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Joint Multi-Period Fermi-LAT and LHAASO Constraints on Axion-Like Particles from Mrk 421 Using Profile Likelihood with Gaussian Copula Correlation

This paper presents a novel multi-epoch joint profile-likelihood framework incorporating Gaussian copula correlations to constrain axion-like particles using simultaneous Fermi-LAT and LHAASO observations of Mrk 421, establishing stringent 95% CL upper limits on the photon coupling $g_{a\gamma}$ across a broad mass range while demonstrating a robust pipeline for future multi-messenger searches.

Original authors: Longhua Qin, Jiancheng Wang, Chuyuan Yang, Huaizhen Li, Ao Wang, Weiwei Na, Hushan Xu, Xiaogu Zhong, Zunli Yuan, Yubin Li, Guangbo Long

Published 2026-06-05

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CC BY 4.0

Original authors: Longhua Qin, Jiancheng Wang, Chuyuan Yang, Huaizhen Li, Ao Wang, Weiwei Na, Hushan Xu, Xiaogu Zhong, Zunli Yuan, Yubin Li, Guangbo Long

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 giant, cosmic highway. On this highway, light (photons) travels from distant, powerful engines called blazars (like the one named Mrk 421) all the way to our telescopes on Earth.

For a long time, scientists thought these light beams traveled in a straight, predictable line. But this paper suggests they might be playing a game of "musical chairs" with invisible, ghost-like particles called Axion-Like Particles (ALPs).

Here is a simple breakdown of what the researchers did and found:

1. The Mystery: Light and Ghosts

ALPs are hypothetical particles that are very light and interact very weakly with normal matter. The theory is that as light travels through magnetic fields in space (like the magnetic fields around the blazar or the vast space between galaxies), it can occasionally turn into an ALP and then turn back into light.

Think of it like a dancer (the photon) who occasionally swaps partners with a ghost (the ALP) while dancing through a crowd. If the dancer swaps partners, the rhythm of the light changes. By looking at the light from Mrk 421, scientists hope to see these "dance moves" and prove the ghosts exist.

2. The Problem: A Noisy Room

Detecting these ghostly swaps is hard because the light gets dimmed and distorted by other things, like a thick fog called the Extragalactic Background Light (EBL). It's like trying to hear a whisper in a room where the air itself is muffling the sound.

Previous studies often treated the "ghost swapping" and the "fog" as two separate problems. But this paper argues that's like trying to fix a car engine while ignoring the flat tire; you need to look at them together. The researchers built a new, more sophisticated computer model that treats the fog and the ghost-swapping as happening at the exact same time.

3. The New Tool: The "Correlation Copula"

The researchers looked at data collected over several years (2021–2024) from two powerful telescopes: Fermi-LAT (which sees lower-energy light) and LHAASO (which sees extremely high-energy light).

Usually, scientists just add up the results from different days as if they were completely unrelated. But the authors realized that even on different days, the data might have tiny, subtle connections (like a shared background noise or a common instrument glitch).

To handle this, they used a statistical tool called a Gaussian Copula.

The Analogy: Imagine you are listening to five different radio stations. Even if they are different stations, they might all pick up a tiny bit of static from the same power line. If you ignore that shared static, you might think you hear a clear signal that isn't there. The Copula is like a filter that acknowledges, "Hey, these five days of data are 3% connected," so they don't accidentally trick themselves into finding a ghost that isn't real.

4. The Models: Two Scenarios

The blazar Mrk 421 is a bit of a puzzle. Sometimes its high-energy light and low-energy light behave differently. To be safe, the team tested two different "maps" of how the light is produced:

The Two-Zone Map: Imagine the blazar has a small, intense inner room and a larger, calmer outer room. The light comes from both.
The One-Zone Map: Imagine the light comes from just one big, chaotic room where everything happens at once.

They ran their "ghost-hunting" simulation through both maps to see if the results changed.

5. The Results: A Very Strict Search

After crunching the numbers, the team found no evidence that ALPs exist in the specific range they were looking for. But in science, "no evidence" is actually a very powerful result because it tells us where not to look.

They set a "speed limit" for how strongly these ALPs could interact with light:

For very light ALPs (less than 1 neV): The interaction must be incredibly weak. If it were any stronger, they would have seen the "dance moves" in the data.
For heavier ALPs (up to 500 neV): The limit is a bit looser, but still very strict.

The Bottom Line:
The researchers used a smarter way of combining data (accounting for the "fog" and the "shared noise") to say: "If these ghost particles exist, they are even more shy and harder to catch than we thought."

They didn't find the ghosts, but they successfully drew a much tighter fence around where those ghosts could be hiding, using the best long-term data we have from the Mrk 421 blazar. This sets a new, high standard for future searches using next-generation telescopes.

Technical Summary: Joint Multi-Period Fermi-LAT and LHAASO Constraints on Axion-Like Particles from Mrk 421

Problem Statement
Axion-like particles (ALPs) are well-motivated extensions of the Standard Model that can induce photon-ALP oscillations in external magnetic fields, potentially altering the spectra of high-energy gamma rays from distant sources. While previous studies have constrained ALP parameters, significant regions of the parameter space remain unconstrained. A major challenge in these searches is the treatment of systematic uncertainties, particularly those related to the Extragalactic Background Light (EBL), which attenuates gamma rays. Traditional approaches often treat EBL absorption and photon-ALP oscillations as independent, multiplicative factors, neglecting their nontrivial interplay. Furthermore, multi-epoch analyses frequently assume observational periods are statistically independent, ignoring weak residual correlations arising from shared source variability, instrumental systematics, and model assumptions. These simplifications can lead to biased constraints or artificially tightened confidence intervals. Additionally, the blazar Mrk 421 exhibits a weak correlation between its GeV (Fermi-LAT) and TeV (LHAASO) emissions, suggesting distinct physical origins that complicate the interpretation of broadband data for ALP searches.

