Model analysis of thermal UV-cutoff effects on the chiral critical surfaceat finite temperature and chemical potential

Jiunn-Wei Chen*

Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan

Hiroaki Kohyama†

Department of Physics, Chung-Yuan Christian University, Chung-Li 32023, Taiwanand Institute of Physics, Academia Sinica, Taipei 115, Taiwan

and Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan

Udit Raha‡

Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan(Received 30 September 2010; revised manuscript received 3 April 2011; published 13 May 2011)

We study the effects of temporal UV-cutoff on the chiral critical surface in hot and dense QCD using a

chiral effective model. Recent lattice QCD simulations indicate that the curvature of the critical surface

might change toward the direction in which the first-order phase transition becomes stronger on increasing

the number of lattice sites. To investigate this effect on the critical surface in an effective model approach,

we use the Nambu–Jona-Lasinio model with finite Matsubara frequency summation. We find that the

nature of the curvature of the critical surface does not alter appreciably as we decrease the summation

number, which is unlike the case that was observed in recent lattice QCD studies. This may suggest that

the dependence of chemical potential on the coupling strength or some additional interacting terms such as

vector interactions could play an important role at finite density.

DOI: 10.1103/PhysRevD.83.094014 PACS numbers: 12.38.Aw, 11.10.Wx, 11.30.Rd, 12.38.Gc

I. INTRODUCTION

The study of the chiral critical point (CP) in the phasediagram of hot and dense quark matter is one of the centralissues in quantum chromodynamics (QCD) [1]. While it iswidely accepted that the QCD phase transitions concerningchiral symmetry restoration and color deconfinement arecrossovers with increasing temperature T for small chemi-cal potential� ’ 0, the order of the phase transitions alongthe � direction for small T is still under considerablespeculation. Model analysis indicates that a first-orderphase transition occurs with increasing � and for small T[2]. The above observations lead us to expect the existenceof a critical point located at the end of the first-order line inthe phase diagram for some intermediate values, TE and�E.

A first principle determination of the phase diagram bysolving QCD itself is difficult due to the strongly interact-ing nature of matter at low energies/temperatures whichsignificantly restricts the range of applicability of pertur-bative calculations. We must then rely on nonperturbativetechniques such as lattice QCD (LQCD) or some low-energy effective theories of QCD. The LQCD simulationsare known to be a viable approach for microscopic

calculations in QCD, and have recently reached a reliablelevel at finite T and� ¼ 0 [3]. However, these simulationsare not yet able to provide a conclusive understanding ofthe QCD phase diagram due to severe limitations posed bythe well-known ‘‘sign-problem’’ at finite�, and difficultiesdealing with small quark masses, though only very ap-proximate methods are available for simulations at small� values [4]. Thus, it is important to develop low-energyeffective models that may show consistency with latticeresults and can be extrapolated into regions not accessiblethrough simulations. Among them, the local Nambu–Jona-Lasinio (NJL) model [5] and its proposed extended versionto include coupling of quarks to Polyakov-loops, the so-called Polyakov-loop NJL (PNJL) model [6], are useful tostudy the quark system at finite T and/or �. Such effectivemodels share the symmetry properties of QCD andsuccessfully describe the observed meson properties andchiral dynamics at low energies (see, e.g., [7–10]).Using the framework of the (P)NJL model we search the

CP and analyze the order of the chiral phase transition byvarying the current quark masses (mu,md,ms). We usuallyset, for simplicity, md ¼ mu and investigate the phasetransitions in the mu-ms-� space. Renormalization group(RG) analysis of chiral models at� ’ 0 conclude that there

is no stable infrared (IR) fixed point for quark flavorsNF >ffiffiffi3

p[11], indicating that the thermal phase transition is of

fluctuation induced first order for two or more flavorsrealized in the chiral limit mu;d;s ¼ 0. However, the tran-

sition becomes a crossover for intermediate quark masses

*jwc@phys.ntu.edu.tw†kohyama@phys.sinica.edu.tw‡udit.raha@unibas.ch

Present affiliation: Department of Physics and Astronomy,University of South Carolina, Columbia, SC 29208, USA

PHYSICAL REVIEW D 83, 094014 (2011)

