Dark Matter in Multi-Singlet Extensions of the Standard Model

This paper investigates how extending the Standard Model with multiple real singlets and varying $\mathcal{Z}_2$ symmetry structures can alleviate the stringent mass constraints on Dark Matter candidates found in single-singlet models, potentially opening up new detectable mass windows for future High-Luminosity LHC searches.

The Gist

The Big Picture: The Invisible Roommate

Imagine the Standard Model of physics as a bustling, well-lit city where we know every building and every person. But we know there is a massive, invisible "Dark Matter" population living in a hidden, dark neighborhood next door. We can't see them, but we know they are there because their gravity holds the city together.

The problem is: How do we find them?

This paper explores a specific theory: What if the "Dark Neighborhood" is just a few extra invisible rooms (called Singlets) added to our city's blueprint? These rooms are connected to the visible city only by a single, narrow hallway called the Higgs Portal. The only way to detect the invisible residents is to see them bumping into the "Higgs" building (the Higgs boson) in that hallway.

The Problem with Just One Room

The authors first looked at the simplest version: adding just one invisible room.

• The Catch: If you try to fit a dark matter particle in this single room, the rules of the universe (specifically, how much dark matter exists and how hard it hits detectors) are very strict.
• The Result: Unless the particle is incredibly heavy (heavier than 3,500 times the mass of a proton) or it has a very specific "resonant" weight (exactly half the weight of the Higgs particle), it gets kicked out of the game.
• The Analogy: It's like trying to park a car in a garage with a very narrow door. If the car is too big or the wrong shape, it won't fit. The only cars that fit are either giant trucks (too heavy for us to see at our current collider) or tiny, perfectly shaped toy cars that only fit if they bounce off the door at a specific angle.

The Two-Room Solution: A New Parking Spot

The authors then asked: "What if we add a second invisible room?"
They explored two ways to build this:

1. Two Independent Rooms: Each room has its own private lock (a different symmetry).
2. One Shared Room: Both rooms share the same lock.

The Discovery (Two Independent Rooms):
When they added a second room with its own lock, a magical new possibility opened up. They found a scenario where:

• Room A contains a light dark matter particle (just a bit heavier than the Higgs particle).
• Room B contains a heavy dark matter particle.

How it works:
Think of the total amount of dark matter in the universe as a fixed amount of water in a bucket.

• In the single-room model, the heavy particle had to hold all the water. This made it very easy to detect (and rule out) because it was so heavy and hit detectors too hard.
• In the two-room model, the heavy particle still holds almost all the water, but the light particle gets a tiny, tiny drop.
• The Magic: Because the light particle only has a "drop" of dark matter to its name, it can be much lighter and interact more strongly without breaking the rules of the detectors. It's like a spy who is so small and quiet that the security guards (Direct Detection experiments) don't notice them, even though they are right there.

This creates a "New Mass Window" where light dark matter particles (around 125–230 GeV) could exist, which was impossible in the single-room model.

The Shared Lock Scenario:
If the two rooms share the same lock, the authors found that the lightest particle can exist anywhere from the Higgs mass up to the TeV scale. The "locks" (symmetries) mix the particles in a way that allows the lightest one to hide its strength from detectors while still contributing to the total dark matter count.

The Three-Room Extension

The authors also looked at adding three rooms.

• Two Light, One Heavy: This behaves like the two-room model (the heavy one does the heavy lifting).
• One Light, Two Heavy: This is interesting. Now, the two heavy particles share the "water bucket." Because they split the responsibility, the rules become slightly more relaxed. The heavy particles don't have to be as strictly constrained as before, opening up even more possibilities for where they could hide.

Can We Catch Them at the LHC?

The Large Hadron Collider (LHC) is like a giant particle smasher. We can't see dark matter directly, so we look for "Mono-X" events: a collision where a visible particle (like a Jet, a Higgs, or a Z boson) flies out, and the dark matter particles zoom off in the opposite direction, leaving a gap in the energy balance (Missing Energy).

