How does a fermion, like an electron, get its mass through the Higgs-mechanism? Can someone explain me this with formulas (Lagrangian)?
I know that the Yukawa interaction has something to do with this, is that right?
Maybe when I'm right, there is a term:
$$g \bar{\Psi} \Phi \Psi?$$
It is about the "the 5-th force."
As you said the Yukawa term introducing the interaction between scalar field $\Phi$ and fermion $\Psi$ field: $$g \bar{\Psi} \Phi \Psi$$
The Higgs mechanism causes the $\Phi$ field condense at a classical expectation value (v.e.v: vacuum expectation value), due to the Higgs potential $U(\Phi)$, so $\Phi$ tend to find a classical minimum, which causes:
$$\Phi(x,t) \to \langle \Phi \rangle=m$$
as a fix value $m$. You can imagine this process as originally $\Phi(x,t)$ is a field variable free to have any real/complex values at any spacetime $(x,t)$ point due to quantum fluctuation. However, the Higgs mechanism causes $\langle \Phi \rangle=m$ finding a (local) classical stable minimum value of the potential $U(\Phi)$.
The remarkable result is that $\Phi(x,t)$ semi-classically now have to take the fix value at $m$ at any spacetime point! (This is the remarkable fact of the 5-th force: Higgs field introduces mass to fermions i.e. quarks, leptons, in the Standard Model. Some people coin the name the 5-th force - a different mechanism from the 4 fundamental forces.)
Add: Some people like to think about (fermions,W$^{\pm}$,Z$^{0}$ bosons) particles moving in the ocean of Higgs fields, thus (fermion,W,Z) particles become massive due to the buoyancy force effects in the Higgs ocean.
The mass $M$ of fermion fields now can be read as
$$g \bar{\Psi} \Phi \Psi \to (g\cdot m) \bar{\Psi} \Psi=M \bar{\Psi} \Psi $$ with fermion mass $M=g\langle \Phi \rangle=g\cdot m$.
Note that now Fermion mass takes the fixed value at $g \langle \Phi \rangle$, BUT there is quantum fluctuation around the v.e.v. ($\langle \Phi \rangle+\delta \Phi $) to cause fermion field interacting with the Higgs fluctuation $\delta \Phi $. You can draw a Feynman diagram to compute its effect.
