$\order(\Gamma)$ Corrections to $W$ pair production in $e^+e^-$ and $\gamma\gamma$ collisions

Several schemes to introduce finite width effects to reactions involving unstable elementary particles are given and the differences between them are investigated. The effects of the different schemes is investigated numerically for $W$ pair production. In $e^+e^-\to W^+W^-$ we find that the effect of the non-resonant graphs cannot be neglected for $\sqrt{s}\geq400\GeV$. There is no difference between the various schemes to add these to the resonant graphs away from threshold, although some violate gauge invariance. On the other hand, in the reaction $\gamma\gamma\to W^+W^-$ the effect of the non-resonant graphs is large everywhere, due to the $t$-channel pole. However, even requiring that the outgoing lepton is observable ($p_\perp > .02\sqrt{s}$) reduces the contribution to about 1\%. Again, the scheme dependence is negligible here.