Sudbury Neutrino Observatory

The SNO Collaboration used $1000$ tonnes of heavy water in a $12\mathrm{m}$ acrylic sphere, surrounded by ordinary water shielding and $9456$ photomultipliers. Weak charged current (CC) breakup of a deuteron selects electron neutrinos; weak neutral current (NC) breakup measures all active neutrino flavors; Electron elastic scattering (ES) has reduced non-electron sensitivity.^[ref]

In 2001, $240.95$ live days, reconstructed radius below $5.50\mathrm{m}$ and effective Electron kinetic energy above $6.75\mathrm{MeV}$ gave a CC neutrino flux of $1.75\times 10^6{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ for the assumed boron-8 spectrum. Source calibrations measured $1.4\%$ instrumental-cut signal loss. SNO's own CC–ES difference was $1.6\sigma$; comparison with Super-Kamiokande's precise ES flux gave $3.3\sigma$ evidence for non-electron active neutrinos. This was not an NC measurement.^[ref]

In 2002 the pure-heavy water analysis used $306.4$ live days and a $5\mathrm{MeV}$ reconstructed kinetic-energy threshold. NC neutrons captured on deuterons emitted $6.25\mathrm{MeV}$ gamma rays, producing secondary Electron light. The NC reaction threshold was $2.2\mathrm{MeV}$, a different quantity. Californium-252 calibrated $29.9\%$ capture efficiency, but only $14.4\%$ detection efficiency after fiducial and energy cuts. Assays constrained backgrounds from radioactive decay and photodisintegration.^[ref]

Energy, radius and direction relative to the Sun statistically separated channels. Assuming the standard boron-8 spectral shape, the NC total active neutrino flux was $5.09_{-0.43}^{+0.44}{(stat.)}_{-0.43}^{+0.46}(syst.)\times 10^6{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, consistent with stellar structure predictions. The non-electron active component differed from zero by $5.3\sigma$. This established active flavour conversion, without individually tagging muon neutrinos and tau neutrinos. It did not exclude every sterile admixture.^[ref]

Why compare inferred neutrino fluxes rather than raw event counts? For channel $j$, exposure time $T$, target population $N_j$, cross section ${\sigma }_{j\alpha }$, and detection efficiency ${ϵ}_{j\alpha }$ including reconstruction and selection, a forward model isHere $B_j$ is expected background; $N_j$ on the left counts events, while $N_j^{\mathrm{target}}$ counts targets. Distinct cross sections, target numbers, thresholds and detection efficiencies prevent a direct CC/NC count ratio from being an electron neutrino fraction. Channel populations themselves can be correlated estimates from overlapping event patterns.

An original reduced linear algebra model isolates the inference. Assume specified spectral shapes and divide each background-corrected channel rate by its own calibrated electron neutrino response. Put ${\varphi }_x={\varphi }_{\mu }+{\varphi }_{\tau }$ and let $r$ be the spectrum- and response-weighted non-electron/electron ES sensitivity ratio. ThenThe equal $\mu$ and $\tau$ responses determine their sum, not their separate neutrino fluxes. Changing the assumed spectra generally changes the integrated coefficients; with arbitrary unknown spectra this small matrix is insufficient.

Choose illustrative inputs, not historical measurements: $r=0.15$, ${\varphi }_e=2$ and ${\varphi }_x=3$, with every neutrino flux in the same arbitrary unit. The forward prediction is $(y_{\mathrm{CC}},y_{\mathrm{NC}},y_{\mathrm{ES}})=(2,5,2.45)$. CC and NC independently invert to ${\varphi }_e=y_{\mathrm{CC}}=2$, ${\varphi }_x=y_{\mathrm{NC}}-y_{\mathrm{CC}}=3$; substituting into ES returns $2+0.15(3)=2.45$. Alternatively, CC and ES give ${\varphi }_x=(y_{\mathrm{ES}}-y_{\mathrm{CC}})/r=3$. Thus the third channel tests the response model and the inferred mixture.

CC alone admits any nonnegative ${\varphi }_x$. CC and ES lose the second independent constraint when $r=0$; for small positive $r$, an ES-minus-CC error is amplified by $1/r$. With chosen independent errors ${\sigma }_{\mathrm{ES}}={\sigma }_{\mathrm{CC}}=0.1$, their propagated standard deviation is $\sqrt{0.1^2+0.1^2}/0.15\simeq 0.943$. If CC and NC instead have those independent errors, the difference has standard deviation $\sqrt{0.1^2+0.1^2}\simeq 0.141$. Actual analyses must retain covariances, including those introduced by statistical channel separation. Complementarity supplies information; calibration establishes what information each channel carries.