Positivity constraints on the pion–nucleon scattering amplitude are derived in this article with the help of general S-matrix arguments, such as analyticity, crossing symmetry, and unitarity, in the upper part of the Mandelstam triangle, R\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{R}$$\end{document}. Scanning inside the region R\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{R}$$\end{document}, the most stringent bounds on the chiral low-energy constants of the pion–nucleon Lagrangian are determined. When just considering the central values of the fit results from covariant baryon chiral perturbation theory using the extended-on-mass-shell scheme, it is found that these bounds are well respected numerically both at the O(p3)\documentclass[12pt]{minimal}
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\begin{document}$$O(p^3)$$\end{document} and the O(p4)\documentclass[12pt]{minimal}
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\begin{document}$$O(p^4)$$\end{document} level. Nevertheless, when taking the errors into account, only the O(p4)\documentclass[12pt]{minimal}
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\begin{document}$$O(p^4)$$\end{document} bounds are obeyed in the full error interval, while the bounds on the O(p3)\documentclass[12pt]{minimal}
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\begin{document}$$O(p^3)$$\end{document} fits are slightly violated. If one disregards the loop contributions, the bounds always fail in certain regions of R\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{R}$$\end{document}. Thus, at a given chiral order these terms are not numerically negligible and one needs to consider all possible contributions, i.e., both tree-level and loop diagrams.We have provided the constraints for special points in R\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}$$\end{document} where the bounds are nearly optimal in terms of just a few chiral couplings, which can easily be implemented and employed to constrain future analyses. Some issues concerned with calculations with an explicit Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} resonance are also discussed.
