A MeV-Scale Dark QCD Solution to the Axion Domain Wall Problem

Abstract: PQ solution to the strong CP problem probably encounters the axion domain wall problem. In this article, we propose a simple and testable solution, assuming that the $U(1)<em>{\rm PQ}$ possesses mixed anomaly to a hidden $SU(N_c)$ color. Then, the axion field receives a new cosine potential from the hidden instantons, which breaks the $Z</em>{N_{\rm DW}}$ subgroup explicitly. The new potential lifts the vacua degeneracy, but also drives the effective $θ$ angle away from the origin, re-incuring the strong CP problem. However, we find that the dark QCD scale within the 0.1 to 3 MeV window survives, maintaining a delicate balance. Two observational signatures are explored: gravitational waves from domain wall collapse, already probed by current PTAs, and di-photon signals from axion-dark-glueball mixing, which require next-generation MeV telescopes. The scenario favors a cold dark QCD sector consistent with dark glueball relic constraints.

• The paper introduces a novel dark QCD mechanism that creates a temperature-dependent cosine potential to explicitly break the axion domain wall symmetry.
• It establishes constraints on the dark QCD transition temperature (0.1–3 MeV) to reconcile the strong CP problem with dark matter relic densities.
• Distinct observational signatures are predicted through gravitational waves from PTA experiments and di-photon signals from next-generation MeV gamma-ray telescopes.

A MeV-Scale Dark QCD Solution to the Axion Domain Wall Problem

Introduction

The strong CP problem—the unnaturally small value of $\Theta$ in QCD implied by neutron EDM constraints—remains a pivotal unsolved question in high-energy theory. The Peccei-Quinn (PQ) mechanism, introducing a light axion from an anomalous $U(1)_{\rm PQ}$ broken at a high scale, elegantly explains the dynamical relaxation of this phase. However, for domain wall number $N_{\rm DW} > 1$, random vacuum selection after PQ breaking leads to stable topological axion domain walls, creating severe cosmological issues unless inflation dilutes them or $N_{\rm DW}=1$. Conventional approaches invoke high-dimensional $U(1)_{\rm PQ}$-breaking operators to lift vacuum degeneracy, but these are highly constrained.

This paper presents an alternative: resolving the axion domain wall problem by introducing a dark $SU(N_c)$ "dark QCD" sector, coupled to the axion via a mixed anomaly. The resulting dark instantons provide an additional, temperature-dependent cosine potential that explicitly breaks the $Z_{N_{\rm DW}}$ symmetry, biases the universe toward a unique vacuum, and ensures cosmological viability. Within a specific dark QCD scale window ($0.1$–$3$ MeV), both the strong CP solution and dark matter relic constraints are satisfied. Experimental prospects are promising, particularly through pulsar timing array (PTA) gravitational wave searches and next-generation MeV gamma-ray telescopes targeting di-photon signals from the dark sector.

Theoretical Framework

The construction extends the PQ mechanism by introducing a mixed $U(1)_{\rm PQ}$ anomaly with a hidden $U(1)_{\rm PQ}$0, so the axion couples to both $U(1)_{\rm PQ}$1 and $U(1)_{\rm PQ}$2 gauge fields via

$U(1)_{\rm PQ}$3

The QCD contribution produces the $U(1)_{\rm PQ}$4-fold degenerate axion potential, while the dark sector generates a secondary cosine term with its own topological susceptibility $U(1)_{\rm PQ}$5, explicitly breaking $U(1)_{\rm PQ}$6 and lifting vacuum degeneracy.

To avoid reintroducing the strong CP problem, $U(1)_{\rm PQ}$7 must remain sufficiently small to ensure the axion vev stays within $U(1)_{\rm PQ}$8 of the QCD CP-conserving vacuum. This imposes a stringent upper limit on the dark QCD phase transition temperature: generically $U(1)_{\rm PQ}$9 MeV, with lower limits from cosmological wall collapse requirements and glueball dark matter abundance. A key feature is the preference for a cold dark sector ($N_{\rm DW} > 1$0) to suppress dark glueball relics.

