aa r X i v : . [ h e p - ph ] J a n YITP-08-84
Isentropic thermodynamics in the PNJL model
Kenji Fukushima
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We discuss the isentropic trajectories on the QCD phase diagram in the temperature and the quarkchemical potential plane using the Nambu–Jona-Lasinio model with the Polyakov loop coupling(PNJL model). We impose a constraint on the strange quark chemical potential so that the strangequark density is zero, which is the case in the ultra relativistic heavy-ion collisions. We compareour numerical results with the truncated estimates by the Taylor expansion in terms of the chemicalpotential to quantify the reliability of the expansion used in the lattice QCD simulation. We finallydiscuss the strange quark chemical potential induced by the strangeness neutrality condition andrelate it to the ratio of the Polyakov loop and the anti-Polyakov loop.
PACS numbers: 12.38.Aw, 11.10.Wx, 11.30.Rd, 12.38.Gc
I. INTRODUCTION
Thermodynamic properties of hot and dense matterout of quarks and gluons are of theoretical and experi-mental importance. To this end the Monte-Carlo methodon the lattice has worked quite successfully to simulatethe matter at high temperature from the first principle,that is, Quantum Chromodynamics (QCD) [1, 2, 3, 4, 5].It is believed that such hot and dense matter has beencreated in the Relativistic Heavy Ion Collider (RHIC) lo-cated at BNL. Since the baryon stopping power in thenucleus-nucleus collision is small at the RHIC (top) en-ergy √ s NN = 200 GeV, the created matter is nearly freefrom the net baryon density, and thus, the correspond-ing baryon chemical potential is much smaller than thetemperature.We are facing two experimental possibilities for thefuture exploration of the QCD phase diagram: One isgoing toward even hotter matter as planned in the LargeHadron Collider (LHC) at CERN. The other one is re-alizing denser matter, which is accessible by collisions atsmaller energy with larger baryon stopping power. Theformation of baryon-rich matter is within the scope of theFacility for Antiproton and Ion Research (FAIR) at GSIand the systematic energy scan is also under discussionin the future plan for RHIC with emphasis on the QCDcritical point search. The present work is focused onthe latter; the thermodynamic properties of matter witha finite quark chemical potential µ whose magnitude iscomparable to the temperature T .The lattice QCD simulation has a serious limitation ifa finite chemical potential is turned on in the Dirac oper-ator [6, 7]. That is, the notorious sign problem preventsus from applying the Monte-Carlo method to a finite-density system. There are a number of proposals to tamethis problem [8], among which the Taylor expansion interms of µ/T seems to work well insofar as µ/T is un-der the radius of convergence [9, 10, 11]. In this way theequation of state (EoS) at finite T and µ is partially avail-able from the direct lattice QCD simulation. Actually theEoS is an indispensable input for the sake of the hydro-dynamic evolution of matter. From the s/n B -constant line, where s is the entropy density and n B is the baryonnumber density, we can draw the isentropic trajectoryalong which the adiabatic system evolves [10, 12, 13].An interesting progress has been made in Ref. [10];the isentropic thermodynamics was investigated with aconstraint that the strange quark density must be zero,i.e. n s = 0. Such a constraint is necessary to emulatematter created by the ultra relativistic heavy-ion colli-sions because thermalization is achieved within the timescale of the strong interaction and the system is far from β -equilibration.In this paper we will utilize the Nambu–Jona-Lasinio model with the Polyakov loop coupling (PNJLmodel) [14, 15, 16, 17, 18, 19, 20, 21] to reveal the isen-tropic thermodynamics with n s = 0 imposed. In contrastto the NJL model study along the same line [22] (see alsoRef. [23] for another type of approach), the PNJL modelhas an advantage that a part of the gluon degrees of free-dom is included, which gives us a hope that the PNJLmodel can lead to a better EoS reflecting the gluody-namics as well as the chiral dynamics. In Ref. [24] wecan already