One-loop order effects from one universal extra dimension on $\lambda\phi^{4}$ theory

TL;DR

This paper investigates the one-loop effects of a single universal extra dimension on the $$ self-energy and vertex functions in $$ $$ theory, revealing how UV divergences are managed and how decoupling is achieved.

Contribution

It provides a detailed analysis of one-loop corrections in $$ theory with extra dimensions, employing Epstein zeta functions and comparing regularization schemes.

Findings

UV divergences are expressed via Epstein zeta and gamma functions.

Counterterms are of canonical dimension four.

Mass-dependent subtraction scheme satisfies decoupling theorem.

Abstract

The self-interacting $\lambda\phi^{4}$ scalar field theory is a warhorse in quantum field theory. Here we explore the one-loop order impact from one universal extra dimension, $S^{1}/\mathbb{Z}_{2}$, to the self-energy and four point vertex functions associated to this theory. Such effects come as an infinite number of UV divergences corresponding to an infinite superposition of excited KK particles around the loop. We show that dimensional regularisation is adequate enough to control them in terms of the product of the one dimensional inhomogenous Epstein zeta function times the gamma function. From the analytical properties of these functions, the UV divergences are extracted and the counterterms defined; the latter turn out to be of canonical dimension four at the Lagrangian level. We use both, the MS-scheme and a mass-dependent subtraction scheme to remove divergences. Only the latter manifestly satisfy the decoupling theorem.