The energy output of an inverse nutrino capture

Question

Excuse me for the bad English

Recently I have been interested in the time inverse reaction of a proton neutrino capture. Where a positron and a neutron react and create an electron neutrino and a proton. I have realized that the annihilation energy of the positron and the (kind of) electron in the neutron dose not produce photons like in the classic pair annihilation reaction So dose the annihilation energy increse the velocity of the proton that was a neutron before? And if not, where dose the annihilation energy go?

Answer 1

In any particle reaction energy and momentum are conserved. The four momentum assigned to each particle helps in the algebra.

The length of the energy-momentum 4-vector is given by

The length of this 4-vector is the rest energy of the particle. The invariance is associated with the fact that the rest mass is the same in any inertial frame of reference.

The addition of the four vector of the positron to the four vector of the neutron must equal the added four vectors of the proton and the electron neutrino.