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

This paper presents a comprehensive phenomenological analysis of the $^{163}$Ho calorimetric electron capture spectrum from the HOLMES experiment, modeling the unfolded data as a sum of Breit-Wigner resonances and shake-off continua to accurately describe spectral features, validate against theoretical calculations, and provide a robust framework for future neutrino mass experiments.

The Gist

The Big Picture: Weighing the Invisible Ghost

Imagine you are trying to weigh a ghost. You can't put it on a scale, and you can't see it. But you know that when this ghost (a neutrino) escapes a room, it takes a tiny bit of energy with it. If you measure exactly how much energy is left behind in the room, you can figure out how heavy the ghost is.

This is the goal of the HOLMES experiment. They are studying a specific atom, Holmium-163, which is like a tiny, unstable bomb. When it explodes (decays), it usually spits out a neutrino. The scientists want to measure the "leftover" energy with extreme precision to calculate the mass of that neutrino.

The Problem: The Room is Too Noisy

The paper describes a major challenge: The "room" (the detector) is incredibly noisy.

When the Holmium atom decays, it doesn't just release the neutrino. It also leaves the remaining atom (now Dysprosium) in a state of total chaos. The electrons inside the atom are jumping around, shaking, and screaming.

• The "Single Hole" Theory: For decades, scientists thought the atom just lost one electron (a "hole"), and the rest of the atom settled down neatly. They thought the energy spectrum would look like a few clean, distinct musical notes.
• The Reality: The new data shows the spectrum is a messy, chaotic jazz improvisation. There are "shake-ups" (electrons jumping to higher seats) and "shake-offs" (electrons getting kicked out of the room entirely). These create a fog of background noise that makes it hard to hear the specific "note" needed to weigh the neutrino.

The Solution: Unfolding the Mess

The HOLMES team used a super-sensitive thermometer (a Transition-Edge Sensor) to listen to these atomic explosions. They collected a massive amount of data—millions of events.

However, their thermometer isn't perfect. It's a bit blurry. If you look at a sharp line through a foggy window, it looks smeared.

1. The Unfolding: The team used a mathematical trick called "unfolding" to digitally remove the blur. Imagine taking a blurry photo of a face and using software to sharpen it until you can see the individual pores. They did this to the energy spectrum to see the true, sharp lines underneath.
2. The Phenomenological Model: Once they had the sharp picture, they didn't just guess what the lines meant. They built a Lego model of the spectrum. They said, "Okay, this bump is a 'Breit-Wigner' resonance (a specific type of atomic note), and this long tail is a 'shake-off' (an electron getting kicked out)."
   • They identified 25 different components (notes and noise) that make up the spectrum.
   • They gave each one a specific shape, width, and intensity.

The "Aha!" Moment: Comparing with Supercomputers

The team compared their Lego model against the most advanced ab initio (first-principles) computer simulations available. These simulations try to calculate the behavior of every single electron from scratch using the laws of quantum mechanics.

• The Result: The computer simulations were close, but not perfect. They missed some of the subtle "fuzziness" and the exact positions of the notes.
• The Takeaway: The HOLMES team's "Lego model" (based on real data) is currently the best map we have. It captures the messy reality better than the pure theory does.

Why Does This Matter? (The End-Point)

The most important part of the spectrum is the very end—the endpoint. This is where the neutrino takes almost all the energy, leaving the detector with almost nothing.

• If the neutrino has mass, the spectrum stops abruptly a tiny bit before the maximum energy.
• If the neutrino has no mass, it goes all the way to the edge.

The paper shows that their new model accurately describes the "tails" of the spectrum leading up to that edge. This is crucial because if you don't understand the noise (the tails) leading up to the finish line, you can't accurately measure where the finish line actually is.

The Analogy: Tuning a Piano in a Storm

Imagine you are trying to tune a piano (the neutrino mass) while a hurricane is blowing outside (the atomic chaos).

• Old View: Scientists thought the hurricane was just a steady wind. They tried to tune the piano assuming the wind was predictable.
• New View: The HOLMES team realized the hurricane is actually a chaotic mix of gusts, rain, and debris (shake-ups and shake-offs).
• The Paper's Contribution: They built a detailed map of the hurricane. They identified exactly how the wind hits the piano keys. Now, even though the storm is still raging, they know exactly how to subtract the wind's noise to hear the true note of the piano.

