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Neutrinos · D01 · The missing energy

Beta Energy Sharing Observatory

Why do electrons from beta decay come out with a whole range of energies? Because a third, unseen particle carries off part of the energy. If the decay produced only the electron and the recoiling nucleus, energy and momentum conservation would give the electron the same energy every time, so recoil cannot explain the spread. Count the third particle in, and energy and momentum are conserved after all.

Replay decays inside an ideal spherical calorimeter. Build up the continuous electron spectrum, compare it with a two-body line, and reveal how an invisible particle carries away energy and momentum.

Interactive modelBeta Energy Sharing Observatory
Frozen electron counts00
Assumed three-body endpoint0 keV0\,\mathrm{keV}
Twice log-likelihood preference00

From single decays to a full spectrum

The missing share.

Measured countsThree-body templateTwo-body template

One line or a continuum?

The spectrum has forty-eight equal energy bins, plus the two overflow bins, which are kept. The lines show the expected count in each bin under the current analysis assumptions.

Read the same counts twice

Three-body deviance
00
Two-body deviance
00
Twice log-likelihood preference
00
Recorded mean energy
00
Assumed three-body endpoint
00
Counts in both flows
00

A positive preference favors the three-body shape. This conditional comparison gives no p-value, discovery significance or fitted mass bound. Where a Gaussian tail underflows numerically, a floor is applied only inside logarithms.

Replay one recorded event

Only electron energy was recorded. Gold and purple arrows, the energy ledger and momentum triangle below are generated truth. Paths and timing are schematic; recoil displacement is magnified.

How the generated energy is shared

Generated momentum triangle

Fixed horizontal-plane projection; vectors share one scale. Cyan electron, purple daughter, gold invisible particle close tip to tail. This is not detector reconstruction.

Recorded spectrum as a table, including both overflow bins
Energy edges in keV; recorded and expected counts
LowHighCountsThree bodiesTwo bodies
Conventions, approximations & primary sources

Fictional parent at rest. Exact relativistic phase space weighted by electron energy times neutrino energy in the parent frame; it approaches the leading allowed beta shape for a heavy daughter, with unit Coulomb factor. No nuclear spin, realistic angular correlations, Coulomb, molecular or radiative corrections. The two-body foil is a kinematic comparison, not a viable weak-decay theory.

Source at the origin; vertical axis upward. The ideal spherical collector has unit efficiency, arbitrary schematic radius and independent Gaussian energy sigma. Open shells expose the full-acceptance model. No material transport, background or detector angle measurement. Large neutrino and light daughter masses exaggerate the mechanism. A fixed event total yields multinomial counts; both normalized response templates retain Gaussian tails.

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Physics tutorial

The invisible share

BackgroundThis Lab replays single decay events. It keeps two things apart: the full energy split known to the generator, and the energy that the one detector actually records.

Why it mattersBeta-decay electrons come out with a continuous range of energies, which does not fit a simple two-body picture.

Start with the essentials

Focus question
Can recoil alone explain the continuum?
One-sentence intuition
In a two-body decay the electron always gets the same energy; with three bodies the energy is shared over a continuous range while every four-vector is still conserved.

Core mathematical model

Energy conservation in the decay

Q=Ke+TD+EνQ=K_e+T_D+E_\nu

The neutrino term is its total energy, including rest energy; the electron and daughter terms are kinetic energies.

The three momenta close into a triangle

pe+pD+pν=0\mathbf p_e+\mathbf p_D+\mathbf p_\nu=\mathbf 0

The parent is at rest. The electron energy alone cannot tell you the other two momentum vectors, which are not measured. The ones shown here are generated truth.

Spectrum shape for a very heavy daughter

dΓdK∝peEeww2−mν2,w=Q−K\frac{d\Gamma}{dK}\propto p_eE_e w\sqrt{w^2-m_\nu^2},\quad w=Q-K

This uses natural units and sets the Coulomb factor to one. It is the heavy-daughter limit of the toy weight used here, not its full density with recoil corrections and not a precision nuclear spectrum.

Two spectrum shapes compared on the same counts

ΔD=2∑iniln⁡p3,ip2,i\Delta D=2\sum_i n_i\ln\frac{p_{3,i}}{p_{2,i}}

A positive value favors the three-body template. Both normalized templates include the underflow and overflow counts produced by the Gaussian response. No distribution for judging significance is supplied.

Common difficulties

Generated truth is not a reconstruction

Typical misconceptionThe calorimeter measures the neutrino trajectory and recoil.

Better mental modelIt records only the electron's energy. The revealed vectors and the energy split come from the generator.

A preference is not a discovery

Typical misconceptionA large difference in deviance is a calibrated significance.

Better mental modelThe two fixed shape hypotheses come with no sampling distribution and no treatment of nuisance parameters, so the difference cannot be read as a significance.

Run the experiment

  1. 01

    Follow an electron

    Choose Three bodies, step through events with the slider, then reveal the generated particles.

    What to observe: The electron reaches the collector while the invisible particle passes straight through. The daughter carries momentum but little energy.
  2. 02

    Remove the third body

    Choose Two-body foil and Light daughter, then compare the spectra and the energy split of each event.

    What to observe: Recoil shifts where the two-body line sits, but with a fixed parent mass it does not turn the line into a continuum.
  3. 03

    Check the detector response

    Choose Resolution trap, then set the analysis sigma to 100 without acquiring again.

    What to observe: The same broad measured line fits a broad detector response. Blurring by the detector is not evidence for an extra particle.
  4. 04

    Change the missing mass

    Choose Heavy invisible particle, then set the analysis mass to 150.

    What to observe: The endpoint and the spectrum move, and the template changes with the assumed mass. This is not a fitted mass bound.

Related experiments and articles

Continue exploring

Neutrino mass in beta-decay spectra

Beta decay shares its energy between an electron and a neutrino. How does the end of the electron spectrum limit the neutrino mass, and what would a peak in a double-beta decay spectrum mean?

  1. You are hereBeta Energy Sharing ObservatoryCompare a continuous electron spectrum with a two-body line, and account for all of the generated energy.
  2. LabBeta Endpoint Mass ObservatoryScan an endpoint and refit the same counts under calibration and response assumptions.
  3. LabDouble Beta Decay ObservatoryFreeze a rare-decay spectrum and separate a conditional bound from background-only sensitivity.

Neutrinos · Reading path

From missing energy to a new particle

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleWhere did the energy go? The puzzle that led to the neutrino
  2. LabBeta Endpoint Mass Observatory
  3. ArticleFermi’s next question: how often should the unseen particle appear?
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