The Beta Decay Endpoint and the Neutrino Mass

The neutrino mass scale is one of the most fundamental open numbers in physics. Cosmology measures the sum of masses through structure formation. Neutrinoless double beta decay (if observed) probes the Majorana mass. The cleanest laboratory measurement, however, is through the kinematics of beta decay — specifically, the shape and endpoint of the electron energy spectrum.

In beta decay, a parent nucleus emits an electron and a neutrino with shared energy: $E_e+E_{\nu }=Q$, where $Q$ is the Q-value of the decay (corrected for the recoil energy of the daughter nucleus). If the neutrino has mass $m_{\nu }$, its minimum energy is $m_{\nu }{c}^2$ — its rest energy. The maximum electron energy is therefore $E_e^{\max }=Q-m_{\nu }{c}^2$, slightly below the value it would have if neutrinos were massless.

This is the endpoint shift. A 1-eV neutrino mass produces a 1-eV shift of the electron spectrum’s upper edge. The measurement is straightforward in principle: count electrons very precisely as a function of energy near the endpoint, and look for the kinematic cutoff.

In practice, the measurement is one of the most demanding in physics. The fraction of beta-decay events near the endpoint is exponentially small, the detector resolution must be far below 1 eV, and the systematic uncertainties — molecular final states, scattering, instrumental effects — must be controlled to corresponding precision.

This post is about the beta-decay endpoint as a probe of neutrino mass: how it works, why it is so demanding, and what the current status is.

The kinematic shift

The differential decay rate for tritium beta decay is: $\frac{dN}{d{E}_e}=KF(Z,E_e)p_e(E_e+m_e{c}^2)(Q-E_e)\sqrt{(Q-E_e{)}^2-m_{\nu }^2{c}^4}$

where $K$ is a constant, $F(Z,E_e)$ is the Fermi function (accounting for Coulomb interaction with the daughter nucleus), $p_e$ is the electron momentum, and the rest follows from phase-space considerations.

For $m_{\nu }=0$, the spectrum near the endpoint has a parabolic shape: $dN/dE\propto (Q-E_e{)}^2$. The number of events drops to zero precisely at $E_e=Q$.

For $m_{\nu }>0$, the shape changes. Near the endpoint, the spectrum is: $\frac{dN}{d{E}_e}\propto (Q-E_e)\sqrt{(Q-E_e{)}^2-m_{\nu }^2{c}^4}$

The square root forces the spectrum to zero at $E_e=Q-m_{\nu }{c}^2$ — earlier than the massless case. The deviation from the massless shape is small but measurable in the last ~30 eV below the nominal endpoint.

The challenge is that the fraction of decays in the last 30 eV is approximately $10^{-13}$ of all decays. Detecting these few events with sub-eV precision requires very high source activity, exquisite energy resolution, and exhaustive systematic-uncertainty control.

Why tritium

Tritium is the standard target because:

Low Q-value (18.574 keV). The fractional distortion from a 1-eV mass scales as $m_{\nu }/Q\sim 5\times 10^{-5}$. For higher-Q-value beta emitters, the fractional distortion is smaller and harder to extract.

Super-allowed transition. The tritium decay ${}^3\text{H}{\to }^3\text{He}+e^{-}+{\bar{\nu }}_e$ is super-allowed: the nuclear matrix element is essentially the unit operator. The resulting spectrum shape is calculable to 1% precision without nuclear-structure uncertainty.

Reasonable half-life (12.3 years). Long enough for stable storage; short enough to give meaningful decay rates.

Available in molecular and atomic forms. Molecular T₂ is easy to produce and condense; atomic T is harder but reduces final-state systematic uncertainty. Both have been used.

The combination of low Q-value, clean nuclear physics, and practical target preparation makes tritium uniquely suited for endpoint measurements. Other isotopes (rhenium-187, nickel-63) have been considered but tritium has been the standard since the 1970s.

Final-state distribution

A subtle but important effect: when a tritium atom in a molecule (T₂) undergoes beta decay, the daughter helium-3 atom is left in a bound state of the resulting ³He-T molecular ion, which can be in various rotational, vibrational, or electronic excited states. Each excited state corresponds to a different effective endpoint energy.

The distribution of final states is calculable via molecular-orbital theory. For T₂, the calculation has been refined for over 40 years and is now considered well-understood. The final-state spectrum has a typical width of approximately 1.5 eV, which broadens the apparent endpoint and contributes to the systematic uncertainty.

The molecular final states are the dominant systematic uncertainty in modern measurements. Reducing this uncertainty requires either better calculations (current effort) or atomic tritium sources (Project 8’s approach).

The MAC-E filter principle

Modern endpoint measurements use the Magnetic Adiabatic Collimation with Electrostatic filter technique, originally developed in the 1980s and refined through the 1990s and 2000s.

