  In the Standard Model Effective Field Theory (SMEFT), operators involving the
top quark are generally difficult to probe, and can generate sizable loop
contributions to the electroweak precision observables, measured by past and
future lepton colliders. Could the high precision of the electroweak
measurements compensate the loop suppression and provide competitive reaches on
these operators? Would the inclusion of these contributions introduce too many
additional parameters for a meaningful global electroweak analysis to be done?
In this paper, we perform a detailed phenomenological study to address these
two important questions. Focusing on eight dimension-6 operators that generate
anomalous couplings between the electroweak gauge bosons and the
third-generation quarks, we calculate their one loop contributions to the
$e^+e^- \to f\bar{f}$ processes both on and off the Z-pole and the $e^-e^+ \to
WW$ process. A global analysis is performed with these eight operators and the
ones that contribute to the above processes at tree level, using the
measurements at LEP, SLC and several low energy experiments. We find that,
while the current electroweak precision measurements are sensitive to the
one-loop effects of the top-quark operators, it is difficult to separate them
from the operators that contribute at the tree level, making a global analysis
rather challenging. Under more assumptions, competitive reaches could be
obtained in a global fit. Another important finding of our study is that the
two operators that generate dipole interactions of the bottom quark have
significant impacts in the Z-pole measurements and should not be omitted. We
also discuss the implication of the recently reported W-boson mass measurement
at CDF to our results. Finally, we estimate the reaches of future lepton
colliders in probing the top-quark operators with precision electroweak
measurements.
