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The standard model has two parallel series of leptons: the charged leptons (electron, muon, and tau) with -1 electric charge and -1/2 weak isospin, and the neutrinos, with 0 electric charge and +1/2 weak isospin.

Why do we not have a third series of particles with non-zero electric charge, but no isospin? I.e., particles with purely electromagnetic interactions, and no weak interaction? Does something else horribly break if such fields are added to the standard model?

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Nothing would break. The theory would be renormalizable.

The reason why we have the peculiar set of particles and forces we do is unknown. One speculation is that they are a random consequence of how extra dimensions happened to compactify after the Big Bang. If there is a multiverse, other universes might have totally different sets of particles and forces.

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