• Current Status: The authors ran simulations using the latest data from the LUX-ZEPLIN (LZ) detector and the ATLAS experiment.
• The Verdict:
  • The "Light" particles in these new models are not yet excluded by current data, but they are very close to the edge.
  • The "Heavy" particles are mostly out of reach for the LHC right now because they are too heavy to be produced easily.
  • The Future: The paper concludes that while we can't see these particles yet, the High-Luminosity LHC (a future upgrade that will smash particles much more frequently) has a very good chance of finding them. Specifically, looking for collisions that produce a Higgs boson plus missing energy seems to be the most promising "fishing spot."

Summary

This paper is a map of the "Dark Neighborhood."

1. One Room: Too restrictive. Only giant monsters or specific resonant toys fit.
2. Two/Three Rooms: By adding more invisible rooms, the rules relax. We can now have light dark matter particles that were previously impossible.
3. The Catch: These light particles are hiding in a very narrow, tricky spot. They are just barely escaping detection by current experiments.
4. The Hope: If we upgrade our detectors (High-Luminosity LHC), we might finally catch a glimpse of these light, invisible roommates hiding in the extra singlet rooms.

Technical Summary

Technical Summary: Dark Matter in Multi-Singlet Extensions of the Standard Model

Problem Statement
The Standard Model (SM) does not account for Dark Matter (DM). While minimal extensions involving a single real scalar singlet coupled to the Higgs via a $Z_2$ symmetry provide a DM candidate, this model is heavily constrained by experimental data. Specifically, for DM masses below approximately 3.5 TeV, the parameter space is excluded by the tension between the observed relic density (requiring a minimum coupling) and direct detection (DD) limits (requiring a maximum coupling), except in the resonant region where the DM mass is near half the Higgs mass ($m_h/2$). Consequently, the minimal single-singlet model is largely inaccessible to current Large Hadron Collider (LHC) searches, as the allowed heavy masses yield negligible production cross-sections. This paper investigates whether extending the SM with multiple real singlets and specific symmetry structures can open new, phenomenologically viable mass windows that are potentially accessible at the LHC.

Methodology
The authors analyze extensions of the SM incorporating two and three real scalar singlet fields ($S_i$). The study focuses on two distinct symmetry scenarios:

1. Independent $Z_2$ Symmetries: Each singlet is stabilized by its own independent $Z_2$ symmetry ($Z_2^{(1)} \times Z_2^{(2)}$ for two singlets; $Z_2^{(1)} \times Z_2^{(2)} \times Z_2^{(3)}$ for three). In this setup, all singlets are stable DM candidates, and there is no mixing between the dark sector scalars and the Higgs.
2. Single $Z_2$ Symmetry: All singlets are odd under a single shared $Z_2$ symmetry. This induces mixing between the gauge eigenstates, resulting in mass eigenstates where only the lightest scalar is stable and constitutes the DM candidate.

The analysis employs the following methodology:

• Theoretical Constraints: The models are subjected to boundedness-from-below conditions (copositivity criteria) and perturbative unitarity constraints derived from $2 \to 2$ scalar scattering amplitudes.
• Experimental Constraints: The parameter space is scanned using micrOMEGAs 6.0/6.1 to calculate the thermal relic density via the freeze-out mechanism. The results are compared against the Planck measurement ($\Omega_{DM}h^2 = 0.120 \pm 0.001$). Direct detection limits are applied using data from XENON1T, DarkSide-50, PICO-60, CRESST-III, PandaX-4T, and the latest LUX-ZEPLIN (LZ) 2024 results. Invisible Higgs branching ratio limits ($BR(h \to inv) < 0.107$) are also enforced.
• LHC Phenomenology: For allowed benchmark points, the authors calculate production cross-sections for mono-X processes ($pp \to S_1S_1 + j, h, Z$) and di-jet processes ($pp \to S_1S_1 b\bar{b}$) at $\sqrt{s} = 13, 13.6,$ and $14$ TeV using MadGraph5 aMC@NLO. These are compared against ATLAS upper limits on visible cross-sections for Run 2 data.

