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Verifying $V_{tb} \simeq 1$ is critical to test the three generationassumption of the Standard Model. So far our best knowledge of $V_{tb}$ isinferred either from the $3\times 3$ unitarity of CKM matrix or from singletop-quark productions upon the assumption of universal weak couplings. Theunitarity could be relaxed in new physics models with extra heavy quarks andthe universality of weak couplings could also be broken if the $Wtb$ couplingis modified in new physics models. In this work we propose to measure $V_{tb}$in the process of $e^+ e^- \to t\bar{t}$ without prior knowledge of the numberof fermion generations or the strength of the $Wtb$ coupling. Using aneffective Lagrangian approach, we perform a model-independent analysis of theinteractions among electroweak gauge bosons and the third generation quarks,i.e. the $Wtb$, $Zt\bar{t}$ and $Zb\bar{b}$ couplings. The electroweak symmetryof the Standard Model specifies a pattern of deviations of the $Z$-$t_L$-$t_L$and $W$-$t_L$-$b_L$ couplings after one imposes the known experimentalconstraint on the $Z$-$b_L$-$b_L$ coupling. We demonstrate that, making use ofthe predicted pattern and the accurate measurements of top-quark mass and widthfrom the energy threshold scan experiments, one can determine $V_{tb}$ from thecross section and the forward-backward asymmetry of top-quark pair productionat an {\it unpolarized} electron-positron collider.

Determining $V_{tb}$ at Electron-Positron Colliders