Do hidden influences or new particles give neutrinos their tiny mass?

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"... Yet so much is still unknown about neutrinos. 
Among the many open questions about neutrinos is how, exactly, they get their mass; the Standard Model of particle physics does not provide an answer. Many theories that explain the small masses of neutrinos also predict the existence of undiscovered particles, so one way to test those theories is to look for those particles.  ...

If neutrinos gain their mass from the Higgs field, they seem to interact with it to an even smaller degree than the electron—much smaller; the mass of the neutrino is about a million times lighter. ...

When a particle interacts with the Higgs field, the Higgs field switches that particle’s “handedness”—a measure of its spin and motion. ...

But as far as scientists have measured, all neutrinos are left-handed. ..."

Do hidden influences give neutrinos their tiny mass? | symmetry magazine The quest to understand the small mass of neutrinos is also a quest to discover new particles.