Sum rules for $e^+e^- \to W^+W^-$ helicity amplitudes from BRS invariance

Abstract

The BRS invariance of the electroweak gauge theory leads to relationships between amplitudes with external massive gauge bosons and amplitudes where some of these gauge bosons are replaced with their corresponding Nambu-Goldstone bosons. Unlike the equivalence theorem, these identities are exact at all energies. In this paper we discuss such identities which relate the process $e^+e^- \to W^+W^-$ to $W^\pmχ^\mp$ and $χ^+χ^-$ production. By using a general form-factor decomposition for $e^+e^- \to W^+W^-$, $e^+e^- \to W^\pm χ^\mp$ and $e^+e^- \to χ^+χ^-$ amplitudes, these identities are expressed as sum rules among scalar form factors. Because these sum rules may be applied order by order in perturbation theory, they provide a powerful test of higher order calculations. By using additional Ward-Takahashi identities we find that the various contributions are divided into separately gauge-invariant subsets, the sum rules applying independently to each subset. After a general discussion of the application of the sum rules we consider the one-loop contributions of scalar-fermions in the Minimal Supersymmetric Standard Model as an illustration.