Momentum space CFT correlators for Hamiltonian truncation

TL;DR

This paper derives formulas for Lorentzian CFT correlators in momentum space, enabling efficient numerical computation of Hamiltonian matrix elements for conformal truncation, demonstrated with 3d phi^4 theory.

Contribution

It provides explicit momentum space formulas for three-point functions with zero momentum middle operators, facilitating Hamiltonian truncation calculations in strongly-coupled QFTs.

Findings

Derived finite-sum formulas involving hypergeometric functions for three-point correlators.

Applied formulas to compute Hamiltonian matrix elements in 3d phi^4 theory.

Enabled efficient numerical evaluation for conformal truncation methods.

Abstract

We consider Lorentzian CFT Wightman functions in momentum space. In particular, we derive a set of reference formulas for computing two- and three-point functions, restricting our attention to three-point functions where the middle operator (corresponding to a Hamiltonian density) carries zero spatial momentum, but otherwise allowing operators to have arbitrary spin. A direct application of our formulas is the computation of Hamiltonian matrix elements within the framework of conformal truncation, a recently proposed method for numerically studying strongly-coupled QFTs in real time and infinite volume. Our momentum space formulas take the form of finite sums over ${}_2F_1$ hypergeometric functions, allowing for efficient numerical evaluation. As a concrete application, we work out matrix elements for 3d $\phi^4$-theory, thus providing the seed ingredients for future truncation studies.