#
Ordinary Muon Capture on ^{136}Ba: Comparative Study Using the Shell Model and pnQRPA

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework

#### 2.1. Bound-Muon S-Orbital Wave Function

#### 2.2. Muon-Capture Rates

#### 2.3. Chiral Two-Body Currents

#### 2.4. Many-Body Methods

## 3. Results and Discussion

**sm-2BC**:**sm-phen**:**qrpa-2BC**:- We used the pnQRPA method as described in Section 2.4 and quenched ${g}_{\mathrm{A}}$ and ${g}_{\mathrm{P}}$ with the 2BC using Equations (10) and (11). We used the PIR scheme and adjusted the isoscalar strength to a value of ${g}_{\mathrm{pp}}^{T=1}=0.86$ in order to achieve the partial isospin restoration, then we adjust the isoscalar strength to the values ${g}_{\mathrm{pp}}^{T=0}=0.65\phantom{\rule{0.277778em}{0ex}}({g}_{\mathrm{pp}}^{T=0}=0.67)$ in order to reproduce the TNDBD half-life ${t}_{1/2}^{\left(2\nu \right)}=(2.18\pm 0.05)\xb7{10}^{21}$ yr [51] using the effective coupling ${g}_{\mathrm{A}}^{\mathrm{eff}}=0.89\phantom{\rule{0.277778em}{0ex}}({g}_{\mathrm{A}}^{\mathrm{eff}}=1.02)$ corresponding to the free nucleon value ${g}_{\mathrm{A}}=1.27$ quenched by the zero-momentum transfer correction ${\delta}_{a}\left(0\right)$ through Equation (10) with parameters $\rho =0.09\phantom{\rule{0.277778em}{0ex}}{\mathrm{fm}}^{-3}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{c}_{D}=-6.08\phantom{\rule{0.277778em}{0ex}}(\rho =0.11\phantom{\rule{0.277778em}{0ex}}{\mathrm{fm}}^{-3}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{c}_{D}=0.30)$.
**qrpa-phen**:- Again, we used the pnQRPA method like above, but we used as the particle–particle strength the value ${g}_{\mathrm{pp}}^{T=0}={g}_{\mathrm{pp}}^{T=1}=0.7$, which was obtained from the extensive survey of the $\beta $-decay and TNDBD half-lives within the mass range $A=$ 100–136 in Reference [52]. We adopted the effective coupling ${g}_{\mathrm{A}}^{\mathrm{eff}}=0.83$ resulting from the so-called linear ${g}_{\mathrm{A}}$ model of the same work. This value is somewhat below the range of values ${g}_{\mathrm{A}}^{\mathrm{eff}}=$ 0.89–1.02 corresponding to the axial vector correction ${\delta}_{a}\left(0\right)$ at zero-momentum transfer. The corresponding effective pseudoscalar coupling is ${g}_{\mathrm{P}}^{\mathrm{eff}}=5.64$, as obtained through the Goldberger–Treiman relation (9). The value ${g}_{\mathrm{A}}^{\mathrm{eff}}=0.83$ can be considered to account for both the missing two-body currents at $q=0$ MeV and the deficiencies of the many-body approach in the spirit of Reference [16]. However, it does not take into account the momentum dependence of the two-body currents.

**qrpa-2BC**and

**qrpa-phen**. It is worth noting that there are three sets of the pnQRPA-computed energies based on the three different values of the (${g}_{\mathrm{pp}}^{T=0}$, ${g}_{\mathrm{pp}}^{T=1}$) pairs used in the pnQRPA calculations. Here, we plotted just one set of energies in the

**qrpa-2BC**scheme, since the two sets of energy are almost identical. From Figure 3, it can be seen that the densities of both the ISM- and pnQRPA-computed states are quite similar, higher than the densities of the measured states. It is, in fact, remarkable that both theories predict low-energy spectra so similarly, with pnQRPA able to reproduce the density of the ISM states. The density of the experimental spectrum is smaller than predicted by the computations, probably due to difficulties in observing some of the states.

