\subsubsection{Longitudinal polarization}

Measurements with longitudinal polarization require maintaining polarization of both ${\rm e}^+$ and ${\rm e}^-$ beams in collisions. At LEP, transverse beam polarization of 40\% was observed and maintained in collisions for more than five hours at Z pole energies ($\sim 45$ GeV per beam) with one collision point, a beam-beam tune shift of 0.04, and a single bunch luminosity of $10^{30}~\cms$~\cite{cite:LEP-beam-beam-pol}. The polarization levels measured during this experiment are displayed in Fig.~\ref{fig:polarization} as a function of time. With the smaller value of $\beta_y^{*}$ and the larger number of bunches, similar polarization levels could be envisioned in collisions at the Z pole with TLEP with a luminosity reduced to around $10^{35}~\cms$, for the same total beam-beam tune shift. A suitable working point will have to be found to optimize the benefits from the much reduced top-up rate, and the adverse effects of the beamstrahlung and the required polarization wigglers on the energy spread.     
