Coarse graining the Bethe–Goldstone equation: Nucleon–nucleon high-momentum components

The $\delta$-shell representation of the nuclear force allows a simplified treatment of nuclear correlations. We show how this applies to the Bethe–Goldstone equation as an integral equation in coordinate space with a few mesh points, which is solved by inversion of a five-dimensional square matrix in the single channel cases and a $10\times 10$ matrix for the tensor-coupled channels. This allows us to readily obtain the high-momentum distribution, for all partial waves, of a back-to-back correlated nucleon pair in nuclear matter. We find that the probability of finding a high-momentum correlated neutron–proton pair is about 18 times that of a proton–proton one, as a result of the strong tensor force, thus confirming in an independent way previous results and measurements.

Figure 1

BG wave functions ${\tilde{u}}_{k,l}^{SJ}\left(r\right)$ for the uncoupled $NN$ partial waves. We also display the free solution ${\widehat{j}}_l\left(kr\right)$ in dashed style and the defect wave function $\mathrm{\Delta }{\tilde{u}}_{k,l}^{SJ}\left(r\right)\equiv {\tilde{u}}_{k,l}^{SJ}\left(r\right)-{\widehat{j}}_l\left(kr\right)$ as dotted lines. The calculations have been done for a relative momentum of the pair $k=140$ MeV.

Figure 2

Correlation function $f_{\mathrm{corr}}\left(r\right)\equiv \frac{{\tilde{u}}_{k,l}^{SJ}\left(r\right)}{{\widehat{j}}_l\left(kr\right)}$ for each uncoupled $NN$ partial wave. The calculations have been done for $k=140$ MeV.

Figure 3

BG wave functions ${\tilde{u}}_{k,l}^{SJ}\left(r\right)$ for the coupled $NN$ partial waves ${}^{3}S_1$–${}^{3}D_1$ and ${}^{3}P_2$–${}^{3}F_2$. We also display the free solution ${\widehat{j}}_l\left(kr\right)$ in dashed style, and the defect wave function $\mathrm{\Delta }{\tilde{u}}_{k,l}^{SJ}\left(r\right)$ as dotted lines. The calculations have been done for a relative momentum of the pair $k=140$ MeV.

Figure 4

Correlation function $f_{\mathrm{corr}}\left(r\right)$ for the coupled $NN$ partial waves ${}^{3}S_1$–${}^{3}D_1$ and ${}^{3}P_2$–${}^{3}F_2$. The calculations have been done for $k=140$ MeV.

Figure 5

Upper figures: BG wave functions ${\tilde{u}}_{k,l}^{SJ}\left(r\right)$ for the $NN$ partial waves ${}^{3}S_1$–${}^{3}D_1$ treating them as fully uncoupled by switching off the nondiagonal strengths, ${\left({\lambda }_i\right)}_{l,l^{\prime }}^{SJ}$ with $l\ne l^{\prime }$. Lines have the same meaning as in Figs. 1 and 3. Lower figures: Correlation function $f_{\mathrm{corr}}\left(r\right)$ for the same partial waves assuming no mixing between them. The calculations have been done for a relative momentum of the pair $k=140$ MeV.

Figure 6

$NN$ high-momentum components ${\left|{\varphi }_{k,lSJ}\left(p\right)\right|}^2$ in the correlated wave function, calculated for relative momentum of the nucleon pair $k=140$ MeV. Panel (a) corresponds to the uncoupled $NN$ partial waves, while panel (b) corresponds to the coupled ones. The high-momentum plotting region is restricted to $p>k_F$.

Figure 7

(a) $NN$ high-momentum components in the correlated wave function for the ${}^{3}S_1$ partial wave. (b) Same high-momentum components for the ${}^{3}D_1$ partial wave. Both calculations have been done for a relative momentum of the nucleon pair $k=140$ MeV. The high-momentum plotting region is restricted to $p>k_F$.

Figure 8

Comparison of $NN$ high-momentum distributions for $np$ and $pp$ pairs with different total spins, $S=0,1$. In panel (a) we show the singlet ($S=0$) high-momentum distribution, while in panel (b) we show the triplet ($S=1$) case. The contributions of different partial waves are accumulatively added in each spin channel. The calculations have been performed for relative momentum of the nucleon pair $k=140$ MeV.

Figure 9

(a) Comparison of total $NN$ high-momentum distributions for $np$ and $pp$ pairs regardless of the total spin $S$. It is also displayed the curve corresponding to $pp$ high-momentum components multiplied by the factor 18, as measured in Ref. [32]. (b) Ratio of the $np$ over $pp$ high-momentum distribution. The central value and extremal given by $18\pm 5$, as measured in Ref. [32], are also displayed as horizontal dotted lines. The calculations have been performed for relative momentum of the nucleon pair $k=140$ MeV.

Figure 10

(a) Comparison of high-momentum distributions for $np$ and $pp$ pairs for three different values of the initial momentum $k$. (b) The same as the panel (a) but with the $pp$ distribution multiplied by a factor 18.