# Cut-touching linear functionals in the conformal bootstrap

## TL;DR

This paper establishes criteria for the validity of cut-touching linear functionals in the conformal bootstrap, enabling their rigorous use in deriving bounds on conformal field theories.

## Contribution

We derive general swapping criteria for cut-touching functionals and verify their applicability to recent functionals used in conformal bootstrap bounds.

## Key findings

- Derived swapping criteria for cut-touching functionals
- Validated criteria on explicit examples of Mazac's functionals
- Enhanced the theoretical foundation for using complex functionals in bootstrap

## Abstract

The modern conformal bootstrap program often employs the method of linear functionals to derive the numerical or analytical bounds on the CFT data. These functionals must have a crucial "swapping" property, allowing to swap infinite summation with the action of the functional in the conformal bootstrap sum rule. Swapping is easy to justify for the popular functionals involving finite sums of derivatives. However, it is far from obvious for "cut-touching" functionals, involving integration over regions where conformal block decomposition does not converge uniformly. Functionals of this type were recently considered by Mazac in his work on analytic derivation of optimal bootstrap bounds. We derive general swapping criteria for the cut-touching functionals, and check in a few explicit examples that Mazac's functionals pass our criteria.