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We perform the most up-to-date comprehensive signal-background analysis forHiggs-pair production in $HH \to b\bar b \gamma\gamma$ channel at the HL-LHCand HL-100 TeV hadron collider, with the goal of probing the self-coupling$\lambda_{3H}$ of the Higgs boson. We simulate all the standard-model signaland background processes with the simulation tools almost as sophisticated aswhat experimentalists are using. We find that even for the most promisingchannel $HH \to b\bar{b}\gamma\gamma$ at the HL-LHC with a luminosity of 3000fb$^{-1}$, the significance is still not high enough to establish the Higgsself-coupling at the standard model (SM) value. Instead, we can only constrainthe self-coupling to $-1.0 < \lambda_{3H} < 7.6 $ at $ 95\% $ confidence levelafter considering the uncertainties associated with the top-Yukawa coupling andthe estimation of backgrounds. Here we also extend the study to the HL-100 TeVhadron collider. With a luminosity of 3 ab$^{-1}$, we find there exists a bulkregion of $2.6 \lsim \lambda_{3H} \lsim 4.8 $ in which one cannot pin down thetrilinear coupling. Otherwise one can measure the coupling with a highprecision. At the SM value, for example, we show that the coupling can bemeasured with about 20 \% accuracy. While assuming 30 ab$^{-1}$, the bulkregion reduces to $3.1 \lsim \lambda_{3H} \lsim 4.3$ and the trilinear couplingcan be measured with about 7 \% accuracy at the SM value. With all thesimulated background events and results, our study can be useful to probe theHiggs potential of various models with extended Higgs sector, such astwo-Higgs-doublet model, MSSM, etc.

Higgs-boson-pair production $H(\rightarrow b\overline{b})H(\rightarrow\gamma\gamma)$ from gluon fusion at the HL-LHC and HL-100 TeV hadron collider