Dual quark condensate and dressed Polyakov loops

Abstract

We construct a new order parameter for finite temperature QCD by considering the quark condensate for U(1)-valued temporal boundary conditions for the fermions. Fourier transformation with respect to the boundary condition defines the dual condensate. This quantity corresponds to an equivalence class of Polyakov loops, thereby being an order parameter for the center symmetry. We explore the duality relation between the quark condensate and these dressed Polyakov loops numerically, using quenched lattice QCD configurations below and above the QCD phase transition. It is demonstrated that the Dirac spectrum responds differently to changing the boundary condition, in a manner that reproduces the expected Polyakov loop pattern. We find the dressed Polyakov loops to be dominated by the lowest Dirac modes, in contrast to thin Polyakov loops investigated earlier.PACS numbers: 12.38. Aw, 11.15.Ha, 11.10.Wx Introductory remarksUnderstanding the nature of confinement has been a challenging task for many years. Several scenarios with different candidates for the relevant gluonic excitations were proposed, but no closed picture has emerged yet (it is even still debated whether confinement is predominantly an infrared or an ultraviolet phenomenon). Also a connection of confinement to chiral symmetry and its breaking has been conjectured, but not been shown either.In recent work [1, 2, 3] we have explored the idea of connecting quantities sensitive to confinement to spectral sums for Dirac and covariant Laplace operators. These ideas were developed further in [4,5,6] where it was shown that also other quantities such as quark propagators and heat kernels may be turned into order parameters for the breaking of center symmetry. Spectral sums provide a natural decomposition into infrared (IR) and ultraviolet (UV) parts and allow one to analyze their respective role in confinement, as studied numerically using quenched [2,3,4,5,6] and dynamical [7] lattice configurations.In this letter we build on those results and develop a new order parameter for center symmetry. In particular we Fourier transform the quark condensate (that turns into the chiral condensate in the massless limit) with respect to a U(1)-valued temporal boundary condition for the fermions, which we parameterize with a phase ϕ ∈ [0, 2π). We show that this duality transformation turns the quark condensate into the expectation value of an equivalence class of Polyakov loops which all have the same winding number n ∈ Z. The winding number n is the conjugate variable to the phase ϕ. To the equivalence class of loops with winding number n = 1, which transforms under center transformations in the same way as the conventional thin Polyakov loop, we refer to as the "dressed Polyakov loop".Since for pure gauge theory the deconfinement transition can be understood as spontaneous breaking of the center symmetry [8], the dressed Polyakov loop is an order parameter for confinement in pure gauge theory. The center transformation property of the dressed Polyakov loop is indep...

Quark Matter in a Strong Magnetic Background

“…An important point is that in the computation of the expectation value, the twisted boundary conditions acts only on the fermion determinant; the gauge fields are taken to be quantized with the canonical periodic boundary condition. Using a lattice regularization, it has been shown in [43] that Eq. (4) can be expanded in terms of loops which wind n times along the compact time direction.…”

“…Besides the Polyakov loop, it is interesting to compute the dressed Polyakov loop [43]. In order to define this quantity, we work in a finite Euclidean volume with temperature extension β = 1/T .…”

“…1 and 2, we plot the dressed Polyakov loop as a function of temperature, for several values of eB. We have normalized Σ 1 multiplying the one defined in [43] by the NJL coupling constant. For small values of eB/m 2 π , the behavior of Σ 1 as temperature is increased, is qualitatively similar to that at eB = 0, which has been discussed within effective models in [21,45].…”

Polyakov loop fluctuations in the Dirac eigenmode expansion

“…In the context of the interplay between confinement and chiral symmetry breaking [15,18,20,21,[23][24][25][26][27][28][29][30], it is important to make a reliable separation of one from another, whereas those phenomena are supposed to be correlated. The apparent coincidence in change of properties of the Polyakov loop fluctuation ratios and the chiral condensate and its susceptibility near the chiral crossover might indicate that there is a tied relation between confinement and chiral symmetry breaking in QCD.…”

Effect of entanglement on dual condensates at finite isospin chemical potential from an entangled Polyakov-Nambu-Jona-Lasinio model

“…Recently, some dual observables are introduced as the new order parameters for center symmetry by using the twisted boundary conditions for quarks [3][4][5][6]. Especially, it is demonstrated in the formalism of lattice QCD [3,4] that the dressed Polyakov loop (DPL) interpolates between the chiral condensate and the thin PL.…”