⚛️ high-energy theory

Multiparticle scalar dark matter with $\mathbb{Z}_N$ symmetry

This paper systematically investigates a $\mathbb{Z}_N$ symmetric model featuring two scalar dark matter components that undergo co-annihilation, analyzing their viability as WIMP or pFIMP candidates within the $\mathbb{Z}_3$ framework while confronting the allowed parameter space against current direct and indirect detection constraints.

Original authors: Subhaditya Bhattacharya, Lipika Kolay, Dipankar Pradhan

Published 2026-09-09

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CC BY 4.0

Original authors: Subhaditya Bhattacharya, Lipika Kolay, Dipankar Pradhan

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

The universe is filled with a substance we cannot see, touch, or smell, yet it holds galaxies together and shapes the cosmos. Astronomers call this dark matter, and while we know it makes up about a quarter of everything that exists, we have never directly detected a single particle of it. The leading idea for decades has been that this invisible matter consists of a single type of heavy particle that interacts with normal matter only through gravity and perhaps a very weak force. However, nature often surprises us with complexity, and it is entirely possible that dark matter is not a single species but a family of different particles, each with its own properties, living together in a hidden sector.

A team of researchers at the Indian Institute of Technology Guwahati has explored this possibility by constructing a theoretical model where two different types of dark matter particles coexist. They focused on a specific mathematical rule, known as a symmetry, that keeps these particles stable and prevents them from decaying into ordinary matter. In their model, the lighter of the two particles behaves like a standard dark matter candidate, interacting with the visible world in a way that allows it to be detected by sensitive underground experiments. The heavier partner, however, is much more elusive. Under certain conditions, this heavier particle becomes so long-lived that it survives for billions of years, effectively acting as a second, stable component of dark matter. The researchers found that the specific rules governing how these two particles interact with each other and with the Higgs boson—a particle that gives mass to others—determine whether both can survive to the present day or if one will eventually vanish.

The study reveals that having two types of dark matter opens up new possibilities for how the universe might have evolved. In many scenarios, the heavier particle would normally decay quickly, but the researchers showed that by carefully adjusting the strength of the forces between the particles, they could make the heavier one stable enough to remain today. This creates a mixed population of dark matter: one part that is relatively easy to find because it bumps into normal atoms, and another part that is incredibly difficult to detect because it barely interacts with anything at all. This second type, which the authors describe as having a "feeble" connection to our world, could still influence the total amount of dark matter in the universe through its interactions with the first type.

The team used powerful computer simulations to test their model against real-world data. They checked their predictions against the limits set by the most sensitive detectors on Earth, such as the XENONnT and LUX-ZEPLIN experiments, which look for dark matter particles hitting atomic nuclei. They also compared their results with observations from space telescopes that search for gamma rays produced when dark matter particles collide and annihilate. Their findings suggest that while a single type of dark matter is tightly constrained by these experiments, a two-particle system offers a much wider range of possibilities. The model allows for a scenario where the lighter particle is a standard "weakly interacting" candidate, while the heavier one is a "pseudofeebly interacting" particle that is nearly invisible. This combination allows the model to fit the observed amount of dark matter in the universe without violating the strict limits set by current detectors.

One of the most significant discoveries in this work is how the two particles can help each other survive. In the early universe, the heavier particle could have converted into the lighter one, or they could have annihilated each other in ways that a single-particle model cannot explain. These processes help regulate the final amount of dark matter left over today. The researchers identified specific conditions where the heavier particle remains stable not because of a separate protective rule, but because the energy required for it to decay is simply not available, or because the forces that would cause it to decay are turned down to almost zero. This means that a single underlying rule can protect two different particles, a finding that simplifies the theoretical landscape of dark matter.

