Two-loop soft anomalous dimension matrix and resummation at next-to-next-to-leading poles

Abstract

We extend the resummation of dimensionally-regulated amplitudes to next-to-next-to-leading poles.This requires the calculation of two-loop anomalous dimension matrices for color mixing through soft gluon exchange. Remarkably, we find that they are proportional to the corresponding one-loop matrices.Using the color generator notation, we reproduce the two-loop single-pole quantities H (2) introduced by Catani for quark and gluon elastic scattering. Our results also make possible threshold and a variety of other resummations at next-to-next-to leading logarithm. All of these considerations apply to 2 → n processes with massless external lines.

From webs to polylogarithms

“…Given that the calculation is lengthy we relegate most of the details to the appendices. These involve many important issues with regards to the integration over the Wilson lines 19 and the simplifications achieved by forming the web at the first stage (appendix B) and the subtracted web (appendix C) at the final stage of the calculation. These steps will be useful for future calculations of other three-loop and higher-order webs [116].…”

“…In recent years we have seen significant progress towards determining the singularities of multi-leg amplitudes with general kinematics beyond the planar limit and beyond one loop [4,19,20,[25][26][27][28][29][30][31][32][33][34]41]. Complete two-loop results are now available for the soft anomalous dimension in both the massless and massive cases.…”

“…The state of the art, as of a few years ago, is two loops [19,20,26,31,33], where up to three Wilson lines are connected. Here we computed explicitly the three-loop contributions to the soft anomalous dimension of a product of nonlightlike Wilson lines from a specific class of webs consisting of multiple gluon exchanges which connect four lines, the 1-2-2-1 and the 1-1-1-3 webs.…”

“…[26,31,33] the terms originating in the two webs of figure 6 are separately logarithmically divergent in this limit, but they conspire to cancel, consistently with the calculation of ref. [19,20] and the dipole formula [35-37, 39, 40]. The implication for three loops is that all webs that span four Wilson lines need to be computed in order to deduce whether there are corrections to the dipole formula.…”

Soft gluon resummation for associated t t ¯ H $$ \mathrm{t}\overline{\mathrm{t}}\mathrm{H} $$ production at the LHC

“…The traditional (Mellinspace) resummation formalism which is applied in this type of calculations has been very well developed and copiously employed for description of the 2 → 2 type processes at the Born level. The universality of resummation concepts warrants their applications to scattering processes with many partons in the final state, as shown in a general analytical treatment developed for arbitrary number of partons [28][29][30]. Recently, the soft gluon resummation technique in the soft collinear effective theory (SCET) framework was applied to pp → ttW ± [31].…”