Soft hadronic production by ECCO in the geometrical branching model

PHYSICAL REVIEW D VOLUME 48. NUMBER 1 1 JULY 1993

Soft hadronic production by ECCO in the geometrical branching model

Jicai Pan and Rudolph C. Hwa Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, Oregon 97403-5203

(Received 1 March 1993)

Soft production of hadrons in hadronic collisions is described in the geometrical branching model and implemented by the eikonal cascade code (ECCO). It is shown that the major global features of multiparticle production can be reproduced by one essential characterization of the dynamics of branching, namely, a scaling law for the mass distribution of daughter clusters. Without further adjustment of any parameters, the event generator can produce local features of multiplicity fluctuations in agreement with the NA22 intermittency data. The scaling exponent v is determined to be 1.522 at v''; =22 GeV, independent of the dimensionality of the intermittency analysis. It is shown that v is approximately independent of the collision energy.

PACS number(s): 13.85.Hd, 12.40.P~

I. INTRODUCTION

The geometrical branching model (GBM), which has been successful in describing all major features of multiparticle production in soft hadronic and nuclear collisions [1,2], has been implemented by an event generator, called ECCO (for eikonal cascade code) [3]. I t has been effective in reproducing not only the global features of the soft production data, but also the local features of multiplicity fluctuations that have come to be known as intermittency [4,5]. In this paper we describe our first major revision of ECCO (version 2.0) so as to reduce the number of free parameters, to improve the kinematical constraints, and to accentuate the one essential dynamical element of the model, i.e., the scale-invariant branching law.

While there exist various other models (e.g., dual parton model [6] and FRITIOF model [7]) that can also describe important features of high-energy collisions, they have not been able to confront successfully the intermittency data [S]. Indeed, we have found that the observed multiplicity fluctuations at various scales of phase-space resolution have provided crucial guidance in the develop- ment of our model. I t is known that the intermittency data of e + e - annihilation processes have been well reproduced by the existing event simulators such as JETSET [9] and HERWIG [lo], which are based on perturbative QCD branching [ 5 ] . Branching is a specific form of successive random processes, which have been found to be necessary to give rise to intermittency [4]. In soft hadronic production branching is an essential mechanism in the GBM, a model that was constructed before the intermittency data became available. I t is therefore natural for the GBM to accommodate intermittency. However, that possibility cannot be demonstrated until the event generator E C C ~ came into being.

Our first version of ECCO [3] was somewhat crude. We did not impose strict energy-momentum conservation at every step of the branching; we only required overall conservation. We needed a singular form for successive rapidity distribution in order to reproduce the observed

features of intermittency. Being framed in the eikonal formalism, it had no difficulty in generating all the global features of particle production. With five parameters we were able to fit all major aspects of the data, including the factorial moments of NA22 [ l l ] . Subsequently, we have extended ECCO to nucleus-nucleus collisions [12], as well as three-dimensional intermittency [13], at the expense of a few extra parameters. We now aim to eliminate some ad hoc assumptions in the code, provide a sensible description of the leading nucleon distribution, demand energy-momentum conservation at every branching step, trace the charges of the clusters, and incorporate resonance production and decay. We show how this can be accomplished with less parameters. More significantly, those parameters are determined by fitting the global features of the data. Local features contained in the intermittency data are reproduced without further adjust- ments of any parameters.

Our focus is still on the soft production part of hadronic collisions, recognizing that h2rd scattering of partons becomes important only for z/s > 100 GeV. The latter can be included in a rather straightforward way at a later pojnt. Thus for now we rely heavily on the NA22 data at z/s =22 GeV for the determination of our parameters and for checking our calculated results.

11. A BRIEF REVIEW OF THE GBM

We give first a short summary of the foundation of the GBM [1,2]. Since hadrons have nontrivial sizes, we put great emphasis on the geometrical aspect of their collisions with one another by using the eikonal formalism as the basis of the model. From elastic scattering one can determine the eikonal function R(b) , in terms of which the cross sections are

0556-2821/93/48(1)/168(11)/$06.00 48 168 @ 1993 The American Physical Society

48 SOFT HADRONIC PRODUCTION BY ~ c c o IN THE . . . 169

where the reality of R ( b ) has been assumed to be a good approximation. For & < 100 GeV geometrical scaling suggests the use of the scaled impact parameter R , where

so that (2.1)-(2.3) may be written in the factorized form as

where o o ( s ) = ~ b i ( s ) . The scale is fixed by identifying oo with o,,. I t then follows from (2.6) that the inelasticity function

satisfies the normalization condition

J o m d ~ 2 g ( ~ i = i . (2.9)

Expanding g( R in a power series, we have

where

T,(R)= [ 2 R ( R ) l p - 2 U R )

P !

