Chirality flip in electrons due to Yukawa interaction

Question

This is the Lagrangian density for a fermions interacting with a Higgs field:

$$\mathcal{L}= i \bar{\Psi} {\gamma}_\mu {\partial}_\mu \Psi - (gv) \bar{\Psi} \Psi + \frac{1}{2} {(\partial h)}^{2} - \lambda {v}^{2} {h}^{2} .$$ The Yukawa potential is $(gv) \bar{\Psi} \Psi$. This potential gives us the interaction between the Higgs field and the fermion, by adding a potential term to the fermionic field.

But in many bibliographies, I see the explanation that the right electron field interacts with the Higgs, "changing" the left electron field and vice versa. ${\Psi}_{L} \to {\Psi}_{R} $. How do we obtain those Feynman diagrams for the Yukawa potential? Where is the explanation

Answer 1

Let us consider the Standard model. The Yukawa interaction terms are generally of the form $\bar\psi\psi\phi$, and due to symmetry properties these terms must be invariant under U$(1)_Y$. To do this, for example, we can combine the SU(2) doublet $Q=(u_L, d_R)^T$ and $d_R$ for Quark to form the term $$\bar Qd_R\Phi.$$Here, since $\Phi$ is a Higgs doublet, it contracts with $\bar Q$ and the above term is properly scalar. Now suppose that the Higgs field has taken the vacuum expectation value $$\Phi=(0,v)^T.$$ In this case, the term above is proportional to the term $$\bar d_L d_R v+\mathrm{h.c.}$$ This is what we want to obtain, and almost same thing works for other Yukawa interactions which satisfies symmetry properties of SM. In short, the terms that can enter the Standard Model are strictly limited by symmetry, so to add any types of Yukawa interaction is not a good idea.

This can also be understood as follows. The terms you have shown correspond to the mass terms of the fermions at low energies. As is well known, terms like $\bar\psi_L\psi_L$ and $\bar\psi_R\psi_R$ are not Lorentz scalars, and Dirac mass term is almost unique choice under the Lorentz symmetry. (Of course Majorana mass is also allowed). Therefore, they should not appear in the Lagrangian.