A note on instantons, $θ$-dependence and strong CP

TL;DR

The note analyzes the standard dilute instanton gas framework to determine the $\theta$-dependence of instanton-dominated observables in QCD with massive quarks, showing that the vacuum energy scales as $E(\theta) = -2 |K| e^{-8\pi^2/g^2} \cos \bar{\theta}$ with $\bar{\theta} = \theta + \alpha$. It emphasizes that the physically relevant connected correlators arise from single-instanton and single-anti-instanton contributions, while higher topological sectors contribute only through disconnected diagrams, and discusses how virial-type corrections can enter. The paper critiques a recent rearrangement of the DIGA sums that takes the infinite-volume limit before summing over topological sectors, arguing that this limit-order yields ill-defined expressions and artificially removes $\theta$-dependence. It concludes that standard instanton calculus, focusing on connected contributions, faithfully reproduces the $\theta$-dependence and CP violation, and situates these results in a broader non-perturbative context including connections to axion physics and supersymmetric theories.

Abstract

I review the standard instanton framework for determining the $θ$-dependence of instanton-dominated correlation functions in QCD. I then contrast these well-established semiclassical results with the recent assertion of [1,2], that $θ$-phases are absent and that strong interactions preserve CP.