Model analysis on thermal UV-cutoff effects on the critical boundary in hot QCD
Jiunn-Wei Chen, Kenji Fukushima, Hiroaki Kohyama, Kazuaki Ohnishi, Udit Raha
aa r X i v : . [ h e p - ph ] D ec YITP-09-107
Model analysis on thermal UV-cutoff effects on the critical boundary in hot QCD
Jiunn-Wei Chen ∗ Department of Physics and Center for Theoretical Sciences,National Taiwan University, Taipei 10617, Taiwan
Kenji Fukushima † Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Hiroaki Kohyama ‡ Institute of Physics, Academia Sinica, Taipei 115, Taiwan andPhysics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan
Kazuaki Ohnishi § and Udit Raha ¶ Department of Physics and Center for Theoretical Sciences,National Taiwan University, Taipei 10617, Taiwan
We study the critical boundary on the quark-mass plane associated with the chiral phase transitionin QCD at finite temperature. We point out that the critical boundaries obtained from the LatticeQCD simulation and the chiral effective model are significantly different; in the (Polyakov-loopcoupled) Nambu–Jona-Lasinio (NJL) model we find that the critical mass is about one order ofmagnitude smaller than the value reported in the Lattice QCD case. It is known in the LatticeQCD study that the critical mass goes smaller in the continuum limit along the temporal direction.To investigate the temporal UV-cutoff effects quantitatively we consider the (P)NJL model withonly finite N τ Matsubara frequencies taken in the summation. We confirm that the critical massin such a UV-tamed NJL model becomes larger with decreasing N τ , which demonstrates a correcttendency to explain the difference between the Lattice QCD and chiral model results. PACS numbers: 12.38.Aw,11.10.Wx,11.30.Rd,12.38.Gc
Introduction
The phase diagram of quark matter asdescribed by Quantum Chromodynamics (QCD) is ofgreat interest in theoretical and experimental physics.Since QCD has been established as the fundamental the-ory of quarks and gluons, it is our ultimate goal to un-derstand all the phenomena related to the strong interac-tions from the first principle theory. It is difficult, how-ever, to surpass perturbative calculations due to largenessof the QCD coupling constant in the low-energy regime.One then has to make a choice among non-perturbativetechniques such as the Lattice QCD (LQCD) simulation,some effective model descriptions sharing the same globalsymmetry as QCD, etc. The Lattice QCD (LQCD) is atheoretical framework of quarks and gluons in discretizedspace-time [1] which is suitable for non-perturbative in-vestigations. The Monte-Carlo simulations in the LQCDhave been recently developing to a reliable level not onlyfor hadron spectrums at zero temperature but also forthermal properties in hot QCD matter [2].Critical phenomena in hot QCD can be studied inthe LQCD simulation [3] and it is an important issue ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]; Present affiliation:GOGA, Inc., Tokyo. ¶ Electronic address: [email protected] to determine the order of the chiral phase transition asa function of the temperature T and the quark chem-ical potential µ . In this paper, we study the criticalboundary in the chiral phase transition on the quark-mass plane, as represented by the “Columbia plot” [4].Figure 1 shows a schematic picture of the Columbia plotat µ = 0 where each point in the parameter space of cur-rent quark masses { m u , m d , m s } is marked by its orderof the phase transition when T is raised. Usually onesets m u = m d ( ≡ m ud ) and studies the order of the phasetransition on the m ud - m s plane. Symmetry argumentsshow that it should be a first order phase transition inthe chiral limit ( m ud = m s = 0) [5], while for inter-mediate quark masses at which both chiral and centersymmetries are badly