Comparing conserved charge fluctuations from lattice QCD to HRG model calculations
Jishnu Goswami, Frithjof Karsch, Christian Schmidt, Swagato Mukherjee, Peter Petreczky
CComparing conserved charge fluctuations from lattice QCDto HRG model calculations ∗ Jishnu Goswami † , Frithjof Karsch, Christian Schmidt Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33615 Bielefeld, Germany
Swagato Mukherjee and Peter Petreczky
Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USAWe present results from lattice QCD calculations for 2 nd and 4 th ordercumulants of conserved charge fluctuations and correlations, and comparethese with various HRG model calculations. We show that differences be-tween HRG and QCD calculations already show up in the second ordercumulants close to the pseudo-critical temperature for the chiral transitionin (2+1)-flavor QCD and quickly grow large at higher temperatures. Wealso show that QCD results for strangeness fluctuations are enhanced overHRG model calculations which are based only on particles listed in theParticle Data Group tables as 3-star resonances. This suggests the im-portance of contributions from additional strange hadron resonances. Wefurthermore argue that additional (repulsive) interactions, introduced ei-ther through excluded volume (mean field) HRG models or the S-matrixapproach, do not improve the quantitative agreement with 2 nd and 4 th or-der cumulants calculated in lattice QCD. HRG based approaches fail todescribe the thermodynamics of strongly interacting matter at or shortlyabove the pseudo-critical temperature of QCD.PACS numbers: 1.15.Ha, 12.38.Gc, 12.38.Mh, 24.60.-k
1. Introduction
The theory of strong interactions, Quantum chromodynamics (QCD),also describe the thermodynamics of strongly interacting matter at finitetemperature and density. It now is understood that at vanishing net baryon-number density the transition from low to high temperature reflects the ∗ 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 ] N ov Jishnu Goswami physics of a true phase transition that occurs at vanishing values of the twolight (up and down) quark masses in QCD and is due to the restoration ofchiral symmetry, which is spontaneously broken in the QCD vacuum and atlow temperatures [1]. QCD with its physical spectrum of light and strangequark masses undergoes a smooth transition from hadronic bound states tothe quark gluon plasma (QGP) at high temperature.Hadron Resonance Gas (HRG) models can be used to describe the ther-modynamics of QCD at low temperature where the degrees of freedom ofQCD matter are hadrons. This model assumes that interactions amonghadrons can be accounted for by production of hadronic resonances whichare added to thermodynamics as additional particles. The simplest imple-mentation of a non-interacting HRG model considers a mixture of ideal Bosegases for mesons and ideal Fermi gases for baryons. The total pressure of ahadronic medium then is obtained as the sum over individual contributionsof partial pressures of different particle species. The HRG model can bejustified using the S-matrix based virial expansion [2]. It has been shownthat using partial wave analysis of the experimental scattering data non-resonant repulsive and attractive interactions of hadrons largely cancel outin the thermodynamic quantities, and thus, the interactions can be indeedwell described by hadronic resonances [3]. This approach has been recentlyrevisited in several papers [4, 5, 6], where also some of the non-resonant(repulsive) interactions were included.HRG models have been used to extract information on thermal condi-tions at the time of freeze-out of hadrons from a high temperature partonicmedium from experimental data on hadron yields [7]. However, a com-parison of lattice QCD calculations of conserved charge fluctuations withcorresponding HRG model calculations shows that the latter deviates fromQCD results more and more with increasing temperature and deviationsare larger for higher order cumulants. This short-coming of simple, non-interacting HRG models has been attempted to compensate by either takinginto account further contributions from repulsive interactions through ther-modynamic calculations with extended hadrons [8] or with a repulsive meanfield [4], since a comprehensive treatment of the repulsive interactions in theS-matrix approach is not yet available. While these modifications of point-like, non-interacting HRG model calculations generically lead to a reductionof cumulants of conserved charge fluctuations, there also is evidence thatstrangeness fluctuations calculated in QCD are larger than those obtainedin HRG model calculations based only on experimentally well established(3-star resonances) hadrons listed by the Particle Data Group (PDG). Onepopular approach to address this issue is to include in HRG model calcula-tions additional strange hadron resonances which are not listed in the PDGtables [9, 10, 11], but are obtained in quark model calculations [12, 13]. This omparing conserved charge fluctuations.... too may be interpreted as an attempt to take care of further interactions.We will present here a comparison of various cumulants of conservedcharge fluctuations, calculated in (2 + 1)-flavor lattice QCD [9, 10], withdifferent HRG model calculations. We mainly focus on the temperaturerange close to the transition temperature which is relevant for setting thebaseline for heavy-ion collision experiments. In the following we refer tothe standard non-interacting HRG as PDG-HRG, where we consider downto 3 star hadrons and hadron resonances listed in PDG2020 [14]. We alsoextended the HRG model based on resonances listed by the PDG by usingadditional hadronic resonances obtained in relativistic quark model calcula-tions (QM-HRG [12, 13]). Furthermore, we discuss modifications of the non-interacting HRG models obtained by including further interactions betweenhadrons either through excluded volume or mean field effects (EV-HRG) oran advanced treatment of the S-matrix approach to the thermodynamics ofstrongly interacting hadrons.
