Heavy QCD Axion in $b\to s$ transition: Enhanced Limits and Projections

We study a "heavy" QCD axion whose coupling to the standard model is dominated by $a G \widetilde{G}$ but with $m_a \gg m_\pi f_\pi / f_a$. This is well motivated theoretically as it can evade the infamous axion quality problem. It also poses interesting challenges for its experimental search due to its suppressed couplings to photons and leptons. Such axion with mass around a GeV is kinematically inaccessible or poorly constrained by most experimental probes except B-factories. We study $B \to K a$ transitions as a powerful probe of the heavy QCD axion by performing necessary 2-loop calculations for the first time. We find some of the existing limits are enhanced. For forthcoming data sets of the Belle II experiment, we provide a projection that $f_a$ of a few TeV is within its future reach, which is relevant to the quality problem.