Improved treatment of the $T_2$ molecular final-states uncertainties for the KATRIN neutrino-mass measurement

This paper presents a refined procedure for estimating uncertainties in the molecular final-state distribution of tritium beta decay, which significantly reduces the associated systematic uncertainty on the squared neutrino mass from 0.02 eV²/c⁴ to 0.0013 eV²/c⁴, thereby enhancing the precision of the KATRIN experiment's neutrino-mass measurement.

The Gist

Imagine the KATRIN experiment as a giant, ultra-precise scale trying to weigh a ghost. That "ghost" is the neutrino, a tiny particle that barely interacts with anything. To find its weight, scientists look at the very end of a specific energy spectrum created when tritium (a heavy form of hydrogen) decays. It's like trying to find the exact weight of a single grain of sand by watching a massive pile of sand slowly fall, focusing only on the very last grain to drop.

However, there's a problem. When the tritium atom decays, it doesn't just turn into a helium atom and a neutrino; it also leaves behind a "molecular cloud" of energy. This cloud is called the Molecular Final-State Distribution (FSD). Think of this cloud as a fog that obscures the view of the last grain of sand. If the scientists don't know exactly how thick or dense this fog is, they can't be sure how heavy the neutrino really is.

In previous measurements, the scientists estimated the uncertainty of this "fog" using a very cautious, guesswork-heavy method. They basically said, "We think the fog might be this thick, but let's assume it could be twice as thick just to be safe." This resulted in a large "safety margin" for their error bars.

The New Approach: Mapping the Fog

This paper introduces a new, much sharper way to measure that fog. Instead of guessing, the authors decided to map the fog's structure in extreme detail. They treated the calculation of the fog not as a black box, but as a complex machine with many moving parts.

Here is how they did it, using some everyday analogies:

1. The "Zoom Lens" (Basis Sets): To calculate the fog, scientists use a mathematical "lens" made of building blocks (called basis functions). In the past, they used a lens with a fixed number of blocks. The new method involves systematically adding more and more blocks to the lens to see if the picture changes. If adding more blocks doesn't change the picture, they know they have a clear view. If it does change, they know they need to keep zooming in. They found that by systematically increasing the number of blocks, they could see exactly where the calculation "settled" or converged.

2. Tuning the Engine (Constants and Approximations): The calculation relies on many fundamental numbers (like the mass of an electron) and shortcuts (approximations) to make the math work. The authors treated these like tuning knobs on a high-performance engine. They turned each knob slightly to see how much it shook the final result.

   • Example: They asked, "What if we use a slightly different value for the mass of the nucleus?" or "What if we ignore a tiny correction for how fast the electron is moving?" By testing each one, they could pinpoint exactly how much each factor contributed to the total uncertainty.

3. The "Pseudo" Blueprint: The original data used for the first KATRIN campaign was built using a mix of different blueprints from various sources, making it impossible to systematically test every single piece. To solve this, the authors built a "Pseudo-KNM1" blueprint. It is a twin of the original, designed to be as identical as possible but built with a single, consistent set of rules. This allowed them to run their "tuning knob" tests without breaking the model.

The Result: A Sharper Picture

By using this new, systematic method, the authors were able to shrink the "safety margin" on the fog's uncertainty dramatically.

• Old Estimate: The uncertainty was estimated at 0.02 eV²/c⁴.
• New Estimate: The uncertainty is now constrained to 0.0013 eV²/c⁴.

This is a massive improvement. It's like going from saying, "The fog might be anywhere between 1 and 10 meters thick," to saying, "The fog is definitely between 1.0 and 1.1 meters thick."

Why This Matters

The paper concludes that the original "fog" calculation used in KATRIN's first two campaigns was actually very accurate, but the way they estimated the error was too conservative. By tightening this error bar, the experiment is now better equipped to reach its ultimate goal: measuring the neutrino's mass with a sensitivity of 0.2 eV/c².

The authors emphasize that this new method isn't just a one-time fix; it's a new standard procedure. For every future KATRIN campaign, they will use this same systematic "tuning" and "zooming" process to ensure that the uncertainty is always calculated as precisely as possible, rather than relying on rough guesses. This ensures that when they finally claim to have measured the neutrino's mass, the result is rock-solid.

Technical Summary

1. Problem Statement

The KArlsruhe TRItium Neutrino (KATRIN) experiment aims to determine the effective electron antineutrino mass ($m_\nu$) by measuring the tritium $\beta$-decay spectrum near its endpoint (approx. 18.6 keV) with a target sensitivity of $0.2 \text{ eV}/c^2$.

A critical systematic uncertainty in this measurement arises from the Molecular Final-State Distribution (FSD). When a tritium molecule ($T_2$) undergoes $\beta$-decay, the daughter molecule ($^3\text{HeT}^+$) is left in various excited rotational, vibrational, and electronic states. This distribution of excitation energies ($V_j$) modifies the shape of the observed electron spectrum.

• Previous Limitation: In the first KATRIN measurement campaigns (KNM1 and KNM2), the uncertainty of the FSD was estimated using a conservative phenomenological approach. By comparing different theoretical calculations, the uncertainty in the variance of the FSD was estimated to be 2%, leading to a contribution to the squared neutrino mass uncertainty of $\Delta m_\nu^2 = 2 \times 10^{-2} \text{ eV}^2/c^4$.
• The Need for Improvement: As KATRIN accumulates more data and reduces statistical errors, this conservative estimate becomes a limiting factor. Furthermore, the phenomenological approach does not allow for the isolation of specific error sources (e.g., basis set convergence, approximations, or fundamental constants), making it difficult to systematically improve the model.

