Potential of the effective Polyakov line action from the underlying lattice gauge theory

Abstract

I adapt a numerical method, previously applied to investigate the Yang-Mills vacuum wave functional, to the problem of extracting the effective Polyakov line action from $\mathrm{SU}(N)$ lattice gauge theories, with or without matter fields. The method can be used to find the variation of the effective Polyakov line action along any trajectory in field configuration space; this information is sufficient to determine the potential term in the action and strongly constrains the possible form of the kinetic term. The technique is illustrated for both pure and gauge Higgs SU(2) lattice gauge theory at finite temperature. A surprise, in the pure gauge theory, is that the potential of the corresponding Polyakov line action contains a nonanalytic (yet center-symmetric) term proportional to $|P{|}^{3}$, where $P$ is the trace of the Polyakov line at a given point, in addition to the expected analytic terms proportional to even powers of $P$.