Modified combinant analysis of the e + e − multiplicity distributions
H.W.Ang, M.Ghaffar, A.H.Chan, M.Rybczyński, G.Wilk, Z.Włodarczyk
aa r X i v : . [ h e p - ph ] J u l July 16, 2019 2:14 WSPC/INSTRUCTION FILE ws-epem-mpla
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
Modifed combinant analysis of the e + e − multiplicity distributions H. W. Ang ∗ , M. Ghaffar, A. H. Chan Department of Physics, National University of SingaporeBlock S12 Level 2 Science Drive 2Singapore [email protected]
M. Rybczy´nski, Z. W lodarczyk
Institute of Physics, Jan Kochanowski University´Swi¸etokrzyska 15, 25-406 Kielce, Poland
G. Wilk
National Centre for Nuclear Research, Department of Fundamental ResearchHo˙za 69, 00-681, Warsaw, Poland
Received (Day Month Year)Revised (Day Month Year)As shown recently, one can obtain additional information from the measured chargedparticle multiplicity distributions, P ( N ), by extracting information from the modifiedcombinants, C j . This information is encoded in their specific oscillatory behavior, whichcan be described only by some combinations of compound distributions such as theBinomial Distribution. This idea has been applied to pp and p ¯ p processes thus far. Inthis note we show that an even stronger effect is observed in the C j deduced from e + e − collisions. We present its possible explanation in terms of the Generalised MultiplicityDistribution (GMD) proposed some time ago. Keywords : Multiparticle Production; Modified Combinants; Modified Multiplicity Dis-tribution.PACS Nos.: 12.40.Ee
1. Introduction
Recently it was shown that the measured multiplicity distributions, P ( N ), containsome additional information on the multiparticle production process, so far undis-closed. The basic idea was to apply the recurrence relation used in countingstatistics when dealing with multiplication effects in point processes. Its impor-tant feature is that it connects all multiplicities by means of some coefficients C j ( modified combinants ), which define the corresponding P ( N ) in the following way: ∗ corresponding author 1 uly 16, 2019 2:14 WSPC/INSTRUCTION FILE ws-epem-mpla H.W. Ang et. al. ( N + 1) P ( N + 1) = h N i N X j =0 C j P ( N − j ) . (1)These coefficients contain the memory of the particle N + 1 about all the N − j previously produced particles and, most important, they can be directly calculatedfrom the experimentally measured P ( N ) by reversing Eq. (1) and putting it in theform of the recurrence formula for C j : h N i C j = ( j + 1) (cid:20) P ( j + 1) P (0) (cid:21) − h N i j − X i =0 C i (cid:20) P ( j − i ) P (0) (cid:21) . (2)Oscillations were previously observed in the modified combinants derived from P ( N )values measured in pp and p ¯ p experiments,
1, 4 whereas the C j values derived from thepopular Negative Binomial Distribution (NBD) are monotonically decreasing withincreasing j . a On the other hand, the Binomial Distribution (BD) gives stronglyoscillating C j (with period two, not observed in the above data). To fit these data,one needs the BD to be compounded with another distribution that can effectivelycontrol the period and amplitude of the resulting oscillations (in Ref 4 it was com-pounded with an NBD).It turns out that in the case of multiplicity distributions of charged particlesproduced in e + e − collisions the observed oscillations are much stronger. Fig. 1shows the results for P ( N ) and for the corresponding C j deduced from the ALEPHexperiment data. The C j ’s can be fitted by using a compound distribution involvingthe Generalized Multiplicity Distribution (GMD) to be introduced in Section 2.The modified combinants are closely related to the combinants C ⋆j introducedin Ref. defined in term of the generating function. G ( z ) = P ∞ N =0 P ( N ) z N , C ⋆j = 1 j ! d j ln G ( z ) dz j (cid:12)(cid:12)(cid:12)(cid:12) z =0 , (3)It can be shown that C j and C ⋆j are related as follows: