Abstract
We investigate the relation between the realization of center symmetry and the dependence on the topological parameter $\theta$ in $SU(N)$ Yang-Mills theories, exploiting trace deformations as a tool to regulate center symmetry breaking in a theory with a small compactified direction. We consider, in particular, $SU(4)$ gauge theory, which admits two possible independent deformations, and study, as a first step, its phase diagram in the deformation plane for two values of the inverse compactified radius going up to $L^{-1} \sim 500$ MeV, comparing the predictions of the effective 1-loop potential of the Polyakov loop with lattice results. The $\theta$-dependence of the various phases is then addressed, up to the fourth order in $\theta$, by numerical simulations: results are found to coincide, within statistical errors, with those of the standard confined phase iff center symmetry is completely restored and independently of the particular way this happens, i.e. either by local suppression of the Polyakov loop traces or by long range disorder.
Highlights
Pure gauge theories, defined on a space-time with one or more compactified directions, possess a symmetry under global transformations, which can be classified as gauge transformations respecting the periodicity but for a global element of the center of the gauge group [e.g., ZN for SUðNÞ gauge theories]; this is known as center symmetry
We will discuss the phase structure of the deformed SUð4Þ gauge theory in the h1-h2 plane and for values of the compactification radius for which center symmetry is broken at h1 1⁄4 h2 1⁄4 0; we will make use of predictions coming from the one-loop effective potential and compare them with results from numerical simulations
We have investigated the relation between center symmetry and θ-dependence in Yang-Mills theories, exploiting trace deformations in order to control the realization of center symmetry breaking in a theory with a small compactified direction
Summary
Pure gauge theories, defined on a space-time with one or more compactified directions, possess a symmetry under global transformations, which can be classified as gauge transformations respecting the periodicity but for a global element of the center of the gauge group [e.g., ZN for SUðNÞ gauge theories]; this is known as center symmetry. Such symmetry regulates most of the phase structure of the pure gauge theory, undergoing spontaneous symmetry breaking for small enough compactification radii, and the Polyakov loop (holonomy) around the compactified direction is a proper order parameter for its realization. A significant role is played by the dependence on the topological parameter θ, which enters the (Euclidean) Lagrangian as follows, Lθ
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