from effective theories for stable particles. The only difference is that the renormalization procedure will involve operators having non-hermitian Wilson coefficients.

The operators that are renormalized by the UV divergences displayed in Eq.(\[dsignaNNLL\]) are the two $(e^+e^-)(e^+e^-)$ operators $$\begin{aligned}
\tilde{\cal O}_V & = &
-\,\big[\,\bar e\,\gamma^\mu\,e\,\big]\,
\big[\,\bar e\,\gamma_\mu\,e\,\big]
\,,
\\[2mm]
\tilde{\cal O}_A & = &
-\,\big[\,\bar e\,\gamma^\mu\,\gamma_5 \, e\,\big]\,
\big[\,\bar e\,\gamma_\mu\,\gamma_5\,e\,\big]
\,,\end{aligned}$$ which give the additional contribution $\tilde\Delta {\cal L} = \tilde C_V \tilde{\cal O}_V + \tilde C_A
\tilde{\cal O}_A$ to the effective theory Lagrangian, $\tilde C_{V/A}$ being the Wilson coefficients. Because in this work we neglect QED effects, the electron and positron act as classic fields and therefore $\tilde C_{V}$ and $\tilde
C_{A}$ run only through mixing due to UV divergences such as in Eq.(\[dsignaNNLL\]). Since only the imaginary parts of the coefficients $\tilde
C_{V/A}$ can contribute to the total cross section through the optical theorem we neglect the real contributions in the following. Using the standard $\msb$ subtraction procedure the (non-hermitian) counterterms of the renormalized $\tilde{\cal O}_{V/A}$ operators read $$\begin{aligned}
\delta \tilde C_V & = &
i\,\frac{N_c m_t^2}{32 \pi^2 \epsilon}\,
\bigg[ (C_V^{\rm born})^2 \frac{\Gamma_t}{m_t} 
  +2 C_V^{\rm born} C_V^{\rm bW,abs}  
\bigg]\,
{\cal V}_c^{(s)}(\nu) 
\nonumber \\[2mm] & & 
+\,i\,\frac{N_c m_t^2}{32 