find the isentropic trajectory evaluated inthe PNJL model and notice that they are qualitativelysimilar to the results in Ref. [22]. We will see that theconstraint n s = 0 causes only little change in the resul-tant isentropic trajectory because the strange quark mass M s is still substantially large around the crossover regionand the strange density is suppressed in any case.The extension from the NJL model to the PNJL modelis not a minor improvement, however. Our nontrivialfinding is, in fact, that the PNJL model is capable of cap-turing the correct behavior of the induced strange quarkchemical potential µ s > n s = 0. Since thenonzero µ s has its origin in confinement physics, as wewill discuss later, the NJL model without any confine-ment effect is of no use but the PNJL model naturallyprovides us with µ s > ℓ and the anti-Polyakov loop ¯ ℓ . As amatter of fact, although it is known that ℓ deviates from¯ ℓ at µ = 0 [9, 17, 25, 26], our present work is the veryfirst demonstration to show that this difference ℓ = ¯ ℓ hasa physical consequence in the best of our knowledge.This paper is organized as follows: We explain themodel definitions in Sec. II. Then, in Sec. III, we showthe numerical results with the constraint n s = 0, followedby the discussions on the validity of the Taylor expansionin Sec. IV. We elucidate the physical meaning of µ s = 0in Sec. V and the summary is in Sec. VI. II. MODEL SETUP
The thermodynamic potential of the three-flavor PNJLmodel consists of three pieces, Ω = Ω vacuum + Ω thermal +Ω Polyakov , namely, the vacuum part (zero-point energyand the condensation energy), the thermal part, and thePolyakov loop potential, respectively. We shall take aclose look at them in order.The vacuum part is exactly the same as in the ordinaryNJL model;Ω vacuum = − X i Z Λ d p (2 π ) ε i ( p )+ g S (cid:16) h ¯ uu i + h ¯ dd i + h ¯ ss i (cid:17) + 4 g D h ¯ uu ih ¯ dd ih ¯ ss i , (1)where i refers to the quark flavor running over u , d , and s . The energy dispersion relations are ε i ( p ) = p p + M i with M i = m i − g S h ¯ q i q i i − g D ǫ ijk h ¯ q j q j ih ¯ q k q k i , where g S represents the four-Fermi coupling constant and g D represents the ’t Hooft interaction strength. There arethree more model parameters: the light current quarkmass m u = m d , the heavy current quark mass m s , andthe ultraviolet cutoff Λ. We here adopt the parameterset by Hatsuda-Kunihiro [27];Λ = 631 . ,g S Λ = 3 . , g D Λ = − . ,m u = m d = 5 . , m s = 135 . , (2)which fits m π , m K , m η ′ , f π , and empirical M ud .The thermal part has a coupling to the Polyakov loop(spatially homogeneous A background) in a form asΩ thermal = − T X i Z d p (2 π ) n ln det h L e − ( ε i ( p ) − µ ) /T i + ln det h L † e − ( ε i ( p )+ µ ) /T io , (3)where the Polyakov loop is a 3 × L ( ~x ) = P exp (cid:20) − i g Z β d x A ( ~x, x ) (cid:21) , (4)and in this paper the Polyakov loop sometimes refers tothe traced one after average, that is, ℓ = 13 (cid:10) tr L (cid:11) , ¯ ℓ = 13 (cid:10) tr L † (cid:11) , (5) as long as no confusion may arise. In a simple mean-fieldapproximation for the Polyakov loop (group) integration,the determinant explicitly reads;det h L e − ( ε − µ ) /T i → − ε − µ ) /T + 3 ℓ e − ( ε − µ ) /T + 3 ¯ ℓ e − ε − µ ) /T , (6)det h L † e − ( ε + µ ) /T i → − ε + µ ) /T + 3 ¯ ℓ e − ( ε + µ ) /T + 3 ℓ e − ε + µ ) /T . (7)It is important to note that a positive µ induces ¯ ℓ > ℓ ,while ¯ ℓ = ℓ at µ = 0. The finite-temperature field theorytells us that the traced Polyakov loop gives the exponen-tial of the free energy cost by a test quark, i.e. ℓ = e − f q /T up to normalization (or energy offset), and the anti-Polyakov loop by a test anti-quark, i.e ¯ ℓ = e − f ¯ q /T [28].Therefore, f q → ∞ and thus ℓ → ℓ nor ¯ ℓ can bestrictly zero due to screening. If the color screening isstronger, the free energy cost is smaller, and the Polyakovloop is larger accordingly. In a medium with µ > ℓ > ℓ [9, 17, 25, 26].We shall choose the Polyakov loop potential as [21]Ω Polyakov = − b · T n
54 e − a/T ℓ ¯ ℓ + ln (cid:2) − ℓ ¯ ℓ − ℓ ¯ ℓ ) + 4( ℓ + ¯ ℓ ) (cid:3)o . (8)Here, there are two parameters a and b in the aboveansatz. We fix a = 664 MeV to reproduce T c ≃
