⚛️ phenomenology

Constraints on dark axion portal: missing energy and fermion EDMs

This paper investigates the dark axion portal model by analyzing missing energy signatures in electron fixed-target experiments like NA64e and LDMX, deriving new constraints on dark photon and ALP production mechanisms, and establishing limits on CP-violating couplings using current fermion electric dipole moment measurements.

Original authors: Sergei N. Gninenko, N. V. Krasnikov, Valery E. Lyubovitskij, Sergey Kuleshov, Alexey S. Zhevlakov, I. V. Voronchikhin, D. V. Kirpichnikov

Published 2026-02-13

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Sergei N. Gninenko, N. V. Krasnikov, Valery E. Lyubovitskij, Sergey Kuleshov, Alexey S. Zhevlakov, I. V. Voronchikhin, D. V. Kirpichnikov

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a massive, bustling city. We know most of the buildings and streets because we can see them with our eyes (this is the Standard Model of physics, the particles we know like electrons and photons). But scientists suspect there's a huge, invisible "Dark Sector" neighborhood right next door, filled with mysterious residents we've never seen.

This paper is like a detective story trying to figure out how to catch a glimpse of these invisible neighbors. The authors are investigating a specific "secret tunnel" connecting our visible city to the dark one, called the Dark Axion Portal.

Here is the breakdown of their investigation in simple terms:

1. The Secret Tunnel (The Dark Axion Portal)

Imagine our visible world has a main highway (the Photon, which is light). The Dark Sector has its own highway (the Dark Photon). Usually, these two roads never touch.

The "Dark Axion Portal" is a magical, invisible bridge (mediated by a particle called an Axion) that connects the two highways.

The Scenario: A beam of high-speed electrons (our "cars") smashes into a target (a wall of lead or aluminum).
The Trick: Instead of just bouncing off, some of these electrons use the magical bridge to spawn a pair of invisible particles: an Axion and a Dark Photon.
The Escape: The Dark Photon immediately decays into even darker, invisible particles (Dark Matter) that slip right through our detectors. The Axion also flies away unseen.

2. The "Missing Energy" Clue

How do we know this happened if we can't see the particles?
Think of it like a bank robbery where the thieves don't leave a trace, but the vault is suddenly lighter.

In the experiment, scientists shoot a beam of electrons at a target and measure exactly how much energy comes out the other side.
If the math says 100 units of energy went in, but only 40 units come out, and the missing 60 units aren't accounted for by any known particle, that's a Missing Energy signature.
The paper calculates exactly how often this "robbery" should happen if the Dark Axion Portal exists.

3. Two Ways to Break In

The paper explores two different ways these invisible particles could be created in the lab:

Method A: The "Bremsstrahlung" (Braking) Trick
Imagine a car slamming on its brakes. Usually, it just stops. But in this quantum world, when an electron slams into a nucleus, it "brakes" so hard that it emits a burst of energy that instantly splits into our invisible Axion-Dark Photon pair. This is the most common way they expect this to happen.
Method B: The "Messenger" Trick (Vector Mesons)
Sometimes, the collision creates a heavy, short-lived particle called a Vector Meson (like a ρ, ω, or J/ψ). Think of this meson as a courier. Usually, the courier delivers a package and vanishes. But in this scenario, the courier secretly swaps its package for the invisible Axion-Dark Photon pair before disappearing.
- The Surprise: The authors found that for the NA64e experiment (a current setup at CERN), this "Messenger" trick is actually a huge deal. It boosts the chance of finding these particles by thousands of times compared to the braking method alone. For the LDMX experiment (a future, more powerful setup), it helps a bit, but not as dramatically.

4. The "Spooky" Magnet Test (Electric Dipole Moments)

The paper also looks at a different kind of clue: Electric Dipole Moments (EDMs).

The Analogy: Imagine an electron is a tiny bar magnet. In our normal world, it's perfectly balanced. But if the Dark Axion Portal exists, it might give the electron a tiny "tilt," making it act like a slightly crooked magnet.
The authors calculated that if the portal exists, it would cause these tiny tilts in electrons, muons, and neutrons.
The Result: By looking at the most precise measurements of these "tilts" we have today, they can rule out certain versions of the Dark Axion Portal. It's like saying, "If the tunnel existed, the magnet would be crooked this much. Since our magnet is straight, that specific version of the tunnel doesn't exist."

