Non-analytic Term in Effective Potential at Finite Temperature for Scalar Field on Compactified Space

We study non-analytic terms, which cannot be written in the form of any positive integer power of field-dependent mass squared, in effective potential at finite temperature in one-loop approximation for a real scalar field on the $D$-dimensional space-time, $S_{\tau}^1\times R^{D-(p+1)}\times\prod_{i=1}^pS_i^1$. The effective potential can be recast into the integral form in the complex plane by using the integral representation for the modified Bessel function of the second kind and the analytical extension for multiple mode summations. The pole structure of the mode summations is clarified and all the non-analytic terms are obtained by the residue theorem. We find that the effective potential has a non-analytic term when the dimension of the flat Euclidean space, $D-(p+1)$ is odd. There appears only one non-analytic term for the given values of $D$ and $p$, for which the non-analytic term exists.