physics point of view $^{58}$. 2. Hybrid inflation from F -terms The most well-known model of SUSY inflation driven by F -terms is of the hybrid type and based on the superpotential [62] W F = κS (ΦΦ − M $^{2}$) . (219) where, S is an absolute gauge singlet while Φ and Φ are two distinct superfields belonging to complex conjugate representation, and κ is an arbitrary constant fixed by the CMB ob- servations $^{59}$. This form of potential is protected from additional destabilizing contributions with higher power of S , if S , Φ and Φ carrying respectively the charges +2, α and − α under R-parity. Then W carries a charge +2 so that the action S = ∫ d $^{2}$θ W + . . . is invariant. The tree level scalar potential derived from Eq. (219) reads where we have denoted by S, φ, φ the scalar components of S, Φ , Φ. It has a form similar to the original hybrid inflation model, though m = 0, and both λ and λ ′ are replaced by only one coupling constant κ $^{2}$. In what follows, it is assumed that φ ∗ = φ along this direction, the D -terms vanish and that the kinetic terms for the superfields are minimal, which is equivalent to a minimal K¨ ahler potential, K = | S | 2 + | Φ | 2 + | Φ | $^{2}$. We can defined two effective real scalar fields canonically normalized, σ ≡ √$_{2Re(}$$_{S}$$_{), and}$ ψ ≡ 2Re(Φ) = 2Re(Φ). The two-field scalar potential can then be put to the form [611] V$_{tree}$ ( S, φ, φ ) = κ $^{2}$| M 2 − φφ | 2 + κ $^{2}$| S | $^{2}$( | φ | 2 + | φ | $^{2}$)2 + D − terms , (220) V$_{tree}$ ( σ, ψ ) = κ 2 ( M 2 − ψ 2 4 ) 2 + κ 2 4 σ $^{2}$ψ2 . (221) Like in the non-SUSY version, the global minimum of the potential is located at S = 0, φφ = M $^{2}$, though at large VEVs, S > S$_{c}$ ≡ M , the potential also possesses a local valley of 58 There are attempts to embed natural inflation realizable at low scales within SUGRA, see [166, 358, 615– 617]. The generic potential has a form: V = Λ $^{4}$(1 + β | φ | 2 + γ | φ | 3 + δ | φ | 4 + . . . ), where β, γ, δ are model dependent coefficients. Challenge for these models is to justify why inflation starts at φ ≈ 0 VEV and ˙ φ ≈ 0. A prior phase of inflation may justify such initial conditions. 59 It is desirable to obtain an effective singlet S superfield arising from a higher gauge theory such as GUT, however to our knowledge it has not been possible to implement this idea, see Sect. IV G 3. Typically S would have other (self)couplings which would effectively ruin the flatness. 100