where the following de"nitions have been used:

(5.8)

x>"k>/p> ,

(5.9)

x> "k> /p> , (5.10)   x>"k>/p> , (5.11)   x> !x>"(k> !k>)/p>"k>/p> , (5.12)      z>"k>/k> "x>/x> , (5.13)     s>"(k!k )K!k>k\ Ks k>/k> "s x>/x> , (5.14)       see Fig. 5.4. The sign `!a in `3/!a refers to the Pomeron `splittinga towards the target hadron (reversed >-diagram); the lower limit for the integral dx> is due to x\ Ks /x> (x\.     The triple-Pomeron contribution (5.6) is by construction expressed via amplitudes ¹F for  parton}parton scattering due to soft Pomeron exchange, each one corresponding to non-perturbative parton dynamics, characterized by restricted parton virtualities Q(Q . We can also take  into account contributions to the triple-Pomeron diagram from semi-hard processes, when some part of the parton cascade mediating the scattering between partons of momenta k and k in J J Fig. C.3 (k and !k ) develops in the perturbative region Q'Q . Then, according to the general   discussion of Section 2, the amplitudes ¹F obtain also contributions from semi-hard sea-type parton}parton scattering ¹F } and from valence quark scattering ¹F } :     (5.15) ¹F"¹F #¹F } #¹F } ,      with (x, s( ,!q )"¹ (s( ,!q )FF (x) exp(!Rq ) ¹F ,  } ,   F ,  }

(5.16)

and



V






x x T s( ,!q exp(!Rq ) dx
¹I }   x , F , Tx  T I ;FM FI (x , x!x ) .   T T

¹F } (x, s( ,!q )" ,  

(5.17)

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149

We have to stress again that we do not consider the possibility of Pomeron}Pomeron coupling in the perturbative region Q'Q . Therefore in our scheme hard parton processes can only  contribute into internal structure of elementary parton}parton scattering amplitudes but do not in#uence the triple-Pomeron coupling. A similar contribution ¹F/F of the >-diagram can be obtained via interchanging x>x\ and  > h h :   (5.18) ¹F/F (x>, x\, s, t)"¹F/F (x\, x>, s, t) .  >  \ One can repeat the above derivation for the case of a general soft multiple scattering process, see Eq. (2.31), where some of the energy invariants s>J , s\J are large [44]. One then obtains "nally the O O general multiple scattering expression (C.22), with the contribution of corresponding pairs of Pomerons being replaced by expressions



1 dq i¹F/F (x>, x\, s, t) , ,  ! 8x>x\s

(5.19)

where x! refers to the summary light cone momentum share of the constituent partons participating in the triple-Pomeron process. So we get

with



 

 

L 1  1  L  dx> dx\  dq , i¹F/ F (s( ,!q, ) i¹   (s, t)"8s
J J J J J FF n! 8s(  J J J L L L L ;FF 1!
x> FF 1!
x\ 
qo , !qo , I H H ,     I H H










(5.20)

(5.21) ¹F/ F "¹F/F #¹F/F #¹F/F .   \  > Here, we allow any number of simple triple-Pomeron diagrams; thus we restrict ourselves to the contributions of double Pomeron iteration in the t-channel rather than to the "rst order in r / .  The Fourier transform ¹I of the amplitude (5.20) is given as




i L i  1  L  dx> dx\  ¹I (s, b)"
¹I F/ F (x>, x\, s, b) J J J J 2s n! 2s F F  J J L L L 1!
x> FF 1!
x\ , ;FF     H H H H with ¹I /F F being the Fourier transform of ¹F/ F ,






(5.22)

(5.23) ¹I F/ F "¹I F/F #¹I F/F #¹I F/F ,   \  > where the Fourier transform ¹I F/F of the triple-Pomeron amplitude ¹F/F is given as  \  \ i V> dx> 1 r /  ¹I F/F (x>, x\, s, b)"  db Im ¹I F (x>, s>, bo !bo )   2s(  \ 2s> 2 \ x> Q V  V\ 1  dx\ dx\  ; dz> i¹I F (x\, s( , b ) (x\!x\!x\) ,   J J    2s(  J J (5.24)

   

with s( "x>x\s (and similarly for ¹I F/F ).  >









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Fig. 5.4. Kinematical variables for the triple Pomeron contribution. Fig. 5.5. Triple Pomeron contribution.

Here we used



1 (qo !qo , !qo , )" db exp(i(qo !qo , !qo , )bo ) .   ,  ,    4

(5.25)

The pro"le function  for hadron}hadron interaction is as usual de"ned as 1    (s, b)" 2Im ¹I   (s, b) , FF FF 2s

(5.26)

which may be evaluated using the AGK cutting rules,



K K  1  dx> dx\  GF/ F (x>, x\, s, b)    (s, b)"
I I I I FF m! I I K



J  1 J  dx > dx \  !GF/ F (x >, x \, s, b) ;
H H H H l! H H J









x !
x > F x !
x \ , ;F H   H   H H

(5.27)

with x  "1!
x! being the momentum fraction of the projectile/target remnant, and with I GF/ F "GF/F #GF/F #GF/F ,   \  >

(5.28)

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151

where GF/F is twice the imaginary part of the Fourier transformed triple-Pomeron amplitude  ! ¹I F/F divided by 2s( ,  ! 1 GF/F (x>, x\, s, b)" 2Im ¹I F/F (x>, x\, s, b) , (5.29)  !  ! 2x>x\s which gives, assuming imaginary amplitudes,

   

r / GF/F (x>, x\, s, b)"!   \ 8 ;



V> dx>  GF (x>, s>, bo !bo )  x> Q V\ 

db  V\

dz>





dx\ GF (x\, s( , b ) GF (x\!x\, s( , b ) ,       

(5.30)

see Fig. 5.5, with x> , s( "x> z>x\s, s( "x> (1!z>)(x\!x\)s . s>"s        x>  The functions GF are de"ned as

(5.31)

1 GF(x, s( , b)" 2Im ¹I F(x, s( , b) , 2s(

(5.32)

with ¹I F being the Fourier transform of ¹F, which gives GF"GF #GF } #GF } ,     

(5.33)

with s( ? \ b exp ! F (x) , s 4 F (s( /s )       dz> dz\EH (z>)EI (z\)
HI (z>z\s( , Q )      b 1 exp ! F (x) , ; 4 F (1/z>z\)   F (1/z>z\)   1 V  x T z>s( , Q GF } (x, s( , b)"
dx dz> EH (z>)
HI    T   x 4   HI 1 b ; exp ! FM I (x , x!x ) , T F (1/z>) 4 F (1/z>)   T  


 






(5.34)



(5.35)

F (z)"R#  ln z .  F 

(5.37)

2   GF (x, s( , b)"  F (s( /s )   1 GF } (x, s( , b)"
  4 HI

 














(5.36)

with

5.2. Cutting enhanced diagrams To treat particle production, we have to investigate the di!erent cuts of an enhanced diagram. We consider the inverted >-diagram here, the same arguments apply to the >-diagram. We employ

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Fig. 5.6. The di!erent cuts of the triple Pomeron inverted >-diagram: none of the lower Pomerons cut (a), one of these Pomerons cut (b), and both Pomerons cut (c). We also indicate their relations with G"GF/F .  \

the cutting rules to Eq. (5.24), 1 (5.38) GF/F " 2Im ¹I F/F "sum over cut diagrams"
GF/F ,  \ 2s(  \  \G G where the index i counts the di!erent cuts. We take into account that the cutting procedure only in#uences the two Pomerons exchanged `in parallela in the triple-Pomeron graph (the lower Pomerons) with the third Pomeron already being cut [44], so that we have three contributions: none of the lower Pomerons cut (i"0, di!raction), one of these Pomerons cut (i"1, screening), and both Pomerons cut (i"2, Pomeron}Pomeron fusion), see Fig. 5.6. So we have , GF/F "GF/F #GF/F #GF/F  \  \  \  \

(5.39)

with

  

  

i GF/F " ¹I F/F ;2 ,  \ 2s(  \ i GF/F " ¹I F/F ;(!2);2;2 ,  \ 2s(  \ i ¹I F/F ;2;2 . GF/F "  \ 2s(  \

(5.40)

Here, we assumed imaginary amplitudes, and we replace as usual a factor i¹I F in Eq. (5.24) by 2Im ¹I F"!2i¹I F for a cut Pomeron, and by (i¹I F)H"i¹I F for an uncut Pomeron being to the left of the cut plane, and by i¹I F for an uncut Pomeron being to the right of the cut plane. Using (see Eq. (5.29)) i ¹I F/F "!GF/F ,   \ 2s(  \

(5.41)

we get GF/F "!1;GF/F ,  \  \ GF/F "#4;GF/F ,  \  \   GF /F "!2;GF/F ,  \  \

(5.42)

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153

which means that each cut contribution is equal to the pro"le function, up to a pre-factor, see Fig. 5.6. The sum of the three contributions is GF/F , as it should be.  \ There is a substantial di!erence between the di!erent cuts of triple-Pomeron contributions: in the case of both lower Pomerons being cut (i"2) all the momentum of the constituent partons, participating in the process, is transferred to secondary hadrons produced, whereas for the cut between these Pomerons (i"0) only the light cone momentum fractions of the cut Pomeron (x>, x\ "s /x> in Eq. (5.30)) are available for hadron production, the momentum share    x\!x\ of the partons, connected to the uncut (virtual) Pomerons is given back to the remnant  state. Correspondingly, the contribution with one of the two lower Pomerons being cut (i"1) de"nes the screening correction to the elementary rescattering with the momentum fractions x>, x\ (considering the "rst of the two Pomerons being cut). It is therefore useful to rewrite the  expression for the pro"le function as



1 K K  dx> dx\ dx( > dx( \  GK /F F (x>, x\, x( >, x( \, s, b)    (s, b)"
I I I I I I I I FF m! I I K 1 J J ;
 dx > dx \  !GF/ F (x >, x \, s, b) H H H H l! J H H









;F x !
x( >!
x > F x !
x( \!
x \ ,   I H   I H I H I H with GF/ F being de"ned earlier, and with GK /F F (x>, x\, x( >, x( \, s, b)"G / (x>, x\, s, b) (x( >) (x( \) I I I I  I I I I  #

GK F/F (x>, x\, x( >, x( \, s, b) ,  NG I I I I N! G where we used

(5.43)

(5.44)

x  "1!
x! . (5.45) I The variables x! are now the momentum fractions for the individual cut contributions, which I de"ne the energy for the production of secondary hadrons resulting from a given elementary interaction. The expressions for the functions GK F/F are obtained from Eqs. (5.27), (5.30), (5.39) and  NG (5.42), by changing the variables properly. Simplest is the case of all Pomerons being cut, no changing of variables is necessary, we have simply GK F/F (x>, x\, x( >, x( \, s, b)"!2GF/F (x>, x\, s, b) (x( >) (x( \)  \  \ V> dx> r /  GF (x>, s>, bo !bo )  " db   4 \ x> Q V   V\ ; dz> dx\ GF (x\, s( , b )GF (x\!x\, s( , b )          ; (x( >) (x( \) .

(5.46)

   

(5.47)

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For none of the two lower Pomerons being cut, we rename x\ into x\and x\!x\ into x( \,   so we get

  


r / 1 GF (x>, x>x\s, bo !bo ) GK F/F (x>, x\, x( >, x( \, s, b)"  db  \  x\  8 ;





dz>

V( \>V\






s dx\ GF x\, x\  z>s, b   x\  





s ;GF x( \#x\!x\, (x( \#x\!x\)  (1!z>)s, b (x( >) .   x\  (5.48) For one of the two lower Pomerons being cut, we rename x\ into x\and x\!x\ into x( \,   and we get

 

r / V> dx>  GF (x>, s>, bo !bo ) GK F/F (x>, x\, x( >, x( \, s, b)"!  db  \   x> 2 Q V\   ; dz> GF (x\, x> z>x\s, b ) GF (x( \, x> (1!z>)x( \s, b ) (x( >) .      (5.49)



The contributions GF/F can be obtained via interchanging x>x\, x( >x( \ and h h in the  >G   above formulas: GF/F (x>, x\, x( >, x( \, s, b)"GF/F (x\, x>, x( \, x( >, s, b) .  >G  \G We de"ne as in the usual eikonal case the virtual emission function
  (x , x , s, b)"
FF J



(5.50)

J 1 J  dx > dx \  !GF/ F (x >, x \, s, b) H H l! H H H H









x !
x > F x !
x \ , ;F H   H   H H which allows to write the pro"le functions Eq. (5.43) as



(5.51)

K K 1  dx> dx\ dx( > dx( \  GK /F F (x>, x\, x( >, x( \, s, b)    (s, b)"
I I I I I I I I FF m! I I K






;
  x !
x( >, x !
x( \, s, b , FF I I I I which may be approximated as

(5.52)



1 K K  dx> dx\  GK /F F (x>, x\, x , x , s, b)    (s, b)"
I I I I FF m! I I K ;
  (x , x , s, b) , FF

(5.53)

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155

with GK /F F (x>, x\, x , x , s, b) " F

1 (x )F (x )    



; dx( > dx( \GK /F F (x>, x\, x( >, x( \, s, b)F (x !x( >)F (x !x( \) .    

(5.54)

Based on Eq. (5.53), we proceed as in the eikonal case. We unitarize the theory by replacing
  by FF
FF in a complete analogy to the eikonal model. The numerical Markov chain procedures have to  be modi"ed slightly due to the fact that the GK functions contain x and x  as arguments. One can no longer restrict oneself to considering one single site of the interaction matrix, since changing x> and x\ modi"es as well x and x  and a!ects therefore the other sites as well (in case of nucleus}nucleus all the sites related to the same projectile and target nucleon). But this does not pose major problems. 5.3. Important features of enhanced diagrams The amplitude corresponding to a single Pomeron exchange as shown in Fig. 5.7(a) behaves as a function of energy approximately as s, where s is the c.m. energy squared for hadron}hadron interaction and  is some e!ective exponent. At the same moment, the amplitude of the so-called >-diagram shown in Fig. 5.7(b) increases asymptotically as s, as can be seen from Eq. (5.30). The amplitude corresponding to the diagram in Fig. 5.7(c) behaves as s. This indicates very important property of enhanced diagrams, namely that they increase with energy much faster than the usual eikonal ones. In the following, we discuss exclusively enhanced >-diagrams for given hadron types h and h ,   and to simplify the notation, we use simply G to refer to the corresponding pro"le function, omitting all indices referring to the initial hadron types and the Pomeron type.

Fig. 5.7. Energy dependence of Pomeron diagrams. Fig. 5.8. The di!erent cuts of the >-diagram.

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In order to calculate contributions of enhanced graphs to the total interaction cross section, one has to consider di!erent cuts of elastic scattering diagrams. We have 1 G, 2Im ¹I (s, t"0)"G #G #G ,    2s

(5.55)

where G refers to the di!erent cut diagrams, as shown in Fig. 5.8. We use our convention G employed already earlier to plot cut Pomerons as dashed and uncut ones as full vertical lines. The diagram in Fig. 5.8(a) gives rise to the process of high mass target di!raction (G ), the diagram in  Fig. 5.8(b) represents the screening correction to the one cut Pomeron process (G ), and the  diagram in Fig. 5.8(c) } the cut Pomeron fusion process (G ). As discussed in the preceding section,  we have G "!1G, G "#4G, G "!2G , (5.56)    the sum being therefore equal to G, as it should be. Since G is negative, the "rst-order contribution to the inelastic cross section is negative, so another very important property of enhanced diagrams is the suppression of the increase of the inelastic cross section with energy. A remarkable feature of enhanced diagrams is connected to their e!ect on the inclusive particle spectra. In particular, if one assumes (for a qualitative discussion) that each cut Pomeron gives rise to a #at rapidity distribution of secondary hadrons, dn/dy" , the sum of all three contributions  gives the screening correction to the inclusive particle spectrum as dn J!1G;0 #4G;1 !2G;2 "0, y'y     dy

(5.57)

dn J!1G; #4G; !2G; "G (0,     dy

(5.58)

and y(y , 

where y is the rapidity position of the triple Pomeron vertex, see Fig. 5.9. The contribution is  negative, because G is negative. Thus enhanced diagrams give rise to screening corrections to secondary hadron spectra only in restricted regions of the kinematical phase space; contributions of di!erent cuts exactly cancel each other in the region of rapidity space where two or more Pomerons are exchanged in parallel (in case of the diagram of Fig. 5.9 for y'y ) [30]. Therefore  another important e!ect of Pomeron}Pomeron interactions is the modi"cation of secondary

Fig. 5.9. E!ect on inclusive spectra.

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hadron spectra, being mainly suppressed in the fragmentation regions of the interaction: close to y"0 from the >-diagrams and close to y for the inverted >-diagrams. That explains the great

 importance of enhanced diagrams for the solution of the unitarity problems inherent for the pure eikonal scheme and for the construction of a consistent unitary approach to hadronic interactions at very high energies (see the discussion in Section 3). Another important e!ect is the considerable increase of #uctuations of hadronic interactions. Let us for a moment consider the indices indicating Pomeron types: G / for the >-diagram as  > discussed above, G / for the corresponding inverted diagram, and G / (s, x>, x\) for the normal   \ Pomeron. The full contribution (so far) is (5.59) G/ "G / #G / #G / .  >   \ Using the fact that G / can be written as the sum of the di!erent cut contributions G / , we get  !G  ! G/ "G / #
G / #
G / ,   \G  >G G G which may be written as #G / #G / #G / #G / #G / . (5.60) G/ "G / #G /  >  \  >  \  >   \ This means that we have three contributions: a modi"ed one cut Pomeron exchange, with probability w , the high mass target (see Fig. 5.8(a)) and projectile di!raction, with probability  w , and the process of Pomeron fusion of Fig. 5.8(c), with probability w , where the     probabilities are given as #G / #G / #G / G G / G / #G /  > , w " /\  > , w  > .  \ "  \ w "       G/ G/ G/ (5.61) The two latter processes result in correspondingly much smaller and much larger values of the hadron multiplicity than for the one Pomeron process. As we mentioned in the beginning of this section, the number of enhanced Pomeron diagrams, which contribute essentially to the interaction characteristics, increases fast with the energy. Therefore, one has to develop a suitable method to take into account the necessary contributions to the forward scattering amplitude, the latter one being related via the optical theorem to the total cross section of the reaction and to the weights for particular con"gurations of the interaction (via the `cuttinga procedure). Such a scheme is still under development and our current goal was the proper treatment of some lowest order enhanced diagrams. Thus we proposed a minimal modi"cation of the standard eikonal scheme, which allowed us to obtain a consistent description of hadronic interactions in the range of c.m. energies from some ten GeV till few thousand TeV. Already this minimal scheme allows to achieve partly the goals mentioned above: the slowing down of the energy increase of the interaction cross section and the non-AGK-type modi"cation of particle spectra, as well as the improvement of the description of the multiplicity and inelasticity #uctuations in hadron}hadron interaction. An important question exists concerning the nature of the triple-Pomeron coupling. As discussed above, we used the perturbative treatment for the part of a parton cascade developing in the region

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of parton virtualities bigger than some cuto! Q , whereas the region of smaller virtualities is  treated phenomenologically, based on the soft Pomeron. There was an argumentation in [145] that the triple-Pomeron coupling is perturbative and therefore can be described on the basis of the QCD techniques. At the same time, it was shown in [167] that such perturbative coupling, corresponding to non-small parton virtualities, would result in negligible contribution to the basic interaction characteristics, in particular, to the proton structure function F . The latter result was  con"rmed experimentally by HERA measurements, where no real shoulder in the behavior of F (x, Q) (predicted in [145]) was found in the limit xP0. Another argument in favor of the  smallness of the perturbative Pomeron}Pomeron coupling comes from the HERA di!ractive data, where the proportion of di!ractive type events (with a large rapidity gap in secondary hadron spectra) appeared to be nearly independent on the virtuality of the virtual photon probe. This implies that the Pomeron self-interaction is rather inherent to the non-perturbative initial condition for the QCD evolution than to the dynamical evolution itself. Therefore we assumed that the Pomerons interact with each other in the non-perturbative region of parton virtualities Q(Q  and considered it as the interaction between soft Pomerons. In our scheme the relatively big value of the soft triple-Pomeron coupling results in the screening corrections which "nally prevent the large increase of parton densities in the small x limit and restore the unitarity, thus leaving a little room for higher twist e!ects in the perturbative part of the interaction.

6. Parton con5gurations In this section, we consider the generation of parton con"gurations in nucleus}nucleus (including proton}proton) scattering for a given interaction con"guration, which has already been determined, as discussed above. So, the numbers m of elementary interactions per nucleon} I nucleon pair k are known, as well as the light cone momentum fractions x> and x\ of each II II elementary interaction of the pair k. A parton con"guration is speci"ed by the number of partons, their types and momenta. 6.1. General procedure of parton generation We showed earlier that the inelastic cross section may be written as





(6.1)
" db
dX> dX\ (m, X>, X\) ,  K where m, X>, X\ represents an interaction con"guration. The function  is known (see Eq. (2.72)) and is interpreted as the probability distribution for interaction con"gurations. For each individual elementary interaction a term GF/F appears in the formula for , where the function  GF/F itself can be expressed in terms of contributions of di!erent parton con"gurations. Namely  the QCD evolution function EGH , which enters into the formula for the elementary interaction /!" contribution GF/F , is the solution of a ladder equation, where adding a ladder rung corresponds to  an integration over the momenta of the corresponding resolvable parton emitted. The complete evolution function is therefore a sum over n-rung ladder contributions, where the latter one can be

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written as an integration over n parton momenta. So we have



 KI R LIIO (m, X>, X\)"  

dp (p ) , (6.2) IIOJ IIOJ I I O J where t is the number of Pomeron types (soft, sea}sea, etc.), and n the number of partons for the IIO th interaction of the pair k in case of Pomeron type . We interpret (p ) IIOJ (m, X>, X\)

(6.3)

as the probability distribution for parton con"gurations for a given interaction con"guration m, X>, X\. The Monte Carlo method provides a convenient tool for treating such multidimensional distributions: with  being known (see Section 2 and the discussion below), one generates parton con"gurations according to this distribution. We want to stress that the parton generation is also based on the master formula Eq. (2.72), no new elements enter. In the following, we want to sketch the generation of parton con"gurations, technical details are provided in the next section. Let us consider a particular elementary interaction with given light cone momentum fractions x> and x\ and given impact parameter di!erence b between the corresponding pair of interacting nucleons, for a "xed primary energy squared s. For the sake of simplicity, we discuss here the procedure without the triple-Pomeron contribution, with the corresponding generalization being done in Section C.4. We have to start with specifying the type of elementary interaction (soft, semi-hard, or valence type). The corresponding probabilities are GF F (x>, x\, s, b)/GF/F (x>, x\, s, b) ,   GF F} (x>, x\, s, b)/GF/F (x>, x\, s, b) ,    GF F} (x>, x\, s, b)/GF/F (x>, x\, s, b) ,    GF F} (x>, x\, s, b)/GF/F (x>, x\, s, b) ,    (6.4) GF F} (x>, x\, s, b)/GF/F (x>, x\, s, b) .    In the case of a soft elementary interaction, no perturbative parton emission takes place. Therefore we are left with the trivial parton con"guration, consisting of the initial active partons}hadron constituents, to which the Pomeron is attached. Let us now consider a semi-hard contribution. We obtain the desired probability distributions from the explicit expressions for GF F} . For given x>, x\, and b we have     GF F} (x>, x\, s, b)J dz> dz\
EG (z>)EH (z\)
GH (z>z\s( , Q )          GH b 1 (6.5) exp ! FF (x>)FF (x\) ;       4 F F (1/(z>z\))   4 F F (1/(z>z\))  

 






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with s( "x>x\s, F F ( )"R #R #  ln , and  F F  d
IJ  (x>x\s( , p )
GH (s( , Q )"K
dx> dx\ dp , dp ,    , IJ ;EGI (x>, Q , M )EHJ (x\, Q , M )(M !Q ) , (6.6) /!"  $ /!"  $ $  representing the perturbative parton}parton cross section, where both initial partons are taken at the virtuality scale Q ; we choose the factorization scale as M "p /4. The integrand of Eq. (6.5)  $ , serves as the probability distribution to generate the momentum fractions x!"x!z! and the  #avors i and j of the initial partons for the parton ladder. The knowledge of the initial conditions } the momentum fractions x! and the starting virtuality  Q for the `"rst partonsa as well as the #avors i and j } allows us to reconstruct the complete  ladder, based on Eq. (6.6) and on the evolution equations (B.28) and (B.29) for EGH . To do so, we /!" generalize the de"nition of the parton}parton cross section
GH to arbitrary virtualities of the   initial partons, de"ning





d
IJ  (x>x\s( , p )
GH (s( , Q , Q )"K
dx> dx\ dp , dp ,     , IJ ;EGI (x>, Q , M ) EHJ (x\, Q , M )(M !max[Q , Q ]) /!"  $ /!"  $ $  

(6.7)

and



d
IH  (x>x\s( , p )
GH (s( , Q , Q )"K
dx> dx\ dp , dp ,    , I ;EGI (x>, Q , M , w>)H(Q , M )(M !max[Q , Q ]) (6.8) /!"  $  $ $   representing the full ladder contribution (
) and the contribution, corresponding to the   ordering of parton virtualities towards the end of the ladder, i.e. to the case of parton j, involved into the highest virtuality Born process (
). We calculate and tabulate
and
initially, so     that we can use them via interpolation to generate partons. The generation of partons is done in an iterative fashion based on the following equations:
GH (s( , Q , Q )"
    I

 

dQ

d G(Q , Q) Q PI( )
IH ( s( , Q, Q )#
HG (s( , Q , Q )        Q 2 G

(6.9)

and

 

dQ

d G(Q , Q) Q PI( )
IH ( s( , Q, Q )#
GH (s( , Q , Q ) . (6.10)
GH (s( , Q , Q )"
         Q 2 G I Here,
GH gives the contribution of the con"guration without any resolvable emission before the  highest virtuality Born process:



d
GH  (s( , p )G(Q , M )H(Q , M )(M !max[Q , Q ]) .
GH (s( , Q , Q )"K dp , dp ,  $  $ $      , The procedure is described in detail in the next section.

(6.11)

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In the case of elementary interactions involving valence quarks, the method is almost identical. In that case, the corresponding momentum fractions x! are the ones of valence quarks, x!"x!,   O to be determined according to the corresponding integrands in the expressions for GF F , GF F} ,    GF F} , see Eqs. (2.48)}(2.50). For example, in the case of both hadron constituents being valence   quarks, one generates momentum fractions x! and valence quark #avors i, j with the distribution  (up to a normalization constant) DF F} GH (x>T x\T s, b)FM F G (x>T , x>!x>T )FM F H (x\T , x\!x\T ) , (6.12)   O O   O O   O O see Eqs. (2.48) and (2.18). One then proceeds to generate parton emissions as discussed above. 6.2. Generating the parton ladder We now discuss in detail the generation of the partons in a ladder, starting from the initial partons (`leg partonsa) with #avors i and j and light cone momentum fractions x> and x\. To   simplify the discussion, we will neglect the e!ects of "nite virtualities and transverse momenta of initial partons in the kinematical formulas so that the 4-momenta k and k of the two leg partons   are purely longitudinal. In the hadron}hadron (nucleus}nucleus) center of mass frame we have: k>"x>(s/2, k\"0, k , "0 ,     (6.13) k\"x\(s/2, k>"0, k , "0 .     The invariant mass squared of the ladder is s( "(k #k ).   One "rst generates all resolvable partons emitted at one side of the ladder before the hardest process (for the de"niteness we start with the leg parton i moving in the forward direction). At each step one decides whether there is any resolvable emission at the forward end of the ladder before the hardest process. An emission is done with the probability prob(forward emission)"(
GH (s( , Q , Q )!
HG (s( , Q , Q ))/
GH (s( , Q , Q ) . (6.14)            In case of an emission, the generation of light cone momentum fraction and momentum transfer squared Q for the current parton branching is done } up to a normalization constant } according to the integrand of (
GH !
HG ),   

1 (6.15) prob( , Q)J G(Q , Q) Q
PGY( )
GYH ( s( , Q, Q )  G    2 Q GY see Eq. (6.9). Here the emitted s-channel parton gets the share 1! of the parent (leg) parton light cone momentum k> and the transverse momentum squared p ,(1! )Q. To leading logarith, mic accuracy, the initial parton virtuality is neglected in the branching probability Eq. (6.15), because of Q ;Q. Generating randomly the polar angle  for the emission, one reconstructs the  4-vector p of the "nal s-channel parton as






p cos  p>"(1! )k>, p\"p /((1! )k>), po " , . (6.16)  ,  , p sin  , The remaining ladder after the parton emission is now characterized by the mass squared s( "(k #k !p)K s( and the initial virtualities Q "Q and Q . The #avor i of the new leg    

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parton is generated according to the corresponding weights in Eq. (6.15), properly normalized given as PGY( )
GYH (s( , Q, Q )    , (6.17) prob(i)" G
PJ ( )
JH (s( , Q, Q ) J G    where
GYH ( s( , Q, Q ) is the parton cross section (6.7) for the new ladder. One then renames s( , i,    and Q into s( , i, and Q and repeats the above procedure.   In case of no forward emission, the generation of all resolvable parton emissions at the forward side of the ladder has been completed. One then proceeds to generate all resolvable parton emissions for the backward side of the ladder, starting from the original leg parton j of virtuality Q "Q , by using a corresponding   recursive algorithm, now based on Eq. (6.10). On the other end of the ladder, we have (after renaming) parton i with the virtuality Q . One decides whether there is any resolvable emission  before the hardest process, where the probability of an emission is given as prob(backward emission)"(
HG (s( , Q , Q )!
GH (s( , Q , Q ))/
HG (s( , Q , Q ) . (6.18)          In case of an emission, the generation of the fraction of the light cone momentum k\, and of  the momentum transfer squared Q is done } up to a normalization constant } according to the integrand of (
HG !
GH ),  

1 (6.19) prob( , Q)J H(Q , Q) Q
PHY( )
HYG ( s( , Q, Q ) ,  H   2 Q HY see Eq. (6.10). The #avor j of the new leg parton is de"ned according to the partial contributions in (6.19). The generation of resolvable parton emissions is completed when the iterative procedure stops, with the probability 1!prob(backward emission) . Note, that all parton emissions are simulated in the original Lorentz frame, where the original leg partons (the initial partons for the perturbative evolution) are moving along the z-axis. The "nal step is the generation of the hardest 2P2 parton scattering process. In the center of mass system of two partons i and j with center-of-mass energy squared s( , we simulate the transverse momentum p for the scattering within the limits (given by the condition M "p /4' , $ , max[Q , Q ])   4max[Q , Q ](p (s( /4 (6.20)   , according to d
GH  (s( , p )G(Q , p /4)H(Q , p ) , prob(p )J ,  ,  , , dp , where the di!erential parton}parton cross section is

(6.21)

d
GH 1  (s( , p )"
MGHIJ(s( , p ) (6.22) , , dp 16s( (1!4p /s( IJ , , with MGHIJ(s( , p ) being the squared matrix elements of the parton subprocesses [168]. ,

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Then we choose a particular subprocess ijPkl according to its contribution to the di!erential cross section (6.22), and reconstruct the 4-momenta p and p of the "nal partons in their center of   mass system as p cos  p\"p /(z(s( ), po " , , p>"z(s( ,  , ,  p sin  , p>"(1!z)(s( , p\"p /((1!z)(s( ),   ,









!p cos  , po " , , !p sin  ,

(6.23)

with z"(1#(1!4p /s( ) (6.24)  , and a random polar angle . We "nally boost the momenta to the original Lorentz frame. 6.3. Time-like parton cascade The above discussion of how to generate parton con"gurations is not yet complete: the emitted partons are in general o!-shell and can therefore radiate further partons. This so-called time-like radiation is taken into account using standard techniques [106}108,169}171], to be discussed in the following. The parton emission from an o!-shell parton is done using the so-called DGLAP evolution equations, which describes the process with the leading logarithmic accuracy. The splitting probability for the initial parton of type j is then given as



(p ) dQ dP "! dz
Q , PI(z) , (6.25) 2 H Q P I with the usual Altarelli}Parisi splitting functions PI(z). Here Q"Q is the virtuality of the parent H H parton j and z is interpreted as the energy fraction carried away by the daughter parton k. The maximum possible virtuality q of the parton j is given by the virtuality of the parent of j. H  One imagines now to decrease the virtuality of j, starting from the maximum value, such that the DGLAP evolution equations give then the probability dP that during a change dQ of the virtuality, a parton splits into two daughter partons k and l } see Fig. 6.1. For the energies of the daughter partons one has E "zE , E "(1!z)E . I H J H Choosing a frame where p , "0, we have H p , "!p , "p , J , I with E(z(1!z)Q!zQ!(1!z)Q)!(Q!Q!Q)#QQ H I H I . H J I  J p " H , E!Q H H

(6.26)

(6.27)

(6.28)

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Fig. 6.1. A branching of a parton j into two partons k and l. The kinematical variables used to describe the branching are the energies E , the virtualities Q, and the transverse momenta p , . G G G

As usual, a cuto! parameter terminates the cascade of parton emissions. We introduce a parameter p , which represents the lower limit for p during the evolution, from which we obtain the lower ,  , limit for the virtualities as Q "4p .

 ,  Let us provide some technical details on the splitting procedure. Using Eq. (6.25) and applying the rejection method proposed in [171] we determine the variables Q and z as well as the #avors H k, l of the daughter partons } see Section B.3. To determine the 4-momentum p of the parent H parton we have to distinguish between three di!erent modes of branching (see Fig. 6.2): 1. If j is the initial parton for the time-like cascade, the energy E is given, and with the obtained H value Q, we can calculate po "(E!Q. The direction of po is obtained from the momentum H H H H H conservation constraint for the summary momentum of all partons produced in the current time-like cascade. 2. If j is the initial parton for the time-like cascade resulted from the Born process, together with a partner parton j, then we know the total energy E #E for the two partons. After obtaining H HY the virtualities Q and Q from the secondary splittings jPk, l and jPk, l we can use the H HY momentum conservation constraint in the parton}parton center of mass frame, which gives po "!po "po and allows to determine po . The direction of po can be chosen randomly. HY H 3. If parton j is produced in a secondary time-like branching, its energy E "zE as well as the H   energy of the second `daughtera E "(1!z)E are known from the previous branching, as HY   well as Q and Q } from the splittings jPk, l and jPk, l, which allows to determine H HY po , "!po , "po using Eq. (6.28) in the frame where the parent parton moves along the z-axis. H H , In some cases one gets p (0; then the current splitting of the parton j is rejected and its , evolution continues. The described leading order algorithm is known to be not accurate enough for secondary hadron production, in particular it gives too high multiplicities of secondaries in e>e\-annihilation. The method can be corrected if one takes into account the phenomenon of color coherence. The latter one appears if one considers some higher order corrections to the simplest leading logarithmic contributions, the latter ones being the basis for the usual Altarelli}Parisi evolution equations. In the corresponding treatment [172}177] } so-called modi"ed leading logarithmic approach } one essentially recovers the original scheme for the time-like parton cascading supplemented by the additional condition, the strict ordering of the emission angles in successive parton branchings. The appearance of the angular ordering can be explained in a qualitative way (see [178]): if

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Fig. 6.2. The three branching modes as explained in the text. The arrows indicate the variables to be determined.

Fig. 6.3. Angular ordering in successive branchings.

a transverse wavelength of an emitted gluon ( f or e in Fig. 6.3) is larger than the separation between the two, this gluon cannot see the color charge of the parent (l) but only the total (much smaller) charge of the two partons (k#l"j) and the radiation is suppressed. In a Monte-Carlo model this can be easily realized by imposing the angular ordering condition via rejection [179]. Thus, for each branching we check whether the angular ordering  ( is valid, where  is the angle of the    previous branching, and reject the current splitting otherwise.

7. Hadronization Till now, our discussion concerned exclusively partons, whereas `in the real worlda one observes "nally hadrons. It is the purpose of this section, to provide the link, i.e. to discuss how to calculate hadron production starting from partonic con"gurations, discussed in the previous sections. Hadron production is related to the structure of cut Pomerons. A cut Pomeron is in principle a sum over squared amplitudes of the type a#bP hadrons, integrated over phase space, with a and b being the nucleon constituents involved in the interaction. So far there was no need to talk about the details of the hadron production, which could be considered to be `integrated outa. In case of soft Pomerons, we used a parameterization of the whole object, based on general asymptotic considerations, which means that all the hadron production is hidden in the few parameters characterizing the soft Pomeron. In case of hard Pomerons, we discussed the explicit partonic structure of the corresponding diagram without talking about hadrons. This is justi"ed based on the assumption that summing over hadronic "nal states is identical to summing over partonic "nal states, both representing complete sets of states. But although our ignorance of

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hadronic states so far was well justi"ed, we "nally have to be speci"c about the hadronic structure of the cut Pomerons, because these are hadronic spectra which are measured experimentally, and not parton con"gurations. Lacking a rigorous theoretical treatment, we are going to use the same strategy as we used already for treating the soft Pomeron: we are going to present a `parameterizationa of the hadronic structure of the cut Pomerons, as simple as possible with no unnecessary details, in agreement with basic laws of physics and basic experimental observations. We do not claim at all to understand the microscopic mechanism, so our parameterization, called `string modela, should not be considered as a microscopic hadronization model. 7.1. Hadronic structure of cut Pomerons In order to develop our multiple scattering theory, we used a simple graphical representation of a cut Pomeron, namely a thick vertical line connecting the external legs representing nucleon components, as shown in Fig. 7.1. This simple diagram hides somewhat the fact that there is a complicated structure hidden in this Pomeron, and the purpose of this section is to discuss in particular the hadronic content of the Pomeron. Let us start our discussion with the soft Pomeron. Based on Veneziano's topological expansion [180}185], one may consider a soft Pomeron as a `cylindera, i.e. the sum of all possible QCD diagrams having a cylindrical topology, see Fig. 7.2. As discussed in detail in Section 2.2, the `nucleon componentsa mentioned earlier, representing the external legs of the diagram, are always quark}anti-quark pairs, indicated by a dashed line (anti-quark) and a full line (quark) in Fig. 7.2. Important for the discussion of particle production are of course cut diagrams, therefore we show in Fig. 7.2 a cut cylinder representing a cut Pomeron: the cut plane is shown as two vertical dotted lines. Let us consider the half-cylinder, for example, the one to the right of the cut, representing an inelastic amplitude. We can unfold this object in order to have a planar representation, as shown in Fig. 7.3. Here, the dotted vertical lines indicate the cuts of the previous "gure, and it is here where the hadronic "nal state hadrons appear. Lacking a theoretical understanding of this hadronic structure, we simply apply a phenomenological procedure, essentially a parameterization. We require the method to be as simple as possible, with a minimum of necessary parameters. A solution coming close to these demands is the so-called string model: each cut line is identi"ed with a classical relativistic string, a Lorentz invariant string breaking procedure provides the transformation into a hadronic "nal state, see Fig. 7.4. The phenomenological microscopic picture which stays behind this procedure was discussed in a number of reviews [73,74,186]: the string end-point partons resulted from the interaction appear

Fig. 7.1. Symbol representing a cut Pomeron.

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Fig. 7.2. Cut soft Pomeron represented as a cut cylinder. The grey areas represent unresolved partons. Fig. 7.3. Planar representation of a half-cylinder obtained from cutting a cylinder diagram (see Fig. 7.2).