The pth-order term will be referred to as the p-cut Pome- ron contribution. At a given R the average value of p is

m

F I R ) = X ~ ~ - J R ) / 2 T p ~ ~ ) = 2 n ( ~ ) / g ( ~ i . (2.12) p= 1 p= 1

After averaging over all R we get

where the phenomenological form for R ( R ) [14],

l - e -"(R),o 7 1 e - l . 1 7 ~ 2 7 (2.14)

has been used in arriving at the numerical value of ( p ). Thus only the first few terms of (2.10) are in reality important, although in our simulation all possible values of p are allowed.

If hard scattering is to be included in our consideration, the eikonal function R ( R ) must then be regarded as consisting of two contributions: Ro( R ) + R l ( s , b ), the soft and hard components, respectively. By restricting the energy to & < 100 GeV, we are essentially asserting R 1 < < a o , so that the effects of hard subprocesses may be ignored.

Without any further dynamical details the eikonal formalism cannot carry us any farther in the description of inelastic processes. In the GBM we model multiparticle production by successive branching processes and combine the eikonal and cascade aspects of the model in the basic formula

where Bf is the probability that n particles are produced in the p-cut Pomeron term. I t is the specification of B/ that has been upgraded at various stages of the develop- ment of the GBM. At the beginning it had an analytical expression in the form of the Furry distribution, which is the solution of the stochastic equation on branching [IS]. However, it provides no information on the momenta of the particles produced. To improve on that the event generator ECCO was developed for which B! represents an alogarithm for branching [3]. I t is that alogarithm that we shall further improve in the next section.

I t should be noted that while geometrical scaling implies the independence of T,( R ) on s, the probability of producing n particles obviously must depend on s. Indeed, (2.15) embodies the central theme of the model: it is an amalgamation of the geometrical attribute, represented by .rr,(R), and the branching dynamics, represented by B f ( s ) , of particle production in hadronic collisions. It also enables us to combine the eikonal formalism with the parton model, since the latter is usually described by the structure functions of hadrons without reference to impact parameter.

As we proceed into the details of specifying BL, we shall find that the R dependence in (2.15) cannot be limit- ed entirely to r , (R ). On physical grounds the portion of incident energy consumed by central production must depend on the amount of overlap in the transverse plane. Also, the uncertainty in pr should also depend on it. Thus Bf in general depends on R , although the component for hard scattering may not.

111. SUCCESSIVE BRANCHING SUBPROCESSES

In a simulation we can, of course, go beyond (2.15) and determine for each event not only the number n of particles produced but also the momenta and charges of those particles. However, the basic structure is as shown in (2.15): each event has a simulated value of R and of p , and the summation 2, and integral S d R ' are effected by adding many such simulated events.

Our input from elastic scattering is R ( R ) given by (2.14). Then in accordance to (2.8) and (2.9) we generate a value of R . For that R we then generate a value for p in accordance to the distribution r p ( R ) , specified by (2.1 1 ) . Although ( p ) is only 1.6, we have obtained p as high as 9, albeit rarely. For a given value of p , we have p parallel branching processes, the produced particles of which are to be added to give the total multiplicity n of that event.

An important point to be emphasized is that the branching process to be discussed below describes the production of particles in the central rapidity region. I t does not include the nucleons or the decay products of the diffractively excited states of the nucleons, all of which are produced by dynamical mechanisms different from the branching process and are to be found in the projectile and target fragmentation regions. In each event there are particles of this type, since they always in-

170 JICAI PAN AND RUDOLPH C. HWA

elude the leading nucleons. In our simulation it is reasonable to identify those particles as being the products of the nonoverlapping regions of the two incident hadrons that behave as spectators of the central nondiffractive production process. This geometrical picture will serve only as the conceptual basis of our consideration below, but will not be taken literally in any de- tailed calculations. I t is mentioned here to indicate how the geometrical aspect of our model accommodates in a physically sensible way the complementary issues associated with the central and fragmentation regions.

The various parts of the branching process are discussed separately below.

