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

TL;DR

This work shows that loop-level soft photon theorems for massless external states can be recast as Ward identities in a celestial CFT_2, necessitating an infinite tower of higher-spin currents built from dipole operators on the celestial sphere. The authors introduce the dipole current as the fundamental symmetry generator for the $\mathcal{O}(\ln\omega)$ sector and extend the construction to higher subleading orders, producing spin-$j$ currents $\bar d^j$ whose correlators with matter encode the all-loop soft behavior. They analyze the resulting algebra, showing that at semiclassical limit $k\to 0$ the global part forms the wedge subalgebra of $w_{1+\infty}$, while quantum corrections at finite $k$ require deformations of the currents and do not alter single-soft Ward identities. The work highlights a rich interplay between infrared physics, celestial holography, and infinite-dimensional symmetries, with implications for constraining amplitudes and potential connections to string-theoretic structures in asymptotically flat spacetimes.

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 external particles are massless. 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 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 $O(\lnω)$ soft theorem for a positive helicity soft photon (with energy $ω$) 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 $O(ω^{2j-1}(\lnω)^{2j})$ 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 $w_{1+\infty}$.

Figures (1)

• Figure 1: $n$-loop Feynman diagrams contributing to $\mathbb{S}_{\text{em}}^{(n-1)}$ in the theory of massless scalar QED.