Chiral mass corrections to decuplet hidden-charm pentaquark states

We calculate the masses of the $J^P={\frac{1}{2}}^{-}$ hidden-charm pentaquark states in the $SU(3)$ decuplet within the heavy pentaquark chiral perturbation theory (HPChPT) up to the next-to-leading order (NLO). The one-loop self-energy corrections from the $\pi$, $K$, and $\eta$ mesons are evaluated. Taking $P_{\psi }^N(4312)$ and $P_{\psi s}^{\mathrm{\Lambda }}(4459)$ as inputs, we predict the masses of the doubly-strange and triply-strange decuplet members $P_{\psi ss}^N$ and $P_{\psi sss}^{\mathrm{\Lambda }}$ to be 4.607 GeV and 4.755 GeV, respectively. The mass splittings between adjacent strangeness layers are nearly constant at approximately 0.148 GeV. This equal-spacing pattern follows directly from the analytical structure of the NLO chiral corrections, while its actual magnitude is set by the low-energy constants (LECs). It indicates that the Gell–Mann–Okubo (GMO) relation, expected at the leading order of SU(3) flavor symmetry breaking, persists to good accuracy at the NLO.