Sketch of the resolution of the axial U(1) problem without chiral anomaly

TL;DR

This paper introduces a novel mechanism explaining the masses of η and η' mesons without relying on the axial U(1) anomaly, resolving the U(1) problem through disconnected meson correlator contributions.

Contribution

It presents a new approach to the U(1) problem by considering the first order disconnected correlator contributions, avoiding explicit chiral anomaly violation.

Findings

Achieves 20% agreement with experimental η and η' masses

Fulfills Weinberg's bound on η mass

Provides the leading chiral Lagrangian for disconnected contributions

Abstract

We propose a mechanism which explains the masses of $\eta$ and $\eta'$ mesons without invoking the explicit violation of $U(1)_A$ symmetry by the chiral anomaly. It is shown that the U(1) problem, the problem for which the prediction of $\eta$ and $\eta'$ masses in the simple chiral perturbation theory largely deviates from the experimental values, is actually resolved by considering the first order contribution of the disconnected meson correlator with respect to the quark mass. The bound of Weinberg $m_\eta^2 \le 3 m_\pi^2$ is fulfilled by considering the negative squared mass of $\eta$ or $\eta'$ which is just the saddle point of the QCD effective potential, and 20% level agreements with experimental data are obtained by just fitting one low energy constant. We provide the leading chiral Lagrangian due to the disconnected contribution in 3-flavor QCD, and also discuss the 2- and 4-flavor cases as well as the consistency of our mechanism with the chiral restoration at high temperature found in lattice calculations.