$text{SU}(5)$ GUT Higgs sector and symmetry breaking

Question

The Higgs sector of minimal $text{SU}(5)$ GUT consists of scalar fields in two representations, a $mathbf{24}$ and a $mathbf{5}$.
All the $mathbf{5}$ non-derivative terms form the potential $V(mathbf{5})$ and all the $mathbf{24}$ Non-derivative terms form the potential $V(mathbf{24})$. But in addition to this, theory has potential of both $mathbf{24}$ and $mathbf{5}$,
$V(mathbf{24} , mathbf{5})$.
How does this $V(mathbf{24} , mathbf{5})$  potential relate to spontaneous symmetry breaking, is it necessary for $text{SU}(5)rightarrow text{SU}(3)times text{SU}(2)times text{U}(1) $ symmetry breaking?
What if we exclude this potential, so the scalar fields of both representations will interact only through the gauge field?

Cosmas Zachos · Answer

One assumes you have duly reviewed Langacker and BEGN 1977.

If the two Higgs multiplets 24 and 5 only interacted through gauge fields at tree level, in general that would develop effective terms for a cross potential at loop level, unless a symmetry excluded them, which none does! In fact, it is very tricky to adjust the parameters of the most general radiatively induced potential to yield the requisite physical v.e.v.s (the "doublet-triplet problem").

The 5, and hence its cross-potential with the 24, is not involved in the superstring breaking SU(5)→SU(3)×SU(2)×U(1); this is the job of the 24. But the next step, EW breaking, requires a Higgs coupling to SU(2)×U(1) but not SU(3), which must come out of the 5. This is its job, even though the 24 is also dragged in EW breaking to a small extent--it cannot stay out.

$text{SU}(5)$ GUT Higgs sector and symmetry breaking

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