Dense quarks, and the fermion sign problem, in aSU(N)matrix model

Abstract

We study the effect of dense quarks in a SU (N ) matrix model of deconfinement. For three or more colors, the quark contribution to the loop potential is complex. After adding the charge conjugate loop, the measure of the matrix integral is real, but not positive definite. In a matrix model, quarks act like a background Z(N ) field; at nonzero density, the background field also has an imaginary part, proportional to the imaginary part of the loop. Consequently, while the expectation values of the loop and its complex conjugate are both real, they are not equal. These results suggest a possible approach to the fermion sign problem in lattice QCD. PACS numbers:At nonzero temperature, numerical simulations in lattice QCD have provided fundamental insight into the transition from a hadronic, to a deconfined, chirally symmetric plasma [1]. At nonzero quark density, however, at present simulations are stymied by the "fermion sign problem" [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Even in the limit of high temperature, and small chemical potential, only approximate methods can be used [18,19,20,21,22,23].In this paper we consider deconfinement in a mean field approximation to a model of thermal Wilson lines [24,25], which is a matrix model [26,27,28,29,30,31,32,33,34,35,36,37]. In Sec. I we discuss general features of SU (N ) matrix models at nonzero quark density [27]. In sec. II, this is briefly contrasted with the (trivial) case of a U (1) model [5]. Numerical results for three colors are presented in Sec. III. In Sec. IV, we conclude with some remarks about some methods which might be of use for dense quarks in lattice QCD. I. SU (N ) MATRIX MODELIn a gauge theory at nonzero temperature, a basic quantity is the thermal Wilson line, L = P exp(ig A 0 dτ ), where g is the gauge coupling, A 0 is the timelike component of the vector potential, and the integral over the imaginary time, τ , runs from 0 to 1/T , where T is the temperature [24]. An effective theory of thermal Wilson lines, interacting with static magnetic fields, can be constructed, and is valid in describing correlations over spatial distances ≫ 1/T [26,28,29,30,31,32,33,35,36,37].Over large distances, we use a mean field approximation to this effective theory. This gives an integral over a single Wilson line, L, with the partition function that of a matrix model:L is an SU (N ) matrix, satisfying L † L = 1 and det L = 1. Under gauge transformations Ω, it transforms as L → Ω † LΩ, so that gauge invariant quantities are formed by taking traces of L. These are Polyakov loops. In the matrix model, the effects of gluons and quarks are represented by potentials, V gl (L) and V qk (L), which are (gauge invariant) functions of the Wilson line. The effects of fluctuations, which are not included in the matrix model, can also be included in a systematic fashion [33,38].The pure glue theory is invariant under a global symmetry of Z(N ) , and so this must be a symmetry of the gluon loop potential, V gl (L). The simplest form for the gluon loop potential is a type of...

Deconfinement in matrix models about the Gross-Witten point

“…The result is that the effective mass, defined properly from (62) and computed with V mf , agrees with that obtained so easily from V eff , (49) and (50).…”

“…IV B. For four colors, it is ≈ 0.63 ± 0.02 at the N = 4 analogy of the Gross-Witten point [50], and decreases with increasing N .…”

“…We do not know if this chiral suppression significantly affects the breaking of Z(N ) symmetry; in mean field theory, this can be analyzed by using a nonlinear sigma model for the chiral fields [50]. For an analysis of Z(N ) breaking terms in terms of linear sigma models, see Moscy, Sannino, and Tuominen [31].…”

“…At infinite N , the value about the Gross-Witten point is given in eq. (50). For N = 3, the effective mass is obtained numerically by computing the curvature of V eff about the non-trivial minimum ℓ 0 , cf.…”

Chiral and deconfinement transitions in a magnetic background using the functional renormalization group with the Polyakov loop

“…Let us finally make a few remarks about the complexity of the effective action. In the mean-field approximation [47][48][49][50] of the PNJL and PQM models at finite µ B , the effective action is complex if one considers it a function of the complex variables Φ andΦ. If one ignores the imaginary part of the effective potential, the effective potential becomes a real function of real variables.…”