Fermion Loops, conserved currents and single-W

The problem of preserving gauge invariance in the calculations involving unstable particles is well known since several years [1–4] . It is connected with the fact that the use of a width (e.g. iΓM) in the denominator of an s-channel unstable boson propagator is necessary to prevent divergences at p = M, but it is in fact an effective way of including only a part of higher order corrections, and it therefore violates gauge invariance. There are various examples in which this violation becomes numerically relevant. For instance, this happens at tree level with four fermion final states for WW production at high energies, where gauge cancellations among the three double resonant (CC03) diagrams become relevant, and for contributions at low e− angle. This last case is relevant for final states like e−e+ → e−ν̄eud̄, e−e+ → e−ν̄eμ+νμ or e−e+ → e−e+νeν̄e with e− undetected, which are commonly referred to as single-W processes. For them, gauge invariance determines the behaviour of the amplitude as a function of t. In the above mentioned examples even small violations of gauge invariance may easily give results which are completely unreliable, and differ from the correct ones not by a few percent but by some large factor. Various gauge restoring methods have been described in the literature and have been used to avoid such inconsistencies. Some of the most used are :