Neutrinos · D03 · SNO and the missing flavors

Solar Neutrino Flavor Observatory

How can you tell a fainter Sun from neutrinos that change flavor on the way? Count the neutrinos two ways. One reaction sees only electron neutrinos; the neutral-current reaction counts all active flavors alike. If electron neutrinos come up short while the all-flavor total does not drop, the Sun is not fainter. Some of its electron neutrinos have turned into other flavors on the way.

Look inside a heavy-water detector and replay the three reactions it can see. Record a simulated exposure, then find where the flux limits from the three channels cross. Can a fainter Sun that emits only electron neutrinos explain the same counts?

Inferred electron flux $\text{—}$

Inferred other active flux $\text{—}$

Inferred total flux $\text{—}$

This teaching model is inspired by SNO: a 12 m vessel and an 18 m sensor sphere, both rounded, with 384 enlarged sensors. The three channels are perfectly separated Poisson counts with a stated response matrix and exactly known backgrounds. They are not SNO data, and SNO’s correlated event extraction is not reproduced. The elastic-scattering flavor weight is fixed, and the neutrino spectrum, thresholds, MSW propagation and detector calibration are left out. The optical examples are sketches, not the events used to sort the total counts into channels. The fits are unconstrained weighted least squares, and the counting bands are not confidence regions or discovery significance.

Electron flavor only

\nu_e+d\to p+p+e^-

Recorded count

\text{—}

Prediction at the inspected point, with background

\text{—}

All active flavors

\nu_\alpha+d\to p+n+\nu_\alpha

Recorded count

\text{—}

Prediction at the inspected point, with background

\text{—}

A weighted flavor mixture

\nu_\alpha+e^-\to\nu_\alpha+e^-

Recorded count

\text{—}

Prediction at the inspected point, with background

\text{—}

Solid bars are observed counts; thin bars are predictions at the inspected flux point. Every channel shares one count scale. Channel labels are ideal here; real SNO extracted overlapping populations with a joint statistical analysis.

Click a flux point, or use arrow keys on the chart. Shading shows one counting-fluctuation scale for each selected channel.

Both axes use $10^6\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$

CCNCESBest fit

Inspected flux point

\text{—}

Reveal simulated source

Response matrix, exact counts & optical hits

Counts per unit exposure and flux. The rows correspond to CC, NC and ES; backgrounds per exposure are shown separately. These are teaching coefficients, not measured SNO efficiencies.

A=\begin{pmatrix}300&0\\100&100\\60&9.24\end{pmatrix},\qquad B=bT\begin{pmatrix}20\\35\\8\end{pmatrix}

Compare with the historical result

SNO’s 2002 paper reported electron and total active fluxes, under its stated boron-8 spectrum assumption. These published values are a separate reference; this simulator neither fits them nor reproduces the collaboration’s analysis.

\begin{aligned}\Phi_e&=1.76\pm0.05\,\mathrm{(stat.)}\pm0.09\,\mathrm{(syst.)}\\\Phi_{\mathrm{NC}}&=5.09^{+0.44}_{-0.43}\,\mathrm{(stat.)}^{+0.46}_{-0.43}\,\mathrm{(syst.)}\end{aligned}

$10^6\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$

SNO · 2002 original paper ↗

This is a separate calculation with BP2000 responses. These controls do not change the SNO exposure recorded above.

Prescribed component averagespp survival

P_{pp}

0.55

Intermediate-component survival

P_{\mathrm{mid}}

0.45

Boron-8 and hep survival

P_{\mathrm{high}}

0.33

Common source scale

f

1

Exposure in days

T

30

Chlorine · Homestake

\nu_e+{}^{37}\mathrm{Cl}\to{}^{37}\mathrm{Ar}+e^-

Capture threshold $0.814\,\mathrm{MeV}$

2.72\,\mathrm{SNU}

Bar scale: 150% of this target’s unchanged BP2000 rate.