C j = j + 1 h N i C ⋆j +1 , (4)By the above relations, C j can be expressed in terms of the generating function G ( z ) of P ( N ) as h N i C j = 1 j ! d j +1 ln G ( z ) dz j +1 (cid:12)(cid:12)(cid:12)(cid:12) z =0 . (5)This relation is particularly useful when C j are calculated from some compoundmultiplicity distribution defined by a generating function G ( z ) for which Eq. (2)would be too difficult to apply. a The only condition is that the data sample under consideration is large enough, otherwise theoscillatory behavior is washed out by fluctuations. uly 16, 2019 2:14 WSPC/INSTRUCTION FILE ws-epem-mpla
Modifed combinant analysis of the e + e − multiplicity distributions P ( N ) N e + + e − , √ s = 91 GeVALEPH, | y | < h N i · C j / . j (cid:1) je + + e − , √ s = 91 GeVALEPH, | y | < Fig. 1. Left panel: data on P ( N ) measured in e + e − collisions by the ALEPH experiment at91 GeV are fitted by the GMD distribution Eq. (6) with parameters: h N i = 12 . k = 3 . k ′ = 0 . C j derived from these data (note thesignificant dependence of the amplitude on rank j . The oscillation amplitude of the plot has beenscaled accordingly making it possible to plot the results. Otherwise the amplitudes would grow ina power-law fashion). As shown in the following sections, they can be fitted by the C j obtainedfrom the P ( N ) derived from GMD.
2. Generalized Multiplicity Distribution - GMD
The GMD was introduced as an alternative to the NBD solution to study multi-plicity distributions. It has the following form: P ( N ) = Γ( N + k )Γ( N − k ′ + 1)Γ ( k ′ + k ) p N − k ′ (1 − p ) k + k ′ , (6)where p = h N i − k ′ h N i + k . (7)The GMD has been successfully applied to p ¯ p reactions and e + e − annihilation. It is based on the stochastic branching equation describing the total multiplicitydistribution of partons inside a jet, dP ( n ) dt = − (cid:16) An + ˜ Am (cid:17) P ( n ) ++ A ( n − P ( n −
1) + ˜
AmP ( n − , (8)where t = 611 N c − N f ln ln (cid:16) Q µ (cid:17) ln (cid:16) Q µ (cid:17) (9)is the QCD evolution parameter, with Q denoting the initial parton invariant mass, Q the hadronization mass and µ the QCD mass scale (in GeV). N c = 3 is thenumber of colours and N f = 4 is the number of flavors. P ( n ) is the probabilitydistribution of n gluons and m quarks in the QCD evolution. In this model, thenumber of quarks is assumed to be fixed. Parameters A and ˜ A denote, respectively,uly 16, 2019 2:14 WSPC/INSTRUCTION FILE ws-epem-mpla H.W. Ang et. al. the average probabilities of the g → gg and q → qg processes (in the version of Eq.(8) used here the contribution of g → q ¯ q process has been neglected). The initialnumber of gluons, k ′ , determines (in the average sense) the initial condition of thegenerating function, which is G ( t = 0 , z ) = z k ′ . The parameter k = m ˜ A/A is related(in the average sense) to the initial number of quarks. The Local Parton HadronDuality
12, 13 was invoked to connect the parton-level results to the experimentaldata, i.e. the hadron spectra were required to be proportional to the correspondingparton spectra. The entire hadronization process is then parameterized by a singleparameter, which determines the overall normalization of the distribution but doesnot affect moments of order one and above.The generating function of the GMD is given by G ( z ) = z k ′ [ z + (1 − z ) κ ] − ( k + k ′ ) , (10)where κ = e At , (11)and the corresponding mean multiplicity is h N i = k ( κ −
1) + κk ′ . (12)This can be derived by noting that generating function (10) can be also calculateddirectly using P ( N ) from Eq. (6), in which case we obtain G ( z ) = ∞ X N = k ′ z N P ( N ) = z k ′ (cid:18) − p − pz (cid:19) k + k ′ . (13)Comparing Eq. (13) with Eq. (10) one gets that p = 1 − κ = 1 − e − At . (14)Using Eq. (7) for p one gets h N i in the form of Eq. (12) b .Note that the distribution P ( N ) described by Eq. (6) is defined for N ≥ k ′ .Hence, both the normalization of the GMD distribution as well as the generatingfunction G ( z ), Eq. (10), are also correspondingly defined for such a range of N . Thisconstraint will not affect the values of C j calculated from Eq. (2) if P (0) >
