Baryon Mass Splittings in Chiral Perturbation Theory

Baryon masses are calculated in chiral perturbation theory at the one--loop-${\cal O}(p^3)$ level in the chiral expansion and to leading order in the heavy baryon expansion. Ultraviolet divergences occur requiring the introduction of counter--terms. Despite this neccessity, no knowledge of the counter terms is required to determine the violations to the Gell--Mann Okubo mass relation for the baryon octet or to the decuplet equal mass--spacing rule, as all divergences cancel {\it exactly} at this order. For the same reason all reference to an arbitrary scale $\mu$ is absent. Neither of these features continue to higher--powers in the chiral expansion. We also discuss critically the absolute neccessity of simultaneously going beyond the leading order heavy baryon expansion, if one goes beyond the one-loop-${\cal O}(p^3)$ level. We point out that these corrections in $1/M_B$ generate new divergences $\propto m^4/M_{10}$. These divergences together with the divergences occuring in one-loop-${\cal O}(p^4)$ graphs of chiral perturbation theory are taken care of by the same set of counter--terms. Because of these unknown counter--terms one cannot predict the baryon mass splittings at the one-loop-${\cal O}(p^4)$ level. We point out another serious problem of going to the one-loop-${\cal O}(p^4)$ level. When the decuplet is off its mass--shell there are additional $\pi N\Delta$ and $\pi\Delta\Delta$ interaction terms. These interactions contribute not only to the divergent terms $\propto (m^4/M_{10})$, but also to nonanalytic terms such as $\propto (m^4/M_{10}){\rm ln}(m/M_{10})$. Thus without a knowledge of the coupling constants appearing in these interactions one cannot carry out a consistent one-loop-${\cal O}(p^4)$ level calculation.