The soft function for color octet production at threshold

TL;DR

This work computes the NNLO soft function for producing a massive color-octet state at threshold, a crucial component for the threshold expansion of heavy-quark pair production cross sections in quark annihilation and gluon fusion channels. By factorizing near-threshold cross sections into color-dependent hard and soft functions and employing Wilson lines, Mellin-space methods, and explicit RG structures, the authors derive the bare and renormalized soft function up to O(αs^2). They present explicit expressions for S^(1)(L) and S^(2)(L) in terms of color factors and logarithms, including a simplified form for gluon-initiated processes, and confirm consistency with RG evolution and soft-splitting limits. The results lay the groundwork for extracting constant terms in NNLO threshold expansions and have potential applications to other massive final states and representations, with the hard functions to be provided in a forthcoming publication.

Abstract

We evaluate the next-to-next-to-leading order soft function for the production of a massive color octet state at rest in the collision of two massless colored partons in either the fundamental or the adjoint representation. The main application of our result is the determination of the threshold expansion of the heavy-quark pair-production cross sections in the quark annihilation and gluon fusion channels. We discuss the factorization necessary for this purpose and explain the relationship between hard functions and virtual amplitudes.

Figures (5)

• Figure 1: Schematic representation of soft factorization. $H$ stands for the hard function, whereas $S$ for the soft function. The double lines denote Wilson lines, whereas the $\otimes$ symbols stand for the insertion of the color structure of the hard matrix element. The sum over different color structures is suppressed.
• Figure 2: Complete set of double-real emission graphs contributing to the soft function. The graphs are divided into: (A) two emissions from eikonal lines, (B) gluon splitting after emission from an eikonal line, (C) massless quark-pair emission.
• Figure 3: Interference diagrams contributing a hypergeometric function to $s_{{\Square}}^{(2)}$.
• Figure 4: Complete set of non-vanishing real-virtual graphs contributing to the soft function. The pairs of graphs in boxes can be combined as explained in the text.
• Figure 5: Real gluon emission (momentum $q$ and color index $a$) from an eikonal line in the presence of a virtual gluon (momentum $k$ and color index $b$).