0. In thisinstance, only the ratio P ( N ) /P (0) is required while the normalization cancels out.However, the C j values calculated from Eq. (5) using the form of the generatingfunction given by Eq. (10) diverge due to the constraint on the range of values N can take. b For t = 0 or equivalently κ = 1 we have h N i = k ′ and h N i increases with Q as h N i ∼ (cid:16) ln Q µ (cid:17) A/ ( N c − N f ). uly 16, 2019 2:14 WSPC/INSTRUCTION FILE ws-epem-mpla Modifed combinant analysis of the e + e − multiplicity distributions
3. Normalization of GMD
For integer values of k ′ , the GMD distribution (6) can be interpreted as an NBD”shifted” by k ′ , where P NBD ( N, k ) denotes the NBD: P GMD ( N, k, k ′ ) = P NBD ( N − k ′ , k + k ′ ) , (15)The normalized form of the GMD as presented in Eq. (6) has a requirement that N ≥ k ′ . However, if we were to extend the domain such that N ∈ [0 , ∞ ), then thenormalized modified GMD becomes P ′ GMD ( N ) = Γ( N + k )Γ( N − k ′ + 1) · Γ (1 − k ′ )Γ( k ) · p N F (1 , k, − k ′ ; p ) , (16)where F ( a, b, c ; z ) is a hypergeometric function. The generating function is thengiven by G ( z ) = ∞ X N =0 z N P ′ GMD ( N ) = F (1 , k, − k ′ ; pz ) F (1 , k, − k ′ ; p ) , (17)It turns out that when we calculate the modified combinants C j using themethod of Eqn (5) with Eqn (17), the resulting C j ’s do not diverge. Differenti-ating the logarithm of Eqn (17) does not result in a factor of z in the denominator,but rather a polynomial in z . The resulting C j does not diverge due to the presenceof the constant term in the polynomial. Nonetheless, for k ′ > c .We have seen the limitations in a model involving the GMD used in isolationfor the purpose of modified combinant analysis. They manifest either in the formof a restricted domain of N (cf. Eq. (6), or in k ′ (cf. Eq. (17)). In the next section,we will propose a modification to the GMD to ensure its continued employment inmodified combinant analysis, while at the same time circumventing the constraintspresented above.
4. Imprints of acceptance
We shall now propose a modification of the initial P ( N ) that will allow for N 85, 2 . 35, 1 . . 91 for | y | < . 0, 1 . 5, 1 . . 5, respectively. Using general Leibniz rule we have that P ( N ) = 1 N ! d N H ( z ) dz N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 = 1 N ! N X i = max { ,N − k ′ } (cid:18) Ni (cid:19) d N − i f ( z ) dz N − i d i g ( z ) dz i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 . (23)Modified combinants h N i C j calculated using the generating function (20) are givenby the sum of the respective modified combinants for the BD and the NBD: h N i C j = 1 j ! d j +1 ln f ( z ) dz j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 + 1 j ! d j +1 ln g ( z ) dz j +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 . (24)We can expect therefore oscillations with period equal to 2, which are superimposedon the monotonically decreasing values: h N i C j = ( − j k ′ (cid:18) α − α (cid:19) j +1 ++ ( k + k ′ ) (cid:20) α ( κ − α ( κ − (cid:21) j +1 . (25)Fig. 2 shows this such approach works very well (however, looking on the exper-imental C j , we can suspect that C j are increasing for small j , this effect has itssource in the second term of Eq. (24)). 5. Scenario of two sources Actually, there is yet another way of treating the e + e − data. The generating function(20) can be formally treated as a generating function of the multiplicity distributionuly 16, 2019 2:14 WSPC/INSTRUCTION FILE ws-epem-mpla H.W. Ang et. al. P ( N ) N e + + e − , √ s = 91 GeVDELPHI, | y | < h N i · C j / . j (cid:1) je + + e − , √ s = 91 GeVDELPHI, | y | < Fig. 4. Left panel: Data on P ( N ) measured in e + e − collisions by DELPHI experiment at 91GeV are fitted by the GMD distribution (6) with parameters: h N i = 13 . k = 4 . k ′ =0 . 