Celestial current algebra from Low’s subleading soft theorem

Andrew Strominger,Elizabeth Himwich

doi:10.1103/physrevd.100.065001

Abstract

The leading soft photon theorem implies that four-dimensional scattering amplitudes are controlled by a two-dimensional (2D) $U(1)$ Kac-Moody symmetry that acts on the celestial sphere at null infinity ($\mathcal{I}$). This celestial $U(1)$ current is realized by components of the electromagnetic vector potential on the boundaries of $\mathcal{I}$. Here, we develop a parallel story for Low's subleading soft photon theorem. It gives rise to a second celestial current, which is realized by vector potential components that are subleading in the large radius expansion about the boundaries of $\mathcal{I}$. The subleading soft photon theorem is reexpressed as a celestial Ward identity for this second current, which involves novel shifts by one unit in the conformal dimension of charged operators.

Highlights

In any four-dimensional (4D) theory with photons, the soft photon theorem implies [1,2,3,4,5] the existence of a two-dimensional (2D) Uð1Þ Kac-Moody symmetry
The consequences of the symmetry become most transparent when 4D scattering amplitudes are reexpressed as correlation functions on the celestial sphere at null infinity (I), on which the 4D Lorentz group acts as the 2D Euclidean conformal group
The Kac-Moody currents act on this celestial sphere and are sourced by electromagnetic charge currents that cross it

Summary

INTRODUCTION

In any four-dimensional (4D) theory with photons, the soft photon theorem implies [1,2,3,4,5] the existence of a two-dimensional (2D) Uð1Þ Kac-Moody symmetry. The celestial Kac-Moody current may be explicitly realized by a sum of the gauge potentials on the S2 boundaries of I, denoted Aðz0Þ below. This story is reviewed in [7]. In this paper we show that the subleading soft theorem implies a second current algebra on the celestial sphere. In attempting to explicitly solve for Aðz1Þ in terms of Aðz0Þ, one encounters an integration function on the sphere This implies that the boundary values of Aðz1Þ are independent fields after all, and turn out to comprise an independent “subleading” current algebra. The current algebra generated on the celestial sphere by boundary values of Aðz1Þ has interesting and unconventional features. Appendix gives some details of the asymptotic expansion about I in Lorenz gauge

MAXWELL EQUATIONS IN LORENZ GAUGE

SUBLEADING SOFT THEOREM AS SUBLEADING GAUGE TRANSFORMATION

Ek ðz zk

CELESTIAL CURRENT WARD IDENTITY