Yukawa Interaction with Chiral Fermions

Question

I understand that an interaction like

$$g\overline\psi_L\psi_R\phi$$

is allowed in the SM (for the sake of this discussion lets ignore gauge charges and focus only on Lorentz invariance).

I want to build a Feynman diagram that has a mass vertex, so I can write an effective coupling, $$g'\overline\psi_L\psi_L\phi$$

However, dealing with the mass vertex has been a pain.

The way I would like to solve this is by use of appropriate projection operators. As I understand this would change the Lagrangian term to something like,

$$ h (P_L\psi)^\dagger \gamma^0 P_R\psi\phi $$

$$ P_L = \frac{1}{2}(1-\gamma^5) , P_R = \frac{1}{2}(1+\gamma^5)$$

The issue arises when I try to calculate this vertex on a Feynman diagram, since it seems like the vertex factor would be $$ ihP_LP_R $$ which is 0 since, $$ P_LP_R = 0 $$.

One way to solve this is by redefining h to have a gamma matrix in it, and then using a vertex factor like $$ P_L h'\gamma^\nu P_R $$ which seems to solve the problem, but also seems to change the scalar nature of $$ \phi $$?

How do I resolve these two things? What is the appropriate vertex factor?

Said another way;

How do you properly include left and right handed projection operators into a Yukawa-type vertex?

Answer 1

I suspect you are trapped in bad notation, which, sadly, is quite common, even though you appreciate the crucial chirality-flipping $\gamma^0$ involved in the overbar.

The more sensible/tasteful representation of your terms, utilized by the better authors, is $$g\overline{\psi_L}\psi_R\phi=g\overline{\psi}P_R\psi_R\phi= g\overline{\psi} \psi_R\phi,$$ just like a mass term, which it produces in the SM, $$ m ~\overline{\psi_L}\psi_R. $$ One hermitianizes terms in the action in the end.

I want to build a Feynman diagram that has a mass vertex, so I can write an effective coupling, $g'\overline{\psi_L}\psi_L\phi$

As the commenters pointed out, your term, if this is what you wrote, is automatically zero, and mercifully does not produce Lorentz-invariant mass terms; I'm not sure why you connect it to a mass vertex. A mass vertex is what is written above!

My first guess for your two-chirality-flip effective interaction, so, then no chirality-flip interaction, is $$igm\phi\overline{\psi_L}(/\!\!\partial ~/\Box) ~~\psi_L= igm \phi~~\overline{\psi}(/\!\!\partial ~/\Box) \psi_L,$$ which doesn't vanish. It only vanishes for infinite momenta, since the chirality-flipping mass is suppressed. This term flips LH to RH, and then, to LH again, in the same breath, so to speak.