# Lorentzian inversion and anomalous dimensions in Mellin space

## TL;DR

This paper develops a Mellin space version of the Lorentzian inversion formula for conformal field theories, enabling precise calculation of operator product expansion data in large N theories.

## Contribution

It introduces a Mellin space approach to the Lorentzian inversion formula, providing explicit calculations of anomalous dimensions up to order 1/N^4 in large N CFTs.

## Key findings

- Analytical match at order 1/N^2 with existing results
- Numerical agreement at order 1/N^4 with literature
- Simplifies computation of OPE data in Mellin space

## Abstract

In this note, we derive a Mellin space version of the Lorentzian inversion formula for CFTs by explicitly integrating over the cross-ratios in $d=2$ and $d=4$ spacetime dimensions. We use the simplicity of the Mellin representation of Witten diagrams and the double discontinuity to find the OPE coefficients and anomalous dimensions of double-trace primaries in large $N$ CFTs to order $\frac{1}{N^4}$. We find that our results match analytically at order $\frac{1}{N^2}$, and numerically at order $\frac{1}{N^4}$ with existing literature.