Effective Potential and Phase Diagram in the Strong-Coupling Lattice QCD with Next-to-Next-to-Leading Order and Polyakov Loop Effects
aa r X i v : . [ h e p - l a t ] O c t Effective Potential and Phase Diagramin the Strong-Coupling Lattice QCDwith Next-to-Next-to-Leading Orderand Polyakov Loop Effects
Takashi Z. Nakano ∗ Department of Physics, Faculty of Science, Kyoto University, Kyoto 606-8502, JapanYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, JapanE-mail: [email protected]
Kohtaroh Miura
INFN-Laboratori Nazionali di Frascati, I-00044, Frascati(RM), Italy
Akira Ohnishi
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We investigate chiral and deconfinement transitions in the strong coupling lattice QCD for colorSU(3). We combine the leading order Polyakov loop effective action of the strong couplingexpansion and the next-to-next-to-leading order (1 / g ) fermionic effective action with one speciesof unrooted staggered fermion. Two approximation schemes are adopted to evaluate the Polyakovloop effects; a Haar measure method (no fluctuation from the mean field) and a Weiss mean-fieldmethod (with fluctuations). The Polyakov loop is found to suppress the chiral condensate andto reduce the chiral transition temperature at m =
0. The chiral transition temperature roughlyreproduces the Monte Carlo results in the region b = N c / g . The XXVIII International Symposium on Lattice Field Theory, Lattice2010June 14-19, 2010Villasimius, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ hase Diagram in the Strong-Coupling Lattice QCD with Polyakov Loop Effects
Takashi Z. Nakano
1. Introduction
The phase transition in Quantum Chromodynamics (QCD) at finite temperature ( T ) and/orquark chemical potential ( m ) is attracting much attention in recent years. Since lattice Monte Carlosimulations have the notorious sign problem at finite m [1], we need to invoke some approximationsin QCD or effective models at finite chemical potential. Strong coupling lattice QCD (SC-LQCD)is one of the most promising approximation schemes in QCD to investigate the chiral and decon-finement phase transitions at finite T and m . The QCD phase transition has two characteristicfeatures; the restoration of the chiral symmetry which is spontaneously broken in vacuum (chiraltransition), and the liberation of color degrees of freedom which is confined at low temperatures(deconfinement transition). The chiral transition at finite T and m has been investigated in SC-LQCD [2, 3, 4, 5]. The deconfinement transition at finite T was qualitatively explained on the basisof the mean-field treatment of the Polyakov loop in the leading order of the strong coupling expan-sion [6]. Higher order corrections on the Polyakov loop action has been investigated recently [7].One of the interesting developments describing the QCD phase transition at finite T and m can befound in the works [8, 9], which include the leading order Polyakov loop effective action (finitecoupling constant) and the strong coupling limit effective action for quarks (infinite coupling con-stant), and enables us to describe the chiral and deconfinement transition in a single framework.To make a step forward towards the true phase diagram, it is necessary to consider Polyakov loopeffects and finite coupling effects with quarks.In this proceedings, we develop a Polyakov loop extended strong coupling lattice QCD (P-SC-LQCD) framework [10] by combining the leading order Polyakov loop effective action and NNLOquark effective action. We derive an analytic expression of the effective potential in P-SC-LQCDat finite T and m , and investigate the chiral and deconfinement phase transitions at m =
2. Effective Potential with Polyakov Loop effects
