Penning-trap measurement of the Q value of electron capture in $^{163}$Ho for the determination of the electron neutrino mass

The Standard Model of particle physics does not account for neutrino mass, yet neutrino oscillation experiments have proven neutrinos possess mass. The absolute scale of this mass, however, remains unknown. This is a crucial question in nuclear and particle physics, cosmology, and beyond-Standard-Model theories. Current approaches to determine the neutrino mass include kinematic studies of weak decays, where the neutrino itself isn't directly detected. This method relies on energy and momentum conservation, and offers the most model-independent approach. The most stringent direct limit of 0.8 eV c² (90% confidence level) comes from the KATRIN collaboration’s analysis of tritium β-decay. Another promising approach utilizes electron capture in $^{163}$Ho, investigated through microcalorimetry by the ECHO and HOLMES collaborations. These experiments aim for sub-eV sensitivity, but require an independently measured Q value for accurate systematic uncertainty assessment. High-precision Penning-trap mass spectrometry offers a direct, independent Q value measurement, crucial for improving the accuracy of neutrino mass determination. This research employs this method to achieve a substantial increase in precision.

Several experiments have attempted to measure the neutrino mass, with varying levels of success and precision. Early calorimetric approaches, like MANU and MIBETA using tritium β-decay, yielded upper limits of 19 and 15 eV c² (90% confidence level), respectively. The KATRIN experiment, using tritium β-decay, currently provides the most stringent direct limit at 0.8 eV c². Complementary approaches using electron capture in $^{163}$Ho are being pursued by ECHO and HOLMES, employing microcalorimetry. While these aim for sub-eV sensitivity, they depend on accurate Q values. Previous Penning-trap measurements of the $^{163}$Ho Q value have been significantly less precise than the microcalorimetry experiments, highlighting the need for an independent and highly accurate Q value determination using Penning-trap mass spectrometry.

The PENTATRAP Penning-trap mass spectrometer was used to measure the free-space cyclotron frequency ratio (R_q⁺) of highly charged ions (HCIs) of $^{163}$Ho and $^{163}$Dy. The experiment used a compact electron beam ion trap (TIP-EBIT) to produce HCIs from a limited sample of $^{163}$Ho (approximately 2 × 10⁵ atoms). A Bradbury–Nielsen gate selected specific charge states (n = 38, 39, 40). The HCIs were then transported to a stack of five Penning traps within a 7 T superconducting magnet. Two traps were equipped with a non-destructive image current detection system to measure the ions' motional frequencies. The measurement involved shuttling ions between traps to alternate between $^{163}$Ho and $^{163}$Dy measurements, enabling simultaneous frequency measurements and mitigating systematic errors. The data analysis involved linearly interpolating the cyclotron frequencies to account for magnetic field drift, and additional uncertainties were considered to account for nonlinear drift behaviour. The Q value was calculated using the equation Q = m_ref(R_q⁺ - 1) + ΔE_b⁺, where m_ref is the reference mass of the $^{163}$Dy HCI, and ΔE_b⁺ is the binding energy difference between the Ho and Dy HCIs, calculated using the configuration interaction (CI) and multiconfiguration Dirac–Hartree–Fock (MCDHF) methods. To further refine the binding energy calculation, a third method, multiconfiguration Dirac-Fock general matrix elements (MCDFGME), was employed as a validation and consistency check against the other two methods.

The PENTATRAP experiment measured the free-space cyclotron frequency ratios for three charge states of $^{163}$Ho and $^{163}$Dy HCIs (n = 38, 39, 40). These ratios, combined with theoretical calculations of the electronic binding energy differences (ΔE_b⁺), yielded Q values of 2,863.4 ± 1.5 eV c² (n = 38), 2,863.2 ± 0.9 eV c² (n = 39), and 2,863.2 ± 0.9 eV c² (n = 40). The agreement between the Q values obtained from different charge states strongly suggests the absence of systematic errors due to metastable electronic states. The final Q value, calculated as the weighted average of the three charge states, is 2,863.2 ± 0.6 eV c⁻². This represents a more than 50-fold improvement in precision compared to the previous state-of-the-art measurement. The systematic uncertainties were carefully analyzed and mitigated through the experimental design and data analysis techniques. The study accounted for various systematic shifts, including relativistic shift, field anharmonicities/imperfections, image charge shift, and dip lineshape, all of which were found to be smaller than 10⁻¹².

This high-precision Q value determination significantly advances the field of neutrino mass measurement. The obtained Q value (2,863.2 ± 0.6 eV c⁻²) is consistent with, but substantially more precise than, previous measurements from both Penning-trap mass spectrometry and microcalorimetry. This improved precision directly translates to enhanced sensitivity in neutrino mass determination experiments using $^{163}$Ho electron capture. The consistency across different charge states minimizes concerns about potential systematic uncertainties associated with metastable electronic states. The results provide a robust and reliable Q value for microcalorimetric experiments such as those conducted by ECHO and HOLMES, enabling more accurate determination of the electron neutrino mass at the sub-electronvolt level.

This study presents a high-precision measurement of the Q value for $^{163}$Ho electron capture using Penning-trap mass spectrometry. The 50-fold improvement in precision (2,863.2 ± 0.6 eV c⁻²) over previous measurements eliminates a significant source of uncertainty in electron neutrino mass experiments based on $^{163}$Ho decay. This advance paves the way for more accurate determinations of the neutrino mass, potentially reaching sub-electronvolt sensitivity. Future work could focus on further refinements of both the experimental techniques and theoretical calculations to further reduce uncertainties and improve the accuracy of the Q value.

While the study significantly reduces uncertainties, there remain some limitations. The theoretical calculation of binding energy differences relies on sophisticated computational methods and approximations; uncertainties associated with these methods might slightly affect the overall Q value precision. The measurement relies on a limited amount of the rare $^{163}$Ho isotope, which might limit the statistical accuracy achievable.