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I'm pretty confident that I'm misunderstanding something here.

The Yukawa coupling that couples the Higgs field to fermions can be written something like: $$ \lambda \bar L H e_r + h.c., $$ where $L,H$ are SU(2) doublets of opposite charge and $e_r$ is uncharged under SU(2) and carries weak hypercharge opposite to $\bar L$ under U(1). So regarding gauge symmetries, everything checks out.

It is my understanding that the SM should have a chiral symmetry as well, so that: $$ \psi_L\rightarrow e^{i\gamma},\psi_R\rightarrow e^{i\eta} $$ leaves the Lagrangian invariant. This doesn't seem to hold for the Yukawa term (after giving higgs vacuum expectation $v$): $$ \lambda\bar L H e_r \supset \lambda e_L v e_R $$ But I still read electroweak theory as being referred to as a chiral theory, and chiral symmetry breaking not being discussed until QCD and Yoichiro Nambu's contribution there.

What's off?

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Higgs Yukawa coupling does preserve chiral symmetry (it's usually called axial symmetry).

What is missing in some text books is that under axial rotation $$ \psi_L\rightarrow e^{-i\theta},\psi_R\rightarrow e^{i\theta} $$ the Higgs field transforms as $$ H\rightarrow e^{-2i\theta}, $$ hence the Yukawa coupling $$ y_e \bar L H e_r + h.c. $$ is invariant under axial rotation.

Upon spontaneous electroweak symmetry breaking, the Higgs field acquires a non-zero VEV $\upsilon$, which DOES break the axial symmetry. In other words the spontaneous electroweak symmetry breaking breaks both the electroweak gauge symmetry and the global axial symmetry.

Moreover, one can entertain chiral masses for fermions if the Higgs VEV is chiral $e^{\theta i\gamma_5}\upsilon$ (see here): $$ m\bar{\psi} e^{\theta i\gamma_5} \psi = m\cos\theta \bar{\psi} \psi + m\sin\theta \bar{\psi} i\gamma_5\psi. $$

However, there is another twist to the story. The ABJ chiral anomaly actually breaks the axial symmetry regardless of the existence of the spontaneous electroweak symmetry breaking. The would-be Nambu-Goldstone massless boson corresponding to the broken axial symmetry becomes pseudo-Nambu-Goldstone boson with non-zero mass due to the chiral anomaly quantum contribution. That is why the axial symmetry properties of the Higgs field and the Yukawa coupling are not usually discussed in some text books.


Added note:

The previous discussion is fine and dandy when we consider Yukawa coupling for isospin down-type particles only, such as electron $e$ or down quark $d$. When it comes to Yukawa coupling for isospin up-type particles like up quark $u$ or neutrino $\nu$, things get a bit hairy: the Higgs fields per se can not be directly coupled to right-handed $u_r$ quark, given the mismatch of hypercharge. In stead, the Yukawa coupling for $u_r$ quark takes the awkward form: $$ y_u \bar Q_l (i\tau_2 H^\star) u_r + h.c. $$ which breaks the axial symmetry (the complex conjugation of $H^\star$ is the culprit), as opposed to the normal $d_r$ quark Yukawa coupling $$ y_d \bar Q_l H d_r + h.c. $$ which preserves the axial symmetry.

On way of savaging the situation is to propose that there are two different Higgs doublets coupled to up quarks $u_r$ and down quarks $d_r$, respectively. And for that matter, this stone can kill another bird: the strong $CP$ problem in QCD. See Peccei–Quinn symmetry for more details.

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  • $\begingroup$ Thanks for a very informative answer! For me this prompts the question as to why the Higgs and both lepton chiralities are charged under the same global chiral symmetry, while you (in reading about chiral symmetries..) should be able to rotate the chiralities independently. I see that it has to be this way in order mantain the symmetry in the Yukawa term, but is there something else to this? $\endgroup$
    – Oyvach
    Commented Dec 4, 2019 at 21:57
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    $\begingroup$ @Oyvach, "but is there something else to this?", in the composite Higgs paradigm, Higgs is the bound state of a fermion and a antifermion. Hence you may infer the chiral properties from the underlying fermion constituencies. $\endgroup$
    – MadMax
    Commented Dec 4, 2019 at 22:10

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