Processes with the t-channel singularity in the physical region: finite beam sizes make cross sections finite

It is known that some high-energy processes have a $t$-channel singularity in the physical region. In this paper we show that this singularity is regularized if one takes into account the finite sizes of the colliding beams, i.e. the realistic situation which takes place at high--energy colliders. On this way we obtain the finite cross section which is linear proportional to the transverse sizes of the colliding beams. As an application of the above result we study the production of the single $W$ boson at $\mu^+\mu^-$ colliders in the reaction $\mu ^- \mu ^+ \to e \bar \nu _e W^+ $. For the total energy $\sqrt {s}\approx 95$ GeV and reasonable parameters of the muon beams the ``beam--size '' dependent part of the cross section for this reaction is of the order of the standard cross section of the $\mu ^- \mu ^+ \to \mu^- \bar \nu _{\mu} W^+ $ process.