Tree-level gluon amplitudes on the celestial sphere

Abstract

Pasterski, Shao and Strominger have recently proposed that massless scattering amplitudes can be mapped to correlators on the celestial sphere at infinity via a Mellin transform. We apply this prescription to arbitrary n-point tree-level gluon amplitudes. The Mellin transforms of MHV amplitudes are given by generalized hypergeometric functions on the Grassmannian Gr(4, n), while generic non-MHV amplitudes are given by more complicated Gelfand A-hypergeometric functions.The Mellin transform maps a plane wave solution for a helicity j field in momentum space to a corresponding conformal primary wave function on the boundary with spin J j , where helicity j and spin J j are mapped onto each other, and the operator dimension takes values in the principal continuous series representation ∆ j = 1 + iλ j [5]. Therefore, A J 1 ⋯Jn (λ j , z j ,z j ) has the structure of a conformal correlator on the celestial sphere, where the symmetry group of diffeomorphisms is the conformal group SL(2, C).

Poincaré constraints on celestial amplitudes

“…Compare with section 4.2 in [33]. 11 A quick way to arrive at the simplification in terms of conformal cross ratios is to employ conformal invariance of f and fix three of the four points to the particular values w1 =w1 = 0, w2 = z,w2 =z, w3 = w3 = 1, w4 =w4 = ∞.…”

Primary fields in celestial CFT

“…Two-dimensional celestial conformal field theory (CCFT) has been recently proposed as a candidate for a holographic description of four-dimensional space-time [1]. 1 In this framework, four-dimensional scattering amplitudes are represented as conformal field correlators, with their Lorentz symmetry realized as the SL(2, C) conformal symmetry group of celestial sphere (CS 2 ) [3][4][5]. The origin of this connection is a natural identification of the kinematic variables describing asymptotic directions of external particles with the points on CS 2 .…”

Correlation functions at the bulk point singularity from the gravitational eikonal S-matrix

“…A more recent proposal that have gained some attention over the last couple of years to map scattering amplitudes in flat space to conformal correlation functions in lower dimensions, (even though so far is not completely clear how it connects to AdS/CFT, some discussions towards this connection has been undergone recently, see for example [12][13][14]), is motivated by the asymptotic BMS symmetries in gravitational theories [15,16], and is based on the observation that the ddimensional conformal symmetry can be linearly realized as a d + 2-dimensional Lorentz symmetry which in turns allows to write the plane waves in flat Minkowskian space as an expansion in conformal primary wave functions [17]. For some progress in this direction see [14,[18][19][20][21][22][23][24][25][26][27] Another approach is to consider scattering processes that under certain conditions are very localized in the middle of AdS and therefore are insensitive to the curvature of the space [3,4,[28][29][30][31].…”