Fig. 7.4. The string model: each cut line (dotted vertical lines) represents a string, which decays into "nal state hadrons (circles). Fig. 7.5. A simple diagram contributing to the semi-hard Pomeron of the `sea}seaa type.

to be connected by a color "eld. With the partons #ying apart, this color "eld is stretched into a tube, which "nally breaks up giving rise to the production of hadrons and to the neutralization of the color "eld. The relation between planar diagrams and strings has also been discussed in several publications [187}189]. We now consider a semi-hard Pomeron of the `sea}seaa type, where we have a hard pQCD process in the middle and a soft evolution at the end, see Fig. 7.5. We generalize the picture introduced above for the soft Pomeron. Again, we assume a cylindrical structure. For the example of Fig. 7.5, we have the picture shown in Fig. 7.6: the shaded areas on the cylinder ends represent the soft Pomerons, whereas in the middle part we draw explicitly the gluon lines on the cylinder surface. We apply the same procedure as for the soft Pomeron: we cut the diagram and present a half-cylinder in a planar fashion, see Fig. 7.6. We observe one di!erence compared to the soft case: there are three partons (dots) on each cut line: apart from the quark and the anti-quark at the end, we have a gluon in the middle. We again apply the string picture, but here we identify a cut line with

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Fig. 7.6. Cylindrical representation of a contribution to the semi-hard Pomeron (left "gure) and planar diagram representing the corresponding half-cylinder (right "gure). Fig. 7.7. The `kinkya string model: the cut line (vertical dotted line) corresponds to a kinky string, which decays into hadrons (circles).

a so-called kinky string, where the internal gluons correspond to internal kinks [186]. The underlying microscopic picture will be presented by three color-connected partons } the gluon connected by the color "eld to the quark and to the anti-quark. The string model provides then a `parameterizationa of hadron production, see Fig. 7.7. The procedure described above can be easily generalized to the case of complicated parton ladders involving many gluons and quark}anti-quark pairs. One should note that the treatment of semi-hard Pomerons is just a straightforward generalization of the string model for soft Pomerons, or one might see it the other way round: the soft string model is a natural limiting case of the kinky string procedure for semi-hard Pomerons. We now need to discuss Pomerons of valence type. In case of `valence}valencea the "rst partons of the parton ladder are valence quarks, there is no soft Pomeron between the parton ladder and the nucleon. The nucleon components representing the external legs are, as usual, quark}antiquark pairs, but the anti-quark plays in fact just the role of a spectator. The simplest possible interaction is the exchange of two gluons, as shown in Fig. 7.8. We follow the scheme used for soft Pomerons and `sea}seaa type semi-hard Pomerons: we draw the diagram on a cylinder, see Fig. 7.9. There is no soft region, the gluons couple directly to the external partons. We cut the cylinder, one gluon being to the right and one gluon to the left of the cut, and then we consider the corresponding half-cylinder presented in a planar fashion, see Fig. 7.9 (right). Here, we have only internal gluons, on the cut line we observe just the external partons, the corresponding string is therefore just an ordinary quark}anti-quark string without internal kinks, as in the case of the soft Pomerons. We apply the usual string breaking procedure to obtain hadrons, see Fig. 7.10. Let us consider a more complicated valence-type diagram, as shown in Fig. 7.11. It is again a contribution to the Pomeron of the `valence}valencea type: the external partons of the parton ladders are the valence quarks of the nucleons. In contrast to the previous example, we have here an emission of s-channel gluons, traversing the cut. As usual, we present the diagram on the cylinder, as shown in Fig. 7.12, where we also show the corresponding planar half-cylinder. In addition to internal gluons, we now observe also external ones, presented as dots on the cut line. As usual, we

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Fig. 7.8. The simplest contribution to the `valence}valencea Pomeron. Fig. 7.9. Cylindrical representation of the diagram of Fig. 7.8 (left "gure) and planar diagram representing the corresponding half-cylinder (right "gure).

Fig. 7.10. The string model: the cut line (vertical dotted line) corresponds to a string, which decays into hadrons (circles). Fig. 7.11. A more complicated contribution to the valence type Pomeron.

identify the cut line with a relativistic kinky string, where each external (s-channel) gluon represents a kink. We then employ the usual string procedure to produce hadrons, as sketched in Fig. 7.13. The general procedure should be clear from the above examples: in any case, no matter what type of Pomeron, we have the following procedure: 1. 2. 3. 4. 5.

drawing of a cylinder diagram; cutting the cylinder; planar presentation of the half-cylinder; identi"cation of a cut line with a kinky string; kinky string hadronization.

The last point, the string hadronization procedure, will be discussed in detail in the following. This work is basically inspired by [190}195] and further developed in [196]. The main di!erences to

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Fig. 7.12. Cylindrical representation of the diagram of Fig. 7.11 (left "gure) and planar diagram representing the corresponding half-cylinder (right "gure). Fig. 7.13. The string model: the cut line (vertical dotted line) corresponds to a string, which decays into hadrons (circles).

[195] are that hadrons are directly obtained from strings instead from low mass clusters, and an intrinsic transverse momentum is added to a string break-up. The main di!erence to the Lund model is to use the area law instead of a fragmentation function. 7.2. Lagrange formalism for strings In this section, we discuss the theory of classical relativistic strings [197}199]. A string can be considered as a point particle with one additional space-like dimension. The trajectory in Minkowski space depends on two parameters: xI"xI(
, ),
"02 , (7.1) with
being a space-like and  a time-like parameter. In order to obtain the equation of motion, we need a Lagrangian. It is obtained by demanding the invariance of the trajectory with respect to gauge transformations of the parameters
and . This way we "nd [1] the Lagrangian of Nambu-Goto: L"!((xx )!xx  ,

(7.2)

with x I"dxI/d, xI"dxI/d
and  being the energy density or string tension. With this Lagrangian we write down the action

 

S"




d


O

d L ,

(7.3)

O  which leads to the Euler}Lagrange equation: R RL R RL # "0 , R
Rx d Rx I I with the initial conditions RL "0,
"0,  , Rx I

(7.4)

(7.5)

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since we have x"0 for " and " . This equation can be solved most easily by a partial   gauge "xing. We have this freedom, since the result is independent on the choice of the parameters. This is done indirectly by imposing the following conditions: x #x"0, x x"0 .

(7.6)

The Euler}Lagrange equation gives us a simple solution, the wave equation: Rx Rx I ! I "0 , R R


(7.7)

with the following boundary conditions: Rx I "0,
"0,  . R


(7.8)

The total momentum of a string is given by [1]



R¸ R¸ d
# d (7.9) Rx Rx ! I I with C being a curve between the two ends of the string (
"0 and
"). This gives for (7.6) and for d"0 pI "!  



pI "  




x I d
.

(7.10)

 We still have to "x completely the gauge since it has been "xed partially only. This can be done with the following conditions for the parameter : nIx "  . (7.11) I Di!erent choices for n and are possible, like n"(1,!1, 0, 0) which is called the transverse gauge. We will use n"(1, 0, 0, 0) which leads to "E/ and another choice "E/ will identify  with the time x , whereas E"
x d
is the total energy of the string. We de"ne `string unitsa    via "1;
and  have thereby the dimension of energy and "E. In `ordinarya units, one has " GeV/fm, with  being approximately 1, so a length of 1 GeV corresponds to 1 fm/+1 fm. The solution of a wave equation is a function which depends on the sum or the di!erence of the two parameters
and . As the second derivative shows up, we have two degrees of freedom to impose the initial conditions on the space-like extension and the speed of the string at "0. One can easily verify that the following Ansatz [193,195] ful"lls the wave equation (7.7):







N>O 1 (7.12) gI( ) d , xI(
, )" f I(
#)#f I(
!)# 2 N\O f I(
)"xI(
, ) , (7.13) O gI(
)"x I(
, ) . (7.14) O We identify the function f (
) with the initial spatial extension and g(
) with the initial speed of the string at the time "0.

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We will consider here a special class of strings, namely those with f"0 (initially point-like) and with a piecewise constant function g, g(
)"v

for E 4
4E , 14k4n (7.15) I I\ I with some integer n. The set E  is a partition of the
-range [0, E], I 0"E (E (2(E (E "E , (7.16)   L\ L and v  represents n constant 4-vectors. Such strings are called kinky strings, with n being the I number of kinks, and the n vectors v being called kink velocities. I In order to use Eq. (7.12), we have to extend the function g beyond the physical range between 0 and . This can be done by using the boundary conditions, which gives g()"g(!) ,

(7.17)

g(#2)"g() ,

(7.18)

So g is a symmetric periodic function, with the period 2. This de"nes g everywhere, and the Eq. (7.12) is the complete solution of the string equation, expressed in terms of the initial condition g ( f is taken to be zero). In case of kinky strings the latter is expressed in terms of the kink velocities v  and the energy partition E . I I 7.3. Identifying partons and kinks We discussed earlier that a cut Pomeron may be identi"ed with two sequences of partons of the type q!g!g!2!g!q ,

(7.19)

representing all the partons on a cut line. We identify such a sequence with a kinky string, by requiring parton"kink ,

(7.20)

which means that we identify the partons of the above sequence with the kinks of a kinky string, such that the partition of the energy is given by the parton energies, E "energy of parton k I and the kink velocities are just the parton velocities,

(7.21)

v "momentum of parton k/E . (7.22) I I We consider massless partons, so that the energy is equal to the absolute value of the parton momentum. Fig. 7.14 shows as an example the evolution of a kinky string representing three partons: a quark, an anti-quark, and a gluon, as a function of the time . One sees that the partons start to move along their original direction with the speed of light. After some time which corresponds to their energy they take the direction of the gluon. One could say that they lose energy to the string itself. The gluon loses energy in two directions, to the quark and to the anti-quark and therefore in half the time. The ends of the string move continuously with the speed of each of the partons until the whole string is contracted again in one point. The cycle starts over.

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Fig. 7.14. Movement of a string with 3 partons on the z}x plane: The partons have momenta of p "(p , p , p )" O V W X (0, 0,!2) GeV/c, p "(2, 0, 0) GeV/c and p  "(0, 0, 3) GeV/c. The "rst half-cycle is "nished after "7 GeV. E O

Another example is shown in Fig. 7.15, where realistic partons coming from a simulation of a e>e\ annihilation process at 14 GeV c.m.s. energy are considered. We will see later how to generate these partons. We observe 6 partons, 2 quarks and 4 gluons, symbolically displayed in the "rst sub-"gure. As the total energy is 14 GeV the cycle has a periodicity of 28 GeV. But one sees that the perturbative gluons play an important role in the beginning of the movement, and later from 2 GeV on, the longitudinal character dominates. As we will see later, a string breaks typically after 1 GeV/ which gives much importance to the perturbative gluons. 7.4. Momentum bands in parameter space As we will see later, it is not necessary for a fragmentation model to know the spatial extension at each instant. Therefore, we concentrate on a description in momentum space which simpli"es the

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Fig. 7.15. Partons in an e>e\ annihilation event with (s"14 GeV in the z}x plane. The "rst "gure shows the momenta in the p }p plane. X V

model even more. By using formula (7.12) we can express the derivatives of x(
, ) in terms of the initial conditions g(
) as (7.23) x (
, )"[g(
#)#g(
!)] ,  x(
, )"[g(
#)!g(
!)] . (7.24)  Since the function g is stepwise constant, we easily identify regions in the parameter space (
, ), where g(
#) is constant or where g(
!) is constant, as shown in Figs. 7.16 and 7.17. These regions are called momentum bands, more precisely R-bands and L-bands, being of great importance for the string breaking. If we overlay the two "gures of 7.16, 7.17 we get Fig. 7.18, which allows us to identify regions, where g(
#) and g(
!) are constant at the same time, namely the intersections of R-bands and L-bands. In these areas x and x are constant, given as (7.25) x (
, )"[v\#v>] ,  (7.26) x(
, )"[v\!v>] ,  with v> and v\ being the velocities of the partons corresponding to the two intersecting bands. Rather than considering a
-range between !R and #R, one may simply consider the physical range from 0 to , and construct the bands via re#ection. As an example, let us follow the L-band corresponding to the parton i, starting at "0. With increasing  one reaches at some stage the border
"0. Here, we have an intersection with the R-band, corresponding to the same

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175

Fig. 7.16. Construction of bands where g(
!) is constant, being referred to as R-bands (`right moving bandsa). Fig. 7.17. Construction of bands where g(
#) is constant, being referred to as L-bands (`left moving bandsa).

Fig. 7.18. The superposition of bands where g(
#) and g(
!) are constant (see Figs. 7.16 and 7.17) gives regions of constant x and x .

parton i, coming from the unphysical region
(0. We now follow this R-band, which corresponds to a re#ection of the above-mentioned L-band, till we hit the border
",2. In the regions where g(
#) and g(
!) have the same value, corresponding to collinear partons or to an overlap of the momentum bands of one and the same parton, one "nds x"0, i.e. there is no spatial extension in the dependence of the parameter
. Therefore the coordinates xI stay unchanged and we recover the speed of the original partons. In particular, this is the case for the whole string at "0, due to f"0. With the string evolving in time, more and more bands of non-collinear partons overlap, which gives xO0; the string is extending as we have seen in Fig. 7.14 until "7. 7.5. Area law In order to consider string breaking, we are going to extend the model in a covariant fashion. We use the method proposed by Artru and Menessier [200], which is based on a simple extension of

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the decay law of unstable particles, where the probability dP to decay within a time interval dt is given as dP" dt ,

(7.27)

with some decay constant . For strings, we use the same formula by replacing the proper time by proper surface in Minkowski space, dP" dA .

(7.28)

By construction, this method is covariant. Since we work in parameter space it is useful to express this dependence as a function of
and , dA"((x x)!xx  d
d .

(7.29)

By using the expressions for x and x and x x"0 and g"0, we "nd (7.30) dA"(!(!2g(
#)g(
!))(2g(
#)g(
!)) d
d   (7.31) "[g(
#)g(
!)] d
d  (7.32) "(1!cos ) d
d ,  with  being the angle between the partons. Consequently, a string cannot break at a point where the momentum bands of the same parton overlap, because in this case the angle  is zero, which leads to dA"0. The maximal contribution is obtained for partons moving in opposite directions. We still have to de"ne how a string breaks and how the sub-strings evolve. At each instant, one knows exactly the momenta of the string by Eqs. (7.25) and (7.26). The con"guration of g(
#) and g(
!) at the time  of the break point is used as initial condition for the two substrings. The  function g(
#) is cut into two pieces between 0 and
and between
and . The two resulting   functions are continued beyond their physical ranges [0,
] and [
, ] by taking them to be   symmetric and periodic with periods 2
and 2(!
). Fig. 7.19 shows this for a breaking at   (
,  ) and a second break point at (
,  ). In principle, the cycle starts over with the two     sub-strings breaking each until the resulting pieces are light enough to form hadrons. However, it is easier to look for many break-points at once. If they are space-like separated, they do not interfere

Fig. 7.19. The breaking of a string. The functions g are found by imposing the symmetry and periodicity conditions.

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177

with each other. For the coordinates in the parameter space this translates into the condition 
!
' !  .    

(7.33)

7.6. Generating break points Having assumed that string breaking occurs according to the area law, dP" dA ,

(7.34)

we now need an algorithm to accomplish this in the framework of the Monte-Carlo method. The most simple way is to sub-divide a given surface into su$ciently small pieces and then to decide according to formula (7.34) if there is a break point or not. This is what we refer to as the naive method, which is of course not e$cient. We will therefore construct another algorithm (the direct one) which is based on P (A)"e\H (7.35)  being the probability of having no break point within the area A. One generates surfaces A , A ,2   according to P (A) as  A "!log(r )/ , (7.36) G G with random numbers r between 0 and 1. The formula does not say anything about the form of the G surfaces A . Actually, several choices are possible as long as they do not violate causality. In the G case of a simple string without any gluons, it is easiest to place the surfaces A from left to right such G that the break points P are the left upper corners of A , as shown in Fig. 7.20: one "rst takes A , G G>  which de"nes the line ¸ . The "rst break point P is generated randomly on this line. The next   surface A has to be placed in a way that does not violate causality. The "rst break point is  therefore used as a constraint for the next one, etc. Finally, if the last surface obtained is too large to be placed on the rest of the string, the procedure is "nished. The advantage of this method is that no

Fig. 7.20. The direct method of searching break points (see text). Fig. 7.21. The area S


 

in parameter (
, ) space, before and after the rede"nition of the outer bands.

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break-points are rejected because the causality principle is obeyed constantly throughout the whole procedure. We generalize the method to work for any number of perturbative gluons in the following way. Since the elementary invariant area dA is proportional to the scalar product of the momenta of two partons, we can easily calculate the area A corresponding to a sub-region S of the (
, )-space, GH GH representing the intersection of the momentum bands of the partons i and j. We "nd



A " GH

(g(
#)g(
!)) d
d"p ) p ,   G H

(7.37)

1 with p and p being the 4-momenta of the two partons. We now construct the break points in the G H parameter space rather than in Minkowski space. One "rst de"nes the area in the parameter space of allowed breakpoints as GH

S "S , (7.38)
  GH with the indices running as indicated in Fig. 7.21. To obtain a unique way of counting the regions, we mark bands which come from a left-moving band at "0 with A. We further observe that the outer bands 1 and 1 as well as 5 and 5, which come from the (anti-)quark momenta, are neighboring. It is therefore useful to rede"ne them as one band 1 and 5 (with double momentum), see Fig. 7.21. For each of these sub-areas S the corresponding area in Minkowski space A is GH GH known ("p p /4). One then generates areas A , A , A ,2 (in Minkowski space) according to G H    Eq. (7.36), and places the corresponding areas S , S , S ,2 (in parameter space) into S from   
  left to right such that the break points P are the left upper corners of S . G G> Let us consider an example of "ve partons (1, 2, 3, 4, 5), see Fig. 7.22. Suppose that we have sampled a surface A . If it is smaller than the "rst region A , we determine S "S ) A /A  Y  Y  Y and we place S into the left side of S and generate the break point P randomly on the right  Y  upper border of S , see Fig. 7.22. If A is greater than A , we subtract A from A : Y  Y Y  A "A !A . (7.39)   Y In the case of the sum of the three areas A "A #A #A being greater than A , the "rst Q YY Y Y  coordinate x of the break point P (see Fig. 7.23) is determined by  x"A /A . (7.40)  Q Otherwise we continue the procedure correspondingly. The y coordinate is determined as



A A Q if 0(r( YY (region S ) , YY A A YY Q A A A #A A YY (r( YY Y (region S ) , Q (7.41) if y" r! YY Y A A A A Q Q Q Y A A #A A #A Q Y YY Y (r(1 r! YY if (region S ) , Y A A A Y Q Q with r being a random number between 0 and 1. This means that after having determined in which of the regions we "nd the break point, it is placed randomly on the world-line which points to the future. After having obtained the break point P we continue the procedure in the same way by  r






 


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179

Fig. 7.22. Placing S into S in case of A (A .  Y  Y Fig. 7.23. The determination of the break point for the case of a string with kinks. The regions are weighted according to the scalar product of their momenta.

Fig. 7.24. Distributions in (xy), x,  log(x/y), and the multiplicity n of break points. The full line represents the naive  method, the dashed one the direct one. The coordinates of the break points are as de"ned in Fig. 7.20.

obeying to the principle of causality. The area to sweep over is then limited by the "rst break point as shown in Fig. 7.23. In Fig. 7.24, we apply our hadronization procedure, referred to as direct method, as discussed above, to calculate the distributions of xy, x, " log(x/y), and the multiplicity n of break points  for a quark}anti-quark string of E "E  "8 GeV. We compare our results with the the naive O O method, where the area of the string is divided into small elements A"8 ) 8 GeV/N, with

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N su$ciently large to not change the results any more. In each of these elements, a break point is found with the probability A. The points which are in the future of another one are rejected. The latter method is a literal realization of the area law. As one can easily see in Fig. 7.24, the two methods agree within statistical errors. 7.7. From string fragments to hadrons So far, we discussed how to break a string into small pieces, i.e. string fragments with invariant masses between 0 and about two GeV. In order to identify string fragments and hadrons, we "rst have to de"ne the #avors ("quark content) of the fragments, and then we have to discuss the question of fragment masses. 7.7.1. Flavors of string fragments A string as a whole has some #avor, carried by the partons at its two extremities. Additional #avor is created (by de"nition) at each break point in the form of a quark}anti-quark or a diquark}anti-diquark pair of a certain #avor. The corresponding probabilities are free parameters of the model. In case of quark}anti-quark formation, we introduce the parameter p , which  gives the probability of #avor u or d. The probability to get an s-quark is therefore 1!2p which  is smaller than p because of the larger mass of the s quark. For diquark}anti-diquark production,  we introduce the corresponding probability p .    7.7.2. Masses of string fragments In the following, we show how to determine the masses of string fragments, characterized by break points in the parameter space. Fig. 7.25 shows an example of two break points for a string with 3 inner kinks. The momentum bands and the regions of the their overlaps are shown: in case of the inner bands, we have three R-bands (2, 3, 4) and three L-bands (6, 7, 8). The bands at the extremities play a special role, since we may have the corresponding R- and L-band as just one band, due to the fact that one of the bands is re#ected immediately. So, we consider two `double bandsa (1 and 5).

Fig. 7.25. How to calculate the mass of a sub-string.

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181

The string momentum is given as



[x d
#x d] , (7.42) ! where C is an arbitrary curve from one border (
"0) to the other (
") in the parameter space. This leads to p "  

1 p "   2



1 (g(
#)#g(
!)) d
# 2



(g(
#)!g(
!)) d .

(7.43)

! ! The momenta of the bands are by de"nition



 g(
!) d
! g(
!) d if R-band , 
 G (7.44) p " 
 G G g(
#) d
# g(
#) d if L-band ,  
 G 
 G where one integrates along an arbitrary curve from one border of the band to the other. An important property: an integration path parallel to a band provides zero contribution. One has to pay attention for the bands at the extremities: integrating only along "0 represents only half the band. We have
p "p . G   G The momenta of the bands are related to the corresponding parton momenta as

(7.45)



p if inner band , (7.46) p "    G p if outer band ,   which one veri"es easily by expressing g in terms of the parton momenta. The di!erence between inner and outer bands is due to the fact that the outer ones (at the extremities) represent in reality two bands. For the example of Fig. 7.25, we have p "p , p "p "p  , p "p "p  , p "p "p  ,    E    E  O    E Summing over the bands, we get

p "p  .  O

(7.47)


p "p #p  #p  #p  #p  "p , (7.48) G O E E E O   G which is the total momentum of the string. For a fragment of the string, the momentum is given as 1 p "   2



1 (g(
#)#g(
!)) d
# 2



(g(
#)!g(
!)) d , (7.49) !Y !Y where the path of the integration C is an arbitrary curve between two breakpoints, or between one break point and a boundary. One may write p "p #p ,   0 *

(7.50)

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with 1 p " 0 2 1 p " * 2

 

!Y

1 g(
!) d
! 2 1 g(
#) d
# 2

 

g(
!) d ,

(7.51)

!Y

g(
#) d , (7.52) !Y !Y where p and p represent sums of momenta of R-bands and L-bands. For the example of Fig. 7.25, 0 * we can choose the path (1PB), (BP2) for the string fragment between the break points 1 and 2. Since the "rst path is parallel to all L-bands (only R-bands contribute) and the second one is parallel to all R-bands (only L-bands contribute), we "nd p "p #p   }  

(7.53)

with p "x p #p #(1!x )p , (7.54)       p "(1!y )pI #pI #y pI , (7.55)       where the factors x , (1!x ), y , (1!y ) represent the fact that the bands at the extremities are     only partially integrated over. The other string fragments are treated correspondingly. For the left string fragment, we may chose the integration path (AP1), for the right one (2PC). So we "nd for the three string fragments (referred to as a, b, c) of Fig. 7.25 the momenta p "y p #p #(1!x )p ,      p "x p #p #(1!x )p #y p #p #(1!y )p ,
          p "x p #p #(1!y )p .       It is easy to verify that the sum of the three sub-strings gives the total momentum of the  p #p #p "
p "P .
 G   G The mass squared of the string fragments is "nally given as m "p ,     for example

(7.56) (7.57) (7.58) string, (7.59)

(7.60)

m"2(p p !x p p #y (p p #p p )!x y p p ) , (7.61)               where we took advantage of the light-cone character of the momenta of the bands (p "0). G 7.7.3. Determination of hadrons So far, we have determined the #avor f and the mass m of each string fragment. In order to identify string fragments with hadrons, we construct a mass table, which de"nes the hadron type as a function of the mass and the #avor of the fragment. For a given #avor f of the fragment, we introduce a sequence mD (mD (2 of masses, such that in case of a fragment mass being within   an interval [mD , mD], one assigns a certain hadron h . The masses mD are determined by the G\ G G G

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masses of the neighboring particles. So we decide for a u!u pair to be a pion if its mass is between 0 and (140#770)/2"455 MeV. So the particle masses give a natural parameterization. This works, however, only up to strange #avor. For charm and bottom #avor we choose with a fraction 1 : 3 between pseudo-scalar and vector-mesons. 7.7.4. Mass corrections An unrealistic feature of our approach, so far, is the fact that stable particles are in general o!-mass-shell. In order to correct this, we employ a slight modi"cation of the break point such that the on-shell mass is imposed. Let us again consider the example of Fig. 7.25. For a given mass, the parameters x and  y describe hyperbolas in the regions of overlapping bands (di!erent ones in di!erent regions).  Fig. 7.26 shows for our example some curves of constant mass for the left sub-string (between the left side and the "rst break point). In the same way we "nd hyperbolas of constant mass for the right sub-string (between the break points 1 and 2). If two neighboring substrings are stable particles, one needs to impose on-shell masses to both fragments, which amounts to "nding the intersection of the two corresponding hyperbolas. If one has to modify the break point according to only one mass condition, with the mass of the second sub-string being still large enough not to represent a stable hadron, a possible break point must lie on the corresponding hyperbola. To completely determine the point, we need a second condition. Apart from the squared mass, another Lorentz invariant variable available is the squared proper time of the break point, de"ned as "(x(
, )) . (7.62)
 
  So the second condition is the requirement that the proper time of the new break point should coincide with the proper time of the original one. To calculate the proper time, we use Eq. (7.12), to obtain






1 N>O  (7.63) g( ) d . 2 N\O The integration is done in the same way as for the masses, it is a summation of the momenta of the bands or of fractions of them. In the case of our example, we "nd in the region (1, 7) "

(7.64) "(x p #p #y p )       " (x p p #y p p #x y p p ) , (7.65)            which represents again a hyperbola in the parameter space, as shown in Fig. 7.27. So "nding a new break point amounts to "nding the intersection of the two curves (hyperbolas) representing constant mass and proper time. 7.8. Transverse momentum Inspired by the uncertainty principle, a transverse momentum is generated at each breaking, which means that 4-vectors p and !p are assigned to the string ends at both sides of the break , , point. First we choose the absolute value k"po  of the transverse momentum according to the ,

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Fig. 7.26. The hyperbolas of constant mass for the two sub-strings if one moves the break-point on the left side. The solutions of the mass conditions are the crossing of the hyperbolas. Fig. 7.27. Lines of constant masses (vertical lines) and constant proper time (horizontal lines).

distribution  , , f (k)Je\IN


(7.66)

with the parameter p
  to be "xed. We require p to be orthogonal to the momenta p and p of , , G H the two intersecting bands where the break point is located. So we have p p "0 , (7.67) , G p p "0 , (7.68) , H p "!k . (7.69) , Technically this is most easily done, if we perform a Lorentz-boost into the center of mass system of the two momenta p and p followed by a rotation such that p is oriented along the z-axis. One G H G de"nes a vector p , having the components , (p ) "0 , (7.70) ,M (p ) "k cos , (7.71) ,V (p ) "k sin , (7.72) ,W (p ) "0 , (7.73) ,X

being a random angle between 0 and 2. The transformation back to the original system gives the 4-vector p . , This operation modi"es, however, the mass of the string. In order to account for this, we consider the transverse momentum as an additional band of the string. It is treated in the same way as the others with the only exception that we do not look for break points in this region. From our example of Fig. 7.25, we obtain for the left string fragment the momentum p "p #y p #p #(1!x )p , ,     

(7.74)

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rather than Eq. (7.56). The modi"cation of the coe$cients of the corresponding hyperbola for the mass correction procedure is obvious. In the case where we have to pass to another region to "nd a modi"ed break point for the mass correction, we have to perform a rotation such that the vector p is transverse to the two momenta of the new region. , 7.9. Fragmentation algorithm In the following, we describe the fragmentation algorithm which is used to obtain a complete set of particles from one string. 1. For a given string, we look for break points. Let n be the number of break points. 2. For each break point, we generate a #avor and a transverse momentum. 3. We choose one break point by random and calculate the masses of the two neighboring substrings. 4. If there is at least one mass in the region of the resonances, we try to modify the break point as discussed to get exactly this mass. If this is not possible, we reject (delete) this break point and go to step 3. 5. If the mass of a sub-string is bigger than the upper limit in the mass table, we fragment this sub-string (go to step 1). In this way, we can deal in an elegant manner with the kinematical constraints. Often, break points are rejected when a sampled transverse momentum is too high, which results in a negative mass squared for a "nal particle. In this case we look for another break point with another transverse momentum until a valid con"guration is found.

8. Parameters We discuss in this section the parameters of the model, how they are determined, and also their values. Parameter "xing is done in a systematic way, starting with the hadronization parameters and the ones determining the time-like cascade, before considering parton}parton-scattering and hadron}hadron scattering. 8.1. Hadronization The breaking probability p is the essential parameter in the hadronization model to
  determine the multiplicity and the form of the rapidity distribution. For p, pp, pA, AA, we use a "xed value, "tted to reproduce the pion multiplicity in p scattering. For e>e\ annihilation, this parameter is considered to be Q dependent as 7.16 GeV 111.1 GeV 855 GeV ! # p (Q)"0.14#
  Q Q Q

(8.1)

in the region 144Q491.2 GeV, and to be constant outside this interval. Fig. 8.1 shows the total multiplicity of charged particles in e>e\ annihilation as a function of energy. The solid line corresponds to the Q dependent p , the dashed line is for p "0.21 (the value for
 
 

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Fig. 8.1. Charged particles multiplicity as a function of energy. The full line is for p parameterized, the dashed one for
  p "0.21.
  Fig. 8.2. Rapidity distributions for the parameters p "0.4 ( full line) and p "0.21 (dashed).
 
 

Q"91 GeV. In Fig. 8.2, we show the corresponding rapidity distributions. The e!ect of making p Q-dependent shows up more in the shape of the rapidity distribution, not in the total
  multiplicity. The parameter p
 , which determines the transverse momenta of the partons at , a string break, is determined by investigating transverse momentum spectra of charged particles. A value of 0.50 GeV provides the best "t to data concerning p, pp, pA, AA, whereas for e>e\ a value of 0.35 GeV is more favorable. The parameter p a!ects strongly kaon production, we use  p "0.44 adjusted to the multiplicity of kaons. Baryon production is determined by p , we     use 0.08 adjusted to proton production in e>e\-annihilation. In Table 8.1, we give the complete list of hadronization parameters, with their default values. So we use absolutely the same parameters for all the reactions p, pp, pA, AA. A perfect "t for e>e\ requires a modi"cation of two parameters, p and p
 .
  , 8.2. Time-like cascade Let us discuss the parameters which determine the time-like cascade, all "xed via studying e>e\ annihilation. For the pQCD parameter, we use the usual leading order value "0.2 GeV. We have also a technical parameter q , which determines the lower mass limit of partons in the   time-like cascade. In Fig. 8.3, we analyze how certain spectra depend on this parameter. We show rapidity, transverse momentum and multiplicity distributions for partons and for charged particles for an e>e\ annihilation at 34 GeV. We show results for di!erent values of q , namely 0.25, 1.0   and 4.0 GeV, which means a lower mass limit of 1, 2 or 4 GeV, respectively. One sees that only parton distributions are sensitive to the choice of this parameter, whereas the corresponding charged particle spectra exhibit rather weak dependence on it. This can be explained from the fact that decreasing q mainly results in production of additional partons with transverse momenta   p &q and such soft collinear partons hardly a!ect the fragmentation procedure, which is in this ,   sense `infrared stablea. The number N of active #avors is taken to be 5 for e>e\ annihilation, D

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Table 8.1 Hadronization parameters Name

Value

Meaning

p

Eq. (8.1) for e>e\ 0.40 for p, pp, pA, AA 0.35 GeV for e>e\ 0.50 GeV for p, pp, pA, AA 0.44 0.08

Break probability


 

p
  , p  p   

Mean transverse momentum at break Probability of u}u or d}d Probability of qq}q q

whereas for p, pp, pA, AA we use for the moment N "3. In Table 8.2 we show the cascade D parameters and their default values. 8.3. Parton}parton scattering There is "rst of all the parameter Q which de"nes the borderline between soft and hard  processes, where one has to choose a reasonable value (say between 1 and 2 GeV). Then we have a couple of parameters characterizing the soft Pomeron: the intercept (0) and  the slope  of the Pomeron trajectory, the vertex value  and the slope R for the      Pomeron}parton coupling, and the characteristic hadronic mass scale s . We have two parameters,   and w , characterizing the coupling between the soft Pomeron and the parton ladder. E   Whereas for s one just chooses some `reasonablea value and R is taken to be zero, one "xes the    other parameters by trying to get a good "t for the total cross section and the slope parameter for proton}proton scattering as a function of the energy as well as the structure function F of deep  inelastic lepton}proton scattering. Concerning the hard scattering part, the resolutions scale p and the K-factor K are "xed such , that the standard parton evolution is reproduced. Finally, we have the triple Pomeron coupling weight r / , which is "xed by as well checking the  energy dependence of the proton}proton total cross section. The values of these parameters are shown in Table 8.3. 8.4. Hadron}hadron scattering Let us "rst discuss the parameters related to the partonic wave function of the hadron h (for numerical applications we only consider nucleons: h"N). The transverse momentum distribution is characterized by the hadronic Regge radius squared R , the longitudinal momentum distribu, tions are given in terms of two exponents, , and . The latter one is taken to be independent     of the hadron type as 2 1 (0)!1, with the usual Reggeon intercept 1 (0)"1/2. The parameter R also a!ects the proton}proton total cross sections, whereas , can be determined by ,   investigating baryon spectra (but it also in#uences the total cross section). There are several more parameters, which have not been mentioned so far: the remnant excitation probability p and the exponent

, which gives a remnant mass      

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Fig. 8.3. Rapidity, transverse momentum and multiplicity distributions for partons (left column) and for charged particles (right column). The curves correspond to the parameters q "0.25, 1.0, 2.0 ( full, dashed, dotted). Only parton   distributions exhibit strong sensitivity to the value of q .  

distribution as (M)\?   .

(8.2)

The minimum string mass m assures that Pomerons with string masses less than this    minimal mass are ignored. The partons de"ning the string ends are assumed to have transverse

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Table 8.2 Time-like cascade parameters Name

Value

Meaning

q   N D

0.2 GeV 0.25 GeV 5 for e>e\ 3 for p, pp, pA, AA

pQCD parameter Transverse momentum cuto! Active #avors

Table 8.3 Parton}parton scattering parameters Name

Value

Meaning

Q  s 

(0) 

     R    E w   p , K r / 

1.5 GeV 1 GeV 1.054 0.21 GeV\ 1.11 GeV\ 0 0.5 0.15 0.25 GeV 1.5 0.0096 GeV\

Soft virtuality cuto! Soft mass scale Pomeron intercept Pomeron slope Pomeron}parton coupling vertex Pomeron}parton coupling slope Pomeron}ladder coupling parameter Pomeron}ladder coupling parameter Resolutions scale K-factor Triple Pomeron coupling constant

Table 8.4 Hadron}hadron scattering parameters (for the case of nucleons) Name

Value

Meaning

R ,

 

,   p   

   m    p ,1# p , 

2 GeV\ 0 1.5 0.45 1.4 0.29 GeV 0.21 GeV 0.35 GeV

Parton}hadron coupling slope Participant exponent Remnant exponent Remnant excitation probability Remnant excitation exponent Minimal string mass Mean p for string ends , Mean p for di!ractive scattering ,

momenta according a Gaussian distribution with a mean value p . Di!ractive scattering is ,1# assumed to transfer transverse momentum according to a Gaussian distribution, with a mean value p . The parameter m is taken to be just slightly bigger than two times the pion mass to ,     allow at least string fragmentation into two pions. The other parameters can be "xed by comparing with experimental inclusive spectra. In Table 8.4, we show the numerical values of the parameters.

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9. Testing time-like cascade and hadronization: electron}positron annihilation Electron}positron annihilation is the simplest possible system to test the time-like cascade as well as the model of fragmentation, since the decay of a virtual photon in electron}positron annihilation gives a quark and an anti-quark, both emitting a cascade of time-like partons, which "nally hadronize. Electron}positron annihilation is therefore used to test both the time-like cascade and the hadronization model, and in particular to "x parameters. The simulation of an electron}positron annihilation event can be divided into three di!erent stages: 1. The annihilation into a virtual photon or a Z boson and its subsequent decay into a quark}anti-quark pair (the basic diagram). 2. The evolution of the quark and the anti-quark into on-shell partons by radiation of perturbative partons (time-like cascade). 3. The transition of the partonic system into hadrons via a fragmentation model (hadronization). These stages are discussed in the following sections. After having described the three stages of electron}positron annihilation, we will be able to test the model against numerous data available. We will show comparisons with experimental results at low energies at PETRA (DESY), by the TASSO collaboration [201]. The center-of-mass energies are 14, 22 and 34 GeV. Higher energies are reached at LEP, where we compare especially with results for 91.2 GeV, the Z mass, where a big number of events has been measured. By comparing with data, we will be able to "x the essential parameters of the hadronization model, namely p ,p ,p and p
 . The free parameters in the parton cascade are the
      , pQCD scaling parameter and q , representing the minimum transverse momentum for   a branching in the cascade. For the pQCD parameter, we use the usual leading order value "0.2 GeV. The in#uence of the technical parameter q has been investigated in detail.   9.1. Basic diagram The "rst-order di!erential cross section for the process (Fig. 9.1) e>e\PH or ZPqq

(9.1)

is given as [178]   d
" [(1#cos )q !2q < <  (s)#(A#<)(A #< ) (s) D D C D  C C D D  2s d cos  #cos !4q A A  (s)#8A < A <  (s)] D C D  C C D D 

(9.2)

with s(s!M ) 8 ,  (s)"  (s!M )# M 8 8 8 s  (s)" ,  (s!M )# M 8 8 8

(9.3) (9.4)

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Fig. 9.1. Electron}positron annihilation.

where  is given as (2G M $ 8 . " 16

Here, is the "ne structure constant, q the quark #avor, M the mass of the Z boson,  its decay D 8 8 width. The vector and axial coupling factors are < "¹ !2q sin  , A "¹ D D D 5 D D

(9.5)

and



 5

 if f", u, c , ¹ "  D ! if f"e, d, s, b .  is the Weinberg mixing angle. The Fermi constant

(9.6)

(2g 5 G " (9.7) $ 8M 5 is expressed via the weak coupling g "4 /sin  and the W-boson mass M . At low energies, 5 5 U s;M , we recover the well known formula 8 4  q . (9.8)
" 3s D The factors  (s) and  (s) correspond to the intermediate Z-boson state and to the photon  Z-boson interference, respectively. Formula (9.2) can now be used to generate an initial quark} anti-quark pair. 9.2. Time-like parton cascade and string formation If one considers a quark and an anti-quark coming from the decay of a virtual photon or Z-boson to be on-shell, this amounts to a lowest order treatment. At high energies, a perturbative correction has to be done. Since it is di$cult to calculate the higher-order Feynman diagrams exactly, one uses the so-called DGLAP evolution equations, which describes the evolution of a parton system with leading logarithmic accuracy. This amounts to successively emitting partons (time-like parton cascade, see Fig. 9.2). This has been discussed in detail in connection with the

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Fig. 9.2. Time-like parton cascade. Fig. 9.3. Forming strings for an e>e\ event.

parton production in proton}proton or nucleus}nucleus collisions. We use exactly the same method here. Even for determining three momenta of the primary quark and anti-quark, we do not need any new input, since this corresponds exactly to the case of the time-like cascade of the two partons involved in the Born scattering in hadronic collisions. The fact that we use one and the same procedure for the time-like parton cascade in all the di!erent reactions, allows us to test elements of hadronic interactions in a much simpler context of elementary electron}positron interactions. The "nal step is the hadronization of the above-mentioned parton con"guration. Here we use the string model, as in case of pp or p scattering. The string formation in e>e\ is much simpler than in proton}proton scattering, where the cut Pomeron is represented as a cylinder. The structure of an e>e\ event is planar in the sense that the whole event can be represented on a plane. So we simply plot the diagram on a plane with only one cut line. In Fig. 9.3(a), we present a half-plane (on one side of the cut) for the amplitude shown in Fig. 9.2. The dotted line represents the cut. There are a couple of partons crossing the cut, indicated by dots. As in the case of proton}proton or photon}proton scattering, we identify the cut line as a kinky relativistic string, with the partons representing the kinks. So in our example, we have a kinky string with six kinks, two external ones and four internal ones. We then apply the usual hadronization procedure, discussed earlier in detail, in order to calculate hadron production from a fragmenting string, see Fig. 9.3(b).

