PoS(LAT2006)140

Twisted mass QCD thermod ynamics:fir st results on apeNEXT

Ernst-Mic hael Ilgenfritz, Michael Müller -Preussker and Andre Sternbec k�

Humboldt-UniversitätzuBerlin, Institut für Physik,Newtonstr. 15 , 12489BerlinE-mail: ilgenfri@physik.hu-berlin.de,mmp@physik.hu-berlin.de,sternbeck@adelaide.edu.au

Karl Jansen and Ines WetzorkeNIC, Platanenallee6, 15378ZeuthenE-mail: Karl.Jansen@desy.de,Ines.Wetzorke@desy.de

Maria Paola Lombar do†

Istituto Nazionaledi Fisica Nucleare, LNF, Via Enrico Fermi 40, I 00044,Frascati(Roma)E-mail: lombardo@lnf.infn.it

Owe PhilipsenWestfälischeWilhelms-Universität Münster, Institut für TheoretischePhysik,Wilhelm-Klemm-Str.9, 48149MünsterE-mail: ophil@uni-muenster.de

Themotivationsfor simulatingQCD thermodynamicswith Wilson fermionsanda twistedmass

termareintroduced.Thetwistedmassapproachprovidesa naturalinfraredcutoff andO�a � im-

provementatmaximaltwist, andcanbeextendedto finite temperature.Ourstrategy for exploring

theQCD phasediagramat finite temperaturein this setup,while takingadvantageof theresults

at T � 0, is explained.Thefirst resultsfor theorderparametersandsusceptibilitieson a 163 � 8

latticearepresented.All dynamicalsimulationsarecarriedoutontheapeNEXTfacility in Rome.

XXIVthInternationalSymposiumonLatticeField TheoryJuly 23-28,2006Tucson,Arizona,USA

�AddresssinceSept.12006: CSSM,Schoolof Chemistry& PhysicsThe University of Adelaide,AUSTRALIA

5005†Speaker.

c�

Copyrightownedby theauthor(s)underthetermsof theCreative CommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/

PoS(LAT2006)140

TwistedmassQCDthermodynamics:first resultsonapeNEXT MariaPaolaLombardo

1. Motivations

Ultimately, wewouldliketo studythefinite temperaturephasetransitionin QCDandtheprop-ertiesof theplasmaphase(QGP),with physicalvaluesof thequarkmassesandin thecontinuumlimit. Thepresentprojectattemptsto setthestagefor suchacompletestudy.

In thisstudyweconsiderQCDwith two degenerateflavors.A crosscheckof thelatticeresultsobtainedwith staggeredandWilson fermion actionwill help controlling discretizationartifacts,while a twistedmasstermis expectedto facilitatethecontinuumandchiral limits.

Considerthephasediagramof two-flavor QCD in thetemperature-massplane:thefirst orderdeconfinementtransitionstemmingfromtheinfinitemass(orquenched)theoryweakenswith lowerquarkmasses,until it turnsinto a smoothcrossover for intermediatequarkmasses.In the chirallimit therehasto bea truephasetransitionagain,but its natureis still underinvestigation[1].

As a first stepof our program,we wish to find the locationof the phaseboundarybetweenhadronicand plasmaphase,i.e. the pseudo-criticaltemperatureand masscombinations,whiletakingadvantageof thepropertiesof twistedmassQCD.Thismeansthatwewill have to exploreathree-dimensionalspaceof temperatureT, barequarkmassm, andtwistedmassparameterµ .

2. Why Twisted Mass QCD Thermodynamics ?

Wilson fermionshave severaladvantagesover staggeredfermions,but they alsohave a moresubtlechiral behavior, andacomplicatedphasestructure,bothatT � 0 [2, 3] andat finite temper-ature[4, 5, 6, 7]. The twistedmassapproachimprovesover thestandardWilson behavior in twoways: first, it preventstheoccurrenceof exceptionalconfigurationsandshouldmake it relativelyeasyto reachmassvaluesof thelight pseudoscalarmesonscloseto thephysicalpionmass;second,oncetheWilsonhoppingparameterκ is setto its critical value,thetwistedmasstermbehavesasaconventionalquarkmass,and,at thesametime,anO� a improvementis automaticallyguaranteed.For recentresultsseeRefs.[8, 9] andfor a review Ref. [10].

