Minimal No-Scale Supergravity

• Minimal no-scale supergravity is a framework featuring a logarithmic Kähler potential that ensures a vanishing tree-level scalar potential even after F-term breaking.
• It predicts a distinctive spectrum with null soft scalar masses at the input scale and universal gaugino masses driven by RG evolution, aligning with collider and dark matter constraints.
• The models naturally accommodate Starobinsky-like inflation, resolving the supergravity η-problem and matching key CMB observations.

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 (Ellis et al., 2010, Ellis et al., 2013, Ellis et al., 2015, Ellis et al., 2020). 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 $p=3$) (Ciupke et al., 2015).

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 $p=3$ (Ciupke et al., 2015, Marsh et al., 2014). 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 (Ellis et al., 2010, Ellis et al., 2017, Forster et al., 2021). 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 (Ellis et al., 2013).

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 (Dall'Agata et al., 2015, Cribiori et al., 2016).

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 (Marsh et al., 2014).

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 (Ellis et al., 2010, Forster et al., 2021).
• 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 $g_\mu - 2$ (Ellis et al., 2010, Ellis et al., 2017, Forster et al., 2021).
• 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 (Ellis et al., 2010, Forster et al., 2021).

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 (Ellis et al., 2013, Ellis et al., 2015, Ellis et al., 2020), 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 (Ellis et al., 2013, Ellis et al., 2015). 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 (Ellis et al., 2017). Extended constructions incorporate shift symmetries, Peccei–Quinn directions, and can be classified in both chiral and linear multiplet descriptions (Ciupke et al., 2015).

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 (Koshimizu et al., 2010).

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 (Dall'Agata et al., 2013, Aldabergenov, 2019).
• Realization via constrained (nilpotent) superfields, eliminating all elementary scalars, yielding “scalar-less” supergravity with only fermions and the graviton in the spectrum (Dall'Agata et al., 2015, Cribiori et al., 2016).
• 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 (Ellis et al., 2013, Ellis et al., 2020).
• 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 (Ellis et al., 2018).
• Compatibility with both “old minimal” and “new minimal” supergravity formalisms for inflationary and non-inflationary applications (Ferrara et al., 2014, Ketov et al., 2013).

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 (Marsh et al., 2014).

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 (Forster et al., 2021).
• Spin-independent dark matter–nucleon cross sections in viable regions are within reach of next-generation direct detection experiments (Ellis et al., 2017).
• A strict upper bound on the gravitino mass $m_{3/2} < 10^3$ TeV emerges in minimal models compatible with Starobinsky inflation (Forster et al., 2021).

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 (Koshimizu et al., 2010, Ellis et al., 2010, Ellis et al., 2013, Ferrara et al., 2013, Ciupke et al., 2015, Ellis et al., 2017, Forster et al., 2021).