9.3. Event shape variables We start our presentation of results by considering the so-called event shape variables, which describe the form of an e>e\ event in general [205}217]. For example, one is interested in knowing whether the particle momenta are essentially aligned along a certain axis, distributed isotropically over the phase space, or lying more or less in a plane. In the Figs. 9.4}9.8, we are going to compare our calculated distributions of several event variables with data. Let us "rst discuss the di!erent event-shape variables, one after the other.

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Fig. 9.4. The sphericity and thrust for the energies 14, 22 and 34 GeV. The data are from the TASSO collaboration [201].

9.3.1. Sphericity The sphericity is de"ned by the eigenvalues of the sphericity tensor, S?@"
(p )?(p )@ , G G G

(9.9)

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Fig. 9.5. Event shape variables at 91.2 GeV. The data (dots) are from the DELPHI collaboration [202].

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Fig. 9.6. Event shape variables for 133 GeV. The data (dots) are from the OPAL collaboration [203].

Fig. 9.7. Event shape variables for 161 GeV. The data (dots) are from the OPAL collaboration [204].

195

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Fig. 9.8. Thrust, minor and major for 91 GeV (full line), 133 GeV (dashed) and 161 GeV (dotted). The data are from the DELPHI [202] (91 GeV) and OPAL [203,204] (133 and 161 GeV) collaborations.

where i sums all particles and p? is the particle four momentum. One "nds three eigenvalues with G ( ( and # # "1. The sphericity is then de"ned as       (9.10) S"( # ) .    For a perfectly isotropic event, one "nds " " "1/3 and therefore S"1. An event    oriented along one axis gives S"3/2(0#0)"0. To test whether an event has planar geometry, one de"nes the aplanarity A" . (9.11)   For events in a plane we will "nd "A"0. The maximum of this value is A"3/2 ) 1/3"1/2 for  an isotropic event, since the eigenvalues are ordered. The three eigenvectors vo of the matrix S?@ can be used to de"ne a coordinate system.  9.3.2. C and D parameters The C-parameter is de"ned by C"3( # # ) ,       with being the eigenvalues of the tensor  p?p@
G G G p  G . M?@"
p  G G The D-parameter is

(9.12)

(9.13)

D"27 . (9.14)    These values measure the multiple jet-structure of events. For small values of C two of the eigenvalues are close to zero, we have a two-jet event. If one of the three eigenvalues is close to zero, the D-parameter is approaching zero as well, we have at least a planar event.

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9.3.3. Thrust The thrust of an event is de"ned as
no ) po  H . (9.15) ¹"max H
po  o H H L The vector no , which maximizes this expression, de"nes the thrust axis. A two-jet event will give  a thrust value of 1 and a thrust axis along the two jets. An isotropic event gives ¹&1/2. One can repeat the same algorithm with the imposed condition no Nno ; this gives an expression  . for the major M with the axis no +  and A third variable, the minor m, is obtained by evaluating the above expression with no Nno  no Nno , the axis being already given. Each of these values describes the extension of the event +  perpendicular to the thrust axis. Similar values for M and m describe therefore a cylindrical event. For this, the oblateness is de"ned as O"M!m which, as the aplanarity, describes a cylindrical event for O"0 as a planar event for higher values. 9.3.4. Jet broadening In each hemisphere, the sum of the transverse momenta of the particles relative to the thrust axis is divided by the sum of the absolute values of the momenta.
po ;no   B " !No Lo   G (9.16) ! 2
po  G G The greater B is, the greater is the mean transverse momentum. One de"nes in addition the > following variables, B "max(B , B ), B "min(B , B ) , (9.17)   > \   > \ B "B #B , B "B !B  , (9.18)  > \   > \ to compare jet broadening in both hemispheres. For small B , one "nds a longitudinal event,   B measures the asymmetry between the two hemispheres.   9.3.5. Heavy jet or hemisphere mass The variable M is de"ned as    p , p

(9.19) M"max G G  No G Lo   No G Lo   and corresponds to the maximal invariant mass squared of the hemispheres. The corresponding formula with `mina instead of `maxa de"nes the variable M . One de"nes as well M "M !M . Usually one analyzes distributions of M /E , M/E or M /E , which             describe the squared masses normalized to the visible energy.










9.3.6. Some comments Fig. 9.4 shows the distributions of sphericity and thrust for the lower energies 14, 22 and 34 GeV. Even though one expects an increasing contribution of perturbative gluons, which is con"rmed by the inclusive hadron distributions (see next section), the events are more longitudinal at higher

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energies, corresponding to the values of thrust close to 1 and to the sphericity SK0. This can be explained by the fact that the leading quarks dominate the event shape. The results for higher energies (Figs. 9.5, 9.6 and 9.7) con"rm the above statements. In Figs. 9.5, 9.6 and 9.7, we show also the distributions of other event variables, like heavy jet mass, Major, etc. In general, our model describes quite well all these event shape variables. 9.4. Charged particle distributions We will now consider the distributions of charged particles, which by de"nition contain all the particles with a decay time smaller than 10\ s, i.e. the spectra contain, for example, products of decay of K , while K! are considered stable. The decay products of strange baryons are also  included in the distributions. In Fig. 9.9, we plot multiplicity distributions of charged particles for three di!erent energies, where one observes an obvious increase of the multiplicity with energy. In Fig. 9.10, we show the distributions of the absolute value of the rapidity for the energies 14, 22, 34, 91.2, 133 and 161 GeV. The rapidity is de"ned as y"0.5 ln((E#p )/(E!p )), where the variable p may be de"ned along X X X the thrust axis or along the sphericity axis. For both, multiplicity and rapidity distributions, the theoretical curves agree well with the data. The multiplicity increases faster than n "a#b ln s as a function of s [219], which is due to  the fact that the maximal height of the rapidity distribution increases with energy, as seen in Fig. 9.10. This comes from radiated gluons, leading to kinky strings, since a #at string without gluons shows an increasing width but a constant height, as one can see in Fig. 9.11. Here, the rapidity distributions of charged particles are plotted for the fragmentation of a #at d!dM string for energies 14, 22 and 34 GeV. One observes that the width of the distributions increases whereas its height does not change, giving rise to a proportionality to ln s. Additional hard gluons with non-collinear momenta increase the multiplicity in the mid-rapidity region. Rather than the rapidity, one may consider the scaled momentum x "2po /E or the scaled G  energy x "2E /E, as well as the `rapidity-likea variable "!ln x . Concerning the distribu# G  tions shown in Fig. 9.12, one sees that the value corresponding to the maximum of the curves

 increases with energy. The x distributions (see Fig. 9.13) show the development of a more and  more pronounced peak at x close to zero, with increasing energy.  Having discussed in detail the variables describing the longitudinal phase space, we now turn to transverse momentum, which can be de"ned according the sphericity axis or to the thrust axis. One writes



for sphericity , vo ) po  p "  , ) p for thrust , no

 o

(9.20)

and



vo ) po  for sphericity , p"  (9.21) , ) po  for thrust . no

 There are mainly two `sourcesa of transverse momentum. The "rst one is the transverse momentum created at each string break. The second one is the transverse momentum from hard gluon

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Fig. 9.9. Multiplicity distributions for charged particles. The data (dots) are from the TASSO [201] (14}34 GeV), ALEPH [218] (91 GeV), and OPAL [203,204] (133, 161 GeV) collaborations.

radiation, which can be much larger than the "rst one. So we "nd large values of p in the event , plane, and smaller ones out of the event plane. Here the event plane is essentially de"ned by the direction of the hardest gluon emitted. Let us have a look at transverse momenta of charged particles coming from a string decay for di!erent energies (see Fig. 9.14). As expected, the curves show the same behavior. In the same "gure,

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Fig. 9.10. Rapidity distributions for charged particles. The data (dots) are from the TASSO [201] (14}34 GeV), ALEPH [218] (91 GeV), and OPAL [203,204] (133, 161 GeV) collaborations.

we show the results for an e>e\ annihilation at 14 GeV. Already at 14 GeV the in#uence of parton radiation is important. Figs. 9.15 and 9.16 show the results for 14}161 GeV. Our results agree well with data from the TASSO, ALEPH, and OPAL collaboration [201,203,204,218]. One can see how transverse momenta increase with the energy as an indication of more hard gluon radiation: the p distributions change little, whereas the p  distributions get much harder at high energies as , , compared to lower ones.

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Fig. 9.11. Rapidity distribution of charged particles for #at strings at 14 GeV (full), 22 GeV (dashed) and 34 GeV (dotted). The height of the distribution does not change, but its width does.

Fig. 9.12. "!ln x distributions of charged particles at the energies 91, 133 and 161 GeV. The data (dots) are from  the ALEPH [218] (91 GeV), and OPAL [203,204] (133, 161 GeV) collaborations.

9.5. Identixed particles In this section we consider inclusive spectra of identi"ed hadrons. This provides a crucial test of the fragmentation model and allows to "x the two hadronization parameters p and p . The     "rst one gives the probability to "nd a pair u!u or d!dM , "xing so the strangeness probability to be (1!2p ). The parameter p determines the multiplicity of baryons.     Let us look at spectra at 29 GeV obtained at SLAC [220,221] (Figs. 9.17 and 9.18) and at 91 GeV at LEP [218] (Figs. 9.19, 9.20 and 9.21). Since the total multiplicity is dominated by small x-values, we show this regions separately for some "gures. The results are in general quite good, however, KH's are underestimated. For charmed particles, there is no production from the string decay due to the large mass of the c!c pair. The corresponding probability p is taken to be zero. Charmed quarks come   therefore directly from the decay of the virtual photon as well as from perturbative parton cascade. This explains as well the drop of D spectrum at small x . 

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Fig. 9.13. x distributions of charged particles. The data (dots) are from the TASSO [201] (14}34 GeV), ALEPH [218]  (91 GeV), and OPAL [203,204] (133, 161 GeV) collaborations.

9.6. Jet rates Jet multiplicities play an important role in e>e\ physics since their measurements proved the validity of perturbative QCD. The "rst 3-jet event was found in 1979. Fig. 9.22 shows this historical event.

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Fig. 9.14. Transverse momentum spectra of charged particles for a #at string at energies 14, 22 and 34 GeV together with a simulation for e>e\ at 14 GeV.

There are several methods to determine the number of jets in an event, but they are all based on some distance y between two particles i and j in momentum space, something like an invariant GH mass. For the JADE algorithm one de"nes [222] 2E E (1!cos  ) GH , (9.22) y " G H GH E    being the angle between the two particles, and for the algorithm DURHAM one has [223] GH 2min(E, E)(1!cos  ) G H GH . y " (9.23) GH E   E is the total visible energy of all the particles which contribute to the jet "nding. The algorithm   works as follows: one determines the pair with the lowest distance y and replaces it with one pseudo-particle having the sum of the momenta of the two particles i and j : pI"pI#pI. One G H repeats this until all pairs of pseudo-particles have a distance greater than y . The number of jets  is then the total number of pseudo-particles. Of course, this depends on the choice of y . Therefore  one displays often the jet multiplicity distribution as a function of y .  Let us compare the jet rates for di!erent energies. Fig. 9.23 shows the jet rates for 91.2 GeV, Fig. 9.24 for 133 and 161 GeV. The greater is y the smaller is the number of jets.  10. Testing the semi-hard Pomeron: photon}proton scattering It is well known that both photon}proton (Hp) scattering and hard processes in proton}proton (pp) collisions can be treated on the basis of perturbative QCD, using the same evolution equations. In both cases, the perturbative partons are "nally coupled softly to the proton(s). This provides a very useful consistency check: any model for (semi)hard proton}proton collisions should be applied to photon}proton scattering, where a wealth of data exists, mainly from deep inelastic

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Fig. 9.15. Transverse momentum spectra of charged particles at 14, 22 and 34 GeV. The spectra of p are steeper than , the ones of p . The data (dots) are from the TASSO collaboration [201]. ,

electron}proton scattering (DIS). In particular the soft coupling to the protons is not calculable from "rst principles, so photon}proton scattering provides a nice opportunity to test the scheme. Let us discuss the relation between Hp and pp scattering. In Fig. 10.1, we show the cut diagram (integrated squared amplitude), representing a contribution to photon}proton scattering: a photon

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Fig. 9.16. Transverse momentum spectra of charged particles at 91, 133 and 161 GeV. The data (dots) are from the ALEPH [218] (91 GeV), and OPAL [203,204] (133, 161 GeV) collaborations.

couples to a quark of the proton, where this quark represents the last one in a `cascadea of partons emitted from the nucleon. In the leading logarithmic approximation (LLA) the virtualities of the partons are ordered such that the largest one is close to the photon [127,128]. Comparing the cut diagram for Hp (Fig 10.1) with the cut diagram representing a semi-hard elementary

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Fig. 9.17. Longitudinal momentum fraction distributions for di!erent identi"ed hadrons at (s"29 GeV. The data are obtained from SLAC [220,221].

proton}proton scattering (Fig. 10.2), we see immediately that the latter one is essentially made of two Hp diagrams, glued together by a Born process. So, understanding Hp implies understanding an elementary nucleon}nucleon interaction as well. Actually, probably everybody agrees with this statement, which is nothing but the factorization hypothesis, proved in QCD [224}227], and the

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Fig. 9.18. Longitudinal momentum fraction distributions for di!erent identi"ed hadrons at (s"29 GeV. The data are obtained from SLAC [220,221].

standard procedure to calculate inclusive cross sections in proton}proton scattering amounts to using input from the DIS structure functions. But one can pro"t much more from studying Hp, for example, concerning the production of hadrons. Not being calculable from "rst principles, the hadronization of parton con"gurations is a delicate issue in any model for proton}proton (or nucleus}nucleus) scattering. So studying Hp provides an excellent possibility to `gaugea the hadronization procedure, such that there is no freedom left on the level of nucleon}nucleon (or nucleus}nucleus) scattering.

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Fig. 9.19. Longitudinal momentum fraction distributions for di!erent identi"ed hadrons at (s"91 GeV. The data are from the ALEPH collaboration [218].

The simple picture, depicted in Fig. 10.1, is correct for large virtualities, but it fails when the photon virtuality becomes small. In that case a virtual photon behaves to a large extent as a hadron and is characterized by some parton content instead of interacting with a proton just as a point-like object. Then the contribution of so-called resolved photon interactions } see Fig. 10.3 } is

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Fig. 9.20. Longitudinal momentum fraction distributions for di!erent identi"ed hadrons at (s"91 GeV. The data are from the ALEPH collaboration [218].

important and has to be taken into account properly for the description of hadron production in DIS. Only then one may deduce the parton momentum distributions of the proton from the measured virtual photon}proton cross section
AHN.

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Fig. 9.21. Longitudinal momentum fraction distributions for di!erent identi"ed hadrons at (s"91 GeV. The data are from the ALEPH collaboration [218].

10.1. Kinematics In the following, we consider photon}proton collisions in the context of electron}proton scattering. We "rst recall the basic kinematic variables, see Fig. 10.4. We use standard conventions:

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Fig. 9.22. The "rst 3-jet event [219].

Fig. 9.23. Jet rates at 91.2 GeV calculated with the DURHAM algorithm as a function of y . The rates for 2, 3, 4 and  5 jets are shown (data from [202]).

k, k, and p are the four-momenta of incoming and outgoing lepton and the target nucleon, q"k!k is the four-momentum of the photon. Then the photon virtuality is Q"!q, and one de"nes the Bjorken x-variable as Q . x " 2(pq)

(10.1)

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Fig. 9.24. Jet rates at 133 GeV [203] (top) and 161 GeV [204] (bottom) for the two algorithms DURHAM and JADE.

Fig. 10.1. The cut diagram representing photon}proton (Hp) scattering. Fig. 10.2. The universality hypothesis implies that the upper and the lower part of the Pomeron diagram are identical to the photon}proton diagram.

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Fig. 10.3. The cut diagram representing resolved photon}proton (Hp) scattering. Fig. 10.4. Kinematics of electron}proton scattering.

The y-variable is de"ned as Q (pq) 2(pq) " " , y" s x s (pk)

(10.2)

which gives the energy fraction of the photon relative to the incident electron in the proton rest frame. In the above formula we used s"(k#p) ,

(10.3)

being the total center-of-mass squared energy and neglected the proton and electron masses, pK0, kK0. For the center-of-mass squared energy of photon}proton interaction, we use s( "(p#q) ,

(10.4)

and we "nally de"ne the variable s "s( #Q"2(pq)"ys ,

(10.5)

which allows us to write x "Q/s .

(10.6)

It is often convenient to take s and Q as the basic kinematical variables instead of x and Q. 10.2. Cross sections The di!erential cross section for deep inelastic electron}proton scattering in the one-photon approximation can be written as [228}231] d
CN(s, x , Q)

" [¸A (y)
AHN(s , Q)#¸A (y)
AHN(s , Q)] , 2 2 * * dQ dx Qx

(10.7)

where
AHN (s , Q) are the cross sections for interactions of transversely (longitudinally) polarized 2* photons of virtuality Q with a proton, is the "ne structure constant and factors ¸AH (y) de"ne the 2*

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#ux of transversely (longitudinally) polarized photons, 1#(1!y) , ¸A (y)" 2 2

¸A (y)"(1!y) . *

(10.8)

The cross sections
AHN (s , Q) are related to the proton structure functions F , F , describing the 2*  * proton structure as seen by a virtual photon probe, as Q [
AHN(s , Q)#
AHN(s , Q)] , F (x , Q)" *  4 2 Q H F (x , Q)"
A N(s , Q) . * 4 *

(10.9)

(10.10)

To the leading logarithmic accuracy one has to take into account a number of processes, contributing to
AHN (s , Q). In the case of transverse polarization of the photon, we have three 2* contributions: the direct coupling of the virtual photon H to a light quark from the proton (`lighta), the direct coupling to a charm quark (`charma), and "nally we have a `resolveda contribution, i.e.
AHN(s , Q)"
AHN (s , Q)#
AHN (s , Q)#
AHN (s , Q) . 2 2  2   2 

(10.11)

The latter one is becoming essential at small Q and large s . For our study, contributions of beauty and top quarks can be neglected. The longitudinal photon cross section receives leading order contributions only from the direct H-coupling to a charm quark. So we have (s , Q) .
AHN(s , Q)"
AHN * *  

(10.12)

Again, contributions of beauty and top quarks can be neglected. Let us list the di!erent contributions. The leading order light quark-H coupling contribution (`lighta) can be expressed via the quark momentum distributions in the proton f (x , Q) as [131] ON 4


AHN (s , Q)" 2  Q


e x f (x , Q) , G GN  M  GZ SBQSBQ

(10.13)



where e is the quark q electric charge squared. The contributions of heavy quarks can be taken O into account via photon}gluon fusion (PGF) process [232],




AHN (s , Q)"e dx\ dp 2*   A ,

d
AHEAA (x\s , Q, p ) 2* , f (x\, M ) , EN $ dp ,

(10.14)

where f (x\, M ) is the gluon momentum distribution in the proton at the factorization scale M , EN $ $ and where the photon}gluon cross section in lowest order is given as 1 d
AHEAA (s , Q, p ) 2* , " MAHEAA (s , Q, p ) , dp 16s s( (1!4(p #m)/s( 2* , , A

(10.15)

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with s( "s !Q, and with the corresponding matrix elements squared given as MAHEAA (s , Q, p )"

2 , Q








t u Q#s(  2ms  4m # # A (Q!2m)# A (s !2Q) , A u t s  tu tu (10.16)






s( Q mQ ! A . MAHEAA (s , Q, p )"8

* , Q s  tu

(10.17)

Here, the variables t, u are expressed via standard Mandelstam variables for parton}parton scattering as t"t!m, u"u!m. According to [232], the factorization scale M has to be A A $ chosen irrespectively to the photon virtuality Q to assure the perturbative stability of the result; we use M "p #m, which coincides at large s( with the virtuality (o!-shellness) of the intermediate $ , A t-channel c-quark t, with m "1.6 GeV being the c-quark mass. A In addition, at small Q and large s , the contribution of resolved photon processes becomes important for the production of parton jets of transverse momenta p 'Q. Here, , d
GH (x>x\s( , p )  , (s , Q)" dx> dx\ dp

AHN 2  , dp , GH ;f H (x>, M, Q) f (x\, M)(p !Q) (10.18) GA A HN N , where s( "s !Q is the c.m. energy squared for H-proton interaction, d
GH /dp is the di!erential  , partonic cross section, f H is the parton momentum distribution in the photon, M, M are the N A GA factorization scales for the proton and photon correspondingly. As in hadron}hadron scattering, we use M"p /4, whereas for the photon we take M"4p . This requires some more explanaN , A , tion. To the leading order accuracy, the factorization scales are rather unde"ned as the di!erence between the results, obtained for di!erent scale choices, is due to higher-order corrections. The scheme would be scale independent only after summing up all order contributions both in the structure functions and in the partonic cross section. High p jet production in H}proton , interaction is known to obtain essential contributions from next to leading order (NLO) direct processes [233}237]. As it was shown in [236], the sum of the leading order resolved H}proton cross section and the NLO direct one exhibits a remarkable independence on the scale M for the A production of parton jets with p 'Q, where Q is the photon virtuality. So our strategy is to , choose M such that it allows to represent e!ectively the full contribution by the leading order A resolved cross section.



10.3. Parton momentum distributions The cross sections mentioned in the preceding section are all expressed via the so-called parton distribution functions f , representing the momentum fraction distribution of parton i inside G? particle a (proton or photon). In this section, we are going to discuss these distribution functions. We are "rst discussing parton distribution functions of the proton. They are represented by the hadronic part of the photon}proton diagram, i.e. the diagram without the external photon line. As mentioned before, this diagram is also a building block of one of the elementary diagrams of pp scattering, and one can therefore repeat literally the argumentation of Section 2. In pp scattering,

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we have (apart from the soft one) four contributions, since on each side the parton ladder couples to the nucleon either via a soft Pomeron or it connects directly to a valence quark. In addition, there is a triple Pomeron diagram. Corresponding we have here three contributions, referred to as `seaa, `triple Pomerona, and `valencea. 10.3.1. Sea contribution The sea contribution contains the perturbative parton cascade, described as a parton ladder with strictly ordered virtualities, and the non-perturbative soft block, dominated by the soft Pomeron asymptotics, see Fig. 10.5. We obtain the momentum distribution f / (x, M ) of the parton i at GN  $ the virtuality scale M in the proton as the convolution of three distributions (see Fig. 10.5): the $ inclusive parton Fock state distribution in the proton FI (x ) (see Eq. (C.18)), N  FI (x )"FN (1!x )FN (x ) , (10.19) N        the distribution for the parton momentum share in the soft Pomeron EH (z) (see Eqs. (B.21) and  (B.22)), and the QCD evolution function EHG (z, Q , M ): /!"  $  dx V dx x x   EHG  FI (x )EH , Q , M , (10.20) f/ (x, M )"
GN  $ N   x /!" x  $ x x V V     H see Fig. 10.5. This equation may be written as

 














 dx x  / (x )EHG , Q , M , (10.21) f / (x, M )"
 $ GN  $ HN   /!" x x  H V  where / (x ) by construction corresponds to the distribution at the initial scale Q , HN     dx x  FI (x )EH  . / (x )" (10.22) HN   N   x x V   Here we use the same expressions for FN (x), FN (x), and EH (z) as in the case of proton}proton      or nucleus}nucleus scattering, see Eqs. (C.20), (C.21), (B.21) and (B.22).






10.3.2. Triple Pomeron contribution We have to also take into account triple-Pomeron contributions f / (x, M ) to gluon and sea GN  $ quark momentum distributions. The latter ones are de"ned by the diagrams of Fig. 5.6 with the upper cut Pomeron being replaced by the `half a of the sea}sea type semihard Pomeron which consists of a soft Pomeron coupled to the triple-Pomeron vertex and the parton ladder describing

Fig. 10.5. The diagram corresponding to the parton distribution functions.

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Fig. 10.6. The triple Pomeron contribution.

the perturbative parton evolution from the initial scale Q to the "nal scale M , see Fig. 10.6. The  $ lower legs are (un)cut Pomerons. Let us consider the part 
/\ } (s) of the contribution of cut triple-Pomeron diagram NN   1 
/\(s)" 2Im ¹/\(s, t"0) , NN NN 2s corresponding to the semihard sea}sea type parton}parton scattering in the upper Pomeron, which is according to Eqs. (5.5) and (5.6) given as

       

r / 
/\ } (s)"!  Im NN   2 ;8





dx>





dx\ 

\V\



dx\ FN (1!x>)FN (1!x\!x\)       

dx\ 1  Im ¹N } (x>, s>, 0)   x\ 2s> 

 

 1 ; dz> dq ,  i¹N(x\, s( ,!q, ) , (10.23)  J J  8s( J J where we used x\ "s /(x> s), s>"x>x\ s, s( "z>s x\/x\ , s( "(1!z>)s x\/x\ , and             ¹N, ¹N } are de"ned in (5.15)}(5.17). Applying the AGK cutting rules, contribution (10.23) can be   written as a sum of three terms: (s)#
/\} (s) , 
/\ } (s)"
/\} (s)#
/\ NN   NN   NN }  NN  

(10.24)

corresponding to the di!ractive type semihard interaction, screening correction to the usual semihard interaction, and double cut Pomeron contribution, with the weights (s)"!1;
/\ } (s) , 
/\ NN }  NN  

(10.25)

(s)"#4;
/\ } (s) , 
/\ NN }  NN  

(10.26)


/\ (s)"!2;
/\ } (s) . NN }  NN  

(10.27)

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Making use of (5.16), (B.20), (2.15), we have



1  dz> dz\ 1   EH (z>) EI (z\) Im ¹N } (x>, s>, 0)" FN (x>)
  2   z> z\     2s> HI      ;
dz> dz\ dp EHK (z>/z>, Q , M ) EIJ (z\/z\, Q , M ) , /!"   $ /!"   $ > X\ KJ X d
KJ  (z>z\s>, p ) (M !Q ) . (10.28) ;K , $  dp , Now, using (10.19) and (10.20), we can rewrite (10.23) as




/\ } (s)"
NN   KJ



;K





dx> dx\ dp f / (x>, M ) f / (x\, M ) , KN  $ JN  $ d
KJ  (x>x\s, p ) (M !Q ) , , $  dp ,

(10.29)

where we denoted x>"x>z>,

s x\"z\  , x> s 

and de"ned








 dx x  / (x )EHG f / (x, M )"
, Q , M ,  $ GN  $ HN   /!" x x  H V 

(10.30)

with

     


  


x r /  dx  EH / (x)"!  dx dx FN (1!x !x )  x       HN  x 2 V    1 i¹N(x , s( ,!q ) ;4 dz dq Im  , J J , 8s( J J x r /  dx  EH "!  dx dx FN (1!x\!x\)  x       8 x V   ; db dz GN(x , s( , b) GN(x , s( , b) ,    



(10.31)

with s( "z s x /x , s( "(1!z)s x /x , where GN(x, s( , b) is given in (5.33)}(5.36).         Now, replacing in (10.29) the interaction with the projectile parton m by the interaction with a virtual photon probe of virtuality M or by the interaction with a hypothetical probe which $ couples directly to a gluon, we immediately see that f / (x, M ) de"nes the (negative) contribuGN  $ tion of the triple-Pomeron diagram to parton structure functions.

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10.3.3. Valence contribution The third contribution, referred to as `valencea, amounts to the case where a valence quark is the "rst parton of the ladder, see Fig. 10.7. Here, the soft pre-evolution, governed by the secondary Reggeon, is typically short and therefore not considered explicitly. One simply uses the parameterized input for valence quark momentum distributions at the initial scale Q ,   (x )"qG (x , Q ) (10.32) GN      with the Gluck}Reya}Vogt parameterization for qG (x, Q ) [238].   10.3.4. Complete proton distribution function The total parton distribution in the proton at the initial scale Q is then de"ned as   (x )" (x )# (x ) , GN  GN   GN   with

(10.33)

(10.34)  (x )"/ (x )#/ (x ) GN   GN   GN   which results for an arbitrary scale M in $ dx x   (x )EHG , Q , M . (10.35) f (x, M )"
GN  /!"  $ GN $ x x   H For M "Q , the semi-hard contribution is a function which peaks at very small values of x and $  then decreases monotonically towards zero for x"1. The valence contribution, on the other hand, has a maximum at large values of x and goes towards zero for small values of x. For moderate values of M , the precise form of f depends crucially on the exponent for the Pomeron}nucleon $ coupling , whereas for large M it is mainly de"ned by the QCD evolution and depends weekly   $ on the initial conditions at the scale Q . 








10.3.5. The photon distribution functions To calculate the resolved photon cross section (10.18), one needs also to know parton momentum distributions of a virtual photon f H (x, M, Q). According to [239}247], f H gets contriA GA GA butions both from vector meson states of the photon and from perturbative point-like photon splitting into a quark}anti-quark pair, f (x, M, Q)"f 4"+(x, M, Q)#f  (x, M, Q) . GA A GA A A GA

(10.36)

Fig. 10.7. The diagram corresponding to the `valencea contribution to the parton distribution functions.

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For the former one, one has [247] f 4"+(x, M, Q)"(Q) [G f  (x, M)# (G!G) f  (x, M)] , GA A G GL A  G S B QL A where the function  is given as

(10.37)

(Q)"(1#Q/m)\ , M with m"0.59 GeV and with M "!1, "1, "0, "0 , S B Q E G"0.836, G"0.250, G"0.543, G"0.543 . S B Q E Pion structure functions f  are de"ned in the same way as proton ones, namely GL dx x    (x ) EHG f  (x, M )"
, Q , M , GL $ GL  /!"  $ x x   H with

(10.38)








(10.39)

(10.40)

  (x )" (x )#  (x ) . (10.41) GL  GL   GL    is a parameterized initial distribution for the valence component from [248] Here   GL     (x )"qG  (x , Q ) , (10.42) GL     L   and the sea component  is given by formulas (10.33), (10.34), (10.22), (10.31) and (10.32) with HL  the subscript p being replaced by  and using the appropriate parameters L ,  in FL , FL ,   L     but keeping all other parameters, characterizing the Pomeron trajectory, unchanged compared to proton case. The point-like contribution f   is given as a convolution of the photon splitting into GA a quark}anti-quark pair (with the Altarelli}Parisi splitting function PAOO (z)"N /2 (z#(1!z)), A N "3 being the number of colors), followed by the QCD evolution of a (anti-)quark from the A initial virtuality q of the splitting till the scale M A dq  dx

A PAOO (x ) f  (x, M, Q)"e A A O GA q x 2 V A x ;
EHG , q, M (q!max[Q , x Q]) (10.43) /!" x A  A   M  A HZ SBQSBQ with x being the share of the virtual photon light cone momentum taken by the (anti-)quark and A with

 






e"(e#e#e) (10.44) O  S B Q being the average light quark charge squared. Here we have chosen the initial scale for the QCD evolution of a t-channel (anti-)quark to be equal to its initial virtuality q"x Q#p /(1!x ), A , A with p being the transverse momentum squared for the splitting in the Hp center of mass , system [249].

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10.4. Structure function F  We have now all elements to calculate the structure function F , based on the formula (10.9),  with all leading order contributions to
AHN as given in Eqs. (10.11)}(10.18) and with the parton 2* momentum distributions in the proton and photon as discussed in the preceding section. The results for F (x, Q) are shown in Fig. 10.8 together with experimental data from H1 [250], ZEUS 

Fig. 10.8. The structure function F for di!erent values of Q together with experimental data from H1 [250], ZEUS  [251] and NMC [252].

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[251] and NMC [252]. The parameters a!ecting the results for F are actually the same ones  which a!ect parton}parton and hadron}hadron scattering. So we "x them in order to have an overall good "t for F and at the same time the energy dependence of the total cross section and of  the slope parameter. It is possible to get a reasonable agreement, which is of course not perfect due to the fact that enhanced diagrams are only treated to lowest order. In Fig. 10.9, we show separately

Fig. 10.9. Same as Fig. 10.8, but here we show separately the `direct-lighta contribution (dashed), the `direct-charma contribution (dotted), and the `resolveda contribution (dashed-dotted).

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the direct light quarks contribution (10.13), as well as the ones of charm quarks (10.14) and of resolved photons (10.18), for Q"1.5, 5, and 25 GeV. It is easy to see that the resolved photon cross section contributes signi"cantly to F (x, Q) for small photon virtualities Q.  10.5. Parton conxgurations: basic formulas In order to have a coherent approach, we base our treatment of particle production on exactly the same formulas as derived earlier for the cross sections. To be more precise, we take the formulas for d
/dx dQ as a basis for treating particle production, which means "rst of all parton production. The di!erential cross section for lepton}nucleon scattering is given in Eq. (10.7). Using (B.28), (B.30), the QCD evolution function EGH , which enters into the formulas for the cross sections /!"
AHN (s , Q), can be expanded as a sum over n-rung ladder contributions, where the latter ones can 2* be written as an integration over the momenta p , p ,2, p of n resolvable "nal partons. Introduc  L ing a multidimensional variable P"p , p ,2, p  , (10.45)   L and considering the symbol
representing
dP , with dP being the invariant phase space . L L L volume for n-parton state, we may write d
JN "

(x , Q, P) . (10.46) dx dQ . After normalization,
(x , Q, P) may be interpreted as the probability distribution for a parton con"guration P for given values of x and Q. The Monte Carlo method provides a convenient tool for treating such multidimensional distributions: with
(x , Q, P) being known (see preceding sections), one generates parton con"gurations P according to this distribution. In addition to x , Q, and P, additional variables occur, specifying a particular contribution to the DIS cross section. One essentially follows the structure of the formula for the cross section. Let us discuss the procedure to generate parton con"gurations in detail. We start with some useful de"nitions. Using the relation (B.28) for the evolution function EGH , /!" any parton momentum distribution at the scale Q can be decomposed into two contributions, corresponding to the case of no resolvable emission in the range of virtualities between Q and  Q and to at least one resolvable emission:






\C dz x EM GH (z, Q , Q) f , Q . (10.47) f (x, Q)"f (x, Q )H(Q , Q)#
/!"  G H H   z z  G V In case of `ia and `ja being quarks, we split the sum
EM GH into two components, H /!"
EM GH "EM #EM (i, j"quarks) , (10.48) /!" ,1 1 H with the so-called non-singlet evolution EM , where only gluons are emitted as "nal s-channel ,1 partons, and the singlet evolution EM representing all the other contributions, see Figs. 10.10(a) 1

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Fig. 10.10. The non-singlet contribution (a), the gluon contribution (b), and the singlet contribution (c).

and (c). The non-singlet evolution (compare with Eq. (B.29)) satis"es the evolution equation EM (x, Q , Q) ,1  / dQ  "  Q /  and the singlet one

















\C dz

x Q PO (z)EM , Q , Q O(Q , Q)# Q PO (x)O(Q , Q) , O ,1     2 O z 2 z V







/ dQ \C dz

x  Q PO (z)EM , Q , Q O 1 z   z 2  Q /  V x #2n PO (z)EM OE , Q , Q O(Q , Q) , D E /!" z   

EM (x, Q , Q)" 1 




(10.49)

(10.50)

with n "3 being the number of active quark #avors. D Now it is convenient to de"ne parton level cross sections, corresponding to di!erent contributions to the deep inelastic scattering process and to di!erent partons, entering the perturbative evolution at the initial scale Q , see Fig. 10.10. Essentially, we include into the cross sections the  perturbative part of parton evolution, whereas the initial conditions, given by parton momentum densities at the initial scale Q , are factorized out. The non-singlet and singlet contributions to the  direct (light) photon}quark interaction are de"ned as






Q 4


AHO (s , Q, Q )" EM , Q , Q , 2,1  ,1 s  s






Q 4


AHO (s , Q, Q )" EM , Q , Q , 21  1  s s

(10.51) (10.52)

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For the direct (light) photon}gluon interaction, we de"ne






Q 4

EM EO , Q , Q .
AHE (s , Q, Q )" 2   /!" s  s

(10.53)

The photon}parton charm production cross section is de"ned as

 

d
AHEAA (x\s , Q, p ) 2* , , (10.54) dp , where the parton (i) may be a quark or a gluon. Finally, we de"ne the parton}parton cross section for resolved processes similarly to (6.7) as
AHG (s , Q, Q )" dx\ dp EGE (x\, Q , M ) 2*    , /!"  $





d
IJ (x>x\s( , p )  ,
GH (s( , Q, q, Q )" dx> dx\ dp
, 2   dp , IJ ;EGI (x>, q, M)EHJ (x\, Q , M )(p !Q) . (10.55) /!" A /!"  N , Based on the above partial photon}parton cross sections, we de"ne now the total photon}parton cross sections, summed over the quark #avors of the quark coupling to the photon with the appropriate squared charge (e) factor. We obtain for the photon}gluon cross section (s , Q, Q )
AHE(s , Q, Q )"e
AHE (s , Q, Q )#e
AHE 2  O 2   A 2   



#
dx f 4"+(x , Q , Q)
HE (x (s !Q), Q, Q , Q ) A HA A  2  A   H dq

dx PAOO (x ) #e A A O q 2

 

;

HE (x s !q, Q, q, Q )(q!max[Q , x Q]) . (10.56) 2  A   A HZSBQS BM Q  The photon}quark cross section for a quark with #avor `ia is given as
AHG(s , Q, Q )"e
AHO (s , Q, Q )#e
AHO (s , Q, Q ) 2  G 2,1  O 21  H #e
A E (s , Q, Q ) A 2   



#
dx f 4"+(x , Q , Q)
HG (x (s !Q), Q, Q , Q ) A HA A  2  A   H dq

dx PAOO (x ) #e A 2 A O q

 

;

HG (x s !q, Q, q, Q )(q!max[Q , x Q]) . (10.57) 2  A   A   M  HZ SBQSBQ The longitudinal photon}parton cross section for a parton of #avor i (quark or gluon) is "nally given as
AHG(s , Q, Q )"e
AHG (s , Q, Q ) . *  A *   

(10.58)

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Finally, we may express the photon}proton cross sections in terms of the above photon}parton cross sections and the parton distribution functions (x)"f (x, Q ) at the scale Q . The transverse   cross section is given as




AHN(s , Q)" dx  (x )
AHE(x s , Q, Q ) 2  EN  2   #
GZSBQS BM Q 








4

ex  (x )O(Q , Q)# dx  (x )
AHG(x s , Q, Q ) , GN   GN  2   Q G (10.59)

the longitudinal cross section is given as




AHN(s , Q)" dx  (x )
AHE(x s , Q, Q ) *  EN  *  



#
dx  (x )
AHG(x s , Q, Q ) ,  GN  *   GZSBQS BM Q  with the quark momentum distributions being a sum of two terms,

(10.60)

 (x )" (x )# (x ) , (10.61) GN  GN   GN   see Eqs. (10.22) and (10.33). The above formulas together with Eq. (10.7) serve as the basis to generate all main variables for the description of deep inelastic scattering. After modeling Q and x we simulate types (a valence quark, a sea quark, or a gluon) and kinematical characteristics for "rst partons, entering the perturbative evolution (at the initial scale Q ), and then, for given initial conditions, generate  corresponding parton con"gurations, based on the particular structure of perturbative cross sections (10.51)}(10.55). The detailed description of this procedure is given in the next sections. The triple-Pomeron contributions are included here in the de"nition of the parton distribution  . At HERA energies, the triple-Pomeron contribution is dominated by the process where the GN  two Pomerons exchanged in parallel are soft ones, and therefore no additional parton production needs to be considered in that case. 10.6. Generating initial conditions for the perturbative evolution We start with the generation of the kinematical variables x and Q according to the di!erential cross section Eq. (10.7) together with the explicit form for the photon}proton cross sections equations (10.59) and (10.60). Then we choose an interaction with the transverse or with the longitudinal polarization component of the photon, with the corresponding weights (10.62) ¸AH (y)
AHN (s , Q)/(¸AH(y)
AHN(s , Q)#¸AH(y)
AHN(s , Q)) . 2* 2 * 2* 2 * After that, we consider virtual photon}proton interaction for given photon virtuality Q and polarization (T, L), and for given c.m. energy squared s( "s !Q for the interaction; we use the photon}proton center of mass system.

