Beta function and infrared renormalons in the exact Wilson renormalization group in Yang-Mills theory

TL;DR

This paper explores the connection between the Wilsonian exact renormalization group approach and the beta function in Yang-Mills theory, analyzing infrared renormalons and non-perturbative effects through a scale-dependent framework.

Contribution

It provides a novel formulation linking the Wilsonian RG flow with the beta function and infrared renormalons in Yang-Mills theory, including explicit one- and two-loop calculations.

Findings

Computed the Yang-Mills beta function to one loop and scalar case to two loops.

Analyzed the role of IR cutoff in non-perturbative effects.

Demonstrated the scale dependence of Wilsonian couplings down to zero frequency.

Abstract

We discuss the relation between the Gell-Mann-Low beta function and the ``flowing couplings'' of the Wilsonian action $S_\L[\phi]$ of the exact renormalization group (RG) at the scale $\L$. This relation involves the ultraviolet region of $\L$ so that the condition of renormalizability is equivalent to the Callan-Symanzik equation. As an illustration, by using the exact RG formulation, we compute the beta function in Yang-Mills theory to one loop (and to two loops for the scalar case). We also study the infrared (IR) renormalons. This formulation is particularly suited for this study since: $i$) $\L$ plays the r\^ole of a IR cutoff in Feynman diagrams and non-perturbative effects could be generated as soon as $\L$ becomes small; $ii$) by a systematical resummation of higher order corrections the Wilsonian flowing couplings enter directly into the Feynman diagrams with a scale given by the internal loop momenta; $iii$) these couplings tend to the running coupling at high frequency, they differ at low frequency and remain finite all the way down to zero frequency.