= Gauge contributions to the F-term supertrace
{title2=$2C-2(4C)+3(2C)=0$}
Canonical gauge interactions preserve the <tree-level supertrace mass sum rule> at a vacuum with vanishing <D-terms>. Put $C=\sum_a g_a^2\|T_av\|^2$ 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 $\tfrac12\sum_aD_a^2$; the fermionic one counts both blocks of the symmetric mixing matrix with entries $\sqrt2g_aT_av$; 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.
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