Landscape, the Scale of SUSY Breaking, and Inflation

TL;DR

The paper investigates how the KKLT framework links the scale of SUSY breaking, the inflationary Hubble scale, and the barrier stabilizing the de Sitter vacuum. It shows that in the simplest KKLT model the gravitino mass is extremely large, $m_{3/2} \\sim 6\times 10^{10}$ GeV, enforcing a bound $H \\lesssim m_{3/2}$ and creating tension with both TeV-scale SUSY and high-scale inflation. To resolve this, the authors propose a racetrack superpotential featuring a supersymmetric Minkowski minimum where $W=0$ and $D_ ho W=0$, yielding $m_{3/2}=0$ at the minimum and allowing decoupling of the inflation scale from SUSY breaking upon uplifting. This framework suggests that high-scale inflation with a light gravitino is achievable while maintaining moduli stabilization, aided by modulus trapping and a barrier height not directly tied to $m_{3/2}$, thereby offering a path to reconcile inflation with low-scale SUSY breaking in the string landscape.

Abstract

We argue that in the simplest version of the KKLT model, the maximal value of the Hubble constant during inflation cannot exceed the present value of the gravitino mass, H< m_{3/2}. This may have important implications for string cosmology and for the scale of the SUSY breaking in this model. If one wants to have inflation on high energy scale, one must develop phenomenological models with an extremely large gravitino mass. On the other hand, if one insists that the gravitino mass should be O(1 TeV), one will need to develop models with a very low scale of inflation. We show, however, that one can avoid these restrictions in a more general class of KKLT models based on the racetrack superpotential with more than one exponent. In this case one can combine a small gravitino mass and low scale of SUSY breaking with the high energy scale of inflation.

Figures (4)

• Figure 1: Thin green line corresponds to AdS stabilized potential for $W_0 =- 10^{{-4}}$, $A=1$, $a =0.1$. Dashed line shows the additional term $C\over \sigma^2$, which appears either due to the contribution of a $\overline{D3}$ brane or of a D7 brane. Thick black line shows the resulting potential including the $C\over \sigma^2$ correction with $C = 2.6\times 10^{{-11}}$, which uplifts the AdS minimum to a dS minimum. All potentials are shown multiplied by $10^{15}$.
• Figure 2: The lowest curve with dS minimum is the one from the KKLT model. The second one describes, e.g., the D3/D7 inflationary potential with the term $V_{\rm infl}={V(\phi)\over \sigma^3}$ added to the KKLT potential; it originates from fluxes on D7 brane. The top curve shows that when the inflationary potential becomes too large, the barrier disappears, and the internal space decompactifies. This explains the constraint $H\lesssim m_{3/2}$.
• Figure 3: The F-term potential (\ref{['pot']}), multiplied by $10^{14}$, for the values of the parameters $A=1,\ B=-1.03,\ a=2\pi/100,\ b=2\pi/99,\ W_0= - 2\times 10^{-4}$. A Minkowski minimum at $V=0$ stabilizes the volume at $\sigma_{cr}\approx 62$. AdS vacuum at $V<0$ stabilizes the volume at $\sigma_{cr}\approx 106$. There is a barrier protecting the Minkowski minimum. The height of the barrier is not correlated with the gravitino mass, which vanishes if the system is trapped in Minkowski vacuum.
• Figure 4: The potential as a function of the complex field $\rho$. The Minkowski minimum occurs at $\alpha = {\rm Im}~\rho =0$, as we have assumed in the analytic investigation.