\subsubsection{The Z invisible width and the number of neutrinos} 


The measurement of the Z decay width into invisible states is of great interest as it constitutes a direct test of the unitarity of the PMNS matrix -- or of the existence of sterile neutrinos, as pointed out in Ref.~\cite{Jarlskog_1990}. It can be performed at the Z pole from the peak hadronic cross section or at larger centre-of-mass energies with radiative return to the Z~\cite{Barbiellini}. As explained below, at TLEP the latter is likely to be more accurate than the former.

The measurement of the peak hadronic cross-section at the Z pole is indeed already dominated by theoretical systematics today, related to the understanding of the low-angle Bhabha-scattering cross section (used for the integrated luminosity determination).  The present measurement, expressed in terms of a number of active neutrinos,
\begin{equation}
N_\nu = 2.984 \pm 0.008,
\end{equation} 
is two standard deviations below the SM value of 3.00. The experimental conditions at TLEP will be adequate to improve the experimental uncertainty considerably, but, to make this measurement worthwhile, a commensurate effort would have to be invested in the theoretical calculations of the small-angle Bhabha-scattering cross section used for normalization. A desirable goal would be to reduce the uncertainty on $N_\nu$ down to 0.001, but it is not clear that it can be achieved from Z peak measurements.

Above the Z peak, the $\epemto {\rm Z}\gamma$ process provides a very clean photon-tagged sample of on-shell Z bosons, with which the Z properties can be measured. From the WW threshold scan alone, the cross section of about 5~pb~\cite{cite:Abbiendi2000hh,cite:Heister2002ut,cite:Achard2003tx,cite:Abdallah2003np} ensures that 10 million ${\rm Z}\gamma$ events will be produced in each TLEP experiment with a ${\rm Z} \to \nnbar$ decay and a high-energy photon in the detector acceptance. The three million ${\rm Z}\gamma$ events with leptonic Z decays will in turn provide a direct measurement of the ratio $\Gamma_{\rm Z}^{\rm inv}/\Gamma_{\rm Z}^{\rm lept}$, in which uncertainties associated with absolute luminosity and photon detection efficiency cancel. The 40 million ${\rm Z}\gamma$ events with either hadronic or leptonic Z decays will also provide a cross check of the systematic uncertainties and backgrounds related to the QED predictions for the energy and angular distributions of the high-energy photon. The invisible Z width will thus be measured with a statistical error corresponding to 0.001 neutrino family. Systematic uncertainties are expected to be at the same level or smaller. 

The data taken at $\sqrt{s} = 240$ and $350$ GeV will contribute to further reduce this uncertainty with the $\epemto {\rm Z}\gamma$ process, and to perform independent cross checks and redundant $\Gamma_{\rm Z}^{\rm inv}$ measurements with {\rm ZZ} and maybe {\rm HZ} production. It is to be determined by the design study whether a dedicated run at a somewhat lower centre-of-mass energy -- with both a larger luminosity and a larger ${\rm Z}\gamma$ cross section -- is more appropriate for this important measurement.\\
\textbf{\textit{An overall, statistics-dominated, uncertainty smaller than 0.001 of a SM neutrino partial width is therefore considered as a reasonable target for the Z invisible width at TLEP.}}
