  The hadron collider phase of the Future Circular Collider (FCC-hh) is a
proton-proton collider operating at a center-of-mass energy of 100 TeV. It is
one of the most ambitious projects planned for the rest of this century and
offers ample opportunities in the hunt for new physics, both through its direct
detection reach as well as through indirect evidence from precision
measurements. Extracting a precision measurement of the Higgs self-coupling
from the Higgs pair production cross-section will play a key role in our
understanding of electroweak symmetry breaking, as the self-coupling gives
insight into the nature of the Higgs potential. With the large data set of in
total 30 $\text{ab}^{-1}$ which is envisioned to be collected during the FCC-hh
runtime the Higgs self-coupling will be determined down to the percent level.
This paper presents prospect studies for Higgs self-coupling measurements in
the $b\bar{b}\gamma\gamma$ and $b\bar{b}\ell\ell + E_{\text{T}}^{\text{miss}}$
final states, with the combined, expected precision on the Higgs self-coupling
modifier $\kappa_{\lambda}$ reaching 3.2-5.7% at 68% confidence level, assuming
all other Higgs couplings follow their Standard Model expectations and
depending on the systematic uncertainties assumed. This high precision is
mostly driven by the $b\bar{b}\gamma\gamma$ final state analysis, while the
$b\bar{b}\ell\ell + E_{\text{T}}^{\text{miss}}$ final state - newly studied for
its FCC-hh prospects in this document - on its own reaches a maximum precision
of roughly 20% on $\kappa_{\lambda}$.
