Dark matter trio in classically conformal theories: WIMP, supercooling, and monopole

This paper explores a classically conformal $SU(2)_X$ gauge theory with a triplet dark scalar, identifying three distinct dark matter scenarios—WIMP, supercooled DM, and monopole—that arise from the model's unique first-order phase transition history and analyzing their viability against current constraints and future experimental sensitivities.

The Gist

Here is an explanation of the paper "Dark matter trio in classically conformal theories," translated into everyday language with creative analogies.

The Big Picture: A Universe Without "Heavy" Rules

Imagine the Standard Model of physics (our current best rulebook for how the universe works) as a house built on a shaky foundation. The "Higgs boson" is like a heavy chandelier hanging from the ceiling. In our current understanding, quantum physics suggests this chandelier should be infinitely heavy, tearing the house apart. This is the "Hierarchy Problem."

The authors of this paper propose a new blueprint: Classically Conformal (CC) Theory.

• The Analogy: Imagine a house where, at the very beginning, there are no heavy beams or chandeliers at all. The structure is perfectly balanced and weightless. The "weight" (mass) only appears later, like a sudden shift in gravity that causes the floor to settle.
• The Result: This solves the stability problem. But the authors ask: If the universe started weightless, where did all the invisible "Dark Matter" come from?

They found a specific model (a hidden sector with a "dark force") that naturally produces three different types of Dark Matter, like a triple-threat team.

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The Cast of Characters

To understand the three scenarios, we need to meet the players in this hidden "Dark Sector":

1. The Dark Force Carriers ($X$ bosons): Think of these as invisible messengers that carry a new force, similar to how photons carry light. They are the main candidates for Dark Matter.
2. The Dark Scalar ($s$): A special particle that acts like a switch. When it flips, it gives mass to the messengers and breaks the symmetry of the dark world.
3. The Monopole: A topological knot or "scar" in the fabric of space, created when the dark world changes state. Think of it like a permanent knot in a rubber band that can never be untied.

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The Three Scenarios: How the Dark Matter Was Born

The paper argues that the history of the universe's cooling process determines which of these three particles becomes the dominant Dark Matter. It's like a cooking recipe where the temperature and speed of cooling change the final dish.

1. The "WIMP" (The Classic Freeze-Out)

• The Analogy: Imagine a crowded dance party (the early hot universe). Everyone is dancing and bumping into each other. As the music slows down (the universe cools), people stop dancing and leave the floor.
• What happens: The Dark Matter particles ($X$) were once dancing with normal matter. As the universe cooled, they stopped interacting and "froze out," leaving a leftover population that we see today.
• The Catch: This is the most famous theory, but current experiments (like underground tanks looking for dark matter) have ruled out the simplest versions of this in the authors' model.

2. The "Supercooled" DM (The Ice Cube Surprise)

• The Analogy: Imagine water in a freezer. Usually, it freezes at 0°C. But if the water is very pure and the freezer is quiet, it can stay liquid down to -10°C. Then, snap! It freezes instantly, releasing a massive burst of heat.
• What happens: In this model, the universe gets "supercooled." It stays in a high-energy state much longer than expected. Then, a sudden "phase transition" (like the snap of freezing) happens.
  • This transition releases a huge amount of energy, reheating the universe.
  • The Dark Matter particles are born after this freeze, but they are so sparse and the universe is so hot that they never get a chance to dance again. They are "frozen in" at a very low density.
• Why it matters: This allows for much heavier Dark Matter particles (thousands of times heavier than the WIMP) that are invisible to current detectors but might be found by future, more sensitive experiments.

3. The Monopole DM (The Cosmic Knots)

• The Analogy: Imagine a long, tangled rope. If you pull the rope tight very quickly, knots get trapped in the fabric. You can't undo them; they are permanent scars in the rope.
• What happens: When the dark universe changed state (the phase transition), the "knots" (Monopoles) formed.
• The Twist: In many old theories, these knots were too heavy or too numerous, making them bad Dark Matter candidates. However, because this model uses a "supercooled" transition, the knots form in a very specific way that makes them the perfect amount of Dark Matter.
• The Result: These massive, ancient knots could be the Dark Matter we are looking for.

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The "First-Order Phase Transition" (The Big Bang of the Dark Sector)

The key to all three scenarios is a First-Order Phase Transition (FOPT).

