Chuan-Xin Cui, Mamiya Kawaguchi, Jin-Yang Li, Shinya Matsuzaki, Akio Tomiya
May 25, 2022
Violation of the $U(1)$ axial symmetry in QCD is stiffer than the chiral
$SU(2)$ breaking, simply because of the presence of the quantum axial anomaly.
Hence it might be expected that if and only if the QCD gauge coupling is turned
off, or sent to zero (the asymptotic-free limit), where the $U(1)$ axial
anomaly goes away, the strength of the $U(1)$ axial breaking trivially
coincides with that of the chiral $SU(2)$ breaking. In this write-up, we find
that a axial-chiral coincidence occurs even with nonzero QCD gauge coupling,
that is 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 with the anomalous chiral Ward-Takahashi identity. The
nontrivial coincidence implies that the $U(1)$ axial anomaly becomes invisible
in the meson susceptibility functions. The invisibility of the $U(1)$ axial
anomaly keeps even at hot QCD, so that the chiral $SU(2)$ symmetry restores
simultaneously with the $U(1)$ axial symmetry at high temperatures. This
simultaneous restoration is independent of the light quark mass, hence is
irrespective to the order of the chiral phase transition. We explicitize what
the nontrivial coincidence can tell us, by working on a chiral effective model.
It turns out that once the strange quark mass gets massive, the topological
susceptibility handles the deviation from the nontrivial coincidence. It is
then clarified that the large discrepancy between the chiral and axial
restorations in the $2+1$ flavors with the physical quark masses is brought by
the significant interference of the topological susceptibility. Thus the
deviation from the nontrivial coincidence monitored by the topological
susceptibility provides a new way of understanding of the chiral and axial
breaking in QCD.