**sm-2BC**calculational scheme). The fifth column lists the OMC rates obtained by using the phenomenological

**sm-phen**calculational scheme. Table 3 has a similar structure, but now there are two sets of

**qrpa-2BC**energies (column 2) corresponding to the two sets of LEC used in our calculations and the set of

**qrpa-phen**energies in column 3. Columns 4–6 list the OMC rates obtained by using the schemes

**qrpa-1BC**,

**qrpa-2BC**, and

**qrpa-phen**.

**qrpa-phen**scheme and not far from our

**qrpa-2BC**calculational scheme. This makes the three computations very comparable, particularly for the OMC of ${}^{136}$Ba, but also for ${}^{76}$Se, where experimental data exist. In Table V of Reference [37], the pnQRPA-computed OMC rates of final states in ${}^{76}$As, below some 1 MeV of excitation like in the present work, were compared with the corresponding experimental ones, and a surprisingly good correspondence was found. There, the total rate for the OMC of the ${0}^{+}$, ${1}^{+}$, ${1}^{-}$, ${2}^{+}$, ${2}^{-}$, ${3}^{+}$, ${3}^{-}$, ${4}^{+}$, and ${4}^{-}$ final states in ${}^{76}$As was $665\times {10}^{3}$ 1/s in the experiment and $675\times {10}^{3}$ 1/s in the pnQRPA. These total OMC rates are in line with the total OMC rates of (674–807) $\times {10}^{3}$ 1/s and $592\times {10}^{3}$ 1/s of our

**qrpa-2BC**and

**qrpa-phen**calculational schemes, respectively. Notably, both in the experiment and in the pnQRPA calculation of Reference [37], the ${1}^{+}$ rate was the largest one with the values $218\times {10}^{3}$ 1/s for the experiment and $237\times {10}^{3}$ 1/s for the pnQRPA, comparable with our (243–303) $\times {10}^{3}$ 1/s and $207\times {10}^{3}$ 1/s in the

**qrpa-2BC**and

**qrpa-phen**calculational schemes. In Reference [37] also, the OMC of ${2}^{-}$ states was strong, some 10 times stronger than in the present calculations, since the role of ${2}^{-}$ states in $pf$-shell nuclei is quite pronounced [4].

**sm-2BC**scheme, 1.4% of the total rate for the

**sm-phen**scheme, 6–7% of the total rate for the

**qrpa-2BC**scheme, and 5.3% of the total rate for the

**qrpa-phen**scheme, thus being below 10% but still non-negligible. This highlights the importance of comparison with the potential future experimental data and the emerging implications for the virtual NDBD transitions below roughly 1 MeV of excitation in the intermediate nucleus of a double-beta triplet of nuclei.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Note

1 | In the present work, we use the convention $c=1$ for compactness of presentation. |

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**Figure 1.**Comparison of the large component of the exact muon wave function for a finite nucleus with a uniform spherical charge distribution (blue line) with a corresponding one for a point-like nucleus (red line) and its Bethe–Salpeter (BS) approximation (black line). The gray band denotes the range inside the nucleus.

**Figure 2.**The two-body currents used in the present work are functions of momentum exchange. The dashed lines denote the currents obtained by $\rho =0.09\phantom{\rule{3.33333pt}{0ex}}{\mathrm{fm}}^{-3}$ and ${c}_{D}=-6.08$, and the dotted lines indicate those obtained with $\rho =0.11\phantom{\rule{3.33333pt}{0ex}}{\mathrm{fm}}^{-3}$ and ${c}_{D}=0.30$. The typical momentum exchange region of the transitions considered in the present work is denoted by a vertical gray band.

**Figure 3.**Excitation energy spectrum of ${}^{136}$Cs. A comparison between the experimental spectrum and those computed by using the ISM and pnQRPA is shown here. The first experimental spectrum was taken from the ENSDF database [53], and the second one was taken from Rebeiro et al. in Reference [54]. Only states with angular momenta $J\le 5$ were considered.

**Table 1.**Values of the weak axial couplings, the Fermi gas density $\rho $, and the LEC ${c}_{D}$ and pnQRPA parameters used in our calculations.

sm-1BC and sm-2BC | sm-phen | qrpa-1BC | qrpa-2BC | qrpa-phen | |||
---|---|---|---|---|---|---|---|

