Maximum scattering angle for 2 body collision

[bayners123]

I've got a question that asks what the maximum scattering angle in $\nu_\mu e \rightarrow \nu_\mu e$ is. The electron is stationary in the lab frame and after the collision has $E >> m_e$.

The answer given is
$$\sqrt{\frac{2 m_e}{E_e}},$$
independent of the neutrino's energy. How can I get to this? I can't think of an approach that doesn't involve the neutrino's energy at all.

 

[mfb]

If you use the neutrino energy in your calculation, it should vanish at some point. You don't have to know that in advance to calculate the scattering angle.

A higher neutrino energy leads to a smaller neutrino deflection angle at the same energy transfer E. It is interesting that the result is independent of the neutrino energy.