Eur. Phys. J. A manuscript No. (will be inserted by the editor) A look at multiparticle production via modified combinants Han Wei Anga,1, Aik Hui Chanb,1, Mahumm Ghaffarc,2, Maciej Rybczynski´ d,3, Grzegorz Wilke,4, Zbigniew Włodarczykf,3 1Department of Physics, National University of Singapore, Singapore 17551 2Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John’s, NL A1B 3X7, Canada 3Institute of Physics, Jan Kochanowski University, 25-406 Kielce, Poland 4National Centre for Nuclear Research, Department of Fundamental Research, 02-093 Warsaw, Poland Received: date / Accepted: date Abstract As shown recently, one can obtain additional in- logical approaches had to be adopted. These can range from formation from the measured charged particle multiplicity dynamical approaches in the form of coloured string inter- distributions, P(N), by investigating the so-called modified actions [1] and dual-parton model [2], to geometrical ap- combinants, Cj, extracted from them. This information is en- proaches [3,4] resulting in the fireball model [5], stochastic coded in the observed specific oscillatory behaviour of Cj, approaches [6–8] modelling high energy collision as branch- which phenomenologically can be described only by some ing [6–8] or clans [9]. combinations of compound distributions based on the Bino- The myriad of stochastic models since proposed have mial Distribution. So far this idea has been checked in pp described the experimental data well with very reasonable and e+e processes (where observed oscillations are spec- − c2=do f values. Amongst the numerous proposed distribu- tacularly strong). In this paper we continue observation of tions, the Negative Binomial Distribution (NBD) and its vari- multiparticle production from the modified combinants per- ants are the most ubiquitous [10]. However, as has been pro- spective by investigating dependencies of the observed os- posed recently [11–14], a good fit to the MD from a statis- cillatory patterns on type of colliding particles, their ener- tical distribution is only one aspect of a full description of gies and the phase space where they are observed. We also the multi-faceted set of information derivable from the MDs. offer some tentative explanation based on different types of A more stringent requirement before any phenomenological compound distributions and stochastic branching processes. model is considered viable is to also reproduce the oscil- latory behaviour seen in the so called modified combinant, C , which can be derived from experimental data. In fact, 1 Introduction j this phenomenon is observed not only in pp collisions dis- cussed in [11–14] but also, as demonstrated recently in [15], Multiplicity distributions (MDs) of high energy collisions in e+e annihilation processes. Such oscillations may be have been extensively studied in the field of multiparticle − therefore indicative of additional information on the multi- production. It is one of the first observables to be deter- particle production process, so far undisclosed. Specifically, mined in new high-energy experiments. This is partly due the periodicity of the oscillations of modified combinants arXiv:1908.11062v3 [hep-ph] 23 Apr 2020 to the ease with which such information can be obtained, derived from experimental data is suggestive. and also because MDs contain useful information on the un- derlying production processes. Due to the inability of per- It is in this spirit that this study sets forth to understand turbative Quantum Chromodynamics (pQCD) to provide a the effects of the collision systems and various experimen- complete theoretical account for the observed MDs incorpo- tal observables on the period and extent of oscillations in rating both the hard and soft processes, various phenomeno- Cj. In Section2, the concept of modified combinant will be reviewed in light of its connection to the earlier con- a e-mail: [email protected] cept of combinant [16–18]. From this link, an attempt is be-mail: [email protected] ce-mail: [email protected] made on the potential interpretation of modified combinant de-mail: [email protected] applied in the context of multi-particle production. Section ee-mail: [email protected] 3 discusses the problem of dependence on collision system fe-mail: [email protected] whereas Section4 discusses the effect various experimental 2 variables have on the modified combinant oscillations and Table 1 Distributions P(n) used in this work: Poisson (PD), Negative summarises the key points observed. Binomial (NBD) and Binomial (BD), their generating functions G(z) Our concluding remarks are