A New Route to the Annihilation of Multi-Wall String Topological Configurations

This paper proposes a new mechanism for the annihilation of cosmologically problematic domain walls attached to cosmic strings in global $U(1)$ symmetry models, demonstrating that radiative corrections from small bare fermion masses can generate a temperature-dependent bias that triggers the network's decay, using a majoron framework with right-handed neutrinos as a representative example.

The Gist

Imagine the universe as a giant, expanding balloon. In the very early moments of its life, some invisible forces decided to "break" a perfect symmetry, much like a perfectly round snowflake cracking as it freezes. This cracking created cosmic "scars" known as topological defects.

The paper focuses on a specific type of scar: a cosmic string (think of it as a thin, infinitely long cosmic thread) that has domain walls (imagine them as soap films or membranes) attached to it. In this scenario, one string is the center, and multiple walls radiate out from it like the spokes of a wheel or the petals of a flower.

The Problem: The Universe's "Heavy Blanket"

Usually, these wall networks are a disaster for cosmology.

• The Analogy: Imagine the universe is a room filled with air (radiation) and people (matter). These cosmic walls are like heavy, thick blankets. As the room expands, the air and people get spread out and become less dense. However, these blankets don't thin out as fast; they stay heavy.
• The Consequence: If left alone, these blankets would eventually become so heavy that they would crush the room, taking over the entire energy of the universe and stopping the formation of stars and galaxies. This is the "Cosmological Domain Wall Problem."

The Usual Fix vs. The New Idea

Traditionally, scientists have tried to solve this by manually adding a "bias" term to the equations—a bit like tilting the floor of the room so the heavy blankets slide off and disappear. But this feels like a "hack" or an arbitrary fix.

This paper proposes a natural, self-cleaning mechanism.

The New Mechanism: The "Tiny Weight" Trick

The authors suggest that the universe doesn't need a manual tilt. Instead, they introduce a tiny, almost invisible ingredient: a small "bare mass" for a specific type of particle (a right-handed neutrino).

Here is how it works, step-by-step:

1. The Setup: The cosmic string is attached to multiple walls. Each wall separates a region of space where a field (the "Majoron") points in a slightly different direction.
2. The Tiny Mass: Because of gravity's subtle effects at the highest energy scales, these particles have a tiny, inherent mass that shouldn't technically exist in a perfect symmetry.
3. The Radiative Effect: This tiny mass interacts with the walls. Through a quantum process called "radiative corrections" (imagine a ripple effect caused by the particle's presence), this tiny mass creates a temperature-dependent pressure.
4. The Result: This pressure acts like a gentle but persistent wind blowing against the heaviest blankets. It creates a difference in energy between the different "sides" of the walls.
5. The Annihilation: Because one side of a wall is now slightly more energetically favorable than the other, the walls start to move. They collapse, merge, and vanish.

The "Odd vs. Even" Puzzle

The paper notes a fascinating detail about how these walls disappear, depending on how many are attached to the string (let's call this number $n$):

• The "Odd" Number Problem: If you have an odd number of walls (like 5), there is a mathematical quirk where two specific walls might initially have equal pressure, so they don't move against each other immediately.
• The Domino Effect: However, as the other walls collapse first, the remaining walls are forced to become neighbors. Once they become neighbors, the pressure difference kicks in, and they collapse too.
• The Final State: Even in the worst-case scenario, the system eventually reduces to a single wall attached to a string. But a single wall on a string is unstable; it's like a tent with only one pole—it collapses on itself immediately.

The Bottom Line

The paper claims that by simply acknowledging that particles have a tiny, natural mass (due to gravitational effects), the universe automatically generates the force needed to clean up these dangerous cosmic scars.

• No manual fixes needed: The "bias" isn't added by hand; it emerges naturally from the physics of the particles.
• Works for all cases: Whether there are 5, 6, or more walls, the mechanism ensures they all eventually annihilate before they can ruin the universe's evolution.
• The Outcome: The universe clears the "heavy blankets," allowing normal cosmic evolution to proceed, potentially leaving behind a stable vacuum and perhaps even helping explain why the universe has more matter than antimatter (leptogenesis) or providing a candidate for dark matter.

In short, the paper argues that a tiny, natural imperfection in particle mass acts as a cosmic janitor, sweeping away the dangerous structures that would otherwise destroy the universe.

Technical Summary

Technical Summary: A New Route to the Annihilation of Multi-Wall String Topological Configurations

Problem Statement
Particle physics models extending the Standard Model (SM) often rely on global symmetries, such as a global $U(1)$, to address open questions. However, quantum gravitational effects at the Planck scale are expected to explicitly break exact global symmetries. In the case of a global $U(1)$, this breaking reduces the symmetry to a discrete subgroup, leading to the formation of cosmic strings attached to multiple domain walls (DWs). These "string-wall networks" pose a severe cosmological problem: the energy density of stable DWs redshifts more slowly than radiation and matter, eventually dominating the Universe's energy budget. This scenario conflicts with Big Bang Nucleosynthesis (BBN) constraints, which require that such networks annihilate before the temperature drops to $T_{BBN} \sim \mathcal{O}(1 \text{ MeV})$.

