Anomalous dimensions at finite conformal spin from OPE inversion

TL;DR

This work uses the OPE inversion formula of Caron-Huot to compute anomalous dimensions of higher-spin operators at finite conformal spin in arbitrary dimensions, systematically contrasting position-space and Mellin-space formulations. The authors isolate log-term contributions to obtain spin-dependent anomalous dimensions $\gamma_{12}(\beta)$ for scalar and spin exchanges, expressing results in closed hypergeometric forms such as ${}_4F_3$ and ${}_5F_4$, and demonstrate exact resummation of large-spin expansions. In four dimensions the spin recursion of blocks can be resummed into compact expressions, while Mellin-space methods yield dimension-agnostic results and straightforward residue-based evaluations. Special cases, including the $\epsilon$-expansion for identical scalars and explicit results in $d=3$ and $d=6$, reproduce known large-spin limits and connect to established literature (Alday et al.). The results underscore the utility of Mellin-space inversion for analytic bootstrap analyses and offer a path toward extensions to tensor operators and AdS-loop computations.

Abstract

We compute anomalous dimensions of higher spin operators in Conformal Field Theory at arbitrary space-time dimension by using the OPE inversion formula of \cite{Caron-Huot:2017vep}, both from the position space representation as well as from the integral (viz. Mellin) representation of the conformal blocks. The Mellin space is advantageous over the position space not only in allowing to write expressions agnostic to the space-time dimension, but also in that it replaces tedious recursion relations in terms of simple sums which are easy to perform. We evaluate the contributions of scalar and spin exchanges in the $t-$channel exactly, in terms of higher order Hypergeometric functions. These relate to a particular exchange of conformal spin $β=Δ+J$ in the $s-$channel through the inversion formula. Our exact results reproduce the special cases for large spin anomalous dimension and OPE coefficients obtained previously in the literature.