Why aren't the mass terms for quarks in the Standard Model Lagrangian gauge invariant?

Question

I'm learning about the section of the Standard Model's lagrangian of Higgs-quarks interactions. This means writing a lagrangian made of a scalar field $\phi$ (the Higgs boson) and spinors $\psi$ (quarks) which is gauge invariant under

$$ SU(3)_C \times SU(2)_L \times U(1)_Y $$

I was told by my professor that the only possible terms with dimensions less or equal to 4 are $\overline{\psi}_L\psi_R\phi$ and $\overline{\psi}_R\psi_L\phi$.

I understand why $\overline{\psi}_L\psi_L\phi$ and $\overline{\psi}_R\psi_R\phi$ are not possible: they would violate the $SU(2)_L$ symmetry since $\phi$ is also a $SU(2)_L$ doublet; I also understand why $\overline{\psi}_L\psi_R$ and $\overline{\psi}_R\psi_L$ are not possible: their $U(1)_Y$ hypercharge is such that it doesn't equal zero after a $U(1)_Y$ transformation of these two terms.

What I don't understand is why $\overline{\psi}_L\psi_L$ and $\overline{\psi}_R\psi_R$ are not possible: both the $SU(2)_L$ and $U(1)_Y$ transforms should cancel out because both are unitary matrices. What am I missing?

Answer 1

What you are missing is you are not writing the bars over the L and R species. Recall a bar contains a $\gamma^0$ at its end!

• Hence, $\overline {\psi_L} =(P_L\psi)^\dagger \gamma^0 =\bar \psi P_R$, an EW doublet, and $\overline {\psi_R} = \bar \psi P_L $, an EW singlet!

Consequently, $$ \overline {\psi_L} ~\psi_L=0, $$ since the $P_R P_L$ concatenation of chiral projectors vanishes identically, and likewise $\overline {\psi_R} ~\psi_R=0 $.

(You have also been sloppy in using the same doublet $\phi$, and not it and its conjugate $\tilde \phi$, to get your singlet Yukawa couplings, $\overline {\psi_L}\cdot \phi \psi_R$ and $\overline {\psi_R}\tilde \phi \cdot \psi_L$ ...)

The surviving terms are $\overline {\psi_R} \psi_L$, and $\overline {\psi_L} \psi_R$, both SU(2) doublets, so not gauge invariant.

I recall the PDG goes out of its way to display fermion chiralities unambiguously. Remember $\overline {\psi_L}~ \gamma^\mu \psi_L= \overline {\psi} P_R\gamma^\mu \psi_L = \overline {\psi} \gamma^\mu \psi_L $ need not vanish, but the cross chirality object must.

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NB. Edit as per @Chiral Anomaly's nomenclature point

Indeed, I caught the majority of the modern texts, as he points out, using $\bar\psi _ L$ as an unfriendly shorthand/synonym for $\overline{ \psi _L}=\bar \psi P_R$. I can't argue with what he sees. I was raised to avoid this needless dyslexic reversal. To spare the reader needless confusion, I excised needless notation.