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An exam question reads;

By considering the helicity of the decay products, and the conservation of angular momentum, show that the high energy positrons in the reaction $\mu^+ \to e^++\nu _e +\nu_\mu^*$are preferentially emitted in the direction of the muon's spin in its rest frame.

In my original attempt to this question I considered the lab frame situation first;

$H_{\nu_\mu^*}=1/2, H_{\nu_e}=-1/2, \implies H_{\mu^+}=H_{e^+}$

$s_{\mu^+}=s_{e^+}+s_{\nu_e}+s_{\nu_\mu^*}$, since the neutrinos will be emitted very close together and back-to-back with the positron and travelling in the same direction as each other, and due to the fact that they have opposite helicities they must have opposite spins, and therefore their spins cancel.

This led me to conclude that the positrons are emitted parallel to the muon's original momentum in the lab frame, which seems totally obvious. I'm not sure how to convert this into the muon's rest frame however.

So, in my second attempt, I tried to just go ahead in the muon's rest frame, but immediately ran into the issue of trying to define the helicity of a stationary particle. Surely this is ill defined? How can this even be done?

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  • $\begingroup$ You came close. Go to the μ's rest frame with its spin going up. Its spin now equals that of the positron's, and the 0 of the antisymmetrized νs' you posited. So the largest projection of it on the momentum of the e+ is when they are aligned, so its momentum is parallel to its spin, no? $\endgroup$ – Cosmas Zachos Apr 25 '17 at 21:45
  • $\begingroup$ @CosmasZachos When you say it like that it seems fairly obvious - does this not violate the conservation of helicity however? This was my main issue when trying to think in the muon rest frame - the initial helicity is 0, so shouldn't the final products have 0 helicity too? $\endgroup$ – arcturus7 Apr 25 '17 at 22:10
  • $\begingroup$ Your exam did not mention any helicity of the parent particle. It asked you to consider conservation of spin and the helicity of the decay products. You consider the helicity of a positron and a di-ν (o). $\endgroup$ – Cosmas Zachos Apr 25 '17 at 22:44
  • $\begingroup$ Hmm. Whilst that is true I'm still not 100% sure I follow - I still don't understand why it isn't an issue... Looks like I'll have to find some new source material for a different perspective to that offered by my notes! $\endgroup$ – arcturus7 Apr 25 '17 at 23:13
  • $\begingroup$ Yes; books rule. That's why they are there. $\endgroup$ – Cosmas Zachos Apr 26 '17 at 0:33

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