Cosmological Dynamics and Vacuum Selection

Lattice and semi-analytic results for $N_{\rm DW} > 1$1 Yang-Mills yield reliable predictions for the temperature-dependent dark topological susceptibility $N_{\rm DW} > 1$2, exhibiting a sharp decline above the dark QCD transition $N_{\rm DW} > 1$3. Detailed balance between $N_{\rm DW} > 1$4, $N_{\rm DW} > 1$5, $N_{\rm DW} > 1$6, $N_{\rm DW} > 1$7, and the explicit PQ violation determines if the domain wall network collapses before Big Bang Nucleosynthesis.

Bias-induced vacuum selection and wall collapse generate observationally relevant gravitational waves. The collapse must occur at $N_{\rm DW} > 1$8 MeV to avoid BBN disruption, which restricts the allowed parameter space to a narrow window. A cooler dark sector, favored by dark glueball cosmology, can dramatically expand the viable range for $N_{\rm DW} > 1$9.

Gravitational Wave Signatures from Domain Wall Collapse

The dominant observational signature arises from the violent collapse of axion domain walls, producing a stochastic gravitational wave background with a peak frequency in the nanohertz range, ideally suited to PTA experiments. The peak amplitude of the GW signal,

$N_{\rm DW}=1$0

is enhanced at lower collapse temperatures and higher axion scales, with domain wall tension and number playing central roles.

Recent 3+1D simulations demonstrate that GW emission persists until $N_{\rm DW}=1$1, substantially boosting signal strength compared to analytic quadrupole estimates. The resulting spectrum displays a doubly broken power law, easily distinguished from SMBH binary backgrounds.

Signals predicted by this model with $N_{\rm DW}=1$5 and $N_{\rm DW}=1$6 GeV are already within the reach of current NANOGrav limits (2607.04137), with SKA poised to enable precision spectral measurements and discrimination from astrophysical backgrounds.

Dark Glueball Phenomenology and Di-Photon Signals

The dark QCD sector is expected to produce composite glueball states. The pseudoscalar $N_{\rm DW}=1$7 glueball mixes with the axion via the topological anomaly, enabling rare decays to diphotons. The decay width

$N_{\rm DW}=1$8

is highly suppressed except for maximal mass ($N_{\rm DW}=1$9 MeV) and enhanced anomaly coefficients. Corresponding lifetimes easily exceed $U(1)_{\rm PQ}$0~s over most of parameter space, evading all current constraints from INTEGRAL/SPI. Only optimistic corners—large couplings or maximal glueball mass—will be accessible to missions such as AMEGO-X, e-ASTROGAM, and COSI.

An additional signature may arise if $U(1)_{\rm PQ}$1–$U(1)_{\rm PQ}$2 mixing is sizable, producing a double-line photon spectrum with energies $U(1)_{\rm PQ}$3 and $U(1)_{\rm PQ}$4. The observability hinges on the size of the dark $U(1)_{\rm PQ}$5-term and experimental sensitivity to $U(1)_{\rm PQ}$6~s.

Implications and Future Prospects

This framework demonstrates that a minimal, dynamically-motivated extension of the PQ solution with a MeV-scale dark QCD sector can resolve the axion domain wall problem while leaving the strong CP solution intact. The most direct evidence will come from PTA GW signals, which probe much of the allowed parameter space.

The di-photon signature from dark glueballs, while theoretically informative, will only be accessible in the most optimistic scenarios and will require next-generation gamma-ray telescopes. The apparently orthogonal dependence of GW and photon signals on $U(1)_{\rm PQ}$7 and $U(1)_{\rm PQ}$8 enables a potential cross-verification strategy if both channels are observed.

The model's theoretical structure can be embedded in broader BSM frameworks, and the temperature/density profile of such a dark sector is of significant cosmological relevance beyond the axion context.

Conclusion

The proposed MeV-scale dark QCD solution introduces a dynamically consistent, observationally testable resolution to the axion domain wall problem. Distinct from higher-dimensional explicit PQ-violating operators, this mechanism exploits a mixed anomaly and MeV-scale hidden confinement to explicitly break the dangerous $U(1)_{\rm PQ}$9 symmetry. The parameter space is sharply delineated by theoretical consistency (avoiding strong CP reintroduction), dark glueball relic density and BBN, and can be explored by PTA gravitational wave observations and future MeV telescopes. Empirical verification will decisively test the viability of this approach and further probe the axion dark sector at the interface of cosmology and particle physics.