Equality of hemisphere soft functions for $e^+e^-$, DIS and $pp$ collisions at $\mathcal{O}(α_s^2)$

TL;DR

The paper shows that hemisphere soft functions governing event shapes in $e^+e^-$, DIS, and DY are identical in perturbation theory up to $O(\alpha_s^2)$, despite differences in Wilson-line directions. By dissecting all contributing amplitudes and their pole structures, the authors demonstrate that the potentially sign-changing imaginary parts cancel in the final soft functions. The result relies on the one-loop soft gluon current and extends to gluon- and multi-jet soft functions, implying that existing $e^+e^-$ results can be reused for DIS and DY at this order. This equality has practical impact by enabling $N^3$LL resummation for a broader class of observables across different collision processes.

Abstract

We present a simple observation about soft amplitudes and soft functions appearing in factorizable cross sections in $ee$, $ep$, and $pp$ collisions that has not clearly been made in previous literature, namely, that the hemisphere soft functions that appear in event shape distributions in $e^+e^-\to$ dijets, deep inelastic scattering (DIS), and in Drell-Yan (DY) processes are equal in perturbation theory up to $\mathcal{O}(α_s^2)$, even though individual amplitudes may have opposite sign imaginary parts due to changing complex pole prescriptions in eikonal propagators for incoming vs. outgoing lines. We also explore potential generalizations of this observation to soft functions for other observables or with more jets in the final state.

Figures (3)

• Figure 1: Amplitudes contributing to the $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(\alpha_s^2)$ soft functions: 1-gluon virtual (1$\mathcal{V}$) and real (1$\mathcal{R}$), 2 real gluon (2$\mathcal{R}$), 1-to-2 splitting (2$\mathcal{S}$), vacuum polarization (2$\mathcal{P}$), and 1-loop real gluon emission (or soft-gluon current) from a three-gluon vertex (2$\mathcal{T}$a) and (2$\mathcal{T}$b). Only (2$\mathcal{T}$b) potentially differs upon changing the directions of the Wilson lines from incoming to outgoing.
• Figure 1: Positions of the three complex $q^-$ poles of $\mathcal{I}_\mathcal{T}^{ep,pp}$ in Eq. \ref{['eq:qminuspoles']}, as a function of $q^+$. In region I where $q^+<0$, the $q^-$ contour can be closed below the real axis, giving zero for the integral Eq. \ref{['eq:ICee']}, while for $q^+>0$ in regions II and III the contour is closed below or above the real axis as shown, yielding the result in Eq. \ref{['eq:IepIppcontour']}.
• Figure 2: Examples of $\mathcal{O}(\alpha_s^2)$ diagrams with 3 Wilson lines and 4 Wilson lines involved. Dots represent Wilson lines not emitting soft gluons.