${g}_{\mathrm{PP}}^{\mathrm{T}=0}$ | - | - | 0.69 | 0.65 | 0.67 | 0.7 | |

${g}_{\mathrm{PP}}^{\mathrm{T}=1}$ | - | - | 0.86 | 0.86 | 0.86 | 0.7 | |

${g}_{\mathrm{ph}}$ | - | - | 1.18 | 1.18 | 1.18 | 1.18 | |

${g}_{A}$ | 1.27 | 0.93 | 1.27 | 1.27 | 1.27 | 0.83 | |

${g}_{P}/{g}_{A}$ | 6.8 | 6.8 | 6.8 | 6.8 | 6.8 | 6.8 | |

$\rho $ | 0.09 | 0.11 | - | - | 0.09 | 0.11 | - |

${c}_{D}$ | −6.08 | 0.30 | - | - | −6.08 | 0.30 | - |

**Table 2.**ISM-computed energies (second column) and OMC rates (third to fifth columns) of the final states (f) of spin J and parity $\pi $ (first column) with angular momenta $J\le 5$. The bottom line summarizes the total OMC rates below some 1 MeV as summed over the OMC rates listed in columns three to five. The lower (upper) limits in column four correspond to the Fermi gas density $\rho =0.09$ fm${}^{-3}$ and the low-energy constant ${c}_{D}=-6.08$ ($\rho =0.11$ fm${}^{-3}$ and ${c}_{D}=0.3$), the rest of the LEC being equal in the two sets.

OMC Rate $({10}^{3}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}1/$s) | ||||
---|---|---|---|---|

${\mathit{J}}_{\mathit{f}}^{\mathit{\pi}}$ | E(MeV) | sm-1BC | sm-2BC | sm-phen |

${5}_{1}^{+}$ | 0.000 | 0.0647 | 0.0661 (0.0836) | 0.0433 |

${3}_{1}^{+}$ | 0.023 | 4.02 | 2.75 (3.36) | 2.60 |

${4}_{1}^{+}$ | 0.039 | 1.50 | 1.36 (1.40) | 1.37 |

${2}_{1}^{+}$ | 0.083 | 10.6 | 5.62 (6.99) | 6.18 |

${3}_{2}^{+}$ | 0.181 | 12.0 | 6.24 (8.08) | 6.66 |

${2}_{2}^{+}$ | 0.225 | 20.1 | 12.8 (15.00) | 13.7 |

${3}_{3}^{+}$ | 0.244 | 4.94 | 2.48 (3.23) | 2.71 |

${4}_{2}^{+}$ | 0.323 | 5.83 | 3.50 (4.17) | 3.78 |

${4}_{3}^{+}$ | 0.498 | 6.00 | 4.34 (4.83) | 4.54 |

${3}_{4}^{+}$ | 0.517 | 31.2 | 16.8 (21.5) | 17.9 |

${5}_{1}^{-}$ | 0.522 | 0.645 | 0.371 (0.451) | 0.404 |

${3}_{1}^{-}$ | 0.545 | 16.1 | 8.85 (11.0) | 9.73 |

${1}_{1}^{+}$ | 0.545 | 9.01 | 4.67 (6.03) | 5.03 |

${4}_{1}^{-}$ | 0.547 | 24.0 | 13.0 (16.7) | 13.7 |

${2}_{3}^{+}$ | 0.615 | 18.2 | 12.5 (14.2) | 13.2 |

${5}_{2}^{-}$ | 0.671 | 0.251 | 0.190 (0.208) | 0.198 |

${1}_{2}^{+}$ | 0.752 | 0.285 | 0.123 (0.163) | 0.146 |

${4}_{2}^{-}$ | 0.760 | 2.22 | 1.31 (1.65) | 1.32 |

${2}_{4}^{+}$ | 0.803 | 2.49 | 1.74 (1.95) | 1.83 |

${4}_{4}^{+}$ | 0.885 | 0.143 | 0.0865 (0.103) | 0.0933 |

${2}_{1}^{-}$ | 1.016 | 78.6 | 41.5 (53.3) | 44.4 |

Sum $({10}^{3}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}1/$s) | 248 | 140 (174) | 150 |

**Table 3.**pnQRPA-computed energies for the

**qrpa-2BC**scheme (second column) and the

**qrpa-phen**scheme (third column) and OMC rates (fourth to sixth columns) of the final states (f) of spin J and parity $\pi $ (first column) with angular momenta $J\le 5$. The bottom line summarizes the total OMC rates below 1 MeV as summed over the OMC rates listed in columns four to six. The two energies in column two and the lower (upper) limits in column five correspond to the Fermi gas density $\rho =0.09$ fm${}^{-3}$ and the low-energy constant ${c}_{D}=-6.08$ ($\rho =0.11$ fm${}^{-3}$ and ${c}_{D}=0.3$), the rest of the LEC being equal in the two sets.

