We study the latest N f = 2 + 1 + 1 and N f = 2 ETMC lattice QCD simulations of the nucleon masses and extract the pion-nucleon sigma term utilizing the Feynman-Hellmann theorem in SU(2) baryon chiral perturbation theory with the extended-on-mass-shell scheme. We find that the lattice QCD data can be described quite well already at the next-to-next-to-leading order. The overall picture remains essentially the same at the next-to-next-to-next-to-leading order. Our final result is σπN = 50.2(1.2)(2.0) MeV, or equivalently, f N u/d = 0.0535 (13)(21), where the first uncertainty is statistical and second is theoretical originated from chiral truncations, which is in agreement with that determined previously from the N f = 2 + 1 and N f = 2 lattice QCD data and that determined by the Cheng-Dashen theorem. In addition, we show that the inclusion of the virtual ∆(1232) does not change qualitatively our results.

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Inspired by the recent discovery of the doubly charmed tetraquark state $$T_{cc}^{+}$$ T cc + by the LHCb Collaboration, we perform a systematic study of masses and strong decays of open charm hexaquark states $${\Sigma }_{c}^{(*)}\Sigma _{c}^{(*)}$$ Σ c ( ∗ ) Σ c ( ∗ ) . Taking into account heavy quark spin symmetry breaking, we predict several bound states of isospin $$I=0$$ I = 0 , $$I=1$$ I = 1 , and $$I=2$$ I = 2 in the one boson exchange model. Moreover, we adopt the effective Lagrangian approach to estimate the decay widths of $${\Sigma }_{c}^{(*)}\Sigma _{c}^{(*)} \rightarrow \Lambda _{c}\Lambda _{c}$$ Σ c ( ∗ ) Σ c ( ∗ ) → Λ c Λ c and their relevant ratios via the triangle diagram mechanism, which range from a few MeV to a few tens of MeV. We strongly recommend future experimental searches for the $${\Sigma }_{c}^{(*)}\Sigma _{c}^{(*)}$$ Σ c ( ∗ ) Σ c ( ∗ ) hexaquark states in the $$\Lambda _c\Lambda _c$$ Λ c Λ c invariant mass distributions.

The three pentaquark states, P c (4312), P c (4440) and P c (4457), discovered by the LHCb Collaboration in 2019, are widely recognized as D( * ) Σ c hadronic molecules. Together with their four D( * ) Σ * c partners dictated by heavy quark spin symmetry they present a complete multiplet of hadronic molecules of D( * ) Σ ( * ) c . The pentaquark states were observed in the J/ψp invariant mass distributions of the Λ b → J/ψpK decay. It is widely recognized that to understand their nature, other discovery channels play an important role. In this work, we investigate two three-body decay modes of the D( * ) Σ ( * ) c molecules. The tree-level modes proceed via off-shell Σ ( * ) c baryons, D( * ) Σ ( * ) c → D( * ) Σ ( * ) c → Λ c π → D( * ) Λ c π, while the triangle-loop modes proceed through D * Σ ( * ) c → J/ψN π, η c N π via DΣ ( * ) c rescattering to J/ψN and η c N . Our results indicate that the decay widths of the P c (4457) and D( * ) Σ * c states into D( * ) Λ c π are several MeV, as a result can be observed in the upcoming Run 3 and Run 4 of LHC. The partial decay widths into D( * ) Λ c π of the P c (4312) and P c (4440) states range from tens to hundreds of keV. In addition, the partial decay widths of D * Σ c molecules into J/ψN π and η c N π are several keV and tens of keV, respectively, and the partial decay widths of D * Σ * c molecules into J/ψN π vary from several keV to tens of keV. In particular, we show that the spin-5/2 D * Σ * c state can be searched for in the J/ψN π and D * Λ c π invariant mass distributions, while the latter one is more favorable. These three-body decay modes of the pentaquark states are of great value to further observations of the pentaquark states and to a better understanding of their nature.

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