Celestial Mellin Amplitudes: Spin-1 and Spin-2 Exchange$\unicode{x2014}$and Beyond

TL;DR

This work develops celestial Mellin amplitudes to incorporate spinning exchanges in Minkowski space, providing closed-form amplitudes for spin-1 and spin-2 exchanges (massless and massive) between external massless scalars. Spinning effects are generated by finite-difference (shift) operators in the Mellin variables acting on scalar exchange amplitudes, enabling a systematic extension to arbitrary integer spin $J$. The authors extract direct-channel OPE data, identifying in the celestial OPE the spin-1 current and spin-2 stress tensor (and their boundary counterparts) and giving explicit leading-OPE coefficients; massless exchanges reduce to celestial conformal partial waves with shadow relations. The formalism builds bridges between celestial CFT and AdS/Mellin techniques, clarifying the operator content of spinning exchanges and providing a practical, recursive method to obtain higher-spin results from the scalar case.

Abstract

We further develop celestial Mellin amplitudes arXiv:2412.11992 to capture the exchange of particles with spin. For massless external scalar fields, we derive closed-form celestial Mellin amplitudes for a spin-1 and spin-2 exchange$\unicode{x2014}$both massless and massive$\unicode{x2014}$and extract the associated OPE data in the direct-channel. For gauge-boson and graviton exchanges, we identify in the celestial OPE the spin-1 current and the spin-2 stress tensor, together with the boundary gauge boson and graviton. Perturbative time-ordered correlators involving spinning fields can be obtained from those of scalars through the action of differential operators on the external points. In Mellin space these actions become finite-difference (shift) operators in the Mellin variables, which we use to extend our results to arbitrary integer spin.

Figures (3)

• Figure 1: Poles in $z$ complex plane and integration contour (purple line) used in the Mellin-Barnes integral \ref{['Mexchs12s13_scalar']}. The poles \ref{['s_12_poles_scalar1']} in $s_{12}$ arise from the pinching of the integration contour between red and blue poles of $z$ as shown in the figure: as $s_{12}$ varies, the blue poles move and collide with red poles. For simplicity of the presentation, in the figure we took the scaling dimensions to lie on the Principal Series $\Delta_i = \frac{d}{2}+i\nu_i$, $\nu_i \in \mathbb{R}$.
• Figure 2: In this work we consider the four-point exchange of fields with spin-1 and 2 (wavy line) between external massless scalars (solid lines) in ($d$+2)-dimensional Minkowski space. The external points $Y_i$ are extrapolated to the celestial sphere according to the prescription \ref{['ccdefn2']}.
• Figure 3: A spin-$J$ exchange can be expressed in terms of the differential operator ${\cal D}$, defined in \ref{['D']}, acting on the spin-$0$ exchange of the same mass $m$, with Gegenbauer polynomial $C^{\left(\frac{d-1}{2}\right)}_J\left(z\right)$. See equation \ref{['spinJcjkspin0']}.