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Ab initio Leading Order Effective Potentials for Elastic Nucleon-Nucleus Scattering

Background: Calculating microscopic effective interactions (optical potentials) for elastic nucleon-nucleus scattering has already in the past led to a large body of work. For first-order calculations a nucleon-nucleon (\textit{NN}) interaction and a one-body density of the nucleus were taken as input to rigorous calculations of microscopic full-folding calculations. Purpose: Based on the spectator expansion of the multiple scattering series we employ a chiral next-to-next-to-leading order (NNLO) nucleon-nucleon interaction on the same footing in the structure as well as in the reaction calculation to obtain an in leading-order consistent effective potential for nucleon-nucleus elastic scattering, which includes the spin of the struck target nucleon. Methods: The first order effective folding potential is computed by first deriving a nonlocal scalar density as well as a spin-projected momentum distribution. Those are then integrated with the off-shell Wolfenstein amplitudes $A$, $C$, and $M$. The resulting nonlocal potential serves as input to a momentum-space Lippmann-Schwinger equation, whose solutions are summed to obtain the nucleon-nucleus scattering observables. Results: We calculate elastic scattering observables for $^4$He, $^6$He, $^8$He, $^{12}$C, and $^{16}$O in the energy regime between 100 and 200 MeV projectile kinetic energy, and compare to available data. We also explore the extension down to about 70 MeV, and study the effect of ignoring the spin of the struck nucleon in the nucleus. Conclusions: In our calculations we contrast elastic scattering off closed-shell and open-shell nuclei. We find that for closed-shell nuclei the approximation of ignoring the spin of the struck target nucleon is excellent. We only see effects of the spin of the struck target nucleon when considering $^6$He and $^8$He, which are nuclei with a $N/Z$ ratio larger than 1.