Theoretical analysis and predictions for the double electron capture of $^{124}$Xe

This paper presents a comprehensive theoretical analysis of the two-neutrino double electron capture in $^{124}$Xe by improving nuclear and atomic structure calculations, which yields refined nuclear matrix elements, predicts specific capture fractions for various decay channels (notably 74% for the KK channel and 24% for the cumulative KL$_1$-KO$_1$ channels), and provides updated atomic relaxation energies for background modeling in liquid Xenon experiments.

The Gist

Imagine the atom as a tiny, bustling city. Inside this city, the nucleus is the city hall, packed with protons and neutrons. Usually, these citizens are very stable, but sometimes, they decide to rearrange themselves to become more comfortable.

This paper is about a very rare and specific "rearrangement" event happening in a city called Xenon-124. In this event, the city hall decides to catch two of its own residents (electrons) from the outer neighborhoods and pull them inside the nucleus. When this happens, the city transforms into a new city called Tellurium-124, and it spits out two tiny, invisible messengers called neutrinos.

Scientists call this the Double Electron Capture (specifically the two-neutrino version, or $2\nu$ECEC). It's like a double-dip in a pool, but instead of water, it's subatomic particles.

Here is what the researchers did, explained simply:

1. Building a Better Blueprint (The Theory)

In the past, scientists tried to predict how often this "double-dip" happens, but their blueprints were a bit rough. They missed some details about how the electrons move and how the nucleus reacts.

The authors of this paper decided to build a much more precise blueprint.

• The "Taylor Expansion" Analogy: Imagine trying to describe the path of a car. A simple description might just say "it goes forward." A better description adds "it accelerates." The best description adds "it accelerates, then turns slightly, then slows down." The authors used a mathematical tool called a "Taylor expansion" to add these extra layers of detail (up to the fourth power of energy). This allowed them to see the "turns and slows" of the decay process that previous models missed.
• The "New Ratios": Because they added these extra details, they discovered new ways to compare different parts of the process (called $\xi$ ratios). Think of these as new checkpoints on a race track that can help scientists understand the race better later on.

2. Looking at the Neighborhood (The Atomic Part)

To calculate how likely this event is, you need to know exactly where the electrons live.

• The "Pauli Blocking" Metaphon: Imagine a crowded elevator. If the elevator is full, you can't just push someone else in; you have to wait for someone to get off. In the nucleus, the "innermost" spots are like a full elevator. The authors realized that the electrons being captured can't just go anywhere; they are blocked by the other electrons already there. They accounted for this "crowding" rule, which changes the calculation.
• Expanding the Search: Previous studies only looked at the two closest neighborhoods to the nucleus (called K and L1 shells). The authors said, "Let's look at all the neighborhoods, even the ones further out (up to the O shell)." They found that while the outer neighborhoods are less likely to be captured, they still contribute to the total event.

3. Simulating the City Hall (The Nuclear Part)

The nucleus is the hardest part to simulate because it's a chaotic crowd of particles. The authors used two different "simulation engines" to predict how the nucleus behaves:

• Engine A (ISM): This is like a detailed, room-by-room simulation. They ran the simulation with different rules (called "Hamiltonians") to see if the results held up. They found that when they included all the possible "intermediate" steps the nucleus takes, the predicted "strength" of the event was lower than what older, simpler models suggested.
• Engine B (pn-QRPA): This is a different type of simulation. They adjusted the settings of this engine until it matched the real-world data we already have. They found that their new, more careful calculation gave a much smaller "strength" value than previous attempts using this engine.

4. The Results: What Did They Find?

By combining their better blueprint, their detailed neighborhood map, and their two simulation engines, they made several predictions:

• The Main Event (KK Channel): They predict that about 74% of the time, the two electrons will be caught from the very closest neighborhood (the K-shell). This is slightly different from the 72.4% used in previous experiments, a small but important tweak.
• The "Next Best" Events: They predict that about 19% of the time, one electron comes from the closest neighborhood and the other from the next closest (KL1).
• The "Cumulative" Prediction: If you add up all the slightly less common events (from KL1 down to KO1), they make up about 24% of the total. This is roughly one-third of the main event.
• The "Relaxation" Energy: When the electrons are caught, the new city (Tellurium) is excited and needs to calm down. It does this by releasing energy (like X-rays). The authors calculated exactly how much energy is released for each type of capture. This is like giving scientists a specific "fingerprint" of energy to look for in their detectors.

Why Does This Matter?

The paper doesn't claim to cure diseases or power cities. Instead, it acts as a refined map for explorers.

Large experiments using liquid Xenon (like those looking for Dark Matter) are constantly watching for this specific "double-dip" event. However, this event can look like "background noise" that confuses the data. By providing a more accurate map of exactly how often this happens, what energy it releases, and which "neighborhoods" the electrons come from, the authors help experimentalists distinguish between a real signal and background noise.

In short, they took a blurry, low-resolution photo of a rare atomic event and turned it into a high-definition, 3D model, helping scientists know exactly what to look for in their detectors.

Technical Summary

Technical Summary: Theoretical Analysis and Predictions for the Double Electron Capture of $^{124}$Xe

Problem Statement
The double electron capture (ECEC) process, particularly the two-neutrino mode ($2\nu$ECEC), is a rare nuclear transition of significant interest for probing nuclear structure and neutrino properties. While the $2\nu\beta^-\beta^-$ decay has been extensively studied, ECEC modes have received less attention due to lower transition probabilities. However, $^{124}$Xe is a unique case as it possesses the largest Q-value among isotopes capable of undergoing ECEC transitions, making it a primary target for large-scale liquid xenon experiments (e.g., those searching for Dark Matter and neutrinoless double-beta decay).

