Partial-wave correlations with uncertainties in the two-nucleon system

The nucleon-nucleon scattering problem is usually analyzed in terms of partial waves and the corresponding coupled-channel phase shifts and mixing angles, but the available experiments induce correlations among the corresponding channels with different quantum numbers. Based on the Granada 2013 database we analyze the meaning and impact of those correlations, taking into account both the purely statistical ones reflecting the primary experimental data uncertainties as well as the systematic ones exhibiting the ambiguities in the form of the potential representing the unknown nuclear force for distances below 3 fm. We find that the combined uncertainties not only display a dominance of systematic over statistical effects, but also show that these correlations are almost compatible with zero. These findings support the frequent practice of determining potentials from separated channel-by-channel direct fits to phase shifts with the combined systematic and statistical effects without the full-fledged partial wave analysis inferred from experimental data but only with much larger uncertainties.

Figure 1

Theoretical distribution function of the correlation coefficient $r$ for a sample of size $N=13$ generated from a bivariate Gaussian population with theoretical correlation coefficients (from left to right) $\rho =-0.75,-0.5,-0.25,0,0.25,0.50,0.75$.

Figure 2

Estimated confidence limits for the correlation $\rho$ as a function of the empirical correlation coefficient of a sample of size $N=13$ for one, two, and three standard deviations corresponding to 68%, 95%, and 99% confidence levels, respectively.

Figure 3

Partial wave correlation coefficients $-1\le r\le 1$ with confidence level as a function of the nucleon laboratory energies for $0\le T_{\mathrm{laboratory}}\le 350$ MeV. The larger (light blue) band corresponds to the systematic uncertainty at the $2\sigma$ confidence level. The medium band (purple) corresponds to the systematic uncertainty at the $1\sigma$ confidence level. The smaller (orange) band corresponds to the statistical uncertainty at the $1\sigma$ confidence level as described in the text. Top: Central phases with total angular momentum $J\le 1$. Middle: Peripheral phases with $J\ge 2$. Bottom: Central-peripheral correlations. We show the results for the 13 HQ potentials quoted in the text (left) and the 6 Granada potentials (right).