Chiral symmetry ($SU(2{)}_L\times SU(2{)}_R$)

Chiral symmetry allows independent flavor rotations of left-handed and right-handed fields. For two massless quark flavors, the non-Abelian chiral transformations form $SU(2{)}_L\times SU(2{)}_R$. The additional vector $U(1)$ symmetry of baryon number acts trivially on the pion field $U$. The low-energy pion field $U\in SU(2)$ transforms as $U\mapsto LU{R}^{†}$. Its derivative Skyrme model terms are invariant under constant $L,R$. A chosen vacuum $U=1$ preserves the diagonal subgroup $L=R$, which acts by conjugation and is the isospin symmetry. An ordinary common nonzero pion mass explicitly breaks the independent left/right symmetry to that vector subgroup; vacuum selection in the massless theory is instead spontaneous symmetry breaking. The singlet axial anomaly is a separate issue and is not part of this two-flavor non-anomalous group.

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.

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.

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 ${\dot{Q}}_{5,a}=m\int P_a{d}^{3}x$.