# Subleading Power Factorization with Radiative Functions

## TL;DR

This paper develops a gauge-invariant factorization framework at subleading power in soft and collinear limits using SCET, introducing universal radiative functions to understand all-order behavior of amplitudes and cross sections.

## Contribution

It derives the complete set of radiative functions at $ ext{O}(\lambda^2)$ in the power expansion using SCET, extending factorization beyond leading power.

## Key findings

- Universal radiative functions are identified for subleading power emissions.
- Factorization is established at all orders in $ ext{O}(\lambda^2)$ in the power expansion.
- Application to $e^+e^- o$ dijets shows how radiative functions contribute to event shape observables.

## Abstract

The study of amplitudes and cross sections in the soft and collinear limits allows for an understanding of their all orders behavior, and the identification of universal structures. At leading power soft emissions are eikonal, and described by Wilson lines. Beyond leading power the eikonal approximation breaks down, soft fermions must be added, and soft radiation resolves the nature of the energetic partons from which they were emitted. For both subleading power soft gluon and quark emissions, we use the soft collinear effective theory (SCET) to derive an all orders gauge invariant bare factorization, at both amplitude and cross section level. This yields universal multilocal matrix elements, which we refer to as radiative functions. These appear from subleading power Lagrangians inserted along the lightcone which dress the leading power Wilson lines. The use of SCET enables us to determine the complete set of radiative functions that appear to $\mathcal{O}(\lambda^2)$ in the power expansion, to all orders in $\alpha_s$. For the particular case of event shape observables in $e^+e^-\to$ dijets we derive how the radiative functions contribute to the factorized cross section to $\mathcal{O}(\lambda^2)$.