We present results from a calculation of the transition temperature in QCD
with two light and one heavier (strange) quark mass on lattices with temporal
extent N_t =4 and 6. Calculations with improved staggered fermions have been
performed with a strange quark mass fixed close to its physical value and for
various light to strange quark mass ratios that correspond to light
pseudo-scalar masses in the range (150-500) MeV. From a combined extrapolation
to the chiral (m_l -> 0) and continuum (aT -> 0) limits we obtain for the
transition temperature at the physical point T_c = 192(7)(4) MeV.
  We also present first results from an ongoing calculation of the QCD equation
of state with almost realistic light and strange quark masses.Comment: 4 pages, 4 figures, to appear in the proceedings of the 19th
  International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions,
  Shanghai, Nov. 200

Aoki Y

Bernard C

Cheng M (RBC-Bielefeld Collaboration)

Frithjof Karsch

Schmidt Ch

the RBC-Bielefeld Collaboration

English

arXiv

Karsch F. Transition temperature in QCD with physical light and strange quark masses. J.Phys.G. 2007;34(8):S627-S630

Karsch, Frithjof

Publications at Bielefeld University

Transition temperature in QCD with physical light and strange quark masses

We present results from a calculation of the transition temperature in QCD with two light (up, down) and one heavier (strange) quark mass as well as for QCD with three degenerate quark masses. Furthermore, we discuss first results from an ongoing calculation of the QCD equation of state with almost realistic light and strange quark masses

KARSCH, F.

