\section{High-energy upgrades}
\label{sec:VHE-LHC}

\subsection{Lepton-collider upgrades to $\sqrt{s} = 500$~GeV}

Both $\epem$ Higgs factories discussed in Section~\ref{sec:Higgs} (TLEP and ILC) have high-energy upgrade options. In the case of TLEP, the centre-of-mass energy  can be increased to $\sqrt{s} = 500$~GeV by tripling the RF length from 600 to 1700~m, thereby increasing the total RF voltage from 12 to 35~GV to compensate for the 31~GeV lost per turn by synchrotron radiation in the 100~km ring. In the case of the ILC, its length can be doubled to reach a centre-of-mass energy of 500 GeV. 

With a 2.5\% momentum acceptance at each interaction point, TLEP-500 would have a one-minute beam lifetime, which would allow for an average luminosity of 90\% of the peak luminosity with the baseline TLEP top-off injection scheme.  With these parameters, a luminosity of $0.5\times10^{34}~\cms$ would be delivered at each interaction point with a beam-beam tune shift of 0.1, for a total luminosity of $2 \times 10^{34}~\cms$ when summed over the four IPs, as displayed in Fig.~\ref{fig:TLEPUpgrade}. Although not included in the TLEP baseline programme at this time, the design study will investigate the feasibility of such an option and define the maximum reachable centre-of-mass energy under reasonable assumptions. 