Generalized Threshold Factorization with Full Collinear Dynamics
Abstract
Soft threshold factorization has been used extensively to study hadronic collisions. It is derived in the limit where the momentum fractions xa,b of both incoming partons approach xa,b 1. We present a generalized threshold factorization theorem for color-singlet processes, which holds in the weaker limit of only xa 1 for generic xb (or vice versa), corresponding to the limit of large rapidity but generic invariant mass of the produced color singlet. It encodes the complete soft and/or collinear singular structure in the partonic momentum fractions to all orders in perturbation theory, including in particular flavor-nondiagonal partonic channels at leading power. It provides a more powerful approximation than the classic soft threshold limit, capturing a much larger set of contributions. We demonstrate this explicitly for the Z and Higgs rapidity spectrum to NNLO, and we use it to predict a nontrivial set of its N3LO contributions. Our factorization theorem provides the relevant resummation of large-x logarithms in the rapidity spectrum required for resummation-improved PDF fits. One of our factorization ingredients is a new beam function closely related to the N-jettiness beam function. As a byproduct, we identify the correct soft threshold factorization for rapidity spectra among the differing results in the literature.