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

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?

• It is possible to write mass terms that are gauge invariant. It's just not possible for them to also agree with the observed maximal parity violation. Sep 6, 2021 at 13:28
• Hot tip: if you used Ψ for SU(2) doublets correctly, and ψ for SU(2) singlets, the whole structure would become instantly evident to you. Sep 7, 2021 at 13:27

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.

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.