A parametrization of the baryon octet and decuplet masses

We construct a general parametrization of the baryon octet and decuplet mass operators including the three-body terms using the unit operator and the symmetry-breaking factors $M^d=\text{diag}(0,1,0)$ and $M^s=$\text{diag}$ (0,0,1)$ in conjunction with the spin operators. Our parametrization has the minimal number of operators needed to describe all the octet and decuplet masses. Investigating the likely size of the three-body terms, we find that contributions of the three-body hypercharge splittings are comparable to those from the one- and two-body isospin splittings and that contributions of the three-body isospin splitting operators are very small. We prove that, in dynamical calculations, one must go to three loops to get the three-body terms. We also find that the suggested hierarchy of sizes for terms in the most general expression for baryon masses that involve multiple factors of $M^d$ and/or $M^s$ does not hold strictly for dynamical calculations in heavy baryon chiral perturbation theory: terms of a given order in a meson loop expansion may appear both with the expected factors of $M^d$ and $M^s$, and with one factor more.