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As one learns in QFT, in Yang-Mills theories non-Abelian gauge transformations give rise to self-interactions of the gauge fields in the quadratic field strength term.

In QCD this produces the 3- and 4-gluon vertices. However, why don't we see the same in electroweak theory? Shouldn't there be 3- and 4-vertices of the weak gauge fields? That's what I would guess, since the $SU(2)_L$ part of the gauge group is non-Abelian. Or are those self-interactions somehow done away with via SSB?

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Electroweak theory does indeed have 3- and 4-gauge Boson vertices. For example, $W^\pm$ Bosons are charged, therefore couple to photons.

You can see the basic Feynman diagram vertices for electroweak and strong interactions on Wikipedia.

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