to be small in gauge mediation as the messenger scale is at most comparable to the See-Saw scale and the running can be made in a very short interval. This is not the cLFV that we are interested in.

In neutrino assisted gauge mediation, neutrino Dirac Yukawa couplings can introduce two-loop generated cLFV effects on $m_L^2$ as a result of gauge-Yukawa or Yukawa mediation, [$$\begin{split}\delta m_{L}^2 = \frac{B_N^2}{(4 \pi)^4} \Big[ \Big({\rm Tr} [Y_{\nu} Y_{\nu}^\dagger]+3{\rm Tr} [Y_U Y_U^\dagger] - 3 g_2^2 - \frac{1}{5} g_1^2 \Big) Y_{\nu}^\dagger Y_{\nu} + 3 Y_{\nu}^\dagger Y_{\nu} Y_{\nu}^\dagger Y_{\nu} \Big]\end{split}$$]{} in the charged lepton mass basis. If the two loop generated slepton mass squared has a nonzero off-diagonal element, it would generate cLFV. Parametrically, this effect is much larger than the expected cLFV in mSUGRA or similar scenarios in which the effect comes from the running above the See-Saw scale. We simply assume that both messengers ${\bf 5}, {\bf \bar{5}}$ and ${\bf 1}$ have the same masses at the See-Saw scale. In principle these two masses can be different and cLFV can arise if the singlet messenger is lighter than ${\bf 5}, {\bf \bar{5}}$. However, this effect is loop suppressed compared to the Yukawa mediation we would 