New aspect of chiral and axial breaking in QCD

Violation of the $U(1)$ axial symmetry in QCD is stricter than the chiral $SU(2)$ breaking, simply because of the presence of the quantum axial anomaly. If the QCD gauge coupling is sent to zero, the strength of the $U(1)$ axial breaking coincides with that of the chiral $SU(2)$ breaking, which we shall in short call an axial-chiral coincidence. This coincidence is trivial since QCD then becomes a non-interacting theory. Actually, there exists another limit in the QCD parameter space, where an axial-chiral coincidence occurs even with nonzero QCD gauge coupling, that can be dubbed a nontrivial coincidence: it is the case with the massive light quarks $(m_l\neq 0)$ and the massless strange quark ($m_s=0$), due to the flavor-singlet nature of the topological susceptibility. This coincidence is robust and tied to the anomalous chiral Ward-Takahashi identity, which is operative even at hot QCD. This implies that the chiral $SU(2)$ symmetry is restored simultaneously with the $U(1)$ axial symmetry at high temperatures. This simultaneous restoration is independent of $m_l (\neq 0)$, hence is irrespective to the order of the chiral phase transition. In this paper, we discuss how the real-life QCD can be evolved from the nontrivial chiral-axial coincidence limit, by working on a Nambu-Jona-Lasinio model with the $U(1)$ axial anomaly contribution properly incorporated. It is shown that at high temperatures the large differences between the restorations of the chiral $SU(2)$ symmetry and the $U(1)$ axial symmetry for two light quarks and a sufficiently large current mass for the strange quark is induced by a significant interference of the topological susceptibility. Thus the deviation from the nontrivial coincidence, which is monitored by the strange quark mass controlling the topological susceptibility, provides a new way of understanding the chiral $SU(2)$ and $U(1)$ axial breaking in QCD.