1550-7998=2011=83(9)=094014(8) 094014-1 � 2011 American Physical Society

because of explicitly broken chiral and center symmetries.Hence, it is naturally expected that there should be a‘‘critical boundary’’ separating the regions of the first-orderphase transition and crossover between small and inter-mediate quark masses. Although LQCD results support theabove model picture qualitatively, there had been a hugequantitative difference between the two kinds of analysis,even at zero chemical potential where the LQCD does notsuffer from the sign problem; the value of the critical massobtained in the (P)NJL model is about 1 order of magni-tude smaller than the value in the LQCD analyses. Inspiredby recent works reporting that the critical mass may be-come smaller when the number of lattice sites is increased[12,13], the present authors studied the critical boundary inthe (P)NJL model with finite Matsubara frequency sum-mation N at zero chemical potential [14]. There it wasfound that the critical mass actually becomes larger if onedecreases the Matsubara summation numberN [see Eq. (7)], thereby showing the correct tendency to explain thequantitative difference between the LQCD and (P)NJLmodel results.

In this paper, our goal is to study the critical boundaryfor all values of the chemical potential, which may even-tually tell us the location of the CP in the QCD phasediagram. As already mentioned, because an ab initio de-termination of the critical point in QCD is a distant hope,nevertheless, it is worth studying the (P)NJL model withfinite frequency summation at finite chemical potentialthat should capture the essential qualitative features ofthe results expected in lattice studies. Note that in themu-ms-� space, the critical boundary becomes a surfacecalled the ‘‘critical surface.’’ More precisely, through thisstudy we would like to investigate the qualitative behaviorof this critical surface whose shape can critically determinewhether the CP exists [15]. We can then compare ourresults with the recent lattice predictions.

II. MODEL SETUP

The NJL model Lagrangian in the three-flavor system iswritten by

L NJL ¼ �qði@� m̂ÞqþL4 þL6; (1)

L 4 ¼ gS2

X8a¼0

½ð �q�aqÞ2 þ ð �qi�5�aqÞ2�; (2)

L 6 ¼ �gD½det �qið1� �5Þqj þ H:c:�: (3)

Here m̂ is the diagonal mass matrix ðmu;md;msÞ inthe flavor space which explicitly breaks the chiral symme-try. L4 is the 4-fermion contact interacting term withcoupling constant gS, and �a is the Gell-Mann matrix in

the flavor space with �0 ¼ffiffiffiffiffiffiffiffi2=3

pdiagð1; 1; 1Þ. L6 is a

6-fermion interaction term called the Kobayashi-Maskawa-t’Hooft interaction whose coupling strength is

gD [16]. The subscripts ði; jÞ indicate the flavor indices andthe determinant runs over the flavor space. This term isintroduced to explicitly break the UAð1Þ symmetry.To study the chiral dynamics, we solve the gap equations

which are derived through differentiating the thermody-namic potential � by the order parameters of the model,

@�

@m�i

¼ 0; i ¼ u; d; s: (4)

The order parameters m�i are the constituent quark masses.

Since, we set md ¼ mu in our analysis, this should lead tothe isospin symmetric result m�

d ¼ m�u, reducing the num-

ber of gap equations from three to two. The thermody-namic potential � is defined by � � � lnZ=ð�VÞ, whereZ is the partition function, �ð� 1=TÞ is the inverse tem-perature, and V is the volume of the thermal system.In the mean-field approximation, after some algebra, we

arrive at the following expressions for the gap equation:

m�u ¼ mu þ 2igSNc trS

u � 2gDN2cðtrSuÞðtrSsÞ;

m�s ¼ ms þ 2igSNc trS

s � 2gDN2cðtrSuÞ2;

(5)

whereNcð¼ 3Þ is the number of colors and trSi is the chiralcondensate written explicitly as

i trSi ¼ 4m�i

Z d3p

ð2�Þ3 ðiTÞX1

n¼�1

i

ði!nÞ2 � E2i

: (6)

Here wn ¼ �Tð2nþ 1Þ is the Matsubara frequency and

Ei ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þm�2

i

qis the energy of the quasiparticle. A de-

tailed calculation for deriving the gap equations (5) isclearly presented in the review paper [8].To study the UV-cutoff effects in the model, we cut the

higher frequency modes in theMatsubara sum as employedin [14],

X1n¼�1

! XN�1

n¼�N

: (7)

This model has five free parameters fmu;ms;�; gS; gDg:two current quark masses, a three-dimensional momentumcutoff, a 4-fermion and a 6-fermion coupling constants.

