Impact of chirality imbalance and nonlocal interactions on the QCD biased axionic domainwall interpretation of NANOGrav 15 year data

This paper demonstrates that within a nonlocal NJL model, a sufficiently large chiral chemical potential can enable the QCD-biased axionic domain-wall scenario to explain the NANOGrav 15-year gravitational wave data even at large $\theta$ angles, where the topological susceptibility exhibits a pronounced width near the critical temperature.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Picture: Listening to the Universe's "Hum"

Imagine the universe is a giant concert hall. For a long time, we thought the only music being played was by massive black holes dancing together. But recently, a group of astronomers (NANOGrav) heard a strange, low-frequency hum coming from everywhere at once. This is called a Stochastic Gravitational Wave Background (SGWB). It's like the universe is humming a bass note that we can't quite place.

The big question is: What instrument is making this sound?

This paper investigates a very specific, exotic theory: Could this hum be the sound of cosmic "cracks" healing themselves?

The Cast of Characters

To understand the theory, we need three main characters:

1. The Axion (The Invisible String): Imagine a cosmic string that stretches across the universe. In the early universe, this string got "stuck" in different positions, creating a patchwork quilt of different vacuum states.
2. Domain Walls (The Cosmic Cracks): Where two different patches of the quilt meet, a "wall" forms. Think of these as fault lines in the fabric of space-time. If these walls are stable, they would eventually take over the universe and crush everything (a cosmological disaster).
3. The Bias (The Push): To save the universe, these walls need to be unstable so they can collapse and disappear. The paper suggests that the QCD force (the glue holding atomic nuclei together) provides a "push" or a bias that makes these walls collapse.

The Problem: The "Too Loud" or "Too Quiet" Signal

When these cosmic walls collapse, they shake the fabric of space-time, creating gravitational waves (the hum).

• The Old Theory: Scientists previously tried to calculate how loud this hum would be. They found a problem:
  • If the "push" (bias) was too weak, the walls wouldn't collapse hard enough to make the hum NANOGrav heard.
  • If the "push" was too strong (specifically at a certain angle called $\theta = \pi$), the walls would collapse with such violence that the hum would be deafeningly loud, far louder than what NANOGrav actually detected.
  • It seemed like this theory was a "Goldilocks" problem: it was either too cold or too hot, but never "just right."

The New Twist: The "Handshake" of Particles

This paper introduces two new ingredients to fix the story: Chirality Imbalance and Nonlocal Interactions.

1. The Chirality Imbalance (The "Spin" of the Crowd)

Imagine a crowded dance floor (the hot plasma of the early universe). Usually, people spin left and right equally. But sometimes, due to chaotic events (sphaleron transitions), everyone starts spinning mostly in one direction. This is chirality imbalance.

• The Analogy: Think of this as a "chiral chemical potential" ($\mu_5$). It's like a strong wind blowing through the dance floor, forcing everyone to spin the same way.
• The Effect: The authors found that when this "wind" is strong, it acts like a catalyst. It changes how the "push" (the bias) works. It can make a weak push stronger or a too-strong push weaker, depending on the situation.

2. Nonlocal Interactions (The "Long-Range" Connection)

Previous studies used a model where particles only talked to their immediate neighbors (like neighbors shouting over a fence). This paper uses a Nonlocal model, where particles can "feel" each other across a distance (like a telepathic connection).

• The Analogy: In the old model, the "cracks" (domain walls) were very sharp and narrow. In this new model, the cracks are wider and fuzzier.
• The Effect: This "fuzziness" changes the peak of the signal. Instead of a sharp, deafening spike in volume, the signal becomes a broader, smoother hill.

