Positivity constraints on the low-energy constants of the chiral pion-nucleon Lagrangian

TL;DR

This work derives positivity constraints on the pion–nucleon scattering amplitude using fixed-$t$ dispersion relations and the upper Mandelstam region, translating them into bounds on chiral low-energy constants within covariant BχPT (EOMS scheme). It shows that loop contributions are essential for satisfying these bounds at $O(p^3)$ and $O(p^4)$, and analyzes the impact of the Delta(1232) resonance, finding that a full Delta-loop treatment is needed for reliable high-energy behavior. Numerically, the bounds are respected when loops are included and LEC uncertainties are accounted for at $O(p^4)$, while some $O(p^3)$ bounds can be violated within 1σ of the LECs. The results provide compact, implementable constraints at special kinematic points and offer guidance for future fits and sigma-term determinations.

Abstract

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 Mandelstam triangle, R. Scanning inside the region R, 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 extended-on-mass-shell scheme, it is found that these bounds are well respected numerically both at O(p^3) and O(p^4) level. Nevertheless, when taking the errors into account, only the O(p^4) bounds are obeyed in the full error interval, while the bounds on O(p^3) fits are slightly violated. If one disregards loop contributions, the bounds always fail in certain regions of R. 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 where the bounds are nearly optimal in terms of just a few chiral couplings, which can be easily implemented and employed to constrain future analyses. Some issues about calculations with an explicit Delta(1232) resonance are also discussed.

Figures (7)

• Figure 1: Mandelstam plane $(\nu,t)$. The Mandelstam triangle is the region contoured by the $s=(m_N+M_\pi)^2$, $u=(m_N+M_\pi)^2$ and $t=4 M_\pi^2$ lines. Our region of study $\mathcal{R}$ is the trapezium formed by the three previous lines and $t=0$, which is marked in red.
• Figure 2: Positivity bound on LECs at $O(p^3)$ level. The fit results from 'Fit I-$O(p^3)$' given in Ref. yao are employed for plotting $f(\alpha,\nu,t)$ at ${\cal O}(p^3)$. Left-hand side: only tree-level; right-hand-side: tree-level + loops. Similar results are obtained if one uses instead the 'WI08' results with $\slashed{\Delta}$-ChPT given in Ref. oller.
• Figure 3: Positivity bound on LECs at $O(p^4)$ level. The fit results from 'Fit I(a)-$O(p^4)$' given in Ref. yao are employed for plotting $f(\alpha,\nu,t)$ up to ${\cal O}(p^4)$ in EOMS-B$\chi$PT. Left-hand side: only tree-level; right-hand side: tree+loop
• Figure 4: Positivity bound on LECs at $O(p^4)$ level. The fit results from 'Fit I(b)-$O(p^4)$' given in Ref. yao are employed for plotting $f(\alpha,\nu,t)$ up to ${\cal O}(p^4)$ in EOMS-B$\chi$PT. Left-hand side: only tree-level; right-hand side: tree+loop
• Figure 5: Positivity bound on LECs at $O(p^3)$ level including the $\Delta(1232)$. The fit results from 'Fit II-$O(p^3)$' given in Ref. yao are employed for plotting $f(\alpha,\nu,t)$. Left-hand side: only tree-level; right-hand side: tree+loop. The analysis 'WI08' with $\Delta$-ChPT in Ref. oller yields a similar outcome.