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Sandbox Physics

Neutrinos · D01 · The missing energy

Beta Energy Sharing Observatory

Replay one decay inside an ideal spherical calorimeter. Sample a continuous electron spectrum, compare a two-body line and reveal how an invisible particle carries energy and momentum.

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

FROM ONE DECAY TO A FROZEN SPECTRUM

The missing share.

Measured countsThree-body templateTwo-body template

One line or a continuum?

Forty-eight equal energy bins, plus two retained overflow bins. Lines are expected bin counts using 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

Positive preference favors the three-body shape. This conditional comparison has no p-value, discovery significance or fitted mass bound. Gaussian tail underflow is floored 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.

Generated energy ledger

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.

Accessible recorded spectrum, including flows
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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CONTINUE EXPLORING

From flavor to evidence

Propagation predicts a flavor probability. Which parts of the experiment turn an interaction into a usable record?

Physics tutorial

The invisible share

BackgroundReplay a single event, then distinguish a generated energy ledger from the energy recorded by one detector.

Why it mattersA continuous electron spectrum challenged a simple two-body energy story.

Start with the essentials

Focus question
Can recoil alone explain the continuum?
One-sentence intuition
Two bodies fix electron energy. Three bodies share it continuously while conserving every four-vector.

Core mathematical model

Close the energy budget

Q=Ke+TD+EνQ=K_e+T_D+E_\nu

Neutrino total energy includes its rest energy; electron and daughter terms are kinetic energies.

A closed momentum triangle

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

The parent is at rest. The two measured-unknown vectors cannot be inferred from electron energy alone. Here they are generated truth.

A heavy-daughter limit

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

Natural units, unit Coulomb factor. This is the limit of the prescribed toy weight, not its full recoil-corrected density or a precision nuclear spectrum.

Compare the same counts

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

Positive values prefer the three-body template. Both normalized templates include Gaussian flows. No significance distribution is supplied.

Common difficulties

Truth is not a reconstruction

Typical misconceptionThe calorimeter measures the neutrino trajectory and recoil.

Better mental modelIt records only electron energy. Revealed vectors and the budget come from the generator.

A preference is not a discovery

Typical misconceptionA large deviance difference is a calibrated significance.

Better mental modelThese fixed shape hypotheses have no supplied sampling distribution or nuisance treatment.

Run the experiment

  1. 01

    Follow an electron

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

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

    Remove the third body

    Choose Two-body foil and Light daughter. Compare spectra and event budgets.

    What to observe: Recoil changes a two-body line position, but does not make a continuum at fixed parent mass.
  3. 03

    Question the response

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

    What to observe: The same broad measured line agrees with a broad response. Detector blur is not evidence of an extra particle.
  4. 04

    Change the missing mass

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

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