270 MeVin the pure gluonic sector and b = 0 .
03 to yield T c ≃
200 MeV for the simultaneous crossovers of deconfine-ment and chiral restoration.
III. RESULTS
We are now ready to proceed to the numerical cal-culations using the PNJL model. Here let us focuson the entropy per baryon density ratio, s/n B . Thisis because the adiabatic hydrodynamic expansion con-serves s/n B along the time evolution. It is easy to con-firm that s/n B is a constant indeed from conservationof the entropy current and the baryon current, that is,(d / d τ )( s/n B ) = u µ ∂ µ ( s/n B ) = 0 readily follows from ∂ µ ( su µ ) = 0 and ∂ µ ( n B u µ ) = 0. Hence, one does nothave to integrate the hydrodynamic equation to drawthe time-evolution path, which is simply inferred froman s/n B -constant line . This is the case insofar as theexpansion is fast enough to make the system thermallyisolated and the entropy production due to dissipation isnegligible. T e m p e r a t u r e [ M e V ]
0 50 100 150 200 250 300 350 400 450250210170130905010
FIG. 1: Chiral susceptibility with respect to light ( u and d )quarks defined by − d Ω / d m u in the unit of Λ as a functionof µ and T . A. Case without constraint on n s We first take a quick look at the results without con-straint on n s . The quark density is specified by the quarkchemical potential (or one third of the baryon chemicalpotential) which is common to all three flavors. In theresults presented in this subsection, thus, the net strangequark density n s is nonzero.We solve the following gap equations self-consistently; ∂ Ω ∂ h ¯ uu i = ∂ Ω ∂ h ¯ ss i = ∂ Ω ∂ℓ = ∂ Ω ∂ ¯ ℓ = 0 (9)with assuming isospin symmetry h ¯ dd i = h ¯ uu i . In thisway we have the chiral condensates, the Polyakov loop,and the anti-Polyakov loop as functions of T and µ . Toillustrate the phase structure, we show the light-quarkchiral susceptibility in a density plot in Fig. 1.We can perceive from Fig. 1 that the critical regionextends in the vicinity of the second-order critical pointlocated at ( T, µ ) = (315 MeV ,
100 MeV) on top of theenhanced strip along the chiral crossover. Because ourapproximation neglects the soft-mode fluctuations, theEoS obtained in this work may miss the singular con-tribution to thermodynamic quantities near the criticalpoint [29, 30, 31]. We already know, however, thatthe PNJL model can reproduce the pressure behaviorat µ = 0, which implies that the soft-mode contributionis not significant there. We can then anticipate that thesingular contribution would become important only ina narrow region surrounding to the critical point (indi-cated by the bright colors on the density plot in Fig. 1).In what follows we shall draw the isentropic trajectoriesonto this phase structure not taking account of soft-modefluctuations.By substituting the solution of the gap equations intoΩ, the thermodynamic functions such as the entropy T e m p e r a t u r e [ M e V ] Quark Chemical Potential [MeV] s/n B =2 s/n B =5 s/n B =10 s/n B =20 s/n B =30 s/n B =40 s/n B =3.5 FIG. 2: Isentropic trajectories on the µ - T plane in the casewithout constraint on n s . The (blue) triangle marks the lo-cation of the critical point. and the baryon number of our current interest obtainby means of the thermodynamic relations; s = − ∂ Ω ∂T , n B = − ∂ Ω ∂µ . (10)Figure 2 shows the numerical results for the isentropictrajectories for various values of s/n B without imposingconstraint on n s .We note that Fig. 2 is reasonably consistent with Fig. 9in Ref. [24] which employs the two-flavor PNJL model.Moreover, the trajectories look qualitatively similar tothe results in the NJL model as in Ref. [22], while thevalue of s/n B associated with each trajectory is greaterin our case than in the NJL model study. This has anintuitive interpretation. The PNJL model is composed ofquasi-quarks and a part of gluons, so that s has steeperbehavior near T c and grows larger above T c as comparedto the NJL model. As for n B , because of the Polyakovloop average, n B has steeper behavior as well, but it doesnot exceed the NJL model value. The ratio s/n B in thePNJL model, therefore, results in mild sensitivity to thesteepness of s and n B near T c , leading to the similartrajectory curves to the NJL model results. In contrast,the magnitude of s/n B corresponding to the trajectorybecomes larger in the PNJL model as a consequence ofthe gluon degrees of freedom contributing to s .Comparing our results to the lattice QCD simulationwith the Taylor expansion [13], we see that our estimateof s/n B (for instance comparing the s/n B = 40 curve inFig. 2 and the s/n B = 45 curve in Ref. [13]) improves anagreement.We remark that all the trajectories in the low densityside must go to µ > M u as T decreases because n B → µ < M u and T →
