Phenomenological refinement of $p$-$d$ elastic scattering descriptions towards the 3NF study in nuclei via the ($p,pd$) reaction

Abstract

The ($p,pd$) reaction is expected to be a powerful tool for probing three-nucleon forces (3NFs) in nuclear medium since it can be essentially regarded as the $p$-$d$ elastic scattering inside nuclei. One of the important points in the theoretical description of the ($p,pd$) reaction is to calculate the $p$-$d$ scattering in a nucleus quantitatively using effective interactions. This work aims to develop a phenomenological approach to improve the quantitativity of the $p$-$d$ scattering cross section in free space calculated with effective interactions. The $p$-$d$ elastic amplitude is decomposed into a 2N part, described using 2N effective interactions, and a residual part, which the 2N part cannot describe. The latter is approximated by a superposition of Legendre polynomials, with coefficients treated as adjustable parameters. These parameters are determined to reproduce experimental $p$-$d$ differential cross-section data at various incident energies. The obtained parameters exhibit smooth energy dependence, which is approximated by quadratic functions. The numerical results with the analytic energy dependence also reproduce the experimental data. The developed approach works well for improving the $p$-$d$ scattering cross section in a wide range of incident energies. This work can be regarded as the first step toward the description of ($p,pd$) reactions taking 3NF effect in nuclear medium into account.

Figures (6)

• Figure 1: Incident-energy dependence of $C_L$. The filled and open circles denote the real and imaginary parts of $C_0$, respectively, while the square symbols correspond to the real and imaginary parts of $C_1$. The approximated behaviors of the filled (open) circles and squares, Eq. \ref{['eq:C_L']}, are plotted by the solid and dotted (dashed and dot-dashed) lines, respectively.
• Figure 2: Differential cross sections of the $p$-$d$ elastic scattering at $T_p^\textrm{L} =$ (a) $108$, (b) $120$, (c) $135$, and (d) $150$ MeV as a function of the c.m. scattering angle of the system. The dots are the experimental data taken from Ref. KErmisch05. The solid lines are the numerical results of Eq. \ref{['eq:CS_pd']} with the parameters evaluated by Eq. \ref{['eq:C_L']}, while the dotted and dashed ones are the results obtained using Eqs. \ref{['eq:CS_pd_2N']} and \ref{['eq:CS_pd_res']}, respectively.
• Figure 3: Same as Fig. \ref{['fig:CS_pd_1']} but at $T_p^\textrm{L} =$ (a) $155$, (b) $170$, (c) $190$, and (d) $250$ MeV. The experimental data are taken from Refs. KKuroda64 ($155$ MeV), KErmisch05 ($170$ and $190$ MeV), and KHatanaka02 ($250$ MeV).
• Figure A1: Same as Fig. \ref{['fig:parameter']} but for the Franey-Love 2N effective interaction.
• Figure A2: Same as Fig. \ref{['fig:CS_pd_1']} but calculated using the Franey-Love 2N effective interaction and Eq. \ref{['eq:C_L']} with the coefficients listed in Table \ref{['tab:coefficientFL']}.