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?


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.

  • $\begingroup$ I was under the impression that the exercise could be solved to give a numerical answer without knowing any mass values except for the mass of the muon. Would this be possible? $\endgroup$ – Vinteuil Apr 17 '18 at 22:35
  • 1
    $\begingroup$ No; this is not the house of elegance: It is the real world. Imagine the science fiction limit where the nucleon were 100 smaller than the muon, instead. Do you see the point? $\endgroup$ – Cosmas Zachos Apr 17 '18 at 22:49

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