Hard-Pion Current-Algebra Calculation of Meson Processes—N-Point Functions

Abstract

Hard-pion techniques are presented for calculating $T$ products of an arbitrary number of vector and axial-vector currents under the assumptions of single $\ensuremath{\sigma}$-, $\ensuremath{\pi}$-, $\ensuremath{\rho}$-, and ${A}_{1}$-meson saturation of intermediate sums, chiral $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$-algebra commutation relations, conservation of vector current (CVC), and partial conservation of axial-vector current (PCAC). The single-meson dominance hypothesis is shown to imply that one calculates the $T$ products, keeping only certain generalized tree and seagull diagrams. Alternatively, the assumption can be replaced by requiring that one calculate with an "effective" interaction Lagrangian ${\mathcal{L}}_{I}$ to lowest nonvanishing order. The conditions that the remaining hypotheses (current commutation relations, CVC, and PCAC) impose on ${\mathcal{L}}_{I}$ are expressed in terms of functional differential equations to determine the form of ${\mathcal{L}}_{I}$. These equations are shown to be consistent with each other and may in fact be integrated order by order. The ${\mathcal{L}}_{I}$ needed to calculate any four-point function is given explicitly.