A note on instantons, $\theta$-dependence and strong CP

TL;DR

This paper reviews the instanton approach to understanding the $ heta$-dependence in QCD and discusses contrasting views on whether strong interactions preserve CP symmetry for all $ heta$ values.

Contribution

It clarifies the role of instantons in $ heta$-dependence and critically examines recent claims denying $ heta$-phases, emphasizing the intrinsic non-triviality of $ heta$-dependence.

Findings

$ heta$-dependence is governed by instantons in weak coupling

Non-perturbative effects influence $ heta$-dependence in strong coupling

Contrasts between semiclassical results and recent assertions are discussed

Abstract

I review the standard instanton framework for determining the $\theta$-dependence of instanton-dominated correlation functions in QCD. I then contrast these well-established semiclassical results with the recent assertion of [1,2], that $\theta$-phases are absent and that strong interactions preserve CP for all values of $\theta$. In all scenarios considered, the $\theta$-dependence is shown to be intrinsically non-trivial, being governed either by instantons in the weak-coupling regime or by alternative, non-perturbative considerations when the theory is strongly coupled.