Cumulants of multiple conserved charges and global conservation laws

Published for SISSA by Springer Received: July 17, 2020 Accepted: September 7, 2020 Published: October 14, 2020 Volodymyr Vovchenko,a Roman V. Poberezhnyukb,c and Volker Kocha a Nuclear Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, U.S.A. b Bogolyubov Institute for Theoretical Physics, Metrolohichna St. 14-b, 03143 Kyiv, Ukraine c Frankfurt Institute for Advanced Studies, Giersch Science Center, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany E-mail: , , Abstract: We analyze the behavior of cumulants of conserved charges in a subvolume of a thermal system with exact global conservation laws by extending a recently developed subensemble acceptance method (SAM) [1] to multiple conserved charges. Explicit expressions for all diagonal and off-diagonal cumulants up to sixth order that relate them to the grand canonical susceptibilities are obtained. The derivation is presented for an arbitrary equation of state with an arbitrary number of different conserved charges. The global conservation effects cancel out in any ratio of two second order cumulants, in any ratio of two third order cumulants, as well as in a ratio of strongly intensive measures Σ and ∆ involving any two conserved charges, making all these quantities particularly suitable for theory-to-experiment comparisons in heavy-ion collisions. We also show that the same cancellation occurs in correlators of a conserved charge, like the electric charge, with any non-conserved quantity such as net proton or net kaon number. The main results of the SAM are illustrated in the framework of the hadron resonance gas model. We also elucidate how net-proton and net-Λ fluctuations are affected by conservation of electric charge and strangeness in addition to baryon number. Keywords: Heavy Ion Phenomenology, QCD Phenomenology ArXiv ePrint: 2007.03850 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP10(2020)089 JHEP10(2020)089 Cumulants of multiple conserved charges and global conservation laws Contents 1 2 Formalism 2.1 Notation 2.2 Subensemble acceptance method 2.3 Second order cumulants 2.4 Third order cumulants 2.5 Results up to sixth order 2.6 Conserved charges in QCD 2.6.1 Single conserved charge B 2.6.2 Two conserved charges B and Q 2.6.3 Three conserved charges B, Q, and S 2.7 Strongly intensive quantities 2.8 Non-conserved quantities 2.8.1 Off-diagonal cumulants involving a single conserved charge 2.8.2 Variance of a non-conserved quantity 3 3 4 6 7 8 10 10 10 12 12 14 15 17 3 Application to the hadron resonance gas model 3.1 HRG model setup 3.2 Second order cumulants of conserved charges 3.3 Third order cumulants of conserved charges 3.4 Fourth order cumulants of conserved charges 3.5 Off-diagonal cumulants involving non-conserved quantities 3.6 Net-proton and net-Λ fluctuations 18 18 19 19 21 22 25 4 Discussion and conclusions 29 A Evaluation of the higher-order cumulants 32 B Deriving QCD cumulants from the general expressions 33 1 Introduction Fluctuations and correlations of conserved charges in statistical systems carry rich information on intrinsic properties of matter. These quantities play a central role in studies of the QCD phase diagram, both in first-principle lattice QCD simulations [2, 3] and in heavy-ion collision experiments [4]. Event-by-event fluctuations of different quantities are used in the search of the QCD critical point [5–7]. Various correlators of conserved charges, on the other hand, carry information on the relevant QCD degrees of freedom, such as the baryon-strangeness correlator [8]. –1– JHEP10(2020)089 1 Introduction –2– JHEP10(2020)089 Fluctuations and correlations of many different quantities, that include both the conserved charges and various hadron number distributions, have been measured in a number of experiments. These include measurements of second order cumulants, both diagonal [9– 12] and off-diagonal [13–15], as well as higher-order fluctuation measures [16–19]. An important question is how to relate the experimental measurements to theoretical predictions. For instance, cumulants of the net-proton number cannot be computed in many of the theories, lattice gauge theory in particular, where only the conserved baryon number is accessible. In such a case one either has to reconstruct net-baryon fluctuations from netproton measurements [20, 21], or directly compare net-proton and net-baryon cumulants, accepting an inevitable systematic error stemming from such an approximation. Another problem is participant (or volume) fluctuations, which is a source of non-dynamical fluctuations affecting comparisons between theory and experiment [22, 23]. Perhaps the most important issue is the choice of statistical ensemble. The vast majority of theories operate in the grand canonical ensemble, where the system can freely exchange conserved charges with a reservoir. Direct comparison of grand canonical susceptibilities with heavy-ion data is commonplace in the literature [24–33]. However, all charges are globally conserved in heavy-ion collisions. This would imply that the canonical ensemble is more appropriate than the grand canonical ensemble. The difference between ensembles does not play a major role if only mean hadron yields are considered in central collisions of heavy ions — due to the thermodynamic equivalence of statistical ensembles for the averages, the difference between hadron abundances evaluated in different statistical ensembles disappears in large systems. However, the thermodynamic equivalence of statistical ensembles does not extend to fluctuations, meaning that values of second and higher order cumulants will depend on the choice of the ensemble, no matter how large the system is. The experimental measurements typically have a limited momentum acceptance, covering only a fraction of the total momentum space. In ref. [34] the necessary conditions to emulate the grand canonical ensemble in heavy-ion collisions have been outlined: measurements should be performed in a rapidity acceptance ∆Yacc which is, on one hand, large enough to capture all the relevant physics, ∆Yacc  ∆Ycor , where ∆Ycor characterizes the correlation range in rapidity, while on the other hand, it covers only a small fraction of the whole momentum space such that global conservation laws can be neglected, ∆Yacc  ∆Y4π . Furthermore, the measurements should cover the entire transverse momentum range. Global conservation effects are non-negligible whenever ∆Yacc is comparable to ∆Y4π . The magnitude of these effects, as well as ways to deal with them, have been studied in the past using a picture of an uncorrelated hadron gas with a single globally conserved charge in a number of papers [35–42]. The analysis in ref. [37] indicated that the effects of global conservation are sizable already for moderate values of the acceptance fraction α ≡ ∆Yacc /∆Y4π . 0.2, especially for higher-orde (...truncated)