PHYSICAL REVIEW D 101, 074502 (2020)

Skewness, kurtosis, and the fifth and sixth order cumulants of net baryon-number distributions from lattice QCD confront high-statistics STAR data

A. Bazavov,1 D. Bollweg ,2 H.-T. Ding ,3 P. Enns,2 J. Goswami ,2 P. Hegde,4 O. Kaczmarek,3,2 F. Karsch,2 R. Larsen,5 Swagato Mukherjee ,5 H. Ohno ,6 P. Petreczky,5 C. Schmidt ,2 S. Sharma,7 and P. Steinbrecher5

(HotQCD Collaboration)

1Department of Computational Mathematics, Science and Engineering and Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA 2Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany 3Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China 4Center for High Energy Physics, Indian Institute of Science, Bangalore 560012, India 5Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA 6Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan 7Department of Theoretical Physics, The Institute of Mathematical Sciences, Chennai 600113, India

(Received 27 January 2020; accepted 16 March 2020; published 3 April 2020)

We present new results on up to sixth-order cumulants of net baryon-number fluctuations at small values of the baryon chemical potential, μB, obtained in lattice QCD calculations with physical values of light and strange quark masses. Representing the Taylor expansions of higher-order cumulants in terms of the ratio of σ2 ¼ χBð μ Þ χBð μ Þ the two lowest-order cumulants, MB= B 1 T; B = 2 T; B , allows for a parameter-free comparison with data on net proton-number cumulants obtained by the STAR Collaboration in the Beam Energy Scan at RHIC. We show that recent high-statisticspffiffiffiffiffiffiffiffi data on skewness and kurtosis ratios of net proton-number distributions, obtained at a beam energy sNN ¼ 54.4 GeV, agree well with lattice QCD results on cumulants of net baryon-number fluctuations close to the pseudocritical temperature, TpcðμBÞ, for the chiral transition in QCD. We also present first results from a next-to-leading-order expansion of fifth- and sixth-order cumulants on the line of the pseudocritical temperatures.

DOI: 10.1103/PhysRevD.101.074502

I. INTRODUCTION temperature, TpcðμBÞ [4–7]. At larger values of the baryon chemical potential it, however, is generally expected that a The phase diagram of strong-interaction matter at non- first-order phase transition line exists, which ends in a zero temperature and nonzero baryon-number density second-order critical point [8,9]. This elusive critical is being explored intensively through numerical calcula- point is searched for in the Beam Energy Scan (BES) tions performed in the framework of lattice-regularized performed at the Relativistic Heavy Ion collider (RHIC) at quantum chromodynamics (QCD) [1], as well as through Brookhaven National Laboratory [10]. However, its exist- ultrarelativistic heavy-ion collisions with varying beam ence as a fundamental property of the theory of strong energies [2]. At vanishing and small values of the chemical interactions (QCD) still awaits confirmation. potentials for conserved charges [baryon number (μB), The pseudocritical line, TpcðμBÞ, which distinguishes the electric charge (μQ), strangeness (μS)] it is well established that the transition from the low-temperature hadronic low- and high-temperature regimes of strong-interaction region to the quark-gluon plasma at high temperature is matter as described by QCD, has been determined quite a smooth transition [3] characterized by a pseudocritical accurately in lattice QCD calculations for baryon chemical potentials up to about twice the pseudocritical temperature, μB ≲ 2Tpcð0Þ ≃ 300 MeV [4–7]. In our recent analysis we found [7] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license.     Further distribution of this work must maintain attribution to μ 2 ’ ðμ Þ¼ 0 1 − κB B þ Oðμ4 Þ ð Þ the author(s) and the published article s title, journal citation, Tpc B Tpc 2 B ; 1 and DOI. Funded by SCOAP3. T

2470-0010=2020=101(7)=074502(13) 074502-1 Published by the American Physical Society A. BAZAVOV et al. PHYS. REV. D 101, 074502 (2020)

0 ¼ð156 5 1 5Þ κB ¼ 0 012ð4Þ σ2 ¼ χBð μ⃗Þ ¼ χBð μ⃗Þ with Tpc . . MeV and 2 . with a variance [ B 2 T; ], skewness [SB 3 T; = Oðμ4 Þ μ ¼ 0 χBð μ⃗Þ3=2 κ ¼ χBð μ⃗Þ χBð μ⃗Þ2 B correction that vanishes within errors. At B the 2 T; ] and kurtosis [ B 4 T; = 2 T; ]are pseudocritical temperature turns out to be in good agree- predicted in QCD equilibrium thermodynamics to be related. ment with the freeze-out temperature determined by the For μS ¼ μQ ¼ 0 one finds ALICE Collaboration at the LHC [11] and the pseudoc- T ðμ Þ 2 3 ritical line, pc B , is also consistent with freeze-out κ σ derivatives of the logarithm of the sNN . GeV [20] are encouraging. As will be dis- QCD partition functions with respect to the chemical cussed in Sec. IV, these data fulfill the above inequality and the difference of the cumulant ratios given in Eq. (3) agrees potentials of conserved charges, μ⃗¼ðμB; μQ; μSÞ, with lattice QCD results even on a quantitative level. 1 ∂n ln ZðT;μ⃗Þ We will present here new results on the density depend- χXð μ⃗Þ¼ ¼ ð Þ n T; 3 ;XB; Q; S; 2 ence of up to sixth-order cumulants of net baryon-number VT ∂μˆ n X fluctuations. We calculate Taylor series at nonzero values of with μˆ ≡ μ=T. These higher-order derivatives become the baryon-number, electric-charge and strangeness chemi- increasingly sensitive to long-range correlations and large cal potentials that involve up to eighth-order cumulants. We fluctuations in the vicinity of a critical point. At least from perform these expansions for the case of strangeness ¼ 0 the theoretical point of view higher-order cumulants thus neutral systems, nS , with a ratio of electric charge ¼ 0 4 are ideally suited to search for a possible critical point in the to baryon number, nQ=nB . , that is representative of – the conditions met in heavy-ion collisions. This allows to QCD phase diagram [14 16]. The BES at RHIC aims at 1 finding evidence for such a critical point through the construct Taylor expansions for nth-order cumulants, χBð μ Þ Oðμ8−nÞ analysis of e.g., higher-order cumulants of net proton- n T; B ,upto B . number fluctuations which are considered to be good For the case of the skewness and kurtosis ratios, σ3 κ σ2 proxies for cumulants of net baryon-number fluctuations. SB B=MB and B B, respectively, we thus can extend Results, obtained with BES-I at RHIC, indicate that earlier NLO calculations and perform next-to-next-to- qualitative changes in the behavior of netp proton-numberffiffiffiffiffiffiffiffi leading-order (NNLO) expansions that allow to better control fluctuations occur at beam energies sNN ∼ 20 GeV truncation effects in the Taylor series. We also present, for the [17,18]. This may hint at the existence of a critical point first time, results from NLO calculations for the hyper- for large values of the baryon chemical potential. skewness and hyper-kurtosis (fifth- and sixth-order cumu- B B B B While the finding of nonmonotonic behavior of higher- lants) ratios χ5 ðT;μBÞ=χ1 ðT;μBÞ and χ6 ðT;μBÞ=χ2 ðT;μBÞ. order cumulants of net proton-number fluctuations generated pWeffiffiffiffiffiffiffiffi show that these ratios are expected to be negative at well-justified excitement [17,18], we still need to establish sNN ¼ 54.4 GeV, in contrast to the preliminary findings that this behavior is caused by thermal fluctuations in the for sixth-order cumulants of net proton-number fluctuations vicinity of a critical point and that these higher-order reported by the STAR Collaboration [20]. cumulants indeed probe thermal conditions at the time of freeze-out. As pointed out in Ref. [19] at least for small values 1 B Rather than specifying in the argument of χn all three of the baryon chemical potential the first four cumulants of chemical potentials, μ⃗, we give in the strangeness neutral case B net baryon-number fluctuations, i.e., mean [MB ≡ χ1 ðT;μ⃗Þ], only the baryon chemical potential.

