# Doubly Heavy Baryons Expanded in $1/m_Q$

## TL;DR

This paper develops a method to calculate the mass spectra and wave functions of doubly heavy baryons using a $1/m_Q$ expansion and a semirelativistic Hamiltonian, providing a foundation for further detailed studies.

## Contribution

It introduces a formulation based on the Foldy-Wouthuysen-Tani transformation to expand baryon properties in $1/m_Q$, linking relativistic and nonrelativistic approaches.

## Key findings

- Derived a $1/m_Q$ expansion for baryon masses and wave functions.
- Examined the $
ho$ mode wave function with nonrelativistic models.
- Proposed a comprehensive framework for doubly heavy baryon analysis.

## Abstract

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 $\lambda$ mode lowest-order equation is examined carefully to obtain a complete set of eigenfunctions. I claim that the $\rho$ 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.