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Roy-Steiner-equation analysis of pion-nucleon scattering

We review the structure of Roy-Steiner equations for pion-nucleon scattering, the solution for the partial waves of the t-channel process $ππ\to \bar N N$, as well as the high-accuracy extraction of the pion-nucleon S-wave scattering lengths from data on pionic hydrogen and deuterium. We then proceed to construct solutions for the lowest partial waves of the s-channel process $πN\to πN$ and demonstrate that accurate solutions can be found if the scattering lengths are imposed as constraints. Detailed error estimates of all input quantities in the solution procedure are performed and explicit parameterizations for the resulting low-energy phase shifts as well as results for subthreshold parameters and higher threshold parameters are presented. Furthermore, we discuss the extraction of the pion-nucleon $σ$-term via the Cheng-Dashen low-energy theorem, including the role of isospin-breaking corrections, to obtain a precision determination consistent with all constraints from analyticity, unitarity, crossing symmetry, and pionic-atom data. We perform the matching to chiral perturbation theory in the subthreshold region and detail the consequences for the chiral convergence of the threshold parameters and the nucleon mass.