Solution of the complex action problem in the Potts model for dense QCD

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...

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“…eq. (14). Finally, both expectation values approach one at large µ, in accord with our discussion in Sec.…”

Section: Figsupporting

confidence: 87%

“…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].…”

Section: Pacs Numbersmentioning

confidence: 99%

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

2005

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“…QCD critical point Philippe de Forcrand, Owe Philipsen (12) (N t = 6), so that a naive a 2 extrapolation would give ∼ 0.4 in the continuum! This very large cutoff effect is consistent with earlier indications [7,14] and with a new study [6], all suggesting that the transition becomes much weaker in the continuum limit.…”

Section: Pos(lattice 2007)178mentioning

confidence: 99%

A QCD chiral critical point at small chemical potential: is it there or not?

Forcrand¹

2008

Proceedings of the XXV International Symposium on Lattice Field Theory — PoS(LATTICE 2007)

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For a QCD chiral critical point to exist, the parameter region of small quark masses for which the finite temperature transition is first-order must expand when the chemical potential is turned on. This can be tested by a Taylor expansion of the critical surface (m u,d , m s ) c (µ). We present a new method to perform this Taylor expansion numerically, which we first test on an effective model of QCD with static, dense quarks. We then present the results for QCD with 3 degenerate flavors. For a lattice with N t = 4 time-slices, the first-order region shrinks as the chemical potential is turned on. This implies that, for physical quark masses, the analytic crossover which occurs at µ = 0 between the hadronic and the plasma regimes remains crossover in the µ-region where a Taylor expansion is reliable, i.e. µ T . We present preliminary results from finer lattices indicating that this situation persists, as does the discrepancy between the curvature of T c (m c (µ = 0), µ) and the experimentally observed freeze-out curve. The XXV International Symposium on Lattice Field Theory

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“…The physically relevant quantities such as Z and O are real and positive, and if we could restructure the ensemble of gauge field configurations into families within which the weights and observables analytically added up to positive values then there would be no sign problem. This is what cluster algorithms can do, and such a restructuring has been achieved for the Z 3 model that approximates QCD at very high quark mass and chemical potential [90]. One strategy for the ultimate solution of the sign problem is to recast the QCD functional integral into a form to which cluster algorithms can be applied [91].…”

Section: Lattice Qcd At High Temperature and Low Densitymentioning

confidence: 99%

QCD at high density/temperature

Alford

2003

Nuclear Physics B - Proceedings Supplements

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Solution of the complex action problem in the Potts model for dense QCD

Cited by 80 publications

References 37 publications

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

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

A QCD chiral critical point at small chemical potential: is it there or not?

QCD at high density/temperature

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