We solve the nuclear two-body and three-body bound states via quantum simulations of pionless effective field theory on a lattice in position space. While the employed lattice remains small, the usage of local Hamiltonians including two- and three-body forces ensures that the number of Pauli terms scales linearly with increasing numbers of lattice sites. We use an adaptive ansatz grown from unitary coupled cluster theory to parametrize the ground states of the deuteron and ${}^3\mathrm{He}$, compute their corresponding energies, and analyze the scaling of the required computational resources. Our quantum simulations reproduce exact benchmarks for ${}^2\mathrm{H}$ and ${}^3\mathrm{He}$ within 100 keV, requiring at most 30 layers in the ansatz and thus resulting in modest circuit depths. Additionally, we find the number of shots required to reach a given precision scales linearly in the lattice size and more mildly in the system size. Based on the agreement with exact benchmarks and mild scaling, we conclude that this can be an efficient, scalable approach for quantum computations of nuclear ground states, particularly to prepare initial states for quantum phase estimation or other filtering algorithms.

Using $(1.0087\pm 0.0044)\times {10}^{10} J/\psi$ events collected with the BESIII detector at the BEPCII storage ring, the reactions ${\mathrm{\Sigma }}^{+}n\to \mathrm{\Lambda }p$ and ${\mathrm{\Sigma }}^{+}n\to {\mathrm{\Sigma }}^{0}p$ are studied, where the ${\mathrm{\Sigma }}^{+}$ baryon is produced in the process $J/\psi \to {\mathrm{\Sigma }}^{+}{\overline{\mathrm{\Sigma }}}^{-}$ and the neutron is a component of the ${}^9\mathrm{Be}, {}^{12}\mathrm{C}$, and ${}^{197}\mathrm{Au}$ nuclei in the beam pipe. Clear signals of these two reactions are observed for the first time. Their cross sections are measured to be $\sigma ({\mathrm{\Sigma }}^{+}+{}^9\mathrm{Be}\to \mathrm{\Lambda }+p+X)=(45.2\pm 12.1_{\mathrm{stat}}\pm 7.2_{\mathrm{sys}})$ mb and $\sigma ({\mathrm{\Sigma }}^{+}+{}^9\mathrm{Be}\to {\mathrm{\Sigma }}^0+p+X)=(29.8\pm 9.7_{\mathrm{stat}}\pm 6.9_{\mathrm{sys}})$ mb for a ${\mathrm{\Sigma }}^{+}$ average momentum of $0.992\mathrm{GeV}/c$, within a range of $\pm 0.015\mathrm{GeV}/c$, where $X$ represents the residual nucleus. This is the first study of ${\mathrm{\Sigma }}^{+}$-nucleon scattering at an electron-positron collider. The measured results serve as a significant element in enhancing the comprehensive understanding of hyperon-nucleon interactions, and also constitute an important factor in resolving the “hyperon puzzle” of neutron stars.

We perform a high-statistics lattice QCD calculation of the low-energy two-nucleon scattering amplitudes. To address discrepancies in the literature, the calculation is performed at a heavy pion mass in the limit that the light quark masses are equal to the physical strange quark mass, $m_{\pi }=m_K\simeq 714\mathrm{MeV}$. Using a state-of-the-art momentum space method, we rule out the presence of a bound di-nucleon in both the isospin 0 (deuteron) and 1 (di-neutron) channels, in contrast with many previous results that made use of compact hexaquark creation operators. To diagnose the discrepancy, we add such hexaquark interpolating operators to our basis and find that they do not affect the determination of the two-nucleon finite-volume spectrum, and thus they do not couple to deeply bound di-nucleons that are missed by the momentum-space operators. Furthermore, we perform a high-statistics calculation of the HAL QCD potential on the same gauge ensembles and find qualitative agreement with our main results. We conclude that di-nucleons do not form bound states at heavy pion masses and that previous identification of deeply bound di-nucleons must have arisen from a misidentification of the spectrum from off-diagonal elements of a correlation function.

