Mellin Amplitudes for Fermionic Conformal Correlators
Josua Faller, Sourav Sarkar, Mritunjay Verma
TL;DR
This work generalizes the Mellin-space description of conformal correlators to spin-1/2 fermions in three dimensions by defining multi-component Mellin amplitudes associated with a chosen tensor-structure basis. It reveals that, unlike purely scalar cases, each exchanged primary can generate two distinct pole series in a given channel, with parity selection potentially removing one series; the residues factorize into products of three-point Mellin amplitudes, connecting higher-point functions to lower-point data. The authors compute explicit tree-level Witten diagrams and conformal Feynman integrals with fermionic legs, illustrating direct and crossed-channel pole structures and their dependence on parity and tensor structure. The results lay groundwork for fermionic Mellin bootstrap, suggest extensions to higher dimensions and loops, and raise questions about canonical Mellin definitions for spinning operators. Overall, the paper provides a concrete, technically detailed framework for analyzing spinning conformal correlators in Mellin space and demonstrates its utility through explicit calculations.
Abstract
We define Mellin amplitudes for the fermion-scalar four point function and the fermion four point function. The Mellin amplitude thus defined has multiple components each associated with a tensor structure. In the case of three spacetime dimensions, we explicitly show that each component factorizes on dynamical poles onto components of the Mellin amplitudes for the corresponding three point functions. The novelty here is that for a given exchanged primary, each component of the Mellin amplitude may in general have more than one series of poles. We present a few examples of Mellin amplitudes for tree-level Witten diagrams and tree-level conformal Feynman integrals with fermionic legs, which illustrate the general properties.