Scaling Relation for a Quantity Related to Particle Production Multiplicities

A scaling relation, d lnlun (s)]/d lns-p(n/lns), for high-energy production of n parti­ cles is proposed. This relation is supposed to be valid for large n and large s. An exten­ sion to present energies is suggested and compared with experiment.

A scaling relation, d lnlu n (s)]/d lns-p(n/lns), for high-energy production of n parti cles is proposed.This relation is supposed to be valid for large n and large s.An exten sion to present energies is suggested and compared with experiment.
Various models of high-energy production yield limiting (in energy) relations for certain experimental quantities.It is a hope that devia tions from these relations are small at present large but finite energies and that we may con front these theoretical ideas with current data.In this note I present a scaling relation for a quantity related to <J n , the cross section for pro ducing n particles of a certain type.The index n may refer to charged particles, negative par ticles, pions, etc.We consider a process where these n particles are produced by an incident state whose center-of-mass energy is Is.Let (2) If this relation is true then the left-hand side, which is a priori a function of two variables, n and Y, approaches at high energies a nontrivial function of their ratio.
The assumptions necessary to establish (2) are the following: (1) Correlations in inclusive production are taken to be of short range. 1 More generally we assume the existence of the thermodynamic limit of the Feynman fluid analog 2 to multiparticle production.Specifically if we assume that the following limit exists: (2) is just the relation between this partition function and the pressure which is a function of the density p.The derivation of the equivalence of the two ways of obtaining the pressure is identical to that in statistical mechanics 5 and will not be reproduced here.
As mentioned previously (2) is to hold for large n and Y.If we wish to test it with presently available data we must decide on what value to assign to s 0 in (1).For present energies the value of s 0 may be crucial for the test of ( 2).An appeal ing suggestion comes from the fluid analog itself.Y is related to the length of the plateau in one particle inclusive production, and the average inelastic multiplicity, (n), is in this analog di rectly proportional to Y. Thus, it is plausible that a proper continuation of ( 2 5) and ( 6) neither the accuracy of the data nor the differences in (n) between the lowest and highest energies permit a definite statement on these relations.It is pos- sible to find phenomenological fits to (Y"satisfying either (5) or ( 6) and over the present range of (n) which appear (within experimental error) to satisfy the other.'3 %e may thus only note the consistency of both relations with present experiment.
Thanks are due many of my colleagues for long discus sion s.The data to which relation (5) was applied were on proton-proton collisions into n negative particles.Prong distributions used were for incident momenta of 28. 5, 50"69, 103, 205, and 303 ' GeV/c.A quadratic approximation was used to obtain the derivatives with respect to (n) and thus these are available only for the four middle energies.The results are presented in Fig, 1.Because of the sizable error bars we may only conclude that (5) is consi.stent with present data.The solid line in Fig. 1 is what we would obtain if a"were given by a Poisson distribution in the negative prongs.
Before closing we should contrast (2) or (5) with a different scaling hypothesis for 0".Koba, Nielsen, and Olesen" have suggested that in the same limit as considered here (6)   This relation is inconsistent with (2) or (5) as it is obtained from different assumptions.There is support for the consistency of (6) with experi-

Scaling
Relation for a Quantity Related to Particle Production Multiplicities Myron Bander* National Accelerator Laboratory, Batavia, lllinois 60510 (Received 20 November 1972) a are at present arbitrary but finite.The limiting relation we propose is .a n,Y -I�� I Y = p a y[I n <Jn ( Y ) ] = p(p ).
with p(z) some finite function of the parameter z.(2)We need an assumption about the rate of decrease of <J n ( Y) with Y fixed and n increasing.The simplest assumption is that CJ n = 0 for n >N(Y) where N(Y) is bounded by a power of Y.(The kinematic limit N=ls/m is not sufficient.)•The stringent requirement that <J n = 0 for n > could be relaxed to a smooth but rapid Not wishing to get involved in delicate details use the above simple assumption. 3In the Feynman fluid analog Q (z, Y) corresponds to the grand canonical partition function, and p (z) to the pressure as a function of the fugacity z . 4In this framework <J n ( Y) is the analog of the partition function in the canonical ensemble and ) to present ener gies is to replace Y by (n).The scaling hypothe sis we propose to test is ( FIG.  1. Plot of dlno"/d(n) versus n/(n) for the reac- tion ppn negative prongs.(n) is the average inelast- ic number of negative prongs.The derivatives are evaluated for 50, 69, 108, and 205 Gev/c.The data used are from Befs.6-10.