Gravitational Positivity Bounds on Higgs-Portal Light Dark Matter

Gravitational positivity bounds are constraints on a renormalizable theory in the presence of a massless graviton, under the assumption that the gravitational theory is ultraviolet-completed by a perturbative string theory. We derive these bounds for the Higgs-portal scalar dark matter model using the forward scattering process $ϕϕ\to ϕϕ$. We find that, in the absence of a dark matter self-coupling, new physics beyond the Higgs-portal dark matter interaction must appear below an energy scale of $10^{10}$ GeV if the dark matter mass is smaller than the Higgs boson mass. The presence of a dark matter self-coupling alters this situation. A hierarchy between the dark matter four-point self-coupling $λ_ϕ$ and a tiny Higgs-portal coupling $λ_{hϕ}$ is required to raise the energy scale at which the new physics appears. If $λ_ϕ/λ_{hϕ} = 10^{12}$, the dark matter model can remain valid up to the grand unified theory (GUT) scale or the typical string scale. In this case, the relic abundance of dark matter in the Universe can be reproduced via the dark freeze-out scenario. A parameter set with $λ_ϕ \sim O(1)$, $λ_{hϕ} \sim 10^{-12}$, and a sub-GeV dark matter mass can accommodate the GUT-scale $Λ$ within the Higgs-portal light dark matter framework.

💡 Research Summary

The paper investigates the constraints imposed by gravitational positivity bounds on the minimal Higgs‑portal scalar dark‑matter model. Assuming that gravity is UV‑completed by a weakly‑coupled perturbative string theory, the authors apply the “gravitational positivity bound” to the forward elastic scattering process φ φ → φ φ, where φ is a real Z₂‑odd scalar singlet that interacts with the Standard Model through the renormalizable Higgs‑portal term λ_{hφ} H†H φ² and possibly a self‑interaction λ_φ φ⁴.

The analysis begins with a twice‑subtracted dispersion relation for the amplitude M(s,t) in the complex s‑plane, taking the forward limit t → 0⁻ while avoiding the t‑channel graviton pole. The amplitude is decomposed into a non‑gravitational part (loop contributions from the Higgs portal and possible dimension‑8 operators) and a gravitational part arising from the massless graviton exchange. The key inequality derived is

 B_non‑grav(Λ) > |B_grav|,

where B_grav is negative in most cases, reflecting the s² dependence of the graviton exchange, and B_non‑grav contains the positive contributions from φ loops and higher‑dimensional operators suppressed by the cutoff Λ.

Using FeynRules, FeynArts, FeynCalc, and Package‑X, the authors compute the one‑loop φ φ → φ φ amplitude without any heavy‑state (EFT) operators. The resulting B_non‑grav scales as 1/Λ⁴, with additional suppression factors for diagrams involving the Higgs VEV. When the self‑coupling λ_φ is set to zero, the positivity bound is violated already at Λ ≈ 10¹⁰ GeV for sub‑Higgs‑mass dark matter (m_φ < m_h). This implies that new physics—additional particles or operators—must appear below that scale to restore the bound.

Introducing a non‑zero λ_φ changes the picture dramatically. The φ⁴ interaction contributes positively to B_non‑grav, and a hierarchy λ_φ / λ_{hφ} ≈ 10¹² is sufficient to push the bound up to the grand‑unified (∼10¹⁶ GeV) or typical string scale (∼10¹⁸ GeV). In this regime the model can remain valid up to very high energies without extra states. The authors point out that such a hierarchy is compatible with the observed dark‑matter relic abundance if the dark freeze‑out mechanism operates: with λ_φ ∼ O(1), λ_{hφ} ∼ 10⁻¹², and m_φ ≲ 1 GeV, the dark sector stays in thermal equilibrium via its strong self‑interaction, then decouples when the portal coupling becomes ineffective, reproducing the correct Ω_DM.

The paper also discusses the role of Regge (higher‑spin) states that appear above the quantum‑gravity scale Λ_QG. Their tree‑level contributions cancel the divergent t‑channel graviton term, leaving at most a small O(α′/M_Pl²) remnant, justifying the neglect of explicit string‑loop effects in the positivity analysis.

Overall, the work demonstrates that gravitational positivity bounds provide a powerful UV‑consistency test for Higgs‑portal dark‑matter models. They not only reproduce the familiar requirement that dimension‑8 operators be positive but also reveal that a sizable self‑coupling can rescue the model up to GUT or string scales, provided the portal coupling is extremely feeble. This insight bridges phenomenological dark‑matter model building with fundamental constraints from quantum gravity and suggests concrete parameter regions (λ_φ ≈ 1, λ_{hφ} ≈ 10⁻¹², sub‑GeV m_φ) that are both cosmologically viable and UV‑consistent. Future work could explore explicit string constructions of the Regge tower, incorporate higher‑order gravitational corrections, and confront the allowed parameter space with upcoming direct‑detection and collider searches.