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Improved indirect limits on muon EDM

Given current discrepancy in muon $g-2$ and future dedicated efforts to measure muon electric dipole moment (EDM) $d_μ$, we assess the indirect constraints imposed on $d_μ$ by the EDM measurements performed with heavy atoms and molecules. We notice that the dominant muon EDM effect arises via the muon-loop induced &#34;light-by-light&#34; $CP$-odd amplitude $\propto{\bf B}{\bf E}^3$, and in the vicinity of a large nucleus the corresponding parameter of expansion can be significant, $eE_{\rm nucl}/m_μ^2 \sim 0.04$. We compute the $d_μ$-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, $d_μ(^{199}{\rm Hg}) < 6\times 10^{-20}e$cm and $d_μ({\rm ThO}) < 2\times 10^{-20}e$cm, constitute approximately three- and nine-fold improvements over the limits on $d_μ$ extracted from the BNL muon beam experiment.