The melting and abundance of open charm hadrons Physics Letters B

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Physics Letters B

www.elsevier.com/locate/physletb

The melting and abundance of open charm hadrons

A. Bazavov

^a

, H.-T. Ding

^b

, P. Hegde

^b

, O. Kaczmarek

^c

, F. Karsch

^c,d

, E. Laermann

^c

,

Y. Maezawa

^c

, Swagato Mukherjee

^d

, H. Ohno

^d,e

, P. Petreczky

^d

, C. Schmidt

^c

, S. Sharma

^c,∗

, W. Soeldner

^f

, M. Wagner

^g

aDepartmentofPhysicsandAstronomy,UniversityofIowa,IowaCity,IA52240,USA

bKeyLaboratoryofQuark&LeptonPhysics(MOE)andInstituteofParticlePhysics,CentralChinaNormalUniversity,Wuhan,430079,China cFakultätfürPhysik,UniversitätBielefeld,D-33615Bielefeld,Germany

dPhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USA

eCenterforComputationalSciences,UniversityofTsukuba,Tsukuba,Ibaraki305-8577,Japan fInstitutfürTheoretischePhysik,UniversitätRegensburg,D-93040Regensburg,Germany gPhysicsDepartment,IndianaUniversity,Bloomington,IN47405,USA

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received22April2014

Receivedinrevisedform14August2014 Accepted14August2014

Availableonline19August2014 Editor:J.-P.Blaizot

Ratiosofcumulantsofconservednetchargefluctuationsaresensitivetothedegreesoffreedomthatare carriersofthecorrespondingquantumnumbersindifferentphasesofstronginteractionmatter. Using latticeQCDwith2+1 dynamicalflavorsandquenched charmquarks wecalculatesecondand fourth ordercumulantsofnetcharmfluctuationsandtheircorrelationswithotherconservedchargessuchas netbaryonnumber,electricchargeandstrangeness.Analyzingappropriateratiosofthesecumulantswe probe the nature of charmeddegrees offreedom inthe vicinity of the QCDchiral crossover region.

We show that for temperaturesabove the chiral crossover transition temperature, charmeddegrees of freedom can no longer be described by an uncorrelated gas of hadrons. This suggests that the dissociation of opencharmhadrons and the emergence ofdeconfined charmstatessets in just near thechiralcrossovertransition.TillthecrossoverregionwecomparetheselatticeQCDresultswithtwo hadron resonance gasmodels—includingonlythe experimentallyestablished charmedresonancesand alsoincludingadditionalstatespredictedbyquarkmodelandlatticeQCDcalculations.Thiscomparison providesevidenceforsofarunobservedcharmedhadronsthatcontributetothethermodynamicsinthe crossoverregion.

©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP³.

1. Introduction

Bound states of heavy quarks, in particular the charmonium state J/ψ anditsexcitationsaswell astheheavierbottomonium states,aresensitiveprobes fordeconfiningfeatures ofthequark–

gluon plasma (QGP) [1]. Different excitations of these states are expectedto dissolve atdifferent temperaturesin the QGP, giving risetoacharacteristicsequentialmeltingpattern[2].Recentlattice QCD calculationsofthermal hadron correlation functionssuggest thatcertainquarkoniumstatessurviveasboundstatesintheQGP wellbeyondthepseudo-criticaltemperatureofthechiralcrossover transitionT_c=(154±⁹)MeV[3];the J/ψ andits pseudo-scalar partner

η

c disappear at about 1.5Tc [4], while the heavier bot- tomoniumgroundstatescansurviveevenupto2Tc [5,6].

*

Correspondingauthor.

E-mailaddress:sayantan@physik.uni-bielefeld.de(S. Sharma).

Lightquarkboundstates,ontheotherhand,dissolvealreadyat orclosetothepseudo-criticaltemperature,Tc,reflectingtheclose relation betweenthe chiralcrossover anddeconfinement oflight quark degreesof freedom. This leads to a sudden change in the bulkthermodynamicobservablesandisevenmoreapparentinthe behavior offluctuationsofconservedcharges,i.e.baryon number, electricchargeorstrangeness[7,8].Thesuddenchangeofratiosof differentmoments(cumulants)ofnet-chargefluctuationsandtheir correlations in the transition region directly reflects the change of degreesof freedom that carry the relevantconserved charges.

Thetotalnumberofhadronicdegreesoffreedom,i.e.thedetailed hadronicmassspectrumalsoinfluencesbulkthermodynamics.For instance,thestrongriseofthetrace anomaly(

−^3P)/T⁴,found in lattice QCD calculationsmaybe indicativeforcontributions of yetunobservedhadronresonances[9].

