The N-quantum approximation in the Bronzan–Lee extended-source static model

Abstract

The Bronzan–Lee model of a soluble field theory with vertex function is analyzed from the viewpoint of the N-quantum approximation. Determination of the Heisenberg field expansions in the single-source one- and two-meson sectors of the model yields three scattering amplitudes, a production amplitude, and the renormalization constants in agreement with other derivations. The solutions to one algebraic and three linear integral equations, the mathematical complexity of which is reminiscent of the Tamm–Dancoff eigenvalue approach in the Lee model with two nontrivial sources, are required for this purpose. Finite results compatible with the conventional renormalization program are obtained from the field expansions and the equations of motion by prescribing that each normal-ordered product of in-fields in an expansion have the same quantum numbers as its Heisenberg field and, so that all relevant terms are included for a sector, all possible combinations of in-fields consistent with the quantum numbers of the sector must occur. In general, this recipe is substituted in place of power counting, which evidently is inappropriate in higher sectors of the Lee model. The investigation of one- and two-meson exchange interactions of two sources in the Lee model via the N-quantum approximation has been carried out, and similar work is contemplated in charged scalar theory and the classic Chew–Low model.