\subsection{Beam polarization}   
\label{sec:beampol}     

\subsubsection{Motivation}     
Polarized beams are useful for several purposes in $\epem$ storage rings. Transverse polarization was used in single beams at LEP for beam energy calibration with 0.1 MeV intrinsic precision~\cite{ement_of_the_W_boson_mass_2005}. This precision will be essential for the TLEP measurements of the Z mass and width, and of the W mass, with the required accuracy. Longitudinal polarization was used in collisions at SLC for the measurement of the left-right asymmetry at the Z pole, $\ALR$, with a $10^{-3}$ accuracy~\cite{altay_Band_Barklow_et_al__2000}, which in turn allowed a determination of the weak mixing angle with an accuracy similar to that of the best LEP unpolarized measurements. It is therefore of great interest to establish both transverse and longitudinal polarization with TLEP, and be able to maintain longitudinal polarization in collisions at the Z pole.    