Fortschritte der Physik / Progress of Physics: Band 28, Heft 3 1980 [Reprint 2021 ed.] 9783112522844, 9783112522837

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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. LÖSCHE, R. R I T S C H L U N D R. R O M P E

H E F T 3 • 1980 • B A N D 28

A K A D E M I E -

V E R L A G EVP 1 0 , - M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an das Zeitungsvertriebsamt. an eine Buchhandlung oder an den AKADEMIE-VERLAG, DDR - 1080 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an des 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 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR - 7010 Leipzig, Postfach 160, oder an den AKADEMIE-VERLAG, DDR - 1080 Berlin, Leipziger Straße 3—4

Zeitschritt „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. A r t u r Lösche, Prof. Dr. Rudolf Ritschl, Prof. Dr. Robert Rompe, im A u f t r a g der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1080 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 22 36221 und 22 36 229; Telex-Nr. 114420; 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 - 1040 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 7400 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik 44 erscheint monatlich. Die 12 Hefte eines Jahres bilden einen B a n d . Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis f ü r die D D R : 120v— M). Preis je H e f t 15,— M (Preis f ü r die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/28/3. (g) 1980 b y Akademie-Verlag Berlin. Printed in the German Democratic Republic. A N (EDV) 57618

Fortschritte der Physik 28, 1 2 3 - 1 7 2 (1980)

A Jet Model Study of Correlations in Hadron-Hadron Reactions Due to Resonance Production K . J. B i e b l , M. K l e i n 1 ) and R . N a h n h a t t e r Institut

für Hochenergiephysik

der AdW

der DDR,

Zeuthen,

DDR2)

Abstract Secondary particle correlations are discussed as a basic feature of hadron induced multiparticle reactions in the 5 to 1500 GeV energy range. Two introductory sections are devoted to definitions of cross sections and to empirical regularities of multiple production. Manybf these regularities can be understood to be consequences of energymomentum conservation and limited transverse momenta (uncorrelated jet model (UJM)). An extension of the U J M is considered, the correlated jet model (CJM), by introducing dynamical short range correlations between the particles which are assumed to be due to resonance production. Many properties of single-particle distributions and correlation functions can be derived already from kinematics of resonance decay. The CJM is used to investigate in detail the origin of angular correlations (azimuthal, GGLP-effect) and of correlations between neutral and charged particles ((jr°) nc , f"(nc)). These correlations are shown to arise almost completely from the production and decay of mainly two-particle resonances like p°, K * and A, the main evidence coming from the use of the invariant mass as the basic variable where only resonances or reflections of them give remarkable effects. Compared to the UJM, the CJM is the first approximation of a successive correlations analysis of the production amplitude squared, i.e. having separated the effects of energy-momentum and quantum number conservation. This might be also interesting in view of the forthcoming accelerator generation.

Contents 1. Introduction

124

2. Cross sections and definitions 2.1. Exclusive cross sections . 2.2. Inclusive cross sections . 2.3. Differences between exclusive and inclusive cross sections 2.4. Grand canonical notation 2.5. Reference systems and variables

126 126 128 129 130

3. Empirical regularities of multiparticle processes . . . 3.1. Mean multiplicities of particles and resonances . . 3.2. Charged multiplicity distribution 3.3. Correlations between neutral and charged particles

131 131 133 135

2

Joint Institute for Nuclear Research, Laboratory of High Energies, Dubna USSR. ) D D R - 1615 Zeuthen.

8

Zeitschrift „Fortschritte der Physik", Heft 3

126

124

K . J . B I E B L , M . K L E I N , R . NAHNHATJER

3.4. Single-particle distributions 3.5. Asymptotic behaviour of single-particle distribution 3.6. Two-particle correlations

136 138 140

4. Uncorrected jet model 4.1. Definitions and distributions 4.2. Grand canonical formulation 4.3. Comparison with data and possible modifications

143 143 144 146

5. Correlated jet model 5.1. Definition 5.2. Single-particle distributions 5.3. Two-particle distributions and correlations

147 147 151 154

6. Experimental results and phenomenology 6.1. Azimuthal angle correlations 6.2. Opening angle correlations and the GGLP-effect 6.3. J e t model with isospin conservation 6.4. Resonance interpretation of neutral-charged correlations

