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.
Highlights
Feature has been noticed in the work by Cardona [24], where the inversion formula has been considered in the collinear approximation to obtain interesting results
The Mellin amplitude is just a constant and we find that the Mellin inversion formula gives a vanishing result
We discuss the region of convergence of the Mellin inversion formula for Witten diagrams in section 2.3 and show that the contact Witten diagram lies outside this region
Summary
We will be interested in large N theories, where these operators are actual composite primaries present in the spectrum In these theories, Mellin space representation is useful since the information about the single-trace primaries encoded in the Mellin amplitude is nicely separated from the double-trace primaries encoded in the measure. Mellin space representation is useful since the information about the single-trace primaries encoded in the Mellin amplitude is nicely separated from the double-trace primaries encoded in the measure This feature has been used to great advantage by Aharony et al [33] to calculate the anomalous dimensions in large N theories. We discuss this technique in appendix B and use it to double check some of our results for the anomalous dimensions of the double-trace primaries
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