Recently it has been shown that the large set of fourth or- dercumulantsofchargefluctuationsandcross-correlationsamong http://dx.doi.org/10.1016/j.physletb.2014.08.034

0370-2693/©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP³.

fluctuationsofconservedchargesallows fora detailedanalysisof thechangefromhadronictopartonicdegreesoffreedomindiffer- entcharge sectors[10]. Forinstance, changes ofdegreesof free- dominthestrangemesonandbaryonsectorsofhadronicmatter canbe analyzedseparately bychoosingappropriate combinations ofcharge fluctuation observables.This ledto the conclusion that adescriptionofstronginteractionmatterintermsofuncorrelated hadronicdegreesoffreedom breaksdownforall strangehadrons inthechiralcrossoverregion,i.e.atT^{160 MeV[10],whichsug-} geststhatstrangenessgetsdissolvedatorcloseto T_c.Thisfinding hasbeenconfirmedwiththeanalysispresentedin[11].

A more intriguing question is what happens to the charmed sector of the hadronic medium at the QCD transition tempera- ture.Whileitseemstobeestablishedthatcharmoniumstates,i.e.

bound states with hidden charm, still exist in the QGP at tem- peratureswellabove Tc,thismaynotbethecaseforheavy–light mesons or baryons, i.e. open charm mesons (D, Ds) [12,13] or charmedbaryons (Λc, Σc, Ξc, Ωc). Toaddress this question we calculatecumulants ofnet-charm fluctuationsaswell as correla- tions between moments of net-charm fluctuationsand moments ofnetbaryonnumber,electriccharge orstrangenessfluctuations.

Motivated by the approach outlined in Ref. [10] we analyze ra- tios of observablesthat may, atlow temperature, be interpreted as contributions of open charm hadrons to the partial mesonic orbaryonic pressure of strong interaction matter. We show that adescriptionofnetcharmfluctuationsintermsofmodelsofun- correlatedhadronsbreaksdownattemperaturesclosetothechiral crossovertemperature.Wefurthermoreshow thatatlowtemper- atures the partial pressure calculated in the open charm sector islarger than expected fromhadron resonancegas (HRG) model calculationsbased onall experimentallymeasured charmedreso- nancesaslistedintheparticledatatables[14].It,however,agrees well withan HRGbased oncharm resonancesfromquark model [15–18]andlatticeQCDcalculations[19–21].Thispointsattheex- istenceandthermodynamicimportance ofadditional,experimen- tallysofarnotestablished,opencharmhadrons.

2. Thecharmedhadronresonancegas

Whilelight quark fluctuationscan be quite well described by a hadron resonance gas [22] built up from experimentally mea- sured resonances that are listed in the particle data tables [14]

itis notatall obvious thatthis sufficesinthe caseofthe heavy open charm resonances. The particle data tables only list a few measuredopencharmresonances.Manymorearepredictedinthe relativisticquark model[15–18] andlattice QCD [20,21] calcula- tions.Infact,thelargesetofexcitedcharmedmesonsandbaryons foundin latticeQCD calculationscloselyresemblesthe excitation spectrumpredictedinquarkmodelcalculations.Itisexpectedthat many new open flavor states will be detected in upcoming ex- periments at Jefferson Laboratory, FAIR and the LHC [16,23–25].

Iftheseresonancesare indeedpartof thecharmedhadron spec- trumofQCD,theybecomeexcitedthermallyandcontributetothe thermodynamicsofthecharmedsectorofahadronresonancegas.

Theywillshowupasintermediatestatesinthehadronizationpro- cessofa quark–gluonplasma formedinheavy ioncollisions and influencetheabundances ofvarious particlespecies[26].Heavy–

lightbound statesalsoplay an importantrole inthe break-upof quarkoniumboundstates.InlatticeQCDcalculationstheircontri- butionbecomes visiblein theanalysisofthe heavyquark poten- tialwheretheycanhelptoexplainthenon-vanishingexpectation valueofthePolyakovloopatlowtemperatures[27,28].

