Approximate Hamiltonian for baryons in heavy-flavor QCD

Building a method for describing gluons in hadrons in the Minkowski space-time, a pilot application of the renormalization group procedure for effective particles (RGPEP) to QCD of bottom and charm flavors is extended from quarkonia to baryons. Using the effective-particle basis in the Fock space, the bound-state eigenvalue problem for baryons is posed in terms of the Hamiltonian obtained by solving the RGPEP equations with accuracy to terms of second order in the expansion in powers of the coupling constant. The eigenvalue problem including the Fock components with effective gluons is reduced to the eigenvalue problem for the component of three effective quarks and no gluons. Namely, we use a hypothesis that all the components with gluons can be approximated by a component with one gluon that is massive, and we take this component into account using second-order perturbation theory. The effective three-quark Hamiltonian contains three quark-quark interaction terms, each of which consists of a Coulomb term with the Breit-Fermi spin couplings and a spin-independent harmonic oscillator term. As in quarkonia, the oscillator frequencies in baryons turn out to be not sensitive to the value of the gluon mass. The quark masses are adjusted at the corresponding scales to reproduce the masses of three lightest charm and bottom quarkonia of spin one. The dynamics in one-flavor baryons involves no free parameters and the resulting estimates for bbb and ccc baryon mass spectra match estimates obtained in quark models and lattice approach to QCD. Masses of ccb and bbc baryons are also estimated. All approximate baryon wave functions have simple oscillator forms. In ccb baryons, charm quarks tend to form diquarks. In order to identify the dynamics of effective gluons beyond the assumption of a mass term, the RGPEP needs to be applied in higher order than second and beyond the perturbative expansion.