Solving the two-body, bound-state Bethe–Salpeter equation

By expanding the solution of the the two-body, bound-state Bethe–Salpeter equation in terms of basis functions that obey the boundary conditions, solutions can be obtained to some, if not many, equations that have heretofore proved intractable. The utility of choosing such basis functions is demonstrated by calculating the zero-energy, bound-state solutions of a spin-0 boson and a spin-1/2 fermion with unequal masses that interact via scalar electrodynamics and are described by the Bethe–Salpeter equation in the ladder approximation. The equation is solved by first making a Wick rotation and then projecting four-dimensional Euclidean space onto the surface of a unit, five-dimensional sphere. Solutions are expanded in terms of basis functions, each of which obeys the boundary conditions and can be expressed in terms of hyperspherical harmonics in five-dimensional space. The Bethe–Salpeter equation is discretized by requiring that the coefficient of each hyperspherical harmonic vanish. All integrations are performed analytically, yielding a generalized matrix eigenvalue equation that is solved numerically. Although the Bethe–Salpeter equation is separable in the zero-energy limit, the feature of Bethe–Salpeter equations that often prevents solutions from being obtained numerically is still present in the equation that is solved.