Non-Perturbative $S$-matrix Renormalization

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.