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So let me elaborate, my problem is that I don't fully understand why the weak interaction is a chiral theory - I would like an answer with the Lagrange density and the Higgs field. I also want to know why symmetry is broken towards left-handed chirality and not right-handed (why left?).

I read that all theories are at their basic level chiral. But as I understand it, at low energies e.g for gluons, the Higgs field combines the two chiralities into a massive Dirac- particle which then does not break symmetry and turns QCD into a vector-theory. The weak interaction then breaks symmetry because its bosons have too high an energy for that and the mass term in the Lagrangian still breaks the symmetry?

I know this sounds very confusing or even wrong, but I did not really understand that and hope someone could elaborate. Why does the weak interaction break symmetry and how can you see that in its Lagrange-density compared to other interactions (e.g. the strong one)?

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    $\begingroup$ Weak interactions are chiral since experimentally they seem to be so $\endgroup$ May 30 at 11:25
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    $\begingroup$ In other words the data can only be fitted with a chiral weak theory. $\endgroup$
    – anna v
    May 30 at 11:29
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    $\begingroup$ +1 to the comments, also note that there may be a deeper theoretical explanation of this, but currently this problem is unsolved. $\endgroup$ May 31 at 10:51
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The low energy Dirac fields aren't "real". They're approximations invented by humans. Fundamentally, there are only Weyl fields with Yukawa couplings between them. All of the forces couple to the Weyl fields, not the Dirac fields. The weak force doesn't violate parity; it couples to anything with weak charge regardless of chirality, but it happens that only one "half" of each Dirac particle has weak charge.

The reason the "halves" have different weak charges but the same strong charge comes down to the fact that the Higgs field has weak charge but not strong charge, and the fact that in Yukawa couplings, two of the three fields have to have matching charges, and the third has to be uncharged.

As far as I know, it's completely unknown why the Higgs field has the charges that it does.

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