Conclusion

This paper is a user manual for the future. It tells future neutrino experiments: "Here is exactly what the Holmium spectrum looks like, here are all the hidden noises, and here is how to model them."

By providing this detailed "phenomenological model," the team has laid the foundation for the next generation of experiments to finally solve the mystery of the neutrino's mass. They have turned a chaotic mess of data into a clear, understandable map.

Technical Summary

1. Problem Statement

The measurement of the absolute neutrino mass is a primary goal in particle physics and cosmology. The HOLMES experiment aims to determine the electron neutrino mass ($m_\nu$) by precisely measuring the endpoint of the electron capture (EC) decay spectrum of Holmium-163 ($^{163}$Ho).

• The Challenge: The theoretical description of the $^{163}$Ho EC spectrum is complex. The standard "single-hole approximation" (modeling the spectrum as a sum of Breit-Wigner resonances corresponding to captured electrons from specific atomic shells) has proven inadequate when compared to high-resolution experimental data.
• Missing Physics: Experimental spectra reveal broad features, asymmetric tails, and structures that cannot be explained by single-hole excitations alone. These are attributed to shake-up (excitation of a spectator electron to a higher bound state) and shake-off (ejection of a spectator electron) processes, as well as multi-electron correlations and Coulomb interactions within the atom.
• The Gap: While ab initio theoretical calculations (e.g., by Brass et al.) have advanced the understanding of these processes, they currently lack the precision to predict exact resonance energies and widths required for a direct fit to experimental data. Conversely, previous phenomenological models often lacked a comprehensive decomposition of the entire spectrum, particularly the low-energy region and the complex tails near the endpoint.

2. Methodology

The authors present a comprehensive phenomenological analysis of high-statistics data collected by the HOLMES experiment using Transition-Edge Sensor (TES) microcalorimeters.

Experimental Setup and Data Collection

• Detectors: Mo/Cu bilayer TES microcalorimeters with gold absorbers, operating at ~40 mK.
• Source: $^{163}$Ho implanted into gold absorbers via ion implantation.
• Data Campaigns: Three distinct campaigns were analyzed:
  1. Calibration: High-statistics measurement of main peaks (M, N, O shells) using X-ray calibration sources (Al, Cl).
  2. Endpoint: High-statistics measurement of the endpoint region (threshold ~300 eV) for neutrino mass sensitivity.
  3. Low Energy: Measurement of the low-energy region (threshold ~30 eV) using a subset of high-activity pixels.
• Total Statistics: Over $6 \times 10^7$ events above 300 eV and more than 1,000 individual spectra combined.

Data Analysis and Unfolding

• Calibration: A Bayesian learning procedure was used to determine precise peak positions ($E_0$) and widths ($\Gamma$) for the main shells (M1, M2, N1, etc.), accounting for detector non-linearity and statistical spread.
• Unfolding: To correct for the varying energy resolutions (5–10 eV FWHM) of individual detectors and avoid distorting intrinsic spectral features, the authors employed an iterative Bayesian unfolding method (using the PyUnfold framework). This reconstructed the "true" energy distribution from the smeared observed data before summation.

Phenomenological Modeling

The unfolded spectrum was fitted using a linear combination of analytical functions representing different atomic de-excitation processes:

1. Breit-Wigner Resonances (BW): Used for discrete peaks. The model employs an asymmetric parametrization (Eqs. 3.1–3.2) to account for:
   • Quantum mechanical interferences.
   • Threshold effects.
   • Atomic phase space suppression on the low-energy side.
2. Shake-Off Spectra (SOF): Used for continuous distributions arising from the ejection of electrons. The model uses an analytical approximation derived from Levinger's work for hydrogenic atoms, convolved with the Breit-Wigner width of the two-hole state.
3. Phase Space Factor: The total model is multiplied by the phase space factor $(Q - E_c)^2$, where $Q$ is the decay energy.