0. In the hadron phase at small T and moderate µ , in any case, the PNJL model de-scription is not realistic as it cannot describe the nucleonand nuclear matter. We should be aware that the PNJL T e m p e r a t u r e [ M e V ] Quark Chemical Potential [MeV] s/n B =2 s/n B =5 s/n B =10 s/n B =20 s/n B =30 s/n B =40 s/n B =3.5 FIG. 3: Isentropic trajectories with n s = 0 imposed, whichis relevant to the relativistic heavy-ion collisions. The (blue)triangle marks the location of the critical point in this case. model works well near and above T c and µ c but not farbelow them. B. Case with constraint on n s We next proceed to the case with imposing n s = 0 toemulate the situation in the high-energy heavy-ion colli-sions. We should determine µ s self-consistently solving n s = − ∂ Ω ∂µ s = 0 , (11)together with other gap equations in Eq. (9).The phase structure is only slightly changed by theconstraint. The critical point moves from ( µ, T ) =(315 MeV ,
100 MeV) to ( µ, T ) = (317 MeV ,
100 MeV).The resultant isentropic trajectory as shown in Fig. 3thus takes a very similar shape to the case without con-straint n s = 0. We further calculate the pressure alongthe isentropic trajectories on Fig. 3 and make a plot ofFig. 4. Here we did not normalize the pressure by T as is often the case, for the density contribution ∝ µ becomes predominant as s/n B goes small.From Fig. 3 we see that the strangeness neutrality con-dition has a noticeable effect on the trajectory only athigh T . This is because the s -quark constituent mass, M s , is still heavy around the chiral crossover with re-spect to u and d quarks, and so n s ≈ T is low and µ is smaller than M s .If µ surpasses M s , we would recognize a difference by theneutrality effect in the low- T region, but then, we haveto consider color superconductivity at such high density,which is not within our current scope.The isentropic thermodynamics hardly changes, as ob-served in Fig. 4, until s/n B .
20. This behavior agreeswith the lattice results [10, 13] in which no significant s/n B dependence has been found. P r e ss u r e [ G e V f m - ] Temperature [MeV] s/n B =2 s/n B =5 s/n B =10 s/n B =20 s/n B =30 s/n B =40 s/n B =3.5 FIG. 4: Pressure along the respective isentropic trajectoriesgiven in Fig. 3.
We could have placed a plot here for the induced µ s as a function of T . For later convenience, however, wepostpone showing it and let us turn to the validity of theTaylor expansion as utilized in the lattice QCD simula-tion. IV. VALIDITY OF THE TAYLOR EXPANSION
All of our calculations result from the PNJL modelin the mean-field approximation and do not rely on theTaylor expansion unlike the lattice QCD simulation. Al-though we have no need to carry the expansion out, itis interesting to compare “full” results of our numericalcalculations and “truncated” ones to verify how nicelythe Taylor expansion works.Within the framework of the PNJL model we havesolved s and n B as functions of µ . Now we shall expandthem into the series like Ref. [32] as s ( µ ) = lim M →∞ M X n =0 d n s (0)d µ n µ n , (12) n B ( µ ) = lim N →∞ N X n =0 d n +1 n B (0)d µ n +1 µ n +1 , (13)which is to be validated if no singularity associated withthe first-order phase transition lies along the µ -direction.It should be mentioned that the derivative in Eqs. (12)and (13) is the total derivative in a sense that it acts onthe implicit µ -dependence in the mean-fields. In this waythe mixing effect can be correctly taken into account inthe model treatment [33]. From symmetry s is an evenfunction of µ and n B is an odd function.Let us explain the numerical procedure in details tomake a comparison between the results with and withouttruncation of the degree in the Taylor expansion. We firstapproximate s ( µ ) and n B ( µ ) by the Taylor expansion T e m p e r a t u r e [ M e V ] Quark Chemical Potential [MeV] s/n B =10 s/n B =20 s/n B =30 s/n B =40 FIG. 5: Comparison between the full results (solid curve) andthe truncated Taylor expansion up to the µ -term in s ( µ ) andthe µ -term in n B ( µ ). The (green) dotted and (blue) dashedcurves represent the estimates by the Taylor series fitted toreproduce the full results in the ranges from zero to 50 MeVand 100 MeV, respectively. Four curves are for s/n B = 40,30, 20, 10 from the left to the right. with sufficiently large number of M and N so that thecoefficients in the first several terms barely change withan increment