29. Right panel: the modified combinants C j deduced from these data on P ( N ) are displayed.Note that in this case they are too spread to be of any further use. This is clear when comparingthem with the respective C j obtained from the same GMD model with the same parameters asused for fitting of P ( N ). The reason is too low statistics in DELPHI experiment (cf. text fordetails). P ( N ) in which N consists of the sum of particles produced from the BD ( N BD )and the NBD ( N NBD ): N = N BD + N NBD . (26)In this case Eq. (23) can be written as P ( N ) = min { N,k ′ } X i =0 P BD ( i ) P NBD ( N − i ) (27)and its associated modified combinants Eq. (25) can be written as h N i C j = h N BD i C ( BD ) j + h N NBD i C ( NBD ) j . (28)The fits shown in Fig. 2 correspond to parameters: k ′ = 1 and p ′ = 0 . k = 4 . p = 0 . 75 for the NBD. It is clear that the implementation ofthe process in Eq. (26) supports our previous interpretation of GMD being a shiftedNBD. 6. Discussion of Results Section 3 highlights the difficulties in deriving the modified combinants of the GMDwhen used by itself. The proposed alternatives to circumvent them are then pre-sented in Section 4. The C j resulting from the original GMD diverges due to afactor of z k ′ in the denominator. This factor has its origins in the assumed initialcondition of k ′ = const . Had the initial condition taken the form of a binomial dis-tribution ? the resulting generating function will match that in Eqn (20) with noneof the restrictions encountered in Sect 3.uly 16, 2019 2:14 WSPC/INSTRUCTION FILE ws-epem-mpla Modifed combinant analysis of the e + e − multiplicity distributions Oscillatory behaviour of modified combinants is observed for all P ( N ) measuredby ALEPH in different rapidity windows. In Fig. 3 we showed the mean values ofthe amplitudes of oscillations (rescaled by factor ξ j ) for even and odd values of j , evaluated from the experimental multiplicity distributions P ( N ) measured invarious rapidity windows | y | < . 0, 1 . 5, 1 . . 5. The averages for both evenand odd j ’s are distinctly different from zero, supporting the fact the oscillationsdo not have their origins in random fluctuations (in which case both values shouldindependently oscillate around zero).It must be noted that had the DELPHI data been used, the analysis of thecorresponding C j ’s would not have been possible. This is illustrated in Fig. 4 wherethe experimentally obtained modified combinants C j are too scattered to discern anyunderlying oscillations due to the low statistics of the DELPHI data (cf footnote a).The measured P (0) from DELPHI is also about twice as large as that from ALEPH.This explains the much smaller oscillation amplitude seen in Fig 4 (cf. Eqn (2)). Forcomparison, the data from ALEPH was obtained over a total of 3 · data events,5 times that of the DELPHI data set. The difference in the unfolding procedureapplied to measured data between ALEPH and DELPHI contributes partly to themore scattered C j data in Fig 4. Nevertheless, the derived C j from both ALEPHand DELPHI exhibits oscillations of period 2, which appears characteristic of theexperimental data used in this paper.Different period of oscillations in C j results from different reaction types. Inthe e + e − annihilation data presented, the derived C j oscillates with period of 2.On the other hand, pp collisions give C j oscillating with a longer period on theorder of 10. The period of C j oscillation has been previously studied 1, 4 under theframework of compound distributions. The oscillatory behaviour is controlled bythe BD component while the NBD component controls the amplitude and period.When interpreted in the framework of cluster production, the number of clustersfollow