Resummation for 2→n processes in single-particle-inclusive kinematics

Abstract

We present a formalism and detailed analytical results for soft-gluon resummation for 2 → n processes in single-particle-inclusive (1PI) kinematics. This generalizes previous work on resummation for 2 → 2 processes in 1PI kinematics. We also present soft anomalous dimensions at one and two loops for certain 2 → 3 processes involving top quarks and Higgs or Z bosons, and we provide some brief numerical results.

Soft Anomalous Dimensions and Resummation in QCD

“…We discuss the factorization and refactorization of the cross section, the renormalization-group evolution (RGE), the eikonal approximation, soft anomalous dimensions, and the resummed cross section. For simplicity we discuss 2 → 2 processes but also explain how this generalizes to 2 → n processes [11]. For specificity we choose 1PI kinematics but also discuss modifications for PIM kinematics.…”

“…If an additional gluon with momentum p g is emitted in the final state, then we can equivalently write s 4 = (p 2 + p g ) 2 − m 2 2 , so as p g goes to 0 (i.e., we have a soft gluon), we again see that s 4 describes the extra energy in the soft emission and that s 4 → 0. We note that we can extend our formulas to the general case of multi-particle final states, i.e., 2 → n processes [11], by replacing m 2 2 with (p 2 + • • • + p n ) 2 in the expressions. With the incoming partons a and b arising from hadrons (e.g., protons/antiprotons) A and B, we define the hadron-level variables…”

“…From Equation (11) we see that the evolution of the soft function is controlled by the soft anomalous dimension, Γ S ab→12 , which is also in general a matrix. In calculating Γ S ab→12 , we use the eikonal approximation, where the Feynman rules for diagrams with soft gluon emission simplify.…”

“…This review discusses soft anomalous dimensions that control soft-gluon threshold resummation in QCD. Resummation follows from factorization properties of the cross section [1][2][3][4][5][6][7][8][9][10][11] and it provides a formalism for calculating contributions to higher-order corrections that are theoretically important and numerically significant. These soft-gluon contributions take the form of logarithms of a variable that is a measure of the available energy for additional radiation in a process.…”

“…is the soft anomalous dimension matrix that controls the evolution of the soft function S ab→12 via Equation (11). In dimensional regularization Z ab→12 has 1/ poles.…”