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EPJWebofConferenceswillbesetbythepublisher DOI:willbesetbythepublisher (cid:13)c Ownedbytheauthors,publishedbyEDPSciences,2017 7 1 0 2 Thermal D mesons from anisotropic lattice QCD n a J 1 AoifeKelly1 and Jon-IvarSkullerud1,a 3 1DepartmentofMathematicalPhysics,NationalUniversityofIrelandMaynooth,Maynooth,CoKildare,Ire- land ] t a l - Abstract.Wepresentresultsforcorrelatorsandspectralfunctionsofopencharmmesons p using2+1flavoursofcloverfermionsonanisotropiclattices.TheDmesonsarefoundto e dissociateclosetothedeconfinementcrossovertemperatureT . Ourpreliminaryresults h c suggestashiftinthethermalDmesonmassbelowT .Mesonscontainingstrangequarks [ c exhibitsmallerthermalmodificationsthanthosecontaininglightquarks. 1 v 5 0 1 Introduction 0 9 The study of heavy quarks, and in particular charm quarks, in high temperature QCD has a long 0 history. Until recently, however, the focus has been almost exclusively on quarkonia, ie the cc¯ and 1. bb¯ systems. Thishaschangedin thepastfew years, with anincreasedexperimentalandtheoretical 0 interestinopencharm,includingDmesonflow[1]andyields[2–4]. 7 Thereareseveralreasonswhyitisimportanttostudyopencharmalongsidecharmonium. Since 1 charm quarks are not created or destroyed thermally to any significant degree at the temperatures : v reached at RHIC and LHC, any decreased yield of charmonium states must be associated with an i X increasedyieldinopencharm(althoughthiscanbehardtoidentifyexperimentally).Ithasalsobeen suggestedthatthedoubleratioR (J/ψ)/R (D)maybeabettermeasureofmediummodifications r AA AA a thanthetraditionalR (J/ψ)asanumberofsystematiceffectsincludingcoldnuclearmattereffects AA canceloutinthisratio. Animportantissuetounderstandinthiscontextistowhatextentopencharmmesonsexperience thermal modifications below T , and whether some bound states may survive above T . It has for c c examplebeensuggestedthattheabundanceofD statesmayincreaserelativetothestatescontaining s lightquarks. Therehavebeenveryfewlattice studiesof opencharmathightemperatureso far[5–7]. These haveusedspatialcorrelatorsandcumulantstoassessthepossiblesurvivalofopencharmboundstates aboveT .Herewewillforthefirsttimepresentresultsfortemporalcorrelatorsandspectralfunctions c ofopencharmmesons. 2 Methods ThisstudyusestheFASTSUMcollaborationensemble[8,9],with2+1flavoursofanisotropicclover fermionsandamean-fieldimprovedanisotropicSymanzikgaugeaction. Thespatiallatticespacing ae-mail:[email protected] EPJWebofConferences Table1.LatticevolumesN3×N ,temperaturesT andnumberofconfigurationsN usedinthiswork.The s τ cfg pseudocriticaltemperatureT wasdeterminedfromtheinflectionpointofthePolyakovloop[9]. c N N T (MeV) T/T N s τ c cfg 24 128 44 0.24 500 24 40 141 0.76 500 24 36 156 0.84 500 24 32 176 0.95 1000 24 28 201 1.09 1000 24 24 235 1.27 1000 24 20 281 1.52 576 24 16 352 1.90 1000 is a = 0.123 fm and the anisotropy ξ = a /a = 3.5. The strange quark mass is tuned to its s s τ physical value, while the light quarks correspond to a pion mass m ≈ 380 MeV. Further details π abouttheensemblesaregivenintable1. TheactionisidenticaltothatusedbytheHadronSpectrum Collaboration[10],andthezerotemperature(N =128)configurationswerekindlyprovidedbythem. τ Forthecharmquarks,wehavealsousedtheparametersfromtheHadronSpectrumCollaboration[11]. BoththeconfigurationsandthecorrelatorsweregeneratedusingtheChromasoftwarepackage[12]. Informationabout hadronic states (including energies, widths and thresholds) in the medium is