2. Second order cumulants of conserved charge fluctuations andcorrelations
Here we will discuss second order cumulants of net baryon-number ( B )and strangeness ( S ) fluctuations. In particular we will compare the secondorder cumulants to various HRG model calculations and discuss to whatextent deviations from HRG model results show up already on the level ofthese low order cumulants.The pressure of the QCD partition function can be written as, P/T = 1 V T ln Z ( V, T, µ B , µ Q , µ S ) . (1)Generalized susceptibilities, i.e. the cumulants of conserved charge fluctua-tions, can be obtained by taking derivatives of the pressure with respect tobaryon ( µ B ), electric charge ( µ Q ) and strangeness ( µ S ) chemical potentials χ T [MeV] cont. extr.N τ = 6 81216PDG-HRGQM-HRGEV-HRG 0 0.05 0.1 0.15 0.2 130 140 150 160 170 180 χ T [MeV] cont. extr. 81216PDG-HRGQM-HRGEV-HRG 0 0.1 0.2 0.3 0.4 0.5 130 140 150 160 170 180 H o t Q C D p r e li m i n a r y χ / χ T [MeV] cont. extr.N τ = 681216PDG-HRGQM-HRGEV-HRG 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 130 140 150 160 170 180 Fig. 1. Second order net baryon-number (left) and strangeness (middle) fluctuationsas well as their ratio (right).
Jishnu Goswami at (cid:126)µ = ( µ B , µ Q , µ S ) = 0, χ BQSlmn = ∂ l + m + n P/T ∂ ( µ B /T ) l ∂ ( µ Q /T ) m ∂ ( µ S /T ) n (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)µ =0 . (2) In Fig. 1 (left, middle) we show results for net baryon-number ( χ B ) andstrangeness ( χ S ) fluctuations (2 nd order cumulants). Results obtained onlattices of size N σ × N τ , with N σ = 4 N τ in (2+1)-flavor QCD simulationsusing the HISQ action [10, 15], are shown for several values of the latticespacing, i.e. several values of temporal lattice extent aN τ = 1 /T . Thesedata have been extrapolated to the continuum limit using a quadratic ansatzfor discretization errors in ( aT ). As discussed above we compare theseresults to HRG model calculations based on hadron spectra listed in thePDG and obtained in quark model calculations, respectively. Some basicformulas for the HRG model calculations are given in Appendices A and B.It is quite evident for χ B and χ S that PDG-HRG curves provide onlya poor description of the QCD results close to the transition region. Theagreement of HRG model and QCD results improves when one includesadditional strange baryon resonances in the spectrum that are predictedin quark model calculations (QM-HRG). However, as can be seen clearly inFig. 1 (right), the non-interacting HRG model calculations do not give a rea-sonable description of the QCD results at temperatures above the pseudo-critical temperature for the chiral transition, T pc = (156 . ± .
5) MeV [15].In particular, HRG results for the ratio χ B /χ S continue to rise above T pc while the QCD results have a maximum close to T pc and then drop towardsthe non-interacting quark gas value, ( χ B /χ S ) T →∞ = 1 /
3. Similarly, it isapparent that temperature derivatives of the 2 nd order cumulants keep ris-ing in HRG model calculations while they reach a maximum for the QCDresults close to T pc .At large temperatures the HRG model results are significantly largerthan the QCD results. This may, partly, be compensated by introducing re-pulsive interactions in the baryon sector of a HRG. When using an excludedvolume of size b (cid:39) . /T , which corresponds to rT (cid:39) . , we find a reduction of χ B of about20% in the transition region. The influence on strangeness fluctuations ismuch smaller, i.e. about 8%, as these are dominated by mesons. We show Note that a constant radius r (cid:39) .
