We use the subleading soft-graviton theorem to construct an operator T_{zz} whose insertion in the four-dimensional tree-level quantum gravity S matrix obeys the Virasoro-Ward identities of the energy momentum tensor of a two-dimensional conformal field theory (CFT_{2}). The celestial sphere at Minkowskian null infinity plays the role of the Euclidean sphere of the CFT_{2}, with the Lorentz group acting as the unbroken SL(2,C) subgroup.

Recently it has been shown that the vacuum state in QED is infinitely degenerate.Moreover a transition among the degenerate vacua is induced in any nontrivial scattering process and determined from the associated soft factor. Conventional computations of scattering amplitudes in QED do not account for this vacuum degeneracy and therefore always give zero. This vanishing of all conventional QED amplitudes is usually attributed to infrared divergences. Here we show that if these vacuum transitions are properly accounted for, the resulting amplitudes are nonzero and infrared finite. Our construction of finite amplitudes is mathematically equivalent to, and amounts to a physical reinterpretation of, the 1970 construction of Faddeev and Kulish.

The operator product expansion (OPE) on the celestial sphere of conformal primary gluons and gravitons is studied. Asymptotic symmetries imply recursion relations between products of operators whose conformal weights differ by half-integers. It is shown, for tree-level Einstein–Yang–Mills theory, that these recursion relations are so constraining that they completely fix the leading celestial OPE coefficients in terms of the Euler beta function. The poles in the beta functions are associated with conformally soft currents.

Celestial amplitudes represent 4D scattering of particles in boost, rather than the usual energy-momentum, eigenstates and hence are sensitive to both UV and IR physics. We show that known UV and IR properties of quantum gravity translate into powerful constraints on the analytic structure of celestial amplitudes. For example the soft UV behavior of quantum gravity is shown to imply that the exact four-particle scattering amplitude is meromorphic in the complex boost weight plane with poles confined to even integers on the negative real axis. Would-be poles on the positive real axis from UV asymptotics are shown to be erased by a flat space analog of the AdS resolution of the bulk point singularity. The residues of the poles on the negative axis are identified with operator coefficients in the IR effective action. Far along the real positive axis, the scattering is argued to grow exponentially according to the black hole area law. Exclusive amplitudes are shown to simply factorize into conformally hard and conformally soft factors. The soft factor contains all IR divergences and is given by a celestial current algebra correlator of Goldstone bosons from spontaneously broken asymptotic symmetries. The hard factor describes the scattering of hard particles together with the boost-eigenstate clouds of soft photons or gravitons required by asymptotic symmetries. These provide an IR safe $$ \mathcal{S} $$ S -matrix for the scattering of hard particles.

Asymptotic particle states in four-dimensional celestial scattering amplitudes are labelled by their SL(2, C) Lorentz/conformal weights (h,h) rather than the usual energy-momentum four-vector. These boost eigenstates involve a superposition of all energies. As such, celestial gluon (or photon) scattering cannot obey the usual (energetically) soft theorems. In this paper we show that tree-level celestial gluon scattering, in theories with sufficiently soft UV behavior, instead obeys conformally soft theorems involving h → 0 orh → 0. Unlike the energetically soft theorem, the conformally soft theorem cannot be derived from low-energy effective field theory.