Two Higgs doublet model with a complex singlet scalar and Multi-critical Point Principle

This paper investigates a Two Higgs Doublet Model extended by a complex singlet scalar where the singlet's imaginary part acts as dark matter, demonstrating that while the tree-level Multiple Point Principle favors large mixing that challenges the required degenerate Higgs mass scenario, viable parameter regions exist that satisfy dark matter constraints and allow for a strong first-order electroweak phase transition driven by thermal loop effects.

The Gist

Imagine the universe as a giant, complex machine built from invisible building blocks. For a long time, scientists had a "Standard Model" blueprint that explained most of these blocks, but it had a few missing pieces. One big missing piece is Dark Matter—the invisible stuff that holds galaxies together but doesn't shine or interact with light. Another mystery is the Higgs field, which gives particles their mass, but we don't fully understand how it's structured.

This paper explores a new blueprint that tries to fix both problems at once by adding two new types of "bricks" to the machine: a second pair of Higgs fields and a mysterious, invisible "singlet" field.

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

1. The Setup: A New House with Two Vacuums

Think of the universe's energy landscape like a hilly terrain. Usually, the universe "settles" in the lowest valley (the vacuum).

• The Problem: In this new model, the terrain has two distinct valleys at the bottom: one where the Higgs fields live (the "Electroweak valley") and another where the invisible singlet field lives (the "Singlet valley").
• The Rule (MPP): The authors applied a rule called the Multiple Point Principle (MPP). Think of this like a strict architect who demands that both valleys must be at the exact same height. If one valley is lower than the other, the universe would fall into it and destroy the other. The rule says, "No, they must be perfectly level."

2. The Conflict: The "Tightrope Walk"

The authors discovered that following this "perfectly level" rule creates a huge conflict with the goal of explaining Dark Matter.

• The Dark Matter Goal: To hide Dark Matter from detectors (like the LUX-ZEPLIN experiment), the three neutral Higgs particles in this model need to have nearly identical weights (masses). Imagine three identical twins. If they are exactly the same weight, they cancel each other out in a way that makes them invisible to detectors. This is called the "degenerate scalar scenario."
• The MPP Goal: To keep the two valleys perfectly level (the MPP rule), the model needs the invisible singlet field to interact strongly with the Higgs fields. This requires the "mixing" between them to be large.
• The Clash: The "hiding" mechanism for Dark Matter works best when that mixing is small. The "leveling" rule (MPP) demands that the mixing be large. It's like trying to balance a seesaw where one side wants to go up and the other wants to go down.

3. The Solution: Finding the Sweet Spots

Despite this tug-of-war, the authors ran the numbers and found that it is still possible to satisfy both rules, but only in two specific "sweet spots":

• Spot A (The Resonance): If the Dark Matter particle has a very specific weight (about half the weight of the Higgs boson), it can "resonate" like a tuning fork. This allows the model to work even with the strong mixing required by the MPP rule.
• Spot B (The Heavyweights): If the Dark Matter particle is extremely heavy (thousands of times heavier than a proton), it naturally avoids detection, regardless of the mixing issue.

4. The Bonus: A Boiling Universe

The paper also looked at the history of the universe, specifically a moment called the Electroweak Phase Transition. This is like the moment water boils and turns into steam.

• The Bad News: The "level valley" rule (MPP) prevents the universe from having a "tree-level" (simple, direct) phase transition. It's like trying to boil water without turning on the stove; it won't happen naturally.
• The Good News: The authors showed that even without the stove, the "heat" of the early universe (thermal loop effects) can still cause a strong, violent phase transition (a big bubble of steam). This is important because a violent transition is a necessary ingredient for a theory called "Electroweak Baryogenesis," which tries to explain why there is more matter than antimatter in the universe.

Summary

The paper proposes a universe where:

1. Two valleys are perfectly level (a strict theoretical rule).
2. Three Higgs particles are nearly identical twins (to hide Dark Matter).
3. These two goals fight each other, making it very hard to build a working model.
4. However, it is still possible if the Dark Matter is either a specific "resonant" weight or extremely heavy.
5. Bonus: Even with these strict rules, the early universe could still have undergone a violent phase change, which is good news for theories about why we exist.

The authors conclude that while the "Multiple Point Principle" makes the model very tight and restrictive, it doesn't break it; viable solutions still exist.

Technical Summary

1. Problem Statement

The Standard Model (SM) lacks a viable Dark Matter (DM) candidate and does not explain the matter-antimatter asymmetry via Electroweak Baryogenesis (EWBG), which requires a strong first-order Electroweak Phase Transition (EWPT).

• The Model: The authors extend the Two-Higgs-Doublet Model (2HDM) by adding a complex singlet scalar ($S$). The imaginary part of $S$ ($\chi$) serves as a WIMP DM candidate, while the real part mixes with the neutral components of the Higgs doublets to form three physical Higgs bosons ($H_1, H_2, H_3$).
• The Conflict:
  1. Direct Detection Constraints: Current experiments (e.g., LZ) place stringent limits on DM-nucleon scattering. In this model, scattering is mediated by $H_{1,2,3}$. To satisfy these limits, the "degenerate scalar scenario" is required, where $H_{1,2,3}$ have nearly identical masses. This degeneracy causes destructive interference in scattering amplitudes due to the orthogonality of the mixing matrix. However, this scenario typically requires small mixing parameters ($\delta_1, \delta_2$) between the doublets and the singlet.
  2. Theoretical Motivation: The authors seek a fundamental principle to explain why these masses are degenerate. They propose the Multiple Point Principle (MPP), specifically the tree-level MPP, which requires the energy densities of the electroweak vacuum and a singlet-dominated vacuum to be degenerate ($\Delta V_0 = 0$).
  3. The Tension: The tree-level MPP condition favors large mixing parameters ($\delta_1, \delta_2$) to create the necessary vacuum separation to satisfy $\Delta V_0 = 0$. Conversely, the degenerate scalar scenario (to evade DM constraints) requires small $\delta_1, \delta_2$. The paper investigates whether these mutually exclusive requirements can be reconciled.

