I have just gone through the exercise of constructing the supersymmetrized QED action.
In the end, I get a reasonable action which matches literature. But after a little analysis, I find that the supersymmetrized gauge interactions (electron-selectron-photino vertices) seem to violate CP. In the two-component Weyl notation, this is:
Where $\lambda$ is the gaugino, $\Phi_L=(\phi_L,e_L)$ is a left chiral superfield with charge $-1$, and $\Phi_R=(\phi_R,e_R)$ is the Dirac-partner chiral superfield with charge $+1$.
I am assuming, that under $CP$, the two chiral superfields transform into one another:
Question: Is my analysis correct? Is is really true that supersymmetric QED violates CP?
Edit: I just realized that in my construction I assumed that $m$ in the superpotential ($m\phi_R\phi_L$) is real. It seems to me that more generally, it is allowed to be complex...