Millicharged Dark Matter Phenomenology

• Millicharged dark matter is a candidate whose constituents acquire a tiny effective charge via kinetic mixing between a massless dark photon and the SM photon.
• Theoretical models predict dark atoms that exhibit elastic and inelastic scattering, with signatures such as hyperfine transitions observable in direct detection experiments.
• Constraints from direct detection, CMB, and exotic isotope searches help define viable regions in the parameter space involving dark fine-structure constants and mass ratios.

Millicharged dark matter refers to dark matter candidates whose fundamental constituents carry a small but nonzero electric charge as a result of kinetic mixing between a massless dark photon and the Standard Model (SM) photon. In leading frameworks, these dark constituents are bound into neutral composite states by forces in a hidden sector, most often through an unbroken U(1) gauge symmetry. The defining property is that individual dark sector particles behave as effectively “millicharged” with respect to electromagnetism, thereby enabling suppressed but non-vanishing interactions with ordinary matter and photons. This scenario is notable for enabling direct detection via weak electromagnetic couplings, for offering distinctive phenomenological signatures in both elastic and inelastic scattering channels, and for providing a minimal portal to the SM without needing additional mediators.

1. Theoretical Structure: Atomic Millicharged Dark Matter

The canonical millicharged atomic dark matter model posits a hidden sector featuring two stable fermions—analogs of the electron and proton—coupled via an unbroken hidden U(1)₍d₎ gauge interaction (Cline et al., 2012). The dark photon (γ′) that mediates this interaction is strictly massless. Binding of the dark fermions into atom-like structures (referred to as “dark hydrogen”) is governed by the dark-sector fine-structure constant α′, with the characteristic Bohr radius

$a_0' \simeq \frac{1}{\alpha' m_e}$

where $m_e$ is the lighter dark fermion mass (dark electron), and $\alpha' = g^2/(4\pi)$ for hidden gauge coupling $g$. Kinetic mixing of the γ′ field with the SM hypercharge $B_{μν}$ (or photon $F_{μν}$) is introduced via

$$\mathcal{L} \supset -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{1}{4}{F'}_{\mu\nu}{F'}^{\mu\nu} + \frac{1}{2} \tilde{\epsilon} F_{\mu\nu}{F'}^{\mu\nu}$$

After diagonalizing the kinetic terms (to leading order in $\tilde{\epsilon}$), the charged dark sector fermions acquire an effective electric charge $\epsilon e$ with

$\epsilon e \equiv \tilde{\epsilon} g$

thus revealing their millicharge.

2. Electromagnetic Interactions and Detection Channels

The presence of a nonzero ε enables two principal classes of SM–dark sector interactions:

• Elastic Scattering: For non-degenerate dark constituents ($m_e \ll m_p$), the neutral dark atom’s effective charge distribution is screened for $q \ll 1/a_0'$, leading to a Fourier-space charge density

$$\tilde{\rho}_H \sim \epsilon e (a_0')^2 \frac{q^2}{2}$$

which cancels the $1/q^2$ from the photon propagator at small $q$, so the scattering becomes contact-like at low energies. The resulting elastic cross section for scattering off a proton is

$\sigma_p = 4\pi \alpha^2 \epsilon^2 \mu^2 (a_0')^4$

with $\mu$ the reduced proton–dark atom mass, and $\alpha$ the electromagnetic fine structure constant.

• Inelastic Transitions via Hyperfine Structure: When the masses of the dark electron and proton are nearly degenerate ($m_e = m_p$), the leading elastic amplitude cancels at Born level due to perfect screening. However, inelastic transitions between hyperfine levels remain accessible, with energy gap

$E_{\text{hf}} \simeq \frac{1}{6} \alpha'^4 m_p$

For representative parameters ($\alpha' \sim 0.06$, $m_p \sim 3$–$6$ GeV), this yields $E_{hf} \sim 15$ keV—of interest for explaining nuclear recoil features such as the CoGeNT excess.

The matrix element for the inelastic (hyperfine) transition is dominated by spin-orbit interactions and takes the schematic form

$$|\mathcal{M}|^2 \sim \frac{C^2 \mu_p}{p q^2} |\vec{v}\times \hat{q}|^2 [F(q)]^2$$

with $C \sim \epsilon e^2/m_p$. Direct detection experiments with sensitivity to weak electromagnetic couplings and keV-scale energy thresholds are directly sensitive to these processes.

