Inequalities for trace anomalies, length of the RG flow, distance between the fixed points and irreversibility

I discuss several issues about the irreversibility of the RG flow and the trace anomalies c, a and Deltaa'. First I argue that in quantum field theory: (i) the scheme-invariant area Deltaa' of the graph of the effective beta function between the fixed points defines the length of the RG flow; (ii) the minimum of Deltaa' in the space of flows connecting the same UV and IR fixed points defines the (oriented) distance between the fixed points and (iii) in even dimensions, the distance between the fixed points is equal to Deltaa = a(UV) - a(IR)- In even dimensions, these statements imply the inequalities 0 less than or equal to Aa less than or equal to Deltaa' and therefore the irreversibility of the RG flow. Another consequence is the inequality a less than or equal to c for free scalars and fermions (but not vectors), which can be checked explicitly. Secondly, I elaborate a more general axiomatic set-up where irreversibility is defined as the statement that there exist no pairs of non-trivial flows connecting interchanged UV and IR fixed points. The axioms, based on the notions of length of the flow, oriented distance between the fixed points and certain 'oriented-triangle inequalities', imply the irreversibility of the RG flow without a global a function. I conjecture that the RG flow is also irreversible in odd dimensions (without a global a function). In support of this, I check the axioms of irreversibility in a class of d = 3 theories where the RG flow is integrable at each order of the large N expansion.