Celestial amplitudes: conformal partial waves and soft limits

Abstract

Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellintransformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.1 The same amplitude in three dimensions was studied in [10].

Poincaré constraints on celestial amplitudes

“…Additionally, as in [15], we note that shifts in conformal dimensions ∆, such as induced by the Poincaré momentum operator, take the dimension value off the complex line of continuous series representation ∆ = 1 + iλ with λ ∈ R. This breaks some of the CFT specific formalism that is employed in the literature. For instance, the formalism of conformal partial wave decomposition for four-point correlators [8,20] relies on the partial wave orthogonality and convergence of the inner product employed, as well as the hermiticity of conformal Casimirs with respect to that inner product -moving the conformal dimensions off the continuous series line generically breaks some of these properties. Related to our comment on the difference in irreducible representations of conformal versus Poincaré group, for partial wave considerations conformal Casimirs should in principle be replaced by Poincaré Casimirs; which suggests that in future work it may be interesting to instead derive a relativistic partial wave decomposition on the celestial sphere (see, e.g., [34,35]).…”

“…The basis of conformal primary wave functions for massless particles of spins zero, one, and two was derived in [6], establishing the map to the celestial sphere in these cases, after gauge fixing, to be given by Mellin transform. Following the PSS prescription, explicit examples of amplitudes were mapped to the celestial sphere for scalar scattering [4,8,9,20], gluon scattering [7,11], and stringy/graviton scattering [12,22,23]. Modification of the PSS prescription, which makes the action of space-time translation simpler, has been proposed and investigated in [10,13,14].…”

“…Modification of the PSS prescription, which makes the action of space-time translation simpler, has been proposed and investigated in [10,13,14]. Conformally soft behavior of operators on the celestial sphere was considered in [16][17][18][19][20][21][22][23][24]. Conformal partial wave decomposition of some four-point amplitudes on the celestial sphere was discussed in [8,20].…”

“…Conformally soft behavior of operators on the celestial sphere was considered in [16][17][18][19][20][21][22][23][24]. Conformal partial wave decomposition of some four-point amplitudes on the celestial sphere was discussed in [8,20]. The representation of Poincaré symmetry generators for massless particles on the celestial sphere was presented in [15].…”

Modified celestial amplitude in Einstein gravity

“…Soft factorization is non-trivial in the Mellin basis because one integrates over the energies of the asymptotic particles. It has been proposed and studied in [12][13][14][15][16][17][18] that soft limit in the Mellin amplitude corresponds to taking λ p → 0 if the p-th particle is going soft. To be more precise, it has been shown that the limiting value of…”