Two-neutrino double-weak decays of $^{126}$Xe and $^{134}$Xe from different many-body methods

Technical Summary: Two-neutrino double-weak decays of 126Xe and 134Xe from different many-body methods

Problem Statement
The paper addresses the need for precise theoretical predictions of the nuclear matrix elements (NMEs) and half-lives for two specific two-neutrino ($2\nu$) double-weak decay processes: the two-neutrino double-electron capture ($2\nu$ECEC) of $^{126}\text{Xe}$ and the two-neutrino double-beta ($2\nu\beta\beta$) decay of $^{134}\text{Xe}$. While $2\nu$ modes are permitted by the Standard Model, they are extremely rare and represent the longest half-lives observed in laboratories. Accurate predictions for these nuclei are critical because $2\nu$ measurements provide essential constraints on the nuclear many-body methods used to predict the NMEs for neutrinoless double-beta decay ($0\nu\beta\beta$), a process that would violate lepton number conservation. Prior to this work, neither $^{126}\text{Xe}$ nor $^{134}\text{Xe}$ had been experimentally detected, and theoretical calculations for these specific transitions were limited or inconsistent.

Methodology
The authors calculate NMEs and corresponding half-lives using four distinct, widely-used many-body methods, explicitly estimating theoretical uncertainties for each:

1. Proton-Neutron Quasiparticle Random-Phase Approximation (pnQRPA):

   • Utilizes spherical nuclei with large no-core single-particle bases (lowest 25 orbitals in a Woods-Saxon potential).
   • Parameters ($g_{ph}$ and $g_{pp}$) are fitted to Gamow-Teller giant resonance energies and neighboring $2\nu$ half-lives ($^{124}\text{Xe}$ and $^{136}\text{Xe}$) using a partial isospin-restoration scheme.
   • Uncertainties are driven by the quenching factor ($q$) of the axial-vector coupling and the particle-particle parameter $g_{pp}$.

2. Nuclear Shell Model (NSM):

   • Employs the configuration interaction method within an active valence space comprising orbitals $1d_{5/2}, 0g_{7/2}, 2s_{1/2}, 1d_{3/2},$ and $0h_{11/2}$.
   • Uses two effective interactions: GCN5082 and QX.
   • NMEs are computed via the Lanczos method, summing over intermediate states until convergence.
   • A quenching factor is applied, inferred from systematic comparisons with measured $2\nu$ decays in the xenon region. The uncertainty in $q$ is identified as the dominant source of theoretical error.

3. Microscopic Interacting Boson Model (IBM-2):

   • Maps the fermionic Hamiltonian to a boson space, fitting parameters to low-lying spectroscopic properties.
   • Calculations employ the closure approximation, evaluated under two scenarios: Single-State Dominance (SSD) and Higher-State Dominance (HSD).
   • A phenomenological quenching factor is introduced to account for omitted correlations and the closure approximation. Three values are considered: bare ($q=1$), moderate ($q=0.788$), and maximum ($q=A^{-0.18}$).

4. Effective Field Theory (EFT) for Heavy Nuclei:

   • Formulated in terms of nucleon and phonon degrees of freedom coupled to an even-even spherical core.
   • Operates under the SSD approximation.
   • Unlike other methods, EFT does not require an ad-hoc quenching factor; low-energy constants (LECs) encode higher-energy physics.
   • Uncertainties are systematically quantified via power counting and truncation errors. The LECs are fitted to hypothetical comparative half-lives ($\log ft$) derived from neighboring isotopic chains.

Key Contributions

• First Comprehensive NSM Study: This work presents the first Nuclear Shell Model study of the $2\nu$ECEC of $^{126}\text{Xe}$ and the $2\nu\beta\beta$ of $^{134}\text{Xe}$.
• Systematic Uncertainty Estimation: The authors place significant emphasis on quantifying theoretical uncertainties for each method, particularly regarding quenching factors and interaction parameters, rather than providing single-point predictions.
• Cross-Method Comparison: The study provides a unified comparison of results across pnQRPA, NSM, IBM-2, and EFT for these specific isotopes, a comparison not previously available in the literature.
• EFT Application: The application of EFT for heavy nuclei to these specific double-weak decays, including a detailed breakdown of truncation and LEC uncertainties.

Results

• Half-Life Predictions:
  • For $^{126}\text{Xe}$ ($2\nu$ECEC), predicted half-lives generally span $\approx 10^{24} - 10^{26}$ years (excluding the IBM-2 SSD case). The pnQRPA, NSM, and EFT results are generally consistent within their respective uncertainties.
  • For $^{134}\text{Xe}$ ($2\nu\beta\beta$), predicted half-lives span $\approx 10^{23} - 10^{25}$ years (excluding IBM-2 SSD).
  • In all calculations, the lower range of the predicted $^{134}\text{Xe}$ half-life is shorter than $T_{1/2} \approx 2 \times 10^{24}$ years.
• Consistency: Across all methods (except the IBM-2 SSD scenario), the predictions are consistent with each other when theoretical uncertainties are included. The results also align with semi-empirical formula predictions and previous calculations for neighboring isotopes.
• Discrepancies: The IBM-2 SSD predictions yield significantly shorter half-lives compared to other methods and are largely disfavored by current experimental lower limits.
• Running Sums: Analysis of the running sums of NMEs reveals that while pnQRPA contributions extend to higher energies ($\approx 15$ MeV) with significant cancellations, and NSM contributions are concentrated at lower energies ($\approx 5-10$ MeV) and are constructive, the final integrated NMEs for both methods converge to similar values.

Significance and Claims
The paper claims that its primary significance lies in providing reliable theoretical benchmarks for future experimental searches. The authors state that:

• The predicted half-lives for $^{134}\text{Xe}$, particularly the lower bounds, may be within the reach of next-generation experiments (projected sensitivity $\approx 1.7 \times 10^{24}$ years).
• For $^{126}\text{Xe}$, the predicted half-lives are typically an order of magnitude longer than those for $^{134}\text{Xe}$, making detection more challenging but still a target for future sensitivity improvements.
• The consistency among different many-body methods (when uncertainties are accounted for) supports the reliability of these approaches for predicting NMEs, which is crucial for interpreting potential neutrinoless decay signals.
• The work highlights that reducing theoretical uncertainties, particularly in the EFT (via GT strength measurements) and in pnQRPA/NSM (via better constraints on quenching), is essential for refining predictions.

The authors do not claim to have detected these decays or to have resolved the neutrinoless decay problem, but rather to have provided the necessary theoretical groundwork to guide and interpret upcoming experimental data.