Initial Particle Instability in Muon Collisions

I consider the process $\mu^+\mu^-\to e\bar{\nu}W^+$ in the case when the effective mass of the $(e\bar\nu)$ system than the muon mass. In this case the momentum transferred from the initial muon to the $e\bar\nu$ system (the virtual neutrino momentum) can be both time--like and space--like. Since the path of integration over $k^2$ goes through a pole at $k^2=0$, it gives a divergent cross section. In the ideal case of large enough beams this divergence disappears if the finite width $\Gamma$ of the initial muon is taken into account. The obtained cross section corresponds to the flux of equivalent neutrino, which coincides with that of muon (with some energy distribution). In practice, the effect of final size of the muon beam reduces this cross section very strong, and the effect is hardly observable.