Split SUSY Hybrid Inflation Model

• The model integrates supersymmetric hybrid inflation with a μ-term mechanism, generating the MSSM μ parameter through gravity-mediated SUSY breaking.
• It employs a radiatively corrected inflaton potential with a linear soft term to match observed spectral indices and predict tensor-to-scalar ratios between 10⁻² and 10⁻³.
• High reheat temperatures create a split SUSY spectrum that supports thermal leptogenesis and yields a viable dark matter candidate via a TeV-scale wino-like neutralino.

The split supersymmetry hybrid inflation model is an advanced framework of inflationary cosmology built on supersymmetric field theory, in which hybrid inflation is realized within a supersymmetric structure that naturally yields a “split supersymmetry” (split SUSY) spectrum. This framework, particularly in its μ-term variant, incorporates both a successful inflationary trajectory and a resolution to several deep problems in beyond-Standard-Model physics—such as the gravitino problem, the high reheat temperature, the μ-problem of the MSSM, thermal leptogenesis, and compatibility with current cosmological and collider data. The model’s phenomenology critically depends on the presence of a renormalizable coupling in the superpotential responsible for both the required high reheating temperature and the split of the soft mass spectrum between heavy scalars and TeV-scale gauginos. Recent data from ACT DR6 has provided strong constraints on the scalar spectral index, which this model accommodates, while preserving the ability to realize leptogenesis and a viable dark matter candidate in the form of a wino-like neutralino (Okada et al., 22 Jul 2025).

1. Theoretical Structure and Superpotential

The defining feature of the split SUSY hybrid inflation (or μ-term hybrid inflation) model is its augmented superpotential: $$W = S \left( \kappa \bar{\Phi} \Phi - \kappa M^2 + \lambda H_u H_d \right)$$ where:

• $S$: gauge singlet superfield (inflaton, real component drives inflation)
• $\Phi, \bar{\Phi}$: chiral superfields breaking a GUT or other gauge symmetry $G$
• $H_u, H_d$: MSSM Higgs doublet superfields
• $\kappa$, $\lambda$: dimensionless couplings; $M$ is the gauge symmetry breaking scale

A global $U(1)_R$ is imposed so that $S$ and $W$ have R-charge 1, others zero. The linearity of $W$ in $S$ ensures cancellation of dangerous Hubble-scale corrections (the “$\eta$ problem”) in the minimal Kähler potential case.

After supersymmetry breaking (arising via gravity mediation), the scalar potential receives a linear soft term: $$V_{\text{soft}} \supset -m_G \kappa M^2 (S + S^\dagger)$$ with $m_G$ the gravitino soft mass. This generates a VEV for $S$, so inserting

$$\langle S \rangle \approx \sqrt{2} \frac{m_G}{\kappa}$$

into the superpotential term $\lambda H_u H_d$ yields the MSSM $\mu$-term: $\mu = \frac{\lambda}{\kappa} m_G \equiv \gamma m_G$ with $\gamma \equiv \lambda / \kappa$. The explicit high-scale origin of $\mu$ and its connection to inflation and SUSY breaking is a central attribute.

2. Inflationary Dynamics, Potential, and Observables

During inflation, the system resides in a vacuum with $\Phi = \bar{\Phi} = 0$. Including loop (Coleman-Weinberg) corrections and the linear soft term, the inflaton potential for $\phi \equiv \sqrt{2}\mathrm{Re}\,S$ is: $$V(\phi) = m^4 \left[ 1 + A \ln(\phi/\phi_*) \right] - 2\sqrt{2} m_G m^2 \phi$$ where $m = \sqrt{\kappa} M$, $A = \frac{1}{4\pi^2}(\lambda^2 + \kappa^2) \ll 1$, and $\phi_*$ is a scale near the CMB pivot.

The linear soft term, controlled by $m_G$, steepens the potential and is crucial for lowering the scalar spectral index $n_s$ from the purely radiatively corrected value ($n_s = 1-1/N$) down to the measured range. The slow-roll parameters ($\epsilon, \eta$) and observable CMB quantities follow: \begin{align*} n_s &\approx 1 - (2/N) f(B), \qquad f(B) = \text{correction from soft term} \ r &\approx (2/\pi^2)(\lambda^2 + \kappa^2)(1-B)^2 [f(B)/N] \end{align*} where $B = 2\sqrt{2} m_G \phi_*/(A m^2)$, and $N$ is the number of e-foldings.

Recent ACT DR6 data fixes $n_s = 0.9743 \pm 0.0034$ (Okada et al., 22 Jul 2025). The split SUSY model can naturally accommodate this, with predicted $r$ in the range $10^{-2}$ to $10^{-3}$, significantly exceeding the negligibly small $r$ of minimal supersymmetric hybrid inflation (Okada et al., 22 Jul 2025, Okada et al., 2015).

