Particle reactions in the center of mass frame

Question

I am having some conceptual difficulties with a seemingly simple phenomena in particle physics. My professor told me that the reaction $e^-\rightarrow \nu_{e^-}+\pi^-$ can not occur in the center of mass frame because the electron does not have enough rest mass energy to account for the pion. However, he also explained that if the electron has enough kinetic energy that the reaction could occur. The problem with my understanding is that this implies that there is a frame of reference in which the electron is moving before it is converted into the two particles. However, if we then transform back into a frame in which the electron is again stationary, it would look like the electron is converted into the two particles and imparts even more kinetic energy to them than the original center of mass frame scenario. Is this because in this new scenario of the electron with kinetic energy there must be a photon involved in the reaction to make up for this energy or something?

Answer 1

However, he also explained that if the electron has enough kinetic energy that the reaction could occur

You have misunderstood something.

It is not enough for the electron to be in a frame where it has large kinetic energy, it has to also interact with some other particle, so a neutrino can go off with the electron quantum number.

This is because an elementary particle cannot decay. The electron is a point particle, it is not composed out of anything, (as a neutron is for example). It will just be sitting there at rest in the center of mass system, or moving in another framework.

If there is enough energy in the center of mass of the whole system , (electron + an interacting particle or field), a pi0 has a probability of appearing. One goes to the center of mass to see what the energy available is, because the reaction of particles are Lorenz transformation invariant.

The reaction could be, for example, $e^-+p > ν_e + π^0 +n$

One has to conserve lepton number baryon number and charge, for the reaction to be possible even if the energy in the center of mass is available for the generation of a $π^0$. The reaction happens through the weak interaction, so it is very improbable to see the channel in electron scattering off protons.