Nonlinear Realizations of Chiral Symmetry

We explore possible realizations of chiral symmetry, based on isotopic multiplets of fields whose transformation rules involve only isotopic-spin matrices and the pion field. The transformation rules are unique, up to possible redefinitions of the pion field. Chiral-invariant Lagrangians can be constructed by forming isotopic-spin-conserving functions of a covariant pion derivative, plus other fields and their covariant derivatives. The resulting models are essentially equivalent to those that have been derived by treating chirality as an ordinary linear symmetry broken by the vacuum, except that we do not have to commit ourselves as to the grouping of hadrons into chiral multiplets; as a result, the unrenormalized value of $\frac{g_A}{g_V}$ need not be unity. We classify the possible choices of the chiral-symmetry-breaking term in the Lagrangian according to their chiral transformation properties, and give the values of the pion-pion scattering lengths for each choice. If the symmetry-breaking term has the simplest possible transformation properties, then the scattering lengths are those previously derived from current algebra. An alternative method of constructing chiral-invariant Lagrangians, using $\rho$ mesons to form covariant derivatives, is also presented. In this formalism, $\rho$ dominance is automatic, and the current-algebra result from the $\rho$-meson coupling constant arises from the independent assumption that $\rho$ mesons couple universally to pions and other particles. Including $\rho$ mesons in the Lagrangian has no effect on the $\pi -\pi$ scattering lengths, because chiral invariance requires that we also include direct pion self-couplings which cancel the $\rho$-exchange diagrams for pion energies near threshold.