gaugino and higgsino interactions generate additional terms on the right hand side of . In the supersymmetric limit, these effects yield [@HH] \_[SUSY]{}(\_1)&=&-g\^4-g\^2 g\^[2]{} -g\^[4]{},\
\_[SUSY]{}(\_3)&=&-g\^4+g\^2 g\^[2]{} -g\^[4]{},\
\_[SUSY]{}(\_4)&=& 2g\^4-2g\^2 g\^[2]{}. \[deltaD\] Hence, $$\delta_{\rm SUSY}\{\mathcal{D}(\lambda_4+\lambda_3+\lambda_1)\}=
-(3 g^4 + 2 g^2 {g^\prime}^2 + {g^\prime}^4)\,.$$ Indeed, when the latter is added to , we see that ${\cal D} (\lambda_4 + \lambda_3 + \lambda_1 )=0$ as expected. Thus, the SUSY relation $\lambda_1=-\lambda_3-\lambda_4
=\tfrac{1}{4}(g^2+g^{\prime\,2})$ is renormalization group invariant when all the Higgs/higgsino/gauge/gaugino interactions are included.

\[sec:necessary\]Basis-independent conditions for the MSSM Higgs potential that are necessary but not sufficient
----------------------------------------------------------------------------------------------------------------

Based on the arguments of section \[sec:Higgs\_MSSM\], no basis-invariant conditions constructed solely from the $Y_{ab}$ and $Z_{ab,cd}$ exist that can guarantee that are satisfied for some choice of basis. Nevertheless, we can establish a weaker invariant condition that is necessary (although not sufficient) for the existence of a supersymmetric THDM scalar potential.[^4]

Consider a basis choice in which: \[weaker\] m\_[11]{}\^2=m\_[22]{}\^2,\_1=\_2, m\_[12]{}\^2=\_5=\_6=\_7=0. Clearly is satisfied by the MSSM Higgs potential in the standard MSSM basis choice for the Higgs fields. However, is not sufficient since the condition $\lambda_4=-\lambda-\lambda_3$ is imposed, nor are the conditions relating the quartic Higgs couplings to gauge couplings imposed. Nevertheless, suppose that 