The two-loop soft function for heavy quark pair production at future linear colliders

TL;DR

The paper computes the two-loop threshold soft function for heavy-quark pair production in $e^+e^-$ annihilation, generalizing the Drell-Yan case to massive quarks. It develops a differential-equations framework with a normal-form basis to obtain the bare soft function to all orders in $\epsilon$, providing a full one-loop result and a complete two-loop calculation split into real-virtual and real-real parts. The authors present the renormalized two-loop soft function, verify consistency with known massless limits via non-Abelian exponentiation, and uncover novel relations linking massive and massless soft functions in the high-energy limit. The results supply essential ingredients for NNLO predictions at future linear colliders and offer insights into the structure of soft Wilson-line matrix elements that could extend to hadron colliders.

Abstract

We report on the calculation of the threshold soft function for heavy quark pair production in e+ e- annihilation at two-loop order. Our main result is a generalization of the familiar Drell-Yan threshold soft function to the case of non-zero primary quark mass. We set up a framework based on the method of differential equations which allows for the straightforward calculation of the bare soft function to arbitrarily high orders in the dimensional regularization parameter. Remarkably, we find that we can obtain the bare two-loop Drell-Yan soft function from the heavy quark soft function to the order in epsilon required for a two-loop calculation by making simple replacements. We expect that our results will be of use, both as an important input for precision physics calculations at linear colliders and, more formally, as a first step towards a better understanding of the connection between vacuum matrix elements of massive soft Wilson lines and vacuum matrix elements of massless soft Wilson lines.

Figures (1)

• Figure 1: Panels $(a)$ and $(b)$ show cut eikonal Feynman diagrams which contribute to $S_{Q\bar{Q}}^{{\rm RR}~(2)}(x,\epsilon)$ and panels $(c)$ and $(d)$ show diagrams which contribute to $S_{Q\bar{Q}}^{{\rm RV}~(2)}(x,\epsilon)$.