Effective Lagrangians, Field Algebra, and Vector-Meson Dominance with Unstable Particles

Abstract

The finite-width mass distribution of the vector and axial-vector mesons is introduced in the phenomenological Lagrangian model. Tree-graph amplitudes becoming complex, unitarity constraints may be imposed. The ${\mathrm{SU}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}}_{2}$ algebra of fields and field-current identities are discussed. The well-known relation between the Schwinger term and the integral over the spectral density is identically obeyed in our tree-graph approach. A tentative unitary solution gives a rather broad ${A}_{1}$ resonance. A discussion of the $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude and of the pion form factor in this approach is given.