  We introduce a toy model implementing the proposal of using a custodial
symmetry to protect the Zbb coupling from large corrections. This
"doublet-extended standard model" adds a weak doublet of fermions (including a
heavy partner of the top quark) to the particle content of the standard model
in order to implement an O(4) x U(1)_X = SU(2)_L x SU(2)_R x P_{LR} x U(1)_X
symmetry that protects the Zbb coupling. This symmetry is softly broken to the
gauged SU(2)_L x U(1)_Y electroweak symmetry by a Dirac mass M for the new
doublet; adjusting the value of M allows us to explore the range of
possibilities between the O(4)-symmetric (M to 0) and standard-model-like (M to
infinity) limits. In this simple model, we find that the experimental limits on
the Zbb coupling favor smaller M while the presence of a potentially sizable
negative contribution to T strongly favors large M. A fit to all precision
electroweak data shows that the heavy partner of the top quark must be heavier
than about 3.4 TeV, making it difficult to search for at LHC. This result
demonstrates that electroweak data strongly limits the amount by which the
custodial symmetry of the top-quark mass generating sector can be enhanced
relative to the standard model. Using an effective field theory calculation, we
illustrate how the leading contributions to alpha T, alpha S and the Zbb
coupling in this model arise from an effective operator coupling right-handed
top-quarks to the Z-boson, and how the effects on these observables are
correlated. We contrast this toy model with extra-dimensional models in which
the extended custodial symmetry is invoked to control the size of additional
contributions to alpha T and the Zbb coupling, while leaving the standard model
contributions essentially unchanged.
