Why do we need neutrino to explain neutron decay? Is there any evidence regarding existence neutrinos in the context of $n\to p + e + \bar{\nu}_e$?
Answer
If the neutron decayed to a two body state (any two body state) the energy spectrum of the products in the neutrons rest frame would be single valued (this is required by the conservation of energy and momentum).
It is not.
Instead the electron energy spectrum is a continuum that runs from that roughly the two-body limit down to as near zero as our instruments can measure. To grab an image from the wikipedia:
So, a third particle is required. That third particle is known to be uncharged (because our detectors are sensitive to charged particles and do not see it). It is also known to be of very low mass because the end-point of the electron energy spectrum is almost exactly what you would expect from the two body decay. The lifetime of the neutron suggests that the interaction that is responsible for it's decay is very weak (and going on a little further in history it obeys the principle of weak universality suggesting that it is the same interaction responsible for the decay of strange hadrons).
The sum of these requirements constrain the properties of the third particle quite a lot, and much observation since then has shown quite conclusively that neutrinos exist.
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