Beta function and flowing couplings in the exact Wilson renormalization group in Yang-Mills theory

We discuss the relation between the Gell-Mann-Low beta function and the ''flowing couplings'' of the Wilsonian action SΛ[φ] of the exact renormalization group (RG) at the scale Λ. This relation involves the ultraviolet region of Λ 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 show that the flowing couplings tend to the running coupling at high frequency, they differ at low frequency and remain finite all the way down to zero frequency. We show also that, by a systematical resummation of higher order corrections, the flowing couplings enter directly into the Feynman diagrams with a scale given by the internal loop momenta. (orig.)