A. Energy consideration

For a given simulated-value of R we must determine the fraction of energy d s that goes into central production, the balance being lost to the leading particles in the fragmentation regions. As discussed immediately above, it is reasonable to give a geometrical basis for the partition. Indeed, beyond impact parameter consideration it is also important to take into account the opacity of the overlap, since it is a way of quantifying inelasticity, which cannot lead to produced particles without energy consumption. Denoting the average energy for central production at R by V(R 1, we argue that it should be proportional to the average number of p-cut Pomerons, p ( R ) , since it is a measure of the absorptivity at R . When weighted by the probability g ( R ) of having an inelastic collision at R , we obtain, from (2.12),

where the proportionaJty factor has been chosen so that at R = O the energy d s is totally absorbed in central production. In the rare case when the simulated R is exactly zero, the produced nucleons would then have negligible energy. Otherwise, the leading particles would share in random partition the balance of the energy

Note that no adjustable parameter is used in the above, and the leading particle energy spectrum is unambiguous- ly specified.

At a later point when hard scattering is to be con- sidered, we would generalize the right-hand side (RHS) of (3.1) to read f lo(R)/ [ f lo(0)+f l , (O)] where f lo (R) is the soft component that is used here, while f l , (R ) is the hard component that is proportional to the jet cross section o j e t

Since f l ( R ) is an average quantity, we have related it in (3.1) to the average energy W ( R ) . From event to event, all at the same R, one would expect some fluctuations from W(R ). However, in the interest of avoiding the introduction of an arbitrary parameter that directly affects the multiplicity fluctuation, we assume the energy for central production for every event at R to be exactly W( R ) without fluctuation. Thus the observed fluctuation will be due entirely to the distribution in R (geometrical)

and to the branching (dynamical). At R we generate a value of p , thus having p parallel

branching processes. We let W( R ) be divided equally among the p cut Pomerons:

Each of them branches independently from one another starting from E( R , p ).

B. Momentum consideration

The first step of the branching process is not the same as the subsequent steps; it is more an expression of the ideas adapted from the parton model. Viewing the incident hadrons as a collection of partons whose rapidities are widely separated into the positive and negative domains in the c.m. frame except for a small fraction in the wee region, most partons are minimally perturbed by the collision. Thus our first step is just to set up the two opposite-going clusters of partons with undetermined effective masses. Whereas the partons in a free nucleon form a cluster with effective mass equal to the nucleon mass, the interaction of the wee partons with small rapidities in the two colliding nucleons breaks up the bags confining the two parton systems, leading to two clusters with effective masses that can be arbitrarily large subject to energy-momentum constraints. We shall determine those masses by requiring appropriate rapidity distribution.

The canonical parton distribution in momentum fraction x, when expressed in terms of rapidity y, exhibits a plateau structure with a width that depends on Ins. Such a distribution is the result of averaging over many events, so for a given event we may assume that the initial cluster rapidities are random within certain bounds. In that way we get a plateau in y after adding the contributions from many events. The bounds are set by kinematics. In the c.m. system of the two clusters the rapidities y , and y 2 are constrained by 0 < y l < y,,, and -y,,, < y, < 0, where

M is the initial rest energy of the cut Pomeron before the partition into the two clusters. M differs from E( R ,p ), which is the corresponding energy in the overall c.m. system, because the cut Pomeron can have nonzero momentum in that system. The latter can be determined only in conjunction with the momenta of the leading particles, a subject which we defer discussing until Sec. I11 E. For now, let it be understood that an appropriate shift in rapidity is to be made so that we can continue with the description of the branching process. We note that the bound in (3.4) is set by using the pion mass as the smallest allowed mass for each of the two clusters.

In most simulated events the values of R and p are such that E ( R , p ) is much less than %'s; however, oc- casionally E ( R , p ) may be large enough such that (3.4) becomes larger than the half-width of the observed rapidity plateau of dN/dy. We therefore need a global bound

SOFT HADRONIC PRODUCTION BY ~ c c o IN THE . . . 171

where is in units of 1 GeV, and po and f are free parameters to be determined by requiring the correct shape for dN/dy after the branching process is finished and after averaging over many events.

To summarize the first step of the branching, we generate for each cut Pomeron at given R and p the rapidities of two clusters at y , ( > 0 ) and y 2 ( <O) randomly in the interval ( -y,,,, +y,,,), where y,,, is the lesser of (3.4) and (3.5). The masses of the clusters m and m 2 are to be calculated after their momenta are determined.

