Radius of Convergence for Perturbation Expansions in the SU3σModel

The dependence of the tree-approximation solution of the S${\mathrm{U}}_3$ $\sigma$ model for mesons on the S${\mathrm{U}}_3$×S${\mathrm{U}}_3$ symmetry-breaking parameters has been investigated in detail. From the explicit relations of the model we establish the existence of a radius of convergence of power series expansions about the symmetry limit. This radius of convergence is an order of magnitude smaller than the value determined by fitting the model to the pseudoscalar nonet masses. The mixing angles and scalar masses implied by the model are in surprisingly good agreement with (fragmentary) experimental data. Using the experimentally determined parameters, it is possible to compare the behavior of the several possible solutions in the limit of S${\mathrm{U}}_3$×S${\mathrm{U}}_3$ symmetry. Of these, only that having S${\mathrm{U}}_3$ symmetry of states and a pseudoscalar octet of massless Goldstone bosons is stable under symmetry-breaking perturbations. This result provides a dynamical reason for this frequently used assumption. The symmetry-breaking parameters turn out to be such that the Lagrangian has approximate S${\mathrm{U}}_2$×S${\mathrm{U}}_2$ symmetry. The physical point can be reached by first turning on either the S${\mathrm{U}}_3$-invariant or the S${\mathrm{U}}_2$×S${\mathrm{U}}_2$-invariant S${\mathrm{U}}_3$×S${\mathrm{U}}_3$ symmetry-breaking operators to their physical values and then using perturbation theory. In each case the perturbation expansion converges.