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This paper is organized as follows. In the next section we TABLE I. Number of gauge field configurations on lattices of 3 3 briefly present our calculational setup, the new statistics size 32 × 8 and 48 × 12 used in the analysis of up to eighth- collected on lattices of size 323 × 8 and 483 × 12 and the order Taylor expansion coefficients. The values of the gauge coupling as well as the strange and light quark mass parameter at general fitting ansatz used for fits at fixed values of Nτ ¼ 8 these temperature values are taken from Ref. [24], where details and 12, joint fits of these data as well as continuum limit 3 on the statistics available on the 24 × 6 lattices were also given. estimates. In Sec. III we present results for Taylor expan- All configurations are separated by 10 time units in rational sions of cumulants of net baryon-number fluctuations that hybrid Monte Carlo simulations [24]. use up to eighth-order cumulants. We compare these results with experimental data for cumulants of net proton-number Nτ ¼ 8 Nτ ¼ 12 fluctuations in Sec. IV. Section V contains our conclusions. T [MeV] No. of conf. T [MeV] No. of conf. Explicit expressions for the first four Taylor expansion coefficients of net baryon-number cumulants are given in 134.64 1 275 380 134.94 256 392 140.45 1 598 555 140.44 368 491 the Appendix. 144.95 1 559 003 144.97 344 010 151.00 1 286 603 151.10 308 680 II. CALCULATIONAL SETUP 156.78 1 602 684 157.13 299 029 162.25 1 437 436 161.94 214 671 Up to fourth-order cumulants of net baryon-number 165.98 1 186 523 165.91 156 111 fluctuations have been calculated previously [19,21,22] 171.02 373 644 170.77 144 633 in a next-to-leading-order Taylor expansion. In particular, 175.64 294 311 175.77 131 248 we performed calculations [19] with the highly improved staggered quark [23] discretization scheme for (2 þ 1)- flavor QCD with a physical strange quark mass and two cumulants, obtained from calculations within a noninter- degenerate, physical light quark masses. Here we extend acting hadron resonance gas (HRG) model that use these calculations by increasing the number of gauge field 3 resonances from the Particle Data Tables [25] (PDG- configurations generated on lattices of size 32 × 8 and HRG) as well as additional resonances calculated within 3 48 × 12 by a factor of 3–5 in the transition region and at the quark model [26,27] (QM-HRG) are given by lines. The least a factor of 2 at other values of the temperature. This latter list contains additional resonances not (yet) observed allows us to calculate up to eighth-order cumulants of net experimentally. baryon-number, net strangeness and net electric-charge B;k We determine the expansion coefficients, χ˜ n ðTÞ, for fluctuations, including also their correlations, at vanishing Taylor series of nth-order cumulants, values of the chemical potentials. These cumulants provide expansion coefficients in Taylor series for net baryon- Xkmax χBð μ⃗Þ χBð μ Þ¼ χ˜ B;kð Þμˆ k ð Þ number cumulants n T; . We calculate NLO expansions n T; B n T B; 4 for fifth- and sixth-order cumulants and obtain NNLO k¼0 results for third- and fourth-order cumulants. In the case of first- and second-order cumulants, i.e., the mean and for the case of vanishing net strangeness density, variance of net baryon-number distributions, we even nS ¼ 0, and an electric-charge to baryon-number ratio, obtain next-to-NNLO (NNNLO) results. The set of gauge nQ=nB ¼ 0.4. Explicit expressions for the NLO expansion field ensembles, which has been used in this analysis, and coefficients of up to sixth-order net baryon-number cumu- the number of gauge field configurations per ensemble on lants are given in Ref. [19]. The explicit form of the higher- lattices with temporal extent Nτ ¼ 8 and 12 are summa- order expansion coefficients are given in the Appendix. rized in Table I. Using the Taylor series for nth-order cumulants, Eq. (4), Results for up to eighth-order diagonal net baryon- we construct cumulant ratios with polynomials of order B B ½ number susceptibilities, χn ≡ χn ðT;0Þ, are given in kmax;lmax , χB Fig. 1. For the quadratic fluctuations, 2 , we also show P B kmax B;k k results for lattices with temporal extent Nτ ¼ 6, which were χ ðT;μ Þ χ˜ ðTÞμˆ B ¼ n B ¼ Pk¼1 n B ð Þ χB Rnm B : 5 already used in Ref. [7]. For the eighth-order cumulant, 8 , χ ðT;μ Þ lmax χ˜ B;lð Þμˆ l m B l¼1 m T B we only show our results for Nτ ¼ 8 as statistical errors on the Nτ ¼ 12 data are still too large. The bands shown in In order to control systematic effects arising from the χBð Þ these figures give a continuum extrapolation for 2 T truncation of the Taylor series expansion for the cumulant using data from calculations for three different lattice B ratios Rnm, we calculate these ratios using different orders spacings (aT ¼ 1=Nτ) and a continuum estimate for of the Taylor expansion for the cumulants appearing in the χBð Þ ¼ 8 χBð Þ 4 T based on Nτ and 12 data sets. For 6 T and numerator and denominator of these ratios. We analyzed χBð Þ ½ 8 T we only show spline interpolations of the data the polynomial ratios for different kmax;lmax as well as obtained on the 323 × 8 lattices. Results for these Taylor expansions of the ratios themselves. We find that the

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0.2 We fit cumulant ratios using a rational polynomial χB 2 ansatz, 0.15 P nmax a ðμˆ ÞT¯ n T fðT;μˆ Þ¼P n¼0 n B ; with T¯ ¼ ; ð6Þ B mmax ¯ m b ðμˆ ÞT T0 0.1 m¼0 m B Tpc=156.5(1.5) MeV cont. extr. where T0 is some arbitrary scale. When using this rational PDG-HRG 0.05 QM-HRG polynomial ansatz for fits at nonzero μB we allow for a Nτ = 6 μ 8 quadratic B dependence of all expansion coefficients, T [MeV] 12 2 anðμˆ BÞ¼an;0 þ an;2μˆ and similarly for bnðμˆ BÞ. When 0 B 130 140 150 160 170 180 performing joint fits of data on lattices with different sizes 2 0.12 and lattice spacings, a, we allow for Oða Þ cutoff correc- T =156.5(1.5) MeV χ4 pc tions that are parametrized in terms of the temporal lattice B cont. est. 0.1 QM-HRG extent Nτ ¼ 1=aT, e.g., PDG-HRG Nτ = 8 0.08 12 1 ð μˆ Þ¼ ð μˆ Þþ ð μˆ Þ ð Þ f T; B h T; B 2 g T; B ; 7 0.06 Nτ

0.04 with gðT;μˆ BÞ and hðT;μˆ BÞ being rational polynomials of the type given in Eq. (6). 0.02 T [MeV] 0 III. CUMULANTS OF NET 130 140 150 160 170 180 BARYON-NUMBER FLUCTUATIONS T =156.5(1.5) MeV 0.20 χ6 pc B Spline : Nτ = 8 A. Mean and variance of net QM-HRG baryon-number fluctuations PDG-HRG 0.10 Nτ = 8 ¼ 12 We have calculated the ratio of the mean, MB χBð μ Þ σ2 ¼ χBð μ Þ 1 T; B , and variance, B 2 T; B , of net baryon- 0 number fluctuations,