broken, only a smooth crossoveris possible. This defines the second-order critical phaseboundary that separates the first-order and crossover re-gions.The “phys. point” in Fig. 1 designates the “real world”with physical current quark masses ( m ud = 5 . m s = 135 . µ = 0. More importantly, the distance between thesecond-order critical boundary and the physical point onthe Columbia plot gives a rough measure of how far theQCD critical point is separated from the µ = 0 situa-tion. Roughly speaking, the farther the physical point islocated away from the critical boundary, the larger µ isnecessary to make the critical surface hit on the physical f ff ∞ ∞ Crossover1st1st S t r a ng e Q u a r k M a ss Light Quark Mass
FIG. 1: Columbia plot at finite T and µ = 0. point.The critical boundary can also be analyzed by meansof the LQCD simulation since the Columbia plot isdrawn at zero baryon density, where the Binder cumu-lant is a useful quantity to identify the order of thephase transition [6, 7, 8]. To draw the Columbia plotwe can use the three-flavor Nambu–Jona-Lasinio (NJL)model [9] which is a model of quarks with the effectiveLagrangian so constructed to have the same global sym-metry as QCD [10, 11]. (For other chiral model studiessee [12, 13].) Gauge symmetry is, however, absent due tothe absence of gluons in the model. It is a crucial aspectin the NJL model to incorporate chiral symmetry as animportant ingredient in understanding the properties oflight hadrons. Further, the model can successfully de-scribe the spontaneous breaking of chiral symmetry at inthe QCD vacuum and its restoration at high tempera-ture and/or density. The three-flavor NJL model gives afirst order phase transition in the chiral limit by virtueof the U A (1) symmetry breaking term [5], while, at in-termediate quark mass, the chiral restoration results ina crossover due to explicit chiral symmetry breaking byfinite quark mass as long as µ ≃
0. Thus, the NJL modelshould be able to reproduce the expected structure of theColumbia plot (i.e. critical boundary in the left-bottomcorner in Fig. 1), which can be compared with the LQCDresults.Although the NJL model successfully describes thequalitative feature of the chiral phase transition, there re-mains a significant difference between the critical bound-aries from the NJL model and the LQCD simulations.In fact, the value of the critical quark mass is about oneorder of magnitude different, which is crucial in the QCDcritical point search. One may have seen a scatter plotof the critical point locations from the LQCD attemptsas well as the chiral model approaches as shown in [14].However, there is a huge discrepancy in the critical massboundary already at zero baryon density depending onthe adopted methods, and then, how can one trust any ofthem at finite density? Let us pick some concrete num- bers up. We define the critical quark mass m c as a pointwhere the m ud = m s symmetric line intersects with thecritical boundary. Then, m c is around 14 MeV in theLQCD simulation [6] but it is only 1 . N τ = 4 (and N τ = 6 respectively) lattice. The con-clusion in [7] is consistent with this qualitatively thoughthe quantitative change is not such drastic. Because thenumber of spatial points is large enough (16, 24, etc)in the LQCD simulations, smallness of N τ in the tem-poral direction is crucial. In other words, the temporalUV cutoff πN τ T has drastic effects on the critical massboundary even at zero density (and hence on the QCDcritical point at finite density as well, which is beyondour current scope in this paper).Here, we shall examine the chiral critical boundary byimposing a finite UV cutoff in the temporal (thermal) di-rection in the NJL model to mimic the LQCD situation.In this way, we can give a quantitative prediction