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

Abstract

The Bethe-Salpeter equation for qq̅ and qqq systems derived in the preceding paper [8] in the instantaneous approximation are solved algebraically for harmonic confinement. The approximate qq̅ spectrum for all flavour is expressible as F(M) =N+3/2, where F( M ) = ( M² - 4m_q²)Ω_M^-1 - Ω_MM^-2γ^-2·(2J.S - 3 - Q_N) + F_QCDΩ_M=8(M m)_q)^½ω˜γ is a mass-dependent FKR-like spring constant ω˜(=0.15 GeV) a universal flavour independent parameter, and γ(≈ 1) a slowly carying quantity. J.S represents the spin-dependent effects and Q_N, a quadratic function of N, comprises some significant momentum dependent corrections, while F_QCD is a small additional correction due to shortrange gluon exchange effects. An identical equation F˜,(M) = N + 3 holds for non-strange qqq excitations, with a very similar definition of F˜(M) in terms of the same parameters ω˜ and m_q. The calculated values of F(M) and F˜(M) in terms of the observed masses M, and m_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 of qq̅ supermultiplet members at the F(M) level, despite huge variations in their actual masses (e.g., P vs V); and likewise for qqq members (e.g.,N_L,Δ_L) at the F˜(M) level, and (ii) fulfilment of the unit spacing rule ΔF =1, ΔF˜ = 1 for successive h.o. supermultiplets. The P-V degeneracy at the F(M) level leads to the prediction ψ - η_c ≈ 100 ± 20 MeV. Finally, the P → ll amplitudes ƒ_Π,k' as well as the principal → e⁺ e^– widths are fairly well reproduced without extra parameters.