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 Schrodinger-like equations with energy-dependent quasipotentials @, . Their nonperturbative covariant structure [leading to perturbative and 0 ( 1 /c2) 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 a4.) 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 @, 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 /c2) field-theoretic results into the highly relativistic regime of bound light quarks.

Relativistic calculation of the meson spectrum: A fully covariant treatment versus standard treatments

“…In addition to obtaining these analytic forms for short and long distances they converted arbitrary. In earlier work [34] we divided the potential in the following way among three relativistic invariants V(r), S, and A. (In our former construction, the additional invariant V was responsible for a possible independent time-like vector interaction.…”

“…The extra structure is automatically inherited from relativistic classical [28], [31] and quantum mechanics [27]. In QED our approach amounts to a "quantum-mechanical transform" [32], [33]of the Bethe Salpeter equation provided by two coupled Dirac equations whose fully covariant interactions are determined by QED in the Feynman Gauge [34], [25]. These "Two-Body Dirac Equations" are legitimate quantum wave equations that can be solved directly [35], [25](without perturbation theory) whose numerical or analytic solutions automatically agree with results generated by ordinary perturbative treatment.…”

Baryon spectrum analysis using Dirac’s covariant constraint dynamics

“…We obtain the following approximation for the three-body eigenvalue equation, which comes from specializing the expanded three body version of Eq. (3.11) as follows: 16) in which the epsilons, representing the c.m. energy of each quark, are given to a good approximation by Eq.…”

“…Numerous three dimensional truncations of the Bethe-Salpeter equation have been proposed for the relativistic two-body problem [6,15]. Some of these types of approximate methods have previously been applied with considerable success to the qq meson spectrum [16]- [22], [1] and [23]- [28].…”

Relativistic correction of the quarkonium melting temperature with a holographic potential

“…To explore this correction, one has to go beyond the Schroedinger equation (2.1) and switch to the two body Dirac equation [21][22][23][24]:…”