Additional Strange Hadrons from QCD Thermodynamics and Strangeness Freezeoutin Heavy Ion Collisions

A. Bazavov,1 H.-T. Ding,2 P. Hegde,2 O. Kaczmarek,3 F. Karsch,3,4 E. Laermann,3 Y. Maezawa,3 Swagato Mukherjee,4

H. Ohno,4,5 P. Petreczky,4 C. Schmidt,3 S. Sharma,3 W. Soeldner,6 and M. Wagner71Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52240, USA

2Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University,Wuhan 430079, China

3Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany4Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA

5Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan6Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

7Physics Department, Indiana University, Bloomington, Indiana 47405, USA(Received 28 April 2014; published 11 August 2014)

We compare lattice QCD results for appropriate combinations of net strangeness fluctuations and theircorrelations with net baryon number fluctuations with predictions from two hadron resonance gas (HRG)models having different strange hadron content. The conventionally used HRG model based onexperimentally established strange hadrons fails to describe the lattice QCD results in the hadronic phaseclose to the QCD crossover. Supplementing the conventional HRG with additional, experimentallyuncharted strange hadrons predicted by quark model calculations and observed in lattice QCD spectrumcalculations leads to good descriptions of strange hadron thermodynamics below the QCD crossover. Weshow that the thermodynamic presence of these additional states gets imprinted in the yields of the ground-state strange hadrons leading to a systematic 5–8 MeV decrease of the chemical freeze-out temperatures ofground-state strange baryons.

DOI: 10.1103/PhysRevLett.113.072001 PACS numbers: 12.38.Aw, 11.15.Ha, 12.38.Gc, 12.38.Mh

Introduction.—With increasing temperature, the stronginteraction among constituents of ordinary nuclear matter,mesons and baryons, results in the copious production ofnew hadronic resonances. The newly produced resonancesaccount for the interaction among hadrons to an extent thatbulk thermodynamic properties becomewell described by agas of uncorrelated hadronic resonances [1]. The hadronresonance gas (HRG) model is extremely successful indescribing the hot hadronic matter created in heavy ionexperiments [2]. Abundances of various hadron speciesmeasured in heavy ion experiments at different beamenergies are well described by thermal distributions char-acterized by a freeze-out temperature and a set of chemicalpotentials ~μ ¼ ðμB; μQ; μSÞ for net baryon number (B),electric charge (Q), and strangeness (S) [3]. Nonetheless,details of the freeze-out pattern may provide evidence for amore complex sequential freeze-out pattern [4,5]. Inparticular, in the case of strange hadrons, arguments havebeen put forward in favor of a greater freeze-out temper-ature than that of nonstrange hadrons [6–8].At the temperature Tc ¼ ð154� 9Þ MeV [9] strong

interaction matter undergoes a chiral crossover to anew phase. In the same crossover region, HRG-baseddescriptions of the fluctuations and correlations of con-served charges for light [10], strange [6,11], as well ascharm [12] degrees of freedom break down. Below Tc, bulk

thermodynamic properties as well as conserved chargedistributions are generally well described by a HRG con-taining all experimentally observed resonances (PDG-HRG) listed in the particle data tables [13]. However,there are also some notable differences between latticeQCD results and the PDG-HRG predictions. At temper-atures below Tc, the trace anomaly, i.e., the differencebetween energy density and 3 times the pressure, is foundto be greater in lattice QCD calculations [14–16] than thatof the PDG-HRG. Fluctuations of net strangeness andcorrelations between net strangeness and baryon numberfluctuations are also larger in QCD compared to those ofthe PDG-HRG [17,18]. It has been argued that the formerprovides evidence for contributions of additional, exper-imentally still-unobserved hadron resonances to thethermodynamics of strong interaction matter [19,20].Indeed, a large number of additional resonances has beenidentified in lattice QCD [21] and quark model calculations[22,23]. The presence of such additional flavored hadronsin a thermal medium enhances fluctuations of the asso-ciated quantum numbers and modifies their correlationswith other quantum numbers. In fact, lattice QCD results onnet charm fluctuations and their correlations with baryonnumber, electric charge, and strangeness fluctuations havealso been compared with the expectations from a HRGcontaining additional, experimentally unobserved charmed