of M and N .We should further specify the fit range of µ ∈ [0 , µ ]to read the Taylor expansion coefficients. In principle, ifthe “exact” calculation were possible, µ could be zeroor there is no µ dependence at all. Even in the mean-field level, however, we are far from the exact calculation.Here we have chosen µ = 50 MeV and µ = 100 MeV.In fact, µ is a parameter which controls the precisionin the determination of the Taylor expansion coefficientsin the same sense as using the multiple-point formula forthe numerical differentiation. One might have thoughtthat µ = 500 MeV, for instance, can cover the wholedensity region in Fig. 3, but such a choice would bringartifact from outside the radius of convergence.Once we fix the Taylor expansion coefficients of s ( µ )and n B ( µ ), then we cut the series at smaller M and N .We will elucidate the leading-order case ( M = 2 and N = 1) and the next to leading-order case ( M = 3 and N = 2) below.Figure 5 shows the s/n B -constant curves with andwithout truncation on the µ - T plane. The solid curvesrepresent the full results without truncation. We havedrawn the (green) dotted curves by choosing N = 2for s ( µ ) and M = 1 for n B ( µ ) and the fit range as µ = 50 MeV. We see that the truncated series canapproximate the curve for s/n B = 40 well, and the curvefor s/n B = 30 is still close to the full estimate, but thedotted curve for s/n B = 20 has a huge deviation fromthe corresponding solid curve. If we determine the Taylorexpansion coefficients in a wider range as µ = 100 MeV,as shown by the (blue) dashed curves, the agreement be-comes better, of course. Not only the curve for s/n B = 40 T e m p e r a t u r e [ M e V ] Quark Chemical Potential [MeV] s/n B =10 s/n B =20 s/n B =30 s/n B =40 FIG. 6: Comparison between the full results (solid curve) andthe truncated Taylor expansion up to the µ -term of s ( µ ) andthe µ -term of n ( µ ). The (green) dotted and (blue) dashedcurves represent the estimates by the Taylor series fitted toreproduce the full results in the ranges from zero to 50 MeVand 100 MeV, respectively. Four curves are for s/n B = 40,30, 20, 10 from the left to the right. but also for s/n B = 30 agrees quite well with the full re-sults. Besides, the curve for s/n B = 20 is significantlyimproved at T &
200 MeV, while it does not fit the fullresults at lower T . It seems that the curve for s/n B = 10is too far from µ to perceive the effect of changing µ .It is intriguing to increase the truncation degree of theTaylor expansion to discuss how much the approximationis improved. We leave the terms up to the µ order in theexpansion of s ( µ ) (i.e. M = 3) and the µ order in the ex-pansion of n B ( µ ) (i.e. N = 2) and have found the curvesin Fig. 6. Although the results at low T become slightlybetter than Fig. 5, it is unexpected that the agreementat high T goes worse! This undesirable poor convergenceturns out to stem solely from the Taylor expansion of n B ( µ ).To examine the problem concretely, we present a plotfor n B ( µ ) in Fig. 7. The truncated results at N = 1(green dotted curve) can well describe the bold solidcurve which is the full data. As we go to the higher order,however, the truncation leads to a larger deviation fromthe full results, implying that the expansion seems to fail.If the numerical accuracy is arbitrarily good as needed, orequivalently, µ → ∞ , the Taylor expansion would workwith very small coefficients for the higher order terms inthe expansion series. In practice, however, the availableaccuracy is limited, and the artifact may enter, whicheventually goes wrong for µ away from the fitted region.Here, at the same time, we should emphasize that thereis no such weird behavior observed in the expansion of s ( µ ); the higher order we take account of, the better con-vergence of s ( µ ) we can reach, as is naturally expected.In view of this, hence, it could be conceivable that theremay be some profound reason why only the expansion of n B ( µ ) is dangerous. n B / T Quark Chemical Potential [MeV] N =1 N =2 N =3Fitted Region R [1,1] R [2,1] FIG. 7: The bold solid curve represents the full data of n B ( µ )in the model calculation which is Taylor expanded in the µ -range of [0 ,
100 MeV]. The thin solid curves represent thetruncated results up to N = 1, N = 2, and N = 3 as indi-cated by the labels. The dotted curves are the Pad´e-improvedresults for N = 2 and N = 3. We can thus learn the following important lessons fromthese analyses using the model.1) The Taylor expansion method does not work in thelow- T and high- µ region. The adiabatic path for s/n B .
