content_image/330316.jpg content_image/330317.jpg content_image/330318.jpg
This accommodates a quark triplet, d$_{R}$ , and a leptonic doublet, ( $_{e}$− $_{L}$, ν$_{L}$ ), respectively in the 5 , and the quark triplets, ( u$_{L}$, d$_{L}$ ) u$_{R}$ , and e $^{+}$, respectively in the 10 . The construction of Q as a combination of two diagonal traceless generators of SU (5) requires that the sum of all the charges of the fermions in the fundamental representation vanishes, and thus explains the quantification of the charges. 

There are 24 gauge bosons of SU (5) and their quantum number under the SM are given by the decomposition under SU (5) ⊃ SU (3) × SU (2) [430]: 

<img src='content_image/330316.jpg'>

which are the photon, the triplet W ± , Z bosons, the so-called X and ¯ X , and the gluon octet. The 12 gauge bosons X and ¯ X carry both SU (3)$_{c}$ and SU (2)$_{L}$ quantum numbers, therefore they violate the baryon quantum number, leading to a proton decay. 

The Higgs sector of SU (5) must contain a Higgs field breaking SU (5) into G$_{SM}$ , and at least another Higgs for the electroweak breaking, as the two scales are different. The smallest representation that contains a singlet under the SM is the adjoint Σ = 24 , see Eq. (149). Its potential is a generalization of the usual mexican hat potential: 

where m 2 Σ > 0 and a, b are constants. The most general potential for Σ would also contain a term of the form TrΣ $^{3}$, if the invariance Σ → − Σ is not assumed. To break SU (5) into the SM, the VEV has to be taken in the direction Σ$_{24}$ = Diag( − 1 , − 1 , − 1 , 3 / 2 , 3 / 2) $^{39}$, though depending on the sign of b , the minimum of the above potential can be obtained for VEVs ∝ Diag(1 , 1 , 1 , 1 , − 4), instead. The breaking in this case is the other maximal subgroup, SU (5) → SU (4) × U (1). The SM Higgs can be embedded into a fundamental, H = 5 , as it contains a component (the H$_{2}$ = ( 1 , 2 )), a singlet under SU (3)$_{c}$ but doublet under SU (2)$_{L}$ has a potential: 2 

<img src='content_image/330317.jpg'>

<img src='content_image/330318.jpg'>

It is important to note that summing the two potentials V = V (Σ) + V ( H ) is not sufficient to construct a realistic theory. As a general rule, the potential of all the Higgs fields contain 

