$n+{}^3\mathrm{H},$ $p+{}^3\mathrm{He},$ $p+{}^3\mathrm{H},$ and $n+{}^3\mathrm{He}$ scattering with the hyperspherical harmonic method

The $n+{}^3\mathrm{H},$ $p+{}^3\mathrm{He},$ $p+{}^3\mathrm{H},$ and $n+{}^3\mathrm{He}$ elastic and charge exchange reactions at low energies are studied by means of the hyperspherical harmonic method. The considered nuclear Hamiltonians include modern two- and three-nucleon interactions, and in particular results are reported in case of chiral two-nucleon potentials, with and without the inclusion of chiral three-nucleon (3N) interactions. A detailed study of the convergence and numerical stability of the method is presented. We have found that the effect the 3N force is in general tiny except for $p+{}^3\mathrm{H}$ scattering below the opening of the $n+{}^3\mathrm{He}$ channel. In such a case, the effect of 3N forces is appreciable and a clear dependence on the cutoff used to regularize the high-momentum tail of the interactions is observed. Such a dependence is related to the presence of the poorly known sharp $0^{+}$ resonance, considered to be the first excited state of ${}^4\mathrm{He}$.

Figure 1

The values of $|{\mathrm{\Delta }}_n({\overline{K}}_1,{\overline{K}}_2,...)|$ obtained for the N3LO500 (left panel) and AV18 (right panel) potentials and the ${}^{1}S_0$, ${}^{3}S_1$, ${}^{3}P_0$, ${}^{1}P_1$, ${}^{3}P_1$, and ${}^{3}P_2$ phase shifts. The quantity $n$ is defined in Eq. (79), i.e., $2n$ is the increase of the grand angular quantum number for each class starting from a given set ${\overline{K}}_1,{\overline{K}}_2,...$. See the main text for more details.

Figure 2

$n+{}^3\mathrm{H}$ total cross section as function of the incoming neutron energy $E_n$ calculated with the NN N3LO interaction of Refs. [38, 39] (light cyan band) or including also the 3N N2LO interaction discussed in the text (darker blue band). The width of the bands reflects the spread of theoretical results using $\mathrm{\Lambda }=500$ or 600 MeV cutoff values. See the main text for more details. The experimental values are taken from Ref. [55].

Figure 3

$p+{}^3\mathrm{He}$ phase shifts as function of the incoming proton energy $E_p$ calculated with the NN N3LO interaction of Refs. [38, 39] (light cyan band) or including also the 3N N2LO interaction discussed in the text (darker blue band). The results of the PSA performed at TUNL have been also reported [65].

Figure 4

$p+{}^3\mathrm{He}$ differential cross section as function of the center-of-mass scattering angle for three different proton energies $E_p$ calculated with the NN N3LO interactions of Refs. [38, 39] (light cyan band) or including also the 3N N2LO interactions (darker blue band). The width of the bands reflects the use of two different cutoff values, $\mathrm{\Lambda }=500$ and 600 MeV. The experimental data are from Refs. [56, 57, 58].

Figure 5

Same as in Fig. 4 but for the $p+{}^3\mathrm{He}$ analyzing power $A_{y0}$. The experimental data are from Refs. [31, 58, 59].

Figure 6

Same as in Fig. 4, but for the $p+{}^3\mathrm{He}$ analyzing power $A_{0y}$. The experimental data are from Refs. [59, 65].

Figure 7

$p+{}^3\mathrm{He}$ analyzing powers at $E_p=5.54$ MeV calculated with the N3LO500/N2LO500 (blue solid lines), N3LO600/N2LO600 (dashed magenta lines), and AV18/IL7 (dot-dashed red lines) interaction models. The experimental data are from Refs. [31, 58, 59].

Figure 8

Same as in Fig. 4 but for the $p+{}^3\mathrm{He}{A}_{yy}$, $A_{xx}$, $A_{xz}$, and $A_{zx}$ spin polarization coefficients at $E_p=5.54$ MeV. The experimental data are from Refs. [59, 65].

Figure 9

$p+{}^3\mathrm{H}{}^{1}S_0$ phase shift calculated with the Minnesota potential as function of the center-of-mass kinetic energy $T_r$. Solid line: present calculation; red dots: RGM calculation of Ref. [121]; crosses: phase shift extracted from the $R$-matrix analysis [24]. The arrow denotes the energy of the $n+{}^3\mathrm{He}$ threshold.

Figure 10

$p+{}^3\mathrm{H}$ phase shifts as function of the center-of-mass kinetic energy $T_r$ calculated with the N3LO500 (red dashed curves), N3LO600 (red dotted curves), N3LO500/N2LO500 (black solid curves), and N3LO600/N2LO600 (red dot-dashed curves) interactions. The experimental phase shifts have been extracted by an $R$-matrix analysis in Ref. [24].

Figure 11

${}^{1}S_{0}p+{}^3\mathrm{H}$ phase shift as function of the center-of-mass kinetic energy $T_r$ calculated with several interactions.

Figure 12

Same as in Fig. 4 but for the $p+{}^3\mathrm{H}$ differential cross section. The experimental data are from Refs. [67, 68, 69, 72, 73, 74].

Figure 13

Same as in Fig. 4 but for the $p+{}^3\mathrm{H}$ proton analyzing power. The experimental data are from Refs. [59, 65].

Figure 14

The same as Fig. 4 but for the $n+{}^3\mathrm{He}$ differential cross section and neutron analyzing power. The experimental data are from Refs. [75, 82, 83].

Figure 15

The same as in Fig. 4 but for the $p+{}^3\mathrm{H}\to n+{}^3\mathrm{He}$ differential cross section and proton analyzing power. The experimental data are from Refs. [92, 93, 94, 96, 97].