deletions | additions       diff --git a/Longitudinal polarization.tex b/Longitudinal polarization.tex  index 09a4fe1..1406d1a 100644  --- a/Longitudinal polarization.tex  +++ b/Longitudinal polarization.tex 
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\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}. $10^{30}~\cms$~\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 maintained in collisions at the Z pole with TLEP with luminosity of around $10^{35}\cms$, $10^{35}~\cms$,  for the same total storage ring beam-beam tune shift.