deletions | additions       diff --git a/Statistical analysis.tex b/Statistical analysis.tex  index a145a0a..ac73bf1 100644  --- a/Statistical analysis.tex  +++ b/Statistical analysis.tex 
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There are nine different functions entering Eq.~\ref{eq:optimal}, and eight form factors $\delta_i$ to be evaluated from a given sample of ${\rm t \bar t}$ events. In principle, all eight form factors and their uncertainties can be determined simultaneously. Experimentalists usually maximize numerically a global likelihood $L$ -- or equivalently, minimize the negative Log-likelihood ($-\log L$) -- with respect to all form factors:  \begin{equation}  \label{eq:likelihood}  L = {\mu^N \over N!}{\rm e}^{-\mu} \times \prod_{k=1}^N p(k),  \end{equation}   where $N$ is the total number of ${\rm t \bar t}$ events observed in the data sample, $\mu$ is the number of events expected for the integrated luminosity ${\cal L}$ of the data sample ($\mu = \sigma_{\rm tot} \times {\cal L}$), and 
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p(k) = {1 \over \sigma_{\rm tot} } { {\rm d}^2\sigma \over {\rm d}x {\rm d}\cos\theta } {\rm , with \ }   \sigma_{\rm tot} = \int {\rm d}x {\rm d}\cos\theta} { {\rm d}^2\sigma \over {\rm d}x {\rm d}\cos\theta } .  \end{equation}  The covariance matrix obtained from the numerical minimization of the negative log-likelihood is then inverted to get the uncertainties on the form factors, $\sigma(\delta_i)$. It can be shown~\cite{Davier_1993} that, in the linear form given in Eq.~\ref{eq:optimal}, this method leads to optimal statistical uncertainties on the form factors. The functions $f_i(x,\cos\theta)$ are therefore called "optimal observables". It turns out~\cite{Diehl_1994} that these the covariance matrix, hence the  statistical uncertainties on the form factors,  can be obtained analytically in the limit of large number of events, which is the case considered in this letter. If the total event rate is included in the derivation of the likelihood, as is the case in Eq.~\ref{eq:likelihood}, the elements of the covariance matrix $V$ are giveb by  \begin{equation}  V_{ij} = {\cal L} \int {\rm d}x {\rm d}\cos\theta} { f_i(x,\cos\theta) \times f_j(x,\cos\theta) \over S^0(x,\cos\theta)}  \end{equation}