# $e^+e^- \rightarrow \mu^+ \mu^-$ cross section with circular polarization?

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.

• 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 Feb 14 at 3:13
• 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...) Feb 14 at 10:55