The study also highlights the importance of the mass difference between the two particles. If the heavier particle is only slightly heavier than the lighter one, they can easily swap identities or annihilate together. However, if the mass difference is large, the heavier particle becomes a distinct, long-lived entity that behaves very differently. The researchers showed that in their model, the heavier particle can be stable even if it is significantly heavier than its partner, provided the interactions between them are tuned correctly. This flexibility allows the model to accommodate a wide range of masses, from the scale of a proton up to the scale of a heavy atom, without conflicting with current observations.

By mapping out the allowed parameters, the authors demonstrated that this two-component model can survive the rigorous tests of modern physics. They showed that the region of possibilities for such a model is larger and more diverse than previously thought, particularly when one of the particles is very hard to detect. This suggests that if dark matter is indeed a family of particles, we might be looking for it in the wrong way if we assume it is just one thing. The work provides a clear roadmap for future experiments, indicating that scientists should look for subtle signs of a second, heavier component that might be hiding in plain sight, interacting only through the faintest of whispers with the matter we can see.

Technical Summary: Multiparticle Scalar Dark Matter with $Z_N$ Symmetry

Problem Statement
The nature of Dark Matter (DM) remains one of the most significant open questions in particle physics and cosmology. While the Standard Model (SM) lacks a suitable candidate, extensions often introduce a single stable particle protected by a discrete symmetry (e.g., $Z_2$ ). However, the possibility of a multiparticle dark sector, where multiple stable components coexist, offers distinct phenomenological advantages, particularly in alleviating constraints from direct and indirect detection experiments. A primary challenge in constructing such models is achieving stability for multiple components using a single discrete symmetry ( $Z_N$ ). Typically, two stable DM components require two distinct symmetries (e.g., $Z_2 \otimes Z'_2$ ). This paper addresses the systematic conditions under which a single $Z_N$ symmetry can yield two stable scalar DM components, specifically focusing on scenarios where the heavier component is stabilized either kinematically or by suppressing specific coupling parameters to render it a long-lived particle (LLP).

Methodology
The authors extend the SM by introducing two complex scalar fields, $\Phi_1$ and $\Phi_2$ , which transform under a single $Z_N$ symmetry. The analysis proceeds through the following steps:

Symmetry and Interaction Analysis: The authors systematically classify the allowed renormalizable interaction terms in the scalar potential based on the charges ( $q_1, q_2$ ) of the scalar fields under $Z_N$ . They categorize four distinct cases based on the relationship between charges and $N$ :
- $q_1 = q_2$ and $q_1 + q_2 = N$
- $q_1 = q_2$ but $q_1 + q_2 \neq N$
- $q_1 \neq q_2$ but $q_1 + q_2 = N$
- $q_1 \neq q_2$ and $q_1 + q_2 \neq N$
Stability Criteria: For cases where the heavier dark sector (DS) particle can decay into the lighter DS particle plus SM particles (or multiple DS particles), stability is enforced by:
- Kinematic Constraints: Ensuring $m_{heavier} < n \cdot m_{lighter}$ (for decays within the dark sector) or $m_{heavier} < n \cdot m_{lighter} + n' \cdot m_{SM}$ .
- Coupling Suppression: Setting specific portal and self-interaction couplings to vanishingly small values (e.g., $\lambda \lesssim 10^{-15}$ ) to suppress tree-level, one-loop, and two-loop decay channels, thereby extending the lifetime of the heavier component beyond the age of the Universe.
Phenomenological Scenarios: The study focuses on $Z_2$ and $Z_3$ symmetries. It explores two main DM classification combinations:
- WIMP-WIMP: Both components are Weakly Interacting Massive Particles.
- WIMP-pFIMP: One component is a WIMP, and the other is a "pseudofeebly interacting massive particle" (pFIMP). The pFIMP interacts feebly with the SM but thermallyizes via interactions with the WIMP.
Numerical Analysis: The authors solve the coupled Boltzmann equations (BEQ) for the relic density using micrOMEGAs and FeynRules. They confront the resulting parameter space with:
- Direct Detection (DD): Constraints from XENONnT, LUX-ZEPLIN (LZ), and PandaX-4T on spin-independent scattering cross-sections.
- Indirect Detection (ID): Constraints from Fermi-LAT, H.E.S.S., and CTA on self-annihilation and semi-annihilation cross-sections.
- Theoretical Bounds: Unitarity, perturbativity, and vacuum stability conditions.

