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Doubly Heavy Baryons Expanded in $1/m_Q$

Starting from the semirelativistic Hamiltonian for a doubly heavy baryon system ($QQq$) with Coulomb and linear confining scalar potentials, and operating the naive Foldy-Wouthuysen-Tani transformation on two heavy quarks, I construct a formulation how to calculate mass spectra and wave functions of doubly heavy baryons. Based on our formulation, their masses and wave functions are expanded in $1/m_Q$ with a heavy quark mass $m_Q$ and the $λ$ mode lowest-order equation is examined carefully to obtain a complete set of eigenfunctions. I claim that the $ρ$ mode wave function should be given by other nonrelativistic methods, e.g., the Godfrey-Isgur model to give a complete wave function for a doubly heavy baryon.