Gauge Invariance and the Unstable Particle.pdf

a r X i v : h e p - p h / 9 7 0 6 5 5 0 v 1 3 0 J u n 1 9 9 7 Gauge Invariance and the Unstable Particle Robin G. Stuart Randall Laboratory of Physics Ann Arbor, Michigan 48109-1120 USA Abstract. It is shown how to construct exactly gauge-invariant S-matrix elements for processes involving unstable gauge particles such as the Z0 boson. The results are applied to derive a physically meaningful expression for the cross-section σ(e+e? → Z0Z0) and thereby provide a solution to the long-standing problem of the unstable particle. I INTRODUCTION A resonance is fundamentally a non-perturbative object and is thus not amenable to the methods of standard perturbation theory. In order to describe physics at the Z0 resonance one is forced, therefore, to employ some sort of non-perturbative procedure. Such a procedure is Dyson summation that sums strings of one-particle irreducible (1PI) self-energy diagrams as a geometric series to all orders in the coupling constant, α and effectively replaces the tree-level Z0 propagator by a dressed propagator, 1 s?M2Z → 1 s?M2Z ∑ n ( ΠZZ(s) s?M2Z )n = 1 s?M2Z ? Π(1)ZZ(s) (1) where ΠZZ(s) is the one-loop Z 0 self-energy. The problem here is that electroweak physics is described by a gauge theory. Results of calculations of physical processes must be exactly gauge-invariant but this comes about through delicate cancella- tions between many different Feynman diagrams each of which is separately gauge- dependent. The cancellation happens at each order in α when all diagrams of a given order are combined. The Z0 self-energy, ΠZZ(s), is gauge-dependent at O(α) and hence the rhs of eq.(1) is gauge-dependent at all orders in α. If the dressed propagator is used in a finite-order calculation the result will be gauge-dependent at some order because the will be no diagrams available to cancel the gauge-dependence beyond the order being calculated. This gauge-dependence should be viewed as an indicator that the approximation scheme being used is in