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227

Let us "rst consider the case of transverse photon polarization. We have to choose between `seaa and `valencea contribution, where the latter one is chosen with the probability



4

1
ex  (x )O(Q , Q) prob(val)" H GN  
A N(s , Q) Q   G 2 GZ SB





#
dx  (x )
AHG(x s , Q, Q ) (10.63)  GN   2     GZ SB and the former one with the probability 1!prob(val), which is the sum of the contributions, corresponding to a gluon or a sea quark from the proton being the "rst ladder parton,



1 4

prob(sea)" H
ex  (x )O(Q , Q) G GN  
A N(s , Q) Q   M  2 GZ SBQSBQ





#
dx  (x )
AHG(x s , Q, Q ) . (10.64)  GN   2     M  GZ ESBQSBQ The next step consists of de"ning the type (#avor) of the "rst ladder parton and its momentum share x in the proton. Here one has to distinguish two possible parton con"gurations for the  interaction: parton cascading without any resolvable parton emission in the ladder (corresponding to a valence or a sea quark of the proton scattered back in the Breit frame), with the relative weights given by the "rst term in the curly brackets in Eqs. (10.63) and (10.64), i.e. 4


ex  (x )O(Q , Q) GN    Q G G , (10.65) prob(no emission)" H prob(val/sea)
A N(s , Q) 2 and the con"gurations with at least one resolvable parton emission, with the weight 1!prob(no emission). In the case of no resolvable emission we have x "x "Q/s and the  #avor of the quark is generated according to the weights e (x ) if val , G GN  (10.66) e (x ) if sea . G GN  Then we are left with a trivial parton con"guration. For a valence quark contribution it consists of an anti-quark, moving along the original proton direction, and the quark, scattered back. In the case of at least one resolvable emission, one generates the type (#avor) i and the light cone momentum fraction x of the "rst parton of the QCD cascade according to the distributions, given  by the expressions in the curly brackets in (10.63) and (10.64),



G  (x )
AHG(x s , Q, Q ) if val , O GN   2   (10.67) prob(i, x )"  if sea ,  (x )
AHG(x s , Q, Q )  GN   2  where G is zero if i"g and otherwise one. O Then one chooses between di!erent types of interactions, contributing to the photon}parton cross section, according to their partial weights in Eqs. (10.56) and (10.57). The direct

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photon}parton contribution is chosen with the weight



[e
AHO #e
AHO #e
AHO ]/
AHG if"quark , G 2,1 O 21 A 2   2 (10.68) ]/
AHE if i"gluon . [e
AHE #e
AHE A 2   2 O 2  For the resolved photon contributions, the weight is given by the last two terms in Eqs. (10.56) and (10.57). The probability for the VDM part is prob(direct)"



prob(VDM)"
dx f 4"+(x , Q , Q)
HG (x (x s !Q), Q, Q , Q )/
AHG , (10.69) A HA A  2  A    2 H where x is the light cone momentum share and i is the type (#avor) of the "rst ladder parton on the  proton side (already determined), whereas the probability for the point-like resolved contribution is

 

dq

prob(point)"e dx PAOO (x ) O A 2 A q ;

HG (x x s !q, Q, q, Q )(q!max[Q , x Q])/
AHG . 2  A    A 2   M  HZ SBQSBQ (10.70) In case of a direct light contribution, one has to generate the con"guration for a parton ladder, strictly ordered in parton virtualities towards the virtual photon. The method is quite analogous to the one of Section 5.2 and is described in the next section. In case of a resolved contribution, we need to de"ne the initial conditions for the other end of the parton ladder, on the photon side, as well as the parton type (a quark of some #avor or a gluon) and the share of the light cone momentum fraction x , taken by the parton from the photon. For the A direct resolved contribution, corresponding to the point-like photon splitting into a quark}antiquark pair, the #avor j, the share x , and the virtuality q of the (anti-)quark, being the "rst ladder A parton, are generated according to 1

PAOO (x )
HG (x x s !q, Q, q, Q )(q!max[Q , x Q]) H . prob( j, x , q)& A 2  A    A O A q 2 (10.71) For the VDM contribution, the "rst ladder parton is taken at the initial virtuality Q and its type  j and momentum share x are chosen according to the distribution A prob( j, x )&f 4"+(x , Q , Q)
HG (x (x s !Q), Q, Q , Q ) , (10.72) A HA A  2  A    with the VDM parton momentum distributions in the photon de"ned in (10.37). In both cases for resolved photon interactions, the simulation of parton con"gurations, corresponding to the ladder of given mass squared s(  (s( "x x s !q for the direct resolved contribution and A  s( "x (x s !Q) for the VDM one), and of given types and virtualities of the leg partons, is done A  exactly in the same way as for proton}proton (nucleus}nucleus) interactions, as described in Section 5. The only di!erence comes from the presence of two di!erent scales M, M and the N A cuto! p 'Q in the parton}parton cross section
GH for resolved DIS processes as given in , 2  Eq. (10.55), when compared to the cross section in Eq. (6.7).

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229

In the case of interaction with the longitudinal photon component, the procedure simpli"es considerably, as one only has to consider direct photon}parton interactions via the parton}gluon fusion process. One starts by choosing between the coupling of the parton ladder to a valence quark (`vala) or to a soft Pomeron (`seaa), the weights are dx
 (x )
AHG(x s , Q, Q )  ; prob(val)"  G GN H  * 
A N(s , Q) *

(10.73)

dx
 (x )
AHG(x s , Q, Q )  .  G GN   *  prob(sea)"
AHN(s , Q) *

(10.74)

The weight for the parton type (#avor) i and the distribution for the light cone momentum fraction x of the "rst parton of the QCD cascade is given by the integrands of (10.73)}(10.74). The "nal step  amounts to generating the con"guration for the parton ladder, strictly ordered in parton virtualities towards the virtual photon, with the largest momentum transfer parton process of parton} gluon fusion type, as discussed in the next section. 10.7. Generating the ladder partons In this section we describe the procedure to generate parton con"gurations, corresponding to direct photon}parton interaction with at least one resolvable emission in the parton cascade. In that case, a parton ladder is strictly ordered in parton virtualities towards the virtual photon and the ladder cross section is given as a sum of the contributions Eqs. (10.51)}(10.54) in the case of transverse photon polarization,



(10.75)

(s , Q, Q )"e
AHG (s , Q, Q ) .
AHG *   A 2   

(10.76)

e
AHO (s , Q, Q )#e
AHO (s , Q, Q ) G 2,1  O 21  (s , Q, Q )" #e
AHO (s , Q, Q ) i"q ,
AHG A 2   2    e
AHE(s , Q, Q )#e
AHEAA (s , Q, Q ) i"g ,  A 2  O or by the cross section Eq. (10.54) for the longitudinal one,

All the photon}parton cross sections are expressed in terms of the QCD evolution functions EM . /!" Using the explicit representation Eqs. (B.30)}(B.32) for EM , one can rewrite the recursive relations /!" Eqs. (B.29), (10.49)}(10.50) in a form, such that the "rst (lowest virtuality) emission in the ladder is treated explicitly, multiplied by a weight factor, given by the contribution of the rest of the ladder (the sum of any number of additional resolvable emissions),



EM GH (x, Q , Q)" /!" 






/ dQ \C dz

x 
Q PI(z)G(Q , Q )EM IH , Q , Q   /!" z  Q z 2 G /  I V



#

/ dQ

 G(Q , Q )H(Q , Q) Q PH(x) ,    Q 2 G / 

 (10.77)

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/ dQ \C dz

x  Q PO (z)O(Q , Q )EM , Q , Q   ,1 z  z 2 O  Q  /  V / dQ

 Q PO (x) , #O(Q , Q)  Q 2 O /  / dQ \C dz x


EM (x, Q , Q)" O(Q , Q ) Q PO (z)EM , Q , Q 1    O 1 Q z z  2 /  I V x #PE (z)EM EO , Q , Q . O /!" z  EM (x, Q , Q)" ,1 

















(10.78)

 (10.79)

With the help of Eqs. (10.77)}(10.79) and (10.51)}(10.54), one can obtain the recursive relations for the cross sections equations (10.75) and (10.76) for an arbitrary virtuality Q of the initial parton i,   / dQ I 

(s , Q, Q )"
G(Q , QI ) dz Q PH (z)
AHH (zs , Q, QI )
AHG 2*    2*  I  2 G  Q  / H #
AHG (s , Q, Q ) , (10.80) 2*    represents the contribution of parton con"gurations with only one resolvable where
AHG 2*   parton emission in the ladder, or with the photon}gluon fusion process without any additional resolvable parton emissions,










4 e Q G EOO (s , Q, Q )" , Q , Q (Q!Q ) (i"quark) ,
AHG 2    /!" s   s




(10.81)



Q 4 e O EEO (s , Q, Q )" , Q , Q (Q!Q )
AHE 2    /!" s   s

 

d
AHEAA (s , Q, p ) 2 , E(Q , M ) , (10.82)  $ dp , H  (s , Q, p ) EAA d
A , E(Q , M ) .
AHG (s , Q, Q )" Ee dp * (10.83) *    G A ,  $ dp , It is noteworthy that to the leading logarithmic accuracy one should not use in
AHG the 2   Born process matrix elements d
AHEOO /dp and d
AHOEO/dp ; the contribution of just one resolv2 , 2 , able parton emission is proportional to EGO , Eq. (B.32), de"ned by the corresponding /!" Altarelli}Parisi kernel PO(z). G Formulas (10.80)}(10.83) allow us to generate the cascade of partons, corresponding to the direct H-quark (gluon) interaction of energy squared s( "s !Q and photon virtuality Q, starting from an initial parton with a #avor i, taken at a scale Q "Q . We use an iterative procedure, similar to   the one of Section 5. At each step one checks whether there is any additional resolvable parton emission before the last one, with the probability #e dp A ,


AHG (s , Q, Q )!
AHG (s , Q, Q )  2*    . prob(forward emission)" 2*  H (s , Q, Q )
A G 2*  

(10.84)

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231

In case of an emission, the #avor j of the new ladder leg parton, the light cone momentum fraction z, taken from the parent parton, and the virtuality QI  are generated according to the !
AHG in (10.80), integrand of
AHG 2*  2*   1

prob( j, z, QI )& G(Q , QI ) Q PH(z)
AHH (zs , Q, QI ) .  2*  QI  2 G

(10.85)

The process is repeated for the new ladder of energy squared s( "zs !Q, with the initial parton i"j, and with virtuality Q "QI  and so on. At each step one decides about emission or not,  which "nally terminates the iteration. Having done the iterative parton emission, we "nally generate the last (highest virtuality) resolvable parton emission or the photon}gluon fusion process (if i"g) in the photon}parton center-of-mass system. The photon}gluon fusion (PGF) process is chosen for i"g with the probability



d
AHEAA (s , Q, p ) 1 , E(QI , M ) e dp 2 prob(PGF)" H , $ dp (s , Q, QI ) A
A E , 2  

(10.86)

for the transverse photon polarization and always for the longitudinal photon polarization. For i"q, we have prob(PGF)"0. In case of PGF, with our choice M "m#p , we generate the $ A , transverse momentum squared of "nal charm quarks in the region QI !m(p (s( !m A ,  A according to d
AHEAA (s , Q, p ) , E(QI , M ) . prob(p )& 2 , $ dp ,

(10.87)

In case of no PGF, we generate the momentum transfer squared for the last resolvable parton emission in the range QI (Q(Q according to 1

prob(Q)& G(QI , Q)O(Q, Q) Q PO(z) , Q 2 G

(10.88)

with




Q z"1! 1! s






Q 1! , s

(10.89)

and "nd the parton transverse momentum squared as p "Q(1!z). , We then reconstruct "nal parton 4-momenta in their center of mass system with a random polar angle for the transverse momentum vector po and boost them to the original Lorentz frame. This , completed the description of the algorithm to generate parton con"gurations, based on exactly the same formulas as for calculations of F before.  The above discussion of how to generate parton con"gurations is not yet complete: the emitted partons are in general o!-shell and can therefore radiate further partons. This so called time-like radiation is taken into account using standard techniques [171], as discussed already in Section 5.

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10.8. Hadron production For the hadronization, we use exactly the same philosophy and even the same procedure as in case of proton}proton (pp) scattering. Hadronization is not considered as a dynamical procedure, rather we consider the hadronic states as being integrated out in the considerations of cross section calculations of the preceding sections. Hadronization means simply a phenomenological procedure to explicitly reintroduce these hadronic states. The procedure employed for pp scattering and to be used here as well is as follows: 1. 2. 3. 4. 5.

drawing a cylinder diagram, cutting the cylinder, planar presentation of half-cylinder, identi"cation of cut line with kinky string, kinky string hadronization (as explained in Section 6).

We are going to explain steps 1}4 for a concrete example of a diagram contributing to photon}proton scattering, where the photon interacts directly with a light quark (contribution `lighta), and where the "rst parton of the ladder on the proton side couples to the proton via a soft Pomeron (contribution `seaa), as shown in Fig. 10.11. The external legs on the lower (proton) side are a quark (full line) and an anti-quark (dashed), representing together the proton constituent participating in the interaction. In Fig. 10.12 (left), we show the result of plotting the diagram on a cylinder. The shaded area on the lower part of the cylinder indicates the soft Pomeron, a complicated non-resolved structure, where we do not specify the microscopic content. The two space-like gluons emerge out of this soft structure. The cut is represented by the two vertical dotted lines on the cylinder. We now consider one of the two half-cylinders, say the left one, and we plot it in a planar fashion, as shown in Fig. 10.12 (right). We observe one internal gluon, and one external one, appearing on the cut line. We now identify the two cut lines with kinky strings such that a parton on the cut line corresponds to a kink: we have one kinky string with one internal kink (gluon) in addition to the two end kinks, and we have one #at string with just two end kinks, but no

Fig. 10.11. A diagram contributing to the photon}proton cross section. Fig. 10.12. Cylindrical representation of the diagram of Fig. 10.11 (left "gure) and planar diagram representing the corresponding half-cylinder (right "gure).

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233

Fig. 10.13. The string model: the cut line (vertical dotted line) corresponds to a string, which decays into hadrons (circles). Fig. 10.14. Distribution of the simulated events of the Q}x plane.

internal one. The strings are then hadronized according to the methods explained in Section 6, see Fig. 10.13. In principle, we have also triple Pomerons contributing to the event topology. However, due to AGK cancellations, such contributions to the inclusive spectra cancel each other in the kinematical region where the two Pomerons are in parallel. Therefore, the average characteristics are correctly described by considering the simple cylinder-type topology corresponding to one Pomeron exchange. 10.9. Results We are now capable to simulate events from deep inelastic scattering. When "xing the parameters, we found that all the ones found in e>e\ can be kept with the exception of the mean transverse momentum of string breaking pD and the break probability p , see the discussion in ,
  Section 8. We show results from ep scattering and compare to the data of the experiments accomplished at HERA. Electrons of an energy 26.7 GeV collide with protons of 820 GeV, which gives a center of mass energy 296 GeV. We made the analysis for the kinematical region 10\(x(10\ and 10(Q(100 GeV. The distribution of the events calculated using our model is shown in Fig. 10.14. We recall the principal variables Q Q " ; x,x " ys 2(pq)

Q y" , xs

(10.90)

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which gives straight lines for y"const. in Fig. 10.14. The sharp borders are due to imposing the experimental cuts 0.03(y, E'12 GeV and  '7.5. In Fig. 10.15, we plot charged particle C distributions for di!erent values of ="s( "2(pq)!Q. The particle spectra look very similar for di!erent = values; the distributions decrease rapidly with x . The dependence of the average p on $ , x shows an overall good agreement with the data from H1 collaboration [253]. $ In Table 10.1 the bins in x and Q are given for the experimental data points in Figs. 10.15}10.20 (see [254]). The bin 0 is the sum of all the others. First, we compare the p distribution in the , photon}proton center of mass system for the di!erent bins } Fig. 10.16. Fig. 10.16(a) shows the comparison of the results for low and high x values for the values of QK10}20 GeV (bins 6 and 3). We "nd harder spectra for smaller x, which is the consequence of the larger kinematical space (in x) for the initial state radiation. Next, in Figs. 10.16(b) and (c), the spectra are compared for two di!erent values of Q and either the energy =&Q/x being "xed (bins 2 and 7, Fig. 10.16(b)), or for a given value of x (bins 6 and 8, Fig. 10.16(c)). The spectra in p are always harder for larger Q, ,

Fig. 10.15. x distribution of charged particles for di!erent values of =: 50}100, 100}150, 150}200 GeV and for the total $ = region of 50}200 GeV. The last diagram shows mean p as a function of x . All the variables are de"ned in the , $ hadronic center of mass system. The data are from H1 collaboration [253].

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Table 10.1 Bins in x and Q for Figs. 10.15}10.20 Bin

x (;10\)

Q (GeV)

x (;10\)

Q (GeV)

= (GeV)

0 1 2 3 4 5 6 7 8 9

0.1}10 0.1}0.2 0.2}0.5 0.2}0.5 0.5}0.8 0.8}1.5 1.5}4.0 0.5}1.4 1.4}3.0 3.0}10

5}50 5}10 6}10 10}20 10}20 10}20 10}20 20}50 20}50 20}50

1.14 0.16 0.29 0.37 0.64 1.1 2.1 0.93 2.1 4.4

18.3 7.7 8.8 13.1 14.0 14.3 15.3 28.6 31.6 34.7

24 975 45 296 31 686 36 893 22 401 13 498 7543 32 390 16 025 8225

Fig. 10.16. p distribution of charged particles for di!erent kinematic regions. The values of the cuts are indicated on the , "gures (column 1 } full line, column 2 } dashed line).

which is now the consequence of the larger kinematical space in p for the initial state radiation. , Two cuts in pseudo-rapidity  for bin 3 are considered in Fig. 10.16(d). We see a harder distribution for 1.5((2.5. Around mid-rapidity, where  is maximal, one "nds higher transverse momenta as this region is dominated by the contribution of the largest virtuality photon process.

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Fig. 10.17. Pseudo rapidity distribution of charged particles for the bins indicated in Table 10.1. The lower left diagram represents no cut results.

Let us now consider pseudo-rapidity distributions of charged particles. Figs. 10.17 and 10.18 show the -distributions for the 9 bins of Table 10.1 for di!erent values of Q and x. In Fig. 10.18, a cut for p '1 GeV has been made to extract the contribution of hard processes. The latter one , results in approximately 10% of the total hadron multiplicity. In Fig. 10.18 we "nd fewer particles

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Fig. 10.18. The same as in Fig. 10.17, but with an additional cut p '1 GeV. ,

at small  which is the consequence of smaller kinematical space (in x) for the initial state radiation and the reduced in#uence of the largest virtuality photon process. The transverse momenta for all particles are generally well described by the model } Figs. 10.19 and 10.20. The fact that we "nd harder spectra for higher Q and lower x is best seen for smaller values of  } see Fig. 10.20.

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Fig. 10.19. Transverse momentum of charged particles for 1.5((2.5.

11. Results for proton}proton scattering In this section we are going to discuss our results for proton}proton interactions in the energy range between roughly 10 and 2000 GeV, which represents the range of validity of our approach.

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Fig. 10.20. Transverse momentum of charged particles for 0.5((1.5.

The lower limit is a fundamental limitation due to the fact that our approach requires hadron production to start after the primary interactions are "nished, which is no longer ful"lled at low energies. The upper limit is due to the fact that above 2000 GeV higher-order screening corrections need to be taken into account.

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11.1. Energy dependence We "rst consider the energy dependence of some elementary quantities in pp scattering in the above mentioned energy range. In Fig. 11.1, the results for the total cross section are shown. The cross section is essentially used to "t the soft Pomeron parameters. Also shown in the "gure is the energy dependence of the number of soft and semi-hard Pomerons. Over the whole energy range shown in the "gure, soft physics is dominating. So for example at RHIC, soft physics dominates by far. Our results for hadron production are based on the Pomeron parameters, de"ned from the cross sections "tting, and on the fragmentation procedure, adjusted on the basis of e>e\ data. In Fig. 11.2, average multiplicities of di!erent hadron species are given as a function of the energy. In Fig. 11.3, we show the energy dependence of the pseudo-rapidity plateau dn /d(0) and of the ! mean squared transverse momentum p. R 11.2. Charged particle and pion spectra In Fig. 11.4, we present rapidity distributions of pions at 100 GeV, in Fig. 11.5 rapidity distributions of pions and charged particles at 200 GeV. The values following the Symbol `I"a represent the integrals, i.e. the average multiplicity. The "rst number is the simulation, the second number (in brackets) represents data. In Fig. 11.6, we show rapidity distributions for positively and negatively charged particles at 53 GeV (cms), where we adopted also for the simulations the experimental de"nition of the rapidity by always taking the pion mass. In Fig. 11.7, we show pseudo-rapidity distributions of charged particles at 200 and 1800 GeV (cms). In Figs. 11.8 and 11.9, we "nally show transverse momentum spectra at di!erent energies between 100 GeV (lab) and 1800 GeV (cms). 11.3. Proton spectra In Fig. 11.10, we plot longitudinal momentum fraction distributions of protons for di!erent values of t at 200 GeV, in Figs. 11.11 and 11.12 as well longitudinal momentum fraction

Fig. 11.1. The total cross section as a function of the energy (s (left "gure): the full line is the simulation, the points represent data [255]. Pomeron numbers as a function of the energy (s (right): soft (dashed) and semi-hard (solid line) Pomerons.

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Fig. 11.2. The average multiplicities of di!erent hadron species, as a function of the energy (s. From top to bottom: all charged particles, >, \, K>, K\, p . The full lines are simulations, the points represent data [256].

distributions at 100}200 GeV, for given values of p or integrated over p . In Fig. 11.13, we show R R transverse momentum spectra of protons for di!erent values of the longitudinal momentum fraction x at 100 and 205 GeV. 11.4. Strange particle spectra In Figs. 11.14 and 11.15, we show transverse momentum and rapidity spectra of lambdas (including ), anti-lambdas (including M ), and kaons. The numbers represent the integrals,   i.e. the average multiplicity. The "rst number is the simulation, the second number (in brackets) represents data.

12. Results for collisions involving nuclei It is well known that secondary interactions play an important role in collisions involving nuclei. Nevertheless, in this report, we do not want to consider any rescattering procedure, we just present bare NEXUS simulations. This seems to us the most honest way to present results. 12.1. Proton}nucleus scattering In Fig. 12.1, we show rapidity spectra of negatively charged particles for di!erent target nuclei. Missing particles in the backward region are certainly due to rescattering. In the forward region,

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Fig. 11.3. Pseudo-rapidity plateau dn/d(0) and mean squared transverse momentum p as a function of the energy R (s. The full lines are simulations, the points represent data [257}259].

Fig. 11.4. Rapidity distributions of pions at 100 GeV. The full lines are simulations, the points represent data [260].

Fig. 11.5. Pseudo-rapidity distributions of pions (>, \) and charged particles (all charged and negatively charged) at 200 GeV. The full lines are simulations, the points represent data [260,261].

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Fig. 11.6. Pseudo-rapidity distributions of positively and negatively charged particles at 53 GeV (cms). The full lines are simulations, the points represent data [262].

Fig. 11.7. Pseudo-rapidity distributions of charged particles at 200 and 1800 GeV (cms). The full lines are simulations, the points represent data [258,263].

Fig. 11.8. Transverse momentum distributions of pions or negatively charged particles at di!erent energies. The full lines are simulations, the points represent data [262,264}266].

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Fig. 11.9. Transverse momentum distributions of charged particles at (from bottom to top) 200, 900, and 1800 GeV (cms). The full lines are simulations, the points represent data [259,267].

Fig. 11.10. Longitudinal momentum fraction distributions of protons for di!erent values of t at 200 GeV. The full lines are simulations, the points represent data [268].

the model works well, except for p#Au, which represents the heaviest target, but in addition one has here a centrality trigger, in contrast to the other reactions. Here, we expect some reduction due to nuclear screening e!ects. The transverse momentum spectra are well reproduced in case of p#S, as shown in Fig. 12.2, whereas for p#Au one sees some deviations for small values of p , see R Fig. 12.3. Let us turn to proton spectra. In Fig. 12.4, we show rapidity spectra of net protons (protons minus anti-protons) for di!erent target nuclei. Since secondary interactions are not considered, we

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Fig. 11.11. Longitudinal momentum fraction distribution of protons at 100 GeV, integrated over p (left) and for R p "0.75 GeV/c (right). The full lines are simulations, the points represent data [265]. R

Fig. 11.12. Longitudinal momentum fraction distributions of protons at 200 GeV, integrated over p (left) and at R 175 GeV, for p "0.75 GeV/c (right). The full lines are simulations, the points represent data [260,265]. R

are missing the pronounced peak around rapidity zero (not visible in the "gure, since we have chosen the range for the y-axis to be [0, 1]). Apart from the target fragmentation region, the model works well. The transverse momentum spectra shown in Fig. 12.5 refer to the target fragmentation region, and therefore the absolute number is too small, whereas the shape of the spectra is quite good. Strange particle spectra are shown in Figs. 12.6 and 12.7. Whereas the simulations agree with the data for the kaons, the spectra are largely underestimated in case of lambdas, in particular in the backward region, where rescattering plays a dominant role. 12.2. Nucleus}nucleus scattering Again, we show results of the bare NEXUS model, without any secondary interactions. Considering rapidity distributions of negatively charged particles, as shown in Fig. 12.8, we observe a strong excess at central rapidities compared to the data. Rescattering will partly cure this, but not completely. For asymmetric systems like for example S#Ag, we observe in addition a missing asymmetry in the shape of the rapidity spectrum, in other words, the simulated spectrum is too symmetric. This is not surprising, since in our approach AGK cancellations apply, which make A#B spectra identical to the p#p ones, up to a factor. Rescattering will not cure this, since it acts

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Fig. 11.13. Transverse momentum spectra of protons for di!erent values of the longitudinal momentum fraction x at 100 and 205 GeV. The full lines are simulations, the points represent data [260,265].

essentially at central rapidities. But we expect another important e!ect due to additional screening e!ects coming from contributions of enhanced Pomeron diagrams. In Fig. 12.9, we show transverse momentum or transverse mass spectra of negatively charged particles for di!erent A#B collisions at 200 GeV (lab). Several rapidity windows are shown; from top to bottom: 3.15}3.65, 3.65}4.15, 4.15}4.65, 4.65}5.15, 5.15}5.65 in case of Pb#Pb and 0.8}2, 2}3, 3}4, 4}4.4 for the other reactions. The lowest curves are properly normalized, the next ones are multiplied by ten, etc. We again observe an excess at certain rapidities, as already discussed above. In addition, in particular for heavy systems, the slopes are too steep, there is clearly some need of secondary interactions to `heat upa the system. This is consistent with the fact that the multiplicity is too high: collective motion should reduce the multiplicity but instead increase the transverse energy per particle. In Fig. 12.10, we show rapidity distributions of net protons (protons minus anti-protons) for di!erent A#B collisions at 200 GeV. For the asymmetric systems one observes clearly the e!ect of missing target nucleons, which should be cured by rescattering. The simulated results for

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Fig. 11.14. Distributions of transverse momentum (left) and rapidity (right) of lambdas plus neutral sigmas (upper) and of anti-lambdas plus neutral anti-sigmas (lower) at 205 GeV. The full lines are simulations, the points represent data [269].

Fig. 11.15. Rapidity distributions of kaons (K , K\, K>) at 205 GeV. The numbers represent y-integrated results: the Q "rst number is the simulated one, the number in brackets the experimental one [270].

symmetric systems are close to the data, rescattering does not contribute much here. But as for pion production, we expect also some changes due to screening e!ects. Transverse momentum spectra, as shown in Fig. 12.11, show a similar behavior as for pions but even more pronounced: the theoretical spectra are much too steep, in particular for heavy systems. In Figs. 12.12}12.15, we show rapidity spectra of strange particles. K\ seem to be correct, whereas K> are in general somewhat too low compared to the data. Lambdas and anti-lambdas are way too low. For these particles, rescattering has to provide most of the particles which are "nally observed.

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Fig. 12.1. Rapidity distributions of charged particles or negative pions for di!erent p#A collisions at 200 GeV (lab). The full lines are simulations, the points represent data [261,271,272].

Fig. 12.2. Transverse momentum distributions of negatively charged particles for di!erent p#S collisions at 200 GeV (lab). The full lines are simulations, the points represent data [272].

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Fig. 12.3. Transverse momentum distributions of negatively charged particles for di!erent p#Au collisions at 200 GeV (lab). The full lines are simulations, the points represent data [272].

Fig. 12.4. Rapidity distributions of net protons for di!erent p#A collisions at 200 GeV (lab). The full lines are simulations, the points represent data [271,272].

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Fig. 12.5. Transverse momentum spectra of net protons for di!erent p#A collisions at 200 GeV (lab). The full lines are simulations, the points represent data [272].

!

Fig. 12.6. Rapidity distributions of net lambdas for di!erent p#A collisions at 200 GeV (lab). The full lines are simulations, the points represent data [272].

Fig. 12.7. Rapidity and p spectra of net lambdas and K for p#Xe collisions at 200 GeV (lab). The full lines are R Q simulations, the points represent data [273].

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Fig. 12.8. Rapidity distributions of negatively charged particles for di!erent A#B collisions at 200 GeV (lab). The full lines are simulations, the points represent data [272,274].

Fig. 12.9. Transverse momentum or transverse mass spectra of negatively charged particles in several rapidity windows (see text) for di!erent A#B collisions at 200 GeV (lab). The full lines are simulations, the points represent data [272,274].

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Fig. 12.10. Rapidity distributions of net protons for di!erent A#B collisions at 200 GeV (lab). The full lines are simulations, the points represent data [272,274].

Fig. 12.11. p distributions of net protons for di!erent A#B collisions at 200 GeV (lab). The full lines are simulations, the R points represent data [272,274].

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Fig. 12.12. Rapidity distributions of kaons for S#S and Pb#Pb. The full lines are simulations, the points represent data [275}277].

13. Summary We presented a new approach for hadronic interactions and for the initial stage of nuclear collisions, which solves several conceptual problems of certain classes of models, which are presently widely used in order to understand experimental data. The main problem of these models is the fact that energy is not conserved in a consistent fashion: the fact that energy needs to be shared between many elementary interactions in case of multiple scattering is well taken into account when calculating particle production, but energy conservation is not taken care of in cross section calculations. Related to this problem is the fact that di!erent elementary interactions in case of multiple scattering are usually not treated equally, so the "rst interaction is usually considered to be quite di!erent compared to the subsequent ones.

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Fig. 12.13. Rapidity distributions of kaons for S#Ag and S#Au. The full lines are simulations, the points represent data [275].

We provided a rigorous treatment of the multiple scattering aspect, such that questions as energy conservation are clearly determined by the rules of "eld theory, both for cross section and particle production calculations. In both cases, energy is properly shared between the di!erent interactions happening in parallel. This is the most important and new aspect of our approach, which we consider to be a "rst necessary step to construct a consistent model for high energy nuclear scattering. Another important aspect of our approach is the hypothesis that particle production is a universal process for all the elementary interactions, from e>e\ annihilation to nucleus}nucleus scattering. That is why we also carefully study e>e\ annihilation and deep inelastic scattering. This allows

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Fig. 12.14. Rapidity distributions of (from top to bottom) lambdas, anti-lambdas, and net lambdas for S#S and Pb#Pb. The full lines are simulations, the points represent data [272,275].

to control reasonably well for example the hadronization procedure, which is not treatable theoretically from "rst principles.

Acknowledgements This work has been funded in part by the IN2P3/CNRS (PICS 580) and the Russian Foundation of Basic Researches (RFBR-98-02-22024). Finally, we would like to thank the members of the SUBATECH theory group for many helpful comments. Furthermore, we are very grateful to E. Reya, B. Kopeliovich, R. Engel, and T. Thouw for interesting and useful discussions.

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Fig. 12.15. Rapidity distributions of (from top to bottom) lambdas, anti-lambdas, and net lambdas for S#Ag and S#Au. The full lines are simulations, the points represent data [272,275].

Appendix A. Kinematics of two-body collisions A.1. Conventions We consider a scattering of a projectile P on a target ¹ (hadron}hadron or parton}parton). We de"ne the incident 4-momenta to be p and p and the transferred momentum q, so that the outgoing momenta are p "p#q and p "p!q, see Fig. A.1. We de"ne as usual the Mandelstam variables s"(p#p),

t"(p !p) .

(A.1)

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Fig. A.1. Two body kinematics.

Usually, we employ light cone momentum variables, connected to the energy and z-component of the particle momentum, as p!"p $p ,  X and we denote the particle 4-vector as p"(p>, p\, po ) . ,

(A.2)

(A.3)

A.2. Proof of the impossibility of longitudinal excitations Here, we present a mathematical proof of the well known fact that longitudinal excitations are impossible at high energies. Consider a collision between two hadrons h and h which leads to two hadrons hI and hI , h(p)#h(p)PhI (p )#hI (p )

(A.4)

with four-momenta p, p, p , p . As usual, we de"ne s"(p#p) and t"(p !p). For the following, we consider always the limit sPR and ignore terms of the order p/s. We expand q"p !p"p!p  as q" p#p#q , , and obtain 2qp 2qp , " ,

" s s

(A.5)

(A.6)

where we used p"0, p"0, pq "pq "0, s"2pp . , , We get q q" (p!p)#q , , s

(A.7)

(A.8)

having used q"p !p"p!p , p"p"0, pp "pp "!q/2 .

(A.9)

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This proves q"q

(A.10) , for sPR and limited q. In other words, momentum transfer is purely transversal at high energies.

Appendix B. Partonic interaction amplitudes B.1. Semi-hard parton}parton scattering Let us derive the mathematical expression corresponding to the contribution of so-called semi-hard parton}parton scattering, see Fig. B.1. Here p, p are the 4-momenta of the constituent partons participating in the process. We denote by k, k the 4-momenta of the "rst partons entering the perturbative evolution, i.e. the initial partons for the perturbative parton cascade (characterized by parton virtualities Q'Q ). Further, we de"ne light cone momentum fractions  p\ k> k\ p> , (B.1) x>" , x\" , x>" , x\"   p\ p> p\ p>     with p! being the total light-cone momenta for the interaction.  At high energies, the dominant contribution to the process comes from the kinematical region where these partons are slow, i.e. x!;x!, so that a relatively small contribution of the pertur bative parton cascade (of the ladder part of the diagram of Fig. B.1) is compensated by the large density of such partons, resulting from the soft pre-evolution [122,145]. Since the initial partons k, k are gluons or sea quarks (contrary to valence quarks) we talk about `sea}seaa contribution. Let us "rst consider the case where the intermediate partons k, k are gluons. Then the amplitude for the diagram of Fig. B.1 can be written as



dk dk i¹EE } (s( , t)"   (2) (2)


iME (p,!k, p#q,!k!q)  HA HHYAAYBBYOOY ;DE (k)DE ((k#q)) iMEE (k, k, k#q, k!q, Q ) HB AO    BOBYOY ;DE (k)DE ((k!q)) iME (p,!k, p!q,!k#q) , HYBY AYOY  HYAY

Fig. B.1. Semi-hard contribution ¹

 }

.

