\subsection{Higgs physics in $\epem$ collisions at $\sqrt{s} = 500$~GeV}
\label{sec:ee500}

The TLEP physics potential at this centre-of-mass energy would be similar to that of the linear colliders ILC and CLIC, which have nominal luminosities that are comparable at $\sqrt{s}=500$~GeV. The ILC TDR~\cite{ILC:Physics} shows that the addition of 500~$\infb$ at 500~GeV to the baseline programme with 250~$\infb$ at 250~GeV and 350~$\infb$ at 350 GeV would improve the precision on all Higgs boson couplings to light fermions and gauge bosons by less than a factor $\sqrt{2}$, quite far from the sub-per-cent precision provided by TLEP at 240~GeV. The measurement of the invisible width of the Higgs boson would not be improved.   

On the other hand, the opening of the $\epemto \ttbar{\rm H}$ process allows the Htt coupling to be measured directly, typically with a precision of 10 to 15\%. However, the improvement with respect to the TLEP measurement at the $\ttbar$ threshold, which has an accuracy of 13\%, is marginal. More importantly, these precisions are not competitive with the HL-LHC projections~\cite{1307.7135,1307.7292}. For example, the CMS collaboration would be able to measure the Htt coupling with an accuracy of 7\% using just the ${\rm H} \to \bbbar, \gamma\gamma$ decay channels and an integrated luminosity of 3~$\inab$. This precision could be improved to 5\% by combining the two LHC experiments, and to significantly better when the ${\rm H} \to \tau\tau$, ZZ, and WW leptonic decay channels are added. 

Similarly, the opening of the $\epemto {\rm ZHH}$ and $\nnbar{\rm HH}$ processes at $\sqrt{s} = 500$~GeV enables a ``measurement'' of the triple Higgs-boson self-coupling, $\lambda_{\rm H}$, with 50 to 80\% precision. Again, these accuracies are not competitive with the HL-LHC projections, for which a 30\% accuracy on $\lambda_{\rm H}$ is envisioned. 

At this stage of the study, it appears that once sufficient $\epem$ data are collected at 250~and 350~GeV, the potential gain in Higgs physics alone is not enough to justify an upgrade to a centre-of-mass energy of 500~GeV. On the other hand, as discussed below, the appearance of some threshold for new physics above 350~GeV could change the picture entirely.