An investigation of the Low equation and the Chew-Mandelstam equations

[摘要] NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.We show that for scalar theories without a cutoff the asymptotic form for large energies [omega] of the perturbation expansion of the Low equation in the one-meson approximation is a double power series in the coupling constant g[superscript 2]and [...]. 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 [...] where [...], and [...] 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 [not equal to] 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 g[superscript 2]. 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 [lamda], and that the perturbation expansion of the solution of their equations for isotopic spin o pions diverges for [lamda] [not equal to] o. For pions with I = 1 we present calculations of the perturbation expansion to sixth order in [lamda].

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Physics