Closed-Loop Calculations Using a Chiral-Invariant Lagrangian

Abstract

Using Feynman rules derived from the chiral-invariant Lagrangian for pions, explicit evaluation is made of all contributions to all Feynman amplitudes in the limit of zero external 4-momenta, through order ${f_{\pi }}^{-6\mathrm{}}$. These calculations are formal in that they involve divergent integrals. For all except the tree-graph diagrams, the results depend explicitly upon the pion "gauge," or Weinberg's function $f\left({\pi }^2\right)$. However, we show that through order ${f_{\pi }}^{-6\mathrm{}}$ there is a unique choice of gauge for which, notwithstanding the divergences, the amplitudes vanish in the soft-pion limit in accord with the soft-pion theorems. We conjecture concerning further application of this special gauge for the removal of divergences.