Another mean field treatment in the strong coupling limit of lattice QCD
aa r X i v : . [ h e p - l a t ] A p r Another mean field treatmentin the strong coupling limit of lattice QCD ∗ Akira Ohnishi † Yukawa 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
Takashi Z. Nakano
Department of Physics, Faculty of Science, Kyoto University, Kyoto 606-8502, JapanYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We discuss the QCD phase diagram in the strong coupling limit of lattice QCD by using a newtype of mean field coming from the next-to-leading order of the large dimensional expansion. TheQCD phase diagram in the strong coupling limit recently obtained by using the monomer-dimer-polymer (MDP) algorithm has some differences in the phase boundary shape from that in the meanfield results. As one of the origin to explain the difference, we consider another type of auxiliaryfield, which corresponds to the point-splitting mesonic composite. Fermion determinant with thismean field under the anti-periodic boundary condition gives rise to a term which interpolates theeffective potentials in the previously proposed zero and finite temperature mean field treatments.While the shift of the transition temperature at zero chemical potential is in the desirable directionand the phase boundary shape is improved, we find that the effects are too large to be compatiblewith the MDP simulation results.
The XXVIII International Symposium on Lattice Field Theory, Lattice2010June 14-19, 2010Villasimius, Italy ∗ Report No.: YITP-11-43 † Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ nother mean field in SCL-LQCD
Akira Ohnishi
1. Introduction
The strong coupling lattice QCD (SC-LQCD) has been successful from the beginning of thelattice QCD formulation. In pure Yang-Mills theory, the strong coupling limit (SCL) expressionof the string tension was proven to follow the area law, and the Monte-Carlo simulation resultson the string tension were qualitatively explained in the strong coupling region by including var-ious plaquette configurations [1]. Recently SC-LQCD for pure Yang-Mills theory is extended tofinite temperatures [2]. SC-LQCD is also powerful in describing the QCD phase diagram. Theeffective potential including fermion contributions were obtained as a function of the chiral con-densate in SCL, and the spontaneous chiral symmetry breaking in vacuum and its restoration atfinite temperature and density have been discussed based on the derived effective potential [3 –6]. Recently, SC-LQCD framework with fermions is extended to include the next-to-leading order(NLO, O ( / g ) ) [7, 8] and the next-to-next-to-leading order (NNLO, O ( / g ) ) [9] contributionsof the strong coupling expansion. The mean field approach in SC-LQCD overestimates the MC re-sults of the transition temperature ( T c ) by around 10 % in SCL, while we overestimate T c by about50-60 % at b = N c / g ∼
4. When we include the Polyakov loop effects in SC-LQCD, it is possi-ble to roughly reproduce T c or the critical coupling ( b c ) at a given temporal lattice size ( N t = / T )in the coupling region b = N c / g . T treatment), and the integral over quarks is simulated by thesum over loop configurations. Since the gauge integral is carried out analytically, the sign problemis weakened. The shape of the phase boundary is somewhat different from that in the mean fieldpredictions; T c for a given m is much lower in the MDP results. It is important to understand theorigin of this deviation, since the MDP simulation is only applicable to SCL (1 / g =
0) and themean field treatment of SC-LQCD is one of the few approaches in which we can discuss cold densematter directly based on non-perturbative QCD.The deviation suggests that some of the approximations adopted in the mean field treatmentof SC-LQCD would not be good enough at finite m . There are two types of approximations inthe mean field approach. One of them is the assumption that the auxiliary chiral field takes aconstant value, and the other is the truncation in the large dimensional (1 / d ) expansion [13]. Inthis proceedings, we discuss the possibility to introduce another type of mean field from the NLOterm of the 1 / d expansion than the chiral condensate in SCL. Specifically, we examine the role ofthe mean field of the type V ± n , x = h n , x ¯ c x c x + ˆ n in the zero T treatment.
