Deep IR GeV–TeV Brane: Warped Extra Dimensions

• Deep Infra-Red GeV–TeV brane is a theoretical framework in extra-dimensional models that uses a three-brane setup to separate the Standard Model and dark matter sectors with a natural hierarchy of scales.
• Gravitational mediation via the radion and Kaluza–Klein gravitons, with couplings controlled by warping factors and brane tensions, enables distinct interactions between the sectors.
• This model predicts novel dark matter annihilation channels, a stochastic gravitational wave background from phase transitions, and offers an elegant solution to hierarchy issues.

A Deep Infra-Red GeV–TeV Brane is a theoretical construct emerging from modern extra-dimensional models, particularly generalizations of the Randall–Sundrum (RS) scenario. In this framework, the visible universe is localized on a brane at an intermediate or high scale (such as the TeV scale), while a separate brane—the Deep Infra-Red (IR) GeV–TeV brane—exists at a much lower energy scale, potentially as low as the GeV regime. The separation and warping of extra dimensions exponentially suppress the energy scales of objects living on the deep IR brane, enabling new physics sectors (notably dark matter) to naturally acquire GeV–TeV masses, with couplings and phenomenology distinct from SM fields localized on the higher-scale TeV brane. The most recent and comprehensive treatments analyze scenarios with at least three branes, gravitationally coupled but allowing the Standard Model (SM) and dark matter (DM) to reside on different branes, and highlight the resulting phenomenological, cosmological, and astrophysical consequences (Donini et al., 4 Sep 2025, Koutroulis et al., 10 Mar 2024).

1. Theoretical Structure: Three-Brane Warped Scenarios

In the generalized Randall–Sundrum (RS) setup, the metric in 5D AdS spacetime with multiple branes is:

$$ds^2 = e^{-2k|y|} \eta_{\mu\nu} dx^\mu dx^\nu - dy^2$$

with $y$ an extra-dimensional coordinate, $k$ the AdS curvature, and branes localized at $y_0$ (UV/Planck), $y_T$ (IR/TeV), and $y_1$ (Deep IR/GeV). The brane scales are defined as

$$\rho_T \equiv 1/z_T \sim \text{TeV}, \quad \rho_1 \equiv 1/z_1 \ll \rho_T$$

allowing for a Planck–TeV–GeV hierarchy without fine-tuning.

The key innovation is the placement of dark matter on the deep IR brane ($z_1$), with the Standard Model on the intermediate (TeV) IR brane ($z_T$). The radion, a scalar mode associated with brane fluctuations and inter-brane separation, is the principal mediator connecting the sectors and acquires a mass and couplings dependent on its brane localization (Koutroulis et al., 10 Mar 2024, Donini et al., 4 Sep 2025).

In the "evanescent brane limit" (where the intermediate brane tension $\sigma_{\rm IR} \propto k_2 - k_1 \to 0$ vanishes as the bulk curvatures to either side become equal, $k_1 \to k_2$), the deep IR brane decouples. However, physically viable constructions operate out of this strict limit, with $\mathcal{O}(1)$ differences between $k_1$ and $k_2$, retaining a nonzero brane tension and a viable brane hierarchy (Donini et al., 4 Sep 2025).

2. Gravitational Mediation and Kaluza–Klein Spectrum

Particle interactions between different branes are mediated exclusively by gravitational excitations: the Kaluza–Klein (KK) tower of massive gravitons and the radion(s). Their wavefunctions and couplings depend sensitively on the geometry and brane tensions.

• The couplings of KK gravitons and radions to fields localized on different branes are suppressed by the respective brane warp factors and brane scales. For a field $\psi$ on brane $z_i$ the coupling typically scales as $1/\rho_i$.
• The radion couples parametrically as $c_r(z_i) \propto 1/\rho_i$, leading to an enhanced radion–DM coupling on the deep IR brane, but a suppressed radion–SM coupling on the TeV brane (Koutroulis et al., 10 Mar 2024).
• Detailed calculations yield, for the canonical radion field $r$,

$$\mathcal{L}_{\rm int} \supset c_r(z_T) r T^{\mu\nu}_{({\rm SM})} + c_r(z_1) r T^{\mu\nu}_{({\rm DM})}$$

with $c_r(z_T) \sim \rho_1/\rho_T^2$ and $c_r(z_1) \sim 1/\rho_1$.

The massive KK graviton modes similarly exhibit couplings suppressed by the brane of localization. The detailed profiles are model-dependent but obey the same general scaling logic. Radion stabilization (e.g., via the Goldberger–Wise mechanism) ensures the radion remains light compared to typical KK scales, and is crucial for consistent phenomenology.

3. Dark Matter Phenomenology on the Deep IR Brane

Dark matter, modeled as a Dirac fermion $\chi$ residing on the deep IR brane, naturally acquires a mass $m_\chi < \rho_1$. Since couplings to the SM are doubly suppressed (by both the warping and the radion's small coupling on $z_T$), the standard WIMP annihilation paradigm is replaced by freeze-in or feebly interacting massive particle (FIMP) scenarios (Koutroulis et al., 10 Mar 2024, Donini et al., 4 Sep 2025).