-0.10 M χBðT;μ Þ RB ðT;μ Þ ≡ B ¼ 1 B ; ð8Þ 12 B σ2 χBð μ Þ B 2 T; B -0.20 T [MeV]

130 140 150 160 170 180 for systems with vanishing net strangeness, nS ¼ 0, and a 2 net electric-charge to net baryon-number density nQ=nB ¼ χ8 Tpc=156.5(1.5) MeV B Spline : Nτ = 8 0.4 on lattices with temporal extent Nτ ¼ 8 and 12. Using 1 PDG-HRG QM-HRG up to eighth-order Taylor expansion coefficients, we can Nτ = 8 Oðμˆ 7 Þ Oðμˆ 6 Þ construct Taylor series up to order B and B for 0 B B χ1 ðT;μBÞ and χ2 ðT;μBÞ, respectively. Truncating these ¼ − 1 series at kmax and lmax kmax , respectively, we con- -1 ½ struct the kmax;lmax polynomial ratios which provide leading-order ([1, 0]), next-to-leading-order ([3, 2]) etc., -2 approximations for the ratio of the mean and variance of the T [MeV] B distribution for net baryon-number fluctuations, R12≡ -3 σ2 ½ 130 140 150 160 170 180 MB= B. Results for different kmax;lmax are shown in Fig. 2. The figure shows results obtained on lattices with FIG. 1. Cumulants of net baryon-number fluctuations from 2 temporal extent Nτ ¼ 8 and 12 at a temperature T ≃ μ ¼ 0 second to eighth order (top to bottom) evaluated at B on lat- 157 3 MeV which is close to the pseudocritical temperature tices of size Nσ × Nτ with Nσ ¼ 4Nτ. For further details see text. at μB ¼ 0. We find that cutoff effects are negligible for μB=T ≤ 1 former are more stable at large μB=T. In the following we and remain comparable to the statistical errors for the ½ will use the ratios of polynomials with kmax;lmax corre- sponding to identical orders (LO, NLO, NNLO, NNNLO) 2As is evident from Table I the temperatures differ slightly of expansions in the cumulants appearing in the numerator for the two lattice sizes: T ¼ 156.76 MeV for Nτ ¼ 8 and and denominator, respectively. T ¼ 157.13 MeV for Nτ ¼ 12, respectively.

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B. Skewness and kurtosis of net baryon-number fluctuations ¼ χBð μ Þ σ2 ¼ While the low-order cumulants MB 1 T; B , B B χ2 ðT;μBÞ and their ratio are in good agreement with HRG model calculations that use noninteracting, point-like hadrons at and below Tpc (see also discussion in Sec. IV), this clearly is not the case for higher-order cumulants. This is apparent in calculations of the skewness and kurtosis ratios,

S σ3 χBðT;μ Þ B ð μ Þ¼ B B ¼ 3 B ð Þ R31 T; B B ; 9 MB χ1 ðT;μBÞ B ≡ σ2 FIG. 2. Expansion of R12 MB= B at a fixed temperature ðμ Þ χBðT;μ Þ close to the pseudocritical line Tpc B versus the baryon B ð μ Þ¼κ σ2 ¼ 4 B ð Þ R42 T; B B B B ; 10 chemical potential. Shown are results from up to NNNLO χ2 ðT;μBÞ expansions on lattices of size 323 × 8 and 483 × 12. which both are unity in noninteracting HRG model calcu- Nτ ¼ 12 data at least up to μB=T ≃ 1.2. This holds true in lations, but are known to be significantly smaller in lattice the entire temperature range T ∈ ½135 MeV∶175 MeV QCD calculations already in the vicinity of the pseudocritical B analyzed by us. Differences in R12 constructed from temperature, Tpcð0Þ, at vanishing values of the baryon NNLO and NNNLO Taylor series of the cumulants are chemical potential. Moreover, in contrast to the cumulant μ ¼ 1 B B B about 2% for B=T . ratio R12, the ratios R31 and R42 show a much stronger B As the temperature dependence of R12 is weak in the temperature dependence and a milder dependence on μB.It B temperature range considered by us and also deviations of thus has been suggested that the ratio R12 is well suited to μ the B dependence from the leading order, linear behavior determine the baryon chemical potential from experimental B B are moderate we find that using [2, 3] rational polynomials data, while the ratios R31 and R42 constrain the temper- in both terms of the fit ansatz given in Eq. (7) are sufficient ature [21,28]. for obtaining good fits to the data. We performed fits Using our results for up to eighth-order cumulants of separately for the NNLO and NNNLO data sets at fixed conserved charge fluctuations and correlations, we can μ ≤ 1 2 values of T and B=T . . The resulting continuum construct NNLO expansions for the third- and fourth-order B B B estimates for R12, evaluated for several values of the cumulants χ3 ðT;μBÞ and χ4 ðT;μBÞ, where again the temperature in the vicinity of the pseudocritical temper- electric-charge and strangeness chemical potentials have ð0Þ ature, Tpc , are shown in Fig. 3. We note that the been fixed by demanding nS ¼ 0 and nQ=nB ¼ 0.4. With variation with temperature is small. As will be discussed this we determine up to NNLO results for the skewness and B μ ≲ 125 B B in Sec. IV the results obtained for R12 at B MeV are kurtosis cumulant ratios R31 and R42. in good agreement with HRG model calculations. For We again first use our high-statistics data obtained on the B;QCD B;HRG ¼ 8 larger values of μB we find, however, R12 >R12 , Nτ lattices to analyze the effect of truncations of the which reflects the large deviations of higher-order cumu- Taylor expansions at finite orders of μB. We used the fit lants, evaluated in QCD at μB ¼ 0, from the corresponding ansatz given in Eq. (6) and performed fits to LO, NLO and B B HRG values. NNLO results for the ratios R31 and R42 in the temperature range ½135 MeV∶175 MeV and for baryon chemical potentials μB ≤ 160 MeV. Results from these fits are shown in Fig. 4 for four values of the temperature in the vicinity of the pseudocritical temperature Tpcð0Þ. The two central T values, T ¼ 155 and 158 MeV, correspond to the lower and upper end of the error band for the pseudocritical temperature at μB ¼ 0. The lowest temperature, T ¼ 152 MeV reflects the lowest T value reached on the pseudocritical line TpcðμBÞ at μB=T ¼ 1. For clarity we show in Fig. 4 the LO results, which are μB independent, only for the lowest temperature. Of course, at all temper- B ature values the LO results coincide with the values of R31 B and R42 at μB ¼ 0. We also note that in the range of B FIG. 3. Continuum estimate for R12 based on NNNLO ex- chemical potentials, 0 ≤ μB=T ≤ 1, the pseudocritical tem- pansion results obtained on lattices of size 323 × 8 and 483 × 12. perature only varies slightly. The data shown in Fig. 4 thus

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B FIG. 5. The μB-dependent correction to R31 compared to one B third of the correction for R42. The inset shows a comparison of B B the second derivatives of R31 and R42=3 with respect to μB=T.

