Large-scale shell-model investigation of $2\nu$ECEC in $^{132}$Ba and $^{78}$Kr

This paper presents a large-scale shell-model investigation of two-neutrino double electron capture in $^{132}$Ba and $^{78}$Kr, providing updated nuclear matrix elements and half-life predictions based on validated effective interactions to support future experimental efforts.

The Gist

The Big Picture: Catching Invisible Ghosts

Imagine the atomic nucleus as a tiny, crowded dance floor. Usually, the dancers (protons and neutrons) are very stable and don't change partners. But sometimes, two dancers decide to switch places at the exact same time. This is a rare event called Double Electron Capture.

In this specific "dance," two protons in the nucleus grab two electrons from the atom's outer shell and turn into neutrons. Because this happens so rarely, it takes a incredibly long time—trillions of years—to see it happen once. Scientists want to find out exactly how long it takes (the half-life) because it helps them understand the fundamental rules of the universe, like the nature of neutrinos (tiny, ghost-like particles).

The authors of this paper are like architects and engineers. They didn't build a new machine to catch these events; instead, they built a super-detailed computer simulation to predict how the dance floor behaves and how long the wait should be.

The Two Stars of the Show: 132Ba and 78Kr

The researchers focused on two specific atoms (nuclei) that are candidates for this rare dance:

1. Barium-132 (132Ba): A heavy atom that scientists suspect might do this dance, but no one has caught it in the act yet. They only know it could happen based on old geological clues.
2. Krypton-78 (78Kr): An atom where scientists have recently confirmed the dance happens, but the measurements are still a bit fuzzy.

How They Did It: The "Lego" Simulation

To predict what happens, the scientists used a method called the Large-Scale Shell Model.

• The Analogy: Imagine trying to predict how a complex structure made of billions of Lego bricks will hold up. You can't just guess; you need to know exactly how every single brick connects to its neighbors.
• The Tool: The scientists used a massive digital "Lego set" (called an effective interaction) that tells the computer how protons and neutrons interact.
  • For Barium-132, they used a set called SN100PN.
  • For Krypton-78, they used a set called GWBXG.

The Upgrade: In their previous work on Krypton, they only looked at the "bottom floor" of the Lego building. In this new study, they expanded the model to include the "upper floors" (higher energy levels). This is like realizing that to understand how a skyscraper sways in the wind, you have to look at the top floors, not just the foundation.

Checking the Blueprint: Did the Simulation Work?

Before trusting their predictions about the rare dance, the scientists had to make sure their simulation was accurate. They did this by checking the "normal" behavior of the atoms involved:

• The Energy Levels: They checked if the computer predicted the correct "vibrations" or energy states of the atoms.
• The Shape: They checked if the atoms were shaped like spheres or slightly squashed eggs (deformation).

The Result: The computer simulation matched real-world experimental data almost perfectly. It was like building a scale model of a bridge and seeing that it held up exactly the same way the real bridge does. This gave them confidence that their predictions for the rare dance were also reliable.

The Main Findings: The "Wait Time" Predictions

1. For Barium-132 (The Mystery Candidate)

Since no one has seen Barium-132 do this dance yet, the scientists provided a theoretical baseline.

• The Prediction: They calculated that if you wait about 7.33 × 10²⁴ years (that's a 7 followed by 24 zeros!), you might see it happen.
• Why it matters: This is a "target" for future experiments. It tells scientists, "Don't look for it in 100 years; you need to build detectors that can wait for trillions of years." Their calculation is much longer than the current minimum limit scientists have set, which means the search is still very much open.

2. For Krypton-78 (The Confirmed Candidate)

Scientists have already seen Krypton-78 do this dance, but the measurements vary.

• The Prediction: The new, more detailed simulation predicts a wait time of 8.78 × 10²² years.
• The Improvement: In their old study (with the smaller Lego set), they predicted a slightly different time. By adding the "upper floors" to their model, their new prediction is much closer to what recent experiments have actually observed. It's like upgrading from a blurry photo to a high-definition image; the picture is now clearer and more accurate.

The "Volume Knob" (The Axial Coupling)

One tricky part of the simulation is that the computer doesn't know every single tiny force in the universe. To fix this, the scientists use a "volume knob" called the effective axial coupling constant ($g_{eff}^A$).

• The Analogy: Imagine you are recording a song, but your microphone misses some of the high notes. You turn up the volume (the knob) to compensate for what the microphone missed.
• The scientists tested different "volume settings" to see how it changed the predicted wait time. Even with different settings, their results remained consistent with what we know so far.

Conclusion: What Did They Learn?

The paper concludes that:

1. The Simulation is Solid: Their computer models are very good at describing how these atoms behave.
2. Barium-132: They have provided the best theoretical guess yet for how long we have to wait to see it decay. This helps experimentalists know how sensitive their detectors need to be.
3. Krypton-78: By looking at a bigger, more complex model, they improved their prediction, making it match real-world data better than before.

In short, these scientists built a better map of the atomic dance floor. They haven't caught the dancers yet (for Barium), but they have a much better idea of where and when to look, and for Krypton, their map is now much more accurate than the old one.

