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Nonanalytic structure of effective potential at finite temperature on compactified space

Makoto Sakamoto¹ and Kazunori Takenaga ^2,*

¹Department of Physics, Kobe University, Rokkodai Nada, Kobe, 657-8501 Japan
²Faculty of Health Science, Kumamoto Health Science University, Izumi-machi, Kita-ku, Kumamoto 861-5598, Japan

^*Contact author: takenaga@kumamoto-hsu.ac.jp

Phys. Rev. D 113, 065014 – Published 18 March, 2026

DOI: https://doi.org/10.1103/28t8-63fy

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 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.

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Phys. Rev. D 113, 065014– Published 18 March, 2026

Vol. 113, Iss. 6 — 15 March 2026

Received 29 January 2026
Accepted 23 February 2026

DOI: https://doi.org/10.1103/28t8-63fy

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

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Nonanalytic structure of effective potential at finite temperature on compactified space

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