Theory of Deviations from Pion Pole Dominance

It is shown how mass dispersion relations can be effectively used to quantitatively determine the effects of extrapolating an amplitude off the mass shell. Light-cone operator-product expansions abstracted from the canonical gluon model are used to determine the asymptotic behavior in the mass variable. This determination is made quantitative via the principles of precocious asymptopia and smooth threshold behavior. These techniques are used to calculate some corrections to pion pole dominance for vertex functions and scattering amplitudes in the weak PCAC (partially conserved axial-vector current) framework. Corrections to kaon pole dominance and to SU (3) symmetry are determined in the course of the analysis. Put differently, some measurable consequences of broken SU (3) ×{} SU (3) symmetry are determined. The canonical equal-time and light-cone operator algebraic structures are presumed to be exact, but the physical states (and hence on-shell) S-matrix elements) may exhibit considerable, but now controllable, deviations from the underlying SU (3) ×{} SU (3) symmetry. As explicit examples of the methods, the Kₗ₃ form factors f_±(q²) and the symmetric and antisymmetric πN scattering amplitudes are discussed in detail. The numerical results are: f₊(0)1.0, ξ(0)≡f_-(0)f₊(0)-0.7, f₊(mK²)+f_-(mK²)(0.7)×fKf_π, ${{α}}₈=[SU(3){-}breaking{{0ex}{0ex}}parameter]{}{-}140$ MeV, ${{σ}}N=({π}N {σ} term){~}450$ MeV, ${g}A{~}1.23$. The results for ${π}N$ scattering confirm and extend the previous weak PCAC analysis, based on Regge theory, but some of the ${K}ₗ₃$ results are now different. The good result ${ξ}(0){~}{-}0.7$ is, however, maintained and is now obtained without any assumption about the ${q}²$ dependence of ${f}_{±{}}$. None of the present results involved assumptions of smooth off-shell behaviors of Green's functions. In all cases the mass dispersion relations were used to compute the effects of off-shell extrapolations.