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non-renormalizable terms (smooth and shifted hybrid inflation), which can be expected to become unstable and suffer from the η -problem when formulated in SUGRA. It ac- tually turns out not to be a major problem, the presence of extra non-renormalizable term changes the shape of the tree-level potential and avoid the η -problem as dis- cussed in the previous section. Moreover, we are allowed to assume non-minimal (non-renormalizable) K¨ ahler potentials, which introduce more parameters and more freedom to flatten the potential [72, 82]. 

For example in the case of a smooth hybrid inflation (with the superpotential given by Eq. (231)) the effective potential in minimal SUGRA is of the form 65 [71] 

<img src='content_image/348134.jpg'>

The last term in this expression comes from the SUGRA corrections. Assuming this expression is valid for relevant scales, it is found that the spectral index is raised from n$_{s}$ ≃ 0 . 97 in global SUSY to larger values n$_{s}$ ∈ [0 . 98 , 1 . 03] depending on the scale ˜ M = √$_{µM}$ [71]. 

Introducing the non-renormalizable corrections to the K¨ ahler potential, K = K$_{min}$ + λS $^{4}$/ (4 M ) + . . . , lead to an effective potential [82, 513] 

<img src='content_image/348135.jpg'>

This allows the model to reduce the tension by increasing the cutoff scale M and reduce of the predicted spectral index to values inside n$_{s}$ ∈ [0 . 95 , 0 . 99]. 

We conclude this section by pointing out that non-renormalizable corrections can also affect significantly other predictions such as the running spectral index, even if inflation takes place well below the scale of new physics [641, 642]. The reason is that if the inflationary valley is flat at tree level, all the dynamics is determined by the lifting of a non-renormalizable term. It was first found in [641] that smooth hybrid inflation could predict a non-negligible amount of running by balancing the non-renormalizable contributions to W and K with the SUGRA effects. This prediction however requires the modification of the superpotential to include a phase of new inflation, since the 

