Unique Determination of Three Proton-Proton Scattering Parameters at 9.69 MeV

The well-established fact that the spin-orbit interaction is of short range compared to one-pion-exchange (OPE) interaction causes the phase-shift combination ${\Delta }_{\mathrm{LS}}=\frac{(-2\mathrm{}{\delta }_{1,0}-3\mathrm{}{\delta }_{1,1}+5\mathrm{}{\delta }_{1,2})}{12}$ to be small compared to ${\Delta }_T=\frac{5(2{\delta }_{1,0}-3\mathrm{}{\delta }_{1,1}+{\delta }_{1,2})}{12}$ at low energy. We find that this makes it impossible to carry through an unambiguous phase-shift analysis of existing data near 10 MeV for ${\delta }_0$ and ${\delta }_{1,J}$, but it also makes it possible to show that the ratio $\frac{{\Delta }_{\mathrm{LS}}}{{\Delta }_T}$ must lie between 0.07 and 0.15 at 9.69 MeV. Using this result, we can then uniquely determine ${{\delta }_0}^E$, ${\Delta }_T$, and ${\Delta }_C=\frac{({\delta }_{1,0}+3\mathrm{}{\delta }_{1,1}+5\mathrm{}{\delta }_{1,2})}{9}$ and obtain ${{\delta }_0}^E=55.69\mathrm{}°\pm 0.28°$, ${\Delta }_T=0.91\mathrm{}°\pm 0.28°$, ${\Delta }_C=-0.020\mathrm{}°\pm 0.029°$ with relative correlations $\mathrm{}\left(0T\right)=-0.96\mathrm{}$, $\mathrm{}\left(0C\right)=0.47\mathrm{}$, $\mathrm{}\left(\mathrm{TC}\right)=-0.30\mathrm{}$. The result is stable against variation of $\frac{{\Delta }_{\mathrm{LS}}}{{\Delta }_T}$ over the full physically allowable range, against the extrapolation needed to include $\left[\frac{A_{\mathrm{yy}}}{A_{\mathrm{xx}}}\right]\left(90\mathrm{}°\right)$ at 11.4 MeV in the data set, against vacuum polarization corrections, and against whether or not phases with $J>\mathrm{}2\mathrm{}$, ${}_{}{}^3{}_{}{}^{}F_2^{}{}_{}{}^{}$, and (marginally) ${\epsilon }_2$ are dropped or given their OPE values. The data require