In the lattice QCD, the partition function and action with one species of unrooted staggeredfermion for color SU ( N c ) in the Euclidean spacetime are given as, Z LQCD = Z D [ c , ¯ c , U n ] e − S LQCD , S LQCD = S F + S G , (2.1) S F = (cid:229) x d (cid:229) r = h h r , x ¯ c x U r , x c x + ˆ r − h − r , x ( h . c . ) i + m (cid:229) x ¯ c x c x , S G = − g (cid:229) P Re U P , (2.2)where c ( ¯ c ) , m , U r , x , and U P denote the quark (antiquark) field, bare quark mass, link variable, andplaquette, respectively. In the staggered fermion, the spinor index is reduced to the staggered phasefactor h r , x = ( h , x , h j , x ) = ( e m , ( − ) x + ··· + x j − ) where m is the quark chemical potential. Note thatwe utilize the lattice unit a = a throughout this proceedings.To treat the chiral and deconfinement phase transitions simultaneously, we evaluate the leadingorder Polyakov loop effects in the pure Yang-Mills sector and NNLO effects in the fermionic sector.In a finite T treatment of SC-LQCD, we first derive an effective action by carrying out spatial link( U j ) integrals, and evaluate the temporal link ( U ) integral later to consider the thermal effects ofquarks [3]. 2 hase Diagram in the Strong-Coupling Lattice QCD with Polyakov Loop Effects Takashi Z. Nakano
We can obtain the leading order Polyakov loop effective action S ( P ) eff from the spatial link inte-gral of the N t plaquettes sequentially connected in the temporal direction as shown in Fig. 1. S ( P ) eff = − (cid:18) g N c (cid:19) N t N c (cid:229) x , j > (cid:16) ¯ P x P x + ˆ j + h . c . (cid:17) , (2.3)where N t and P x = tr c (cid:213) t U ( t , x ) / N c represent the temporal lattice size and Polyakov loop, re-spectively. This effective action shows the nearest-neighbor interaction of the Polyakov loop. Thefactor 1 / g N c arises from the spatial link integral. By comparison, we derive NLO and NNLOquark effective action by the cluster expansion in Ref. [5]. From a naive counting of the strong x t ¯ P x P x +ˆ j spatialintegrallink x x + ˆ j Figure 1:
Leading order of the Polyakov loop effects in the strong coupling expansion. The squares in theleft and loops in the right represent the temporal plaquettes and the Polyakov loops, respectively. coupling expansion order, the leading order Polyakov loop action composed of the plaquettes is inthe higher order ( O ( / g N t ) ), compared with the NNLO terms ( O ( / g ) ) stem from the quark sec-tor. To describe the chiral and deconfinement phase transitions, we set the starting point of chiraland deconfinement dynamics as the effective action of O ( / g ) and O ( / g N t ) in the quark andPolyakov loop sectors, respectively. The Polyakov and NNLO quark effective action correspondsto an extension in the quark sector from this starting point.From the effective action described above, we obtain an approximate QCD partition functionand define the effective potential F eff as F eff ≡ − N t L d log Z LQCD ≃ − N t L d log (cid:20) Z D [ c , ¯ c , U ] e − S NNLO − S ( P ) eff (cid:21) = F q ( F ; m , T ) + U g ( ℓ, ¯ ℓ ) + F ( X ) eff ( F ) , (2.4)where L and d (= ) are the spatial lattice size and the spatial dimension, respectively. Here S NNLO , F q , U g , and F ( X ) eff represent the NNLO quark effective action, quark free energy, pure glu-onic potential, and effective potential including only the auxiliary fields F , respectively. In thiswork, we first reduce the effective action to the bilinear form for the quark fields by introducingseveral auxiliary fields, and assume the mean-field values for F later. We obtain F q and U g by eval-uating the Grassmann ( c , ¯ c ) and temporal link ( U ) integrals, In this proceedings, we evaluate thetemporal link integral in two kinds of methods, the Haar measure and Weiss mean-field methods.We shall now derive the effective potential with Polyakov loop effects in the Haar measuremethod (H-method). In the H-method, we replace the Polyakov loop with a mean-field