Stacking the Deck: Gambling on a Light QCD Axion

TL;DR

The paper proposes a soft $\mathbb{Z}_N$-breaking extension of the Z_N axion to populate ultra-light QCD axions by coupling a reheaton to the Higgs, thereby reheating only our sector and lifting the discrete $1/N$ tuning. Concretely, a small parameter $\epsilon_\phi$ arising from Higgs-VEV shifts decouples the axion mass from its decay constant, yielding $m_a^2/(m_a^2)_{\rm QCD}=(-1)^{N+1}\epsilon_N+\epsilon_\phi$ and enabling light axions for generic $N$. The cosmology features a two-stage misalignment with three regimes: shuffled, rigged, and folded; high $T_{\rm rh}$ leads to an initial oscillation around $\theta=\pi$ followed by a second stage near $\theta=0$, while low $T_{\rm rh}$ reduces dynamics to ALP-like behavior. Depending on $f_a$ and $T_{\rm rh}$, the axion can account for all dark matter across broad regions of parameter space, significantly expanding viable targets beyond the canonical QCD axion line. The approach offers a testable framework connecting reheating dynamics, Higgs portal physics, and axion cosmology with potential astrophysical and laboratory constraints.

Abstract

We consider axions lighter than what their QCD couplings might otherwise suggest. Starting with a $\mathbb{Z}_N$-axion, we introduce a small explicit $\mathbb{Z}_N$ symmetry-breaking coupling between the Standard Model Higgs boson and a reheaton. This small explicit breaking allows us to populate a large portion of the light axion $m_a$-$f_a$ plane, removes the $1/N$ tuning in the $\mathbb{Z}_N$-axion, and explains why only our sector was reheated. Due to finite temperature effects, axions of this sort undergo either ``rigged" misalignment, where the axion misalignment angle is effectively $π$ regardless of its initial value; or ``shuffled" misalignment, where the initial angle is effectively randomized.

Figures (6)

