We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension ∆. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series ∆ ∈ 1 + iℝ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S2) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT.

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Asymptotic symmetries and celestial CFT

We argue that, in a theory of quantum gravity in a four dimensional asymptotically flat spacetime, all information about massless excitations can be obtained from an infinitesimal neighbourhood of the past boundary of future null infinity and does not require observations over all of future null infinity. Moreover, all information about the state that can be obtained through observations near a cut of future null infinity can also be obtained from observations near any earlier cut although the converse is not true. We provide independent arguments for these two assertions. Similar statements hold for past null infinity. These statements have immediate implications for the information paradox since they suggest that the fine-grained von Neumann entropy of the state defined on a segment $(-\infty,u)$ of future null infinity is independent of u. This is very different from the oft-discussed Page curve that this entropy is sometimes expected to obey. We contrast our results with recent discussions of the Page curve in the context of black hole evaporation, and also discuss the relation of our results to other proposals for holography in flat space.

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The Holographic Nature of Null Infinity

All 4D gauge and gravitational theories in asymptotically flat spacetimes contain an infinite number of non-trivial symmetries. They can be succinctly characterized by generalized 2D currents acting on the celestial sphere. A complete classification of these symmetries and their algebras is an open problem. Here we construct two towers of such 2D currents from positive-helicity photons, gluons, or gravitons with integer conformal weights. These generate the symmetries associated to an infinite tower of conformally soft theorems. The current algebra commutators are explicitly derived from the poles in the OPE coefficients, and found to comprise a rich closed subalgebra of the complete symmetry algebra.

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Holographic symmetry algebras for gauge theory and gravity

We elaborate on the proposal of flat holography in which four-dimensional physics is encoded in two-dimensional celestial conformal field theory (CCFT). The symmetry underlying CCFT is the extended BMS symmetry of (asymptotically) flat space­ time. We use soft and collinear theorems of Einstein-Yang-Mills theory to derive the OPEs of BMS field operators generating superrotations and supertranslations. The energy­ momentum tensor, given by a shadow transform of a soft graviton operator, implements superrotations in the Virasoro subalgebra of 𝔟𝔪𝔰4. Supertranslations can be obtained from a single translation generator along the light-cone direction by commuting it with the energy-momentum tensor. This operator also originates from a soft graviton and generates a flow of conformal dimensions. All supertranslations can be assembled into a single primary conformal field operator on celestial sphere.

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Extended BMS algebra of celestial CFT

A central feature of scattering amplitudes in gravity or gauge theory is the existence of a variety of energetically soft theorems which put constraints on the amplitudes. Celestial amplitudes which are obtained from momentum-space amplitudes by a Mellin transform over the external particle energies cannot obey the usual energetically soft theorems. Instead, the symmetries of the celestial sphere imply that the scattering of conformally soft particles whose conformal weights under the 4D Lorentz group SL(2, ℂ) are taken to zero obey special relations. Such conformally soft theorems have recently been found for gauge theory. Here, I show conformally soft factorization of celestial amplitudes for gravity and identify it as the celestial analogue of Weinberg’s soft graviton theorem.

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Conformally soft theorem in gravity

Scattering amplitudes in d  +  2 dimensions can be expressed in terms of a conformal basis, for which the S-matrix behaves as a CFT correlation function on the celestial d-sphere. We explain how compact expressions for the full tree-level S-matrix of gauge theory, gravity and other QFTs extend to this conformal basis, and are easily derived from ambitwistor strings. Using these formulae and their worldsheet origins, we prove various tree-level ‘conformal soft theorems’ in gauge theory and gravity in any dimension; these arise from limits where the scaling dimension of an external state in the scattering process takes special values. These conformally soft limits are obscure from standard methods, but they are easily derived with ambitwistor strings. Additionally, we make an identification between the residues of conformally soft vertex operator insertions in ambitwistor strings and charges generating asymptotic symmetries.

/pdf/celestial-amplitudes-and-conformal-soft-theorems-11tki8bzbo.pdf

Celestial amplitudes and conformal soft theorems

Scattering amplitudes in $d+2$ dimensions can be expressed in terms of a conformal basis, for which the S-matrix behaves as a CFT correlation function on the celestial $d$-sphere. We explain how compact expressions for the full tree-level S-matrix of gauge theory, gravity and other QFTs extend to this conformal basis, and are easily derived from ambitwistor strings. Using these formulae and their worldsheet origins, we prove various tree-level 'conformal soft theorems' in gauge theory and gravity in any dimension; these arise from limits where the scaling dimension of an external state in the scattering process takes special values. These conformally soft limits are obscure from standard methods, but they are easily derived with ambitwistor strings. Additionally, we make an identification between the residues of conformally soft vertex operator insertions in ambitwistor strings and charges generating asymptotic symmetries.

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Multi-collinear factorization limits provide a window to study how locality and unitarity of scattering amplitudes can emerge dynamically from celestial CFT, the conjectured holographic dual to gauge and gravitational theories in flat space. To this end, we first use asymptotic symmetries to commence a systematic study of conformal and Kac-Moody descendants in the OPE of celestial gluons. Recursive application of these OPEs then equips us with a novel holographic method of computing the multi-collinear limits of gluon amplitudes. We perform this computation for some of the simplest helicity assignments of the collinear particles. The prediction from the OPE matches with Mellin transforms of the expressions in the literature to all orders in conformal descendants. In a similar vein, we conclude by studying multi-collinear limits of graviton amplitudes in the leading approximation of sequential double-collinear limits, again finding a consistency check against the leading order OPE of celestial gravitons.

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Descendants in celestial CFT and emergent multi-collinear factorization

We present all-multiplicity formulas for the tree-level scattering of gluons and gravitons in the maximal helicity violating (MHV) helicity configuration, calculated in certain chiral strong fields. The strong backgrounds we consider are self-dual plane waves in gauge theory and general relativity, which are treated exactly and admit a well-defined S matrix. The gauge theory background-coupled MHV amplitude is simply a dressed analog of the familiar Parke-Taylor formula, but the gravitational version has nontrivial new structures due to graviton tails. Both formulas have just one residual integral rather than the n-2 expected at n points from space-time perturbation theory; this simplification arises from the integrability of self-dual backgrounds and their corresponding twistor description. The resulting formulas pass several consistency checks and limit to the well-known expressions for MHV scattering of gluons and gravitons when the background becomes trivial.

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Maximal-Helicity-Violating Scattering of Gluons and Gravitons in Chiral Strong Fields

Using the ambitwistor string, we compute tree-level celestial amplitudes for biadjoint scalars, Yang-Mills and gravity to all multiplicities. They are presented in compact CHY-like formulas with operator-valued scattering equations and numerators acting on a generalized hypergeometric function. With these we extend the celestial double copy to tree-level amplitudes with arbitrary number of external states. We also show how color-kinematics duality is implemented in celestial amplitudes and its interpretation in terms of a generalized twisted cohomology theory.

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Atul Sharma

Papers

Celestial amplitudes and conformal soft theorems

Celestial amplitudes and conformal soft theorems

Descendants in celestial CFT and emergent multi-collinear factorization

Maximal-Helicity-Violating Scattering of Gluons and Gravitons in Chiral Strong Fields

Celestial double copy from the worldsheet