UNT (University of North Texas) Digital Library

BROOKHBUEN N A T I  0 N A L L A B O R A T O  IZY BNL-77488-2007-CP Transition temperature in QCD with physical light and strange quark masses Frithj of Karsch Brookhaven National Laboratory, Upton, NY 11973 USA Presented at the 19th International Conference on Ultra-Relacivistic Nucleus-Nucleus Collisions Shanghai, China Novemberl4-20, 2006 January 2007 Physics Department Brookhaven National Laboratory P.O. Box 5000 Upton, NY 11 973-5000 www.bnl.gov Notice: This inmuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U S .  Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This preprint is intended for publication in a journal or proceedings. Since changes may be made before publication, it may not be cited or reproduced without the author’s permission. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party's use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. BNL-NT-07/7 . . , _:. . Transition temperature in QCD with physical light and strange quark masses F'rithjof Karsch (for the RBC-Bielefeld collaboration) Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA E-mail: karsch@bnl . gov Abstract. We present results from a calculation of the transition teinperat.ure in QCD with two liglit.(up, down) and one heavier (strange) quark inass as well as for QCD with three degenerate quark inasses. Furthermore, we discuss first. results from an ongoing calculation of the QCD equation of state with almost realistic light and strange quark masses. 1. Introduct ion Lattice calculatrioiis provide a unique first principle approach to the thermodymmics of QCD. They lime the potential to produce definit.ive answers for basic bulk t heriiiodyiianiic properties a.s well as the phase diagram of QCD. To reach this a.nibitious goal oiie 'has t.o cont,rol lattice discretizat,ion errors t1ia.t 1ea.d to systematic devia.tions of 1a.t t.iw calculations from contiiiuuin physics, as well as finite volume effects t.liat. that. can, obscure the thmodyiiarnic limit. We present. here results from a detailed. analysis of bhe transition tteiiipera.tue in QCD with an alniost. physical value of the stsrange quark mass mid severa.1 values of the light. quark nia.ss. This allows to perform a,n * ext4rapolatjioii to the chiral limit. as well as t.0 trhe physical point. Calculations have been performed for t.wo different- \dues of the la,t,tice cut-off which allows' to estimate the lattice discretization errors and perform an extrapolation tmo the continuum limit. The flavor dependence of these results are emmined through an ana.logous analysis of QCD with three degenemte qmrk flavors. We also report on results from an ongoing study of the QCD equation of state wit.11 almost. physical light. and stxange quark niasses. All these cdculations have bcmi performed wit.11 a.n 0 (u?)  tree-level iinproved gmge action a.nd an iniprowl st,aggerecl ferinioii action (p4fat3) that. also improves bulk t1iermodyna.niic observables at 0 (a,2).  For furt.Iier details on the calculational set-up we refer to [l, 21. 2. T h e  t ransi t ion temperatmure In order tro determine the transition temperature and its dependence on the light) quark mass we have performed calculations wit,h a fixed value of t,he bare strange quark mass Transition temperature in QCD with physical light and strange quark masses 2 0.54 0.52 0.50 0.48 0.46 ' 0.44 0.42 0.40 .. . . , . . . . 1 . . . . . . . . ,_ ...t Tc ro n,=2+1 N,=4 (squares, triangles) 6 (circles) mps ro 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 .p' p-3.94, m8=0.01625 s [1=3.76. m,=0.02 A 11=3.68. m,=0.026 (3-3.61, rn,=0,0325 0 p-3.518, m,=0.0382 D p=3.41. rn,=0.052 A p=3.382. rn,=0.052 ' p13.351. rn,=0.0591 R ,? Figure 1. The transition temperature (lek) calculated on lattices with teiiiporal extent N, = 4, G and the static quark potential (right) in units of TO calculated on 1G3 x 32 lattices. V ( T )  has been calculated for a wide range of gauge coupling {3 iii which the lattice cut-off varies by a factor 4. This range also covers the the relevant parameter range of the finite temperature calculations. a.nd several (dcgenerat.e) values for the two light, quark masses t,liat. correspond to light. pseudo-scalar inasses (mPs) ranging from 7 - 7 ~ ~ ~  N 150 MeV to  nxps CY 500 MeV. For noli-zero d u e s  of tjhe quark masses the transition to the high t.emperature pliase of QCD is signaled by peaks in response functions. We have analyzed Polyakov loop a.nd chira.1 susc.eptibilities to locate the transition point for the different quark mass d u e s  011 lattices with temporal extent N, = 4 and 6 [l]. The spatial extent of t,he lattice has be,eii varied between iV0/N, = 2 and 4. We generally observe only a 1vea.k volume dependence of the transition point and also find that the a.bove susceptibilities a,s well as quark nuiiiber susceptibilities in tthe light and strange quark sector [3] yield, within sta.t.istica1 errors, consistent- results for the ttransition point. In ordcr t.o convert. the results for the transition tempera.t.iirc t.0 ph>.sical valiies oiir riecds to set. the s d e  through the calcula,tion of zero t.eiiipera,t.ure observables. Uiililce 1ia.clroiiic. observables in the light. quark see tor, which require a careful handling of c.hira.1 er;t.rapolat,ions, are known to be difficult to determine a.ad in some mses are iiiflueiiced by large discxkization errors, the paranieters of the statsic quark poteiit.ia1 liavc beer1 found tro show only a mild dependence of t,he lattice cut.