The $Z_3$ soft breaking in the I(2+1)HDM and its cosmological probes

This paper investigates the phenomenological and cosmological implications of introducing a $Z_3$ soft-breaking term in the I(2+1)HDM, analyzing scenarios where the resulting long-lived neutral scalar serves as a dark matter candidate or produces distinctive displaced vertex signatures at the International Linear Collider.

The Gist

Imagine the universe as a giant, complex machine. For decades, scientists have been trying to understand how this machine works, specifically how it gives things mass (like why you have weight) and what makes up the invisible "dark matter" that holds galaxies together.

This paper explores a new, slightly tweaked version of the machine's blueprint. Here is the story in simple terms, using some everyday analogies.

1. The Setup: The "Three-Headed" Machine

In our current understanding (the Standard Model), the universe has one "Higgs field" (think of it as a cosmic molasses that gives particles mass). It's like a single chef in a kitchen.

But this paper suggests there might actually be three chefs (three Higgs fields) working in the kitchen.

• Chef 1 (Active): This is the one we know. It's the "active" chef that interacts with everything and gives us the Higgs boson we discovered in 2012.
• Chef 2 & 3 (Inert): These two are "inert." They don't talk to normal matter (like you or me). They are shy, invisible, and only interact with each other.

Because these two "inert" chefs are so shy, the lightest particle they create could be Dark Matter. It's like a ghost in the machine: it's there, it has mass, but you can't see it or touch it.

2. The Twist: Breaking the Rules (Soft Breaking)

In the original version of this idea, the two inert chefs were perfect twins. They had the exact same weight and properties. This created a problem: if they were too similar, they would interact too strongly with normal matter, and experiments looking for Dark Matter (like XENON1T) would have already found them and ruled them out.

So, the authors introduce a "Soft Breaking."

• The Analogy: Imagine the two inert chefs are twins, but one of them decides to wear a slightly different hat. They are still twins, but now they are slightly different.
• The Result: This tiny difference (caused by a "soft-breaking term") makes one twin (let's call him A1) slightly heavier than the other (H1).

3. The Two Scenarios: The Ghost and the Long-Lived Traveler

Because of this tiny difference, the paper explores two fascinating possibilities for what happens to the heavier twin, A1:

Scenario A: The "Cosmic Ghost" (Two-Component Dark Matter)
If the difference between the twins is extremely small, the heavier twin (A1) becomes incredibly stable. It doesn't want to decay (die) for a very, very long time—longer than the current age of the universe.

• The Metaphor: Imagine a traveler who is so slow that they haven't finished their journey yet, even though the universe is billions of years old.
• The Implication: In this case, we have two types of Dark Matter living side-by-side: the light twin (H1) and the heavy, long-lived twin (A1). Both are ghosts haunting the universe.

Scenario B: The "Disappearing Act" (Collider Signatures)
If the difference between the twins is a bit larger, the heavy twin (A1) becomes unstable. It wants to decay, but because the "rule-breaking" is so small, it takes a long time to happen.

• The Metaphor: Imagine a magician who takes a long time to pull a rabbit out of a hat. He starts the trick, wanders around the stage for a few seconds, and then pulls out the rabbit.
• The Implication: If we smash particles together in a giant machine (like the proposed International Linear Collider, or ILC), we could create this heavy twin. It would travel a short distance inside the detector before suddenly decaying into the light twin (Dark Matter) and some visible particles (like electrons or jets).
• The "Smoking Gun": This would look like a "displaced vertex"—a spot in the detector where particles appear out of nowhere, away from the main collision point. It's like seeing a ghost appear in the middle of a room, rather than at the door.

4. The Search: Hunting for the "6-Lepton" Signal

The authors calculated what would happen if we looked for these particles at a future particle collider.

• They found that if we smash electrons and positrons together, we might see a very specific, rare event: 6 leptons (particles like electrons or muons) appearing out of nowhere, plus a huge amount of "missing energy" (the invisible Dark Matter escaping).
• The Analogy: It's like throwing two balls together and having six other balls fly out in a perfect pattern, while one invisible ball steals all the momentum and runs away.
• They also looked for signals with 4 leptons and 2 jets (sprays of particles), which are also very clean and easy to spot against the background noise.

5. Why Does This Matter?

This paper is exciting because it offers a way to solve two problems at once:

1. Dark Matter: It explains what Dark Matter could be (the inert twins).
2. Experimental Survival: It explains why we haven't found Dark Matter yet (the twins are slightly different, hiding from current detectors).
3. Future Discovery: It tells us exactly what to look for in future experiments (like the ILC): a "disappearing act" where a particle travels a short distance before turning into Dark Matter and leaving a trail of 6 leptons.