In this first reportwe searchfor the transitionsbetweenthehadronicandplasmaregimesbyvaryingtheWilsonhoppingparameterκ relatedto thebarequarkmassby κ � 1�� 2m � 8 atfixedβ andfixedtwistedmassparameterµ .

3. Strategy and simulations

The simulationswereperformedon a 163 8 lattice with an improved versionof the HMCalgorithmasdetailedin Ref. [11] andwith a Symanziktree-level improved gaugeaction. Theyusedapproximatively threemonths crateof apeNEXT[12]. We chooseto work at β � 3� 75 andβ � 3� 9 in orderto take advantageof theT � 0 results[8, 9]. In principle,we canthencrossthe(pseudo-)criticalline at a fixed temperatureby tuning thequarkmass,eitherby varyingκ and/orµ .

For this strategy to besuccessful,thesimulationparametersNt andβ needto satisfy:

Tchiralc � Tsimulation � 1�� Nta� β � � Tquenched

c � (3.1)

For Tsimulation � Tquenchedc thehadronicphasecannotexist, while for Tsimulation � Tchiral

c theQGPcannotexist, irrespective of themassvalue.

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PoS(LAT2006)140

TwistedmassQCDthermodynamics:first resultsonapeNEXT MariaPaolaLombardo

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Plaquette, Beta = 3.75

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CG, Beta = 3.75d (Plaquette) / d(kappa) (X 700), Beta = 3.75

Figure 1: � P� asa function of κ (left diagram);ConjugateGradient(CG) Iterationssuperimposedwith∂ � P�∂κ (magnified)asa functionof κ at fixedβ � 3� 75 andµ � 0� 005(right diagram).Theresultsindicatea

thermaltransitionor crossoveratκt � 0� 165�1 � .

We fixed Nt � 8 by taking into accountthe lattice spacingfrom the T � 0 studies,namelya� 3� 75�� 0� 12 fm, a� 3� 9�� 0� 095 fm, aswell asthe known critical temperatures,Tc

chiral � 170MeV andTc

quenched � 270MeV.

In thefirst setof runsreportedherewefixedµ � 0� 005andvariedthehoppingparameterκ .

4. Results at β � 3� 75

At β � 3� 75, the T � 0 resultsshow that the minimum pion masswhich can be reachedwith our twisted massparameterµ � 0� 005 is mπ � 400MeV extrapolatingexisting resultsatµ � 0� 005 [8]. Our first goal hereis merely to checkwhethera thermalphasetransitionor acrossover canbefoundin therequiredrangeκt � κc.

Figure1 (left) shows a scanof theaverageplaquetteasa functionof κ . Thesteepestslopeoftheplaquetteaswell astheincreaseof thenumberof CGiterationsneededfor theinversionshownin Figure1 (right), bothsuggestacrossover or phasetransitionat [8]

κt � β � 3� 75� µ � 0� 005�� 0� 165� 1�� (4.1)

Henceκt � κc � T � 0�� 0� 1669,asrequired.

5. Results at β = 3.9

After theexploratorystudyat β � 3� 75, thechoiceβ � 3� 9 bringsuscloserto thecontinuumlimit. Resultsfor T � 0 at thisβ areavailableatanumberof valuesfor thetwistedmassparameterµ , seeRef. [9]. Theminimumpion massat T � 0 for our µ � 0� 005, inferredfrom theseresults,is about350MeV.

As adirectcomparisonbetweentheresultsatthetwo β valuesweshow in Figure2 thenumberof ConjugateGradientiterationsrequiredfor the inversion– which is an indicatorof criticality –asa functionof κ .

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PoS(LAT2006)140

TwistedmassQCDthermodynamics:first resultsonapeNEXT MariaPaolaLombardo

4500

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# C

G It

erat

ions

�

K

Beta = 3.9

Beta = 3.75

Figure 2: Thenumberof ConjugateGradientiterationsasa functionof κ for thetwo β -values.Thesolidlinesarea smoothinterpolationto guidetheeye.