In order to explore the significance of a potentially large ad- ditionalsetofopen charmresonancesin thermodynamiccalcula- tionsatlowtemperaturewehaveconstructedHRG modelsbased

on different sets of open charm resonances. In addition to the HRGmodelthat isbasedonall experimentallyobservedcharmed hadrons (PDG-HRG), we also construct an HRG model based on a setofcharmedhadronscalculatedina quarkmodel (QM-HRG) whereweusedthecharmedmeson[17]andcharmedbaryon[18]

spectrumcalculatedbyEbertet al.¹

One may wonder whether all the resonances calculated in a quark model exist or are stable and long-lived enough to con- tributetoe.g.thepressureofcharmedhadrons.However,ashighly excited states with masses much larger than the ground state energy in a given quark flavor channel are strongly Boltzmann suppressed, they play no significant role in thermodynamics. For thisreasonwe alsoneed not considermultiple charmedbaryons or open charm hybrid states that have been identified in lattice QCD calculations [20,21] but generally have masses more than (0.8–1) GeV above thoseof the groundstate resonances. We ex- ploretheimpactofsuchheavystatesbyintroducingdifferentcut- offstothemaximummassuptowhichopencharmresonancesare takenintoaccountintheHRGmodel.Forinstance,QM-HRG-3in- cludesallcharmedhadronresonancesdeterminedinquark model calculationsthathavemasseslessthan3 GeV.

We calculate the open charm meson (MC(T,

μ

)) and baryon (B_C(T,

μ

))pressureinunitsofT⁴,suchthat thetotalcharmcon- tributiontothepressureiswrittenas PC(T,

μ

)/T⁴=^MC(T,

μ

)+ BC(T,

μ

). As the charmed states are all heavy compared to the scale of the temperatures relevant for the discussion of the thermodynamics in the vicinity of the QCD crossover transition, a Boltzmannapproximationisappropriateforallcharmedhadrons,

MC

(

T

, μ ) =

¹ 2

π

²

i∈C-mesons

gi

mi

T

2

K2

(

mi

/

T

)

×

cosh

(

Qi

μ ˆ

Q

+

Si

μ ˆ

S

+

Ci

μ ˆ

C

),

BC

(

T

, μ ) =

¹

2

π

²

i∈^C-baryons g_i

m_i T

2

K2

(

m_i

/

T

)

×

^cosh

(

Bi

μ ˆ

B

+

^Qi

μ ˆ

Q

+

^Si

μ ˆ

S

+

^Ci

μ ˆ

C

).

(1) Here,

μ

₌(

μ

B,

μ

Q,

μ

S,

μ

C),

μ

ˆ _≡

μ

/T and gi are thedegeneracy factorsforthedifferentstateswithelectriccharge Qi,strangeness S_iandcharmC_i.

Results from calculations of open charm meson and baryon pressuresusingdifferentHRG modelsareshowninFig. 1.Thein- fluenceofadditionalstatespredictedbythequarkmodelisclearly visiblealreadyintheQCDcrossovertransitionregion.AtTc,differ- encesbetweenPDG-HRG (dashedlines) andQM-HRG(solidlines) inthebaryonsector areaslargeas40% whiletheyarenegligible in themeson sector. Thisreflects that the experimentally known mesonspectrumismorecompletethanthebaryonspectrum.

In the open charm meson sector, the well established excita- tionscoveramassrangeofabout700 MeVabovethegroundstate D,D_s-mesons.Inthecharmedbaryon sectormuchlessisknown, for instance, experimentally well known excitations of Ξc range upto 350 MeVabovethegroundstateandinthedoublystrange charmedbaryonsectoronlytwo Ωc statesseparatedby 100 MeV arewellestablished.

As a consequence of the limited knowledge of the charmed baryon spectrum compared to the open charm meson spectrum, the ratio of partial pressures in the baryon and meson sectors differs strongly betweenthe PDG-HRG andthe QM-HRG. This is

1 Thethermodynamicconsiderationspresentedherearemainlysensitivetothe numberofadditionalhadronsincludedinthecalculationsandnottotheprecise valuesoftheirmasses.ThuslatticeQCDresultsonthecharmedbaryonspectra[21]

alsoleadtosimilarconclusions.

Fig. 1.Partialpressureofopencharmmesons(Mc,bottom),baryons(Bc,middle) andtheratioBC/MC (top)inagasofuncorrelatedhadrons,usingallopencharm resonanceslistedintheparticledatatable(PDG-HRG,dashedlines)[14]andusing additionalcharmresonancescalculatedinarelativisticquarkmodel(QM-HRG,solid lines)[17,18].AlsoshownareresultsfromHRGmodelcalculationswheretheopen charmresonancespectrumiscutoffatmass3 GeV(QM-HRG-3)and3.5 GeV(QM- HRG-3.5).Attemperaturesbelow160 MeVthelattercoincideswiththecomplete QM-HRGmodelresultstobetterthan1%.

shown in Fig. 1 (top). Significant differences between the QM- HRG-3 andPDG-HRGresultsalso indicatethat almost halfofthe enhanced contributions actually comes from additional charmed baryonsthatare lighterthantheheaviest PDGstate.Similar con- clusions can be drawn when analyzing partial pressures in the strange-charmedhadronsectorortheelectricallychargedcharmed hadronsectors.