In Fig. 6 we show results for the skewness and kurtosis B B ratios R31ðT;μBÞ and R42ðT;μBÞ obtained at μB ¼ 0 on lattices with temporal extent Nτ ¼ 8 and 12. Obviously results for Nτ ¼ 12 are systematically below those for Nτ ¼ 8. This is in accordance with the observed shift of the pseudocritical temperatures [7] to smaller values with increasing Nτ or, equivalently, decreasing lattice spacing B ð μ Þ ≡ σ3 at fixed temperature aT ¼ 1=Nτ. When performing joint FIG. 4. The cumulant ratios R31 T; B SB B=MB (top) and B ð μ Þ ≡ κ σ2 μ fits to the Nτ ¼ 8 and 12 data, using the ansatz given in R42 T; B B B (bottom) versus B=T for four different values of the temperature calculated from LO, NLO and NNLO B Taylor expansions of the cumulants χn ðT;μBÞ on lattices with temporal extent Nτ ¼ 8. cover the entire parameter range of relevance for the calculation of these cumulant ratios on the pseudocritical line for μB=T ≲ 1. In Ref. [19] we showed that the skewness and kurtosis B B ratios R31 and R42 are almost identical at leading order, Oðμ0 Þ B B . The NLO correction to the kurtosis ratio R42, however, is about a factor of 3 larger than that for the B skewness ratio R31. Figure 4 suggests that these relations are still well respected by the NNLO results. The slope of B R42ðT;μBÞ as a function of μˆ B at fixed T is significantly B larger than that of R31ðT;μBÞ and, in fact, it is still consistent with being about a factor of 3 larger. This is shown in Fig. 5 where we compare the μB-dependent parts B B of R31 and R42=3. Also shown in this figure are the second B B derivatives of R31ðT;μBÞ and R42ðT;μBÞ=3 with respect to μB=T. B Compared to the lower-order ratio R12 higher-order B corrections in the Taylor expansion of R31 are significantly larger. In the temperature range shown in Fig. 4 corrections 5 to the NLO results, arising from the NNLO, Oðμ Þ, B B FIG. 6. Continuum estimates for the skewness ratio, R31 ≡ corrections in the Taylor expansions of the cumulants σ3 B ≡ κ σ2 μ ¼ SB B=MB (top), and kurtosis ratio R42 B B (bottom) at B χBð μ Þ μ ¼ 0 8 3 3 T; B , are about 5% at B=T . and rise to about 0 based on results obtained on lattices of size 32 × 8 and B 3 10% at μB=T ¼ 1. Consequently truncation effects in R42 48 × 12, respectively. The inset in the bottom figure shows the B B are about a factor of 3 larger. difference R42 − R31 at μB ¼ 0 as function of T.

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SHσ5 χBðT;μ Þ B ð μ Þ ≡ B B ¼ 5 B R51 T; B B ; MB χ1 ðT;μBÞ χBðT;μ Þ B ð μ Þ ≡ κHσ4 ¼ 6 B ð Þ R62 T; B B B B : 11 χ2 ðT;μBÞ Unlike the ratios for skewness and kurtosis cumulants, the corresponding ratios involving fifth- and sixth-order cumulants are negative already at μB ¼ 0 in a broad temperature interval in the vicinity of Tpcð0Þ and become smaller with increasing μB. This reflects the properties of the sixth- and eighth-order cumulants shown in Fig. 1. B B The μB dependence of the cumulant ratios R51 and R62 follows a pattern similar to that of the skewness and kurtosis ratios. In particular, at LO both ratios are almost identical B and the NLO correction to R62 is about a factor of 3 larger B than that for R51. Like in the case of the corresponding relations for the skewness and kurtosis ratios these relations simply result from the structure of Taylor expansions for odd and even cumulants [19]. The relations are exact for expansions at vanishing μQ and μS and apparently they are not much altered in the strangeness neutral case nS ¼ 0 with nQ=nB ¼ 0.4. A fit to the Nτ ¼ 8 lattice QCD results B B for the difference R62 − R51 at μB ¼ 0 yields 0.029(9). FIG. 7. Continuum estimates for the skewness (top) and While statistical errors are strongly correlated between kurtosis (bottom) ratios obtained from joint fits to data obtained the fifth- and sixth-order cumulants they are large for each on lattices with temporal extent Nτ ¼ 8 and 12. of these cumulants individually. For this reason we only present results for these cumulants obtained on lattices with Eq. (7), we find that within our current statistical errors on temporal extent Nτ ¼ 8 and evaluate the NLO corrections the Nτ ¼ 12 data we cannot resolve any T or μ =T B only for μ =T ≤ 0.8. NLO results for RB ðT;μ Þ and dependence of cutoff effects. It thus suffices to use a B 51 B B ð μ Þ constant ansatz for the cutoff corrections, i.e., we use R62 T; B are shown in Fig. 8. Obviously NLO corrections for these ratios are negative gðT;μ Þ¼a0 0 and a [3, 4] rational polynomial for the B ; and substantially larger than those in the skewness and continuum limit result fðT;μBÞ. A joint fit to the Nτ ¼ 8 B kurtosis ratios. In the vicinity of the pseudocritical temper- and 12 data yields a0;0 ¼ 3.2ð1.5Þ for R31ðT;μBÞ and B ature the difference between LO and NLO results at a0 0 ¼ 3.2ð3.0Þ for R ðT;μ Þ. The resulting continuum ; 42 B μ =T ¼ 0.8 is about an order of magnitude larger in limit estimates at μ ¼ 0 are also shown in Fig. 6. B B RB ðT;μ Þ than in RB ðT;μ Þ. This is also the case when The inset in Fig. 6 (bottom) shows the continuum 51 B 31 B comparing RB ðT;μ Þ with RB ðT;μ Þ. estimate for the difference RB − RB at μ ¼ 0 as a 62 B 42 B 42 31 B The magnitude and sign of the NLO corrections to fifth- function of T. At temperatures below T ≃ 150 MeV this and sixth-order cumulants in relation to corresponding difference is consistent with being zero. In the crossover results for the third- and fourth-order cumulants is evident region, T ð0Þ¼156.5ð1.5Þ MeV we find that the differ- pc from the structure of the corresponding Taylor expansion B ð Þ − B ð Þ¼0 008ð3Þ ence is slightly positive, R42 Tpc R31 Tpc . . coefficients. It is easy to see this in Taylor expansions B B Continuum estimates for R31ðT;μ Þ and R42ðT;μ Þ at B B performed at μQ ¼ μS ¼ 0. In this case one has, for two values of the temperature, corresponding to the current instance, μ ¼ 0 error band for the pseudocritical temperature at B are     shown in Fig. 7. χB μ 2 χB μ 4 χBðT;μ Þ¼χB þ 6 B þ 8 B þ; ð12Þ 4 B 4 2 T 24 T   C. Hyper-skewness and hyper-kurtosis χB μ 2 of net baryon-number fluctuations χBðT;μ Þ¼χB þ 8 B þ ð13Þ 6 B 6 2 T The fifth- and sixth-order cumulants are related to the corresponding fifth- and sixth-order standardized moments, As can be deduced from Fig. 1, despite the large errors on H H B B B i.e., the hyper-skewness, S , and hyper-kurtosis, κ .We current results for χ8 , the cumulants χ6 and χ8 are both consider here the cumulant ratios for fifth- and sixth-order negative in the vicinity of the pseudocritical temperature; cumulants of net baryon-number fluctuations, however the absolute value of the eighth-order cumulant is

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B FIG. 9. The cumulant ratio R12ðT;μBÞ evaluated on the pseudocritical line TpcðμBÞ for the case nS ¼ 0 and nQ=nB ¼ 0.4. Also shown is the corresponding result obtained in HRG model calculations. In the latter case the width of the line reflects differences resulting from using particle spectra for a noninter- acting HRG listed in the Particle Data Tables as well as resulting within quark model calculations.

