Statistical error analysis for phenomenological nucleon-nucleon potentials

Nucleon-nucleon potentials are common in nuclear physics and are determined from a finite number of experimental data with limited precision sampling the scattering process. We study the statistical assumptions implicit in the standard least-squares ${\chi }^2$ fitting procedure and apply, along with more conventional tests, a tail-sensitive quantile-quantile test as a simple and confident tool to verify the normality of residuals. We show that the fulfillment of normality tests is linked to a judicious and consistent selection of a nucleon-nucleon database. These considerations prove crucial to a proper statistical error analysis and uncertainty propagation. We illustrate these issues by analyzing about 8000 proton-proton and neutron-proton scattering published data. This enables the construction of potentials meeting all statistical requirements necessary for statistical uncertainty estimates in nuclear structure calculations.

Figure 1

Abundance plots for $pp$- (top panel) and $np$- (bottom panel) scattering data. Full database (left panel). Standard $3\sigma$ criterion (middle panel). Self-consistent $3\sigma$ criterion (right panel). We show accepted data (blue), rejected data (red), and recovered data (green).

Figure 2

Normalized histogram of the resulting residuals after fitting the potential parameters to the complete $pp$ and $np$ database (blue boxes with solid borders) and to the $3\sigma$ consistent database (red boxes with dashed borders). The $N(0,1)$ standard normal probability distribution function (green solid line) is plotted for comparison.

Figure 3

Quantile-quantile plot of different random samples against the standard normal distribution. Blue crosses are sampled from the $N(0,1)$ distribution, red diagonal crosses from $N(0,1.5)$, green asterisks from $N(-1,1)$ and yellow squares from the exponential distribution $E\left(1.5\right)$.

Figure 4

Quantile-quantile plot of the residuals obtained from fitting the $3\sigma$ consistent database against the standard normal distribution. The deviations at the tails, which are not detected using the Kolmogorov-Smirnov test, are clearly visible with this graphical tool.

Figure 5

Rotated quantile-quantile plot of the residuals obtained (blue points) from fitting the complete database with the OPE-$\delta$-shell potential (upper left panel), the $3\sigma$ self-consistent database fitted with the OPE-$\delta$-shell potential (upper right panel), the $\chi$TPE-$\delta$-shell potential (lower left panel), and the OPE-Gaussian potential (lower right panel). $95\%$ confidence bands of the TS (red dashed lines) and KS (green dotted lines) tests are included.

Figure 6

Correlation matrix ${\mathcal{C}}_{ij}$ for the short distance parameters in the partial wave basis ${\left(V_i\right)}_{l,l^{\prime }}^{LSJ}$, see Eq. (18). We show the OPE-DS (upper panel) and the $\chi$TPE-DS (middle panel) potentials. The points $r_i=\Delta r_{\pi }(i+1)$ are grouped within every partial wave. The ordering of parameters is as in the parameter tables in Refs. [38, 39] and [48] for OPE-DS 46 parameters and the $\chi$TPE 30+3 parameters (the last three are the chiral constants $c_1,c_3,c_4$) respectively. The OPE-Gaussian case (lower panel) also contains the parameter $a$. We grade gradually from $100\%$ correlation, ${\mathcal{C}}_{ij}=1$ (red), $0\%$ correlation, ${\mathcal{C}}_{ij}=0$ (yellow) and $100\%$ anti-correlation, ${\mathcal{C}}_{ij}=-1$ (blue).

Figure 7

Lowest $np$ and $pp$ partial-wave potentials (in MeV) and their errors (solid band) as a function of the internucleon separation (in fm) for the present OPE+Gaussian analysis (blue band), Reid93 [15] (red dashed), NijmII [15] (green dotted), and AV18 [16] (light-blue dashed-dotted) as a function of the internucleon distance $r$ (in fm).

Figure 8

$\mathit{NN}$ potentials (in MeV) in the operator basis with errors (solid band) as a function of the internucleon separation (in fm) for the present OPE+Gaussian analysis (blue band), Reid93 [15] (red dashed), NijmII [15] (green dotted), and AV18 [16] (light-blue dashed-dotted) as a function of the internucleon distance $r$ (in fm).

Figure 9

Lowest $np$ and $pp$ phase shifts (in degrees) and their errors for the present OPE+Gaussian analysis (blue band), Reid93 [15] (red dashed), NijmII [15] (green dotted), and AV18 [16] (light-blue dashed-dotted) as a function of the LAB energy (in MeV).

Figure 10

$np$ (left) and $pp$ (right) Wolfenstein parameters (in fm) as a function of the center-of-mass angle (in degrees) and for $E_{\mathrm{LAB}}=50$ MeV. We compare our fit (blue band) with Reid93 [15] (red dashed), NijmII [15] (green dotted), and AV18 [16] (light-blue dashed-dotted).

Figure 11

Same as in Fig. 10 but for $E_{\mathrm{LAB}}=100$ MeV.

Figure 12

Same as in Fig. 10 but for $E_{\mathrm{LAB}}=200$ MeV.

Figure 13

Same as in Fig. 10 but for $E_{\mathrm{LAB}}=350$ MeV.