abouthow much the critical boundary moves by the finite N τ effects. In terms of Fourier transformed variables thesummation over finite- N τ temporal sites amounts tothe Matsubara summation over finite- N τ frequencies.Hereafter, throughout this paper, we will drop the sub-script and simply write as N for notational simplicity.In what follows, we present a brief description of thismodified NJL model and discuss the machinery of howto deal with finite N in this model treatment. Later, weshall also present some results using the Polyakov-loopcoupled NJL (PNJL) model [9, 15, 16, 17, 18], forthe PNJL model is a much better description of QCDthermodynamics near the critical temperature. Modified NJL model
The three-flavor NJL model La-grangian is, L NJL = ¯ q (i ∂/ − ˆ m ) q + L + L , (1) L = g S X a =0 h (¯ qλ a q ) + (¯ q i γ λ a q ) i , (2) L = − g D [det ¯ q i (1 − γ ) q j + h.c. ] . (3)The current quark mass matrix ˆ m in the kinetic termtakes the diagonal form ˆ m = ( m u , m d , m s ), and we set m u = m d ( ≡ m ud ). In the above L expresses thefour-fermion contact interaction with the coupling con-stant g S , where λ a is the Gell-Mann matrix in flavorspace, and L is the Kobayashi-Maskawa-’t Hooft inter-action [19, 20, 21] with the coupling strength g D . The de-terminant runs over the flavor indices, which leads to six-fermion interaction terms in the three-flavor case. Thisinteraction explicitly breaks the U A (1) symmetry, whosemicroscopic origin comes from instantons [21].We should solve the gap equations which we can getby differentiating the thermodynamic potential Ω withrespect to the constituent quark masses; ∂ Ω ∂ m ∗ u = 0 , ∂ Ω ∂ m ∗ s = 0 . (4)Here m ∗ u and m ∗ s are the constituent masses for u and s quarks. The thermodynamic potential is defined by Ω = − ln Z/ ( βV ), with the partition function Z , the inversetemperature β = 1 /T , and the volume of the system V .In the mean-field approximation for the NJL La-grangian (1) we obtain the following gap equations, m ∗ u = m u + 2i g S N c tr S u − g D N c (tr S u )(tr S s ) ,m ∗ s = m s + 2i g S N c tr S s − g D N c (tr S u ) . (5)Here N c = 3 is the number of colors and tr S i is the chiralcondensate given explicitly byi tr S i = 4 m ∗ i Z d p (2 π ) (i T ) ∞ X n = −∞ i(i ω n ) − E i , (6)where the Matsubara frequency is ω n = (2 n + 1) πT andthe quasi-particle energy is E i = p p + m ∗ i . A detailedderivation of the gap equations is clearly given in [10, 11].To study the effect of the finite cutoff in the temporaldirection on the critical boundary, we then perform thefollowing modification in the Matsubara frequency sum; ∞ X n = −∞ −→ N X n = − N +1 . (7)We shall reveal the critical boundary in the m ud - m s planefor various values of N . To this end, we adjust the modelparameters to fit the physical quantities for a given N (each parameter set representing a modified NJL modelwith a different temporal cutoff). This is reminiscent ofthe familiar Wilsonian renormalization group (RG) flow.After the high frequency modes are integrated out, inprinciple, the coefficients of the operators run on the RGflow. Here we remark that we neglect interaction oper-ators in other channels than those in Eqs. (1)-(3)) sincetheir effects are small in the mean-field calculations per-formed below (see e.g., [10] and refs. therein).The following are the model parameters in the three-flavor NJL model;current quark masses m ud , m s ,three-momentum cutoff Λ ,four-point coupling constant g S ,six-point coupling constant g D .As for the light current quark masses, we fix m ud = 5 . m ∗ u = m ∗ d . The remaining parameters are de-termined by a fit to the four physical quantities: the pion N m s (MeV) Λ (MeV) g S Λ g D Λ
15 134 . . .
16 12 . . . .
02 11 . . . .
82 10 . . . .
75 9 . ∞ . . .
