Weinberg Soft Theorems from Weinberg Adiabatic Modes

TL;DR

The paper develops a unified framework in which Weinberg soft theorems for photons and gravitons arise as Ward identities of adiabatic large-gauge/diffeomorphism modes in asymptotically flat spacetimes. By constructing gauge- and coordinate-appropriate adiabatic modes, it derives conserved Noether currents and shows how their conservation reproduces the leading soft theorems, with higher-order results following from the leading term. It then extends the analysis to a finite-radius setting, deriving charges on a sphere and elucidating antipodal matching through the inclusion of dressing fields, thus connecting to the BMS-based pictures and ensuring consistency at finite separations and for massive states. The work bridges cosmological consistency relations and flat-space soft theorems, clarifying the role of adiabatic modes and providing a finite-distance interpretation of asymptotic symmetries that could generalize to higher dimensions and curved backgrounds.

Abstract

Soft theorems for the scattering of low energy photons and gravitons and cosmological consistency conditions on the squeezed-limit correlation functions are both understood to be consequences of invariance under large gauge transformations. We apply the same method used in cosmology -- based on the identification of an infinite set of "adiabatic modes" and the corresponding conserved currents -- to derive flat space soft theorems for electrodynamics and gravity. We discuss how the recent derivations based on the asymptotic symmetry groups (BMS) can be continued to a finite size sphere surrounding the scattering event, when the soft photon or graviton has a finite momentum. We give a finite distance derivation of the antipodal matching condition previously imposed between future and past null infinities, and explain why all but one radiative degrees of freedom decouple in the soft limit. In contrast to earlier works on BMS, we work with adiabatic modes which correspond to large gauge transformations that are $r$-dependent.

Figures (4)

• Figure 1: The insertion of the quadratic piece of the electric current in the external lines.
• Figure 2: The world-volume of the sphere of radius $R$ on which the charge ${\mathscr Q}$ is defined. Ingoing and outgoing massless particles and radiation enter and exit the enclosed region $M$ respectively through the shaded areas $(-T_2,-T_1)$ and $(T_1,T_2)$. For large enough $R$ massive states enter well before $-T_2$ and exit well after $T_2$.
• Figure 3: Electric field of a pair of massless electron and positron resulting from a two step decay of a $Z$ boson at the origin.
• Figure 4: A massless charge moving inside the sphere $R$ along the third axis. The electric field is confined in a plane.