The transverse momenta of the clusters are generated according to the distribution

where f, is the normalization constant, and a, and a l are free parameters adjusted to fit the experimental distribution dN/dp; at the end of branching. The dependence on R reflects our view that the uncertainty in p T bears a conjugate relationship to the degree of overlap in the transverse plane. The azimuthal angles 4 of the two clusters are opposite (differing by x) , but otherwise arbitrary.

After generating the values of y, pT , and 4 for each of the two clusters, and applying energy-momentum conservation to the first step of branching with the initial rest energy of the system being M, the masses m l and m , of the two initial clusters can now be determined.

For all subsequent steps of branching, we reverse the above process by generating masses first and then calcu- lating the rapidities. We impose our key input in the branching dynamics by requiring the daughter masses m , and m to satisfy the scaling law

where m is the mass of the mother cluster and y is a parameter to be adjusted. After generating the values of m , and m 2 , and then generating ( P ~ , , # ~ 1, i = 1,2, in the same

way as described above for the first step, we now determine the rapidities y l and y, in the mother's rest frame by use of energy-momentum conservation. Note that (3.41, (3.5), and the associated random distribution in y were used only for the first step, where the parton model provided the guidance on how the initial clusters are set up. All subsequent steps of branching follow essentially only two rules: the scaling law (3.7) and energy- momentum conservation. This is the essence of this version of ECCO.

In the classical description of a branching process one usually uses time as the evolution parameter. Here masses play the role of the evolution parameter along a particular branch. When the cluster masses are low enough, we may identify them with resonance masses. However, when masses are large, their role in our branching process is similar to the virtuality in the branching process of e +e - annihilation. Whereas virtuality is well defined in hard processes, no such quantity is available in soft processes, since perturbation theory does

not apply. Nevertheless, our model provides a series ex- pansion (2.15) and a tree diagram for each branching process, and the only sensible evolution parameter is the cluster mass. I t is not an observable, but it evolves smoothly into a particle mass that is observable. In the conclusion we shall give a physical interpretation for the clusters.

C. Resonance consideration

We repeat the above procedure for branching until a cluster mass reaches the range 0.8 < m < 2.8 GeV. At that point we regard the cluster as a generic resonance, and abandon (3.6) in favor of an isotropic decay distribution, while still keeping (3.7) for the daughter masses. For 2m,<m <0 .8 GeV, we abandon even (3.7) and re- quire that the cluster decays isotropically into two pions only. Even though the cluster may be a vector meson, the indeterminacy of its polarization is the reason for our choice of isotropic decay. Any mass generated in the range 0 < m < 2m, is regarded as that of a pion and set equal to 0.14 GeV with no further branching thereafter. Although this procedure does not account for each and every known resonance specifically, it does terminate the branching process with either p or direct x production, which are the dominant modes observed.

D. Charge consideration

In this version of ECCO we keep track of the charges of all clusters, which are allowed only to be either +, or -, or 0. Charge conservation is imposed at each step of branching. To get started, the leading particle is assigned the charge of the incident particle (in the same direction) with a probability that is proportional to the degree of nonoverlap, i.e., 1 - R ( R ) / f l ( O ) . The charge of the initial cluster is therefore known after the charge of the leading particle is determined. As that cluster branches into m , and m , clusters, their charges are given equal probability to have one of the possible assignments. For example, if the mother cluster has charge +, then the daughter clusters can have either (+,O) or (0,+) assignments. If the mother cluster has charge 0, then the daughters can have equal probabilities in the three possible assignments: ( + , - 1, ( - , + ), or (0,O).

E. Leading particles

The energy that the leading particles carry away is WL(R), given in (3.21, in the overall c.m. system. They are partitioned into two distinct clusters, going forwards with energy Ef and backwards with energy Eb. We re- quire that their proportions be totally random, subject only to the condition

When R is large so that the transverse overlap is small, we expect diffractive excitation with a wider spread in pT than in the case when R is small, since a large amount of angular momentum is involved. In the interest of intro- ducing no additional free parameters, we adopt the same p, distribution as (3.6) for the leading clusters. For the

172 JICAI PAN AND RUDOLPH C. HWA 48

masses of those clusters there is a distribution known from experiments on diffractive excitation. Since our aim in the present version of ECCO is on the nondiffractive part of soft production only, we make the simple choice of using just the nucleon mass for the clusters, which is sensible at small R where most particles are produced, but not so good at large R (a point to be improved later when we focus on diffractive dissociation processes).

