Study of the Violation of KNO Scaling and the Validity of Modified KNO Scaling Based on the Cluster Model

Abstract

The violation of Koba-Nielsen-Olesen (KNO) scaling observed in the recent ffP collider experiments is explained by means of the cluster model for the multi-particle production processes. The energy dependence of the KNO function is well reproduced on the assumption that hadrons are emitted through three clusters in the statistic thermodynamical way. It is also shown that modified Koba-Nielsen-Olesen (MKNO) scaling is valid in the framework of the cluster model. Our model yields that the scaling parameter of MKNO scaling should be one. Since the late 1960's, several scaling laws have been proposed with the help of the experimental studies.!) KNO scaling,2) which is a sort of scaling in the mean,3) especially has been considered to be valid for the wide energy regions. The energy independent KNO scaling function is a good measure for the study of multi-particle production mechanism. However, the experimental data show only an approximate KNO scaling behaviour for the multiplicity distributions in the different energy regions_ It is possible to find a slight energy dependence of the KNO functions in a strict sense. And hence, several authors 4 ),5) proposed MKNO scaling in which the KNO scaling variable z = nc I is modified to z' = (nc- a) I in view of the analysis of correlations. Here nc( < nc» represents charged particle multiplicity (its mean value) in the final state, and a is a process dependent parameter. Recently it has become evident from the CERN -SPS ffP Collider experiments 6 ),7) that KNO scaling is apparently violated. Therefore, now it is quite interesting to examine theoretically whether MKNO scaling is valid at this energy instead of KNO scaling. In this paper we account qualitatively the violation of KNO scaling and examine the possibility of MKNO scaling from the viewpoint of the cluster model. In a previous paperS) (hereafter referred to as 1) we analyzed the cluster model with Gaussian cluster number distribution, and got a conclusion that it is sufficient to take a fixed cluster number for the narrow multiplicity distribution in hadron-hadron collisions. Our aim of this paper is not to obtain a perfect fit to the data by using a complicated model with many parameters, but to derive the characteristic energy dependence for the KNO function and its moments with the use of statistic thermodynamical method. In this paper, therefore, we introduce only one parameter, i.e., the cluster number m. According to our analysis, the cluster model with m=3 well reproduces the energy dependent multiplicity distributions up to SPS­ Collider energy region. We assume approximately that three clusters have the same average mass, while this mass relation can be modified in order to improve the data fit. However, our result shown in Figs. 1 (a) ~ (c) suggests that this assumption yields the gross feature of cluster mass relation. It should be noticed that m=3 is different from the conclusion of I where we assumed an exact precocious scaling.