Positron decay direction from muon

[Malamala]

Hello! I read that in the rest frame of a positive muon, decay positrons are preferentially emitted in the direction of the muon spin. Why is that the case? The decay is $\mu^+\to e^+\nu_e\bar{\nu_\mu}$. Assuming that the positron is emitted at almost the speed of light, it will be a left handed particle. The 2 muons are right handed and left handed. So we have a 3 body decay, with the 3 particles having their spin aligned or anti-aligned with their momentum. But I am not sure how can I get from here that the positron is more likely to be along the muon spin. Can someone help me? Thank you!

 

[Astronuc]

The 2 muons are right handed and left handed.

I believe one is referring to two neutrinos, or rather neutrino and antineutrino.

A muon decay positron is also correlated with the angular momentum of the muon in a more complicated way. Here, the positron with maximum energy takes the angular momentum of the decaying muon, since the left-handed electron neutrino and the right-handed muon antineutrino angular momenta cancel.

See slide 6 in https://www.triumf.ca/sites/default/files/sslec_201310.pdf

See also - https://www.hep.lu.se/staff/c.jarlskog/nffl_7184.pdf

 

[Malamala]

I believe one is referring to two neutrinos, or rather neutrino and antineutrino.

See slide 6 in https://www.triumf.ca/sites/default/files/sslec_201310.pdf

See also - https://www.hep.lu.se/staff/c.jarlskog/nffl_7184.pdf

Thank you so much for this! The slide 6 illustration is useful, however I am not totally clear I understand. In the upper illustration, it is clear why the spin and momenta are the way they are, as there is no other way. But why does the configuration in the bottom plot has to be like that? Assuming helicity and chirality eigenstates are the same, can't we switch the position of $e^+$ and $\bar{\nu}_\mu$, and still conserve the overall spin? In that case the positron would be emitted in the opposite direction from the muon spin.

 

[ChrisVer]

So the main thing there is the word "preferential". If you were to violate a principle like angular momentum, the world would have been "unique". So you can't answer to your question by trying to reach a spin-violation configuration.

See https://arxiv.org/pdf/hep-ph/9909265.pdf [C.1 and e.g. equation 30 or Fig.2]
or the whole Sec.1.5 here: https://www.psi.ch/sites/default/fi...apter_1_to_5_up_to_Section_5_5_23_03_2018.pdf