Past exam of the sub-atomic physics course of the University of Oxford / 2016 / 4 / c / Solution

An extra active generation with the same weak neutral current couplings would increase the Z boson total decay width if any new particle–antiparticle pair were light enough. A simple sufficient kinematic condition for closing each such two-body channel isThe corresponding within-generation W boson channels close whenIf both members of each doublet already exceed $m_Z/2$, these two sum constraints follow automatically. Mixing with lighter generations can also open charged-current channels involving one heavy and one light particle, depending on the mixing amplitudes; that information is not specified.

These are threshold estimates, not precise confidence limits. Close to threshold, two-body decay phase space suppresses a new partial decay width, and a quantitative lower mass bound requires the experimental width uncertainties and the fermion's couplings. The given observations yield no upper mass bound. A sterile neutrino with no standard active coupling would not obey the stated $Z$-width inference, whereas the question specifies the same quantum numbers as the existing generations.