All loop soft photon theorems and higher spin currents on the celestial sphere

TL;DR

This paper explores the implications of all-loop soft photon theorems for asymptotic symmetries on the celestial sphere, introducing higher spin currents and their algebraic structures in quantum field theory.

Contribution

It introduces a framework to interpret loop-level soft photon theorems as Ward identities involving higher spin currents on the celestial sphere, extending the symmetry analysis beyond tree level.

Findings

Loop level soft theorems imply the existence of higher spin currents.

Higher spin currents form a wedge subalgebra of the $w_{1+ ablafty}$ algebra.

Introduction of dipole currents as celestial sphere operators related to charged particles.

Abstract

Soft factorization theorems can be reinterpreted as Ward identities for (asymptotic) symmetries of scattering amplitudes in asymptotically flat space-time. In this paper we study the symmetries implied by the all loop soft photon theorems when all the charged particles are highly energetic and the relation $\omega << m << E$ holds where $E$ is the typical energy of a charged particle, $m$ is the typical mass and $\omega$ is the soft photon energy. Loop level soft theorems are qualitatively different from the tree level soft theorems because loop level soft factors contain multi-particle sums. If we want to interpret them as Ward identities or define celestial OPE between between soft and hard operators then we need to introduce additional fields which live on the celestial sphere but do not appear as asymptotic states in any scattering experiment. For example, if we want to interpret the one-loop exact $\mathcal{O}(\ln\omega)$ soft theorem for a positive helicity soft photon (with energy $\omega$) as a Ward identity then we need to introduce a pair of antiholomorphic currents on the celestial sphere which transform as a doublet under the $SL(2,\mathbb{R})_{R}$. We call them dipole currents because the corresponding charges measure the monopole and the dipole moment of an electrically charged particle on the celestial sphere. More generally, the soft photon theorem at $\mathcal{O}(\omega^{2j-1}(\ln\omega)^{2j})$ for every $j\in \frac{1}{2}\mathbb{Z}_+$ gives rise to $(2j+1)$ antiholomorphic currents which transform in the spin-$j$ representation of the $SL(2,\mathbb{R})_{R}$. These currents exist in the quantum theory because they follow from loop level soft theorems. We argue that under certain circumstances the (classical) algebra of the higher spin currents is the wedge subalgebra of the $w_{1+\infty}$.