For every event with a generated WL(R) we further generate E ~ = ~ V ~ ( R ) , ~ ~ = ( l - r ) p ~ ( ~ ) [where r is a random number in (0,1)] and p T according to (3.6). We can then calculate the longitudinal momenta pL of the two leading particles. With random azimuthal angles, we thus determine the momentum four-vectors of the forward and backward leading particles, p j and p l .

By energy-momentum conservation we now know the overall P v of all the centrally produced particles, where P O = W( R 1, and P = -pf -pb . For an event at R having p cut Pomerons, the momentum for each cut Pomeron is

The rest energy of each cut Pomeron is therefore

I t is this M that we have used in (3.4) to initiate branching. From the momentum Pi we can then determine the shift in rapidities of the produced particles relative to the c.m. system. At each step of the branching the rapidities of the daughter clusters, determined in the mother rest frame, are shifted by the rapidity of the mother cluster in the rest frame of the grandmother, and so forth.

IV. RESULTS OF SIMULATION

In the present version of ECCO there are five parameters: a,, a , , fa, f l , and Y. Although the total number is the same as before 131, they parametrize only three distributions: d ~ / d ~ i ( ~ ) , y,,,(s), and D(m, ,m , ,m) . More importantly, we have incorporated a superior description of branching kinematics without free parameters. The first four parameters essentially put the generated parti-

FIG. 1. Average charge multiplicity vs c.m. energy. The data are from [17], and the solid line is from ECCO.

FIG. 2. Normalized moments vs c.m. energy. The data are from [17], and the solid lines are from ECCO.

cles in the appropriate part of the phase space. The only key parameter that controls the dynamical behavior of particle production is y in (3.7). We believe that this is the most attractive feature of ECCO now.

The five parameters are determined principally by fitting the global features of the multiparticle data. They are (a) ( n ) , h v s s , (b) C, v s s for q=2 ,..., 5, (c) ( n ) ~ , vs n / ( n ) , ( d ) d n / d y v s y , a n d ( e ) d a / d p + .

Since we treat only soft production,the data used are restricted to the energy range 1 0 < g s <65 GeV. That means that they are from experiments at Fermilab, Ser- pukov, and the CERN Intersecting Storage Rings (ISR) [16], as summarized in [17,18], and NA22 [I 1,191.

In Figs. 1-5 we show the five sets of data on (a)-(el. The solid lines are our simultaneous fits of all five figures, using the following values for the parameters:

k " " l " " l " ' ' " " ' ~

-

-

- - ISR ECCO GeV

- - - - - - i A

: A - 62.6 \

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 o - ~ 0 1 2 3 4

n/cn>

FIG. 3. Multiplicity distribution in a Koba-Nielsen-Olesen (KNO) plot. The data are from [16].

SOFT HADRONIC PRODUCTION BY ~ c c o IN THE . . . 173

FIG. 4. Rapidity distribution at E,,,, =22 GeV. The data are from [19].

We regard the fits as excellent, considering the vast number of data points over a wide range of energy. In Figs. 1 and 2 we have included the two energy points below 10 GeV, not because we believe the validity of our model at such low energies, but because the extrapolations of our calculations result in good fits for free. These global features, especially ( n ), P, , and C, , at any given energy are determined mainly by the parameter y , which controls the number of cascade steps in evolving from the initial M to the final m,. That in turn affects the average multiplicity and the fluctuations from it. It is remarkable

103 ! ' " ' " " " ' " ' ' ~ ISR ECCO GeV

0 - - - 44.5

FIG. 5. Transverse momentum distribution. The data are from [IS].

that one parameter can effect a good fit of all those features over the entire energy range.

It should be remarked that we do not regard the input function for O ( R ) given in (2.14) as adjustable; if the pa- rametrization of W R ) is regarded as determined by fitting, then u,,(s), d u / d t , and u,,,(s) must also be included as data that have been fitted. What we are dealing with here is a highly constrained system of data, and our results are not order-of-magnitude rough fits.

