[Sniffnoy]

So, I'm not actually a particle physicist. My understanding had been (based on something I'd read somewhere -- should try to find it again) that there is some problem caused by just declaring "neutrinos just have innate masses, they're not from the Higgs mechanism", but I could be mistaken. Obviously, if that is mistaken, then as you say it merely a question rather than a contradiction. Should try to dig that up though.

Edit: Doing some quick searching seems to indicate that giving neutrinos a bare mass term would violate electroweak gauge invariance? I don't know enough to evaluate that claim, or TBH really even to understand it. But I believe that's what I was thinking of, so maybe you can say how true and/or pertinent that is.

[T-A]

> giving neutrinos a bare mass term would violate electroweak gauge invariance?

Giving any standard model fermion a bare mass term would violate electroweak gauge invariance. That was one of the problems with Glashow's electroweak model from 1961 [1]: he had the right symmetry group, but all particles had to be massless in order not to break it. Weinberg's contribution was to combine Glashow's proposal with Higgs' mass generation mechanism. It is done exactly the same way for any electroweak fermion doublet (as long as you are happy with the default choice of Dirac mass terms for all of them), be it up quark and down quark or neutrino and electron.

[1] https://www2.physik.uni-muenchen.de/lehre/vorlesungen/sose_2...

[Sniffnoy]

Huh. Why do other sources seem to say that's only the case for bosons? Or am I conflating two distinct problems? Sorry, once again, not a physicist.

But if that's correct then I'm confused what your objection is to what I said earlier. If a bare mass would violate electroweak gauge invariance, then instead the mass should come from the Higgs mechanism, but that has the problem of, where are all the right-handed neutrinos, then? Am I missing something here? If you can't just give the neutrinos a bare mass and call it a day (at least not w/o causing significant problems), but do in fact have to make a more significant modification like inventing sterile neutrinos or making them Majorana particles, I'd call that a "contradiction" rather than merely a "question", because no hypothesis so far is a good fit for all of what we see (searches for sterile neutrinos have come up empty, neutrinoless double beta decay remains undetected, and I assume nobody's ever observed violations of electroweak gauge invariance!). Or I guess there are more out-there hypotheses that are consistent with what we see in that they've yet to really be tested, but, y'know, nothing that's been really tested AFAIK.