The Two-loop Hemisphere Soft Function

Abstract

The hemisphere soft function is calculated to order (\alpha_s^2). This is the first multi-scale soft function calculated to two loops. The renormalization scale dependence of the result agrees exactly with the prediction from effective field theory. This fixes the unknown coefficients of the singular parts of the two-loop thrust and heavy-jet mass distributions. There are four such coefficients, for 2 event shapes and 2 color structures, which are shown to be in excellent agreement with previous numerical extraction. The asymptotic behavior of the soft function has double logs in the (C_F C_A) color structure, which agree with non-global log calculations, but also has sub-leading single logs for both the (C_F C_A) and (C_F T_F n_f) color structures. The general form of the soft function is complicated, does not factorize in a simple way, and disagrees with the Hoang-Kluth ansatz. The exact hemisphere soft function will remove one source of uncertainty on the (\alpha_s) fits from (e^+e^-) event shapes.