2. Strong coupling limit of lattice QCD with another mean field
In this section, we first review the framework of the mean field treatment of SCL-LQCDbriefly, and compare the obtained phase diagram with that in the MDP simulation. Next we in-troduce another kind of mean field. Throughout this proceedings, we set the lattice spacing as a =
1. 2 nother mean field in SCL-LQCD
Akira Ohnishi
We consider the lattice QCD action with one species of unrooted staggered fermion for colorSU ( N c = ) , Z LQCD = Z D [ c , ¯ c , U n ] e − S F − S G , (2.1) S F = (cid:229) x d (cid:229) n = (cid:2) h n , x ¯ c x U n , x c x + ˆ n − h − n , x ¯ c x + ˆ n U † n , x c x (cid:3) + m (cid:229) x ¯ c x c x , (2.2)where c ( ¯ c ) , m , U n , x , h n , x = exp ( d n m )( − ) x + ··· + x j − denote the quark (antiquark) field, the barequark mass, link variable, and the staggered phase factor, respectively. The pure Yang-Mills action S G is proportional to 1 / g , and disappears in SCL, g → ¥ . T m MDPMF (Std.)MF(present) 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 T / T c m / m c MDPMF (Std.)MF (present)KMOO
Figure 1:
Phase boundaries in the MDP simulation (dashed lines) [11], the standard mean field treatment(dotted lines) [6], and the present treatment (solid lines). Left (right) panel shows the phase boundary in thelattice unit (normalized by T c ( m = ) and m c ( T = ) ). In the finite T treatment of SCL-LQCD, referred to as the standard mean field MF (Std.) inthe later discussion, we first integrate out spatial link variables, and obtain the effective action ofquarks and the temporal links [6], S ( Std . ) eff = (cid:229) x h e m ¯ c x U , x c x + ˆ0 − e − m ¯ c x + ˆ0 U †0 , x c x i − N c (cid:229) n , x M x M x + ˆ n + m (cid:229) x M x + O ( / √ d ) , (2.3)where d = M x = ¯ c x c x represents a mesonic composite. We next in-troduce an auxiliary field for the chiral condensate to make the effective action bilinear in fermions,and integrate fermions and temporal link variables, for which the anti-periodic and periodic bound-ary conditions are imposed, respectively. The effective potential as a function of the chiral auxiliaryfield s is obtained in the mean field approximation as, F ( Std . ) eff = d N c s − T log (cid:20) sinh (( N c + ) E q / T ) sinh ( E q / T ) + ( N c m / T ) (cid:21) , (2.4)where E q = arcsinh ( d s / N c + m ) is the one-dimensional quark excitation energy. The phasediagram in MF (Std.) is shown by the dotted curve in Fig. 1. Compared with the MDP results,we overestimate the transition temperature at m = T =
0. In between, the transition temperature stays high in the3 nother mean field in SCL-LQCD
Akira Ohnishi chemical potential region m < . s is constant, and the other one is the truncation in the 1 / d expansion. In the 1 / d expansion [13], the M x M x + ˆ n term containing four quarks and sum over di-mensions is assumed to be finite at large d . Then the quark field scales as d − / , and the NLOterms containing six quarks are proportional to 1 / √ d . Baryonic composite action belongs to NLO,and it is natural to expect that the baryonic action would affect the phase boundary at finite m .Baryonic composite effects have been discussed at finite T in Refs. [14], but the adopted bosoniza-tion method [15] is not fully compatible with the chiral symmetry. Further studies are necessary todiscuss the chiral phase transition seriously.By comparison with MF (Std.), both spatial and temporal link variables are integrated out firstin the