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For the Dirac equation with two equal quark masses $m$, the equations for $q$ and $\bar{q}$ give the divergence of the nonsinglet axial current as $im\bar{q}{\gamma }_5{\tau }_{a}q=m{P}_a$. The color chiral anomaly is absent in this nonsinglet identity because the quark flavor generator is traceless. Equivalently, the mass term in the Hamiltonian is $m\int S{d}^{3}x$, and the nonsinglet axial charge algebra of quark densities gives ${\dot{Q}}_{5,a}=m\int P_a{d}^{3}x$.

The gauge-covariant kinetic term for two massless quarks separates into left- and right-handed parts. Independent constant unitary quark flavor rotations preserve these terms, giving classical $U(2{)}_L\times U(2{)}_R$ internal symmetry in addition to local color $SU(3)$. The singlet axial $U(1{)}_A$ has a chiral anomaly, while the nonsinglet chiral symmetry and vector baryon number remain exact quantum symmetries, up to their shared discrete identifications. Equal nonzero quark masses reduce the continuous flavor symmetry to $SU(2{)}_V\times U(1{)}_V$.

Let $S=\bar{q}q$, $P_a=i\bar{q}{\gamma }_5{\tau }_{a}q$ and $Q_{5,a}=\int q^{†}{\gamma }_5{\tau }_{a}q/2d^{3}x$. The canonical anticommutation relations imply $[Q_{5,a},q]=-{\gamma }_5{\tau }_{a}q/2$ and $[Q_{5,a},\bar{q}]=-\bar{q}{\gamma }_5{\tau }_a/2$. The Leibniz rule for the commutator, together with $\{{\tau }_a,{\tau }_b\}=2{\delta }_{ab}$ and ${\gamma }_5^2=1$, gives the displayed transformations. These are equal-time operator relations; common nonzero quark masses remove conservation of the charges without changing their canonical action on the fields.

A nonzero vacuum value of the scalar quark density breaks the massless nonsinglet chiral symmetry to its diagonal vector subgroup. Indeed, the nonsinglet axial charge algebra of quark densities gives $⟨[Q_{5,a},P_b]⟩=-i{\delta }_{ab}⟨\bar{q}q⟩$, whereas invariance of the vacuum under every conserved axial charge would make these expectations vanish. The three broken generators produce three massless Goldstone bosons in the exact chiral limit. Small equal quark masses explicitly break that symmetry and turn the pions into pseudo-Goldstone bosons, with squared masses linear in the quark mass at leading order.

A quark mass is the coefficient of its Dirac mass term. This term couples left- and right-handed quarks, so nonzero quark masses explicitly break independent rotations of the two chiralities. For two equal masses, the vector flavor symmetry survives, while the nonsinglet axial current obeys the flavor-nonsinglet axial Ward identity with equal quark masses. In the renormalized theory the mass parameter depends on the subtraction scale and scheme, while measurable hadron masses are physical observables.