Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs

TL;DR

This paper develops Polyakov-Mellin bootstrap techniques for 1D CFTs, enabling analytical computation of CFT data at one loop, exploring effective theories in AdS2, and establishing connections between Regge behavior and large-twist limits.

Contribution

It introduces a novel PM bootstrap framework for 1D CFTs, including $O(N)$ theories, and provides explicit analytic results and constraints on CFT data at one loop and in the Regge limit.

Findings

Analytic one-loop CFT data for 1D theories in AdS2.

Universal constraints on large-twist CFT data in Regge-bounded theories.

A new basis of transcendental functions for fixing four-point correlators.

Abstract

We develop the technology for Polyakov-Mellin (PM) bootstrap in one-dimensional conformal field theories (CFT$_1$). By adding appropriate contact terms, we bootstrap various effective field theories in AdS$_2$ and analytically compute the CFT data to one loop. The computation can be extended to higher orders in perturbation theory, if we ignore mixing, for any external dimension. We develop PM bootstrap for $O(N)$ theories and derive the necessary contact terms for such theories (which also involves a new higher gradient contact term absent for $N=1$). We perform cross-checks which include considering the diagonal limit of the $2d$ Ising model in terms of the $1d$ PM blocks. As an independent check of the validity of the results obtained with PM bootstrap, we propose a suitable basis of transcendental functions, which allows to fix the four-point correlators of identical scalar primaries completely, up to a finite number of ambiguities related to the number of contact terms in the PM basis. We perform this analysis both at tree level (with and without exchanges) and at one loop. We also derive expressions for the corresponding CFT data in terms of harmonic sums. Finally, we consider the Regge limit of one-dimensional correlators and derive a precise connection between the latter and the large-twist limit of CFT data. Exploiting this result, we study the crossing equation in the three OPE limits and derive some universal constraints for the large-twist limit of CFT data in Regge-bounded theories with a finite number of exchanges.