Measuring the Parameters

The PMNS framework has six oscillation parameters. Pinning down each one required matching a source and baseline to the relevant mass splitting and angle. The result is a remarkably consistent global picture assembled from very different experiments.

The atmospheric sector: θ₂₃ and the large splitting

The first measurement came from atmospheric neutrinos. Cosmic rays striking the upper atmosphere produce muon and electron neutrinos; Super-Kamiokande, by comparing neutrinos arriving from overhead with those from through the Earth, found in 1998 a distance-dependent disappearance of muon neutrinos. This established the large “atmospheric” splitting $\Delta m^2_{31}$ and a large mixing angle $\theta_{23}$, close to maximal. Accelerator experiments with controlled muon- neutrino beams over hundreds of kilometres later sharpened these values.

The solar sector: θ₁₂ and the small splitting

The “solar” parameters came from two complementary directions. Solar-neutrino experiments — the radiochemical detectors, Super-Kamiokande, SNO, and Borexino — measured the energy-dependent conversion of electron neutrinos from the Sun, governed by $\theta_{12}$ and the MSW effect. SNO’s neutral-current measurement (2002) clinched the conversion interpretation.

The matching terrestrial measurement came from KamLAND, which watched reactor antineutrinos disappear over an average baseline of ~180 km. KamLAND saw not just a deficit but the characteristic spectral distortion of oscillation, fixing the small splitting $\Delta m^2_{21}$ with reactor neutrinos under full human control. The agreement between solar and reactor results was a striking cross-check of the whole framework.

The last angle: θ₁₃

The smallest angle, $\theta_{13}$, resisted measurement the longest, and its value was not even known to be non-zero. The decisive measurement came from reactor experiments at short baselines — about 1–2 km, tuned to the atmospheric splitting — which look for a small disappearance of reactor antineutrinos governed by $\theta_{13}$.

In 2012 the Daya Bay experiment in China measured a non-zero $\theta_{13}$ at 5.2σ, reporting $\sin^2 2\theta_{13} \approx 0.092$, with the RENO experiment confirming it shortly after. This was a turning point: $\theta_{13}$ turned out to be relatively large, which — because the CP phase enters oscillation multiplied by $\theta_{13}$ — made the search for leptonic CP violation experimentally feasible.

A consistent global picture

Taken together, these experiments overdetermine the framework: the same parameters fit atmospheric, solar, reactor, and accelerator data simultaneously. Global fits now report the three angles and both mass splittings to a few percent. Two things remain unsettled — the sign of the atmospheric splitting (the mass ordering) and the value of the CP phase $\delta_{\mathrm{CP}}$ — which is where the field’s current effort is concentrated.

References

1. 1.Y. Fukuda et al. (Super-Kamiokande Collaboration), Evidence for Oscillation of Atmospheric Neutrinos, Phys. Rev. Lett. (1998). DOI: 10.1103/PhysRevLett.81.1562.
2. 2.Q. R. Ahmad et al. (SNO Collaboration), Direct Evidence for Neutrino Flavor Transformation from Neutral-Current Interactions in the Sudbury Neutrino Observatory, Phys. Rev. Lett. (2002). DOI: 10.1103/PhysRevLett.89.011301.
3. 3.K. Eguchi et al. (KamLAND Collaboration), First Results from KamLAND: Evidence for Reactor Antineutrino Disappearance, Phys. Rev. Lett. (2003). DOI: 10.1103/PhysRevLett.90.021802.
4. 4.F. P. An et al. (Daya Bay Collaboration), Observation of Electron-Antineutrino Disappearance at Daya Bay, Phys. Rev. Lett. (2012). DOI: 10.1103/PhysRevLett.108.171803.