5. The Verdict

The authors ran the numbers for two major experiments:

NA64e (CERN): They used data from 937 billion electrons hitting a target. They found that if the Dark Axion Portal exists with certain strengths, they should have seen it by now. They ruled out a large chunk of possibilities.
LDMX (Future Fermilab): They predicted that this future experiment will be incredibly sensitive. If the portal exists, LDMX will likely find it, especially if we count the "Messenger" trick (Vector Mesons).

In Summary:
This paper is a roadmap for hunting invisible dark matter. It tells us:

Where to look: In the "missing energy" of electron beams.
How to look: By watching for invisible particles escaping through a specific "Axion bridge."
What to expect: The "Messenger" trick (Vector Mesons) is a game-changer for current experiments, making them much more powerful than we thought.
The Constraints: We are getting closer to the truth. If the "tilted magnet" (EDM) measurements stay perfect, the Dark Axion Portal might be smaller or more hidden than we hoped. But if we see missing energy, we might just have found the door to the Dark Sector.

1. Problem Statement

The paper investigates the phenomenology of the Dark Axion Portal, a theoretical framework introducing a new interaction between the Standard Model (SM) photon, a dark photon ( $\gamma_D$ ), and an Axion-Like Particle (ALP, denoted as $a$ ). The primary goals are:

To explore the production and detection of invisible dark sector states in electron fixed-target experiments.
To derive constraints on the portal's effective coupling constant ( $g_{a\gamma\gamma_D}$ ) using missing energy signatures.
To establish limits on CP-violating couplings by analyzing their contributions to the Electric Dipole Moments (EDMs) of SM fermions (electron, muon, and neutron).

The specific scenario assumes the dark photon acts as a mediator between the SM and a hidden dark sector, decaying predominantly into stable, invisible dark fermions ( $\chi$ ), i.e., $\gamma_D \to \bar{\chi}\chi$ .

2. Methodology

A. Theoretical Framework

The authors utilize an effective Lagrangian describing the non-renormalizable dark axion portal interaction:
$\mathcal{L}_{\text{portal}} \supset \frac{g_{a\gamma\gamma_D}}{2} a F_{\mu\nu} \tilde{F}'^{\mu\nu}$
where $F_{\mu\nu}$ is the SM photon field strength and $\tilde{F}'^{\mu\nu}$ is the dual field strength of the dark photon.
The model also includes:

A dark fermion $\chi$ coupled to the dark photon via a hidden $U_D(1)$ gauge coupling $g_D$ .
CP-violating Yukawa-like couplings between the ALP and SM fermions ( $g_f^a a \bar{f}f$ ).
CP-conserving kinetic mixing between the dark photon and SM fermions ( $e\epsilon_f A'_\mu \bar{f}\gamma^\mu f$ ).

B. Production Mechanisms in Fixed-Target Experiments

The study focuses on two distinct production mechanisms for the invisible $a\gamma_D$ pair in electron-nucleus scattering ( $eN \to eN a \gamma_D$ ):

Bremsstrahlung-like Emission: An off-shell photon emitted by the electron converts directly into an ALP and a dark photon ( $\gamma^* \to a\gamma_D$ ).
Exclusive Vector Meson Photoproduction: An off-shell photon converts into a vector meson ( $V = \rho, \omega, \phi, J/\psi$ ) via the reaction $\gamma^* N \to N V$ . The meson then decays invisibly via mixing with the off-shell photon: $V \to \gamma^* \to a\gamma_D$ . The Vector Meson Dominance (VMD) model is employed to calculate these decay widths.

C. Experimental Benchmarks

The analysis targets two specific electron fixed-target experiments:

NA64e (CERN): Uses a 100 GeV electron beam on a lead-scintillator target. The study utilizes current data ( $E_{OT} \approx 9.37 \times 10^{11}$ electrons on target) and projects future sensitivities ( $E_{OT} \approx 5 \times 10^{12}$ and $10^{13}$ ).
LDMX (Fermilab): A proposed experiment using an 8 GeV electron beam on a thin aluminum target, aiming for $E_{OT} \approx 10^{16}$ .

The signature for both is missing energy/momentum: a single scattered electron is detected with energy significantly lower than the beam energy ( $E_{rec}^e \lesssim 0.5 E_e$ for NA64e, $\lesssim 0.3 E_e$ for LDMX), while the $a\gamma_D$ system escapes undetected.

D. Computational Tools

CalcHEP: Used to compute the $2 \to 4$ bremsstrahlung cross-sections ( $eN \to eN a \gamma_D$ ) with custom implementations of the ALP and dark photon.
VEGAS Algorithm: Used for multidimensional phase-space integration to ensure high numerical precision.
FeynCalc/Mathematica: Used to calculate the decay widths for vector mesons ( $V \to a\gamma_D$ ) and the loop-induced contributions to fermion EDMs.

