The momentum-space conformal bootstrap in 2d

TL;DR

This paper introduces a novel momentum-space bootstrap equation for 2D conformal field theories, leveraging the analyticity of commutators, leading to new analytic functionals and applications.

Contribution

It derives a new asymmetric bootstrap equation in momentum space for 2D CFTs, providing closed-form expressions and reproducing known analytic functionals.

Findings

Reproduces known sets of analytic functionals

Provides a new closed-form expression for a functional with zeros at double-twist dimensions

Demonstrates applications of the new crossing equation

Abstract

A new bootstrap equation in 2-dimensional conformal field theory is derived starting from the momentum-space representation of the correlation functions. Since Wightman functions are not crossing-symmetric, the analyticity properties of the commutator are leveraged instead to obtain a relation between two distinct operator product expansions. The procedure requires evaluating a 4-point function with two light-like momenta. The result is an asymmetric equation valid for arbitrary theories in 1d and 2d. The new crossing equation admits two simple projections onto orthogonal bases of Jacobi polynomials, reproducing known sets of analytic functionals. One of these, with zeros at double-twist dimensions, was known only as a contour integral. The closed-form expression found in this work is new. We provide a few examples of applications of the new crossing equation.