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

TL;DR

The paper addresses how to incorporate finite width effects for unstable particles in two-resonance processes, focusing on $e^+e^- \to W^+W^-$ and $\gamma\gamma \to W^+W^-$. It compares several gauge-invariant and gauge-variant schemes for including ${\cal O}(\Gamma)$ corrections, including the resonant, off-shell, naive, overall, and polescheme approaches, and analyzes threshold behavior and non-resonant contributions. The main findings show that non-resonant graphs matter for $\sqrt{s} \gtrsim 400$ GeV in $e^+e^-$ collisions, while away from threshold most schemes agree within a few percent; in $\gamma\gamma$ collisions, non-resonant effects are large only without cuts but are suppressed to about 1% with a modest $p_T$ cut, with negligible scheme dependence. These results guide precision predictions for LEP II and future colliders, highlighting when simple resonant approximations suffice and when threshold-region effects demand more careful treatment, potentially including resummation or bound-state considerations.

Abstract

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 $γγ\to W^+W^-$ 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.

Figures (10)

• Figure 1: Resonant diagrams for $e^+e^-\to \mu^- \bar{\nu}_\mu u \bar{d}$
• Figure 2: Two of the four non-resonant diagrams for $e^+e^-\to \mu^- \bar{\nu}_\mu u \bar{d}$
• Figure 3: Resonant diagrams for $\gamma\gamma\to \ell^- \bar{\nu}_\ell W^+$
• Figure 4: Non-resonant diagrams for $\gamma\gamma\to \ell^- \bar{\nu}_\ell W^+$
• Figure 5: Cross sections for $e^+e^-\to \mu^- \bar{\nu}_\mu u \bar{d}$ in the different schemes defined in the text. The "overall" scheme yields results which are indistinguishable from those of the "naive" scheme.