A new idea for relating the asymmetric dark matter mass scale to the proton mass

This paper proposes a new mechanism linking the asymmetric dark matter mass scale to the proton mass by introducing an extended $SU(3)_1 \times SU(3)_2$ color group with a spontaneously broken $\mathbb{Z}_2$ symmetry that relates the QCD and dark gauge couplings, resulting in comparable confinement scales and a rich TeV-scale spectrum.

The Gist

Here is an explanation of the paper, broken down into simple concepts with creative analogies.

The Big Mystery: Why is Dark Matter So Heavy?

Imagine the universe is a giant party. There are two groups of guests:

1. The Visible Guests (Ordinary Matter): These are the atoms, stars, and planets we can see. They make up about 15% of the "mass" of the party.
2. The Invisible Guests (Dark Matter): These are the mysterious guests we can't see, but we know they are there because they are holding the dance floor together with gravity. They make up about 85% of the mass.

The Coincidence: Scientists have noticed something weird. The total weight of the invisible guests is almost exactly 5.4 times the total weight of the visible guests.

In the past, scientists thought this was just a lucky accident. It's like flipping a coin and getting heads 5.4 times in a row. But usually, in physics, when two things are so closely related, there's a hidden reason.

The Problem with Previous Explanations:
Some scientists tried to explain this by saying, "Okay, let's just assume there are 5.4 times more dark matter particles than visible ones." But that just pushes the mystery back a step: Why are there 5.4 times more? And why does a single dark matter particle weigh about the same as a proton (a piece of ordinary matter)?

This paper proposes a new idea: The dark matter particles and the protons are actually "cousins" from the same family.

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The New Idea: The "Mirror" and the "Broken Symmetry"

The authors suggest that our universe has a hidden "shadow" version of itself, specifically regarding the force that holds atomic nuclei together (called the Strong Force or QCD).

1. The Twin Forces (The Mirror)

Imagine you have two identical factories running side-by-side.

• Factory A (Our World): This is our normal Strong Force. It binds quarks together to make protons.
• Factory B (The Dark World): This is a "Dark Strong Force." It binds "dark quarks" together to make "dark protons" (Dark Matter).

In a perfect world, these two factories would be identical. They would produce products of the exact same weight. This is called a $Z_2$ symmetry (like a mirror reflection).

2. The Broken Mirror

But here's the twist: The mirror is slightly cracked. The two factories aren't exactly identical anymore.

• The authors propose that at very high energies (like right after the Big Bang), these two forces were connected.
• As the universe cooled down, a "switch" was flipped (spontaneous symmetry breaking).
• This switch made the Dark Factory run slightly differently than our Factory.

The Result: Because the Dark Factory runs slightly faster or slower in its internal mechanics, the "Dark Protons" end up being slightly heavier than our "Normal Protons."

The Analogy: Think of two identical twins. One twin (Us) eats a normal diet. The other twin (Dark Matter) eats a slightly different diet. They are still the same species, but one is a bit bigger. The paper calculates that if you tweak the "diet" (the physics parameters) just right, the Dark Twin ends up being about 5.4 times heavier in total mass, perfectly matching what we observe in the universe.

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How They Did It (The Mechanics)

To make this work, the authors built a complex mathematical model with three main ingredients:

1. Extended Color Groups: They added extra "colors" to the universe's physics. Imagine our world has 3 primary colors (Red, Green, Blue). Their model adds a second set of 3 colors for the dark world, and a third set that acts as a bridge between them.
2. The Heavy "Bridge" Particles: To break the mirror symmetry, they introduced new, heavy particles (like heavy fermions and scalars). These particles are so heavy that we haven't seen them yet, but they act like the "glue" that connects our world to the dark world before the symmetry breaks.
3. The Axion (The Peacekeeper): A major bonus of this model is that it naturally solves a different, very old problem in physics called the "Strong CP Problem" (which asks why the universe doesn't violate certain time-reversal rules). Their model automatically includes a particle called an Axion that fixes this problem. Think of the Axion as a "peacekeeper" that ensures the laws of physics stay consistent in both the visible and dark sectors.

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What Does This Mean for Us?

1. It's Testable:
Unlike some theories that live only in math, this one predicts new particles.

• The Prediction: There should be heavy "glue" particles (called Colorons) and heavy "dark quarks" floating around at energy levels we might reach with future particle colliders (like a super-charged version of the Large Hadron Collider).
• The Search: If we build a bigger collider, we might smash atoms hard enough to create these heavy particles. If we find them, it proves our "Dark Twin" theory is real.