3. Reheating, High Temperature, and the Gravitino Problem

Reheating proceeds dominantly through inflaton decay to Higgsino pairs, with decay width (Okada et al., 22 Jul 2025, Okada et al., 2015): $$\Gamma(\phi \to \tilde{H}_u \tilde{H}_d) = \frac{\lambda^2}{8\pi} m_\phi, \quad m_\phi = \sqrt{2} \kappa M$$ The reheating temperature is consequently extremely high: $T_r \sim 10^{12}\ \text{GeV}$ Such high $T_r$ thermally produces gravitinos in abundance, which would disrupt conventional low-scale SUSY scenarios. Reconciliation is achieved by setting the gravitino (and scalar) masses above $10^{7}$–$10^{8}$ GeV so that they decay before lightest supersymmetric particle (LSP) freeze-out, thus avoiding cosmological conflicts. This is the origin of the necessity for “split” supersymmetry in the hybrid inflation framework when $\mu$-term is present.

4. Split Supersymmetry Spectrum and Phenomenology

The high reheating temperature enforces a split in the SUSY-breaking pattern:

• Scalar partners (squarks/sleptons) and Higgs soft masses: $m_0 \sim m_G \gtrsim 5 \times 10^7$ GeV
• Gauginos: can be $\sim$TeV-scale (due to anomaly mediation or other mechanisms)
• $\mu$ term: $\mu \gtrsim m_G$, by construction

This arrangement allows a light wino-like LSP with $m_{\chi^0} \sim 2$ TeV, matching the thermal relic abundance expected for dark matter (Okada et al., 22 Jul 2025, Okada et al., 2015). Simultaneously, the gluino may be accessible at colliders, and the Higgs mass constraint is naturally satisfied for this SUSY spectrum with $\tan\beta \approx 1.7$. The model predicts the gravitino decays rapidly enough to avoid Big Bang Nucleosynthesis constraints.

5. Leptogenesis and Baryogenesis

The elevated reheating temperature ($T_r \gtrsim 10^{12}$ GeV) is favorable for implementing thermal leptogenesis. Majorana masses for right-handed neutrinos arise through $G$-invariant non-renormalizable operators (e.g., $L^c L^c \bar{\Phi}\Phi / m_P$) after symmetry breaking. Thermal leptogenesis proceeds via decay of these heavy right-handed neutrinos, with the generated lepton asymmetry converted to baryon asymmetry via electroweak sphalerons. The split SUSY spectrum does not compromise the viability of this scenario (Okada et al., 22 Jul 2025).

6. Comparison with Standard Hybrid Inflation and Empirical Status

Key Feature	Standard SUSY Hybrid Inflation	Split SUSY Hybrid ($\mu$-Term) Inflation
Reheat temperature	$10^6$–$10^9$ GeV	$\gtrsim 10^{12}$ GeV
$n_s$ (for $N\sim 50$)	$0.98$ (all soft terms omitted)	$0.97\pm0.004$ (includes $m_G$-induced linear term)
Tensor-to-scalar ratio $r$	$<10^{-11}$	$0.001$–$0.01$
Soft SUSY spectrum	All fields $\sim$m_{soft} (TeV to 100 TeV)	Scalars $\geq 10^8$ GeV, gauginos $\sim$TeV
LSP	Neutralino, various possibilities	~2 TeV wino, viable DM candidate
Viability of thermal leptogenesis	Typically more constrained	Naturally allowed by $T_r$

The split SUSY hybrid inflation model is thus distinguished by: high reheating temperature necessitating a split spectrum; predictive inflationary observables matching recent results; clear mechanisms of μ-term generation and dark matter composition; compatibility with baryogenesis; and potentially detectable gravitational wave signals ($r \gtrsim 10^{-3}$) (Okada et al., 22 Jul 2025, Okada et al., 2015).

7. Experimental Tests and Future Prospects

The key predictions of the split supersymmetry hybrid inflation model are subject to falsification or refinement by ongoing and future experiments:

• Precise CMB measurements (e.g., LiteBIRD, CMB-S4) could detect or further constrain $r$ values in the $10^{-3}$–$10^{-2}$ range.
• Searches for TeV-scale winos at colliders or indirect detection experiments are directly relevant.
• The pattern of gaugino and scalar masses, as well as possible long-lived gluinos, is a haLLMark signature.
• Proton decay, cosmological constraints from the relic LSP and consistency with baryogenesis, and other rare decays could further corroborate or constrain the split SUSY hybrid inflation scenario.

The structure, predictions, and necessity for scale separation in this model are intrinsic consequences of the interplay among the inflationary potential, reheating processes, and SUSY-breaking mediation in the presence of the special μ-term superpotential coupling. The model’s observational and theoretical status is both robust and sharply testable in the coming decade (Okada et al., 22 Jul 2025, Okada et al., 2015).