I'm trying to learn the spinor-helicity formalism from Schwartz's QFT book.

His equation 27.44 is describes the annihilation of an electron(1)-positron(2) pair to a muon(3)-antimuon(4) pair. He writes,

$$iM(1^-2^+3^-4^+)=(-ie)^2 \langle 2\gamma^{\mu}1]\frac{-i g_{\mu\nu}}{s}\langle 3 \gamma^{\nu} 4] = 2 \frac{ie^2}{s} [41] \langle 23 \rangle \tag{27.44}.$$

On the RHS, we have $1]$, $4]$ and $2\rangle$, $3\rangle$ which correspond to right-handed particles 1 and 4, left-handed particles 2 and 3.

However on the LHS, the labels 1 and 2 appear to label them oppositely. Furthermore, considering the symbol on the LHS is defined for all incoming momenta, then we seem to have the opposite helicity label for 3 and 4 also.

If anyone could explain, I'd appreciate that thanks.

  • $\begingroup$ Why do you feel like the RHS labels the particles inversely? $\endgroup$ – Prahar Mar 20 '16 at 22:27
  • $\begingroup$ Well a $1]$ for instance means a right-handed electron. But on the left it is labelled $1^-$ which means negative helicity. I thought right-handed meant the same as positive helicity. $\endgroup$ – Kris Mar 20 '16 at 22:39

There is no problem here. You may refer to (27.42) in Schwartz's book, which is usually called Fierz identity. Or by direct matrix calculation you can get the same result.


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