calculated in QCD on the pseudocritical line given in Eq. (1). The pseudocritical line shows only a rather weak depend- μ Oðμ4 Þ ðμ Þ ence on B. The B correction to Tpc B is found to be zero within errors [19].ForμB ≤ Tpcð0Þ it changes from B ð μ Þ B ð μ Þ FIG. 8. The cumulant ratios R51 T; B and R62 T; B versus ¼ 156 5ð1 5Þ μ T . . to 154.5(2.0) MeV. This range of temper- B=T from LO and NLO Taylor expansions of the cumulants atures is well covered by the results for cumulant ratios as a calculated on lattices with temporal extent Nτ ¼ 8. function of μB evaluated at fixed values of the temperature that have been shown in the previous section. about an order of magnitude larger. This results in the much In Fig. 9 we show results for RB ðT ðμ Þ; μ Þ on the larger NLO correction to the expansion of χBðT;μ Þ. 12 pc B B 6 B pseudocritical line and compare with results obtained from Although the expansions of all cumulants χBðT;μ Þ will n B noninteracting HRG model calculations that utilize hadron have the same radius of convergence it is apparent that resonance gas spectra as listed in the Particle Data expansions for higher-order cumulants will converge Tables [25] as well as spectra calculated in quark models more slowly. Higher-order corrections to χBðT;μ Þ and 5 B [26,27]. As can be seen in Fig. 9 HRG model calculations for χBð μ Þ 6 T; B will thus be needed to arrive at firm conclusions RB agree well with QCD results obtained on the pseudoc- μ ≃ 1 12 on the behavior of these cumulants close to B=T .For ritical line up to about μ =T ≃ 0.8 or μ ≃ 125 MeV. This μ ≃ 0 3 B B B=T . ; however, the NLO correction is about an order suggests that the use of low-order HRG cumulants, in of magnitude smaller and thus of similar magnitude as the particular the mean of hadron distributions (hadron yields) χBð μ Þ χBð μ Þ μ ≃ 1 NNLO correction to 3 T; B and 4 T; B at B=T . that are used experimentally to determine freeze-out param- For small values of the baryon chemical potential and eters, may be appropriate at small values of the baryon μ ¼ μ ¼ 0 S Q we thus may extend the result on the ordering chemical potential or small net baryon-number densities. of cumulant ratios stated in Eq. (3) and also include results The HRG model estimates of freeze-out parameters for the fifth- and sixth-order cumulant ratios, [12] suggest that the range of baryon chemical potentials μB=T ≲ 1 corresponds to thermal conditions at freeze-out B B B B χ6 ðT;μ⃗Þ χ5 ðT;μ⃗Þ χ4 ðT;μ⃗Þ χ3 ðT;μ⃗Þ generated in heavy-ion experiments at beam energies < < < : ð14Þ pffiffiffiffiffiffiffiffi χBð μ⃗Þ χBð μ⃗Þ χBð μ⃗Þ χBð μ⃗Þ ≳ 27 2 T; 1 T; 2 T; 1 T; spNNffiffiffiffiffiffiffiffi GeV. Figure 9 suggests that below this value of sNN HRG model determinations of baryon chemical potentials differ from QCD determinations by more than IV. BARYON-NUMBER FLUCTUATIONS ON THE 10%. It thus may be useful to eliminate μB in favor of a PSEUDOCRITICAL LINE AND THE CUMULANTS directly accessible physical observable, e.g., RB . OF NET PROTON-NUMBER FLUCTUATIONS 12 At least for μB ≲ 200 MeV truncation errors in the Taylor In this section we compare results on higher-order expansion of the first two cumulants, the mean and variance, cumulants of net proton-number fluctuations, obtained by as well as lattice discretization errors are small. The con- B the STAR Collaboration during BES-I at RHIC [18,20], with tinuum limit extrapolation for R12ðTpcðμBÞ; μBÞ,shownin our results for cumulants of net baryon-number fluctuations Fig. 9 thus does not suffer from truncation errors in the Taylor

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3 RB ðT;μ Þ ≡ S σ =M B B FIG. 10. The cumulant ratios (bands) 31 B B B B FIG. 11. The cumulant ratios R51ðT;μBÞ and R62ðT;μBÞ versus B ð μ Þ ≡ κ σ2 B ð μ Þ ≡ σ2 B and R42 T; B B B versus R12 T; B MB= B on the R12ðT;μBÞ evaluated on the pseudocritical line. Data are pre- pseudocritical line calculated from NNLO Taylor series. Data P liminary results for the cumulant ratio R62 of net proton-numberpffiffiffiffiffiffiffiffi are results on cumulant ratios of net proton-number fluctuations fluctuations obtained by the STAR Collaboration at sNN ¼ 200 obtained by the STAR Collaborationpffiffiffiffiffiffiffiffi [18]. Also shown are and 54.4 GeV for the 0–40% centrality class [20]. preliminary results obtained at sNN ¼ 54.4 GeV [20]. Dashed lines show joint fits to the data as described in the text. As the experimentally determined skewness ratio of net series at least up to μB=T ¼ 1.2. It is a monotonically rising proton-number fluctuations has a rather weak dependence 3 P B function of μB. This allows to replace the chemical potential on R12 and also the QCD result for R31 has a weak B B in an analysis of higher-order cumulant ratios in favor of R12. dependence on R12, it obviously is not of much importance We have done so for the comparison of higher-order cumulant for the comparison of data and lattice QCD calculations P B ratios calculated in lattice QCD on the pseudocritical line with whether R12 equals R12 or only is a proxy within say experimental data on cumulants of net proton-number fluc- 10–20%. More relevant is the question to what extent the P 5 B tuations. In Fig. 10 we show the skewness and kurtosis ratios, magnitude of R31 is a good approximation for R31. A direct B B B P B R31 and R42, on the pseudocritical line as a function of R12, comparison between R31 and R31, as shown in Fig. 10, which also has been evaluated on the pseudocritical line. suggests that freeze-out happens in the vicinity of but below Similar results for the hyper-skewness and hyper-kurtosis the pseudocritical temperature. In fact, as can be seen in B B ratios are shown in Fig. 11. Figs. 4 and 7, the ratios R31 and R42 are decreasing B ¼ 0 75 P In Fig. 10 we show lattice QCD results up to R12 . , functions of the temperature. Experimental data for R31 which corresponds to μ ¼ T ðμ Þ ≃ 154.5 MeV. The B B pc B lying above the theoretical band for R31, evaluated on the width of the bands shown in the figure reflect the error pseudocritical line, thus suggest that freeze-out happens at ðμ Þ on Tpc B as given in Eq. (1) as well as the error on the a lower temperature. B B NNLO and continuum limit estimates for R31 and R42. Note Although errors on experimental results for the kurtosis P that the upper ends of these error bands correspond to the ratio R42 are large, they are thermodynamically consistent lower temperature, i.e., T ¼ 155 MeV at μB ¼ 0 and with the data on the skewness ratio as pointed out already in T ≃ 152.5 MeV at μB=T ¼ 1. our earlier analysis [19]. This gets further support through 6 Also shown in this figure are results for the skewness and recent high-statisticspffiffiffiffiffiffiffiffi data obtained by the STAR kurtosis ratios of net proton-number fluctuations obtained Collaboration at sNN ¼ 54.4 GeV [20]. These data are P by the STAR Collaboration [18,20]. These ratios are plotted shown in Fig. 10 at R12 ¼ 0.4672ð2Þ. For this value of the versus the measured ratio of the mean and variance of net P beam energy the kurtosis ratio R42 is found to be smaller proton-number fluctuations, which is taken as a proxy for 4 B the net baryon-number cumulant ratio R12. 5Many caveats for a direct comparison between net baryon- number fluctuations calculated in equilibrium thermodynamics and net proton-number fluctuations measured in heavy-ion 3Note that this will no longer be the case when one comes collisions have been discussed in the literature [10,13]. The B close to a critical point, where χ2 is expected to diverge lattice QCD results shown in Fig. 10 thus may be considered only B and R12ðTpcðμBÞ; μBÞ thus would approach zero. as a starting point for a more refined analysis of the experimental 4In a noninteracting HRG with vanishing strangeness and data that may take into account effects arising from experimental electric-charge chemical potential the ratios of the mean and acceptance cuts, the small size of the hot and dense medium, variance of net proton-number fluctuations and net baryon- nonequilibrium effectspffiffiffiffiffiffiffiffi etc. 6 ¼ 54 4 number fluctuations are identical. In the case of a strangeness Thep statisticsffiffiffiffiffiffiffiffi at sNN . GeV is a factor of 3.4 larger ¼ 0 ¼ 0 4 ¼ 200 – neutral (nS with nQ=nB . ), noninteracting HRG, how- than at spNNffiffiffiffiffiffiffiffi GeV and a factor of 17 30 larger than for ever, the latter is about 10% smaller. the other sNN data sets shown in Fig. 10.

