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

TL;DR

This paper analyzes conformal blocks and the Regge limit in CFTs, especially focusing on Minkowski conformal blocks and their application to SYK-like models, revealing bounds on effective spin and connections to string modes.

Contribution

It establishes a precise relationship between Minkowski and Euclidean conformal blocks without explicit analytic continuation, and applies this to SYK-like models to explore bounds on effective spin.

Findings

Bound on effective spin: ff  2 for CFTs with gravity duals

Nearly saturated ff  2 in SYK-like models

Methodology for diagonalizing dynamical equations using harmonic analysis

Abstract

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 Euclidean 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 $F^{(M)}(u,v)\sim {u}^{-\delta}\,$, $\delta>0$, in the limit $u \rightarrow 0$ and $v\rightarrow 1$. Here, $\delta=(\ell_{eff}-1)/2$, with $\ell_{eff}$ serving 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. We provide a direct treatment in diagonalizing dynamical equations via harmonic analysis over physical scattering regions. As an example these methods are applied to the SYK model.