Renormalization group flow for Yang-Mills fields interacting with matter

TL;DR

This paper applies the Wilson Renormalization Group method to an SU(2) gauge theory with fermionic matter, demonstrating gauge symmetry preservation, calculating the beta-function, and establishing perturbative renormalizability.

Contribution

It introduces a boundary condition approach that maintains gauge invariance in RG flow, avoiding fine-tuning issues and providing a framework for analyzing matter effects on gauge couplings.

Findings

Fermionic matter modifies the asymptotic behavior of couplings.

Beta-function computed and matches known results.

Gluon transversality is verified.

Abstract

We show an application of the Wilson Renormalization Group (RG) method to a SU(2 ) gauge field theory in interaction with a massive fermionic doublet. By choosing suitable boundary conditions to the RG equation, i.e. by requiring the relevant monomials not present in the classical action to satisfy the Slavnov-Taylor identities once the cutoffs are removed, we succeed in implementing the local gauge symmetry. In this way the so called fine-tuning problem, due to the assignation of boundary conditions in terms of the bare parameters, is avoided. In this framework, loop expansion is equivalent to the iterative solution of the RG equation; we perform one loop calculations in order to determine how much the fermionic matter modifies the asymptotic form of the couplings. Then we compute the beta-function and we check gluon transversality. Finally, a proof of perturbative renormalizability is shown.