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Analyses of pion-nucleon elastic scattering amplitudes up to $O(p^4)$ in extended-on-mass-shell subtraction scheme

We extend the analysis of elastic pion-nucleon scattering up to $O(p^4)$ level using extended-on-mass-shell subtraction scheme within the framework of covariant baryon chiral perturbation theory. Numerical fits to partial wave phase shift data up to $\sqrt{s}=1.13$ GeV are performed to pin down the free low energy constants. A good description to the existing phase shift data is achieved. We find a good convergence for the chiral series at $O(p^4)$, considerably improved with respect to the $O(p^3)$-level analyses found in previous literature. Also, the leading order contribution from explicit $Δ(1232)$ resonance and partially-included $Δ(1232)$ loop contribution are included to describe phase shift data up to $\sqrt{s}=1.20$ GeV. As phenomenological applications, we investigate chiral correction to the Goldberger-Treiman relation %$Δ_{GT}$ and find that it converges rapidly, and the $O(p^3)$ correction is found to be very small: $\simeq 0.2%$. We also get a reasonable prediction of pion-nucleon sigma term $σ_{πN}$ up to $O(p^4)$ by performing fits including both the pion-nucleon partial wave phase shift data and the lattice QCD data. We report that $σ_{πN}=52\pm7$ MeV from the fit without $Δ(1232)$, and $σ_{πN}=45\pm6$ MeV from the fit with explicit $Δ(1232)$.