A Toolbox for $q_T$ and $0$-Jettiness Subtractions at N$^3$LO

TL;DR

This paper derives the leading-power singular terms at three loops for $q_T$ and 0-jettiness, providing essential subtraction terms for N$^3$LO calculations in particle physics, and advances the understanding of beam and soft functions for resummation.

Contribution

It presents the complete set of differential subtraction terms at N$^3$LO for $q_T$ and $ ext{0-jettiness}$, including three-loop beam and soft functions, and estimates unknown boundary coefficients.

Findings

Derived three-loop singular terms for $q_T$ and $ ext{0-jettiness}$.

Provided the full structure of beam and soft functions at three loops.

Estimated size of unknown boundary coefficients beyond threshold.

Abstract

We derive the leading-power singular terms at three loops for both $q_T$ and 0-jettiness, $\cal{T}_0$, for generic color-singlet processes. Our results provide the complete set of differential subtraction terms for $q_T$ and $\cal{T}_0$ subtractions at N$^3$LO, which are an important ingredient for matching N$^3$LO calculations with parton showers. We obtain the full three-loop structure of the relevant beam and soft functions, which are necessary ingredients for the resummation of $q_T$ and $\cal{T}_0$ at N$^3$LL$'$ and N$^4$LL order, and which constitute important building blocks in other contexts as well. The nonlogarithmic boundary coefficients of the beam functions, which contribute to the integrated subtraction terms, are not yet fully known at three loops. By exploiting consistency relations between different factorization limits, we derive results for the $q_T$ and $\cal{T}_0$ beam function coefficients at N$^3$LO in the $z\to 1$ threshold limit, and we also estimate the size of the unknown terms beyond threshold.