Improved indirect limits on muon EDM

Given current discrepancy in muon $g-2$ and future dedicated efforts to measure muon electric dipole moment (EDM) $d_\mu$, we assess the indirect constraints imposed on $d_\mu$ by the EDM measurements performed with heavy atoms and molecules. We notice that the dominant muon EDM effect arises via the muon-loop induced "light-by-light" $CP$-odd amplitude $\propto{\bf B}{\bf E}^3$, and in the vicinity of a large nucleus the corresponding parameter of expansion can be significant, $eE_{\rm nucl}/m_\mu^2 \sim 0.04$. We compute the $d_\mu$-induced Schiff moment of the $^{199}$Hg nucleus, and the linear combination of $d_e$ and semileptonic $C_S$ operator (dominant in this case) that determine the $CP$-odd effects in ThO molecule. The results, $d_\mu(^{199}{\rm Hg}) < 6\times 10^{-20}e$cm and $d_\mu({\rm ThO}) < 2\times 10^{-20}e$cm, constitute approximately three- and nine-fold improvements over the limits on $d_\mu$ extracted from the BNL muon beam experiment.