Neutrino Mass: Oscillation, Beta Decay, Cosmology

This article compares the three main ways scientists measure neutrino mass: oscillation experiments, beta decay endpoint measurements, and cosmological observations. Each method probes a different aspect of the neutrino’s elusive mass, and together they reveal that neutrinos are far lighter than any other fermion, but not zero.

Formula

The calculator above estimates your running pace given a time and a fixed distance of 5 km. The formula is simply:

Pace (min/km) = Total time (min) ÷ Distance (km)

Speed (km/h) = Distance (km) ÷ (Total time (min) ÷ 60)

Where t is the total time in minutes, and the distance is fixed at 5 km. The pace is the number of minutes it takes to cover one kilometer, and the speed is the average kilometers covered per hour.

Worked Example

Suppose you run 5 km in 25 minutes. Using the formula:

Pace = 25 min ÷ 5 km = 5.0 min/km

Speed = 5 km ÷ (25 min ÷ 60) = 5 ÷ 0.4167 = 12.0 km/h

The calculator with the default time of 25 minutes gives exactly these results: a pace of 5.0 min/km and a speed of 12.0 km/h.

Background: Why Neutrino Mass Matters

In the Standard Model of particle physics, neutrinos were originally assumed to be massless. They are produced in weak nuclear processes, such as beta decay, and they interact so weakly that they pass through ordinary matter almost undisturbed. Yet in the late 1990s, experiments studying neutrinos from the Sun and from cosmic rays showed that neutrinos change flavor as they travel—a phenomenon called neutrino oscillation. This is only possible if neutrinos have different masses, because the oscillation probability depends on the differences of the squared masses. The discovery of neutrino oscillation earned the 2015 Nobel Prize in Physics, but it only measures mass differences, not the absolute mass scale.

How Oscillation Experiments Work

Neutrino oscillation experiments measure the probability that a neutrino of one flavor (electron, muon, or tau) transforms into another over a given distance. The key parameters are the two mass-squared differences, Δm²₂� and Δm²₃₂, and the mixing angles. These experiments are extremely precise: the solar neutrino experiments (e.g., Super-Kamiokande, SNO) and reactor experiments (e.g., KamLAND) have determined Δm²₂� to about 7.5×10�� eV², while atmospheric and accelerator experiments (e.g., MINOS, T2K, NOvA) measure |Δm²₃₂| to about 2.5×10�³ eV². However, oscillation experiments cannot tell you the absolute mass of the lightest neutrino—they only give the differences. If the lightest neutrino is very heavy (say 1 eV), the differences are small perturbations; if it is very light (say 0.01 eV), the differences dominate. So oscillation data alone leave the absolute scale unknown.

Beta Decay Endpoint: A Direct Measurement

The most direct laboratory method to measure neutrino mass is the study of the electron energy spectrum in beta decay. In a beta decay, a nucleus converts a neutron into a proton, emitting an electron and an antineutrino. The electron’s energy spectrum has a maximum endpoint energy, which is shared between the electron and the neutrino. If the neutrino has mass, the endpoint is slightly reduced, and the shape of the spectrum near the endpoint is distorted. The most sensitive experiment to date is the KATRIN experiment in Germany, which uses tritium (a radioactive isotope of hydrogen) and measures the electron spectrum near the endpoint. KATRIN has set an upper limit on the effective electron neutrino mass of about 0.8 eV (at 90% confidence) as of 2022. This is a direct, model-independent measurement—it does not rely on cosmology or assumptions about the early universe. The challenge is that the effect of a sub-eV neutrino mass is tiny, requiring extremely high energy resolution and a very strong tritium source.

Cosmology: The Most Sensitive Probe

Cosmological observations provide the most stringent constraints on the sum of neutrino masses, Σmν. Neutrinos were abundant in the early universe, and their gravitational influence left imprints on the cosmic microwave background (CMB) and the large-scale structure of galaxies. Massive neutrinos suppress the growth of structure on small scales, because they are fast-moving and free-stream out of overdense regions. By measuring the CMB power spectrum (from Planck) and galaxy clustering (from surveys like BOSS), cosmologists can infer the total neutrino mass. The current upper limit is around Σmν < 0.12 eV (95% confidence) from Planck plus BOSS data, which is far more stringent than KATRIN's limit. However, this bound is model-dependent: it assumes a standard ΛCDM cosmological model, and it can be relaxed if there is extra dark radiation or if neutrinos have non-standard interactions. Also, cosmology measures the sum of all three masses, not individual masses.

