Uniqueness of theSMatrix for a Class of Chiral Lagrangians

We construct the general pion-nucleon Lagrangian for a certain infinite class of nonlinear realizations of chiral $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$; the symmetry-breaking term is the isoscalar component of a chiral tensor of rank $n$. By explicit calculation, the $\pi -\pi$ scattering lengths are shown to be the same for all choices of the nonlinear realization. For both $\pi -\pi$ and $\pi -N$ scattering, we find that for fixed $n$ the $S$ matrix, diagram by diagram and in the tree approximation, is unique, and that it remains unique if $\rho$ mesons are introduced.