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hadrons predicted by quark model calculations [12]. Suchcomparisons have provided evidence for the thermody-namic importance of additional charmed hadrons in thevicinity of the QCD crossover [12].In this Letter, we show that discrepancies between lattice

QCD results and PDG-HRG predictions for strangenessfluctuations and correlations below the QCD crossover canbe quantitatively explained through the inclusion of addi-tional, experimentally unobserved strange hadrons. Thethermodynamic presence of these additional strangehadrons also gets imprinted on the yields of ground-statestrange baryons, resulting in observable consequences forthe chemical freeze-out of strangeness in heavy ioncollision experiments.Hadron resonance gas models.—The partial pressure of

all open strange hadrons can be separated into mesonic andbaryonic components PS;X

tot ¼ PS;XM þ PS;X

B ,

PS;XM=BðT; ~μÞ ¼

T4

2π2X

i∈Xgi

�mi

T

�2

K2ðmi=TÞ

× cosh ðBiμ̂B þQiμ̂Q þ Siμ̂SÞ: ð1Þ

Here, M (B) labels the partial pressure of open strangemesons (baryons), gi is the degeneracy factor for hadronsof mass mi, and μ̂q ≡ μq=T, with q ¼ B, Q, S. The sum istaken over all open strange mesons or baryons listed in theparticle data tables (X ¼ PDG) or a larger set includingadditional open strange mesons [23] and baryons [22]predicted by quark models (X ¼ QM). Throughout thiswork, we refer to the HRG model containing theseadditional, quark model predicted, experimentally undis-covered hadrons as the QM-HRG. In Eq. (1), the classical,Boltzmann approximation has been used, which is knownto be appropriate for all strange hadrons at temperaturesT ≲ Tc [11].The masses and, more importantly, the number of

additional states are quite similar in the quark modelcalculations and the strange hadron spectrum of latticeQCD [21]. HRG models based on either one, thus, givevery similar results. As the lattice computations of thestrange hadron spectrum have so far been performed withunphysically heavy up and down quark masses, fordefiniteness we have chosen to compare our finite temper-ature lattice results with the quark model predictions thatgenerally reproduce the masses of the experimentallyknown states rather well.Figure 1 compares partial pressures of open strange

mesons and baryons calculated within PDG-HRG andQM-HRG models. The additional strange baryons presentin the QM-HRG lead to a large enhancement of the partialbaryonic pressure relative to that obtained from the PDG-HRG model. In the mesonic sector, changes are below 5%even at T ¼ 170 MeV, i.e., above the applicability range ofany HRG [11]. This simply reflects that a large part of the

open strange mesons is accounted for in the PDG-HRGmodel, and the additional strange mesons contributing tothe QM-HRG model are too heavy to alter the pressuresignificantly.Strangeness fluctuations and correlations.—We calcu-

late cumulants of strangeness fluctuations and theircorrelations with baryon number and electric charge in(2þ 1)-flavor QCD using the highly improved staggeredquark action [24]. In these calculations the strange quarkmass (ms) is tuned to its physical value and the masses ofdegenerate up and down quarks have been fixed toml ¼ ms=20. In the continuum limit, the latter correspondsto a pion mass of about 160 MeV. In the relevant temper-ature range, 145 MeV ≤ T ≤ 170 MeV, we have analyzedð10–16Þ × 103 configurations, separated by 10 time units inrational hybrid Monte Carlo updates, on lattices of size6 × 243 and 8 × 323. Some additional calculations on8 × 323 lattices have been performed with physical lightquarks, ml ¼ ms=27, to confirm that the quark massdependence of observables of interest is indeed small.To analyze the composition of the thermal medium in

terms of the quantum numbers of the effective degrees offreedom, we consider generalized susceptibilities of theconserved charges,

χBQSklm ¼ ∂ðkþlþmÞ½PðT; μ̂B; μ̂Q; μ̂SÞ=T4�

∂μ̂kB∂μ̂lQ∂μ̂mS����~μ¼0

; ð2Þ

where P denotes the total pressure of the hot medium. Forbrevity, we drop the superscript when the correspondingsubscript is zero.The correlation of net strangeness with net baryon

number fluctuations normalized to the second cumulantof net strangeness fluctuations, χBS11 =χ

