Fortschritte der Physik / Progress of Physics: Band 25, Heft 3 1977 [Reprint 2021 ed.] 9783112519400, 9783112519394

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM A U F T R A G E D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DER DEUTSCHEN DEMOKRATISCHEN

REPUBLIK

VON F. K A S C H L U H N , A. L Ö S C H E , R. R I T S C H I . U N D R. R O M P E

H E F T 3 • 1 9 7 7 - B A N D 25

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EVP 1 0 , - M 31728

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BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie-Verlag, DDR-108 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR-701 Leipzig, Postfach 160, oder an den Akademie-Verlag, DDR-108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der Physik" Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritschl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3 - 4 ; F e m r u f : 2200441; Telex-Nr. 114420; Postscheckkonto: Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R -104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VEB Druckhaus „Maxim Gorki", D D R - 7 4 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik 41 erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band: 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je Heft IS,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/2S/3. (c) 1977 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57618

Fortschritte der Physik 25, 123-165 (1977)

Statistical Bootstrap Approach to Hadronic Matter and Multiparticle Reactions E.-M. Ilgenfritz, J. Kripfganz and H.-J.

Möhring

Sektion Physik der Karl-Marx- Universität Leipzig,

DDR

Abstract We present the main ideas behind the statistical bootstrap model and recent developments within this model related to the description of fireball cascade decay. Mathematical methods developed in this model might be useful in other phenomenological schemes of strong interaction physics; they are described in detail. We discuss the present status of applications of the model to various hadronic reactions. When discussing the relations of the statistical bootstrap model to other models of hadron physics we point out possibly fruitful analogies and dynamical mechanisms which are modelled by the bootstrap dynamics under definite conditions. This offers interpretations for the critical temperature typical for the model and indicates futher fields of application.

1. Introduction Since the time when Fermi's statistical model [J] was proposed methods and notions borrowed from statistical mechanics have helped to reveal the regularities of multihadronic production processes. I t is believed t h a t these features are to a certain extent independent of the details of the dynamics of strong interactions. At present there is no theory from wich all features of strong interactions could be deduced. Even if one would know a basic theory one were necessarily dealing with a very complex relativistic many body problem even in most simple processes. In order to describe reaction amplitudes with reasonable effort and to reproduce the empirical regularities one cannot dispense with a statistical treatment of the underlying dynamics. At present statistical models of different complexity play an important heuristic role and bridge between dynamical concepts and data analysis. Fermi's model [1] rests on the assumption t h a t the energy brought in by the colliding particles is available in the center of mass system to establish instantaneously (within the collision time of order i0 ~ 10 -23 s) a perfect gas of final state particles in equlibrium within the interaction volume of order V0 ~ m^r3. There are several empirical facts of multiparticle production which cannot correctly described within this model: — slowly rising average multiplicity with increasing energy ;— leading particle effect 1 — transverse momentum cut-off — multiplicity ratios between different final state particles — resonance structures. 10

