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I'm trying to work through problem 5.4 of Schwartz (QFT and the Standard Model), which involves calculating the spin components of the matrix corresponding to the $e^+e^- \rightarrow \mu^+ \mu^-$ cross section.

The problem starts by assuming two incoming electrons $e^-$ and $e^+$ as before, with momenta $p_1^\mu = (E, 0, 0, E)$ and $p_2^\mu = (E, 0, 0, -E)$. These electrons annihilate and produce a spin-1 photon, for which the circular polarizations are $\epsilon^1 = \frac{1}{\sqrt{2}}(0, 1, i, 0)$ and $\epsilon^2 = \frac{1}{\sqrt{2}}(0, 1, -i, 0)$.

The electrons then produce two muons with momenta for the $\mu^-$ and $\mu^+$ are (respectively) $p_3^\mu = E(1, 0, \sin{\theta}, \cos{\theta})$ and $p_4^\mu = E(1, 0, -\sin{\theta}, -\cos{\theta})$. The muon direction of motion, with respect to the electron direction of motion, corresponds to a rotation of $\theta$ about the $x$-axis. So the corresponding spin-1 polarizations are $\overline{\epsilon}^1 = \frac{1}{\sqrt{2}}(0, 1, i\cos{\theta}, -i\sin{\theta})$ and $\overline{\epsilon}^2 = \frac{1}{\sqrt{2}}(0, 1, -i\cos{\theta}, i\sin{\theta})$. I've checked that these are mutually orthogonal, as well as orthogonal to the muon direction fo motion.

The problem then comes from calculating the matrix elements $\mathcal{M_1} = \epsilon^{1*}\overline{\epsilon}_1 + \epsilon^{1*}\overline{\epsilon}_2$ and $\mathcal{M_2} = \epsilon^{2*}\overline{\epsilon}_1 + \epsilon^{2*}\overline{\epsilon}_2$. Since $\overline{\epsilon}_1 + \overline{\epsilon}_2 = \frac{1}{\sqrt{2}}(0, 2, 0, 0)$, then we no longer have any dependence on $\theta$. But the differential cross section does have a theta dependence of $(1 + \cos{\theta})^2$. So I'm a little confused as to where my thought process has gone wrong.

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  • $\begingroup$ Polarizations need to satisfy the completeness also $\sum_{i=1}^2\epsilon_\mu^i \epsilon_\nu^j=-g_{\mu\nu}+\frac{p^\mu p^\nu}{|P|^2}$. The second set of polarizations seem to me wrong $\endgroup$ Feb 14 at 3:13
  • $\begingroup$ That's interesting - even the linear polarization vectors given in the main text (section 5.3) of Schwartz do not obey this completeness relation! (Though I agree that physically it must hold...) $\endgroup$
    – rrose
    Feb 14 at 10:55

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