Figure 3 (left) shows a collection of resultsfor the expectationvalue of the plaquette P! .The errorsaresmallerthanthe symbol, the solid line is a Bezier interpolationto guidethe eye.We performedlocal fits to a straightline P!�� a � bκ within subsequentintervalsof width ∆κ �0� 002,andwe show in thesamediagramthetangentin theinflectionpoint. Theparametersb areusedasestimatorsof thederivative of thePlaquettew.r.t. to κ andareshown in Figure3 (right).Theseresultsindicatea phasetransitionor crossover aroundκ � 0� 1597locatedaccordingto themaximal slopebmax. To make this predictionmore quantitative, our statisticswas enhancedto

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uette"

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#

K

Figure 3: � P� asa functionof κ (left diagram);∂ � P�∂κ asfunctionof κ at fixedβ � 3� 9 andµ � 0� 005(right

diagram)

O� 10000 HMC trajectorieson a selectedsampleof points: κ � 0� 1586� 0� 1591� 0� 1593� 0� 1597in the candidatecritical region at β � 3� 9. The results(Figure4) suggesta long autocorrelationtime in the critical region. Given theseautocorrelations,our resultsarevery preliminary. Mostprobablytheerrorsareunderestimated,but still theplotsmayserve asindicatorsfor the locationof acrossover or transition.

Figure5 shows theresultsfor thePolyakov loop: thesteepeningis clearlyseen,mostlythanksto thelatest,high statisticsresults.ThePolyakov loop increasesin thesameκ interval astheoneobservedfor theplaquette.ThePolyakov loop histogramsof theHMC resultsafterthermalizationarenarrow in thetwo purephases,andbroadenaroundatκ � 0� 1597,indicatinganincreaseof the

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PoS(LAT2006)140

TwistedmassQCDthermodynamics:first resultsonapeNEXT MariaPaolaLombardo

0

0.005

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ReP

ol$

#HMC

k=0.1586k=0.1591k=0.1593k=0.1597

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0 1000 2000 3000 4000 5000 6000 7000

Re

Pol

yako

v

%

Discarded #HMC

k=0.1586k=0.1591k=0.1593k=0.1597

Figure 4: HMC evolution anderroranalysisfor thehigh statisticsrunsat β � 3� 9 : binnedaverages(left)asafunctionof theHMC trajectories,andresultsanderrorsasa functionof thediscardedHMC trajectories(right).

fluctuationsandacritical behavior. Weseenotwo-statesignal(two peaksin thehistograms),whichseemsto excludea first ordertransition,but only a finite sizeanalysiscanassesswith confidencethenatureof thecritical behavior. Thesteepestslopeof theplaquetteandof thePolyakov loop,aswell asthebroadeningof theprobabilitydistributionssuggestacrossover or phasetransitionatκt :

κt � β � 3� 9� µ � 0� 005�� 0� 1597� 5 (5.1)

and,asit wasalsoobservedat β � 3� 75,

κt � κc � T � 0&� 0� 16085� (5.2)

Althoughwe postponea detailedanalysisof thespectrumandrelatedobservablesto our on-goinghigh statisticsstudy[13], we have collecteda few resultsfor thepion propagator.

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cy

(

Re Polyakov

k = 0.1550k = 0.1586k = 0.1591k = 0.1593k = 0.1597k = 0.1625k = 0.1605

Figure 5: The real part of the Polyakov loop as a function of κ (left diagram),and the Polyakov loophistograms(right diagram)at β � 3� 9 and µ � 0� 005. The dataset is the sameas in Figure2, with theinclusionof somehigh statisticsresults.Both diagramsareconsistentwith a critical point or crossover atκt � 0� 1597

�5 � .

5

PoS(LAT2006)140

TwistedmassQCDthermodynamics:first resultsonapeNEXT MariaPaolaLombardo

The (zeromomentum)pion propagatorG� t is measuredat a selectedsampleof couplings,andis fitted to astandardhyperboliccosineform

G� t �� A cosh M t ) Nt

2(5.3)

in thetime interval [2:6].

0.001

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ion

(t)

*

t

k = 0.1540k = 0.1580k = 0.1600k = 0.1610k = 0.1620

0.01

0.1

2 3 4 5 6t

k = 0.1540k = 0.1580k = 0.1620k = 0.1610

Figure 6: ThepionpropagatorG�t � for selectedκ values.Thesolid linesarethesimplefits describedin the

text. Theright figureshowsa subsetof thesamedatapointsasin theleft figureon adifferentscale.

We show thequality of theresults,with thefits themselvessuperimposed,in Fig. 6 (left andright, noteadifferentscalebetweenthetwo). Theresultingfit parameters,A(mplitude)andM(ass),arecollectedin Fig. 7 . They indicatethattheeffective pionmassdecreaseswhile approachingthethermaltransitionfrom below, while theamplitudeof thepropagatorincreases,following thetrendof theaverageplaquette.

