the unitarity of the underlying theory. In fact, this issue is analogous to the so-called strong phases in QCD amplitudes that are the basis for the search for CKM CP-violation in a number of B meson decays [@BaBarBook].

In this paper we extend this approach and investigate the role of absorptive parts related to the top quark decay in electroweak loop corrections to the NRQCD matching conditions that contribute to the NNLL total cross section.[^3] For simplicity we neglect the bottom quark mass and set $V_{tb}=1$, and approximate the W boson and bottom quark as stable particles. We demonstrate that these electroweak corrections properly account for the interference of the dominant double-resonant process $e^+e^-\to t\bar
t\to bW^+\bar b W^-$ with the $v^2$-suppressed single-resonant amplitudes for $e^+e^-\to b W^+\bar t\to bW^+\bar b W^-$ and $e^+e^-\to t\bar b W^-\to bW^+\bar b W^-$. We also show that absorptive parts that do not contribute to the $bW^+\bar b W^-$ final state, and are therefore also not accounted for in line-shape measurements, can be excluded in a gauge invariant way. This requires to include also fields for the electrons and positrons from the initial state into the effective theory, which act like classic fields for 