  Contact interactions of a muon, an electron and two photons can contribute to
the decay $\mu \to e \gamma \gamma$, but also to the conversion of a muon into
an electron in the electric field of a nucleus. We calculate the $\mu \to e$
conversion rate, and show that for the coefficients of operators involving the
combination $FF \propto |\vec{E}|^2$ (as opposed to $F\tilde{F} \propto \vec{E}
\cdot \vec{B}$), the current bound on $\mu \to e$ conversion is more sensitive
than the bound on $\mu \to e \gamma \gamma$.