3. Model Parameters and Allowed Parameter Space

The phenomenology is determined primarily by the dark fine-structure constant $\alpha'$, the dark electron and proton masses ($m_e$, $m_p$), and the kinetic mixing parameter $\epsilon$. The allowed parameter space is mapped using the combination

$$\beta \equiv \frac{\epsilon^2}{\alpha'^4} \left(\frac{(1+x)^4}{x^4}\right)$$

where $x = m_e/m_p$. Regions consistent with cosmological, astrophysical, and direct detection bounds require $\epsilon \lesssim 10^{-2}$ (subject to other parameters). The relic abundance is assumed to arise from an unspecified mechanism (dark-sector asymmetry), as thermal freeze-out cannot produce the correct abundance in this scenario. Constraints include:

• Direct detection upper limits on $\epsilon$ over wide $m_{DM}$ ranges (e.g., from Xenon100).
• CMB bounds on DM–photon interactions.
• Bounds from exotic isotope searches and neutron star survival can be respected for viable choices of $\epsilon$, $\alpha'$, and dark-sector composition.

The effective cross section for detection is reduced by a $(Z/A)^2$ factor, reflecting that only protons couple electromagnetically.

4. Elastic–Inelastic Dichotomy: Degenerate and Non-Degenerate Regimes

The two limiting regimes yield distinct experimental signatures:

Regime	Dominant Process	Cross Section / Feature	Significance
$m_e \ll m_p$ (asymmetric)	Elastic scattering	$\sigma_p = 4\pi \alpha^2 \epsilon^2 \mu^2 (a_0')^4$	Accessible to direct detection, wide mass range
$m_e = m_p$ (degenerate)	Inelastic (hyperfine)	$E_{\text{hf}} \simeq (1/6) \alpha'^4 m_p$, $\epsilon \sim 10^{-2}$ for CoGeNT	Potential explanation for CoGeNT excess

The general, nondegenerate case is similar to ordinary (neutral) hydrogen, with elastic nuclear scattering dominating at direct detection experiments. In the degenerate case, inelastic upscattering to hyperfine excited states provides the leading signal, especially if standard elastic channels are suppressed.

5. Cosmological and Astrophysical Constraints

The scenario is subject to multiple external constraints, but their interplay allows substantial viable parameter space:

• Direct detection: Xenon100 and similar experiments limit $\epsilon$ to below $10^{-2}$ (context-dependent), as signals approach current sensitivity over broad dark atom mass intervals.
• CMB and large-scale structure: Bounds on long-range DM–photon interactions constrain parameter space, but can be avoided by choosing dark ionization fractions and masses appropriately.
• Exotic isotope searches: Potential constraints from dark matter capture in baryonic nuclei are respected for small enough $\epsilon$ and for scenarios where the dark sector ionization fraction is negligible.

The minimal nature of the model (no neutral massive Z′ mediator) limits the number of free parameters and enhances predictivity.

6. Implications for Direct Detection and Phenomenology

This framework is motivated by its capacity for:

• Generating direct-detection signals through suppressed but non-vanishing electromagnetic couplings.
• Providing minimal, calculable portal interactions—only the kinetic mixing of a massless dark photon with the SM gauge boson.
• Realizing elastic and inelastic scattering channels at rates near current experimental sensitivity, with predicted recoil spectra and energy scales dictated by hyperfine structure or elastic form factors.

Elastic cross sections can be within an order of magnitude of current experimental bounds over a large swath of the $m_{DM}$–$\epsilon$ plane, while inelastic channels can explain hints such as the CoGeNT recoil excess under suitable degeneracy conditions.

7. Synthesis and Model Minimality

Atomic millicharged dark matter models, as developed in (Cline et al., 2012), illustrate a minimal but phenomenologically rich scenario. Their distinctiveness lies in the following aspects:

• A massless U(1) gauge interaction binds dark fermions, with composite “atoms” structurally similar to hydrogen.
• Kinetic mixing alone generates SM couplings, with no need for heavy mediators.
• Both elastic and inelastic detection channels are operative, contingent on the mass hierarchy of dark constituents.
• Robust regions of parameter space simultaneously explain DM abundance (presumed asymmetric), yield detectable signatures, and obey cosmological and astrophysical constraints.

The theoretical cleanliness and testability of these models make them critical benchmarks for the broader search for dark matter via suppressed electromagnetic portals.