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## 5. Other inflationary models within SUSY GUTs 

Inspite of the challenge to seek an origin of a gauge singlet inflaton within the GUT group, there have been many examples to embed the waterfall field within SUSY GUTs. Here we briefly sketch some examples of the embedding. 

• Shifted and smooth hybrid inflation: 

As we discussed in Sec. IV E 7, the shifted hybrid inflation model was built embedded in the Pati-Salam subgroup G$_{P}$$_{S}$ ≡ SU (4)$_{C}$ × SU (2)$_{L}$ × SU (2)$_{R}$ of SO (10) [67, 692]. The idea is to break the gauge group during inflation in a single step which also avoids the monopole problem arising in the breaking of the Pati-Salam group down to the MSSM group, breaking that can be done using a pair of Higgs superfields, H c = ( ¯ 4 , 1 , 2 ) and ¯ H c = ( 4 , 1 , 2 ). These Higgs fields also give through their VEVs a Majorana mass to the right-handed neutrino directions, inducing the see-saw mechanism to account for the tiny mass of the left-handed neutrinos. This pair of Higgs fields is also assumed to be the one coupled to the inflaton field, realizing the symmetry breaking during inflation, and supporting inflation through their F -terms. The vanishing of the D -term part of the potential constraints, | ¯ H $^{c}$| = | H $^{c}$| . The inflationary potential is embedded in a potential where there are many more fields, which are relevant for the computation of one-loop corrected inflationary potential. The actual potential also is invariant under the global U (1)$_{R}$ of conventional SUSY and under a Peccei-Quinn symmetry, U (1)$_{PQ}$. Like for the original F -term inflation, the R -symmetry prevents from non-linear terms for S , such as S $^{2}$, S3 terms in the superpotential, which would ruin the flat direction for the inflaton at tree level. 

In Ref. [76], the authors embedded this shifted model within an explicit model based on SO(10)and also implemented the Dimopoulos-Wilczek mechanism [437], and addressed the MSSM µ -problem. The inflationary superpotential is given by [67, 76] 

<img src='content_image/338133.jpg'>

