Calculation of the $nd$ Scattering Lengths by a Realistic Nonlocal Gaussian Potential

The calculation of the quartet and doublet $nd$ scattering length, $^{4}a_{nd}$ and $^{2}a_{nd}$, is very difficult because of the deuteron distortion effect, especially for $^{2} a_{nd}$. Even for $^{4}a_{nd}$, it is not easy to keep the numerical accracy since the many channels couple. The well- known corellation between the triton binding energy and $^{2}a_{nd}$ is not completely understood in the calculations including three-body forces. Motivated by the successful application of the our quark-model baryon-baryon interaction fss2 to the triton binding energy without the three-body forces, we reexamine $^{2}a_{nd}$ and $^ {4}a_{nd}$ to study the effect of the nonlocality. We use a nonlocal Gaussian potential based on fss2, which is constructed to make few-baryon calculations much easier than the original interaction. We calculate the eigen-phase shift ${\delta}$ and plot $k\cot{\delta}$ versus $k^{2}$. The charge independence breaking is ignored. The energy region examined is from $E_ {c.m.}$=50 keV to 1 MeV. For the quartet scattering length, $k\cot{\delta}$ is almost linear with respect to $k^{2}$ above $E_{c.m.}$=100 keV. Below 100 keV, the numerical accuracy seems not to be maintained. We have obtained $^{4}a_{nd}$=6.3 fm, which is close to the experimental value $^{4}a^{\rm exp}_{nd}=6.34 \pm $ 0.03 fm. For the doublet scattering length, $^{2}a_{nd}$ is expected to be about 0.8 fm (vs. $^{2}a^{\rm exp}_{nd}=0.65 \pm 0.03$ fm), but we need much wider model space than $J_{\rm max} =2$ and fine mesh points to keep the numerical accuracy of the eigen-phase shifts.