containedinthespectralfunctionρ(ω;T),whichformesonicstatesisrelatedtotheeuclideancorre- latorG(τ;T)accordingto cosh[ω(τ−1/2T)] G(τ;T)= ρ(ω;T)K(τ,ω;T)dω, K(τ,ω;T)= , (1) Z sinh(ω/2T) Determiningρ(ω) froma given(noisy)G(τ)cannotbedoneexactly; wewillhereemploythemax- imum entropy method (MEM) to obtain the most likely spectral function for the given data. The spectralfunctioniswrittenintermsofadefaultmodelm(ω),whichencodespriorinformation,anda setofN basisfunctionsu (ω)as b k Nb ρ(ω)=m(ω)exp[ b u (ω)]. (2) k k Xk=1 ThestandardimplementationofMEMemploysthesingularvaluedecomposition(SVD)ofthekernel K, K(ω,τ )= K =(UΞV) , (3) i j ij ji andthebasisfunctionsu arechosentobethecolumnvectorsofU correspondingtothe N ≤ N k s data singularvaluesinthediagonalmatrixΞ. WewillinsteaduseaFourierbasis[13]whichwasfoundto yieldmorereliableresultsforthedataathand.Theresultshavebeencross-checkedusingthestandard SVDbasisandtheSVDbasiswithextendedsearchspace[14],andfoundtobeconsistentwithinthe uncertaintiesinherentintherespectivemethods. ThesystematicuncertaintyoftheMEMcanbeavoidedbystudyingthereconstructedcorrelator, definedas ∞ G (τ;T,T )= ρ(ω;T )K(τ,ω,T)dω, (4) r r Z r 0 whereT isareferencetemperaturewherethespectralfunctioncanbereliablyconstructed,usually r chosentobethelowestavailabletemperature.Itisclearthatiftherearenomediummodifications,ie ρ(ω;T)=ρ(ω;T )thenG (τ;T,T )=G(τ;T)(althoughtheconverseisnotnecessarilythecase). r r r CONF12 ThereconstructedcorrelatorcanalsobecomputeddirectlyfromtheunderlyingcorrelatorG(τ;T ) r withouthavingtoextractanyspectralfunctions[15]. Using cosh ω(τ−N/2) m−1cosh ω(τ+nN+mN/2) = , (5) sin(cid:2)h(ωN/2) (cid:3) Xn=0 (cid:2)sinh(ωmN/2) (cid:3) where 1 1 N T = , T = , r =m∈N, (6) r a N a N N τ τ r wefind m−1 G (τ;T,T )= G(τ+nN,T ). (7) r r r Xn=0 InthisstudywewilluseT = 44MeV(N = 128)asthereferencetemperaturethroughout,andwill r τ employ(7)tocomputethereconstructedcorrelator,paddingG(τ;T )withzerosinthemiddlewhere r necessary. 3 Results We have computed correlators and spectral functions in the pseudoscalar and vector channels for light–charm(D,D∗)andstrange–charm(D ,D∗)mesons.Figure1showsthespectralfunctionsatour s s lowesttemperaturein allfourchannels. We also show the groundandfirst excitedstates computed 4 lc, PS (D) 3 lc, V (D*) sc, PS (D) s sc, V (D*) s ) ω 2 ( ρ 1 0 2 3 4 ω (GeV) Figure1.Zerotemperaturespectralfunctionρ(ω)inallfourchannels. Theverticallinesshowthegroundstates andfirstexcitedstatescomputedbytheHadSpecCollaboration[16]. bytheHadronSpectrumCollaboration[16]usingavariationalanalysis. Comparingthetwo,wesee that the groundstate is very well reproducedby the MEM, while we are not able to reproduce the EPJWebofConferences 1.05 PS, lc (D) PS, sc (D) 1 c e Gr0.95 / G TT//TT == 11..990 cc 0.9 TT//TT == 11..5522 cc TT//TT = = 11..2277 cc TT//TT == 11..0099 0.85 cc 1.1 V, lc (D*) V, sc (D*) TT//TTcc == 00..9955 s TT//TT == 00..8844 cc TT//TT == 00..7766 cc 1.05 c e Gr / G 1 0.95 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 τ [fm] τ [fm] Figure2. OpencharmcorrelatorsG(τ;T)dividedbythereconstructed correlatorG (τ;T,T )computed from r r eq.