contained in Section5 to- and modified combinants Cj emerging from them. gether with a tentative proposal of employing the character- istics of oscillations in experimental modified combinants to distinguish between different collision types. Some ex- P(N) G(z) Cj planatory material is presented in appendices: Appendix A presents the relationship between Cj and the Kq and Fq mo- l N ments that are more familiar to the particle physics commu- PD exp( l) exp[l(z 1)] d j0 N! − − nity whereas Appendix B shows the possible origin of the observed oscillations of C based on the stochastic approach j k N+k 1 N k 1 p k j+1 to the particle production processes. NBD N− p (1 p) 1 −pz N p − − h i 2 Modified Combinant and Combinant j+1 K N K N K K p BD N p (1 p) − (pz + 1 p) −N p 1 − − h i − Statistical distributions describing charged particle multiplic- ity are normally expressed in terms of their generating func- ¥ N tion, G(z) = ∑N=0 P(N)z , or in terms of their probability function P(N). One other way to characterise a statistical Combinants were first introduced to quantify the extent any distribution is a recurrent form involving only adjacent val- distributions deviate from a Poisson distribution. For the Pois- ues of P(N) for the production of N and (N + 1) particles, son distribution C0 = 1 and Cj>1 = 0. In this way, any non- zero Cj at higher orders indicate a deviation from the Pois- (N + 1)P(N + 1) = g(N)P(N): (1) son distribution. From Eq. (3), two obvious interpretations for C follow. Cast in this form, every P(N) value is assumed to be deter- j First, there is a one-to-one map between C to C via Eq. mined only by the next lower P(N 1) value. In other words, ∗j j − (4). Modified combinants can be interpreted as a proxy to the link to other P(N j)’s for j > 1 is indirect. In addition, − the extent of deviation from a Poisson distribution at differ- the eventual algebraic form of the P(N) is determined by the ent higher orders. Secondly, C ’s are the normalized weights function g(N). In its simplest form, one can assume g(N) to j in the series for the value of (N + 1)P(N + 1). This can be linear in N, such that be interpreted as the "memory" which P(N + 1) has of the g(N) = a + bN: (2) P(N j) term. In other words, the modified combinants are − the weights in which all earlier P(N j) values has on the Some prominent distributions have been defined in this form. − For example, when b = 0 one gets the Poisson Distribution current probability. In this interpretation the links between P(N + 1) to all P(N j) values are clearly established. (PD). The Binomial Distribution (BD) arises for b < 0 and − b > 0 results in the Negative Binomial Distribution (NBD). One further notes that since Cj’s are expressed in terms While conceptualising a phenomenological model, the form of the probability function in Eq. (3), it may be reasonable of g(N) can be modified accordingly to describe the experi- to attempt casting the modified combinant in terms of the N mental data, cf., for example, [19, 20]. generating function G(z) = ∑N P(N)z . Such an expression However, the direct dependence of P(N + 1) on only is immensely useful should a theoretical distribution avail P(N), as seen in Eq.1, seems unnecessarily restrictive. This itself to describe experimental data. In this case, Cj can be constraint can be further relaxed, by writing the probability expressed as follows: function connecting all smaller values of P(N j) as follows − j+1 [16], 1 d lnG(z) N Cj = j+1 : (5) N h i j! dz z=0 (N + 1)P(N + 1) = N CjP(N j): (3) h i ∑ − j=0 Modified combinants for some prominent distributions are The coefficients Cj are known as the modified combinants shown in Tab.1. Note that the generating functions of NBD and forms the core of this study. They are related to the com- and BD are in fact some quasi-power functions of z and as binants C∗j first defined for the study of boson production such can be written in the form of the corresponding Tsallis models [17, 18] by the following relation [11]: distribution [21], ( j + 1) 1 Cj = C∗j+1: (4) 1 q N G(z) = expq[ N (1 z)] = [1 + (q 1) N (1 z)] − (6) h i h i − − h i − 3 + where e + e−, √s = 91 GeV ALEPH, y < 2 1 p | | q 1 = = for NBD; (7) − k (1 p) N − h i 1 p ) q 1 = = for BD; (8) N − −K − N ( h i P whereas for q 1 in both cases we obtain G(z) for PD. ! Eqs. (7) and (8) allow to write Cj for all three distributions differentiated by the above choice of the paramater q in one formula, N 1 (q 1) N j + Cj = − h i : (9) e + e−, √s = 91 GeV (q 1) N + 1 (q 1) N + 1 ALEPH, y < 2 − h i − h i | | Note that while for the PD and NBD coefficients Cj are j 85 .