Relative to no flavor change: $0.358$
Expected extracted atoms: $4.78$

Gallium · GALLEX / GNO / SAGE

\nu_e+{}^{71}\mathrm{Ga}\to{}^{71}\mathrm{Ge}+e^-

Capture threshold $0.233\,\mathrm{MeV}$

63.06\,\mathrm{SNU}

Bar scale: 150% of this target’s unchanged BP2000 rate.

Relative to no flavor change: $0.493$
Expected extracted atoms: $67.75$

pp componentIntermediate componentsHigh-energy components

Equal illustrative populations of target isotope atoms; constant production, radioactive loss during exposure and 90% extraction. Values are expectations, not recorded integer counts. $N_{\mathrm{target}}=10^{30}$

Inspect every component and the calculation

Rounded BP2000 Table 7 components, multiplied by the prescribed survival and source scale. All rates in SNU.
Source	Chlorine baseline	Chlorine current	Gallium baseline	Gallium current
$pp$	0.00	0.00	69.70	38.34
$pep$	0.22	0.10	2.80	1.26
$\mathrm{hep}$	0.04	0.01	0.10	0.03
${}^7\mathrm{Be}$	1.15	0.52	34.20	15.39
${}^8\mathrm{B}$	5.76	1.90	12.10	3.99
${}^{13}\mathrm{N}$	0.09	0.04	3.40	1.53
${}^{15}\mathrm{O}$	0.33	0.15	5.50	2.48
${}^{17}\mathrm{F}$	0.00	0.00	0.10	0.05

R_d=f\sum_i c_{di}P_i,\qquad 1\,\mathrm{SNU}=10^{-36}\,\mathrm{atom}^{-1}\,\mathrm{s}^{-1}

N_{\mathrm{ext}}=\epsilon\frac{q}{\lambda}(1-e^{-\lambda T}),\quad\lambda=\frac{\ln2}{t_{1/2}}

Production is in atoms per day. Half-lives: argon 35.0 days, germanium 11.43 days. The intermediate group combines pep, beryllium-7 and CNO; the high group combines boron-8 and hep. These prescribed averages are not exact energy intervals or a numerical matter-propagation solution.

Separate historical measurements · 2001 snapshot

BP2000 Table 8 reports these measurements. They stay fixed while you move the controls; no joint fit, discovery significance or present-day best value is claimed.

\begin{aligned}R_{\mathrm{Cl}}&=2.56\pm0.23\,\mathrm{SNU}\\R_{\mathrm{GALLEX+GNO}}&=74.1^{+6.7}_{-7.8}\,\mathrm{SNU}\\R_{\mathrm{SAGE}}&=75.4^{+7.8}_{-7.4}\,\mathrm{SNU}\end{aligned}

Fixed historical spectrum and rounded integrated coefficients; no solar luminosity constraint, correlated flux/cross-section uncertainties, extraction fluctuations or counter backgrounds. Radiochemistry integrates captures and loses event direction and capture time.

Physics tutorial

One source. Three different questions.

BackgroundRadiochemical experiments found too few solar electron neutrinos. SNO used heavy water to ask an additional question: how many active neutrinos arrived, regardless of flavor? Comparing the charged-current, neutral-current and elastic-scattering rates told a faint source apart from a change of flavor.

Why it mattersEach channel picks out a band on a plane whose axes are the two flux components. A single rate that is sensitive to electron flavor cannot tell a weaker source from a change of flavor; channels with different sensitivities can.

Start with the essentials

Focus question: Can the same electron-channel count come from two Suns with different total active fluxes?
One-sentence intuition: The neutral-current rate counts all active flavors together, so a low electron-flavor rate does not have to mean a low total active flux.

Core mathematical model

Two flux components

\Phi_x=\Phi_\mu+\Phi_\tau,\qquad\Phi=\Phi_e+\Phi_x

The second component contains the other two active flavors. This model cannot distinguish muon from tau flavor, and does not include sterile states.

