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Helicity: projection of spin onto motion. Since neutrinos are massive, I can always move to a reference frame where their motion is towards the opposite direction, meaning I should reverse their helicity. Yet it's said "neutrinos can only be left-handed"; so how do you deal with this change of reference frames? Is the quote actually wrong?

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    $\begingroup$ I had a theory TA who said, "I think of mass as the amplitude to flip helicity" . He also said, "$1 = \pi = i = -1$", at least for realtime computation of amplitudes in class. $\endgroup$
    – JEB
    Apr 13, 2023 at 14:25
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    $\begingroup$ The correct statement is "neutrinos observed by weak interactions are only left-chiral". Your flawed version sets itself up for misconceptions.... $\endgroup$ Apr 13, 2023 at 14:45

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Yep. You're right. Since neutrinos have mass, they cannot be purely left handed if they are the same kind of fermion as everything else in the standard model. This idea - that neutrinos were purely left handed, came from before the neutrinos were known to have mass. In fact, it worked out quite nicely back then. Neutrinos were only made as left-handed particles in weak interactions, and if you believed they could only exist as left-handed particles, there was no way to write a mass term for them in the standard model. All good - they don't have mass because they're only left handed, and the weak interaction only couples left handed neutrinos to left handed electrons (only the left-handed Weyl spinors transform under SU(2) in the standard model) because right handed neutrinos don't exist.

But then neutrino masses were discovered in the form of neutino mixing :(. We could easily give neutrinos a mass by begrudgingly allowing them to have a right handed spinor and giving them the same kind of mass term every other particle has in the standard model. But that seems a bit unsatisfying - then there's no underlying reason for the small neutrino masses - they just happen to have a much smaller coupling than other standard model particles. So people have introduced fancier theories, like the seesaw mechanism and majorana neutrinos which allow neutrinos to have mass, and explain why that mass is small. This model makes neutrinos a fundamentally different kind of fermion to all the other ones in the standard model. This is why I think neutrinos are the #1 particle to look out for for the next step in our understanding of the universe, and why experiments like DUNE, KATRIN, and neutrinoless double beta decay experiments meant to improve our understanding of neutrino masses are so important.

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    $\begingroup$ "if there's right handed neutrinos, the weak coupling to right handed neutrinos is a totally free parameter that the universe just decided to make exactly zero." No, the coupling is zero simply because they're uncharged. It's no more mysterious than particles with no electric charge having exactly zero coupling to the electromagnetic field, and it's exactly analogous to all of the other right-handed fermions. The small neutrino mass is a puzzle but this isn't. $\endgroup$
    – benrg
    Apr 13, 2023 at 17:26
  • $\begingroup$ @benrg You're right I had this wrong. I recall one can formulate the standard model where you write it in terms of Dirac spinors instead of Weyl spinors and the weak force is written with a left handed projection operator and you can just as well add a small term with a right handed projection operator. I think experiments work to constrain that term. You can also just write the SM in terms of Weyl spinors and assert that SU(2) only transforms a doublet of left handed spinors. Do I have any of this right? I adjusted the answer; you may take issue with end of first paragraph still though. $\endgroup$
    – AXensen
    Apr 13, 2023 at 17:39
  • $\begingroup$ @benrg are all right-handed fermions uncharged? do electrons not interact in half of all \reference frames? $\endgroup$
    – user253751
    Apr 14, 2023 at 4:58
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    $\begingroup$ @user253751 Uncharged with regard to weak interaction. $\endgroup$ Apr 14, 2023 at 16:09
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    $\begingroup$ @user253751 The (charged) weak interaction coupling is based on chirality, which is not as simple as helicity (which just says whether the spin points in the momentum direction or not). See en.wikipedia.org/wiki/Chirality_(physics) The "Chiral theories" section mentions the weak interaction. Chirality does not depend on the reference frame, only helicity does. $\endgroup$ Apr 15, 2023 at 18:43
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The helicity of a particle is conserved, but is not an invariant under a Lorentz transformation, i.e., a change of reference frame.

The chirality, on the other hand, is an Lorentz invariant and does not change under a reference frame change.

It just so happens that for massless particles, helicity and chirality are coincidental. This also nicely shows that neutrinos ought to have rest mass, although it is indeed very small, since we have only seen left-handed neutrinos and right-handed antineutrinos.

I recommend the Wikipedia article on helicity for more details.

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    $\begingroup$ The fact that helicity has different properties for massive and massless particles has a simple qualitative explanation. Helicity is the projection of angular momentum onto linear momentum. Massive particles always move slower than the speed of light, and helicity depends on the relative velocity between particle and observer. In contrast, massless particles always move with the speed of light, which is the same for all observers – hence the helicity of a massless particle is Lorentz-invariant. $\endgroup$
    – printf
    Apr 14, 2023 at 2:56

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