Neutrinos · D15 · Search for an absent particle

Double Beta Decay Observatory

In double beta decay, what would a peak in the two electrons’ total energy mean, and what if none appears? That peak would mark a decay with no neutrinos. The two electrons would then carry all the decay energy, so their combined energy would always land on the same value. If no peak appears, the empty spectrum gives a lower limit on the half-life, but only once you spell out the exposure (how much isotope, for how long), the energy resolution, the background and the detector response.

Open a cryogenic detector array, record an energy spectrum and look for a hypothetical peak at the full decay energy. Compare germanium with tellurium and see how energy resolution, exposure and background change the search.

Frozen window counts $0$

Signal upper mean · 90% credible $0$

Conditional half-life lower bound $0\,\mathrm{yr}$

Search for neutrinoless double beta decay

Look for a peak at the released energy

Isotope · element mass exposureIsotope exposure

\mathcal{E}

200\,\mathrm{kg\,yr}

Acquired energy FWHM

\Delta E

3\,\mathrm{keV}

Injected flat background index

b

0.001\,\frac{1}{\mathrm{keV\,kg\,yr}}

Injected line yield

s_0

0\,\frac{1}{100\,\mathrm{kg\,yr}}

Reanalyze the same spectrum

Window half width

w

3\,\mathrm{keV}

Window center offset

\delta

0\,\mathrm{keV}

Assumed flat background index

b_{\mathrm{a}}

0.001\,\frac{1}{\mathrm{keV\,kg\,yr}}

To see the background trap, change the assumed index to 0.020 without acquiring again. The ordinary-decay template and the response stay fixed.

Changes to the acquisition settings need a new record. Changing the energy window or the background assumption reuses the recorded spectrum.

This is a teaching simulation, not LEGEND or CUORE data. A normalized beta-distribution curve stands in for the ordinary-decay spectrum, the detector response is an ideal Gaussian and the background is flat in measured energy; nuclear and detector physics are not modeled in detail. The 90% Bayesian upper bound on the signal uses a flat prior on nonnegative signal and treats background, efficiency and response as exactly known. The median sensitivity expected from background alone is shown separately from the observed result. The Lab gives no discovery significance, no calibrated frequentist coverage and no numerical limit on neutrino mass.

Observed countsAssumed backgroundReveal generating means

The ordinary continuum

Counts per keV on a logarithmic scale. Underflow and overflow are retained in the table and CSV.

Look around the line

Observed one-keV bins and assumed ordinary-plus-flat background. Shading is the selected window. Click or use arrow keys to move its center.

What signal can remain?

Normalized posterior for the signal mean in this window. Gold marks the 90% credible upper limit on that mean. No discovery significance is calculated.

Choose the window before the search

Median half-life sensitivity with background only, plotted against the window half width. The Poisson median makes the curve jump in small steps. The window is not tuned to the observed counts.

This record · conditional inference

Energy release: $0$
Window counts: $0$
Assumed background mean: $0$
Ordinary-decay leakage: $0$
Line acceptance: $0$
90% credible signal upper mean: $0$
Conditional half-life lower bound: $0$

Known response and ordinary-decay rate; assumed flat background. Changing that assumption is a sensitivity exercise, not a background measurement.

No signal · design sensitivity

Background-only median count: $0$
Median half-life sensitivity: $0$
Suggested half width: $0$
Injected half-life hypothesis: $0$

The recommendation depends only on the background model and signal acceptance. It is chosen over the displayed discrete widths; it is not a continuous optimum.

Full spectrum as a table, including overflow

Measured energy edges in keV; counts; assumed background mean
Low	High	Counts	Background	Window

Conventions, approximations & primary sources

The germanium mode shows cylindrical source-as-detector crystals; tellurium uses generic calorimeter blocks. Both share a conceptual cryostat and shielding layout, not replicas of LEGEND or CUORE. The vertical coordinate points upward; the highlighted crystal contains a local illustrative two-electron deposition. Paths are magnified and do not generate the energy spectrum. No charge, heat, light, veto or pulse-shape transport is simulated.

Rounded energy releases are 2,039 and 2,528 keV. Element-mass exposure uses 88% enrichment and 80% efficiency; mass numbers replace molar masses. Ordinary half-lives are illustrative 2 and 0.7 sextillion years. A normalized symmetric beta-distribution summed-energy proxy omits Coulomb factors, electron masses, nuclear structure and radiative corrections. A constant Gaussian FWHM applies to both the proxy and hypothetical line. Background is already defined in measured energy, flat between zero and the energy release plus 300 keV. Full Gaussian tails are retained.

