Dimensional Regularization and Mellin Summation in High-Temperature Calculations

The infinite sums often encountered in thermal Feynman diagrams are commonly computed using the function coth, or one with similar properties, to generate poles in the complex plane whose residues correspond to the terms in the sum [1]. This transforms the summation into a contour integration and conveniently splits the zero-temperature and thermal contributions. The method ceases to be ideal when calculating the high temperature asymptotic expansion of such sums. Cancellations occur between the thermal and non-thermal parts suggesting that the calculation could be streamlined. Here we propose a more concise method using dimensional regularization and the Mellin transform pair. The sums we shall consider occur in one-loop calculations and though these are well understood, the aim of this work is to outline a convenient and general method for their evaluation in the high temperature limit. We recall first that the Mellin transform pair [2] can be written in the form