Following [7], we set mu ¼ mphysu ¼ 5:5 MeV fixed for all

values of N, while the remaining four parameters are fittedeach time by using the following physical observables:

TABLE I. The various fitted parameters for different N [14].

N ms (MeV) � (MeV) gS�2 gD�

5

15 134.7 631.4 4.16 12.51

20 135.0 631.4 4.02 11.56

50 135.3 631.4 3.82 10.14

100 135.4 631.4 3.75 9.69

1 135.7 631.4 3.67 9.29

JIUNN-WEI CHEN, HIROAKI KOHYAMA, AND UDIT RAHA PHYSICAL REVIEW D 83, 094014 (2011)

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m� ¼ 138 MeV; f� ¼ 93 MeV;

mK ¼ 495:7 MeV; m�0 ¼ 958 MeV:

The parameter fitting for various N has been done in [14],and in this analysis we employ the same values, which areagain displayed in Table I for the convenience of thereader.

III. CHIRAL CRITICAL SURFACE

The main purpose of this paper is to determine the chiralcritical surface in the (P)NJL model with finite Matsubarasummation. The critical surface is the set of all critical pointsin the mu-ms-� space which are analyzed by scanning thespace for discontinuities of the chiral condensate. It shouldbe noted that for each value of N, we treat both the currentquark massesmu and ms as free parameters in obtaining thecritical surface once the other parameters, namely, �, gS,and gD are determined by fitting to the physical parameters,as shown in Table I. Of course, we will eventually beinterested in the case of the real (physical) current quark

massesmphysu ’ 5:5 MeV andm

physs ’ 135 MeV, in order to

determine the possible existence/nonexistence of the CPthrough our model analysis. Because our motivation is tomake a direct comparison of our results with that of LQCD,where the simulations are mainly performed in themu ¼ ms

symmetric case at finite �, we shall also consider this case.In the actual numerical calculations, we scouted out thecritical masses ðmuc;mscÞ for each � by searching for dis-continuities in the solutions of the gap equations (5) in theentire mu-ms-� space, in general.

The LQCD and model studies indicate a crossover real-ized at � ¼ 0 for physical current quark masses. Thismeans that the curvature of the critical surface will tellus whether the CP is favored in the phase diagram. To bemore concrete, if the region of the first-order phase tran-sition expands with increasing �, the physical quark massline will intersect with the critical surface and this will endup as a CP. If on the other hand, the first-order phasetransition region shrinks with �, there is less chance ofan appearance of a CP and a crossover transition will befavored for the whole range of T and �. Thus the sign ofthe curvature is indeed important, and the main purpose ofthis paper is to study whether the sign can change bydecreasing N from the usual case of N ¼ 1.

In the LQCD calculations, the curvature of the criticalsurface along the mu ¼ ms symmetric line is analyzed byobtaining the critical mass mc as a function of the ratio ofthe critical chemical potential �c and the critical tempera-ture Tc, through the following Taylor expansion formula:

mNc ð�cÞ

mNc ð�c ¼ 0Þ ¼ 1þ X

k¼1

cNk

��c

�Tc

�2k; (8)

which so far yielded the following results: c41 ¼ �3:3ð3Þ,c42 ¼ �47ð20Þ for Nt ¼ 4, and c61 ¼ 7ð14Þ, �17ð18Þ

(preliminary) for a leading order (LO) and next-to-leadingorder (NLO) extrapolation in �2, respectively, for Nt ¼ 6,where Nt represents the number of the lattice sites in thetemporal direction [13]. These results are graphically rep-resented in Figs. 3 and 6 in the following sections whichindicate that the sign of the curvature has not yet beendetermined from lattice simulations.

IV. NJL MODEL RESULTS

In this section, we demonstrate the numerical method-ology in obtaining the phase diagram and chiral criticalsurface for the NJL model.

