via leptogenesis cannot take place. This is another reason why it is desirable to go beyond SU (5). The next to the simplest gauge group candidate for the unification is SO (10) [438] (see some recent reviews [439–442]), for which the entire generation of fermions can be fit into a single representation, 16 , of SO (10) which decompose under SU (5) × U (1) into ¯ 5 + 10 + 1 . It can therefore accommodate all the fermions of the MSSM along with an additional sin- glet fermion, with the quantum numbers of a right-handed neutrino which can fit into the fundamental spinorial representation 16 [347–350]. The SM leptons are now identified as a fourth color of quarks after the inclu- sion of right handed neutrino. The direct product of spinor representation gives, 16 ⊕ 16 = 10 ⊕ 120 ⊕ 126 , therefore the Higgs sector must contain either of these three, 10 , 120 , or 126 , in order to generate mass terms for the fermions, for detailed discussion see Refs. [38, 39, 431]. In particular when 126 develops a VEV, it gives rise to the Ma- jorana right handed neutrino masses as large as the GUT scale, in order to generate the neutrino masses one would then have to invoke the see saw mechanism [350, 443] $^{40}$. This also opens up naturally the possibility to realize baryogenesis via leptogenesis [327]. On the proton decay front, SO (10) is also attractive as it contains a (gauged) B − L , the kernel of which is a Z$_{2}$ that automatically plays the role of R-parity. This requires however to break SO (10) only with safe representations, such as 10 , 45 , 54 , 120 , 126 , 210 , . . . but not 16 , 144 , 560 , . . . [445]. Finally in the doublet triplet front, the situation is improved compared to the SU (5) case by employing the Dimopoulos-Wilczek scenario [446]. 2. Symmetry breaking in SUSY GUT The most simple superpotential to break a GUT symmetry is simply the generalization of the Higgs, one particular type of superpotential which can break G$_{GUT}$ can be written as: W = λX (TrΣ 2 − M 2 $_{Σ}$) , (154) 40 The SO (10) can be broken down to the MSSM via many routes, but the most popular one is through left- right symmetric group, SO (10) → SU (4)$_{c}$ × SU (2)$_{L}$ × SU (2)$_{R}$ → SU (3)$_{c}$ × SU (2)$_{L}$ × SU (2)$_{R}$ × U $^{(1)$_{B}$}$− L → SU (3)$_{c}$ × SU (2)$_{L}$ × U (1)$_{Y}$ (see for e.g. [444] and references therein). This can happen by a scalar belonging to 210 or a combination of the fields belonging to 45 + 54 . The next stage of breaking, i.e. left-right symmetry, can happen via the Higgs of 126 dimensional representation or 16 of SO (10), see for a recent review [38]. 69