valueand take into account the Haar measure in the Polyakov loop potential instead of carrying out thetemporal link integral. The contribution to the effective action is, S ( P ) eff ≃ − b p L d ¯ ℓℓ , (2.5)3 hase Diagram in the Strong-Coupling Lattice QCD with Polyakov Loop Effects Takashi Z. Nakano where b p = ( / g N c ) N t N c d , ℓ = h P x i and ¯ ℓ = h ¯ P x i . We assume the mean fields ℓ and ¯ ℓ are constantand isotropic. The temporal link integral is represented by using the Haar measure in the Polyakovgauge [2], which is a static and diagonalized gauge for temporal link variables, Z d U = Z d ℓ d ¯ ℓ · h − ℓ ¯ ℓ + (cid:0) ℓ + ¯ ℓ (cid:1) − (cid:0) ℓ ¯ ℓ (cid:1) i , (2.6)where U ( x ) = (cid:213) t U ( x , t ) . This Haar measure shows the Jacobian in the transformation from thetemporal link variables ( U ) to the Polyakov loop ( ℓ, ¯ ℓ ). F q and U g are given as, F q = − N c E q − T log R ( E q − ˜ m , N c ℓ, N c ¯ ℓ ) − T log R ( E q + ˜ m , N c ¯ ℓ, N c ℓ ) − N c log Z c , (2.7) R ( x , L , ¯ L ) ≡ + Le − x / T + ¯ Le − x / T + e − x / T , (2.8) U g = − T b p ¯ ℓℓ − T log h − ℓ ¯ ℓ + (cid:0) ℓ + ¯ ℓ (cid:1) − (cid:0) ℓ ¯ ℓ (cid:1) i , (2.9)where b p = ( / g N c ) / T N c d , and E q is the quark excitation energy. Here we have replaced the N t with 1 / T and omitted irrelevant constants. F q includes the vacuum, quark, and antiquark freeenergies and the contribution of the wave function renormalization factor Z c . The quark free energyincludes one- and two-quark excitations ( e − ( E − ˜ m ) / T , e − ( E − ˜ m ) / T ) . In the confined phase ( ℓ ∼ ¯ ℓ ∼ U g does not include the fluctuation of the Polyakov loop sincewe treat the Polyakov loop as the mean field without the temporal link integral. In the Polyakov-loop extended Nambu-Jona-Lasino model, this pure gluonic potential is incorporated to express theproperties of the deconfinement phase transition [11].Now we shall evaluate the Polyakov loop effects in the Weiss mean-field method (W-method).The W-method includes some part of the fluctuation effects of the Polyakov loop. We first bosonizethe Polyakov loop action by using the Extended Hubbard-Stratonovich (EHS) transformation [5],which is a procedure to bosonize the product of different types of composites. Then, the Polyakovloop action in Eq. (2.3) is linearized as, S ( P ) eff ≈ (cid:18) g N c (cid:19) N t N c (cid:229) x , j > (cid:0) ¯ ℓℓ − ¯ P x ℓ − ¯ ℓ P x (cid:1) ≃ b p L d ¯ ℓℓ − b p (cid:229) x (cid:0) ¯ P x ℓ + ¯ ℓ P x (cid:1) , (2.10)where b p , P x , and ¯ P x are defined before. ℓ and ¯ ℓ represent the auxiliary fields for the Polyakov loop, ( ℓ = h P x i , ¯ ℓ = h ¯ P x i ) . In ” ≃ ” of Eq. (2.10), we assume constant and isotropic values for auxiliaryfields ℓ and ¯ ℓ . We obtain F q and U g by evaluating the Grassmann ( c , ¯ c ) and temporal link ( U )integrals, F q = − T log ( Z P / L ) − N c log Z c , (2.11) Z P = Z d U det c h ( E q / T ) + U e ˜ m / T + U †0 e − ˜ m / T i exp h h tr U †0 + ¯ h tr U i , (2.12) U g = T b p ¯ ℓℓ − T log L . (2.13)We have defined h = b p ℓ/ N c and ¯ h = b p ¯ ℓ/ N c . In Eq. (2.12), we can perform the temporallink integral and obtain the analytic expression. Z P and L are functions of the combination ofthe modified Bessel functions. In Ref. [10], we show the explicit expression of them. Compared4 hase Diagram in the Strong-Coupling Lattice QCD with Polyakov Loop Effects Takashi Z. Nakano with the effective potential in the H-method where the quark contributions are represented as thevacuum, quark and antiquark parts , Eq. (2.7), the effective potential is more complicated.Note that the pure gluonic potential U g includes the dependence on ¯ ℓ/ℓ explicitly (i.e. thedependence on m ). When the quark chemical potential is zero ( m = ( ℓ = ¯ ℓ ) . In comparison, when the quark chemical potential is finite( m = ( ℓ = ¯ ℓ ) [12].
