An Investigation of the Low Equation and the Chew-Mandelstam Equations

Abstract

We show that for scalar theories without a cutoff the asymptotic form for large energies ω of the perturbation expansion of the Low equation in the one-meson approximation is a double power series in the coupling constant g² and lnω. The method applied by Gell-Mann and Low to the photon propagator in electrodynamics is used to show that if the crossing matrix has only one negative eigenvalue this power series reduces to a series in a single variable g = g₁ / 1 - g₁θ/π lnω where g₁ = g₁(g²), and θ is a constant. The series in y is evaluated for several interactions and for some crossing matrices with no physical interpretation; for the former the series is a simple algebraic function, while for the latter it usually diverges for all values of y ≠ o. We obtain the exact solution of the one-meson approximation for the symmetric scalar pion-nucleon interaction; it is a multiple-valued function of g². We compare perturbation approximation to the determinantal function of Baker and the cotangent of the phase shift, with the numerical solution of Salzman, for the symmetric pseudoscalar theory with a cutoff; they are found to be often accurate to a few percent. We show that Chew and Mandelstam's approximate equations for pion-pion scattering have no solution for positive coupling λ, and that the perturbation expansion of the solution of their equations for isotopic spin o pions diverges for λ ≠ o. For pions with I = 1 we present calculations of the perturbation expansion to sixth order in λ.