Expected counts in the three channels

\begin{aligned}\lambda_{\mathrm{CC}}&=300T\Phi_e+20bT\\\lambda_{\mathrm{NC}}&=100T(\Phi_e+\Phi_x)+35bT\\\lambda_{\mathrm{ES}}&=60T(\Phi_e+0.154\Phi_x)+8bT\end{aligned}

Flux is in millions per square centimeter per second, and exposure is relative. The coefficients and known backgrounds are chosen for teaching. The fixed scattering weight stands in for a response that really depends on the spectrum; it is not a universal ratio.

Sampling the counts

N_i\sim\operatorname{Poisson}(\lambda_i)

Each of the three channels, perfectly separated here, is sampled independently. The real SNO analysis had to fit overlapping event populations with correlated uncertainties. The optical examples are not used to assign these total counts to channels.

Fitting only the selected channels

Q=\sum_{i\in\mathcal S}\frac{[N_i-B_i-T(A\boldsymbol\Phi)_i]^2}{\max(N_i,1)}

The unconstrained weighted least-squares fit uses each observed count as its variance estimate, but never less than one. Its local covariance and residual scores describe this approximation; neither is a calibrated confidence region or significance.

Light travel time to each sensor

t_j=\frac{n\,|\mathbf r_j-\mathbf r_v|}{c}

Every ray in the scene ends at a stored sensor position, 9 meters from the center. The clock starts when the electron effectively emits light, and the water index is 1.33. Refraction at the acrylic, dispersion, scattering and the neutron capture delay are left out.

Common difficulties

A neutral-current flash is not a direct neutrino image

Typical misconceptionA neutron emits the light shown in the NC scene.

Better mental modelThe freed neutron is captured, the capture gamma rays scatter electrons, and those electrons emit the Cherenkov light. The illustration is averaged over directions, starts at that effective emission point and leaves out the capture delay and the neutron’s diffusion.

A negative fit result warns about the approximation

Typical misconceptionA negative other-flavor estimate is a physical negative flux.

Better mental modelThe fit is deliberately unconstrained, so near zero, counting fluctuations can push the estimate outside the physical range. A physical inference needs a proper treatment of that boundary, which this Lab does not provide.

Run the experiment

01
Compare the reactions
Select CC, NC and ES above the detector. Drag along the light-flight timeline and check the side and interaction views.
What to observe: CC and ES show broadened cones of light from electrons. NC shows a capture flash averaged over directions, because the neutron is neutral and is not the particle that emits the Cherenkov light.
02
Make two similar electron counts
Acquire Faint Sun, then Changed flavors. Compare CC and NC on the shared count scale.
What to observe: Both presets have the same electron-flavor flux. The larger total active flux raises the NC count; losing electron flavor is not the same as dimming the source.
03
Remove a channel
Turn off NC, then ES. Keep only CC in the fit, then restore a second channel.
What to observe: One rate only confines the fluxes to a line, not a single point. With this response, CC plus ES can also separate the two components, but its error on the other-flavor flux is larger than with CC plus NC.
04
Test an electron-only explanation
Look at the best electron-only hypothesis and compare its predicted bars with the recorded counts.
What to observe: That hypothesis gets its own best fit to the selected channels. When the counts need another active component, matching one channel leaves the others unmatched.
05
Separate truth from inference
Reveal the simulated source, vary the seed, and acquire a long exposure. Export the CSV.
What to observe: The truth marker plays no part in the fit. Counts fluctuate, so the best fit need not land exactly on the source. The export contains every displayed count, response, illustrative hit and sensor coordinate.

Solar Neutrino Flavor Observatory

Comparing the channels reveals the missing flavor

Charged current

Neutral current

Elastic scattering

Where the channel constraints cross

Two targets see the same Sun differently

Chlorine · Homestake

Gallium · GALLEX / GNO / SAGE

One source. Three different questions.

Core mathematical model

Two flux components

Expected counts in the three channels

Sampling the counts

Fitting only the selected channels

Light travel time to each sensor

Common difficulties

A neutral-current flash is not a direct neutrino image

A negative fit result warns about the approximation

Run the experiment

Compare the reactions

Make two similar electron counts

Remove a channel

Test an electron-only explanation

Separate truth from inference

Related experiments and articles

How flavor change is predicted and tested

What happened to the Sun’s neutrinos?