Cumulants of net-charge distribution from particle-antiparticle sources
CCumulants of net-charge distribution fromparticle-antiparticle sources ∗ Igor Altsybeev
Saint-Petersburg State UniversityUniversitetskaya nab. 7/9, St. Petersburg, 199034, Russia [email protected]
It is shown how high-order cumulants of net-charge distribution inhadronic collisions at LHC energies can be expressed via lower-order termsunder the assumption that particle-antiparticle pairs are produced in inde-pendent local processes. It is argued and tested with HIJING model thatthis assumption is typically valid for net-proton fluctuations in case whenno critical behaviour is present in the system. Values estimated in such away can be considered as baselines for direct measurements of high-ordernet-charge fluctuations in real data.
1. Introduction
In heavy-ion collision experiments, measurements of high-order fluctua-tions of conserved quantities, such as net-charge, net-baryon, net-strangeness,are of great importance since they should increase in the vicinity of the crit-ical point of the QCD diagram [1] and may serve as a signature of thetransition between the hadronic and partonic phases. These expectationsare confirmed also by lattice QCD calculations [2]. At LHC energies, asmooth crossover between a hadron gas and the QGP is expected [2, 3].Studies of the net-particle cumulant ratios are of a special interest becauseof their direct connection to susceptibilities theoretically calculable in thelattice QCD. In particular, net-proton fluctuations have been extensivelystudied experimentally [4, 5].The net-charge is defined as ∆ N = N + − N − , where N + and N − arethe numbers of positively and negatively charged particles measured in anevent within rapidity acceptance Y (i.e. y ∈ − Y / , Y / κ (∆ N ) = (cid:10)(cid:0) ∆ N − (cid:104) ∆ N (cid:105) (cid:1) (cid:11) = (cid:104) (∆ N ) (cid:105) − (cid:104) ∆ N (cid:105) , (1) ∗ Presented at Excited QCD 2020. (1) a r X i v : . [ nu c l - t h ] S e p Cumulants˙from˙sources˙Altsybeev printed on September 15, 2020 where angular brackets denote averaging over events. Expressions for higher-order cumulants are increasingly more complicated.If fluctuations of both N + and N − are Poissonian, ∆ N has the Skellamdistribution, with cumulants κ r (∆ N ) = (cid:104) N + (cid:105) + ( − r (cid:104) N − (cid:105) , r = 1 , , ... ,such that, for instance, ratio κ /κ = 1. The Poissonian particle productionis usually considered as a baseline model. However, in reality the cumulantsof particle distributions are very sensitive to two so-called non-dynamical contributions that are not related to criticality in the system. The firstcontribution comes from the fluctuations of a number of emitting sources– the so-called “volume fluctuations” (VF) [6, 7]. The second contributionis due to charge conservation laws, for example, from neutral resonancesdecaying into pairs of oppositely charged particles. These two effects makeinterpretation of experimental measurements of the cumulants highly non-trivial, especially for higher-order cumulants. Both of them should be takeninto account when one tries to extract signals of critical behaviour frommeasured observables [6, 8, 9]. At LHC energies, however, it is possibleto construct a simple baseline model that include both these effects, if oneassumes production of oppositely charged pairs that are nearly uncorrelatedin rapidity. This is demonstrated in Section 2, and tested with HIJING eventgenerator in Section 3 for the case of net-proton fluctuations. More detailscan be found in [10].Yet another caveat about ordinary cumulants κ r is that they get trivialcontributions to all orders due to self-correlations. It was shown in [11, 12]that self-correlations can be removed systematically by constructing facto-rial cumulants. This is briefly considered in Section 4.
