Testing Lepton Wave Function Factorization in $^{71}\text{Ge}$ Electron Capture

Technical Summary: Testing Lepton Wave Function Factorization in $^{71}$Ge Electron Capture

Problem Statement
High-purity germanium detectors are critical for rare-event searches (neutrino and dark matter) but suffer from background activation producing the isotope $^{71}$Ge. This isotope undergoes electron capture (EC) to the ground state of $^{71}$Ga, a process that serves as both a background source and a calibration tool. A robust theoretical description of this EC process is essential for interpreting experimental data. While the standard theory of orbital EC typically factorizes the leptonic wave functions from the nuclear matrix element (expanding them at leading order within the nuclear volume), recent discussions suggest this factorization may be insufficient for interpreting certain experimental observations, particularly in the context of the "gallium anomaly." The paper investigates whether non-factorization effects—specifically the interplay between the nuclear transition density and exact leptonic wave functions—significantly alter the predicted EC rates and ratios for $^{71}$Ge.

Methodology
The authors develop a general framework for predicting $^{71}$Ge EC rates using exact Dirac-Hartree-Fock-Slater (DHFS) radial electron wave functions obtained from the RADIAL package, rather than relying on simplified approximations.

• Formalism: The capture rate is calculated as an incoherent sum over neutrino partial waves. The transition amplitude is treated as the exact spatial overlap of initial and final states over the weak interaction operator, integrating the nuclear transition density ($\rho^{TD}_{1,1}$) with the leptonic wave functions (electron and neutrino).
• Non-Factorized Treatment: Unlike the standard factorized approach where the nuclear structure cancels out in ratios, this work retains the full radial dependence of the transition density. The authors analyze the radial behavior of the leptonic kernels (products of electron and neutrino wave functions) for various atomic shells (K, L, M).
• Phenomenological Constraints: To address the complexity of calculating the nuclear transition density from first principles for mid-size nuclei, the authors adopt phenomenologically constrained transition densities (specifically Double-Gaussian and modified Double-Gaussian parametrizations) previously proposed in Ref. [7] to resolve the gallium anomaly.
• Atomic Corrections: The calculation includes exchange and overlap corrections ($B_x$) and shake-up/shake-off effects ($S_x$), evaluated using the specific electron wave functions derived in this work.

Key Contributions

1. Explicit Radial Amplitudes: The paper derives explicit radial amplitudes for all relevant atomic shells (K, $L_{1-3}$, $M_{1-5}$) by evaluating the dominant neutrino channel contributions. It identifies that $s_{1/2}$ and $p_{1/2}$ orbitals are dominated by the zero-order Bessel function ($j_0$), while higher shells involve higher-order Bessel functions ($j_1, j_2$, etc.) which are significantly suppressed near the origin.
2. Analysis of Factorization Validity: The authors demonstrate that the dominant shells (K, $L_1$, $L_2$, $M_1$, $M_2$) possess leptonic amplitudes that are effectively flat within the nuclear radius. Consequently, these shells probe similar radial moments of the nuclear transition density. In contrast, subdominant shells like $L_3$ exhibit a different radial dependence ($r^2$), making their effective strength sensitive to the specific shape of the transition density.
3. Re-evaluation of Ratios: The study re-evaluates the L/K, M/K, and M/L capture ratios using both the standard factorized leading-order (LO) approximation and the complete non-factorized scheme with the phenomenological transition densities.

Results

• Impact of Non-Factorization: The introduction of the full non-factorized scheme, even with transition densities that vary significantly in shape (e.g., extending beyond the nuclear radius), barely affects the predictions for the dominant L/K and M/K ratios. The differences between the factorized and non-factorized schemes remain within the theoretical uncertainties for these dominant ratios.
• Subdominant Shells: The non-factorized approach allows for the prediction of individual contributions from subdominant shells (e.g., $L_3$, $M_3$). While the specific profile of the transition density can modify these subdominant predictions by up to a factor of two, these contributions are highly suppressed (orders of magnitude smaller than dominant shells) and remain unmeasurable with current or near-future experimental sensitivity.
• Comparison with Data:
  • The theoretical predictions for the L/K ratio in the factorized scheme show a $3.6\sigma$ discrepancy with the current experimental world average.
  • Discrepancies for M/K and M/L ratios are approximately $2\sigma$.
  • However, the theoretical predictions are in excellent agreement with the latest experimental measurement by CONUS+ [29], which reports a precise M/K ratio of $0.0214(4)$ and an L/K ratio of $0.119(1)$.
  • The tension between the theoretical prediction and the older experimental world average persists even when accounting for the non-factorization effects (reducing the discrepancy only slightly from $3.6\sigma$ to $3.3\sigma$).

Significance and Claims
The paper concludes that the standard factorized allowed-LO approximation is robust for interpreting current measurements of $^{71}$Ge electron capture ratios. The analysis confirms that nuclear structure effects, mediated through the transition density, do not significantly alter the dominant capture ratios (L/K, M/K) because the dominant atomic shells share a similar radial dependence that effectively cancels out nuclear structure variations in the ratio.

The authors assert that the observed $3.6\sigma$ tension between their theoretical predictions and the older experimental world averages is unlikely to be resolved by nuclear structure effects or non-factorization mechanisms. Instead, the agreement with the recent CONUS+ data suggests that the older datasets may contain unaccounted systematic issues. The paper emphasizes that new high-precision measurements are essential to clarify the experimental picture, as the current data remains complex and contradictory. The work provides an up-to-date assessment of the theoretical and experimental status of $^{71}$Ge EC, validating the use of standard factorized models for dominant shells while offering a rigorous framework for future investigations into subdominant effects.