Dark Matter and Dark Energy in Three-Higgs Doublet Model

This paper proposes a Three-Higgs Doublet Model incorporating $Z_2$-odd CP-even scalars and shift symmetry to simultaneously accommodate dark matter and dark energy, demonstrating that the model satisfies observational relic density constraints and maintains radiative stability across a wide energy range.

The Gist

Imagine the universe as a giant, invisible ocean. We can only see the tiny islands floating on top (stars, planets, us), but we know the ocean itself is made of two mysterious, invisible ingredients: Dark Matter (which acts like heavy glue holding galaxies together) and Dark Energy (which acts like a mysterious wind pushing the universe apart).

For a long time, scientists have tried to build a "recipe book" (a physics model) that explains these invisible ingredients using the same ingredients that make up our visible world. This paper attempts to write a new chapter in that recipe book called the Three-Higgs Doublet Model.

Here is a simple breakdown of what the authors did, using everyday analogies:

1. The Setup: A Three-Person Band

In the standard recipe (the Standard Model), there is one "Higgs field" (think of it as the main singer) that gives particles their mass.

• The Old Idea: Scientists previously added a "silent partner" (an Inert Doublet) to explain Dark Matter. This partner doesn't sing (doesn't interact with light) but stays around to hold things together.
• The New Idea: This paper adds a second silent partner. Now, instead of a duo, we have a trio:
  1. The Main Singer (H3): The normal Higgs boson we found in 2012.
  2. The Dark Matter Partner (H2): A heavy, invisible particle that acts like the "glue."
  3. The Dark Energy Partner (H1): An incredibly light, ghost-like particle that acts like the "wind" pushing the universe apart.

2. The Rules of the Game (Symmetries)

To make sure these invisible particles don't just disappear or turn into normal matter, the authors set up two strict "rules of the road":

• The "No-Exit" Rule (Z2 Symmetry): This rule acts like a bouncer at a club. It stops the Dark Matter particle from decaying (dying) into normal particles. It ensures the Dark Matter stays stable and hangs around forever, just like we need it to.
• The "Ghost" Rule (Shift Symmetry): The Dark Energy particle is so light (almost weightless) that if it interacted too much with other things, it would become heavy and break the laws of physics. To prevent this, the authors gave it a special "ghost" ability: it can shift its position without changing its energy. This keeps it light and "inert" (inactive) for now, mimicking the effect of Dark Energy.

3. The Heavy Lifting: Calculating the Mass

The authors wanted to know: If this model is real, how heavy must the Dark Matter be to match what we see in the universe?

They used a powerful computer program (called micrOMEGAs) to run simulations. Think of this as a cosmic video game where they adjust the "weight" of the Dark Matter particle and see if the universe looks right.

• The Result: They found a "Goldilocks zone." If the Dark Matter particle is too light, it disappears too fast. If it's too heavy, there's too much of it.
• The Sweet Spot: They found that if the Dark Matter weighs between 536 and 548 GeV (a specific unit of mass), the math works perfectly. It matches the amount of Dark Matter astronomers actually observe in the sky.

4. The "Fine-Tuning" Problem

Here is the tricky part. The Dark Energy particle is supposed to be incredibly light (like a feather). But in quantum physics, heavy particles usually try to "push" light particles to become heavy, too. It's like trying to keep a feather floating in a hurricane; the wind (heavy particles) wants to blow it away.

The authors checked if their "Ghost Rule" (Shift Symmetry) was strong enough to keep the Dark Energy particle light.

• The Problem: Without the rule, the math says the particle should get heavy, which breaks the model.
• The Solution: They showed that if the "Ghost Rule" is strictly enforced, the particle stays light. They argued this is "technically natural" because if you turn the rule off completely, the particle becomes perfectly symmetrical. It's like saying, "It's okay for the feather to be light, because the only reason it's heavy is if we break the rule."

5. The Future: A Time Traveler's View

The paper admits that this model works great for today's universe (the current epoch). However, they suggest that in the early universe (right after the Big Bang), the rules might have been different.

• They kept the "Three-Person Band" structure in their math to allow for the possibility that, back in the day, the Dark Energy and Dark Matter might have interacted more strongly.
• They calculated how these rules change as energy levels go up (using something called Renormalization Group Equations), essentially mapping out how the model evolves from the Big Bang to today.

Summary

This paper proposes a new way to organize the universe's invisible ingredients. It suggests that Dark Matter and Dark Energy are actually two different "siblings" in a family of three Higgs particles.

• Dark Matter is the heavy, stable sibling that holds galaxies together.
• Dark Energy is the ultra-light, ghost-like sibling that pushes the universe apart.
• The Main Higgs is the normal sibling we already know.

The authors proved that if you set the mass of the Dark Matter sibling to a very specific weight (around 540 GeV), the model perfectly matches our observations of the universe today. They also showed that the model is mathematically stable, provided the "Ghost Rule" for Dark Energy is strictly followed.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics fails to explain two major cosmological observations: the existence of Dark Matter (DM) and the accelerated expansion of the universe driven by Dark Energy (DE). While the Inert Doublet Model (IDM) successfully incorporates DM candidates via a $Z_2$ symmetry, it does not naturally accommodate DE. Furthermore, DE models often suffer from severe fine-tuning problems due to the ultra-light mass scale required ($m \sim 10^{-33}$ eV), which makes them unstable against radiative corrections.

The authors aim to address these issues by extending the Three-Higgs Doublet Model (3HDM), specifically the I(2+1)IDM variant, to simultaneously host both DM and DE candidates within a unified scalar sector. The primary challenges are:

1. Ensuring the stability of the DM candidate.
2. Mimicking DE (quintessence) without violating observational bounds.
3. Solving the fine-tuning problem associated with the ultra-light DE scalar mass via radiative stability analysis.

