Application of Hard-Pion Three-Point Functions

Some comparisons with experiment of the hard-pion three-point functions obtained from $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ current algebra are given. Available experimental data involving $\pi$, $\rho$, $A_1$, and $\sigma$ mesons are examined. The hard-pion current-algebra method is used to calculate the decays $\rho \to \pi +\pi$, $A_1\to \pi +\rho$, and $\sigma \to \pi +\pi$, and the electromagnetic form factor of the pion. Peripheral processes such as $\pi +N\to \rho +N$ are also examined as a test of meson-vertex functions for spacelike momentum transfers. Here, to reproduce correctly the momentum-transfer dependence at the nucleon vertex, a new extrapolation for the pionic nucleon form factor is introduced, using the Goldberger-Treiman relation. The results of the above calculations are found to be consistent with the present experimental situation. Current-algebra predictions for the $\gamma +N\to A_1+N$ cross section and the decays $A_1\to \pi +\gamma$ and $A_1\to \pi +\sigma$ are given. A cross section of about 0.1 μb is obtained for the $A_1$ photoproduction, which is on the verge of being detectable.