Double soft graviton factors from the gravitational Wilson line
Karan Fernandes, Feng-Li Lin, Chris D. White
TL;DR
This work extends the gravitational Wilson line (GWL) program to two-soft-graviton emissions by combining external emission dynamics, off-shell three-graviton vertices, and internal (LBK) contributions within a Schwinger-proper-time framework. It reproduces the leading double-soft graviton theorem and derives new universal subleading terms, ensuring gauge invariance even with nonzero initial separations of hard particles. A central outcome is the packaging of the double-soft physics into an exponential dressing operator $\tilde{S}_n({\{p_i\}}k,l,h)$, which yields a squeezed coherent state for the gravitational dressing and lays groundwork for radiative observables in binary-scattering EFTs. The results connect with infrared factorization, memory effects, and potential double-copy relations, offering a versatile framework for subleading soft dynamics in gravity and beyond. Key objects include $\tilde{S}_n = \exp[\phi_1 + \phi_2 + \bar{\Delta}_2 + \bar{\Upsilon}_2]$ and the double-soft amplitudes $\mathcal{M}^{\mu\nu}({\{p_i\}},k)$ and $\mathcal{M}^{\mu\nu\rho\sigma}({\{p_i\}},k,l)$.
Abstract
The description of low-energy (``soft") gravitons using universal theorems continues to attract attention. In this paper, we consider the emission of two soft gravitons, using a previously developed formalism that describes (next-to) soft graviton emission in terms of generalised Wilson lines (GWLs). Based on Schwinger's proper time methods, the GWL allows for a systematic accounting of graviton emission from external hard particles in the amplitude, as well as from three-graviton vertices located off the individual worldlines. By combining these effects, previously derived results for the leading double soft graviton theorem are recovered. Still, the formalism allows us to go further in deriving new universal double soft graviton terms at subleading order in the momentum expansion. We further demonstrate how gauge invariance can be utilized to account for double soft graviton emissions within the non-radiative amplitude, including the effects of non-zero initial positions of the hard particles. Our results can be packaged into an exponential dressing operator, and we comment on possible applications to the effective field theory for binary scattering processes.