Reconciling axion quality with post-inflation cosmology

Technical Summary: Reconciling Axion Quality with Post-Inflation Cosmology

Problem Statement
The paper addresses the tension between the "axion quality problem" and the requirements of post-inflationary Peccei-Quinn (PQ) breaking scenarios. In post-inflationary models, the PQ symmetry breaks after cosmic inflation, leading to the formation of axion strings and domain walls. To avoid the domain-wall problem (where a stable network dominates the energy density), the model must produce a minimal string configuration ($N_{DW}=1$) such that the string-wall network is unstable. Simultaneously, the model must protect the accidental global $U(1)_{PQ}$ symmetry from explicit breaking by higher-dimensional operators to a level where the axion potential minimum does not shift by more than $\langle a \rangle/f_a \lesssim 10^{-10}$. Furthermore, UV completions addressing the quality problem often introduce new stable particles. In post-inflationary scenarios, these dangerous relics (particularly fractionally charged ones) cannot be inflated away and must be either unstable or cosmologically subdominant. Existing models often fail to satisfy all these constraints simultaneously without introducing Landau poles in Standard Model (SM) gauge couplings below the PQ-breaking scale.

Methodology and Framework
The authors propose a framework based on an $SU(N)_L \times SU(N)_R$ gauge symmetry spontaneously broken to the diagonal subgroup $SU(N)_{L+R}$ by a bifundamental scalar field $Y$.

• Axion Quality: The $U(1)_{PQ}$ symmetry arises accidentally from the renormalizable scalar potential. The leading operator violating this symmetry is the determinant, $\det Y$, which has dimension $N$. This ensures that PQ violation is suppressed by $(\Lambda_{PQ})^{-(N-4)}$, becoming exact in the large $N$ limit.
• Topological Stability: The symmetry breaking pattern $SU(N)_L \times SU(N)_R \times U(1)_{PQ} \to SU(N)_{L+R}$ realizes the Lazarides-Shafi gauge identification, producing minimal axion strings ($N_{DW}=1$) to which a single domain wall attaches, ensuring the network decays.
• Fermion Sector and Relics: The model introduces QCD-colored fermions ($Q$) and three QCD-singlet fermions ($\Psi_a$). Unlike previous constructions where these carried zero hypercharge (leading to stable fractionally charged relics), the authors assign specific hypercharges to the $\Psi_a$ fields. This allows for dimension-six operators that transfer conserved charges from heavier to lighter species, ensuring all unstable particles decay before Big Bang Nucleosynthesis (BBN).
• Stable Baryons: After $SU(N)_{L+R}$ confinement, the lightest baryon carrying the residual $U(1)_{Q-\Psi}$ symmetry is cosmologically stable. The authors analyze two mass orderings ($m_Q < m_\Psi$ and $m_\Psi < m_Q$) to determine the nature of this relic (colored vs. color-singlet, charged vs. neutral).
• Baryon Suppression: To ensure the stable baryon does not overclose the universe, the authors invoke the "Casimir bottleneck" mechanism (citing Refs. [56, 57]). In large $N$ gauge theories, the formation of baryons is strongly suppressed relative to mesons due to the dynamics of cluster assembly.

Key Contributions and Results

1. Resolution of the Quality/Cosmology Tension: The authors demonstrate that for specific values of $N$ (specifically $N=15, 18, 21$), the model satisfies the PQ quality bound, avoids the domain-wall problem, and suppresses the abundance of stable relics to sub-dominant levels (below 10% of Dark Matter density).
2. Hypercharge Assignment and Decay: By assigning hypercharges to the $\Psi_a$ fields, the model avoids stable fractionally charged relics. The dimension-six operators facilitate decays of heavy states, ensuring no long-lived charged particles survive to BBN.
3. Neutral Baryon Candidates:
   • In the $m_Q < m_\Psi$ ordering, the stable relic is a QCD-colored, electrically neutral baryon ($B_Q$) which is screened to a color singlet by gluons.
   • In the $m_\Psi < m_Q$ ordering, a specific hypercharge splitting can select an electrically neutral, QCD-singlet mixed-flavor baryon ($B_0$) as the lightest state. This $B_0$ is identified as the "cleanest" relic candidate.
4. Perturbativity and Landau Poles: The paper performs a two-loop renormalization group (RG) analysis to ensure that the new fermion content does not drive the SM gauge couplings ($SU(3)_C$ and $U(1)_Y$) to Landau poles below the PQ-breaking scale $\Lambda_{PQ}$.
   • For $N \le 12$, the suppression of PQ violation and baryon formation is insufficient to satisfy all constraints simultaneously.
   • For $N=15, 18, 21$, viable parameter spaces exist where the axion constitutes the majority of Dark Matter, and the axion mass remains predictable.
5. Benchmark Scenarios:
   • $N=15, 18$: Allow for a separation between the PQ-breaking scale $\Lambda_{PQ}$ and the flavor-transfer scale $\Lambda_6$ (e.g., $\Lambda_6 = 10^{12}$ GeV), facilitating rapid decays.
   • $N=21$: Allows for a common scale $\Lambda_6 = \Lambda_{PQ} = 10^{14}$ GeV with order-one Yukawa couplings, while maintaining perturbativity and satisfying all cosmological bounds.

Significance and Claims
The paper claims to provide a concrete, UV-complete realization of a high-quality QCD axion model specifically designed for post-inflationary PQ breaking. Its significance lies in reconciling three traditionally conflicting requirements:

1. High PQ Quality: Achieved via the accidental symmetry protected by large operator dimension $N$.
2. Cosmological Viability: Achieved by ensuring an unstable string-wall network and the absence of dangerous stable relics (via decay channels and the Casimir bottleneck suppression of baryon formation).
3. Perturbativity: Achieved by carefully selecting $N$ and hypercharge assignments to avoid low-scale Landau poles.

The authors emphasize that in this post-inflationary scenario, the axion mass and Dark Matter abundance are calculable and predictable, as the initial misalignment angle is statistically fixed rather than a free parameter. The model relies on the phenomenological modeling of large $N$ confinement dynamics (specifically the Casimir bottleneck) to estimate relic densities, acknowledging that these are benchmark estimates rather than first-principles predictions. The work establishes that $N=15, 18,$ and $21$ represent the minimal viable solutions where all constraints are jointly satisfied.