We formulate a renormalizable pionless effective field theory ($\cancel{\pi }\mathrm{EFT}$) with a nonperturbative treatment of the Coulomb interaction up to next-to-leading order (NLO) for few-nucleon systems. We extract scattering observables for charged clusters by employing two-, three-, and four-body contact interactions and using the stochastic variational method with a Coulomb-corrected harmonic oscillator trap. Our NLO results yield a $pd$ spin-quartet scattering length and effective range of $a_{pd}^{3/2}=12.76\left(29\right)\mathrm{fm}$ and $r_{pd}^{3/2}=1.17\left(7\right)\mathrm{fm}$; for $dd$ scattering in the spin-quintet channel we find $a_{dd}^2=6.26\left(3\right)\mathrm{fm}$ and $r_{dd}^2=1.41\left(7\right)\mathrm{fm}$; and for $p^3\mathrm{He}$ scattering the spin-singlet and spin-triplet channels are characterized by $a_{p^3\mathrm{He}}^0=11.26\left(4\right)\mathrm{fm}, r_{p^3\mathrm{He}}^0=1.65\left(26\right)\mathrm{fm}$ and $a_{p^3\mathrm{He}}^1=9.06\left(4\right)\mathrm{fm}, r_{p^3\mathrm{He}}^1=1.36\left(25\right)\mathrm{fm}$, respectively. Our predictions exhibit mild cutoff dependence and agree well with existing experimental phase shift analyses and potential model calculations. This demonstrates the predictive power of $\cancel{\pi }\mathrm{EFT}$ for charged few-nucleon systems.

Benchmarking simulation codes for electron transport and scattering in matter is a crucial step for estimating uncertainties in many applications. However, experimental data for electron energies of a few MeV are too scarce to make such comparisons. We report here the measurement and the quantitative analysis of backscattering probabilities of electrons in the energy range 0.1 to 3.4 MeV impinging on a Ce-doped ${\mathrm{YAlO}}^3$ (YAP:Ce) scintillator. The setup consists of a $2\times 2\pi$ calorimeter which enables, in particular, the inclusion of large incidence angles. The results are used to benchmark various scattering models incorporated in Geant4, showing relative deviations smaller than 5% between experiment and simulations. They demonstrate the current rather high reliability of the simulations when employing appropriate electromagnetic physics lists.

The ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction governs the carbon-to-oxygen ratio set during helium burning, shaping white-dwarf structure and Type Ia supernova yields. At the astrophysical energy $E\approx 300\mathrm{keV}$, the cross section is controlled by the subthreshold $1^{-}$ (7.12 MeV) and $2^{+}$ (6.92 MeV) states, whose contributions depend on their asymptotic normalization coefficients (ANCs) $C_1$ and $C_2$, respectively. I perform a Bayesian analysis of the $S_{E1}$ and $S_{E2}$ factors using calibrated $R$-matrix mappings and contemporary ANC constraints for the $1^{-}, 2^{+}$, and $0^{+}$ ground state. For $S_{E1}$, a flat prior on the $1^{-}$ ANC yields a broad posterior (MAP $\approx 95.5\mathrm{keV}\mathrm{b}$), while Gaussian priors centered at the low and high anchors produce narrower posteriors with MAPs $\approx 75.4$ and $\approx 93.2\mathrm{keV}\mathrm{b}$, respectively. For $S_{E2}$, I employ a calibrated quadratic map in $(C_2,C_0)$ that captures the interference between the subthreshold $2^{+}$ capture and the external direct capture; the resulting posteriors reflect how the joint $(C_2,C_0)$ posterior loads the interference corridor. The resulting $S\left(300\mathrm{keV}\right)$ range is consistent with stellar-evolution models that produce low-spin, high-mass ($\approx 50-70M_{\odot }$) black holes. Overall, the Bayesian framework yields statistically robust posteriors for $S_{E1}$ and $S_{E2}$, quantifying correlations and uncertainties relevant for stellar modeling. This work links the microscopic subthreshold ANCs ($C_1,C_2,C_0$) to a macroscopic astrophysical outcome—the formation of massive, low-spin black holes—thereby connecting nuclear reaction experiments and theory with stellar evolution, black-hole demographics, and gravitational-wave observations.

