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Abstract The formalism of relativistic partial wave expansion is developed for four-point celestial amplitudes of massless external particles. In particular, relativistic partial waves are found as eigenfunctions to the product representation of celestial Poincaré Casimir operators with appropriate eigenvalues. The requirement of hermiticity of Casimir operators is used to fix the corresponding integral inner product, and orthogonality of the obtained relativistic partial waves is verified explicitly. The completeness relation, as well as the relativistic partial wave expansion follow. Example celestial amplitudes of scalars, gluons, gravitons and open superstring gluons are expanded on the basis of relativistic partial waves for demonstration. A connection with the formulation of relativistic partial waves in the bulk of Minkowski space is made in appendices.

Abstract A natural extension of the Pasterski-Shao-Strominger (PSS) prescription is described, enabling the map of Minkowski space amplitudes with massive spinning external legs to the celestial sphere to be performed. An integral representation for the conformal primary wave function (CPW) of massive spinning bosons on the celestial sphere is derived explicitly for spin-one and -two. By analogy with the spin-zero case, the spinning bulk-to-boundary propagator on Euclidean AdS is employed to extend the massive CPW integral representation to arbitrary integer spin, and to describe the appropriate inverse transform of massive spinning CPWs back to the plane wave basis in Minkowski space. Subsequently, a massive spin-s momentum operator representation on the celestial sphere is determined, and used in conjunction with known Lorentz generators to derive Poincaré symmetry constraints on generic massive spinning two-, three- and four-point celestial amplitude structures. Finally, as a consistency check, three-point Minkowski space amplitudes of two massless scalars and a spin-one or -two massive boson are explicitly mapped to the celestial sphere, and the resulting three-point function coefficients are confirmed to be in exact agreement with the results obtained from Poincaré symmetry constraints.

Abstract The functional structure of celestial amplitudes as constrained by Poincare symmetry is investigated in 2, 3, and 4-point cases for massless external particles of various spin, as well as massive external scalars. Functional constraints and recurrence relations are found (akin to the findings in [24]) that must be obeyed by the respective permissible correlator structures and function coefficients. In specific three-point cases involving massive scalars the resulting recurrence relations can be solved, e.g., reproducing purely from symmetry a three-point function coefficient known in the literature. Additionally, as a byproduct of the analysis, the three-point function coefficient for gluons in Minkowski signature is obtained from an amplitude map to the celestial sphere.

Abstract Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin-transformed 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.

Abstract A Cachazo-He-Yuan (CHY) type formula is derived for the leading gluon, bi-adjoint scalar ϕ 3, Yang-Mills-scalar and non-linear sigma model m-soft factors S m in arbitrary dimension. The general formula is used to evaluate explicit examples for up to three soft legs analytically and up to four soft legs numerically via comparison with amplitude ratios under soft kinematics. A structural pattern for gluon m-soft factor is inferred and a simpler formula for its calculation is conjectured. In four dimensions, a Cachazo-Svrček-Witten (CSW) recursive procedure producing the leading m-soft gluon factor in spinor helicity formalism is developed as an alternative, and Britto-Cachazo-Feng-Witten (BCFW) recursion is used to obtain the leading four-soft gluon factor for all analytically distinct helicity configurations.

We provide corrections to equations (5.31), (5.32) and (5.33) in the original manuscript. The corrections remove typos due to which some terms in the results were omitted. The concepts, derivation strategy and conclusions remain unchanged.