(B.2)

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where the amplitude MEE (k, k, k#q, k!q, Q ) represents the perturbative contribution of    BOBYOY the parton ladder with the initial partons of momenta k, k and with the momentum transfer along the ladder q (hard parton}parton scattering), the amplitude ME (p, k, p#q, k!q) corresponds  HA to the non-perturbative soft interaction between the constituent parton with the 4-momentum p and the gluon with the 4-momentum k, and DE (k) is the non-perturbative gluon propagator, HB DE (k)"iDI E(k)  (k) with  (k) being the usual gluon polarization tensor in the axial gauge; HB HB HB , , ,2 denotes symbolically the combination of color and Lorentz indexes for the intermediate gluons. As by construction partons of large virtualities Q'Q can only appear in the parton ladder  part of the diagram of Fig. B.1, we assume that the integral dk converges in the region of restricted virtualities k&!s with s K1 GeV being the typical hadronic mass scale, i.e. the   combination ME (p,!k, p#q,!k!q) DE (k)DE ((k#q)) (B.3)  HA HB AO drops down fast with increasing k; this implies that the transverse momentum k is also restricted , to the region k 4s . Similar arguments apply for k. Further we make the usual assumption that ,  longitudinal polarizations in the gluon propagators DE (k) are canceled in the convolution with HB the soft contribution ME even for "nite gluon virtualities k [135]. Finally we assume that in the  considered limit x!;x! the amplitude ME (p, k, p#q, k!q) is governed by the non-pertur  HA bative soft Pomeron exchange between the constituent parton p and the gluon k, which implies in particular that it has the singlet structure in the color and Lorentz indexes: ME (p, k, p#q, k!q) & H . (B.4)  HA A Then, for small momentum transfer q in the process of Fig. B.1 the intermediate gluons of momenta k, k, k#q, k!q can be considered as real (on-shell) ones with respect to the perturbative parton evolution in the ladder, characterized by large momentum transfers Q'Q . Then we obtain  1
MEE (k, k, k#q, k!q, Q )    BOBYOY K E HHYBBYOOY ; (k) (k#q) (k) (k!q)K¹EE (s( , q, Q ) , (B.5) HB BO HYBY BYOY      where the averaging over the spins and the colors of the initial gluons k, k is incorporated in the factor K, ¹EE (s( , t, Q ) is de"ned in (2.16), and s( "(k#k)Kk>k\"x>x\s. E          Now, using (B.4) and (B.5) we can rewrite (B.2) as



i¹EE } (s( , t)"  

dk> dk\ dk dk> dk\dk , , i¹EE (s( , q, Q )      2(2) 2(2)

 



; !i
ME (p,!k, p#q,!k!q) DI E(k)DI E((k#q))  HH H



; !i
ME (p,!k, p!q,!k#q) DI E(k) DI E((k!q)) . (B.6)  HYHY HY It is convenient to perform separately the integrations over k\, k>, k , k keeping in mind that the , , only dependence on those variables in Eq. (B.6) appears in the non-perturbative contributions in

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the square brackets. Let us consider the "rst of those contributions, corresponding to the upper soft blob in Fig. B.1 together with the intermediate gluon propagators (for the lower blob the derivation is identical). Being described by the soft Pomeron exchange, the amplitude ME (p,!k, p#q,!k!q) is an analytical function of the energy invariants s( "(p!k)K  HH  !p>k\ and u( "(p#k#q)Kp>k\ with the singularities in the complex k\-plane, corre sponding to the values s( "s!i0 for all real s which are greater than some threshold value  s for the Pomeron asymptotics to be applied, as well as for u( "u!i0, where u'u with    some threshold value u [44]. Thus one has the singularities in the variable k\ in the upper half of  the complex plane at k\K(!s!i0)/p> and in the lower half of the complex plane at k\K(u!i0)/p>. Then one can use the standard trick to rotate the integration contour C in the variable k\ such that the new contour C encloses the left-hand singularities in k\, corresponding to the right-hand singularities in the variable s( [44]. Then one ends up with the integral over the  discontinuity of the amplitude ME on the left-hand cut in the variable k\, which is up to a minus  sign equal to the discontinuity on the right-hand cut in s( :  > dk\
ME (p,!k, p#q,!k!q) DI E(k) DI E((k#q))  HH \ H \Q I> dk\ disc ( 
ME (p,!k, p#q,!k!q) DI E(k) DI E((k#q)) " (B.7) Q  HH \ H \Q I> dk\ 2i Im
ME (p,!k, p#q,!k!q) DI E(k) DI E((k#q)) . (B.8) "  HH \ H Now, using  dk\" dk/k> and recalling our assumption that the integral over k gets dominant contribution from the region k&!s , we may write  dk\ dk 1 , Im
ME (p,!k, p#q,!k!q) DI E(k)DI E((k#q)) " Im ¹E (s( , q)  HH   (2) k> H (B.9)



 



















with p> x> "s . (B.10)  k>  x>  The integrations over k in the vicinity of k"!s and over k 4s are supposed to just  ,  contribute to the redetermination of the Pomeron}gluon coupling of the usual soft Pomeron amplitude, and therefore we parameterize the amplitude ¹E as (compare with Eq. (2.5))  s( ?.  s @E exp(  (s( /s ) t) 1!  , (B.11) ¹E (s( , t)"8s (t)         E s s(  with s(



"s









 (z)"R #  ln z , (B.12)     where we used  for the Pomeron}gluon coupling and we neglected the radius of the Pomeron} E gluon vertex assuming that the coupling is local in the soft Pomeron; the factor (1!s /s( )@E is 

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included to ensure that the Pomeron has su$ciently large mass, which is the necessary condition for applying Regge description for the soft evolution. As we shall see below the parameter  determines the gluon momentum distribution in the Pomeron at x!Px!. E  Finally, using the above results we obtain

 

dk> dk\ Im ¹E (s( , t) Im ¹E (s(  , t) i¹EE (s( , t, Q ) i¹EE } (s( , t)"            k> k\







 dz> dz\ s s  , t Im ¹E  , t i¹EE (z>z\s( , t, Q ) , Im ¹E  z>  z\    z> z\  where the following de"nitions have been used: "

(B.13) (B.14)

x! x> s x\ s "  , s(  "s "  , s( "x>x\s"z>z\s( . (B.15) z!"  , s( "s   x> z>   x\ z\     x!   In the case of the intermediate parton k being a (anti-)quark, we assume that it originates from local gluon splitting in the soft Pomeron. Thus we neglect the slope of the (non-perturbative) gluon}quark vertex. Using the usual Altarelli}Parisi kernel PO (z)"(z#(1!z)) for the gluon E  light cone momentum partition between the quark and the anti-quark, we get the corresponding amplitude ¹OE } (s( , t) as    dz> dz\ s s  , t Im ¹E  , t i¹OE (z>z\s( , t, Q ) , i¹OE } (s( , t)" Im ¹O (B.16)    z>  z\    z> z\  where the imaginary part of the amplitude Im ¹O for the soft Pomeron exchange between the  constituent parton and the quark q3u, d, s, u , dM , s  is expressed via Im ¹E as 











Im ¹O (s( , t)" d PO ( ) Im ¹E ( s( , t) , E     OE

(B.17)

with  representing the quark}gluon vertex value and being the ratio of the quark and the OE parent gluon light cone momentum, "k>/k>; the mass squared of the Pomeron between the E initial constituent parton and the gluon is then p> " s( . (B.18) (p!k )Ks  E  k> E The full amplitude for the semi-hard scattering is the sum of the di!erent terms discussed above, i.e. i¹ } (s( , t)"
i¹HI } (s( , t)     HI  dz> dz\ s s  , t Im ¹I  , t i¹HI (z>z\s( , t, Q ) , Im ¹H (B.19) "
 z>  z\    z> z\ HI  where j, k denote the types (#avors) of the initial partons for the perturbative evolution (quarks or gluons).









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The discontinuity of the amplitude ¹ } (s( , t) on the right-hand cut in the variable s( de"nes the   contribution of the cut diagram of Fig. B.1. Cutting procedure amounts here to replace the hard parton}parton scattering amplitude i¹HI (s( , t, Q ) in (B.19) by 2Im ¹HI (s( , t, Q ), whereas           the contributions of the soft parton cascades Im ¹H stay unchanged as they are already de"ned by  the discontinuities in the corresponding variables s( and s(  . So the cut diagram contribution   is just 2Im ¹ } (s( , t). At t"0 it de"nes the cross section for the semi-hard parton}parton   scattering:



1 
} (s( )" 2Im ¹ } (s( , 0)"
dz> dz\ EH (z>)EI (z\)
HI (z>z\s( , Q ) , (B.20)          2s( HI  where we used the explicit expressions (2.16), (B.11), (B.17) for ¹HI , ¹H , and the functions    EH are de"ned as  (B.21) EE (z)"8s   z\? (1!z)@E     E  z EO (z)" . (B.22) d PO ( )EE  OE E  X It is easy to see that EH (z) has the meaning of the momentum distribution of parton j at the scale  Q for an elementary interaction, i.e. parton distribution in the soft Pomeron. Introducing the gluon  splitting probability w and the coupling  via   E   "w  ,  "(1!w ) , (B.23) OE E   E E   E the light cone momentum conservation reads










 dz
zEH (z)"8s   dz z\? (1!z)@E     E   H and therefore 1"



1 (3! (0)# )  E  " . E 8s  (2! (0))(1# )     E

(B.24)

(B.25)

B.2. Parton evolution In this section, we discuss the properties of the evolution function E , describing the /!" perturbative evolution of partons. The evolution function EHK (z, Q , Q) satis"es the usual DGLAP equation /!"   dz

x dEHK (Q , Q, x) Q PI K(z)EHI /!"  "
, Q , Q (B.26) I /!" z 2 z  d ln Q I V with the initial condition EHK (Q , Q , z)" K (1!z). Here PI K(z) are the usual Altarelli}Parisi /!"   H I splitting functions, regularized at zP1 by the contribution of virtual emissions. One can introduce the concept of `resolvablea parton emission, i.e. an emission of a "nal (s-channel) parton with a "nite share of the parent parton light cone momentum (1!z)'" p /Q (with "nite relative transverse momentum p "Q(1!z)'p ) [106] and use the , , ,








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so-called Sudakov form factor, corresponding to the contribution of any number of virtual and unresolvable emissions (i.e. emissions with (1!z)() } see Fig. B.2.







/ dq 

dz Q PI I (z) (B.27) I q 2  \C / This can also be interpreted as the probability of no resolvable emission between Q and Q.  Then EHK can be expressed via EM HK , corresponding to the sum of any number (but at least one) /!" /!" resolvable emissions, allowed by the kinematics: I(Q , Q)"exp 

EHK (z, Q , Q)" K (1!z)H(Q , Q)#EM HK (z, Q , Q) , /!"  H  /!"  where EM HK (z, Q , Q) satis"es the integral equation /!"  / dQ \C dz

x 
Q PK(z)EM HI , Q , Q EM HK (x, Q , Q)" /!" z   /!"  z 2 I  Q  /  I V

#H(Q , Q ) Q PK(x) K(Q , Q) .   2 H 



 






(B.28)

 (B.29)

Here PI(z) are the Altarelli}Parisi splitting functions for real emissions, i.e. without -function and H regularization terms at zP1. Eq. (B.29) can be solved iteratively, expressing EM HK as the contribution of at most n (nPR) /!" resolvable emissions (of an ordered ladder with at most n ladder rungs) } see Fig. B.3: EM HK (Q , Q, x)" lim ELHK(Q , Q, x) , /!"  /!"  L

(B.30)

with



 






/ dQ \C dz

x 
Q PK(z)EL\HI , Q , Q /!" Q z 2 I z   /  I V ;K(Q , Q)#EHK(x, Q , Q) , (B.31)  /!"  / dQ

 H(Q , Q )K(Q , Q) Q PK(z) . (B.32) EHK(x, Q , Q)"    /!"  Q 2 H /  So the procedure amounts to only considering resolvable emissions, but to multiply each propagator with H, which is the reason for the appearance of H in Eqs. (B.29), (B.31) and (B.32). ELHK(x, Q , Q)" /!" 



B.3. Time-like parton splitting We discuss here the algorithm for Monte Carlo generation of time-like parton emission on the basis of Eq. (6.25). The standard procedure is to apply the Monte Carlo rejection method [171]. We consider the splitting of a parton j with a maximal virtuality Q given by the preceding process. For the H  proposal function, we de"ne the limits in z for given Q using an approximate formula H p Kz(1!z)Q!zQ!(1!z)Q (B.33) , H J I

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Fig. B.2. Unresolvable emissions are summed over, providing the so-called Sudakov form factor. Fig. B.3. The calculation of EM . /!"

instead of the exact one, Eq. (6.28), with the lowest possible virtualities for daughter partons Q"Q"p , which gives I J ,  1 1 4p z (Q)" $ 1! ,  .

  H (B.34) 2 2 Q H Further, we de"ne upper limits,







1 1 , PM E (z)"3 # E z 1!z N PM O (z)" D , E 2 4 2 PM E (z)" , O 3 1!z

(B.35)

for the splitting functions, PE (z)"3  E

(1!z(1!z)) , z(1!z)

N
PG (z)" D (z#(1!z)) , E 2 GZSBQS BM Q  4 1#z PE (z)" , O 3 1!z

1 2

(B.36)

with N being the number of active quark #avors. Integrating these three functions PM I (z) over D H z from z "z (Q) to z "z (Q) as



 H



 H X  dz PM I(z) , (B.37) II(Q)" H H H X 



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one obtains






z 1!z

 IE (Q)"3 ln  #ln E H z 1!z



 n IO (Q)" D (z !z ) ,

 E H 2 






,

(B.38) (B.39)



1!z

 . (B.40) IO (Q)" ln O H  1!z

 De"ning I (Q) as H H I (Q)"IE (Q)#IO (Q) E#IO (Q) O , (B.41) H H E H E H H O H H we propose the value Q based upon the probability distribution H 1

, (B.42) f (Q)"!  I (Q) H H 2 H H Q H with " (p )" (p ). The #avor k of the daughter parton is then chosen according to

 Q ,  Q ,  partial contributions II(Q) in (B.41), and the value of z according to the functions PM I(z). H H H The proposed values for Q"Q, k, and z are accepted according to the probability H

(p ) Q , wI , (B.43) H

 with p "z(1!z)Q and , H wE "(1!z(1!z)) , (B.44) E wO "z#(1!z) , (B.45) E wE "(1#z)/2 . (B.46) O Otherwise, the proposal is rejected and one looks for another splitting with Q "Q. H  H Appendix C. Hadron}hadron amplitudes In this appendix, we discuss the hadron}hadron scattering amplitude ¹   , where h and  FF h represent any pair of hadrons.  C.1. Neglecting valence quark scatterings We start with the general expression for hadron}hadron scattering amplitude, Eq. (2.31),

 



 1 L dk dk dq J J J NL(p, k ,2, k , q ,2, q )  i¹   (s, t)"
 L  L FF n! (2) (2) (2) F L J L L ;  [i¹ / (s( , q)]NL (p, k ,2, k ,!q ,2,!q ) (2) 
q !q , F  L  L  J J G J I






(C.1)

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with t"q, s"(p#p)Kp>p\, and with p, p being the 4-momenta of the initial hadrons. We consider for simplicity identical parton constituents (neglecting valence quark scatterings) and take ¹ / to be the sum of the soft Pomeron exchange amplitude (see Eq. (2.5)) and the semi-hard sea}sea  scattering amplitude (see Eq. (2.20)): (C.2) ¹ / (s( , q)"¹ (s( , q)#¹ } (s( , q),  J J  J J   J J with s( "(k #k )Kk>k \. The momenta k , k and q denote correspondingly the 4-momenta of J J J J J J J J the initial partons for the lth scattering and the 4-momentum transfer in that partial process. The factor 1/n! takes into account the identical nature of the n scattering contributions. NL(p, k ,2, k , q ,2, q ) denotes the contribution of the vertex for n-parton coupling to the F  L  L hadron h. We assume that the initial partons k , k are characterized by small virtualities k&!s , J J J  k&!s , and therefore by small transverse momenta k, (s , k, (s , so that the general J  J  J  results of the analysis made in [25,44] are applicable for the hadron}parton vertices NL. Using F dq " dq> dq\dq , , (C.3) dk " dk> dk\dk , , J J J  J J J J  J we can perform the integrations over k\, k , , q\ and k>, k, , q> separately for the upper and the J J J J J J lower vertexes by making use of qK!q, , k\, q\;k \, k >, q>;k> , (C.4) J J J J J J J J as well as the fact that the integrals dk>, dk \ are restricted by the physical region J J 0(k>4p>,
k>4p> , (C.5) J J J (similar for k \) [44]. We shall consider explicitly the upper vertex as for the lower one the J derivation is identical. The integrals over k\, q\ are de"ned by the discontinuities of the analytic J J amplitude NL with respect to the singularities in the corresponding energy invariants F s>"(p!k )K!p>k\ ,     s>"(p!k !2!k ) , L  L

(C.6)

and s> "(p#q )Kp>q\ , O    s>L\ "(p#q #2#q ) . (C.7) O  L\ As the processes corresponding to large values of s>, s>J need an explicit treatment (the so-called J O enhanced diagrams, see Section 5), we only get contributions from the region of large k>&p>, J so that






p> p> #2# (M , s>K!p>(k\#2#k\)&s  J  J  k> k>  J

(C.8)

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where M is some minimal mass for the Pomeron asymptotics to be applied. The similar argument  holds for the momenta q\, such that J s>J Kp>(q\#2#q\)(M . (C.9) O  J  Using dk dk\" J , J k> J

ds> dq\" OJ , J p>

(C.10)

we get

      

L dk L\ dq\ J  J NL(p, k ,2, k , q ,2, q )  F  L  L (2) 2 J J L dk dk>dk , L\ ds>J J J J (s>)  O (s>) "  J OJ 2(2)k> 2p> J J J

 







 

L ;disc > 2 >L >O 2 >OL\ NL (p, k ,2, k , q ,2, q ) 1!
x> Q  Q Q  Q F  L  L H H  L dx> 1 L J FL(x>,2x>, q ,2, q ) 1!
x> .  , (C.11)  , , L  L H x> F  (p>)L\  J J H The only di!erence of the formula Eq. (C.11) from the traditional expression for FL in [25,44] is the F fact that we keep explicitly the integrations over the light cone momentum shares of the constituent partons x>"k>/p>. Further we assume that the dependences on the light cone momentum J J fractions x> and on the momentum transfers along the Pomerons qK!q, factorize in FL, and F J J J we use the Gaussian parameterization for the latter one,








L (C.12) FL(x>,2x>, q, ,2, q, )"FI L(x>,2x>) exp !R
q, , L F  L F H F  L  H where the parameter R is known as the hadron Regge slope [44]. Based on the above discussion F and a corresponding treatment of the lower part of the diagram, Eq. (B.2) can be rewritten as

  

 L dx> dx\  1 L\ dq , J J J FI L(x>, x>)FI L(x\, x\)   i¹   (s, t)"
FF x> x\ F  2 L F  2 L n! 8s  J J J J L L L L ;  [i¹ / (s( ,!q, ) exp(![R #R ]q, )] 1!
x>  1!
x\  J J F F J H H H H J (C.13)









Formula (C.13) can be also obtained using the parton momentum Fock state expansion of the hadron eigenstate [103]








 1  I I  dx f (x ,2x ) 1!
x a>(x )2a>(x )0 , h"
J I  I H  I k!  J I H

(C.14)

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where f (x ,2x ) is the probability amplitude for the hadron h to consist of k constituent partons I  I with the light cone momentum fractions x ,2, x and a>(x) is the creation operator for a parton  I with the fraction x. f (x ,2x ) satis"es the normalization condition I  I  1  I I
 dx  f (x ,2x ) 1!
x "1 . (C.15) J I  I H k!  J I H Then, for the contribution of n pair-like scatterings between the parton constituents of the projectile and target hadrons one obtains Eq. (C.16), as shown in [103], with















1 k!  I  1 I  dx  f (x ,2x ) 1!
x FI L(x ,2x )"
(C.16) J I  I F  L H k! n! (k!n)! n!  JL> IL H representing the `inclusivea momentum distributions of n `participatinga parton constituents, involved in the scattering process. From the normalization condition (C.15) follows the momentum conservation constraint





dx xFI (x)"1 . (C.17) F  To get further simpli"cations, we assume that FI L  (x ,2x ) can be represented in a factorized L F F   form as a product of the contributions FF (x ), depending on the momentum shares x of the   J J `participatinga or `activea parton constituents, and on the function FF (1!
L x ), represent  H H ing the contribution of all `spectatora partons, sharing the remaining share 1!
x of the initial H H light cone momentum (see Fig. C.1). So we have






L L FI L(x ,2x )"  FF (x )FF 1!
x . (C.18) F  L   J   H J H The participating parton constituents are assumed to be quark}anti-quark pairs (not necessarily of identical #avors), such that the baryon numbers of the projectile and of the target are conserved. So we have x"x #x  with x and x  being the light-cone momentum fractions of the quark and the O O O O anti-quark. The function FF may thus be written as  



FF (x)" dx dx  FM F (x , x  ) (x!x !x  ) . O O   O O O O  

(C.19)

In case of soft or semi-hard Pomerons, FM F is taken as a product of two asymptotics x\?O , i"q, q , G   so we have FF (x)" x\?  ,   F

(C.20)

Fig. C.1. Nucleon Fock state.

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269

with "2 !1. The parameter de"nes the probability to slow down the constituent   O O (`dresseda) (anti-)quark; it is related to the Regge intercept of the qq -trajectory [76]: " O de"nes the probability to slow down the initial hadron

1 (0)K0.5. The remnant function FF   quark con"guration; it is assumed to be of the form FF (x)"x?F  ,   with an adjustable parameter F . Using (C.18)}(C.21), Eq. (C.13) can be rewritten as   L 1  1  L  dx> dx\  i¹   (s, t)"8s
dq , i¹F/F (x>, x\, s,!q, )  J J FF J J J J n! 8s(  J J J L L L L ;FF 1!
x> FF 1!
x\ 
qo , !qo     I H H , I H H with









 






(C.21)



¹F/F (x>, x\, s,!q, )"¹ / (s( ,!q, ) FF (x>) FF (x\) exp(![R #R ]q, )    J   J F J J J  J J F J representing the contributions of `elementary interactions plus external legsa.

(C.22)

(C.23)

C.2. Including valence quark hard scatterings To include valence quark hard scatterings one has to replace the inclusive parton momentum distributions 1 FI L(x , , x ) n! F  2 L

(C.24)

in (C.13) by the momentum distributions 1 (C.25) FI LLT G 2GLT (x  ,2, x LT , x T ,2, x ) , T L T L > n ! (n!n )! F T T corresponding to the case of n partons being valence quarks with #avors i ,2, i T (taken at the T  L virtuality scale Q ) and other n!n partons being usual non-valence participants (quark}anti T quark pairs). One has as well to take into account di!erent contributions for scatterings between a pair of valence quarks or between a valence quark and a non-valence participant. In particular, for a single hard scattering on a valence quark we have to use FI (x )"qG (x ) , (C.26) F T  J where qG is the momentum distribution of a valence quark of #avor i at the scale Q . In order to   conserve the initial hadron baryon content and to keep the simple factorized structure (C.18), we associate a `quasi-spectatora anti-quark to each valence quark interaction, de"ning the joint contribution FM G (x , x  ) of the valence quark with the #avor i and the anti-quark. Thus we have   T O FI LLT G 2GLT (x  ,2, x LT , x T> ,2, x ) F T T L L LT " J





VTJ

dx FM FGJ (x J , x !x J ) J   T J T








L L  FF (x )FF 1!
x , I   K   KLT > I

(C.27)

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where x is the sum of the momentum fractions of lth valence quark and the corresponding J anti-quark; we allow here formally any number of valence quark participants (based on the fact that multiple valence type processes give negligible contribution to the scattering amplitude). By construction the integral over x  of the function FI G(x , x  ) gives the inclusive momentum F T O O distribution of the valence quark i. Thus the function FM G has to meet the condition  



\VT



dx  FM FG (x , x  )FF (1!x !x  )"qG (x , Q ) . T O  T  O   T O  

(C.28)

Assuming that the anti-quark momentum distribution behaves as (x  )\?0 , and using the aboveO mentioned parameterization for FF , we get   FM FG (x , x  )"N\ qG (x , Q )(1!x )?1 \\?  (x  )\?1 ,  T  T O   T O

(C.29)

with the normalization factor (1#

) (1! 1 )   . N" (2#

! 1 )  

(C.30)

Now we can write the normalization condition (C.17) for `activea (participating in an interaction) partons as









 V dx xFF (x) FF (1!x)#
x dx dx FM FG (x , x!x )FF (1!x)"1 ,     T T   T T     G 

(C.31)

which gives (3! # F )     ,  "(1!x ) F T (2! ) (1# F )    

(C.32)

with



 x "
x dx qG (x , Q ) T T T  T  G 

(C.33)

being the average light cone momentum fraction carried by valence quarks. C.3. Enhanced diagrams In this appendix, we demonstrate how triple Pomeron contributions appear naturally in the Gribov}Regge formalism under certain kinematical conditions, and we derive a formula for the hadron}hadron scattering amplitude in this case. To introduce enhanced type diagrams let us come back to the process of double soft Pomeron exchange, which is a particular case of the diagram of Fig 2.7. The corresponding contribution to the elastic scattering amplitude is given in Eqs. (2.31), (C.11) with n"2 and with ¹ / being 

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

replaced by ¹



271

:



1 dk dk dk dk dq      (s>)(s\)(s>)(s\)(s> )(s\ ) i¹ (s, t)"   FF     O O 2 (2) (2) (2) (2) (2) ;disc > > >O N (p, k , k , q , q!q ) Q Q Q F      ;  [i¹ (s( , q)] disc \ \ \O N (p, k , k ,!q ,!q#q )  J J Q Q Q F     J

(C.34)

see Fig. C.2. We are now interested in the contribution with some of the invariants s>"(p!k )K!p>k\ ,    s>"(p!k !k )K!p>(k\#k\) ,      (C.35) s> "(p#q )Kp>q\ ,   O being large, implying k\, q\ to be not too small. As was shown in [44], one can restrict the G  integration region to s>4s> , because s><s> , (k\
(C.36)

s s s s s k\&!  , k >&!  , q>&!  , q\&  ,  (C.37) G G   k> k \ k \ k> k> G G    and correspondingly k>&k> and the invariants s>, s>, s> are of the same order. The vertex     O N for large s> can be described by the soft Pomeron asymptotics and we may write F O (p, k , k , q , q!q ) as a product of the Pomeron}hadron coupling N(p, k, q), disc > > >O N     F Q Q Q F twice imaginary part of the soft Pomeron exchange amplitude 2Im ¹ ((k!k ), q) with   k "k #k , and a term <.(k , k , k , q, q , q!q ) describing the coupling of the three        

Fig. C.2. Double Pomeron exchange. Fig. C.3. Three Pomeron coupling.

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Pomerons. So we get



dk dk dq\    (s>)(s>)(s> ) disc > > > N(p, k , k , q , q!q )   O Q Q QO F     (2) (2) 2



"

dk dk dk dk dq\    (s>)(s>)(s> )  (2) (k !k !k )   O (2) (2)    (2) (2) 2

;N (p, k, q) 2Im ¹ ((k!k ), q)</(k , k , k , q, q , q!q ) F        dk> dk dk dk> dk dk , , (s>) disc > N(p, k, q)    2Im ¹ ((k!k ), q) "  Q F   2k> (2) 2k>(2)  dk> dk dk , d(k !k !q !q)       ; (s>)(s>)(s> )   O 2k>(2) 2(k> !k>)    ;</(k , k , k !k , q, q , q!q ) (C.38)       see Fig. C.3, with s>"(p!k)K!p>k\ and dq\"(1/k>) d(k !q !q). Now we can per     form the integrations over k, k , k , (k !q !q) assuming their convergence in the region     k&!s , as well as over k , k , , k , 4s to transform Eq. (C.28) to the form  G  ,   dx> 1  dx> FF (x>)FF (1!x>) exp(!R q ) Im ¹ (s>,!q )     F , x> 2s>  ,  dz> ; (x> !x>)p>    with













x>"k>/p> , x> "k> /p> ,   x>"k>/p> ,   x> !x>"(k> !k>)/p>"k>/p> ,      z>"k>/k> "x>/x> ,     s>"(k!k )K!k>k\ Ks k>/k> "s x>/x> .       We used (see Eqs. (C.11)}(C.18))



dk dk , (s>) disc > N(p, k, q)"FF (x>)FF (1!x>) exp(!Rq )  Q F     F , 2(2)

(C.40) (C.41) (C.42) (C.43) (C.44) (C.45)

(C.46)

 The triple-Pomeron vertex is assumed to have non-planar structure, corresponding to having the two lower Pomerons `in parallela; the planar triple-Pomeron vertex would correspond to subsequent emission of these Pomerons and gives no contribution in the high energy limit [155].

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and the de"nition


dk dk , dk dk , d[(k !k !q !q)]        (2) (2) 2

;(s>)(s>)(s> )</(k , k , k !k , q, q , q!q ) .   O       (C.47) We have also taken into account the fact that the triple Pomeron vertex </ has a scalar structure, and we therefore suppose that it can only depend on the invariants k, (k #q ), (k !q !q)&!s , qK!q , q K!q, , q K!(q !q , ) (C.48) ,  G      ,    and on the partition of the light cone momentum k> between the two lower Pomerons,  z>"k>/k> "(p#k )/(p#k ) .     Furthermore, we did assume the #at distribution in z> in order to obtain k>&k>&k> /2 (see the    discussion above). We use the Gaussian parameterization for the q-dependence of


(s, t)" i¹/\ F F

 dx> dx\ FF (1!x>)FF (1!x\) i¹F/F (x>, x\, s, t)    \ x> x\   

(C.50)

with







r / V> dx> 1  i¹F/F (x>, x\, s, t)"8x>x\s  Im ¹F (x>, s>,!q )  > , 2  \ x> 2s> Q V  V\ 1  dx\ dx\  ; dz> dq , dq , i¹F (x\, s( ,!q, )     J J J 8s(  J J ; (x\!x\!x\) (qo !qo , !qo , ) ,   ,  

 





 (C.51)

with ¹F(x, s( ,!q )"¹F (x, s( ,!q )"¹ (s( ,!q )FF (x) exp(!Rq ) ,  ,  ,   F ,

(C.52)

and s( "x> z>x\s, s( "x> (1!z>)x\s . (C.53)       The sign `!a in `3/!a refers to the Pomeron `splittinga towards the target hadron (reversed >-diagram); the lower limit for the integral dx> is due to x\ Ks /x> (x\.    

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C.4. Parton generation for triple-Pomeron diagrams The inclusion of the triple-Pomeron contributions results only in slight modi"cation of the standard procedure. Now the full contribution of an elementary interaction is  (C.54) GF/F (x>, x\, s, b)#

GK F/F (x>, x\, x , x , s, b) ,   NG N G where x , x  are the corresponding remnant light cone momentum fractions, and with the contributions of di!erent cuts of triple-Pomeron diagrams being obtained from Eq. (5.54), together with Eqs. (5.54) and (5.47)}(5.49) as

 

1 r / GK F/F (x>, x\, x , x , s, b)"  db GF (x>, x>x\s, bo !bo )  x\   \ 8

 






V   V( \>V\ s dx( \ dz> dx\ GF x\, x\  z>s, b    x\     s ;GF x( \#x\!x\, (x( \#x\!x\)  (1!z>)s, b   x\ 

;




F (x !x( \) ;   F (x )  



(C.55)

and

 

r / V> dx>  GF (x>, s>, bo !bo ) GK F/F (x>, x\, x , x , s, b)"!  db  \   2 \ x> Q V Q   ; dz>GF (x\, x> z>x\s, b )    V  ; dx( \ GF (x( \, x> (1!z>)x( \s, b )    F (x !x( \) ;   (x ) F  

 

(C.56)

and

   

r / V> dx>  GF (x>, s>, bo !bo ) GK F/F (x>, x\, x , x , s, b)"  db  \   4 x> Q QV\   V\ ; dz> dx\ GF (x\, s( , b ) GF (x\!x\, s( , b ) (C.57)          (similarly for GK F/F ). Thus, the elementary process splits into a single elementary scattering  >G contribution } with the weight  GK F/F "GF/F #

GK F/F ,    NG N G

(C.58)

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275

and the contribution with all three Pomerons being cut } with the weight . (C.59) GK F/F #GK F/F  >  \ Choosing the "rst one, one proceeds in the usual way, with the functions GF/F , GF F , GF F} ,     GF F} , GF F} in Eq. (6.4) being replaced by GK F/F ,GK F F ,GK F F} ,GK F F} ,GK F F} , where we introduce     K         the functions GF F via ( #GK F/F #GK F/F #GK F/F , (C.60) GK F F "GF F #GK F/F ( (  >(  \(  \(  \( with J being `softa, `sea}seaa, `val}seaa or `sea}vala. Here GK F/F is given by Eq. (C.55) with  \   GF (x>, x>x\s, bo !bo ) being replaced by its soft contribution GF (x>, x>x\s, bo !bo ) (see    is given in (C.56) with both GF (x>, s>, bo !bo ) and Eq. (5.34)), whereas GK F/F  \  GF (x\, x> z>x\s, b ) being represented by the soft contributions GF F . The other functions    are de"ned similarly to the `softa case } considering corresponding `seaa and `valencea GK F/F  >G( contributions in the cut Pomerons GF F in Eq. (C.55) and (C.56). For the contribution corresponding to the cases of all three Pomerons being cut, we split the processes into three separate cut Pomeron pieces. Each piece is characterized by the function GM F F (x >, x \, s, b ), where b Kb/2 and G!   GM F F (x >, x \, s, b )"GF (x >, x >x \s, b ), x >"x>, x \"s /(x> s) , (C.61) \     (C.62) GM F F (x >, x \, s, b )"GF (x \, x >x \s, b ), x >"z>x> , x \"x\ , \     (C.63) GM F F (x >, x \, s, b )"GF (x \, x >x \s, b ), x >"(1!z>)x> , x \"x\!x\ ,    \  where the variables x> , z>, x\ are generated according to the integrand of Eq. (C.57) (similar for   GM F F ). After that each contribution GM F F (x >, x \, s, b ) is treated separately in the usual way, G> G!  starting from Eq. (6.4), with the functions GF/F , GF F , GF F} , GF F} , GF F} being replaced by         GM F F (x >, x \, s, b ) and by the corresponding partial contributions of soft GF , `sea}seaa type G!   GF } , and `valence}seaa-type parton scattering GF } (see Eqs. (5.34)}(5.36)), but without     `valence}valencea contribution. Appendix D. Calculation of
and H D.1. Calculation of
NN Here, we present the detailed calculation of
for proton}proton collisions. As shown earlier,
may be written as NN P >2>P,  1 1  2  dx> dx\
(x>, x\s, b)"
2
H H NN r ! r ! , P,   H P  P >2>P, P !G ,  ;  !G  2 M ,M 2 M  M, P > >P,\ >



;F

 









x>!
x> F x\!
x\ , H   H H H

(D.1)

276

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

with F (x)"x?  (x) (1!x) ,   and with G being of the form GH G (x>, x\, s, b)" (x>x\)@G , GH H H G H H with and  being s- and b-dependent parameters. We obtain G G   (! )P (! )P,  2 , I 2 (x>)I 2 (x\)
(x>, x\, s, b)"
2
NN P  P, P  P, r ! r !  , P  P,  with

(D.3)

(D.4)



P >2>P, P >2>P, P  dx  x@ 2  x!
x , x@,, F M M   H H H M  M, P >2>P,\ > H



P >2>P, ?   [dx xCH ] x!
x  x!
x  1! x!
x H H H H H H H H H

I  2 , (x)" P  P




(D.2)



(D.5)

which amounts to I  2 , (x)" P  P













,

(D.6)

with



    "  H 

for 4r ,  for r ( 4r #r ,   

(D.7)

 for r #2#r ( 4r #2#r . ,  ,\  , We de"ne new variables, x H u " , H x!x !2!x  H\ dx H du " , H x!x !2!x  H\ which have the following property: H\ H\ x!2!x x!2!x ? " H\ ,  (1!u )"  ? x!2!x x ?\ ? ? and therefore H\ x "xu  (1!u ) , H H ? ? H\ dx "x du  (1!u ) . H H ? ?

(D.8)

(D.9)

(D.10)

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

277

This leads to





P >2>P, H\  du uCH x>CH  (1!u )>CH H H ? H ?

I  2 , (x)" P  P





P >2>P, ?  x  (1!u ) . H H

(D.11)

De"ning

"

P >2>P,  "

#r I #2#r I #
H     , ,   H

(D.12)

and P >2>P,   "

#
J H   JH>

#(r ! )I #r I #2#r I       , ,

#(r #r ! )I #r I #2#r I      , , "   



#(r #2#r ! )I    , ,

with

if 4r ,  if r ( 4r #r ,   

(D.13)

if 'r #2#r ,  ,\

I "#1 ,

(D.14)

 "#1 ,

(D.15)

we "nd



P >2>P,  du uCH (1!u )AH H H H H The u-integration can be done, I  2 , (x)"x? P  P

(D.16)



(D.17)



(1#)(1#) , du uC(1!u)A" (2##)

 and we get

P >2>P, (1# )(1# ) H H I  2 , (x)"x?  P  P (2# # ) H H H Using the relation 1# # " , we get H H H\ P >2>P, (1# ) H I  2 , (x)"x?(1# )P 2(1# )P,   , P  P (1# ) H\ H (1#  2 , ) P > >P , "x?(1# )P 2 (1# )P,  , (1# )  or (    I  2 , (x)"x?  >P @I  >2>P, @I , (I )P 2 (I )P, . P  P  , (  #r I #2#r I )     , ,

(D.18)

(D.19) (D.20)

(D.21)

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H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

The "nal expression for


NN

is therefore





 (1#

)    
(x>, x\, s, b)"x? 
2
NN (1#

#r I #2#r I )     , , P  P,  I I (! x@ (I ))P (! x@, (I ))P,  ,  2 , ; r ! r !  , with x"x>x\. This is the expression shown in Eq. (3.15).

(D.22)

D.2. Calculation of H The function H is de"ned as




K  1 K  dx> dx\  G(x>, x\, s, b) H(z>, z\)"
I I m! I I I K I K K ; 1!z>!
x> 1!z\!
x\ . I I I I Using the expression






(D.23)

, (D.24) G(x>, x\, s, b)"
(x>x\)@G , I I G I I G where and  are functions of the impact parameter b and the energy squared s, G G (D.25)

"( G # HG )s@"G >A"G @e\@B"G , " G " , (D.26)  " G # G b#HG !

" "   G " with HG O0 and HG O0 only if G "0, and using the same method as for the calculation of " " "
(x>, x\), one may write NN    ( )P ( )P,
 2 , J 2 (z>)J 2 (z\) (D.27) H(z>, z\)"
2 P  P, r ! P  P, P  P,  r ! , GFHFI 2 P > >P,  with



J  2 , (z)" P  P




P >2>P, P >2>P,  [dx xCH ] 1!z!
x H H H H H



(D.28)

and



    "  H   ,

for 4r ,  for r ( 4r #r ,    for r #2#r ( 4r #2#r .  ,\  ,

(D.29)

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

279

One may use the function to obtain








P >2>P, P >2>P, C  [dx xCH ] (1!z)!
x . H H H H H Introducing x "x , we de"ne new variables, ? ?> x HY u " , HY 1!z!x !2!x  HY\ dx HY du " , HY 1!z!x !2!x  HY\ which have the following property: J  2 , (z)" P  P

(D.30)

(D.31)

1!z!2!x HY\ HY\ 1!z!2!x HY\ , ? "  (1!u )"  ? 1!z 1!z!2!x ?\ ? ? and therefore HY\ x "(1!z)u  (1!u ) , HY HY ? ? HY\ dx "(1!z) du  (1!u ) . HY HY ? ? This leads to





(D.33)

P >2>P, \ HY\  du uCHY> (1!z)>CHY>  (1!u )>CHY> ? HY HY HY ? C P >2>P, \  (1!u ) ; (1!z) . HY HY

J (z)" PQR



(D.32)



 (D.34)

De"ning P >2>P, \

" #
 "r I #2#r I !1  HY>   , , HY

(D.35)

and P >2>P, \  " #
 HY  J> JHY>  #(r !1! )I #r I #2#r I      , ,  #(r #r !1! )I #r I #2#r I      , , "  



 #(r #2#r !1! )I   , ,

(D.36) if 4r !1 ,  if r !1( 4r #r !1 ,    if 'r #2#r !1 ,  ,\ (D.37)

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H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

with I "#1 ,

(D.38)

 "#1 ,

(D.39)

we "nd



P >2>P, \  du uCHY> (1!u )AHY . HY HY HY HY The u-integration can be done, J  2 , (z)"(1!z)?Y P  P

(D.40)



(D.41)



(1#)(1#) du uC(1!u)A" (2##)

 and we get

)(1# ) P >2>P, \ (1# HY> HY .  J  2 , (z)"(1!z)?Y P  P (2# # ) HY> HY HY Using the relation 1# # " , we get, if r O0, HY> HY HY\  P >2>P, \ (1# ) HY  J  2 , (z)"(1!z)?Y(I )P \(I )P 2(I )P,   , P  P (1# ) HY\ HY (1#  2 , ) P > >P \ "(1!z)?Y(I )P \(I )P 2(I )P,  ,  (1# )  (I )  "(1!z)?Y(I )P \(I )P 2(I )P, .   , (rI #2#r I )  , , If r "0 and r O0, we get   (I )  , J  2 , (z)"(1!z)?Y(I )P \(I )P 2(I )P,   , (r I #2#r I ) P  P   , , and so on, which corresponds "nally to (I )P 2(I )P, ,  . J  2 , (z)"(1!z)P @I  >2>P, @I , \ P  P (r I #2#r I )   , , The "nal expression for H is therefore

(D.42)

(D.43) (D.44) (D.45)

(D.46)

(D.47)

  [(1!z>)(1!z\)]P @I  >2>P, @I , \ H(z>, z\)"
2
(r I #2#r I ) P  P,    , , GFHFI P >2>P, $ ( (I ))P ( (I ))P, ,  2 , . ;  r ! r !  ,

(D.48)

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

281

D.3. Calculation of
 The expression for the virtual emissions in case of nucleus}nucleus collisions is given as

 

  




 JI  1 JI     dx > dx \   !G(x > , x \ , s, b)
(X>, X\, s, b)"
2
IH IH IH IH  l ! I I H J  J  I H  ; F x>!
x >  F x\!
x \ , (D.49)   H   G IH IH H G LIG OIH where X>"x>2x>, X\"x\2x\ and (k) and (k) represent the projectile or target    nucleon linked to pair k. Using the expression






, G(x > , x \ , s, b)"
(x > x \ )@G , IH IH G IH IH G GFHFI %GIH

(D.50)

where and  are functions of the impact parameter b and the energy squared s, G G

"( G # HG )s@"G >A"G @e\@B"G , G " " ,  " G # G b#HG !