Within nuclear density-functional theory, we calculated spectroscopic magnetic-dipole and electric-quadrupole moments for various quasiparticle configurations of odd-$N$, even-$Z, 83\le N\le 125$ nuclei ranging from gadolinium to osmium. By tagging the blocked quasiparticles with single-particle states of the semimagic dysprosium isotope, we efficiently computed 22 prolate and 22 oblate states for each of the 154 nuclei and tracked them across the entire major neutron shell. We compared this extensive set of theoretical results with experimental data for 82 states in the region. Breaking rotational, time-reversal, and signature symmetries, we aligned the intrinsic angular momenta along the axis of axial symmetry, thereby enabling full shape- and spin-self-consistent polarizations. The spectroscopic moments were then obtained by restoring rotational symmetry. We conducted a detailed analysis of the pattern of agreement and disagreement between theory and experiment in individual nuclei. For the magnetic-dipole moments, agreement with the data varies and is characterized by an overall average and rms deviation of $0.11{\mu }_N$ and $0.35{\mu }_N$, respectively. For the electric-quadrupole moments, a good corresponding agreement of 0.16 and 0.29 b was observed.

We study the evolution of fission modes across the Th isotopic chain using a microscopic framework combining the time-dependent generator coordinate method and finite-temperature covariant density-functional theory. Theoretical fission-fragment charge distributions agree well with experiments, and reveal a rapid symmetric-to-asymmetric transition from $A=222$ to 234. By analyzing the collective potential-energy surfaces and time evolution of collective probability density distributions, we demonstrate that this fission mode transition is strongly correlated with the rapidly deepening asymmetric fission valley—a phenomenon driven by the reduction of deformation energies of both the heavy and light fragments formed in the asymmetric fission valley. Further analysis attributes the decrease of light-fragment deformation energies to the onset of a coexisting large-deformed minimum in neutron-rich Kr and Sr isotopes (the dominant isotopes for light asymmetric peak), which arises from a deformed proton $Z=38$ shell closure near ${\beta }_2\approx 0.46$. Notably, we identify, for the first time, the pivotal role of the light fragment and its shape coexistence structure on the fission mode transition in Th isotopes in a fully microscopic framework.

The problem of the two-body nature of the mechanism in multinucleon transfer (MNT) process is very important, especially when it comes to the production of unknown isotopes. In this work, we conduct a comprehensive study on the correlations of multiple physical quantities in the MNT reaction ${}^{238}\mathrm{U} + {}^{238}\mathrm{U}$: colliding energy, kinetic energies of emitting fragments, impact parameter, proton/neutron transfer number, and scattering angle. Based on our discovered correlation behaviors, the emission properties of the ${}^{244}\mathrm{U}$ fragment are emphasized, which would contribute significantly to the production of the actinide nuclei near shell closure $N=152$.

Located just one neutron removed from doubly magic, stable ${}^{48}\mathrm{Ca}$, the adjacent isotope ${}^{47}\mathrm{Ca}$ has been studied extensively with neutron-removing transfer reactions and interpreted within a shell-model picture. Neutron-adding transfer reactions onto the stable ${}^{46}\mathrm{Ca}$, however, have not been performed in nearly 60 years and never with the aid of high-resolution $\gamma$-ray spectroscopy. Here, we report on the ${}^{12}\mathrm{C}({}^{46}\mathrm{Ca},{}^{47}\mathrm{Ca}+\gamma )X$ one-neutron pickup reaction from ${}^{12}\mathrm{C}$ onto ${}^{46}\mathrm{Ca}$ at more than 50 MeV/nucleon, a regime of orbital angular momentum mismatch that selectively populates ${}^{47}\mathrm{Ca}$ final states based on $\ell \ge 3$ neutron configurations. Newly observed excited states and $\gamma$-ray transitions are discussed in comparison to $fp$ shell-model calculations in a variety of configuration spaces. It is shown that the nature of the observed spectrum of final states can be well interpreted, albeit with a significant fragmentation of the $\nu 0f_{5/2}$ strength observed in the measurement that is not borne out in the structure calculations.

In a recent Letter [Phys. Rev. Lett. 130, 052501 (2023)], we suggested the quasi-SU(3) coupling across the $N=50$ and 82 shell gaps as the main source driving shape evolution and phase transitions. The present article extends the discussion in the Letter. We demonstrate that by varying the coupling strengths of the quasi-SU(3) partner orbits, $(1g_{9/2},$ $2d_{5/2})$ for protons and $(1h_{11/2}$, $2f_{7/2})$ for neutrons, we can well reproduce the features of the IBM U(5), O(6), SU(3), and $\overline{\mathrm{SU}\left(3\right)}$ symmetries and the Iachello E(5), X(5), and $\overline{\mathrm{X}\left(5\right)}$ with critical behavior. Therefore, all known dynamical and critical-point symmetries are realized in our shell-model parameter space. Moreover, in the phase diagram of the parameter space, we find that the line connecting U(5), E(5), and O(6) is characterized by the vanishing spectroscopic quadrupole moment of the first $2^{+}$ state [i.e., $Q_s\left(2_1^{+}\right)=0]$. It gives a new physical meaning to the central axis of the extended Casten triangle and implies that any shape evolution passing through this line can be naturally considered as a prolate-oblate shape phase transition. Thus, we have found microscopic origins for all symmetries originally suggested by algebraic or geometric models. By visualizing the shape using the constrained Hartree-Fock-Bogolyubov (HFB) method based on our shell-model results, we conclude that ${}^{134}\mathrm{Ba}$ is likely the softest nucleus in the nuclear chart, having no definite shape at all in its ground state.