zero T treatment. The effective action including the NLO terms in the 1 / d expansion is givenas, S eff = − N c (cid:229) n , x M x M x + ˆ n + m (cid:229) x M x + (cid:229) n , x (cid:2) h n , x ¯ B x B x + ˆ n − h − n , x ¯ B x + ˆ n B x (cid:3) + O ( / ( d + )) , (2.5)where B x = e abc c ax c bx c cx / O ( / √ d ) , in the 1 / d expansion; it contains six fermions ( (cid:181) d − / ) and sum over space-timedimensions. This effective action with higher order terms in the large dimensional expansion isused in the MDP simulation [12]. In Ref. [5], the phase transition at finite m and zero T wasinvestigated by introducing the auxiliary baryon field b ∼ B in the mean field approximation forthe chiral condensate, but the phase transition at finite T is not well described. This shortcomingcould come from the lack of the anti-periodic boundary condition, which is decisive in the chiraltransition at finite T .In order to impose the anti-periodic boundary condition for quarks in the effective actionEq. (2.5), we here consider another type of auxiliary field, which corresponds to the point-splittingmesonic composite, V + n , x = h n , x ¯ c x c x + ˆ n , V − n , x = h − n , x ¯ c x + ˆ n c x . (2.6)We note that the baryonic term in Eq. (2.5) can be rewritten by using the anti-commuting propertyof the Grassmann variables as h n , x ¯ B x B x + ˆ n = ( V + n , x ) / , h − n , x ¯ B x + ˆ n B x = ( V − n , x ) / . (2.7)By applying the extended Hubbard-Stratonovich transformation [8],exp ( ∓ a V ) ≃ exp h − a ( ¯ y ( ) y ( ) − V y ( ) ± ¯ y ( ) V ) i , (2.8)exp ( ay ( ) V ) ≃ exp h − a ( ¯ y ( ) y ( ) − V y ( ) − ¯ y ( ) V y ( ) ) i . (2.9)we obtain the effective action for quarks and auxiliary fields, S eff = N t L d F ( X ) eff ( s , y ( k ) ± n , ¯ y ( k ) ± n ) + (cid:229) n , x [ Z + n V + n , x − Z − n V − n , x ] + m q (cid:229) x M x , (2.10) Z ± n = a (cid:16) ¯ y ( ) ± n ∓ y ( ) ± n ∓ ¯ y ( ) ± n y ( ) ± n (cid:17) . (2.11)4 nother mean field in SCL-LQCD Akira Ohnishi
Under the assumption that auxiliary fields take constant values, we can evaluate the Matsubaraproduct with the anti-periodic boundary condition for quarks. Equilibrium condition for y ( k ) ± n and¯ y ( k ) ± n is used to reduce the number of independent variables. For example, equilibrium values of y ( k )+ n and ¯ y ( k )+ n are related as, ¯ y ( )+ n = − y ( )+ n ≡ j + n and ¯ y ( )+ n = − y ( )+ n = j + n . We also utilize therotational and reflection symmetry, j + j = j − j ≡ j s ( j = , , ) . The effective potential is found tobe as follows. F eff = d + N c s + a ( j + + j − ) + a d j s + V q ( a = / ) , (2.12) V q = − N c T L d (cid:229) k (cid:20) E k T + log (cid:16) + e − ( E k − ˜ m ) / T (cid:17) + log (cid:16) + e − ( E k + ˜ m ) / T (cid:17)(cid:21) − N c log Z c , (2.13) Z ± = aj ± , Z c = p Z + Z − , ˜ m = m + log ( Z + / Z − ) , (2.14) E k = arcsinh ( e k / Z c ) , e k = q m q + Z s sin k , m q = d + N c s + m , Z s = aj s . (2.15)This effective potential has interesting features; quarks couple with auxiliary fields via the con-stituent quark mass m q and the wave function renormalization factor Z ± , s , and it contains the mo-mentum integral with a lattice type replacement, k → sin k .It should be noted that the composite V ± n is not gauge invariant, and the mean field introducedhere should be regarded as the one in a fixed gauge. We expect that this feature may not be serious,since we are discussing the dynamics in the link integrated effective action Eq. (2.5).
