Conformally Soft Photons and Gravitons

TL;DR

Reframes the 4D S-matrix as a celestial CFT correlator on the sphere at null infinity and introduces conformally soft photons and gravitons with h or hbar equal to zero. It constructs Goldstone modes for large gauge transformations and BMS supertranslations, together with logarithmic partners that generate celestial currents, including a U(1) Kac-Moody current and a supertranslation current. The work derives canonical pairings between Goldstone and conformally soft modes, develops their OPE and shadow structures, and connects soft theorems to celestial symmetry algebras and a 2D stress tensor on the celestial sphere. These results provide a concrete framework linking asymptotic symmetries to 4D scattering data and suggest strong constraints on celestial correlators in quantum gravity.

Abstract

The four-dimensional $S$-matrix is reconsidered as a correlator on the celestial sphere at null infinity. Asymptotic particle states can be characterized by the point at which they enter or exit the celestial sphere as well as their $SL(2,\mathbb C)$ Lorentz quantum numbers: namely their conformal scaling dimension and spin $h\pm \bar h$ instead of the energy and momentum. This characterization precludes the notion of a soft particle whose energy is taken to zero. We propose it should be replaced by the notion of a conformally soft particle with $h=0$ or $\bar h=0$. For photons we explicitly construct conformally soft $SL(2,\mathbb C)$ currents with dimensions $(1,0)$ and identify them with the generator of a $U(1)$ Kac-Moody symmetry on the celestial sphere. For gravity the generator of celestial conformal symmetry is constructed from a $(2,0)$ $SL(2,\mathbb C)$ primary wavefunction. Interestingly, BMS supertranslations are generated by a spin-one weight $(\frac{3}{2},\frac{1}{2})$ operator, which nevertheless shares holomorphic characteristics of a conformally soft operator. This is because the right hand side of its OPE with a weight $(h,\bar h)$ operator ${\cal O}_{h,\bar h}$ involves the shifted operator ${\cal O}_{h+\frac{1}{2},\bar h+ \frac{1}{2}}$. This OPE relation looks quite unusual from the celestial CFT$_2$ perspective but is equivalent to the leading soft graviton theorem and may usefully constrain celestial correlators in quantum gravity.

Figures (1)

• Figure 1: Wavefunction of the conformally soft photon. A radiative shock wave with initial data at $v=0$ emerges from past null infinity ${\mathcal{I}}^-$, impinges on the origin, and reemerges at $u=0$ at future null infinity ${\mathcal{I}}^+$. Coulombic fields are produced by and confined to the future of the incoming shockwave, and annihilated by and confined to the past of the outgoing shock wave.