Relativistic quasiparticle random-phase approximation calculation of total muon capture rates

The relativistic proton-neutron quasiparticle random phase approximation ($\mathit{pn}$-RQRPA) is applied in the calculation of total muon capture rates on a large set of nuclei from ${}^{12}\mathrm{C}$ to ${}^{244}\mathrm{Pu}$, for which experimental values are available. The microscopic theoretical framework is based on the relativistic Hartree-Bogoliubov (RHB) model for the nuclear ground state, and transitions to excited states are calculated using the $\mathit{pn}$-RQRPA. The calculation is fully consistent, i.e., the same interactions are used both in the RHB equations that determine the quasiparticle basis, and in the matrix equations of the $\mathit{pn}$-RQRPA. The calculated capture rates are sensitive to the in-medium quenching of the axial-vector coupling constant. By reducing this constant from its free-nucleon value $g_A=1.262$ by 10% for all multipole transitions, the calculation reproduces the experimental muon capture rates to better than 10% accuracy.

Figure 1

The square of the $1s$ muon wave function in the Coulomb potentials of self-consistent ground-state charge densities of ${}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}$, ${}^{120}\mathrm{Sn}$ and ${}^{208}\mathrm{Pb}$ (solid curves), compared to eigenfunctions of the Coulomb potential for the corresponding point charge Z (dashed curves). The figures also include the calculated charge densities of the four nuclei, scaled by arbitrary factors.

Figure 2

Ratio of the calculated and experimental total muon capture rates, as function of the proton number $Z$. The theoretical values are calculated with muon $1s$ wave functions determined by self-consistent ground-state charge densities (filled circle symbols), and by the corresponding point-charge Coulomb potentials (squares).

Figure 3

Ratio of the calculated and experimental total muon capture rates, as function of the proton number $Z$. Circles correspond to rates calculated with the free-nucleon weak form factors Eqs. (10)–(13) [21], and diamonds denote values obtained by quenching the free-nucleon axial-vector coupling constant $g_A=1.262$ to $g_A=1.135$ for all operators, i.e., in all multipole channels.

Figure 4

Relative contributions of different multipole transitions to the RHB plus RQRPA total muon capture rates in ${}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}$, ${}^{120}\mathrm{Sn}$ and ${}^{208}\mathrm{Pb}$.

Figure 5

Total muon capture rates on Ca, Cr and Ni isotopes. Experimental rates (filled symbols) are compared to theoretical values (empty symbols), calculated using the fully consistent RHB plus RQRPA framework with the DD-ME2 universal effective interaction, and with the quenching of the axial-vector coupling constant $g_A=1.262\to g_A=1.135$ for all multipole operators.