Conventional solutions involve introducing an ad hoc bias term into the scalar potential to render the walls unstable. Alternatively, recent work on $Z_2$ DWs suggested that a small bare fermion mass term could radiatively generate the necessary bias. This paper investigates whether this radiative mechanism can be extended to the more complex and general setting of multi-wall string networks arising from explicit $U(1)$ breaking.

Methodology
The authors analyze the annihilation dynamics within a Majoron framework, where the SM is extended by three right-handed neutrinos (RHNs) and a complex scalar $\Phi$ charged under a global $U(1)_L$ (lepton number) symmetry. The methodology proceeds as follows:

1. Gravitational Breaking: The authors incorporate Planck-suppressed operators induced by quantum gravity (e.g., via wormhole nucleation) that explicitly break the continuous $U(1)_L$ symmetry down to a discrete $Z_n$ subgroup. This generates a potential for the Goldstone boson (majoron) of the form $V \sim \cos(n\chi/v)$, creating $n$ discrete vacua and a network where each string is attached to $n$ domain walls.
2. Introduction of Bare Mass: The authors introduce a small, explicit bare Majorana mass term ($M_N$) for a single flavor of RHN in the Lagrangian. This term explicitly breaks $U(1)_L$ by two units.
3. Radiative Bias Generation: The authors calculate the one-loop Coleman-Weinberg (CW) effective potential and thermal corrections generated by the RHNs. Because the RHN mass depends on the vacuum expectation value (vev) of $\Phi$, which varies across the different discrete vacua ($k = 0, \dots, n-1$), the loop corrections generate a $k$-dependent energy shift.
4. Bias Analysis: The authors derive the bias potential $\Delta V$ between adjacent vacua. They demonstrate that the interference between the bare mass $M_N$ and the Yukawa coupling term ($y\Phi$) creates a phase-dependent mass term $|M_{R,k}|^2$. This results in a bias proportional to $\cos(2\pi k/n)$, which lifts the degeneracy between vacua.
5. Dynamics Simulation: The authors analyze the evolution of the network, comparing the bias energy density ($\Delta V$) against the wall energy density ($\rho_w \sim \sigma_w H$). They examine specific benchmark points for $n=5$ and $n=6$ to determine the annihilation temperature $T_{ann}$.

Key Contributions and Results

• Radiative Mechanism for Multi-Wall Networks: The paper establishes that a small bare fermion mass term, previously studied in $Z_2$ contexts, can successfully generate a temperature-dependent bias capable of triggering the annihilation of complex string-wall networks with $n > 2$ walls.
• Vacuum Bias Structure: The authors show that the bias arises from the non-trivial phase dependence of the RHN mass in different vacua. The leading-order bias is proportional to $M_N (yv)^3 [\cos(2\pi k_1/n) - \cos(2\pi k_2/n)]$.
• Handling the $\Delta V = 0$ Condition: A critical finding is that for certain pairs of vacua (specifically where $k_1 + k_2 = n$), the cosine terms cancel, resulting in zero bias ($\Delta V = 0$).
  • For odd $n$ (e.g., $n=5$), an adjacent pair satisfying this condition exists immediately.
  • For even $n$, such a pair may form dynamically as other walls annihilate.
  • The authors argue that even if a zero-bias wall survives initially, the resulting configuration is a string attached to a single domain wall. Such a configuration is topologically unstable and will self-annihilate shortly thereafter, ensuring the complete resolution of the DW problem without requiring additional parameters (like a phase $\delta$ in the mass term).
• Benchmark Points:
  • Case $n=5$: With parameters $M_N = 10^4$ GeV, $y=0.01$, and $v=2\times 10^{14}$ GeV, the network forms at $T_f \sim 2\times 10^{13}$ GeV and annihilates at $T_{ann} \sim 10^{11}$ GeV, well above the BBN scale.
  • Case $n=6$: For similar parameters, the bias term dominates the wall energy density even at the formation epoch ($T_f$), preventing the formation of a stable network entirely.
• Dominance of CW Corrections: The study finds that the temperature-independent Coleman-Weinberg corrections are the dominant source of the bias, while thermal corrections remain subdominant for the benchmark points considered.

Significance and Claims
The paper claims to propose a "novel annihilation mechanism" that does not rely on ad hoc bias terms in the scalar potential. Instead, it utilizes the radiative effects of a physically motivated small bare fermion mass term, which is plausible in theories where global symmetries are broken by gravity.

The authors emphasize that this mechanism works irrespective of the number of walls ($n$) attached to a string. They highlight that the annihilation dynamics in the $U(1)$ case are more intricate than in simple $Z_2$ scenarios due to the nontrivial vacuum structure, yet the mechanism robustly drives the system toward the true vacuum. Furthermore, the framework suggests potential secondary cosmological implications, such as spontaneous leptogenesis (due to the complex phase-dependent RHN masses) and the majoron serving as a dark matter candidate, though the primary focus remains on the resolution of the cosmological DW problem.

The work concludes that the inclusion of a tiny bare mass for right-handed neutrinos provides a natural origin for the bias required to eliminate the cosmological dangers associated with string-wall networks in global $U(1)$ models.