3. Key Contributions

Inclusion of Vector Meson Decays: Unlike previous studies focusing solely on bremsstrahlung, this paper rigorously quantifies the contribution of exclusive vector meson photoproduction ( $V \to a\gamma_D$ ) to the missing energy signal.
EDM Constraints on CP-Odd Couplings: The authors derive novel constraints on the product of CP-odd and CP-even couplings ( $g_f^a \cdot g_{a\gamma\gamma_D} \cdot \epsilon_f$ ) by calculating one-loop Bar-Zee diagrams. This links the dark axion portal directly to precision low-energy observables (EDMs).
Updated NA64e Sensitivity: The paper derives new exclusion limits based on the latest NA64e dataset and provides projected sensitivity curves for future runs, explicitly accounting for the vector meson contribution.
Comparative Analysis: A detailed comparison of the discovery potential between NA64e and LDMX across the sub-GeV mass range ( $1 \text{ MeV} \lesssim m_{\gamma_D} \lesssim 1 \text{ GeV}$ ).

4. Results

A. Missing Energy Sensitivity

Vector Meson Impact: The contribution of vector meson decays is highly dependent on the experiment and mass range.
- For LDMX, vector mesons provide an $\mathcal{O}(1)$ enhancement to sensitivity in the sub-GeV region.
- For NA64e, vector meson decays (particularly $J/\psi$ and $\phi$ ) boost sensitivity by several orders of magnitude for dark photon masses $m_{\gamma_D} \lesssim 1 \text{ GeV}$ .
Exclusion Limits (NA64e):
- Current data ( $9.37 \times 10^{11}$ EOT) excludes couplings in the range $8 \times 10^{-3} \text{ GeV}^{-1} \lesssim g_{a\gamma\gamma_D} \lesssim 1.5 \times 10^{-2} \text{ GeV}^{-1}$ for $m_{\gamma_D} \lesssim 1 \text{ GeV}$ .
- Projected sensitivity for $5 \times 10^{12}$ EOT reaches $g_{a\gamma\gamma_D} \lesssim (3.5 - 5) \times 10^{-3} \text{ GeV}^{-1}$ .
LDMX Potential: With $10^{16}$ EOT, LDMX is projected to probe couplings as low as $g_{a\gamma\gamma_D} \sim 10^{-4} \text{ GeV}^{-1}$ for light dark photons ( $m_{\gamma_D} \sim \text{MeV}$ ).

B. Astrophysical and Collider Constraints

Supernova SN1987A: The paper confirms that SN1987A cooling constraints exclude a significant portion of the parameter space for $500 \text{ MeV} \lesssim m_{\gamma_D} \lesssim 1 \text{ GeV}$ .
BaBar and ATLAS: Existing BaBar limits on invisible $\Upsilon$ decays and monophoton signatures, as well as projected ATLAS muon missing momentum limits, are shown to be less sensitive than the projected NA64e and LDMX results in the sub-GeV regime.

C. EDM Constraints

The study establishes upper bounds on the coupling combination $|g_f^a g_{a\gamma\gamma_D} e\epsilon_f|$ based on current EDM limits:

Electron EDM: The most stringent constraint, limiting the combination to $\lesssim 10^{-15} \text{ GeV}^{-1}$ .
Muon EDM: Limits are at the level of $\lesssim 10^{-5} \text{ GeV}^{-1}$ .
Neutron EDM: Limits are at the level of $\lesssim 10^{-12} \text{ GeV}^{-1}$ .
These constraints are logarithmically dependent on the dark photon mass and the UV cutoff scale ( $\Lambda$ ).

5. Significance

This work significantly advances the search for dark sector physics by:

Broadening the Search Strategy: Demonstrating that vector meson photoproduction is a critical, often dominant, channel for detecting dark axion portal signatures in fixed-target experiments, particularly for NA64e.
Connecting High and Low Energy Physics: Bridging the gap between high-energy missing energy searches and precision low-energy measurements (EDMs), showing that EDMs provide powerful, independent constraints on the CP-violating structure of the dark axion portal.
Guiding Future Experiments: Providing concrete sensitivity projections that justify the design and operation of LDMX and future NA64e runs, highlighting their potential to probe unexplored regions of the dark photon-ALP parameter space.

The paper concludes that the dark axion portal remains a viable candidate for explaining dark matter, and the proposed experimental strategies offer a robust path toward discovery or stringent exclusion.