Higher order cumulants of electric charge and strangeness fluctuations on the crossover line
HHigher order cumulants of electric charge and strangenessfluctuations on the crossover line. ∗ D. Bollweg † , J. Goswami, F. Karsch, C. Schmidt Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33615 Bielefeld, Germany
S. Mukherjee
Physics Department, Brookhaven National Laboratory, Upton, New York 11973,USAWe present lattice QCD calculations of higher order cumulants of elec-tric charge distributions for small baryon chemical potentials µ B by usingup to NNNLO Taylor expansions. Ratios of these cumulants are evaluatedon the pseudo-critical line, T pc ( µ B ), of the chiral transition and comparedto corresponding measurements in heavy ion collision experiments by theSTAR and PHENIX Collaborations. We demonstrate that these compar-isons give strong constraints on freeze-out parameters. Furthermore, weuse strangeness fluctuation observables to compute the ratio µ S /µ B on thecrossover line and compare it to µ S /µ B at freeze-out stemming from fits tostrange baryon yields measured by the STAR Collaboration.PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.Mh, 24.60.-k
1. Introduction
Uncovering the phase structure of QCD poses a long-standing, openchallenge in heavy ion research. Large efforts - in the form of relativisticheavy ion collision experiments - are made to find signs of a critical pointproposed by many model calculations. This point would provide an impor-tant landmark in the largely unknown phase diagram of QCD and signalsitself for instance through divergences in conserved charge fluctuations thatcouple to the order parameter. Remnants of this divergence might be seen ∗ Presented at Criticality in QCD and the Hadron Resonance Gas, 29-31 July 2020,Wroclaw Poland † speaker (1) a r X i v : . [ h e p - l a t ] O c t D. Bollweg printed on October 30, 2020 in measurements of higher order cumulants of conserved charge distribu-tions in heavy ion collisions if freeze-out occurs in the vicinity of the criticalpoint. Baryon number and strangeness fluctuations are studied in heavy ioncollision experiments through the measurement of Event-by-Event fluctua-tions of proxy particle species, such as proton or kaon numbers, respectively.Electric charge fluctuations on the other hand can be measured without re-sorting to proxies, making them particularly attractive for comparisons withlattice QCD calculations. Here, we want to provide thermal QCD baselinesfor higher order cumulants of electric charge fluctuations via state-of-the-art lattice QCD calculations and contrast them with results from the STARand PHENIX experiments. Furthermore, we will also use strangeness fluc-tuation observables to construct µ S /µ B on the crossover line.
2. Setup
In previous studies [1], we presented calculations of higher order cu-mulants of net-baryon number fluctuations based on a high statistics dataset of (2+1)-flavor HISQ gauge field configurations with physical light andstrange quark masses. We use degenerate light quark masses and a lightto strange quark mass ratio m l /m s = 1 /
27. Here we add to this analysisthe corresponding higher order cumulants of electric charge and strangenessfluctuations. By computing up to eighth order generalized susceptibilities χ BQSijk ( T, (cid:126)µ ) = 1
V T ∂ i + j + k ln Z ( T, (cid:126)µ ) ∂ ˆ µ iB ∂ ˆ µ jQ ∂ ˆ µ kS , with ˆ µ X = µ X T (1)we are able to construct Taylor series coefficients ˜ χ X,kn ( T ) for n -th orderstrangeness and electric charge cumulants χ Xn ( T, µ B ) = k max (cid:88) k =0 ˜ χ X,kn ( T )ˆ µ kB , with X = Q, S. (2)In order to match the conditions present in heavy ion collisions, we constrainthe Taylor series such that the ratio of charge density n Q to baryon density n B is n Q /n B = 0 . n S = 0. This is achievedby expanding electric charge and strangeness chemical potentials in µ B withcoefficients q i and s i chosen such that n Q /n B = 0 . , n s = 0 hold at eachorder, ˆ µ Q ( T, µ B ) = (cid:88) i q i +1 ( T )ˆ µ i +1 