well if the couplings are weak enough to evade experimental bounds
Electron interactions by now are measured to precisions on the order of ~10^-9, and they agree with theoretical predictions. This means that such hypothetical couplings are weaker than that, and can only induce neutrino masses below ~1 meV.
maybe neutrino masses are below ~1 meV, or like the higgs, its couplings to neutrinos are higher than for charged leptons
the feng boson or light neutral boson, is also postulated to solve anomalous magnetic dipole moment muon and muon radius puzzle
Higgs-like coupling (IOW: fermion to scalar field coupling) would require existence of RH neutrinos.
feng proposes his boson is a spin-1 vector boson with axial and vectorlike currents
my point about the higgs is that it couples to top quarks is quite different from electrons
Nonzero-spin fields with nonzero VEV break Lorentz invariance.
perhaps this boson has zero VEV to preserve Lorentz invariance
Then how it generates neutrino mass?
if there is more than 1 higgs field, perhaps neutrinos couple to a second set of higgs
Depends on how "Higgs-like" it has to be. I don't see anything wrong in principle with a scalar field that couples only to left handed particles and thereby generates mass. It wouldn't be very Higgs-like, compared to the SM, but shouldn't be mathematically impossible.
Fuzzy dark matter.
W. Hu, R. Barkana, and A. Gruzinov, “Cold and fuzzy dark matter,” Phys. Rev. Lett. 85, 1158–1161 (2000), https://arxiv.org/abs/astro-ph/0003365
One of the big conundrums is that the assumption of dark matter works extremely well on large scales for explaining the structure of the cosmos, but seems to fail to work on the scale of galaxies. One interesting idea how to fix this (without giving up on the established theory of gravity as MOND does) is to assume that dark matter consists of massive but extremely light particles, whose de Broglie wavelength is of the order of thousands of parsecs. This has the consequence that at around the scale of that wavelength the behaviour of this dark matter changes.
Now Edward Witten et al. have argued in more detail that this works really well, and consistent with existing null results on direct detection:
Lam Hui, Jeremiah P. Ostriker, Scott Tremaine, Edward Witten, "On the hypothesis that cosmological dark matter is composed of ultra-light bosons" https://arxiv.org/abs/1610.08297
does this fuzzy dark matter couple to higgs?
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