Mass and width of unstable gauge bosons

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It has been proposed that the mass of theZ 0 boson should be defined as the real part, and the width of theZ 0 as the imaginary part of the complex poleM c , for the reason thatM c is gauge invariant (g.i.). Clearly, this definition is only useful if we can calculate theS-matrix up to some accuracy in a g.i. way. This is not at all obvious, since the use of theZ 0 dressed propagator mixes up the order of the diagrams, and thus order by order gauge invariance of theS-matrix, in this formulation, becomes an ill-defined concept.

By means of an infinitesimal change of the gauge breaking term, we investigate what happens to perturbative gauge invariance of theS-matrix in the non-perturbative energy region of theZ 0 resonance.

Within the language of perturbation theory, we find that in fact there exists a Ward identity valid in the complex plane, i.e. for complex values of the momenta of the external legs, from which we can derive the gauge invariance ofM c