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Contributions to $K^0$-$\bar K^0$ mixing from exchanges of flavor violating axion, up to and including one loop in ChPT.

Contributions to $K^0$-$\bar K^0$ mixing from exchanges of flavor violating axion, up to and including one loop in ChPT.

Contributions to $K^0$-$\bar K^0$ mixing from exchanges of flavor violating axion, up to and including one loop in ChPT.

Contributions to $K^0$-$\bar K^0$ mixing from exchanges of flavor violating axion, up to and including one loop in ChPT.

Matching of the axion-mediated contributions to $B-\bar B$ mixing, diagrams $a_1)$ and $a_2)$, onto the dimension-six local operators in the OPE, diagram $b)$, where we also indicate the operators that receive the contributions. Single (double) lines represent the $d$ quark ($b$ quark).

Radiative contribution to the flavor mixing of the couplings of the axion to the quark-doublets.

Summary of the most important bounds for the different flavor sectors and for vectorial (red) and axial-vectorial (blue) couplings. On the lower axis we indicate the corresponding values for the effective axion mass defined by $m_{i. {\rm eff}} \equiv 4.69 \eV \times 10^6 \GeV/F_i$. Also shown as vertical gray lines are the bounds on axion couplings to electrons $F_e$ (95\%CL), nucleons $F_N$, and photons $F_\gamma$ (95\%CL), see Section \ref{sec:diag} for details.