On the evaluation of Matsubara sums

Abstract

Given a connected (multi)graph G, consisting of V vertices and I lines, we consider a class of multidimensional sums constructed in the following way: - orient the lines of the graph in some (arbitrary) fashion - assign to each line i a positive variable q_i and an integer summation variable n_i - assign to each vertex v an integer variable N_v - construct the following rational function: -- the denominator is a product of factors (n^2+q^2), one for each line of the graph; -- the numerator is a product of Kronecker deltas, one for each vertex of the graph. For each vertex, the Kronecker delta imposes a linear constraint among the summation variables n_i of the lines incident upon the vertex, requiring that the sum of the variables n_i of the lines coming out of vertex minus the sum of the variables n_i of the lines coming into the vertex be equal to the integer variable N assigned to that vertex. - sum over all the n_i variables from minus infinity to infinity The sums thus constructed, called Matsubara sums, are functions of the I real positive variables q_i and the V integer variables N_v. It is shown any Matsubara sum can be evaluated in closed form by applying a linear operator to an integral closely associated with the sum.