Asymptotic expansions of two-loop Feynman diagrams in the Sudakov limit

Abstract

Recently presented explicit formulae for asymptotic expansions of Feynman diagrams in the Sudakov limit are applied to typical two-loop diagrams. For a diagram with one non-zero mass these formulae provide an algorithm for analytical calculation of all powers and logarithms, i.e. coefficients in the corresponding expansion $(Q^2)^{-2} \sum_{n,j=0} c_{nj} t^{-n} \ln^j t$, with $t=Q^2/m^2$ and $j \leq 4$. Results for the coefficients at several first powers are presented. For a diagram with two non-zero masses, results for all the logarithms and the leading power, i.e. the coefficients $c_{nj}$ for n=0 and j=4,3,2,1,0 are obtained. A typical feature of these explicit formulae (written through a sum over a specific family of subgraphs of a given graph, similar to asymptotic expansions for off-shell limits of momenta and masses) is an interplay between ultraviolet, collinear and infrared divergences which represent themselves as poles in the parameter $\eps=(4-d)/2$ of dimensional regularization. In particular, in the case of the second diagram, which is free from the divergences, individual terms of the asymptotic expansion involve all the three kinds of divergences resulting in poles, up to $1/\eps^4$, which are successfully canceled in the sum.