A. Phase diagram

In Fig. 1, we display the phase diagrams for simulationsat the physical quark mass values of the current quarksobtained from fitting for N ¼ 15, 50, and 1, respectively.Here we use the parameters shown in Table I, where mu iskept fixed at 5.5 MeVand ms is around 135 MeV. We havealso studied the model for other values ofN, however, therewere no significant qualitative differences and we prefer toselect just the above three representative cases for graph-ical clarity. Here, it may be noteworthy mentioning that forvalues of N < 15, it becomes a matter of numerical chal-lenge to perform simulations to determine the phaseboundaries. So, henceforth, we shall be displaying ourresults only for the above three cases. Note that the caseN ¼ 1 corresponds to the traditional NJL model. In draw-ing the phase diagrams, we apply the same criterion em-ployed in [10] where the phase transition or crossover isdefined by the condition

0

50

100

150

200

250

0 100 200 300 400

Tem

pera

ture

[M

eV]

Quark Chemical Potential [MeV]

Critical Point

N=∞N=50

N=15

FIG. 1. Position of the CPs on the T-� phase diagrams fromthe NJL model with finite Matsubara summation and the currentquark masses being equal to the respective physical quarkmasses obtained by fitting for N ¼ 15, 50, 1. The dotted andthe solid lines represent crossovers and first-order phase tran-sitions, respectively. The N ¼ 1 case corresponds to the tradi-tional NJL model.

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h �uuih �uuiT0

��������T¼Tð�Þ¼ 1

2; (9)

h �uuiT0being the expectation value of the chiral condensate

for the up-quark at temperature T0 and � ¼ 0. We chooseT0 ¼ 0 for the N ¼ 1 traditional model, while we setT0 ¼ 50 MeV for the case with finite N since the modelis ill-defined for small T, as discussed in [14]. This is whywe choose not to display the results for the small T regionwhere the curves are no longer physically reliable.

Here it is seen that the region below each of the curvesthat represents the chiral symmetry broken phase expandswith decreasing N. This comes from the fact that thecoupling constants become larger with decreasingN, beingconsistent with RG arguments that the coupling strengthbecomes smaller when one considers the physics at highermomenta, i.e., for larger N. When the coupling strengthgrows the chiral condensate tends to enlarge, which can beeasily seen from Eq. (5). Thus, it is naturally understoodwhy the transition temperature increases with a smallerchoice of the summation number N.

Before proceeding to the critical surface, let us discusshow the phase diagram varies with changing current quarkmasses. Since we treat the current quark masses as freeparameters in drawing the critical surface, it is worthdisplaying the phase diagrams for different current quarkmasses, as shown in Fig. 2 for the case N ¼ 1. Here, inparticular, we have chosen the mu ¼ msð� mÞ symmetricdirection to get a general idea of the nature of phasediagram.

We see that the CP moves toward lower � and higherT with decreasing m, and it eventually hits the � ¼ 0 axisat m ¼ 1:1 MeV. For small current quark masses thetransition is of the first order, which is consistent with the

symmetry arguments presented in [11]. This suggests thatthe critical chemical potential �c and the critical tempera-ture Tc could be thought of as functions of the currentquark masses, or conversely, one can express the criticalmassmc as a function of�c and Tc, namely,mcð�c; TcÞ, orequivalently, mcð�c=�TcÞ. This leads to the very idea ofthe chiral critical surface or the Columbia plot which is ourcentral issue. Note that for other values of N, although theregion of the broken phase expands with decreasing N, thequalitative behavior seen in Fig. 2 is found to be more orless the same.For the sake of comparison with the LQCD Taylor

expansion formula (8), we define the following functionRðN; xÞ:

RðN; xÞ � mNc ðxÞ

mNc ðx ¼ 0Þ ¼ 1þ X

k¼1

cNk x2k; (10)

with x ¼ �c=�Tc. Thus the ratio of the critical mass mc

can be written as the function of N and x. For example,mN

c ð0Þ takes the values 1.1, 1.3, and 2.2 MeV, respectively,for N ¼ 1, 50, and 15, and hence mN

c ð0Þ is strongly Ndependent. It may also be worth showing the correspond-ing values of Tc in the above three cases: Tc ¼ 135, 153,and 190 MeV for N ¼ 1, 50, and 15.

