Analytic Hard-Pion Methods: TheA1ρπSystem

We use current algebra and analyticity to study vertex functions occurring in the $A_1\rho \pi$ system. Employing the conserved vector current and partially conserved axial-vector current relations, and the $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ algebra of currents, we generate Ward identities which relate two- and three-point functions of vector and axial-vector currents. Extracting the pion poles from these vertex functions and exposing their isospin content, we define form factors whose analytic properties may readily be studied and, in particular, deduce from the Ward identities a relation involving the pion form factor. With suitable low-energy approximations which maintain the correct cut structure, we use this relation to calculate an effective-range formula for the pion form factor, and consequently from unitarity, the $p$-wave $\pi \pi$ phase shift. Our results are generally in agreement with experiment. Using ${\mathrm{A}}_1$ dominance, we are able to obtain analytic effective-range formulas for the form factors appearing in the vector-current matrix element of $\pi$, $A_1$ mesons. From these form factors, measurable in the reaction $e^{+}{e}^{-}\to \pi A_1$, we calculate the $A_1\to \rho \pi$ width and the $A_1-\rho$ spin correlation. Finally, we extend our methods to encompass both $\pi \pi$ and $\pi A_1$ cut contributions, and derive a set of coupled integral equations which we solve approximately for the $\pi \pi$ and $\pi A_1$ form factors. We conclude with a general observation on the complementary roles played by current algebra and by unitarity.