Soft Collinear Effective Theory for Heavy QCD Axions

We develop a soft-collinear effective theory (SCET) framework for heavy QCD axion, considering two low-energy realizations and taking $B\to Ka$ as a benchmark mode. In the first realization, $aG\widetilde G$ is assumed to be the only independent axion interaction at the scale $\mu\sim m_b$. We show that eliminating the redundant flavor-changing derivative-gluon operator in the weak effective theory generates a new dimension-seven axion operator identified as $\mathrm{O}_{\partial ag}$. We match this operator onto SCET and derive the corresponding leading-power soft and spectator-scattering contributions to $B\to Ka$. We obtain a factorized expression for the spectator contribution in terms of perturbative hard kernels and the $B$- and $K$-meson light-cone distribution amplitudes. The spectator contribution arises at the same order in the power expansion as the soft-overlap term and amounts to approximately $25\%$ of the soft contribution. In the second realization, the Wilson coefficient of $aG\widetilde G$ is assumed to be present above the electroweak scale. Renormalization-group evolution and matching then induce a direct $b\to sa$ operator, which subsequently results in a dominant soft form-factor contribution, whereas the gluonic spectator term turns out to be numerically subleading ($\sim 6-7\%$). We thus identify the conditions under which spectator scattering becomes relevant for heavy-axion production in rare $B$-meson decays. Finally, we derive the corresponding bounds on the axion decay constant $f_a$ for both realizations and compare their phenomenological implications.