Before we show what ECCO ~ r e d i c t s in local features. it is instructive and illuminating to exhibit a sample tree showing the cascade of clusters for a particular cut Pcmeron. In Fig. 6 we show a simulated event a t g s = 100 GeV (chosen high enough to allow several steps of branching for illustrative purpose). The vertical axis shows the mass of a cluster at a vertex (in GeV in loga- rithmic scale) and the horizontal axis marks the rapidity of the cluster in the c.m. system. The initiating mass M of the tree in Fig. 6 is 32 GeV at a nonzero rapidity. It is the only cut Pomeron ( p = l ) with R=0.77, and W ( ~ ) / f i =0.35. Thep, and 4 of each cluster are generated but not shown in the figure. Evidently, there is significant overlap in y for particles produced from the right-going and left-going clusters. Thus for a given interval in y there may be particles in it that take very different routes in the tree to get there. That is a mechanism for intermittency. Note that the genealogy of the particles produced is very different from that of successive bin splitting such as in the a model [4], or of successive string breaking such as in the FRITIOF model [7] or the dual parton model [6] , although the latter also allows string overlap in a cut Pomeron. In Fig. 6 there is no or- dering in y for clusters of the same or different genera- tions, except for the two extremes at the ends. It is the sum of manv such exclusive v distributions for 6X lo4 events that add up to the inclusive distribution dn /dy shown in Fig. 4 (which is for g s =22 GeV).

FIG. 6. Cascade tree diagram for a simulated event at E,,,, = 100 GeV. Each vertex represents a branching with the coordinates of the vertex indicating the mass and rapidity of the mother cluster, those of the daughters being given by the coordinates of the two vertices down the tree.


Having determined the parameters from the global where p (y ' ) is the rapidity density, which is not flat. F i s features, we have no more freedom in adjusting the code. as defined in (4.3) except that Tr lnp, . Since the distri- Thus we can proceed to the calculation of the factorial bution in p T is sharply peaked at pT=O, it is necessary to moments without ambiguity. Recall [4] consider a smoothed-out variable such as F, in terms of

1 (n,(n,-1) . . . (n , -q+1) which the transverse-momentum distribution is flat. F =-2 (4.2) These variables will be partitioned into small bins of sizes

M i = , ( n j ) q 67, 84, and 6F.

where n, is the multiplicity in the jth bin and JZ/1 is the to- The results of our simulation by ECCO enable us to calculate F, in different dimensions. In Figs. 7(a), 7(b), and tal number of bins. We shall consider three-dimensional

(3D) phase space, or any lower-dimensional projection of 7(c) we show the one-dimensional ( I D ) analysis of F, as

it. The three kinematical variables are 9, $, and F, where functions of 67, S$, and SF, respectively, in log-log plots.

the tilde makes them the normalized cumulative variables In Fig. 8 are the 2D results, while Fig. 9 shows the 3D re-

[20,211, e.g.9 sult. We have used 6 X lo4 events to minimize the fluctuation of the simulated result. Even so, we have found

7=JYy p ( y y . ) d y l / ~ y ~ ) ( y r ) d y l , (4.3) that F, in the small bin regions for q = 4 and 5 fluctuate mln considerably, but are consistent with the data [ l l ] within

FIG. 7. One-dimensional intermittency plots. The data are FIG. 8. Two-dimensional intermittency plots. The data are from [I I]. from [ l 11.

SOFT HADRONIC PRODUCTION BY ~ c c o IN T H E . . . 175

FIG. 9. Three-dimensional intermittency plots. The data are from [ l 11.

NA22 8 . X ECCO

P4 - 6 - ~=1 .522

4 .

2 .

0 1 2 3 4 5 6

9

FIG. 11. Plot of /3, from which the scaling exponent v is determined. The solid line is a fit of the ECCO result from Fig.

10. The dashed line is from [23]. The data are from [ l l ] .

FIG. 10. Ochs-Wosiek plots of ECCO results at 22 GeV. FIG. 12. Ochs-Wosiek plots of ECCO results at 50 GeV


errors. The results for q = 2 , 3 and the larger bin region of q =4,5 are stable. Evidently, the excellent agreement between our calculated results and the data gives support to the soundness of the GBM in general and the reliabili- ty of E C C ~ in particular. To our knowledge the data in Figs. 7-9 have not been fitted by any model before except in Refs. [3,13]; now with this new version of ECCO we do not even adjust any parameters in order to obtain good agreement with the intermittency data.

Finally, we consider the scaling law

by examining the Ochs-Wosiek plot [22] for the E C C ~

data on all dimensional analyses. In Fig. 10 we see that there is universality in the slopes independent of ID, 2D, or 3D analyses. The values of 0, extracted from the slopes are shown by X in Fig. 11. As shown by the solid line they can be well fitted by the formula

with the scaling exponent being

In F3 Ec.,,.= I00 GeV I

FIG. 13. Ochs-Wosiek plots of ECCO results at 100 GeV.

FIG. 14. The scaling exponent v as a function of energy.

