Curing the ills of Higgsless models: TheSparameter and unitarity

We consider various constraints on Higgsless models of electroweak symmetry breaking based on a bulk $\mathrm{S}\mathrm{U}(2{)}_L\times \mathrm{S}\mathrm{U}(2{)}_R\times \mathrm{U}(1{)}_{B-L}$ gauge group in warped space. First we show that the $S$ parameter which is positive if fermions are localized on the Planck brane can be lowered (or made vanishing) by changing the localization of the light fermions. If the wave function of the light fermions is almost flat their coupling to the gauge boson Kaluza-Klein (KK) modes will be close to vanishing, and therefore contributions to the $S$ parameter will be suppressed. At the same time the experimental bounds on such $Z^{\prime }$ and $W^{\prime }$ gauge bosons become very weak, and their masses can be lowered to make sure that perturbative unitarity is not violated in this theory before reaching energies of several TeV. The biggest difficulty of these models is to incorporate a heavy top quark mass without violating any of the experimental bounds on bottom quark gauge couplings. In the simplest models of fermion masses a sufficiently heavy top quark also implies an unacceptably large correction to the $Zb\overline{b}$ vertex and a large splitting between the KK modes of the top and bottom quarks, yielding large loop corrections to the $T$ parameter. We present possible directions for model building where perhaps these constraints could be obeyed as well.