deletions | additions       diff --git a/Z partial widths and peak cross section.tex b/Z partial widths and peak cross section.tex  index 3429d43..fed98ea 100644  --- a/Z partial widths and peak cross section.tex  +++ b/Z partial widths and peak cross section.tex 
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\subsubsection{The Z hadronic and leptonic partial widths}  Determination of the Z partial widths requires measurements of branching ratios at the Z peak, peak --  in particular the ratio of branching fractions of the Z boson into lepton to and into  hadrons --  and the peak hadronic cross-section. cross section.  The hadronic to leptonic hadronic-to-leptonic  ratio was measured at LEP to be $R_\ell \begin{equation}   R_\ell  = \frac{\Gamma_{had}}{\Gamma_{\ell}}= \frac{\Gamma_{\rm had}}{\Gamma_{\ell}}=  20.767\pm 0.025$ 0.025,   \end{equation}  with acommon  systematic error uncertainty  of 0.007. The experimental error uncertainty  was dominated by the statistics of leptonic decays; decays, and other uncertainties related to the event  selectionsystematics  will tend to decrease with statistics. The remaining systematics are systematic uncertainty were  related to the t-channel $t$-channel  contribution in the electron channel (this could be removed (which would vanish  by using the sole use of  the muon channel only) channel)  and to the detailed modelling of final state final-state  radiation or emission of additional lepton pairs: here pairs. Here,  theory should be considerably helped by the large sample of leptonic Z decays available for the study of these rare processes. \\ {\em We consider that a A  relative precision of $< 10^{-5}$ less than $10^{-5}$  is considered to be  a reasonable targetgoal  for the ratio of the hadronic to leptonic Z hadronic-to-leptonic  partial widths at TLEP-Z }. TLEP}.  \subsubsection{The peak hadronic cross section and the number of neutrinos}  This measurement of the peak hadronic cross-section is already dominated by theoretical systematics today,  related to the understanding of the low angle Bhabha cross-section. low-angle Bhabha-scattering cross section (needed for the integrated luminosity measurement).  It is a measurement of great interest as it determines the Z invisible width, a direct test of the unitarity of the PMNS matrix -- or of the existence of sterile neutrinos. The present measurement expressed in terms of a number of active neutrinos $N_\nu \begin{equation}   N_\nu  = 2.984 \pm 0.008$ 0.008,   \end{equation}  is ~2$\sigma$ below the SM value of 3. The experimental conditions at TLEP will be adequate to improve the experimental uncertainty considerably but a commensurate effort would have to be invested in the theoretical calculations of the small angle Bhabha scattering used for luminosity normalization to make this measurement worthwhile. {\em A desirable goal would be to reduce the uncertainty on $N_\nu$ down to $\pm 0.001$ but it is not clear that it can be achieved from Z peak measurements}.