Closed-form thermal threshold functions for the proper-time renormalisation group

Abstract

We derive closed-form expressions for the thermal threshold functions of the finite-temperature proper-time renormalisation group (PTRG): until now these have been evaluated numerically, Matsubara mode by Matsubara mode. For the standard one-parameter regulator family, Poisson resummation of the Matsubara sum yields a rapidly convergent winding-number series of modified Bessel functions, and a single algebraic identity reduces every higher threshold function to this same closed form at a shifted kernel parameter. The sharp proper-time regulator, recovered as the exact $m\to\infty$ endpoint of the family with a controlled $O(1/m)$ approach, factorises into a field-dependent and a purely thermal piece. Built on them, the local potential approximation (LPA) and its refinement to a running anomalous dimension (LPA$'$) for the $O(N)$-symmetric theory reduce to established results, all cross-checked against the exact Wetterich equation with the optimised regulator. Two findings go beyond reduction. First, the known zero-temperature anomalous-dimension construction is extended here to finite temperature. Second, because the closed form holds for any real regulator parameter, the refined truncation's fixed point can for the first time be tracked as a continuous function of the regulator, rather than at a handful of isolated points, and is found to vary smoothly and remain bounded, with no special or pathological point anywhere on the line.