electroweak loop would require a factor $g^4$ in addition to the $1/m_t^2$ suppression for dimensional reasons, such an operator could not contribute at NNLL order either [@Beneke2].

It remains to discuss $g^2$ corrections to the operators contributing at the LL level. For the bilinear quark operators in the effective theory Lagrangian one obtains the terms shown in Eq.(\[Lke\]). Concerning the instability of the top quark only the time dilatation correction is obtained, and the ${\cal O}(\alpha_s^2)$ and one-loop electroweak corrections to the on-shell top quark width [@Blokland1; @Denner1] have to be accounted for. The ${\cal O}(\alpha_s^2)$ correction to the top width is easy to implement together with the Born and one-loop QCD results in the width term $\Gamma_t$ and will not be discussed further in this work. On the other hand, for the Coulomb potential all dominant $g^2$ corrections cancel due to SU(3) gauge invariance because in the first approximation one can neglect the gluon momentum flowing into the vertex correction [@Modritsch1; @WiseTTbarSusy]. The mechanism is equivalent to the gauge cancellation discussed above for the time-like $A^0$ gluon. The order $g^4$ matching conditions of the production operators ${\cal O}_{V,{\mbox{\scriptsize\boldmath $p$}}}$ and ${\cal O}_{A,{\mbox{\scriptsize\boldmath $p$}}}$, on the other hand, do not 