In summary: The authors are proposing that the universe has a secret family of invisible particles. They are almost identical twins, but a tiny rule-breaking difference makes one of them a long-lived traveler. If we build the right machine, we might catch this traveler in the act of disappearing, finally solving the mystery of Dark Matter.

Technical Summary

1. Problem Statement

The Standard Model (SM) fails to explain Dark Matter (DM), baryogenesis, and neutrino mass origins. While the Inert Doublet Model (IDM) provides a viable DM candidate via a $Z_2$ symmetry, extending this to a 3-Higgs Doublet Model (3HDM) with a $Z_3$ symmetry offers a "Hermaphrodite DM" scenario. In the exact $Z_3$ limit, the lightest CP-even ($H_1$) and CP-odd ($A_1$) inert scalars are degenerate in mass and both stable, creating a two-component DM scenario.

However, this exact degeneracy leads to a fatal phenomenological flaw: the $Z H_1 A_1$ vertex is non-zero at tree level, leading to a large DM-nucleus scattering cross-section that is ruled out by Direct Detection (DD) experiments (e.g., LUX-ZEPLIN). Furthermore, the exact degeneracy makes it difficult to distinguish the two DM components experimentally.

The paper investigates a softly broken $Z_3$ symmetry in an I(2+1)HDM (one active doublet, two inert doublets). The goal is to determine if introducing a small soft-breaking term ($\mu_{12}^2$) can:

1. Suppress the $Z H_1 A_1$ coupling to satisfy DD constraints.
2. Create a mass splitting between $H_1$ and $A_1$ such that $A_1$ becomes either a long-lived cosmological DM candidate or a decaying particle with a displaced vertex at colliders.

2. Methodology

Model Framework:

• Lagrangian: The model utilizes three Higgs doublets ($\phi_1, \phi_2, \phi_3$) with a $Z_3$ symmetry where $\phi_1 \to \omega \phi_1$, $\phi_2 \to \omega^2 \phi_2$, and $\phi_3 \to \phi_3$.
• Symmetry Breaking: A soft-breaking term $V_{Z3}' = -\mu_{12}^2 (\phi_1^\dagger \phi_2) + h.c.$ is introduced. This term is small enough to allow for a long-lived $A_1$ but breaks the exact degeneracy.
• Vacuum: Only $\phi_3$ acquires a Vacuum Expectation Value (VEV), acting as the SM-like Higgs ($h$). $\phi_1$ and $\phi_2$ are inert.

Theoretical Constraints & Calculations:

• Tree-Level Suppression: To avoid DD exclusion, the authors impose the condition $\theta_h = \pi/4$ (mixing angle), which forces the tree-level $Z H_1 A_1$ vertex to vanish.
• Loop-Induced Decay: The decay $A_1 \to H_1 f \bar{f}$ (where $f$ are SM fermions) is induced at the one-loop level via the soft-breaking parameter $\mu_{12}^2$. The authors calculated the effective vertex $g_{A_1 H_1 Z^*}$ using FeynCalc and LoopTools, considering diagrams with charged scalars, neutral scalars, and $Z/W$ bosons.
• Cosmological Constraints:
  • Relic Density: Calculated using micrOMEGAs 5.2.4, considering contributions from both $H_1$ and $A_1$ depending on the lifetime of $A_1$.
  • Direct Detection (DD): Constraints from XENON1T and LUX-ZEPLIN (2023) on the spin-independent WIMP-nucleon cross-section.
  • Indirect Detection (ID): Constraints from Fermi-LAT on DM annihilation into $b\bar{b}$ and $\tau^+\tau^-$.
  • Higgs Invisible Decays: Constraints on $BR(h \to \text{invisibles}) < 0.08-0.15$.

Phenomenological Probes:

• Parameter Scan: A random scan was performed over physical observables (masses $m_{H_1}, m_{H_2}$, mass splittings $\Delta_h, \Delta_n$, and couplings $g_1, g_2$) to identify Benchmark Points (BPs) satisfying all constraints.
• Collider Signatures (ILC): The study focuses on the International Linear Collider (ILC) at $\sqrt{s} = 1000$ GeV.
  • Processes: $e^+e^- \to 2l + \cancel{E}_T$, $4l + 2j + \cancel{E}_T$, and $6l + \cancel{E}_T$.
  • Simulation: Used MadGraph for parton-level generation and MadAnalysis for differential distributions.
  • Displaced Vertices: Analyzed the decay probability of $A_1$ inside the detector based on its lifetime ($\tau_{A_1}$) and boost factor.

