Exact high temperature expansion of the one-loop thermodynamic potential with complex chemical potential

We present a derivation of an exact high temperature expansion for a one-loop thermodynamic potential $\mathrm{\Omega }(\tilde{\mu })$ with complex chemical potential $\tilde{\mu }$. The result is given in terms of a single sum, the coefficients of which are analytical functions of $\tilde{\mu }$ consisting of polynomials and polygamma functions, decoupled from the physical expansion parameter $\beta m$. The analytic structure of the coefficients permits us to explicitly calculate the thermodynamic potential for the imaginary chemical potential and analytically continue the domain to the complex $\tilde{\mu }$ plane. Furthermore, our representation of $\mathrm{\Omega }(\tilde{\mu })$ is particularly well suited for the Landau-Ginzburg type of phase transition analysis. This fact, along with the possibility of interpreting the imaginary chemical potential as an effective generalized-statistics phase, allows us to investigate the singular origin of the $m^3$ term appearing only in the bosonic thermodynamic potential.

Figure 1

The contours of integration in the complex $k$ plane. Dots represent the poles of the cotangent function, while $\mathbf{x}$ represents the summation-induced pole of the polylogarithm functions for $\phi =0$.