20 deduced from the expansion may be totally differentfrom the true results. This value of s/n B happens tobe close to the threshold below which the pressure startsmoving apart from that at µ = 0 as seen in Fig. 4. Thisfact implies that the Taylor expansion may well workonly in the regime where the density effect is not suchappreciable.2) Even at high T where the Taylor expansion is be-lieved to work nicely, the expansion of the baryon density n B may be problematic suffering from the uncertainty inthe higher order terms than O ( µ ). One of the simplestremedies for this pathological expansion is the Pad´e im-provement as addressed in Ref. [11]. We can see in Fig. 7that the Pad´e-improved results, R [1 ,
1] for N = 2 and R [2 ,
1] for N = 3, certainly come close to the full curve.[ R [1 ,
1] denotes c µ (1 + c µ ) / (1 + c µ ) with c , c , c fixed to yield the original series up to N = 2 and R [2 , s does not have such a kind of expansion problemat all. V. STRANGE QUARK CHEMICALPOTENTIAL
As promised in Sec. III, the final topic discussed inthis paper is the induced chemical potential for strangequarks to keep the strangeness neutrality. In general thecondition of n s = 0 requires a positive µ s . This can beintuitively understood as follows.The strong interaction does not change the number T e m p e r a t u r e [ M e V ] Strange Quark Chemical Potential [MeV] s/n B =2 s/n B =5 s/n B =10 s/n B =20 s/n B =30 s/n B =40 s/n B =3.5 FIG. 8: Induced µ s necessary to keep n s = 0 for various valuesof s/n B in the PNJL model calculation. of strange quarks but can make strange particles in aprocess, for example, such as π − + p → K + Λ and π − + p → K + + Σ − etc. The strangeness of Λ and Σ ,Σ ± is negative one, meaning that they contain positiveone strange quark. Therefore, if p and n are abundantat finite baryon chemical potential, the strong interac-tion pushes strange quarks into strange baryons , thatresults in µ s >
0. [For more phenomenological details seeRef. [34] and references therein, and see also Ref. [35].]This sort of dynamics is completely missing in the NJLmodel without color confinement. If one solves n s = 0in the three-flavor NJL model, it ends up with µ s = 0,which is unphysical. In fact a positive µ s has been con-cluded in Ref. [10] as it should be. In this section we willsee that the PNJL model has a crucial advantage in de-scribing the induced µ s correctly because it encompassesthe confinement physics.Let us first show our numerical results in Fig. 8. A non-zero and positive µ s certainly appears in the PNJL modelunlike the NJL model. The numerical values are quali-tatively consistent with the lattice results in Ref. [10],though the quantitative comparison is not straightfor-ward. We would say that this demonstration of µ s > µ s .We recall that the Polyakov loop coupling takes a formofln det (cid:2) L e − ( ε − µ ) /T (cid:3) + ln det (cid:2) L † e − ( ε + µ ) /T (cid:3) , (14)in each flavor sector. As long as µ is small Eq. (14) iswell approximated as ≃ − ε/T (cid:0) ℓ e µ/T + ¯ ℓ e − µ/T (cid:1) . (15)For light flavors at µ >
0, therefore, the source weightfor anti-quarks is larger than that for quarks which yields¯ ℓ > ℓ , as is consistent with the argument given belowEq. (7). It might be a bit confusing but ℓ e µ/T is thesource for anti-quarks because the derivative of ¯ ℓ ℓ withrespect to ℓ gives the anti-Polyakov loop ¯ ℓ . Then, in theheavy flavor sector, the neutrality condition means in thesame approximation, ∂∂µ s (cid:0) ℓ e µ s /T + ¯ ℓ e − µ s /T (cid:1) = 0 , (16)which immediately leads to µ s = T (cid:0) ¯ ℓ/ℓ (cid:1) . (17)Interestingly enough, the above equation (17) holds ap-proximately in the entire µ - T plane with only a 3% vio-lation at worst! This is an intriguing relation discoveredin a heuristic manner in the PNJL model and could betested in the future lattice simulation.One might wonder why the PNJL model could yielda positive µ s though it does not properly describe theconfined baryons such as p and n . As a matter of fact,at sufficiently high temperature, the thermal system maywell consist of mesons and quarks rather than baryons, inwhich π − + p → K + Λ, for instance, should be replacedby π − + u → K + s in terms of quarks. The Polyakovloop mediates the mesonic correlation through the coloraverage, so that this kind of