3. Calculationofcharmfluctuationsin(2+^{1)-flavorlatticeQCD}

Inorder todetect changesin therelevant degreesoffreedom thatare thecarriersofcharmquantumnumbersatlowandhigh temperaturesaswell asto studytheir propertieswe calculatedi- mensionlessgeneralizedsusceptibilitiesofconservedcharges,

χ

_klmn^{B Q SC}

= ∂

^(k+l+m+n)

[

^P

( μ ˆ

B

, μ ˆ

Q

, μ ˆ

S

, μ ˆ

C

)/

T⁴

]

∂ μ ˆ

^k_B

∂ μ ˆ

^l_Q

μ ˆ

^m_S

∂ μ ˆ

ⁿ_C

μ=⁰

.

(2) Here P denotesthetotalpressure ofthe system.Inthefollowing wealsousetheconventiontodropasuperscriptin

χ

_klmn^{B Q SC} when thecorrespondingsubscriptiszero.

Forour analysisof netcharm fluctuationswe use gauge field configurationsgeneratedwiththehighlyimprovedstaggeredquark (HISQ)action[29].UseoftheHISQactioninthecharmsectorsin- cludestheso-called -termandthusmakesourcalculationsfreeof tree-levelorder(amc)⁴ discretizationerrors[29],wheremc isthe bare charmquark mass inunits ofthe latticespacing. Thesedy- namical(2+^{1)-flavorQCD}calculationshavebeencarriedoutwith astrangequarkmass(ms)thathasbeentunedtoitsphysicalvalue andlight(u,d)quarkswithmassml/ms=¹/20.Inthecontinuum limit,thelattercorrespondstoalightpseudo-scalarmassofabout 160 MeV.Thecharmquark sectoristreatedwithin thequenched approximation,neglectingtheeffectsofcharmquarkloops.Within thetemperaturerangerelevantforthepresentstudy,thequenched approximationforthecharmquarksisverywelljustified.Various lattice QCD calculations using dynamical charm have confirmed that contributions ofdynamical charm quarksto bulk thermody- namicquantities,includingthe gluonicpartofthetrace anomaly

aswell asthe susceptibilities oflight,strange andcharm quarks, remain negligible even up to temperatures as high as 300 MeV [30,31].Wenotethatthesequantitiesdirectlyprobetheinfluence ofvirtualquarkpairsonobservablescalculatedatafixedvalueof the temperature.Unlikeinthesecasesthereisnosimpleobserv- ableknownthatwouldallowustodirectlycalculatethepressure atfixed temperature.Thismaybethe reasonfordifferencesseen in currentcalculationsofthepressure [30,31] usingquenchedor dynamical charm.In thiswork, we onlyuseobservables that are of the formertype andalsodo not require anymultiplicative or additiverenormalization.

The line of constant physics for the charm quark has been determined at zerotemperature by calculating thespin-averaged charmoniummass[32], ¹₄(mηc+^3mJ/ψ).Forthispurposeweused gauge fieldconfigurationsgeneratedbyhotQCDonlatticesofsize 32⁴ and 32³ ·^{48 in the range of gauge couplings, 6}.39≤β = 10/g²≤⁷.15 [3,22].Onfinite temperaturelatticeswithtemporal extent Nτ =^{8, this covers the} temperature range² 156.8 MeV≤ T≤³³⁰.2 MeV.Ontheselatticesandfortheslightlylarger-than- physicallight quarkmassvalue usedinourcalculationsthetran- sitiontemperatureis158(3) MeV,i.e.about4 MeVlargerthanthe continuum extrapolatedresultsatthephysicalvaluesofthelight andstrangequarkmasses[3].Weconsiderthisdifferenceofabout 3%asthetypicalsystematicerrorforalltemperaturevaluesquoted forouranalysis,whichisnotextrapolatedtothephysicalpointin thecontinuumlimit.

Thelineofconstant physicsforthecharmquarksectoriswell parametrizedby

m_ca

=

^c0R

(β) +

^c2R³

(β)

1

+

^d2R²

(β) ,

(3)

with R(β) denoting thetwo-loop β-functionof massless 3-flavor QCD and c0=⁵⁶.0, c2=¹.16×^{106, d}2 =⁸.67×^{103. On this} line the charm quark mass varies by less than 5%. The ratio of charmandstrangequarkmasses,mc/ms,variesbyabout10%,with mc/ms=¹².42 atβ=⁶.39 andmc/ms=¹¹.28 atβ=⁷.15.