3. Chiral and Deconfinement Phase Transitions b =2N c /g =4.0, m =0, m =0.05 s /N c s /N c (w/o PL) ℓ b =2N c /g =4.0, m =0, m=0.05-d( s /N c )/dTd ℓ /dT Figure 2:
Left panel: Chiral condensate and Polyakov loop in P-SC-LQCD (solid lines), and chiral con-densate in SC-LQCD without the Polyakov loop effects (dashed line) as functions of T at m = d s / dT (dashed line) and d ℓ/ dT (solid line) in theW-method. T c , m = b =2N c /g w/ PL (Weiss)w/ PL (Haar)w/o PL b c (T=1/N t , m =0)(MC)T c (SCL, m =0)(MC) Figure 3:
Comparison of chiral transition temperature between P-SC-LQCD (solid line) and SC-LQCDwithout the Polyakov loop (dashed line). We show the results of Weiss mean-filed method (bolid solid line)and Haar measure method (thin solid line) in P-SC-LQCD. We define the critical temperature as the peakof − d s / dT . The triangles represent the results of the critical temperature ( T c , m = , open triangle) and thecritical coupling ( b c , filled triangles) obtained in Monte-Carlo simulations with one species of unrootedstaggered fermion [13, 14] hase Diagram in the Strong-Coupling Lattice QCD with Polyakov Loop Effects Takashi Z. Nakano
In Fig. 2, we show the chiral condensate ( s ), Polyakov loop, d s / dT and d ℓ/ dT as functionsof T . The chiral and deconfinement phase transitions seems to take place at almost the same T .We find that Polyakov loop suppresses the chiral condensate and reduces the chiral transitiontemperature in P-SC-LQCD. In left panel of Fig. 2 and Fig. 3, we compare the chiral condensate s and the chiral transition temperature T c , c with and without Polyakov loop effects. We find thatboth s and T c , c in P-SC-LQCD are smaller than those without Polyakov loop effects; the chiralcondensate becomes smaller when the Polyakov loop takes a finite value, and the chiral symmetryrestoration takes place at lower T by the Polyakov loop effects. In SC-LQCD without Polyakovloop effects, we find the contribution only from color-singlet states, then we implicitly assume thatthe quarks are confined at any T . In the W-method, we have one- and two-quark contribution aswell when the Polyakov loop takes a finite value. Quark excitation generally breaks the chiralcondensates, then it promotes the chiral symmetry to be restored at lower T . A similar behavior isfound also in the H-method. While qualitative behaviors are the same in both of the methods, theW-method exhibits a little larger T c than the H-method. Physically, the temporal link integral in theWeiss mean-field method favors the color-singlet states and therefore suppress the quark excitation.In our previous work on NLO and NNLO SC-LQCD [5], the critical temperature is calculatedto be larger than MC results, and NNLO effects on T c , c are found to be small. In Fig. 3, we show T c , c in two treatments of P-SC-LQCD in comparison with the MC results. The MC results, espe-cially in the region b .
4, are roughly explained in P-SC-LQCD developed in this work. Namely,the Polyakov loop and and finite coupling (NLO and NNLO) effects dominantly contribute to thereduction of T c , c . This observation implies that introducing the Polyakov loop, the deconfinementorder parameter, is essential to explain the QCD phase transition temperature.
4. Concluding Remarks
In this proceedings, we have derived an analytic expression of the effective potential at finitetemperature and chemical potential in the strong coupling lattice QCD with the Polaykov loopeffects using one species of unrooted staggered fermion. The chiral and deconfinement transitionsat m = O ( / g N t ) , and we have evaluated the temporal link ( U ) integral in two methods. Oneis the Haar measure method (H-method), where we replace the Polyakov loop with its constantmean-field without the U integral. The deconfinement dynamics is taken into account via the Haarmeasure. Another is the Weiss mean-field method (W-method), where we bosonize the effectiveaction and carry out the temporal link integral explicitly, then the fluctuation effects are included.We have found that one- and two-quark excitations are allowed in both methods, when the Polyakovloop takes a finite value at high T .We find that the Polyakov loop reduces the chiral transition temperature and the obtainedtransition temperatures roughly explain the MC results in the region b .
4. The chiral and de-confinement transitions are found to take place at similar temperatures. The W- and H-methodsexhibit qualitatively the same results, while the Polyakov loop effects are found to be weaker inthe W-method. This is because the temporal link integral in the W-method favors the color-singletstates and leads to the suppression of the quark excitation.6 hase Diagram in the Strong-Coupling Lattice QCD with Polyakov Loop Effects
Takashi Z. Nakano
As future works, we should investigate the QCD phase diagram in finite T and m on the basisof the effective potentials derived in this work. Results in this direction are discussed in part inRef. [15]. Acknowlegements
We would like to thank Owe Philipsen and Yoshimasa Hidaka for useful discussion. Thiswork was supported in part by Grants-in-Aid for Scientific Research from MEXT and JSPS (Nos.22-3314), the Yukawa International Program for Quark-hadron Sciences (YIPQS), and by Grants-in-Aid for the global COE program ‘The Next Generation of Physics, Spun from Universality andEmergence’ from MEXT.
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