B , (3)ˆ µ S ( T, µ B ) = (cid:88) i s i +1 ( T )ˆ µ i +1 B . lectric charge fluctuations printed on October 30, 2020 Explicit formulas for q j and s j for j ≤ χ X,kn will be given in an upcomingpublication. Finally, we form cumulant ratios R Xnm ( T, µ B ) = χ Xn ( T, µ B ) χ Xm ( T, µ B ) = (cid:80) k max k =0 ˜ χ X,kn ( T )ˆ µ kB (cid:80) l max l =0 ˜ χ X,lm ( T )ˆ µ lB , (4)in order to cancel the volume factor in (1) that is unknown in heavy ioncollision experiments. In this work, we will focus on the mean-to-varianceratio R X = M X σ X , the skewness ratio R X = S X σ X M X and the kurtosis ratio R X = κ X σ X . We evaluate (4) for nine temperatures ranging from 135 MeVto 175 MeV and normalized chemical potentials ˆ µ B ranging from 0 to 2 insteps of 0.01. This produces nine slices in the ( T, ˆ µ B )-plane that trace out
130 140 150 160 170 180 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.05 0.1 0.15 0.2 0.25 N t =8T [MeV] µ/TR Q12
Fig. 1. R Q for N t = 8 in the ( T, ˆ µ B )-plane. the R Xnm ( T, ˆ µ B ) surface. An example for this is shown in Fig. 1 for thecase of R Q for N t = 8 lattices. The data for the different lattice sizes isthen jointly fitted via low-order polynomial Ans¨atze to obtain continuumextrapolations. The details of the fitting procedure will be described in aforthcoming publication. D. Bollweg printed on October 30, 2020
3. Electric charge fluctuations on the crossover line
The mean to variance ratio R Q = M Q σ Q , shown across the ( T, ˆ µ B )-planein Fig. 1, has been calculated to NNNLO in µ B . Apart from a small area inthe low T and high ˆ µ B region, it shows nearly perfect linear behavior in ˆ µ B direction with very small variation along the T direction. This qualifies it tobe used as a baryometer when comparing this quantity with experimentalresults. In Fig. 2, we show R Q for the three different lattices sizes attemperatures close to T pc, as well as its continuum extrapolation evaluatedon the crossover line T pc ( µ B ) = T pc, (cid:32) − κ B,f (cid:18) µ B T pc, (cid:19) (cid:33) with T pc, = 156 . ± . κ B,f = 0 . ± .
004 [3]. This figure also R Q µ B [MeV]T=156 MeV, N t =8T=156 MeV, N t =12T=156 MeV, N t =16Cont. ext. - T pc (µ B ) Fig. 2. R Q for different N t and continuum extrapolation along T pc ( µ B ). Horizontallines show results from STAR [4] at different beam energies. contains the mean to variance ratios measured by STAR at different beamenergies depicted as horizontal lines. The intersections of the experimentallydetermined values with the lattice result on the crossover line are highlightedwith dashed vertical lines. These marks provide a mapping between beamenergies and freeze-out chemical potentials if that freeze-out happens onthe crossover line. The numerical values extracted from this comparison arelisted in Table 1. lectric charge fluctuations printed on October 30, 2020 √ s NN [GeV] µ B,f [MeV]200 19.4(1)62.4 58(1)39 92(2)27 131(3)
Table 1. Chemical potentials extracted from comparing results from STAR [4] withlattice QCD.
We also calculated the skewness ratio R Q ( T, µ B ) = S Q σ Q M Q to NNLO in µ B . As seen in Fig. 3, the roles of T and µ B are reversed when com-pared to R Q . The skewness ratio shows a strong variation with temper-ature but only a mild dependence on chemical potential. In T direction, R Q possesses a (mirrored) sigmoidal shape and decreases with increasingtemperature while in µ B direction, R Q decreases only mildly with increas-ing µ B . R Q therefore functions as a thermometer when comparing lat-tice results with measurements from heavy ion collisions. By evaluatingthe continuum estimate on the crossover line, the two effects cancel suchthat R Q ( T pc ( µ B ) , µ B ) remains almost constant when varying µ B . We find R Q ( T pc ( µ B ) , µ B ) = 1 . µ B <
150 MeV. The current continuum es-
Q31
T [MeV]Cont. estimateN t =8N t =12N t =16 oldN t =16 new -0.16-0.12-0.08-0.04 0 0.04 0 20 40 60 80 100 120 140 R Q31 (T,µ B )-R Q31 (T,0) µ B [MeV]T=156 MeV, N t =8N t =12 Fig. 3.
