Transient Bias for CP Domain Wall Decay and Dark Matter

The Gist

The Big Mystery: The "Strong CP Problem"

Imagine the universe has a set of rules, like a game. One of these rules is called "CP symmetry," which basically means the game should look the same whether you play it forward or backward, or whether you swap left and right.

In the world of particle physics (specifically the strong force that holds atoms together), the rules should allow for a tiny bit of "cheating" (a violation of this symmetry). However, when scientists look at the universe, they see absolutely no cheating happening. It's as if the game is perfectly balanced, which is incredibly strange and puzzling. This is known as the Strong CP Problem.

The Proposed Solution: Spontaneous Symmetry Breaking

One popular idea to fix this is Spontaneous CP Violation (SCPV). Think of it like a pencil balanced perfectly on its tip. The laws of physics (the table) are perfectly symmetrical, but the pencil (the universe) eventually has to fall over to one side. Once it falls, the symmetry is "broken," and the universe picks a specific direction.

In this theory, the universe "falls" into a state that explains why we see CP violation in some places (like in the CKM matrix) but not in the strong force.

The New Problem: The "Domain Wall" Disaster

Here is where the story gets tricky. If this "falling over" happens after the universe expands rapidly (after inflation), different patches of the universe might fall in different directions.

• Patch A falls "North."
• Patch B falls "South."

Where these patches meet, they form a boundary called a Domain Wall. Imagine a giant, invisible sheet of energy separating two regions of the universe.

The Disaster: These walls are heavy and stable. If they exist, they would act like a cosmic anchor, eventually taking over the entire universe's energy and crushing everything. To prevent this, physicists usually say, "The universe must have fallen over before it expanded," which limits how hot the early universe could have been. This creates a lot of restrictions on how the universe evolved.

The Paper's Solution: The "Temporary Nudge"

The authors of this paper propose a clever way to get rid of these walls without breaking the rules of the game permanently. They introduce a new character: a new scalar field (let's call it "S").

Think of the universe as a ball rolling in a bowl with two identical valleys (the North and South options). Usually, the ball gets stuck in one valley, creating a wall where the two sides meet.

The authors suggest that in the very early universe, the "S" field acts like a giant, temporary hand that pushes the bowl slightly to one side.

1. The Push: This hand creates a "bias" (a tilt). It makes one valley slightly lower than the other.
2. The Collapse: Because one side is lower, the "North" walls and "South" walls feel a pressure difference. The "North" side starts eating the "South" side (or vice versa), causing the walls to crumble and disappear.
3. The Retreat: As the universe expands and cools, the "S" field moves back to its original position. The "hand" lets go, and the bowl becomes perfectly symmetrical again.

The Result: The walls are gone, but the fundamental rules of the universe remain perfectly symmetrical. The "tilt" was only a temporary glitch, not a permanent change to the laws of physics.

The Bonus Prize: Dark Matter

Here is the twist: The "S" field doesn't just disappear after it does its job. Once the walls are gone and the "hand" lets go, the "S" field starts vibrating or oscillating around its center point.

Think of this like a plucked guitar string that keeps vibrating long after the initial pluck. These vibrations don't interact much with normal matter, but they have mass. The authors calculate that these vibrations naturally survive until today and make up exactly the amount of Dark Matter we observe in the universe.

Summary of the Mechanism

1. The Setup: A new field (S) has a huge value in the early universe.
2. The Crisis: CP symmetry breaks, creating dangerous domain walls.
3. The Fix: The new field (S) creates a temporary "tilt" in the energy landscape. This tilt pushes the walls to collapse.
4. The Cleanup: The tilt disappears, leaving the universe's laws intact.
5. The Legacy: The new field (S) settles into a stable vibration, becoming the invisible Dark Matter that holds galaxies together.

Why This Matters

This paper solves two major headaches at once:

1. It explains how to get rid of the dangerous domain walls without forcing the universe to be "cold" or restricted in its history.
2. It provides a natural origin for Dark Matter using the exact same field that fixed the wall problem.

It's like finding a single key that unlocks two different doors: one leading to a stable universe, and the other to the mystery of Dark Matter.

Technical Summary

Technical Summary: Transient Bias for CP Domain Wall Decay and Dark Matter

1. The Problem
Spontaneous CP violation (SCPV) offers an elegant solution to the strong CP problem by ensuring the QCD $\bar{\theta}$ parameter vanishes at the fundamental level, with observed CP violation arising from complex vacuum expectation values (VEVs) of scalar fields. However, if SCPV occurs after cosmic inflation, different Hubble patches settle into degenerate CP vacua related by CP transformation. This leads to the formation of CP domain walls. Unlike standard domain walls, these CP domain walls are stable and, if they dominate the energy density of the Universe, would lead to a cosmological disaster.

The standard resolution requires CP symmetry to be broken before or during inflation and never restored, which imposes a stringent upper bound on the post-inflation reheating temperature, severely restricting thermal history scenarios like thermal leptogenesis. Introducing a permanent explicit CP-breaking term to lift the vacuum degeneracy and decay the walls is generally undesirable, as it can induce a large contribution to $\bar{\theta}$, undermining the solution to the strong CP problem. Furthermore, if CP is a gauged spacetime symmetry (as expected in quantum gravity), explicit breaking operators are forbidden.

2. Methodology and Setup
The authors propose a dynamical mechanism that removes CP domain walls without introducing permanent explicit CP violation. The model introduces a new light complex scalar field, $S$, which interacts with the CP-breaking scalar field, $\eta$, via higher-dimensional operators suppressed by the Planck scale ($M_{Pl}$).

• Potential Structure: The scalar potential includes a CP-symmetric form for $\eta$ and $S$, with a mixing term of the form $\frac{\lambda}{m!\ell! M_{Pl}^{m+\ell-4}} S^m \eta^\ell + \text{h.c.}$
• Inflationary Era: During inflation, the field $S$ acquires a large, homogeneous complex field value $\langle |S_{inf}| \rangle$ due to a negative mass term induced by the inflaton potential. The phase of $S$ is randomly assigned but remains frozen due to Hubble friction.
• Radiation-Domination Era: As the Universe reheats, $\eta$ is stabilized at the origin by thermal masses. The field $S$ follows a scaling solution, maintaining a large field value.
• CP Breaking and Bias Generation: When the temperature drops below the CP-breaking scale ($T \sim v_{CP}$), $\eta$ acquires a complex VEV, spontaneously breaking CP and forming domain walls. At this stage, the large VEV of $S$ and its non-trivial phase induce a "transient bias" in the potential of $\eta$ through the mixing term. This bias lifts the degeneracy between the CP vacua, creating a pressure difference across the domain walls.
• Decay and Oscillation: The pressure difference drives the domain walls to collapse. As the Universe expands further, the Hubble parameter drops below the mass of $S$ ($H \lesssim m_S$). The field $S$ begins to oscillate around the origin, the induced bias vanishes, and the low-energy CP structure remains intact, preserving the solution to the strong CP problem.

3. Key Contributions and Results

• Dynamical Removal of Domain Walls: The paper derives the conditions under which the volume pressure generated by the transient bias ($p_V$) overcomes the tension force of the domain walls ($p_T$). The authors identify viable parameter spaces (specifically for interaction powers $n=6, m=11, \ell=1$) where the bias is sufficient to trigger wall decay before the bias disappears.
• Constraints on Backreaction: The study carefully analyzes the backreaction of $\eta$ on $S$. It establishes constraints on the coupling $\lambda$ to ensure that the mixing term does not dominate the self-interaction of $S$ before the domain walls decay, which would otherwise alter the scaling solution and prevent the necessary bias generation.
• Dark Matter Connection: The coherent oscillation of the scalar field $S$ around the origin, initiated when $H \sim m_S$, naturally survives as a cold dark matter candidate. The authors calculate the relic abundance $\Omega_\chi h^2$ and demonstrate that the same field responsible for eliminating the domain walls can account for the observed dark matter density within the viable parameter space.
• UV Completion: The paper proposes a supersymmetric (SUSY) UV completion using chiral superfields and specific $U(1)$ and $R$-symmetries to justify the non-generic form of the potential and the absence of lower-dimensional operators. This framework addresses the origin of the required higher-dimensional interactions.

4. Significance and Claims
The authors claim that this mechanism provides a robust solution to the cosmological domain wall problem inherent in post-inflationary SCPV scenarios without compromising the solution to the strong CP problem. By dynamically mimicking an explicit CP violation only during a limited cosmological epoch, the model avoids permanent imprints on the underlying theory.

Furthermore, the work establishes a novel link between the resolution of the strong CP problem, the elimination of cosmological defects, and the origin of dark matter, all mediated by a single scalar field. The authors note that while their mechanism preserves the fundamental CP symmetry, the question of whether it can resolve topologically protected domain walls in gauged-CP frameworks (where explicit breaking is strictly forbidden) remains an open issue for future investigation. Additionally, they suggest that the phase of $S$ could potentially play a role in baryogenesis, though this is presented as a possibility for unified cosmological frameworks rather than a proven result of the current work.