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\label{sec:Higgs}  \section{\ref{sec:Higgs} Precise Measurements of the Higgs boson properties}  The primary goal of a Higgs factory is to measure the Higgs boson properties with a precision deemed adequate to be sensitive to Physics beyond the standard model. Tree-level couplings of the Higgs boson to fermions and gauge bosons are expected be modified with respect to the standard-model prediction, with a magnitude rapidly decreasing with the new Physics scale $\Lambda$, typically like $1/\Lambda^2$. For $\Lambda = 1$ TeV, departures up to 5\% are expected. To discover new Physics through its effects on the Higgs boson couplings with a significance of 5$\sigma$, it is therefore necessary to measure these couplings to fermions and gauge bosons with a precision of 1\%, and actually at the per-mil level if the if $\Lambda$ is to be larger than 1 TeV. The number of Higgs bosons expected to be produced, hence the integrated luminosity, is therefore a key element in the choice of the right Higgs factory for the future of high-energy physics: a per-mil accuracy cannot be reached with less than a million Higgs bosons. The Higgs production cross section, through the Higgs-strahlung process $\epemto {\tm HZ}$ and the $\rm WW}$ or ${\rm ZZ}$ fusion processes, is displayed in Fig.~\ref{fig:HiggsCross}.