Momentum-space conformal blocks on the light cone

TL;DR

This paper derives a new momentum-space conformal block expansion for scalar four-point functions in conformal field theory, providing explicit formulas and applications to free theories.

Contribution

It introduces a novel expansion of momentum-space conformal blocks on the light cone, with explicit polynomial forms and an inversion formula for OPE coefficients.

Findings

Conformal blocks are polynomials in the cosine of the scattering angle.

Explicit closed-form expressions for conformal blocks in arbitrary dimensions.

Application of the inversion formula to compute OPE coefficients in free theories.

Abstract

We study the momentum-space 4-point correlation function of identical scalar operators in conformal field theory. Working specifically with null momenta, we show that its imaginary part admits an expansion in conformal blocks. The blocks are polynomials in the cosine of the scattering angle, with degree $\ell$ corresponding to the spin of the intermediate operator. The coefficients of these polynomials are obtained in a closed-form expression for arbitrary spacetime dimension $d > 2$. If the scaling dimension of the intermediate operator is large, the conformal block reduces to a Gegenbauer polynomial $\mathcal{C}_\ell^{(d-2)/2}$. If on the contrary the scaling dimension saturates the unitarity bound, the block is different Gegenbauer polynomial $\mathcal{C}_\ell^{(d-3)/2}$. These results are then used as an inversion formula to compute OPE coefficients in a free theory example.