S2 is a sensitive probe

of the strangeness carrying degrees of freedom [25].Consistent continuum extrapolations for this ratio havebeen obtained with two different staggered fermion dis-cretization schemes [17,18]. In Fig. 2 (top) we show ourpresent, refined results obtained for lattices with temporal

PBS,QM/PB

S,PDG

PtotS,QM/Ptot

S,PDG

PMS,QM/PM

S,PDG

1.0

1.1

1.2

1.3

120 130 140 150 160 170

T [MeV]

FIG. 1 (color online). Ratios of partial pressures of open strangehadrons (PS;X

tot ), mesons (PS;XM ), and baryons (PS;X

B ) calculated inHRG with particle spectra from the particle data table, X ¼ PDG,and with additional resonances predicted by the relativistic quarkmodel, X ¼ QM, respectively (see text).

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extents Nτ ¼ 6 and 8, which are in agreement with theearlier results, together with an improved estimate for thecontinuum result based on the enlarged statistics for theselattices. In the crossover region and also at lower temper-atures in the hadronic regime, the lattice QCD results for−χBS11 =χS2 are significantly greater than those of the PDG-HRG model predictions.In the validity range of HRG models, the BS correlation

χBS11 is a weighted sum of partial pressures of strangebaryons, while the quadratic strangeness fluctuations χS2 aredominated by the contribution from strange mesons. Thelarger value of −χBS11 =χS2 found in lattice QCD calculationscompared to that of PDG-HRG model calculations, thus,reflects the stronger increase of PS;QM

B =PS;PDGB compared to

PS;QMM =PS;PDG

M (see Fig. 1). As a consequence, the QM-HRG model provides a better description of QCD thermo-dynamics in the hadronic phase. This can be more directlyverified by considering the ratio of two observables, whichin a HRG model give PS

M and PSB, respectively. There is a

large set of “pressure observables” that can be constructedfor this purpose by using second- and fourth-order cumu-lants of strangeness fluctuations and correlations with netbaryon number [11]. They will all give identical results in agas of uncorrelated hadrons, but differ otherwise. Inparticular, they can yield widely different results in a freequark gas. We use two linearly independent pressure

observables for the open strange meson (MS1 , M

S2) and

baryon (BS1 , B

S2) partial pressures, respectively,

MS1 ¼ χS2 − χBS22 ;

MS2 ¼

1

12ðχS4 þ 11χS2Þ þ

1

2ðχBS11 þ χBS13 Þ;

BS1 ¼ −

1

6ð11χBS11 þ 6χBS22 þ χBS13 Þ;

BS2 ¼

1

12ðχS4 − χS2Þ −

1

3ð4χBS11 − χBS13 Þ: ð3Þ

Three independent ratios BSi =M

Sj are shown in Fig. 2

(bottom). They start to coincide in the crossover regiongiving identical results only below T ≲ 155 MeV. Thisreconfirms that a description of QCD thermodynamics interms of an uncorrelated gas of hadrons is valid till thechiral crossover temperature Tc. Below Tc, the informationone extracts from BS

i =MSj agrees with that of −χBS11 =χS2 . In

the hadronic regime, the ratios calculated in lattice QCD aresignificantly greater than those calculated in the PDG-HRGmodel. QM-HRG model calculations are in good agree-ment with lattice QCD. These results provide evidence forthe existence of additional strange baryons and theirthermodynamic importance below the QCD crossover.Implications for strangeness freeze-out.—Since the ini-

tial nuclei in a heavy ion collision are net strangeness free,the HRG at the chemical freeze-out must also be strange-ness neutral. Obviously, for such a strangeness neutralmedium, all three thermal parameters T, μB, and μS are notindependent; the strangeness chemical potential can beexpressed as a function of T and μB. While μSðT; μBÞ isunique in QCD, for a HRG it clearly depends on the relativeabundances of the open strange baryons and mesons. Forfixed T and μB, a strangeness neutral HRG having a largerrelative abundance of strange baryons over open strangemesons naturally leads to a larger value of μS.Calculations of μSðT; μBÞ in a strangeness neutral HRG

are straightforward. For QCD, this can be obtained fromlattice QCD computations of μS=μB using next-to-leading-order Taylor expansion of the net strangeness density[3,26]. The ratio μS=μB¼s1ðTÞþs3ðTÞðμB=TÞ2þOðμ4BÞis closely related to the ratio χBS11 =χ