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P P P than R31. The magnitude of this difference, R42 − R31 ¼ As pointed out in the previous section the NLO correc- B −0.12ð5Þ, is in good agreement with the corresponding tions for the hyper-skewness ratio R51 are a factor of 3 B lattice QCD result on the pseudocritical line. For the range smaller than those for the hyper-kurtosis ratio R62. B R12 ¼ 0.45ð5Þ, which corresponds to μB ¼ 80–100 MeV, Truncation errors for the former series are thus expected B or μB=T ¼ 0.57ð7Þ, we find from a fit to the difference of R42 to be less severe. Furthermore, this ratio will also be easier B B B and R31, R42 − R31 ¼ −0.08ð3Þ. At these values of the to determine experimentally with smaller statistical errors. B baryon chemical potential (or for R12 ≃ 0.5)theNNLO It thus would be an important check on the thermodynamic results for the skewness and kurtosis ratios, presented in the consistency of higher-orderpffiffiffiffiffiffiffiffi cumulants to compare exper- P ≥ 54 4 previous section, seem to suffer little from truncation effects imental data on R51 at sNN . GeV with the NLO in the Taylor expansions. lattice QCD calculations presented here. Also shown in Fig. 10 with dashed lines is a jointpffiffiffiffiffiffiffiffi fit to P P the experimental data on R31 and R42 [18] for sNN ≥ V. SUMMARY AND CONCLUSIONS 19.6 GeV using a quadratic ansatz, already used in We have presented new results on the μB dependence of Ref. [19], up to sixth-order cumulants using our latest results on up to eighth-order cumulants calculated at vanishing chemical P P 2 R31 ¼ S0 þ S2ðR12Þ ; potentials. Using simulation results obtained on lattices of 3 3 P P 2 size 32 × 8 and 48 × 12 we further presented continuum R ¼ K0 þ K2ðR Þ ; ð15Þ 42 12 limit estimates for up to fourth-order cumulant ratios. For pffiffiffiffiffiffiffiffi this analysis we used results from NNLO expansions of ≡ ¼ 54 4 with K0 S0. Including the new data at sNN . GeV cumulants in the baryon chemical potential for strangeness yields a fit, consistent with Ref. [19], but further constrains neutral systems, nS ¼ 0 at an electric-charge to baryon- the parameters. One finds S0 ≡ K0 ¼ 0.761ð20Þ, S2 ¼ number ratio n =n ¼ 0.4. Systematic effects arising from −0 077ð70Þ ¼ −0 54ð22Þ Q B . , and K2 . . From the continuum the truncation of Taylor series for the skewness and kurtosis estimates of RB and RB at μ ¼ 0 shown in Fig. 6 one 31 42 B ratios were shown to be small for μB=T ≤ 1, i.e., for the finds that the value of S0 corresponds to a freeze-out range of chemical potentials that can bep probedffiffiffiffiffiffiffiffi in heavy-ion temperature of 153.5(2.0) MeV. This temperature range is collisions in a range of beam energies sNN ≥ 27 GeV. A consistent with an earlier determination of the freeze-out comparison of the results on ratios of up to fourth-order temperature that was based on a comparison of the mean-to- cumulants of net baryon-number fluctuations calculated in variance ratio of net electric-charge and net proton-number equilibrium QCD thermodynamics with corresponding ratios obtained by the STAR and PHENIX collaborations cumulants of net proton-number fluctuations yields quite [29,30] with corresponding lattice QCD calculations for net good agreement. This suggests that the latter are consistent electric-charge and net baryon-number cumulant ratios [31]. with reflecting the imprint of thermodynamical fluctuations B B We also note that the ratio of the curvature of R42 and R31 on at a temperature close to but below the pseudocritical the pseudocritical (freeze-out) line tends to be larger than 3, temperatures TpcðμBÞ. The particularly good agreement which also has been noted in our previous analysis of the between lattice QCD calculations and the high-statistics skewness and kurtosis ratios [19]. pexperimentalffiffiffiffiffiffiffiffi data for up to fourth-order cumulants at While the experimental data on the skewness and kurtosis sNN ≥ 54.4 GeV suggests that this conclusion could be cumulantpffiffiffiffiffiffiffiffi ratios of net proton-number fluctuations, obtained further strengthened, if data with similarly high statistics at s ≥ 27 GeV, are consistent with results on net NN also becomespffiffiffiffiffiffiffiffi available at other beam energies in the baryon-number cumulants calculated within equilibrium range sNN ≥ 27 GeV. QCD thermodynamics, this is not the case for the prelimi- We also presented first results from a NLO calculation of nary data on sixth-order cumulants presented by thepffiffiffiffiffiffiffiffi STAR fifth- and sixth-order cumulants and showed that the hyper- ¼ B B Collaboration [20]. The still preliminary data at sNN skewness and hyper-kurtosis ratios R51 and R62 are 200 – and 54.4 GeV,taken from the 0 40% centrality class, are negative at low values of μB=T and temperatures in the shown in Fig. 11 together with the NLO lattice QCD ðμ Þ pffiffiffiffiffiffiffiffi vicinity of Tpc B . This is at odds withpffiffiffiffiffiffiffiffi preliminary data calculations. At both values of s deviations from the NN obtained by the STAR Collaboration at sNN ≥ 54.4 GeV P NLO lattice QCD results are large and of similar magnitude. for the sixth-order cumulant ratio, R62, of net proton- B ≃ 0 5 While it is conceivable that the NLO results at R12 . (or number fluctuations, which was found to be positive and μ ≃ 0 6 B=T . ) will receive sizable corrections at NNLO, this is close to unity. However, on the one hand corrections to the B ≃ 0 15 μ ≃ 0 3 not the case at R12 . (or B=T . ). It thus seems LO result for these cumulants, calculated in lattice QCD, impossible to describe both data points within QCD equi- are large already at μB ≃ 0.5. This makes a calculation of librium thermodynamics. We also note that a large positive NNLO corrections for these cumulants desirable. On the χB 10 is needed, for such ap contributionffiffiffiffiffiffiffiffi to render the hyper- other hand, the experimental determination of sixth-order kurtosis ratio positive at sNN ¼ 54.4 GeV. cumulant ratios is known to require high statistics and