OMC Rate $({10}^{3}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}1/$s) | |||||
---|---|---|---|---|---|

${\mathit{J}}_{\mathit{f}}^{\mathit{\pi}}$ | E(MeV)qrpa-2BC | E(MeV)qrpa-phen | qrpa-1BC | qrpa-2BC | qrpa-phen |

${5}_{1}^{+}$ | $0.000$ | $0.000$ | $0.902$ | $0.491\phantom{\rule{0.166667em}{0ex}}\left(0.601\right)$ | $0.483$ |

${3}_{1}^{+}$ | $0.110\phantom{\rule{0.166667em}{0ex}}\left(0.107\right)$ | $0.102$ | $3.04$ | $1.74\phantom{\rule{0.166667em}{0ex}}\left(2.19\right)$ | $1.51$ |

${2}_{1}^{+}$ | $0.124\phantom{\rule{0.166667em}{0ex}}\left(0.122\right)$ | $0.120$ | 133 | $102\phantom{\rule{0.166667em}{0ex}}\left(111\right)$ | $93.3$ |

${4}_{1}^{+}$ | $0.139\phantom{\rule{0.166667em}{0ex}}\left(0.144\right)$ | $0.154$ | $10.0$ | $8.81\phantom{\rule{0.166667em}{0ex}}\left(9.34\right)$ | $8.96$ |

${1}_{1}^{+}$ | $0.227\phantom{\rule{0.166667em}{0ex}}\left(0.213\right)$ | $0.193$ | 443 | $243.0\phantom{\rule{0.166667em}{0ex}}\left(303.4\right)$ | $206.9$ |

${4}_{2}^{+}$ | $0.180\phantom{\rule{0.166667em}{0ex}}\left(0.179\right)$ | $0.203$ | $12.4$ | $8.76\phantom{\rule{0.166667em}{0ex}}\left(9.61\right)$ | $8.25$ |

${3}_{2}^{+}$ | $0.249\phantom{\rule{0.166667em}{0ex}}\left(0.254\right)$ | $0.264$ | $11.6$ | $7.93\phantom{\rule{0.166667em}{0ex}}\left(10.8\right)$ | $3.49$ |

${3}_{3}^{+}$ | $0.268\phantom{\rule{0.166667em}{0ex}}\left(0.273\right)$ | $0.281$ | 156 | $85.6\phantom{\rule{0.166667em}{0ex}}\left(108\right)$ | $77.5$ |

${3}_{4}^{+}$ | $0.330\phantom{\rule{0.166667em}{0ex}}\left(0.332\right)$ | $0.338$ | $12.2$ | $9.50\phantom{\rule{0.166667em}{0ex}}\left(11.9\right)$ | $5.34$ |

${2}_{2}^{+}$ | $0.340\phantom{\rule{0.166667em}{0ex}}\left(0.346\right)$ | $0.367$ | $88.3$ | $49.0\phantom{\rule{0.166667em}{0ex}}\left(60.2\right)$ | $50.1$ |

${3}_{1}^{-}$ | $0.461\phantom{\rule{0.166667em}{0ex}}\left(0.459\right)$ | $0.458$ | $48.0$ | $28.9\phantom{\rule{0.166667em}{0ex}}\left(34.4\right)$ | $25.8$ |

${4}_{3}^{+}$ | $0.471\phantom{\rule{0.166667em}{0ex}}\left(0.477\right)$ | $0.494$ | $4.19$ | $3.24\phantom{\rule{0.166667em}{0ex}}\left(3.60\right)$ | $3.18$ |

${5}_{1}^{-}$ | $0.505\phantom{\rule{0.166667em}{0ex}}\left(0.509\right)$ | $0.515$ | $1.20$ | $0.825\phantom{\rule{0.166667em}{0ex}}\left(0.933\right)$ | $0.775$ |

${4}_{1}^{-}$ | $0.553\phantom{\rule{0.166667em}{0ex}}\left(0.555\right)$ | $0.558$ | $1.60$ | $1.04\phantom{\rule{0.166667em}{0ex}}\left(1.19\right)$ | $0.596$ |

