= c_{1,2}^{\rm QCD} \,,
    \nonumber\\
   -\frac32\,\lambda_t\,C_i &=& c_{i-6}^{\rm EW} \,;
    \quad i=7,\dots,10 \,.\end{aligned}$$

The isospin-violating effects induced by the coefficients $c_i^{\rm EW}$ and $\widetilde c_i^{\rm EW}$ are the main focus of this paper. In the SM, the matching conditions for the corresponding electroweak penguin coefficients at the weak scale, and to leading order in perturbation theory, are $C_8(m_W)=C_{10}(m_W)=0$, $C_7(m_W)\approx 0$, and [@Bur93] $$\label{C9match}
   C_9(m_W) \approx -\frac{\alpha}{12\pi}\,
   \frac{x_t}{\sin^2\!\theta_W} \left( 1 + \frac{3\ln x_t}{x_t-1}
   \right) ,$$ where $x_t=(m_t/m_W)^2$. For simplicity, we show only the large electroweak contribution to $C_9(m_W)$ and omit a common, renormalization-scheme dependent electromagnetic contribution to $C_7(m_W)$ and $C_9(m_W)$, which is negligible compared with the contribution in (\[C9match\]). In this approximation, the SM matching conditions for the electroweak penguin coefficients in our basis read $$\begin{aligned}
\label{SMinit}
   c_3^{\rm EW,SM}(m_W) &=& \frac{\alpha}{8\pi}\,
    \frac{\lambda_t\,x_t}{\sin^2\!\theta_W} \left( 1
    + \frac{3\ln x_t}{x_t-1} \right)
    \approx -5.5\times 10^{-4} \,, \nonumber\\
   c_{i\ne 3}^{\rm EW,SM}(m_W) 
   &=& \widetilde c_i^{\rm EW,SM}(m_W) = 0 \,.\end{aligned}$$ For the numerical estimate we have used $m_t=\overline{m}_t(m_t)
=170$GeV, $\alpha=1/129$, and $\lambda_t=V_{tb}^* V_{ts}=-0.04$.

In Sections \[sec:tree\] and \[sec:loop\], we will calculate the Wilson coefficients $c_i^q$ at a high scale, which for simplicity will be identified with the electroweak scale. In phenomenological applications, however, one usually prefers working with 