Low-energy chiral two-pion exchange potential with statistical uncertainties

We present a new phenomenological nucleon-nucleon $\left(NN\right)$ chiral potential fitted to $925 pp$ and $1743 np$ scattering data selected from the Granada-$2013 NN$ database up to a laboratory energy of 125 MeV with 20 short-distance parameters and three chiral constants $c_1,c_3$, and $c_4$ with ${\chi }^2/\nu =1.02$. Special attention is given to testing the normality of the residuals which allows for a sound propagation of statistical errors from the experimental data to the potential parameters, phase shifts, scattering amplitudes, and counterterms. This fit allows for a new determination of the chiral constants $c_1,c_3$, and $c_4$ compatible with previous determinations from $NN$ data. This new interaction is found to be softer than other high-quality potentials by undertaking a Weinberg eigenvalue analysis. We further explore the interplay between the error analysis and the assumed form of the short-distance interaction. The present work shows that it is possible to fit $NN$ scattering with a two-pion exchange (TPE) chiral potential fulfilling all necessary statistical requirements up to 125 MeV and shows unequivocal nonvanishing $D$-wave short-distance pieces.

Figure 1

Correlation ellipses of the chiral constants $c_1,c_3$, and $c_4$ determined from a fit to the self-consistent database of [13] with a DS-$\chi \mathrm{TPE}$ potential including data with $T_{\mathrm{LAB}}\le 350$ [14] (blue solid lines) and $T_{\mathrm{LAB}}\le 125$ (red dashed lines). The concentric ellipses give, from the smallest to the largest one, the $68\%,95\%$, and $99\%$ confidence regions respectively. The points and crosses correspond to the determinations listed in Table VI of [14].

Figure 2

Rotated quantile-quantile plot for the residuals of the low energy DS-$\chi \mathrm{TPE}$ potential for $E_{\mathrm{LAB}}\le 125\mathrm{MeV}$ (blue crosses). The Kolmogorov-Smirnov (dotted green line) and tail sensitive (dashed red line) confidence bands with an $\alpha =0.05$ significance level are also included.

Figure 3

Trajectories (upper panel) and modulus (lower panel) of the largest repulsive Weinberg eigenvalue in the ${}^1{S}_0$ channel as a function of CM momentum $k$ for high-quality potentials. The potentials shown, ordered from largest to smallest modulus, are Reid93 [6], NijmII [6], AV18 [9], Gauss-OPE [15], DS-OPE [12, 13], DS-$\chi \mathrm{TPE}$ [14, 46], and the present work. We show the $\left|\eta \right|=1$ circle and line as a black solid line for reference.

Figure 4

Phase shifts for the low energy DS-$\chi \mathrm{TPE}$ potential fitted to experimental data with $T_{\mathrm{LAB}}\le 125\mathrm{MeV}$ with statistical error bands (blue bands). The corresponding phase shifts and error bands for the full DS-$\chi \mathrm{TPE}$ reproducing data with $T_{\mathrm{LAB}}\le 350$ MeV [14] are also included for comparison (red bands).

Figure 5

$np$ (left) and $pp$ (right) Wolfenstein parameters (in fm) as a function of the CM angle (in degrees) and for $T_{\mathrm{LAB}}=50\mathrm{MeV}$. We compare the low energy DS-$\chi \mathrm{TPE}$ potential (blue band) with full DS-$\chi \mathrm{TPE}$ potential [14] (red band), the Nijmegen PWA [5] (dotted, light blue line), and the AV18 potential [9] (dashed-dotted, green line).

Figure 6

Same as Fig. 5 but for $T_{\mathrm{LAB}}=100\mathrm{MeV}$.

Figure 7

Same as Fig. 5 but for $T_{\mathrm{LAB}}=200\mathrm{MeV}$.

Figure 8

Short-distance phase shifts with statistical error bands in degrees (blue bands) for the $\delta$-shell $\chi \mathrm{TPE}$-potential fitted up to a maximum $E_{\mathrm{LAB}}=125\mathrm{MeV}$ obtained by eliminating all contributions of the potential with $r\ge 1.8\mathrm{fm}$ and keeping just the delta-shells potential (see main text). The complete phase shifts (red bands) are drawn for comparison.

Figure 9

Rotated quantile-quantile plot for the residuals of the lo- energy optimized momentum space chiral potential of Ref. [34] (dark blue crosses). The scaled (yellow crosses) and standardized (light blue crosses) values are also shown. The Kolmogorov-Smirnov (dotted green line) and tail-sensitive (dashed red line) confidence bands with an $\alpha =0.05$ significance level are also included.

Figure 10

Values of ${\chi }^2/N$ for the local chiral potentials of Ref. [35] as a function of the maximum laboratory energy (in MeV). We distinguish the different contributions building their LO, NLO, and NNLO for $np$ and $pp$.