Applications of dispersive sum rules: $\epsilon$-expansion and holography

TL;DR

This paper develops dispersive sum rules in Mellin space for conformal field theories, applying them to the Wilson-Fisher model and holographic CFTs to derive known and new results without assuming analyticity at low spin.

Contribution

It introduces a new family of sum rules that suppress double twist operators, applicable to both Wilson-Fisher and holographic CFTs, with novel predictions and derivations.

Findings

Re-derived Wilson-Fisher results to order ε^4

Predicted new results for Wilson-Fisher model

Calculated anomalous dimensions in holographic CFTs

Abstract

We use Mellin space dispersion relations together with Polyakov conditions to derive a family of sum rules for Conformal Field Theories (CFTs). The defining property of these sum rules is suppression of the contribution of the double twist operators. Firstly, we apply these sum rules to the Wilson-Fisher model in $d=4-\epsilon$ dimensions. We re-derive many of the known results to order $\epsilon^4$ and we make new predictions. No assumption of analyticity down to spin $0$ was made. Secondly, we study holographic CFTs. We use dispersive sum rules to obtain tree-level and one-loop anomalous dimensions. Finally, we briefly discuss the contribution of heavy operators to the sum rules in UV complete holographic theories.