$(e^+e^-)(t \,\bar t)$ effective theory operators relevant for the $bW^+\bar bW^-$ final state are computed. In Sec.\[sectionrenormalization\] the resulting NNLL corrections for the total cross section are determined and the renormalization of the $(e^+e^-)(e^+e^-)$ effective theory operators needed to account for the phase space divergences is discussed. In particular, we compute and solve the anomalous dimensions and determine their contribution to the total cross section. Sec.\[sectionanalysis\] contains a brief numerical analysis and in Sec.\[sectionconclusion\] we conclude.

Power Counting and Matching Conditions {#sectionpowercounting}
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In this work we are interested in the total cross section and not in any differential information on the top decay final states. We therefore include all effects related to the top quark decay as non-hermitian matching conditions of effective theory operators that describe the non-relativistic top and antitop dynamics and their interactions with soft and ultrasoft gluons. We employ gauge invariant operators, and the matching conditions are computed for on-shell external lines. This allows to maintain gauge invariance in a transparent way.

To illustrate the power counting needed to classify the order at which these electroweak effects can contribute let us recall the matching conditions for the bilinear quark field operators. They are obtained by matching 