Dark matter and dark radiation from chiral $U(1)$ gauge symmetry

This paper proposes a chiral $U(1)$ gauge dark sector model where anomaly-free conditions necessitate at least five fermions, leading to stable dark matter candidates and dark radiation that constrain kinetic mixing parameters and suggest testable scenarios for direct detection and future lepton colliders.

The Gist

Imagine the universe as a giant, bustling city. We know a lot about the "Main Street" of this city—the visible stars, planets, and people (what physicists call the Standard Model). But we also know there's a massive, invisible neighborhood called Dark Matter that holds the city together with its gravity, even though we can't see it or touch it.

This paper is like a blueprint for a new, secret neighborhood in that city, designed to explain exactly what this invisible stuff is and how it might interact with our world.

Here is the story of their discovery, broken down into simple concepts:

1. The "Secret Club" with a Strict Door Policy

The authors propose a hidden sector of the universe governed by a special rule called Chiral U(1) Gauge Symmetry. Think of this as a secret club with a very strict bouncer.

• The Rule: To keep the club's math from breaking (a problem called "anomaly cancellation"), the bouncer demands that you must have at least five members to get in. You can't have 3 or 4; it has to be 5.
• The Members: These five members are invisible particles. Some of them are heavy and slow (potential Dark Matter), while others are light and fast, zooming around like ghosts (called Dark Radiation).

2. The "Two-Ingredient" Recipe for Dark Matter

The paper suggests that Dark Matter isn't just one thing; it's a cocktail of two different types of particles:

1. The "Majorana" Guest: This is a shy particle that is its own antiparticle. It's very good at hiding from our detectors.
2. The "Dirac" Guest: This is a more "normal" particle (like an electron) that has an opposite twin. It's easier to spot but also easier to get caught.

The Problem: If the "Dirac" guest is the main ingredient, it interacts too strongly with our detectors (like the XENONnT experiment), and we would have seen it by now.
The Solution: The authors found a "Goldilocks" scenario.

• Scenario A: The "Majorana" guest is the boss (the main Dark Matter). The "Dirac" guest is just a tiny sidekick. This works because the boss is hard to catch, and the sidekick is too small to notice.
• Scenario B: The "Dirac" guest is the boss. But to avoid getting caught, the connection between the secret club and our city (called Kinetic Mixing) must be incredibly weak—about one-millionth of a percent. It's like the secret club is so far away that we can barely hear them whispering.

3. The "Heat" Problem (Dark Radiation)

Imagine the secret club and our city were once in the same room, sharing heat (thermal equilibrium). If they stayed connected too long, the secret club would have heated up the room too much, messing up the temperature of the universe (measured by something called $\Delta N_{eff}$).

• The Constraint: Recent measurements from the Cosmic Microwave Background (the "afterglow" of the Big Bang) say the secret club must have left the room early.
• The Result: Because they left early, there can only be a few "ghosts" (massless particles) left behind. If there were too many ghosts, the universe would be too hot. This forces the model to have exactly the right number of heavy Dark Matter particles and very few light ones.

4. The "Invisible Messenger" (The Dark Photon)

How do we know this secret club exists? The authors suggest a "messenger" particle called the Dark Photon.

• Think of the Dark Photon as a courier that can run between the Secret Club and Main Street.
• Usually, couriers drop off packages we can see. But this courier is special: it drops off "invisible packages" (Dark Matter) that vanish into thin air.
• The Hunt: Scientists are looking for this courier at giant particle accelerators (like the future CEPC or FCC-ee). They are looking for a specific signal: a flash of light (a photon) appearing out of nowhere, followed by a sudden loss of energy (because the courier took the invisible package away).

5. The Verdict: What Does This Mean for Us?

• If we find nothing: It might mean the connection between our world and the dark world is so weak (like that 1 in a million whisper) that we need much bigger, more sensitive microscopes to hear it.
• If we find something: We might see a "missing energy" event at a collider, or we might detect a tiny wobble in the cosmic background radiation that proves the existence of this hidden neighborhood.

In a nutshell:
The universe has a hidden neighborhood with a strict 5-person rule. It contains a mix of shy and normal invisible particles. To explain why we haven't found them yet, the connection between our world and theirs must be very weak, or the shy particles must be the leaders. Future experiments will try to catch a glimpse of the "invisible courier" that links these two worlds.

Technical Summary

1. Problem Statement

The Standard Model (SM) lacks a viable candidate for Dark Matter (DM), and direct detection experiments have increasingly constrained simple DM models. This motivates the exploration of "Dark Sectors" with limited interactions to the SM.

• The Challenge: Constructing a consistent dark sector model that simultaneously explains:
  1. The nature of Dark Matter (relic abundance and stability).
  2. Dark Radiation constraints (specifically the effective number of relativistic species, $\Delta N_{\text{eff}}$, constrained by CMB data like ACT DR6).
  3. Direct detection limits (spin-independent and spin-dependent cross-sections).
  4. Collider signatures for dark photons.
• Specific Constraint: Anomaly-free chiral $U(1)$ gauge theories require a minimum of five chiral fermions. The authors investigate how this specific particle content interacts with cosmological thermal history to satisfy current observational bounds.

2. Methodology

The authors construct a minimal Chiral $U(1)_X$ Dark Sector model and analyze its phenomenology through the following steps:

• Model Construction:

  • Gauge Symmetry: A chiral $U(1)_X$ symmetry acting on five left-handed fermions.
  • Anomaly Cancellation: The charges are fixed to satisfy $U(1)_X^3$ and $U(1)_X \times \text{grav}^2$ anomaly cancellation conditions. The minimal solution involves charges: $\{-9, -5, -1, 7, 8\}$.
  • Symmetry Breaking: A dark Higgs field $\phi_q$ acquires a Vacuum Expectation Value (VEV), breaking $U(1)_X$ and generating masses for the gauge boson ($Z'$) and fermions via Yukawa couplings.
  • Mass Generation: Depending on the Higgs charge, fermions acquire either Dirac masses, Majorana masses, or both. This leads to a spectrum of massive (DM candidates) and massless (Dark Radiation) fermions.
  • Portal: The dark sector connects to the SM via Kinetic Mixing ($\epsilon$) between the $U(1)_X$ gauge boson and the SM hypercharge boson.