Comparing the Three Methods

Each method has its strengths and weaknesses:

• Oscillation: Measures mass differences with high precision, but not the absolute scale. It is the only method that proves neutrinos have mass.
• Beta decay: Direct, model-independent, but limited to about 0.2 eV sensitivity in practice (KATRIN’s design goal). It measures the effective electron neutrino mass, which is a combination of the three mass eigenstates.
• Cosmology: Most sensitive, but model-dependent. It measures the sum of masses, and the result depends on assumptions about the universe’s expansion history and structure formation.

The three methods are complementary. If neutrinos are hierarchical (one mass much larger than the others), then the lightest mass is nearly zero, and the sum is roughly the largest mass. If they are quasi-degenerate (all three masses similar), then the sum is about three times the common mass. Combining oscillation data (which gives the differences) with a cosmological upper limit on the sum can place bounds on the individual masses. For example, if Σmν < 0.12 eV, and the mass differences are known, then the lightest neutrino mass must be less than about 0.03 eV, and the masses are hierarchical.

Common Problems and Misconceptions

A common misconception is that oscillation experiments can determine the absolute neutrino mass. They cannot—they only measure differences. Another misconception is that the beta decay endpoint gives the mass of the electron neutrino directly. Actually, it measures an effective mass that is a weighted average of the three mass eigenstates, because the neutrino is produced as a flavor state, which is a superposition of mass states. Also, people often confuse the mass-squared differences with the masses themselves. The mass-squared differences are on the order of 10�� and 10�³ eV², which implies masses are at most a few tenths of an eV, but the absolute scale could be anywhere from nearly zero to about 1 eV.

Another issue is the unit of mass used: particle physicists use electronvolts (eV), where 1 eV = 1.78×10�³� kg. For comparison, the electron mass is about 511,000 eV, so neutrinos are at least a million times lighter than electrons.

How to Choose Which Result to Trust

If you want a direct, model-independent answer, the beta decay limit from KATRIN is the gold standard, but it only gives an upper limit. If you want the tightest constraint, cosmology gives the best number, but you must accept the cosmological model. For a complete picture, you need all three. The current status is that no experiment has yet measured a nonzero neutrino mass directly; all we have are upper limits. The next generation of experiments, such as the Project 8 experiment (which aims to use atomic tritium to reach 0.04 eV sensitivity) and future cosmological surveys (like Euclid or the Dark Energy Spectroscopic Instrument), may finally pin down the absolute mass scale.

Frequently Asked Questions

What is the difference between neutrino mass and mass-squared difference?

Neutrino mass is the actual mass of a neutrino in eV, while the mass-squared difference is the difference between the squares of two mass eigenstates. Oscillation experiments measure the latter, not the former. For example, if the masses are 0.01 eV and 0.02 eV, the mass-squared difference is (0.02² – 0.01²) = 0.0003 eV², which is much smaller than the masses themselves.

Why can’t oscillation experiments measure the absolute neutrino mass?

Because the oscillation probability depends only on the differences of the squared masses, not on the absolute scale. If you add a constant to all three masses, the oscillation pattern remains the same. This is analogous to measuring the distance between two points on a line—you know the gap, but not where the line starts.

What does the KATRIN experiment measure exactly?

KATRIN measures the electron energy spectrum from tritium beta decay near the endpoint. The shape of the spectrum near the endpoint is sensitive to the effective electron neutrino mass, which is a combination of the three mass eigenstates weighted by the electron flavor mixing matrix. KATRIN’s latest limit is m(νe) < 0.8 eV at 90% confidence.

How does cosmology measure neutrino mass?

Cosmology measures the sum of neutrino masses by observing the effect of massive neutrinos on the growth of cosmic structure. Neutrinos suppress the clustering of matter on small scales, which leaves a signature in the matter power spectrum and the CMB lensing. By comparing observations to theoretical models with different neutrino masses, cosmologists can infer Σmν. The current limit is Σmν < 0.12 eV.

Which method is the most reliable?

Beta decay is the most reliable in the sense that it is direct and model-independent, but it has the weakest sensitivity. Cosmology is the most sensitive but relies on assumptions about the universe. Oscillation is the most precise for differences but cannot give the absolute scale. The most reliable overall answer comes from combining all three, which currently points to a sum of masses below 0.12 eV, implying the lightest neutrino is nearly massless.

Could neutrinos be massless after all?

No. The observation of neutrino oscillation is definitive proof that neutrinos have mass, because a massless neutrino cannot change flavor. The only way to explain oscillations is if neutrinos have different masses, and at least two of them are nonzero. The absolute scale is still unknown, but it is certainly not zero.

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in t | Time (minutes) | min | 1 | 0
out pace | Pace (min/km) | min/km | t / 5 | 2
out speed | Speed (km/h) | km/h | 5 / (t / 60) | 2