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arXiv · 1603.00722

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

M. A. Shpot·M. P. Chaudhary·R. B. Paris
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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.

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M. A. Shpot, M. P. Chaudhary, R. B. Paris. 2016-03-02. Integrals of products of Hurwitz zeta functions and the Casimir effect in $\phi^4$ field theories. https://doi.org/10.7153/jca-09-11

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