---
title: Minimal No-Scale Supergravity
url: https://www.emergentmind.com/topics/minimal-no-scale-supergravity
type: topic
---

# Minimal No-Scale Supergravity

Minimal No-Scale Supergravity is a class of $\mathcal{N}=1$ supergravity theories characterized by the vanishing (or positivity) of the scalar potential even after spontaneous supersymmetry breaking. Originating from string compactifications and constructed to possess a specific “no-scale” Kähler potential, these models provide a technically natural resolution to the “$\eta$-problem” of supergravity inflation, enable efficient mediation of supersymmetry breaking with a vanishing tree-level cosmological constant, and are directly relevant to both particle physics and cosmological applications such as Starobinsky-like inflation. Minimal no-scale models are notable for their simple parameterization, predictive power, and close geometric connection to effective low-energy limits of string theory.

## 1. Defining Features and Kähler Structure

Minimal no-scale supergravity models employ a logarithmic Kähler potential such that certain dangerous term cancellations occur in the scalar potential, producing flat directions. The prototypical Kähler potential is
\[
K = -3 \ln(T + T^* - |\phi|^2/3) \, ,
\]
where $T$ represents a (volume-type) modulus and $\phi$ a matter fields multiplet [1004.5399, 1305.1247, 1507.02308, 2009.01709]. More generally, for $n$ complex chiral fields $T_i$, the shift-symmetric no-scale structure requires that $K$ depends only on the real part: $K = -p \ln Y(T_i + \overline{T}_i)$, with $p$ a real parameter (usually $p=3$) [1509.00855].

The scalar potential in supergravity,
\[
V = e^K (K^{i \bar j} D_i W D_{\bar j} \overline{W} - 3 |W|^2),
\]
exhibits the hallmark “no-scale” cancellation: for the above $K$ and $W$ independent of $T$, $K^{T \bar T} K_T K_{\bar T} = p$, so $V$ identically vanishes for $p=3$ [1509.00855, 1411.6625]. In essence, supersymmetry breaking in these models does not lift the vacuum energy at tree level due to the geometric form of $K$.

## 2. Supersymmetry Breaking and the Vacuum Structure

Minimal no-scale models generically have F-term supersymmetry breaking sourced by the no-scale modulus $T$. The expected pattern of soft breaking parameters at a high scale $M_{in}$ is
\[
m_0 = A_0 = B_0 = 0,
\]
while the universal gaugino mass $m_{1/2}$ is unconstrained [1004.5399, 1702.00379, 2109.10802]. Even after adding a constant term to the superpotential to break supersymmetry and give a gravitino mass $m_{3/2}$, the potential remains manifestly semi-positive-definite or flat at tree level [1310.4770].

An alternative realization involves constrained superfields (e.g., nilpotent goldstino), in which all elementary scalars are absent and the cosmological constant remains fully controlled by the superpotential parameters and goldstino auxiliary field [1509.06345, 1611.01490].

The vacuum is Minkowski classical and typically features a flat (modulus) direction. Lifting this flatness may be achieved through radiative or non-perturbative corrections, matter couplings, or subleading modifications to the superpotential and Kähler potential [1411.6625].

## 3. Phenomenology and Parameter Space

When embedded cosmologically or phenomenologically, minimal no-scale models offer a compressed and predictive spectrum:
- All soft scalar masses vanish at $M_{in}$, so low-energy sfermion and gaugino masses are generated via RG evolution driven by $m_{1/2}$ alone [1004.5399, 2109.10802].
- The allowed parameter space typically forms an “L-shaped” or triangular strip in the $(m_{1/2}, M_{in})$ plane, constrained by electroweak symmetry breaking, LEP/LHC Higgs mass bounds, rare processes ($b \to s \gamma$), relic abundance, and $g_\mu - 2$ [1004.5399, 1702.00379, 2109.10802].
- The predicted superpartner mass spectra (e.g., gluinos up to 1.5 TeV, squarks up to 1.3 TeV) lie within or near current/future collider sensitivity [1004.5399, 2109.10802].

The dark matter relic abundance can be accommodated via coannihilation channels (LSP-neutralino being nearly degenerate with stau or chargino), and the Higgs mass remains compatible with MSSM radiative corrections for viable $(m_{1/2}, M_{in})$.