${2}_{3}^{+}$ | $0.533\phantom{\rule{0.166667em}{0ex}}\left(0.538\right)$ | $0.561$ | $87.1$ | $60.0\phantom{\rule{0.166667em}{0ex}}\left(68.3\right)$ | $57.7$ |

${5}_{2}^{-}$ | $0.621\phantom{\rule{0.166667em}{0ex}}\left(0.624\right)$ | $0.637$ | $0.017$ | $0.0135\phantom{\rule{0.166667em}{0ex}}\left(0.0149\right)$ | $0.0178$ |

${4}_{2}^{-}$ | $0.681\phantom{\rule{0.166667em}{0ex}}\left(0.686\right)$ | $0.695$ | $43.1$ | $24.1\phantom{\rule{0.166667em}{0ex}}\left(30.6\right)$ | $20.5$ |

${2}_{1}^{-}$ | $0.750\phantom{\rule{0.166667em}{0ex}}\left(0.725\right)$ | $0.704$ | $27.3$ | $26.6\phantom{\rule{0.166667em}{0ex}}\left(26.0\right)$ | $14.2$ |

${3}_{2}^{-}$ | $0.896\phantom{\rule{0.166667em}{0ex}}\left(0.901\right)$ | $0.926$ | $20.5$ | $12.7\phantom{\rule{0.166667em}{0ex}}\left(15.1\right)$ | $12.9$ |

Sum $({10}^{3}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}1/$s) | 1103 | $674\phantom{\rule{0.166667em}{0ex}}\left(807\right)$ | 592 |

**Table 4.**Total OMC rates of each multipole ${J}^{\pi}$ as summed over the rates listed in Table 2 for the

**sm-2BC**and

**sm-phen**schemes and in Table 3 for the

**qrpa-2BC**and

**qrpa-phen**schemes. The lower (upper) limits in the second and fourth columns correspond to the Fermi gas density $\rho =0.09$ fm${}^{-3}$ and the low-energy constant ${c}_{D}=-6.08$ ($\rho =0.11$ fm${}^{-3}$ and ${c}_{D}=0.3$).

OMC Rate $({10}^{3}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}1/$s) | ||||
---|---|---|---|---|

${\mathit{J}}^{\mathit{\pi}}$ | sm-2BC | sm-phen | qrpa-2BC | qrpa-phen |

${5}^{+}$ | 0.1 (0.1) | 0.0 | 0.5 (0.6) | 0.5 |

${4}^{+}$ | 9.3 (10.5) | 9.8 | 20.8 (22.6) | 20.4 |

${3}^{+}$ | 28.3 (36.2) | 29.9 | 104.8 (132.9) | 87.8 |

${2}^{+}$ | 32.7 (38.1) | 34.9 | 211.0 (239.5) | 201.1 |

${1}^{+}$ | 4.8 (6.2) | 5.2 | 243.0 (303.4) | 206.9 |

${5}^{-}$ | 0.6 (0.7) | 0.6 | 0.8 (0.9) | 0.8 |

${4}^{-}$ | 14.3 (18.4) | 15.0 | 25.1 (31.8) | 21.2 |

${3}^{-}$ | 8.9 (11.0) | 9.7 | 41.6 (49.5) | 38.7 |

${2}^{-}$ | 41.5 (53.3) | 44.4 | 26.6 (26.0) | 14.2 |

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**MDPI and ACS Style**

Gimeno, P.; Jokiniemi, L.; Kotila, J.; Ramalho, M.; Suhonen, J.
Ordinary Muon Capture on ^{136}Ba: Comparative Study Using the Shell Model and pnQRPA. *Universe* **2023**, *9*, 270.
https://doi.org/10.3390/universe9060270

**AMA Style**

Gimeno P, Jokiniemi L, Kotila J, Ramalho M, Suhonen J.
Ordinary Muon Capture on ^{136}Ba: Comparative Study Using the Shell Model and pnQRPA. *Universe*. 2023; 9(6):270.
https://doi.org/10.3390/universe9060270

**Chicago/Turabian Style**

Gimeno, Patricia, Lotta Jokiniemi, Jenni Kotila, Marlom Ramalho, and Jouni Suhonen.
2023. "Ordinary Muon Capture on ^{136}Ba: Comparative Study Using the Shell Model and pnQRPA" *Universe* 9, no. 6: 270.
https://doi.org/10.3390/universe9060270