Stable Calculations for Unstable Particles: Restoring Gauge Invariance

TL;DR

Unstable vector-boson propagators with energy-dependent widths generically violate gauge invariance, which jeopardizes precision predictions for high-energy scattering. The authors show that the most consistent restoration is to add fermionic one-loop corrections to the triple-boson vertex, yielding results that satisfy all Ward identities. They derive the relevant vertex correction and demonstrate, in the process e^+e^- → e^- ν̄_e u d̄, that current conservation is restored and high-energy behaviour returns to the gauge-invariant limit. The work compares schemes numerically and provides practical recommendations for LEP2 Monte Carlo computations, while noting that decays into bosons can complicate gauge invariance.

Abstract

We discuss theoretical and phenomenological aspects of the use of boson propagators with energy-dependent widths in predictions for high-energy scattering processes. In general, gauge invariance is violated in such calculations. We discuss several approaches to restore gauge invariance, necessary for a reliable result. The most promising method is the addition of the relevant parts of the fermionic corrections, which fulfills all Ward identities. The numerical difference between this and other approaches is studied. A number of recommendations are given for LEP2 computations.

Figures (3)

• Figure 1: The four diagrams of the process $e^-e^+\to e^-\bar{\nu}_e u \bar{d}$ which are considered here.
• Figure 2: The extra fermionic diagrams needed to cancel the gauge-breaking terms.
• Figure 3: The effect of gauge-breaking terms in $e^-e^+\to e^-\bar{\nu}_e u \bar{d}$ as a function of the virtuality of the photon.