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

TL;DR

This paper calculates the mass spectra and wave functions of the $(bcq)$ baryons using the relativistic Bethe-Salpeter equation, considering diquark mixing effects, and finds that mixing is minimal, supporting the diquark model assumptions.

Contribution

It introduces a detailed relativistic calculation of $(bcq)$ baryons incorporating diquark mixing effects, providing new insights into their structure and mass spectra.

Findings

Mixing effects between $ ext{Xi}_{bc}$ and $ ext{Xi}_{bc}'$ are small (~1%).

The diquark model effectively describes the baryon states with definite diquark spins.

Numerical solutions yield specific mass spectra consistent with theoretical expectations.

Abstract

Mass spectra and wave functions of the $J^P=\frac{1}{2}^+$ $(bcq)$ baryons are calculated by the relativistic Bethe-Salpeter equation\,(BSE) with considering the mixing effects between the $1^+$ and $0^+$ $(bc)$-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 $0^+$ 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 $0^+$ and $1^+$ diquarks. The general wave function of the $\frac{1}{2}^+$ $(bcq)$ baryons is constructed and solved to obtain the corresponding mass spectra. Especially, by using the obtained wave functions, the mixing effects between $\Xi_{bc}(\Omega_{bc})$ and $\Xi_{bc}'(\Omega'_{bc})$ in ground states are computed and determined to be small ($\sim \!1\%$). The numerical results indicate that it is a good choice to take $\Xi_{bc}$ and $\Xi'_{bc}$ as the baryon states with the inside $(bc)$-diquarks occupying the definite spin.