Are the mass matrices the same if Higgs corresponding to different Cartan generators get a vev?

Question

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.

Answer 1

The 4 generators of $SU(5)$ are not all "equivalent". In general, the generators of the group/algebra satisfy a defining equation of the form $$[T^i,T^j]=f^{ijk} T^k$$ so depending on the structure constants $f^{ijk} $,it is possible for example that $$[T^1,T^2]\neq[T^2,T^3]\quad\text{etc,}$$ so it is important which generators are broken.

In terms of Dynkin diagrams, the Dynkin diagram of SU(5) is

If it is the generator that corresponds to the first node that breaks,

then we are left with an unbroken $SU(4)$ group. If on the other hand, the generator corresponding to the second node gets broken,

then we are left with a $U(1)\times SU(3)$ unbroken group.

Note that so far the discussion is about groups only and we haven't even mention the word representation. Once the remaining group has been established, then it makes sense to ask how particular representations decompose under this breaking. So for example, in the first case above, one would like to ask: "When SU(5) breaks to SU(4) how does the $5$ (or the $\bar5$) representation of SU(5) decompose into representations of SU(4)?"

I have myself posted a question asking for these steps to be explained here: Decomposing a representation under a subgroup but without much success.