• The Metaphor: Think of boiling water. Bubbles form, grow, and merge until the whole pot is steam.
• In this paper: The universe didn't just slowly cool down; it formed "bubbles" of the new, lower-energy state. These bubbles expanded and crashed into each other.
• The Sound: When these bubbles crash, they create ripples in space-time called Gravitational Waves.
  • If the transition was "supercooled" (Scenario 2), the bubbles crash with incredible force, creating a loud "bang" of gravitational waves.
  • Future telescopes (like LISA) might be able to "hear" this sound from the early universe, proving this theory is real.

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What Does This Mean for Us?

1. It solves a math problem: It fixes the "heavy chandelier" issue in the Standard Model without needing new, complicated rules.
2. It offers three answers: Instead of guessing which Dark Matter theory is right, this model says, "It depends on how the universe cooled."
   • If it cooled normally: WIMP (likely ruled out).
   • If it supercooled: Supercooled DM (heavy, hard to find).
   • If it supercooled fast: Monopoles (massive cosmic knots).
3. It gives us a roadmap:
   • Direct Detection: We might need to look for heavier particles or different types of interactions.
   • Gravitational Waves: We might hear the "sound" of the dark universe freezing.
   • Particle Colliders: We might find the "switch" particle ($s$) at the Large Hadron Collider (LHC) or future colliders, which would decay slowly and leave a unique signature.

The Bottom Line

This paper suggests that the universe's Dark Matter isn't just one thing. It's a trio of possibilities born from a specific, elegant mathematical framework where the universe "supercooled" before freezing. It turns the search for Dark Matter into a detective story where the clues are hidden in the sound of the early universe and the behavior of heavy, invisible particles.

Technical Summary

Here is a detailed technical summary of the paper "Dark matter trio in classically conformal theories: WIMP, supercooling, and monopole" by Ke-Pan Xie and Cheng-Hao Zhan.

1. Problem Statement

The Standard Model (SM) suffers from the gauge hierarchy problem, where the Higgs boson mass receives quadratically divergent quantum corrections. Classically Conformal (CC) theories offer a solution by postulating that the Lagrangian contains no dimensionful parameters at the tree level, preserving conformal invariance. Symmetry breaking is then generated radiatively via the Coleman-Weinberg (CW) mechanism.

While CC theories have been studied for solving the hierarchy problem and generating Dark Matter (DM), most existing models focus on fundamental scalar representations or Abelian gauge groups. There is a need to explore CC theories with non-Abelian gauge groups (specifically $SU(2)_X$) and adjoint (triplet) scalar representations. Such a setup introduces unique features:

• Stability of vector DM candidates due to the residual unbroken $U(1)_X$ symmetry.
• The potential existence of topological solitons ('t Hooft-Polyakov monopoles) as DM candidates.
• Distinct cosmological evolution histories involving supercooled first-order phase transitions (FOPTs) that differ significantly from standard thermal histories.

2. Methodology

The authors construct and analyze a specific CC $SU(2)_X$ gauge theory model:

• Model Construction:

  • Gauge Group: A hidden $SU(2)_X$ with gauge fields $X^a_\mu$.
  • Scalar Sector: A real scalar triplet $\phi^a$ (adjoint representation) under $SU(2)_X$ and a singlet under the SM.
  • Potential: The tree-level potential contains only dimensionless couplings ($\lambda_h, \lambda_s, \lambda_{hs}$).
  • Symmetry Breaking: The CW potential generated by $SU(2)_X$ gauge boson loops triggers the breaking of $SU(2)_X \to U(1)_X$ at a high scale $w$. This subsequently triggers Electroweak (EW) symmetry breaking via the Higgs portal coupling $\lambda_{hs}$.
  • Particle Spectrum: The model yields massive vector bosons $X^\pm$ (DM candidates), a massless dark photon $A'$, and two mixed scalar mass eigenstates ($h$ and $s$).

• Thermal History Analysis:

  • The authors calculate the finite-temperature effective potential to determine the cosmological evolution.
  • They classify the evolution into four distinct types (Type-N1, N2, I1, I2) based on the timing of the $SU(2)_X$ FOPT relative to the QCD phase transition and the EW scale.
  • Key parameters calculated include the latent heat ratio ($\alpha$), the transition duration ($\beta/H$), and the reheating temperature ($T_{rh}$).

• Dark Matter Production Mechanisms:
  The paper investigates three distinct DM production scenarios arising from the unique thermal history:

  1. WIMP Freeze-out: Standard thermal freeze-out of $X^\pm$ if they re-thermalize after the FOPT.
  2. Supercooled DM: A "freeze-in" mechanism where $X^\pm$ never reach thermal equilibrium after an ultra-supercooled FOPT. The abundance is set by the initial entropy dilution and subsequent non-thermal production.
  3. Monopole DM: Production of 't Hooft-Polyakov monopoles via the Kibble-Zurek mechanism during the FOPT, assuming no subsequent annihilation due to the lack of a dark photon thermal bath.

