Asymptotic chiral invariance

We discuss the conditions under which chiral symmetries become exact at asymptotic momenta, utilizing the homogeneous Callan-Symanzik equations so that momenta are not restricted to the deep Euclidean domain as others have done previously. In particular, we consider as our prototype the chiral SU(2) ⊗ SU(2) $\sigma$ model of Gell-Mann and Levy without fermions, where we allow for both spontaneous symmetry breaking and explicit breaking of the symmetry via a term in the Lagrangian linear in the $\sigma$ field. To check our general results, we calculate to two-loop order in perturbation theory the coefficients of the homogeneous Callan-Symanzik equations along with the interesting amplitudes of the theory and finally show that under suitable conditions the symmetry-broken theory and the symmetry-conserved theory both approach the same massless, symmetric theory as the momenta become asymptotic.