" "   G " with HG O0 and HG O0 only if G "0. The remnant function F is given as " " "   F (x)"x?  (x)(1!x) .   We have (see calculation of
) NN   1 JI 1 JI 2
2  !G(s, x > , x \ , b)"2
2  (!G !2!G ) IH IH IH ,IH l ! l ! I H I H JI  JI   PI >2>P,I PI  1 "2
2
2  !G 2  . !G IM ,IM, r !2r ! I ,I M  PI M, PI >2>P,\I > P,I  So Eq. (D.49) can be written as

 

(D.51) (D.52)

(D.53)

(D.54) (D.55)



 1  PI >2>P,I    dx > dx \
(X>, X\, s, b)"
2
IH IH  r !2r ! ,I I P 2P, I I H P 2P, PI  PI >2>P,I ;   ! (x >  x \  )@ 2  ! (x > , x \ , )@,  IM IM , IM IM 2 I M  M, PI > >P,\I >  ; F x>!
x >  F x\!
x \ , (D.56)   H   G IH IH H G LIG OIH which leads to













 (! )PI (! )P,I  2 , I>(X>)I\(X\) , 
(X>, X\, s, b)"
2
0 0  r ! r ! 2 2 I ,I P P, I P P,



(D.57)

282

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

where








 PI >2>P,I .N IN (X)"   dx (x )CIH  F x !
x , 0 IH IH   G IH N I H G G IG with R"r , P>"A, P\"B, >(k)"(k), and \(k)"(k). Using the property HI  .N  "  , I G GNIG one can write






(D.58)

(D.59)



PI >2>P,I .N   dx (x )CIH F x !
x . (D.60) IN (X)"  IH IH   G 0 IH G GNIG H GNIG Let us rename the x linked to the remnant i as x , x ,2, x NG 2 N,G , where rN is per de"nition IH   P > >P NG the number of Pomerons of type p linked to remnant i. So we get for the term in brackets








PNG >2>PN,G PNG >2>PN,G  dx x CJ F x!
x . J J   G J J J This is exactly the corresponding I for proton}proton scattering. So we have

(D.61)

.N IN (X)"  x?  >PNG @I  >2>PN,G @I , (I )PNG 2(I )PN,G g(rN I #2#rN I ) , 0 G  , G  ,G , G

(D.62)

(1#

)   g(z)" , (1#

#z)  

(D.63)

I "#1 .

(D.64)

with

and

Since we have rN "
r , NG NI GNIG we "nd "nally

(D.65)


(X>, X\, s, b) 

 (! )PI (! )P,I ,  2  "
2
r ! r ! 2 2  ,  , I ,I P P P P I  ;  (x>)?   ((I )(x>)@I  )PI 2((I )(x>)@I , )P,I g
r I #2#r I G  G , G I  ,I , G LIG LIG





 

;  (x\)?   ((I )(x\)@I  )PI 2((I )(x\)@I , )P,I g
r I #2#r I . H  H , H I  ,I , H OIH OIH (D.66)

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

283

D.4. Exponentiation of
 We "rst replace in Eq. (D.66) the function g(z) by the `exponentiateda function g (z), 










r I #2#r I "exp !
r I #2#r I  N I  ,I , I  ,I , G IG G IG "  (e\C @I  )PI 2(e\C @I , )P,I , GNIG and we obtain g 

N


 (X>, X\, s, b)   "  (x>)?   (x\)?  G H G H  (! )PI   ;
   (D (x>)@I  )PI   (D (x\)@I  )PI  G  H r ! 2 I P P I G LIG H OIH











(D.67)

2  (! )P,I  ,    (D (x>)@I , )P,I   (D (x\)@I , )P,I ;
, G , H r ! 2 ,I G LIG H OIH P, P, I

(D.68)

with D "(I ) e\C @I H . H H And using again the property (D.59), we can write

(D.69)


 (X>, X\, s, b)"  (x>)?   (x\)?  H  G H G  (! D (x> x\ )@I  )PI   LI OI  ;
r ! 2 I P P I 2  (! D (x> x\ )@I , )P,I , , LI OI  . (D.70) ;
r ! 2 ,I P, P, I Now the sum can be performed and we get the "nal expression for the `exponentiateda
,  
 (X>, X\, s, b)"  (x>)?   (x\)?   e\%I V>LI V\OI  , H G  H I G with , GI (x)"
 x@I G , G G

(D.71)

284

H.J. Drescher et al. / Physics Reports 350 (2001) 93}289

where

 " (I ) e\C @I G , G G G I " #1 , G G

(D.72) (D.73)

and

"( G # HG )s@"G >A"G @e\@B"G , G " " ,  " G # G b#HG !

" "   G " with HG O0 and HG O0 only if G "0. " " "

(D.74) (D.75)

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THE BOGOLIUBOV MODEL OF WEAKLY IMPERFECT BOSE GAS

V.A. ZAGREBNOV, J.-B. BRU UniversiteH de la MeH diterraneH e (Aix-Marseille II) and Centre de Physique TheH orique, CNRS-Luminy-Case 907, 13288 Marseille, Cedex 09, France

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

Physics Reports 350 (2001) 291}434

The Bogoliubov model of weakly imperfect Bose gas Valentin A. Zagrebnov*, Jean-Bernard Bru Universite& de la Me& diterrane& e (Aix-Marseille II) and Centre de Physique The& orique, CNRS-Luminy-Case 907, 13288 Marseille, Cedex 09, France Received October 2000; editor: A.A. Maradudin Dedicated to AndreH Verbeure on the occasion of his 60th birthday Contents 1. Introduction 1.1. Historical remarks 1.2. Bose}Einstein condensation and super#uidity 1.3. The Bogoliubov model of weakly imperfect Bose gas (WIBG) 1.4. Outline 2. The Bogoliubov theory of super#uidity 2.1. Second quantization, perfect Bose gas and conventional Bose}Einstein condensation 2.2. The weakly imperfect Bose gas and Bogoliubov approximation 2.3. Spectrum of elementary excitations and super#uidity 2.4. Beyond the Bogoliubov theory 3. The Bogoliubov theory, revisited 3.1. Generalized Bose}Einstein condensation and the "rst Bogoliubov ansatz 3.2. Quantum #uctuations and the second Bogoliubov ansatz 3.3. Instability of the weakly imperfect Bose gas 3.4. From the weakly imperfect Bose gas to the Bogoliubov theory

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4. Exact solution of the weakly imperfect Bose gas: pressure 4.1. Diagonal part of the Bogoliubov model 4.2. Stability domain: (non) triviality of the weakly imperfect Bose gas and condensation 4.3. Theorem about exactness of the Bogoliubov approximation 4.4. Proofs 5. Nonconventional and conventional condensations in the weakly imperfect Bose gas 5.1. Nonconventional Bose condensation 5.2. Conventional Bose}Einstein condensation 5.3. Complementary remarks 6. Concluding remarks 6.1. The weakly imperfect Bose gas versus the Bogoliubov theory 6.2. The nonconventional condensation versus conventional BE condensation Acknowledgements

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* Corresponding author. Tel.: (#33) 4 91 26 95 04; fax: (#33) 4 91 26 95 53. E-mail addresses: [email protected] (V.A. Zagrebnov), [email protected] (J.-B. Bru). 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 1 3 2 - 0

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434 Appendix A. Classi"cation of Bose condensations A.1. The van den Berg}Lewis}Pule` classi"cation of conventional generalized condensations A.2. Nonconventional and conventional Bose condensations Appendix B. The Bogoliubov approximation B.1. De"nition of the Bogoliubov approximation B.2. Application to the weakly imperfect Bose gas

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Appendix C. The Gri$ths lemma Appendix D. The Bogoliubov (convexity) inequality Appendix E. The correlation inequalities for quantum states E.1. States for "nite systems E.2. KMS-condition and correlation inequalities Appendix F. Quasi-free states Appendix G. Stable and superstable interactions References

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Abstract We present a systematic account of known rigorous results about the Bogoliubov model of weakly imperfect Bose gas (WIBG). This model is a basis of the celebrated Bogoliubov theory of super#uidity, although the physical phenomenon is, of course, more complicated than the model. The theory is based on two Bogoliubov's ansaK tze: the "rst truncates the full Hamiltonian of the interacting bosons to produce the WIBG, whereas the second substitutes some operators by c-numbers (the Bogoliubov approximation). After some historical remarks, and physical and mathematical motivations of this Bogoliubov treatment of the WIBG, we turn to revision of the Bogoliubov's ansaK tze from the point of view of rigorous quantum statistical mechanics. Since the exact calculation of the pressure and the behaviour of the Bose condensate in the WIBG are available, we review these results stressing the di!erence between them and the Bogliubov theory. One of the main features of the mathematical analysis of the WIBG is that it takes into account quantum #uctuations ignored by the second Bogoliubov ansatz. It is these #uctuations which are responsible for indirect attraction between bosons in the fundamental mode. The latter is the origin of a nonconventional Bose condensation in this mode, which has a dynamical nature. A (generalized) conventional Bose}Einstein condensation appears in the WIBG only in the second stage as a result of the standard mechanism of the total particle density saturation. It coexists with the nonconventional condensation. We give also a review of some models related to the WIBG and to the Bogoliubov theory, where a similar two-stage Bose condensation may take place. They indicate possibilities to go beyond the Bogoliubov theory and the Hamiltonian for the WIBG.  2001 Elsevier Science B.V. All rights reserved. PACS: 05.30.Jp; 02.70.Lq; 03.75.Fi; 67.40.Db Keywords: Bogoliubov theory of super#uidity; Bogoliubov weakly imperfect gas; Bose}Einstein condensation; Generalized condensations (conventional, nonconventional)

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1. Introduction `La seule vraie connaissance est la connaissance des faits.a Bu4on, Histoire naturelle The main object of the present review is the Bogoliubov model of weakly imperfect Bose gas (WIBG). The corresponding Hamiltonian H
is a truncation of the full Hamiltonian H
of a Bose system with pair interactions between particles. For Bogoliubov the Hamiltonian H
was a starting point for his celebrated microscopic theory of super#uidity [1}3]. This theory includes another famous Bogoliubov trick: approximation of some operators in H
by c-numbers. Then the corresponding Hamiltonian H
(c) becomes bilinear. Its diagonalization by the Bogoliubov u}v transformation shows that for a suitable choice of c-numbers the system H
(c) has a Landau-type excitation spectrum [4,5]. The latter is considered as an explanation of superyuidity of the WIBG and, "nally, as a microscopic theory of this phenomenon in liquid He. Notice that both of Bogoliubov's ingenious inventions: E truncation of the full Hamiltonian; E approximation of operators by c-numbers, have behind them a fundamental hypothesis about the existence in the systems H
and H
of a Bose}Einstein condensation in one speci"c mode [6}8]. At this point we touch upon one of the most fascinating problems of contemporary mathematical physics } the proof of the existence of a condensation of bosons in interacting systems. The present paper is motivated by a recent progress in rigorous results concerning the Bogoliubov model of WIBG [9}16]. Since these results touch upon one of the fundamental parts of the existing theory of Bose systems, we believe that this review gives more insight in understanding of interacting quantum models. 1.1. Historical remarks In 1947 Bogoliubov published three revolutionary papers on the theory of interacting Bose gas [1}3]. The famous paper [1] is in fact the English translation of the Russian version [2] presented to the Session of the Physical and Mathematical Departments of the Academy of Sciences of the USSR on October 21, 1946. The third paper [3] contains a lecture delivered on March 20, 1947, on a Seminar of the Physical Department of the Moscow State University. In this (almost unknown) paper Bogoliubov scrutinizes his theory and formulates it in a transparent, elegant form. Only in 1970 the paper was reproduced in [17]. In this re"ned form the Bogoliubov theory was presented in 1949 in his famous `Lectures on Quantum Statisticsa, but in the Ukrainian language! The English edition appeared only in 1967 [18] and the Russian one in [19]. Bogoliubov formulated his theory for temperature ¹"0 K. Throughout the papers [1}3,17,18], he stressed that his aim is to give a microscopic explanation of the phenomenological Landau's theory of super#uidity [4,5]. It was Landau who, for the "rst time, understood that the properties of two substances He and He, which rest liquid (under normal pressure) even for ¹P0 K, can be explained only by a new kind of quantum arguments. The Landau phenomenology was based on the following assumptions [4,5]:

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(a) at ¹"0 K, in quantum boson liquids, like He, a collective (coherent) behaviour dominates the individual movement of atoms, i.e. the properties of these liquids can be entirely described by the spectrum of collective excitation; (b) for liquid He, this spectrum has two branches: `phononsa for long-wavelength excitations and `rotonsa for relatively short-wavelength collective excitations. Therefore, according to Landau, the quantum liquid, as He, at low temperatures, is nothing but a `gas of collective elementary excitationsa or `quasi-particlesa. The second assumption ensures that this liquid is superyuid, i.e. the structure of the spectrum of excitations ensures that the liquid can move in a capillary without friction when the #ow velocity is smaller than some critical value de"ned by the spectrum. Moreover, this structure of spectrum agrees with the low-temperature thermodynamic properties of liquid He, such as speci"c heat and others [4,5,20]. Bogoliubov treated Landau's assumptions in the framework of mathematical physics. From this point of view, the key problem is to show that quantum mechanical behaviour of the macroscopic system of about 10 atoms of He is such that the spectrum of the corresponding microscopic Hamiltonian is close to the one proposed by Landau. It was a courageous programme, since calculations of the spectrum even for a few-body quantum system (e.g. atomic or molecular spectra beyond the hydrogen atom) pose enormous problems. Faced with this oppressive di$culty, Bogoliubov was guided by Landau's benchmark that (at least) the low-energy part of the spectrum of about 10 atoms of He is de"ned by coherent collective movements of the system instead of individual ones. In the next (after [1}3]) paper [21] about kinetic equation in his theory of super#uidity, Bogoliubov alludes to low-energy excitations in crystals as a sound example of collective excitations. In fact, the crystalline order of atoms suppresses their individual displacements in favour of coherent collective movements which are nothing but the sound waves or phonons. For low-energy excitations, i.e. in harmonic approximation, spectrum of the corresponding macroscopic Hamiltonian for the microscopic system of atoms can be calculated exactly! Since the system in question, i.e. He, is liquid (and not solid!), the starting point of the Bogoliubov programme was to "nd a physical (or mathematical) mechanism which as in crystals, favours the collective motions of the `helium jellya, via some kind of ordering or coherence. Since the atoms of He are bosons (in contrast to He, which are fermions), a plausible conjecture relates to the Bose}Einstein (BE) condensation predicted for the perfect Bose gas (PBG) by Einstein [6] in 1925. It was Fritz London [22] who suggested, in 1938, that the transition to super#uidity in liquid He might be an example of BE condensation. (Notice that super#uid phases in He were discovered only in 1972!) The basis of his arguments was that E the atoms of He are bosons; E the transition of the normal liquid He (called He I) to superyuid phase He II (discovered by Kapitza [23] and Allen and Misener [24] in 1938) takes place at a temperature ¹ "2.17 K H (along the vapour pressure curve); if the liquid He is treated as a PBG, its temperature of BE condensation ¹ would be very close to ¹ : ¹ "3.14 K;  H  E the BE condensation in the PBG corresponds to a macroscopic occupation of the ground state which is related to collective coherent properties of the condensate. Inspired by these observations, Tisza and London [25] tried to argue that the super#uidity is related to the motion of the BE condensate, moving as a whole. These attempts raised many

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objections. Since the spectrum of PBG Hamiltonian is known explicitly, the most obvious objection is that this spectrum does not satisfy the Landau criterion of superyuidity! In [1}3,17,18,21,26], Bogoliubov stressed that, in spite of a long-range coherence of condensate, elementary excitations in the PBG correspond to movements of individual atoms, i.e. `quasiparticlesa simply coincide with particles. Therefore, accepting the idea that the BE condensation plays a crucial role in decoding the nature of super#uidity, he insisted that `an energy level scheme based on the solution of the quantum mechanical many-body problem with interactions, must be founda [18, Part 3.4]. To summarize, the BE condensation together with interaction between bosons will transform individual excitations of the PBG into collective excitations of the `helium jellya with a Landau-type spectrum. 1.2. Bose}Einstein condensation and superyuidity Since the concept of the BE condensation is the xrst important ingredient of the Bogoliubov theory, let us discuss it in more detail. The phenomenon of macroscopic accumulation of boson particles in the ground state of the PBG was described "rst by Einstein [6] in 1925. He based his arguments on a combination of Bose statistics and the classical expression for the density of states in phase space. This combination plus di$culties in understanding transition in "nite volume provoked, in 1927, the Uhlenbeck criticism of the Einstein analysis [27]. It was only in 1937, when Kramers pointed out the importance of the thermodynamic limit for the sharp manifestation of transition into condensed phase, that Uhlenbeck withdrew his objection and pointed out that the Einstein prediction is correct in the thermodynamic limit [28]. In 1938, Fritz London [29] developed the concept of macroscopic occupation of the ground state and showed that it implies long-range coherence properties of the BE condensate. After [29], the physical mechanism of this conventional BE condensation of the PBG became so transparent that, at present, it is an essential part of any standard textbook on statistical physics, see, e.g. [30}33]. Let a PBG be enclosed in a three-dimensional cubic box of volume <. Then the structure of the one-particle density of states (in the thermodynamic limit
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representations of canonical commutation relations. The "rst results in this direction were due to Araki and Woods [34], Lewis and Pule` [35], Cannon [36]; see also [37]. It is appropriate to mention here that the conventional BE condensation with macroscopic occupation of the ground state due to the saturation mechanism, is a rather subtle and fascinating matter, for an authoritative review about PBG, see the works of Zi! et al. [7] and van den Berg et al. [8,38]. In [8], the notion of generalized BE condensation is extensively discussed. The concept of generalized (conventional) BE condensation was introduced by Girardeau [39], in 1960. He was motivated by the observation that, in one-dimensional model of impenetrable bosons, there is a sort of generalized condensation but no macroscopic occupation of the ground state. In fact, in 1963, Schultz [40] (and then Lenard [41] by another method) showed that this is not the case. In spite of that, the theoretical importance of this concept was realized by van den Berg et al. In [42] van den Berg and Lewis discussed rigorously and in detail the BE condensation in the PBG due to the standard saturation mechanism in a prism of volume < with sides
the geometrical details, were given by van den Berg in [44,45]. In [46,47], van den Berg et al. showed that the generalized BE condensation of type II or III can be provoked in the PBG by a scaled external xeld. What is behind the generalized BE condensations in the PBG? The van den Berg}Lewis}Pule` analysis shows that it is the structure of the one-particle spectrum (spectrum of Laplacian) which distinguishes the type I, type II or type III condensations. On the other hand, thermodynamic properties (e.g. pressure in the thermodynamic limit or density of condensed particles) are independent of type of condensation [8]. This means that the generalized BE condensation is a kind of `xnite-sizea ewect due to restricted geometry or (scaled) external "eld, which gets visible above the saturation density .(
).  Since in the recent experiments with trapped Bose gases, we have both: a restricted geometry and an external "eld, see, e.g. [48,49], it is not clear which type of conventional BE condensation one gets in this case! Moreover, in spite of extreme dilution of atoms in the trap, it has become evident that one cannot explain experimental data (e.g. spatial distribution of condensed atoms) without taking into account the interaction between them [48}50]. But if one takes into account an indirect interaction between bosons, a nonconventional Bose condensation may come into existence [12]. To explain it, notice that by cooling a quantum gas or by increasing its density, one can make the thermal de Broglie wavelength comparable to the average separation distance between particles. In this case, the overlapping of wave functions of individual atoms is so strong (degenerate quantum gas)

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that one cannot ignore anymore their statistics. In other words, even in the case of perfect gas, i.e. no direct interparticle interaction, the quantum particles are not independent . Calculation of the two-point correlation function [30] shows an ewective repulsion between fermions but an ewective attraction between bosons. This is the essential physics behind the conventional BE condensation: the only phase transition that occurs in the absence of (direct) interactions between particles! Therefore, an ewective attraction is present in PBG. It makes this system coherent (BE condensation) at low temperatures, or high densities, and unstable: the grand partition function diverges for positive chemical potentials. Since bosons are very sensible with respect to attraction, any direct attractive interaction between particles leads to the collapse (fortunately, our world is essentially fermionic: the matter is stable! [51]). But, if one counterbalances the direct attractive interaction by a repulsion stabilizing the system, the original attraction may be an origin of a new mechanism of condensation which we call a nonconventional (Bose) condensation. Thouless [52], in 1961, presented an instructive `back-of-the-envelopea calculation, which shows that this new kind of condensation occurs in the Huang}Yang}Luttinger model [53]. In 1998, a nonconventional condensation was discovered in the Bogoliubov weakly imperfect Bose gas (WIBG) by the authors of the present paper [12,15,16]. The di!erence between conventional and nonconventional condensations re#ects the di!erence in the mechanism of their formation. The conventional BE condensation is a consequence of the balance between entropy and kinetic energy, whereas the nonconventional condensation results from the balance between entropy and interaction energy. Rigorous analysis of the Huang}Yang}Luttinger (HYL) model and the proof of a prediction of Thouless [52] that the nonconventional (type I) condensation in the HYL model should appear with a jump is due to van den Berg et al. [54}57]. A similar jump behaviour of the nonconventional condensation was discovered in the WIBG [15]. Moreover, we proved that in this system condensation occurs in two stages: with increasing density (or chemical potential), the nonconventional (type I) condensation appears "rst with a jump, then, after a new critical density, the conventional (type I) BE condensation gradually springs up [16]. Therefore, conventional and nonconventional condensation may coexist. In [58,59] we consider a model where a nonconventional (type I) condensation coexists with a conventional (type III, i.e. nonextensive) BE condensation. As we see, the notion of the condensate of Bosons has to be reexamined when we consider interacting systems. Therefore, it is appropriate to clarify this notion, since the Bogoliubov theory insists on the necessity of interaction to give a microscopic explanation of super#uidity. Intuitively, the Bose condensate is a fraction of particles which does not move, `frozen in momentum spacea, with p"0. This picture poses no problem for the PBG, where the operator of number of particles in a given mode is the integral of motion. Since single-particle quantum states are not proper for interacting system, Onsager and Penrose [60,61], in 1956, tried to work out a de"nition of condensate in this case. Their proposal was to identify condensation with an o!-diagonal longrange order (ODLRO), related to asymptotics of one-particle density matrix. This criterion shows that the mean-value of particle operator for the mode p"0 can still be used as a characterization of a BE condensation. Moreover, they have given 8% as an estimate of the fraction of particles in liquid He having p"0 at ¹"0 K. Hohenberg and Platzman [62], in 1966, proposed to use deep-inelastic neutron scattering for experimental observation of the Bose condensate in He. Cowley and Woods [63], in 1968, were

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the "rst to announce that they observed about 17% of condensate at ¹"1.1 K. After that, di!erent groups repeated and improved experiments in this direction. The `Dubna}Obninsk groupa, for the "rst time, not only reexamined the fraction of the condensate: (3.6$1.4)% at ¹"1.2 K [64], but also measured its temperature dependence [65]. These data give: (2.2$0.2)% for the condensate at ¹"0 K and ¹ "(2.24$0.04) K for the critical temperature. Recall that the  -point for super#uidity corresponds to ¹ "2.18 K! Hence, the appearance of the condensate H with a good accuracy coincides with the appearance of super#uidity, while the latter at ¹"0 K attains 100%. Of course, they were only the "rst (far from de"nite!) experimental estimates of the Bose condensate fraction in the liquid He. After these pioneering attempts, now (as we have learned from referee) it is universally agreed by the `super#uid He communitya that the best estimate of the condensate fraction is about 9% at ¹"0 K; for details of the history and of the methods see [68] and the recent review by Sokol in [49]. Notice that this value of condensate is surprisingly close to the above Penrose}Onsager estimate [60,61]. While the experimental evidence of the condensate in the liquid He bolsters one of the main hypotheses of the Bogoliubov theory, we have to mention another experimental aspect directly related to superyuidity. Henshaw and Wood [66], in 1961, measured the spectrum of elementary excitations in the liquid He by a weak-inelastic neutron scattering. Their data gave a sound evidence that a main branch of the excitations indeed has a Landau-type form with connected `phononsa and `rotonsa parts, as it is predicted by the Bogoliubov theory! But there were at least two points that cause di$culties to "t these data with the theory of super#uidity: E Application of the Landau criterion of super#uidity to the Henshaw}Woods spectrum gives for the critical velocity v +60 m/s whereas super#uidity in capillaries disappears when velocity is of  the order of few cm/s. Moreover, it depends sensitively on the diameter of the channel. E The Henshaw}Woods spectrum does not change drastically when the temperature crosses ¹ : H the Landau criterion still gives v '0 for ¹'¹ , whereas there is no super#uidity for these  H temperatures. The attempts to explain these `mis"ttingsa are concentrated around the idea that Landau-type spectrum discovered by Henshaw and Woods is only a part of a plethora of other types of `elementarya excitations not covered by the Bogoliubov theory, see, e.g. [67] and the book [68]. Of course, it would be too nam ve to imagine that the Bogoliubov theory is a theory of `real super#uiditya or an ultimate theory of He. It is, "rst of all, a mathematical approach to microscopic theory of many-body interacting boson systems, which solves principal di$culties on the way leading to calculation of the ground-state energy and of the spectrum of such systems at ¹"0 K, see Section 2.4. Moreover, the Bogoliubov phonon}maxon}roton dispersion branch is obviously only a part of the spectrum of the full (or even truncated) quantum-mechanical Hamiltonian of this system, for discussion see [84,88] and Section 6.1. For example, being a `quadratic theorya, the Bogoliubov approach does not take into account interaction between the above excitations, which is known to lead to decay of excitations, to di!erent kinds of hybridizations, including appearance of pair-excitations branches, bound states of excitations etc. It is not the aim of the present paper to go into details of the hierarchy of excitations in the liquid He; we refer curious readers to, e.g. a very detailed book by Gri$n [68]. For `nam vea readers we mention only that the above limit on the critical velocity of super#uid He in capillaries is

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determined essentially by excitations of vertices, that is obviously out of the scope of the standard Bogoliubov theory. 1.3. The Bogoliubov model of weakly imperfect Bose gas (WIBG) With this brief history of Bose condensation in mind we return to Bogoliubov's determination to give a microscopic theory of Landau-type spectrum of elementary excitations in He II. Since the BE condensation in the PBG does not produce the Landau-type spectrum (and by consequence, any super#uidity), Bogoliubov turned his attention to interacting Bose systems. He insisted that due regard must be extended to the boson interaction together with the BE condensation. Then, he immediately discovered that this problem cannot be tackled by standard methods of perturbation theory. It is the BE condensation that causes the di$culty. Since for the unperturbed system (i.e. for the PBG) the ground state corresponds to Bose condensate of particles either in the mode with momentum p"0 (BE condensation) or in a band around this mode (generalized BE condensation), any perturbation due to interaction is larger than kinetic energy in this state. Therefore, if we take only the kinetic energy operator of bosons as unperturbed part of the Hamiltonian, then we must face up to a theory of singular perturbations. Since these singularities are due to interactions with condensed particles, Bogoliubov formulated his xrst ansatz as a truncation of the full Hamiltonian which takes into account only interaction of particles in the condensate, and interaction of condensed particles with particles out of the condensate. This truncated Hamiltonian, (see [1] or [18], Eq. (3.81) [17], Eq. (28)) he named as `nearly perfecta [1] or `weakly imperfecta [18] Bose}Einstein gas. Below we call this truncated Hamiltonian the Bogoliubov model of weakly imperfect Bose gas (WIBG). To proceed further, Bogoliubov formulated the second ansatz. Combining the macroscopic occupation of the ground state (i.e. the BE condensate in the mode p"0) with canonical commutation relations of creation/annihilation operators in this mode, he proposed to treat the particle in the condensate classically. Due to this prescription the condensate acts as a classical particle reservoir (or external classical complex xeld) which noncondensate particles can enter and leave via scattering, see [1}3,18]. Notice that this famous Bogoliubov prescription amounts to neglecting the dynamical quantum behaviour of condensate, i.e. the quantum yuctuations. The corresponding, approximating to the WIBG, Hamiltonian with c-numbers instead of zero-mode creation/annihilation operators, has the following properties: E it is a bilinear form in creation/annihilation operators of noncondensate particles; E it does not conserve (noncondensate) particles. The "rst allows us to diagonalize the approximating Hamiltonian of the WIBG by canonical Bogoliubov u}v transformation to obtain the spectrum of this system in the explicit form. After a judicious choice of c-numbers involved in this Hamiltonian, Bogoliubov got the Landau-type gapless spectrum of elementary excitations with the phonon-like branch for small wave vectors and with a `rotona minimum in domain close to the Henshaw}Woods experimental data. The u}v transformation shows that elementary excitations are a noninteracting gas of bosons. Bogoliubov stresses that all the above is due to approximations (the xrst and the second ansaK tze) involved in his

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theory [1}3,18]. Having taken into account the terms disregarded by the "rst ansatz, one goes beyond the Bogoliubov theory, which gives, e.g. an interaction between elementary excitations. This programme was initiated by Beliaev in 1958 [69,70]. His approach forms the basis of the systematic application of the quantum "eld theory methods to boson systems with condensate, including anomalous propagators representing two particles going into or out of the condensate, Dyson}Beliaev equations, etc. Since the condensate is treated in the framework of the second Bogoliubov ansatz, the Beliaev theory needs the formalism of the grand-canonical ensemble with the chemical potential de"ned by the density of particles and interaction. In particular, Beliaev bolstered up the Bogoliubov hypothesis about the elementary excitations interaction by demonstrating that in the long-wavelength limit kP0 the spectrum of the excitations contains an imaginary part (decay) which goes as &k [70]. This approach was developed further in 1959 by Hugenholtz and Pines [71,72], and was thoroughly reexamined in 1963 by Gavoret and Nozie`res [73,74]. They showed that the singular character of interaction in the Bose systems with condensate remarked by Bogoliubov manifests itself in divergences of perturbation theory at small momenta (infrared divergences). Gavoret and Nozie`res [74], scrutinized this question inspired by the higher-order calculations by Beliaev [70] and Hugenholtz}Pines [71]. They showed that E infrared divergences cancel out in all physical quantities; E the energy gap in elementary excitation spectrum vanishes to all order of perturbation theory; for further developments see [68,75]. The last observation (known as the Hugenholtz}Pines}Gavoret}Nozie`res theorem) was discovered by Bogoliubov under the name 1/q theorem in his `Quasi-averages in Problems of Statistical Mechanicsa published in 1963 ([26], Part I, Section 2.8, and [76]). Instead of perturbation theory Bogoliubov's proof of this theorem is based on local gauge transformations. This method clari"es why one is faced with the appearance of a gap in the elementary excitations spectrum when one tries to `improvea the perturbation theory, or the Bogoliubov theory `mismatchinga the approximations, see [77}80]. Bogoliubov showed [26,76] that the appearance of a gap in some models is a consequence of the lack of local gauge invariance of these models, see [81}84] for discussions. The Bogoliubov theory [1}3,18] as well as further developments by Beliaev, Hugenholtz}Pines, Gavoret}Nozie`res were restricted to the zero temperature ¹"0. The "rst "nite-temperature calculations were made in 1957 by Lee et al. [85] for a gas of hard spheres, see also [86]. For ¹O0 the thermally induced depletion of condensate is now important since it may spoil the validity of both the Bogoliubov ansaK tze. To avoid the `mismatchinga one has to treat the condensate in some consistent fashion. It was Popov (see, e.g. [87,88]) who proposed in 1965 a generalization of the Bogoliubov theory for ¹O0 that gives the elementary excitation spectrum similar to that for ¹"0, but now with temperature-dependent condensate. For further details of the Beliaev}Popov theory we refer readers to a recent review by Shi and Gri$n [89]. 1.4. Outline In the present review we try to summarize recent rigorous results about the Bogoliubov model of the weakly imperfect Bose gas (WIBG).

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The motivation comes from the following. Whilst the Bogoliubov truncation of the full Hamiltonian (the 1st ansatz), which leads to the WIBG-model, re#ects his vision of the ro( le of di!erent interactions in the nonideal Bose gas, his second ansatz is based on two hypotheses: E The "rst one is that the Bose}Einstein (BE) condensation does exist in this nonideal system and it occurs only in one mode (ground state) of the perfect Bose gas (PBG). E The second one is that we may treat this conventional condensation `classicallya, i.e. approximating the creation/annihilation operators, corresponding to this mode, by c-numbers. Neither the "rst, nor the second of these hypotheses have been ever justi"ed rigorously for the WIBG. Moreover, Bogoliubov himself stressed many times in [3,18,21] that, since the perturbation theory around the PBG is highly singular, a Bose gas with any interaction, however weak, may be qualitatively rather diwerent from the noninteracting system. Therefore, the assumption that in the Bogoliubov WIBG we do have a Bose condensation must be proven. In the Bogoliubov theory, which starts with the full Hamiltonian, this problem is still open: the existence of the BE condensate in one mode (ground state) of the PBG was meant ad hoc, see an instructive discussion by Lieb [90] and Section 2.4. A challenge to mathematical physics was to prove this fact at least for the truncated WIBG-model related to the Bogoliubov theory. The second hypothesis is also too straightforward. First of all, it sounds as `approximationa of unbounded creation/annihilation operators by some bounded c-numbers. Second, it excludes all quantum yuctuations related to the condensed mode, which may be a cause of indirect ewective interactions between bosons in this mode as well as outside of it, not taken into account by direct interaction between zero- and nonzero-mode particles in the WIBG. It is well known that #uctuations of the condensate in the PBG are `abnormala, see [7]. Therefore, e!ective interactions via mediation of the #uctuating condensate could be (and they are, see [13,14]) important in the WIBG if condensation in this system exists, and if it is similar to the one in the PBG. For the "rst time, the problem of exactness of the Bogoliubov approximation for a non-ideal Bose gas with superstable interaction [91] was considered rigorously in 1968 by Ginibre [92] and then by Bu!et et al. [93]. A mathematical justi"cation of this procedure for the Bogoliubov model of the WIBG was provided only recently in [15]. There we also proved that, the condensate in the WIBG starts as a nonconventional Bose condensate due to an e!ective attraction between bosons in the ground-state mode. This interaction results from the quantum #uctuations in this mode [13,14]. A conventional (generalized) Bose}Einstein condensation comes into existence after the nonconventional one [16]. Since there are very few rigorous results concerning the Bogoliubov model of the WIBG and the Bogoliubov theory, we believe that the present review will "ll this gap, at least partially. We suppose that the important ro( le of quantum #uctuations, which are behind the thermodynamic behaviour of the WIBG, is instructive for further attempts in theoretical understanding of interacting Bose systems. This review is a status report on the WIBG from the point of view of rigorous results. We hope that it will discourage the performance of sloppy manipulations with Bose condensations, quantum #uctuations and di!erent kinds of ansaK tze. We believe that example of the exact solution of the WIBG-model (on the thermodynamic level) will provide a basis for further mathematical and theoretical studies of similar problems. As any exact solution, this one should be considered as a caricature on physical reality, cf., for example, the well-known exact solution of the two-dimensional Ising model. The evident value of such models is in a con"dence that one is able to control the properties of the model exactly. This

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allows to check rigorously fundamental principles related to a modelized phenomenon. Using again a magnetic analogy we recall the rigorous Peierls}Dobrushin argument proving spontaneous magnetization in the Ising model at low temperatures, which has nothing to do with magnetization in Fe, Ni or Co. Similarly, we do not claim that properties of the WIBG are directly related to recent experimental data on liquid helium, see [67], or on vapour of trapped alkali metals, see [48}50]. As far as physics is concerned the provenance of the WIBG-model hints at its probable (?) relevance to the limit of low temperatures and densities. But, since this model does not coincide with (another caricature) the PBG, the rigorously established mechanism of e!ective attraction between bosons due to quantum #uctuations, implying the nonconventional condensation, undoubtedly merits attention. Is it relevant to physics? The answer of the referee of the present paper is `it does nota. The authors' answer is `we don't knowa. But what is de"nitely interesting is breaking the `common wisdoma proof of existence of the nonconventional condensation, which appears not due to the standard saturation mechanism of the BE condensation, but due to the interaction, see Section 3. As with any review the present one has evidently many omissions. One of the major omissions is the lack of any serious discussion of the rigorous results on dynamics of (collective) quantum #uctuations in the WIBG and on inhomogeneous Bose condensations (conventional and nonconventional). The latter might have interest for a justi"cation of the Gross}Pitaevskii equation, which plays an important ro( le in analysis of experimental data concerning the Bose condensate of trapped alkali metals [48}50,94}99]. The only excuse for the authors is that there are yet very few rigorous results in these directions, see, e.g. the paper [100] about mathematical status of collective quantum #uctuations in boson systems, or [101,102] and the recent paper [103] giving the proof of exactness of the Gross}Pitaevskii energy functional for low-density bosons in a trap, as well as discussion in Section 6.