Formostofourcalculationsweusedatasetsonlatticesofsize 32³·^{8.Asubset ofthese}configurationshasalreadybeenusedfor theanalysisofstrangenessfluctuations[10].Thesedatasetshave beenenlargedandnowcontainupto16 700configurationsatthe lowest temperature,separatedby10timeunitsinrationalhybrid MonteCarloupdates. Someadditionalcalculationshavebeenper- formedoncoarser24³·6 lattices,withfixedmc/ms=^12,inorder tocheckcut-offeffectsalsointhecharmquarksector.Wesumma- rizethestatisticsexploitedinthiscalculationinTable 1.Wecalcu- lateallthemomentsofnetcharmfluctuationsneededtoconstruct uptofourthordercumulantsthatcorrelatenet-charmfluctuations withnet baryon number,electriccharge andstrangenessfluctua- tions.Asthecalculationofcharmfluctuationsisfastwecanafford touseoneachgaugefieldconfigurationupto6000Gaussiandis- tributed random sourcevectors forthe inversionof thecharmed fermion matrix.This leaves uswithstatistical errors that mainly arisefromfluctuationsinthelightandstrangequarksectorswhere wehaveused1500randomsourcevectorsfortheinversionofthe correspondingfermionmatrices.

4. Partialpressureofopencharmhadronsfromfluctuationsand correlations

Our analysis of higher order cumulants of net charm fluctu- ations and their correlations with net baryon number, electric charge and strangeness, closely follows the concepts developed

2 Atfinitelatticespacing fK hasbeenusedtosetthetemperaturescale[22].

Table 1

Numberofconfigurationsanalyzedat differentvaluesof thetemperatureandondifferentsizelattices.

Nτ=8 Nτ=6

T[MeV] # conf T[MeV] # conf

156.8 16 700

162.0 9520 162.3 7820

165.9 9000 166.7 3590

168.6 6130 170.2 5140

173.5 5510

178.3 5500

184.8 5730

189.6 4930

196.0 6000

207.3 1800

237.1 1600

273.9 1600

330.2 1600

forouranalysisof strangenessfluctuations[10]. Thelarge charm quarkmass,mc^{T, howeverleadsto some} simplifications.First ofall, fortemperaturesa few times theQCD transitiontempera- ture, Boltzmannstatistics isstill a good approximation fora free charm quark gas. In the high temperature phase we can thus compareourresults withcumulantsderived from afree massive quark–antiquarkgasintheBoltzmannapproximation,

P_c,free

(

m_c

/

T

, μ /

T

)

T⁴

=

³

π

²

m_c T

2

K₂

(

m_c

/

T

)

cosh

μ ˆ

B

3

+

²

3

μ ˆ

Q

+ ˆ μ

C

,

(4)

whereweusedexplicitlythequantumnumbersofcharmquarks.

Anothersimplificationoccursatlow temperatures,where weex- pect a hadron resonance gas to provide a good description of cumulants ofnet charge fluctuations. At thesetemperatures, the pressure of the hadronic medium receives contributions from different open charm mesons and baryons. Using the fact that thesehadronscarryintegerconservedchargesforbaryonnumber (|^B|≤^{1),electriccharge(}|^Q|≤^2),strangeness(|^S|≤^2)andcharm (|^C|≤^{3),wecanseparatethetotalopencharm}contributiontothe pressure in terms of different mesonic (M_C) and baryonic (B_C,i withi≡ |^C|=¹,2,3)sectors,

PC

(

T

, μ )

T⁴

=

MC

(

T

, μ ) +

BC

(

T

, μ )

=

^MC

(

T

, μ ) +

3

i=¹

B_C,i

(

T

, μ ).

(5)

Inthiswork,we further motivatethe decompositionofthe open charm pressure interms of partial pressures in different electric chargeandstrangenesssectors. In suchcases,we decompose the correspondingpartialpressuresas

PC,X

(

T

, μ )

T⁴

=

^MC,|^X|=¹

(

T

, μ ) +

^BC,|^X|=¹

(

T

, μ )

+

^BC,|X|=2

(

T

, μ ),

X

=

^Q

,

S

.

(6) Due to the large charm quark mass, the masses of charmed baryons with |^C|=^{2 or 3 are} substantially larger than those of the |^C|=^{1 hadrons; e.g.} =^mC=²−^mC=¹¹.2 GeV. Even at T ^{200 MeV, i.e.well beyond the validity range of any HRG} model,thecontributionofa |^C|=^{2 hadronto P}C(T,

μ

)/T⁴ thus issuppressedbya factorexp(−/T)^{10−3 relativeto thatofa} corresponding|^C|=^{1 hadron.Thelatterthuswilldominatetheto-} talpartialcharm pressure, P_C(T,

μ

)/T^4MC(T,

μ

)+^BC,1(T,

μ

).

