Minimum energy of a neutrino for charged current reaction with nucleus

Question

I am struggling with a practice problem regarding the following reaction:

\begin{equation} \nu_{\mu}+n\rightarrow\mu^{-}+p \end{equation}

The question is "What is the minimum energy of the muon neutrino for the reaction to occur on a neutron initially at rest?". The question is supposed to be solved by assuming that $m_{p}=m_{n}$ without quoting their values. Furthermore, the mass of the muon is provided and the neutrino mass is neglected.

In my attempt to answer the question I have obtained the following relationship from conservation of the energy-momentum four-vector:

\begin{equation} \sqrt{2E_{\nu_{\mu}}m_{n}+m_{n}^{2}}=m_{\mu}+m_{p} \end{equation}

In order, to solve the question I am trying to eliminate $m_{n}$ and $m_{p}$ from this equation. How would one do this?

Answer 1

You are almost there. Drop the label of the nucleon mass, as instructed, and square both sides of your inequality (bounding the labile magnitude of the 4-momentum of the r.h.s. evaluated in the cm frame by the fixed magnitude of the 4-momentum of the l.h.s.), $$ E_\nu \geq m_\mu (1+m_\mu/2m) > m_\mu , $$ since $m_\mu/m \sim 0.11$, not quite a big deal, all in all.