In Fig. 11 we also show the values of 0, extracted from the NA22 data together with their errors. Evidently, our ECCO results are well with the error bars. Equation (4.5) was first discovered in connection with the intermittency behavior of phase transition in the Ginzburg-Landau theory [23]. The value of v for that problem was found to be 1.304, as indicated by the dashed line in Fig. 11. It has been suggested [23] that the value of v be used to indicate the presence or absence of quark-hadron phase transition in high-energy collisions. Our result from ECCO for pp collision at 22 GeV as given in (4.6) clearly implies the absence of such a transition.

With ECCO at hand, we can now explore the energy dependence of multiplicity fluctuation. Since the scaling exponent succinctly quantifies the degree of such fluctuations, its energy dependence should then provide a simple picture of the general behavior. In Figs. 12 and 13 we show the Ochs-Wosiek plots for d; =50 and 100 GeV, respectively. For each value of q we draw a straight line by eye to represent the average slope. The resultant values of 0, are fitted again by (4.5), yielding the scaling exponents v as shown in Fig. 14 together with their errors. We see that v has no significant dependence on energy within errors. This is an interesting result, since it implies an additional scaling behavior that is predicted by ECCO, but remains to be checked by experiment.

V. CONCLUSION

With the successful description of multiparticle production process by EccO, as judged by the body of data that the event generator can fit, it is appropriate to review the essential features of the model and restate the physics of hadronic collisions in a way that may be useful for comparison with other models.

The first part of GBM is the eikonal formalism. That seems to be a rock solid foundation for hadronic collisions, in which the geometrical sizes of the hadrons play an essential role. The next part is the cut-Pomeron ex- pansion, which makes contact with the S-matrix ap- proach to multiparticle production; in both cases the summation of a series of terms exponentiates the proba-

48 - SOFT HADRONIC PRODUCTION BY ~ c c o IN T H E .

bility of no inelastic collision and imposes unitarity. Each cut Pomeron is then treated in the model as a branching process.

A cluster that branches may be regarded as a collection of partons that have no net color and separate into two colorless subcollections. The cluster mass is the invariant square of the sum of all the four-momenta of the partons in the collection. The dynamics of the strong interaction is in the creation, absorption, and rearrangement of all the partons into subsets, which, if colorless, split from one another without restraining force. The process is too complicated to be tracked in detail, but can be described by a simple scaling law on the mass distribution, (3.7), which turns out to be sufficient to specify the dynamics of branching. If the number of partons in a collection is large, the probability that they can be arranged into two colorless subcollections is presumably independent of the specific number of partons involved, but depends on the fraction of splitting and on the clustering of the parts. That is the origin of the scale invariant form of (3.7) that depends only on the mass fractions m / m and m , /m.

The basic difference between GBM-ECCO and dual parton model (DPM) [6] and FRITIOF [7] is that we assume the existence of partons in hadrons before their collision as in the parton model; most of the partons in one hadron pass through the opposite-going ones in the other hadron without much interaction except for the ones that are close in rapidity. In D P M strings are stretched between forward-going diquark and the held-back quark of the other hadron; in FRITIOF hadrons are excited into string- like vortex lines upon collision. In both cases the stretch- ing and breaking of strings lead to the production of particles. In our case the partons that are there rearrange themselves through short-range interaction (in rapidity) into colorless clusters and undergo successive branchings, which are successive random processes that give rise to intermittency. Indeed, we have shown in this work how the intermittency data for hadronic collisions can be fitted by ECCO without tuning any adjustable parameters. That, we believe, is an attribute of our model that is

[I] W. R. Chen and R. C. Hwa, Phys. Rev. D 36, 760 (1987); 39, 179 (1989); R. C. Hwa, ibid. 37, 1830 (1988).

[2] R. C. Hwa and X. N. Wang, Phys. Rev. D 39,2561 (1989); 42, 1459 (1990).

[3] R. C. Hwa and J. Pan, Phys. Rev. D 45, 106 (1992). [4] A. Bialas and R. Peschanski, Nucl. Phys. B273, 703

(1986); 308, 857 (1988). [S] For a recent review see Fluctuations and Fractal Structure,

edited by R. C. Hwa, W. Ochs, and N. Schmitz (World Scientific, Singapore, 1992).

[6] For a review see A. Capella, V. Sukhatme, C. I. Tan, and J. Tran Thanh Van, Phys. Rep. (to be published).

