Renormalization group patterns and c-theorem in more than two dimensions

We elaborate on a previous attempt to prove the irreversibility of the renormalization group flow in more than two dimensions. This involves the construction of a monotonically decreasing c-function using a spectral representation. The missing step of the proof is a good definition of this function at the fixed points. We argue that for all kinds of perturbative flows the c-function is well defined and the c-theorem holds in any dimension. We provide examples in multicritical and multicomponent scalar theories for dimension 2 < d < 4. We also discuss the non-perturbative flows in the yet unsettled case of the O(N) sigma model for 2 ≤ d ≤ 4 and large N.