Anomalous $U(1)_A$ couplings and the Columbia plot

TL;DR

This paper investigates how anomalous $U(1)_A$ symmetry breaking influences the order of the chiral phase transition in QCD-like theories, revealing conditions under which a second order transition can occur for three massless flavors.

Contribution

It introduces an extended linear sigma model analysis showing the role of $U(1)_A$ anomaly terms in altering the chiral transition order, especially near the Columbia phase diagram.

Findings

Second order chiral transition possible for three massless flavors.

Spontaneous $CP$ breaking at negative strange quark mass and $ heta=\pi$.

Relation between $U(1)_A$ anomaly and phase transition order clarified.

Abstract

When the quark masses are lighter than those in QCD, the standard lore is that a chiral transition of first order must emerge for three, light flavors. Recently, however, numerical simulations on the lattice suggest that the chiral transition is of second order in the chiral limit. Using an extended linear sigma model in the mean field approximation, we study the relation between terms which break the anomalous, $U(1)_A$ symmetry and the order of the chiral phase transition, especially how a chiral transition of second order can arise for three, massless flavors. We note that in an (unphysical) region of the "Columbia" phase diagram, when the strange quark mass is light and negative, corresponding to topological angle $\theta=\pi$, the $CP$ symmetry is spontaneously broken.