156 156 160 164 166

7. Summary

169

8. References

170

A Jet Model Study of Correlations due to Resonance Production 1. Introduction An intensive investigation of high-energy hadron hadron reactions in the last 10 years lead to the accumulation of a large amount of d a t a for multiparticle production. Due to the absence of a basic field theory of strong interactions our understanding of these data has been limited to phenomenological considerations. Although the Mueller-Regge analysis gave an explanation for the scaling laws of inclusive cross sections [1] it could not predict the detailed form of the distribution functions. Present gauge theories for quark interaction (QCD) [2] allow some calculations from first principles. As long as the problem of quark binding in hadronic states (confinement) is not solved, however, these calculations need some phenomenological pieces. Thus many phenomenological models have been constructed from the regularities of the data, which certainly are important piers of the long bridge between the raw data and the theoretical understanding of the dynamics of the interaction. The aim of these models is to separate first kinematical effects due to energy-momentum conservation (phase space), to isolate different production mechanisms and to investigate their interrelation, in order to bring the experimental results into a form which more easily can be treated theoretically. In the next years experimental results on multiparticle production will become available at much higher energies provided by a new accelerator generation. Cosmic ray physics suggests t h a t we may have to expect completely new and interesting phenomena [3]. I n analysing these phenomena one has to take into account all effects known from lower energies. Therefore it seems to be worthwhile at this time to review some models describing the main effects a t present energies. On of the oldest and most successful models is the uncorrected jet model [4] (UJM) which strictly respects* global energy-momentum conservation and assumes the final state particles (e.g. pions) to be produced dynamically independent with a strong (gaussian or exponential) cutoff of the transverse momenta. The final state is therefore, up to an energy-momentum ¿-function, a coherent state characterized by a production ampli-

Correlations in Hadron-Hadron Reactions

125

tude strength ftp). This quantity may be regarded as the Fourier transform of an effective source function j(x) of the pion field which is formed during the collision, process. A similar coherent state arises in the emission of independent radiation quanta from a classical electromagnetic current density j ^ x ) . In the U J M the coherent state is further modified by an additional factor for the production of those particles which are the fragments of the beam and target particles whereas the coherent state is responsible for particles in the central region. The uncorrelated jet model was able to describe inclusive single-particle distributions, but it failed in the case of two-particle distributions (and higher ones) since the model has no (dynamical) correlations besides the kinematic ones. Therefore, in correlation studies, the U J M was used mainly as a background subtracted from the data. Refined correlation studies \3, 5, 6] in the last few years showed that one of the most important sources of these dynamical correlations is the production of resonances decaying subsequently into the observed stable final state particles. From the point of view of the quark model, or of the dual model, the meson resonances are of the same nature as the stable pseudoscalar mesons (qq bound states). Thus we have to expect that resonances are produced initially by the same basic mechanism as the stable mesons itself. The decay of the resonances leads to additional particle distributions which together with the distributions of the "directly" produced stable particles yield the observed distributions. Assuming that also the resonances are produced independently, the U J M should be modified accordingly, containing now the source functions ftp) for directly produced particles and /res(Pres) for resonances in a coherent state and also some conversion mechanism for resonance decay. Starting from these physical ideas many generalizations of the UJM have been constructed [7] which differ in whether they take into account interferences of various parts of the final state or not, whether they neglect kinematical effects, or even consider clusters other than resonances. Here we concentrate on one generalization of the UJM, the correlated jet model [5, 9] (CJM), which has strict energy momentum conservation, and neglecting interference effects [20] take into account independent production of single particles and resonances. Two-particle resonances are considered to be the dominant source of dynamical correlations. The straight forward generalization to three-particle resonances is indicated and in a few cases treated numerically. The interplay of directly produced particles with particles from different resonances yields complicated net distributions (resonance reflections). The correlated jet model serves to disentagle these effects in the data and to isolate the resonance production in a more appropriate manner than by simply subtracting an U J M background. Results are obtained free of kinematical modifications "and directly for the correlations pieces of \Tn\2, {T„ being the «-particle production amplitudes). Therefore the CJM studied here can be understood as a special form of a more general correlation analysis of the set \Tn\2 having separated already the energy momentum ¿-functions and thus the main kinematical effects. This method could be useful for the general phenomenological analysis of data from the new accelerators. The CJM allows one to investigate the contribution of pions from resonance decay in various kinematical regions of the single particle distribution and the correlation function. Special expressions for angular correlations are constructed giving clear signals of high mass resonance likes / and ¡7 which in the mass plot are almost unvisible. The main result of these investigations is that short-range correlations arise dominantly from resonance decay. In section 2 we first give the definitions of various exclusive and inclusive cross sections and of the variables used. Section 3 summarizes the main experimental properties of multiparticle processes. After a short review of the U J M in section 4 we consider in section 5 the main analytic and physical properties of the CJM. Comparisons of the CJM 8*

