Neutrinos · D06 · A deficit in the spectrum

Reactor Near–Far Spectra

How can detectors near and far from a reactor reveal neutrinos disappearing on the way? Compare the far detector's spectrum with the near one's. A dip at the far site can mean neutrinos disappeared on the way, but a slightly wrong energy calibration, or an error in scaling one detector's rate against the other's, can fake a similar deficit. The near detector pins down the flux both detectors share; it does not cancel every difference in response or background.

Record prompt-energy spectra at a near and a far detector, then fit the disappearance amplitude. Compare the effects of energy resolution, exposure and calibration, and refit the same recorded counts under different assumptions.

Near counts · including tails $0$

Far counts · including tails $0$

Fitted disappearance amplitude $0$

Compare a near and a far detector

Measure the fraction that disappears

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.

Changing an acquisition setting replaces the counts only when you acquire again. The two analysis controls refit the existing record right away.

This teaching simulation has one point-like reactor, one target at the near site and four equal targets at the far site. A smooth illustrative source spectrum, the leading-order energy relation for inverse beta decay, a Gaussian response, a fixed mass splitting and a known background replace real reactor and detector models. A joint Poisson fit adjusts a common flux and a constrained relative normalization. The intervals are deviance contours that hold only under these assumptions; their coverage is not calibrated and no discovery significance is given. The Lab does not reproduce Daya Bay, RENO, Double Chooz or JUNO data.

Near countsFar countsFitted meansRefitted 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: $0$

Bin counts and fitted means as a table

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? →

Physics tutorial

Measure a deficit by making a comparison

BackgroundA reactor emits electron antineutrinos. Over kilometer distances a small fraction disappears, by an amount that depends on energy, and this changes the prompt spectrum. The near detector helps pin down the source normalization that both detectors share.

Why it mattersDaya Bay, RENO and Double Chooz compared detectors at different distances and kept the differences between detectors under control. This Lab shows the spectral comparison on its own, with a deliberately simpler source, geometry and response.

Start with the essentials

Focus question: Could the same deficit at the far detector come from disappearance, from calibration or from the relative normalization?
One-sentence intuition: A near detector pins down the flux that both detectors share. It does not cancel every difference in response, every background uncertainty or every error in the source shape.

Core mathematical model

Disappearance at short baselines, to leading order

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 mass splitting is fixed at a positive 0.00245 eV squared. Baselines are entered in kilometers and converted to meters. Solar and matter corrections are left out, so this is a leading-order approximation, not a full three-flavor fit.

Expected counts in each energy bin

\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

Integrating a Gaussian response over each bin gives the response probabilities, with underflow and overflow included. The far station has four times the target of the near station. The common flux multiplier is free, while the logarithm of the far relative multiplier is constrained. The source and response are illustrative.

Fit statistic for the 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

For a bin with zero counts, the log term is taken as zero. At each scanned amplitude, the common flux and the relative log-normalization are fitted again. All other parameters, including the calibration and the known background, stay at their assumed values.

Interval from a deviance rise of one

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

The shaded interval uses this descriptive threshold. The amplitude scan has physical and numerical boundaries, and neither its coverage at finite counts nor its behavior near those boundaries has been calibrated. No discovery significance is drawn from it.

Common difficulties

The near detector also sees disappearance

Typical misconceptionThe near spectrum is simply the original reactor spectrum.

Better mental modelThe near station is still some distance from the reactor, so the model includes disappearance there too. It also has its own response and background.

A fit cannot make up for effects left out

Typical misconceptionA narrow interval proves the model is right.

Better mental modelThe contour assumes a given source shape, a known background and a fixed calibration. Effects outside the model can bias even a precise result.

Run the experiment

01
Record both spectra
Start with Near–far. Look at both spectra and the corrected ratio.
What to observe: The near and far counts differ for three reasons: the flux thins out with distance, the target masses differ and some antineutrinos disappear. Reading the ratio requires understanding all three.
02
Increase the exposure
Choose More counts and compare the width of the profile.
What to observe: More events usually tighten the statistical limit. The uncertainty in the relative normalization still limits the comparison.
03
Change an assumption, not the data
Increase Relative normalization uncertainty without acquiring again.
What to observe: The normalization can now absorb more of the deficit, so the interval changes on the same data.
04
Try the calibration trap
Choose Calibration trap, then set Assumed far energy-scale offset to two percent.
What to observe: Matching the injected calibration changes the fit, though no sampled count changes. The deliberately wrong model can give a biased result.
05
Compare the truth with the fit
Reveal the generating means, then export the counts and the profile scan.
What to observe: The simulation truth is shown only as a separate reference. The fitted curves come from the sampled spectra; the fit never reads the injected amplitude.

Reactor Near–Far Spectra

The dip is small, so the comparison must be careful.

Near: constrain the common flux

Far: resolve the missing fraction

Compare shape after distance and target correction

The fit is a landscape

Inspect one energy bin

Read the conditional result

Measure a deficit by making a comparison

Core mathematical model

Disappearance at short baselines, to leading order

Expected counts in each energy bin

Fit statistic for the sampled counts

Interval from a deviance rise of one

Common difficulties

The near detector also sees disappearance

A fit cannot make up for effects left out

Run the experiment

Record both spectra

Increase the exposure

Change an assumption, not the data

Try the calibration trap

Compare the truth with the fit

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