CP-odd interaction of axion with matter

In a world with an axion the precise measurement of gravitational forces and checks of the equivalence principle at small distances, $\lambda \sim 1\hspace{0.5em}\mathrm{cm}$ and less, could provide an additional test of $\mathrm{CP}$ symmetry. Using the chiral approach, we connect the strength of the $\mathrm{CP}$-odd axion-nucleon vertex with the $\mathrm{CP}$-odd pion-nucleon coupling for color electric dipole moments of quarks as a low-energy source of $\mathrm{CP}$ violation. Strong limits on $\mathrm{CP}$-odd coupling $g_{\pi \mathrm{NN}}$ coming from atomic electric dipole moments give the best limits on the long range force mediated by axions, $F_{\mathrm{axion}}{/F}_{\mathrm{gravity}}<2\mathrm{}\cdot {10}^{-10}(1\mathrm{}\mathrm{cm}/\lambda {)}^2.$