  The full off-shell one loop renormalization for all divergent amplitudes up
to dimension 6 in the Abelian Higgs-Kibble model, supplemented with a maximally
power counting violating higher-dimensional gauge-invariant derivative
interaction $\sim g ~ \phi^\dagger \phi (D^\mu \phi)^\dagger D_\mu \phi$, is
presented. This allows one to perform the complete renormalization of
radiatively generated dimension 6 operators in the model at hand. We describe
in details the technical tools required in order to disentangle the
contribution to UV divergences parameterized by (generalized) non-polynomial
field redefinitions. We also discuss how to extract the dependence of the
$\beta$-function coefficients on the non-renormalizable coupling $g$ in one
loop approximation, as well as the cohomological techniques (contractible
pairs) required to efficiently separate the mixing of contributions associated
to different higher-dimensional operators in a spontaneously broken effective
field theory.
