A Bethe-Salpeter basis for meson and baryon spectra under harmonic confinement

Abstract

The Bethe-Salpeter equation forq−q andqqq systems derived in the preceding paper [8] in the instantaneous approximation are solved algebraically for harmonic confinement. The approximateq−q spectrum for all flavour is expressible asF(M)=N+3/2, where

$$\begin{gathered} F\left( M \right) = \left( {M^2 - 4m_q^2 } \right)\Omega _{_{\rm M} }^{ - 1} - \Omega _{\rm M} M^{ - 2} \gamma ^{ - 2} \hfill \\ \cdot \left( {2J \cdot S - 3 - Q_N } \right) + F_{QCD} \hfill \\ \end{gathered} $$

$\Omega _{\rm M} = 8(Mm)_q )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \tilde \omega \gamma$ is a mass-dependent FKR-like spring constant$\tilde \omega$(=0.15 GeV) a universal flavourindependent parameter, and γ(≈1) a slowly varying quantity. J·S represents the spin-dependent effects andQ _N, a quadratic function ofN, comprises some significant momentum dependent corrections, whileF _QCD is a small additional correction due to shortrange gluon exchange effects. An identical equation$\bar F$(M)=N+3 holds for non-strangeqqq excitations, with a very similar definition of$\bar F$(M) in terms of the same parameters$\tilde \omega$ andm _q. The calculated values ofF(M) and$\bar F$ (M) in terms of theobserved masses,M, andm _ud=0.28,m _s=0.35,m _c=1.40 (all in GeV), conform rather well to the principal features of the predictions, viz. (i) spin and flavour degeneracy ofq $\bar q$ supermultiplet members at theF(M) level, despite huge variations in their actual masses (e. g. P vs V); and likewise forqqq members (e.g.,N _L,Δ _L) at the$\bar F$ (M) level, and (ii) fulfilment of the unit spacing rule ΔF=1, Δ$\bar F$=1 for successive h.o. supermultiplets. TheP—V degeneracy at theF(M) level leads to the prediction ψ−η _c ≈100±20 MeV. Finally, theP→l $\bar l$ amplitudesf _π,k , as well as the principalV→e ⁺ e⁻ widths are fairly well reproduced without extra parameters.