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

TL;DR

This paper explores the irreversibility of the renormalization group (RG) flow in quantum field theory, linking trace anomalies, flow length, and fixed point distances, and proposes axioms suggesting irreversibility even in odd dimensions.

Contribution

It introduces a scheme-invariant measure of RG flow length and fixed point distance, and formulates axioms implying RG irreversibility without requiring a global a function.

Findings

Distance between fixed points equals Delta(a) in even dimensions

Inequalities Delta(a) <= Delta(a') imply RG irreversibility

Axioms support RG irreversibility in certain 3D theories

Abstract

I discuss several issues about the irreversibility of the RG flow and the trace anomalies c, a and a'. First I argue that in quantum field theory: i) the scheme-invariant area Delta(a') of the graph of the effective beta function between the fixed points defines the length of the RG flow; ii) the minimum of Delta(a') in the space of flows connecting the same UV and IR fixed points defines the (oriented) distance between the fixed points; iii) in even dimensions, the distance between the fixed points is equal to Delta(a)=a_UV-a_IR. In even dimensions, these statements imply the inequalities 0 =< Delta(a)=< Delta(a') and therefore the irreversibility of the RG flow. Another consequence is the inequality a =< 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 irreversible also 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.