Key Contributions and Results

Systematic Classification of $Z_N$ Models: The paper provides a comprehensive table (Table 1) detailing allowed interaction terms for $Z_2, Z_3,$ and $Z_4$ symmetries based on field charges. It demonstrates that for $Z_2$ and $Z_3$ , achieving two stable DM components often requires suppressing specific couplings, effectively elevating the symmetry to a higher group (e.g., $Z_3 \to Z_6$ ) or mimicking a product symmetry ( $Z_3 \otimes Z'_3$ ).
$Z_3$ Symmetry Analysis: The core of the work focuses on the $Z_3$ $Z_{3}$ symmetric model with two complex scalars. The authors identify six distinct scenarios (A–F) arising from different choices of suppressed couplings.
- Scenario A: Effectively mimics a $Z_3 \otimes Z'_3$ model. In the WIMP-pFIMP limit, the pFIMP detection relies on loop-mediated interactions, significantly relaxing direct detection bounds compared to the WIMP-WIMP case.
- Scenario B: Retains specific interaction terms (e.g., $\chi_1^2 \chi_2^2$ ) that open up "semi-conversion" channels ( $\chi_2 \chi_2 \to \chi_1^* h$ ). This allows for correct relic density even with large mass splittings between the two components, a feature not present in simpler $Z_2$ models.
WIMP-pFIMP Viability: The study highlights that the WIMP-pFIMP scenario offers a larger allowed parameter space than WIMP-WIMP. The pFIMP component, being feebly coupled to the SM, evades stringent direct detection limits while still contributing to the relic density via conversion and semi-conversion processes with the WIMP.
Mass Splitting and Signatures: Unlike previous $Z_2$ models where mass splitting was constrained to be small, the $Z_3$ scenarios (particularly Scenario B) allow for mass splittings up to $\sim 100$ GeV for a $\sim 500$ GeV WIMP. This large splitting could lead to distinguishable signatures in collider searches, such as kinks or double-hump missing energy distributions.
Relic Density Mechanisms: The analysis confirms that semi-annihilation and semi-conversion processes are crucial for satisfying the observed relic density ( $\Omega_{DM}h^2 \approx 0.12$ ) without violating indirect detection bounds, particularly in regions away from the Higgs resonance.

Significance and Claims
The paper claims to systematically address the possibility of obtaining two scalar DM components under a single $Z_N$ symmetry, a topic previously lacking a unified treatment. The authors emphasize that:

Symmetry Reduction: Stabilizing the heavier component in a single $Z_N$ framework often necessitates suppressing couplings, which effectively imposes a higher symmetry (e.g., $Z_6$ or $Z_8$ ) or a product symmetry structure on the Lagrangian.
Phenomenological Distinction: The choice of which couplings to suppress leads to distinct phenomenological outcomes. Specifically, the $Z_3$ model allows for "semi-conversion" channels that are absent in $Z_2$ models, enabling viable parameter spaces with large mass splittings between DM components.
Observational Prospects: The WIMP-pFIMP limit in these models provides a viable alternative to standard WIMP scenarios, offering a parameter space that is less constrained by current direct detection experiments while remaining testable by future indirect detection and collider searches. The authors note that the distinction between single and two-component cases, as well as between WIMP-WIMP and WIMP-pFIMP scenarios, is manifest in the allowed parameter space and the nature of the relic density depletion mechanisms.

The work concludes that multiparticle DM frameworks under a single symmetry offer rich dynamical possibilities, segregating the depletion processes (governing relic density) from the direct/indirect interaction processes, thereby allowing for viable models that satisfy current observational bounds.

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