Modern nuclear theory provides qualitative insights into the fundamental mechanisms of nuclear fission and is increasingly capable of making reliable quantitative predictions. Most quantities of interest pertain to the primary fission fragments, whose subsequent decay is typically modeled using statistical reaction models. Consequently, a key objective of fission theory is to inform these models by predicting the initial conditions of the primary fragments. In this work, we employ a framework that combines joint angular momentum and particle number projection with time-dependent configuration mixing to calculate the angular momentum distributions of primary fragments. Focusing on the benchmark cases of neutron-induced fission of ${}^{235}\mathrm{U}$ and ${}^{239}\mathrm{Pu}$, we predict—for the first time—microscopic angular momentum distributions for all fragments observed in experiments. Our results reveal a pronounced sawtooth pattern in the average angular momentum as a function of fragment mass, consistent with recent measurements. Additionally, we observe substantial variations in angular momentum distributions along isobaric chains, indicating that commonly used empirical formulas lack sufficient accuracy. We also quantify a strong correlation between the angular momentum and the deformation of the fragments at scission, and a weak correlation in the magnitude of the angular momentum between fragment partners. The generated data will enable estimation of the impact of microscopic distributions on fission spectra, paving the way toward fission modeling based on microscopic inputs.

This Letter deals with the so-called nuclear polarization correction to the 1S levels in light to intermediate muonic atoms. An easy to use recipe to compute it is given. The calculation includes the effect of the nucleon polarization, i.e. the contribution from inelastic states in the hadronic range, and Coulomb corrections beyond the leading-logarithm approximation to both nuclear and nucleon polarization.

One method of determining the free neutron lifetime involves the absolute counting of neutrons and trapped decay protons. In such experiments, a cold neutron beam traverses a segmented proton trap inside a superconducting solenoid while the neutron flux is continuously monitored. Protons that are born within the fiducial volume of the trap are confined radially by the magnetic field and axially by the electrostatic potential supplied by trap electrodes. They are periodically released and counted, and the ratio of the absolute number of neutrons to protons is proportional to the neutron lifetime. Systematic error can be introduced if protons in the trap are lost, gained, or misidentified. The influence of molecular hydrogen interactions is of particular interest because of its ubiquitous presence in ultrahigh vacuum systems. To understand how it could affect the neutron lifetime, measurements were performed on the production and detection of molecular hydrogen in an apparatus used to measure the neutron lifetime. We demonstrate that charge exchange with molecular hydrogen can occur with trapped protons, and we determine the efficiency with which the molecular hydrogen ions in the trap are detected. Finally, we comment on the potential impact on a neutron lifetime experiment using this beam technique. We find that the result of the beam neutron lifetime performed at NIST is unlikely to have been significantly affected by charge exchange with molecular hydrogen.

