On the link between finite QFT and standard RG approaches

TL;DR

The paper addresses how a finite QFT formulation based on differential equations akin to the Callan-Symanzik equations can be recast in a fully renormalized language and shown to be equivalent to the standard renormalization-group approach. Using a massive $\phi^4$ toy model, the authors derive the CS system with renormalized quantities and demonstrate that, under an on-shell mass renormalization and with $\mu^2 = m^2$, the CS system matches the Callan–Symanzik–Ovsyannikov RG equation term-by-term. They define finite RG-functions $\tilde{\beta}$, $\tilde{\gamma}_m$, etc., through renormalization factors and show how the $\theta$-operation yields finite relations among $\Gamma^{(n)}$, connecting the two frameworks. The study also discusses scheme-dependence, the limitations to massive theories, and implications for extending the equivalence to more realistic theories such as QCD, outlining future computational directions to test higher-order consistencies.

Abstract

A finite formulation of quantum field theory based on a system of differential equations reminiscent of the Callan-Symanzik equations is discussed. This system of equations was previously formulated in the bare language. We rederive it in a fully renormalized language. For the latter, within a simple $φ^4$ toy model, it is shown that with a specific choice of renormalization conditions - namely, the on-shell scheme for the renormalized mass - this class of finite renormalization prescriptions is equivalent to the standard renormalization-group equation written in the Callan-Symanzik-Ovsyannikov form.