Bethe-Salpeter equation for heavy baryonsωQ(*)in the diquark picture

In the heavy quark limit, the heavy baryons ${\omega }_Q^{\left(*\right)}$ $(\omega$ stands for $\Sigma ,$ $\Xi$ or $\Omega$ and $Q=b$ or $c)$ are regarded as composed of a heavy quark and an axial vector, light diquark with good spin and isospin quantum numbers. Based on this diquark picture we establish the Bethe-Salpeter (BS) equation for ${\omega }_Q^{\left(*\right)}$ in the limit where the heavy quark has infinite mass, $m_Q\to \infty$. It is found that in this limit there are three components in the BS wave function for ${\omega }_Q^{\left(*\right)}$. Assuming the kernel to consist of a scalar confinement term and a one-gluon-exchange term we derive three coupled integral equations for the three BS scalar functions in the covariant instantaneous approximation. Numerical solutions for the three BS scalar functions are presented, including a discussion of their dependence on the various input parameters. These solutions are applied to calculate the Isgur-Wise functions $\xi \left(\omega \right)$ and $\zeta \left(\omega \right)$ for the weak transitions ${\Omega }_b^{\left(*\right)}\to {\Omega }_c^{\left(*\right)}$. Using these we give predictions for the Cabibbo-allowed nonleptonic decay widths and up-down asymmetries for ${\Omega }_b\to {\Omega }_c^{\left(*\right)}$ plus a pseudoscalar or vector meson.