Mass spectra, wave functions and mixing effects of the (bcq) baryons

Mass spectra and wave functions of the $$J^P=\frac{1}{2}^+$$ J P = 1 2 + ( bcq ) baryons are calculated by the relativistic Bethe–Salpeter equation (BSE) with considering the mixing effects between the $$1^+$$ 1 + and $$0^+$$ 0 + ( bc )-diquarks inside. Based on the diquark picture, the three-body problem of baryons is transformed into two two-body problems. The BSE and wave functions of the $$0^+$$ 0 + diquark are given, and then solved numerically to obtain the effective mass spectra and form factors. Also we present the wave functions at zero point for the ( bc )-diquark. Considering the obtained diquark form factors, the ( bcq ) baryons are then described by the BSE as the bound state of a diquark and a light quark, where the interaction kernel includes the inner transitions between the $$0^+$$ 0 + and $$1^+$$ 1 + diquarks. The general wave function of the $$\frac{1}{2}^+$$ 1 2 + ( bcq ) baryons is constructed and solved to obtain the corresponding mass spectra. Especially, by using the obtained wave functions, the mixing effects between $$\Xi _{bc}(\Omega _{bc})$$ Ξ bc ( Ω bc ) and $$\Xi _{bc}'(\Omega '_{bc})$$ Ξ bc ′ ( Ω bc ′ ) in ground states are computed and determined to be small ( $$\sim \!1\%$$ ∼ 1 % ). The numerical results indicate that it is a good choice to take $$\Xi _{bc}$$ Ξ bc and $$\Xi '_{bc}$$ Ξ bc ′ as the baryon states with the inside ( bc )-diquarks occupying the definite spin.