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I'm hopelessly perplexed about the $SU(5)$ doublet-triplet splitting.

I read that the color triplet part of the Higgs quintuplet needs to have extremely high mass compared to the "electroweak" Higgs boson.

My qualms are:

  1. In the SM Higgs mechanism, the Higgs boson becomes massive after the Higgs field doublet acquires a non-zero VEV, thus giving mass to the gauge bosons. In $SU(5)$, it is clear that the gluons cannot get a mass, so I take it to entail that the 'triplet' sector of the 5-component VEV has to be zero. Therefore how can the color-triplet acquire its mass?

  2. I'm also confused from a degree of freedom perspective. In the SM, the Higgs complex doublet has 4 dof; after SSB, we have 1 dof for the neutral Higgs boson, + 3 dof absorbed by the bosons' mass terms. In $SU(5)$ the number of bosons' mass term stays the same. Not sure what the dof analysis could be.

  3. Lastly: SSB happens at an energy level corresponding to the Higgs mass. If the Higgs color-triplet form $SU(5)$ was so much more massive, then wouldn't it correspond to a symmetric being broken earlier? Yet $SU(3)$ is still unbroken at our current, very low, energy.

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  • $\begingroup$ Might appreciate Magg, M., & Shafi, Q. (1980): Symmetry breaking patterns in SU (5). Zeitschrift für Physik C Particles and Fields, 4, 63-66. $\endgroup$
    – Cosmas Zachos
    Commented Jul 27, 2023 at 14:42
  • $\begingroup$ Your 2 is an answerable question, and maybe should be separate. Otherwise, you are all but asking for a tutorial on the t-d problem, beyond BEGN & Langacker's Physics Reports review. In terms of counting, the 24 and the 5 Higgs multiplets mix, so there are, in all, 15 Goldstone bosons, all other dof being massive. $\endgroup$
    – Cosmas Zachos
    Commented Jul 27, 2023 at 14:57

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