Accidental degeneracy in the Bethe-Salpeter equation

Abstract

The Bethe-Salpeter equation for the bound state of the pion-nucleon system has been studied in the ladder approximation; the propagation time of the exchanged nucleon is neglected. By using the two-component formalism, the spinor equation is first reduced to a pair of simultaneous integral equations in momentum space. Following Fock, we transform these equations into ones in a four-dimensional hyperspace and the solutions are obtained in terms of series of O(4) harmonics. As a simple illustration of our method, we have also considered the Bethe-Salpeter equation for the scalar-meson system. We find that the pion-nucleon Bethe-Salpeter equation shows an accidental degeneracy in the discrete-energy spectra similar to that in the solutions of the Dirac equation for the hydrogen atom, provided the coupling constant does not exceed a certain critical limit. The scalar problem exhibits at small binding energies a Schrodinger-type degeneracy. Convergence criteria for the pion-nucleon Bethe-Salpeter eigensolutions as series in O(4) harmonics have been discussed; it is found that no solution exists when the coupling constant exceeds a certain critical value. In our approximation scheme, there are no abnormal solutions as are encountered in the fully covariant treatment of the equation.