39 fm has been used in [16, 17], which is close toour value of r in the pseudo-critical region. omparing conserved charge fluctuations.... χ T [MeV] cont. extr.N τ = 6 81216S-matrixPDG-HRGQM-HRGEV-HRG χ T [MeV] cont. extr.N τ = 6 81216PDG-HRGQM-HRGEV-HRG Fig. 2. Second order cumulants of net baryon-number fluctuations correlated withnet electric charge (BQ) and strangeness (BS) fluctuations, respectively. H o t Q C D p r e li m i n a r y - χ / χ T [MeV] cont. extr.N τ = 6 81216PDG-HRGQM-HRGEV-HRG 0.1 0.15 0.2 0.25 0.3 130 140 150 160 170 180 H o t Q C D p r e li m i n a r y χ / χ T [MeV] cont. extr.N τ = 681216PDG-HRGQM-HRGEV-HRG H o t Q C D p r e li m i n a r y χ / χ T [MeV] cont. extr.N τ = 6 81216S-matrixPDG-HRGQM-HRGEV-HRG Fig. 3. Continuum extrapolated results for 2 nd order cumulant ratios. results for the QM-HRG with excluded volume effects for baryons (EV-HRG) in Fig. 1. It is apparent that the hadronic interaction consideredhere is not sufficient to describe the QCD data. They rather tend to worsenthe agreement between HRG and QCD calculations achieved by introducingadditional strange baryon resonances. In Fig. 2 we show QCD results for correlations among conserved chargefluctuations and compare with HRG model calculations as discussed abovefor the 2 nd order cumulants of conserved charge fluctuations. The generalpicture is the same. Additional strange baryon resonances seem to be neededto improve agreement between HRG model calculations for BS -correlationsand corresponding QCD results, and the inclusion of repulsive interactionsamong baryons through excluded volume effects seems to deteriorate thisagreement. Also shown in Fig. 2 is the result of a S-matrix calculation [5]that takes into account resonance decays in the ∆ ++ ↔ N ∗ π channel (seealso Appendix C). As can be seen also the S-matrix approach leads to areduction of correlations between net baryon-number and electric charge.The contribution of doubly charged ∆ ++ resonances thus seems to be sup-pressed. Jishnu Goswami χ BS /χ S χ B /χ QS χ BQ /χ Q -0.241(4) 1.10(4) 0.059(14) Table 1. Continuum extrapolated results for three independent ratios of 2 nd ordercumulants at the pseudo-critical temperature T pc . The three conserved charges, (
B, Q, S ), give rise to 6 second order cumu-lants of charge fluctuations and cross-correlations. In the isospin symmetriclimit of degenerate up and down quark masses, which usually is used in lat-tice QCD calculations, only 4 of these cumulants are independent as isospinsymmetry imposes the two constraints χ S = 2 χ QS − χ BS , χ B = 2 χ BQ − χ BS . (3)This gives rise to three independent cumulant ratios, for instance the set ofthree ratios of second order cumulants shown in Fig. 3. In Table 1 we giveresults for continuum extrapolations of these three ratios at the pseudo-critical temperature, T pc = 156 . .
5) MeV, for the chiral transition in (2 +1)-flavor QCD. Note, for instance, that due to the first constraint in Eq. 3,the ratio χ B /χ S shown in Fig. 1 (right) is related to the two ratios χ BS /χ S and χ B /χ QS shown in Fig. 3 and given in Table 1 at T pc , χ B χ S = 12 χ B χ QS (cid:32) χ BS χ S (cid:33) , (4)and ( χ B /χ S ) T pc = 0 .
