operators ${\cal O}_{{\mbox{\scriptsize\boldmath $p$}},1}^j$ and ${{\cal
    O}_{{\mbox{\scriptsize\boldmath $p$}},1}^j}^{\!\!\!\dagger}$ [@hmst1]. In dimensional regularization the result reads $$\begin{aligned}
\Delta\sigma_{\rm tot}^{\Gamma,1} & = &
2\,N_c\,\mbox{Im}\bigg\{\,
2i\,\big[\,C_V^{\rm born}\,C_V^{\rm bW,abs}+C_A^{\rm
  born}\,C_A^{\rm bW,abs}\,\big]\,G^0(a,v,m_t,\nu)
\nonumber\\[2mm] & &
+\,\big[\,(C_V^{\rm born})^2+(C_A^{\rm born})^2\,\big]\,
\delta G^0_\Gamma(a,v,m_t,\nu)
\,\bigg\}
\,,
\label{dsignaNNLL}\end{aligned}$$ where $a\equiv -{\cal V}_c^{(s)}(\nu)/4\pi=C_F\alpha_s(m_t\nu)$. The term $G^0$ is the zero-distance S-wave Green function of the non-relativistic Schrödinger equation which is obtained from the LL order terms in the Lagrangian shown in Eqs.(\[Lke\]) and (\[Lpot\]). In dimensional regularization it has the form [@hmst1] $$\begin{aligned}
 G^0(a,v,m_t,\nu) & = &
 \frac{m_t^2}{4\pi}\left\{\,
 i\,v - a\left[\,\ln\left(\frac{-i\,v}{\nu}\right)
 -\frac{1}{2}+\ln 2+\gamma_E+\psi\left(1\!-\!\frac{i\,a}{2\,v}\right)\,\right]
 \,\right\}
 \nonumber \\ & &
 +\,\frac{m_t^2\,a}{4 \pi}\,\,\frac{1}{4\,\epsilon}
\,,
\label{deltaGCoul}\end{aligned}$$ where $$\begin{aligned}
v & = &
\sqrt{\frac{\sqrt{s}-2(m_t+\delta m_t)+i\Gamma_t}{m_t}}
\,,\end{aligned}$$ $\sqrt{s}$ being the c.m.energy. The term $\delta G^0_\Gamma$ represents the corrections originating from the time dilatation correction in Eq.(\[Lke\]) and reads $$\begin{aligned}
\delta G^0_\Gamma(a,v,m_t,\nu) & = &
-i\,\frac{\Gamma_t}{2m_t}\,\bigg[\,
1+\frac{v}{2}\frac{\partial}{\partial v} + a\frac{\partial}{\partial a}
\,\bigg]\,G^0(a,v,m_t,\nu)
\,.
\label{deltaGGamma}\end{aligned}$$ Note that the Wilson coefficients $C_{V/A}$ do not run at LL order, so only the matching conditions at $\nu=1$ appear in Eq.(\[dsignaNNLL\]).

It is straightforward to check that the terms proportional to $C_{V/A}^{\rm bW,abs}$ in Eq.(\[dsignaNNLL\]) are in agreement with the full theory matrix elements from the interference between the double-resonant amplitudes for the process 