Effect of Initial Particle Instability in Muon Collisions

Abstract

We study $\mu^+\mu^-$ collision with production of some state $F$ (for example, $F= W^+$) and $(e\bar\nu)$ system with the effective mass, which is less than muon mass. In this case the momentum $k$, transferred from $\mu$ to $e\bar\nu$ system (virtual neutrino momentum), can be time--like. The path of integration over $k^2$ goes through the pole at $k^2=0$. It gives divergent cross section. This divergence disappears if the finite width $\Gamma$ of initial $\mu$ is taken into account. It gives the factor $\propto\Gamma^{-1}$ in cross section. The compensating factor $\Gamma$ appears from $\mu\to e\bar\nu \nu$ subprocess. The resulting cross section is finite but high enough.