Soft Theorems from Effective Field Theory

TL;DR

This work develops a comprehensive SCET-based framework to understand soft theorems in gauge theories beyond the leading order. It proves the tree-level Low-Burnett-Kroll (LBK) subleading soft theorem for well-separated external particles by exploiting gauge invariance and reparametrization invariance (RPI) within SCET, and shows how LBK fails once collinear loops or collinear emissions are present. The authors derive a general one-loop subleading soft theorem that incorporates hard-loop, soft-loop, and collinear-loop contributions, including the effects of one-loop splitting amplitudes and soft-collinear fusion terms, and they illustrate these structures with explicit amplitude examples. They also connect RPI to infinite-dimensional asymptotic symmetries, linking the effective theory’s symmetry structure to the S-matrix’s infrared behavior. The results demonstrate the power of effective-field-theory methods to organize, constrain, and extend perturbative S-matrix understanding in gauge theories, with potential applications to gravity and higher-order perturbative corrections.

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.

Figures (9)

• Figure 1: a) Illustration of an amplitude with $N$ energetic external lines with soft gluon attachments encoded by a soft amplitude $S$. b) Illustration of the dominant modes for a two-jet event, where we have a hard amplitude $C_H$, two directions with splittings generating collinear amplitudes ${\cal I}_{n_1}$ and ${\cal I}_{n_2}$, and a soft amplitude $S$.
• Figure 2: The leading-power color-ordered SCET Feynman rules for emission of a soft gluon off of an $n$-collinear fermion or gluon in Feynman-'t Hooft gauge. The collinear fermion is denoted by the solid line, the collinear gluon by the wave with a line through it, and the soft gluon is the wavy line.
• Figure 3: Subleading power Feynman rules for the coupling of a soft gluon to a collinear fermion. The $\times$ symbol denotes the subleading Lagrangian insertion, $(1)$ denotes it is from ${\cal L}^{(1)}$, and $p_{s\perp}^\mu$ is the component of the off-shell collinear fermion's momentum from subsequent soft gluon emissions.
• Figure 4: Sub-subleading power Feynman rules for the coupling of a soft gluon to a collinear fermion. The $\times$ symbol denotes the sub-subleading Lagrangian insertion, $(2)$ denotes it is from ${\cal L}^{(2)}$, and $p_{s\perp}^\mu$ is the component of the off-shell collinear fermion's momentum from subsequent soft gluon emissions. For simplicity, here the vertex rule assumes that the outgoing fermion is on-shell.
• Figure 5: Examples of one-loop diagrams with soft loops that enter for computing the correction to the subleading soft theorem. $n_i$ and $n_j$ are two collinear directions with $n_i\cdot n_j \gg \lambda^2$ and $\stackrel{(1)}{\times}$ denotes the coupling of the soft gluon via the subleading SCET Lagrangian, ${\cal L}^{(1)}$. Using RPI to set $p_i^\perp=0$, all such soft loop graphs vanish.