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

TL;DR

This work delivers the analytic NNLO ($2$-loop) transverse-momentum-dependent soft function for heavy-quark pair production near threshold in hadron collisions. It develops a two-step factorization framework that combines NRQCD/pNRQCD insights at threshold with small-$q_T$ factorization, and computes the NNLO contributions by systematically evaluating double-real and real-virtual pieces, aided by master-integral reductions and Mellin–Barnes techniques. The main result is a simple, universal color-octet soft function in $b_T$-space, while the color-singlet case vanishes due to scaleless integrals; the function satisfies the renormalization group equation with the soft anomalous dimension and is suitable for $q_T$-slicing NNLO calculations and N$^3$LL resummation at small $q_T$. This NNLO soft function completes a crucial ingredient for precise predictions of color-octet $S$-wave quarkonium hadroproduction and provides a robust methodological template for analogous two-loop soft-function computations.

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.

Figures (5)

• Figure 1: All possible soft configurations at the amplitude level up to NNLO. Thick black lines represent Wilson lines, and soft partons are shown in red.
• Figure 2: The two-loop one Wilson-line soft function. Each thick black line represents a Wilson line, and soft partons are shown in red. Each dashed line indicates a Cutkosky cut. The blue bubble represents the combined contributions from quark-antiquark, ghosts, gluons, as depicted in figures \ref{['fig:softampRR1']}, \ref{['fig:softampRR2']}, and \ref{['fig:softampRR3']}.
• Figure 3: Same as figure \ref{['fig:oneWL']}, but for the two-Wilson-line case.
• Figure 4: Same as figure \ref{['fig:oneWL']}, but for the three-Wilson-line case.
• Figure 5: Same as figure \ref{['fig:oneWL']}, but for the four-Wilson-line case.