 Gauge contributions to the F-term supertrace

Canonical gauge interactions preserve the tree-level supertrace mass sum rule at a vacuum with vanishing D-terms. Put $C={\sum }_a{g}_a^2\Vert T_{a}v{\Vert }^2$ for the scalar expectation vector $v$. The extra real-scalar squared-mass trace is $2C$, the extra Weyl spinor squared-mass trace from gaugino mixing is $4C$, and the gauge boson squared-mass trace is $2C$. Their spin-weighted supertrace is $2C-2(4C)+3(2C)=0$. The scalar contribution comes from differentiating $\frac{1}{2}{\sum }_a{D}_a^2$; the fermionic one counts both blocks of the symmetric mixing matrix with entries $\sqrt{2}{g}_a{T}_{a}v$; the vector one follows from the covariant scalar kinetic term. Any zero-mass Goldstone bosons in the unreduced scalar Hessian do not change its trace.