Break-up channels in muon capture on ${}^3\mathrm{He}$

The ${\mu }^{-}+{}^2\mathrm{H}\to {\nu }_{\mu }+n+n$, ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+{}^3\mathrm{H}$, ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+d$, and ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+n+p$ capture reactions are studied with various realistic potentials under full inclusion of final-state interactions. Our results for the two- and three-body break-up of ${}^3\mathrm{He}$ are calculated with a variety of nucleon-nucleon potentials, among which is the AV18 potential, augmented by the Urbana IX three-nucleon potential. Most of our results are based on the single-nucleon weak-current operator. As a first step, we tested our calculation in the case of the ${\mu }^{-}+{}^2\mathrm{H}\to {\nu }_{\mu }+n+n$ and ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+{}^3\mathrm{H}$ reactions, for which theoretical predictions obtained in a comparable framework are available. Additionally, we have been able to obtain for the first time a realistic estimate for the total rates of the muon capture reactions on ${}^3\mathrm{He}$ in the break-up channels: 544 ${\mathrm{s}}^{-1}$ and 154 ${\mathrm{s}}^{-1}$ for the $n+d$ and $n+n+p$ channels, respectively. Our results are compared with the most recent experimental data, finding a rough agreement for the total capture rates, but failing to reproduce the differential capture rates.

Figure 1

Differential capture rate $d{\Gamma }_d^F/d{E}_{\nu }$ for the ${\mu }^{-}+{}^2\mathrm{H}\to {\nu }_{\mu }+n+n$ process, calculated with the Bonn B potential [23] in the three-dimensional formalism of Ref. [22] and using the single-nucleon current operator from Eqs. (2.7) and (2.8) for (left panel) $F=\frac{1}{2}$ and (right panel) $F=\frac{3}{2}$ as a function of the neutrino energy $E_{\nu }$. The dashed curves show the plane-wave results and the solid curves are used for the full results. Note that the average nucleon mass is used in the kinematics and in solving the Lippmann–Schwinger equations (see text for more details).

Figure 2

The same as in Fig. 1 but given in the form of $d{\Gamma }_d^F/dp$ and shown as a function of the magnitude of the relative neutron-neutron momentum $p$.

Figure 3

Differential capture rate $d{\Gamma }_d^F/dp$ of the ${\mu }^{-}+{}^2\mathrm{H}\to {\nu }_{\mu }+n+n$ process calculated with the Bonn B potential [23] in the three-dimensional formalism of Ref. [22] for (left panel) $F=\frac{1}{2}$ and (right panel) $F=\frac{3}{2}$ as a function of the relative neutron-neutron momentum $p$. The dashed (solid) curves show the full results obtained with the single-nucleon current operator without (with) the relativistic corrections. Note that the average nucleon mass is used in the kinematics and in solving the Lippmann–Schwinger equations (see text for more details).

Figure 4

Differential capture rate $d{\Gamma }_d^F/dp$ of the ${\mu }^{-}+{}^2\mathrm{H}\to {\nu }_{\mu }+n+n$ process calculated by using standard PWD with various nucleon-nucleon potentials: the AV18 potential [13] (solid curves), the Bonn B potential [23] (dashed curves), and the set of chiral NNLO potentials from Ref. [24] (bands) for (left panel) $F=\frac{1}{2}$ and (right panel) $F=\frac{3}{2}$ as a function of the relative neutron-neutron momentum $p$. Note that the bands are very narrow and thus appear practically as a curve. All the partial-wave states with $j\le 4$ have been included in the calculations with the single-nucleon current operator containing the relativistic corrections. Note that the average nucleon mass is used in the kinematics and in solving the Lippmann–Schwinger equations (see text for more details).

Figure 5

The kinematically allowed region in the $E_{\nu }\text{−}{E}_d$ plane calculated relativistically (solid curve) and nonrelativistically (dashed curve) for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+d$ process.

Figure 6

The kinematically allowed region in the $E_{\nu }\text{−}{E}_p$ plane calculated relativistically (solid curve) and nonrelativistically (dashed curve) for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+n+p$ process.

Figure 7

The differential capture rates ${d\Gamma }_{nd}/d{E}_{\nu }$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+d$ process calculated with the AV18 potential [13] and the single-nucleon current operator as a function of the muon neutrino energy, using (left panel) the symmetrized plane wave and (right panel) a full solution of Eq. (5.9) with $V_4^{\left(1\right)}=0$. The curves representing results of the calculations employing all partial-wave states with $j\le 3$ $\left(j\le 4\right)$ in the 2N subsystem are depicted with dashed (solid) curves. The maximal total 3N angular momentum is $J_{\max }=\frac{9}{2}$.