S2 shown in Fig. 2. At

leading order, it only receives a small correction fromnonzero electric charge chemical potential μQ=μB,

�μSμB

�

LO≡ s1ðTÞ ¼ −

χBS11χS2

−χQS11

χS2

μQμB

: ð4Þ

The next-to-leading-order correction s3ðTÞðμB=TÞ2 [3] issmall for μB ≲ 200 MeV. We show the leading order resultin Fig. 3. At a given temperature, the strangeness neutralityconstraint gives rise to a larger value, consistent with latticeQCD results, of μS=μB for the QM-HRG compared to thePDG-HRG. In other words, for a given μB=T, the required

cont. est.PDG-HRGQM-HRG

0.15

0.20

0.25

0.30 - χ11BS/χ2

S

Nτ=6: open symbolsNτ=8: filled symbols

B1S/M1

S

B2S/M2

S

B2S/M1

S

0.15

0.25

0.35

0.45

140 150 160 170 180 190

T [MeV]

FIG. 2 (color online). Top: BS correlations normalized to thesecond cumulant of net strangeness fluctuations. Results are from(2þ 1)-flavor lattice QCD calculations performed with a strangeto light quark mass ratio ms=ml ¼ 20 (squares) and ms=ml ¼ 27(diamonds). The band depicts the improved estimate for thecontinuum result facilitated by the high statistics Nτ ¼ 6 and 8data. Bottom: Ratios of baryonic (BS

i ) to mesonic (MSi ) pressure

observables defined in Eq. (3). The dotted and solid lines showresults from PDG-HRG and QM-HRG model calculations,respectively. The shaded region denotes the continuum extrapo-lated chiral crossover temperature at physical values of quarkmasses Tc ¼ ð154� 9Þ MeV [9].

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value of μS=μB necessary to guarantee strangeness neutral-ity is achieved at a lower temperature in the QCD andQM-HRG models than in the PDG-HRG model.The relative yields of strange anti-baryons (H̄S) to

baryons (HS) at freeze-out are controlled by the freeze-out parameters (Tf, μfB, μ

fS),

RH ≡ H̄S

HS¼ e−2ðμ

fB=T

fÞð1−ðμfS=μfBÞjSjÞ: ð5Þ

(For simplicity of the argument we will ignore here theinfluence of a nonvanishing electric charge chemicalpotential. As shown in Fig. 3 a nonzero μQ=T has onlya small influence on μS=μB.) While this relation does notexplicitly depend on the content and spectra of hadrons in aHRG, the presence of additional strange hadrons implicitlyenters through the strangeness neutrality constraint. Asdiscussed before, for different HRG models at fixed T andμB strangeness neutrality leads to different values of μS=μB.Once μfB=T

f and μfS=μfB are fixed through experimental

yields of strange hadrons, it is obvious from Fig. 3 that agiven value of μfS=μ

fB is realized at a larger temperature in

the PDG-HRG model than in the QM-HRG model.Ratios of the freeze-out parameters μfB=T

f and μfS=μfB can

be obtained by fitting the experimentally measured valuesof the strange baryon ratios RΛ ¼ Λ̄=Λ, RΞ ¼ Ξþ=Ξ− andRΩ ¼ Ωþ=Ω− to Eq. (5). Fits of these strange anti-baryonto baryon yields to Eq. (5) result in ðμfS=μfB; μfB=TfÞ ¼ð0.213ð10Þ; 1.213ð30ÞÞ for the NA57 results at

