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Would the existence of a fourth (sterile) neutrino imply the existence of a fourth generation of quarks and a fourth lepton, beyond electrons, muons and tau particles?

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No.

First, sterile neutrinos don't participate in the weak interactions. They are thus are SU(2) singlets. Therefore no charged electron-like counterpart is required let alone new quarks.

In contrast known left-handed neutrinos interact weakly. That's because they are components of SU(2) doublet with other component being electrically charged lepton. Classically if there were new weakly interacting neutrino one extra charged lepton would be enough and no new quark generation would be required. However in QFT that would lead to the non-cancellation of the gauge anomalies unless you introduce new fermions that cancel those.

So to summarize, new sterile neutrino requires nothing more. New weakly interacting neutrino requires the whole bunch.

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