# Muon 3 Body Decay

I'm trying to calculate the maximum energy of the electron in the decay muon >electron + electronantineutrino +muonneutrino in the reference frame of the muon having no kinetic energy.

$m_m$=mass of muon

$m_n$=mass of neutrino

$p_n$=neutrino momentum

$p_n1$=electron anti neutrino momentum

$p_n2$=muon neutrino momentum

$E_e$=Electron energy

Using energy conservation

E²before=E²after

$m_m^2c^4=m_e^2c^4+2m_n^2c^4+p_e^2c²+p_n1^2c^2+p_n2^2c^2$

The mass of neutrino's is negligible

$m_m^2c^4=m_e^2c^4+p_e^2c^2+p_n1^2c^2+p_n2^2c^2$

Max energy of electron occurs when the least amount of energy is given to the neutrino's, this will happen when both electrons move together in the opposite direction to the electron, so that all their momentum is used to cancel out the electron's momentum, with the neutrino's sharing the momentum.

$p_n1=p_n2=p_n$

$2p_n=-p_e$

$p_e²=4p_n²$

$p_n²=p_e²/4$

$m_m^2c^4=m_e^2c^4+p_e^2c^2+p_e^2c^2/4+p_e^2c^2/4$

$m_m^2c^4=m_e^2c^4+p_e^2c^2+p_e²c²/2$

$m_m^2c^4=m_e^2c^4+3p_e^2c^2/2$

$2m_m^2c^4/3=2m_e^2c^4/3 +p_e^2c^2$

$2m_m^2c^4/3+m_e^2c^4/3=m_e^2c^4+p_e^2c^2$

$2m_m^2c^4/3+m_e^2c^4/3=E_e^2$

$m_e^2c^4/3$ is negligible.

$2m_m^2c^4/3=Ee^2$

$sqrt(2)m_mc^2/sqrt(3)=Ee$

$E_e = sqrt(2/3)$ the energy of the muon, however I know this is wrong as the energy should be 0.5*energy of muon.

Could someone people point out where I've gone wrong?

Using energy conservation E²before=E²after
E after is of course the sum of energies:
and Energy conservation means, first of all, that E before == E after.
When squaring this equation you need to take care that the right-hand-side thereby becomes the square of a sum of energies: (E before)² == (E after)²;