074502-10 SKEWNESS, KURTOSIS, AND THE FIFTH AND … PHYS. REV. D 101, 074502 (2020) current experimental data may be statistics limited. We (iii) the INCITE program at Argonne Leadership pointed out that a measurement of ratios of fifth- and first- Computing Facility, a U.S. Department of Energy Office order cumulants would be very helpful as this ratio can be of Science User Facility operated under Contract No. DE- better controlled experimentally and suffers less from AC02-06CH11357; and (iv) the USQCD resources at the truncation effects in NLO lattice QCD calculations. Thomas Jefferson National Accelerator Facility. This All data from our calculations, presented in the figures of research also used computing resources made available this paper, can be found at https://pub.uni-bielefeld.de/ through: (i) a PRACE grant at CINECA, Italy; (ii) the record/2941824 [32]. Gauss Center at NIC-Jülich, Germany; and (iii) the Nuclear Science Computing Center at Central China Normal ACKNOWLEDGMENTS University and (iv) the GPU-cluster at Bielefeld This work was supported by: (i) the U.S. Department of University, Germany. Energy, Office of Science, Office of Nuclear Physics through the Contract No. DE-SC0012704; (ii) the U.S. APPENDIX: TAYLOR EXPANSION Department of Energy, Office of Science, Office of Nuclear COEFFICIENTS OF NET BARYON-NUMBER Physics and Office of Advanced Scientific Computing CUMULANTS Research within the framework of Scientific Discovery through Advance Computing (SciDAC) award Computing We give here explicit expressions for the first four the Properties of Matter with Leadership Computing expansion coefficients in the Taylor series for net Resources; (iii) the U.S. Department of Energy, Office baryon-number cumulants in strangeness neutral systems ¼ 0 of Science, Office of Nuclear Physics within the framework (nS ) with a fixed ratio of electric-charge to baryon- ¼ 0 4 of the Beam Energy Scan Theory (BEST) Topical number densities (nQ=nB . ) as defined in Eq. (4). Collaboration; (iv) the U.S. National Science Foundation These constraints determine the strangeness and electric- under Grant No. PHY-1812332; (v) the Deutsche charge chemical potentials (μS, μQ) in terms of the baryon Forschungsgemeinschaft (DFG, German Research chemical potential μB [24], Foundation)—Project No. 315477589-TRR 211; (vi) the μˆ ð μ Þ¼ ð Þμˆ þ ð Þμˆ 3 þ ð Þμˆ 5 þ … Grant No. 05P2018 (ErUM-FSP T01) of the German Q T; B q1 T B q3 T B q5 T B ; Bundesministerium für Bildung und Forschung; (vii) the 3 5 μˆ ðT;μ Þ¼s1ðTÞμˆ þ s3ðTÞμˆ þ s5ðTÞμˆ þ … ðA1Þ European Union H2020-MSCA-ITN-2018-813942 S B B B B (EuroPLEx); (viii) the National Natural Science Explicit expressions for the expansion coefficients q and s Foundation of China under Grants No. 11775096 and i i up to i ¼ 5 were given in Appendix B of Ref. [24]. Results No. 11535012 (HTD); (ix) the Early Career Research for i ¼ 7 can easily be generated following the procedure Award of the Science and Engineering Research Board outlined in that Appendix. of the Government of India (PH); and (x) a Ramanujan The expansion coefficients of the cumulant series Fellowship of the Department of Science and Technology, χBðT;μ Þ Government of India (SS). This research used awards of n B defined in Eq. (4) are given in terms of the computer time provided by: (i) the INCITE program at Oak expansion coefficients of the pressure series, Ridge Leadership Computing Facility, a DOE Office of P X 1 Science User Facility operated under Contract No. DE- ¼ χBQSμˆ i μˆ j μˆ k: ðA2Þ T4 i!j!k! ijk B Q S AC05-00OR22725; (ii) the ALCC program at National i;j;k Energy Research Scientific Computing Center, a U.S. B;k Department of Energy Office of Science User Facility For n even, one obtains for the expansion coefficients χ˜ n , operated under Contract No. DE-AC02-05CH11231; appearing in Eq. (4)

χ˜ B;0 ¼ χBQS n n00 ; χ˜ B;2 ¼ðχBQS þ 2 χBQS þ 2 χBQS þ 2 χBQS þ 2 χBQS þ 2 χBQSÞ 2 n nþ2;00 s1 n02 q1 n20 s1 nþ1;01 q1 nþ1;10 q1s1 n11 = ; χ˜ B;4 ¼ð24 χBQS þ 4 χBQS þ 24 χBQS þ 24 χBQS þ 4 3 χBQS þ 24 χBQS þ 6 2 2 χBQS þ 4 3 χBQS n s1s3 n02 s1 n04 q3s1 n11 q1s3 n11 q1s1 n13 q1q3 n20 q1s1 n22 q1s1 n31 þ 4 χBQS þ 24 χBQS þ 4 3 χBQS þ 24 χBQS þ 12 2 χBQS þ 12 2 χBQS þ 4 3 χBQS q1 n40 s3 nþ1;01 s1 nþ1;03 q3 nþ1;10 q1s1 nþ1;12 q1s1 nþ1;21 q1 nþ1;30 þ 6 2 χBQS þ 12 χBQS þ 6 2 χBQS þ 4 χBQS þ 4 χBQS þ χBQS Þ 24 s1 nþ2;02 q1s1 nþ2;11 q1 nþ2;20 s1 nþ3;01 q1 nþ3;10 nþ4;00 = ;

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χ˜ B;6 ¼ð360 2 χBQS þ 720 χBQS þ 120 3 χBQS þ 6 χBQS þ 720 χBQS þ 720 χBQS þ 720 χBQS n s3 n02 s1s5 n02 s1s3 n04 s1 n06 q5s1 n11 q3s3 n11 q1s5 n11 þ 120 3 χBQS þ 360 2 χBQS þ 6 5 χBQS þ 360 2 χBQS þ 720 χBQS þ 360 2 χBQS q3s1 n13 q1s1s3 n13 q1s1 n15 q3 n20 q1q5 n20 q1q3s1 n22 þ 360 2 χBQS þ 15 2 4 χBQS þ 360 2 χBQS þ 120 3 χBQS þ 20 3 3 χBQS þ 120 3 χBQS q1s1s3 n22 q1s1 n24 q1q3s1 n31 q1s3 n31 q1s1 n33 q1q3 n40 þ 15 4 2 χBQS þ 6 5 χBQS þ 6 χBQS þ 720 χBQS þ 360 2 χBQS þ 6 5 χBQS þ 720 χBQS q1s1 n42 q1s1 n51 q1 n60 s5 nþ1;01 s1s3 nþ1;03 s1 nþ1;05 q5 nþ1;10 þ 360 2 χBQS þ 720 χBQS þ 30 4 χBQS þ 720 χBQS þ 360 2 χBQS q3s1 nþ1;12 q1s1s3 nþ1;12 q1s1 nþ1;14 q1q3s1 nþ1;21 q1s3 nþ1;21 þ 60 2 3 χBQS þ 360 2 χBQS þ 60 3 2 χBQS þ 30 4 χBQS þ 6 5 χBQS þ 360 χBQS q1s1 nþ1;23 q1q3 nþ1;30 q1s1 nþ1;32 q1s1 nþ1;41 q1 nþ1;50 s1s3 nþ2;02 þ 15 4 χBQS þ 360 χBQS þ 360 χBQS þ 60 3 χBQS þ 360 χBQS þ 90 2 2 χBQS s1 nþ2;04 q3s1 nþ2;11 q1s3 nþ2;11 q1s1 nþ2;13 q1q3 nþ2;20 q1s1 nþ2;22 þ 60 3 χBQS þ 15 4 χBQS þ 120 χBQS þ 20 3 χBQS þ 120 χBQS þ 60 2 χBQS q1s1 nþ2;31 q1 nþ2;40 s3 nþ3;01 s1 nþ3;03 q3 nþ3;10 q1s1 nþ3;12 þ 60 2 χBQS þ 20 3 χBQS þ 15 2 χBQS þ 30 χBQS þ 15 2 χBQS q1s1 nþ3;21 q1 nþ3;30 s1 nþ4;02 q1s1 nþ4;11 q1 nþ4;20 þ 6 χBQS þ 6 χBQS þ χBQS Þ 720 s1 nþ5;01 q1 nþ5;10 nþ6;00 = :