-oR [4]. We t.hus hwe . performed extensive studies of the static quark potential, V ( r ) ,  011 zero t.eniperat,ure 1a.ttices and extracted the sca.le parameter T O  defined through the slope of V ( r ) ,  i.e. ( r 2 d V ( r ) / d r )  Using the results obtained for TO in lattice units at. t.he crossover couplings, which have been determined in finite teniperature simulations on lattices with temporal extent N, = 4 and 6, the transition temperatures can then be calcula,ted a,s T,ro rO/N,a. Results for ??,TO and the static quark pot.ent.ia1 in units of 'ro are sl-iown in Fig. 1. Results for T,ro have t h i i  been extrapolated t,o the continuum limit. using an a1isat.z that. incorporates t-lie quark inass depeiideiicz of T, t.liroug1i the = 1.65. r i r o  Transition temperature in QCD with physical light and strange quark masses 3 10 8 6 4 2 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 # RBC-BI preliminary . E . m a  .I? . i .I TJMeV] m ,  1.0 1.5 2.0 2.5 3.0 3.5 4.0 100 200 300 400 500 600 700 Figure  2. The trace anomaly, ( E  - 3 p ) / T 4 ,  calculated in (2+1)-flavor QCD 011 lattices with temporal extent N, = 4 and 6, respectively. Fig.2(right) shows the resulting pressure from Eq. 3 as well as earlier results obtained for 3-flavor QCD (see text). cxlcii1at.w-1 va.lues of' the lightest pseudo-scalar mass as well as a. qua.dra.t.ic dependence 011 the liit.t.icc> c-ut-of€': (Tc~.O), i7l, , i i~~.N, = (TcrO)O,m,,m + A ( m p ~ r 0 ) ~  + B/fv: + (1) Here d has been va.ried between d = 1 and 2 to allow for uncertainties in the extrapolation to t.lie chiral limit. [l]. The continuum extrapolation is shown as the lower curve in Fig. 1 (left.). For the txansition temperature at the physical point we find. T,rO = 0.457(7)?; -++ Tc = 192(7)(4) MeV , (2) where we have used ro = 0.469(7) fm [5] to  set the scale for T,. A simila,r analysis has been performed for QCD with three degeiierat>e quark nia.sses [a] .  In this ca.se the transition becomes first order for small but, non-zero va.lues of the qua.rli mass. We find for the transition temperature in the chiral limit. T,:ro = 0.419(9) while in the sa.me limit, the t.ransition temperatme in (Z+l)-fla.vor QCD is T,-.ro = 0.444(6)( 12). We note that. Tcro in 3-fla.vor QCD is about. 5% siiia-ller t.1ia.n in (2+1)-flavor QCD. Mol-eover, the difference between the transition t.empera.t.ure in t.he chira.1 limit. and a.t. t,lie physical point. is only about. 3%. This is a direct, oonsecluence of t.lic weak cluark imss dependence of the transition temperatme. which has been observed carlier [SI. A similarly 1vea.k quark mass dependence lias also I m i i  observed in stxidies performed wit.11 a.iiother version, of an improved stsa.ggered fermion action (ascltad) on latkices with teniporal extent. N, = 4, 6 and 8 [7]. This a.na.lysis? however, lead t:o a 10% siiialler transition temperature, T,rO = 0.402(29) tha.n the value reported here. To resolve the origin of this discrepancy needs further investigations. . 3. The QCD equation of state We currently study the equation of state in (2fl)-flavor QCD [8] on lattices wit,h t.enipora.1 extent ATT = 4 and 6 using the same calculatrional set-up used for the st.udy of l?/n?i,sit io/i. tenipemture in QCD with physical light and strange quark masses 4 t hc t.raiisit.ion t.einperatJure. Calculations are performed along a. line of c.onstant. physics characterized by a. fixed strange pseudo-scdar mass nzB, N 660 MeV and a fixed ratio of mSs a,iid the kaon inass, m B s / m K  N 1.3. The light pseudo-sca.lar (pion) niass on this t2r2iljcct80ry is about, 220 MeV. In Fig. 2 we show results for the trace a.noma,ly, OE(T) f c - 3y in units of' T" and t.lie pressure ( p )  which is deduced from Or ( T )  using standard t,heri~iodyna.iiiic: relat.ions. We also included iii Fig. 2(right) results for the pressure obtained earlier [9] for 3-flavor QCD with larger quark masses that have been held fixed in units of T .  For these earlier ca.lcula.tions the light pseudo-scalar mass was about 770 MeV in the tmnsition region sild ivould have been a.bout 2 GeV in the region a.round T = 3T,! Tlie resu1t.s presented here for ( E  - 3 p ) / T 4  and p / T 4  in (2+l)-fla.vor QCD are entirely coiisist.ent with results obtained with the asqt,ad action [lo]. In pa.rt,icular, results for p/T4 obt.a.ined with the p4fat3 and ascltad act,ions, which bot.11 have a, O(u') iinprovecl ideal gas limit,, show little cut-off dependence in the entirc t,eniperat,ure regime a.iialyzed. This is in coiitmst, to calculations performed wit,li a va,ria.nt. of standsrd st a.ggered ferinions [ 111 which unlike p4fat3 and asqtad act,ioiis lead to la.rg-e cut,-off dist,ort,ioiis of t>he high-?' limit on lattices with t.empora1 extent N, = 4 and 6. We also note that we find 11.0 evidence for a significant broadening of the transition region when going closer tro the contiiiuum limit. Whether this happens for lmger N,, as a recent analysis of' the tra.nsition temperature on lattices with t#emporal extent, up to N, = 10 may suggest, [12], has to be examined in more detail. Acknowledgments References [l] M. Cheng et al. (RBC-Bielefeld collaboration), Phys. Rev. D 74, 054507 (2006) [2 ]  h4. Cheiig e t  al. (RBC-Bielefeld collaboration), hep-lat/OG12001, to be published in Phys. Rev. D [3] Ch. Scliii-iidt., liep-lat/O701019. [4] C. Aubiii et al., Pliys. Rev. D70 (2004) 094505. [5] A. Gray et al., Phys. Rev. D72 (2005) 094507. [GI F. Karsch, E. Laermaiin, A. Peikert, Nucl. Phys. B 605 (2001) 579. [7] C. Beriiard e t  al., Phys. Rev. D 71, 034504 (2005). [e] M. Cheiig et al. (RBC-Bielefeld collaboration), in preparation [9] F. Karsch, E. Laerinaiiii and A. Peilert, Phys. Lett. B 478, 447 (2000) [IO] C. Beriiard et al., hep-lat/OG11031 [ll] Y. Aolri, Z. Fodor, S. D. Katz arid K. K .  Szabo, ,JHEP 0601, 089 (2006). [12] Y. Aolri, Z. Fodoi. S. D. Katz a id  K .  E(. Szabo, Phys. Lett. B 643, 46 (2006) 

TRANSITION TEMPERATURE IN QCD WITH PHYSICAL LIGHT AND STRANGE QUARK MASSES.

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