One of the notable coincidences of the Standard Model is that every particle with rest mass has a parity variant with weak force couplings to the W and Z bosons (i.e. the Standard Model fermions, the W boson, the Z boson and the Higgs boson), while every Standard Model particle that lacks rest mass does not have weak force couplings to the W and Z bosons (photons and gluons). Photons couple to W bosons, but only due to the W boson's electrical charge with no weak force contribution. Hypothetical massless gravitons, while they couple to the mass-energy of all particles do not have weak force interactions either.
Obviously, it is possible to have a particle that has weak force charge but not electromagnetic charge (we call them neutrinos).
In electro-weak unification, electric charge Q=weak isospin (T3) + Weak Hypercharge (Yw)/2. If a particle must have weak isospin (i.e. weak force couplings) equal to 0 in both right and left parities (or lack a left parity, as in the case of a hypothetical sterile neutrino) in order to lack mass then Q=Yw/2 for such a particle. The right handed version of the quarks and charged leptons have such quantum numbers, but all of them have a parity counterpart with weak isospin.
An electrically neutral sterile neutrino would have to have weak-hypercharge and weak isospin of zero. But, if a sterile neutrino has a zero weak isospin and a the same weak hypercharge as a "fertile" neutrino, then it would have electric charge = +/- 0.5, filling a gap in the progression of possible fundamental particle charges that currently has values of -1, -2/3, -1/3, 0, +1/3, + 2/3 and +1.
But, given that it also appears to be empirically the case that all fundamental particles with non-integer values of electric charge Q are confined into particles with integer electric charge (which in quarks is assured by confinement plus the requirement that particles have neutral color charge and hence must come in 3 color balanced baryons or color and anti-color balanced leptons), one would expect that any particle with an electric charge of +/- 0.5 would have to be confined in particles with either electric charge of +/- 1, or electric charge of 0 and bound by the exchange of a confining two-color gluon-type boson and that bosonic binding energy would give rise to mass. Also, one would expect that in either case, that the composite particle would be a boson since any combination of two fermion spins is an integer and any combination of two boson integer spins is an integer. So, even if one had a plausible fundamental particle with zero mass and an electric charge, it might only manifest as a massive particle.
Of course, the Standard Model already has massive bosons with charges of +/- 1 (W bosons) and zero (Z bosons). These do have an interaction W+ + W- => Z0 + Photon but the W+, W-, photon part of the interaction can be considered an electro-magnetic interaction. But, the W+, W- and Z vertex would seem to be a weak force self-interaction which a composite particle with zero weak force isospin should lack.
In the absence of experimentally motivating exceptions, of course, it is hard to know if either the weak force interaction is necessary for fundamental particle mass, or the Q is always observed in integer quantities rules are mere coincidences, or a genuine laws of nature.
