Neutrinos · D15 · Search for an absent particle

Double Beta Decay Observatory

Open a cryogenic detector array, freeze an energy spectrum and search for a hypothetical line. Compare germanium and tellurium, resolution, exposure and backgrounds.

Frozen window counts $0$

Signal upper mean · 90% credible $0$

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

SHIELD · COUNT · CONSTRAIN

Look for the line.

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}}

READ 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}}

Background trap: change the assumed index to 0.020 without acquiring. The ordinary-decay template and response stay fixed.

Acquisition settings require a new record. Window and background assumptions reuse the frozen spectrum.

Teaching simulation, not LEGEND or CUORE data. A normalized beta-shaped ordinary-decay proxy, ideal Gaussian response and flat measured-energy background replace nuclear and detector modeling. The 90% Bayesian signal upper bound uses a flat nonnegative-signal prior with fixed known background, efficiency and response. Background-only median sensitivity is separate from the observed result. No discovery significance, calibrated frequentist coverage or numerical neutrino-mass limit is claimed.

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 inside this window. Gold marks the 90% credible upper mean. No discovery significance is calculated.

Choose the window before the search

Background-only median half-life sensitivity versus half width. It uses a Poisson median, so the curve has small jumps. No observed-count optimization.

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.

Accessible full spectrum, 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 bound becomes a mass statement only with an assumed mechanism and nuclear inputs. This Lab provides no numerical effective-mass conversion.

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

CONTINUE EXPLORING

From flavor to evidence

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

Start with the essentials

Focus question: When does a null spectrum become a half-life constraint?
One-sentence intuition: One frozen record and a background-only design study answer different questions.

Core mathematical model

Exposure becomes a count

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

Exposure is in kilograms of the element times years. Enrichment, efficiency and the signal fraction inside the window enter separately. Atomic masses are approximated by mass numbers.

A conditional posterior

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

The prior is flat on nonnegative signal counts. Background and response are fixed assumptions. Integrating the posterior defines the 90% credible upper mean; this is not a discovery test.

The quiet limit

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

Zero counts still allow a positive signal mean. With this prior the same result holds for zero observed counts and any fixed background.

Half-life is not a universal mass ruler

(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 convention assumes light Majorana-neutrino exchange. Phase space, nuclear matrix elements and axial coupling depend on the chosen nucleus and conventions. Other mechanisms change the interpretation.

Common difficulties

Credibility is conditional

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

Better mental modelThe prior and fixed-background likelihood define a posterior probability. No coverage calibration or systematic uncertainty treatment is supplied.

No peak is not zero mass

Typical misconceptionAn absent line proves neutrinos are Dirac particles or massless.

Better mental modelA half-life bound depends on exposure and response. Its mass interpretation also assumes a decay mechanism and nuclear inputs.

Run the experiment

01
Start without a signal
Keep No signal, inspect the endpoint zoom and the posterior.
What to observe: A random count does not establish a forbidden decay; the conditional signal upper mean remains positive.
02
Keep the data frozen
Move the window and assumed background without acquiring.
What to observe: The spectrum is unchanged. Only the selected counts, acceptance and inference change.
03
Reveal a false excess
Choose Background trap, then raise the assumed background index to 0.020.
What to observe: The same counts change interpretation when the incorrect background assumption is corrected.
04
Compare designs
Try tellurium and blurred response; inspect the full spectrum and window-design curve.
What to observe: Nucleus count, ordinary-decay leakage and line acceptance compete. The recommended width uses background-only sensitivity, not the observed spectrum.