Breaking of Chiral Symmetry for Pseudoscalar Mesons

The content of the generator-divergence commutators proposed by Gell-Mann, Oakes, and Renner is analyzed using the pole-dominance and smoothness approximations for the two- and three-point functions in the theory. This leads to very general sum rules which include both Hamiltonian and vacuum symmetry breaking. There are many different solutions to these sum rules. In particular, two interesting limiting cases, one with the Hamiltonian approximately $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$-invariant and the other with the vacuum approximately $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$-invariant, are found to be allowed. When $\eta -{\eta }^{\prime }$ mixing is included and the smoothness conditions imposed, a consistent solution exists which is in good agreement with present experimental results. We consider the applications of this model to the $K_{l3\mathrm{}}$ form factors, and compare our results with experiment as well as other theoretical work.