In the Wikipedia article on the electroweak interaction, they show the Lagrangian before and after symmetry breaking, and to my untrained eye they look totally different, in part because the 4 gauge fields are now different ones! But I thought the symmetry breaking was entirely a result of the Higgs field's non-zero average value due to the sombrero potential term in its own Lagrangian. So is the "before" Lagrangian actually just the "after" Lagrangian with zero plugged in for the Higgs magnitude? In which case, could we write the "before" Lagrangian in terms of the $W^\pm, Z, A$ fields instead of $W_i, B$, because these are just linear combinations of each other? And do we just write it in the latter form because it shows the full symmetry or something?
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$\begingroup$ "... before electroweak symmetry breaking becomes manifest"; the emphasis is on manifest. The "after" variables display SB manifestly. How did you miss that? $\endgroup$– Cosmas ZachosCommented Apr 16, 2021 at 22:28
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$\begingroup$ @CosmasZachos That's a yes, right? I wasn't knocking the article, it's incredibly informative, it's just my lack of standard model background makes me doubt my understanding. $\endgroup$– Adam HerbstCommented Apr 16, 2021 at 23:14
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$\begingroup$ The "before" and "after" expressions just represent the change of variables making the SSB manifest, i.e. absorbing three Higgs field degrees of freedom into three gauge fields, rearranging them, and resulting in the customary manifestly SSB-"Higgsed" physical fields. The unabsorbed component of the Higgs field is the scalar physical particle observed a few years ago. $\endgroup$– Cosmas ZachosCommented Apr 17, 2021 at 15:18
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$\begingroup$ @CosmasZachos Got it, that helps, thanks for the clarification $\endgroup$– Adam HerbstCommented Apr 17, 2021 at 23:05
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