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cross-terms that are allowed by symmetries. The correct approach is to consider a field content and minimize the multi-dimensional potential to be sure that the minima give the right phenomenology. Embedding the SM Higgs into a fundamental leads to the well-known doublet-triplet splitting problem, as a mass term for H$_{5}$ would give the same mass to the SM doublet and the additional triplet H$_{3}$ = ( 3 , 1 ) contained in H . This is catastrophic as H$_{3}$ possesses the right quantum number to contribute to the proton decay, thus requiring that its mass is higher than ∼ 10 15 GeV. Interestingly one possible solution of the second problem is to allow cross-terms between H and Σ. 

Non-SUSY SU (5) unification has many problems. The three gauge couplings do not exactly meet at the unification scale at a single value. In addition, the unification scale M$_{GUT}$ ∼ 10 $^{14}$GeV is now incompatible with the most recent observations of the proton decay [436]. 

SUSY SU (5) introduces new low mass particles in the spectrum and the existence of a SUSY breaking scale affect the running of the gauge couplings, which allow for a better unification at a larger scale M$_{GUT}$ ≃ 2 × 10 16 GeV. Now the proton decay is even more suppressed in the dimensional 6 channel. However the proton decay can now happen through dimensional 4 operators, that would be more dramatic. In order to suppress them one requires the R-parity (see the discussion in Sect. III C 1). 

One of the major problems of SUSY SU (5) is the doublet-triplet splitting. Part of the superpotential which generates the masses to the fermions is given by: 

<img src='content_image/336610.jpg'>

where Σ belongs to 45 , H belongs to 5 and ¯ H to ¯ 5 . Out of three degenerate vacua, the right one with the SM gauge group is given by; Diag( 〈 Σ 〉 ) = (2 , 2 , 2 , − 3 , − 3) M$_{GUT}$ . Substituting the VEV in the above superpotential generates an effective superpotential; 

<img src='content_image/336611.jpg'>

Since, the Higgs mass ought to be around ∼ 100 GeV, the cancellation in µ ∼ 3 M$_{GUT}$ has to be within one part in 10 12 to match the observed expectation. This fine tuning is also related to the µ -problem in MSSM. This fine tuning can be replaced by a see saw mechanism in the case of SUSY SO (10), where the triplet can be made naturally heavy as compared to the doublet [437]. On the cosmological front, SU (5) conserves B − L , and baryogenesis 

