Soft Theorems from Effective Field Theory

TL;DR

This paper uses soft-collinear effective theory (SCET) to analyze the behavior of massless gauge theory amplitudes in soft limits, extending the Low-Burnett-Kroll theorem to include one-loop corrections and highlighting the role of gauge invariance and symmetries.

Contribution

It proves the Low-Burnett-Kroll theorem at tree-level using SCET and extends it to a new one-loop subleading soft theorem, emphasizing gauge invariance and reparametrization invariance.

Findings

Tree-level Low-Burnett-Kroll theorem proven using gauge-invariant SCET matrix elements.

Explicit demonstration of on-shell corrections spoiling the theorem at loop level.

Generalization of the theorem to include one-loop subleading soft emissions.

Abstract

The singular limits of massless gauge theory amplitudes are described by an effective theory, called soft-collinear effective theory (SCET), which has been applied most successfully to make all-orders predictions for observables in collider physics and weak decays. At tree-level, the emission of a soft gauge boson at subleading order in its energy is given by the Low-Burnett-Kroll theorem, with the angular momentum operator acting on a lower-point amplitude. For well separated particles at tree-level, we prove the Low-Burnett-Kroll theorem using matrix elements of subleading SCET Lagrangian and operator insertions which are individually gauge invariant. These contributions are uniquely determined by gauge invariance and the reparametrization invariance (RPI) symmetry of SCET. RPI in SCET is connected to the infinite-dimensional asymptotic symmetries of the S-matrix. The Low-Burnett-Kroll theorem is generically spoiled by on-shell corrections, including collinear loops and collinear emissions. We demonstrate this explicitly both at tree-level and at one-loop. The effective theory correctly describes these configurations, and we generalize the Low-Burnett-Kroll theorem into a new one-loop subleading soft theorem for amplitudes. Our analysis is presented in a manner that illustrates the wider utility of using effective theory techniques to understand the perturbative S-matrix.