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A related post was What is polarisation, spin, helicity, chirality and parity?

In $m_\mu<<E$ region, $m_\mu$ could be treated as massless particles and the conservation of helicity indicated the polarized cross section dependent on $\theta$.

In $m_\mu\approx E$ region, $m_\mu$ could not be ignored but that the conservation of spin made the polarized cross section independent on $\theta$.

However, notice that in both case, $m_e<<E$ the electron was in the relativistic region, and it was a bit hard to see why in one case the computation used helicity and in another case the computation used spin, where the treatment of the electron's amplitude was not explained.

Why not use the spin conservation for $m_\mu<<E$ and helicity conservation for $m_\mu\approx E$? What happened to the helicity and spin conservation in the $e^+e^-\rightarrow \mu^+ \mu^-$?

ShoutOutAndCalculate
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    Linked. – Cosmas Zachos Oct 23 '21 at 15:47
  • @CosmasZachos Peskin chapter 5 – ShoutOutAndCalculate Oct 23 '21 at 22:10
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    But... m < is below threshold, as P&S remind you! the ruction can't go for lack of energy. In fact, at threshold, the cross section vanishes. They also explain that associating helicity with chirality breaks down in the non relativistic case, near threshold. Keep reading... – Cosmas Zachos Oct 23 '21 at 22:50
  • @CosmasZachos I guess the question was how did the $\theta$ dependence transitional vanished when taking $E\rightarrow m_\mu$... Where did he mention "$m_\mu<E$" again? The chapter 5 seemed to be quite contained and they concluded the calculation in section 5.3. – ShoutOutAndCalculate Oct 25 '21 at 06:10
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    You learned that in the NR limit there is no association between helicity and chirality for the muons. He reminds you (5.12) → (5.33) that your chimerical m << so, m < cannot support muon production, and, at threshold, the cross section vanishes. At that very point, the muons are stationary, so θ is not defined. At slightly higher energies, spin can point any which way, as in the figure, so the direction of the muon momenta is decoupled from it, as per the properties of the Dirac equation solutions. – Cosmas Zachos Oct 25 '21 at 14:08
  • @CosmasZachos I think you might had a different version of the book. In the beginning of section 5.3, quote"When $E$ is barely larger than $m_\mu$", as $m_\mu <E$. There were discussion about the bounded states but that's different story. He didn't mention the case where $E=m_\mu$ since he was not interested. – ShoutOutAndCalculate Oct 25 '21 at 17:37
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    These are issues you brought up, and I'm telling you what his (5.12) → (5.33) mean. He need not define threshold and its implications for you: aren't they evident? – Cosmas Zachos Oct 25 '21 at 17:54
  • @CosmasZachos It was not evident, $E=m_\mu$ physically had so many issues even from the experimental point of the view so he only discussed it in terms of the bounded states.(and quote "can still from $\mu^+\mu^-$ pairs in electromagnetic bounded states") The process where the dependence on $\theta$ vanished was hide in those relativistic and non relativistic "approximations" in Eq. 5.11 where it gradually demonstrated how the (massless) relativistic particle's helicity different from that of the non relativistic ones. There ought be be some sort of symmetry representation's generator g ->1. – ShoutOutAndCalculate Oct 25 '21 at 18:30

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