⚛️ high-energy theory

Cosmological Constraints on Long-Lived Particles Using Dimension-Six Effective Operators

This study investigates cosmological constraints on long-lived particles decaying into dark matter via dimension-six effective operators, analyzing how such decays affect Big Bang Nucleosynthesis and the effective number of neutrino species to delineate the allowed parameter space consistent with observational data.

Original authors: Mickael V. S. de Farias, Rodrigo Holanda, Matheus M. A. Paixao, Farinaldo S. Queiroz, Priscila V. dos Santos

Published 2026-06-30✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Mickael V. S. de Farias, Rodrigo Holanda, Matheus M. A. Paixao, Farinaldo S. Queiroz, Priscila V. dos Santos

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is a giant, cooling soup. In the very beginning, it was incredibly hot and dense, filled with particles zipping around at near-light speed. As it expanded and cooled, these particles settled down to form the stars, galaxies, and us.

This paper investigates a specific, mysterious ingredient in that cosmic soup: Long-Lived Particles (LLPs).

The Mystery of the "Slow Decayer"

In our everyday world, most unstable things decay quickly. A banana rots in days; a radioactive atom might decay in seconds. But in the subatomic world, some particles are "long-lived." They hang around for a surprisingly long time—sometimes longer than the age of the entire universe—before finally breaking apart.

The authors of this paper are asking: What happens if these long-lived particles finally decide to decay during the early history of the universe?

Specifically, they look at a scenario where a heavy, long-lived particle (let’s call it Parent X) decays into two things:

Dark Matter (a mysterious, invisible substance that makes up most of the matter in the universe).
A Photon (a particle of light).

The "Cosmic Budget" Problem

Here is the core conflict the paper addresses. Cosmologists have a puzzle called the "Hubble Tension."

If you measure how fast the universe is expanding by looking at very old light (the Cosmic Microwave Background), you get one number.
If you measure it by looking at nearby stars and galaxies, you get a different, faster number.

These two numbers don’t match. It’s like two accountants looking at the same bank statement and getting different totals. One way to fix this mismatch is to suggest that there was extra "radiation" or energy floating around in the early universe that we didn’t account for. This extra energy would make the early universe expand slightly faster, potentially reconciling the two measurements.

The authors propose that the decay of these Long-Lived Particles could provide that missing extra energy. When Parent X decays, it releases energy that acts like "dark radiation," effectively adding to the universe’s early energy budget.

The Rules of the Road (Constraints)

However, you can’t just add as much extra energy as you want. The early universe is very sensitive. If you inject too much energy at the wrong time, you break things. The paper uses three main "cosmic speed limits" to see if this idea works:

Big Bang Nucleosynthesis (BBN): In the first few minutes, the universe was cooking up the first light elements (like hydrogen, helium, and lithium). If the Long-Lived Particles decayed too late (after about 10,000 seconds), the energy they released would have scrambled this cooking process, creating the wrong amounts of helium or lithium. We know how much helium and lithium exist today, so this puts a strict limit on how long these particles can live.
Structure Formation: Dark matter is the glue that holds galaxies together. If the dark matter produced by these decays is moving too fast (like hot gas), it won’t clump together properly to form galaxies. The paper calculates how much of the total dark matter can come from this process without ruining the formation of galaxies.
The Cosmic Microwave Background (CMB): This is the afterglow of the Big Bang. If the extra radiation from the decays is too strong, it would distort the pattern of this light in ways we don’t see.

The "Dimension-Six" Shortcut

To describe how these particles decay, physicists use mathematical tools called "Effective Field Theories." Think of these as simplified rulebooks for particle interactions.

Dimension-5 operators are like simple, direct interactions. The problem? They make the particles decay too fast. To make them live long enough to fit the cosmic timeline, you’d need to assume the physics happens at energy scales higher than the Planck scale (the absolute limit of our current understanding), which is theoretically awkward.
Dimension-6 operators are slightly more complex interactions. The authors show that using these allows the particles to live long enough to be interesting, while keeping the energy scales within a more reasonable range (below $10^{15}$ GeV). This makes the theory much more plausible.

The Two Outcomes

The paper concludes with two main takeaways, depending on what you want to achieve:

If you want to solve the Hubble Tension: You need the decays to add a specific amount of extra radiation ( $\Delta N_{eff}$ between 0.1 and 0.3). This requires the dark matter produced to be very light (less than 4 keV, which is incredibly light—about 100,000 times lighter than an electron).
If you want to stay safe from observational limits: If you want to ensure you don’t break the rules of BBN or galaxy formation, you need the dark matter to be heavier. The heavier the dark matter, the less "radiation-like" it behaves, and the safer it is from these cosmic constraints.

In a Nutshell

This paper is like a cosmic detective story. The detectives (physicists) are looking for a suspect (Long-Lived Particles) that might have left a fingerprint (extra radiation) in the early universe to explain why our measurements of the universe’s expansion don’t match.

They used a specific mathematical tool (Dimension-6 operators) to model the suspect’s behavior. They found that the suspect could be responsible for the extra radiation, but only if the suspect’s byproduct (Dark Matter) is extremely light. If the byproduct is heavier, the suspect is innocent of causing the extra radiation, but still allowed to exist without breaking the laws of the early universe.

