The Effective Bootstrap

TL;DR

This paper introduces an efficient conformal bootstrap method that simplifies calculations by integrating out higher-dimensional operators, and applies it to 3D Ising, O(n) models, and 4D CFTs, yielding new operator dimension results.

Contribution

It presents a novel, effective bootstrap approach that reduces computational complexity and demonstrates its utility on various models, including new scalar operator dimensions.

Findings

Validated the method on known models with consistent results.

Obtained new scalar operator dimensions for O(n) models.

Achieved faster and simpler bootstrap computations.

Abstract

We study the numerical bounds obtained using a conformal-bootstrap method - advocated in ref. [1] but never implemented so far - where different points in the plane of conformal cross ratios $z$ and $\bar z$ are sampled. In contrast to the most used method based on derivatives evaluated at the symmetric point $z=\bar z =1/2$, we can consistently "integrate out" higher-dimensional operators and get a reduced simpler, and faster to solve, set of bootstrap equations. We test this "effective" bootstrap by studying the 3D Ising and $O(n)$ vector models and bounds on generic 4D CFTs, for which extensive results are already available in the literature. We also determine the scaling dimensions of certain scalar operators in the $O(n)$ vector models, with $n=2,3,4$, which have not yet been computed using bootstrap techniques.