2. The Bogoliubov theory of super6uidity `To construct a complete molecular theory of super#uidity it is necessary to consider the liquid helium as being a system of interacting atoms.a Bogoliubov, Lectures on Quantum Statistics. `Faced with this failure, theorists retreated into the corner of low density gases with weak interaction.a Lieb, The Bose Fluid. To "x notations and to de"ne explicitly Bogoliubov's two ansaK tze we brie#y review in this section his microscopic approach to the super#uidity theory. We start by the perfect Bose gas (PBG) for the simplest case of the system enclosed in a cubic box with periodic boundary conditions to explain the phenomenon of the conventional Bose}Einstein condensation in the zeromode single-particle state, see Section 2.1. In Section 2.2 we discuss the "rst ansatz of Bogoliubov, i.e. the truncation which leads to the Bogoliubov Hamiltonian of the WIBG. Then we discuss the Bogoliubov approximation, i.e. Bogoliubov's second ansatz. In Section 2.3 we review the Landau criterion of super#uidity and the spectrum of elementary excitations results from the Bogoliubov theory. A brief survey of theories that go beyond the Bogoliubov theory is presented in Section 2.4.

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2.1. Second quantization, perfect Bose gas and conventional Bose}Einstein condensation Consider a system of bosons of mass m enclosed in a cubic box
"¸;¸;¸L1B of side length ¸. (In what follows we shall stick essentially to the physical dimensionality d"3 with comments (if necessary) concerning the cases dO3.) Without loss of generality, we may suppose periodic boundary conditions (b.c.) for wave functions on the surface R
. We denote by <,
"¸B the volume of the box
. Then the one-particle Hilbert space H coincides with the space of (periodic) square-integrable functions ¸(
) in
. The set of one-particle quantum states  (x)"<\e IV
H (i.e. eigenfuncI IZ ) forms an tions of the one-particle self-adjoint SchroK dinger operator ¹
"(! /2m)   
 orthonormal basis in H"¸(
). Here the dual set of wave vectors





2
n ? , n "0,$1,$2,2, "1, 2, 3 .
H" k31B: k " ? ? ¸

(2.1)

is de"ned by the periodic b.c. on R
, i.e. k"2
n/¸, n39. The standard device used in the quantum many-body problems is to pass to the second quantization, for details see, e.g. [104}106] and standard texts on quantum statistical mechanics like [18,37,107,108] or [109]. We wish to emphasize that it is not essential for the theory of the PBG but it is a very convenient device for our discussion of the Bogoliubov theory of the WIBG. We recall that the boson Fock space F (H) over the one-particle space H is a Hilbert space constructed as the orthogonal direct sum   F (H)"  HL"  (¸(
L)) , 

L L where

(2.2)

HL,(¸(
L)) (2.3) 

are symmetrized n-particle Hilbert spaces appropriate for bosons, and HL"" is the set of complex numbers. Space (2.2) is endowed by the scalar product  (, )F "M #  (L, L)HL , L where



(L, L)HL "


L

(2.4)

dx 2dx M L(x ,2, x )L(x ,2, x ) ,  L  L  L

for any L, L3(¸(
L)) . Recall that for any f3¸(
) one can introduce annihilation and 

(conjugate) creation operators, a( f ) and aH( f ), acting on F (H) as (a( f ))L(x ,2, x )"(n#1  L






dx f (x)()L>(x, x ,2, x ) ,  L

1 L (aH( f ))L(x ,2, x )"  f (x )()L\(x ,2, x( ,2, x ) ,  L  H L (n H H

(2.5)

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305

where x( means that the argument x is omitted (a `holea). One can check that operators (2.5) verify H H on F (H) the canonical commutation relations (CCR): [a( f ), aH( f )]"( f, g)H  .

(2.6)

Notice that "(1, 0, 0,2) is a vacuum vector in the space F (H), since by de"nition a( f )"0 ∀f3H ,

(2.7)

whereas J  aH( f )"(0,2, 0, J, 0,2) , (2.8) H H where J is de"ned by the set f J according to (2.5). Finally, we denote by H H a "a( ) and aH"aH( ) , (2.9) I I I I the usual annihilation and creation Bose operators in modes k3
H, i.e. in one-particle quantum states  (x)
H . I IZ With these notations the second-quantized Hamiltonian ¹
,(¹
) of the PBG in the box
can be written (formally) as ¹
"   aHa . (2.10) I I I
H IZ Here  " k/2m is the one-particle energy spectrum of free bosons in the modes k3
H. Similarly, I one de"nes the total particle-number operator (2.11) N
"  aHa , I I
H IZ where N ,aHa is the occupation-number operator in the mode k3
H. Operators (2.10) and (2.11) I I I are self-adjoint on some appropriate domains dom(¹
) and dom(N
) dense in the space F (H). By the CCR (2.6) and (2.7) and (2.8) one gets that the spectrum of the occupation-number operator is equal to (N )"0, 1, 2,2 . Vectors I (aH)LI (2.12)  I "  I , n 3 (N ) , L I I
H (n ! IZ I form a (canonical) orthonormal basis in F (H) and they are proper for the operators N , q3
H, O and (2.10), (2.11): N  I "n  I , O L O L







N
 I "  n  I , L L O OZ
H



¹
 I "   n  I . L L O O OZ
H

(2.13)

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In particular, restriction of set (2.12) on the n-particle sector HL of the Fock space F (H): L  " L  LI 

LI LI k3
H

(2.14)

is an orthonormal basis in HL. By ¹L HL and NL HL we denote corresponding
"¹
U
"N
U restrictions of operators (2.10) and (2.11) on the n-particle sector HL. Finally, formulae (2.5) get in the basis (2.12) a particular simple form: a (2, n ,2)"(n (2, n !1,2) , O O O O (2.15) aH(2, n ,2)"(n #1(2, n #1,2) . O O O O Let the system, we are concerned with, consist of n bosons enclosed in
which are in thermal equilibrium with a reservoir of the temperature
"k ¹,\. Then the canonical Gibbs state is de"ned by the canonical density matrix in HL: e\@2L
D (¹L ,
)" @ Z.
(, n)

(2.16)

where Z.
(, n)"¹rHL e\@2L


(2.17)

is the PBG canonical partition function related to the density of the free energy: 1 f
[¹L ln Z.
(, n) .
]"! <

(2.18)

) belongs to the ¹r-class of bounded operators on One can verify that for '0 the operator D (¹L @
HL, see, e.g. [91]. Therefore, (2.17) and (2.18) are correctly de"ned, but explicit calculation of the free-energy density even in the thermodynamic limit:
(2.19)

which is a ¹r-class operator on this space for '0 and (0, see below. Then the PBG grand partition function .
(, )"¹rF e\@2
\I,


(2.20)

is related to the grand-canonical pressure by 1 p
[¹
]" ln .
(, ) . <

(2.21)

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307

To calculate the trace in (2.20) one can use basis (2.12) and properties (2.13):  .
(, )"   e\@CI \ILI . (2.22) IZ
H LI  Since for the periodic b.c. inf
H  "0, the grand partition function of the PBG is bounded for IZ I (0 and 1 p
[¹
]"  ln (1!e\@CI \I)\ . (2.23) <
H IZ Using again basis (2.12) and (2.13), one readily gets for the grand-canonical mean value of the occupation-number operator in the mode k: 1 . N 
(, ),¹rF (D (¹
!N
)N )" @ I I 2 e@CI \I!1

(2.24)

Therefore, the average of the total particle density (cf. (2.13)) 1 1 .
(, )"  "R p
[¹
] , (2.25) I I <
H e@C \I!1 IZ is again valid only for (0. Now we are in a position to consider the question of the Bose}Einstein condensation. First, notice that for allowed values of chemical potential, (0, there is no macroscopic occupation of any mode k3
H:

  N I <

(, )"0, k3
H, (0 . 2
Here we denote the thermodynamic limit
1 p.(, )" (2
)



1

1 .(, (0)" (2
)

dk ln (1!e\@CI \I)\ ,



1

dk(e@CI \I!1)\ .

If one introduces the standard thermal wavelength  by @ 2
   , , @ m






(2.26)

(2.27)

and the function  zQ g (z),  , ? s? Q

04z41, '0 ,

(2.28)

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then 1 p.(, )" g (e@I) ,   @ 1 .(, (0)" g (e@I) .   @ The crucial observation now comes from the fact that, since

(2.29)

lim .
(, )"0 , I\ lim .
(, )"#R , (2.30) \ I and function .
(, ), (Eq. (2.25)) is strictly monotonic for (0, the xnite-volume equation ".
(, ), '0 ,

(2.31)

has unique solution .
(
, )(0 for any '0. On the other hand, the monotonic function g (z)  attends the maximum at z"1. We denote by 1 .(
),sup .(, )" lim .(, )" g (1)    @ I I\ the critical density for the PBG. Then the inxnite-volume equation (cf. Eq. (2.31)), ".(, )

(2.32)

(2.33)

has unique solution .(
, )40 for 4. (
), and it has no solutions for '.(
)! To solve this   paradox (which leads to the BE condensation) one has to return to individual terms in the sum (2.25) corresponding to the particle density in each mode k3
H. Let us single out the one-particle ground-state term corresponding to the mode k"0. Then Eq. (2.31) reads as 1 1 1 # " < e\@I!1 <

1 . (2.34) e@C \I!1 IZ
H

Let (.(
) in (2.34). Since for any (0 the sum in (2.34) tends (for
I

lim .
(
, )".(
, )(0, (.(
) . 


(2.35)

Let '. (
) in (2.34). Notice that by (2.1) 

 1 " (2
) . (2.36) inf  " I I
* 2m <
H

IZ  Therefore, the Darboux sum in (2.34) converges for
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309

in such a way that lim


  N I <

2


(, .
(
, ))"lim





1 1 1 "lim ! < e\@I.
FM!1 <


"!.(
) .  From (2.37) we "nd for the asymptotics of the chemical potential:

1 I IZ
H e@C \I FM!1 

.


 (2.37)

1 ,
  N I <

(, .
(
, '.(
)))". (
)"!.(
) , (2.39)    2
which is density of the conventional BE condensation in one mode. Indeed, as mentioned above, (see Eq. (2.36)) for the next to the k"0 mode and for
1 1 1 m 1 sup N 
(, .
(
, '. (
)))" K . (2.40) I 2  < e@CI
* \I.
FM!1 2
  ¸ <
H

IZ  Finally, for ".(
) by virtue of the arguments (2.34)}(2.37) one gets instead of (2.38) the  asymptotics: const .
(
, ".(
))K , 2/3((1 . 
(2.41)

This is a textbook presentation of the BE condensation, see [30}32]. The mathematical study of the in"nite-volume quantum Gibbs states was initiated in 1963 by Araki and Woods [34]. The complete theory of the grand-canonical and canonical Gibbs states and calculation of the corresponding generating functional of the cyclic representation of the CCR is due to Lewis and Pule` [35] and Cannon [36], see also [37]. 2.2. The weakly imperfect Bose gas and Bogoliubov approximation As above let us consider a system of identical bosons of mass m in a cubic box
L1 with periodic boundary conditions (torus
). If Hypothesis (A)

(x)"(x)3¸(1)

denotes an absolutely integrable (real) two-body interaction potential, then its (real) Fourier transform



v(q)"

1

dx (x)e\ OV, q31 ,

(2.42)

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Fig. 1. Kinetic-energy and interaction vertices of the full Hamiltonian.

exists and it is a continuous function vanishing at in"nity. To ensure stability, i.e. convergence of the grand-canonical partition function, following Bogoliubov [1}3,18] we suppose below that Hypothesis (B)

v(0)5v(q)50 and v(0)'0 ,

see, e.g. [91] and Appendix G. Notice that (2.42) for real (x)"(x) implies v(q)"v(q). The Hamiltonian for n of these interacting bosons is de"ned by a self-adjoint (s.-a.) extension








L

 1 L H #   (x !x ) HL ! (2.43)
"   G H 2m 2 U H GH G$H on the boson space HL of symmetric functions on the torus
, see, e.g. [37,91]. Here  (x)" 9 (x#n¸), x3
, is the restriction of the potential (x) on the torus
, i.e. LZ 



v(q)"




dx  (x) e\ OV . 

Then the second-quantized Hamiltonian (2.43) acting on the boson Fock space F (H) has the form 1 H
"¹
#;
,   aHa #  v(q)aH aH a  a  . I I I 2< I \O I >O I I
H IZ I I OZ
H The corresponding interacting vertex is illustrated in Fig. 1. Notice that .(
"0)"0 (see (2.32)) then by virtue of (2.37) and (2.39) one has 

(2.44)

lim . (
)"! lim .(
)" , (2.45)   > > F F i.e. at zero temperature all particles are in condensate. This might be easily expected, since for the n-particle ground state of the PBG one gets  ( , ¹
)F "min ( , ¹
)F "0 ,   RZF

(2.46)

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311

i.e. by (2.12) we obtain that the normalized ground state has the form (aH )L " 

" I  . 2  L (n!

(2.47)

This means that all of n bosons are accumulated in the one-particle ground state which corresponds to the mode k"0. Notice that the interaction energy in this state is equal to




1 ( , ;
)F , ,    2<



 v(q)aH aH a  a  O0 . I \O I >O I I  I I OZ
H

(2.48)

Let (in spite of the evident con#ict with (2.48)) one suppose that even for v(q)O0 the overwhelming majority of particles rests in the state , which is now not a ground state of (2.44). Then, according  to Bogoliubov, the system (2.44) behaves as two interacting subsystems: the "rst one corresponds to the condensate, whereas the second one to a small amount of particles outside the condensate. Since (a , a )F "n , I  I  I

(2.49)

Bogoliubov proposed (the xrst ansatz) to truncate Hamiltonian (2.44) in such a way that one disregards in (2.44) the terms of the third and higher order in operators a , aH . This truncation I$ I$ gives the Bogoliubov Hamiltonian for the WIBG, see Eq. (3.81) in [18]: H
"¹
#;
"¹
#;
"#;
, ,

(2.50)

where ¹
"   aHa , I I I IZ
H v(0) v(0) aHa # aHa ;
""   2< <   1 ;
," 2<

(2.51)

IZ




H



1 aHa # I I 2<

 v(k)(aHaH a #aHa a ) . I \I   I \I IZ
H




H

IZ



v(k)aHa (aHa #aH a ) ,   I I \I \I

(2.52)

(2.53)

Vertices corresponding to the interaction in the WIBG are illustrated by Figs. 2 and 3. The vertex with three dotted lines is excluded by the momentum conservation. Notice that even after this truncation the problem of the Hamiltonian (2.50) remains rather di$cult. Still the perturbation ;
of the state is singular: one obviously has ( , ;
)O0 and ( , ¹
)"0. Therefore,      one poses the same question as above: why do we have to believe that the conventional BE condensation in the mode k"0 persists in the presence of the truncated interaction ;
? Bogoliubov does not answer this question. The response was found only recently [12,15] and it will be our preoccupation in Sections 4 and 5. Instead, he proposed a neat ingenious trick (the second ansatz) to treat the Hamiltonian H
, presupposing that the conventional BE condensation in the mode k"0 does exist!

312

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

Fig. 2. Diagonal-interaction vertices corresponding to ;
".

Fig. 3. Nondiagonal-interaction vertices corresponding to ;
,.

His motivation comes from the observation that operators a , aH enter H
only in the form   a /(<, aH/(<. These operators for large <:   (i) almost commute, [a /(<, aH/(<]"1/< ,   (ii) are nontrivial in the state ,  aHa n ,   " ,  <  F <






(2.54)

(2.55)

if n&<, as it is in the case of BE condensation in the mode k"0. Therefore, they could be replaced in a macroscopic system by complex numbers: (2.56) a /(
V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

313

respect to the gauge transformations ; "e P,
, 31, i.e. P ; ¹
;H"¹
, ; H
;H"H
, ; H
;H"H
, P P P P P P and (here aI,a or aH ) I I I ; aI;H"e! PaI P I P I implies (in contrast to (2.56) for cO0) that

(2.57)

(2.58)

aI
(, )"aI
(, )"aI
(, )"0, k3
H . (2.59) I 2 I & I & Let H
(cI) be the Hamiltonian of the WIBG after the Bogoliubov approximation (2.56): H
(cI)"¹
(cI)#;
(cI)"¹
(cI)#;
"(cI)#;
,(cI) , ¹
(cI),   aHa ; I I I IZ
H

1 1 ;
"(cI), v(0)c<#v(0)c  aHa #  v(k)c[aHa #aH a ] ; I I 2
H

I I \I \I 2
H

IZ  IZ  1 ;
,(cI),  v(k)[caHaH #c a a ] ; (2.60) I \I I \I 2
H

IZ  with cI,c or c . Then the interaction ;
(cI) does not commute with the particle-number operator N
, i.e. it will no longer conserve particles because of creation and annihilation of pairs due to nondiagonal terms caHaH and c a a in ;
,(cI) (2.60). Therefore, one cannot treat the I \I I \I approximating Hamiltonian (2.60) in canonical ensemble. There are two ways around this problem. The "rst is the method proposed by Bogoliubov in the same year as his "rst paper; see [3,18,21,90,112,113]. Since N "aHa is a non-negative self-adjoint    operator, the operator (N #I)\ is correctly de"ned and bounded. Let   "aH(N #I)\a , H"aH(N #I)\a , k3
H . (2.61) I   I I I   Then for kO0 these operators satisfy the CCR, and the Hamiltonian H
(2.50) can be rewritten in the form N v(0) N H
"   H #  v(0)  H # N (N !I)#   v(k)(H #H  ) I I I I I I I \I \I < 2<   2<
H

IZ
H IZ
H

IZ  1 N(N !I) N(N !I)   #  v(k) HH  #    . (2.62) I \I I \I 2
H

< < IZ  Here we use (N #I)\a "a N\. Now the Bogoliubov approximation for Hamiltonian (2.62)     corresponds to




N  Pc, <

N(N !I)   Pc . <



(2.63)

This gives the same result as in (2.60) after the canonical gauge transformation to boson operators a "a e\  A, k3
H0 . I I

(2.64)

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Notice that, since by (2.58) the operators (2.61) conserve the number of particles, Hamiltonian (2.62) inherits this property even after the Bogoliubov approximation (2.63). Hence, now we may treat this approximating Hamiltonian in canonical ensemble with some parameter c. To exclude this uncertainty Bogoliubov proposed to eliminate the operator N from (2.62) at the cost of further  approximations, see [3,18] and discussion in [21]. Since N
,  aHa "  H , N "N
!N
, I I I I  IZ
H

IZ
H

he used approximation N N #  2< <




H

(2.65)

N
H K I I 2<

(2.66)

IZ  in the sum of the second and the third terms of (2.62) and N N  K
P < <

(2.67)

in the third, the fourth and the "fth terms to arrive at an approximating Hamiltonian for the canonical ensemble with density :  N
1 HI
()"  ( #v(k))H #  v(k)(HH #  )#v(0) ! v(0) . (2.68) I I I 2
H

I \I I \I 2< 2
H

IZ  IZ  Since the last two terms are constants in the canonical ensemble, we rest with a bilinear form (2.68) in Bose operators 
H . I IZ  The second method was essentially advocated by Beliaev [69,70], Hugenholtz and Pines [71,81,114,115], Tserkovnikov [116] and Tolmachev [84]. Since the Bogoliubov approximation (2.56) leads to nonconservation of particles, they proposed simply to use a grand-canonical ensemble from the very beginning. This means that instead of H
one has to consider H
()"H
!N
.

(2.69)

Then Bogoliubov's second ansatz gives the approximating Hamiltonian in the form (cf. (2.60)) 1 H
(cI, )"  [ !#v(0)c]aHa #  v(k)c[aHa #aH a ] I I I I I \I \I 2
H

IZ
H

IZ  1 1  v(k)[caHaH #c a a ]!c<# v(0)c< , (2.70) # I \I I \I 2 2
H

IZ  where cI"c or c . In fact the turn to the grand-canonical ensemble would not be very astonishing, since properties of the PBG are also more transparent in this ensemble. After the gauge transformation (2.64) it becomes clear that, as above, Hamiltonian (2.70) depends only on c. Then the procedure for determining c, proposed in [69,71,81,84] gives the variation principle:






R R 1 ( , H
(cI, ) )F " v(0)c
(2.71)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

where is the ground state (2.47) of the PBG with n"c<. Hence one obtains  "v(0)c .

315

(2.72)

For a given total particle density  the chemical potential should be excluded from (2.72) by the subsidiary condition, which de"nes c as a function of . If one does this in the xrst approximation [69,71,81], then one gets c", which returns us to (2.68), see discussion in Section 2.4. It is needless to say that calculations done with approximating Hamiltonian by this last method give the same results as those in canonical ensemble [3,18,21,112,113]. Below we shall consider the WIBG in the grand-canonical ensemble (, ). 2.3. Spectrum of elementary excitations and superyuidity Since the ultimate aim of the Bogoliubov theory is a microscopic explanation of the super#uidity a` la Landau, one has to calculate the spectrum of excitations either of the Hamiltonian HI
() (2.68) or the Hamiltonian H
(cI, ) (Eq. (2.70)) with condition (2.72). Both give the same result, so we consider only the grand-canonical ensemble. Notice that the Hamiltonian H
(cI, ) is a bilinear form in boson operators aI
H . So, one I IZ  can diagonalize it by the Bogoliubov canonical u}v transformation to a new set of boson operators bI
H . First, we perform the canonical gauge transformation (2.64) to a I
H . Then I IZ  I IZ  a "u b !* bH , I I I I \I a H"u bH!* b . (2.73) I I I I \I Here the real coe$cients u "u
H and * "*
H satisfy the relations I \I IZ  I \I IZ  u!*"1 (2.74) I I to ensure the CCR for bI
H , and I IZ   h u#*" I , (2.75) 2u * " I , I I E I I E I I where E "(f !h , I I I f " !#c[v(0)#v(k)] , I I h "cv(k) , I to ensure diagonalization of the form (2.70). Eqs. (2.74) and (2.75) imply that









(2.76)



1 f 1 f I #1 , I !1 . *" (2.77) u" I 2 E I 2 E I I Notice that the Bogoliubov transformation (2.73) is trivial if v(k)"0, and it makes sense only if f 5h . This constraints c and  to satisfy the inequality I I 4v(0)c# min  . I IZ
H

(2.78)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

Fig. 4. Interaction potential (x) and its Fourier transform v(q).

In terms of the new boson operators bI
H Hamiltonian (2.70) becomes I IZ  1 v(0) c< , H
(cI, )"  E bHb #  (E !f )!c# I I I 2
H I I 2
H

IZ  IZ  where

(2.79)

(2.80) E "(( !#cv(0))( !#cv(0)#2cv(k)) I I I is the spectrum of elementary excitations of (2.70). Now, if we take into account the "rst-order calculation (2.72) for the chemical potential (cf. condition (2.78)), we obtain the famous Bogoliubov spectrum: (2.81) E "( ( #2v(k)c) . I I I Recall that by Hypotheses (A) and (B) (see Section 2.2) the two-body potential (x) is spherically symmetric, i.e. (x)"(x). Then the Fourier transform v(q)"4






dr r(r)

sin(qr) ,v(q) q

(2.82)  is also spherically symmetric, (see Fig. 4). Suppose in addition the absolute integrability of x(x)3¸(1). Then one gets from (2.82) the expansion v(q)"v(0)#qv(0)#o(q) , (2.83)  where v(0)40, and v(q)4const q\. Now we return to the Bogoliubov spectrum (2.81). First, let c"0 (no condensation). Then the Bogoliubov theory is trivial: H
(cI"0, )"¹
!N
,

(2.84)

i.e. it coincides with a PBG with excluded mode k"0. This kind of PBG has in the thermodynamic limit the same properties as the standard PBG (see [117] and Section 4). In this case the Bogoliubov spectrum E " I I

(2.85)

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317

Fig. 5. The Bogoliubov spectrum E . I

is identical to the spectrum of the PBG. Now let cO0. Then (see Fig. 5) by (2.83) one gets



E " I






  k" wk for kP0> , v(0)c m

(2.86)  " k/2m for kP#R . I So, the Bogoliubov spectrum is gapless and phonon-like for small k, whereas for large wave vectors it behaves like single-particle excitations  in the PBG. Since v(q) attains its maximum at I q"0, one can choose the potential (x) in such a way that ( ( #2cv(k)))"0 at k"k O0 . (2.87) I I  In this case the Bogoliubov spectrum has local (`rotona) minimum at k , see Fig. 5. On the other  hand, by (2.81) one gets


 



   E 5k , v (c)k , (2.88) min ( #2v(k)c) I  I 2m I see Fig. 5. These properties of the Bogoliubov spectrum give a microscopic interpretation of the Landau theory of super#uidity of quantum liquids [1,3,18,20,90,118,119]. Recall that the latter reposed on the following principle: E quantum liquid remains yuid even for zero temperature; E at low temperatures collective behaviour dominates the individual movement of atoms, i.e. apart from translations (#ow) the state of this liquid is entirely described by the spectrum of collective (elementary) excitations; E by virtue of thermodynamic data [20,120] (e.g. speci"c heat capacity) this spectrum for He should be phonon-like for the long-wave length collective excitations and should be above a straight line with positive slope with (`rotona) minimum in the vicinity of k K2 As \.  The Bogoliubov theory shows that within some approximations the WIBG behaves as this quantum liquid with collective excitations described by the boson operators bI
H and by the I IZ  gapless Bogoliubov spectrum (2.81) with properties (2.86)}(2.88).

318

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

Fig. 6. Friction due to excitation k by the walls of the channel.

Suppose that a mass, M, of this Bogoliubov quantum liquid with velocity v moves in a channel (see Fig. 6). Let it be in its ground state E in the rest system of the liquid. Then in the laboratory  system this #uid will have energy E "E #Mv/2. Since any interaction of the quantum liquid   with the `outside worlda manifests as elementary excitations, the friction has to produce them, decreasing the initial energy E . For the Bogoliubov quantum liquid (2.79), this means appearance  of excitations by action of creation of Bose operators bI
H on vacuum I of the correspondI IZ  ing Fock space F I . To make the e!ect of the friction more sound, suppose that, created by this process, excited state bHI (with one elementary excitation) corresponds to the wave vector I k parallel to the velocity v. Then the new energy E and the new momentum P"Mv of the #ow Y are related with initial conditions E and P"Mv by  E "E #E ,  Y I Mv"Mv# k . (2.89) From (2.89) one readily gets (2.90)  k(v#v)"E . I  If excitation bHI results by the friction, then v must be less than the initial velocity v. By virtue I of (2.90) this means that the initial velocity must be bounded from below: E v' I 'v .

k

(2.91)

For the Bogoliubov quantum liquid one has (cf. (2.89))



'0 if c'0 , E (2.92) inf I "v (c)" 

k 0 if c"0 . I Therefore, if v (c)'0 and if the initial velocity of the #ow  v(v (c) ,  then the #uid will not be able to lose any momentum to the walls of the channel! Hence, it will display super#uid behaviour. This is the famous Landau's criterion of superyuidity of 1941 [4,5] in application to the Bogoliubov theory.

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

319

Notice that for the PBG (independent of the presence or absence of the BE condensate) one has  inf I "0 . (2.93)

k I Consequently, by (2.91) the #ow of the PBG in a channel is always able to lose energy via friction for any vO0. As it was claimed by Bogoliubov [1}3], the coherency due to the presence of the BE condensation is not enough to make the PBG super#uid. The spectrum of elementary excitations in this case is not collective: it corresponds to the individual movement of particles. The same we have for the Bogoliubov quantum #uid when the density of the BE condensate c"0, see (2.85). The crucial ro( le of interaction is that it transforms the coherency due to the BE condensation into collective behaviour of the system. In particular, it modi"es spectrum of individual excitations  into the Bogoliubov spectrum E . They are drastically di!erent for kP0, i.e. where interaction I I dominates the kinetic energy, cf. discussion in Section 1. Since in the "rst approximation one has (2.72) and c" (see end of Section 2.2), the pressure p"!R ( , H
(cI, "v(0)c) )F "v(0) ,   4  and hence one has the velocity of sound de"ned by

(2.94)

s"R p" v(0)/m . (2.95) M From (2.88), (2.92) and (2.95) one would expect that the critical velocity for super#uidity should be of order of magnitude of the velocity of sound, which is supported independently by the Feynman theory of He [118,119]. 2.4. Beyond the Bogoliubov theory Since the Bogoliubov theory is founded on two ansa( tze: E truncation of the full Hamiltonian, E Bogoliubov's approximation, i.e. substitution of the zero-mode creation/annihilation operators by judiciously chosen c-numbers, one could imagine several directions for improving the Bogoliubov theory. If one does not aspire to deal with the full Hamiltonian (2.44) but on the other hand does not like the idea of the Bogoliubov truncation either, then one could try the Bogoliubov approximation directly in the full Hamiltonian (2.44). With the substitution (2.56), the interaction ;
in (2.44) changes to ;
(cI) with number of terms, depending on how many nonzero modes are in the vertex (see Fig. 7): ;
(cI),; #; #; #; #; #; #; ,        where 1 ; "  v(q)aH aH a  a  ,  2< I \O I >O I I I I OZ
H

1 ; "  v(q)aHaH a  c, ; ";H ,   2(< O I \O I 
H

I OZ 

(2.96)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

Fig. 7. Full interaction with a classical reservoir of condensed particles.

1 ; "  2 1 ; "  2



OZ


H






H

v(q)aHaH c, ; ";H ,  O \O  (v(0)#v(q))aHa c, O O

OZ  1 ; " <cv(0) .  2

(2.97)

Notice that Bogoliubov's part of interaction (2.50)}(2.53) corresponds in (2.97) to ;
"(cI)"; #; ,   (2.98) ;
,(cI)"; #; .   The interactions ; #; and ; #; , and hence ;
(cI), do not conserve the number of     particles, i.e. they are not gauge invariant: e P,
;
(cI)e\ P,
O;
(cI) . Therefore, following [26,69}71,76] we consider the Bogoliubov approximation in the grandcanonical ensemble, that gives (cf. (2.96) and (2.97)) H
(cI, )"  ( !)N !<c#;
(cI) . I I IZ
H

(2.99)

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321

After canonical transformation to boson operators a "e\  Aa , a H"e  AaH, k3
H0 , one I I I I "nds that Hamiltonian (2.99) depends only on c. To avoid a discrepancy between prescription (2.56) and aI/(<
(, )"0 , &  see (2.57)}(2.59), which persists even in the thermodynamic limit lim aI/(<
(, )"0 ,  &


(2.100)

Bogoliubov proposed ([26], Part I, and [76]) to include in the Hamiltonian H
a term breaking the gauge invariance in the mode k"0: (2.101) H
(I)"H
!(<(a #aH), "e  J3" .   Then the Bogoliubov quasi-average (i.e. expectation in pure Gibbs phase de"ned by the gauge "arg  ([26,76]; Part I) should "t with (2.56): (2.102) lim aI/(<
I (, )"ce! P , lim  & J  J  JP
in the condensed phase. Application of the Bogoliubov approximation to (2.101) changes (2.99) by a constant term. The system described by Hamiltonian (2.99) is a generalization of (2.70), known as a model system with a (uniform) condensate, see [26,69,70,76]. It was Beliaev [69,70], who considered this generalization of the Bogoliubov theory for the "rst time by systematic application of the methods of quantum "eld theory. The new feature associated with the existence of a BE condensate reservoir is that the Dyson equation should be replaced by the pair of Dyson}Beliaev equations for normal and anomalous propagator (Green's functions). The latter represents two particles going into or out of the condensate (cf. Fig. 7). The model with a condensate contains two parameters  and c, which are determined by the following procedure, cf. the case of H
(cI, ). For given  and c one "rst calculates the ground state  and eigenvalue  E (c, ) of (2.99) and the corresponding mean-value of particles outside the condensate:  (2.103) N
 (c, )"( , N
 )F .   Then one determines c by the condition 

(2.104) R  (E (c, )/<)"0 , A  and the chemical potential  (for given total density  and
"0) from the subsidiary equation

 

"c#

N
<



(c, ) ,

(2.105)



see [26,69}76,81,84]. So the real problem now is to calculate, in some approximation, the ground-state eigenvalue and the particle density. Beliaev veri"ed [69] that in the xrst order in

322

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

interaction the only contribution to the ground-state energy comes from ; , see (2.97), i.e. the trial  ground-state wave function  " , see (2.47). Therefore,   (2.106) E (c, )"
(2.107)

compare with (2.72). Moreover, the particle density outside the condensate (depletion fraction) in the lowest order of the perturbation theory is

  N
<

1 (c, )" < 

 * . I R IZ
H

By (2.77) and (2.107) one gets

(2.108)

 

(2.109)

"c .

(2.110)

N
<

(c, "cv(0))&o[(v(0))] . R Hence, in the "rst order we have by (2.105)

With (2.107) and (2.110) Beliaev found that in the "rst order, the self-energy parts in Dyson}Beliaev equations give exactly the Bogoliubov spectrum (2.81) with phonon velocity equal to s; see (2.95). In [70] Beliaev showed that in the next approximation the spectrum remains gapless and phonon-like in the long-wavelength limit kP0, while the interactions ; , ; of (2.97) make the   elementary excitations unstable with damping which goes, in this limit, as &k. These calculations demonstrate explicitly that the singularity of the perturbation theory indicated by Bogoliubov as a general feature of models with a condensate [3,18,21], manifests in infrared divergences (for small energies) of Feynman}Beliaev diagrams. This leads to a non-analytic dependence of the groundstate energy E (c, ) and other observables on the parameter (v(0)), cf. (2.109).  As we see, the model with condensate (2.99) is a kind of self-consistent theory since the condensate is supposed as an a priori datum (classical external "eld in the mode k"0) which has to be determined from subsidiary conditions that are `missinga in the frame of this model. They can be found from Green-function equations ([69], Eqs. (6.11)}(6.13)) variational principle ([71], Eq. (3.5)), minimizing E (c, ), or a dynamical equation for the zero mode ([26], Part I, Section 2.7,  Eq. (7.11); [116], Eq. (3)) equivalent to the variational principle. Following [26,81] we would like to warn against the `mismatchinga of approximations in di!erent parts of this self-consistent scheme. For example, if one tentatively improves the approximation of  (2.107) by a more precise expression, then we lose the cancellation in (2.80) necessary to close the gap and to recover the Bogoliubov spectrum (2.81)! However, Hugenholtz and Pines in 1959 [71], showed that in a consistent calculation, the spectral gap vanished to all order in perturbation theory, which is in fact an expansion in the potential v(k) and in the density , cf. (2.107) and (2.109). In 1960, Bogoliubov showed ([26], Part I, Section 2.8, [76]); that this Hugenholtz}Pines theorem follows from his 1/q theorem, which he proved in a nonperturbative way. In the above context, this means that to improve expression (2.80) for the spectrum E it is not I su$cient only to substitute a more accurate than (2.107) approximation for . In this higher order

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

323

of accuracy, besides ; #; #; #; one has to take into account the contribution     ; #; (see (2.97)) as it was done by Beliaev [70].   Notice that the Hugenholtz}Pines theorem says nothing about the asymptotics of the spectrum E P0 for kP0. It was Gavoret and Nozie`res [73,74], who extended the above theorem showing I that in all order of the perturbation theory this asymptotics is linear with the slope which coincides with the sound velocity, see also [26,84] for applications of the 1/q theorem in this case. Notice that the Bogoliubov theory and the model with a condensate are formulated on the level of approximating Hamiltonian, i.e. for zero temperature
"0. If one would try naively to substitute the variational principles (2.71) and (2.104) for
"0 by the corresponding one for thermodynamic potential, then relations (2.72) and (2.107) will be disbalanced (the BE condensate depletion), and we obtain a gap in the spectrum. The "rst consistent theory of the model with condensate for
O0 was developed by Popov in 1965, see [87,88] and also Tserkovnikov [116]. For a nice detailed discussion of the Popov generalization of the Bogoliubov theory on
O0 we refer the reader to a recent review article [89]. Notice that in the present paper we review the exact solution of the WIBG. Therefore, we have no need for any self-consistent scheme or ansatz depending on the temperature. Another way to relax the xrst Bogoliubov ansatz is a less severe truncation. This truncation was proposed by Zubarev and Tserkovnikov in 1958 [77] inspired by their studies of the Bardeen} Cooper}Schri!er (BCS) model where pair excitations play a central ro( le. Later this idea was developed in several papers, see, e.g. [78}80,83,111,121]. The corresponding truncated Hamiltonian is known as the Pair Hamiltonian which has the form H.
"¹
#;.
"#;.
,,¹
#;.
 .

(2.111)

Here ;.
" is the diagonal part of the interaction in H.
: 1 1 ;.
""  v(0)aH aH a  a  #  v(k !k )aH a  aH a  , I I I I  I I I I 2< 2<
H
H  I I Z
H I Z I Z !I and ;.
, is the corresponding nondiagonal part: 1 ;.
," 2<

(2.112)

 v(k!k)aHaH a a . (2.113) I \I \IY IY IZ
HIYZ
HI

From (2.112) and (2.113) it is clear that the full interaction ;
in (2.44) is truncated in the following way: "rst, put q"0 or q"k !k and then k "k, k "!k, q"k!k. Another evident     remark is that interaction ;.
 contains the Bogoliubov interacting terms ;
", (2.52) and ;
,, (2.53). To obtain (2.52) one has to truncate both of the sums (2.112) by the constraints k "0, k O0 or k O0, k "0. Similarly, one gets (2.53) truncating (2.113) by the conditions     k"0, kO0 or kO0, k"0. There are very few rigorous mathematical results on the model H.
 (2.111). Starting on [77,78] numerous articles on this model use di!erent kinds of approximations or variational schemes. Inspired by the success in the rigorous study of the BCS model, the early papers [77}80,111] used either a BCS-type variational principle or the Approximating Hamiltonian Method (AHM) which indeed gives the exact result for the BCS model, see [122,123] (translated in [26], Part II; [124}128]).

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

The main idea of the AHM grew from Bogoliubov's approximation ansatz of 1947 applied to combinations of operators A and B , k3
H, by I I A "aHa ! , B "a a ! , (2.114) I I I I I \I I I where " 31
H and "I3"
H are set of trial parameters. In terms of operators I IZ I IZ (2.114) the Hamiltonian H.
()"H.
!N
takes the form H.
()"H.
(, ; )#=.
(, ) ,

(2.115)

where H.
(, ; )"E.
(, ; )#  IZ
H





1 f . aHa # (h. aHaH #h. a a ) , I I I 2 I I \I I \I I

(2.116)

and v(0) 1  AH A  #  v(k !k )AH A  =.
(, )" I I   I I 2< 2<
H
H
H

I I Z I Z I Z !I 1 #  v(k!k)BHB . (2.117) I IY 2<
H
H

IZ IYZ I Here E.
(, ; ) corresponds to zero- and linear terms in development of the operator H.
() around the point  and I" or  : I I I I 1 1  v(0)    #  v(k !k )    E.
(, ; )"! I I  I I 2< 2<
H
H 
H      I I Z I Z I Z !I 1 !  v(k!k)  , (2.118) I IY 2<
H
H

IZ  IYZ I and



v(0) f . " !# I I <







1 1   ! #  v(k!k) , IY IY 2 <
H

IYZ
IYZ !I

1 h. " I 2<

 v(k!k) . (2.119) IY IYZ
HI

By virtue of the Bogoliubov inequality (Appendix D) one can estimate the di!erence of pressures corresponding to H.
, (2.111) and to the approximating Hamiltonian H.
(, ): 1 1 . =.
(, ) .
 4p
[H.
(, )]!p
[H.
]4 =.
(, ) .
 & & ?A < <

(2.120)

If the interaction satis"es v(q)50 (see Hypothesis (B) of Section 2.2), then v(0)  AH A  50 . I I 2< I I Z
H

(2.121)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

325

If, in addition, one supposes that (x)50 (pure repulsion), then v(q) is a positively dexned function on
H: 1  v(k!k)z z 50, z 3" . I IY I 2<
H IIYZ Therefore, the operator (cf. (2.117))

(2.122)

1 2<

1  v(k !k )AH A  #  v(k!k)BHB 50 (2.123)   I I I IY 2<
H
H
H
H

I Z I Z !I IZ IYZ I is non-negative. By virtue of (2.121) the same would be true for the expectation on the left-hand side of estimate (2.120), at least for
1 <



1 <





 v(k)AHA #  v(2k)AHA "0 I I I \I .