126

K. J. Biebl, M. Klein, E. Nattnhauer

with data are performed in sect. 6. Particular attention is given to two types of correlation data which until now are of special importance for discriminating between different models; angular correlations and correlations between neutral and charged particles. Section 7 summarizes the results. 2. Cross sections and definitions 2.1. Exclusive cross sections For simplicity we take into account in this section only one kind of scalar particles. The extension to different kinds of particles is straightforward. First we want to consider exclusive reactions with n finalstate particles a + b

Cj . . . c„.

(2.1)

The $-matrix elements for such reactions are given b y P i„> /tisinh?/). The kinematical limits of the variables x and y are — 1 ^ x +1 and \y\ < In ( ' t f s / f i ) respectively. Quantities often used in correlation studies are the azimuthal angle 0 L defined by COS0 1 =

J>ll Px, l I J P J . I I IPI.1

|

,',

(2.29)

3 ) This quantity if often called the transverse mass. One should note, however, that [i is the mass of a particle having only longitudinal momentum and energy E which suggests the name used above.

Correlations in Hadron-Hadron Reactions

131

the opening angle 6 defined by cos0 = / V ^ T I Pi I \Pa\

(2-30>

and the two particle mass

M' = mf + m2 2 + 2 / ^ cosh ( V l - y2) - 2]pLl\

|pj.,| c o s & ± .

(2.31)

For the following considerations it is furthermore usefull to define the dynamical different regions of phase space: — the target fragmentation region

-1

^ xe < - 2 / u J f

(2.32)

- I n (|/s/,mc) < yc < - I n {]/slfic) + X,

2

— the beam fragmentation region +2fiJ^ +

(z) [24, 25] which approximately is given by y>(z) = 7i • z • exp (—7i • z 2 /4).

(3.5)

Correlations in Hadron-Hadron Reactions

135

The experimental data reveal KNO scaling according to eq. (3.4) fiir 7ip and p ± p reactions above EN > 40 GeV, Ejs>1 GeV and EP > 50 GeV for charged pions. An analogeous scaling behaviour was found for neutral pions [26] and for K° and A0 [14]. At ISR energies, however, there is some recent indication of deviations from KNO scaling in pp reactions [26]. It may come out therefore that the universal KNO-behaviour of onc is only a low energy phenomenon giving, somewhat accidently, a reasonable parametrization of a„c below pla,b >=» 1000 GeV/c. KNO scaling requires the energy independence of the socalled y-moments [16, 22] of the multiplicity distribution, e.g.

» =

O -a O o

o

M/GeV

Fig. 10. Mass-dependent correlation function

9*

C(M) for 205 GeV/c pp interaction

(from ref.

{44})

142

K. J. Biebl, M. Klein, R. Nahnhauer

In fig. 10 the correlation function is shown in dependence on the two-pion-mass for 205GeV/c pp-interactions [44], One observes for unlike charged pion pairs a clear signal in the p°-resonance region. Furthermore, threshold enhancements are seen both for unlike and like charged pion combinations. These low-mass bumps have been attributes to interference phenomena [43], resonance reflections [ 9 , 44] or clusterization effects [42]. However, they may originate also from other 'non-dynamical' effects [46] as discussed below. The two-particle correlation function a t fixed multiplicity n is defined as is small compared to the large contribution only due to n

differences between inclusive and semi-inclusive single particle densities, reflecting no direct dynamical correlations. This underlines the necessity of correlation studies a t fixed multiplicities. On the other hand, data on inclusive correlations should be inter-

143

Correlations in Hadron-Hadron Reactions

preted very carefully. Based on inclusive data it seems to be hard to estimate dynamical effects quantitatively. Therefore also mass-dependent correlation functions should be studied in fixed multiplicity final states. This was done for a 7r + p-experiment a t 8 GeV/c [46]. Fig. 12 shows the results for 7r+7t~-pairs. The inclusive correlation function (fig. 12 a) has a similar shape as found in pp-interactions at 205 GeV/c — a broad threshold bump and a peak in the p°-mass region. The threshold enhancement diminishes and signals appear at the masses of the p° and f-meson in the semi-inclusive reaction (fig. 12b). In the exclusive finalstate (fig. 12c) ^P

o o 0.04 o 2 'fc V *M o.oo o 1

0.04

T

h i

0.68

TT*p

* ¿charged • X °

-Xi«i.

