Comparable Dark Matter and Baryon energy densities from Dark Grand Unification

We investigate a theory of $SU(9)$ dark grand unification, where dark matter consists of asymmetric dark baryons from the $Sp(4)D$ dark QCD sector. By unifying the dark color gauge group with the Standard Model gauge group, the asymmetry generation in both sectors originates from a common process that preserves a $U(1){D-(B-L)}$ symmetry, resulting in comparable number densities. Furthermore, thanks to dark grand unification, the $Sp(4)_D$ dark QCD sector shares a similar matter content with the QCD sector, leading to comparable running of the gauge couplings from the scale as high as $10^{15}$ GeV. This predicts a dark color confinement scale and thus dark baryon masses around the GeV scale, comparable to visible baryon masses. Together with the similar number densities, the model provides an explanation for the observed similarity between the energy densities of dark matter and baryons, $ρ_D \approx 5,ρ_B$. The model also features some novel phenomenology, including a GeV-scale flavored dark QCD sector with diquark dark baryons and light dark mesons. The interaction between the dark sector and the visible sector occurs via a new $Z’$ boson with a mass of $\mathcal{O}(10)$ TeV, which could be searched for at future hadron colliders. We also briefly discuss an $SU(8)$ dark grand unified theory featuring an $SU(3)_D$ dark QCD sector.

💡 Research Summary

The paper tackles the long‑standing “dark‑baryon coincidence” – the observed similarity of the dark‑matter and baryon energy densities, ρ_D ≈ 5 ρ_B – by constructing a unified framework in which both sectors share a common origin. The authors propose an SU(9) dark grand unified theory (GUT) that embeds the Standard Model (SM) gauge group together with a dark color group Sp(4)_D. The choice of Sp(4) is motivated by its quadratic Casimir C₂=3, identical to that of QCD’s SU(3)_C, ensuring that the one‑loop beta‑function coefficients for the two strong interactions are nearly the same. Consequently, after symmetry breaking the gauge couplings of SU(3)_C and Sp(4)_D run almost in parallel from a unification scale as high as 10¹⁵ GeV, leading to confinement scales Λ_QCD and Λ_D that are both of order a few GeV. This automatically yields dark baryon masses comparable to ordinary nucleon masses.

A crucial ingredient is a global U(1){D‑(B‑L)} symmetry that remains unbroken throughout the early‑Universe dynamics. The authors assume that a high‑scale process (e.g., decay of heavy X‑bosons) generates a B‑L asymmetry which, because of the conserved U(1){D‑(B‑L)}, is transferred equally to the dark sector. Thus the number densities of dark baryons (n_D) and visible baryons (n_B) are equal, while their masses are similar, reproducing the observed ratio ρ_D/ρ_B ≈ 5 without fine‑tuning.

The model’s field content is built by extending the familiar SU(5) GUT representations into SU(9). Each SM generation is embedded in a 36‑dimensional antisymmetric tensor plus a 9‑dimensional fundamental. To cancel SU(9) gauge anomalies, four additional 9̅ representations are introduced. The fermions decompose under Sp(4)_D × SU(3)_C × SU(2)_W × U(1)_Y into several multiplets: χ_c (Sp(4) quartet, color triplet), χ_w (Sp(4) quartet, weak doublet), χ_f (Sp(4) singlet), and their partners ψ_c, ψ_w, ψ_f. The χ_c–ψ_c and χ_w–ψ_w pairs acquire Dirac masses after symmetry breaking via Yukawa couplings to scalar fields in the 36·36̅ and 9·9̅ sectors. An extra gauged U(1)5 (identified with U(1){D‑(B‑L)}) provides a massive Z′ boson with a mass of order 10 TeV, mediating interactions between the two sectors.

Anomalies specific to Sp(4) are addressed by gauging an auxiliary Sp(4)′ subgroup within a global SU(4)′ symmetry and identifying the diagonal Sp(4)_D as the dark color gauge group. The Witten non‑perturbative anomaly is cancelled by adding an even number of fundamental Weyl fermions (χ_cf). The resulting spectrum is free of gauge and mixed anomalies.

Running of the gauge couplings is computed at one loop, showing that both strong couplings meet at the GUT scale and evolve to confinement at Λ ≈ 1–3 GeV. A two‑loop analysis in the appendix confirms the stability of this result. The dark sector therefore contains GeV‑scale dark mesons (π_D, ρ_D) and dark baryons made of two Sp(4) “quarks” (di‑quark dark baryons). The lightest dark baryon is stable due to the conserved U(1)_D, while the symmetric component of dark mesons annihilates efficiently into SM particles via the Z′ portal.

Cosmologically, the model predicts that after the freeze‑out of the Z′ mediated interactions, the dark sector decouples from the SM plasma at temperatures around a few hundred MeV, preserving the equal asymmetries. The symmetric relic of dark mesons is depleted through annihilation into SM fermions, leaving only the asymmetric dark baryon abundance, which matches the observed dark‑matter density.

Phenomenologically, several signatures are highlighted:

• Z′ searches: A 10 TeV Z′ can be probed at future 100 TeV hadron colliders through dilepton or dijet resonances. Current LHC limits (≈5 TeV) already constrain the coupling strength.
• Dark meson production: Light dark mesons may be produced in rare meson decays (e.g., B → K π_D) or in fixed‑target experiments, offering opportunities for intensity‑frontier searches.
• Direct detection: Elastic scattering of dark baryons off nuclei proceeds via Z′ exchange, yielding spin‑independent cross sections near the current XENONnT/LZ bounds for reasonable gauge couplings.
• Self‑interactions: The GeV‑scale dark baryons have sizable self‑scattering cross sections (σ/m ∼ 0.1–1 cm²/g), potentially alleviating small‑scale structure anomalies.
• Flavor constraints: Loops involving χ_w and ψ_w contribute to flavor‑changing neutral currents; the model respects existing K–\bar K and B–\bar B mixing limits for the chosen parameter space.

The authors also discuss an alternative SU(8) GUT with an SU(3)_D dark color group. They show that SU(3)_D suffers from more severe anomaly cancellation issues and requires additional matter to align the running, making the Sp(4) construction more economical.

In conclusion, the SU(9) dark GUT with Sp(4)_D provides a coherent explanation for the dark‑baryon coincidence: a unified gauge origin ensures identical initial couplings, a shared global symmetry guarantees equal asymmetries, and similar matter content yields comparable confinement scales. The model is theoretically consistent, predicts a rich phenomenology, and offers clear targets for upcoming collider, direct‑detection, and intensity‑frontier experiments.