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Phenomenological Modeling of the $^{163}$Ho Calorimetric Electron Capture Spectrum from the HOLMES Experiment

F. Ahrens, B. K. Alpert, D. T. Becker, D. A. Bennett, E. Bogoni, M. Borghesi, P. Campana, R. Carobene, A. Cattaneo, A. Cian, H. A. Corti, N. Crescini, M. De Gerone, W. B. Doriese, M. Faverzani, L. Ferrari Barusso, E. Ferri, J. Fowler, G. Gallucci, S. Gamba, J. D. Gard, H. Garrone, F. Gatti, A. Giachero, M. Gobbo, A. Irace, U. Köster, D. Labranca, M. Lusignoli, F. Malnati, F. Mantegazzini, B. Margesin, J. A. B. Mates, E. Maugeri, R. Mezzena, E. Monticone, R. Moretti, A. Nucciotti, G. C. O'Neil, L. Origo, G. Pessina, S. Ragazzi, M. Rajteri, C. D. Reintsema, D. R. Schmidt, D. S. Swetz, Z. Talip, J. N. Ullom, L. R. Vale

Abstract

We present a comprehensive phenomenological analysis of the calorimetric electron capture (EC) decay spectrum of $^{163}$Ho as measured by the HOLMES experiment. Using high-statistics data, we unfold the instrumental energy resolution from the measured spectrum and model it as a sum of Breit-Wigner resonances and shake-off continua, providing a complete set of parameters for each component. Our approach enables the identification and tentative interpretation of all observed spectral features, including weak and overlapping structures, in terms of atomic de-excitation processes. We compare our phenomenological model with recent ab initio theoretical calculations, finding good agreement for both the main peaks and the spectral tails, despite the limitations of current theoretical and experimental precision. The model delivers an accurate description of the endpoint region, which is crucial for neutrino mass determination, and allows for a realistic treatment of backgrounds such as pile-up and tails of low-energy components. Furthermore, our decomposition facilitates the generation of Monte Carlo toy spectra for sensitivity studies and provides a framework for investigating systematic uncertainties related to solid-state and detector effects. This work establishes a robust foundation for future calorimetric neutrino mass experiments employing $^{163}$Ho, supporting both data analysis and experimental design.

Phenomenological Modeling of the $^{163}$Ho Calorimetric Electron Capture Spectrum from the HOLMES Experiment

Abstract

We present a comprehensive phenomenological analysis of the calorimetric electron capture (EC) decay spectrum of Ho as measured by the HOLMES experiment. Using high-statistics data, we unfold the instrumental energy resolution from the measured spectrum and model it as a sum of Breit-Wigner resonances and shake-off continua, providing a complete set of parameters for each component. Our approach enables the identification and tentative interpretation of all observed spectral features, including weak and overlapping structures, in terms of atomic de-excitation processes. We compare our phenomenological model with recent ab initio theoretical calculations, finding good agreement for both the main peaks and the spectral tails, despite the limitations of current theoretical and experimental precision. The model delivers an accurate description of the endpoint region, which is crucial for neutrino mass determination, and allows for a realistic treatment of backgrounds such as pile-up and tails of low-energy components. Furthermore, our decomposition facilitates the generation of Monte Carlo toy spectra for sensitivity studies and provides a framework for investigating systematic uncertainties related to solid-state and detector effects. This work establishes a robust foundation for future calorimetric neutrino mass experiments employing Ho, supporting both data analysis and experimental design.

Paper Structure

This paper contains 14 sections, 8 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Aim of the present work
  3. Beyond the single hole approximation
  4. Previous theoretical work and experimental data
  5. Methods
  6. Experimental set-up
  7. Data analysis
  8. Results
  9. Models
  10. Spectrum fit
  11. Discussion
  12. Conclusions
  13. Parameters for the fitted components
  14. Fit parameter correlations

Figures (11)

  • Figure 1: Calorimetric EC decay spectrum in the single-hole approximation, as described in derujula_calorimetric_1982. Dashed lines indicate the individual Breit-Wigner resonances from Eq. \ref{['eq:E_c-distr']}, shown without the phase space factor, ( i.e., omitting the prefactor before the summation in Eq. (\ref{['eq:E_c-distr']})). The right panel displays a zoom of the end-point region for $m_{\nu} = 0$ eV (green) and $m_{\nu} = 0.1$ eV (cyan).
  • Figure 2: Schematic representation of atomic excitations following EC: single-hole creation (left), double-hole via shake-up (center), and double-hole via shake-off (right). Here, $n=4,5,6$ denotes the principal quantum numbers of the core holes.
  • Figure 3: Superposition of the 51 calibrated spectra used to determine the positions of the main EC peaks (right). The peak labeled N? cannot be attributed to any single-hole excitations in Eq. \ref{['eq:E_c-distr']}. The spectra are calibrated using the energies of the Al and Cl K$\alpha$1 peaks, as indicated. The left panels show the distribution of the determined peak positions $E_0$ for the M1 (top) and N1 (bottom) lines across the 51 calibrated spectra. Error bars have been enlarged by a factor of 5 for clarity. The peak positions are plotted as $\Delta E = E_0 - 2040$ eV and $\Delta E = E_0 - 411$ eV for the M1 and N1 lines, respectively.
  • Figure 4: Final calorimetric spectrum of the EC $^{163}$Ho decay. The blue histogram was measured in the neutrino mass campaign alpert_most_2025 and has an energy threshold at 300 eV. The spectrum below 300 eV is not used. The violet portion of the histogram below 300 eV comes from the last campaign and has an energy threshold of about 30 eV. The histogram has been scaled to precisely overlap to the blue one, using the ratio of the integrals above 300 eV.
  • Figure 5: Unfolded spectrum (black dots) with the fitted model (red curve) after the STAN fit. Uncertainties on the unfolded spectrum are shown as a gray band, which is barely visible due to the small size of the statistical and systematic errors. The band is primarily visible below 300 eV and near the endpoint, where the statistical fluctuations in the data are larger. The red line and reddish band represent the mean and standard deviation of the STAN-generated model, based on the posterior distributions. Coloured curves indicate the 25 fitted components: symmetric Breit-Wigner resonances (violet), asymmetric Breit-Wigner resonances (green), and shake-off spectra (red). The dashed dark blue line denotes the constant background term.
  • ...and 6 more figures