  The European Strategy for Particle Physics (ESSP) submitted in 2013 a
deliberation document to the CERN council explaining that a lepton collider
with "energies of 500\,GeV or higher could explore the Higgs properties
further, for example the [Yukawa] coupling to the top quark, the [trilinear]
self-coupling and the total width.". In view of the forthcoming ESPP update in
2020, variations on this qualitative theme have been applied, inaccurately, to
the case of the ILC, to argue that an upgrade to 500\,GeV would allow the
measurement of the Higgs potential and would increase the potential for new
particle searches. As a consequence, the strategic question was raised again
whether the FCC-ee design study ought to consider a 500 GeV energy upgrade. In
this note, we revisit the ESSP 2013 statement quantitatively and find [...]
that 500 GeV is not a particularly useful energy for the lepton colliders under
consideration, especially for the FCC-ee. A 5 sigma demonstration of the
existence of the Higgs self-coupling is within reach at the energies foreseen
for the FCC-ee, with a moderate change of configuration, which certainly
deserves consideration.
