Two-body Dirac equations for meson spectroscopy.

Abstract

Recently we used Dirac's constraint mechanics and supersymmetries to derive two coupled compatible 16-component Dirac equations that govern two relativistic spinning particles interacting through world scalar and vector potentials. They reduce exactly to four decoupled four-component local Schr\odinger-like equations with energy-dependent quasipotentials ${\ensuremath{\Phi}}_{w}$. Their nonperturbative covariant structure [leading to perturbative and O(1/${c}^{2}$) expansions that agree with field-theoretic approaches] suit these equations ideally for phenomenological applications in which the potentials have some links with relativistic field theories. (These equations are exactly solvable for singlet positronium producing a spectrum correct through order ${\ensuremath{\alpha}}^{4}$.) Here we use our equations to extend the validity of various one- or two-parameter models for the heavy-quark static potential to the relativistic light-quark regime. These models include the leading-log model (for all length scales) of Adler and Piran and Richardson's potential modified by flavor-dependent vacuum corrections.They significantly improve the good results that we obtained using Richardson's potential alone. Both nonperturbative and perturbative properties of the constraint approach are responsible for the spin-dependent consequences of the potential that result in a good overall fit to the meson masses. The nonperturbative structure dictated by the compatibility of our two Dirac equations enforces an approximate chiral symmetry that may account for the goodness of our pion fit. Perturbatively, for weak potentials, the upper-upper components of our equations reduce to the appropriate Todorov equation and then for low velocities to the Breit Hamiltonian. Thus, our approach reproduces the semirelativistic spin-dependent consequences of a quantum field theory. We strengthen this connection by deriving the Todorov inhomogeneous quasipotential equation for ${\ensuremath{\Phi}}_{w}$ from the Bethe-Salpeter equation using an operator generalization of Sazdjian's quantum-mechanical transform of the Bethe-Salpeter equation. Consequently our covariant compatible coupled Dirac equations provide a nonperturbative framework for extrapolating O(1/${c}^{2}$) field-theoretic results into the highly relativistic regime of bound light quarks.