# Some new results for the one-loop mass correction to the compactified $\lambda\phi^{4}$ theory

## TL;DR

This paper calculates the one-loop mass correction for a self-interacting scalar field in a compactified Euclidean space, extending known results to Dirichlet and Neumann boundary conditions and applying findings to Ginzburg-Landau models.

## Contribution

It provides new calculations of one-loop mass corrections for scalar fields with Dirichlet and Neumann boundary conditions in compactified spaces, filling a gap in existing literature.

## Key findings

- Derived mass correction formulas for Dirichlet and Neumann boundary conditions.
- Applied the results to analyze critical temperature in Ginzburg-Landau models.
- Extended understanding of quantum corrections in compactified field theories.

## Abstract

In this work we consider the one-loop effective action of a self-interacting $\lambda\phi^{4}$ field propagating in a $D$ dimensional Euclidean space endowed with $d\leq D$ compact dimensions. The main purpose of this paper is to compute the corrections to the mass of the field due to the presence of the compactified dimensions. Although results for the one-loop correction to the mass of a $\lambda\phi^{4}$ field are very well known for compactified toroidal spaces, where the field obeys periodic boundary conditions, similar results do not appear to be readily available for cases in which the scalar field is subject to Dirichlet and Neumann boundary conditions. We apply the results for the one-loop mass correction to the study of the critical temperature in Ginzburg-Landau models.