Similarlythebaryoncontributionstothechargedandstrangepar- tialcharmpressureswillbedominatedby|^C|=^{1 baryonsonly.}

The dominanceof the|^C|=^{1 sectorin all fluctuationobserv-} ablesinvolvingopencharmhadronsisimmediatelyapparentfrom thetemperaturedependenceofsecondandfourthordercumulants ofnet-charmfluctuations,

χ

₂^C and

χ

₄^C,aswellasthecorrelations betweenmoments ofnet baryon numberandcharm fluctuations (BC-correlations). As long as the strong interaction medium can be described by a gas of uncorrelatedhadronsthese observables havesimple interpretationsin termsofpartialpressure contribu- tions MC andB_C,i evaluatedat

μ

=^0,

χ

_n^C

=

MC

+

BC,1

+

2ⁿBC,2

+

3ⁿBC,3

MC

+

BC,1

,

χ

_mn^BC

=

BC,1

+

2ⁿBC,2

+

3ⁿBC,3

BC,1

,

(7) where n,m>0 andn orn+^{m are even,} respectively. Here and in the following we often omit the arguments of the functions M_C(T,0),B_C,i(T,0).

The quantity (

χ

₄^C −

χ

₂^C)/12 is an upper bound for the con- tribution to the pressure fromthe |^C|>1 channels in the open charm sector. Forall temperaturevaluesanalyzed by us, we find thatthisquantityislessthan0.2%of

χ

₂^C.Infact,fortemperatures T≤200 MeV thedifferencevanisheswithinerrors.Thismayeasily be understoodasthisdifference isonlysensitivetocontributions ofbaryonswithcharm|^C|=²,3;i.e.

χ

₄^C−

χ

₂^C=^12BC,2+^72BC,3 in a gas of uncorrelated hadrons. We thus conclude that up to negligiblecorrectionsallcumulantsofnet-charmfluctuations,

χ

_n^C, with n>0 and even, directly give the total open charm contri- butionto thepressureinan HRG, PC≡^PC(T,0)

χ

₂^C.Moreover, eachoftheoff-diagonalBC-correlations,

χ

_nm^BC,withn+^m>0 and even,approximates wellthe partialpressureofcharmedbaryons, BC≡^BC(T,0)

χ

_mn^BC. In Fig. 2(right)we show lattice QCD data for

χ

₄^C/

χ

₂^C.Inthecrossoverregionthisratioisclosetounity.This confirms that atlow temperaturethe charmfluctuations

χ

₂^C and

χ

₄^C indeed are equally good representatives for the open charm partialpressure.

5. Meltingofopencharmhadrons

In order to determine the validity range of an uncorrelated hadronresonancegasmodeldescriptionoftheopen charmsector ofQCD,withoutusingdetailsoftheopencharmhadronspectrum, weanalyzeratiosofcumulantsofcorrelationsbetweennetcharm fluctuationsandnet-baryonnumberfluctuations(BC-correlations) aswellascumulantsofnetcharmfluctuations(

χ

_n^C).

As motivated in the previous section, a consequence of the dominance of the |^C|=^{1 charmed baryon sector in thermody-} namicconsiderations is that,to a goodapproximation, BC-corre- lationsinthehadronicphaseobeysimplerelationsas

χ

_nm^BC

χ

₁₁^BC

,

n

+

^m

>

2 and even

.

(8) The ratio of any two of these susceptibilities, i.e.

χ

_nm^BC/

χ

_kl^BC

thus will be unity in a hadron resonance gas irrespective of its composition and the details of the baryon resonance spectrum.

In Fig. 2 (left) we show the ratio

χ

₁₃^BC/

χ

₂₂^BC. It clearly suggests that above the crossover region, an uncorrelated gasof charmed baryonsdoesnolonger providean appropriatedescription ofthe BC-correlations.Alsoshowninthisfigureistheratio

χ

₁₁^BC/

χ

₁₃^BC.It isconsistent withunity foralltemperatures becausethe relation

χ

_1n^BC₌

χ

₁₁^BC not only holds ina non-interacting charmedhadron gas (Eq. (8)), but alsois validin an uncorrelated charmedquark gas, asis easily seen from Eq.(4).Higher orderderivatives with respect to baryon chemical potentials,on the other hand,distin- guish between the hadronic and partonic phases. E.g., one finds

Fig. 2.Thelefthandfigureshowstworatiosoffourthorderbaryon-charm(BC)correlations.Inanuncorrelatedhadrongasbothratiosreceivecontributionsonlyfrom charmedbaryons.Similarly,fortherighthandfiguretheratioχ4^C/χ2^C isdominatedbyand(χ2^C−χ22^BC)/(χ4^C−χ13^BC)onlyreceivescontributionsfromopencharmmesons.

