Positivity, low twist dominance and CSDR for CFTs

TL;DR

This paper introduces a crossing symmetric dispersion relation for CFT four-point functions, revealing low twist dominance and positivity properties, with implications for bootstrap methods and universal coefficient ratios.

Contribution

It develops a novel crossing symmetric dispersion relation for CFT correlators and uncovers low twist dominance and positivity features, extending insights to epsilon expansion.

Findings

Low twist dominance maximized near 2d Ising and Yang-Lee models

Positivity of Taylor coefficients explained by CSDR and LTD

Universal ratios of coefficients predicted and supported by results

Abstract

We consider a crossing symmetric dispersion relation (CSDR) for CFT four point correlation with identical scalar operators, which is manifestly symmetric under the cross-ratios $u,v$ interchange. This representation has several features in common with the CSDR for quantum field theories. It enables a study of the expansion of the correlation function around $u=v=1/4$, which is used in the numerical conformal bootstrap program. We elucidate several remarkable features of the dispersive representation using the four point correlation function of $\Phi_{1,2}$ operators in 2d minimal models as a test-bed. When the dimension of the external scalar operator ($\Delta_\sigma$) is less than $\frac{1}{2}$, the CSDR gets contribution from only a single tower of global primary operators with the second tower being projected out. We find that there is a notion of low twist dominance (LTD) which, as a function of $\Delta_\sigma$, is maximized near the 2d Ising model as well as the non-unitary Yang-Lee model. The CSDR and LTD further explain positivity of the Taylor expansion coefficients of the correlation function around the crossing symmetric point and lead to universal predictions for specific ratios of these coefficients. These results carry over to the epsilon expansion in $4-\epsilon$ dimensions. We also conduct a preliminary investigation of geometric function theory ideas, namely the Bieberbach-Rogosinski bounds.