Polarization structure and spin covariance of massive vector-boson amplitudes in QCD

Nearly thirty years ago, Bern, Dixon and Kosower computed all helicity amplitudes for the annihilation of an electron-positron pair into four QCD partons through an electroweak vector boson. More recently, the leading-color two-loop amplitudes for the same process were obtained. When such amplitudes are expressed in the massless spinor-helicity formalism, they effectively correspond to the decay of a transversely polarized vector boson. However, for several reasons, it is highly desirable to extend these calculations to the case where the polarization of the vector boson is longitudinal. Due to the complexity of such computations, repeating them to obtain the result for the “missing” polarization of the electroweak boson is a significant undertaking even at one loop. Besides, when attempting new higher-loop computations, it is beneficial to identify the minimal set of quantities (e.g. form factors) that must be determined to obtain the full amount of physically-relevant information. In this paper, we show that amplitudes involving vector-boson decays to massless leptons—although they appear to project onto the transverse polarization—still encode the full information about all polarization states of the vector boson, including the longitudinal one. This follows from the little-group (spin) covariance of the amplitude, which allows us—largely through simple replacement rules—to rewrite the helicity amplitudes as a matrix with open SU(2) spin indices in the massive spinor-helicity (or spin-spinor) formalism. Therefore, knowledge of an amplitude for any polarization component suffices to reconstruct the full covariant matrix.