B For the expansion coefficients of cumulants χn ðT;μBÞ, with n odd, one obtains

χ˜ B;1 ¼ χBQS þ χBQS þ χBQS n s1 n01 q1 n10 nþ1;00; χ˜ B;3 ¼ð6 χBQS þ 3 χBQS þ 6 χBQS þ 3 2 χBQS þ 3 2 χBQS þ 3 χBQS n s3 n01 s1 n03 q3 n10 q1s1 n12 q1s1 n21 q1 n30 þ 3 2 χBQS þ 6 χBQS þ 3 2 χBQS þ 3 χBQS þ 3 χBQS þ χBQS Þ 6 s1 nþ1;02 q1s1 nþ1;11 q1 nþ1;20 s1 nþ2;01 q1 nþ2;10 nþ3;00 = ; χ˜ B;5 ¼ð120 χBQS þ 60 2 χBQS þ 5 χBQS þ 120 χBQS þ 60 2 χBQS þ 120 χBQS þ 5 4 χBQS n s5 n01 s1s3 n03 s1 n05 q5 n10 q3s1 n12 q1s1s3 n12 q1s1 n14 þ 120 χBQS þ 60 2 χBQS þ 10 2 3 χBQS þ 60 2 χBQS þ 10 3 2 χBQS þ 5 4 χBQS þ 5 χBQS q1q3s1 n21 q1s3 n21 q1s1 n23 q1q3 n30 q1s1 n32 q1s1 n41 q1 n50 þ 120 χBQS þ 5 4 χBQS þ 120 χBQS þ 120 χBQS þ 20 3 χBQS þ 120 χBQS s1s3 nþ1;02 s1 nþ1;04 q3s1 nþ1;11 q1s3 nþ1;11 q1s1 nþ1;13 q1q3 nþ1;20 þ 30 2 2 χBQS þ 20 3 χBQS þ 5 4 χBQS þ 60 χBQS þ 10 3 χBQS þ 60 χBQS þ 30 2 χBQS q1s1 nþ1;22 q1s1 nþ1;31 q1 nþ1;40 s3 nþ2;01 s1 nþ2;03 q3 nþ2;10 q1s1 nþ2;12 þ 30 2 χBQS þ 10 3 χBQS þ 10 2 χBQS þ 20 χBQS þ 10 2 χBQS q1s1 nþ2;21 q1 nþ2;30 s1 nþ3;02 q1s1 nþ3;11 q1 nþ3;20 þ 5 χBQS þ 5 χBQS þ χBQS Þ 120 s1 nþ4;01 q1 nþ4;10 nþ5;00 = ;

χ˜ B;7 ¼ð5040 χBQS þ 2520 2 χBQS þ 2520 2 χBQS þ 210 4 χBQS þ 7 χBQS þ 5040 χBQS þ 2520 2 χBQS n s7 n01 s1s3 n03 s1s5 n03 s1s3 n05 s1 n07 q7 n10 q5s1 n12 þ 5040 χBQS þ 2520 2 χBQS þ 5040 χBQS þ 210 4 χBQS þ 840 3 χBQS þ 7 6 χBQS q3s1s3 n12 q1s3 n12 q1s1s5 n12 q3s1 n14 q1s1s3 n14 q1s1 n16 þ 2520 2 χBQS þ 5040 χBQS þ 5040 χBQS þ 2520 2 χBQS þ 840 3 χBQS þ 1260 2 2 χBQS q3s1 n21 q1q5s1 n21 q1q3s3 n21 q1s5 n21 q1q3s1 n23 q1s1s3 n23 þ 21 2 5 χBQS þ 2520 2 χBQS þ 2520 2 χBQS þ 1260 2 2 χBQS þ 840 3 χBQS þ 35 3 4 χBQS q1s1 n25 q1q3 n30 q1q5 n30 q1q3s1 n32 q1s1s3 n32 q1s1 n34 þ 840 3 χBQS þ 210 4 χBQS þ 35 4 3 χBQS þ 210 4 χBQS þ 21 5 2 χBQS þ 7 6 χBQS þ 7 χBQS q1q3s1 n41 q1s3 n41 q1s1 n43 q1q3 n50 q1s1 n52 q1s1 n61 q1 n70 þ 2520 2 χBQS þ 5040 χBQS þ 840 3 χBQS þ 7 6 χBQS þ 5040 χBQS þ 5040 χBQS s3 nþ1;02 s1s5 nþ1;02 s1s3 nþ1;04 s1 nþ1;06 q5s1 nþ1;11 q3s3 nþ1;11 þ 5040 χBQS þ 840 3 χBQS þ 2520 2 χBQS þ 42 5 χBQS þ 2520 2 χBQS þ 5040 χBQS q1s5 nþ1;11 q3s1 nþ1;13 q1s1s3 nþ1;13 q1s1 nþ1;15 q3 nþ1;20 q1q5 nþ1;20 þ 2520 2 χBQS þ 2520 2 χBQS þ 105 2 4 χBQS þ 2520 2 χBQS þ 840 3 χBQS q1q3s1 nþ1;22 q1s1s3 nþ1;22 q1s1 nþ1;24 q1q3s1 nþ1;31 q1s3 nþ1;31 þ 140 3 3 χBQS þ 840 3 χBQS þ 105 4 2 χBQS þ 42 5 χBQS þ 7 6 χBQS þ 2520 χBQS q1s1 nþ1;33 q1q3 nþ1;40 q1s1 nþ1;42 q1s1 nþ1;51 q1 nþ1;60 s5 nþ2;01 þ 1260 2 χBQS þ 21 5 χBQS þ 2520 χBQS þ 1260 2 χBQS þ 2520 χBQS þ 105 4 χBQS s1s3 nþ2;03 s1 nþ2;05 q5 nþ2;10 q3s1 nþ2;12 q1s1s3 nþ2;12 q1s1 nþ2;14 þ 2520 χBQS þ 1260 2 χBQS þ 210 2 3 χBQS þ 1260 2 χBQS þ 210 3 2 χBQS þ 105 4 χBQS q1q3s1 nþ2;21 q1s3 nþ2;21 q1s1 nþ2;23 q1q3 nþ2;30 q1s1 nþ2;32 q1s1 nþ2;41

074502-12 SKEWNESS, KURTOSIS, AND THE FIFTH AND … PHYS. REV. D 101, 074502 (2020)

þ 21 5 χBQS þ 840 χBQS þ 35 4 χBQS þ 840 χBQS þ 840 χBQS þ 140 3 χBQS q1 nþ2;50 s1s3 nþ3;02 s1 nþ3;04 q3s1 nþ3;11 q1s3 nþ3;11 q1s1 nþ3;13 þ 840 χBQS þ 210 2 2 χBQS þ 140 3 χBQS þ 35 4 χBQS þ 210 χBQS þ 35 3 χBQS q1q3 nþ3;20 q1s1 nþ3;22 q1s1 nþ3;31 q1 nþ3;40 s3 nþ4;01 s1 nþ4;03 þ 210 χBQS þ 105 2 χBQS þ 105 2 χBQS þ 35 3 χBQS þ 21 2 χBQS þ 42 χBQS q3 nþ4;10 q1s1 nþ4;12 q1s1 nþ4;21 q1 nþ4;30 s1 nþ5;02 q1s1 nþ5;11 þ 21 2 χBQS þ 7 χBQS þ 7 χBQS þ χBQS Þ 5040 q1 nþ5;20 s1 nþ6;01 q1 nþ6;10 nþ7;00 = :

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