content_image/352920.jpg content_image/352921.jpg
then λ = 0. This is due to the fact that SU (5) preserves B − L . 

• SO (10): 

In this case all matter fields of one generation are included in the spinorial represen- tation 16 of SO (10). Hence the superpotential terms in Eq. (310) are [ 16 ] 6 under SO (10), which does not provide a singlet. A gauge invariant operator will be obtained by multiplying with a 126-plet Higgs. This implies that in SO (10) the lowest order gauge invariant superpotential term with 6 matter fields arises at n = 7 level: 

<img src='content_image/352920.jpg'>

Once 126$_{H}$ acquires a VEV, S 0(10) can break down to a lower ranked subgroup, for instance SU (5). This will induce an effective n = 6 non-renormalizable term as in Eq. (310) with 

<img src='content_image/352921.jpg'>

Hence, in the case of SO (10), we can expect λ ∼ O (10 − 2 − 10 − $^{1}$) depending on the scale where SO(10) gets broken. 

By embedding MSSM in SO (10) naturally implies that λ ≪ 1. Smaller values of n$_{s}$ (within the range 0 . 92 ≤ n$_{s}$ ≤ 1) point to smaller λ , as can be seen from figure 6. This, according to Eq. (332), implies a scale of SO (10) breaking, i.e. 〈 126$_{H}$ 〉 , which is closer to the GUT scale. Further note that embedding the MSSM within SO (10) also provides an advantage for obtaining a right handed neutrino. It was concluded in Ref. [714] that if we include the RH neutrinos, then udd direction is preferred over LLe . 

## 5. Gauged inflaton in SM × U $^{(1)$_{B}$}$− L 

If we augment MSSM with three right-handed (RH) neutrino multiplets, then it is possible to realize neutrinos of Dirac type with an appropriate Yukawa coupling. Whether the nature of neutrino is Dirac or Majorana can be determined in the future neutrinoless double beta decay experiment. 

