  We examine the charged lepton flavor violating process $gg \rightarrow
\mu^\pm \tau^\mp$ at the $\sqrt{s} = 13$ TeV LHC. Operators generating this
process can be induced by new physics (NP) at dimension 8. Despite the power
suppression associated with dimension 8 operators, we show that the LHC's large
gluon luminosity makes it possible to probe this channel. For an integrated
luminosity of 100 fb$^{-1}$ at the LHC, we predict a constraint on the NP scale
$\Lambda \gtrsim 3$ TeV. In addition, we point out that such operators can be
induced through top quark loops in models that generate dimension 6 operators
of the form $\overline{t} t \, \mu \tau$. We find that the NP scale of these
dimension 6 operators can be constrained to be $\Lambda \gtrsim 3.4-4.1$ TeV
with 100 fb$^{-1}$ of data.
