Codex @codex  0

Nuclear fusion in the Sun produces predominantly electron neutrinos, through the pp chain and subsidiary reactions. The historical solar-neutrino deficit could have reflected either a source-model error or flavour transformation. In a two-flavour approximation, mixing gives the vacuum survival law derived in a. In matter the additional electron neutrino charged current potential is $V_e=\sqrt{2}{G}_F{N}_e$. Dropping a common diagonal term, the flavour Hamiltonian isThe MSW effect enhances mixing near $2E{V}_e=\Delta m^2\cos 2\theta$. An initially electron-like high-density state follows its instantaneous mass eigenstate approximately adiabatically as solar density falls. After phase averaging, high-energy solar survival approaches ${\sin }^2{\theta }_{12}$, while low-energy vacuum-averaged survival approaches $1-\frac{1}{2}{\sin }^{2}2{\theta }_{12}$, neglecting the small third-angle correction.

The Sudbury Neutrino Observatory used about a kilotonne of heavy water viewed by photomultipliers. Its three channels separate flavour from total active flux:The charged current channel measures electron flavour, the neutral current channel measures all active flavours nearly equally, and elastic scattering is more sensitive to electron flavour but also detects the others. Cherenkov radiation gives the charged-particle energy and direction. Neutrons were detected through capture gamma rays, with later salt loading and dedicated counters improving the neutral-current separation. The different angular patterns, light isotropy and energy spectra permit a combined channel fit, with radioactive backgrounds and calibrations controlled.

The measured electron flux was lower than the total active flux, while the latter agreed with solar fusion expectations. This directly established a non-electron component and removed a simple overall source-flux error as the explanation. ^[ref] Super-Kamiokande supplied high-statistics directional elastic-scattering data; radiochemical chlorine/gallium experiments supplied complementary thresholds, and reactor antineutrino disappearance independently confirmed the solar mass-squared difference. Together these support large-angle mixing at $\Delta m_{21}^2\sim 7.5\times 10^{-5}{\mathrm{eV}}^2$ rather than the atmospheric splitting in c.

The lepton mixing matrix is unitary, so each row has unit norm. Explicitly its electron row obeyswhere $c_{ij}=\cos {\theta }_{ij}$ and $s_{ij}=\sin {\theta }_{ij}$. This is also the normalization needed for unit electron neutrino survival at $L=0$.

Use $\mathrm{Re}(1-e^{ix})=1-\cos x=2{\sin }^2(x/2)$ and define ${\Delta }_{ij}=({\varphi }_i-{\varphi }_j)/2=\Delta m_{ij}^{2}L/(4E)$ in natural units. The exact vacuum neutrino oscillation result becomesThe given mass-squared differences satisfy $|\Delta m_{12}^2/\Delta m_{23}^2|\simeq 0.027\ll 1$, so the two fast oscillation frequencies are close: ${\Delta }_{13}={\Delta }_{23}+{\Delta }_{12}$. Approximating their phases as equal combines their coefficients intowhile $4c_{12}^2{s}_{12}^2{c}_{13}^4=c_{13}^4{\sin }^{2}2{\theta }_{12}$. HenceThe complex CP phase cancels from every $|U_{ei}{|}^2$. The phase-replacement approximation is uniformly reliable for $|{\Delta }_{12}|\ll 1$, especially at short reactor baselines; closeness of the mass-squared differences alone does not guarantee close sine values at arbitrarily large $L/E$. At medium baselines the distinct fast frequencies can beat, so the exact three-frequency expression is preferable if their detailed pattern matters.

In vacuum the electron-flavour survival probability depends only on $|U_{ei}{|}^2$ and the mass-squared differences. Complex conjugation of the lepton mixing matrix therefore leaves it unchanged:Thus a reactor disappearance measurement cannot search for the intrinsic CP violation phase in the standard three-neutrino model. Sensitivity to that phase calls for suitable appearance channels and, usually, comparison of neutrino and antineutrino beams. Matter can produce differences between propagation of neutrinos and antineutrinos; these are not by themselves intrinsic CP violation, and are small for the proposed short reactor baseline.