Finite Size Effects in Thermal Field Theory

TL;DR

This paper analyzes how finite size effects and boundary surfaces influence the one-loop renormalization of a thermal scalar field theory, identifying local and surface divergences and proposing strategies to handle them.

Contribution

It provides a detailed analysis of surface divergences in thermal field theory with boundaries and discusses methods to renormalize these divergences.

Findings

Surface divergences require boundary-specific counterterms.

Different strategies can address surface divergences effectively.

Infrared divergences in Neumann boundary conditions are briefly discussed.

Abstract

We consider a neutral self-interacting massive scalar field defined in a d-dimensional Euclidean space. Assuming thermal equilibrium, we discuss the one-loop perturbative renormalization of this theory in the presence of rigid boundary surfaces (two parallel hyperplanes), which break translational symmetry. In order to identify the singular parts of the one-loop two-point and four-point Schwinger functions, we use a combination of dimensional and zeta-function analytic regularization procedures. The infinities which occur in both the regularized one-loop two-point and four-point Schwinger functions fall into two distinct classes: local divergences that could be renormalized with the introduction of the usual bulk counterterms, and surface divergences that demand countertems concentrated on the boundaries. We present the detailed form of the surface divergences and discuss different strategies that one can assume to solve the problem of the surface divergences. We also briefly mention how to overcome the difficulties generated by infrared divergences in the case of Neumann-Neumann boundary conditions.