Max Electron Energy from Muon Relativistic Decay

[eep]

Consider the process of muon decay: muon (at rest) -> electron + 2 neutrinos. Assuming the neutrinos are massless and the muon decays from rest, what is the maximum possible electron energy? We are given the mass of the muon, the mass of the electron, and are told to treat the neutrinos as being massless (E = pc).

My intuition tells me that the electron will have maximum energy when it is moving opposite to the two neutrinos, both moving in the same direction. I'm not too sure how to show this mathematically, however. Obviously, momentum and energy need to be conserved but by doing this I end up with 3 equations (Energy conservation, parallel momentum conservation, perpendicular momentum conservation) but six unknowns (speed of electron in parallel direction, speed of election in perp direction, momentum of first neutrino in parallel direction, perp direction, momentum of second neutrino in parallel, perp direction).

I figure by doing some boosts into different frames I can get more equations, but is there a simpler way of approaching this? I can post more details if needed but I think the question is pretty straightforward.

 

[Meir Achuz]

$$2p+\sqrt{4p^2+m^2}=M$$.