If the Higgs mechanism makes the neutrinos' masses, then it has a very serious oddity. The neutrinos' low masses translate into very low Higgs-particle couplings for them.
The Higgs-particle field's vacuum value is about 246 GeV. Calculating Higgs-particle couplings for the elementary fermions, the top quark has 0.7 and the electron has 2*10^(-6), but the heaviest neutrino has 2*10^(-13).
That is rather bizarrely tiny.
But there is a possible solution, the Seesaw mechanism
WP. It states that neutrinos have an additional kind of mass, their "Majorana masses". This kind of mass is alongside the masses induced by the Higgs particle, the "Dirac masses". These masses then mix, with some of the resulting masses being the very low observed masses:
m(observed) = m(Dirac)^2 / m(Majorana)
If m(Dirac) is about 10 GeV, then to get 5*10^(-11) GeV, one needs m(Majorana) = 2*10^(12) GeV.
That's close to Grand Unified Theory mass scales of around 2*10^(16) GeV (gauge unification: Standard Model + TeV-scale SUSY).
So there may be some connection with GUT's.
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The neutrino masses form this 5*5 matrix:
(0 m')
(m M)
with 0, m, and M being 3*3 ones. m' is the hermitian conjugate of m. The 3 comes from the 3 generations. The 6 comes from including both left-handed and right-handed neutrinos.
Multiply on the right by (I, -M^(-1).m), and one gets (-m'.M^(-1).m, 0)
Thus, one gets mass matrix -m'.M^(-1).m
The up-like quarks, the down-like quarks, and the electronlike leptons also have 3*3 mass matrices.
The mass matrices for the up-like quarks and the down-like quarks are almost aligned, as is evident from their near-diagonal generation mixing, while that is clearly not the case for the electronlike leptons and the neutrinos. If the neutrinos' Dirac matrices are almost aligned, then their Majorana masses must be rather badly misaligned.