• The dominant annihilation channel is $\chi\bar\chi \to rr$ (radion pair), with a p-wave suppressed cross section (proportional to $v^2$).
• The radion mass hierarchy ($m_r < m_\chi < \rho_1$) is critical to prevent DM decay and to ensure that DM annihilation can deplete the relic abundance efficiently.
• Direct detection signatures are naturally weak, as the radion–SM coupling is suppressed by $\sim m_f \rho_1/\rho_T^2 \ll 1$, where $m_f$ is the mass of SM fermions (Koutroulis et al., 10 Mar 2024).

The model thus achieves viable relic density, evades stringent LHC bounds on new mediators (particularly because KK graviton and radion masses and couplings are adjustable via brane parameters), and is largely insulated from direct and indirect detection constraints.

4. Astrophysical and Cosmological Implications

A unique feature of warped three-brane scenarios is the possibility of cosmological phase transitions on the deep IR brane (Koutroulis et al., 10 Mar 2024). If $\rho_1 \lesssim 3$ GeV, the spontaneous condensation of the radion during a first-order confinement/deconfinement phase transition generates a stochastic gravitational wave background peaking at nanoHz frequencies. This prediction is testable by Pulsar Timing Array (PTA) experiments, providing a direct probe of GeV-scale brane physics.

For $$0.15~\mathrm{GeV} \lesssim m_\chi \lesssim 2~\mathrm{GeV}$$ the model can simultaneously explain the observed DM relic abundance and accommodate a detectable PTA signal, with all relevant experimental constraints satisfied.

5. Generalizations Beyond the Evanescent Brane Limit

Earlier phenomenological analyses commonly worked in the evanescent brane limit ($k_2 \to k_1$), where the intermediate brane becomes degenerate with the bulk. However, physical restrictions require that the brane tension $\sigma_{\rm IR} \propto k_2 - k_1$ not vanish. When one departs from this idealized limit:

• The primary results (dark matter relic abundance, mediator spectra) remain robust for order unity differences in $k_1$ and $k_2$, provided the warp factor hierarchy is retained (Donini et al., 4 Sep 2025).
• Couplings of radions and KK gravitons to brane-localized fields are rescaled by a factor of roughly $(k_2/k_1)^{1/2}$, but the analytic structure of the amplitude, cross sections, and mode profiles otherwise follow the same pattern.
• The parameter space for successful dark matter genesis enlarges, with ample regions compatible with cosmological, astrophysical, and collider bounds.

A distinctive implication is that the Deep Infra-Red brane enables recovery of the correct relic dark matter abundance via purely gravitational channels, even when non-gravitational cross-brane interactions are completely absent.

6. Summary Table: Three-Brane Warped Scenarios—Key Features

Feature	Description	Geophysical/Energy Scale
Brane hierarchy	UV (Planck), IR (TeV), Deep IR (GeV–sub-GeV) branes	$\rho_T \sim$ TeV, $\rho_1 \ll \rho_T$
Dark matter localization	$\chi$ (Dirac fermion) on Deep IR brane ($m_\chi < \rho_1$)	$m_\chi$ in GeV regime
Mediator	Radion (scalar fluctuation of brane separations), KK gravitons	$m_r < m_\chi$, gravitational interactions
Annihilation/trapping	$\chi\bar\chi \to rr$ annihilation (p-wave; $\sigma v \propto v^2$)	Efficient relic depletion, faint detection
Direct detection	Suppressed, $c_r(z_T) \ll 1$ for SM–DM interactions	Below current reach for all $m_f$
Phenomenological signals	Gravitational wave background from radion phase transition	nHz (PTA)
Limit regime	Out of evanescent limit: $k_1 \neq k_2$, $\sigma_{\rm IR} \neq 0$	Rescales couplings by $(k_2/k_1)^{1/2}$

7. Broader Implications and Context

The Deep Infra-Red GeV–TeV brane concept enables new classes of DM candidates and gravitational mediators with naturally suppressed couplings to the Standard Model, providing an economical solution to both the Higgs hierarchy problem (via SM localization on the TeV brane) and the dark sector relic abundance (via the deep IR brane) (Koutroulis et al., 10 Mar 2024, Donini et al., 4 Sep 2025). This framework relaxes or eliminates severe bounds from LHC searches for new particles that afflict traditional two-brane constructions, as cross-brane couplings can be tuned or vanishing, and mediator spectra adjusted.

Cosmological signatures—especially gravitational wave backgrounds from GeV-scale phase transitions—provide rare experimental windows into the physics of deep IR branes, opening the possibility of probing these models through observables beyond the canonical WIMP, axion, or standard cosmological portals. The approach also illustrates the utility of multibrane generalizations for simultaneously addressing multiple open questions in particle physics and cosmology, without introducing artificial hierarchies or excessive fine-tuning in the extra-dimensional sector.