Irreducible nonlocality of optical model potentials based on realistic NN interactions

We investigate the nonlocal structure of optical model potentials for nucleon-nucleus scattering based on microscopic approaches. To this purpose, in medium folding optical potentials are calculated in momentum space and their corresponding coordinate-space counterpart are examined, paying special attention to their nonlocal shape. The nucleon-nucleon effective interaction consists of the actual full off-shell $g$ matrix in Brueckner-Hartree-Fock approximation. The nonlocality of effective interactions is preserved throughout all stages in the calculation. Argonne $v_{18}$ bare potential and chiral next-to-next-to-next-to-leading order bare interaction are used as starting point. The study is focused on proton elastic scattering off ${}^{40}\mathrm{Ca}$ at beam energies between 30 and 800 MeV. We find that the gradual suppression of high-momentum contributions of the optical potential results in quite different-looking coordinate-space counterparts. Despite this nonuniqueness in their nonlocal structure, the implied scattering observables remain unchanged for momentum cutoff above a critical one, which depends on incident energy of the projectile. We find that coordinate-space potentials with momentum cutoffs at the critical value yield the least structured nonlocal behavior. Implications of these findings are discussed.

Figure 1

(a) Real and (b) imaginary parts of the $s$-wave central optical potential as functions of the relative momenta $k$ and $k^{\prime }$. Potential corresponding to $p+{}^{40}\mathrm{Ca}$ scattering at 200 MeV. Results based on AV18 bare interaction. Color code in units of MeV fm.

Figure 2

The same as Fig. 1, but for N3LO interaction.

Figure 3

Contour plots for the real (left) and imaginary (right) $s$-wave central optical potential as functions of the relative distance $r$ and $r^{\prime }$. Potential for $p+{}^{40}\mathrm{Ca}$ scattering at 200 MeV. Results based on AV18 bare interaction. (a), (b), and (c) represent NA potentials with cutoffs at $\mathrm{\Lambda }=12$, 7 and $4{\mathrm{fm}}^{-1}$, respectively. All three potentials yield the same scattering observables. Color bar in units of MeV ${\mathrm{fm}}^{-1}$.

Figure 4

The same as Fig. 3, but for N3LO bare interaction.

Figure 5

Nonlocal behavior of the $s$-wave $r^{\prime }U(r^{\prime },r)r$ as function of $(r^{\prime }-r)$, satisfying $r+r^{\prime }=\mathrm{constant}$. Solid, dashed and dotted curves correspond to results for $\mathrm{\Lambda }=12$, 7, and $4{\mathrm{fm}}^{-1}$, respectively. See text for descriptions of each panel.

Figure 6

The same as Fig. 5, but for N3LO.

Figure 7

Nonlocal behavior of the $s$-wave optical potential near (a) bulk and (b) surface for cutoff $\mathrm{\Lambda }=4{\mathrm{fm}}^{-1}$. Potential for $p+{}^{40}\mathrm{Ca}$ elastic scattering at 200 MeV beam energy. Solid and dotted curves represent results for the real and imaginary components, respectively. Black and gray curves denote results based on AV18 and N3LO bare interactions, respectively.

Figure 8

(a) Differential cross section and (b) analyzing power as functions of the momentum transfer for ${}^{40}\mathrm{Ca}(p,p)$ scattering at 200 MeV. Black (gray) curves correspond to results based on AV18 (N3LO) bare interaction. Solid curves correspond to $\mathrm{\Lambda }=12$, 7, and $4{\mathrm{fm}}^{-1}$, whereas dashed curves correspond to $\mathrm{\Lambda }=3.5{\mathrm{fm}}^{-1}$.

Figure 9

Reaction cross section as a function of the cutoff $\mathrm{\Lambda }$ applied to the optical model potential for $p+{}^{40}\mathrm{Ca}$ elastic scattering. Black and gray curves represent results based on AV18 and N3LO bare interactions. Labels on each curve refer to the beam energy. Squares denote the critical cutoff.

Figure 10

Critical cutoff ${\mathrm{\Lambda }}_A$ for the nonreducible $k$ domain as function of the beam energy $E$ in the case of proton elastic scattering from ${}^{40}\mathrm{Ca}$.

Figure 11

(a) Differential cross section and (b) analyzing power as functions of the center-of-mass scattering angle for ${}^{40}\mathrm{Ca}(p,p)$ scattering at 30.3 MeV. Overlapping curves in the cases AV18, N3LO, and PB correspond to results based on $\mathrm{\Lambda }=6$, 4, and 2.6 ${\mathrm{fm}}^{-1}$. Data from Refs. [29, 30].

Figure 12

Contour plots for the real (l.h.s.) and imaginary (r.h.s.) $s$-wave central optical potential as functions of the relative distance $r$ and $r^{\prime }$. Potential for $p+{}^{40}\mathrm{Ca}$ scattering at 30.3 MeV. Results based on AV18 bare interaction. (a), (b), and (c) represent results based on $\mathrm{\Lambda }=6$, 4, and 2.$6{\mathrm{fm}}^{-1}$, respectively. All three potentials yield the same scattering observables. Color bar in units of MeV ${\mathrm{fm}}^{-1}$.

Figure 13

The same as Fig. 12, but for N3LO bare interaction.

Figure 14

The same as Fig. 12, but based on NSM approach.

Figure 15

The same as Fig. 12, but based on Perey-Buck-type nonlocal parametrization of Ref. [14].

Figure 16

Surface plots of real (l.h.s.) and imaginary (r.h.s.) $s$-wave reduced central optical potential $rU(r,r^{\prime })r^{\prime }$ as functions of the relative distance $r$ and $r^{\prime }$. Potential for $p+{}^{40}\mathrm{Ca}$ scattering at 800, 200, and 30.3 MeV. Results based on AV18 bare interaction. Color bars and vertical axes are in units of MeV ${\mathrm{fm}}^{-1}$.

Figure 17

(a) Differential cross section and (b) analyzing power as functions of the momentum transfer for ${}^{40}\mathrm{Ca}(p,p)$ scattering at 800 MeV. Overlapping solid curves correspond to results based on $\mathrm{\Lambda }=12$, 10, and $8{\mathrm{fm}}^{-1}$. Dashed curves correspond to $\mathrm{\Lambda }=7.5{\mathrm{fm}}^{-1}$. Data taken from Refs. [31, 32].