Effect of Initial Particle Instability in Muon Collisions

We study μ+μ− collision with production of some state F (for example, F = W+) and (eν̄) system with the effective mass, which is less than muon mass. In this case the momentum k, transferred from μ to eν̄ system (virtual neutrino momentum), can be time–like. The path of integration over k2 goes through the pole at k2 = 0. It gives divergent cross section. This divergence disappears if the finite width Γ of initial μ is taken into account. It gives the factor ∝ Γ−1 in cross section. The compensating factor Γ appears from μ → eν̄ν subprocess. The resulting cross section is finite but high enough. The modern discussions about muon colliders (see e.g. [1]) provide new place for studying of effects related to particle instability. For the definiteness, we consider the process (momenta of particles are shown in brackets): μ−(p1)μ (p2) → e(q1)ν̄(q2)W(p3). (1) We use the following notations: M is the W boson mass, m is the muon mass, s ≡ 4E = (p1 + p2); x = M/s; q = q1 + q2, k = p1 − q ≡ p3 − p2; we neglect electron mass. 1. We study the specific kinematical region where the effective mass of eν̄ system is less than muon mass, q < m. In this case the transferred momentum k can be time–like, its maximal value is positive: