Domain Walls from $\Sigma(36 \times 3)$, $\Delta(54)$ and $\Delta(27)$ potentials

This paper classifies the distinct domain walls and calculates their tensions arising from the degenerate minima of scalar potentials invariant under the $\Sigma(36 \times 3)$ symmetry and its subgroups $\Delta(54)$ and $\Delta(27)$, with and without imposed CP symmetries.

The Gist

Imagine the universe as a giant, invisible landscape. In the very early moments after the Big Bang, this landscape was smooth and uniform. But as the universe cooled down, it was like a pot of water freezing into ice. Just as water doesn't freeze into a single perfect crystal instantly, the universe didn't settle into one single state everywhere at once. Instead, different regions "froze" into different patterns.

Where these different patterns meet, you get a crack or a seam in the fabric of space-time. In physics, we call these Domain Walls. They are like invisible, two-dimensional sheets stretching across the cosmos, separating regions that made different choices about how to settle down.

This paper is a detailed map and a stress-test of these walls, specifically looking at three very complex, exotic types of "rules" (mathematical symmetries) that the universe might have followed: $\Sigma(36 \times 3)$, $\Delta(54)$, and $\Delta(27)$.

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

1. The Rules of the Game (The Symmetries)

Think of these symmetries as the rules of a dance.

• $\Delta(27)$ is a basic dance with a few specific moves.
• $\Delta(54)$ is a more complex dance that includes all the moves of $\Delta(27)$ plus some extra twists.
• $\Sigma(36 \times 3)$ is the most elaborate dance of all, containing the moves of the other two plus even more intricate steps.

The authors are studying what happens when the dancers (particles) try to find their "resting position" (vacuum) while following these specific dance rules. Because the rules are so complex, the dancers can end up in several different "resting spots" that all feel equally comfortable. These are the degenerate minima.

2. The Problem: Too Many Choices

When the universe cools, different regions pick different resting spots.

• Region A picks "Spot 1."
• Region B picks "Spot 2."
• Region C picks "Spot 3."

When Region A and Region B meet, they have to transition from one spot to the other. This transition creates a Domain Wall.

The paper asks: How "expensive" is it to build these walls?
In physics, this "cost" is called Tension. Think of tension like the tightness of a rubber band.

• A low tension wall is like a loose, floppy sheet of tissue paper. It's easy to move and might disappear quickly.
• A high tension wall is like a thick, heavy steel plate. It's hard to move, carries a lot of energy, and could potentially cause problems for the universe (like creating too much gravitational pull or messing up the formation of galaxies).

3. The Twist: Mirror Images (CP Symmetry)

The authors also looked at a special rule called CP Symmetry. You can think of this as a mirror.

• If you have a "Left-Handed" dancer and a "Right-Handed" dancer, and the rules treat them exactly the same, they are CP symmetric.
• If the rules treat them differently, the symmetry is broken.

The paper explores what happens when we add these "mirror" rules to our dance.

• Without mirrors: The dancers have many different resting spots, and the walls between them are all unique.
• With mirrors: Some resting spots become "twins" (mirror images of each other). The walls between these twins are different from the walls between non-twins.

4. What They Found (The Results)

The authors built a computer program to simulate these dances and calculate the "tension" (the cost) of the walls.

• The "General" Case ($\Delta(54)$ without mirrors): They found four distinct types of resting spots (labeled A, A', B, C). The walls between them have different "weights." Some are light (easy to break), some are heavy.
• The "Mirror" Cases: When they added mirror rules:
  • Some resting spots merged together (like two friends deciding to sit at the same table).
  • This changed the landscape. Now, instead of four types of walls, you might have fewer types, but the "weight" of the walls connecting the new merged groups is different.
  • Crucial Finding: They discovered that walls connecting a "Left-Handed" spot to a "Right-Handed" spot (connected by a mirror) have a different tension than walls connecting two "Left-Handed" spots.