The Solution: Finding the "Sweet Spot"

By combining the "wind" (chirality imbalance) with the "fuzzy cracks" (nonlocal interactions), the authors found a way to make the theory work:

1. For the "Too Loud" case ($\theta \approx \pi$): The old model predicted a massive spike in volume that didn't match the data. The new model shows that because the "cracks" are wider (nonlocal) and the "wind" is blowing (chirality), that massive spike gets spread out. The volume drops from "deafening" to "just right," matching the NANOGrav hum perfectly.
2. For the "Too Quiet" case (Large $\theta$): Previously, if the angle was large, the push was too weak to make any sound. But the authors found that if the "wind" (chirality imbalance) is strong enough, it boosts the push. Suddenly, the walls collapse with enough energy to create the hum we hear, even in regions where it was previously thought impossible.

The Conclusion

The paper essentially says: "We found a way to tune the volume."

By accounting for the "spin" of particles in the early universe and how they interact over distances, the theory of collapsing cosmic walls can now perfectly explain the low-frequency hum detected by NANOGrav. It turns out that the universe's "cracks" didn't just snap; they healed in a way that created a perfect, observable song, provided we look at the physics with the right level of detail.

In short: The universe is humming, and this paper explains how the "music" of collapsing cosmic walls can be tuned to match the song we hear, thanks to some invisible "wind" and "long-distance connections" in the early universe.

Technical Summary

Here is a detailed technical summary of the paper "Impact of chirality imbalance and nonlocal interactions on the QCD-biased axionic domain-wall interpretation of NANOGrav 15-year data" by Ruotong Zhao and Zhao Zhang.

1. Problem Statement

The NANOGrav 15-year (NG15) dataset has reported evidence for a stochastic gravitational wave background (SGWB) in the nanohertz (nHz) frequency range. While supermassive black hole binaries are a primary candidate, new physics sources are being investigated. One promising scenario involves the collapse of axion-like particle (ALP) domain walls (DWs) induced by a bias potential from Quantum Chromodynamics (QCD).

Previous studies (specifically Ref. [75]) utilizing a local Nambu–Jona-Lasinio (NJL) model suggested that this scenario faces a significant challenge:

• In hot QCD matter, local CP-odd domains can form, characterized by a non-zero $\theta$ angle.
• The QCD bias potential is proportional to the topological susceptibility ($\chi_t$).
• Previous calculations showed that for large $\theta$ (specifically near $\pi$), $\chi_t$ is either suppressed (preventing DW collapse) or enhanced to such a degree at the critical temperature ($T_c$) that the resulting gravitational wave (GW) signal exceeds NG15 observational limits.
• Crucially, these studies ignored the chirality imbalance (quantified by the chiral chemical potential $\mu_5$), which naturally arises in hot QCD via sphaleron transitions and the axial anomaly.

The Core Question: Does the inclusion of a finite chirality imbalance ($\mu_5$) and a more realistic nonlocal QCD interaction model alter the topological susceptibility $\chi_t$ sufficiently to allow the ALP domain wall collapse scenario to remain consistent with NG15 data?

2. Methodology

The authors employ a theoretical framework combining cosmological GW generation with non-perturbative QCD modeling.

• Physical Model:

  • ALP Domain Walls: The study assumes ALPs exist prior to the QCD phase transition. A small $Z_2$ symmetry breaking potential (induced by the QCD bias) makes the domain walls unstable, leading to their collapse and the emission of GWs.
  • GW Signal Strength: The signal strength $\alpha_*$ is calculated based on the ratio of the released latent heat (from DW collapse) to the radiation energy density. This depends directly on the magnitude of the bias potential $\Delta V \propto |\chi_t|$.

• QCD Framework (Nonlocal NJL Model):

  • Instead of the local NJL model used in previous works, the authors utilize a Nonlocal Two-Flavor NJL (NNJL) model with 't Hooft interactions.
  • Rationale: The local NJL model fails to reproduce lattice QCD results at finite $\mu_5$ and does not adequately capture non-perturbative features like confinement dynamics and nonlocal quark-gluon coupling.
  • Key Parameters: The model incorporates:
    • Temperature ($T$)
    • CP-violating angle ($\theta$)
    • Chiral chemical potential ($\mu_5$), representing the chirality imbalance.
  • Form Factors: Three different sets of momentum-dependent form factors ($R(p)$ and $C(p)$) are tested (labeled NNJL I, II, and III) to ensure robustness.

