\subsubsection{Transverse polarization}

Transverse beam polarization builds up naturally in a storage ring by the Sokolov-Ternov effect. A transverse polarization in excess of 5-10\%, which was obtained for beam energies up to 61 GeV per beam at LEP, is sufficient for beam energy calibration purposes. It is generally accepted that this upper limit is determined by the energy spread, which becomes commensurate with the fractional part of the spin-tune  $\nu_s = E_{\rm beam} {\rm [GeV]} /0.440665$. Given that the energy spread scales as $E^2_{\rm beam} / \sqrt{\rho}$, where $\rho$ is the ring bending radius, it is expected that beam polarization sufficient for energy calibration should be readily available up to and above  the WW threshold (i.e., 81~GeV per beam) at TLEP. A new machine with a better control of the orbit should, however, be able to increase this limit. For example, a full 3D spin tracking simulation of the electron machine of the Large Hadron-electron Collider (LHeC) project in the 27 km LHC tunnel predicts 20\% polarization at a beam energy of 65 GeV for typical machine misalignments~\cite{1206.2913}. 

At LEP, the natural polarization building time amounted to five hours at the Z peak. This time is predicted to increase like the third power of the ring bending radius, hence will reach the unpractical value of 150 hours at TLEP. Asymmetric "polarization" wigglers were in use in LEP, and their effect on the polarization time and the beam energy spread, as well as other depolarizing sources, is analyzed in Ref.~\cite{cite:Blondel-Jowett-LEP606}. Such polarization wigglers could be used to reduce the polarization time at TLEP, while keeping the energy spread to a reasonable value. As an example, the use of the LEP polarization wigglers in TLEP with a central pole field of 0.6~T would reduce the polarization time to 18 hours at the Z peak, while keeping the beam energy spread below 48~MeV -- a value at which polarization could routinely be obtained in LEP at with a beam energy of 55~GeV. In these conditions, a level of polarization sufficient to perform resonant depolarization could be reached in a couple hours.  Energy calibrations would then be performed every ten minutes if at least twelve bunches of electrons and of positrons were kept "single" (i.e., not colliding) in the machine. For a beam energy of 80~GeV, the polarization time would be 9 hours in TLEP, and the use of wigglers should not be necessary.       