Baryon Mass Splittings in Chiral Perturbation Theory

Baryon masses are calculated in chiral perturbation theory at the one–loop-$\mathcal{O}(p^3)$ 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-$\mathcal{O}(p^3)$ level. We point out that these corrections in $1\mathrm{/}{M}_B$ generate new divergences $\propto m^4/M_{10}$. These divergences together with the divergences occuring in one-loop-$\mathcal{O}(p^4)$ 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-$\mathcal{O}(p^4)$ level. We point out another serious problem of going to the one-loop-${\cal O}(p^4)$level\mathrm{.}Whenthedecupletisoffitsmass--shellthereareadditional$\pi N\Delta$and$\pi\Delta\Delta$ interaction terms. These interactions contribute not only to the divergent terms $\propto (m^4\mathrm{/}{M}_{10})$, but also to nonanalytic terms such as $\propto (m^4\mathrm{/}{M}_{10})\mathrm{l}\mathrm{n}(m\mathrm{/}{M}_{10})$. Thus without a knowledge of the coupling constants appearing in these interactions one cannot carry out a consistent one-loop-$\mathcal{O}(p^4)$ level calculation.