• Figure 1: Plot of the lightness of the axion mass i.e.$\sqrt{\epsilon_\phi}$ as a function of the reheating temperature $T_{\rm rh}$, for different choices of $\Lambda_{\rm NP}$ and $m_\phi$. The red and turquoise lines show the parameter space for $\Lambda_{\rm NP}=100\,{\rm TeV}$ and $3\,{\rm TeV}$ respectively. The solid lines indicate the minimum value of $\epsilon_\phi$ that can be obtained for a given $\Lambda_{\rm NP}$ with $m_\phi/\Lambda_{\rm NP}=1\,$ whereas the dashed red and dashed turquoise lines are for $m_\phi=10\,{\rm TeV}$ and $0.5\,{\rm TeV}$ respectively. The values of $\epsilon_\phi$ in the turquoise shaded excluded region require $m_\phi\gtrsim \Lambda_{\rm NP}$ for $\Lambda_{\rm NP}=3\,{\rm TeV}$, where the validity of the EFT breaks down. The sudden drop in the values of $\epsilon_\phi$ around $T_{\rm rh}\sim 0.2\,{\rm GeV}$, is due to the change in $g_*(T)$ close to the QCD phase transition.
• Figure 2: Shuffled misalignment. The evolution of the axion angle $\theta$ ( top) and its normalized energy density $\rho_a \, t_c^2/f_a^2$ ( bottom) as a function of time $t$, for $f_a\simeq 2\times 10^{15}\,{\rm GeV}$ and $\epsilon_\phi=10^{-7}$. Top: The evolution of $\theta(t)$ for three very similar initial conditions: $\theta_{\rm rh} = \pi/4$ (blue), and $\theta_{\rm rh} = \pi/4(1\pm10\%)$ (red and green, respectively). In yellow we plot the evolution of a canonical QCD axion with same $f_a$ and $\theta_{\rm rh}=\pi/4$. The dashed (solid) black line shows $t_{\rm osc,\pi}$ ($t_{\rm osc,0}$), when the axion begins to oscillate around $\theta=\pi$ ($\theta=0$). The gray dashed vertical line marks $t_c$. Note that, due to the periodicity of the axion potential, the vertical axis in the top panel must be taken to be modulo $2\pi$. Therefore the final oscillatory stage, shown to be around $\theta = 2\pi$, can equivalently be interpreted to take place around $\theta=0$. Bottom: The evolution of the energy density of the axion. The solid lines correspond to the total energy density, while the thin lines to the kinetic energy term only. The colors have the same meaning as in the top panel, and so do the vertical lines.
• Figure 3: Rigged misalignment. The evolution of the axion angle $\theta$ ( top) and its normalized energy density $\rho_a \, t_c^2/f_a^2$ ( bottom) as a function of time $t$, for $f_a\simeq 2\times 10^{15}\,{\rm GeV}$ and $\epsilon_\phi=0.1$. Top: The evolution of $\theta(t)$ for three very different initial conditions: $\theta_{\rm rh}= \pi/4$ (solid blue), $\theta_{\rm rh} = \pi/2$ (solid red), and $\theta_{\rm rh} = 3\pi/4$ (dashed green). The vertical and horizontal lines have the same meaning as those in Fig. \ref{['fig:case_1a']}. Note that, due to the periodicity of the axion potential, the vertical axis in the top panel must be taken to be modulo $2\pi$. Therefore, the blue and green curves' oscillations around $\theta = 2\pi$ are the same as those of the red curve around $\theta = 0$. Bottom: The evolution of the energy density of the axion shifted by a constant such that when $t<t_c$, $\theta = \pi$ has zero energy and when $t>t_c$, $\theta = 0$ has zero energy. The colors have the same meaning as in the top panel, and so do the vertical lines. The black horizontal line is $\rho(\theta=\pi) - \rho(\theta=0)$. The fact that, regardless of initial conditions, the axion total energy density at $t=t_c$ jumps to black line demonstrates that it is restarting the misalignment mechanism with an angle rigged to $\theta = \pi$.
• Figure 4: The $m_a$--$f_a$ parameter space of our model, for the $T_{\rm rh} \gtrsim T_c$ case. Here we have taken $T_{\rm rh} = 1~\,{\rm GeV}$. The dashed green and pink horizontal lines correspond to $f_a = (f_a)_{\rm crit}$ and $f_a = M_{\rm Pl}$, respectively. The blue line corresponds to the canonical QCD axion. The brown region is ruled out by various astrophysical and cosmological bounds. The dashed green vertical line corresponds to $m_a = H_c$, which represents the boundary between the shuffled and rigged regimes of the two-stage misalignment mechanism of our model. Since $\epsilon_\phi < 1$, both regimes lie to the left of the blue line. In the purple region $\epsilon_\phi \lesssim (\epsilon_\phi)_{\rm min}$ as given by Eq. \ref{['eq:ephi_Trh']}. For this plot, $(\epsilon_\phi)_{\rm min} \approx 9 \times 10^{-20}$, for $m_\phi = \Lambda_{\rm NP} = 3\,\,{\rm TeV}$, and $T_{\rm rh} = 1\,\,{\rm GeV}$. The black line corresponds to those values of $f_a$ and $m_a$ for which the axion can account for the entirety of the dark matter. In the shuffled regime, one can still get the right dark matter abundance above or below this line by tuning $\theta_{\rm osc,0}$ in the ways indicated in the plots. In the rigged regime, all points above (below) the black line, shown by the light red shading, correspond to an under-abundance (overabundance) of axion dark matter. We have omitted the experimental bounds on this parameter space to avoid overcrowding the plot; they can be seen in Fig. \ref{['fig:DM_small_Trh']}.
• Figure 5: The parameter space of our model when $T_{\rm rh} \lesssim T_c$. In this case, there is no first stage of oscillations around $\theta = \pi$, and thus there is neither a shuffled nor a rigged regime. The dashed pink and blue lines and brown region have the same meaning as those in Fig. \ref{['fig:DM_large_Trh']}. Since $\epsilon_\phi < 1$, our model lies to the left of the blue line. The dashed green vertical lines correspond to $m_a = 3 H_{\rm rh}$, for $T_{\rm rh} = 10\,\,{\rm MeV}$ and $T_{\rm rh} = 100\,\,{\rm MeV}$. These lines represent the boundary between when the axion oscillates some time afterward reheating (left of the line) or the more model dependent case where it starts oscillating before or at reheating (right of the line). In the purple region $\epsilon_\phi \lesssim (\epsilon_\phi)_{\rm min}$ as given by Eq. \ref{['eq:ephi_Trh']}. For this plot, $(\epsilon_\phi)_{\rm min} \approx 3 \times 10^{-24}$, for $m_\phi = \Lambda_{\rm NP} = 3\,\,{\rm TeV}$, and $T_{\rm rh} = 10\,\,{\rm MeV}$. The black line corresponds to those values of $f_a$ and $m_a$ for which the axion can account for the entirety of the dark matter. One can still get the right dark matter abundance above or below this line depending on the value of $\theta_{\rm rh}$. The gray shaded regions represent combined exclusions from various model dependent astrophysical, cosmological, and laboratory experimental results Hook:2017psmZhang:2021mksSchulthess:2022pbpBlum:2014vsaGue:2025nxqZhang:2022ewzJEDI:2022hxaRoussy:2020ilyAbel:2017rtmZhang:2023lemAlda:2024xxaBanerjee:2023bjcGomez-Banon:2024ouxKumamoto:2024wjdBalkin:2022qerHook:2017psmLucente:2022vuoSpringmann:2024retCaloni:2022uyaBaryakhtar:2020gaoBaryakhtar:2020gaoStott:2020gjjUnal:2020jiyHoof:2024qukWitte:2024drgCaputo:2024oqcIwamoto:1984irBuschmann:2021juvSpringmann:2024mjpSpringmann:2024ret.