Anomalous Dimensions from Crossing Kernels

TL;DR

This work develops a robust framework to extract crossed-channel corrections to CFT data by recasting crossing kernels in a Wilson-polynomial basis and evaluating their spectral integrals as Wilson functions, yielding analytic-in-spin OPE data for double-twist operators across general dimensions and external spins. The authors provide closed-form expressions for leading and subleading twists, including cases with spinning external legs, and connect these results to Mellin-space 6j symbols and to higher-spin symmetry breaking. They show that large-spin results can be obtained either from the spectral-integral approach or via shadow-projected Mellin representations, with consistent outcomes and agreement with the Lorentzian inversion formula. Applications to CFTs with slightly broken higher-spin symmetry illustrate the utility of the method for computing 1/N corrections in critical boson/fermion theories and related 3D models, including simplifications in special dimensions and spins.

Abstract

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.

Figures (3)

• Figure 1: ${\sf s}$-channel decomposition of the exchange of an operator of twist $\tau^\prime$ and spin $\ell^\prime$ (+descendents) in the crossed channel.
• Figure 2: The plot of the expression \ref{['ScalarAnalytic']} obtained by evaluating the spectral integral is in tick blue, the expression \ref{['ScalarExch']} is in dashed orange and the first three terms of the large spin expansion in $1/\mathfrak{J}$ are the grey dots. In this plot we took $d=4$ and $\tau^\prime=4$, for $\Delta=2+7/11$ (LHS) and $\Delta=2+9/11$ (RHS). The expression \ref{['ScalarExch']} displays some oscillating singular behaviour which diminishes as $\Delta$ approaches $\Delta=3$. For $\Delta\geq3$ no non-analytic behaviour in $1/\mathfrak{J}$ is observed. Remarkably the analytic result in spin matches the first three terms in the large spin expansion up to very low spin with a very small error that is indistinguishable in the graphs!
• Figure 3: The plot of the expression \ref{['ScalarAnalytic']} obtained by evaluating the spectral integral is in tick blue, the expression \ref{['ScalarExch']} in dashed orange and the first three terms of the large spin expansion in $1/\mathfrak{J}$ in gray dots. We considered $d=4$ and $\tau^\prime=5/2$, with $\Delta=2+10/11$ (LHS) and $\Delta=3+2/11$ (RHS). In this case the expression \ref{['ScalarExch']} precisely coincides with analytic in spin result \ref{['ScalarAnalytic']} obtained by evaluating the spectral integral and with the first three terms of the asymptotic $1/\mathfrak{J}$ expansion. A small deviation can be observed for $\ell<1$.