So far, we've added one new spin-1 gauge boson (a particle like the photon of light), producing a new force between neutrinos. The first question that springs to mind, is how strong is this force. We can guess this by assuming it unifies with the other forces at the usual grand unified theory energy of about 10^16 GeV, and then having it screened by the pairs of neutrinos in the Quantum Vacuum. It turns out that is makes the force between neutrinos about 1/60, a sixtieth of the strength of the electromagnetic force.

  The next question is why haven't we observed it in existing experiments. Well it turns out that in order to emit a spin-1 particle, a neutrino has to flips it spin, and this can only happen if it also flips it direction of motion. All neutrino experiments work with very high energy neutrinos so, and at these energy coupling to the axial force, will be highly suppressed.

  We'll assume that the axi-photon is massless, then the axial charges have to be total conserved it all interactions. Consider beta decay.

   Since a W- particle has transformed to a electron with zero axial charge and a neutrino with +1 axial charge. It follows that the W- weak boson must have +1 axial charge. Seemly the W+ must have
negative axial charge. Because the axial charge reverses with reversing the spin. We can for the first time offer an explanation as to why the weak force is always left handed. The conservation of the axial charges between neutrino, requires it to be so. There could also be a right handed version, but it would have different charges, and interact between different particles (the right handed neutrinos)
and so have a very different mass (needs to be over about 3TeV to fit with experiments).

  How about the quarks, well if a left handed down quark can turn into a up quark and a W- particle. Then the up quark must have a axial charge that is +1 units, greater than the down, quark, we
don't yet know the charge on the down quark, but we can work it out, by require something called the anomaly diagrams cancel. For the minute lets assume the charge on the (left handed) up charge is +1/2,  and it -1/2 for the down quark. The weak force is favour universal (the same for each generation), so the axial charges have to be the same for each generation of quarks and leptons.

   Now it turns out there is a problem, as far as out experiments have shown, protons and neutrons have no trouble flipping there spins, e.g. under a magnetic field. But the axial charge is supposed to reverse when the spin reverses. We can get around this only if we introduce a second set of up and down quarks per generation, with opposite axial charge. I call these tera-quarks after Sheldon Glashow paper which also added right hand acting quarks, (he added tera-leptons too, but i don't). One can either make these tera-quarks, monstrously massive, so out of reach of experiments, or they might just explain the sigma (555) meson, which has the opposite parity to normal and isn't well explained in the standard quark model.
Anyway, if the quarks also carry axial charge in non integer amounts, if axial charges, and conserved, (like any locally gauge invariant charge must be). Then it turn out that the proton has to be absolutely stable. They just aren't any particles less massive than the proton with non integer axial charges it can decay into. This saves a lot of grand unified theories which are  beginning to flounder because experimentalists haven't found any examples of protons decaying in nature.

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