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...

Finite temperature lattice QCD is probed by varying the temporal boundary conditions of the fermions. We develop the emerging physical behavior in a study of the quenched case and subsequently present first results for a fully dynamical calculation comparing ensembles below and above the phase transition. We show that for low temperature spectral quantities of the Dirac operator are insensitive to boundary conditions, while in the deconfined phase a non-trivial response to a variation of the boundary conditions sets in.

We study quenched SU(2) lattice gauge theory with adjoint fermions in a wide range of temperatures. We focus on spectral quantities of the Dirac operator and use the temporal fermionic boundary conditions as a tool to probe the system. We determine the deconfinement temperature through the Polyakov loop, and the chiral symmetry restoration temperature for adjoint fermions through the gap in the Dirac spectrum. This chiral transition temperature is about four times larger than the deconfinement temperature. In between the two transitions we find that the system is characterized by a non-vanishing chiral condensate which differs for periodic and anti-periodic fermion boundary conditions. Only for the latter (physical) boundary conditions, the condensate vanishes at the chiral transition. The behavior between the two transitions suggests that deconfinement manifests itself as the onset of a dependence of spectral quantities of the Dirac operator on boundary conditions. This picture is supported further by our results for the dual chiral condensate.

We propose a new renormalization scheme of the running coupling constant in general gauge theories using the Wilson loops. The renormalized coupling constant is obtained from the Creutz ratio in lattice simulations and the corresponding perturbative coefficient at the leading order. The latter can be calculated by adopting the zeta-function resummation techniques. We perform a benchmark test of our scheme in quenched QCD with the plaquette gauge action. The running of the coupling constant is determined by applying the step-scaling procedure. Using several methods to improve the statistical accuracy, we show that the running coupling constant can be determined in a wide range of energy scales with relatively small number of gauge configurations. * ) Here we take the continuum limit and estimate the systematic error in the same way as we did in subsection 4.3.

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