# Non-Perturbative $S$-matrix Renormalization

## TL;DR

This paper introduces a novel renormalization group flow equation for the $S$-matrix that simplifies the analysis of scattering amplitudes and offers advantages over traditional off-shell methods in non-perturbative quantum field theory.

## Contribution

It presents a new flow equation directly for scattering amplitudes, avoiding Hessian inversion and simplifying on-shell conditions compared to Wetterich's approach.

## Key findings

- The flow equation is polynomial and easier to implement.
- It captures the quantum equations of motion more straightforwardly.
- Potential for non-perturbative analysis of quantum field theories.

## Abstract

We propose a renormalization group flow equation for a functional that generates $S$-matrix elements and which captures similarities to the well-known Wetterich and Polchinski equations. While the latter ones respectively involve the effective action and Schwinger functional, which are genuine off-shell objects, the presented flow equation has the advantage of working more directly with observables, i.e. scattering amplitudes. Compared to the Wetterich equation, our flow equation also greatly simplifies the notion of going on-shell, in the sense of satisfying the quantum equations of motion. In addition, unlike the Wetterich equation, it is polynomial and does not require a Hessian inversion. The approach is a promising direction for non-perturbative quantum field theories, allowing one to work more directly with scattering amplitudes.