Massive Celestial Fermions

TL;DR

This work extends celestial CFT by constructing conformal primary wavefunctions for massive fermions, starting with explicit Dirac spinor primaries and deriving their transformation of momentum-space amplitudes into celestial amplitudes. It establishes shadow relations, delta-function normalizability on the principal series, and a complete basis structure for spin-1/2 primaries, then builds higher-spin primaries (spin-1 and spin-$\tfrac{3}{2}$) through Clebsch-Gordan composition, providing a general prescription to generate all massive fermionic and bosonic conformal primaries from spin-$\tfrac{1}{2}$. It also derives spin-$\tfrac{1}{2}$ momentum generators on the celestial sphere, showing non-diagonal action and a consistent Poincaré algebra with Lorentz generators. The results enable a systematic CCFT treatment of massive fermionic amplitudes and supply a concrete framework for analyzing higher-spin celestial correlators via spin-1/2 building blocks. Overall, the paper broadens the CCFT toolkit for massive fields and lays groundwork for future explorations of massive-half-integer and integer-spin dynamics in celestial holography.

Abstract

In an effort to further the study of amplitudes in the celestial CFT (CCFT), we construct conformal primary wavefunctions for massive fermions. Upon explicitly calculating the wavefunctions for Dirac fermions, we deduce the corresponding transformation of momentum space amplitudes to celestial amplitudes. The shadow wavefunctions are shown to have opposite spin and conformal dimension $2-Δ$. The Dirac conformal primary wavefunctions are delta function normalizable with respect to the Dirac inner product provided they lie on the principal series with conformal dimension $Δ= 1+iλ$ for $λ\in\mathbb{R}$. It is shown that there are two choices of a complete basis: single spin $J=\frac{1}{2}$ or $J=-\frac{1}{2}$ and $λ\in\mathbb{R}$ or multiple spin $J=\pm\frac{1}{2}$ and $λ\in\mathbb{R}_{+\cup 0}$. The massless limit of the Dirac conformal primary wavefunctions is shown to agree with previous literature. The momentum generators on the celestial sphere are derived and, along with the Lorentz generators, form a representation of the Poincare algebra. Finally, we show that the massive spin-$1$ conformal primary wavefunctions can be constructed from the Dirac conformal primary wavefunctions using the standard Clebsch-Gordan coefficients. We use this procedure to write the massive spin-$\frac{3}{2}$, Rarita-Schwinger, conformal primary wavefunctions. This provides a prescription for constructing all massive fermionic and bosonic conformal primary wavefunctions starting from spin-$\frac{1}{2}$.