Indirect upper limits on ${\mathrm{\ell }}_i\to {\mathrm{\ell }}_j\gamma \gamma$ from ${\mathrm{\ell }}_i\to {\mathrm{\ell }}_j\gamma$

We perform an effective field theory analysis to correlate the charged lepton flavor violating processes ${\mathrm{\ell }}_i\to {\mathrm{\ell }}_j\gamma \gamma$ and ${\mathrm{\ell }}_i\to {\mathrm{\ell }}_j\gamma$. Using the current upper bounds on the rate for ${\mathrm{\ell }}_i\to {\mathrm{\ell }}_j\gamma$, we derive model-independent upper limits on the rates for ${\mathrm{\ell }}_i\to {\mathrm{\ell }}_j\gamma \gamma$. Our indirect limits are about three orders of magnitude stronger than the direct bounds from current searches for $\mu \to e\gamma \gamma$, and four orders of magnitude better than current bounds for $\tau \to \mathrm{\ell }\gamma \gamma$. We also stress the relevance of Belle II or a Super Tau Charm Facility to discover the rare decay $\tau \to \mathrm{\ell }\gamma \gamma$.