Dark Photon Model in Warped Extra Dimensions

• Dark Photon-Mediated Model is a framework where a U(1) gauge boson links the dark sector with the SM using warped extra dimensions and dilaton-controlled mass generation.
• It employs mechanisms like extra-dimensional localization, kinetic mixing, and pseudo-Dirac inelastic dark matter to naturally achieve a GeV-scale dark photon.
• The dual AdS/CFT description connects strong dynamics to observable signatures in dark matter annihilation and collider experiments, addressing hierarchy and UV-completeness.

A dark photon-mediated model is a theoretical framework in which a new U(1) gauge boson ("dark photon") acts as the force carrier between particles in a dark sector and the Standard Model (SM). These models leverage various mechanisms—extra-dimensional localization, strong dynamics, symmetry breaking, and kinetic mixing—to generate the dark photon mass, determine its couplings, and address outstanding problems in particle physics, including the origin of mass hierarchies and the nature of dark matter.

1. Warped Extra Dimensions and Geometric Naturalness

In the warped extra-dimensional realization, the dark sector is embedded into a slice of five-dimensional (5D) anti-de Sitter space (AdS₅) bounded by ultraviolet (UV) and infrared (IR) branes. The dark photon arises as the lightest Kaluza-Klein mode of a bulk U(1)′ gauge field propagating in an AdS₅ metric of the form

$ds^2 = e^{-2ky}\eta_{\mu\nu}dx^\mu dx^\nu + dy^2$

with the extra-dimensional coordinate $y$ running from the UV brane ($y=0$, scale~$M_{Pl}$) to the IR brane ($y=L$, scale~TeV). The hierarchy between the Planck and TeV scales is thus naturally generated by the exponential warp factor, with $m_{IR} \sim k e^{-kL}$.

The dark photon's mass is induced by a combination of a y-dependent dilaton profile $\phi(y)$ which enters exponentially into the gauge kinetic term,

$$S^{A'} = \int d^5x \sqrt{-g} \left[ -\frac{1}{4} e^{-2\phi(y)} F^{\prime}_{MN}F^{\prime MN} \right],$$

and the enforcement of symmetry-breaking boundary conditions (e.g., Dirichlet) for U(1)′ on the UV brane. The resulting dark photon mode profile is strongly IR-localized,

$$f_A^{(0)}(y) \propto e^{- \langle \phi(y) \rangle },$$

with analytic mass formulas for examples:

• Linear dilaton case:

$m_0 \simeq e^{(1-b)kL} m_{IR}$

for a profile $\langle \phi \rangle = a - bk y$.

• Exponential profile case:

$$m_0 \simeq \sqrt{\frac{2\beta}{kL}} e^{-\beta/2} k, \quad \beta \equiv 2v/f_\Phi^{3/2}$$

For suitable choices of parameters ($b > 1$, $\beta \sim 80\mbox{--}100$), $m_0$ lies naturally in the GeV range—even though all fundamental scales originate at the Planck scale (Gherghetta et al., 2010).

2. Dark Matter Couplings and Inelasticity

The dark matter (DM) candidate is either an IR-brane-localized fermion or a bulk fermion in the 5D warped geometry, charged under U(1)′. The SM–dark sector interaction strength is determined by nontrivial wavefunction overlaps: $$g_{rs\ell} = g_5' \int dy\, e^{-3ky} f^{(r)}(y) f^{(s)}(y) f_A^{(\ell)}(y)$$ where $f^{(r,s)}$ and $f_A^{(\ell)}$ are the mode functions for the dark fermions and the dark photon, respectively.

Adding a localized Majorana mass term for the bulk dark fermion on the UV brane splits the Dirac state into two quasi-degenerate Majorana states (pseudo-Dirac), suppressing direct diagonal couplings to the dark photon and realizing a scenario suitable for inelastic dark matter:

• Only off-diagonal (flavor-flipping) interactions mediated by the dark photon are significant,
• The mass splitting $\delta m$ is exponentially suppressed, as the Majorana term overlaps weakly with the bulk-localized dark fermion. This inelastic structure helps the model evade direct detection constraints (Gherghetta et al., 2010).

