Analytic NNLO transverse-momentum-dependent soft function for heavy quark pair hadroproduction at threshold

TL;DR

This paper analytically computes the NNLO transverse-momentum-dependent soft function for heavy quark pair production at threshold, enabling precise predictions and resummation for quarkonium production at colliders.

Contribution

It provides the first complete NNLO calculation of the TMD soft function for heavy quark pairs, crucial for accurate quarkonium production predictions.

Findings

Analytical NNLO soft function with simple form

Facilitates NNLO calculations for quarkonium production

Enables N$^3$LL resummation at small transverse momentum

Abstract

The transverse-momentum-dependent (TMD) soft function for non-relativistic heavy quark pair production at hadron colliders is analytically computed at next-to-next-to-leading order (NNLO) in the strong coupling expansion. We present the details of our computational approach and analyze the general two-loop structure of the soft function. The final result, which takes a particularly simple form, provides the last missing ingredient for a complete NNLO calculation of color-octet $S$-wave quarkonium hadroproduction--including charmonium, bottomonium, and toponium--using the $q_T$-slicing formalism. It also enables next-to-next-to-next-to-leading-logarithmic (N$^3$LL) resummation at small transverse momentum for the same process.