I don't understand how to solve this:
A $\pi^+$ decays into a muon and neutrino. Find the pion's energy if
- max $E_\nu$ / min $E_\nu$ = 100/1;
- $m_\nu = 0$
- $m_\pi*c^2 = 140\text{ peta-eV}$
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Here's the deal. The energy of the decay products depends only on the emission angles in the laboratory frame (they always come out back-to-back in the decay frame). First, recall that you must conserve four momentum at the vertex, and that you can look up the mass the muon (and you've been given masses for the pion and the neutrino [1] ). So, draw the kinematics for the most and least energetic neutrino emissions (this is the part the problem expects you to think about, the rest is mechanical computation); express the neutrino energies in terms of only the masses and the pion energy; form the ratio, equate it too 100 and solve for $E_\pi$. Done. [1] What happens if you use the pion mass from the PDG, instead? If you allow that the neutrino actually has some mass less than 1 eV? Are these approximations reasonable? |
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