= Solution
For <electron-positron annihilation into a muon pair> the gauge diagrams are s-channel <photon> and Z exchange. For <electron-positron annihilation into electron neutrinos> there is s-channel Z exchange and t-channel W exchange, with <interference>. For <electron-positron annihilation into W-boson pairs> there are s-channel <photon> and Z diagrams and a t-channel electron-neutrino diagram. All three gauge amplitudes are required for the proper high-energy cancellation. Electron-Yukawa-suppressed s-channel Higgs diagrams can also produce the massive <muon> or W pair; they vanish in the usual zero-electron-mass approximation used for the leading gauge comparison.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2012/paper-1-lepton-topologies.png]
{title=Lepton and W-pair gauge diagrams}
There is \b[no energy-independent ordering]. For unpolarized beams and inclusive Born <cross sections> the useful regimes are:
* Below the <muon> threshold only the <neutrino> channel is open, ignoring its tiny mass threshold. Immediately above $2m_\mu$, <muon> <phase space> initially vanishes.
* Well above $2m_\mu$ but well below the weak-boson masses, <photon> exchange gives $\sigma_{\mu\mu}\simeq4\pi\alpha^2/(3s)$, whereas $\sigma_{\nu_e\bar\nu_e}=O(G_F^2s)$. Hence the <muon> channel dominates and real W-pair production is closed.
* Near the Z pole, W pairs are still closed and the Z resonance dominates both open channels. The ratio is approximately $\Gamma(Z\to\nu_e\bar\nu_e)/\Gamma(Z\to\mu^+\mu^-)\simeq2$, from their neutral-current couplings: <neutrinos> rank above <muons>. This is for one <neutrino> flavour.
* At $\sqrt s=2m_W\simeq160.806\,\mathrm{GeV}$ the on-shell W-pair channel opens from zero <phase space>. Its position in the ordering changes across threshold and must be obtained from the summed amplitudes, rather than inferred solely from a weak/electromagnetic label.
* Far above the weak masses, forward t-channel W exchange makes the inclusive electron-neutrino <cross section> approach a nonzero scale of order $g^4/m_W^2$. The W-pair <cross section> behaves as $O[g^4\log(s/m_W^2)/s]$ and the <muon> <cross section> as $O(g^4/s)$. Thus asymptotically the decreasing order is $\nu_e\bar\nu_e$, then $W^+W^-$, then $\mu^+\mu^-$. Fixed angular cuts, <beam spin polarization> and radiative corrections modify these comparisons.
The Z-pole ratio follows $[\tfrac12]^2/[(-\tfrac12+\sin^2\theta_W)^2+\sin^4\theta_W]\simeq1.99$. Treating every weak reaction as uniformly slower misses both the resonance and forward exchange.
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