Massive celestial fermions
Sruthi A. Narayanan
Abstract
A bstract 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 − ∆. The Dirac conformal primary wave- functions are delta function normalizable with respect to the Dirac inner product provided they lie on the principal series with conformal dimension ∆ = 1 + iλ for λ ∈ ℝ. It is shown that there are two choices of a complete basis: single spin $$ J=\frac{1}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> or $$ J=-\frac{1}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> and λ ∈ ℝ or multiple spin $$ J=\pm \frac{1}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> and λ ∈ ℝ +∪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 Poincaré 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- $$ \frac{3}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> , Rarita-Schwinger, conformal primary wavefunctions. This provides a prescription for constructing all massive fermionic and bosonic conformal primary wavefunctions starting from spin- $$ \frac{1}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> .