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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.

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  • $\begingroup$ Why do you feel like the RHS labels the particles inversely? $\endgroup$
    – Prahar
    Commented Mar 20, 2016 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
    Commented Mar 20, 2016 at 22:39
  • $\begingroup$ I have the same problem here. Did you find a solution for your question? Because upon rescalling 1] and 4] we would get that they have the same helicity, which isn't true looking at the form on the left. If we consider other formulas in the book, like (27.60), we see that upon rescaling the helicity spinors in the right, they correspond to the helicty on the left where we are considering all incoming momenta. But the particular case of the formula in the question doesn't agree with doing the rescaling and analysing the corresponding helicity in the left side, like in other cases in the chapter $\endgroup$
    – Slayer147
    Commented Nov 21, 2019 at 23:20
  • $\begingroup$ @Slayer147 I was learning this for an exam and haven't looked at it since. The answer given by Kangle might be correct but I haven't verified. $\endgroup$
    – Kris
    Commented Nov 22, 2019 at 15:36

1 Answer 1

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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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