Principle: an electron emitted from the source enters a region of strong magnetic field (~5 T) where it spirals along the field lines. As it travels into a much weaker field region (~0.001 T), the spiral expands while preserving the magnetic moment $\mu =E_{\perp }/B$. This means the transverse energy decreases while the longitudinal energy is preserved. By the time the electron reaches the central spectrometer, its momentum is almost entirely along the field axis.

A retarding electrostatic potential is applied. Only electrons with longitudinal energy greater than the retarding potential pass through to the detector. By varying the retarding voltage, the integrated electron rate above each threshold is measured.

The MAC-E filter combines:

• Large solid-angle acceptance at the source (high statistics)
• Sharp energy resolution at the spectrometer (high precision)

KATRIN’s spectrometer, with a 23-metre diameter and the world’s largest ultra-high-vacuum tank for that purpose, has an energy resolution of approximately 1 eV at 18.6 keV — among the best ever achieved for an electrostatic spectrometer.

Current status

The MAC-E filter chain that started with Mainz and Troitsk in the 1990s has now reached KATRIN. The current bound (2024 release): $m_{\nu }<0.45\text{ eV}\text{ (90\% C.L.)}$

This is approximately a factor of 5 better than the Mainz/Troitsk combined limit and represents the best direct-measurement bound to date.

KATRIN expects to continue running through the late 2020s and reach approximately 0.2 eV ultimately — the practical limit of the MAC-E filter approach.

What’s next

To go below 0.2 eV requires fundamentally different detection technology. Two main approaches:

Project 8 uses cyclotron radiation emission spectroscopy. A magnetic field traps the beta-decay electron in a small volume; the electron emits cyclotron radiation at a frequency that depends on its kinetic energy. Frequency measurement can be much more precise than electrostatic filtering. With atomic tritium (eliminating the molecular final-state uncertainty), Project 8 aims for 0.04 eV sensitivity.

Holmium-163 electron capture. ECHo and HOLMES use the inverse process — electron capture instead of beta decay — in ${}^{163}$Ho. The electron-capture spectrum has a distinctive endpoint shape that can be measured with similar techniques. This offers an independent measurement at a different Q-value (2.8 keV) and energy regime.

Both approaches are in development; first results expected late 2020s.

Why direct measurement matters

The neutrino mass scale is also constrained by:

• Cosmology: $\sum m_{\nu }<0.12$ eV at 95% C.L. from CMB+BAO (model-dependent)
• Neutrinoless double beta decay: probes the effective Majorana mass, sensitive to phases and ordering
• Tritium beta decay: the cleanest, model-independent direct measurement

Each approach has strengths and weaknesses. Cosmology is the tightest but depends on assumed cosmological models. Neutrinoless double beta decay only works if neutrinos are Majorana. Beta-decay endpoint is direct, model-independent, and works for any mass-generation mechanism.

The three approaches are complementary. If they all agree on the same value, it’s a clean confirmation of consistent physics. If they disagree, it points to either new physics or incorrect modeling assumptions. Currently they are consistent — within their respective uncertainties, all bounds are at or below 0.5-1 eV scale.

By 2030, with KATRIN final results, Project 8 first results, and improved cosmological measurements, the picture should be substantially clearer. The neutrino mass — that ancient open quantity — may soon have a direct laboratory value.

A long programme

The first beta-decay endpoint measurement was done in the 1930s, motivated by Pauli’s neutrino postulate. Initial bounds were of order $10^4$ eV. Each subsequent generation of experiments improved the bound by an order of magnitude or so:

• 1930s-1950s: Endpoint resolution improved through Geiger-Müller and proportional counters; bound ~ 100 eV
• 1960s-1970s: Magnetic and electrostatic spectrometers; bound ~ 50 eV
• 1980s: Improved spectrometer energy resolution; bound ~ 30 eV
• 1990s-2000s: Mainz and Troitsk MAC-E filters; bound ~ 2 eV
• 2019-2024: KATRIN; bound ~ 0.5 eV
• 2030s: Project 8 atomic tritium; expected reach ~ 0.04 eV

Each generation took 20-30 years. Each improved the bound by a factor of 5-10. The cumulative improvement over the 90-year programme is approximately 6 orders of magnitude.

The fundamental physics has not changed: it is still tritium beta decay, still kinematic energy conservation, still the same endpoint cutoff that Pauli implicitly invoked. What has changed is the experimental technique, the systematics control, and the patience to extract sub-eV signals from $10^{14}$-event datasets.

Eventually — perhaps within the next decade — the absolute neutrino mass will be measured rather than just bounded. The endpoint shift, predicted before the neutrino was even directly detected, will yield a number. That number will be one of the most important measurements in physics: a direct, model-independent measurement of one of the fundamental scales of the Standard Model.

Until then, the bounds tighten. Each generation builds on the last. The endpoint, that mathematical cutoff in the spectrum, has been the target for almost a century and remains the target today.