Microscopic calculations based on chiral two- and three-nucleon forces for proton- and ${}^4\mathrm{He}$-nucleus scattering

We investigate the effects of chiral three-nucleon force (3NF) on proton scattering at 65 MeV and ${}^4\mathrm{He}$ scattering at 72 MeV/nucleon from heavier targets, using the standard microscopic framework composed of the Brueckner-Hartree-Fock (BHF) method and the $g$-matrix folding model. For nuclear matter, the $g$ matrix is evaluated from chiral two-nucleon force (2NF) of ${\mathrm{N}}^3\mathrm{LO}$ and chiral 3NF of NNLO by using the BHF method. Because the $g$ matrix thus obtained is numerical and nonlocal, an optimum local form is determined from the on-shell and near-on-shell components of $g$ matrix that are important for elastic scattering. For elastic scattering, the optical potentials are calculated by folding the local chiral $g$ matrix with projectile and target densities. This microscopic framework reproduces the experimental data without introducing any adjustable parameter. Chiral-3NF effects are small for proton scattering, but sizable for ${}^4\mathrm{He}$ scattering at middle angles where the data are available. Chiral 3NF, mainly in the $2\pi$-exchange diagram, makes the folding potential less attractive and more absorptive for all the scattering.

Figure 1

Differential cross sections for neutron-proton scattering at $E_{\mathrm{in}}\simeq 65$ MeV in free space. The solid (dashed) curves represent the results of the original (Gaussian) chiral $t$ matrix. Experimental data are taken from Refs. [38, 39].

Figure 2

$k_{\mathrm{F}}$ dependence of ${\mathcal{U}}^{ST}$ at $E_{\mathrm{in}}=65$ MeV for $({\mathrm{a})}^1\mathrm{E}$, (b) ${}^3\mathrm{E}$, (c) ${}^1\mathrm{O}$, and (d) ${}^3\mathrm{O}$. Squares (circles) mean the results of the original chiral $g$ matrix with (without) 3NF. The solid (dashed) lines represent the results of the Gaussian chiral $g$ matrix with (without) 3NF. For ${}^3\mathrm{O}$, the imaginary part is shifted down by 10 MeV.

Figure 3

Angular distribution of (a) differential cross sections and (b) vector analyzing powers for proton elastic scattering at 65 MeV. The solid (dashed) lines denote the results of chiral $g$ matrix with (without) 3NF effects. Each cross section is multiplied by the factor shown in the figure, while each vector analyzing power is shifted up by the number shown in the figure. Experimental data are taken from Ref. [45].

Figure 4

Angular distribution of differential cross sections for ${}^4\mathrm{He}$ scattering at 72MeV/nucleon from ${}^{58}\mathrm{Ni}$ and ${}^{208}\mathrm{Pb}$ targets. The solid (dashed) lines denote the results of the chiral $g$ matrix with (without) 3NF effects. Each cross section is multiplied by the factor shown in the figure. Experimental data are taken from Ref. [46].

Figure 5

$R$ dependence of the central part of the folding potential for ${}^4\mathrm{He}+{}^{208}\mathrm{Pb}$ elastic scattering at $E=72$ MeV/nucleon. These (a) and (b) correspond to the real and imaginary parts of $U_{\mathrm{CE}}\left(R\right)$, respectively. The solid (dashed) lines represent the results of chiral $g$ matrix with (without) chiral 3NF.