Comprehensive Constraints on ALP Couplings from future $e^+e^-$ Colliders, Muon $g-2$, Thermal Dark Matter and Higgs Measurements

In this article, we present projected 95\% C.L. limits on Axion-Like Particle (ALP) couplings from ALP production at a future $e^+e^-$ collider operating at $\sqrt{s} = 250~\text{GeV}$ with integrated luminosity $L = 0.5~\text{ab}^{-1}$. We constrain the effective couplings $g_{\gamma\gamma}$, $g_{Z\gamma}$, $g_{ZZ}$, and $g_{WW}$ over the ALP mass range $20~\text{GeV} \leq m_a \leq 100~\text{GeV}$, finding projected bounds at the level of $\mathcal{O}(10^{-1})~\text{TeV}^{-1}$ for $g_{\gamma\gamma}/f_a$. Given that the latest muon anomalous magnetic moment measurement ($\Delta a_\mu$) shows no statistically significant deviation from the Standard Model prediction, we reinterpret the ALP contributions to $\Delta a_\mu$ as a stringent consistency requirement. We then derive the corresponding allowed regions for $g_{\gamma\gamma}$ and the ALP--muon coupling $C_{\mu\mu}$, and apply them to a fermionic dark matter scenario in which the relic density depends on both the dark matter mass $m_\chi$ and $m_a$. The same parameter space is further constrained by Higgs signal strength measurements through $h \to \gamma\gamma$ and $h \to Z\gamma$. A comparative analysis with existing experimental and theoretical bounds highlights the complementarity of $\Delta a_\mu$, dark matter, and Higgs observables in restricting ALP couplings, demonstrating that even in the absence of a $\Delta a_\mu$ anomaly, these constraints provide essential guidance for viable ALP parameter space.