H
H & IZ IZ

(2.124)

and



 v(0)BHB "0 . (2.125) I I .
 & IZ
H The < denominators in (2.124) and (2.125) make these hypotheses plausible. If we admit them, then estimate (2.120) gives a variational problem with the approximating Hamiltonian H.
(, ): lim


R I p
[H.
(, )]"0 , ?

(2.126)

and (2.127) R I p
[H.
(, )]"0, R p
[H.
(, )]"0 , A AI corresponding to a minimum of the trial pressure p
[H.
(, )]. This leads to self-consistency equations: ,  "aHa 
.  I I I & ?A

(2.128)

and . (2.129)  "a a 
.  ,  "aHaH 
.  I I \I & ?A I \I I & ?A Since the approximating Hamiltonian, see (2.116), is bilinear, one can diagonalize it by the Bogoliubov u}v transformation (2.73) with











1 f .  1 f .  u" I #1 , *" I !1 , I 2 E.  I 2 E.  I I where E. "(( f . )!h. , k3
H . I I I

(2.130)

(2.131)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

Therefore, equations (2.128) and (2.129) take the explicit form






 

1 1 f .   " I cth E.  !1 , I 2 E.  2 I I

(2.132)

and











1 h.  1 1 1 h.  (2.133)  "! I cth E.  ,  "! I cth E.  . I I 2 E.  2 I 2 I 2 E.  I I Now, to check the exactness of the approximating Hamiltonian (2.116) one has to prove that the right-hand side in the estimate is zero in the limit
lim E. "(4. (
)v(0) , (2.136) I  I see [77,79,80]. Result (2.136) does not satisfy the Hugenholtz}Pines-theorem or the Bogoliubov 1/q-theorem about gapless excitations in the Bose systems with condensate ([71,76], and [26], Part I, Section 2.8). In spite of nonrigorous status of (2.136), it should discourage us from attempting to go beyond the Bogoliubov Hamiltonian via the Pair Hamiltonian H.
. In fact, the proof of the 1/q-theorem shows that the gap in the spectrum of elementary excitations of the Pair Hamiltonian is not an artifact of uncontrolled approximations or a &&fortuitous result'' but it results from a breaking of the local gauge symmetry in the truncated theory. Our discussion above shows that the gapless spectrum in the Bogoliubov theory is due to a delicate balance between Bogoliubov's two ansa( tze and the tuning of the value of the chemical potential, see (2.72), (2.80) and (2.107). Bogoliubov's proof of the 1/q-theorem (implying the

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

327

Hugenholtz}Pines theorem) is nonperturbative and it is based on the local gauge transformations: a(x)Pe PVa(x) , aH(x)Pe\ PVaH(x) .

(2.137)

Here x3
(torus), (x) is a continuous function on
and aI(x)
are boson-"eld operators VZ (cf. Section 2.1) a(x)"  a (x) , I I IZ
H aH(x)"  a (x) . I I IZ
H Then the full grand-canonical Hamiltonian (2.44) can be rewritten in the form



H
()"H
!N
" #

1 2

  dx






(2.138)



 dx aH(x) ! ! a(x) 2m


dy aH(x)aH(y)(x!y)a(y)a(x) .

(2.139)




Notice that this Hamiltonian is invariant with respect to the local gauge transformation (2.137), which leads to local particle number conservation, i.e. the continuity equation R (x)#j(x)"0 , R relating the density operator (x)"aH(x)a(x) with the current density

(2.140)

 aH(x)(a)(x)!(aH)(x)a(x) . j(x)" 2mi Notice that, for example, the nondiagonal part (2.113) of H.
: 1 ;.
," 2<

   dx

dx







dz aH(x#z)aH(x#z)(z)a(x)a(x)




is not locally gauge invariant, cf. (2.137). One sees that a particle absorbed at the point x can be recreated at the point x#z. The same concerns interaction (2.112). Therefore, the appearance of gap (2.136) is a consequence of the lack of local gauge invariance or of local conservation of particle density (2.140). Similarly, the nondiagonal part ;
, (2.53) of the WIBG combined with the term v(0)(aH)(a )/2   from (2.52) gives the interaction 1 ;I
," 2< #

  


dx 

1 2<







  

dx (x !x )aH(x )aH(x )     

dx aH(x )  






dx aH(x )  




dx 









dx a(x )  




dx a(x )  

dx (x !x )a(x )a(x ) ,     

(2.141)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

which is not locally gauge invariant, see (2.137); and the same concerns the ;
", (2.52). Hence one has to anticipate a gap in the spectrum of elementary excitations in the exact solution of the Bogoliubov WIBG with the Hamiltonian H
, (2.50). This conclusion is con"rmed in [15,16], see Sections 4 and 5. Another temptation to go beyond the Bogoliubov theory is related to the full Hamiltonian (2.44) or H
in (2.139). Instead of considering the variational principle based on the Pair Hamiltonian part H.
 (2.111) of the full problem, see [90], one can try to work it out for H
. In 1965, Robinson [129] formulated such a principle for ¹"0, on a subset of the in"nite-volume ground states, which coincide with translation-invariant quasi-free states. Then it has been generalized to ¹O0 in [121]. Recall that the in"nite-volume quasi-free states  (!) in the set of translation-invariant (locally  normal) states are completely determined by the following functions on the boson-"eld operators (2.138) (see Appendix F and [37]):  (aI(x))"cI , (2.142)  2 (aH(x)a(y)), (aH(x)a(y))! (aH(x)) (a(y)),d(x!y) , (2.143)     2 (a(x)a(y)), (a(x)a(y))! (a(x)) (a(y)),w(x!y) . (2.144)     Here c3", d is a real-valued function, and w is a complex-valued function, both depending only on x!y. Then the thermodynamic functional, cf. (F.5) in Appendix F, g ()"e()!\s()!(n()!) , (2.145) I that we require for the variational principle, is the free-energy density f ()"e()!\s() with the particle density constraint n()" taken via the Lagrange multiplier . For " the  functional can be calculated explicitly using the local normality and (F.2)}(F.4) (Appendix F): 1 n( )" (aH(0)a(0))"d(0)#c"   (2
) 1 s( )"  (2
)



1



1

dk dK (k)#c ,

dk [1#r( (k)] ln[1#r( (k)]!r( (k) ln[r( (k)] ,





  

 1 e( )" aH(0) ! a (0) #   2m 2

1

dx (x) (aH(x)aH(0)a(0)a(x)) 

(2.146) (2.147) (2.148)

where r( (k)"((dK (k)#)!w( (k)! .   Using (2.142)}(2.145) we obtain for the energy-density functional (2.148): 1 e( )"  (2
) #



1 c dk  dK (k)# n( )v(0)# I  2 (2
) 1

1 c 2 (2
)



1 c  dk w( (k)v(k)# 2 (2
) 1



1



1

dk dK (k)v(k)

dk w( (k)v(k)

(2.149)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

#

#

1 1 2 (2
) 1 1 2 (2
)

    dk

1

1

dk

1

329

dk dK (k)v(k!k)dK (k) dk w( (k)v(k!k)w( (k) .

(2.150)

1

To determine the equilibrium quasi-free state one has to minimize g ( ) with respect to the I  parameters c, "arg c, dK (k), w( (k), (k)"arg w( (k) and . The solution of the set of equations:

R g "0, R g "0, g "0, g "0, g "0, A I P I

dK (k) I

w( (k) I

(k) I

R g "0 I I

(2.151)

provided it exists and gives a global minimum, determines the canonical equilibrium quasi-free state  (!), (!). Notice that it is possible that (2.151) have no solution at all. For example, for  @M the PBG, i.e. v(q),0, the minimum of g ( ) occurs on the hyperplane r( (k)"dK (k) which is not I  a solution of Eq. (2.151). Retaining only those terms linear in dK (k) and w( (k) in the expression for the energy-density functional (2.150), one obtains the simplixed form of the set of equations (2.151). As it is shown in [121] the solution of these equations gives the quasi-free  (!)"lim

(!), where the A  "nite-volume state 
(!)"!
I is generated by H
(cI), see (2.60), with c satisfying (2.72). A & A  Therefore the equilibrium state corresponding to the Bogoliubov theory is a particular case of a linear approximation of the full variational problem on the set of quasi-free states. Since the rigorous results for the full Hamiltonian (2.43), (2.44) (for d"3) are very rare, we would like to conclude this section by one of the very recent papers [130]. It concerns the proof of the (put forward many decades ago, see [90]) formula for the asymptotics of the ground state energy per particle (n"<)





1 e ()"lim  n


inf ( , HL (2.152)
)HL HL ,R,RZ for dilute Bose gas. Let (x)"(x"r) be non-negative spherically symmetric potential decreasing faster than r\ at in"nity. The scattering length a of the potential is de"ned by





u (r) , a" lim r!  u (r)  P> where u (r) solves the zero-energy scattering equation 

 ! u (r)#(r)u (r)"0  m 

(2.153)

for two-particle problem (i.e. for the reduced mass m/2) and for u (0)"0. Lieb and Yngvason [130]  have recently proved that the low-density asymptotic formula m e ()  "1 lim (2.154) 2
 a  M?  is exact. Since this formula is valid for the Lieb}Liniger exact solution of the one-dimensional Bose gas with a pair-wise repulsive -function potential [131,132] (which presumably does not manifest

330

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

a BE condensation), it seems that BE condensation is not needed for the asymptotic limit (2.154) as well. On the other hand, Bogoliubov [1}3] argued in favour of the existence of the BE condensation and of a similar formula in this regime for the WIBG, see discussions in [90,131,130]. Notice that earlier, see, e.g. [69,78,85,114,133}135] several derivations of (2.154) were presented, even including higher-order terms (cf. discussion at the beginning of this section): 128 e ()  "1# (a)#8(4
/3!3)(a) ln(a)#O(a) , 15
 4
a

(2.155)

where we have denoted /2m by  for short. They all rely on di!erent kinds of assumptions about the ground state or on speci"c summations of perturbation series that have never been proved to be exact, for a review see [90]. At that time the only rigorous estimate of e ()/4
a from below and  from above was due to Dyson [139]. After forty years Lieb and Yngvason [130,136] succeeded in re"ning the Dyson analysis for non-negative spherically symmetric potential (decreasing faster than r\ at in"nity) with a'0, to prove (2.154). Together with the study of the one-dimensional Bose gas with a pair-wise repulsive -function potential [131,132] this in particular shows that the Bogoliubov theory (which a priori presupposed the existence of the BE condensation) may give a correct answer for ground-state energy for asymptotics (2.154) even without having to decide about the BE condensation. The analogue of the formula (2.154) for a two-dimensional Bose gas gives ln(a)\ instead of a in the denominator, see [137]. Notice that in contrast to the three-dimensional case this ground-state energy density is not proportional to the density of particles! Recently, using the ideas of [134], Lieb and Solovej extended this analysis to a `jelliuma model of the one-component charged Bose gas [138]. For the case of the Coulomb interaction and high densities the "rst (mean-"eld) term in the above energy density representation (2.155) is zero by charge neutrality. Therefore, it is the next term that gives the "rst nontrivial input. In 1961 Foldy [140] used the Bogoliubov theory to calculate it in the high-density limit, where the Bogoliubov method was claimed to be exact for this problem. Lieb and Solovej proved that it is true. This gives a rigorous example of veri"cation of the Bogoliubov theory in a three-dimensional boson system.

3. The Bogoliubov theory, revisited `Il y a deux choses que l'expeH rience doit apprendre: la premie`re, c'est qu'il faut beaucoup corriger; la seconde, c'est qu'il ne faut pas trop corriger.a Delacroix `But of equal importance is understanding, with a little hindsight, the failure of the theory, for its success led to regrettable tendency to take its predictions as gospel.a Lieb, The Bose Fluid. In this section we take one step back in the Bogoliubov theory to return to the level of the xrst Bogoliubov ansatz. This ansatz suggests to us the Bogoliubov truncated Hamiltonian (2.50)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

331

as a starting point of this theory. Since this ansatz is based on the hypothesis concerning the BE condensation in zero-mode and on its stability with respect to interactions (2.44) or (2.50), we present in Section 3.1 some rigorous results which cast doubt on this hypothesis. If, nevertheless, one accepts Hamiltonian (2.50) for the WIBG, then one is faced with di$culties with the second Bogoliubov ansatz. In Sections 3.2 and 3.3 we discuss the Bogoliubov prescription concerning substitution of the zero-mode annihilation and creation operators by c-numbers and concerning their choice. In fact this substitution eliminates quantum yuctuations, which are mediators of supplementary ewective interactions between bosons of the WIBG. This interaction in the zeromode is attractive, whereas for bosons outside this mode it is repulsive. Then we show that the WIBG is unstable for positive chemical potentials: the grand-canonical partition function diverges for "cv(0)'0, which is the choice of the Bogoliubov theory, see (2.72) and (2.107). In the last Section 3.4 we revise the Bogoliubov theory in the light of these rigorous results. 3.1. Generalized Bose}Einstein condensation and the xrst Bogoliubov ansatz Recall that the starting point of the Bogoliubov theory is based on two hypotheses (Section 2.2): (a) there is a BE condensation in one mode; (b) this condensation is stable with respect to weak interaction between particles. These hypotheses inspired Bogoliubov on his xrst ansatz: the truncation of the full Hamiltonian (2.44) to Hamiltonian (2.50) of the WIBG. First we scrutinize the (most innocent) hypothesis (a), which concerns the PBG. As we discussed in Section 1.2, the conventional BE condensation is a subtle matter even for the perfect Bose gas (PBG). Let us take, instead of the cubic box
"¸;¸;¸ with 
"<, a prism
"¸ ;¸ ;¸    of the same volume with sides of length ¸ "
1 1 1  #  . (3.1) I I e@C e@C \I!1 \I!1 < 
H


H

IZ  L $L L  IZ  LH $H   For '. (
) Eq. (3.1) gives asymptotic behaviour of the solution .
(
, '.(
)) and distributions   of bosons in the mode k"0 and in its vicinity. 1. Let  (1/2. Then (cf. (2.36)) for
 "O(<\? ) . I

IZ  This means that

1 1 lim sup "0 . I < e@C !1
IZ
H

(3.2)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

Therefore, the Darboux sum in (3.1) for kO0 is well de"ned and it converges to the integral .(, 0)"lim .
(, .
(
, '.(
))) , 
see (2.26). Since '.(
)".(, 0), Eq. (3.1) implies that  1 .
(
, '.(
))K ,
lim


1 N  (, .
(
, '.(
)))". (
),!.(
)'0 ,    < I 2
1 N  (, .
(
, '.(
)))"0 .  < I$ 2


2. Now let  "1/2. Then one gets  (2
) 1 inf  " , (3.4) I 2m <
H

IZ  which corresponds to the mode with (n "1, n "0, n "0), see (3.1). Since  (1/2 and  (1/2,      we get 1 1 sup "0 . < e@CI !1
IZ
H LH $H  

Then the Darboux sum corresponding to the last term in (3.1) converges to the integral .(, ) for all (0. Since this integral attains its sup .(, )".(
) for P0\, from (3.1) one obtains for I  '.(
) that  1 1 1 1 #  "!.(
) . (3.5) lim .
.
 < e\@I FM!1 < 
H


e@CI \I FM!1    IZ  L $L L  By virtue of  "1/2 the left-hand side of (3.5) implies  A 1 .
(
, '.(
))"! #o , A'0 , (3.6)  < < lim








for









1 (2
) \ 1 ,  "  n #A .
2m  

H

e@CI \I FM!1 2 IZ L  L !! (3.5) implies lim

1 

 L !!2






(2
) \ "!.(
) , n #A   2m

(3.7)

(3.8)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

333

where A"A(
, ) is a unique root of this equation. Moreover, limit (3.7) shows that for '.(
):  (2
) \ for k3
H: (n , 0, 0)

\ n #A 1  2m  lim N 
(, .
(
, ))" . (3.9) I 2
< 0 for k3
H: n O0, j"2 or 3

H








This means that for the long prism with  "1/2 there is macroscopic occupation of an inxnite  number of low-lying levels k3
H: (n , 0, 0) including the zero-mode k"0.  3. Let  '1/2, i.e. we deal with a highly anisotropic prism in the direction j"1. Then we have  (2
) 1 , 2 '1 , (3.10) inf  " I  2m <?
H

IZ  which corresponds to the mode with (n "1, n "0, n "0). Since for any (0 the right-hand    side of (3.1) converges to the integral .(, ) which is monotonically increasing up to .(
), when  P0\, the solution .
(
, '.(
)) of (3.1) has to have the asymptotics  B 1 .
(
, '.(
))" #o , B(0, '0 , (3.11) 



for Q@I.
FM  e\Q@
KL 4? . < Q L !!2 Notice that 1 <

(3.12)

1 e\
HL "  e\
KH , (3.13)  ( 2 2 K!! L !! where "(2
/m)s<\? . Taking into account (3.11)}(3.13) we "nd that only the term with "0 is important in (3.12) in the limit



 

 


 

2
 \ @ Q4B  .
"2(1! ) .  Then one gets

.

(3.14)

(3.15)




0(!.(
)" 



2
 \  \ d e@ K\ , m 

(3.16)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

where B"B(
"\, )(0 is the unique root of Eq. (3.16), or




m 
!.(
)" ,  2  (B(
, )

(3.17)

since



I"



dy e\Wy\"(
.

 By virtue of (3.11), (3.15) and (3.17) we get that for '.(
) and



 

 

 2
n  m/2   # , k3
H: (n , 0, 0) ,   2m
  !.
(
, )K I 2m

2
n  2
n   # 
(3.18)

m/2  # , k3
H: n O0 . H   (!.(
))<\?  

Since  '1/2 and  # # "1, asymptotics (3.18) implies     1 lim N 
(, .
(
, '. (
)))"0, k3
H , I 2 
<

(3.19)

while 1 m/2  N 
(, .
(
, '.(
)))" , k3
H:(n , 0, 0) , I 2     <\? (!. (
))
 and (cf. (3.1), (3.12) and (3.14)) lim

1 . (
)"!.(
)"lim   <


(3.20)

 N 
(, .
(
, ))'0 . I 2

IZ L  This means that for the extremely long prism with  '1/2 for '.(
) there is conventional BE   condensation with density . (
)'0, whereas there are no macroscopically occupied levels in
H.  Cases  "1/2 and  '1/2 provide examples of generalized condensation in Girardeau's   sense [39]. This gives a motivation for the following van den Berg}Lewis}Pule`'s classixcation of the generalized BE condensation [8,42,44] (see Appendix A): 
H

E the condensation is called type I when a xnite number of single-particle levels are macroscopically occupied; E it is of type II when an inxnite number of levels are macroscopically occupied; E it is called type III, or the nonextensive condensation, when none of the levels are macroscopically occupied whereas one has 1  N "! (
) . lim lim I  < 
H IZ X,I,XB

B>


V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

335

Remark 3.1. Returning back to the Bogoliubov theory we have to recognize that the hypothesis (a) about the one-mode BE condensation loses its grip if we pass to the PBG in a long prism. The latter is not an exotic situation if one considers super#uidity of extremely thin "lms of He, see [20,120]. In [46,47] it was shown that the type III condensation can be caused in the PBG by a weak external xeld or by a speci"c choice of the boundary conditions and geometry [44,45]. But recently other examples of this nonextensive BE condensation were discovered for bosons in an isotropic box
with repulsive interaction which spreads out the conventional BE condensation of type I into type III [58,59,141]. To avoid the in#uence of the shape of
on the BE condensation let us take the cubic box
"¸;¸;¸. Imposing the periodic boundary conditions on R
, we consider an interacting boson system with the Hamiltonian: 1  v(0)aHaHa a , v(0)'0 . (3.21) H'
"   aHa # I I I I I I I 2<
H
H IZ IZ Interaction in (3.21) results from a severe truncation of the full interaction (2.44). Below we shall demonstrate that even this small part of the total interaction can drastically change the distribution of condensed particles spreading out the BE condensation in the one mode k"0 into nonextensive BE condensation of type III. First we show that the interaction in (3.21) is so gentle that it does not even modify the pressure of the perfect system. Theorem 3.2. For model (3.21) the thermodynamic limit of the pressure coincides with the pressure of the PBG: lim p
[H'
]"lim p
[¹
]"p.(, ) .


(3.22)




Proof. Notice that aHaHa a "N!N . Therefore, the grand partition function for the model I I I I I I (3.21) takes the form (cf. (2.22))  '
(, )"¹rF e\@&'
\I,
"   e\@ CI \ILI >T4LI \LI  , (3.23)
H IZ LI  which converges for all 31. Then for the pressure in the "nite volume we have the estimates 1  4

 ln  e\@ CI \ILI >T4LI \LI  . p
[¹
]5p
[H'
]5 <
H IZ LI  By virtue of n!n 4[ln <] (here [x] denotes the integer part of x) one gets I I  4

 4

 e\@ CI \ILI >T4LI \LI  5  e\@ CI \ILI >T4  4 

LI  LI  1!e\@CI \I  4 \ . "e\@T4  4  1!e\@CI \I





(3.24)

(3.25)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

For (0 the left-hand side of (3.25) converges (uniformly in k3
H) to (1!e\@CI \I)\ when




1 1!e\@CI \I  4 \ 1  ln e\@T4  4  "lim  ln(1!e\@CI \I)\ <
H 1!e\@CI \I
<
H IZ IZ "lim p
[¹
] ,


which proves equality (3.22). Since lim \ lim
p
[¹
] is bounded, one extends p.(, ) to "0 by I continuity. 䊐 As usual, the total density of particles for model (3.21) is de"ned in the grand-canonical ensemble for a "nite volume < by (cf. (2.25))

 

'
(, )"

N
<

1 (, )"  N  '
(, )"R p
[H'
] . I & I < &'
IZ
H

(3.26)

Corollary 3.3. Due to v(0)'0 the pressure p
[H'
] is dexned for all chemical potentials 31, see (3.23). Moreover, it is obviously a convex function of this parameter. The limit function p.(, ) is dexned only for 40 and is also a convex function, but only for (0. Since (3.22) is a statement about convergence of convex functions of 3(!R, 0), by the Grizths lemma (Appendix C) we get convergence of the corresponding derivatives (cf. (2.25) and (3.26) ): lim '
(, )"lim R p
[H'
]"R lim p
[H'
]"R p.(, )".(, ) , I I
I

on the very same interval 3(!R, 0). So, in the thermodynamic limit the total particle density '(, ) coincides with .(, ). Since lim \ .(, ) is bounded for d'2, one dexnes I '(, "0)".(, "0) by continuity. From this corollary one deduces that, as in the case of the PBG, '(, ) attains its maximum ' (
)"lim \ '(, )". (
) at "0, i.e. for densities '' (
)".(
) one must have a BE     I condensation. On the other hand, for the expectation of the ground-state occupation number at "0 we get from (3.21)

 

(1/(<) (n /(<)e\@TL (4  L   (, "0)" . (1/(<) e\@TL (4 (< &'
L  N

(3.27)

Since the numerator and the denominator on the right-hand side of (3.27) are nothing but Darboux sums for the corresponding integrals in the limit
 dx xe\@TV 1 N  '
(, "0)"  " 2  dx e\@TV (<  &  1



2
. v(0)

(3.28)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

337

This means that there is no macroscopic occupation of the mode k"0 because N  '
(, "0)  & grows only as < for
1 N  ' (, )"0 ∀k3
H , < I &


(3.29)

i.e. there is no macroscopic occupation of any mode k3
H. Proof. By the Bogoliubov convexity inequality for pressure (see Appendix D) we get v(0) 2<









v(0)  (N!N ) 4p
[¹
]!p
[H'
]4  (N!N ) . I I I 2<
H I &'
2
IZ
H IZ By virtue of v(0)'0, N!N 5(N !I) and Theorem 3.2, one has I I I 1  (N !I) '
(, 40)"0 . lim I &
<
H IZ Notice that by the Cauchy}Schwarz inequality for the Gibbs state we get

(3.30)

N !I '
4(N !I) '
. I & I & Then






1  1 1 N !I '
4 (N !I) '
4  (N !I) '
. & I & O & < I < <
H OZ The estimates (3.30) and (3.31) give the assertion (3.29). 䊐 04

(3.31)

Resuming Corollary 3.3 and Theorem 3.4 we must admit that model (3.21) gives an example of nonextensive (i.e. type III) BE condensation ' (
)"!' (
). Namely, for '' (
)".(
)     one has lim


1 N  ' (, '
(
, '' (
)))"0, k3
H ,  < I &


(3.32)

where '
(
, ) is de"ned by the equation "'
(, ), and at the same time "lim '
(, '
(
, '' (
))) 
1 "' (
)# lim lim  N  '
(, '
(
, '' (
))) .  I &  < >

H

B IZ  XIXB

(3.33)

Remark 3.5. Returning back to the Bogoliubov theory we again recognize that there is a problem with the hypotheses (a) and (b) about the one-mode (i.e. type I) BE condensation. Even if it exists in PBG, a very gentle interaction (e.g. (3.21)), which does not change the pressure of the PBG, may drastically change the asymptotics of the occupation number expectations, cf. (3.28) and (3.29).

338

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In model (3.21) the interaction spreads out the type I condensation in one mode k"0 into the type III BE condensation, which means that the Bogoliubov hypothesis (b) is a priori doubtful. The perturbation of model (3.21) is a function of only occupation-number operators, i.e. this interaction commutes with the kinetic-energy operator. This class of interactions is called diagonal. In the series of papers [54}57,142], van den Berg, Dorlas, Lewis and Pule` gave an exhausting analysis of a whole class of diagonal models (including the exact solution of Huang}Yang}Luttinger model (4.8)) showing how subtle and various the BE condensation may be in the presence of this type of interaction. Since the interaction is diagonal, their analysis is based on a re"ned Large Deviation Principle of the classical probability. If one accepts the "rst Bogoliubov ansatz, one faces a more complicated nondiagonal interaction (2.53), which does not commute with ¹
. Therefore, the existence of the k"0 mode BE condensation in the Bogoliubov WIBG (2.50) is screaming for a rigorous analysis. 3.2. Quantum yuctuations and the second Bogoliubov ansatz Let us accept the xrst Bogoliubov ansatz, i.e. truncated Hamiltonian (2.50). Then, anticipating that the BE condensation persists in the same mode k"0 as for the PBG, Bogoliubov proposes to pass to the approximating Hamiltonian (2.60) with judiciously chosen c-numbers instead of operators aI/(<. It is the second Bogoliubov ansatz.  As we mentioned in Section 2.4, this ansatz is a basis of theories with a uniform condensate, which is considered as a classical particle reservoir creating and absorbing bosons with kO0, i.e. as a classical external xeld. Therefore, the Bogoliubov approximation aI/(
 v(k)(aHaH a #aHa a ) , (3.34) I \I   I \I IZ
H

there is an ewective direct interaction of bosons with kO0 (see Fig. 8) due to transformations of a couple of bosons with k"0 into a couple with k, !k, and similarly, an e!ective direct interaction of bosons in the mode k"0, see [13]. The simplest way to calculate the corresponding two e!ective coupling constants g
and NO is to use the FroK hlich transformation originally proposed to deduce the BCS model of g


V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

339

Fig. 8. E!ective interaction in modes kO0.

superconductivity, see, e.g. [143]. This transformation consists of a unitary transformation of a Hamiltonian H HI "e\ 1He 1 ,

(3.35)

with S"SH being a self-adjoint operator. By developing e 1 and e\ 1, one obtains commutator series: HI "H#i[H, S]![[H, S], S]#2 (3.36)  In the case of the Bogoliubov Hamiltonian H
(), we have to search for such an operator S that the nondiagonal part ;
, will be cancelled, giving instead the terms of kind (g
/<)aHaHa a and      (g
/<)aHaH a a . To this end, we de"ne the self-adjoint operator S by NO N \N O \O S"  ((k)aHaH a #(k)aHa a ) , (3.37) I \I   I \I IZ
H

where (k) is determined in order to cancel ;
,. Thus, by analogy with perturbation theory, to evaluate g
and g
, we have to calculate (3.36) to the second order in v(k).  IO Therefore, we have HI
"e\ 1H
()e 1"HI
#HI
#2   "H
()#i[H
(), S]![[H
(), S], S]#2 , 

(3.38)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

and the "rst-order term in v(k) is equal to HI
";
"#;
,#i[¹
(), S] .  Thus, to calculate (k) we get the equation ;
,#i[¹
(), S]"0 .

(3.39) (3.40)

Straightforward calculation of commutator gives i[¹
(), S]"2i   (k)aH aHa ! (k)aHa a , I \I I  I  \I I IZ
H

and so, we have to de"ne (k) by (k),iv(k)/4< , (3.41) I for every k3
H0 . By (3.38) and (3.41), the second-order term in v(k) is equal (see (3.38)) to





i HI
"i ;
"#;
,# [¹
(), S], S .  2

(3.42)

By virtue of (3.40) we "nd that i 1 ;
,# [¹
(), S]" ;
, . 2 2 Hence i HI
"i[;
", S]# [;
,, S] .  2

(3.43)

After some straightforward calculations, one "nds that i[;
", S] does not give any term of the kinds (g
/<)aHaHa a and (g
/<)aHaH a a for p, q3
H, p, qO0, which is not surprising.      NO N \N O \O Indeed, one can remark that vertices corresponding to ;
" and S cannot produce vertices of the kind (g
/<)aHaHa a or (g
/<)aHaH a a . Moreover, a direct calculation of the second term      NO N \N O \O of (3.43) gives

with

i iv(q) [;
,, S]"  ((k)[aHa a , aHaH a ]#(k)[aH aHa , aHa a ])  O \O I \I  \O O   \I I 2
H 4< IOZ 

(3.44)

[aHa a , aHaH a ]"2aHa (1#aH a #aHa ) !2(1#2aHa )aHaH a a  \O O I \I    \I \I I I OI   I \I \O O "![aH aHa , aHa a ]H . (3.45) \O O   \I I Therefore, by (3.44) and (3.45) we get the e!ective interaction of bosons in the mode k"0 of the form [v(k)] (aHa #a aH) HI
' "!      
H 8< I IZ  aHa [v(k)] aHa [v(k)] [v(k)] "!    !    !  . < 4< 4< <
H 2< I I I IZ
H

IZ  IZ
H

(3.46)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

341

Fig. 9. E!ective interaction in the mode k"0.

Thus, the coupling constant g
corresponding to the e!ective vertex of Fig. 9 is equal to  1 [v(k)] g
"!  , (3.47)  <
H 4 I IZ  and 1 [v(k)] dk (0 . (3.48) g ,lim g
"!   4(2
) 1 
I On the other hand, commutator (3.44) contains also the following terms:








1 (1#2aHa ) v(k)v(q) 1   aHaH a a .  # (3.49) I \I \O O  8  <
H

O I IOZ  It is known that in order to "nd the e!ective BCS interaction between two electrons due to the phonons exchange, one has to project the result of the FroK hlich transformation on the state which is `emptya for phonons, i.e. on the fundamental phonon state, see, e.g. [143]. Since e!ective interaction between bosons with kO0 is due to an exchange via condensate, we project HI
'' on the  coherent state for the mode k"0, i.e. on the state " (c)  where   1
(c),e\4A  ((
342

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

We denote by H the one-dimensional Hilbert space spanned by (x)"1/(< (the one  particle constant function corresponding to the mode k"0, see Section 2.1) and by H, the  orthogonal complement (H"¸(
)"H H,), then F ,F (H ) and F ,F (H,)      are the boson Fock space constructed on H and H,, respectively, see Section 2.1. Notice that the   boson Fock space F (H) is naturally isomorphic to the tensor product F (H )F (H,),   see Appendix B. Then






aHa (c),   (c)  < 

"c ,

F

and the operator of the e!ective interaction due to the coherent condensate state is de"ned by the bilinear form






v(p)v(q) 1 1 (1#2c)  # (  , aHaH a a  )F .  N \N \O O  8  < 
H

N O NOZ  Hence, we obtain that the coupling constant g
, which corresponds to e!ective interaction NO presented in Fig. 8 is equal to 1 ( , HI
'' )F "    <






1 1 (1#2c) 1 g
" v(p)v(q) # . NO 8  <  N O If we suppose that lim


(3.50)

1 aHa  ("
\, )" (
, ) ,  <   &


(3.51)

is the density of the condensate in the mode k"0 for the Gibbs state de"ned by H
, then we get that the e!ective interaction between particles outside the mode k"0 is proportional to density (3.51):






v(p)v(q) 1 1 g ,lim g
"  (
, ) # 50 . (3.52) NO NO  4  
N O Notice that if  (
, )O0, then repulsion g P#R for qP0 or pP0.  NO Thus, the nondiagonal part ;
, of the Bogoliubov Hamiltonian implies an e!ective attraction between particles in the mode k"0 and an e!ective repulsion of particles outside the condensate (mode kO0). To estimate the interaction of particles in the mode k"0 for the WIBG, one has to take into account besides (3.46) a direct repulsion corresponding to the "rst term in the diagonal part ;
" (2.52). Thus, we have to evaluate the sign of






v(0) v(0) 1 #g " !  2 2 4(2
)



1

dk

[v(k)] .  I

(3.53)

Therefore, the inequality






v(0) #g (0  2

(3.54)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

343

gives a su$cient condition that e!ective attraction prevails in zero-mode the bare repulsive term in ;
". Remark 3.6. The above calculations show that even in the case, when the BE condensation persists the interaction ;
"#;
, in the WIBG (2.50), the quantum #uctuations of the condensate are not negligible. They are responsible for e!ective attraction of bosons in the zero-mode. Since the Bose statistics allows accumulation of particles, this attraction may be a source of instability of the WIBG or a new kind of condensation. In fact, this is proved in [15]. Therefore, the second Bogoliubov ansatz missed these phenomena caused by quantum #uctuations. 3.3. Instability of the weakly imperfect Bose gas We recall that a `judiciousa choice of the constant cI in the second Bogoliubov ansatz corresponds to the variational principle (2.71) or to the "rst-order perturbation calculation (2.107), which gives the same relation "v(0)c50. So, if the BE condensation in k"0 does exist, one must get '0. The aim of this section is to show that this inequality is in a conyict with stability of the Bogoliubov Hamiltonian for the WIBG (2.50); see [9]. First we prove the following assertion: Theorem 3.7. Let H
""¹
#;
" be diagonal part of the Bogoliubov Hamiltonian (2.50). Then (a) for any 40 one has p "(, )"lim p
[H
"]"p.(, ) ,

(3.55)




with p.(, ) (2.26) corresponding to the PBG pressure, (b) for '0 one has p "(, )"#R .

(3.56)

Proof. By de"nition of the Hamiltonian H
""¹
#;
" (2.50) and v(k)50 one gets v(0) v(0) 1 H
""¹
# N N
! (N #N )#  <  2<  <




H

v(k)N N 5¹
.  I

(3.57)

IZ  Therefore, by direct calculation of the ¹rF we deduce that

1 1 p
"(, )"p
[H
"]" ln ¹rF e\@&
"\I,
" ln 
"(, ) , < <

(3.58)

with





 
"(, )"  e\@T4L \L \IL   (1!e\@CI \I> T>TI 4L )\ . L  IZ
H

Hence the last two equalities imply 1 p
"(, )5p
. (, ), <

 ln[(1!e \@CI \I )\] . IZ
H

(3.59)

(3.60)

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V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

Note that p .
(, ) is the pressure of a perfect Bose gas with excluded mode k"0 and . Therefore by (3.59), p
. (, )(#R only for ( I
* p
"(, )(#R for ( inf  " , ,I,
* I
H

(3.61) IZ  lim p
"(, )"#R . IC,I,
* Moreover, estimate (3.57) implies a second inequality 1  ln[(1!e \@CI \I )\] , p
[H
"]4p
[¹
]" <
H IZ for (0. Since for (0 one has lim p .
(, )"lim p
[¹
]"p.(, ) ,


(3.62)

(3.63)




in the thermodynamic limit, (3.60)}(3.63) imply (3.55) for (0. Then taking limit P0\ one gets (3.55) for "0. Moreover, inequality (3.60) implies in the thermodynamic limit (3.56). 䊐 Now we can prove that the WIBG exists only for 40. Theorem 3.8. For any 40 one has p (, )"lim p
[H
]5p.(, ) ,

(3.64)




whereas for '0 p (, )"#R .

(3.65)

Proof. By the Bogoliubov inequality (Appendix D), we know that for any 40, 1 1 ;
,
4p
[H
"]!p
[H
]4 ;
,
" "0 , & & < <

(3.66)

the latter because ;
, is nondiagonal whereas H
" is diagonal. Then by Theorem 3.7 and (3.66) we obtain (3.64) in the thermodynamic limit. Divergence (3.65) comes directly from (3.61) and (3.66). 䊐 Remark 3.9. The last observation shows that only the trivial choice c"0 in the second Bogoliubov ansatz is compatible with thermodynamic properties of the WIBG, which is unstable for '0 by (3.65). In fact one can say more about the WIBG for 40. The next statement is due to [9]. Theorem 3.10. If 4!1/2(0), , then the pressure of the WIBG coincides with the pressure of H the PBG: p (, 4 )"p.(, 4 ) . H H

(3.67)

V.A. Zagrebnov, J.-B. Bru / Physics Reports 350 (2001) 291} 434

345

Proof. Regrouping terms in (2.50) one gets 1 H
"HI
# 2<

 v(k)(aHa #aH a )H(aHa #aH a )5HI
,  I \I   I \I  IZ
H

(3.68)

where






v(k) v(0) v(0) 1 HI
"   ! # N N # N ! (0)N . (3.69) I  2< <  I 2<  2
H

IZ  Hence, by the Bogoliubov inequality for (3.68) and for its diagonal part (3.69) (see Appendix D) we obtain in the thermodynamic limit for 40 p (, )4lim p
[HI
]" sup G(, ;  ) . 
M Y

(3.70)

Here










v(0) 1  # # (0)  #p.(, !v(0) ) . G(, ;  ), !     2 2

(3.71)

If 4!(0), then sup  G(, ;  )"p.(, ). Therefore, by Theorem 3.8 and inequality (3.70)   M Y we get p (, )"p.(, ) for (
"\, )3 H "
50, 4!(0), . I  H

䊐

(3.72)

So, the properties of the WIBG for very negative chemical potential (4! ) are in a H sense trivial. The interaction ;
is not able to change them with respect to the PBG, cf. the model (3.21) of Section 3.1. The question of behaviour of the WIBG in the interval [ , 0] is less trivial. It H has been answered in our papers [12,14}16]. This will be discussed in Sections 4 and 5. 3.4. From the weakly imperfect Bose gas to the Bogoliubov theory Resuming the observations of Sections 3.1}3.3, one has to admit that in various respects the Bogoliubov model of the WIBG is not very appropriate as the model of super#uidity. In Sections 4 and 5 we discuss an exact solution of the WIBG manifesting properties rather far from those following from the Bogoliubov theory. This obviously means that the second Bogoliubov ansatz gives a solution of something other than the Bogoliubov WIBG. The xrst main feature of the second Bogoliubov ansatz is that the Bogoliubov approximation aI/(