The two-loop hemisphere soft function

TL;DR

This paper presents the first two-loop calculation of the hemisphere soft function, improving theoretical precision for event shape analyses and confirming predictions from effective field theory.

Contribution

It provides the first multi-scale soft function calculation at two loops, fixing unknown coefficients and comparing with effective field theory predictions.

Findings

Renormalization scale dependence matches EFT predictions.

Coefficients agree with previous numerical extractions.

Soft function exhibits complex behavior with double and single logs.

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 CF CA color structure, which agree with non-global log calculations, but also has sub-leading single logs for both the CF CA and CF TF nf 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.