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

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 $ω« m « E$ holds where $E$ is the typical energy of a charged particle, $m$ is the typical mass and $ω$ 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 $\mathcal{O}(\lnω)$ 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 $\mathcal{O}(ω^{2j-1}(\lnω)^{2j})$ 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 $w{1+\infty}$.

💡 Research Summary

The paper investigates the symmetries encoded in the all‑loop soft photon theorems of quantum electrodynamics when the scattering process occurs in the high‑energy regime defined by ω ≪ m ≪ E (soft photon energy ≪ particle mass ≪ typical hard particle energy). In this limit the charged particles can be treated as effectively massless for the purpose of the soft expansion, and the soft factor S_em can be organized as an infinite series in powers of ω and logarithms, S_em = ∑_{n≥−1} S^{(n)} ω^{n}(ln ω)^{n+1}. The authors adopt the known results that S^{(−1)} (the leading Weinberg pole) is tree‑level exact, while S^{(0)} (the O(ln ω) term) is one‑loop exact, and they extend the analysis to higher orders S^{(n)} (n≥0) which are (n+1)‑loop exact. In the high‑energy limit the classical contributions to S^{(n)} are subleading, so the remaining terms are purely quantum.

To reinterpret these soft theorems as Ward identities of a two‑dimensional celestial conformal field theory (CCFT), the authors transform the scattering amplitudes into the conformal primary basis via Mellin transforms of creation/annihilation operators. In this basis the soft photon insertion becomes a celestial operator S_{1−2j}(w, \bar w) with conformal dimensions (h, \bar h) = (1−j, −j) for each half‑integer j ≥ ½. The operator S_{1−2j} is antiholomorphic and transforms as a spin‑j representation of the right‑moving SL(2,ℝ)_R.

For j = ½ the soft theorem yields a pair of antiholomorphic currents J^{(½)}{±}( \bar w) which the authors call “dipole currents.” Their charges measure the monopole and dipole moments of charged particles on the celestial sphere. The OPE of these currents takes the form
 J^{(½)}{+}( \bar w) J^{(½)}{−}(z) ∼ k/( \bar w − z)² + …
where k is a real parameter determined by the underlying QED loop dynamics. When k = 0 the currents commute (Abelian algebra); when k ≪ 1 the non‑trivial term generates a non‑Abelian algebra that matches the wedge subalgebra of the infinite‑dimensional w{1+∞} algebra.

Higher‑spin currents for j = 1, 3/2, … are constructed as normal‑ordered products of the dipole currents, e.g. J^{(1)} ∼ :J^{(½)}J^{(½)}:. Their OPEs are fixed by the same parameter k, and the resulting classical Poisson brackets reproduce the w_{1+∞} wedge algebra in the small‑k regime. Thus an infinite tower of antiholomorphic currents emerges, each transforming in a finite‑dimensional SL(2,ℝ)_R representation, and together they form a robust symmetry of the celestial CFT that survives all quantum corrections in the high‑energy limit.

The paper emphasizes that the high‑energy limit is essential: ω must be the smallest scale, ensuring that the soft expansion is IR‑finite, while the massless limit cannot be taken because in massless QED the physical charge vanishes in the infrared and the soft symmetries would be ill‑defined. The authors also discuss the physical meaning of the constant k, suggesting it encodes loop‑induced electromagnetic interactions (e.g., charge‑to‑mass ratios) and controls the deformation from an Abelian to a w_{1+∞}‑type algebra.

In summary, the work provides a novel bridge between all‑loop soft photon theorems and celestial holography. It introduces dipole and higher‑spin antiholomorphic currents on the celestial sphere, shows how they realize an infinite‑dimensional symmetry algebra (the wedge of w_{1+∞}) that is exact at all loop orders in the high‑energy regime, and opens new avenues for constraining scattering amplitudes and exploring connections between flat‑space holography and string theory.