The soft S $$ \mathcal{S} $$ -matrix in gravity
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Springer
Received: July 7, 2020 Accepted: August 24, 2020 Published: September 21, 2020
E. Himwich,a S.A. Narayanan,a,1 M. Pate,a,b N. Paula and A. Stromingera a
Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, U.S.A. b Society of Fellows, Harvard University, Cambridge, MA 02138, U.S.A.
E-mail: [email protected], sruthi [email protected], [email protected], [email protected], [email protected] Abstract: The gravitational S-matrix defined with an infrared (IR) cutoff factorizes into hard and soft factors. The soft factor is universal and contains all the IR and collinear divergences. Here we show, in a momentum space basis, that the intricate expression for the soft factor is fully reproduced by two boundary currents, which live on the celestial sphere. The first of these is the supertranslation current, which generates spacetime supertranslations. The second is its symplectic partner, the Goldstone current for spontaneously broken supertranslations. The current algebra has an off-diagonal level structure involving the gravitational cusp anomalous dimension and the logarithm of the IR cutoff. It is further shown that the gravitational memory effect is contained as an IR safe observable within the soft S-matrix. Keywords: Scattering Amplitudes, Space-Time Symmetries, Gauge Symmetry ArXiv ePrint: 2005.13433
1
Corresponding author.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)129
JHEP09(2020)129
The soft S-matrix in gravity
Contents 1
2 Preliminaries
3
3 Supertranslations and Weinberg’s soft graviton theorem
3
4 Currents on the celestial sphere 4.1 Gravitational Wilson lines 4.2 Summary of current algebra
4 5 5
5 Infrared divergences from virtual gravitons
6
6 Current algebra correlators and the soft S-matrix 6.1 Gravity 6.2 Gravitational memory
7 8 8
7 Soft S-matrix for massive particles
9
1
Introduction
The scattering amplitudes of four-dimensional asymptotically flat quantum gravity transform covariantly under asymptotic symmetries, which include the infinite-dimensional symmetry group of a two-dimensional conformal field theory [1, 2]. In particular, SL(2, C) Lorentz transformations act as the global conformal symmetry on the celestial sphere CS at null infinity. This observation lends scattering amplitudes a natural reinterpretation as correlation functions of a holographically dual “celestial conformal field theory” (CCFT) on the celestial sphere [3, 4]: hout|S|ini → hO1 · · · On iCS .
(1.1)
The asymptotic symmetries additionally include supertranslations, whose associated conservation laws are equivalent to Weinberg’s soft graviton theorem [5, 6]. In U(1) gauge theory, the soft photon theorem is the conservation laws following from the asymptotic large gauge symmetries [7–9]. These symmetries are realized as current algebras on the celestial sphere. This structure both governs the form of and explains the origin of infrared (IR) divergences: the latter are needed in order to se
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