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In the search for neutrinos, Cowan and Reines discovered the electron anti-neutrino and named it as such. Why is the particle they discovered the anti-variety?

The reason we call electrons 'electrons' and not 'anti-electrons' is because the electrons are the variety we see all around us, as opposed to positrons. Since this makes it seem arbitrary, I figured it would have been more logical for the particle they discovered to just be a neutrino. This makes me believe that it's not just arbitrary, so what exactly was the reason?

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To maintain lepton number as a conserved quantity.

Consider, in detail, what's going on in a beta decay (well, I'm going to ignore the nuclear context). The reaction is then $$ n \longrightarrow p^+ + e^- + \nu \,,$$ where you should take the symbol $\nu$ to mean some neutrino (without prejudice about matter-type or anti-matter-type for the moment).

There are zero leptons in the LHS of the equation, and on the RHS there is the electron (which counts as one lepton) and the neutrino. If we want lepton number to be conserved in the reaction (we do) then that has to have lepton number -1, which makes it an anti-particle. So, written properly the reaction is $$ n \longrightarrow p^+ + e^- + \bar{\nu} \,.$$

Later still it was recognized as necessary to assign a flavor to the neutrino and the symbol becomes $\bar{\nu}_e$, but that's another story.

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