PMNS mixing matrix
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If a neutrino's flavor is a recipe blending three pure mass states, then somewhere there must be a master recipe book that says exactly how much of each mass state goes into each flavor. That recipe book is the PMNS mixing matrix. It is a compact grid of nine numbers — three rows for the flavors (electron, muon, tau), three columns for the mass states — and each entry tells you how strongly a given mass state contributes to a given flavor.
The name honours the physicists Pontecorvo, Maki, Nakagawa, and Sakata, who developed the idea. The matrix is not free to be anything: because probabilities must add up to one, its rows and columns obey strict balancing rules (mathematically, it is 'unitary'). Once you know its entries, you can predict everything about oscillation — how an electron neutrino's chance of staying an electron neutrino rises and falls with distance, how often it becomes a muon or tau neutrino, and so on. In practice the nine numbers are summarized by just a few independent parameters: three mixing angles and one phase that can produce a matter-antimatter asymmetry.
The PMNS matrix is the neutrino-world counterpart of the CKM matrix that governs quark mixing, but the two look strikingly different. The CKM matrix is nearly diagonal — quarks mix only a little — whereas the PMNS matrix has large off-diagonal entries, meaning neutrinos mix a lot. Why leptons mix so generously while quarks mix so stingily is an unsolved puzzle, a clue that may point toward whatever deeper theory lies beyond the Standard Model.
Reading the matrix's electron row tells you what an electron neutrino is made of: large helpings of the first and second mass states and a small dash of the third. That small third entry, called theta-13, was the last mixing angle to be measured, finally pinned down by reactor experiments in 2012.
Each row of the PMNS matrix is the mass-state recipe for one flavor.
The PMNS matrix has at least one phase that could let neutrinos and antineutrinos oscillate differently — a form of CP violation in leptons. Whether that phase is non-zero is a major question that experiments are still working to settle.