parts of the full set of electroweak one-loop diagrams indicated in Figs.\[figcuts\] and the QCD matching corrections known from previous work [@hmst1; @c12loop; @Hoangc12loopQED]. These corrections lead to an energy-independent multiplicative modification of the cross section normalization which is, however, not subject of the investigations in this work. The results for the real parts of the full set of electroweak one-loop diagrams were given in [@GuthKuehn].

Time-Ordered Product and Renormalization {#sectionrenormalization}
========================================

Using the optical theorem the NNLL order corrections to the total cross section that come from the absorptive one-loop electroweak matching conditions for the operators ${\cal O}_{V/A,{\mbox{\scriptsize\boldmath $p$}}}$ and from the time dilatation corrections can be computed from the imaginary part of the $(e^+e^-)(e^+e^-)$ forward scattering amplitude, $$\begin{aligned}
\sigma_{\rm tot} & \sim &
\frac{1}{s}\, 
\mbox{Im}\left[\,\Big(C_V^2(\nu)+C_A^2(\nu)\Big)\,L^{lk}\,{\cal  A}_1^{lk}\,\right]
\,,\end{aligned}$$ where ($k+k^\prime=(\sqrt{s},0)$ and ${\bf \hat e}={\mathbf k}/|{\mathbf k}|$) $$\begin{aligned}
L^{lk} &=&
       \frac{1}{4}\,\sum\limits_{e^\pm\rm spins}\,
       \Big[\,\bar v_{e^+}(k^\prime)\,\gamma^l\,(\gamma_5)\,u_{e^-}(k)\,\Big]
       \,\Big[\,\bar u_{e^-}(k)\,\gamma^k\,(\gamma_5)\,v_{e^+}(k^\prime)\,\Big]
\nonumber
\\[2mm]
       &=& \frac{1}{2}\,(k+k^\prime)^2\,(\delta^{lk}-\hat e^l  \hat e^k)\end{aligned}$$ is the spin-averaged lepton tensor and ($\hat{q}\equiv(\sqrt{s}-2m_t,0)$) $$\begin{aligned}
{\cal A}_1^{lk} &=& 
i\, \sum\limits_{\mbox{\scriptsize\boldmath $p$},\mbox{\scriptsize\boldmath $p'$}} \int\! d^4x\:
e^{-i\hat q \cdot x}\:
\Big\langle\, 0\,\Big|\,T\,
{{\cal O}_{\mbox{\scriptsize\boldmath $p$},1}^l}^{\!\!\!\dagger} (0)\, 
{\cal O}_{\mbox{\scriptsize\boldmath $p'$},1}^k (x)\Big|\,0\,\Big\rangle
\nonumber\\[2mm] & = &
2\,N_c\,\delta^{lk}\,G^0(a,v,m_t,\nu)\end{aligned}$$ is the time-ordered product of the $t\bar t$ production and annihilation 