Nonanalytic Structure of Effective Potential at Finite Temperature on Compactified Space

TL;DR

This paper analyzes the nonanalytic terms in the finite-temperature effective potential on compactified spaces, revealing their origins and how they depend on boundary conditions and dimensionality.

Contribution

It provides a mode recombination formula that separates analytic and nonanalytic parts of the effective potential, clarifying the structure of nonanalytic terms at finite temperature.

Findings

Nonanalytic terms originate from zero Matsubara frequency modes.

Odd spatial dimensions produce odd powers of mass M; even dimensions produce log M terms.

Fermions with general boundary conditions do not exhibit these nonanalytic terms.

Abstract

We thoroughly investigate nonanalytic terms in the finite-temperature effective potential in one-loop approximation on a $D$-dimensional spacetime, $S_{\tau}\times R^{D-(p+1)}\times \prod_{i=1}^p S_i^1$, 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 from the modes with zero Matsubara frequency. For a real scalar field with periodic boundary conditions, if the number of noncompacted spatial dimensions is odd (even), there are odd powers of $M$ ($\log M$ terms) but no $\log M$ terms (no odd powers of $M$). 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.