the BD while the number of particles per cluster obeys the NBD. The periodof C j oscillation has been shown to be related to the mean multiplicity of the NBD. 7. Concluding remarks In our paper, we have demonstrated that modified combinants are valuable addi-tional tools for the examination between different models, in conjunction with themore traditional approach of analysis of multiplicity distribution. A careful analysisof the modified combinants C j deduced from the experimentally measured P ( N )can provide additional information on the dynamics of particle production, therebyallowing us to reduce the number of potential explanations presented amongst thevarious models. In this context, C j deduced from experimental measurements of P ( N ) becomes an important addition to the toolkit in the field of particle physicsphenomenology.A detailed discussion of the sensitivity of modified combinants C j to statistics ofevents and the associated uncertainties of measurements is given in Ref. 4 (see alsouly 16, 2019 2:14 WSPC/INSTRUCTION FILE ws-epem-mpla H.W. Ang et. al. Ref. ( )). Notwithstanding the sensitivity of oscillations of modified combinantsto systematic uncertainties in the measurements of P ( N ), the oscillatory signalobserved in the modified conbinants derived from ALEPH data is shown to bestatistically significant. The oscillations are not artefacts of random fluctuations asseen in Fig. 3. These observations justify the study of oscillations in C j to discernfiner details in experimentally measured multiplicity distributions.There has been a large number of papers suggesting universality in the mecha-nisms of hadron production in e + e − annihilation, pp and p ¯ p collisions. Such univer-sality arises from observations of the average multiplicities and relative dispersionsin the different types of processes (cf. for example, Sect. 19 of Ref. and ref-erences therein). A detailed analysis of the modified combinants derived from theexperimental P ( n )’s reveals differences between the various processes. In e + e − an-nihilation, the C j ’s oscillate with a period of 2 with amplitudes increasing as apower-law. On the other hand, pp and p ¯ p collisions produce C j ’s oscillating withapproximately 10 times the period of their e + e − counterparts, with decaying am-plitudes. 1, 4 Further analysis in this aspect will be welcome.In what concerns the e + e − results discussed here, the most plausible interpreta-tion lies with the GMD approach (with some modifications discussed above). TheGMD contains in its structure a BD, both in the original version resulting in Fig. 1and in modified version resulting in Fig. 2. In such cases, any distributions will showoscillatory behaviour in the derived C j ’s when compounded with the BD. However,this problem seems at the moment still open and subject to future investigations. Acknowledgments This research was supported in part by the National Science Center (NCN) un-der contracts 2016/23/B/ST2/00692 (MR) and 2016/22/M/ST2/00176 (GW).M.Ghaffar would like to thank NUS for the hospitality where part of this workwas done. H.W. Ang would like to thank the NUS Research Scholarship for sup-porting this study. We would also like to thank Q. Leong for his critical discussionand Dr Enrico Sessolo for reading the manuscript. References 1. G. Wilk and Z. W lodarczyk, J. Phys. G , 015022 (2017).2. G. Wilk and Z. Wlodarczyk, Int. J. Mod. Phys. A , 1830008 (2018).3. M. Rybczy´nski, G. Wilk and Z. W lodarczyk, EPJ Web Conf. , 03002 (2019).4. M. Rybczy´nski, G. Wilk and Z. W lodarczyk, Phys. Rev. D , 094045 (2019).5. B. Saleh and M. Teich, Proc. IEEE , 229 (1982).6. ALEPH Collaboration, Z. Phys. C , 15 (1995).7. S. K. Kauffmann and M. Gyulassy, Phys. Rev. Lett. , 298 (1978).8. C. K. Chew, D. Kiang and H. Zhou, Phys. Lett. B , 411 (1987).9. A. H. Chan and C. K. Chew, Phys. 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