In this paper we present our results on the decay of ${}^{64}\mathrm{Se}$. It is the heaviest $T_z=-2$ nucleus that both $\beta$ decays and has a stable mirror partner ($T_z=+2$), thus allowing comparison with charge exchange reaction studies. The $\beta$ decays of ${}^{64}\mathrm{Se}$ and its descendants were studied at the RIKEN Nishina Center (Japan) following their production in the fragmentation of ${}^{78}\mathrm{Kr}$ on a beryllium target. $\beta$-Delayed $\gamma$-ray and particle radiation was identified for each of the nuclei in the decay chain allowing us to obtain decay schemes for ${}^{64}\mathrm{Se}$, ${}^{64}\mathrm{As}$, and ${}^{63}\mathrm{Ge}$. Thus new excited states could be found for the descendant nuclei, including the interesting case of the $N=Z$ nucleus ${}^{64}\mathrm{Ge}$. Furthermore, we observed for the first time the $\beta$-delayed proton emission of ${}^{64}\mathrm{Se}$ and ${}^{64}\mathrm{As}$. Based on these results we obtained proton branching ratios of $48.0\left(9\right)\%$ in ${}^{64}\mathrm{Se}$ decay and $4.4\left(1\right)\%$ in ${}^{64}\mathrm{As}$ decay. We obtained a half-life value of $22.5\left(6\right)\mathrm{ms}$ for ${}^{64}\mathrm{Se}$ decay and half-lives slightly more precise than those in the literature for each nucleus involved in the decay chain. Using our results on the excited levels of ${}^{64}\mathrm{As}$ and the mass excess in the literature for ${}^{63}\mathrm{Ge}$ we obtained $-39588\left(50\right)\mathrm{keV}$ for the mass excess of ${}^{64}\mathrm{As}$. Then based on the isobaric mass multiplet equation we obtained the mass excess of $-27429\left(88\right)\mathrm{keV}$ for ${}^{64}\mathrm{Se}$ by extrapolation. The mirror process of ${}^{64}\mathrm{Se}\beta$ decay, the charge exchange reaction ${}^{64}\mathrm{Zn}{({}^3\mathrm{He},t)}^{64}\mathrm{Ga}$, has already been measured allowing us to study the mirror symmetry through the comparison of the weak force ($\beta$a decay) and strong force (charge exchange reaction). An interpretation of the decay schemes based on the idea of the antisnalogue tate is proposed.

Optical potentials are a standard tool in the study of nuclear reactions, as they describe the interaction between a target nucleus and a projectile. The use of phenomenological optical potentials built using experimental data on stable isotopes is widespread. Although successful in their dedicated domain, it is unclear whether these phenomenological potentials can provide reliable predictions for unstable isotopes. To address this problem, optical potentials based on microscopic nuclear structure input calculations prove to be crucial and are an important current line of research. In this work we present an explicit implementation of the Feshbach formalism for the systematic derivation of optical potentials using input from nuclear structure models. Numerical tools for the derivation of Green's functions associated with nonlocal potentials are presented. The new optical potential, based on the valence shell model, is applied to the calculations of $n+{}^{24}\mathrm{Mg}$ elastic scattering and yields a close agreement with the experimental data.

We develop a calculation method for describing the direct and pre-equilibrium processes in neutron-induced reactions based on the framework of noniterative finite amplitude method (FAM) and distorted-wave Born approximation (DWBA). The noniterative FAM is used to derive equations of quasiparticle random-phase approximation for neutron-induced inelastic scatterings to both the discrete and continuum states in a consistent manner. The Skyrme force is employed as an interaction between the projectile neutron with nucleons inside the target nucleus. We apply this method to the neutron-induced reaction on ${}^{208}\mathrm{Pb}$. We demonstrate that the calculated differential inelastic scattering cross-sections to low-lying states reproduce available experimental data without any phenomenological parameters that are often introduced in conventional DWBA calculations. The calculated double differential cross-section to the continuum state also agrees with the experimental data in the energy region relevant to the direct and pre-equilibrium processes. These results are used to investigate the spin distribution of the populated states in the residual nucleus.

We present a method to calculate neutron scattering cross sections for deformed nuclei using many-body wave functions described with multiple reference states. Nuclear states are calculated with the generator coordinate method using a low energy effective Hamiltonian. Using these states, a nonlocal and energy dependent optical potential is consistently constructed, allowing one to directly investigate the role of nuclear structure properties in nuclear scattering. The case of neutron scattering on ${}^{24}\mathrm{Mg}$ is presented. The results are compared to experiment and to phenomenological optical potentials at energies below 13 MeV, demonstrating the importance of low-energy collectivity in elastic and nonelastic scattering.

Background: The use of theranostic isotope pairs is an emerging field in contemporary medicine, yet their clinical availability is often constrained by production and supply chain challenges. Among them, ${}^{67}\mathrm{Cu}$ has gained significant interest as the therapeutic counterpart in the ${}^{67}\mathrm{Cu}\text{/}{}^{64}\mathrm{Cu}$ theranostic pair. Traditional production methods rely on reactor-based or charged-particle accelerator routes, both of which face limitations in scalability and isotopic purity.

