In curved space time, there is a spin-connection term $\overline{\psi}\gamma^\mu\sigma^{ab}\omega^{ab}_\mu\psi$.
Here's my apparent problem. If there were no Higgs field and no gravity, all particles would be massless. And hence the left-handed and right handed electrons would uncouple and be seen as separate massless fermions with no connection to each other. It is only the higgs coupling that combines these in a pair with mass. As the higgs interaction turns the left-handed electron into a right handed one.
But the spin-connection term, for example, (as far as I can tell) mixes the left and right handed electrons. Even though, apparenly, these two particles have nothing to do with each other. (e.g. This predicts a left handed neutrino passing by a rotating black hole will experience some kind of torsion effects and can turn into a right-handed neutrino).
Worse still if we consider the Cabbibo mass mixing matrix, there is not pairing of the left-handed electron and right-handed but a pairing of the triplet of 3 generations of charged fermions with their counterparts.
e.g. we would get a term like $\overline{\psi}^A(m^{AB}+\gamma^\mu\sigma^{ab}\omega^{ab}_\mu)\psi^B$
Where $m$ is a mass mixing matrix e.g. for up-quarks.
Hence I don't see the justification for assuming that the spin-connection term pairs, say, left-handed up-quarks to right-handed up-quarks. Ignoring the Higgs term it could just as well pair left handed up-quarks to left-handed electrons. (e.g. this would predict a left handed quark passing by a rotating black hole would turn into a left-handed electron. Since we can't do the experiment how can we rule that out?)
So it seems to me there is a some unexplained coincidence occuring where the Higgs interactions are pairing up particles identically to the way the spin-connecton pairs up particles.
Or is there even an experiment to show how the chirality of a fermion is affected by gravitational torsion from a rotating gravitational mass? (Or perhaps some equivalent accelerating frame?)
Can you explain this?
My view is that a left-handed neturino in a gravitational field should stay a neutrino left-handed neutrino and there should be no mixing. Perhaps that could be achieved by replacing the term a pesudo-vector term: $\overline{\psi}\gamma_5\gamma^\mu\sigma^{ab}\omega^{ab}_\mu\psi$ (Just a guess)??