Baryon masses at second order in large-N chiral perturbation theory.

Abstract

We consider flavor breaking in the octet and decuplet baryon masses at second order in large-$N$ chiral perturbation theory, where $N$ is the number of QCD colors. We assume that $\frac{1}{N}\ensuremath{\sim}\frac{1}{{N}_{F}}\ensuremath{\sim}\frac{{m}_{s}}{\ensuremath{\Lambda}}\ensuremath{\gg}\frac{{m}_{u,d}}{\ensuremath{\Lambda}}$, ${\ensuremath{\alpha}}_{\mathrm{EM}}$, where ${N}_{F}$ is the number of light quark flavors, and $\frac{{m}_{u,d,s}}{\ensuremath{\Lambda}}$ are the parameters controlling $\mathrm{SU}({N}_{F})$ flavor breaking in chiral perturbation theory. We consistently include nonanalytic contributions to the baryon masses at orders ${m}_{q}^{\frac{3}{2}}$, ${m}_{q}^{2}\mathrm{ln}{m}_{q}$, and $\frac{({m}_{q}\mathrm{ln}{m}_{q})}{N}$. The ${m}_{q}^{\frac{3}{2}}$ corrections are small for the relations that follow from $\mathrm{SU}({N}_{F})$ symmetry alone, but the corrections to the large-$N$ relations are large and have the wrong sign. Chiral power counting and large-$N$ consistency allow a two-loop contribution at order ${m}_{q}^{2}\mathrm{ln}{m}_{q}$, and a nontrivial explicit calculation is required to show that this contribution vanishes. At second order in the expansion, there are eight relations that are nontrivial consequences of the $\frac{1}{N}$ expansion, all of which are well satisfied within the experimental errors. The average deviation at this order is 7 MeV for the $\ensuremath{\Delta}I=0$ mass differences and 0.35 MeV for the $\ensuremath{\Delta}I\ensuremath{\ne}0$ mass differences, consistent with the expectation that the error is of order $\frac{1}{{N}^{2}}\ensuremath{\sim}10%$.