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does anybody know how to compare (at tree level) the two reactions

[tex]e^- + e^+ \to \gamma^\ast \to e^- + e^+[/tex]

and

[tex]e^- + e^+ \to Z^\ast \to \nu + \bar{\nu} \;\;\; \text{and} \;\;\; e^- + e^+ \to W^\ast \to \nu_e + \bar{\nu}_e[/tex]

The first process is the so-called Bhabha scattering whereas the second one is the neutrino-antineutrino pair creation from virtual W- and Z exchange.

In order to compare similar processes one should restrict to electron neutrinos in the final state of the Z boson; then both processes are rather similar in terms of their Feynman diagrams; the only difference is that in Bhabha scattering the exchanged particle (in both s- and t-channel) is a photon whereas in the second process the s-channel contribution comes from the Z, the t-channel contribution from the W, respectively.

My problems are the following:

1) the total cross section for Bhabha scattering diverges due to the long range Coulomb force; it has the well-known forward singularity

[tex]\frac{d\sigma_{e^-e^+ \to e^-e^+}}{d\Omega} \sim \frac{1}{\sin^4 \frac{\theta}{2}}[/tex]

so one can't compare cross-sections directly

2) for the latter process I can't find any low-energy matrix element; in the literature only the high-energy regime is described.

Does anybody know

- how to compare the matrix elements, cross sections or the branching ratio for the two processes?

- whether there is a different possibility to compare photon- and W-/Z-boson-exchange at low energies?

- whether there are experimental results down to the MeV range?

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# Neutrino pair creation in electron positron scattering

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