Baryon $\sigma$ terms in SU(3) BChPT x 1/Nc

ChPT and the $1/N_c$ expansion provide systematic frameworks for the strong interactions at low energy. A combined framework of both expansions has been developed and applied for baryons with three light-quark-flavors. The small scale expansion of the combined approach is identified as the $\xi$-expansion, in which the power counting of the expansions is linked according to $O(p)=O(1/N_c)=O(\xi)$. The physical baryon masses as well as lattice QCD baryon masses for different quark mass masses are analyzed to $O(\xi^3)$ in that framework. $\sigma$ terms are addressed using the Feynman Hellmann theorem. For the nucleon, a useful connection between the deviation of the Gell-Mann-Okubo relation and the $\sigma$ term $\sigma_{8N}$ associated with the scalar density $\bar u u+\bar d d-2\bar s s$ is identified. In particular, the deviation from the tree level relation $\sigma_{8N}=\frac 13(2 m_N-m_\Sigma-m_\Xi)$, which gives rise to the so called $\sigma$-term puzzle, is studied in the $\xi$-expansion. A large correction non-analytic in $\xi$ results for that relation, making plausible the resolution of the puzzle. Issues with the determination of the strangeness $\sigma$ terms are discussed, emphasizing the need for lattice calculations at smaller $m_s$ for better understanding the range of validity of the effective theory. The analysis presented here leads to $\sigma_{\pi N}=69(10)$~MeV and $\sigma_{\pi \Delta}=60(10)$~MeV.