3. Fourth order cumulants of conserved charge fluctuations andcorrelations
In a non-interacting HRG (PDG-HRG or QM-HRG) ratios of cumulantsinvolving net baryon-number fluctuations that differ only by an even number χ / χ T [MeV] cont. extr.N τ = 6812HRGEV-HRG 0 0.2 0.4 0.6 0.8 1 1.2 130 140 150 160 170 180 H o t Q C D p r e li m i n a r y χ / χ T [MeV]cont. extr.N τ = 6812HRGEV-HRG 0 0.2 0.4 0.6 0.8 1 1.2 130 140 150 160 170 180 0 0.2 0.4 0.6 0.8 1 1.2 130 140 150 160 170 180 H o t Q C D p r e li m i n a r y χ / χ T [MeV]cont. extr.N τ = 6812HRGEV-HRGS-matrix 0 0.2 0.4 0.6 0.8 1 1.2 130 140 150 160 170 180 Fig. 4. Ratios of some fourth and second order cumulants. omparing conserved charge fluctuations.... of derivatives with respect to the baryon chemical potential are unity, e.g.for fourth order cumulants χ B /χ B = χ BS /χ BS = χ BQ /χ BQ = 1. Thisreflects that all known hadrons with non-zero baryon number have | B | = 1.This, of course, does not hold in QCD at high temperatures where quarkscarry non-integer baryon number. As a consequence the above ratios areall found to be smaller than unity in lattice QCD calculations. They areshown in Fig. 4.At low temperatures the deviations from unity follow a trend also presentin HRG model calculations that incorporate excluded volume effects. Thisalso is shown in Fig. 4. Although the agreement of these model calculationswith lattice QCD data seems to be reasonable below the pseudo-criticaltemperature, we note that this is to some extent accidental as the EV-HRGcalculations do not provide an adequate description for neither the numera-tor nor the denominator of these ratios. E.g., in the case of quadratic ( χ B )and quartic ( χ B ) net baryon-number fluctuations the EV-HRG calculationsboth underestimate the QCD results. We also note that in the present im-plementation of the S-matrix approach the ratio χ BQ /χ BQ remains unitylike it is the case in non-interacting HRG models.
4. Conclusions
Qualitative features of 2 nd and 4 th order cumulants of conserved chargefluctuations and correlations, calculated in lattice QCD, are reasonably welldescribed by non-interacting HRG models up to the pseudo-critical temper-ature for the QCD transition. At higher temperatures significant deviationsquickly set in, being large already for temperatures about 10% higher than T pc and being larger for 4 th than for 2 nd order cumulants. In order to reacha somewhat satisfactory description of strangeness fluctuations the addi-tion of strange resonances calculated in quark models (QM-HRG model) isneeded, which may be viewed as taking care of additional interactions, rep-resented for instance in less well-established resonances listed in the PDG,which are not reflected in HRG models based on down to 3-star resonancesonly. Including further interactions through e.g. excluded volume HRGmodels (EV-HRG) or S-matrix approaches does not seem to lead to a fur-ther quantitative improvement of many of the cumulants considered here. Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft (DFG)- project number 315477589 - TRR 211, the European Union H2020-MSCA-ITN-2018-813942 (EuroPLEx), and the U.S. Department of Energy, Officeof Science, Office of Nuclear Physics through i) the Contract No. DE-
Jishnu Goswami
SC0012704 and (ii) within the framework of the Beam Energy Scan Theory(BEST) Topical Collaboration, and (iii) the Office of Nuclear Physics andOffice of Advanced Scientific Computing Research within the framework ofScientific Discovery through Advance Computing (SciDAC) award ”Com-puting the Properties of Matter with Leadership Computing Resources”.
