A universal flow invariant in quantum field theory

TL;DR

This paper derives a general formula for a flow invariant in quantum field theory, explaining its relation to central charges and addressing open questions about scale invariance breaking.

Contribution

It provides a universal formula for the flow invariant applicable even with stress tensor improvements and clarifies conditions where it equals c_{UV}-c_{IR}.

Findings

Derived a general formula for the flow invariant.

Identified conditions for the flow invariant to equal c_{UV}-c_{IR}.

Applied results to non-unitary theories and the Standard Model.

Abstract

A flow invariant is a quantity depending only on the UV and IR conformal fixed points and not on the flow connecting them. Typically, its value is related to the central charges a and c. In classically-conformal field theories, scale invariance is broken by quantum effects and the flow invariant a_{UV}-a_{IR} is measured by the area of the graph of the beta function between the fixed points. There exists a theoretical explanation of this fact. On the other hand, when scale invariance is broken at the classical level, it is empirically known that the flow invariant equals c_{UV}-c_{IR} in massive free-field theories, but a theoretical argument explaining why it is so is still missing. A number of related open questions are answered here. A general formula of the flow invariant is found, which holds also when the stress tensor has improvement terms. The conditions under which the flow invariant equals c_{UV}-c_{IR} are identified. Several non-unitary theories are used as a laboratory, but the conclusions are general and an application to the Standard Model is addressed. The analysis of the results suggests some new minimum principles, which might point towards a better understanding of quantum field theory.