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.