 Solution

ID: past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2012/paper-1/3/d/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.

There is 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 $2 m_{μ}$ , muon phase space initially vanishes.
Well above $2 m_{μ}$ but well below the weak-boson masses, photon exchange gives $σ_{μμ} ≃ 4 π α^{2} / (3 s)$ , whereas $σ_{ν_{e} \overset{ν}{ˉ}_{e}} = O (G_{F}^{2} s)$ . 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 $Γ (Z \to ν_{e} \overset{ν}{ˉ}_{e}) /Γ (Z \to μ^{+} μ^{-}) ≃ 2$ , from their neutral-current couplings: neutrinos rank above muons. This is for one neutrino flavour.
At $s = 2 m_{W} ≃ 160.806 GeV$ 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 $O [g^{4} lo g (s / m_{W}^{2}) / s]$ and the muon cross section as $O (g^{4} / s)$ . Thus asymptotically the decreasing order is $ν_{e} \overset{ν}{ˉ}_{e}$ , then $W^{+} W^{-}$ , then $μ^{+} μ^{-}$ . Fixed angular cuts, beam spin polarization and radiative corrections modify these comparisons.

The Z-pole ratio follows

[\frac{1}{2}]^{2} / [(- \frac{1}{2} + sin^{2} θ_{W})^{2} + sin^{4} θ_{W}] ≃ 1.99

. Treating every weak reaction as uniformly slower misses both the resonance and forward exchange.

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