3. Key Contributions

1. Resolution of the DD Problem: The paper demonstrates that a small soft-breaking term ($\mu_{12}^2$) can suppress the tree-level $Z H_1 A_1$ coupling to zero while inducing a loop-level decay. This allows the model to evade stringent DD limits that would otherwise rule out the exact $Z_3$ symmetric "Hermaphrodite" scenario.
2. Dual Cosmological Scenarios: The authors identify two distinct regimes based on the magnitude of $\mu_{12}^2$:
   • Regime 1 (Long-lived $A_1$): If $\mu_{12}^2$ is extremely small ($\sim 10^{-5} - 10^{-7}$ GeV$^2$), $A_1$ has a lifetime comparable to the age of the universe. It acts as a second DM component, contributing to the total relic density alongside $H_1$.
   • Regime 2 (Decaying $A_1$): If $\mu_{12}^2$ is slightly larger ($\sim 1 - 10$ GeV$^2$), $A_1$ becomes unstable but has a macroscopic decay length. It decays inside the detector, producing displaced vertices.
3. Distinctive Collider Signatures: The paper proposes spectacular signatures for the ILC involving multi-lepton final states (up to 6 leptons) plus missing transverse energy and displaced vertices. Specifically, the process $e^+e^- \to A_1 A_1 \ell^+ \ell^-$ followed by $A_1 \to H_1 \ell^+ \ell^-$ (or jets) creates unique kinematic features.

4. Results

Cosmological & Parameter Space:

• Benchmark Points (BPs): Three scenarios were identified based on the mass of the lightest inert scalar ($m_{H_1}$):
  • Scenario A: $20 < m_{H_1} < 35$ GeV.
  • Scenario B: $50 < m_{H_1} < 65$ GeV (Optimal for $\Delta_n \approx 24$ GeV).
  • Scenario C: $70 < m_{H_1} < 80$ GeV.
• Relic Density: For small mass splittings ($\Delta_h < 1$ GeV), both $H_1$ and $A_1$ can contribute to the relic density. As the splitting increases, $A_1$ becomes unstable, and $H_1$ saturates the relic density.
• Direct Detection: The loop-induced coupling $g_3$ (effective $A_1 H_1 Z$) is sufficiently suppressed for $\Delta_h \lesssim 0.1$ GeV, allowing the model to pass LUX-ZEPLIN 2023 constraints.

Collider Phenomenology (ILC):

• Cross-sections: For a representative BP in Scenario B ($m_{H_1}=60$ GeV, $\Delta_h=5$ GeV, $\sqrt{s}=1000$ GeV):
  • $\sigma(e^+e^- \to 2l + 2H_1) \approx 10.8$ fb.
  • $\sigma(e^+e^- \to 4l + 2j + 2H_1) \approx 0.44$ fb.
  • $\sigma(e^+e^- \to 6l + 2H_1) \approx 0.031$ fb.
• Displaced Vertices: For $\Delta_h \approx 5$ GeV, the average decay length of $A_1$ is $\sim 1.3 \times 10^{-2}$ m, placing the decay well within the inner tracker of a typical detector.
• Kinematic Distributions: The study shows distinct peaks in missing transverse energy ($\cancel{E}_T$), lepton transverse momentum ($p_T$), and invariant mass distributions ($M_{ll}$) that differentiate this signal from SM backgrounds. The "6 lepton" channel is particularly clean and "spectacular."

5. Significance

This work provides a robust extension of the Inert Doublet Model that reconciles the "Hermaphrodite DM" concept with current experimental limits. Its significance lies in:

• Theoretical Viability: It offers a mechanism to suppress dangerous tree-level couplings via symmetry breaking while retaining the rich phenomenology of a multi-component DM sector.
• Experimental Testability: Unlike many DM models that are only probed via invisible decays or direct detection, this model predicts visible, displaced signatures at future lepton colliders.
• Discovery Potential: The proposed multi-lepton + displaced vertex signatures at the ILC are highly distinctive and low-background, offering a clear path to discovering the specific structure of the inert sector and the nature of the soft-breaking term.
• Cosmological Flexibility: It bridges the gap between stable multi-component DM and decaying long-lived particles, showing how a single parameter ($\mu_{12}^2$) dictates the cosmological history and collider phenomenology of the model.