Q_t-Q_e-\frac{1}{4s_w^2}
\,\bigg]
\,,
\label{Cvborn}
\\[2mm]
C_A^{\rm born}(\nu=1) & = &
-\,\frac{\alpha\pi}{m_t^2(4c_w^2-x)}\,\bigg[\,
Q_t-\frac{1}{4s_w^2}
\,\bigg]
\,,
\label{Caborn}\end{aligned}$$ where $\nu$ is the vNRQCD renormalization scaling parameter.[^6] Except for the QED beam effects, which we will not consider here, the electroweak effects in Eqs.(\[Gamma\],\[Cvborn\],\[Caborn\]) are the only ones at leading order. In particular, at this order there are no electroweak effects contributing to the Coulomb potential acting between the $t\bar t$ pair, $$\begin{aligned}
{\cal L}_{\rm pot} & = &
-\sum_{{\mbox{\scriptsize\boldmath $p$}},{\mbox{\scriptsize\boldmath $p$}}^\prime} \frac{{\cal V}_c^{(s)}(\nu)}{({\mathbf p}-{\mathbf p}^\prime)^2}\,
\psip{{\mbox{\scriptsize\boldmath $p$}}^\prime}^\dagger \psip{{\mbox{\scriptsize\boldmath $p$}}} \chip{-{\mbox{\scriptsize\boldmath $p$}}^\prime}^\dagger\chip{-{\mbox{\scriptsize\boldmath $p$}}}
\,,
\label{Lpot}\end{aligned}$$ where ${\cal V}_c^{(s)}(\nu)=-4\pi C_F \alpha_s(m_t\nu)$ is the LL Coulomb Wilson coefficient for a color singlet heavy quark pair.

For the NLL order approximation it has been frequently stated that there are no new operators that can contribute and that power counting tells that we only need to consider ${\cal O}(\alpha_s)$ QCD corrections to the LL matching conditions in Eq. (\[Lke\]) to account for all electroweak effects [@Fadin2; @Melnikov1]. However, it was also noted in [@HoangTeubnerdist] that the mismatch between the $t\bar t$ phase space in the full and the effective theory leads to additional NLL matching corrections for unstable top quarks. We come back to the role played by 