Minkowski Conformal Blocks and the Regge Limit for SYK-like Models

We discuss scattering in a CFT via the conformal partial-wave analysis and the Regge limit. The focus of this paper is on understanding an OPE with Minkowski conformal blocks. Starting with a t-channel OPE, it leads to an expansion for an s-channel scattering amplitude in terms of t-channel exchanges. By contrasting with Euclideans conformal blocks we see a precise relationship between conformal blocks in the two limits without preforming an explicit analytic continuation. We discuss a generic feature for a CFT correlation function having singular growth $F^{M}(u,v)\sim u^{-(\ell_{eff}-1)/2}$, in the limit $u\rightarrow 0$ and $v\rightarrow 1$. Here, $\ell_{eff}$ serves as an effective spin and it can be determined through an OPE. In particular, it is bounded from above, $\ell_{eff}\leq 2$, for all CFTs with a gravity dual, and it can be associated with string modes interpolating the graviton in AdS. This singularity is historically referred to as the Pomeron. This bound is nearly saturated by SYK-like effective $d=1$ CFT, and its stringy and thermal corrections have piqued current interests. Our analysis has been facilitated by dealing with Wightman functions, leading to a simpler treatment diagonalizing dynamical equations via appropriate harmonic analysis, e.g., the BFKL equation for $\mathcal{N}=4$ SYM and also ladder sum for SYK-like models.