2. It Explains the "Coincidence":
Instead of the 5.4 ratio being a random accident, it's a natural consequence of the two forces being related but slightly different. It's like asking, "Why is a pound of feathers heavier than a pound of lead?" (Wait, bad analogy).
Better analogy: "Why is a gallon of water heavier than a gallon of oil?" Because they are both liquids (related), but they have different densities (the symmetry breaking). The paper explains why the densities are what they are.

Summary in One Sentence

This paper suggests that Dark Matter isn't a random stranger; it's a slightly heavier "cousin" of ordinary matter, born from a hidden, broken mirror symmetry in the forces that hold the universe together, and we might be able to find proof of this relationship in future experiments.

Technical Summary

Here is a detailed technical summary of the paper "A new idea for relating the asymmetric dark matter mass scale to the proton mass" by Cox, Pérez, and Volkas.

1. Problem Statement

The paper addresses the "cosmological coincidence" problem: the observed ratio of dark matter (DM) to baryonic matter energy densities is $\Omega_D \simeq 5.4 \Omega_b$. While Asymmetric Dark Matter (ADM) models explain this by positing that both densities arise from particle-antiparticle asymmetries generated in the early universe, they typically fail to explain why the DM mass ($m_{DM}$) is of the same order as the proton mass ($m_p \sim \text{few GeV}$).

Standard ADM models assume similar number densities ($n_{DM} \sim n_b$) and simply deduce $m_{DM} \sim m_p$ from the density ratio. However, this shifts the coincidence problem to the unexplained question: Why is the DM mass scale naturally close to the QCD confinement scale? The authors propose a mechanism to dynamically relate the confinement scale of a dark QCD sector ($\Lambda_D$) to the visible QCD scale ($\Lambda_{QCD}$), thereby explaining the mass coincidence.

2. Methodology and Model Construction

The authors propose an extension of the Standard Model (SM) gauge group involving an extended color sector and a discrete symmetry.

• Gauge Structure: The model introduces an extended local symmetry group:
  $$SU(3)_1 \times SU(3)_2 \times SU(3)_D \times SU(2)_L \times U(1)_Y$$

  • SM Fermions: Charged only under $SU(3)_1$.
  • Dark Sector: Contains a new set of fermions ($\psi_D$) charged under $SU(3)_D$.
  • Mirror/Exchange Sector: Contains a new set of fermions ($\psi_2$) charged under $SU(3)_2$.

• Symmetry Breaking and Unification:

  • QCD Identification: The visible QCD group $SU(3)_c$ is identified as the diagonal subgroup of $SU(3)_1 \times SU(3)_2$. This breaking occurs at a scale $u_3$ via bifundamental scalar fields ($\Sigma_{12}$ and $\Sigma_{1D}$).
  • $Z_2$ Exchange Symmetry: A discrete $Z_2$ symmetry exchanges the fields and couplings of the $SU(3)_2$ sector with the $SU(3)_D$ sector ($\psi_2 \leftrightarrow \psi_D$, $\Sigma_{12} \leftrightarrow \Sigma_{1D}$).

• Coupling Relations:
  At the symmetry breaking scale $u_3$, the couplings satisfy:
  $$\frac{1}{\alpha_c} = \frac{1}{\alpha_1} + \frac{1}{\alpha_2}$$
  Due to the $Z_2$ symmetry, $\alpha_2 = \alpha_D$ (in the symmetric limit). This leads to the relation:
  $$\frac{1}{\alpha_c} = \frac{1}{\alpha_1} + \frac{1}{\alpha_D}$$
  Since $\alpha_c < \alpha_1, \alpha_D$, the visible coupling is weaker than the dark coupling at the UV scale.

• Mass Generation and $Z_2$ Breaking:
  To satisfy collider constraints, the colored fermion $\psi_2$ must be heavy ($m_{\psi_2} \gtrsim 1.5$ TeV), while the dark fermion $\psi_D$ must be light ($m_{\psi_D} \ll \Lambda_D$) to ensure the DM mass is set by $\Lambda_D$.

  • The authors implement Spontaneous $Z_2$ Breaking using two scalar singlets ($\phi_2, \phi_D$).
  • $\phi_2$ acquires a Vacuum Expectation Value (VEV), giving mass to $\psi_2$.
  • $\phi_D$ does not acquire a VEV, leaving $\psi_D$ massless (or very light).
  • This setup naturally generates the required mass hierarchy without explicit soft breaking terms.

