55 pages, 2 figures. v3: Finite spin expressions included in general d. Connection underlined between Wilson polynomials, the spectral function in the conformal partial wave expansion, and the Wilson function. References added. v4: Improved presentation and typos fixed. v5: typos fixed
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.
On Conformal Block, Crossing Kernel and Multi-variable Hypergeometric Functions
在本文中，我们给出了一般时空维中具有外标量的共形块在Appell第四超几何函数${\bf{F}}_4$上的有限求和的另一种表示。我们还构造了它的推广到具有连续自旋的非定域主交换算子及其相应的Mellin表示，这是相关的洛伦兹时空。利用这些结果，我们应用Lorentz反演公式计算了一般时空维中的交叉核，得到的表达式可以表示为具有正确双迹算子奇异结构的Kampe de Feriet超几何函数的双无限求和.我们还包括一些互补的AdS空间中的计算，证明了共形块的正交性和执行的分解。
