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

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 $a\ge 2$ is also presented. As an application we calculate the $O(g)$ weak-coupling expansion coefficient $c_1(\epsilon )$ of the Casimir energy for a film with Dirichlet-Dirichlet boundary conditions, first stated by Symanzik [Schrödinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1-44] in the framework of $g{\varphi }_{4-\epsilon }^4$ theory.

To appear in Journal of Classical Analysis

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• The `Friction' of Vacuum, and other Fluctuation-Induced Forces
• Fluctuation-induced forces in periodic slabs: Breakdown of epsilon expansion at the bulk critical point and revised field theory
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Cited by in corpus (2)

• Thermodynamics of the independent harmonic oscillators with different frequencies in the Tsallis statistics
• Multiple quantum harmonic oscillators in the Tsallis statistics