5. Why Should We Care?

Why do we care about invisible walls in the early universe?

• Gravitational Waves: When these heavy walls eventually collapse or wiggle, they shake the fabric of space-time, creating gravitational waves (ripples in the universe).
• The NANOGrav Signal: Recently, astronomers detected a strange hum of gravitational waves coming from everywhere in the sky. Some scientists think this hum might be the echo of these very Domain Walls collapsing!
• Solving the Flavor Puzzle: The universe has three "generations" of particles (like three copies of electrons, but with different weights). We don't know why. These complex symmetries ($\Delta(54)$, etc.) are candidates for explaining why there are three generations.

The Bottom Line

This paper is like a structural engineer's report for the universe's early history.
They took three very complex blueprints (symmetries), figured out all the possible ways the universe could have "settled down," and calculated exactly how strong the cracks (Domain Walls) would be between those settlements.

They found that the "strength" of these cracks depends heavily on whether the universe followed simple rules or complex rules involving mirrors. This helps scientists predict what kind of gravitational waves we should be looking for today, potentially helping us solve the mystery of why our universe looks the way it does.

Technical Summary

Here is a detailed technical summary of the paper "Domain Walls from $\Sigma(36 \times 3)$, $\Delta(54)$ and $\Delta(27)$ potentials".

1. Problem Statement

The paper addresses the cosmological consequences of spontaneous symmetry breaking in models utilizing non-Abelian discrete symmetries to solve the flavor problem (specifically the origin of fermion mass hierarchies and mixing patterns).

• Context: While discrete symmetries like $A_4$ and $S_4$ are well-studied, the groups $\Delta(27)$, $\Delta(54)$, and $\Sigma(36 \times 3)$ offer more complex structures capable of generating geometrical CP violation (non-rotatable complex phases in vacuum expectation values).
• The Issue: The spontaneous breaking of these discrete symmetries in the early universe leads to the formation of Domain Walls (DWs). These are 2-dimensional topological defects. If stable, they can dominate the energy density of the universe, conflicting with observational data (e.g., Big Bang Nucleosynthesis).
• Goal: The authors aim to systematically classify the distinct types of domain walls that form between the degenerate vacua of scalar potentials invariant under these specific groups and to calculate their tensions (energy per unit area). Understanding the tension and topology is crucial for predicting the gravitational wave signatures these defects might produce.

2. Methodology

The authors employ a combination of group theory analysis and numerical simulation:

• Symmetry Analysis & Potential Construction:
  • They define the generators for $\Delta(27)$, $\Delta(54)$, and $\Sigma(36 \times 3)$ acting on a triplet of scalar fields.
  • They construct the most general renormalizable scalar potentials ($V$) invariant under these symmetries.
  • They analyze the effects of imposing specific CP symmetries (denoted $S_0$ through $S_{11}$) on the $\Delta(54)$ potential, which constrains the complex coupling constants (specifically $\lambda_4$).
• Vacuum Structure Analysis:
  • They identify the orbits of minima (sets of degenerate vacuum expectation values, or VEVs) for different regions of the parameter space.
  • They determine how these orbits merge or split when the symmetry is enlarged (e.g., from $\Delta(54)$ to $\Sigma(36 \times 3)$) or when specific CP symmetries are imposed.
• Numerical Calculation of Tension:
  • They developed a custom program using the CosmoTransitions Python module.
  • The code numerically minimizes the potential for 3 complex fields (6 degrees of freedom).
  • It computes the domain wall profile and calculates the tension ($\sigma$) by varying the relative phase between two degenerate minima to find the minimum energy configuration.
  • They benchmarked results for specific parameter sets where specific orbits are the true global minima.

