Integrals of products of Hurwitz zeta functions and the Casimir effect in $\phi^4$ field theories

TL;DR

This paper evaluates specific integrals involving Hurwitz zeta functions and applies these results to compute the first-order weak-coupling coefficient of the Casimir energy in a quantum field theory with boundary conditions.

Contribution

It introduces new integral evaluations involving Hurwitz zeta functions and applies them to calculate Casimir energy coefficients in $ ext{g}oldsymbol{ ightarrow}$phi^4 theory.

Findings

Explicit integral formulas for products of Hurwitz zeta functions.

Calculation of the $O(g)$ coefficient of Casimir energy in a $ ext{g}oldsymbol{ ightarrow}$phi^4 framework.

Extension of results to integer values of the parameter $a$.

Abstract

We evaluate two integrals over $x\in [0,1]$ involving products of the function $\zeta_1(a,x)\equiv \zeta(a,x)-x^{-a}$ for $\Re (a)>1$, where $\zeta(a,x)$ is the Hurwitz zeta function. The evaluation of these integrals for the particular case of integer $a\geq 2$ is also presented. As an application we calculate the $O(g)$ weak-coupling expansion coefficient $c_{1}(\varepsilon)$ of the Casimir energy for a film with Dirichlet-Dirichlet boundary conditions, first stated by Symanzik [Schr\"odinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1-44] in the framework of $g\phi^4_{4-\varepsilon}$ theory.