2. Methodology

The authors propose a new, rigorous procedure to estimate FSD uncertainties by analyzing the specific theoretical inputs and approximations used in the ab initio calculations.

A. The "Pseudo-KNM1" FSD

A major obstacle to applying a systematic convergence study to the original KNM1 FSD was that it was constructed using a "patchwork" of input data from various literature sources, employing different basis sets and parameters that could not be fully reconstructed.

• Solution: The authors generated a pseudo-KNM1 FSD. This dataset is nearly identical to the original KNM1 FSD in terms of physical output but is constructed using a single, consistent computational framework (the H2SOLVm code).
• Key Feature: This allows for the systematic variation of the basis-set convergence parameter ($\Omega$), which controls the number of explicitly correlated basis functions used to solve the electronic Schrödinger equation.

B. The Uncertainty Analysis Procedure

The new method involves the following steps:

1. Reference Data: An Asimov Monte Carlo dataset is generated using the pseudo-KNM1 FSD with a true neutrino mass of $m_\nu^2 = 0$.
2. Test FSDs: A series of "test" FSDs are generated, each modifying a specific source of uncertainty (e.g., changing the basis set size $\Omega$, altering the reduced mass, or omitting theoretical corrections).
3. Fitting: The reference Monte Carlo data is fitted using the model containing the test FSD.
4. Shift Calculation: The deviation of the fitted $m_\nu^2$ from the true value ($0$) quantifies the impact of that specific uncertainty source.
5. Convergence Study: By varying $\Omega$, the authors observe the convergence behavior of $m_\nu^2$. The difference between the result at a converged $\Omega$ (e.g., $\Omega=10$) and the result at the effective $\Omega$ of the original KNM1 FSD provides the uncertainty estimate.

3. Key Contributions

• Shift from Phenomenology to First-Principles: The paper moves away from estimating uncertainty based on the agreement between different historical calculations to a method based on systematic convergence studies of the underlying quantum mechanical parameters.
• Decomposition of Uncertainties: The study isolates and quantifies individual contributions to the total FSD uncertainty, including:
  • Basis-set convergence ($\Omega$).
  • Choice of basis set parameters (bases) for excited states.
  • Inclusion of theoretical corrections (adiabatic, relativistic, radiative).
  • Choice of reduced mass (nuclear vs. effective).
  • Binning effects and energy scale adjustments.
  • Approximations (sudden approximation, constant fractional recoil).
• Reconstruction of Input Data: The paper provides a detailed reconstruction of the basis sets and parameters used in previous calculations (Appendix B), clarifying ambiguities in the literature data used for the original KNM1 FSD.

4. Results

The study quantified the systematic uncertainty contributions for the conditions of the first KATRIN campaign (KNM1). The individual contributions to the uncertainty in the squared neutrino mass ($\Delta m_\nu^2$) are:

Source of Uncertainty	Contribution ($\text{eV}^2/c^4$)
Basis-set $\Omega$ convergence	$2 \times 10^{-4}$
Bases for electronically excited states	$3 \times 10^{-5}$
Reduced mass (daughter)	$1.5 \times 10^{-4}$
Binning effects	$2 \times 10^{-4}$
Corrections to sudden approximation	$4 \times 10^{-4}$
Theoretical corrections (Adiabatic, Relativistic, Radiative)	$1.2 \times 10^{-3}$ (dominant)
Total FSD Uncertainty	$1.3 \times 10^{-3}$

• Comparison: The new total uncertainty of $1.3 \times 10^{-3} \text{ eV}^2/c^4$ is approximately 15 times smaller than the previous conservative estimate of $2 \times 10^{-2} \text{ eV}^2/c^4$.
• Dominant Factor: The largest remaining uncertainty stems from the theoretical corrections (adiabatic, relativistic, and radiative) applied to the Born-Oppenheimer potential curves, particularly for electronically excited states where data is less precise.
• Validation: The study confirms that the original KNM1 FSD was highly accurate; the large previous uncertainty estimate was an artifact of the conservative phenomenological method rather than a flaw in the FSD calculation itself.

5. Significance

• Enabling Future Sensitivity: By reducing the FSD systematic uncertainty by an order of magnitude, this work removes a major bottleneck for KATRIN. It ensures that the experiment's sensitivity is not artificially limited by an overestimated systematic error, allowing the target sensitivity of $0.2 \text{ eV}/c^2$ to be reached as statistical errors decrease.
• Methodological Blueprint: The proposed procedure provides a robust framework for future neutrino mass experiments. It demonstrates how to rigorously propagate theoretical calculation uncertainties (basis sets, approximations) directly into the experimental observable ($m_\nu^2$).
• Future Applications: The authors note that a new, fully consistent FSD (not requiring a "pseudo" version) is currently being developed for future KATRIN campaigns. This new FSD will utilize the methods established in this paper to provide even more precise uncertainty budgets and will extend the analysis to larger energy intervals (e.g., 60 eV below the endpoint).

In conclusion, this paper represents a critical advancement in the precision physics of the KATRIN experiment, transforming the treatment of molecular final-state uncertainties from a conservative guess into a rigorously quantified, systematic component of the error budget.