Resonant ALP-Portal Dark Matter Annihilation as a Solution to the $B^{\pm }\to K^{\pm }\nu \bar{\nu }$ Excess

The Belle II collaboration recently reported a $2.7\sigma$ excess in the rare decay $B^{\pm }\to K^{\pm }\nu \bar{\nu }$, potentially signaling new physics. We propose an axion-like particle (ALP)-portal dark matter (DM) framework to explain this anomaly while satisfying the observed DM relic abundance. By invoking a resonant annihilation mechanism ($m_a\sim 2m_{\chi }$), we demonstrate that the ALP-mediated interactions between the Standard Model and DM sectors simultaneously account for the $B^{\pm }\to K^{\pm }\nu \bar{\nu }$ anomaly and thermal freeze-out dynamics. Two distinct scenarios-long-lived ALPs decaying outside detectors (displaced diphotons) and ALPs decaying invisibly to DM pairs (missing energy)-are examined. While the displaced diphotons scenario is excluded by kaon decay bounds ($K^{\pm }\to {\pi }^{\pm }+\text{inv.}$), the invisible decay channel remains unconstrained and aligns with Belle II’s missing energy signature. Using the coupled Boltzmann equation formalism, we rigorously incorporate early kinetic decoupling effects, revealing deviations up to a factor of 20 from traditional relic density predictions in resonance regions. For the missing energy scenario, the viable parameter space features ALP-SM and ALP-DM couplings: $g_{aWW}(g_{a\gamma \gamma })\in (7.13\times {10}^{-5}-9.60\times {10}^{-5}){\text{GeV}}^{-1}$ (from $B^{\pm }\to K^{\pm }a$) and $g_{a\chi \chi }\in (7.12\times {10}^{-5}-7.73\times {10}^{-3}){\text{GeV}}^{-1}$ (for resonant annihilation), accommodating ALP masses $m_a\in (0.6,4.8)\text{GeV}$. Therefore, this work establishes the ALP portal as a viable bridge between the $B^{\pm }\to K^{\pm }\nu \bar{\nu }$ anomaly and thermal DM production, emphasizing precision calculations of thermal decoupling in resonance regimes.