Left: R Q ( T, µ B = 0) for different lattices sizes. Right: µ B dependence of R Q for N t = 8 ,
12 lattices. timate of R Q , based on our N t = 8 and N t = 12 data, evaluated on thecrossover line is shown as a red curve in Fig. 4 together with the skewnessratio measured by the PHENIX experiment [5] in a pseudorapidity range of | η | ≤ .
35 and 27 MeV ≤ √ s NN ≤
200 MeV depicted as blue data points.Here we used the µ B -values determined by comparing to the STAR resultsand given in Table 1 to plot the PHENIX data for R Q . Fig. 4 shows thatthe PHENIX results are consistent with lattice QCD calculations on thecrossover line. The kurtosis ratio R Q = κ Q σ Q has also been calculated to D. Bollweg printed on October 30, 2020 R Q µ B [MeV]Cont. estimate. - T pc (µ B )PHENIX Fig. 4. Comparison of the continuum estimate of R Q ( T pc ( µ B ) , µ B ) with resultsfrom PHENIX [5]. NNLO in µ B . Its behavior in the ( T, µ B )-plane is very similar to R Q , albeitwith a different magnitude. The results are shown in Fig. 5. The error of R Q is noticeably smaller than that of the skewness ratio since it does notcontain the noisy Baryon-electric charge correlations that plague R Q . Weestimate R Q ( T pc ) = 0 . µ B <
150 MeV. Unfortunately, measure-ments of the kurtosis ratio in heavy ion collision experiments come withlarge uncertainties such that a meaningful comparison can not be made atpresent.
Q42,0 T pc T [MeV]cont. estimateN t =8N t =12N t =16 -0.12-0.08-0.04 0 0.04 0 20 40 60 80 100 120 140 R Q42 (T,µ B )-R Q42 (T,0) µ B [MeV]T=156 MeV, N t =8N t =12 Fig. 5.
Left: R Q ( T, µ B = 0) for different lattices sizes. Right: µ B dependence of R Q for N t = 8 ,
12 lattices. lectric charge fluctuations printed on October 30, 2020
4. Strangeness fluctuations
While mean-to-variance, skewness and kurtosis ratios of strangeness fluc-tuations are also accessible via lattice QCD following the discussion pre-sented in the previous section, we want to explore here a different appli-cation of strangeness observables that provides a helpful consistency checkfor estimated freeze-out parameters. Recall that the strangeness neutralityconstraint fully determines the strangeness chemical potential µ S ( T, µ B ) asseen in (3). Rewriting this equation to obtain µ S /µ B gives µ S µ B = s ( T ) + s ( T ) (cid:16) µ B T (cid:17) + O (cid:18)(cid:16) µ B T (cid:17) (cid:19) . (5)This ratio is almost exclusively determined by s ( T ) ≈ − χ BS χ S . At thepseudo-critical temperature T pc ( µ B = 0), we find s ( T pc, ) = 0 . − χ BS χ S > . ± . s is smaller than s by an order of magnitude, as shown in Fig. 6. µ S /µ B is also sensitive to the strangeness content in hadron resonance gas models.The PDG-HRG, which contains the hadron states listed in the particle databooklet, shows a clear deviation from lattice QCD. This deviation shrinks ifthe QM-HRG, which contains additional, not yet observed hadronic statespredicted by the quark model, is used instead. In a non-interacting hadronresonance gas, µ S /µ B enters also in the ratio of anti-strange baryon tostrange baryon yields ¯ B/B given by¯ BB ( √ s ) = exp (cid:18) − µ B T (2 − | S | µ S µ B ) (cid:19) . (6)Assuming that HRG relations provide a good approximation for particleyields generated at the time of freeze out we may fit the experimentallymeasured yield ratios for different particle species in | S | via (6). The ratio µ S /µ B at T = T f can then be extracted from such a fit. We performedthis fit on the Λ, Ξ and Ω yield data published by STAR in [7] and [8] andcompare the result with µ S /µ B from lattice QCD evaluated on the pseudo-critical line in Fig. 7. Apart from the data point at √ s NN = 200 GeV, thestrangeness to baryon chemical potential ratio at freeze-out, shown as redpoints, agrees very well with the lattice QCD result on the pseudo-criticalline.
