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We construct a general parametrization of the baryon octet and decuplet massoperators including the three-body terms using the unit operator and thesymmetry-breaking factors $M^d=\textrm{diag} (0,1,0)$ and $M^s=\textrm{diag}(0,0,1)$ in conjunction with the spin operators. Our parametrization has theminimal number of operators needed to describe all the octet and decupletmasses. Investigating the likely size of the three-body terms, we find thatcontributions of the three-body hypercharge splittings are comparable to thosefrom the one- and two-body isospin splittings and that contributions of thethree-body isospin splitting operators are very small. We prove that, indynamical 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 mostgeneral expression for baryon masses that involve multiple factors of $M^d$and/or $M^s$ does not hold strictly for dynamical calculations in heavy baryonchiral perturbation theory: terms of a given order in a meson loop expansionmay appear both with the expected factors of $M^d$ and $M^s$, and with onefactor more.

A parametrization of the baryon octet and decuplet masses