When building a supersymmetric theory people typically write the superpotential as a products of fields. However, there is another ingredient that could in principle be used - the covariant derivative. Since it still transforms properly under supersymmetry it seems like a perfectly valid operator. Furthermore such terms will be SUSY invariant since we integrate over superspace.

As an example I've seen it written that the most general renormalizable superpotential for a single chargeless field is given by,

\begin{equation}

W = \alpha + \beta \Phi + m\Phi ^2 + \lambda \Phi ^3

\end{equation}

However, there could in principle also be a term,

\begin{equation}

{\cal D} ^\alpha \Phi {\cal D} _\alpha \Phi

\end{equation}

where $ {\cal D} ^\alpha \equiv \partial ^\alpha + i ( \sigma ^\mu \bar{\theta} ) _\alpha \partial _\mu $. This would give Lagrangian contribution:

\begin{equation}

{\cal L} \supset \int \,d^2\theta {\cal D} ^\alpha \Phi {\cal D} _\alpha \Phi + h.c. = F ( y ) F ( y ) + h.c.

\end{equation}

and since $ F $ is real this would potentially wreck havoc on the scalar potential.

Is there some reason they can be omitted?