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

Abstract

A large number of treatments of the meson spectrum have been tried that consider mesons as quark -anti quark bound states. Recently, we used relativistic quantum "constraint" mechanics to introduce a fully covariant treatment defined by two coupled Dirac equations. For field-theoretic interactions, this procedure functions as a "quantum mechanical transform of Bethe-Salpeter equation". Here, we test its spectral fits against those provided by an assortment of models: Wisconsin model, Iowa State model, Brayshaw model, and the popular semi-relativistic treatment of Godfrey and Isgur. We find that the fit provided by the two-body Dirac model for the entire meson spectrum competes with the best fits to partial spectra provided by the others and does so with the smallest number of interaction functions without additional cutoff parameters necessary to make other approaches numerically tractable. We discuss the distinguishing features of our model that may account for the relative overall success of its fits. Note especially that in our approach for QCD, the resulting pion mass and associated Goldstone behavior depend sensitively on the preservation of relativistic couplings that are crucial for its success when solved nonperturbatively for the analogous two-body bound-states of QED. * hcrater@utsi.edu

Tests of two-body Dirac equation wave functions in the decays of quarkonium and positronium into two photons

“…As seen in the table, only the fully coupled system of equations (lower-lower and upper-upper for the singlet, the same in addition to tensor coupling for the triplet) produces accurate results to the require precision. l s j n N c Perturbative (eV) Numerical (eV) Diff/µα 4 0 0 0 1 [7] is spoiled if the we ignore these couplings, leading to m π ∼ 850 MeV, m ρ ∼ 1060 MeV. The same relativistic structure in the constraint equations responsible for the success of the Sommerfeld-Balmer formula for positronium spin singlet states appears to be important for bringing the pion mass down to its observed value.…”

“…(2.24) and (2.25), the corresponding Schrödinger-like equations decouple [6], [34], [7]. We obtain [7] [p 2 + 2m 18) where the prime symbol stands for d/dr. We have used the abbreviations…”

“…They compare quite favorably with the results of Godfrey and Isgur [33], but with only two parametric functions (A, S) as opposed to the six or so used in their approach. In the table below we reproduce a portion of the entire spectrum given in [7]. The quark masses and potential parameters are given by m u ∼ 55 MeV , m c ∼ 1.…”

“…We reinterpret the static V AP covariantly by replacing the nonrelativistic r by x 2 ⊥ ≡ r, and parceling out the static potential V AP into the invariant functions A(r) and S(r) [7] as follows:…”

“…Part of the purpose of this review section is to outline some of the numerous tests made so far on the formalism. We give the Pauli forms of the Two-Body Dirac equations of constraint dynamics that we used in [7] to describe the entire meson spectrum (exceptions being light quark isoscalars such as the η, ω, η ′ and their orbital and radial excitations). We review those aspects of the formalism which give one confidence in the accurate accounts for all bound states from the excited states of bottomonium to the pion.…”