Local gauge-invariant vector operators in the adjoint $SU(2)$ Higgs model

In this work, we scrutinize local gauge-invariant vector operators of dimension four in the adjoint $SU(2)$ Higgs model, which are candidates for interpolating fields of the fundamental excitations of the model due to the so-called FMS mechanism. We use the equations of motion and the properties of the BRST operator to derive a Ward identity that allows us to determine whether a given operator can propagate. To corroborate this analysis, we explicitly compute the two-point function of the non-propagating operator at the one-loop level.