(Non-)Perturbative Dynamics of a Light QCD Axion: Dark Matter and the Strong CP Problem

Abstract

Considerable theoretical efforts have gone into expanding the reach of the QCD axion beyond its canonical mass--decay-constant relation. The $Z_\mathcal{N}$ QCD axion model reduces the QCD axion mass naturally, by invoking a discrete $Z_\mathcal{N}$ symmetry through which the axion field is coupled to $\mathcal{N}$ copies of the Standard Model. Before the QCD phase transition at temperature $T_{\rm QCD}$, the $Z_\mathcal{N}$ potential has a minimum at misalignment angle $θ=π$. At $T_{\rm QCD}$, $θ=π$ becomes a maximum; the axion potential becomes exponentially suppressed and develops $\mathcal{N}$ minima -- only one of which actually solves the strong CP problem. Before $T_{\rm QCD}$, $θ$ relaxes towards $π$. After $T_{\rm QCD}$, the axion field starts from around the hilltop and may have sufficient kinetic energy to overcome the newly suppressed potential barriers. Such a field evolution leads to nonlinear effects via the self-interactions near the hilltop, which can cause the exponential growth of fluctuations and backreaction on the coherent motion. This behavior can influence the relic density of the field and the minimum in which it settles. We conduct the first lattice simulations of the $Z_{\mathcal{N}}$ QCD axion using ${\mathcal C}$osmo${\mathcal L}$attice to accurately calculate dark matter abundances and find nonlinear dynamics reduce the abundance by up to a factor of two. We furthermore find that the probability of solving the strong CP problem tends to diverge considerably from the naive expectation of $1/\mathcal{N}$.

Figures (8)

• Figure 1: Temperature evolution of the $Z_{\mathcal{N}}$ potential for $\mathcal{N}=3$.
• Figure 2: Contours of the critical amplitude $\theta_c$, which is the minimum initial amplitude $|\pi-\theta_i|$ necessary for the axion to reach the CP-conserving minimum after the QCD phase transition. The thin yellow lines show different values of $Z_{\mathcal{N}}$ with $Z_{\mathcal{N}}=27$ near the white dwarfs constraint and $Z_{\mathcal{N}}=5$ above the canonical QCD axion line, whereas $Z_{\mathcal{N}}=3$ is too close to the QCD axion line to be visible. The shaded regions show astrophysical constraints from supernovae Springmann:2024ret, the solar core Hook:2017psm, white dwarf composition Balkin:2022qer, neutron star cooling Gomez-Banon:2024ouxKumamoto:2024wjd, stellar black hole spins Baryakhtar:2020gao, and axion-nucleon couplings from GW170817 Zhang:2021mks. Regions above the purple dashed lines are within the reach of CASPEr-Electric JacksonKimball:2017elr and the experiment involving the piezoaxionic effect Arvanitaki:2021wjk.
• Figure 3: The black dots show the final minima that the axion settles in as a function of the QCD time/temperature variation (upper/lower panel) from the zero-mode analysis with parameters specified at the top axes. The horizontal gray lines indicate the location of the potential minima, while the CP-conserving minimum is at the blue dashed lines.
• Figure 4: The average probability of solving the strong CP problem, accounting for variations in the initial misalignment angle $\theta_i$ and the uncertainty in the QCD phase transition temperature $T_{\rm QCD}$. The timing of the QCD phase transition is varied up to an axion oscillation cycle or over the theoretical 2% uncertainty in $T_{\rm QCD}$, whichever is smaller. Here, the initial amplitude $|\pi-\theta_i|$ is restricted to values above the critical angle $\theta_c$, below which the probability vanishes identically. The solid black line indicates the naı ve expectation, $P=1/\mathcal{N}$.
• Figure 5: Same as Fig. \ref{['fig:prob1']}, but with $|\pi - \theta_i|$ allowed to vary over the full range $(0, \pi)$, now including the ranges of $\theta_i$ where probability vanishes. The $y$-axis is on logarithmic scale due to the substantially lower probabilities.