3. Effective potential surface and phase boundary
We shall now examine the effective potential Eq. (2.15). For simplicity, we ignore the effectsof j s which connects spatially separated quarks, and we discuss the results only in the chiral limit, m =
0. We first consider the F eff in vacuum, ( T , m ) = ( , ) . At m =
0, the temporal forwardand backward auxiliary fields take the same equilibrium value, j + = j − . When j ± are zero, thevacuum effective potential becomes the logarithmic type, F eff → b s s / − N c log m q , which is theleading order effective potential of the 1 / d expansion in the zero T treatment [3]. For finite j ± ,the effective potential becomes the arcsinh type, which is typical in the finite T treatment. Thus thepresent effective potential may be regarded as an interpolating one of the zero and finite T effectivepotentials.In Fig. 2, we show the effective potential F eff as a function of s and j ≡ √ j + j − at several ( T , m ) . In vacuum, equilibrium is realized at finite s and j ± =
0, while j ± ( s ) grows (decreases)as T increases. The phase transition at finite T and zero m is the second order. The transitiontemperature at m = T c = .
92) is smaller than those in MDP ( T c ≃ .
4) and MF (Std.) ( T c = / T c may be understood as the contribution from single quarks. In MF (Std),coherently moving three quarks contribute to the effective potential as a baryon as seen in theBoltzmann factor of exp ( − N c m / T ) in Eq. (2.4). By comparison, the mean field j ± allows a singlequark excitation as found in the Boltzmann factor exp ( − m / T ) in Eq. (2.13).At finite m and T =
0, the energy surface is separated by the ridge at E q = ˜ m , and the vacuumconfiguration (finite s and zero j ) jumps to the high density configuration (zero s and finite j ).5 nother mean field in SCL-LQCD Akira Ohnishi
0 1 2 3 0 1 2 3 4 5-2 0 2F eff s j F eff (T, m )=(0,0) 0 1 2 3 0 1 2 3 4 5-2 0 2F eff s j F eff (T, m )=(0.8,0) 0 1 2 3 0 1 2 3 4 5-2 0 2F eff s j F eff (T, m )=(1,0) 0 1 2 3 0 1 2 3 4 5-2 0 2F eff s j F eff (T, m )=(0,1) 0 1 2 3 0 1 2 3 4 5-2 0 2F eff s j F eff (T, m )=(0,1.2) 0 1 2 3 0 1 2 3 4 5-2 0 2F eff s j F eff (T, m )=(0.8,0.5) Figure 2:
Effective potential surface in the ( s , j = √ j + j − ) plane. The solid line connects the points whichgive the lowest F eff for a given j value, and the filled circles show the equilibrium. The transition chemical potential at T = m c = .
08) is larger than those in MDP ( m c ≃ .
59) andMF (Std.) ( m c = . j ± with thosein the MDP simulation and MF (Std.). The new mean fields j ± shift T c and m c from those inMF (Std.) in right directions, and when we normalize T and m by T c ( m = ) and m c ( T = ) ,the phase boundary shape is significantly improved and becomes is similar to that in the finite T treatment with baryonic composite effects [14]. However, their effects are too much in the shifts of T c and m c .
4. Summary
In this proceedings, we have discussed how we can understand the phase boundary in theMonomer-Dimer-Polymer (MDP) simulation [11] in the strong coupling limit (SCL) of latticeQCD for color SU(3) with unrooted staggered fermion in an analytical method based on the meanfield treatment. Since the MDP simulation is available at present only in SCL, it is important tounderstand it in a method which is applicable to finite coupling cases. Here we have examined anew type of mean field j ± , which connects the fermion in a different temporal variable in the zero T treatment, where both spatial and temporal link integrals are carried out first. This enables us toimpose the anti-periodic boundary condition of quarks, and may be important to describe the phase6 nother mean field in SCL-LQCD Akira Ohnishi transition at finite T . Actually we can describe the finite T phase transition in the zero T treatment.The new mean field shifts the transition temperature at m = T = Acknowledgments
We would like to thank Alejandro Vaquero and Professor Vicente Azcoiti for usefuldiscussions. This work was supported in part by Grants-in-Aid for Scientific Research from JSPS(Nos. 22-3314), the Yukawa International Program for Quark-hadron Sciences (YIPQS), and byGrants-in-Aid for the global COE program “The Next Generation of Physics, Spun fromUniversality and Emergence” from MEXT.
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