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parameter range to generate a large running imposes a short (less than 60 e-foldings) phase of smooth inflation. A model independent study of SUSY hybrid type models with potentials flat at tree level, and lifted by radiative corrections was considered in [642], 2 N 

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where the last term contains an arbitrary non-renormalizable operator. It was found that typically running of the spectral tilt is negligible, α$_{s}$ ≪ 1, in renormalizable models if a large number of e-foldings is realized. However they also pointed out that if the non-renormalizable mass scale, M is larger than the inflationary scale can generate a large running O ( − 0 . 05). 

• Extensions of hybrid F -term inflation: 

Many models have been proposed, that generalize or extend the idea of the hybrid inflaton driven by F -terms. In Ref. [620, 621], the realization of hybrid inflation has been illustrated in conjunction with solving the µ -problem of the MSSM within the NMSSM. They proposed that the waterfall field coupled to the inflaton also induce the mass term for the electroweak Higgs pairs of the MSSM: 

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At the cost of adding two scalar singlets to the MSSM, this model is found to solve the µ -problem, prevents from domain walls to form during the electro-weak symmetry breaking (see [643]), and give rise to a phase of inflation of the (non-SUSY) hybrid type with an effective potential of the form V$_{0}$ + m $^{2}$φ$^{2}$. 

Other examples of extensions of the hybrid model via F -terms are the smooth and shifted scalar potentials obtained with only renormalizable operators [70, 83]. They were named “new smooth” and “new shifted” hybrid inflation, and require the intro- duction of additional fields interacting with the inflaton and the waterfall field. 

For example, new shifted hybrid inflation is still based on a singlet S of the Pati-Salam gauge group G$_{P}$$_{S}$ coupled to Φ, ¯ Φ in ( ¯ 4 , 1 , 2 ) and ( 4 , 1 , 2 ). But the model also assumes the introduction of new superfields Ψ, ¯ Ψ both in ( 15 , 1 , 3 ) to which the inflaton S would be coupled. The introduction of these extra-fields has motivations [69] from the fermion spectrum, in particular the predicted mass of the bottom quark, that becomes 

