Dispersion Relations for Electroweak Observables in Composite Higgs Models

We derive dispersion relations for the electroweak oblique observables measured at LEP in the context of $SO(5)/SO(4)$ composite Higgs models. It is shown how these relations can be used and must be modified when modeling the spectral functions through a low-energy effective description of the strong dynamics. The dispersion relation for the parameter $\epsilon_3$ is then used to estimate the contribution from spin-1 resonances at the 1-loop level. Finally, it is shown that the sign of the contribution to the $\hat S$ parameter from the lowest-lying spin-1 states is not necessarily positive definite, but depends on the energy scale at which the asymptotic behavior of current correlators is attained.