2. Decomposition of cumulants for two-particle sources
Creation of oppositely charged particle pairs is governed by a local chargeconservation. The simplest case of a pair production process is a two-bodyneutral resonance decay, where integer +1 and − p and p pair. Such p - p pairs are produced mainly in stringbreaking (each of them may be produced directly or via a decay of a short-lived resonance). Moreover, a probability of production of two or more umulants˙from˙sources˙Altsybeev printed on September 15, 2020umulants˙from˙sources˙Altsybeev printed on September 15, 2020
Creation of oppositely charged particle pairs is governed by a local chargeconservation. The simplest case of a pair production process is a two-bodyneutral resonance decay, where integer +1 and − p and p pair. Such p - p pairs are produced mainly in stringbreaking (each of them may be produced directly or via a decay of a short-lived resonance). Moreover, a probability of production of two or more umulants˙from˙sources˙Altsybeev printed on September 15, 2020umulants˙from˙sources˙Altsybeev printed on September 15, 2020 baryon pairs from adjacent parts of the same string is low. Registrationof p - p pairs from jets in a low transverse momentum ( p T ) range (typicallyone takes p T (cid:46) c ) should be low as well. Therefore, if there areno processes other than resonance decays and string fragmentation, the p - p pairs visible in an event may be considered as nearly independent. Thisallows one to write simplifying expressions for the cumulants of net-chargefluctuations as it is described below.Decompositions of cumulants for a system of N S independent sourcesup to the fourth order are provided in [6] and up to the eighth order –in [10]. At LHC energies, where (cid:104) N + (cid:105) = (cid:104) N − (cid:105) , the second and the fourthcumulants of net-charge distribution decompose as κ (∆ N ) = k (∆ n ) (cid:104) N S (cid:105) , (2) κ (∆ N ) = k (∆ n ) (cid:104) N S (cid:105) + 3 k (∆ n ) K ( N S ) , (3)where ∆ n = n + − n − is a net-charge of a single source, and different no-tations for cumulants κ , k and K serve only for better visual distinctionwhich distribution they are referred to. The ratio of the fourth to the sec-ond cumulant reads as κ κ (∆ N ) = k k (∆ n ) + 3 k (∆ n ) K ( N S ) (cid:104) N S (cid:105) . (4)The VF enter this equation via the second term that is proportional to thevariance of the number of sources. The formulae above are valid for anytypes of sources, in particular, in [6] “wounded nucleons” are considered.Instead, we may treat the sources as particle-antiparticle pairs , for instance, p - p . Note that these sources may be correlated to a certain extent (forexample, due to radial and azimuthal flow) provided that swapping of thecharges in each produced pair does not affect the physics of the whole event.The fourth cumulant for a single two-particle source simplifies to k (∆ n ) = k (∆ n ) − k (∆ n ) . (5)We may now recall the argument that p - p pairs are nearly uncorrelated inrapidity and the fact that the distribution of p ( p ) is nearly flat at mid-rapidity | y | (cid:46) κ κ (∆ N ) = 1 + 3 κ (∆ N ) R ( N + ) , (6) Cumulants˙from˙sources˙Altsybeev printed on September 15, 2020 where R ( N + ) = (cid:104) N + ( N + − (cid:105) / (cid:104) N + (cid:105) − Y (equivalently, R ( N − ) could be used instead). Values of the cumulant ratio calculatedwith (6) could be considered as baselines for experimental measurements ofthe ratios (instead of, for instance, the Skellam baseline). Possible signalsfrom critical phenomena would be indicated by some deviations from thesebaselines.
3. Application to realistic model
Validity of the assumptions about charged pair production done abovewas put into test using HIJING monte-carlo generator, which simulatesmultiple jet production and fragmentation of quark-gluon strings [13]. Forthat, analysis of net-proton fluctuations in Pb-Pb collisions simulated inHIJING was performed [10]. Protons and antiprotons within y ∈ ( − ,
2) andtransverse momentum range 0.6–2 GeV/ c were selected. Figure 1 (a) showsthe dependence on rapidity acceptance Y of the κ /κ ratios calculateddirectly (circles) and by expression (6) (lines) in several centrality classes.Centrality was determined using multiplicity distribution in two symmetric3 < | η | < p - p pairs as nearlyindependent sources is approximately valid in HIJING. Slopes of the lines Y N ) D ( k / k HIJING Pb-Pb circles - direct calc. + V.F. k lines - calc. based on p T ∈ c net-proton centrality: Y N ) D ( k / k HIJING Pb-Pb circles - direct calc. + V.F. k lines - calc. based on p T ∈ c net-proton centrality: Fig. 1. Dependence of the net-proton κ /κ ratio on the size of the rapidity accep-tance Y in HIJING in Pb-Pb events at √ s NN = 2 .
76 TeV [10]. Direct calculationsare shown by circles, analytical calculations with (6) – by dashed lines. Panel (a) –results in several centrality classes of the class width 10% are shown, (b) – depen-dence on the width of centrality class (20%, 10% and 5%) is demonstrated. Notethat in each graph there are point-by-point correlations as Y increases. umulants˙from˙sources˙Altsybeev printed on September 15, 2020umulants˙from˙sources˙Altsybeev printed on September 15, 2020
76 TeV [10]. Direct calculationsare shown by circles, analytical calculations with (6) – by dashed lines. Panel (a) –results in several centrality classes of the class width 10% are shown, (b) – depen-dence on the width of centrality class (20%, 10% and 5%) is demonstrated. Notethat in each graph there are point-by-point correlations as Y increases. umulants˙from˙sources˙Altsybeev printed on September 15, 2020umulants˙from˙sources˙Altsybeev printed on September 15, 2020 for different centrality classes reflect changes in VF via the second term in(6). Panel (b) demonstrates a decrease of κ /κ values with the width of acentrality class (when the width changes from 20% down to 5%), which isexplained by a reduction of the volume fluctuations with the narrowing ofthe class. It was checked also that the robust variance R ( N + ) as a functionof Y stays constant, which is essential for calculations with (6). More detailsof this study, in particular, a decomposition expression for the κ /κ (∆ N )ratio can be found in [10].