6. Summary and Outlook

WehavestudiedQCDwith two flavorsof dynamicalWilsonfermionsincludingatwistedmasstermona163 8 latticeat two valuesof thetemperature:β � 3� 75correspondingto T � 205MeVandβ � 3� 9 correspondingto T � 259MeV.

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Figure 7: Amplitudeof thepropagatorandeffectivepionmassfrom thehyperboliccosinefits asa functionof κ .

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PoS(LAT2006)140

TwistedmassQCDthermodynamics:first resultsonapeNEXT MariaPaolaLombardo

In eithercaseswe have simulatedO� 10 valuesof barequarkmasses,by varyingthehoppingparameterκ at constantµ � 0� 005. Wehave observeda behavior consistentwith a crossover (andnotexcludingarealtransition)atacritical valueof κ , whichwedenotedasκt , whichis lessthanthecritical κ atT � 0. Thisbehavior is similar to thatobservedwith ordinaryWilson fermions[4, 5].

In ourcurrentstudy[13] ontheapeNEXTmachinesatDESYandINFN wehave to studynexttheµ dependenceof our results.To this end,we arerepeatinga κ scanat a larger µ value.At thesametime we want to take full advantageof the twistedmassapproachby working at full twistwith κ � κc � β � T � 0 . In this casethephasetransitionwill becrossedby tuningthetwistedmassµ . Model studiesalongthelinesof Ref. [6] will bemostusefulto guideoursimulations.

Acknowledgments: It is apleasureto thankMike Creutz,RobertoFrezzotti,AgnesMocsy, Gian-CarloRossiandCarstenUrbachfor interestingandhelpful discussions.Wewish alsoto thanktheapeNEXTCollaboration,andin particularAlessandroLonardo,Davide Rossetti,HubertSimma,RaffaeleTripiccioneandPiero Vicini, for their crucial help andsupport,aswell asGiampietroTecchiolli for grantingaccessto theAmaroapeNEXTprototype.

References

[1] O. Philipsen,PoSLAT2005, 016(2006)[PoSJHW2005, 012(2006)][arXiv:hep-lat/0510077];U.Heller, theseProceedings.

[2] E.-M. Ilgenfritz, W. Kerler, M. Müller-Preussker, A. Sternbeck,andH. Stüben,“A numericalreinvestigationof theAoki phasewith N(f) = 2 Wilson fermionsat zerotemperature”,Phys.Rev. D69, 074511(2004)arXiv:hep-lat/0309057.

[3] F. Farchionietal., “Twistedmassquarksandthephasestructureof latticeQCD”, Eur. Phys.J.C 39,421(2005)arXiv:hep-lat/0406039.

[4] A. Ali Khanet al. [CP-PACSCollaboration],Phys.Rev. D 63, 034502(2001)[arXiv:hep-lat/0008011].

[5] A. Ali Khanet al. [CP-PACScollaboration],Phys.Rev. D 64, 074510(2001)[arXiv:hep-lat/0103028].

[6] M. Creutz,“Wilson fermionsat finite temperature”,arXiv:hep-lat/9608024.:

[7] E.-M. Ilgenfritz, W. Kerler, M. Müller-Preussker, A. SternbeckandH. Stüben,“ProbingtheAokiphasewith N(f) = 2 Wilson fermionsat finite temperature”,arXiv:hep-lat/0511059.

[8] F. Farchionietal., PoSLAT2005, 072(2006)[arXiv:hep-lat/0509131].

[9] K. JansenandC. Urbach[ETM Collaboration],“First resultswith two light flavoursof quarkswithmaximallytwistedmass”,arXiv:hep-lat/0610015.

[10] A. Shindler, “TwistedmasslatticeQCD:Recentdevelopmentsandresults”,PoSLAT2005, 014(2006)arXiv:hep-lat/0511002.

[11] C. Urbach,K. Jansen,A. ShindlerandU. Wenger, “HMC algorithmwith multiple time scaleintegrationandmasspreconditioning”,Comput.Phys.Commun.174, 87 (2006)arXiv:hep-lat/0506011.

[12] F. Belletti et al. [apeNEXTCollaboration],Nucl. Instrum.Meth.A 559, 90 (2006).

[13] E.-M. Ilgenfritz, K. Jansen,M.-P. Lombardo,M. Müller-Preussker, M. Petschlies,O. Philipsen,A. Sternbeck,andL. Zeidlewicz, in progress.

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