Baryon mass splittings in chiral perturbation theory

Baryon masses are calculated in chiral perturbation theory at the one-loop O(${\mathit{p}}^3$) level in chiral expansion and to leading order in the heavy baryon expansion. Ultraviolet divergences occur requiring the introduction of counterterms. Despite this necessity, no knowledge of the counterterms is required to determine the violations of the Gell-Mann–Okubo mass relation for the baryon octet or of the decuplet equal-mass-spacing rule, as all divergences cancel exactly at this order. For the same reason all references to an arbitrary scale μ are absent. Neither of these features continue to higher powers in the chiral expansion. We also discuss critically the absolute necessity of simultaneously going beyond the leading-order heavy baryon expansion, if one goes beyond the one-loop O(${\mathit{p}}^3$) level. We point out that these corrections in 1/${\mathit{M}}_{\mathit{B}}$ generate new divergences ∝${\mathit{m}}^4$/${\mathit{M}}_{10}$. These divergences together with the divergences occurring in one-loop O(${\mathit{p}}^4$) graphs of chiral perturbation theory are taken care of by the same set of counterterms. Because of these unknown counterterms one cannot predict the baryon mass splittings at the one-loop O(${\mathit{p}}^4$) level even if the parameters of all ${\mathit{scrL}}_1^{\mathrm{\pi }\mathit{N}}$ terms are known. We point out another serious problem of going to the one-loop O(${\mathit{p}}^4$) level. When the decuplet is off its mass shell there are additional πNΔ and πΔΔ interaction terms. These interactions contribute not only to the divergent terms ∝(${\mathit{m}}^4$/${\mathit{M}}_{10}$), but also to nonanalytic terms such as ∝(${\mathit{m}}^4$/${\mathit{M}}_{10}$)ln(m/${\mathit{M}}_{10}$). Thus without knowledge of the coupling constants appearing in these interactions one cannot carry out a consistent one-loop O(${\mathit{p}}^4$) level calculation.