  The global picture of the Higgs potential in the bottom-up approach is still
unknown. A large deviation as big as O(1) fluctuations of the Higgs self
couplings is still a viable option for the New Physics. An interesting New
Physics scenario which can be linked to a large Higgs self coupling is the
baryogenesis based on the strong first order phase transition. We revisit the
strong first order phase transition in two classes of Beyond the Standard
Models, namely the Higgs portal with the singlet scalar under the Standard
Model gauge group with Z2 symmetry and the effective field theory approach with
higher-dimensional operators. We numerically investigate a few important issues
in the validity of the effective potential, caused by the breakdown of the
high-temperature approximation, and in the criteria for the strong first order
phase transition. We illustrate that these issues can lead to O(1)
uncertainties in the precision of the Higgs self couplings, which are relevant
when discussing sensitivity limits of different future colliders. We also find
that the quartic coupling of the above two classes of scenarios compatible with
the strong first order electroweak phase transition where the cubic coupling is
not negligible, can achieve a $2\sigma$ sensitivity at the 100 TeV pp-collider.
From this novel observation, we show that the correlation between the Higgs
cubic coupling and the quartic coupling will be useful for differentiating
various underlying New Physics scenarios and discuss its prospect for the
future colliders. Throughout our numerical investigation, the contribution from
Goldstone boson is not included.
