make some preliminary considerations. Section 3 considers the pure seesaw case by separately analyzing the flavour conserving (FC) and flavour violating (FV) amplitudes contributing to lepton EDMs. In Section 4 the framework of seesaw and $SU(5)$ is discussed along the same lines. Concluding remarks are drawn in Section 5. Finally, in Appendix A and B we collect the RGE in the case of the seesaw, without and with a minimal $SU(5)$ unification respectively.

Framework and Method {#sec2}
====================

In this section we recall the expression for the lepton EDMs in the mass insertion approximation [@massins; @moroi; @prs; @fms; @ms1] to display the supersymmetric mass parameters that are constrained by the present and projected searches for $d_e$ and $d_\mu$. We then draw some preliminary considerations to introduce our procedure to calculate the radiative 1-loop contribution to these mass parameters. The relevant RGE can be found in the Appendix.

We adopt here the following conventions for the $6 \times 6$ slepton mass matrix in the lepton flavour (LF) basis where the charged lepton mass matrix, $m_{\ell}$, is diagonal: (

[cc]{} \_L\^& \_R\^

)    (

[cc]{} m\_[LL]{}\^2 & A\_e\^v\_d - m\_ A\_e v\_d - \^\* m\_ & m\_[RR]{}\^2

)   (

[c]{} \_L \_R