Key Contributions and Results

• Two Singlets with Independent $Z_2$ Symmetries:

  • A new allowed mass region is identified where one DM candidate ($S_1$) is light ($m_{S_1} \in [124.8, 230.0]$ GeV) and the other ($S_2$) is heavy ($m_{S_2} \in [4321, 9977]$ GeV).
  • In this scenario, the heavy particle $S_2$ accounts for nearly the entire observed relic density, satisfying the tension between relic density and DD constraints. The light particle $S_1$ contributes a negligible fraction ($\Omega_{S_1}h^2 \sim 10^{-7}$).
  • Because the DD constraint on $S_1$ scales with its abundance fraction ($\sigma_{SI} \times \Omega_{S_1}/\Omega_{DM}$), $S_1$ can have a large portal coupling ($\kappa_{H1} \sim 4-10$) and a large scattering cross-section without violating DD limits.
  • However, the authors note that with the LZ 2024 results, most points in this new region fall within the experimental uncertainty band, suggesting they are on the verge of exclusion.

• Two Singlets with a Single $Z_2$ Symmetry:

  • This scenario allows the lightest scalar DM candidate to exist across the entire mass range from $m_h/2$ to the TeV scale.
  • The mixing between singlets redefines the couplings such that the portal coupling governing DD ($\kappa_{H1}$) can be kept small to satisfy experimental bounds, while the other couplings ($\kappa_{H2}, \kappa_{H12}$) can be large to ensure the correct relic density. This decoupling of the relic density and DD constraints opens a significantly larger parameter space compared to the independent symmetry case.

• Three Singlets with Independent $Z_2$ Symmetries:

  • The authors scan "one-light-two-heavy" and "two-light-one-heavy" configurations.
  • The "one-light-two-heavy" case ($m_{S_1} < m_{S_2} < m_{S_3}$) offers a unique feature: the two heavy particles ($S_2, S_3$) share the relic density. Consequently, the DD exclusion for these heavy states is determined by their respective abundance fractions ($\sigma_{SI} \times \Omega_{S_{2,3}}/\Omega_{DM}$).
  • This sharing of the relic density alleviates the tension between relic density and DD constraints for the heavy states, slightly weakening the bounds on their masses and portal couplings compared to the single-heavy-particle scenarios.

LHC Search Prospects
The paper evaluates the detectability of the light DM candidates ($m_{S_1} \sim 125-220$ GeV) at the LHC:

• Mono-Jet: Current ATLAS limits exclude the benchmark points by more than one order of magnitude.
• Mono-Higgs: The points are excluded by slightly less than one order of magnitude.
• Mono-Z: The points are orders of magnitude away from current exclusion limits.
• Conclusion on LHC: While current Run 2 data does not exclude these models, the authors state there is a "good chance" to probe these models in specific final states (particularly mono-Higgs) at the High-Luminosity LHC (HL-LHC) stage, given the large portal couplings allowed in these multi-singlet scenarios.

Significance
The paper demonstrates that adding multiple singlets to the SM, particularly with specific symmetry assignments, can fundamentally alter the phenomenology of Higgs-portal Dark Matter. Unlike the minimal single-singlet model, which is effectively blind to the LHC for non-resonant masses, multi-singlet extensions can accommodate light DM candidates ($m \sim 100$ GeV) with large couplings. These models remain viable under current constraints (though increasingly tight with LZ 2024) and offer a concrete target for future HL-LHC searches, specifically in mono-Higgs and mono-jet channels. The work highlights that the interplay between multiple DM candidates sharing the relic density is a crucial mechanism for evading direct detection limits while maintaining testability at colliders.