67 9 . N . mass m π , the pion decay constant f π , the kaon mass m K ,and the η ′ mass m η ′ , whose empirical values are m π = 138 MeV , f π = 93 MeV ,m K = 495 . , m η ′ = 958 MeV . (8) Numerical results
The limit of N = ∞ invariablycorresponds to the standard framework of the NJL modeland we then use the same parameter set fixed in [11]. Forfinite N , we have picked up a temperature T = 50 MeVat which we have carried out our fitting procedure. Thischoice is just because of technical convenience. Besides,calculations at T = 50 MeV should be sufficiently closeto those at zero temperature since the constituent quarkmass is then several times larger than the temperature.The fitted parameters in this way for N = 15, 20, 50, 100at T = 50 MeV are displayed in Tab. I. C on s tit u e n t Q u a r k M a ss [ M e V ] Temperature[MeV] N=15N=20N=50N=100N= ∞ mm *s*u FIG. 2: Constituent quark masses m ∗ u (lower) and m ∗ s (upper)as a function of T for various values of N . As seen from the table, the couplings g S and g D be-come larger with decreasing N . If we fixed the couplingconstants, the effect of smaller N would be to reducethe constituent quark masses. Therefore, to keep m ∗ i un-changed, we need larger g S and g D to compensate forthis reduction (see Eq. (5)). Note that, as expected, thewhole behavior of m ∗ u , m ∗ s as a function of T monotoni-cally approach the standard (i.e. N = ∞ ) case with in-creasing N . In Fig. 2, we only show the numerical resultsabove T = 40 MeV because the small- T region is badlyaffected by finite N and the calculations are no longerreliable there. We can understand this intuitively; 1 /T isthe extent along the imaginary-time (thermal) directionand thus, the extent becomes longer as T goes smaller.Hence, at lower T , reliable estimates need more N . Thismeans, fixing the value of N , the choice of the small- T regions would automatically lead to unphysical results. S t r a ng e Q u a r k M a ss [ M e V ] Light Quark Mass[MeV]N=15N=20N=50N= ∞ FIG. 3: Critical boundaries on the m ud - m s plane in the NJLmodel (left panel) as compared with the LQCD results with N τ = 4 (right panel). In Fig. 3, we display the result for the critical bound-ary in our modified NJL model with the finite Matsubarafrequency summations: N = 15, 20, 50, ∞ , along withthe results from the LQCD simulation with N τ = 4 [6]for comparison. We find that the first-order regionwidens as N decreases. This observation qualitativelyagrees with the arguments in the LQCD simulationsthat the first-order region tends to get bigger thanits genuine size due to the finite UV-cutoff effects.Quantitatively, however, the critical boundaries still lookfar from the LQCD results at N τ = 4. We comment thatnot only the temporal UV-cutoff but also the spatial(three-momentum) cutoff leads to similar alteration inthe critical boundary i.e., we have found that lowering Λalso pushes the critical mass up, though such effects areonly minor. The PNJL model extension
It is also worth studyingthe critical boundary using the PNJL model [9, 15, 16,17, 18]. This is because, in view of the results in [22, 23],the location of the critical point is significantly movedtoward higher T by the inclusion of the Polyakov loopwhich suppresses unphysical quark excitations. Further-more, the PNJL model is quite successful to reproducethe LQCD observables for QCD thermodynamics suchas the quark number susceptibility, the interaction mea-sure, the sound velocity, etc. Because the PNJL modelnicely grasps the essential nature of the QCD phase tran-sitions, it would be rather mysterious if only the criticalboundary on the m ud - m s shows a huge discrepancy.The model is defined by the following Lagrangian [15, 16], L PNJL = L + L + L + U (Φ , Φ ∗ , T ) , (9) L = ¯ q (i ∂/ − i γ A − ˆ m ) q , (10) U (Φ , Φ ∗ , T ) = − bT (cid:8)
