I'm trying to understand what happens when a Higgs field in the adjoint representation of a given gauge group gets a vacuum expecation value (vev).
Normally, the fermions do not couple to adjoint Higgs, but as a toy-model assume a $SU(5)$ theory with fermions $f_5$, $f_{\bar 5}$ in the $5$ and in the $\bar 5$ representation, respectively. The adjoint of $SU(5)$ is $24$ dimensional and therefore we denote the Higgs fields by $\phi_{24}$. Then we can write a $SU(5)$ invariant Yukawa term $$ f_5 f_{\bar 5} \phi_{24} $$
Now, assume that a Higgs field corresponding to a Cartan generator $H_i$ (=diagonal generator) gets a vev. There are four such generators in $24$.
Which fermions do get a mass after symmetry breaking and is there any difference if the Higgs field corresponding to $H_1$ or the Higgs field corresponding to $H_2$ gets a vev?
In tensor notation, at least for me, it does not seem to make any difference which of the four Higgs fields inside $24$ that correspond to the Cartan generators get a vev. Is this correct?
My problem with this observation is that the subgroup we are breaking to depends heavily on the choice of the Higgs field. The subgroup after symmetry breaking when $H_1$ gets a vev is in general completely different than if $H_2$ gets a vev. Therefore, I assume, the mass matrices should be different.