process is to be taken intoaccount in the PNJL model. Moreover this process viaquarks is rather realistic because in the hadronic phasethe cross section of kaons is small and n s = 0 would nolonger hold at the later stage of evolution [35].From Eq. (17) it is easy to confirm that µ s → T grows large. If T exceeds T c , both the Polyakov loopand the anti-Polyakov loop come close to unity, and thustheir ratio is nearly one, the logarithm of which is zero.This naturally coincides with the intuition that decon-fined quarks have no correlation and µ s = 0 correspondsto n s = 0 just like in the NJL model. In contrast tothe high- T situation, the confined phase at small T has ℓ ≃ ¯ ℓ ≃
0. The ratio of ℓ and ¯ ℓ could be any number. If¯ ℓ goes to zero much slower than ℓ , the ratio can becomearbitrarily large.In the best of our knowledge Eq. (17) is the very firstdemonstration that the discrepancy between ℓ and ¯ ℓ atfinite µ has a physically significant consequence. Theimportance of ℓ = ¯ ℓ has been overlooked maybe becausethe difference, ¯ ℓ − ℓ , is negligibly small (only a few % atmost) as compared to ( ℓ + ¯ ℓ ). The essential point is thatwe consider not the difference but the ratio, ¯ ℓ/ℓ , whichmay take a huge value when ℓ ≃ ℓ ≃ µ s on the µ - T plane in Fig. 9. Thefunctional shape is quite characteristic at small tempera-ture. Let us disclose the origin of this triangle-peak struc-ture at low T separating the µ -region into three pieces:i) Quark-regime — At small µ up to the peak position,the induced µ s rises linearly along with µ . In this re- Q u a r k C h e m i ca l P o t e n ti a l [ M e V ] T e m p e r a t u r e [ M e V ]
0 100 200 300 400 290250210170130905010080604020 ( T /2) log( l − / l ) = µ s FIG. 9: Induced µ s (or logarithm of the ratio between ℓ and¯ ℓ ) on the µ - T plane. gion the mesonic correlation is the governing dynamicsas already explained above.ii) Diquark-regime — The turning point of µ where µ s starts decreasing corresponds to the chemical poten-tial with which the diquark excitation ¯ ℓ e − ε − µ ) /T isenergetically more favorable than the quark excitation ℓ e − ( ε − µ ) /T . Using Eq. (17) we can derive the condition µ > ε for the diquark excitation overcoming the quarkexcitation. Since the constituent quark mass is 336 MeVin this model, the threshold should be µ ∼
110 MeV.This estimate is really consistent with our numerical re-sults shown in Fig. 9. In this µ -region µ s decreases be-cause diquarks behave like anti-quarks in color space. Inother words the mesonic (quark–anti-quark) correlationis taken over by the baryonic (quark–diquark) correla-tion which carries nonzero baryon number and so µ s ispartially canceled by this effect.iii) Baryonic-regime — If µ is greater than ε the colorsinglet contribution, e − ǫ − µ ) /T , is dominant, which isinterpreted as the baryonic excitation. In this regime thePolyakov loop is decoupled from the dynamics and thereis no confinement effect, even though the Polyakov loopis zero. (Such a state is recently named the “quarkyonicphase” [36, 37, 38, 39].) Then µ s = 0 suffices for n s = 0. VI. SUMMARY
We have calculated the isentropic trajectories on thephase diagram in the µ - T plane using the PNJL model.Our results are quantitatively consistent with the latticeresults in the high- T and low- µ region where the latticedata is available by means of the Taylor expansion.To test whether the Taylor expansion is under the-oretical control or not, we have expanded our numeri-cal results of the entropy s and the baryon density n B into polynomial series in terms of µ . We have confirmedthat the Taylor expansion can access the trajectoriesfor s/n B = 40, 30, and 20, but cannot for s/n B = 10at which the isentropic thermodynamics differs substan-tially from that at zero density. We have also realizedthat n B ( µ ) has pathological behavior if expanded at hightemperature, which can be cure by the Pad´e approxima-tion.Finally we have discussed the induced strange quarkchemical potential µ s to keep n s = 0 in the system ruledby the strong interaction. We have found the interestingrelation between µ s and the ratio of the Polyakov loop ℓ and the anti-Polyakov loop ¯ ℓ . This prediction could beconfirmed in the lattice QCD simulation.In this work we neglected the effect of the soft-modefluctuations around