Bound-state,Bethe—Salpeter solutions with conjugate pairs of complex coupling constants

Solutions are obtained to the Bethe-Salpeter equation describing bound states of two massive scalars interacting via the exchange of a third, massive scalar. Covariance of the equation implies that the interaction is retarded, and in part because the energy appears more than once in the equation, a Hamiltonian for the bound state does not exist. Thus in contrast to the Schrodinger equation, the Bethe-Salpeter equation is solved by specifying the energy and solving for the coupling constant as an eigenvalue. Although the Bethe-Salpeter equation is derived from a Lagrangian with real coupling constants, depending on the value of the energy and the masses of the scalars, some values of the coupling constant that satisfy the Bethe-Salpeter equation are complex and always occur in conjugate pairs. The unexpected existence of solutions with real energy and a complex coupling constant raises the possibility that there are also resonance solutions with real values of the coupling constant and complex energy. Supported by a grant from the Ohio Supercomputer Center. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.