B. Critical surface

In Fig. 3, we show the numerical results formN

c ðxÞ=mNc ð0Þ as a function of xð¼ �c=�TcÞ, along with

the corresponding results obtained in the recent LQCDsimulations for Nt ¼ 4 and 6 [13], respectively. We seethat the slope of the curves tends to go up only very slightlyon decreasing N from N ¼ 1 down to N ¼ 15. On theother hand, the results from lattice simulations do not yieldconclusive results so far, as clearly revealed from Fig. 3,

0.96

0.98

1

1.02

0 0.02 0.04 0.06 0.08 0.1

mc(

x)/m

c(0)

x=µc/(πTc)

LQCD(Nt=4)

LQCD(Nt=6, LO)

LQCD(Nt=6, NLO)

N=∞N=50N=15

FIG. 3. mNc ðxÞ=mN

c ð0Þ vs xð¼ �c=�TcÞ plot for different Nvalues from the NJL model, along with the corresponding resultsobtained from lattice simulations for Nt ¼ 4 and Nt ¼ 6. Theactual values of mN

c ð0Þ are 1.1, 1.3, and 2.2 MeV for N ¼ 1, 50,and 15, respectively.

0

50

100

150

0 100 200 300

Tem

pera

ture

[M

eV]

Quark Chemical Potential [MeV]

CP

m=1.1MeV

m=3.0MeV

m=5.0MeV

FIG. 2. Position of the CPs in the phase diagrams from thetraditional N ¼ 1 NJL model with mð¼ mu ¼ msÞ ¼1:1; 3; 5 MeV. The dotted and the solid lines represent cross-overs and first-order phase transitions, respectively.

JIUNN-WEI CHEN, HIROAKI KOHYAMA, AND UDIT RAHA PHYSICAL REVIEW D 83, 094014 (2011)

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where the sign of the curvature has not yet been con-strained for the Nt ¼ 6 case. Thus, in comparison withthe lattice studies, we find that the qualitative behavior ofour model results hardly changes by using different Nvalues in our analysis. This means that there is indeed astark quantitative contrast between the NJL model resultsand the lattice predictions.

One particular aspect of the above results deserves acloser inspection: here we draw Fig. 3 based on the idea ofEq. (10) in which theN dependence of the curvature comesfrom the coefficients cNk . The slope critical curves as seenin Fig. 3 have a negligibleN dependence for small x, whichmean that, if we were to assume a numerical expansion asEq. (10) applicable to the critical curves, cNk for small kwould be nearly independent of N as well. A splitting ofthe curves indeed takes place on increasing to large xwhensome N dependence may creep in for larger k values.However, although there appears some deviations forhigher x (or �), the qualitative behavior of the criticalsurface does not change appreciably with N at least forup to some �ð�300 MeVÞ as we shall see below.

We are now in the position to actually display our resultsfor the chiral critical surface on the Columbia plot for thegeneral case of real current quark masses. In Fig. 4, wedisplay our results obtained for the N ¼ 1 and 15 cases. Itis clearly seen that, although the values of the critical masschange about factor of 2, the slope of the curvature does notdiffer as we change the value of N. Thus, the region of thefirst-order phase transition expands with respect to � forall N values, since the effect of variation of N is rather toonominal to change the sign of the curvature of the criticalsurface. This would then mean that the NJL model willalways favor the CP with physical current quark masses atsome finite value of�, e.g., we obtain the CP when� goesup around �c ¼ 324ð342Þ MeV for N ¼ 1ð15Þ, as exhib-ited in the phase diagram in Fig. 1.

V. THE PNJL MODEL EXTENSION

It is also intriguing to study the critical surface in thePNJL model, because of the closer resemblance to QCDwhich treats the chiral and deconfinement phase transitionssimultaneously. In the PNJLmodel, the order parameter forthe deconfinement phase transition is the Polyakov-loopand it is described by a global mean-field being similar tothe chiral condensation in the traditional NJL model. TheLagrangian of the PNJL model is written as [10],

L PNJL ¼ L0 þL4 þL6 þUð�;��; TÞ; (11)

L 0 ¼ �qði@� i�4A4 � m̂Þq; (12)

