5. Non-minimal kinetic terms and the SUGRA η problem 63 Under this symmetry, a superfield S → S + iC , where C is a constant. 104 • Safe non-minimal K¨ ahler potentials have also been proposed [630–633] making use of the shift symmetry 63 [613, 634] to protect the K¨ ahler potential of the form K (Φ , ¯ Φ) → K (Φ + ¯ Φ). This symmetry generates an exactly flat direction for an inflaton field and a non-invariance of the superpotential induces some slope to its potential to allow slow-roll at the loop level. • The first one is to constrain the parameter of the leading corrections, imposing κ$_{S}$ ∼ 10 − 3 which is sufficient to keep the model safe, but without much physical motivation. For a generic inflationary model it is not possible to compute κ$_{s}$ at all (see the discussion on the footnote below). In this model | S | ≪ M$_{P}$ ensures that higher order terms are negligible [611]. the kinetic terms K $^{ij}$∂$_{µ}$ Φ$_{i}$ ∂ $^{µ}$Φ ∗ j are non-minimal because K ij = δ $^{ij}$. One obtains in particular ( ∂$_{SS}$ ∗ K $_{)}$− 1 ∼ 1 − 4 κ$_{S}$ | S | $^{2}$/M2 $_{P}$+ . . . that leads to a problematic κ$_{S}$ × O (1) H contribution to the inflaton mass, and therefore on the second slow-roll parameter η , unless some suppression is invoked. Several mechanisms have been proposed to tackle this η -problem. For K¨ ahler potentials such as Note that these corrections affect the dynamics at large field, though at small VEVs, the radiative corrections are the dominant origin of the dynamics. This is generically the case during the last 60 e-foldings of inflation [516]. | S | 2 + | Ψ | 2 + | Ψ | 2 [512], V F − SUGRA tree = κ $^{2}$exp ( s 2 + ψ 2 2 M 2 P ) × {( ψ 2 4 − M 2 ) 2 ( 1 − s 2 2 M 2 P + s 4 4 M 4 P ) + s $^{2}$ψ2 4 [ 1 + 1 M 2 P 1 4 ψ 2 − M 2 )] 2 } . (229) K = | S | 2 + | Φ | 2 + | Φ | 2 + κ$_{S}$ | S | $^{4}$/M2 P + . . . , (230)