Appendix A
Non-Interacting HRG Model
The pressure of a non-interacting hadron gas can be written as a sum ofcontributions for mesons ( M ) and baryons ( B ) and the corresponding anti-particles ( ¯ M , ¯ B ), P = P M + P ¯ M + P B + P ¯ B , (A.1) P/T = (cid:88) H ∈ B, ¯ B g π ( m H /T ) ∞ (cid:88) k =1 ( − k +1 k K (cid:18) km H T (cid:19) exp[ k (cid:126)C H .(cid:126)µ/T ]+ (cid:88) H ∈ M, ¯ M g π ( m H /T ) ∞ (cid:88) k =1 k K (cid:18) km H T (cid:19) exp[ k (cid:126)C H .(cid:126)µ/T ] , (A.2)where (cid:126)C H = ( B H , Q H , S H ) represents the conserved charges, i.e. baryonnumber, electric charge and strangeness number of the hadron H , and µ B , µ Q , µ S are the baryon, electric charge and strangeness chemical po-tentials, respectively. K is the modified Bessel function of second kind.Generalized susceptibility can be obtained from Eq.(2), χ BQSlmn = (cid:88) H ∈ B, ¯ B g H π ( m H /T ) B lH Q mH S nH K (cid:18) m H T (cid:19) (A.3)+ (cid:88) H ∈ M, ¯ M g H π ( m H /T ) ∞ (cid:88) k =1 ( kQ H ) m ( kS H ) n k K (cid:18) km H T (cid:19) In Eq.(A.3) the first term corresponds to the baryon sector, where we onlyused the Boltzmann approximation to the Fermi sum given in Eq. A.2,and the second term corresponds to the meson sector. Note that the secondterm will drop out from the Eq.(A.3) for any baryonic observables as baryonnumber( B ) is 0 for mesons. Appendix B
Excluded Volume HRG Model omparing conserved charge fluctuations.... In excluded volume we only consider the interaction between baryons, BB ,and anti-baryons, ¯ B ¯ B . The meson-meson, M M , and meson-baryon,
M B ( ¯ B ),as well as baryon-antibaryon, B ¯ B , interactions are neglected. Hence, theexcluded volume will only modify P B and P ¯ B independently and the totalpressure Eq.(A.1), can be replaced by, P = P M + P ¯ M + P intB + P int ¯ B , (B.1)where the interacting baryon or anti-baryon pressure can be written as,ˆ P intB/ ¯ B = (cid:88) H ∈ B/ ¯ B ˆ P idH ( T, (cid:126)µ ) exp[ − b (cid:48) ˆ P intB/ ¯ B ] , (B.2)(B.3)with b (cid:48) = bT and ˆ P = P/T . This equation may be solved iteratively,ˆ P intB/ ¯ B = (cid:88) H ∈ B/ ¯ B ˆ P idH ( T, (cid:126)µ ) − b (cid:48) (cid:20) (cid:88) H,H (cid:48) ∈ B/ ¯ B ˆ P idnm ( T, (cid:126)µ ) (cid:21) (B.4)+(3 b (cid:48) / (cid:20) (cid:88) H,H (cid:48) ,H (cid:48)(cid:48) ∈ B/ ¯ B ˆ P idlmn ( T, (cid:126)µ ) (cid:21) + ..... For the baryon species H , ˆ P idH ( T, (cid:126)µ ) can be written from Eq.(A.2) usingBoltzmann approximation as,ˆ P idH = g H π ( m H /T ) K (cid:18) m H T (cid:19) exp[ (cid:126)C H .(cid:126)µ/T ] . (B.5)The term linear in b (cid:48) appearing in Eq.(B.5) acts as a repulsive term. It de-creases the pressure and is related to the second virial coefficient. Theterm quadratic in b acts as an attractive term. However, since ˆ P idH ∼ exp( − m H /T ), only the term linear in b will survive for m H (cid:29) T at high tem-perature, i.e. excluded volume effects are predominantly repulsive. More-over, since for low temperatures ˆ P idH →
0, ˆ P intH will also approach ˆ P idH .Generalized susceptibilities can be obtained by taking derivatives ofEq.(B.3) with respect to chemical potentials ( µ B , µ Q and µ S ). We also notethat the resulting equations in the mean field approach [4] are quite similarto those obtained in the excluded volume approach. The difference is thatin mean field approach the repulsive interactions have between used onlyfor the baryon octet and decuplet, while in the excluded volume approachone generally considers interaction between all baryons. Jishnu Goswami
Appendix C
S-matrix formalism
In the S-matrix formalism as used here by us, we only considered the decayand production of N ∗ and ∆ ++ to pion and nucleon in a hot hadron gas.The pressure can then be separated into two parts; one part arises from theinteraction of pion and nucleon, the other part describes the contributionfrom all other particles,ˆ P = ˆ P id + (cid:88) I z =3 / , / ˆ P int , (C.1)ˆ P int = g π (cid:90) ∞ m th d(cid:15) K ( (cid:15)/T ) ( (cid:15)/T ) dδ IJ πd(cid:15) , (C.2)where ˆ P id is same as in Eq.(A.2), but without the contribution from ∆ ++ and N ∗ resonances. Their contribution to the pressure is included in theˆ P int . Here we follow the notation and steps of [3, 5] for calculating thecumulants of net baryon-number and electric charge correlations, χ BQnm , inthe S-matrix approach. REFERENCES [1] for a recent review see: O. Philipsen, PoS
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