Polarization of a vector boson produced in decay of a heavy fermion in an arbitrary frame

Abstract

We consider processes where an electroweak gauge boson ($W$ and $Z$) is produced in the decays of a heavy fermion. The polarization state of the produced gauge boson is given in terms of a vector polarization and a rank-2 tensor polarization. In the rest frame of the mother fermion, these are given directly by the dynamical parameters of the underlying theory. In a frame where the mother particle is moving, the polarization parameters of the gauge boson are dependent additionally upon kinematical factors. We show that these kinematical factors depend only on the magnitude of the velocity of the mother particle and derive analytical expressions for them. We apply the results to pair production of heavy fermions at the LHC with one of the fermions decaying to a gauge boson ($W$, $Z$) and a light Standard Model fermion. We construct estimators of laboratory frame values of polarization parameters of the produced gauge boson. These estimators can be used to estimate the laboratory frame values of polarization parameters of the produced gauge boson without a detailed simulation of the entire process. We validate our expressions with detailed Monte Carlo simulations in the context of beyond the Standard Model scenarios which have a vector-like top partner. We also indicate how to include finite width effects of the heavy fermion in some special cases.