The fitting procedure involved a two-step approach:

• Step 1: Incremental introduction of components (O, N, M groups) using Minuit2 optimization.
• Step 2: A simultaneous Bayesian estimation of all 103 free parameters (25 spectral components + background) using Hamiltonian Markov Chain Monte Carlo (HMC) in STAN.

3. Key Contributions

• First High-Statistics Phenomenological Decomposition: The paper provides the first complete analytical description of the $^{163}$Ho EC spectrum based on high-statistics HOLMES data, covering the energy range from ~30 eV to the endpoint ($Q \approx 2863$ eV).
• Identification of 25 Components: The model successfully identifies and fits 19 Breit-Wigner peaks and 6 shake-off continua. This includes:
  • Standard single-hole excitations (M1, M2, N1, N2, O1, O2).
  • Double-hole excitations (shake-up) involving core holes in shells not directly accessible by EC (e.g., M3, M4, M5, N3, N4, N5) due to intra-orbital Coulomb scattering.
  • Shake-off continua associated with specific double-hole configurations.
• Parameterization of Asymmetry: The introduction of specific asymmetry parameters ($\delta_{AS}$, $p$, $E_{th}$) for the Breit-Wigner peaks allows for a more accurate description of the spectral tails than previous symmetric models.
• Bridge Between Theory and Experiment: The work establishes a robust framework to compare experimental data with ab initio calculations, highlighting where theory succeeds (general shape, multiplet splitting) and where it requires empirical adjustment (exact energies and intensities).

4. Results

• Spectral Fit: The phenomenological model provides an excellent fit to the unfolded spectrum (Fig. 5), with residuals generally within statistical uncertainties.
• Component Interpretation:
  • Table 2 lists the interpretation of all 25 fitted components. For example, the peak at 412.9 eV is interpreted as a double-hole excitation (N1 + N6) or a shake-off process, resolving ambiguities present in simpler models.
  • The model confirms the presence of inter-core relaxation processes, where Coulomb repulsion transfers angular momentum, creating holes in shells (like M4, M5) that are not directly accessible via single-electron capture.
• Comparison with Ab Initio Theory:
  • The phenomenological model aligns well with the ab initio calculations by Brass et al. [22, 23] regarding the general structure and the presence of multiplets.
  • However, the ab initio results show deviations in peak positions (by several eV) and do not fully capture the natural broadening without phenomenological convolution.
  • Crucially, both the phenomenological model and ab initio calculations suggest the endpoint region is free of sharp peaks, consisting mainly of smooth tails and the M1 resonance tail.
• Neutrino Mass Sensitivity:
  • Applying the new model to the endpoint region yields an upper limit of $m_\nu < 31$ eV/c$^2$ (90% credibility) and an endpoint energy $Q = (2851 \pm 6)$ eV, consistent with previous HOLMES results.
  • The model reveals that shake-off processes (specifically M1N2) and the asymmetry of the M1 resonance significantly enhance the spectral intensity near the endpoint compared to the single-hole approximation. This increases the statistical sensitivity for neutrino mass determination.

5. Significance

• Foundation for Future Experiments: This work establishes a robust, analytical framework essential for future $^{163}$Ho neutrino mass experiments (e.g., HOLMES, ECHo, NuMECS). It provides the necessary "toy spectra" generators for sensitivity studies and Monte Carlo simulations.
• Systematic Uncertainty Control: By decomposing the spectrum into specific atomic de-excitation processes, the model allows researchers to better assess systematic uncertainties related to solid-state effects and detector responses, which are critical for reaching the sub-eV sensitivity required for neutrino mass discovery.
• Atomic Physics Insights: The analysis provides empirical evidence for complex multi-electron dynamics (shake-up, shake-off, and inter-core Coulomb scattering) in heavy atoms, validating and refining theoretical ab initio approaches.
• Background Modeling: The model offers a realistic treatment of backgrounds, including pile-up and the tails of low-energy components, which are crucial for accurately modeling the endpoint region where the neutrino mass signature resides.

In summary, this paper moves beyond the single-hole approximation to provide a complete, data-driven phenomenological model of the $^{163}$Ho spectrum, resolving long-standing discrepancies between theory and experiment and paving the way for the next generation of neutrino mass measurements.