We determine the curvature of the phase transition line in the mu-T plane
through an analysis of various observables, including the Polyakov loop, the
quark number susceptibilities and the susceptibility of the chiral condensate.
The second derivative of these quantities with respect to mu was calculated.
The measurements were carried out on N_T = 4,6,8 and 10 lattices generated with
a Symanzik improved gauge and stout-link improved 2+1 flavour staggered fermion
action using physical quark masses.Comment: Talk presented at the XXVI International Symposium on Lattice Field
  Theory, July 14 - 19, 2008, Williamsburg, Virginia, USA. 7 pages, 6 figure

G. Endrődi

K. K. Szabó

S. D. Katz

Z. Fodor

English

arXiv

Gergely Endrodi

Zoltan Fodor

Sandor D. Katz

Kalman Szabo

Crossref

The curvature of the QCD phase transition line

Endrödi G, Fodor Z, Katz SD, Szabo K. The curvature of the QCD phase transition line. In:  Proceedings of The XXVI International Symposium on Lattice Field Theory  — PoS(LATTICE 2008). Trieste, Italy: Sissa Medialab;  2009: 205

Endrödi, Gergely 

Fodor, Zoltan

Katz, Sandor D.

Szabo, Kalman

Publications at Bielefeld University

Endrodi, G

Fodor, Z

Katz, S D

Szabo, K K

ELTE Digital Institutional Repository (EDIT)

