Indirect upper limits on $\ell_i\to\ell_j\gamma\gamma$ from $\ell_i\to\ell_j\gamma$

We perform an effective field theory analysis to correlate the charged lepton flavor violating processes $\ell_i\to\ell_j\gamma\gamma$ and $\ell_i\to\ell_j\gamma$. Using the current upper bounds on the rate for $\ell_i\to\ell_j\gamma$, we derive model-independent upper limits on the rates for $\ell_i\to\ell_j\gamma\gamma$. 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 $\tau\to\ell\gamma\gamma$. We also stress the relevance of Belle II or a Super Tau Charm Facility to discover the rare decay $\tau\to\ell\gamma\gamma$.