3. Key Contributions

A. Dynamical Relation of Confinement Scales

The core contribution is the demonstration that the specific gauge structure and symmetry breaking pattern lead to a predictable relationship between $\Lambda_{QCD}$ and $\Lambda_D$.

• Renormalization Group Evolution (RGE): The authors solve the two-loop RGEs for the couplings.
• Mechanism: Because $\alpha_c < \alpha_D$ at the scale $u_3$, and the beta-functions are similar (depending on the number of flavors $N_f$), the visible coupling $\alpha_c$ runs slower than the dark coupling $\alpha_D$ (or vice versa depending on thresholds).
• Result: This evolution naturally produces a ratio of confinement scales $R = \Lambda_D / \Lambda_{QCD}$ that is of order unity ($O(1)$) to ten ($O(10)$) across a wide range of parameter space. This explains why $m_{DM} \sim m_p$.

B. Solution to the Strong CP Problem

The model naturally incorporates an axion solution to the Strong CP problem:

• The spontaneous breaking of the global axial symmetry $U(1)_{2A}$ (associated with the $\psi_2$ mass generation) produces a pseudo-Goldstone boson.
• Due to the $Z_2$ symmetry and the structure of the topological $\theta$ terms, this boson is identified as the QCD axion.
• The model realizes a KSVZ-type axion, where the axion couples to the heavy colored fermions ($\psi_2$).
• The axion decay constant $f_a \sim u_\phi$ is constrained by astrophysics to be $> 10^8$ GeV, which is consistent with the model's parameters.

C. Phenomenological Viability

The paper analyzes constraints from:

• Colliders: Bounds on stable colored particles ($\psi_2$) and heavy gluons (colorons) constrain $M_{\psi_2} \gtrsim 1.5$ TeV and $u_3 \gtrsim 0.85$ TeV.
• Self-Interacting Dark Matter (SIDM): Constraints on DM self-interaction cross-sections exclude regions where $\Lambda_D < \Lambda_{QCD}$ (specifically $R \lesssim 0.8$).
• Naturalness: The model remains natural up to scales of $\sim 40$ TeV.

4. Results

• Confinement Scale Ratio ($R$):
  • For a benchmark with $N_f = 6$ (matching SM quarks) and specific coupling parameters ($x=0.1$), the model predicts $R = \Lambda_D / \Lambda_{QCD} \approx 1 - 10$.
  • This implies $m_{DM} \approx (1-10) m_p$, perfectly fitting the ADM paradigm.
  • The ratio is robust against variations in the symmetry breaking scale $u_3$ and the mass of $\psi_2$, provided they are within the viable parameter space.
• Particle Spectrum:
  • The model predicts a rich spectrum of new particles at the TeV scale, including heavy colored scalars, vector-like fermions ($\psi_2$), and heavy gluons (colorons).
  • These particles are potentially detectable at future high-energy colliders (e.g., FCC, HL-LHC).
• Axion Phenomenology:
  • The axion decay constant is tied to the symmetry breaking scale $u_\phi$.
  • To satisfy neutron star cooling bounds ($f_a \gtrsim 10^8$ GeV), the Yukawa coupling $y_\psi$ must be small ($y_\psi \lesssim 10^{-4}$), which is technically natural.
  • The model requires a pre-inflationary axion scenario to avoid overproducing axion dark matter, ensuring the axion is a sub-component of the total DM.

5. Significance

This paper provides a theoretically motivated, dynamical explanation for the "coincidence" between the dark matter mass and the proton mass.

1. Solves the Mass Hierarchy Puzzle: Instead of treating the $m_{DM} \sim m_p$ relation as an accidental input, the model derives it from the fundamental gauge structure and symmetry breaking of the early universe.
2. Unification of Problems: It simultaneously addresses the Dark Matter mass scale, the Strong CP problem (via the axion), and the baryon-DM density ratio (via ADM).
3. Testability: Unlike many ADM models that are difficult to probe, this model predicts specific TeV-scale signatures (colorons, heavy fermions) that can be tested at current and future colliders.
4. Robustness: The mechanism works across a significant region of parameter space, making it a viable and adaptable framework for asymmetric dark matter.

In conclusion, the authors successfully construct a minimal extension of the SM that links the visible and dark sectors through a broken $Z_2$ symmetry, naturally generating the observed dark matter mass scale while solving the Strong CP problem.