  The quantum anomaly that breaks the U(1) axial symmetry of massless
multi-flavored QCD leaves behind a discrete flavor-singlet chiral invariance.
With massive quarks, this residual symmetry has a close connection with the
strong CP-violating parameter theta. One result is that if the lightest quarks
are degenerate, then a first order transition will occur when theta passes
through pi. The resulting framework helps clarify when the rooting prescription
for extrapolating in the number of flavors is valid.