• Thermal History Analysis:

  • The authors model the production of the dark sector via inverse decays ($f\bar{f} \to Z'$) and scattering.
  • They calculate the decoupling temperature ($T_1$) where the dark sector temperature ($T_{\text{DS}}$) diverges from the SM temperature ($T$).
  • They compute the contribution to $\Delta N_{\text{eff}}$ based on the number of massless fermions and the temperature ratio $T_{\text{DS}}/T$.

• Phenomenological Constraints:

  • Cosmology: Comparison with ACT DR6 limits ($\Delta N_{\text{eff}} < 0.17$).
  • Direct Detection: Calculation of Spin-Independent (SI) and Spin-Dependent (SD) scattering cross-sections with nucleons, constrained by XENONnT, PandaX-4T, and LZ.
  • Relic Density: Calculation of thermal freeze-out using micrOMEGAs to match $\Omega_{\text{DM}}h^2 \approx 0.12$.
  • Colliders: Analysis of invisible Higgs decays and $e^+e^- \to \gamma Z'$ processes at future lepton colliders (ILC, CEPC, FCC-ee).

3. Key Contributions

• Two-Component Dark Matter Scenario: The paper demonstrates that the anomaly-free requirement (5 fermions) combined with $\Delta N_{\text{eff}}$ constraints naturally leads to a two-component DM scenario consisting of a Majorana fermion and a Dirac fermion.
• Accidental Symmetry: The model possesses an accidental $Z_2 \times Z'_2$ symmetry that ensures the stability of the massive fermions, making them viable DM candidates without ad-hoc stability assumptions.
• Kinetic Mixing Regimes: The authors identify two distinct viable regimes based on the dominant DM component:
  1. Dirac-Dominated: Requires extremely small kinetic mixing ($\epsilon \sim 10^{-6}$) to evade direct detection limits while maintaining thermal freeze-out.
  2. Majorana-Dominated: Allows for larger kinetic mixing ($\epsilon \sim 10^{-3}$), as Majorana fermions evade SI constraints. However, the Dirac sub-component must be suppressed to avoid detection.
• Invisible Dark Photon Search: The paper provides specific discovery potentials for invisible dark photons at future $Z$-factories (CEPC, FCC-ee) considering the tight cosmological constraints on thermalization.

4. Results

• Dark Radiation Constraints:

  • To satisfy $\Delta N_{\text{eff}} < 0.17$, the number of massless fermions in the dark sector must be $\le 2$.
  • This forces the model to have exactly two massive DM candidates (one Dirac, one Majorana) and two massless fermions (dark radiation).
  • If the dark sector thermalizes with the SM, the decoupling temperature must be high ($T_1 \gtrsim 0.6 - 2$ GeV) to suppress the dark radiation contribution.

• Direct Detection Limits:

  • Dirac DM: If the Dirac fermion is the main component, the kinetic mixing $\epsilon$ must be $\approx 10^{-6}$. A larger $\epsilon$ leads to SI cross-sections exceeding LZ limits. A smaller $\epsilon$ prevents thermalization, failing to produce the correct relic density via freeze-out.
  • Majorana DM: If the Majorana fermion dominates, $\epsilon$ can be larger ($\sim 10^{-3}$). The Dirac component acts as a sub-component; its abundance must be low enough that its effective SI cross-section remains below LZ limits.

• Collider Prospects:

  • Invisible Higgs Decay: The branching ratio $BR(h \to Z'Z')$ is negligible ($\sim 10^{-11}$) for the parameter space of interest.
  • $e^+e^- \to \gamma Z'$: Future $Z$-factories (CEPC/FCC-ee) can probe the invisible dark photon signal.
    • For $M_{Z'} = 30$ GeV, kinetic mixing $\epsilon_{DP} \gtrsim 4 \times 10^{-4}$ is testable.
    • For $M_{Z'} = 5$ GeV, the region is largely excluded by $\Delta N_{\text{eff}}$ constraints if the sector is thermalized, making collider detection difficult unless the sector never thermalizes.

5. Significance

• Theoretical Consistency: The work provides a concrete, anomaly-free realization of a chiral dark sector that naturally links the particle content (5 fermions) to cosmological observables ($\Delta N_{\text{eff}}$).
• Resolution of Tension: It resolves the tension between direct detection limits and thermal freeze-out by proposing a specific two-component DM structure where the dominant component (Majorana) evades SI limits, while the sub-component (Dirac) is suppressed.
• Experimental Guidance: The paper offers clear targets for future experiments:
  • CMB: Future experiments (CMB-S4, DESI) can test the predicted $\Delta N_{\text{eff}}$ values.
  • Colliders: It defines the sensitivity reach of FCC-ee and CEPC for invisible dark photons, specifically highlighting the $e^+e^- \to \gamma Z'$ channel as a primary probe for $M_{Z'} \sim 10-30$ GeV.
• Model Building: It illustrates how "accidental symmetries" in minimal chiral models can naturally provide DM stability without imposing arbitrary discrete symmetries.

In summary, the paper establishes that a minimal chiral $U(1)$ dark sector, constrained by anomaly cancellation and CMB data, predicts a specific two-component dark matter scenario with distinct signatures for direct detection and future lepton colliders.