arXiv:0901.3018v2  [hep-lat]  22 Jan 2009The curvature of the QCD phase transition lineG. Endr ődi ∗1, Z. Fodor 1,2, S.D. Katz1,2, K.K. Szabó 21Institute for Theoretical Physics, Eötvös University, H-1117 Budapest, Hungary.2Department of Physics, University of Wuppertal, D-42097 Wuppertal, Germany.E-mail: endrodi@general.elte.huWe determine the curvature of the phase transition line in the µ −T plane through an analysisof various observables, including the Polyakov loop, the quark number susceptibilities and thesusceptibility of the chiral condensate. The second derivative of these quantities with respect toµ was calculated. The measurements were carried out onNT = 4,6,8 and 10 lattices generatedwith a Symanzik improved gauge and stout-link improved 2+1 flavour staggered fermion actionusing physical quark masses.The XXVI International Symposium on Lattice Field TheoryJuly 14 - 19, 2008Williamsburg, Virginia, USA∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/The curvature of the QCD phase transition line G. Endr̋odi1. IntroductionQuantum chromodynamics (QCD) is the theory of strong interac ions. While at low temper-atures the dominant degrees of freedom are hadrons, due to one f the most important propertiesof QCD, asymptotic freedom, at high temperatures it describes a different phase of matter calledquark-gluon plasma (QGP). The phase transition between thehadronic phase of matter and QGPcan be investigated by lattice simulations. The transitionat zero chemical potential – which repre-sents the case of equal number of quarks and antiquarks – is ofhuge importance, since it is relevantfor the early Universe and also for high energy collisions. The region of the phase diagram corre-sponding to small and moderateµ is also interesting, since the cooling down of the QGP after aheavy ion collision occurs in this area.Simulations at nonvanishing chemical potential are burdene by a significant problem: thefermion determinant here becomes complex, and as a result makes importance sampling impossi-ble. A possible solution for this issue is to expand observables into a Taylor-series inµ , where thecoefficients can be calculated at vanishing chemical potential. Due to the symmetry of the partitionfunction describing the system, the first term in these expansions always vanishes. The second termon the other hand is related to the curvature of the transitioline.The 2+1 flavour QCD transition was found to be an analytic crossover [1] (instead of a first-order phase transition), which usually results in different transition temperatures for different ob-servables [4]. Our aim is to determine the curvatureκ of the phase transition line for differentobservables. These include the Polyakov loop, the strange quark number susceptibility, the chiralcondensate and the chiral susceptibility.2. Possible scenariosFigure 1: Two possible scenarios about the QCD phase diagram around zero µ . Transition temperaturesdetermined by different observables may converge to (left side, compare to first lattice indications onNT =4 [2]) or diverge from (right side, first lattice indicationsonNT = 4 [3]) each other as the chemical potentialis increased. This tendency can also have an effect on the existence (and if it exists, the position) of the QCDcritical endpoint.2The curvature of the QCD phase transition line G. Endr̋odiAs an effect of the crossover character of the transition at zero µ , the critical temperaturedetermined from the chiral susceptibility (Tc χψ̄ψ) ≈ 151 MeV) and that from the strange quarknumber susceptibility (Tc(χs)≈ 175 MeV) are quite different [4]. This situation can change at finitechemical potential according to two possible scenarios. The strengthening of the transition couldcause these two values to approach each other, while if the transi ion remains a weak crossover,the two transition lines can diverge (see figure 1). The two cases are separated by the sign of thedifferenceκ(χs)− κ(χψ̄ψ) of the curvatures for the two observables. A positive sign (left side)would indicate that the crossover region shrinks, which might suggest that a critical endpoint existson the phase diagram. On the other hand a negative sign would ca se the crossover region toexpand, which may result in the abscence of the critical point.Supposing that there exists a critical endpoint around baryonic chemical potentialµB ≈ 360MeV, the difference of the curvatures should be around∆κ ≈ 0.02 (see definition below). Hereand also in the following underµ we mean theµB baryonic chemical potential. There are severalworks in the literature in which the value of the curvature was computed [5, 6, 7, 8, 9]. In theseNT = 4 works the curvature was found to be in the intervalκ = 0.003. . .0.01. On the other hand,the NT = 6 results of [3] indicate that the curvature might become less pronounced towards thecontinuum limit.3. The curvatureLet us parametrize the transition line in the vicinity of thev rtical µ = 0 axis asTc(µ) =Tc(1−κ ·µ2/T2c). This implies that the curvature can be written asκ =−TcdTc(µ)d(µ2)∣∣∣∣µ=0(3.1)We illustrate our first procedure to measure the derivative in question using the quantityχs/T2(see definition in section 6). Since 0≤ χs/T2 ≤ 1 stands independently ofµ , we can defineTc(µ)via the relationχs/T2(Tc,µ) = 0.5. In the following we use the short notationO ≡ χs/T2.The total derivative of the observableO(T,µ2) may be written as dO = (∂O/∂T) · dT +(∂O/∂ (µ2))·dµ2. So, since along theTc(µ) line dO = 0 by definition, for the derivative in thedefinition of the curvature this reads asdTcdµ2=−(∂O∂ µ2)/(∂O∂T)(3.2)This way it is enough to measure these two derivatives at an arbitrary temperature. For morecomplicated quantities this procedure gives only an approximate value for the curvature. In thesecases one should carry out measurements at various temperatures; thenTc(µ) can be defined asthe inflection point (for e.g.χs/T2) or the maximum value (for e.g.