5. Summary
We presented NNNLO calculations of the mean to variance ratio of elec-tric charge fluctuations obtained from state of the art lattice QCD calcula-
D. Bollweg printed on October 30, 2020 pc (µ B =0) s n T [MeV] N t =8N t =12N t =16Cont. ext.PDG-HRGQM-HRG Fig. 6. Temperature dependence of the expansion coefficients s ( T ) and s ( T ) for N t = 8 ,
12 and 16 lattices and PDG-HRG and QM-HRG curves for s . tions and demonstrated how R Q can be used to extract freeze-out chemi-cal potentials by comparing to measurements of this ratio by STAR. Fur-thermore, we estimated the skewness ratio R Q to NNLO in µ B and found R Q ( T pc ( µ B )) = 1 . µ B <
150 MeV. This is in agreement with theskewness ratio measured by PHENIX for beam energies 27 MeV ≤ √ s NN ≤
200 MeV. For the kurtosis ratio R Q , we found R Q ( T pc ) = 0 . µ S /µ B on the pseudo-critical line andcompared it to µ S /µ B at freeze-out determined by fitting strange hadronyields measured by STAR. Again, we found the results at freeze-out to beconsistent with lattice QCD results on the pseudo-critical line.
6. Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) - project number 315477589 - TRR 211; theGerman Bundesministerium f¨ur Bildung und Forschung through Grant No.05P2018 (ErUM-FSP T01) and the European Union H2020-MSCA-ITN-2018-813942 (EuroPLEx). It furthermore received support from the U.S.Department of Energy, Office of Science, Office of Nuclear Physics through lectric charge fluctuations printed on October 30, 2020 PBM et al : lower limits NN [GeV] : µ S / µ B R B (T pc ) HotQCDYields : Λ , Ξ , Ω Fig. 7. Comparison of µ S /µ B from strange baryon yield fits and lattice QCD onthe pseudo-critical line. The green box shows the lower limit on − χ BS /χ S from[6]. (i) the Contract No. DE-SC0012704 and (ii) within the framework of theBeam Energy Scan Theory (BEST) Topical Collaboration, and (iii) theOffice of Nuclear Physics and Office of Advanced Scientific Computing Re-search within the framework of Scientific Discovery through Advance Com-puting (SciDAC) award Computing the Properties of Matter with Leader-ship Computing Resources. REFERENCES [1] A. Bazavov et al. Skewness, kurtosis, and the fifth and sixth order cumulantsof net baryon-number distributions from lattice QCD confront high-statisticsSTAR data. Phys. Rev. D , 101:074502, 2020.[2] A. Bazavov et al. QCD equation of state to O ( µ B ) from lattice QCD. Phys.Rev. D , 95:054504, 2017.[3] A. Bazavov et al. Chiral crossover in QCD at zero and non-zero chemicalpotentials.
Phys. Lett. , B795:15–21, 2019.[4] L. Adamczyk et al. Beam energy dependence of moments of the net-chargemultiplicity distributions in Au + Au collisions at RHIC.
Phys. Rev. Lett. ,0 D. Bollweg printed on October 30, 2020 √ s NN = 7 . − −
200 GeV.
Phys. Rev.C , 93:011901, 2016.[6] P. Braun-Munzinger et al. Confronting fluctuations of conserved charges in cen-tral nuclear collisions at the LHC with predictions from Lattice QCD.