the critical point. This is one impor-tant direction of the future extension. Another interest-ing direction is the origin of the poor convergence in theexpansion of n B ( µ ). This is not fully understood in thepresent work; other thermodynamic quantities like s ( µ )have a smooth expansion but only n B ( µ ) fails. It would be also interesting, if the baryon number susceptibilitydoes not have such a problem of expansion (we guessso), to validate the idea that the QCD critical point isto be deduced from the radius of convergence using themodel. Although these issues are all beyond our currentscope, we believe that the present research contributes toopening these extensions. Acknowledgments
The author thanks H. Iida, L. Levkova, T. T. Taka-hashi for discussions. He thanks the Institute for NuclearTheory at the University of Washington for its hospitalityand the Department of Energy for partial support duringthe completion of this work. He is supported by JapaneseMEXT grant No. 20740134 and also supported in part byYukawa International Program for Quark Hadron Sci-ences. [1] C. W. Bernard et al. [MILC Collaboration], Phys. Rev.D , 6861 (1997) [arXiv:hep-lat/9612025].[2] Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, JHEP , 089 (2006) [arXiv:hep-lat/0510084].[3] Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo, Phys.Lett. B , 46 (2006) [arXiv:hep-lat/0609068].[4] M. Cheng et al. , Phys. Rev. D , 014511 (2008)[arXiv:0710.0354 [hep-lat]].[5] C. DeTar, PoS LAT2008 , 001 (2008) [arXiv:0811.2429[hep-lat]].[6] C. Schmidt (for RBC-Bielefeld and H. Collaborations),arXiv:0810.4024 [hep-lat].[7] S. Ejiri, PoS
LATTICE2008 , 002 (2008)[arXiv:0812.1534 [hep-lat]].[8] S. Muroya, A. Nakamura, C. Nonaka andT. Takaishi, Prog. Theor. Phys. , 615 (2003)[arXiv:hep-lat/0306031].[9] C. R. Allton et al. , Phys. Rev. D , 054508 (2005)[arXiv:hep-lat/0501030].[10] C. Bernard et al. , Phys. Rev. D , 014503 (2008)[arXiv:0710.1330 [hep-lat]].[11] R. V. Gavai and S. Gupta, Phys. Rev. D , 114503(2008) [arXiv:0806.2233 [hep-lat]].[12] S. Ejiri, F. Karsch, E. Laermann and C. Schmidt, Phys.Rev. D , 054506 (2006) [arXiv:hep-lat/0512040].[13] C. Miao and C. Schmidt [RBC-Bielefeld Collaboration],arXiv:0810.0375 [hep-lat].[14] A. Gocksch and M. Ogilvie, Phys. Rev. D , 877 (1985).[15] E. M. Ilgenfritz and J. Kripfganz, Z. Phys. C , 79(1985).[16] K. Fukushima, Phys. Lett. B , 277 (2004)[arXiv:hep-ph/0310121].[17] C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D ,014019 (2006) [arXiv:hep-ph/0506234].[18] K. Kashiwa, H. Kouno, M. Matsuzaki and M. Yahiro,Phys. Lett. B , 26 (2008) [arXiv:0710.2180 [hep-ph]].[19] W. j. Fu, Z. Zhang and Y. x. Liu, Phys. Rev. D , 014006 (2008) [arXiv:0711.0154 [hep-ph]].[20] M. Ciminale, R. Gatto, N. D. Ippolito, G. Nardulliand M. Ruggieri, Phys. Rev. D , 054023 (2008)[arXiv:0711.3397 [hep-ph]].[21] K. Fukushima, Phys. Rev. D , 114028 (2008)[Erratum-ibid. D , 039902 (2008)] [arXiv:0803.3318[hep-ph]].[22] O. Scavenius, A. Mocsy, I. N. Mishustin andD. H. Rischke, Phys. Rev. C , 045202 (2001)[arXiv:nucl-th/0007030].[23] M. Bluhm, B. Kampfer, R. Schulze, D. Seiptand U. Heinz, Phys. Rev. C , 034901 (2007)[arXiv:0705.0397 [hep-ph]].[24] T. Kahara and K. Tuominen, Phys. Rev. D , 034015(2008) [arXiv:0803.2598 [hep-ph]].[25] A. Dumitru, R. D. Pisarski and D. Zschiesche, Phys. Rev.D , 065008 (2005) [arXiv:hep-ph/0505256].[26] K. Fukushima and Y. Hidaka, Phys. Rev. D , 036002(2007) [arXiv:hep-ph/0610323].[27] T. Hatsuda and T. Kunihiro, Phys. Rept. , 221 (1994)[arXiv:hep-ph/9401310].[28] B. Svetitsky, Phys. Rept. , 1 (1986).[29] C. Nonaka and M. Asakawa, Phys. Rev. C , 044904(2005) [arXiv:nucl-th/0410078].[30] B. J. Schaefer and J. Wambach, Phys. Rev. D , 085015(2007) [arXiv:hep-ph/0603256].[31] K. Kamikado and K. Fukushima, work in progress.[32] C. Ratti, S. Roessner and W. Weise, Phys. Lett. B ,57 (2007) [arXiv:hep-ph/0701091].[33] C. Sasaki, B. Friman and K. Redlich, Phys. Rev. D ,074013 (2007) [arXiv:hep-ph/0611147].[34] J. Letessier and J. Rafelski, “Hadrons and quark - gluonplasma,” Camb. Monogr. Part. Phys. Nucl. Phys. Cos-mol. , 1 (2002).[35] L. V. Bravina et al. , Phys. Rev. C , 014907 (2008)[arXiv:0804.1484 [hep-ph]].[36] L. McLerran and R. D. Pisarski, Nucl. Phys. A , 83 (2007) [arXiv:0706.2191 [hep-ph]].[37] L. Y. Glozman and R. F. Wagenbrunn, Phys. Rev. D77