Uð�;��; TÞ ¼ �bTf54e�a=T��� þ ln½1� 6���

þ 4ð�3 þ��3Þ � 3ð���Þ2�g; (13)

where U is the Polyakov-loop effective potential and �and �� are the traced Polyakov- and the anti-Polyakov-loop, respectively. They are defined by � ¼ ð1=NcÞ trL,�� ¼ ð1=NcÞ trLy with L ¼ P exp½iR�

0 d�A4� and A4 ¼iA0. There are several candidates for the Polyakov-looppotential in defining the PNJL model [6,10,17], and weadopt the strong-coupling inspired form of Eq. (13) follow-ing [10]. In the above expressions, the parameter a solelyparametrizes the strength of the Polyakov-loop condensatefor the deconfinement phase transition, while the parame-ter b controls the relative strength of the mixing betweenthe Polyakov-loop and chiral condensates, with a smallervalue of b signifying chiral phase transition dominatingover deconfinement. Here, the parameters a and b are set asa ¼ 664 MeV, and b ���3 ¼ 0:03. Regarding the rest ofthe model parameters, it is legitimate to use the same onesfixed in the NJL model because the Polyakov-loop exten-sion is likely to affect the system only at finite temperaturescomparable to the critical temperature Tc. At much lowertemperatures the chiral condensate is only very marginallymodified by the Polyakov loops.

0 1

2 3

4 5 0

5 10

15 20

25

0

50

100

150

200

250

300

Che

mic

al p

oten

tial [

MeV

]

Light Quark Mass [MeV] Strange Q

uark M

ass [M

eV]

(a)N=∞

0 1

2 3

4 5 0

5 10

15 20

25

0

50

100

150

200

250

300

Che

mic

al p

oten

tial [

MeV

]

Light Quark Mass [MeV] Strange Q

uark M

ass [M

eV]

(b)N=15

FIG. 4. The chiral critical phase surfaces for (a) N ¼ 1 and(b) N ¼ 15 from the NJL model in the mu-ms-� space.

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VI. PNJL MODEL RESULTS

Let us now discuss the results in the PNJL model withfinite frequency summation.

In Fig. 5, we present the phase diagrams resulting fromthe PNJL model at physical quark masses withN ¼ 15, 50,and 1. Here we see that the curves are very similar to theones in Fig. 1, however, the critical temperatures are abouta factor of 2 or more larger than those obtained via the NJLmodel. This is almost the same quantitative difference as isobserved between the traditional (N ¼ 1) NJL and PNJLmodels [18].

We again display the corresponding curves for the ratiomN

c ðxÞ=mNc ð0Þ and compare the results with the LQCD

simulations in the small ðx < 0:1Þ region in Fig. 6. Wesee that the nature of the results exhibit similar qualitativecharacteristics with the ones obtained in the NJL model;the ratio does not change appreciably, although mN

c ð0Þ isstrongly dependent on N. However, the ratios are numeri-cally larger than that obtained with the NJL case. Thisresult can be interpreted as an effect of the Polyakov-loopstending to suppress unphysical quark excitations below Tc

[10].Finally, in Fig. 7, the critical surfaces in the PNJL model

with N ¼ 1 and 15 are displayed. We confirm that theregion of the first-order transition expands with respect to� in these two figures, quite similar to that found previ-ously in the NJL case. It is interesting to note that suchqualitative similarity between the NJL and PNJL modelresults is a priori nontrivial, since the deconfinment phasetransition order parameters, namely, the Polyakov-loops�and��, respectively, may cause the system with chiral and

0.96

0.98

1

1.02

1.04

1.06

0 0.02 0.04 0.06 0.08 0.1

mc(

x)/m

c(0)

x=µc/(πTc)

LQCD(Nt=4)

N=15N=50N=∞

LQCD(Nt=6, LO)

LQCD(Nt=6, NLO)

FIG. 6. mNc ðxÞ=mN

c ð0Þ vs xð¼ �c=�TcÞ plot for different Nvalues from the PNJL model, along with the correspondingresults obtained from lattice simulations for Nt ¼ 4 andNt ¼ 6. The actual values of mN

c ð0Þ are 1.3, 1.6, and 2.6 MeVfor N ¼ 1, 50, and 15, respectively.

