Anomalous Dimensions from Crossing Kernels

In this note we consider the problem of extracting the corrections to CFT data induced by the exchange of a primary operator and its descendents in the crossed channel. We show how those corrections which are analytic in spin can be systematically extracted from crossing kernels. To this end, we underline a connection between: Wilson polynomials (which naturally appear when considering the crossing kernels given recently in arXiv:1804.09334), the spectral integral in the conformal partial wave expansion, and Wilson functions. Using this connection, we determine closed form expressions for the OPE data when the external operators in 4pt correlation functions have spins $J_1$-$J_2$-$0$-$0$, and in particular the anomalous dimensions of double-twist operators of the type $[\mathcal{O}_{J_1}\mathcal{O}_{J_2}]_{n,\ell}$ in $d$ dimensions and for both leading and sub-leading twist. The OPE data are expressed in terms of Wilson functions, which naturally appear as a spectral integral of a Wilson polynomial. As a consequence, our expressions are manifestly analytic in spin and are valid up to finite spin. We present some applications to CFTs with slightly broken higher-spin symmetry. The Mellin Barnes integral representation for $6j$ symbols of the conformal group in general $d$ and its relation with the crossing kernels are also discussed.