(7)withT =0.24T . r c firstexcitedstateaccuratelywithourcurrentdata.ThewigglesseenintheD spectralfunctionarean s artefactoftheFourierbasis. Infigure2weshowthecorrelatorsG(τ)dividedbythereconstructedcorrelatorsG (τ)atthesame r temperature,forallfourchannels. Thisratioshouldbe 1if thereare nothermalmodifications. We observethatthecorrelatoratT =0.76T isconsistentwithnomodificationsinallchannels,butthere c arethermalmodificationsalreadyatT =0.84T ,inparticularintheDmesonchannel.AboveT ,the c c modificationsbecomesignificantinallchannels. Wealsoseethatexceptforthehighesttemperature (T = 1.9T ),themodificationsinthestrange–charmsectoraresmallerthanthoseinthelight–charm c sector,whichmaylendsupporttothehypothesisthatD yieldsmaybeincreasedrelativetoDyields. s We now turn to the spectral functions, which are shown in figures3 and 4 for the pseudoscalar andvectorchannelsrespectively. We seenosignofanysurvivingboundstateaboveT , suggesting c that all open charm states dissociate near T . It may appear that the dissociation occurs already at c T = 0.95T ,butitshouldbenotedthatthetransitiontothequark–gluonplasmaisabroadcrossover c forourparameters,andthatT =0.95T lieswithinthecrossoverregion[9]. c Interestingly, there are suggestions, in particular in the vector channel, of a thermal mass shift belowT , with thegroundstatepeakpositionatT = 0.76T beingconsiderablyhigherthanatzero c c temperature. Sofar,thisfeatureappearstoberobustwithrespecttovariationsinthebasisfunctions anddefaultmodel,butworkisstillinprogresstoconfirmthis. Aswillbeapparentfromfig.2,thestatisticaluncertaintiesinourcorrelatorsarerelativelylarge,at O(1%).Typically,permilleerrorsorsmallerarerequiredtounambiguouslyidentifyspectralfeatures. The fairly broad features seen in figs 3 and 4 for T & T may hence be a reflection of our limited c CONF12 4 5 ρ(ω) 23 TTTTTTTT/////// =TTTTTTT ccccccc0 ======= 1111000.......9520987279546 ρ(ω) 43 TTTTTTTT/////// =TTTTTTT ccccccc0 ======= 1111000.......9520987279546 2 lc, PS (D) sc, PS (D) s 1 1 00 1 2 ω (GeV) 3 4 5 00 1 m0 ω (GemV1) 3 4 5 Figure3. Pseudoscalarchannelspectralfunctionρ(ω)forlight–charm(left)andstrange–charm(right)mesons. Theverticallinesdenotethegroundandfirstexcitedstatemassesfromref.[16]. 4 5 ρ(ω) 23 TTTTTTTT/////// =TTTTTTT ccccccc0 ======= 1111000.......9520987279546 ρ(ω) 43 TTTTTTTT/////// =TTTTTTT ccccccc0 ======= 1111000.......9520987279546 2 lc, V (D*) sc, V (D*) s 1 1 00 1 2 ω (GeV) 3 4 5 00 1 m0ω (GeVm)1 3 4 5 Figure4. Vectorchannelspectralfunctionρ(ω)forlight–charm(left)andstrange–charm(right)mesons. The verticallinesdenotethegroundandfirstexcitedstatemassesfromref.[16]. precision.Wearecurrentlyworkingonimprovingonthisbyusingmultiplesourcesperconfiguration forourcorrelators. 4 Summary and outlook Wehavecarriedoutthefirstlatticecalculationoftemporalcorrelatorsandspectralfunctionsofopen charmmesons, both aboveand belowthe deconfinementcrossover. We find clear evidenceof ther- malmodificationsalreadybelowthecrossover,whichmayincludea thermalmassshift. Abovethe crossover,wefindnoevidenceofanysurvivingboundstates. Thethermalmodificationsaresmaller forD andD∗mesonsthanforDandD∗mesons,whichmaypointtoanenhancedyieldofD relative s s s to D mesonsinheavyioncollisions, in line withtheoreticalpredictionsandemergingexperimental evidence[4]. EPJWebofConferences Wearecurrentlyworkingonimprovingourstatisticsusingmultiplesources,whichwouldenable amorepreciseandreliableextractionofspectralfunctions.Wearealsocarryingoutacompletestudy ofMEMsystematics;sofar,ourmainresultsdonotappeartobeaffectedbythesesystematics. Weare alsoplanningtocross-checktheMEMresultsusingalternativemethodssuchastheBRmethod[17]. Acknowledgments We thank AlexanderRothkopf for providingaccess to his MEM code and for numerousinvaluable discussions. WealsothankGertAartsandChrisAlltonforfruitfuldiscussions. Weacknowledgethe useofcomputationalresourcesprovidedbyICHECandtheSTFCfundedDiRACfacility. 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