Precise determination of charge-dependent pion-nucleon-nucleon coupling constants

We undertake a covariance error analysis of the pion-nucleon-nucleon coupling constants from the Granada-2013 $np$ and $pp$ database, comprising a total of 6720 scattering data below laboratory energy of 350 MeV. Assuming a unique pion-nucleon coupling constant in the one-pion exchange potential above a boundary radius $r_c=3\mathrm{fm}$ we obtain $f^2=0.0763\left(1\right)$. The effects of charge symmetry breaking on the ${}^{3}P_0$, ${}^{3}P_1$, and ${}^{3}P_2$ partial waves are analyzed and we find $f_p^2=0.0761\left(4\right)$, $f_0^2=0.0790\left(9\right)$, and $f_c^2=0.0772\left(6\right)$ with a strong anticorrelation between $f_c^2$ and $f_0^2$. We successfully test normality for the residuals of the fit. Potential tails in terms of different boundary radii as well as chiral two-pion-exchange contributions as sources of systematic uncertainty are also investigated.

Figure 1

Phase shifts obtained from a partial-waves analysis to $pp$ and $np$ data and statistical uncertainties. Blue band from Ref. [34] and red band from a fit with charge symmetry breaking on the ${}^{3}P_0$, ${}^{3}P_1$, and ${}^{3}P_2$ partial waves.

Figure 2

Correlation ellipses and scatter diagrams for the coupling constants $f_c^2$, $f_p^2$, and $f_0^2$ appearing in the OPE potential from a PWA with (yellow line and blue dots) and without (green line and red diamonds) charge independence on the $P$ waves and a $3\sigma$ consistent database. The black diagonal line indicates $f_c^2=f_p^2=f_0^2$.

Figure 3

Rotated quantile-quantile plots for the fits introduced in this work. All points should be inside the confidence band to state that residuals of the fit follow a normal distribution $N(0,1)$, in which case the fit is self-consistent a posteriori. Left panel, assuming a charge-independent pion-nucleon constant used as a fitting parameter and charge symmetry breaking only on the ${}^{1}S_0$ partial wave parameters. Central panel, assuming three different charge-dependent pion-nucleon constants used as a fitting parameters and charge symmetry breaking only on the ${}^{1}S_0$ partial wave parameters. Right panel, assuming three different charge-dependent pion-nucleon constants used as a fitting parameters and charge symmetry breaking on the ${}^{1}S_0$ and $P$ partial wave parameters

Figure 4

Relative error in $f^2$ as a function of the number of data ordered according to a decreasing Hessian value.

Figure 5

Relative error in $f^2$ as a function of the number of data ordered according to their date of publication. Every point represents the number of $np+pp$ scattering data extracted from the Granada-2013 NN database and starting in 1960 forward in 5-year steps. We also show the statistical estimate $\epsilon \left(f^2\right)=1/\sqrt{N_{\mathrm{Par}}}$ (black, dashed) and the fitting $\epsilon \left(f^2\right)=30/N_{\mathrm{Par}}$ (red, solid).

Figure 6

Proton-proton and neutron-neutron interaction above 3 fm due to exchange of a neutral pion for different spin polarization states. The bands correspond to the statistical uncertainties from a fit to 6713 $np+pp$ scattering data below $T_{\mathrm{LAB}}=350\mathrm{MeV}$ with ${\chi }^2/\nu =1.025$.