Detailed description of exclusive muon capture rates using realistic two-body forces

Starting from state-by-state calculations of exclusive rates of the ordinary muon capture, we evaluated total ${\mu }^{-}$ capture rates for a set of light- and medium-weight nuclear isotopes. We employed a version of the proton-neutron quasiparticle random-phase approximation ($pn$-QRPA, for short) which uses as realistic nuclear forces the Bonn C-D one-boson exchange potential. Special attention was paid on the percentage contribution to the total ${\mu }^{-}$ capture rate of specific low-spin multipolarities resulting by summing over the corresponding multipole transitions. The nuclear method used offers the possibility of estimating separately the individual contributions to the total and partial rates of the polar-vector and axial-vector components of the weak-interaction Hamiltonian for each accessible final state of the daughter nucleus. One of our main goals is to provide a reliable description of the charge-changing transitions matrix elements entering the description of other similar semileptonic nuclear processes like the charged-current neutrino-nucleus reactions, the electron capture on nuclei, the single ${\beta }^{\pm }$-decay mode, etc., which play important role in currently interesting laboratory and astrophysical applications like the neutrino detection through lepton-nucleus interaction probes and neutrino nucleosynthesis. Such results can also be useful in various ongoing muon capture experiments at Paul Scherrer Institute (PSI), Fermilab, Japan Proton Accelerator Research Complex, and Research Center for Nuclear Physics, Osaka University.

Figure 1

Comparison of the theoretical excitation spectrum, resulting from the solution of the $pn$-QRPA eigenvalue problem, with the low-lying (up to about 3 MeV) experimental one for ${}^{56}\mathrm{Mn}$ and ${}^{90}\mathrm{Y}$ nuclei (for the other spectra, see Refs. [28, 60]). The agreement is quite good at least for low excitation energies.

Figure 2

Individual contribution of the polar-vector ${\mathrm{\Lambda }}_V$ (a) and axial-vector ${\mathrm{\Lambda }}_A$ (b) to the total muon capture rate (c) as a function of the excitation energy $\omega$ for the ${}^{28}\mathrm{Si}$ and ${}^{32}\mathrm{S}$ nuclei.

Figure 3

The same as Fig. 2 but for the nuclei ${}^{48}\mathrm{Ti}$ and ${}^{56}\mathrm{Fe}$.

Figure 4

The same as Fig. 2, but for the nuclei ${}^{66}\mathrm{Zn}$ and ${}^{90}\mathrm{Zr}$.

Figure 5

Partial muon capture rates ${\mathrm{\Lambda }}_{J^{\pi }}$ of different multipole transitions in ${}^{28}\mathrm{Si}$ and ${}^{32}\mathrm{S}$ isotopes. In both isotopes the pronounced contributions are the $J^{\pi }=1^{-}$ and $J^{\pi }=1^{+}$ multipolarity.

Figure 6

Contribution of multipole transition rates ${\mathrm{\Lambda }}_{J^{\pi }}$ (up to $J^{\pi }=4^{\pm }$) with the total muon capture rate in ${}^{48}\mathrm{Ti}$, ${}^{56}\mathrm{Fe}$, ${}^{66}\mathrm{Zn}$, and ${}^{90}\mathrm{Zr}$ isotopes with (solid histograms) and without (double dashed histograms) quenching effect. The dominance of $J^{\pi }=1^{-}$ and $1^{+}$ multipolarities is obvious in all nuclei.

Figure 7

Ratio of the calculated and experimental total muon capture rates as a function of $Z$. Circles and X symbols correspond to rates calculated with free nucleon $g_A$ and quenched value of $g_A$, respectively.