Technical Summary

Technical Summary: Large-scale shell-model investigation of 2νECEC in 132Ba and 78Kr

Problem Statement
The study of second-order weak interaction processes, specifically two-neutrino double electron capture (2νECEC), is essential for probing fundamental neutrino properties and testing nuclear structure models. While the two-neutrino double-beta ($2\nu\beta\beta$) decay is well-established, the 2νECEC mode has been less explored due to its typically longer half-lives and lower Q-values. Reliable theoretical predictions of Nuclear Matrix Elements (NMEs) and half-lives are critical for guiding future experimental searches, particularly for candidate nuclei like $^{132}$Ba, where only a lower limit on the half-life exists, and $^{78}$Kr, which has recently been observed. Existing theoretical estimates exhibit significant model dependence, necessitating robust, independently validated methods to constrain these rare decay modes.

Methodology
The authors employ large-scale nuclear shell-model (NSM) calculations to investigate the 2νECEC process in $^{132}$Ba and $^{78}$Kr.

• Effective Interactions: The study utilizes the SN100PN effective interaction for $^{132}$Ba and the GWBXG interaction for $^{78}$Kr.
• Model Spaces:
  • For $^{132}$Ba (parent), the model space includes the $0g_{7/2}1d_{2s}0h_{11/2}$ orbitals for both protons and neutrons, with $^{100}$Sn as an inert core. Truncations were applied to ensure computational feasibility, restricting configurations to specific particle-hole limits.
  • For $^{78}$Kr, the study extends previous work by expanding the neutron model space. While earlier calculations were limited to the $N=50$ shell closure, the current work includes one-particle-one-hole (1p-1h) excitations across $N=50$ into the $0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, and $2s_{1/2}$ orbitals.
• Computational Tools: The NUSHELLX code was used for one-body transition densities (OBTDs), and the KSHELL code calculated energy spectra and transition probabilities.
• Validation: The reliability of the interactions was first verified by comparing calculated spectroscopic properties (low-lying energy spectra and reduced quadrupole transition probabilities $B(E2)$) of the parent, intermediate, and granddaughter nuclei against experimental data.
• NME Calculation: The Gamow-Teller (GT) NMEs were calculated using the standard one-body GT operator. The cumulative contribution of the NME with respect to the $1^+$ state energies in the intermediate nuclei ($^{132}$Cs and $^{78}$Br) was analyzed to ensure convergence. Effective axial-vector coupling constants ($g^A_{eff}$) were applied to account for missing many-body correlations and two-body currents.

Key Contributions

1. First Shell-Model NME for $^{132}$Ba: This work presents the first shell-model calculation of the NME for the 2νECEC process in $^{132}$Ba.
2. Improved Model Space for $^{78}$Kr: The study provides an updated and improved shell-model estimate for $^{78}$Kr by extending the neutron model space beyond the $N=50$ shell closure, addressing limitations in previous configuration-limited studies.
3. Spectroscopic Validation: The authors demonstrate that the employed effective interactions successfully reproduce the experimental spectroscopic properties (energy levels and $B(E2)$ strengths) of the nuclei involved in the decay chains, establishing a reliable basis for the NME calculations.
4. Sensitivity Analysis: A sensitivity analysis was performed on the $^{132}$Ba calculation regarding the unconfirmed energy of the lowest $1^+$ state in the intermediate nucleus $^{132}$Cs.

Results

• Spectroscopy: The calculations show remarkable agreement with experimental data for $^{132}$Ba, $^{132}$Cs, $^{132}$Xe, $^{78}$Kr, $^{78}$Br, and $^{78}$Se. For $^{132}$Ba, the results suggest a moderately prolate deformation ($\beta_2 \approx +0.18$), while $^{78}$Kr exhibits an oblate intrinsic shape with shape mixing.
• $^{132}$Ba NME and Half-Life:
  • The cumulative NME saturates at a value of 0.0654 around 6.3 MeV in the intermediate nucleus.
  • Using $g^A_{eff} = 0.94$, the predicted half-life is $7.33 \times 10^{24}$ yr. This is approximately three orders of magnitude higher than the current experimental lower limit ($> 2.2 \times 10^{21}$ yr).
  • The calculation is robust provided the lowest $1^+$ state in $^{132}$Cs remains below 100 keV; significant suppression of the NME occurs only if this state lies above 500 keV.
• $^{78}$Kr NME and Half-Life:
  • The cumulative NME reaches a final value of 0.2311 after saturation, an improvement over the previous value of 0.1997 obtained with a restricted model space.
  • Using $g^A_{eff} = 0.8$, the predicted half-life is $8.78 \times 10^{22}$ yr.
  • This result is in reasonable agreement with the recently measured experimental value of $[1.9^{+1.3}_{-0.7} \pm 0.3] \times 10^{22}$ yr.
  • The study also estimates half-lives for competing positron-emitting channels ($2\nu\beta^+\beta^+$ and $2\nu EC\beta^+$) as $4.68 \times 10^{26}$ yr and $1.19 \times 10^{23}$ yr, respectively.

Significance and Claims
The authors claim that this work provides a baseline theoretical prediction for $^{132}$Ba to assist future experimental efforts in constraining this rare decay mode. For $^{78}$Kr, the study offers an updated and improved shell-model estimate relative to earlier studies with more restricted model spaces. The results are presented as reasonable within typical theoretical uncertainties, acknowledging that the use of effective axial couplings partially absorbs missing many-body effects (such as two-body currents) that are not explicitly included in the standard one-body operator. The paper emphasizes that while spectroscopic agreement validates the wave functions, the decay operator requires renormalization via $g^A_{eff}$. The authors conclude that their findings support the reliability of the underlying nuclear wave functions for evaluating 2νECEC NMEs, though they note that future work with larger model spaces and improved isospin-symmetry interactions could further refine these predictions.