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The Higgs field that gives (some) particles mass was theorized to solve problem(s) with the weak force; the neutrino only interacts with the weak force, so....

Also, it is a fermion, these otherwise always have mass, correct?

I heard somewhere that the Higgs mechanism was associated only with giving mass to charged fermions... But why? Z bosons (although bosons) derive their mass from the Higgs, right? Even though they are uncharged?

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A perfect storm of misconceptions.

I heard somewhere that the Higgs mechanism was associated only with giving mass to charged fermions.

You heard wrong, and you should toss the book (or science malefactor) stating that with extreme prejudice.

The Higgs Yukawa couplings (not mechanism!) may give mass consistent with the L-isospin gauge symmetry to both upper and lower members of fermion gauge doublets. This is the second job of the Higgs. So the Yukawa couplings and SSB that give mass to both the u and d quarks may give mass to both charged and neutral leptons.

At the time the electroweak gauge model was written down, in the late 60s, there was no indication of neutrino mass, and only the "active", left-handed neutrino component was observed and accounted for, so people naturally assumed there was no right-handed neutrino, because that was possible, and the simplest (minimal) possibility. So a superfluous hypothetical neutrino mass term did not enter Weinberg's or Salam's models; this led to the virulent misconception that the gauge theory somehow does not like or permit one.

When neutrino masses were observed, an up-mass-like term was natural to write down.

The mechanism giving gauge bosons a mass is completely different: the Higgs mechanism, the first and major job of the Higgs; so discussions on the Z boson in this context are red herrings.

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