Purpose: This work investigates the photonuclear production of ${}^{67}\mathrm{Cu}$ via the ${}^{68}\mathrm{Zn}(\gamma ,p){}^{67}\mathrm{Cu}$ reaction, using high-energy bremsstrahlung photons from an electron accelerator. In addition to quantifying production yields and radionuclidic purity, the work provides spectrum-weighted average cross sections for multiple emission channels, serving as a test of reaction modeling under broad excitation conditions. The results inform both the optimization of ${}^{67}\mathrm{Cu}$ production for theranostic applications and the improvement of nuclear reaction theory in an underexplored energy regime.

Method: Natural zinc targets were irradiated using bremsstrahlung photons from an 855 MeV electron beam at the Mainzer Mikrotron (MAMI). The photon energy spectrum was measured via tagging and extended beyond the tagged range using Geant4 simulations. Reaction products were identified via gamma-ray spectroscopy, and spectrum-weighted average cross sections were derived. Theoretical cross sections were calculated using talys, incorporating giant dipole resonance and quasideuteron components of the photon absorption cross section. A model sensitivity analysis was performed to assess the influence of level density and photon strength function variations on the predicted cross sections.

Results: A production yield of 0.410(16) MBq $\mu {\mathrm{A}}^{-1}{\mathrm{h}}^{-1}{\mathrm{g}}^{-1}$ was measured for ${}^{67}\mathrm{Cu}$, with corresponding average cross sections determined from the bremsstrahlung spectrum. Yields and average cross sections were also measured for coproduced isotopes including ${}^{64}\mathrm{Cu}, {}^{61}\mathrm{Cu}, {}^{60}\mathrm{Cu}, {}^{69m}\mathrm{Zn}, {}^{65}\mathrm{Zn}, {}^{63}\mathrm{Zn}$, and ${}^{62}\mathrm{Zn}$. The relatively low production of nontarget isotopes indicates that the ${}^{68}\mathrm{Zn}(\gamma ,p){}^{67}\mathrm{Cu}$ reaction proceeds with favorable radionuclidic selectivity under the applied irradiation conditions. talys calculations showed good agreement with the ($\gamma ,p$) channel, but neutron-emission and multi-particle channels were systematically overestimated. Model sensitivity points to assumptions in the photon strength function and level density, as well as the absence of explicit pion-resonance dynamics at high energies, as likely contributors to this behavior.

Conclusions: Photonuclear reactions provide a promising approach for high-purity ${}^{67}\mathrm{Cu}$ production. Theoretical discrepancies highlight limitations in current reaction models, especially for ($\gamma ,\mathit{xn}$) channels at high energy. Beyond this, the scalability of electron-driven methods, enabled by existing high-current infrastructure and compact low-energy linacs, offers flexibility for both centralized and decentralized isotope production. Continued development of accelerator technologies and nuclear models will be essential for expanding access to photonuclear-produced medical isotopes.

We present a calculation of the nuclear matrix element for the two-neutrino double-beta ($2\nu \beta \beta$) decay of ${}^{48}\mathrm{Ca}\to {}^{48}\mathrm{Ti}$ using a post-Hartree-Fock (HF) density functional theory–based no-core configuration-interaction framework developed by our group [Phys. Rev. C 94, 024306 (2016)]. In the present calculation, we utilize a variant of the approach that restores rotational symmetry and mixes states projected from self-consistent mean-field configurations obtained by solving the HF equations with the density-independent local Skyrme interaction. Our calculations yield $|{\mathcal{M}}_{2\nu \beta \beta }|=0.056\left(6\right){\mathrm{MeV}}^{-1}$ for the nuclear matrix element describing this process. This result is in very good agreement with shell-model studies—for example, with the calculations by Horoi et al. [Phys. Rev. C 75, 034303 (2007)], which yielded $0.054\left(0.064\right){\mathrm{MeV}}^{-1}$ for the GXPF1A (GXPF1) interactions, respectively. It is also in a reasonable agreement with the most recent experimental estimate from the review by Barabash, which is $0.068\left(6\right){\mathrm{MeV}}^{-1}$, assuming quenching $q{g}_{\text{A}}\approx 1$. The consistency of our prediction with the shell-model results increases our confidence in the nuclear modeling of this second-order, very rare process which is of paramount importance for further modeling of the neutrinoless double-beta ($0\nu \beta \beta$) decay process.