  After observing the Higgs boson by the ATLAS and CMS experiments at the LHC,
accurate measurements of its properties, which allow us to study the
electroweak symmetry breaking mechanism, become a high priority for particle
physics. The most promising of extracting the Higgs self-coupling at hadron
colliders is by examining the double Higgs production, especially in the $b
\bar{b} \gamma \gamma$ channel. In this work, we presented full loop
calculation for both SM and New Physics effects of the Higgs pair production to
next-to-leading-order (NLO), including loop-induced processes $gg\to HH$,
$gg\to HHg$, and $qg \to qHH$. We also included the calculation of the
corrections from diagrams with only one QCD coupling in $qg \to qHH$, which was
neglected in the previous studies. With the latest observed limit on the HH
production cross-section, we studied the constraints on the effective Higgs
couplings for the LHC at center-of-mass energies of 14 TeV and a provisional
100 TeV proton collider within the Future-Circular-Collider (FCC) project. To
obtain results better than using total cross-section alone, we focused on the
$b \bar{b} \gamma \gamma$ channel and divided the differential cross-section
into low and high bins based on the total invariant mass and $p_{T}$ spectra.
The new physics effects are further constrained by including extra kinematic
information. However, some degeneracy persists, as shown in previous studies,
especially in determining the Higgs trilinear coupling. Our analysis shows that
the degeneracy is reduced by including the full NLO corrections.