Figure 8

The differential capture rates ${d\Gamma }_{nd}/d{E}_{\nu }$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+d$ process calculated with the single-nucleon current operator and different types of 3N dynamics: plane wave (dash-dotted curve), symmetrized plane wave (dotted curve), full solution of Eq. (5.9) without (dashed curve) and with 3N force (solid curve). The calculations are based on the AV18 nucleon-nucleon potential [13] and the Urbana IX 3N force [14] and employ all partial-wave states with $j\le 3$ and $J\le \frac{9}{2}$.

Figure 9

The double-differential capture rates ${d^2\Gamma }_{nd}/\left(d{E}_{\nu }d{\Omega }_{q_0}\right)$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+d$ process calculated with the single-nucleon current operator and using the plane wave (dotted curve), symmetrized plane wave (dashed curve), and full solution of Eq. (5.9) but with $V_4^{\left(1\right)}=0$ (solid curve) for two values of the neutrino energy. The calculations are based on the AV18 nucleon-nucleon potential [13] and employ all partial-wave states with $j\le 3$ and $J\le \frac{9}{2}$.

Figure 10

The differential capture rates ${d\Gamma }_{nnp}/d{E}_{\nu }$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+n+p$ process calculated with the AV18 potential [13] and using a full solution of Eq. (5.9) with $V_4^{\left(1\right)}=0$. The curves representing results of the calculations employing all partial-wave states with $j\le 3$ $\left(j\le 4\right)$ in the 2N subsystem are depicted with dashed (solid) curves. The maximal total 3N angular momentum is $J_{\max }=\frac{9}{2}$.

Figure 11

The differential capture rates ${d\Gamma }_{nnp}/d{E}_{\nu }$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+n+p$ process calculated with full solutions of Eq. (5.9) with $V_4^{\left(1\right)}=0$ (dashed curve) and with $V_4^{\left(1\right)}\ne 0$ (solid curve). The calculations are based on the AV18 nucleon-nucleon potential [13] and the Urbana IX 3N force [14] and employ all partial-wave states with $j\le 3$ and $J\le \frac{9}{2}$.

Figure 12

The events for two selected bins corresponding to Fig. 14 with (left panel) the central deuteron energy $E_d=15.5$ MeV and (right panel) 20.5 MeV, generated with 60 uniformly distributed $E_{\nu }$ points in the $[0,E_{\nu }^{\text{max},nnp}]$ interval and 72 uniformly distributed ${\theta }_{q_0}$ values of the relative momentum ${\mathbf{q}}_0$ (in $[0,\pi ])$ as explained in the text. For the selected examples the number of events is approximately 130.

Figure 13

The events for two selected bins corresponding to Fig. 15 with (left panel) the central proton energy $E_p=25.5$ MeV and (right panel) 35.5 MeV, generated with 60 uniformly distributed $E_{\nu }$ points, 36 uniformly distributed ${\theta }_q$ values of the relative momentum $\mathbf{q}$, and 32 values of the magnitude of $\mathbf{q}$ (see text for a detailed explanation). For these two examples the number of events is approximately 1000.

Figure 14

The capture rates $\langle {d\Gamma }_{nd}/d{E}_d\rangle$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+d$ process averaged over 1 MeV deuteron energy bins are compared with the experimental data given in Table VI of Ref. [15]. In the left (right) panel the experimental data are evaluated using method I (method II) of Ref. [15]. The notation for the curves is the same as that of Fig. 8.

Figure 15

The capture rates $\langle {d\Gamma }_{nnp}/d{E}_p\rangle$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+n+p$ process averaged over 1 MeV proton energy bins are compared with the experimental data shown in Table V of Ref. [15]. In the left (right) panel the experimental data are evaluated using method I (method II) of Ref. [15]. The notation for the curves is the same of Fig. 11.

Figure 16

Convergence of the full results (without a 3N force) with respect to the total angular momentum of the final 3N system corresponding to (left panel) Figs. 14 and (right panel) 15. Curves show results of calculations with $J\le \frac{1}{2}$ (double dashed), $J\le \frac{3}{2}$ (dash-dotted), $J\le \frac{5}{2}$ (dotted), $J\le \frac{7}{2}$ (dashed), and $J\le \frac{9}{2}$ (solid).

Figure 17

The capture rates $\langle {d\Gamma }_{nnp}/d{E}_p\rangle$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+n+p$ process averaged over 5 MeV proton energy bins are compared with the experimental data shown in Table I of Ref. [17]. Circles and triangles are used to represent data taken at two different detectors. The notation for the curves is the same as for Fig. 11.

Figure 18

The capture rates $\langle {d\Gamma }_{nd}/d{E}_d\rangle$ for the ${\mu }^{-}+{}^3\mathrm{He}\to {\nu }_{\mu }+n+d$ process averaged over 1 MeV deuteron energy bins are compared with the experimental data given in Table III of Ref. [17]. The notation for the curves is the same of Fig. 8.