54 e − a/T ΦΦ ∗ + ln (cid:2) − ∗ + 4(Φ + Φ ∗ ) − ∗ ) (cid:3)(cid:9) , (11)where U is the effective potential in terms of the Polyakovloop. Here Φ and Φ ∗ are the traced Polyakov loop andthe anti-Polyakov loop which are order parameters fordeconfinement. They are defined by Φ = (1 /N c )tr L ,Φ ∗ = (1 /N c )tr L † with L = P exp[i R β d τ A ]. We setthe parameters a and b as a = 664 MeV, b · Λ − = 0 . S t r a ng e Q u a r k M a ss [ M e V ] Light Quark Mass[MeV]N=15N=20N=50N= ∞ FIG. 4: Critical boundaries on the m ud - m s plane in the PNJLmodel (left panel) as compared with the LQCD results with N τ = 4 (right panel). Because the Polyakov-loop part is relevant only atfinite T (as high as comparable with T c ), we can simplyemploy the same parameter sets, as fixed previouslyin the NJL model case with finite N (see Tab. I).Then, we show the results for the critical boundarydetermined by the same procedure in Fig. 4. Hereagain, we see expanding behavior of the first-orderregion with decreasing N . Although the PNJL resultsare quantitatively different from the NJL ones, thequalitative tendency is the same and, besides, thedifference is not really substantial. The fact that thePNJL model results are such close to the NJL ones isin fact non-trivial. The Polyakov loop generally governsthe thermal excitation of quarks near T c and particularlyunphysical quarks below T c are prohibited by small Φand Φ ∗ . This leads to the consequence that the criticalpoint temperature is lifted up to twice as high. From thecomparison between Figs. 3 and 4, gives us confidencethat the smallness of the first-order region is not simplyan NJL-model artifact with unphysical quark excitations. Concluding remarks
Our model analysis yields in-triguing results that the critical masses become largerwith truncation in the Matsubara frequency summation.This effect is not large enough to give a full account forthe quantitative difference in the critical boundaries be-tween the LQCD and NJL model studies. Nevertheless,it raises an interesting possibility that the critical massesin the LQCD simulation become smaller if the LQCDsimulation accommodates larger N τ . This is an impor-tant message since the LQCD simulation at N τ = 4, 6,and even 8 may have a potential danger that the LQCDsimulation overestimates the strength of the first-orderphase transition, and thus the critical point emerges atsmall density due to lattice artifacts.An interesting extension of our study is to determinethe chiral phase transition boundary on the plane with T and the baryon (quark) chemical potential µ . It is a com-mon folklore that the QCD critical point should appearat some T E and µ E . However, it is highly non-trivial todetermine the order of the phase transition at non-zero µ .Recent LQCD simulations at small µ (i.e. expansion as apower of µ/T ) disfavor the existence of the critical point in the QCD phase diagram [6, 7]. There have been the-oretical efforts to understand this LQCD finding in thecontext of the chiral effective model approaches [24, 25].However, the errors of the LQCD simulations grow biggerat higher µ and, to be worse, the (P)NJL-model analysescan be trusted only at a qualitative level so far once µ gets large. It is still inconclusive, therefore, whether thecritical point actually exists or not. Our model analysisshows that increasing N τ leads to smaller critical quarkmasses. This result might indicate to further minify thepossibility for finding the QCD critical point using theLQCD simulation unless the discretization errors are un-der good theoretical control.JWC, KO and UR are supported by the NSC andNCTS of Taiwan. KF is supported in part by JapaneseMEXT grant No. 20740134 and the Yukawa InternationalProgram for Quark Hadron Sciences. UR also thanksINT, Seattle for the kind hospitality during the course ofthis work. [1] K. G. Wilson, Phys. Rev. D , 2445 (1974).[2] For a recent review, see; O. Philipsen, Prog. Theor. Phys.Suppl. , 206 (2008) [arXiv:0808.0672 [hep-ph]].[3] S. Ejiri et al. , arXiv:0909.5122 [hep-lat].[4] F. R. Brown et al. , Phys. Rev. Lett. , 2491 (1990).[5] R. D. Pisarski and F. Wilczek, Phys. Rev. D , 338(1984).[6] P. de Forcrand and O. Philipsen, JHEP , 077 (2007)[arXiv:hep-lat/0607017].[7] P. de Forcrand, S. Kim and O. Philipsen, PoS LAT2007 , 178 (2007) [arXiv:0711.0262 [hep-lat]]; P. deForcrand and O. Philipsen, JHEP , 012 (2008)[arXiv:0808.1096 [hep-lat]].[8] G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, PoS
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