3. Key Contributions

• Systematic Classification of Domain Walls: The paper provides a complete taxonomy of domain walls for $\Delta(27)$, $\Delta(54)$, and $\Sigma(36 \times 3)$, distinguishing between walls connecting minima within the same orbit and those connecting different orbits.
• Impact of CP Symmetries: It explicitly demonstrates how imposing different CP transformations alters the vacuum structure:
  • Trivial CP ($S_0$) merges orbits $A$ and $A'$.
  • Specific CP symmetries ($S_2, S_3$) merge orbit $A$ with $C$ or $A'$ with $C$.
  • Full $\Sigma(36 \times 3)$ symmetry merges $A \leftrightarrow A'$ and $B \leftrightarrow C$.
• Topological Distinction: The authors identify that in certain cases (e.g., $\Delta(54)$ with trivial CP), there are two topologically distinct types of domain walls between the same set of minima:
  1. Walls related by group elements (symmetry-connected).
  2. Walls related by CP conjugation (distinct tension).
• Quantitative Tension Data: The paper provides the first numerical estimates of domain wall tensions for these specific groups, offering benchmark values for future cosmological simulations.

4. Key Results

A. Vacuum Orbits and Merging

The minima fall into four primary alignments (orbits):

• A: $(\omega, 1, 1)$ type.
• A': $(\omega^2, 1, 1)$ type.
• B: $(1, 0, 0)$ type.
• C: $(1, 1, 1)$ type.

Merging Rules:

• $\Delta(54)$ (No CP): 4 distinct orbits ($A, A', B, C$).
• $\Delta(54) + S_0$ (Trivial CP): $A$ and $A'$ merge.
• $\Delta(54) + S_2$: $A$ and $C$ merge.
• $\Delta(54) + S_3$: $A'$ and $C$ merge.
• $\Sigma(36 \times 3)$: $A \leftrightarrow A'$ merge AND $B \leftrightarrow C$ merge.

B. Domain Wall Tensions ($\sigma$)

The numerical results (with $m^2=1, \lambda_1=1$) reveal distinct tension values depending on the connection type:

Case	Connection Type	Tension ($\sigma$)	Observation
$\Delta(54)$	$A \leftrightarrow A$	5.55	Single type per orbit.
	$B \leftrightarrow B$	7.05
	$C \leftrightarrow C$	8.63
$\Delta(54) + S_0$	$A \leftrightarrow A$ (or $A' \leftrightarrow A'$)	4.24	Reduced due to merging.
	$A \leftrightarrow A'$ (CP conjugate)	2.76	Distinct lower tension.
$\Sigma(36 \times 3)$	$B \leftrightarrow B$ (or $C \leftrightarrow C$)	7.05
	$B \leftrightarrow C$ (Group element $d$)	4.58	Distinct lower tension.

Key Finding: When orbits merge due to symmetry or CP, the tension between minima within the original separate orbits (now degenerate) is often lower than the tension between minima of the same original orbit. This implies a hierarchy of wall stability and potential decay channels.

5. Significance

• Cosmological Probes: The calculated tensions are critical inputs for simulating the evolution of the domain wall network. The decay of these walls produces a stochastic gravitational wave (GW) background.
• Distinguishing Models: The specific pattern of tensions and the existence of multiple wall types (e.g., CP-conjugate walls with different tensions) provide a unique "fingerprint" for these non-Abelian symmetries. This could allow future GW observatories (like Pulsar Timing Arrays or space-based interferometers) to distinguish between models based on $\Delta(54)$, $\Sigma(36 \times 3)$, and simpler groups like $A_4$.
• Geometric CP Violation: The work reinforces the role of these specific groups in generating CP violation geometrically, linking particle physics flavor models directly to early-universe cosmology.
• Stability Constraints: By quantifying the tensions, the paper aids in assessing whether these models require small explicit symmetry breaking to ensure the domain walls collapse before Big Bang Nucleosynthesis, avoiding the "domain wall problem."

In summary, this paper bridges the gap between abstract flavor symmetry models and observable cosmological phenomena by providing a rigorous classification and quantitative analysis of the topological defects they inevitably produce.