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].

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).

Symbol alphabets of n-particle amplitudes in $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4, n) as well as certain algebraic functions of cluster variables. In this paper we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C ∙ Z = 0 to associate functions on Gr(m, n) to parameterizations of certain cells of Gr(k, n) indexed by plabic graphs. For m = 4 and n = 8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-particle amplitude from four plabic graphs.