\section{Precise measurements of the Higgs boson properties}
\label{sec:Higgs}
The primary goal of a Higgs factory is to measure the Higgs boson properties with the best possible precision as to be sensitive to physics beyond the Standard Model at the highest possible scale. Tree-level couplings of the Higgs boson to fermions and gauge bosons are expected to 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~\cite{ILC:Physics,Gupta_Rzehak_Wells_2012}. 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 at least 1\%, and at the per-mil level to reach sensitivity to $\Lambda$ larger than 1 TeV, as suggested at by the negative results of the searches at the LHC. 

The number of Higgs bosons expected to be produced, hence the integrated luminosity delivered by the collider, are therefore key elements 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 (obtained with the {\tt HZHA} generator~\cite{cite:HZHA}), through the Higgs-strahlung process $\epemto {\rm HZ}$ and the ${\rm WW}$ or ${\rm ZZ}$ fusion processes, is displayed in Fig.~\ref{fig:HiggsCross}. A possible operational centre-of-mass energy is around 255 GeV, where the total production cross section is maximal and amounts to 210~fb. 