Complete reduction of fermion-antifermion Bethe-Salpeter equation with static kernel. II

Abstract

Recent work with the Bethe-Salpeter (BS) equation for a spin- 1 2 fermion-antifermion system is developed further and extended in a number of ways. Static kernels are considered, as previously, but in the present case, the structure of the kernels serves to suppress completely all pair terms. The reduction of the 16 coupled integro-differential equations involving the BS bispinor wavefunctions can be carried out much more simply than before. The effective Hamiltonians of the eigenvalue equations which result for single scalar and vector wavefunctions are not only in formally closed operator form, correct to all orders in the coupling constant and v c , but have an explicit and simple structure. They depend at most linearly on the static potential, and also on powers of momentum and simple functions of the relativistic free-particle energies. Previous work is further generalized by the consideration of static four-vector interactions (vector gluons or photons) in an arbitrary “gauge” for equal mass particles and antiparticles on the one hand, and the fourth (scalar) component of a vector potential for unequal mass particle and antiparticle on the other hand, the latter in order to study the mass dependence of the effective interactions and of the wavefunctions. Spin-orbit, spin-spin, and tensor interactions are expected to arise from the Breit-like four-vector interactions in conventional v c expansions, but the present results are correct to all orders in v c . Furthermore, such types of interactions also arise, as previously, for the case of a Coulomb-like potential, for which they are not, at first guess, expected. In the unequal mass case, an interesting new type of effective tensor interaction is also generated which serves to mix triplet and singlet states. Various special “gauges” are considered in the equal mass case. The general structure of the BS amplitude is also discussed, and specifically developed in some detail in perturbation theory for the ground state of positronium. While the current work should be considered as a general examination of the properties of a large class of BS equations of the Coulomb-Dirac or Breit-type, two areas for specific applications immediately come to mind: the calculation of positronium energy levels to very high order and the study of quark-antiquark binding involving both like and unlike quarks.