# How is the chirality for the weak interaction conserved for non-relativistic neutrinos?

In this article, one can read that the neutrinos in the cosmic neutrino background have a speed of about 1/50 of the speed of light, which is clearly non-relativistic.

From the viewpoint of, say, cosmic rays emerging from the sun, these slow neutrinos can change helicity. A left-handed neutrino then changes into a right-handed one.

Now, the weak interaction affects only left-handed particles and right-handed anti-particles. Does this mean that a left-handed neutrino, as seen from an observer at rest wrt co-moving coordinates, participating in a weak process, cannot participate in the same process, as seen from the frame in which high-energy cosmic rays are at rest and from which the neutrino seems right-handed? How is the chirality of the weak interaction conserved? Are non-relativistic neutrinos behaving like, say, electrons, which can have both right-handed and left-handed versions but still behave chirally?

• I suspect you are confusing helicity with chirality. Commented Sep 10, 2023 at 15:20
• Left handed means left chiral, not with negative helicity. Are you firm on that? Commented Sep 10, 2023 at 16:01
• @CosmasZachos Yes, I'm aware of that. So a negative and positive helicity can both be connected to a negative and positive helicity? A left-handed neutrino stays a left-handed neutrino for ever, even when it changes chirality? Commented Sep 10, 2023 at 16:47
• @CosmasZachos Did electrons initially have a fixed chirality too? Like the neutrinos? Commented Sep 10, 2023 at 16:56
• Electors of both chiralities can be equally created by photons. Commented Sep 10, 2023 at 17:18

## 1 Answer

I am unclear what you are asking, but here is the connection of slow fermions' helicity and chirality. From (6.38), for ultra relativistic fermions, $$\kappa= \frac{p}{m+E}$$ goes to 1, so positive helicity ones are almost pure right-chiral, and negative helicity ones are almost pure left-chiral.

Not your case. For speed v ~1/50, you have $$\kappa\approx 0.01$$, so, by contrast, positive helicity fermions are almost equal parts right-chiral and left-chiral, and similarly for negative helicity.

This holds for both electrons and Dirac neutrinos, of the same speed.

From the viewpoint of, say, cosmic rays emerging from the sun, these slow neutrinos can change helicity. A left-handed neutrino then changes into a right-handed one.

No, really slow neutrinos produced by weak interactions change chirality, as per above. In their frame, only about half will interact weakly.

Are non-relativistic neutrinos behaving like, say, electrons, which can have both right-handed and left-handed versions but still behave chirally?

Indeed, slow Dirac neutrinos and electrons share in that feature: about half of them are liable to interact weakly in that frame. However, note there is at least a ratio of a million between electron and neutrino masses, so you must scale them appropriately. Electrons here are a mental crutch...

PS. Solar neutrinos, however, are ultrarelativistic, since their energy is ~ 300keV...