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Why do we need neutrino to explain neutron decay? Is there any evidence regarding existence neutrinos in the context of $n\to p + e + \bar{\nu}_e$?

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up vote 5 down vote accepted

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:

enter image description here

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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I would add that not only do neutrinos exist but we have made neutrino beams in the accelerators and the interactions with nuclei recorded in the experiments agree with all the definitions of the netrino as a particle with interactions. –  anna v Jul 7 '12 at 3:54
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To add to dmckee's fine answer, a large amount of the reverse process, p+e-> n occurs in a very short time during a supernova explosion, as all the electrons are shoved into the nucleus and inverse beta-decay becomes energetically favorable. The supernova 1987A was accompanied by a burst of neutrinos, which shows that the inverse beta decay emits neutrinos. The theory matches the experiment, which gives strong support both the the more controversial supernova models, but also to the older well accepted neutrino theory of beta-decay.

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We can detect neutrinos though, there are three types and they are massive (in the order of eV or meV).

  • Did it not cross your mind that the down quark is heavier than the up quark?

If you're going to disprove the existence of neutrinos, you are going to have to do more than some basic energy conservation; you will have to completely rewrite the standard model. A lot of electroweak physics involves neutrino interactions.

  • What is your substitute theory?

You need two fermions lines in an interaction vertex (ignoring the 3 and 4 boson vertices).

In fact I'm not sure what your calculation "proves" at all.

i guess All you can do is state an energy difference is the rest masses and give a random electron momentum

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