Jordan Frame Supergravity and Inflation in NMSSM

TL;DR

The authors derive the full $N=1$ supergravity action in a general Jordan frame with non-minimal coupling $Φ(z, \bar z) R$, revealing a modified Kähler structure and explicit frame-function dependence in both bosonic and fermionic sectors. They apply the formalism to embed the NMSSM and study Higgs-type inflation via Einhorn-Jones-like constructions, showing that while canonical kinetic terms can be achieved in the Jordan frame, the singlet direction develops a tachyonic instability that disrupts slow-roll inflation. The work provides a symmetry-driven route from a superconformal theory to Jordan-frame supergravity and establishes a framework for assessing frame-dependent cosmology and potential stabilization mechanisms. The results indicate that, in this setup, simple NMSSM-inspired inflation requires additional modifications to stabilize the singlet sector, but the Jordan-frame formulation offers a clear path for building and evaluating realistic supergravity inflationary models.

Abstract

We present a complete explicit N=1, d=4 supergravity action in an arbitrary Jordan frame with non-minimal scalar-curvature coupling of the form $Φ(z, \bar z)\, R$. The action is derived by suitably gauge-fixing the superconformal action. The theory has a modified Kaehler geometry, and it exhibits a significant dependence on the frame function $Φ(z, \bar z)$ and its derivatives over scalars, in the bosonic as well as in the fermionic part of the action. Under certain simple conditions, the scalar kinetic terms in the Jordan frame have a canonical form. We consider an embedding of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) gauge theory into supergravity, clarifying the Higgs inflation model recently proposed by Einhorn and Jones. We find that the conditions for canonical kinetic terms are satisfied for the NMSSM scalars in the Jordan frame, which leads to a simple action. However, we find that the gauge singlet field experiences a strong tachyonic instability during inflation in this model. Thus, a modification of the model is required to support the Higgs-type inflation.

Figures (4)

• Figure 1: The $F$-term potential $V^{F}$ in the Einstein frame. The inflationary trajectory $s=0$ is unstable.
• Figure 2: During inflation at large $h$ the angular variable $\beta$ is stabilized at $\beta =\pi /4$, corresponding to $h_{1}=h_2$. For $g^2,g^{\prime 2}\gg \lambda ^2$, this stabilization is preserved even after the end of inflation.
• Figure 3: During inflation at large $h$ the angular variable $\beta$ is stabilized at $\beta =\pi /4$, corresponding to $h_{1}=h_2$. For $g^2,g^{\prime 2}\ll \lambda^2$, at the end of inflation the curvature of the potential in $\beta$-direction becomes large and negative, much greater than the curvature in the inflaton direction. This leads to tachyonic instability, generation of large fluctuations of the field $\beta$, and spontaneous symmetry breaking.
• Figure 4: "Tachyonic preheating" effect at the end of inflation (for $g^2,g^{\prime 2}\ll \lambda^2$).