3. Mass Generation and the Role of the Dilaton

The explicit mass of the dark photon is governed by the overlap of its IR-localized wavefunction with the UV brane, where the U(1)′ breaking is imposed. Two main mechanisms are treated:

• For a linear dilaton profile:

$$m_0 \simeq e^{(1-b)kL} m_{IR}, \quad m_{IR} \sim \mathrm{TeV}$$

• For a dynamically generated (exponential) dilaton profile:

$m_0 \simeq \sqrt{\frac{2\beta}{kL}} e^{-\beta/2} k$

Both cases result in exponential suppression relative to the IR scale, achieving a GeV-scale dark photon for plausible parameter choices. The dilaton profile thus precisely controls the localization, and hence the coupling of the dark photon to each brane—a crucial handle for tuning phenomenological signatures and ensuring compatibility with experimental constraints.

4. Dual Strongly Coupled Description via AdS/CFT

The warped construction possesses a dual 4D interpretation via the AdS/CFT correspondence:

• The 5D bulk theory with warped geometry is equivalent to a strongly coupled, nearly conformal field theory in 4D, which confines at the IR (TeV) scale.
• Both the dark photon (the lightest gauge boson Kaluza-Klein resonance) and dark matter (bound states or fermionic composites) are interpreted as low-lying composites of the new strong sector.
• The appearance of hierarchies, small mass-splittings, and suppressed couplings are understood as emergent properties of strong dynamics.
• The SM Higgs and dark sector can naturally co-localize on the IR brane, solving both the electroweak hierarchy and the dark matter scale coincidence problem in a unified setup.

5. Kinetic Mixing Portal to the Standard Model

Communication between the dark sector and the SM is via kinetic mixing,

$$\mathcal{L} \supset -\frac{\epsilon}{2} F'_{\mu\nu} F^{\mu\nu}$$

with $\epsilon \ll 1$ induced by IR-brane-localized operators or higher-dimensional effects. This mixing:

• Allows the dark photon to decay predominantly into SM leptons (for $m_{A'} < 2 m_\pi$), yielding signals accessible at colliders and fixed-target experiments,
• Is naturally small (without tuning), as the overlap of the IR-localized dark photon with the UV brane (where SM resides) is exponentially suppressed.

6. Phenomenological Implications and Experimental Signatures

The warped dark photon model predicts a spectrum and couplings with distinctive experimental signatures:

• A GeV-scale, visibly decaying dark photon with highly suppressed SM-visible couplings, but substantial couplings to the (TeV-scale) dark matter,
• Dark matter annihilation dominated by dark photon-mediated channels, predominantly to leptons, addressing cosmic-ray and astrophysical anomalies,
• Pseudo-Dirac inelastic dark matter scenarios that can relax the constraints from null direct detection,
• Strong predictive power for indirect and collider signatures, highly sensitive to the dark photon's mass and kinetic mixing parameter.

The model is further testable by direct searches for GeV-scale vector bosons decaying to leptons, and by future probes of inelastic dark matter and lepton-rich final states.

7. Theoretical Cohesion: Hierarchy Problem and UV-Completeness

The construction simultaneously solves the gauge hierarchy problem: the exponential warp factor achieves Planck–TeV scale separation non-supersymmetrically, and all mass parameters (including the GeV-scale dark photon) are naturally explained with order-one input in the UV. The 5D-to-4D duality also ensures that the description is UV-complete within the effective field theory framework, robustly linking the hierarchy solution to the dynamics of both the electroweak sector and the dark sector (Gherghetta et al., 2010).

---

This model sets a geometric and dynamical standard for naturalness in dark sector constructions, offering a high degree of phenomenological flexibility while maintaining UV sensitivity only in higher-dimensional fundamental parameters. It illustrates the technical synergy between extra-dimensional model-building, AdS/CFT duality, and dark photon phenomenology.