Resummation at finite conformal spin

TL;DR

The paper extends the Lorentzian OPE inversion approach to arbitrary space-time dimension and finite conformal spin $β$, enabling analytic computation of anomalous dimensions and OPE coefficient corrections for double-twist operators. By employing the Mellin representation of conformal blocks, each exchange contribution is written in terms of generalized Wilson polynomials, yielding exact, dimension-independent expressions for both scalar and spinning exchanges. The results are compactly captured by Wilson functions and balanced hypergeometric functions (e.g., ${}_7F_6$), with explicit simplifications in two and four dimensions. This framework provides a dimension-agnostic, systematically resum-able method to extract CFT data at finite spin and offers pathways to include descendants and external-spin operators in future work.

Abstract

We generalize the computation of anomalous dimension and correction to OPE coefficients at finite conformal spin considered recently in \cite{arXiv:1806.10919, arXiv:1808.00612} to arbitrary space-time dimensions. By using the inversion formula of Caron-Huot and the integral (Mellin) representation of conformal blocks, we show that the contribution from individual exchanges to anomalous dimensions and corrections to the OPE coefficients for "double-twist" operators $[\mathcal{O}_1\mathcal{O}_2]_{Δ,J}$ in $s-$channel can be written at finite conformal spin in terms of generalized Wilson polynomials. This approach is democratic {\it wrt} space-time dimensions, thus generalizing the earlier findings to cases where closed form expressions of the conformal blocks are not available.