[7] B. Andersson, G. Gustafson, and B. Nilsson-Almqvist, Nucl. Phys. B281, 289 (1987); B. Nilsson-Almqvist and El. Stenlund, Phys. Commun. 43, 387 (1987).

[8] I. V. Ajineko et al., Phys. Lett. B 222, 306 (1989); W. Kit- tel, in Multiparticle Production, Proceedings of the Perugia Workshop, Perugia, Italy, 1988, edited by R. C. Hwa, G. Pancheri, and Y. Srivastava (World Scientific, Singapore,

unique at this point. There are two other more recent event simulators that

should be mentioned in comparison. One is HIJING [24] and the other parton cascade model [25]. Although they can both be reduced to pp collisions, they are basically constructed for heavy-ion collisions. HIJING emphasizes minijet production with the soft part of the production process being adapted from the FRITIOF model 171. The parton cascade model studies the time evolution of the parton distribution using perturbative QCD, a procedure that is more reliable for hard collisions between partons than for soft interaction. Both models can make significant claims about the simulation of hadron production in high-energy nuclear collisions when hard interac- tions dominate over soft ones. The aim of GBM-ECCO is to describe soft interaction at energies where hard collisions are unimportant. Thus the two approaches are complementary.

A concise summary of the intermittency behavior is the scaling exponent v, which is independent of the di- mension of the phase space in which the factorial moments are analyzed. It describes the scaling behavior of the multiplicity fluctuation that is self-similar under changes of resolution scale. As we have found in this paper, it is also scaling in the sense of energy independencz in pp collision. However, we also anticipate that as z /s increases beyond 100 GeV, the production of minijets by hard scattering will become important. The effect on v both experimentally and theoretically will be of great interest to determine.

For soft hadronic production we believe that GBM- ECCO has captured the essence of the dynamical process involved.

ACKNOWLEDGMENTS

We are grateful to W. Kittel and F. Botterweck for providing us with the data of NA22. This work was sup- ported in part by the U.S. Department of Energy under Grant No. DE-FG-06-9 lER40637.

1989). [9] M. Bengtsson and T. Sjostrand, Nucl. Phys. B289, 810

(1987); Comput. Phys. Commun. 43, 367 (1987). [lo] G. Marchesini and B. Webber, Nucl. Phys. B310, 461

(1988). [ l l ] W. Kittel, in Intermittency in High Energy Collisions,

Proceedings of the Santa Fe Workshop, 1990, edited by F. Cooper, R. C. Hwa, and I. Sarcevic (World Scientific, Singapore, 1988), p. 83; M. Adamus et al., Z. Phys. C 37, 215 (1988); I. V. Ajineko et al., Phys. Lett. B 222, 306 (1989); 235, 373 (1990); F. Botterweck et al., in Fluctua- tions and Fractal Structure 151.

[12] R. C. Hwa and J. Pan, Phys. Rev. D 46, 2941 (1992). [13] J. Pan and R. C. Hwa, Phys. Rev. D 46,4890 (1992). [14] A. W. Chao and C. N. Yang, Phys. Rev. D 8,2063 (1973). [15] R . C. Hwa, in Hadronic Multiparticle Production, edited

by P. Carruthers (World Scientific, Singapore, 1988). [16] A. Breakstone et al., Phys. Rev. D 30, 528 (1984). [17] G. J. Alner et al., Phys. Lett. 160B, 193 (1985); 167B, 476


(1986). [22] W. Ochs and J. Wosiek, Phys. Lett. B 214, 617 (1988). [18] G. Giacomelli and M. Jacob, Phys. Rep. 55, 1 (1978). [23] R. C. Hwa and M. T. Nazirov, Phys. Rev. Lett. 69, 741 [19] M. Adamus et a l . , Z. Phys. C 39, 31 1 (1988). (1992); R. C. Hwa and J. Pan, Phys. Lett. B 297, 35 (1992); [20] W. Ochs, Phys. Lett. B 247, 101 (1990); Z. Phys. C 50, 339 R. C. Hwa, Phys. Rev. D 47,2773 (1993).

(1991); and in Fluctuations and Fractal Structure [ 5 ] . [24] X. N. Wang and M. Gyulassy, Phys. Rev. D 44, 3501 [21] A. Bialas and M. Gazdizicki, Phys. Lett. B 252, 483 (1991).

(1990). [25] K. Geiger and B. Miiller, Nucl. Phys. B369, 600 (1992).

Documents

Soft hadronic production by ECCO in the geometrical branching model