# More Analytic Bootstrap: Nonperturbative Effects and Fermions

## TL;DR

This paper advances the analytic bootstrap by incorporating nonperturbative effects and extending calculations to fermions, improving precision in 3d models and predicting operator behaviors.

## Contribution

It introduces nonperturbative effects into the Lorentzian inversion formula and extends lightcone bootstrap to fermion four-point functions in 3d.

## Key findings

- Nonperturbative effects are crucial at finite spin for precision bootstrap.
- Reproduced the 3d Ising stress tensor with 10^{-5} accuracy.
- Predicted asymptotic behavior of fermion double-twist operators.

## Abstract

We develop the analytic bootstrap in several directions. First, we discuss the appearance of nonperturbative effects in the Lorentzian inversion formula, which are exponentially suppressed at large spin but important at finite spin. We show that these effects are important for precision applications of the analytic bootstrap in the context of the 3d Ising and O(2) models. In the former they allow us to reproduce the spin-2 stress tensor with error at the $10^{-5}$ level while in the latter requiring that we reproduce the stress tensor allows us to predict the coupling to the leading charge-2 operator. We also extend perturbative calculations in the lightcone bootstrap to fermion 4-point functions in 3d, predicting the leading and subleading asymptotic behavior for the double-twist operators built out of two fermions.