• Calculation of $\chi_t$:

  • The topological susceptibility is defined as the second derivative of the thermodynamic potential with respect to $\theta$.
  • The authors solve the gap equations for the scalar and pseudoscalar condensates ($\sigma'_0, \eta'_0$) at finite $T, \mu_5, \theta$.
  • They compute the derivatives of these condensates with respect to $\theta$ to evaluate $\chi_t(T, \theta, \mu_5)$.

3. Key Contributions

1. Inclusion of Chirality Imbalance ($\mu_5$): This is the first study to simultaneously consider the effects of a finite $\theta$ angle and a finite chiral chemical potential $\mu_5$ on the QCD bias relevant to GW generation.
2. Adoption of Nonlocal NJL Model: By moving beyond the local approximation, the authors provide a more accurate description of the QCD vacuum structure, particularly regarding the behavior of condensates near the critical temperature and the critical point $\theta = \pi$.
3. Re-evaluation of the "Forbidden" $\theta$ Regions: The study challenges the conclusion that large $\theta$ values are incompatible with NG15 data, demonstrating that $\mu_5$ can catalyze $\chi_t$ to viable levels.

4. Key Results

• Catalytic Effect of $\mu_5$:

  • The chirality imbalance significantly enhances the chiral condensate and, consequently, the topological susceptibility $\chi_t$.
  • For large $\mu_5$ (e.g., 500 MeV), $\chi_t$ is roughly twice as large as the $\mu_5=0$ case at low temperatures.
  • This enhancement allows the QCD bias to overcome the suppression caused by the $\theta$ parameter in certain ranges.

• Revised $\theta$ Windows for NG15 Compatibility:

  • **Small $\theta$ ($0 \le \theta < 0.5\pi$):** The range of$\theta$compatible with NG15 data widens as$\mu_5$ increases.
  • Large $\theta$ ($0.5\pi < \theta \le \pi$):
    • In the local NJL model, the region near $\theta = \pi$ was problematic due to an overly sharp, divergent spike in $\chi_t$ at $T_c$, leading to GW signals too strong for NG15.
    • In the NNJL model, the peak of $|\chi_t|$ at $T_c$ for $\theta = \pi$ is significantly broader (less divergent) than in the local model.
    • Furthermore, for $\theta$ near $\pi$, a secondary peak in $\chi_t$ emerges near $T_c$ that is consistent with NG15 data.
    • Crucially, a new "middle" window emerges for large $\mu_5$ (e.g., $0.6\pi \le \theta \le 0.89\pi$) where the enhanced$\chi_t$allows the GW signal to fall within the NG15 2$\sigma$ contour.

• Behavior at Critical Temperature ($T_c$):

  • At $\theta = \pi$, the NNJL model predicts a second-order phase transition for CP restoration. The resulting $|\chi_t|$ peak is broad enough that the GW signal strength near $T_c$ is compatible with observations, unlike the "overly large" signal predicted by local models.

5. Significance and Conclusion

The paper concludes that the interpretation of the NANOGrav nHz signal as originating from QCD-biased axionic domain wall collapse is still viable, provided that:

1. Chirality imbalance is present: A sufficiently large chiral chemical potential $\mu_5$ (potentially generated by QCD sphalerons in the early universe) acts as a catalyst, enhancing $\chi_t$ and expanding the allowed parameter space for $\theta$.
2. Nonlocal interactions are considered: The nonlocal NJL model provides a more realistic description of the QCD phase transition, smoothing out the divergent spikes in $\chi_t$ seen in local models and allowing for a broader range of temperatures and angles to produce observable GWs.

Implications:

• The study resolves the tension found in previous local NJL studies where large $\theta$ values were ruled out.
• It suggests that the early universe could have hosted local CP-odd domains with significant chirality imbalance, which played a crucial role in the dynamics of axion domain walls and the resulting gravitational wave background.
• Future work should investigate the dynamical generation of $\mu_5$ and the inclusion of strange quarks and Polyakov-loop dynamics (PNNJL) to refine these predictions.