Thehorizontallinesontherighthandsideofbothfiguresshowtheinfinitetemperaturenon-interactingcharmquarkgaslimitsoftherespectivequantities.Theshaded regionindicatesthechiralcrossovertemperatureatthe physicalpionmassinthecontinuumlimit, Tc=(154±9)MeV,determinedfromthemaximum ofthechiral susceptibility[3].Calculationshavebeenperformedonlatticesofsize32³·8 (filledsymbols)and24³·6 (opensymbols).

Fig. 3.Ratiosofbaryon-electriccharge(B Q),baryon-strangeness(B S)andbaryon- charm(BC)correlationscalculatedonlatticesofsize32³·^{8.InthecaseofB Qand} B Scorrelationsweshowresultsfromthe(2+^1)-flavorcalculationswhereBandQ donotcontainanycharmcontribution.ThesedataaretakenfromRefs.[10,33].The shadedregionshowsthechiralcrossoverregionasinFig. 2.Horizontallinesonthe rightsideshowcorrespondingresultsforanuncorrelatedquarkgas.Itshouldbe notedthatthislimitingvalueisnotdefinedforχ₃₁^{B Q}/χ₁₁^{B Q} sincethedenominator aswellasthenumeratorvanishesinperturbationtheoryuptoO(g⁴).

thatfornbeingodd,

χ

_n1^BC/

χ

₁₁^BC₌1 inahadrongasand3¹⁻ⁿ inan uncorrelatedcharmquarkgas.

Subtracting any of the BC-correlations from the quadratic or quarticcharmfluctuationsprovidesanapproximationfortheopen charmmesonpressureina gasofuncorrelatedhadrons. Wethus expectforinstance,therelation

M_C

= χ

₄^C

− χ

₁₃^BC

= χ

₂^C

− χ

₂₂^BC (9) to hold at low temperatures. Their ratio thus should be unity at low temperatures as long as the HRG description is valid.

Fig. 2 (right) showsthe ratioof the two observables introduced inEq.(9).Itisobviousfromthefigurethatalsointhemesonsec- tor,an HRGmodeldescriptionbreaksdowninthecrossoverregion atorcloseto Tc.

The behavior seen in Fig. 2 for correlations between net charm fluctuations and net baryon number fluctuations, in fact, is quite similar to the behavior seen in the strangeness sector (B S-correlations) [10] aswell asin the light quark sector which dominates the correlations between net electric charge and net baryonnumber(B Q-correlations)[22].InFig. 3 weshow acom- parisonofratiosofcumulantsofsuchcorrelations.Forthe B Sand B Q correlationswith the lighter quarks we have two additional

datapoints below156 MeV.Inthecharm sectorwe chooseara- tio of cumulantsinvolving higher order derivativesin the charm sector as correlations involving only first order derivatives have large statistical errors. These ratios all should be unity in a gas of uncorrelated hadrons. It is apparent from Fig. 3 that such a description breaks down for charge correlations involving light, strange, orcharmquarksinorjustabovethe chiralcrossoverre- gion.

6. Abundanceofopencharmhadrons

We now turn to the analysis of ratios of charge correlations andfluctuationsthatare,incontrasttotheratiosshowninFig. 2, sensitivetosomedetailsoftheopencharmhadronspectrum.We constructpartialpressure componentsfortheelectrically charged charmed mesons and the strange-charm mesons, MQ C

χ

₁₃^{Q C}₋

χ

₁₁₂^{B Q C} andMSC

χ

₁₃^SC₋

χ

₁₁₂^{B SC},respectively.We alsoconsiderthe partialpressureofallopencharmmesonsM_C=

χ

₄^C−

χ

₁₃^BC asmo- tivatedinEq.(9).Usingtheseobservablesweconstructratioswith cumulants, whichinan HRGreceive contributions onlyfromdif- ferentcharmedbaryonsectorsinthenumerator,

R^BC₁₃

= χ

₁₃^BC

M_C

,

R₁₃^{Q C}

= χ

₁₁₂^{B Q C}

M_{Q C}

,

R₁₃^SC

= − χ

₁₁₂^{B SC}

M_SC

.

(10) In an HRG, the first ratiojust givesthe ratioof charmedbaryon andmesonpressure, (R₁₃^BC)H R G=^BC/M_C.Inthe twoother cases, thenumeratorisaweightedsumofpartialcharmedbaryonpres- sures in charge sectors |^X|=^{1 and} |^X|=^{2 with X}=^{Q and S,} respectively.TheseratiosareshowninFig. 4.

