Discrete \texorpdfstring{$θ$}{theta} Projection: A Gauge-Protected Solution to the Strong CP Problem Without Axions

Abstract

We address the strong CP problem: why the physical QCD angle theta-bar must be extraordinarily small given the stringent bounds on the neutron electric dipole moment. Peccei-Quinn axion models can relax theta-bar dynamically, but rely on an approximate global symmetry expected to be violated by quantum gravity and face severe astrophysical and cosmological constraints. We propose Discrete theta Projection, an axionless, gauge-protected resolution obtained by gauging a finite cyclic subgroup $Z_N $of the $2π$ shift symmetry of theta. Coupling QCD to a compact, local and gapped topological sector orbifolds the path integral, identifying theta values that differ by $2π/N$ and admitting only instanton sectors whose topological charge lies in $Z_N$. In the large four-volume limit the vacuum energy becomes the lower envelope of the orbifold images, so the theory dynamically selects the branch closest to the CP-symmetric point, enforcing $|\barθ| \le π/N$ without assuming any prior smallness. Because the discrete shift is gauged, continuous renormalization of theta is forbidden; the construction can be formulated via higher-form/two-group structure with integer-quantized couplings fixed by anomaly inflow, ensuring radiative and gravitational stability and satisfying mixed gauge-gravity consistency conditions. The framework predicts a neutron EDM suppressed by $1/N$, no axion signatures, no domain-wall/isocurvature issues, and lattice diagnostics: piecewise-analytic theta dependence with cusps at odd fractions of the reduced period and a global curvature scaling as $1/N^2$. We provide the EFT construction, a nonperturbative proof of vacuum projection, a full anomaly analysis, and UV embeddings (including discrete clockwork chains) that generate large effective N while preserving integrality and consistency throughout.

Figures (13)

• Figure 1: Orbifolding of the $\theta$-circle under discrete shift symmetry. Points separated by $2\pi/N$ along the original $\theta$-circle are identified, yielding a smaller fundamental domain.
• Figure 2: Discrete $\theta$-vacua (labeled by an integer $k$ mod $N$) related by the $\theta \to \theta + 2\pi/N$ shift. In the ungauged theory, these are distinct vacua leading to a $\mathbb{Z}_N$ degeneracy. A domain wall (dashed segment) interpolates between neighboring vacua that differ by a discrete $\theta$-shift. Crossing the domain wall changes the topological angle by $2\pi/N$ and transfers one unit of topological charge. Gauging the $\mathbb{Z}_N$ shift symmetry identifies all these vacua as a single state, and the domain wall becomes unobservable.
• Figure 3: Schematic multi-branch vacuum energy $E(\theta)$ obtained as the lower envelope of shifted branch functions (here exemplified by $E_0[1-\cos(\theta+\tfrac{2\pi k}{N})]$) for $N=3$ and $N=5$. Vertical dashed lines indicate the transition (cusp) locations at $\theta=\tfrac{(2\ell+1)\pi}{N}$. The ground state always lies in the principal cell $[-\pi/N,\pi/N]$, realizing $|\theta_{\rm eff}|\le \pi/N$.
• Figure 4: Predicted neutron EDM $|d_n|$ as a function of the discrete symmetry order $N$ in the D$\theta$P framework (blue band, reflecting the range $d_n\approx(1$--$3)\times10^{-16}\,\bar{\theta}\,e\,$cm). The red dashed line indicates the current experimental $90\%$ C.L. limit $|d_n|<1.8\times10^{-26}$$e$·cm Abel:2020pzs. For $N\gtrsim10^{10}$ the D$\theta$P prediction lies well below the current bound.
• Figure 5: Comparison of the PQ axion potential and the discrete-$\theta$ projected potential. Blue dashed curve:$V_{PQ}(\theta)/\chi_t = 1-\cos\theta$, the standard cosine potential (for illustration we set $\bar{\theta}=0$ and $N_{DW}=1$ so that $\theta=a/f_a$). The axion has a single minimum at $\theta=0$ and oscillates with full $2\pi$ periodicity, implying a mass $m_a^2=\chi_t/f_a^2$ and multi-vacuum structure only if $N_{DW}>1$. Red solid curve:$V_{D\theta P}(\theta)/\chi_t$ for a gauged $Z_3$ (discrete $\theta$-periodicity $2\pi/3$). The physical vacuum energy is the lower envelope of the $1-\cos\theta$ branches shifted by $\pm 2\pi/3$, resulting in a unique minimum at $\theta=0$ and a maximal $|\theta|=\pi/3$ in any vacuum. In the $Z_N$ case, the effective $\bar{\theta}$ is confined to $[-\pi/N,\pi/N]$, and would-be $\theta$ vacua are identified by the gauge symmetry (no degenerate minima).

Theorems & Definitions (1)

• Lemma 3.1