Optical potentials derived from nucleon-nucleon chiral potentials at ${\mathrm{N}}^4\mathrm{LO}$

Background: Elastic scattering is probably the main event in the interactions of nucleons with nuclei. Even if this process has been extensively studied in the past years, a consistent description, i.e., starting from microscopic two- and many-body forces connected by the same symmetries and principles, is still under development.

Purpose: In a previous paper [M. Vorabbi, P. Finelli, and C. Giusti, Phys. Rev. C 93, 034619 (2016)] we derived a theoretical optical potential from $NN$ chiral potentials at fourth order (${\mathrm{N}}^3\mathrm{LO}$). In the present work we use $NN$ chiral potentials at fifth order (${\mathrm{N}}^4\mathrm{LO}$), with the purpose to check the convergence and to assess the theoretical errors associated with the truncation of the chiral expansion in the construction of an optical potential.

Methods: Within the same framework and with the same approximations as the previous paper [M. Vorabbi, P. Finelli, and C. Giusti, Phys. Rev. C 93, 034619 (2016)], the optical potential is derived as the first-order term within the spectator expansion of the nonrelativistic multiple scattering theory and adopting the impulse approximation and the optimum factorization approximation.

Results: The $pp$ and $np$ Wolfenstein amplitudes and the cross section, analyzing power, and spin rotation of elastic proton scattering from ${}^{16}\mathrm{O}$, ${}^{12}\mathrm{C}$, and ${}^{40}\mathrm{Ca}$ nuclei are presented at an incident proton energy of 200 MeV. The results obtained with different versions of chiral potentials at ${\mathrm{N}}^4\mathrm{LO}$ are compared.

Conclusions: Our results indicate that convergence has been reached at ${\mathrm{N}}^4\mathrm{LO}$. The agreement with the experimental data is comparable with the agreement obtained in the previous paper [M. Vorabbi, P. Finelli, and C. Giusti, Phys. Rev. C 93, 034619 (2016)]. We confirm that building an optical potential within chiral perturbation theory is a promising approach for describing elastic proton-nucleus scattering.

Figure 1

Real (left) and imaginary (right) parts of $pp$ and $pna$ and $c$ Wolfenstein amplitudes as functions of the center-of-mass $NN$ angle $\varphi$. All the amplitudes are computed at 200 MeV using one of the EKM [6, 7] potentials (red bands determined by $R=0.9$ fm) and one of the EMN [8, 9] potentials (cyan bands) which uses a momentum cutoff $\mathrm{\Lambda }=500$ MeV. To estimate theoretical errors, we used Eq. (8) with ${\mathrm{\Lambda }}_b=600$ MeV. Empirical data are taken from Ref. [24].

Figure 2

Scattering observables (differential cross section $d\sigma /d\mathrm{\Omega }$, analyzing power $A_y$, and spin rotation $Q$) as a function of the center-of-mass scattering angle $\theta$ for elastic proton scattering on ${}^{16}\mathrm{O}$ computed at 200 MeV (laboratory energy). We employ one of the EKM [6, 7] potentials (red bands determined by $R=0.9$ fm) and one of the EMN [8, 9] potentials (cyan bands) which uses a momentum cutoff $\mathrm{\Lambda }=500$ MeV. To estimate theoretical errors, we used Eq. (8) with ${\mathrm{\Lambda }}_b=600$ MeV. Coulomb distortion is included as explained in Ref. [2]. Empirical data are taken from Refs. [42, 43].

Figure 3

The same as in Fig. 2 for ${}^{12}\mathrm{C}$ at an energy of 200 MeV. Empirical data are taken from Refs. [42, 43].

Figure 4

The same as in Fig. 2 for ${}^{40}\mathrm{Ca}$ at an energy of 200 MeV. Empirical data are taken from Refs. [42, 43].

Figure 5

Scattering observables as a function of the center-of-mass scattering angle $\theta$ for elastic proton scattering on ${}^{16}\mathrm{O}$ computed at 200 MeV (laboratory energy) with the EKM potential [6, 7] at different orders: green bands are the ${\mathrm{N}}^2\mathrm{LO}$ results, and blue and red bands are the $\mathrm{N}{}^3\mathrm{LO}$ and $\mathrm{N}{}^4\mathrm{LO}$ results, respectively. Empirical data are taken from Refs. [42, 43].