Nonanalytic Structure of Effective Potential at Finite Temperature on Compactified Space

TL;DR

This paper analyzes nonanalytic terms in the one-loop finite-temperature effective potential on a D-dimensional spacetime with compactified spatial dimensions, using a mode-recombination framework to separate analytic and nonanalytic contributions. It shows that nonanalyticity comes in two forms—power-type and logarithmic-type—and that their emergence is driven by zero modes, with fermions (lacking zero modes) exhibiting no nonanalytic terms. The authors derive a new compact expression for the potential and provide explicit parity-based classifications for scalars under periodic spatial boundary conditions, revealing that the two nonanalytic types never occur simultaneously and depend on whether $D-(p+1)$ is even or odd. For fermions with arbitrary boundary conditions, the zero-mode absence implies no nonanalytic terms for any $D$ and $p+1$, a result with implications for finite-temperature phase structure in theories with compactified extra dimensions. Overall, the work clarifies the nonanalytic structure of the finite-temperature potential and its dependence on boundary conditions and dimensionality, offering a systematic residue-based method for extracting these contributions.

Abstract

We thoroughly investigate nonanalytic terms in the finite-temperature effective potential in one-loop approximation on a $D$-dimensional spacetime, $S_τ\times R^{D-(p+1)}\times \prod_{i=1}^p S_i^1$, using a mode recombination formula. Such nonanalytic terms cannot be expressed as positive powers of field-dependent mass squared. The formula provides a clear separation of the effective potential into a part that contains the nonanalytic terms and a part that is purely analytic, and clarifies the origin of the nonanalytic terms. We obtain all the nonanalytic terms and show that only two types of nonanalytic terms arise: power-type and logarithmic-one. For a real scalar field with periodic boundary conditions, the manner of the emergence of these terms is highly characteristic; the two types never appear simultaneously. By contrast, for fermions with general boundary conditions, we find that neither of the two types appears. These results clarify the nonanalytic structure of the finite-temperature effective potential on the spacetime with compactified spatial dimensions.