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In this section, we review how the hybrid inflation superpotential (6.1) can be combined with a minimal K¨ahler potential and a non-minimal K¨ahler potential expansion in terms of higher dimensional effective operators. The section is based on [31, 105].

Let us consider the SUSY hybrid inflation superpotential (6.1) combined with a K¨ahler potential as an expansion up to mass dimension four in the scalar component fields given by K =|Φ|2+|H|2+ κΦ 4 Λ2|Φ| 4 + κH 4 Λ2|H| 4 + κΦH Λ2 |Φ| 2 |H|2+. . . , (6.11) where in the following, we set the suppression scale of the effective operators to Λ = 1.

The minimal K¨ahler potential, which gives rise to canonical kinetic terms without kinetic mixing is just given by a sum of the absolute values squared of the scalar components in all chiral supermultiplets. Thus, as the simplest SUGRA extension of the standard SUSY hybrid inflation, let us first study a minimal K¨ahler potential by switching off all higher dimensional operators setting κΦ =κH =κΦH =. . .= 0.

As stated in Ch. 5, we want to study the viability of inflation models in SUGRA regarding the η-problem and the problem of moduli stabilization. In the simple case at hand, there are no additional moduli fields involved, so we only have to study the SUGRA corrections to the tree level inflaton mass. Using Eq. (3.67), the F-term scalar potential obtained from Eq. (6.11) reads

VF = e|Φ|2+|H|2_|(WΦ+W KΦ)|2+|(WH +W KH)|2−3|W|2

, (6.12)

since Ki¯j _=δi¯j_.

As explained in Sec. 5.1, the exponential factor in Eq. (6.12) contributes the most serious potential source of the η-problem in SUGRA inflation. It turns out that with a minimal K¨ahler potential, to leading order in the SUGRA expansion, the mass squared terms for the inflaton exactly cancel. This is a very special and desirable feature of minimal SUSY hybrid inflation since mysteriously, theη-problem is not present in this simple case, as has been pointed out in [97, 105]. Let us push on to see how this comes about. Expanding Eq. (6.12), one obtains to leading order

VF _≃4κ2_|Φ_|2_|H_|2+κ2 _|H_|2₋M22 1 +_|H_|2+ |Φ| 4 2 +|Φ| 2 |H_|2 +. . . , (6.13) where the positive mass squared contribution for _|Φ_| comes from the exponential in (6.12) and the negative one from the sum of terms within the the squared brackets. They cancel and there is no dangerous tree level mass term for the inflaton at leading order.

Just as in the case of global SUSY hybrid inflation, the waterfall direction obtains its large mass squared contribution for |Φ| > |Φc|. This stabilizes it at zero during the inflationary epoch. With a canonically normalized inflaton direction φ = √2_|Φ_|, the resulting one-loop effective potential in the inflationary minimum_|H_|= 0 is approximately given by V_effmin(φ)_≃κ2M4 1 + φ 4 8 +. . . +V_loopmin(φ). (6.14)

6.1 Supersymmetric Hybrid Inflation 63

The one-loop correction Vmin

loop(φ) is again given by Eq. (6.4), however, with the mass ma- trices calculated using the SUGRA formulae displayed in Sec. 3.2. Note that the difference to the global SUSY potential is the φ4_{-dependence of the tree level potential in Eq. (6.14)} generated by SUGRA corrections. Since typically in the hybrid models we are interested in φ≪1, these corrections are subdominant in the relevant inflationary field space. As κ

is increased, the field values at a fixed number of e-folds increases and the SUGRA effects become more important. This can be seen, e.g., in Fig. 6.1, where the spectral index is driven to larger values for increasing κ.

Next, let us explore the effect of the higher dimensional operators in the K¨ahler potential of Eq. (6.11). Due to the resulting non-diagonal K¨ahler metric, the F-term potential contains many additional terms. Since this is not very enlightening, we only give the expansion VF _≃4κ2_|Φ_|2_|H_|2+κ2 _|H_|2₋M22 1₋κΦ|Φ|2+γH|H|2+γΦ| Φ_|4 2 +. . . , (6.15) where the new parameters have been defined as

γH = 1−κΦH, γΦ = 1− 7 2κΦ+ 2κ 2 Φ. (6.16)

Given the tree level F-term potential, we can again calculate the one-loop effective potential in the inflationary trajectory, φ _≫φc and |H|= 0, given by

V_effnon-min(φ)≃κ2M4 1−κΦ φ2 2 +γΦ φ4 8 +. . . +V_loopnon-min(φ). (6.17) Note, that the effective operator corresponding to κΦ induces a (negative) mass squared term at tree level. Thus, fulfilling the slow-roll conditions requires a tuning to make these parameters somewhat small. A parameter κΦ =O(1) would be unacceptable because it is inconsistent with slow-roll inflation. As it should be, switching off all higher dimensional operators by setting κΦ = κH = κΦH = . . .= 0, Eq. (6.17) just reproduces the potential

resulting from the minimal K¨ahler potential in Eq. (6.14). Since the K¨ahler potential is in principle arbitrary, we consider the minimal K¨ahler potential to be a specific version of the more general expansion of Eq. (6.11) where all higher order expansion parameters have been set to zero.

In order to study the effect of the crucial model parameter κΦ, we fix the number of e-folds of observable inflation to Ne= 60. From the critical value φc, we use the slow-roll EOM Eq. (2.16) applied to the potential Eq. (6.17) to calculate back to the field value at which scales corresponding to the present horizon size crossed the de Sitter horizon during inflation. As before, the WMAP normalization on P_R in Eq. (2.41) then fixes the scale M = _O(10−3_{) for the relevant parameters. We have implemented this calculation} numerically.

0.001 0.002 0.005 0.010 0.020 0.050 0.94 0.96 0.98 1.00 1.02 ns κ

Figure 6.1: Predicted spectral index depending on the fundamental coupling parameter κfor

different values ofκΦ. The solution for minimal K¨ahler potential is represented by the solid line (κΦ= 0), while higher dimensional operators are taken into account in the dashed (κΦ= 5·10−3) and dotted (κΦ= 10−2) lines. The gray shaded region corresponds to the region preferred by the latest WMAP data at 68% CL.

The resulting spectral index for different values ofκΦdepending on the model parameter

κ is displayed in Fig. 6.1, where for convenience we have set all other model parameters to zero. The case of a minimal K¨ahler potential is then just obtained by setting κΦ = 0, which corresponds to the solid line in the plot. This reproduces the results of Ref. [105]. For κΦ ≃ 10−3−10−2, the predicted value can lie well inside the region preferred by the seven-year WMAP data at 68% CL which is highlighted in grey. Successively larger values ofκΦ can effectively reduce the spectral tilt which is due to the fact that it contributes to a negative squared mass pushingηto more negative values. Looking at Eq. (2.38), this leads to a smaller spectral tilt ns. As in the global SUSY result, the predicted tensor-to-scalar ratio is again very small.