P " ,

M, .

• p 2tT*W~

1

I

Mf

Mf

.

©

© i 1.48

0.68 M {IT*it " ] I

1.48

V.

068

© 1.48

GeV

Fig. 12. Mass-dependent two-particle correlation function for a) 7u+p --> JiTinelastic b) 7i+p -> 4charsed + c) rc f p —> p27ï+7t~

pairs of tile reactions a t 8 GeV/c (from ref. [46])

one observes close to threshold only a small spike, a broad negative minimum at 0.5 GeV and again strong resonance signals. I t is concluded t h a t resonances are a source of correlations and that the threshold bump in inclusive mass-dependent lation functions is mainly of 'non-dynamical' origin due to the second term of eq.

about strong corre(3.18).

4. U n c o r r e l a t e d j e t m o d e l

4.1. Definition and distributions The simplest model which describes the most important properties of inclusive and exclusive cross sections mentioned in sect. 2.3. is the uncorrelated jet model (UJM) [4]. I t takes into account exact energy-momentum conservation and assumes dynamically independent particle production. All correlations in this model are therefore of kinematical origin only. The model starts with the definition of a factorizable production amplitude squared for an exclusive final state a + b —> cx . . . c„ \Tn(p1...pn)\*=nb(Pb) [4, 9, 48].

144

K . J. BIEBL, M . KLEIN, R . NAHNHATJEH

The uncorrelated jet model is characterized also by the representation of the scattered wave in the form Iab) = d'(P" -

W) =

pa" -

pb») exp [ / dpf(p) a+(p)\ |0>

(4.2)

with + (p ±, + p A ,)» = tf + pf +

cosh ( Vl - y t ) .

(5.10)

The transversal cutoff is provided by a cutoff in ¡u12 or fi\ 2 . (There may be also a polynomial in pi* according to the spin of the resonance which can be neglected if the resonance is produced unpolarized). For resonance production we have a Breit-Wigner type dependence in the mass M. The simplest assumption is a Breit-Wigner function 6(1, 2) in M multiplied by a transversal cutoff factor in ¡u12 being the same as the cutoff function in To be specific we give as an example the functions j + (z2/2) & 2 ] is the generating function. It is broader than the Poisson distribution of the UJM but not of KNO type [23] since we have no long range correlations in the model (cf. sect. 3.2). I t is obvious how these formulae are generalized if there are higher cluster functions i, 2) =

- V ) (1 +

z* /

cPpit

X 9-2(2/1 -

f

niPu) d(Vl

-

y i . P w P i . )

y2)

[(1

-

V

-

V )

(1 -

-

^")]

p ( 2 )

-

1

(5.15)

K. J. Biebl, M. Klein, R. Nahnhauer

152

(see eq. (4.20) in the UJM), where ic 2 ± / a; i ± = (pd^i) e x P (±(2/2 ~ //1)) and the y2 — y1 integration is essentially the integration over the two particle mass M(x^ = (E^ j/s). The range of integration is kinematically limited by the condition + x2± 1 i.e. there must be some momentum left for the remaining particles besides p1 and p2. The short-range correlations of cp2 restrict the integration to a range where | y2 — y1| is finite, this being valid in particular for resonance contributions in 0, xu «a 0) and p2 also, only the first factor (1 — x^ — x2+) in eq. (5, 15) is different from 1. The presence of correlations implies a modified 2 dependence of Qi via the exponent p(z) — zpx + (z2/2) p2. This does not concern the inclusive distribution (z = 1) but the exclusive and semiinclusive distributions which follow from eq. (5.15) replacing 2 by zn with n\in) = (znpl + znip2)Hpl + p2). The variation of p„ = p(zn) with n is for n > (n) weaker than in the UJM. On the other hand, the different weights z„ and z„2 of the contributions cp! and (p2, respectively, imply that