χψ̄ψ ) of the observable. Thissecond procedure – which is clearly more CPU-demanding – is an ongoing project. Nevertheless,to calculate the derivative∂O/∂ (µ2) we need to measure new operators. These are detailed in thenext section.3The curvature of the QCD phase transition line G. Endr̋odi4. TechniqueLet us consider the partition function in its usual formZ =∫DUe−Sg(U)(detM)Nf /4 (4.1)and denote the derivative with respect toµu,d by ′. The derivatives ofZ are easily calculated to be(logZ )′ = 〈nu,d〉 and(logZ )′′ = 〈χu,d〉, where the light quark number densitynu,d and the lightquark number susceptibilityχu,d are the following combinations:nu,d =Nf4Tr(M−1M′)χu,d = n2u,d +Nf4Tr(M−1M′−M−1M′M−1M′)Using these definitions the second derivative of any (possibly µu,d-dependent) observable can bedetermined as∂ 2〈O〉∂ µ2u,d= 〈Oχu,d〉− 〈O〉〈χu,d〉+ 〈2O ′nu,d +O ′′〉 (4.2)For observables that do not depend explicitly onµu,d (like the Polyakov loopL or χs) the thirdterm in (4.2) vanishes. Forµu,d-dependent observables the derivativesO ′ andO ′′ were calculatednumerically, using a purely imaginary chemical potential.5. Simulation setupWe used a Symanzik improved gauge and stout-link improved staggered fermionic lattice ac-tion in order to reduce taste violation. The configurations were generated with an exact RHMCalgorithm. We determined the line of constant physics (LCP)using physical masses for the lightquarksmu,d as well as for the strange quarkms. The LCP was fixed by setting the ratiomK/ fK andmK/mπ to their physical values. We used four different lattice spacingsNT = 4,6,8,10 and aspectratiosNS/NT of 4 and 3. The scale was fixed byfK and its unambiguity checked by calculatingmK,fπ andr0. For measuring the operators necessary for the above mention d derivative the randomnoise estimator method was used. The number of random vectors was set to 80, in order for theerror coming from the random estimator method and that from the finiteness of the statistics to beof the same extent. The details of the simulation setup can befound in [4] or [10].6. ObservablesIn order to determine the derivative∂Tc/∂ (µ2) expressed in (3.2), one needs to select someobservableO. For this role we used the following quantities:The Polyakov loop is defined asL =1N3S∑xTrNT−1∏t=0U4(x, t) (6.1)4The curvature of the QCD phase transition line G. Endr̋odiIn order to be able to extract continuum limit results, an appro riate renormalization is necessary.For L this meansLr = Lexp(V(r0)/2T), whereV(r) is theT = 0 static potential [4].The strange susceptibilityχs =TV∂ 2 logZ∂ µ2u,d(6.2)needs no renormalization, since it is connected to a conserved current.The chiral condensate can also be expressed as a derivative of the partition function:ψ̄ψ =TV∂ logZ∂m(6.3)It can be renormalized by subtracting the additive divergences, and then multiplying by the quarkmass, so that the multiplicative factors also cancel (here the factorm4π is used to get a dimensionlesscombination):ψ̄ψr = (ψ̄ψ − ψ̄ψ(T = 0)) ·m·1m4π(6.4)Finally, the definition of the chiral susceptibility isχψ̄ψ =TV∂ 2 logZ∂m2(6.5)and the renormalization is the following:χψ̄ψr = (χψ̄ψ − χψ̄ψ(T = 0)) ·m2 ·1T4(6.6)7. ResultsAs already mentioned, in our approach we select an arbitraryemperature to calculate the cur-vature of the transition line. In order to increase statistics, we can also average measurements atdifferent temperatures. This procedure is illustrated on figure 2 for the case of the strange suscep-tibility and the chiral condensate.Having obtained the renormalized quantities forNT = 4,6,8 and 10, we are in a position tocarry out the continuum extrapolation. The results shown onfigure 3 are only preliminary; notethat the extrapolated values have quite large statistical errors.8. SummaryWhile the curvature values for̄ψψr are smaller than those forχs/T2 at every lattice spacing,they become consistent in thea→ 0 limit. Our aim in the first place is to determine the differencκ(ψ̄ψr)−κ(χs/T2), so increasing the statisctics in order to clarify the situat on is very important.Nevertheless we report the continuum results:κ(ψ̄ψr) = 0.0034(40) andκ(χs/T2) = 0.0013(51),and also for the Polyakov loop and the chiral susceptibility: κ(Lr) =−0.0095(93) andκ(χψ̄ψr) =−0.0018(34).As the next step we are looking forward to perform the above described procedure to measure∂O/∂ (µ2) for the whole temperature interval aroundTc. This way it becomes possible to determinethe critical temperature for the given observables (by locating their inflection point or maximumpoint) at nonvanishing chemical potential. Using this technique and much larger statistics we planto give a final answer for the curvature of the QCD transition lne.5The curvature of the QCD phase transition line G. Endr̋odiFigure 2: The curvature determined using the normalized strange susceptibility (left side) and the renor-malized chiral condensate (right side), respectively. Independent measurements at different temperatures (βvalues) can be averaged to decrease statistical errors. Results fromNT = 8 lattices are shown.Figure 3: The curvature of the transition line as a function of the lattice spacing squared, for the strangesusceptibility (left side) and the chiral condensate (right side). As characteristic to the applied action, theNT = 4 results are out of the scaling regime, so the linear extrapolation is carried out using only the finerlattices. As it can be seen, larger statistics is needed in order to extract a reliable continuum result.References[1] Y. Aoki, G. Endr̋odi, Z. Fodor, S.D. Katz and K.K. Szabó,The order of the quantum chromodynamicsphase transition predicted by the standard model of particle physics, Nature443 (2006) 675-678[hep-lat/0611014].[2] Z. Fodor and S. D. Katz,Lattice determination of the critical point of QCD at finite Tand mu JHEP0203, 014 (2002) [arXiv:hep-lat/0106002].[3] P. de Forcrand, S. Kim and O. Philipsen,A QCD chiral critical point at small chemical potential: is it6The curvature of the QCD phase transition line G. Endr̋odithere or not?, PoSLAT2007, 178 (2007) [arXiv:0711.0262 [hep-lat]].[4] Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabó,The QCD transition temperature: Results with physicalmasses in the continuum limit, Phys. Lett.B643 (2006) 46-54 [hep-lat/0609068].[5] C. R. Allton et al., The QCD thermal phase transition in the presence of a small chemical potential,Phys. Rev. D66 (2002) 074507 [arXiv:hep-lat/0204010].[6] P. de Forcrand and O. Philipsen,The QCD phase diagram for small densities from imaginarychemical potential, Nucl. Phys. B642 (2002) 290 [arXiv:hep-lat/0205016].[7] M. D’Elia and M. P. Lombardo,Finite density QCD via imaginary chemical potential, Phys. Rev. D67 (2003) 014505 [arXiv:hep-lat/0209146].[8] P. de Forcrand and O. Philipsen,The QCD phase diagram for three degenerate flavors and smallbaryon density,Nucl. Phys. B673 (2003) 170 [arXiv:hep-lat/0307020].[9] Z. Fodor and S. D. Katz,Critical point of QCD at finite T and mu, lattice results for physical quarkmasses, JHEP0404 (2004) 050 [arXiv:hep-lat/0402006].[10] Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabo,The equation of state in lattice QCD: With physicalquark masses towards the continuum limit, JHEP01 (2006) 089 [hep-lat/0510084].7

The Curvature of the QCD phase transition line

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The curvature of the QCD phase transition line

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