# Fermion Conformal Bootstrap in 4d

## TL;DR

This paper uses numerical conformal bootstrap methods to establish bounds on operator dimensions and OPE coefficients in 4d non-supersymmetric CFTs, revealing a new fake primary effect affecting these bounds.

## Contribution

It introduces a novel analysis of four-point functions of Weyl spinors in 4d CFTs and uncovers a previously unobserved fake primary effect influencing bootstrap bounds.

## Key findings

- Discovered universal bounds on operator dimensions and OPE coefficients.
- Identified a fake primary effect causing discontinuities in bootstrap bounds.
- Developed a new numerical method for approximating 4d conformal blocks.

## Abstract

We apply numerical conformal bootstrap techniques to the four-point function of a Weyl spinor in 4d non-supersymmetric CFTs. We find universal bounds on operator dimensions and OPE coefficients, including bounds on operators in mixed symmetry representations of the Lorentz group, which were inaccessible in previous bootstrap studies. We find discontinuities in some of the bounds on operator dimensions, and we show that they arise due to a generic yet previously unobserved fake primary effect, which is related to the existence of poles in conformal blocks. We show that this effect is also responsible for similar discontinuities found in four-fermion bootstrap in 3d, as well as in the mixed-correlator analysis of the 3d Ising CFT. As an important byproduct of our work, we develop a practical technology for numerical approximation of general 4d conformal blocks.