4. Factorial cumulants
The fourth-order factorial cumulant of net-charge distribution is given by f = κ − (cid:0) (cid:104) N Q (cid:105) − (cid:104) N (cid:105)(cid:104) Q (cid:105) − (cid:104) N Q (cid:105)(cid:104) Q (cid:105) + 2 (cid:104) N (cid:105)(cid:104) Q (cid:105) (cid:1) + 8 (cid:0) (cid:104) Q (cid:105) − (cid:104) Q (cid:105) (cid:1) + 3 (cid:0) (cid:104) N (cid:105) − (cid:104) N (cid:105) (cid:1) − (cid:104) N (cid:105) , (7)where N = N + + N − and ∆ N is denoted as Q for clarity [12]. It is in-teresting to check the behaviour of this observable in realistic models. Asan example, Pb-Pb collisions from HIJING were analyzed in the presentwork. Values of the factorial cumulant f of net-proton distribution and theconventional κ /κ ratios are shown in Figure 2 as a function of the accep-tance width Y in two centrality classes 70-80% and 80-90% . The κ /κ Results are shown for two peripheral centrality classes only, since for more centralclasses statistical uncertainties for f are much larger. Y N ) D ( f , k / k , 70-80% k / k , 80-90% k / k , 70-80% f , 80-90% f HIJING Pb-Pb p T ∈ c net-proton Fig. 2. Dependence net-proton cumulant ratio κ /κ (closed markers) and factorialcumulant f (open markers) on Y in HIJING. Pb-Pb collisions at √ s NN = 2 . Cumulants˙from˙sources˙Altsybeev printed on September 15, 2020 ratios are the same as in Fig. 1, and the values are above unity (i.e. abovethe Skellam baseline) due to the VF, as it was discussed above. Moreover,values in class 70-80% are higher than in 80-90% since the VF in the formerclass are larger. In contrast, factorial cumulants f are compatible withzero for both centralities. This is because factorial cumulants of order k re-move contributions of lower orders r < k , which means, in particular, thatnet-proton f should be suppressed in HIJING. Factorial cumulants are alsomuch less sensitive to the VF than ordinary cumulants [12]. However, indistinction from ordinary cumulants, factorial cumulants cannot be directlycompared with the lattice data [11, 12], therefore their usefulness in studiesof the QCD diagram is under question.
5. Summary
It was shown that high-order cumulants of net-charge distribution canbe decomposed into lower-order terms under the assumption of independentproduction of particle-antiparticle pairs. At LHC energies, this should bea good approximation for net-proton fluctuations at mid-rapidity in case ifthere is no critical behaviour in the system, as it was demonstrated for the κ /κ ratio in HIJING. Such reduced expressions for high-order cumulantscan be considered as baselines for direct experimental measurements. Itwas shown also that the fourth net-proton factorial cumulant in HIJINGis compatible with zero, which also indicates that there are no sources ofgenuine high-order net-proton correlations in this generator. Acknowledgements
This work is supported by the Russian Science Foundation, grant 17-72-20045. REFERENCES [1] M.A. Stephanov, K. Rajagopal and E.V. Shuryak, Phys. Rev. D60 (1999)114028, hep-ph/9903292.[2] A. Bazavov et al., Phys. Rev. D85 (2012) 054503, 1111.1710.[3] S. Borsanyi et al., JHEP 10 (2018) 205, 1805.04445.[4] STAR, X. Luo, PoS CPOD2014 (2015) 019, 1503.02558.[5] ALICE, S. Acharya et al., (2019), 1910.14396.[6] P. Braun-Munzinger, A. Rustamov and J. Stachel, Nucl. Phys. A 960 (2017)114, 1612.00702.[7] T. Sugiura, T. Nonaka and S. Esumi, Phys. Rev. C100 (2019) 044904,1903.02314. umulants˙from˙sources˙Altsybeev printed on September 15, 2020umulants˙from˙sources˙Altsybeev printed on September 15, 2020