• Phenomenological Constraints:

  • The authors map the viable parameter space ($m_X, m_s$) against current experimental limits (Direct Detection: LZ, PandaX-4T; Collider: LHC, Beam-dumps; Astrophysics: SN1987A).
  • They project future sensitivities for Long-Lived Particle (LLP) searches and Gravitational Wave (GW) observatories (LISA, BBO).

3. Key Contributions

• Identification of a "Dark Matter Trio": The paper uniquely identifies three viable DM candidates within a single minimal CC framework:
  1. Vector WIMP: The charged gauge boson $X^\pm$.
  2. Supercooled Vector DM: $X^\pm$ produced via freeze-in in a supercooled universe.
  3. Topological Monopole: 't Hooft-Polyakov monopoles arising from the adjoint scalar breaking pattern.
• Resolution of the Monopole Overproduction Problem: Contrary to previous literature suggesting monopoles are always overproduced or subdominant, this work demonstrates that the extreme supercooling inherent to CC theories (large $\beta/H$) allows for a viable monopole DM scenario by suppressing the bubble nucleation rate and adjusting the density.
• Detailed Thermal History Classification: The authors provide a comprehensive map of the cosmological evolution in the $(m_X, m_s)$ plane, distinguishing between "Normal" (Type-N) and "Inverted" (Type-I) patterns, and identifying regions where FOPTs do not occur (crossover/second-order).
• Phenomenological Viability: They establish that while the WIMP scenario is largely excluded by current direct detection limits, the Supercooled DM and Monopole DM scenarios remain viable and offer distinct signatures for future experiments.

4. Key Results

• Parameter Space:
  • WIMP Region: Requires $m_X \sim \text{TeV}$ and $g_X \sim 10^{-1}$. This region is excluded by current direct detection experiments (LZ, PandaX-4T) due to the Higgs portal coupling.
  • Supercooled DM Region: Characterized by lighter couplings ($g_X \sim 10^{-2} - 10^{-3}$) and heavier masses ($m_X \sim 10 - 100 \text{ TeV}$). The reheating temperature $T_{rh}$ is high, allowing for heavy DM. This scenario is currently unconstrained but accessible via future LLP searches (FASER, SHiP) targeting the long-lived scalar $s$.
  • Monopole DM Region: Viable only in the $m_s < m_h$ regime with extremely rapid transitions ($\beta/H \sim 10^9$). The monopole mass is predicted to be $M_{mp} \sim 10^{12} \text{ GeV}$.
• Gravitational Waves:
  • In the $m_s > m_h$ regime, the FOPT is strong enough to generate a stochastic GW background detectable by LISA and BBO.
  • In the $m_s < m_h$ regime (Type-I1), the transition is so strong and short-lived that the GW signal is effectively undetectable by current or near-future interferometers.
• Experimental Projections:
  • LLP Searches: The scalar $s$ becomes long-lived in the non-WIMP regions due to suppressed mixing ($\theta$). Future experiments like FASER, CODEX-b, and SHiP can probe the supercooled and monopole parameter spaces.
  • Colliders: The HL-LHC and a future 10 TeV Muon Collider can probe the mixing angle $\theta$ for the $m_s > m_h$ regime, particularly via Vector Boson Fusion (VBF) production of $s$.

5. Significance

This work significantly advances the understanding of Dark Matter in Classically Conformal theories by:

1. Unifying Scenarios: It demonstrates that a single minimal model can accommodate three distinct DM paradigms (WIMP, Supercooled, Monopole), depending on the specific mass hierarchy and coupling strengths.
2. Challenging Dogma: It overturns the assumption that monopoles are non-viable DM candidates in such models, showing that the unique dynamics of CC theories (supercooling) can naturally suppress their abundance to the observed level.
3. Guiding Future Searches: It provides clear, testable predictions for next-generation experiments. Specifically, it highlights the importance of Gravitational Wave astronomy for the $m_s > m_h$ regime and Long-Lived Particle searches for the $m_s < m_h$ regime, offering a roadmap to distinguish between these DM candidates.
4. Theoretical Consistency: It reinforces the viability of the CC principle as a solution to the hierarchy problem that simultaneously addresses the nature of Dark Matter without introducing arbitrary mass scales.