0

50

100

150

200

250

0 100 200 300 400

Tem

pera

ture

[M

eV]

Quark Chemical Potential [MeV]

N=15N=50

N=∞

Critical Point

FIG. 5. Position of the CPs on the T-� phase diagrams fromthe PNJL model with finite Matsubara summation and thecurrent quark masses being equal to the respective physicalquark masses obtained by fitting for N ¼ 15, 50, and 1. Thedotted and the solid lines represent crossovers and first-orderphase transitions, respectively. The N ¼ 1 case corresponds tothe traditional PNJL model.

0 1

2 3

4 5 0

5 10

15 20

25 0

50

100

150

200

250

Che

mic

al p

oten

tial [

MeV

]

Light Quark Mass [MeV] Strange Q

uark M

ass [M

eV]

(a)N=∞

0 1

2 3

4 5 0

5 10

15 20

25

0

50

100

150

200

250

Che

mic

al p

oten

tial[

MeV

]

Light Quark Mass [MeV] Strange Q

uark M

ass [M

eV]

(b)N=15

FIG. 7. The chiral critical phase surfaces for (a) N ¼ 1 and(b) N ¼ 15 from the PNJL model in the mu-ms-� space.

JIUNN-WEI CHEN, HIROAKI KOHYAMA, AND UDIT RAHA PHYSICAL REVIEW D 83, 094014 (2011)

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deconfinment phase transitions to deviate significantly(both qualitatively and quantitatively) from a system withonly chiral phase transition, especially in the vicinity of theCP.

VII. SUMMARYAND DISCUSSION

In this paper, we studied the UV-cutoff effects on thechiral critical surface with two light and one heavy flavorsusing the (P)NJL model. We found that although the criti-cal masses may strongly depend on the choice of N, the‘‘normalized’’ critical surface, denoted by the functionRðN; xÞ, is not appreciably affected by changing N whilerestricting to the small x region where lattice simulationdata are currently available. Note, however, that thereappears a nominal deviation for larger x. Our generalconclusion is that the critical surfaces show rapidly ex-panding behavior for high �, namely, at high x, as clearlyevident from Figs. 4 and 7, in which case the ‘‘physical’’line corresponding to the physical current quark mass linesalways intersects the critical surface due to its monotoni-cally expanding nature. In other words, the NJL modelalways seems to favor the existence of the CP scenario,irrespective of the choice of N. While, on the other hand,the current lattice simulations are not decisive, at the mo-ment being beset with larger lattice cutoff effects thanfinite-density effects, making continuum extrapolationsdoubtful. However, there is still room for further preciselattice calculations in the future with a greater number oflattice sites and development of new techniques for finite-density simulations that may be necessary to make definiteconclusions.

As a final note, we point out the possibility of thetemperature and density dependence of the coupling

constants that may become important at high energies, aswe expect the couplings to run with respect to the energyscale. However, to test the effects of the temporal UV-cutoff, we have used constant values of the couplings gSand gD which were fixed from the very outset by fitting tophysical quantities at small temperature and chemical po-tential. These dependencies so arising may turn out to havecrucial effects on the critical surface especially when con-sidering the system at high densities, where it is expectedto be dominated by nonhadronic states. Thus, the locationof the CP is indeed sensitive to the nature and magnitude ofthe coupling constants. In fact, it was actually found in [19]that the UAð1Þ anomaly strength modeled through thechemical potential dependent coupling gD may changethe curvature of the critical surface, as well as its sign,resulting in a characteristic ‘‘back-bending’’ of the criticalsurface as a function of �. This reflects the fact that thedensity dependence of the coupling strengths plays crucialrole when investigating the chiral critical surface.

ACKNOWLEDGMENTS

We are grateful to P. de Forcrand for suggesting that weperform this calculation. H.K. wants to thank T. Inagakiand D. Kimura for fruitful discussions. H.K. is supportedby the National Science Council (NSC) of Taiwan, GrantNo. NSC-99-2811-M-033-017. J.W. C. and U. R. are sup-ported by the NSC and NCTS of Taiwan. U. R. is alsosupported in part by the U.S. National Science Foundation,Grant No. PHYS-0758114. He thanks the Institute ofMathematical Science, Chennai, and the Indian Instituteof Technology, Bombay, for their kind hospitality duringthe progress of this work.

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