2. Methodology

The study employs a theoretical framework combining particle physics phenomenology with cosmological constraints:

• Model Construction (I(2+1)IDM):

  • The model introduces three Higgs doublets: $H_1$, $H_2$ (Inert), and $H_3$ (Active/SM-like).
  • Symmetries:
    • A global $Z_2$ symmetry ($H_1, H_2 \to -H_1, -H_2$) is imposed to stabilize the DM candidate (the lightest CP-even inert scalar, $H^0_2$) by forbidding its decay into SM fermions.
    • A Shift Symmetry ($H^0_1 \to H^0_1 + c$) is imposed on the DE scalar ($H^0_1$) to suppress polynomial interactions, mimicking a cosmological constant and protecting its ultra-light mass.
  • Potential: A general renormalizable scalar potential $V_H$ is defined with quartic couplings ($\lambda_{ij}$) and soft-breaking terms. The vacuum expectation values (VEVs) are set such that only $H_3$ acquires a VEV ($v \approx 246$ GeV), while $H_1$ and $H_2$ remain inert ($v_1=v_2=0$).

• Dark Matter Analysis:

  • The relic density ($\Omega h^2$) is calculated using micrOMEGAs (v6.1.15).
  • The tool numerically solves the Boltzmann equation to determine the thermal freeze-out of the DM candidate.
  • Key parameters analyzed include the DM mass ($m_{DM}$), the mass splitting between CP-even and CP-odd scalars ($\delta = m_{H^0_2} - m_{A^0_2}$), and the mass splitting between the two inert doublets ($\Delta = m_{H^0_2} - m_{H^0_1}$).

• Radiative Stability & RGEs:

  • To address the fine-tuning of the DE mass, the authors compute one-loop mass corrections using the Coleman-Weinberg potential.
  • Renormalization Group Equations (RGEs) for the quartic coupling $\lambda_{31}$ (coupling the DE scalar to the SM Higgs) are computed at one-loop and two-loop orders using the SARAH package.
  • The study investigates the limit where gauge couplings $g_1, g_2 \to 0$ in the current epoch, effectively treating the DE doublet as a gauge singlet.

3. Key Contributions

1. Unified Framework: The paper successfully constructs a 3HDM where a single theoretical structure accommodates both a WIMP-like Dark Matter candidate and a Quintessence-like Dark Energy field.
2. Radiative Stability via Shift Symmetry: The authors demonstrate that the ultra-light mass of the DE scalar is technically natural under 't Hooft's naturalness criterion. By imposing a shift symmetry, setting the coupling $\lambda_{31} \to 0$ restores a symmetry, making the small coupling value stable against quantum corrections if the symmetry is strictly enforced.
3. Phenomenological Viability: The study identifies a specific "viable window" for the DM mass that satisfies Planck observational data without requiring discontinuous or fine-tuned coupling constants.
4. Epoch-Dependent Decoupling: The paper proposes a mechanism where the DE doublet behaves as a gauge singlet in the current epoch ($g_{1,2} \to 0$) to avoid radiative instability, while retaining the doublet structure to allow for potential DM-DE interactions in the early universe.

4. Results

• Dark Matter Relic Density:

  • Using micrOMEGAs, the authors found that the observed relic density ($\Omega h^2 = 0.119 \pm 0.0027$) can be achieved for a DM mass range of $536 \text{ GeV} \le m_{DM} \le 548 \text{ GeV}$.
  • This result holds for a mass splitting $\delta \approx 0.1$ GeV and a large inter-doublet splitting $\Delta > 50$ GeV.
  • In the limit of large $\Delta$, the model effectively decouples to the standard Inert Doublet Model (IDM), and the results align with established literature (specifically Case H in Keus et al., 2015).
  • For $m_{DM} < 536$ GeV, the relic density is too low; for $m_{DM} > 549$ GeV, the required coupling constant becomes unnaturally large, conflicting with experimental constraints.

• Radiative Constraints on DE:

  • The one-loop mass correction calculation yields a constraint on the effective quartic coupling: $|\lambda_{31}| \le 10^{-86}$.
  • While this appears to be extreme fine-tuning, the authors argue it is technically natural because $\lambda_{31} \to 0$ restores the shift symmetry.
  • RGE Analysis: The computed $\beta$-functions show that if gauge couplings ($g_1, g_2$) are non-zero, they generate additive terms that would drive $\lambda_{31}$ away from zero, violating the stability condition.
  • Conclusion on Stability: The model is only radiatively stable in the current epoch if the gauge couplings are negligible ($g_{1,2} \to 0$), effectively decoupling the DE scalar from electroweak bosons.

5. Significance

• Theoretical Advancement: This work bridges the gap between particle physics extensions (3HDM) and cosmological observations, offering a concrete mechanism to treat DM and DE within the same scalar sector.
• Naturalness Solution: It provides a robust argument for the naturalness of ultra-light scalars (DE) using shift symmetry, a concept often used in quintessence but rarely integrated into multi-Higgs models with DM.
• Future Directions: The paper lays the groundwork for studying DM-DE interactions in the early universe. By relaxing the $g_{1,2} \to 0$ limit and softly breaking the $Z_2$ symmetry at high energy scales, future work could explore how these sectors interacted during the early stages of the universe, potentially explaining the coincidence problem or early-universe dynamics.
• Experimental Predictions: The identification of the $536-548$ GeV mass window provides a specific target for future collider searches (e.g., High-Luminosity LHC) for heavy inert scalars.

In summary, the paper presents a viable, radiatively stable extension of the Standard Model that unifies Dark Matter and Dark Energy, resolving fine-tuning issues through symmetry arguments and identifying a narrow, testable mass range for the Dark Matter candidate.