A table summarizing soft breaking parameter properties in minimal no-scale SUGRA versus other SUGRA scenarios:

| Parameter      | Minimal No-Scale        | CMSSM/Generic Supergravity         |
|----------------|------------------------|-------------------------------------|
| $m_0$          | $0$ at $M_{in}$        | Free parameter                     |
| $A_0$          | $0$ at $M_{in}$        | Free parameter                     |
| $B_0$          | $0$ at $M_{in}$        | Free parameter                     |
| $m_{1/2}$      | Nonzero, universal     | Nonzero, universal                 |
| $M_{in}$       | $\gtrsim M_{GUT}$      | Model-dependent                    |

## 4. Connections to Inflation and String Theory

Minimal no-scale supergravity is a preferred setting for embedding cosmological inflation. Its Kähler potential and superpotential structure correspond naturally to plateau-type inflationary potentials, notably Starobinsky-like $R+R^2$ inflation [1305.1247, 1507.02308, 2009.01709], yielding
\[
V(\chi) = \mu^2 e^{-\sqrt{2/3}\chi} \sinh^2(\chi/\sqrt{6}),
\]
with predictions $n_s\sim0.965$, $r\sim0.0035$, matching Planck CMB data [1305.1247, 1507.02308]. The avoidance of the supergravity “$\eta$-problem” is direct—dangerous corrections to the inflaton mass are canceled by the no-scale structure.

The models are formally motivated by string compactifications: the no-scale Kähler potential emerges generically as the tree-level effective theory of moduli in heterotic and type II orientifolds (e.g. $K = -3 \ln(T + T^*)$ for a volume modulus $T$). Modular weights can be assigned for “untwisted” and “twisted” matter to orchestrate soft breaking terms in GUTs [1702.00379]. Extended constructions incorporate shift symmetries, Peccei–Quinn directions, and can be classified in both chiral and linear multiplet descriptions [1509.00855].

## 5. Reheating, Preheating, and Microscopic Dynamics

After inflation, the dynamics of energy transfer and reheating in minimal no-scale SUGRA models involve independent preheating channels:
- Direct perturbative decay of the inflaton into gauge sector fields and gauginos, with a characteristic decay rate $\Gamma \sim 3.89 \times 10^3$ GeV, yields a reheating temperature $T_R \sim 4.45 \times 10^{10}$ GeV.
- Instant preheating or non-thermal production: inflaton scattering produces right-handed sneutrinos (via NMSSM-like superpotential couplings), which decay into Higgs and other MSSM particles; this channel produces a similar reheating temperature [1001.3733].

A salient point is that both these independent transfer channels yield similar $T_R$ and together are essential for ensuring efficient population of all MSSM species and for connecting supersymmetry breaking to observable sectors.

## 6. Extensions, Variants, and Stability

Minimal no-scale supergravity supports several controlled extensions:
- Inclusion of D-term breaking via gauged axionic shift symmetries, leading to controlled symmetry breaking and a massless scalar dilaton [1308.5685, 1903.11829].
- Realization via constrained (nilpotent) superfields, eliminating all elementary scalars, yielding “scalar-less” supergravity with only fermions and the graviton in the spectrum [1509.06345, 1611.01490].
- Embedding in extended GUT frameworks ($SU(5)$, $SO(10)$, flipped $SU(5)\times U(1)$), allowing cosmological neutrino mass generation, baryogenesis, and cold dark matter production [1310.4770, 2009.01709].
- Generalization to multi-field and de Sitter vacua, employing Minkowski “endpoint” pairings and quartic stabilization in the Kähler potential to ensure holomorphy and metastable minima [1809.10114].
- Compatibility with both “old minimal” and “new minimal” supergravity formalisms for inflationary and non-inflationary applications [1407.6164, 1309.7494].

Stability analyses show that metastable dS vacua may be constructed via small superpotential deformations, generically decoupling $N-2$ massive superfields and tuning only $O(1)$ parameters [1411.6625].

## 7. Current Phenomenological Status and Experimental Constraints

Comprehensive scans of the minimal no-scale SUGRA parameter space, accounting for LHC sparticle searches, Higgs mass, dark matter relic density, and the anomalous muon magnetic moment, delineate allowed regions with predictive signatures:
- The muon $g-2$ anomaly may be accommodated only in certain regions with negative universal gaugino mass parameter $k$, yielding light sleptons/charginos near current LHC exclusion boundaries [2109.10802].
- Spin-independent dark matter–nucleon cross sections in viable regions are within reach of next-generation direct detection experiments [1702.00379].
- A strict upper bound on the gravitino mass $m_{3/2} < 10^3$ TeV emerges in minimal models compatible with Starobinsky inflation [2109.10802].

The current and future collider experiments (e.g., LHC Run 3), as well as precision dark matter searches, provide strong testing grounds for minimal no-scale scenarios.

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In conclusion, minimal no-scale supergravity is a class of supergravity models built upon a unique logarithmic Kähler geometry and a superpotential accommodating F-term breaking with vanishing tree-level scalar potential, often extended to accommodate GUT, cosmological inflation, and string-theoretic origin. Their predictive and constrained phenomenology is under continuous scrutiny in cosmological and collider experiments, with multi-channel reheating, controlled metastability, and a robust connection to high-scale physics at the forefront of present research [1001.3733, 1004.5399, 1305.1247, 1307.7696, 1509.00855, 1702.00379, 2109.10802].

Source: https://www.emergentmind.com/topics/minimal-no-scale-supergravity