Zeitschrift „Fortschritte der Physik", Heft 3

124

E . - M . ILGENFRITZ, J . KMPFGANZ and H . - J . MOHRING

Before these faults became apparent the original statistical model has been oriticized because of its physical assumptions, mainly that an equilibrium state should be reached during the colision time. To models which attempt simply to describe transition probabilities (an ensemble of identical collisions) and not the time evolution of the process the above objection does not apply. The assumption of many hadrons coexisting within F 0 without any interaction is more severe. P O M E R A N C H U K ' S [2] and later L A N D A U ' S hydrodynamical [3] model were attempts to consider the final state fully developed into free particles as a result of expansion and break-up of strongly interacting pre-mater. Here the break-up stage can be defined to occur at a temperature T 0 ~ mn independent of the primary energy. This temperature will show up in (transverse) momentum spectra and particle ratios. Collective motion mainly in longitudinal direction is separately described as a result of hydrodynamical expansion [3]. For this purpose one has to choose a specific equation of state of pre-matter. The existence of a hydrodynamical regime requires indeed local equilibrium established in pre-matter. This kind of statistical models attempts to give a microscopic view of the process in space-time. There are other models constructed to accomodate some of the empirical features mentioned above, e.g. jet and resonance structure. They postulate production amplitudes factorized into functions depending on individual momenta of particles or resonances with an appropriate p1 behaviour built in. The uncorrected jet model [4] and its descendants [5] belong to this class of models. In 1965 HAGEDOKN proposed the thermodynamical model [6] of particle production. I n this model the thermal motion inside the ideal fireball is combined with collective motion of excited hadronic matter. B y the collision of hadrons pieces of their matter become decelerated and excited. Their collective motion is described in terms of empirical velocity distributions. The radiation of real final state particles from these volume elements is treated thermodynamically. In order to do this and, particularity, to specify the relation between energy density and temperature a microscopic model of hadron matter is needed. This is accomplished by imposing a selfconsistency (bootstrap) condition onto the level density of excited hadronic states. This ingredient of the thermodynamical model has developed into what is called now statistical bootstrap model (SBM). There is an excellent review of the S B M within the thermodynamical interpretation by HAGEDORN [7]. Since that time the thermodynamical model has been reinterpreted as a model for multiperipheral or independent (uncorrelated) production of clusters (fireballs) and their subsequent decay [£]. This trend has been corroborated by analyses of particle correlations [0]. I n order to specify the decay distributions of clusters into particles the cascade decay of clusters was considered in accordance with the assumptions of the SBM. The assumed virtual compositeness of fireballs (clusters) of fireballs etc. turns into reality during the decay. In this way the decay of fireballs (clusters, considered as average resonances) became the main concern of the SBM. Most of the technical developments to be reviewed here connected with the development of this point of view. Some further remarks on statistical approaches in general might be in order. There are two different points of view followed in statistical approaches in particle physics: 1. Description of hadronic states, explanation of hadron structure and hadronic processes by refering to compositeness of hadrons and assuming some kind of thermodynamical equilibrium. 2. Description of multiparticle final states in terms of statistics of competing channels in particle momentum space. Hadron constituents in the first case can be considered either to be elementary (more

125

Statistical Bootstrap Approach

„elementary" than hadrons) in models of the quark parton variety or to be hadrons themselves in bootstrap models. There is no absoltue contradiction between these two attitudes: Within an excited hadron there might exist under some circumstances effectively elementary hadronic complexes which can become free during decay. Some models like the hydrodynamical model [3] do not specify the nature of the constituents forming the pre-matter before decay into particles. A problem for all statistical models having elementary constituents and trying to describe the space-time evolution is to describe the transition into the asymptotic particle states. The SBM on the other hand belongs to both above mentioned categories. It describes both the compositeness and the decay into the asymptotic final particle channels. The reason for the applicability of thermodynamics for describing these excited hadron (fireball) states within the model rests in the high degeneracy of states predicted by the model. The assumption of compositeness out of fireballs forces the mass spectrum to rise exponentially

M

e(M)~exp-i. 1

(1.1)

o

An average over all these individual hadronic states within some mass range is what is described by the SBM and called „fireball". Whether or how this mixed state is prepared in hadronic reactions remains unanswered by the model itself. Now we will assume to have prepared a mixed state properly described by the microcanonical ensemble of all hadronic states within a given mass intervall or by the canonical ensemble according to a temperature T = / H There are two ways to give a thermodynamical description. The Hamiltonian approach to construct the partition function and the level density as

z = Tr e-i>H = / dEe~PEe(E) 00

o is not feasible. Using the ^-matrix formulation of equilibrium statistical mechanics due to D A S H E N , M A und B E R N S T E I N [10] one can calculate in principle the logarithm of the partition function from the set of all scattering amplitudes oo log Z =

l o g Z0

+

f er>* T r (s+

J

\

L

s )

dE.

(1.3)

C£J /connected

0 This is the most complete relativistic generalization of the method used by B E T H and U H L E N B E C K [ 2 1 ] for representing the second virial coefficient by the phase shifts associated with the interaction potential,

,

n I Jf *

, * *

8s

(1.4,

If we assume that all interactions are saturated by resonance formation and treat these in the narrow width approximation (width F we can represent the interaction contribution to the partition function by taking effectively elementary resonances as additional constituents in a free system. In two recent papers the conditions for the effective elementariness have been analyzed critically by D A S H E N and RAJARAMAN [ 1 2 ] . In order to avoid consequences of the Levinson theorem which might invalidate this procedure one has to have T e , on the other hand, where e , determined by the inverse interaction range, R, is an energy where de eventually returns to zero. Using this approach one treats in the SBM an highly excited hadron as a free gas of all kinds of hadrons, including all resonances, within a typical hadron volume. In the 10*

(1.2)

126

E.-M. Ilgenfritz, J. Kbipfganz and H.-J. Móheing

microcanonical formulation proposed by Fkatjtschi [2