Neutrinos · D06 · A deficit in the spectrum

Reactor Near–Far Spectra

Acquire two prompt-energy spectra, then fit their disappearance amplitude. Compare resolution, exposure and calibration using the same frozen counts.

Near counts · including tails $0$

Far counts · including tails $0$

Fitted disappearance amplitude $0$

TWO DETECTORS · ONE QUESTION

Find the missing fraction.

Near baseline

L_n

0.5\,\mathrm{km}

Far baseline

L_f

1.65\,\mathrm{km}

Exposure multiplier

T

4\,\times

Resolution coefficient

a

8\,\%

$\sigma_E=a\sqrt{E_p/\mathrm{MeV}}\,\mathrm{MeV}$
The percent slider sets the dimensionless coefficient. Both detectors share this resolution.

Injected simulation & background

Injected disappearance amplitude

A

0.085\,

Injected far energy-scale offset

\epsilon_{\mathrm{true}}

0\,\%

Known flat background scale

b

2\,\%

Background percentage is relative to 9,000 reference counts per exposure, in each detector. It is not the local background fraction.

REFIT THE SAME COUNTS

Assumed far energy-scale offset

\epsilon_{\mathrm{fit}}

0\,\%

Relative normalization uncertainty

\sigma_{\eta}

1\,\%

Try Calibration trap, then set the assumed offset to match the injection. The counts do not change.

Acquisition changes replace the counts only when you acquire. The two analysis controls refit the existing record immediately.

Teaching simulation with one point reactor, one near target and four equal far targets. A smooth illustrative source, leading-order inverse-beta energy mapping, Gaussian response, fixed mass splitting and known background replace real reactor and detector models. Joint Poisson fits profile a common flux and a relative normalization pull. Intervals are conditional deviance contours; no coverage calibration or discovery significance. This does not reproduce Daya Bay, RENO, Double Chooz or JUNO data.

Near countsFar countsFitted meansProfiled no-disappearance referenceReveal generating means

Near: constrain the common flux

Dots are sampled counts. Vertical bars show the square root of the count as a descriptive scale, not exact Poisson intervals. Lines include the known background.

Far: resolve the missing fraction

Dots are sampled counts. Vertical bars show the square root of the count as a descriptive scale, not exact Poisson intervals. Lines include the known background.

Compare shape after distance and target correction

Divide the background-subtracted far counts by the near counts and the geometric target factor. Dashed: the assumed unoscillated response ratio before a normalization pull. Delta-method bars are approximate; bins with nonpositive near denominators are omitted. The vertical axis is clipped to the displayed range.

The fit is a landscape

At each amplitude, refit common flux and far relative normalization. The minimum is gold; shading marks a deviance rise of one. Values above twelve are clipped. Pink appears only when generating means are revealed.

Inspect one energy bin

$0$

Near counts: $0$
Far counts: $0$

Select a spectrum bin with a click or arrow keys. The record remains unchanged.

Read the conditional result

Deviance-rise-one interval: $0$
Common flux multiplier: $0$
Far relative normalization: $0$
Minimum joint deviance: $0$

Accessible bin counts and fitted means

Prompt-energy bins in MeV, including response tails
Energy	Near	Far	Near fit	Far fit

Conventions, model boundaries & primary sources

One point reactor feeds both stations. Four equal far target modules compensate partly for inverse-square dilution. Scene coordinates, cavern sizes and routes are schematic; baseline numbers control the model. Phototubes and liquid illustrate apparatus, without simulated optical hits or neutron transport. All four far modules use one nominal baseline.

The positive fixed splitting is 0.00245 eV squared. The leading short-baseline disappearance amplitude is scanned from zero to 0.30; solar oscillations, matter, isotope evolution, source-shape errors and nonlinear response are omitted. A smooth illustrative source times leading-order IBD phase space is integrated from threshold to 10 MeV. Gaussian prompt-energy tails are retained in two overflow bins and included in the likelihood. The model sets 9,000 unoscillated near counts per exposure at a half-kilometer baseline.

Poisson sampling occurs after response convolution. The background is a known flat mean, not a measured subtraction. Both spectra constrain one freely fitted common flux. A Gaussian penalty applies to the logarithm of the far relative normalization; its width is the control value. Calibration is fixed to the analysis setting, not profiled. The interval conditions on all other model assumptions and is not an experimentally validated confidence statement.

Injected amplitude： $0$

Read the near–far story →How did a candidate become a count? →

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: Can the same far deficit arise from disappearance, calibration or relative normalization?
One-sentence intuition: A near detector constrains a common flux. It does not cancel every response mismatch, background uncertainty or source-shape error.

Core mathematical model

Leading short-baseline disappearance

P_{ee}(E,L)=1-A\sin^2\!\left(1.267\,\frac{\Delta m^2}{\mathrm{eV}^2}\frac{L/\mathrm{m}}{E/\mathrm{MeV}}\right),\qquad A\simeq\sin^2 2\theta_{13}

The fixed positive splitting is 0.00245 eV squared. Baseline inputs use kilometers and are converted to meters. Solar and matter corrections are omitted; this is a leading approximation, not a full three-flavor fit.

Turn a prediction into bin means

\mu_{di}=\alpha f_d\,\frac{C_d T}{L_d^2}\int \Phi(E)\sigma(E)P_{ee}(E,L_d)R_{di}(E)\,dE+B_i

Response probabilities come from Gaussian bin integrals, including underflow and overflow. Target ratio is four at the far station. The common flux multiplier is free; the far relative multiplier has a constrained logarithm. The source and response are illustrative.

Compare the actual sampled counts

D=2\sum_{d,i}\left[\mu_{di}-n_{di}+n_{di}\ln\frac{n_{di}}{\mu_{di}}\right]+\frac{\eta^2}{\sigma_\eta^2},\qquad f_n=1,\quad f_f=e^\eta

The zero-count log term is defined as zero. At each scanned amplitude, the common flux and relative log-normalization are refitted. All other parameters are conditioned upon, including calibration and the known background.

A contour is not automatically a confidence statement

D_{\mathrm{prof}}(A)-D_{\mathrm{min}}=1

The shaded interval uses this descriptive threshold. The amplitude scan has physical and numerical boundaries; finite-count coverage and boundary behavior have not been calibrated. No discovery significance is inferred.

Common difficulties

Near is not unoscillated

Typical misconceptionThe near spectrum is automatically the original reactor spectrum.

Better mental modelEven the near station has a finite baseline and is modeled with disappearance. It also has a response and a background.

A fit cannot fix every omitted effect

Typical misconceptionA narrow interval proves the model is right.

Better mental modelThe contour conditions on a prescribed source shape, known background and fixed calibration. Unmodeled effects can bias a precise result.

Run the experiment

01
Acquire a comparison
Start with Near–far. Inspect both spectra and the corrected ratio.
What to observe: Near and far counts are different because of dilution, target mass and disappearance. A ratio requires all three to be understood.
02
Increase the exposure
Select More counts. Compare the profile width.
What to observe: More observations generally sharpen the statistical constraint. The relative normalization pull still limits the comparison.
03
Keep the counts fixed
Increase relative normalization uncertainty without acquiring again.
What to observe: The nuisance can absorb more of the deficit, changing the profile interval on the same data.
04
Fall into the calibration trap
Choose Calibration trap, then set the assumed far offset to two percent.
What to observe: Matching the injected calibration changes the fit without changing any sampled count. The intentionally incorrect model can be biased.
05
Separate reference from observation
Reveal generating means and export counts plus the profile scan.
What to observe: The simulation truth is a separate reference. The fitted curves come from the sampled spectra, not a hidden restoration of the injected amplitude.