CP-violation in SUSY QED?

Question

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:

$$\mathcal{L}_\text{int}=\ldots+\sqrt{2}e(\phi_Le_L^\dagger\lambda^\dagger+\phi_L^*e_L\lambda)-\sqrt{2}e(\phi_R e_R^\dagger\lambda^\dagger+\phi_R^*e_R\lambda)$$

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:

$$CP:\qquad\Phi_L\leftrightarrow\Phi_R$$

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...

Answer 1

I am in error. The supersymmetric gauge interactions displayed in my original post does not violate CP. The error is in the transformation rule I gave above. Under CP, the chiral superfields should transform:

$$CP:\qquad \Phi_L\rightarrow\Phi_L^*,\enspace \Phi_R\rightarrow\Phi_R^*$$

Therefore these interactions only violate P. This is something that I can come to grips with (rather than CP violation). It would be nice to get a double check on my claim.