The Unitary Architecture of Renormalization

TL;DR

This paper develops a new on-shell renormalization approach in massless four-dimensional models, deriving all-loop recursion relations from unitarity that determine the beta function and anomalous dimensions.

Contribution

It introduces a bootstrap framework linking unitarity to renormalization, providing a novel recursive method for calculating quantum corrections in scalar field theories.

Findings

Derived all-loop recursion relations from unitarity.

Connected on-shell renormalization with standard perturbation theory.

Fixed the beta function and anomalous dimensions through initial conditions.

Abstract

We set up a bootstrap problem for renormalization. Working in the massless four-dimensional O$(N)$ model and the $\lambda \phi^4$ theory, we prove that unitarity leads to all-loop recursion relations between coefficients of scattering amplitudes with different multiplicities. These turn out to be equivalent to the identities imposed by renormalization of the coupling and the wavefunction through subleading logarithmic order, except with different initial conditions. Matching the initial conditions thus fixes the beta function and wavefunction anomalous dimension to these orders. We explain how to connect this new on-shell renormalization picture with the standard renormalized perturbation theory, highlighting a rich interplay between finiteness, dimensional regularization, and unitarity cuts.