Maximally supersymmetric brane actions have been considered extensively in the literature, starting with [63, 64, 65, 66]. However, as far as we are aware of, theories of the DBI and more generally $P(X,\varphi )$ type have not been constructed in minimal $\mathcal{N}=1$, $d=4$ supersymmetry, which is the framework most relevant for phenomenology. For related work in $d=3$ space-time dimensions, see [67].

We also could have considered terms with three $\Phi$s and one ${\Phi }^{†}$, or terms with four $\Phi$s, plus their hermitian conjugates. Following the same logic as above, and making use of integration by parts, it is straightforward to see that the terms with four $\Phi$s (for example, $\overline{D}D\Phi \overline{D}D\Phi {\Phi }^2$) are all zero after integrating over $d^4\theta$. For the case of three $\Phi$s and one ${\Phi }^{†}$, there are only two independent terms, ${\overline{D}}^2{\Phi }^{†}{D}^2\Phi {\Phi }^2{|}_{\theta \theta \overline{\theta }\overline{\theta }}=A^2\square A\square A^{*}$ and ${\overline{D}}^2{\Phi }^{†}D\Phi D\Phi \Phi {|}_{\theta \theta \overline{\theta }\overline{\theta }}=A(\partial A{)}^2\square A^{*}$, and since neither contains $X^2$ we do not consider these terms any further.

Had we used the superfield extension (23) instead of (20), we would have found exactly the same fermionic fluctuations, and hence also the wrong-sign spatial gradient term.