Quantitative corrections to mass sum rules involving baryons containing heavy quarks

Quantitative corrections are estimated to three of Franklin's sum rules involving the masses of baryons containing at least one charmed quark and to three analogous sum rules for baryons containing at least one bottom quark. The corrections arise from three-body contributions to baryon interaction energies and are calculated from a semiempirical formula for the colormagnetic contributions to baryon masses. About 20 years ago, Franklin [1] derived sum rules relating the masses of baryons containing at least one charmed quark. Most of these baryon mass relations have not yet been tested by experimental measurements. Franklin restricted himself to ground-state baryons without radial or orbital excitations, and so do we in this work. The Franklin sum rules are based on the constituent quark model plus the assumption that the interaction energy of quarks in a baryon consists of only one-and two-body terms. However, according to quantum mechanics, the interaction energy should contain a three-body term even if the forces between quarks consist only of two-body terms [2–5]. These three-body terms in baryons give rise to corrections to Franklin's sum rules. It is the purpose of this Brief Report to estimate the magnitude of these corrections in certain selected cases. Franklin's sum rules fall into two categories: (i) formulas involving the mass splittings of baryon isospin multiplets, and (ii) formulas relating the masses of different isospin multiplets. In this work we neglect isospin breaking and consider only mass relations of type (ii). Furthermore, our method allows us to calculate corrections to only three of Franklin's formulas in category (ii). These three formulas involve mass differences among baryons with spin 3/2 and spin 1/2 with the same quark content. These mass differences arise from spin-dependent forces and, to a good approximation, are caused by the colormagnetic interaction of QCD [5]. We also obtain corrections to three analogous formulas in which c quarks are replaced by b quarks. For some baryons, the spin-dependent mass splittings are known from experiment [5]; in other cases the splittings can be estimated from a semiempirical mass formula [6–8] with parameters chosen so as to get a best fit to the observed splittings. In the case in which the splittings are known from experiment, the magnitudes of three-body interaction energies have been deduced [5]. The semiempirical mass formula incorporates some three-body effects as a built-in feature. As a consequence, we can use the semiempirical formula to get a good estimate …