Every disjoint bin is an independent Poisson count. Integer windows use complete one-keV bins. The posterior assumes exact ordinary-decay normalization, response and flat background; those uncertainties are not marginalized. The 90% bound is a Bayesian credible statement under the displayed prior, not a guaranteed confidence construction. Searching many windows is exploratory and provides no discovery claim. No real experiment sensitivity is reproduced.

A half-life limit says something about mass only after you assume a decay mechanism and nuclear inputs. This Lab does not convert it into an effective mass.

Why search with several nuclei? →What would an absent neutrino mean? →

Start with the essentials

Focus question: How does a spectrum with no visible peak become a limit on the half-life?
One-sentence intuition: The limit from one recorded spectrum and the sensitivity expected from background alone answer different questions.

Core mathematical model

Expected signal count from exposure and half-life

N_{\mathrm{sig}}=\frac{\ln 2}{T_{1/2}^{0\nu}}\frac{10^3N_A f}{A}\,\mathcal{E}\,\epsilon\,a_W

Exposure is the mass of the element in kilograms times the years of running. Enrichment, efficiency and the fraction of the signal inside the window are separate factors. Mass numbers stand in for the atomic masses.

Posterior for the signal with a fixed background

p(s\mid n,B)\propto(s+B)^n e^{-(s+B)}\,\Theta(s)

The prior is flat for any nonnegative signal count, and the background and response are held fixed. Integrating this posterior gives the 90% credible upper limit on the mean signal. It is not a test for discovery.

The limit when no counts are seen

n=0,\ B=0:\quad s_{90}=-\ln(0.1)

Seeing zero counts still leaves room for a positive mean signal. With this prior, zero observed counts give the same limit whatever the fixed background is.

Converting a half-life into a neutrino mass

(T_{1/2}^{0\nu})^{-1}=G^{0\nu}g_A^4|M^{0\nu}|^2\frac{|m_{\beta\beta}|^2}{m_e^2}

This form assumes the decay happens through exchange of a light Majorana neutrino. The phase space, nuclear matrix element and axial coupling depend on the nucleus and on the convention used. A different mechanism would change how a half-life translates into a mass.

Common difficulties

A credible limit rests on its assumptions

Typical misconceptionA 90% credible bound guarantees 90% frequentist coverage.

Better mental modelA credible bound is a posterior probability set by the prior and the fixed-background likelihood. The Lab does not calibrate coverage or treat systematic uncertainties.

No peak does not mean zero mass

Typical misconceptionIf no peak shows up, neutrinos must be Dirac particles or massless.

Better mental modelA missing peak gives a half-life limit, and that limit depends on exposure and detector response. Turning it into a statement about mass also requires a decay mechanism and nuclear inputs.

Run the experiment

01
Start without a signal
Keep No signal and look at the endpoint zoom and the posterior.
What to observe: A few random counts do not establish the forbidden decay. The upper limit on the mean signal, given the assumed background, stays above zero.
02
Change the analysis, not the data
Move the window and change the assumed background without acquiring again.
What to observe: The spectrum stays the same. Only the counts in the window, the signal acceptance and the resulting limit change.
03
Expose a false excess
Choose Background trap, then raise the assumed background index to 0.020.
What to observe: Once the wrong background assumption is corrected, the same counts lead to a different conclusion.
04
Compare designs
Choose Try tellurium and Blur the endpoint, then look at the full spectrum and the window-design curve.
What to observe: The number of nuclei, ordinary decays leaking into the window and the fraction of the peak inside it pull in different directions. The recommended width comes from the sensitivity with background alone, not from the observed spectrum.

Double Beta Decay Observatory

A peak that may not exist

The ordinary continuum

Look around the line

What signal can remain?

Choose the window before the search

This record · conditional inference

No signal · design sensitivity

Read a rare-decay search

Core mathematical model

Expected signal count from exposure and half-life

Posterior for the signal with a fixed background

The limit when no counts are seen

Converting a half-life into a neutrino mass

Common difficulties

A credible limit rests on its assumptions

No peak does not mean zero mass

Run the experiment

Start without a signal

Change the analysis, not the data

Expose a false excess

Compare designs

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Neutrino mass in beta-decay spectra

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