2. Methodology

• Model Setup: The authors define the scalar potential $V_0$ for the 2HDM extended by a complex singlet. They derive the tadpole conditions, mass matrices for neutral scalars, and the DM mass ($m_\chi$).
• Imposing Tree-Level MPP: They impose the condition that the potential energy at the electroweak vacuum $(v_1, v_2, v_S)$ equals the energy at the singlet vacuum $(0, 0, v'_S)$. This creates a non-trivial constraint linking the vacuum expectation values (VEVs) and coupling constants.
• Numerical Analysis:
  • They scan the parameter space, fixing physical inputs (Higgs masses, DM mass, mixing angles) and varying the singlet VEV ($v_S$) and soft-breaking parameter ($a_1$).
  • They utilize the code micrOMEGAs to calculate DM relic density and spin-independent scattering cross-sections.
  • They utilize CosmoTransitions with the Parwani resummation scheme to calculate the strength of the EWPT ($v_C/T_C$) using the one-loop finite-temperature effective potential.
• Benchmark Points: They construct specific benchmark points (BP1–BP6) with varying degrees of Higgs mass degeneracy (from 0.5 GeV to 0.1 GeV splitting) to test the interplay between MPP and DM constraints.

3. Key Contributions

• Theoretical Tension Identification: The paper explicitly demonstrates that the tree-level MPP and the degenerate scalar scenario impose opposing constraints on the doublet-singlet mixing parameters ($\delta_1, \delta_2$).
  • MPP $\implies$ Large $\delta_{1,2}$ (to maximize $|v_S - v'_S|$).
  • Degenerate Scenario $\implies$ Small $\delta_{1,2}$ (to suppress DM scattering).
• Resolution of the Tension: Despite the tension, the authors prove that viable parameter regions exist. They show that the MPP condition effectively fixes $v_S$ for a given mass splitting pattern, limiting the ability to arbitrarily reduce $\delta_{1,2}$ by increasing mass degeneracy.
• EWPT Mechanism: The authors clarify that while tree-level MPP forbids a tree-level-driven first-order phase transition (due to exact vacuum degeneracy at $T=0$), a strong first-order EWPT can still be generated purely by thermal loop effects (bosonic thermal contributions).

4. Key Results

• Parameter Space Viability:
  • The MPP condition forces the singlet VEV ($v_S$) to be very small (typically $< 1$ GeV) to satisfy $\Delta V_0 = 0$ with large mixing parameters.
  • Consequently, the suppression of the DM scattering cross-section is less effective than in the pure degenerate scenario without MPP.
  • However, viable regions are found where constraints are satisfied simultaneously in two specific regimes:
    1. Resonance Region: DM mass $m_\chi \approx 62.5$ GeV (near $m_H/2$), where annihilation is enhanced via s-channel Higgs resonance, reducing relic density.
    2. Heavy DM Region: $m_\chi = \mathcal{O}(1\text{--}10)$ TeV, where the scattering cross-section naturally drops and relic density is low.
• Mass Degeneracy Limits: Increasing the mass degeneracy of the Higgs bosons (from 0.5 GeV to 0.1 GeV splitting) does not significantly improve the suppression of the DM direct detection rate. This is because the MPP condition fixes the relationship between $v_S$ and the mass splitting, locking the mixing parameters $\delta_{1,2}$ to values that are too large for perfect cancellation.
• Electroweak Phase Transition:
  • All benchmark points satisfy the condition for a strong first-order EWPT ($v_C/T_C \gtrsim 1$).
  • The transition is driven by thermal loop effects (cubic terms in the effective potential) rather than the tree-level potential structure.
  • The strength of the transition is robust across different benchmark points because the bosonic mass spectrum remains nearly degenerate.

5. Significance

• Theoretical Consistency: The work provides a theoretical justification (via MPP) for the mass degeneracy required to hide DM from direct detection, moving beyond "fine-tuning by hand."
• Phenomenological Viability: It demonstrates that a model satisfying deep theoretical principles (MPP) can still be consistent with current experimental null results from DM searches (LZ) and collider data, provided the DM mass is in specific resonant or heavy regimes.
• Baryogenesis: It establishes that the MPP does not preclude successful Electroweak Baryogenesis. Even though the tree-level potential is degenerate, thermal corrections are sufficient to drive the strong first-order phase transition necessary for baryogenesis.
• Future Directions: The paper highlights that future direct detection experiments will probe the "heavy DM" and "resonance" regions identified, offering a clear path to test the 2HDMS+MPP scenario.

In conclusion, the paper successfully navigates the conflict between the Multiple Point Principle and Dark Matter constraints, showing that while the MPP makes the "degenerate scalar scenario" harder to realize, it does not render the model impossible. The model remains a viable candidate for explaining Dark Matter and the matter-antimatter asymmetry.