Schwinger Model with a Dynamical Axion

Abstract

One of the major open puzzles in the Standard Model of particle physics is the strong CP problem: although Quantum Chromodynamics allows a CP-violating topological $θ$-term, experiments constrain its value to be extremely small. The Peccei--Quinn mechanism resolves this problem by promoting the $θ$-angle to a dynamical field-introducing the axion -- whose dynamics relax the effective angle $θ_\text{eff}$ to a CP-conserving minimum. Here, we investigate the resulting axion physics in a Hamiltonian lattice gauge theory (LGT) by coupling a quantized axion field to the massive Schwinger model with a topological $θ$-term. Using infinite matrix product state techniques, we compute the ground-state properties of the resulting theory and demonstrate that the axion dynamically relaxes $θ_\text{eff}$ to the minimum of the vacuum energy. Consequently, the ground-state energy becomes independent of $θ$, demonstrating the axion-mediated solution to the strong CP problem within a fully dynamical LGT. We further analyze CP restoration and extract the axion mass from the topological susceptibility and excitation spectrum. Our results provide a nonperturbative demonstration of axion dynamics in a quantum LGT amenable to investigation on modern quantum hardware.

Figures (7)

• Figure 1: Schematic illustration of the effect on the ground-state energy of the Schwinger model $E_0^\text{Sch}$ induced by adding an axion field. Initially, the system is described by Hamiltonian \ref{['eqn:ThethaSchwingerModel']}, whose ground-state energy depends on $\theta$. By promoting $\theta$ to a dynamical field $\hat{\theta}_\text{eff}$, which introduces the axion field, the expected value $\langle \hat{\theta}_\text{eff} \rangle$ is driven to the minimum of the ground-state energy of the Schwinger model. This results in the ground-state energy of the Hamiltonian \ref{['eqn:FullAxionHamiltoinan']} being independent of $\theta$.
• Figure 2: iDMRG calculations of the ground-state energy dependence of Hamiltonian \ref{['eqn:FullAxionHamiltoinan']} without dynamical axion fields on the topological $\theta$-angle at (a) $g^2=1$ and $s\in \{\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\}$, (b) $g^2=1$ and $s\in \{1, 2, 3\}$, (c) $g^2=5, 10$ and $s\in \{\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\}$, (d) $g^2=5, 10$ and $s\in \{1, 2, 3\}$, with increasing $s$ going from lighter to darker shades in the curves. For half-integer spin on plots (a) and (c), as $s$ or $g^2$ increases, the minima move closer to the expected $\theta=0,2\pi$, although for $g^2=1$ the shown truncations still lead to sizable shifts of the minima. For integer spin on plots (b) and (d), the minima are located at the expected values for $g^2=5,10$, while there is no minimum at $\theta=2\pi$ for $g^2=1$. The minima at $\theta =0, 2\pi$ do not show the desired $2\pi$ periodicity, but as $s$ increases, the shape of the axion potential moves closer to showing a period of $2\pi$.
• Figure 3: iDMRG calculations of the (a) dependence of the ground-state energy of Hamiltonian \ref{['eqn:FullAxionHamiltoinan']} on the topological $\theta$-angle and (b) the expectation value of the axion field $\hat{a}$ as a function of $\theta$, for the $s=1$ QLM. An equivalent study is done for the $s=\frac{3}{2}$ QLM in plots (c) and (d). In (a) and (c), the energy shows no dependence on the $\theta$ angle, as the axion drives the effective angle to the minimum of the axion potential $\theta_\text{min}$. The only exception is in (c) for $g^2=1$, where a slight deviation from a flat line is observed, due to the flatness of the axion potential in this case. On (b) and (d), the overlap between $\langle \hat{a} \rangle$ and $-\frac{f_a\theta}{2\pi}$ confirms that $\langle\hat{\theta}_\text{eff}\rangle=0$ for $s=1$. On the other hand, for $s=\frac{3}{2}$ we have $\langle\hat{\theta}_\text{eff}\rangle=\theta_\text{min}$, a constant value for $g^2=5,10$, which grows closer to zero as we increase $g^2$. Similarly, because of the axion potential's flatness, the behavior is much more irregular for $g^2=1$.
• Figure 4: Expectation value of the electric field $\langle \hat{E}\rangle$ as a function of $\theta$ for the massive Schwinger model with axions (dashed lines) and without axions (solid lines) for $g^2\in\{1,5,10\}$ and a QLM truncation of $s=1$. For the case without axions, CP symmetry is violated and $\langle\hat{E}\rangle \neq 0$ for $\theta\neq0$, while introducing the axions leads to $\langle \hat{\theta}_\text{eff}\rangle =0$, restoring CP symmetry, as evident from $\langle\hat{E}\rangle = 0$.
• Figure S1: iDMRG calculations of (a) the ground-state energy of Hamiltonian \ref{['eqn:FullAxionHamiltoinan']} on the topological $\theta$-angle, (b) the expectation value of the axion field $\hat{a}$ as a function of $\theta$, for the spin $s=1$ QLM, $N_\text{max}=32$ and $m_a^2=0.5$. The ground-state energy is no longer a constant, and there is no axion cancellation, as a result of the breaking of the axion mechanism for a nonzero bare mass.