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For not violating the gauge invarinace, of course, the gluon is massless. In respect of Spontaneous Symmetry breaking(SSB), however, I wonder why the gluon is massless, compared the $W^+$, $W^-$, $Z$ boson got masses due to Higgs boson.

In GUT or beyond Standard Model, they predict the all fundamental force was unified in high energy level and turns out Standard Model in present energy level. Like electroweak -> E.M + Weak interaction with 3 gauge bosons got masses as i said above, I think gluon must have had a mass in same Higgs mechanism. But It is not.

What is the error in my thought?

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    $\begingroup$ Welcome to SE. I am not a GUT expert, not remotely, but in the Brout-Englert-Higgs mechanism, the SSB breaks generators of the gauge group with the $W^\pm,Z,B$ bosons being the corresponding gauge bosons. Therefore, they, or rather certain linear combinations, get mass. In GUT you are looking for a large gauge group that contains the SM groups as subgroups, very loosely speaking. QCD would still have a different gauge group than the electroweak model, hence the SSB of the Higgs would not break generators of SU(3), so why would you expect that the gluons now get masses from the Higgs? $\endgroup$
    – Koschi
    Commented Jan 27, 2023 at 13:28
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    $\begingroup$ Your error might be logical. One "observes" the color group is not broken spontaneously, and then one constructs a GUT with this property, contriving a Higgs sector expressly to prevent SSBreaking of the SU(3) color group. This is a tentative effort to speculate about viable theories at higher energy scales consistent with the initial "observation" of unbroken color. Are you asking how this contrivance was arranged, technically? $\endgroup$ Commented Jan 27, 2023 at 15:16
  • $\begingroup$ @CosmasZachos Yes, same question i guess, Ultimately, i wonder why Higgs sector prevent SSB of SU(3) color group. In other words, how was strong force (gluon) split from the unified force without Higgs mechanism or some mechanism with gluon massless state. $\endgroup$
    – Sasha Shin
    Commented Jan 28, 2023 at 0:36
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    $\begingroup$ @Sasha Shin The Higgs field that breaks GUT to standard model, is different from the Higgs field that breaks electroweak symmetry within standard model. The GUT-Higgs gives mass to so-called X and Y bosons that change leptons to quarks, while leaving gluons and electroweak bosons massless. Then standard-model-Higgs gives mass to W and Z, while leaving photon massless. $\endgroup$ Commented Jan 28, 2023 at 2:31
  • $\begingroup$ @MitchellPorter This is what i want exactly! Even though my question is strange to understand, thank you for reading my exact intention. $\endgroup$
    – Sasha Shin
    Commented Jan 28, 2023 at 8:51

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As commented, one "observes" the color group is not broken spontaneously, in real life, and then one constructs a GUT with this property, contriving a Higgs sector expressly to prevent SSBreaking of the SU(3) color group. You then clarified you want the "how" this was contrived.

This is achieved, with difficulty, by the potential (16) of this review. The parameters are tweaked, unnaturally, to get the 5 Higgs H pick up a v.e.v. 13 orders of magnitude smaller than that of the 24 Φ which broke SU(5) to SU(3)×SU(2)×U(1). It is an ingenious arrangement that only breaks SU(2)×U(1) and leaves SU(3) unbroken. The triggering v.e.v. is dialed to the last component of the 5.

I'm not sure how much technical detail you are pining for. The hyper-details are in section 4 of Buras, Ellis, Gaillard, & Nanopoulos, 1978.

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