Methodology
The authors present a comprehensive framework to constrain ALP parameters ( $m_a$ and $g_{a\gamma}$ ) using five simultaneous observational epochs of Mrk 421 from March 2021 to March 2024, combining data from Fermi-LAT and LHAASO.

Propagation Formalism: Unlike factorized approximations, the study employs a self-consistent photon-propagation framework where EBL absorption is incorporated directly into the photon-ALP mixing Hamiltonian via an absorptive term ( $i/2\lambda_\gamma$ ). The EBL intensity is treated with a normalization factor ( $K_{EBL}$ ) as a nuisance parameter with a Gaussian prior ( $\mu=1.0, \sigma=0.2$ ) to account for uncertainties.
Source Modeling: Two distinct jet emission models are tested to address the weak GeV-TeV correlation in Mrk 421:
- A two-zone hybrid model featuring a compact inner zone (flaring VHE emission) and an extended outer zone (quiescent emission).
- A single-zone hadronic model, physically plausible given the weak correlation and distinct variability amplitudes.
  Both models utilize helical and tangled magnetic field components and propagate photons through the source, intergalactic medium (IGMF), and Galactic magnetic field (GMF).
Statistical Framework: The authors develop a joint profile-likelihood analysis. To address the limitation of treating epochs as independent, they introduce a Gaussian copula formalism with a conservative correlation coefficient ( $\rho = 0.03$ ). This models weak inter-epoch correlations through transformed $\Delta\chi^2$ excess distributions. The analysis optimizes nuisance parameters (intrinsic spectral parameters and $K_{EBL}$ ) globally under the no-ALP hypothesis before profiling the ALP mass and coupling.
Data Processing: Photon survival probabilities are precomputed on a four-dimensional grid (energy, ALP parameters, EBL normalization) using the gammaALPs code and interpolated during likelihood evaluation. Intrinsic spectral models are selected per epoch using the Akaike Information Criterion corrected for finite sample size (AICc).

Key Contributions

Unified Propagation Model: The paper implements a self-consistent treatment of photon-ALP mixing and EBL absorption within the propagation equation, avoiding the biases of multiplicative factorization.
Correlated Multi-Epoch Analysis: The introduction of a Gaussian copula to handle weak correlations between observational periods provides a statistically consistent method for combining multi-epoch datasets, preventing artificial tightening of confidence intervals.
Broadband Constraints: The study leverages the unique broadband coverage of Fermi-LAT (GeV) and LHAASO (multi-TeV to PeV) to probe ALP parameters across a wide mass range (0.1–500 neV).
Robust Modeling: By testing both two-zone and single-zone hadronic scenarios, the work accounts for the complex emission physics of Mrk 421, ensuring constraints are not overly dependent on a single emission assumption.

Results
Using the joint profile-likelihood framework, the authors derive 95% confidence level (CL) upper limits on the photon-ALP coupling ( $g_{a\gamma}$ ):

Low-Mass Regime ( $m_a \lesssim 1$ neV): The constraints are highly stringent, with a lower bound of $5.0 \times 10^{-14}$ GeV $^{-1}$ and an upper limit of $8.33 \times 10^{-14}$ GeV $^{-1}$ . This region is sensitive to CAST constraints.
Intermediate to High Mass ($0.1 - 500$ neV):
- For the two-zone hybrid model, the conservative upper limit is $g_{a\gamma} < 6.50 \times 10^{-12}$ GeV $^{-1}$ .
- For the single-zone hadronic model, the conservative upper limit is $g_{a\gamma} < 7.30 \times 10^{-12}$ GeV $^{-1}$ .
Behavior: The exclusion contours exhibit a characteristic "jump-plateau" feature. A sharp transition occurs around $m_a \approx 1$ neV, where the upper limit increases by roughly an order of magnitude, followed by a gradual relaxation at higher masses. The lower bound remains stable across the entire mass range.
Robustness: Tests with alternative correlation coefficients ( $\rho = 0, 0.15, 0.3$ ) show that variations in the constraints remain within less than one order of magnitude, confirming the stability of the results.

Significance
The paper claims that these results benefit significantly from LHAASO's broadband very-high-energy coverage and long-term monitoring of Mrk 421. The developed framework offers a robust and scalable pipeline suitable for future multi-messenger ALP searches with next-generation gamma-ray observatories. By explicitly modeling weak correlations and treating EBL absorption self-consistently, the study provides more reliable and conservative constraints on the ALP parameter space, addressing systematic biases present in previous factorized approaches. The work demonstrates that the weak GeV-TeV correlation in Mrk 421 necessitates complex emission models (two-zone or hadronic) for accurate ALP constraints, and that the proposed statistical methodology effectively handles the associated uncertainties.

1. The Mystery: Light and Ghosts

2. The Problem: A Noisy Room

3. The New Tool: The "Correlation Copula"

4. The Models: Two Scenarios

5. The Results: A Very Strict Search

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