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?
Best Answer
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.