Previous theoretical calculations for the $2\nu$ECEC half-life of $^{124}$Xe suffered from limitations in both nuclear and atomic modeling. Specifically, earlier works often relied on the closure approximation for nuclear matrix elements (NMEs), considered only K and L$_1$ shell captures, and utilized less rigorous treatments of atomic screening and Pauli blocking. With recent experimental measurements of the $^{124}$Xe half-life becoming available, a more rigorous and complete theoretical description is required to interpret these data and predict unobserved decay channels.

Methodology
The authors provide a comprehensive theoretical framework improving both the formalism and the specific calculations for $^{124}$Xe:

1. Improved Formalism: The decay rate is derived using a Taylor expansion method, retaining terms up to the fourth power in lepton energies. This approach introduces higher-order corrections and defines new NME ratios, $\xi_{31}^{2\nu\text{ECEC}}$ and $\xi_{51}^{2\nu\text{ECEC}}$, which depend on the energy denominators of intermediate $1^+$ states.
2. Nuclear Structure Calculations:
   • Interacting Shell Model (ISM): Calculations were performed in the $jj55$ model space (orbitals $0g_{7/2}, 1d_{5/2}, 1d_{3/2}, 2s_{1/2}, 0h_{11/2}$). Unlike previous studies that assumed single-state dominance (SSD), this work performs a full summation over intermediate $1^+$ states using a strength function approach. Two effective Hamiltonians (GCN5082 and SVD) were employed with quenching factors ($q_H$) calibrated to reproduce $2\nu\beta^-\beta^-$ data for $^{136}$Xe. Truncations allowing up to four nucleon excitations from lower orbitals were included.
   • Proton-Neutron Quasiparticle Random Phase Approximation (pn-QRPA): Calculations utilized isospin restoration. The particle-particle interaction strength ($g_{pp}^{T=0}$) was adjusted to reproduce the experimental half-life of $^{124}$Xe. The study utilized realistic nucleon-nucleon potentials (Argonne V18 and CD-Bonn) and accounted for BCS vacuum overlap.
3. Atomic Structure and Phase Space Factors (PSFs):
   • The Dirac-Hartree-Fock-Slater (DHFS) self-consistent framework was used to generate electron wave functions.
   • Improvements include rigorous treatment of atomic screening, diffuse nuclear surface corrections, realistic nuclear charge density, and electron exchange-correlation effects.
   • Crucially, the calculation accounts for Pauli blocking of innermost nucleon states and extends the capture summation to all available s-wave electrons (up to the O$_1$ subshell), rather than limiting to K and L$_1$ shells as in prior studies.
   • Atomic relaxation energies for the final $^{124}$Te atom were computed to assist in background modeling.

Key Results

• Nuclear Matrix Elements (NMEs):
  • The ISM calculations yield NMEs that are robust against changes in truncation and effective Hamiltonian. The full summation over intermediate states (without SSD) reduced the NME for the GCN5082 Hamiltonian by approximately 25% compared to truncated approaches.
  • The pn-QRPA results, when constrained by the experimental half-life, yield NME values significantly lower than previous pn-QRPA calculations (which often relied on different adjustment strategies for $g_{pp}$). The new NMEs for $^{124}$Xe are found to be comparable to those for the $2\nu\beta^-\beta^-$ decay of $^{128,130}$Te, reflecting similar pairing features.
• Half-Life Predictions:
  • Using the ISM, the calculated total half-life for $^{124}$Xe agrees with the experimental value of $(1.1 \pm 0.2_{\text{stat}} \pm 0.1_{\text{sys}}) \times 10^{22}$ yr within a factor of two.
  • The inclusion of Taylor expansion terms and the refined PSFs contributed to bringing theoretical predictions closer to experimental data.
• Capture Fractions and Channels:
  • The predicted capture fraction for the dominant KK channel is 74.1%, differing from the 72.4% used in previous experimental analyses. This difference arises from the exclusive consideration of s-wave electrons in the theoretical model versus the inclusion of all electrons in the experimental signal model.
  • The next most probable channel is KL$_1$ with a fraction of ~19%.
  • The cumulative capture fraction for the KL$_1$ through KO$_1$ channels is predicted to be approximately 24%.
• Atomic Relaxation Energies:
  • Precise relaxation energies were calculated for various hole configurations (e.g., KK at 64.62 keV, KL$_1$ at 37.05 keV). These values are intended for use in background modeling for liquid xenon experiments.

Significance and Claims
The paper claims to provide the first complete theoretical description of the $2\nu$ECEC in $^{124}$Xe that simultaneously improves the nuclear structure treatment (via full summation and isospin restoration) and the atomic physics (via DHFS, Pauli blocking, and extended orbital capture).

The authors assert that their ISM results are consistent with experimental data, validating the reliability of the calculated NMEs and the newly predicted capture fractions. A primary contribution is the prediction that the cumulative KL$_1$-KO$_1$ channels (representing ~24% of decays) should be experimentally observable in the energy interval of 31.93–37.05 keV. Furthermore, the work suggests that the ratio of partial half-lives (e.g., $T_{KK}^{1/2}/T_{KL_1}^{1/2}$) could serve as an experimental constraint on the $\xi_{31}^{2\nu\text{ECEC}}$ parameter, offering a new avenue to probe the interplay between low- and higher-lying intermediate states. The study emphasizes that previous theoretical estimates of half-lives were often incompatible with the most accurate PSFs due to the use of closure approximations and incomplete orbital summations.