ffiffiffis

p ¼17.3 GeV [27] and (0.254(7),0.697(20)) for the STARpreliminary results at

ffiffiffis

p ¼ 39 GeV [28]. In Fig. 4 weshow comparisons of these (μfS=μ

fB, μ

fB=T

f) values with thelattice QCD, QM-HRG, and PDG-HRG predictions forμS=μB at μB=T ¼ μfB=T

f. By varying the temperatureranges, one can match the values of μS=μB to μfS=μ

fB

and, thus, determine the freeze-out temperatures Tf. As

expected from Fig. 3, the QM-HRG predictions are in goodagreement with lattice QCD results and lead to almostidentical values for Tf. The PDG-HRG-based analysis,however, results in freeze-out temperatures for strangebaryons that are larger by about 8 (5) MeV for the smaller(larger) value of Tf.Conclusions.—By comparing lattice QCD results for

various observables of strangeness fluctuations and corre-lations with predictions from PDG-HRG and QM-HRGmodels, we have provided evidence that additional, exper-imentally unobserved strange hadrons become thermody-namically relevant in the vicinity of the QCD crossover.We have also shown that the thermodynamic relevance ofthese additional strange hadrons modifies the yields of theground-state strange hadrons in heavy ion collisions. Thisleads to significant reductions in the chemical freeze-outtemperature of strange hadrons. Compared to the PDG-HRG, the QM-HRG model provides a more completedescription of the lattice QCD results on thermodynamicsof strange hadrons at moderate values of the baryonchemical potential. This suggests that the QM-HRG modelis probably the preferable choice for the determination ofthe freeze-out parameters also at greater values of thebaryon chemical potential beyond the validity of thepresent lattice QCD calculations.Finally, we note that the observation regarding the

thermodynamic relevance of additional strange hadrons hintsthat an improved HRG model including further, unobservedlight quark hadrons may resolve the current discrepancybetween lattice QCD results for the trace anomaly and theresults obtained within the PDG-HRG model.

This work was supported in part through ContractNo. DE-AC02-98CH10886 with the U.S. Department ofEnergy, through the Scientific Discovery throughAdvanced Computing (SciDAC) program funded by theU.S. Department of Energy, Office of Science, Advanced

cont. est.PDG-HRGQM-HRG

QM-HRG: -χ11BS/χ2

S

0.15

0.20

0.25

0.30

140 150 160 170 180 190

(µS/µB)LO

Nτ=6: open symbolsNτ=8: filled symbols

T [MeV]

FIG. 3 (color online). The leading-order result for the ratio ofstrangeness and baryon chemical potentials versus temperature.Data and HRG model results are for a strangeness neutral thermalsystem having a ratio of net electric charge to net baryon numberdensity NQ=NB ¼ 0.4. The dashed line shows the QM-HRGresult for vanishing electric charge chemical potential. All othercurves and labels are as in Fig. 2.

LQCD: T=155(5) MeVLQCD: T=145(2) MeVPDG-HRGQM-HRG39 GeV (STAR prlim.)17.3 GeV (NA57)

0.15

0.20

0.25

0.30

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

µS/µB

µB/T

T=170 MeV

T=166 MeV

T=162 MeV

T=155 MeV

T=152.5 MeV

T=150 MeV

T=161 MeV

T=158 MeV

T=155 MeV

T=149 MeV

T=147 MeV

T=145 MeV

FIG. 4 (color online). Values of (μfS=μfB, μ

fB=T

f) extracted fromfits to multiple strange hadrons yields (see text) are compared toμS=μB predictions, obtained by imposing strangeness neutrality,from lattice QCD calculations (shaded bands) as well as fromQM-HRG (solid lines) and PDG-HRG (dotted lines) models. Thepredictions are shown for μB=T ¼ μfB=T

f. For each case, thetemperature ranges are chosen such that the predicted valuesreproduce μfS=μ

fB.

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Scientific Computing Research and Nuclear Physics, theBMBF under Grants No. 05P12PBCTA and No. 56268409,the DFG under Grant No. GRK 881, the EU under GrantNo. 283286, and the GSI BILAER Grant. Numericalcalculations were performed using GPU clusters at JLab,Bielefeld University, Paderborn University, and IndianaUniversity. We acknowledge the support of Nvidia throughthe Cuda Research Center at Bielefeld University.

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