deletions | additions       diff --git a/Z partial widths and peak cross section.tex b/Z partial widths and peak cross section.tex  index fed98ea..ec83b76 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 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-scattering cross section (needed (used  for the integrated luminosity measurement). determination).  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 measurement,  expressed in terms of a number of active neutrinos \begin{equation}  N_\nu = 2.984 \pm 0.008,  \end{equation}   is ~2$\sigma$ two standard deviations  below the SM value of 3. 3.00.  The experimental conditions at TLEP will be adequate to improve the experimental uncertainty considerably considerably,  but a commensurate effort would have to be invested in the theoretical calculations of the small angle Bhabha scattering small-angle Bhabha-scattering cross section  usedfor luminosity normalization  to make this measurement worthwhile. worthwhile.\\  {\em A desirable goal would be to reduce the uncertainty on $N_\nu$ down to $\pm 0.001$ 0.001$,  but it is not clear that it can be achieved from Z peak measurements}. measurements.}