Technical Summary: Cosmological Constraints on Long-Lived Particles Using Dimension-Six Effective Operators

Problem and Motivation
The paper addresses the interplay between particle physics and cosmology, specifically focusing on Long-Lived Particles (LLPs) that decay into dark matter (DM). While colliders probe short-lived particles, cosmological observations constrain particles with lifetimes exceeding one second. The authors investigate the scenario where an LLP ( $X_1$ ) decays into a lighter dark matter particle ( $X_2$ ) and a photon ( $X_1 \to X_2 + \gamma$ ). If the LLP lifetime exceeds $10^4$ seconds, the decay products can disrupt Big Bang Nucleosynthesis (BBN) and alter the effective number of relativistic species, $N_{\text{eff}}$ . The study aims to determine the parameter space in which such decays remain consistent with cosmological bounds from BBN, structure formation, the Cosmic Microwave Background (CMB), and Baryon Acoustic Oscillations (BAO), while exploring whether this mechanism can contribute to dark radiation to alleviate the Hubble tension.

Methodology
The authors employ an Effective Field Theory (EFT) approach, utilizing dimension-six effective operators to describe the interaction between the visible and dark sectors. This choice is motivated by the need to suppress decay widths to achieve long lifetimes without requiring energy scales above the Planck mass, which would be necessary for dimension-five operators. The analysis considers two specific cases for the dark matter candidate:

Fermion Dark Matter: Described by the operator $\mathcal{L}_{\text{eff}} \propto \frac{1}{\Lambda^2} \bar{X}_1 \gamma^\mu \partial^\nu X_2 F_{\mu\nu}$ .
Vector Dark Matter: Described by the operator $\mathcal{L}_{\text{eff}} \propto \frac{1}{\Lambda^2} G_{\mu\nu} X^{\nu\alpha} F_{\mu\alpha}$ .

The methodology involves deriving the decay widths ( $\Gamma$ ) and lifetimes ( $\tau$ ) for these operators as functions of the cutoff scale $\Lambda$ and particle masses ( $m_{X_1}, m_{X_2}$ ). The authors then calculate the contribution to $\Delta N_{\text{eff}}$ arising from the relativistic kinetic energy of the produced DM particles. This contribution is linked to the fraction of DM produced via LLP decay ( $f$ ) and the mass ratio $m_{X_1}/m_{X_2}$ . Finally, these theoretical predictions are constrained against observational data.

Key Contributions and Results
The paper derives explicit relations between the cutoff scale $\Lambda$ , particle masses, and $\Delta N_{\text{eff}}$ . The key results are categorized by the nature of the dark matter and the imposed cosmological constraints:

Cosmological Bounds:
- BBN: Requires the LLP lifetime $\tau \lesssim 10^4$ s to avoid over-producing or destroying light elements (He-4, Li-7, Deuterium).
- Structure Formation: Limits the fraction of hot dark matter produced by LLP decays to $f < 0.009$ (based on DESI constraints on neutrino masses).
- CMB/BAO: Imposes $\Delta N_{\text{eff}} < 0.3$ . The authors adopt this limit to ensure consistency with sound horizon measurements, noting that while $\Delta N_{\text{eff}}$ correlates with $H_0$ , large values are ruled out by BAO data.
Fermion Dark Matter:
- For a high cutoff scale $\Lambda = 10^{16}$ GeV, DM masses must be $m_{X_2} > 1$ TeV to satisfy $\Delta N_{\text{eff}} < 0.1$ .
- For a lower cutoff scale $\Lambda = 10^8$ GeV, viable DM masses drop to the MeV range. Specifically, sub-MeV fermion DM can yield $\Delta N_{\text{eff}} = 0.1–0.3$ while satisfying BBN constraints.
Vector Dark Matter:
- For $\Lambda = 10^{16}$ GeV, DM masses must be $m_{X_2} > 10$ GeV to keep $\Delta N_{\text{eff}} < 0.1$ .
- For $\Lambda = 10^8$ GeV, the constraint tightens significantly; to achieve $\Delta N_{\text{eff}} < 0.1$ , $m_{X_2}$ must be $> 10$ keV. Conversely, if one seeks to alleviate the Hubble tension via $\Delta N_{\text{eff}} = 0.1–0.3$ , the DM mass must be $< 4$ keV.

Significance and Claims
The authors claim that using dimension-six operators offers a distinct advantage over dimension-five operators by allowing for weak-scale or sub-TeV dark matter masses without violating cosmological bounds. The paper highlights a "rich interplay" between particle physics parameters (masses, $\Lambda$ ) and cosmological observables ( $N_{\text{eff}}$ , $H_0$ ).

The study concludes with two distinct outcomes depending on the cosmological goal:

If an upper bound on $\Delta N_{\text{eff}}$ (e.g., $< 0.1$ ) is imposed to satisfy observational limits, it results in a lower bound on the dark matter mass.
If LLP decays are invoked as a source of dark radiation to alleviate the Hubble tension (requiring $\Delta N_{\text{eff}} \sim 0.1–0.3$ ), it results in an upper bound on the dark matter mass.

The authors assert that cosmological observations provide powerful probes for the production mechanism of dark matter via LLP decays, effectively constraining the viable parameter space for both fermionic and vectorial dark matter candidates.