HRG model predictions for these ratios strongly depend on the relative abundanceofthe charmedbaryonsover open charm mesons. Shownin Fig. 4are resultsobtained fromthe PDG-HRG calculation (dashed lines) and the QM-HRG (solid lines). Clearly in the temperature range of the QCD crossover transition, the lattice QCD data for these ratios are much above the PDG-HRG model results. In all the cases,the deviation from the PDG-HRG at T=^{160 MeV is 40% orlarger. As discussed inSection 2,this} maynotbetoosurprisingasonlyafewcharmedbaryonshaveso far beenlistedin theparticledatatables. The latticeQCD results insteadshowgoodagreementwithan HRGconstructedfromopen charmmesonandbaryonspectracalculatedinarelativisticquark model[17,18].ThedifferenceinPDG-HRGandQM-HRGmodelcal- culations mainly arises from the baryon sector (see Fig. 1). The observablesshowninFig. 4 thusprovidefirst-principles evidence forasubstantialcontributionofexperimentallysofarunobserved

Fig. 4.Thermodynamiccontributionsofallcharmedbaryons,R^BC₁₃ (top),allcharged charmedbaryons, R^{Q C}₁₃ (middle)andallstrangecharmedbaryons, R₁₃^SC (bottom) relativetothatofcorrespondingcharmedmesons(seeEq.(10)).Thedashedlines (PDG-HRG) arepredictionsfor anuncorrelated hadron gas using onlythe PDG states.Thesolidlines(QM-HRG) aresimilarHRG predictionsincludingalso the statespredictedbythequarkmodelofRefs.[17,18].Thedottedlines(QM-HRG- 3)arethesameQMpredictions,butonlyincludingstateshavingmasses<3 GeV.

TheshadedregionshowstheQCDcrossoverregionasinFig. 2.Thehorizontallines ontherighthandsidedenotetheinfinitetemperaturenon-interactingcharmquark gaslimitsfortherespectivequantities.ThelatticeQCDdatahavebeenobtainedon latticesofsize32³·8 (filledsymbols)and24³·6 (opensymbols).

charmedbaryonstothepressureofahadronresonancegas.³ This is also consistent witha large set of additional charmedbaryon resonancesthatarepredictedinlatticeQCDcalculations[21].

7.Conclusions

We have calculated second and fourth order cumulants of netcharmfluctuationsandtheir correlationswithfluctuationsof other conserved charges, i.e. baryon number, electric charge and strangeness.Ratiosofsuchcumulantsindicatethatadescriptionof thethermodynamicsofopen charm degreesoffreedom interms ofan uncorrelated charmedhadron gas is validonly up to tem- peraturesclosetothechiralcrossovertransitiontemperature.This suggeststhat open charm hadronsstart to dissolve alreadyclose tothechiralcrossover.Moreover,observablesthataresensitiveto theratio ofthe partial open charm mesonand baryon pressures aswellastheircounterpartsintheelectricallychargedcharmsec- torandthe strange-charm sector suggest that a large numberof sofarexperimentallynotmeasuredopencharmhadronswillcon- tributetobulkthermodynamicsclosetothemeltingtemperature.

Thisshould betakenintoaccountwhenanalyzingthehadroniza- tionofcharmedhadronsinheavyioncollisionexperiments.

Sofarour analysishasbeenperformedby treating thecharm quark sector in quenched approximation using fully dynamical (2+^{1)-flavorgaugefield}configurationsasthermalheatbath.This, infact, seemstobeappropriateforthesituationmetinheavyion

3 It should be obvious that this contribution to the pressure nonetheless is stronglysuppressedrelativetothecontributionofthenon-charmedsectorinHRG models.

collisions,wherecharmquarksarenotgeneratedthermallybutare embedded intothethermalheatbathoflightandstrangequarks through hard collisions at early stages of the collision. We also donotexpect thatthecumulantratiosanalyzedherewillchange significantly by treating also the charm sector dynamically. This, however,shouldbeverifiedinfuturecalculations.

Acknowledgements

This work has been supported in part through contract DE- AC02-98CH10886withtheU.S.DepartmentofEnergy,throughSci- entific Discoverythrough Advanced Computing(SciDAC) program fundedby U.S.DepartmentofEnergy,OfficeofScience,Advanced Scientific ComputingResearchandNuclearPhysics,theBMBFun- der grant 05P12PBCTA, the DFG under grant GRK881, EU under grant 283286 and the GSI BILAER grant. Numerical calculations havebeenperformedusingGPU-clustersatJLab,BielefeldUniver- sity,PaderbornUniversity,andIndianaUniversity.Weacknowledge thesupportofNvidiathroughthe CUDAresearch centeratBiele- feldUniversity.

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