An Exponential Regulator for Rapidity Divergences

TL;DR

This paper introduces a new regularization framework for rapidity divergences in high-energy physics, connecting threshold and transverse momentum resummation through a multi-differential factorization approach.

Contribution

It proposes an alternative non-perturbative regularization method for rapidity divergences using multi-differential phase-space factorization, unifying threshold and transverse momentum resummation.

Findings

Demonstrates the factorization as a mother theory for both resummations.

Shows how to calculate rapidity renormalized functions directly.

Provides a higher loop scheme for anomalous dimension calculations.

Abstract

Finding an efficient and compelling regularization of soft and collinear degrees of freedom at the same invariant mass scale, but separated in rapidity is a persistent problem in high-energy factorization. In the course of a calculation, one encounters divergences unregulated by dimensional regularization, often called rapidity divergences. Once regulated, a general framework exists for their renormalization, the rapidity renormalization group (RRG), leading to fully resummed calculations of transverse momentum (to the jet axis) sensitive quantities. We examine how this regularization can be implemented via a multi-differential factorization of the soft-collinear phase-space, leading to an (in principle) alternative non-perturbative regularization of rapidity divergences. As an example, we examine the fully-differential factorization of a color singlet's momentum spectrum in a hadron-hadron collision at threshold. We show how this factorization acts as a mother theory to both traditional threshold and transverse momentum resummation, recovering the classical results for both resummations. Examining the refactorization of the transverse momentum beam functions in the threshold region, we show that one can directly calculate the rapidity renormalized function, while shedding light on the structure of joint resummation. Finally, we show how using modern bootstrap techniques, the transverse momentum spectrum is determined by an expansion about the threshold factorization, leading to a viable higher loop scheme for calculating the relevant anomalous dimensions for the transverse momentum spectrum.