  Examining the Higgs sector at high energy scales through off-shell Higgs
production can potentially shed light on the naturalness problem of the Higgs
mass. We propose such a study at the LHC by utilizing a representative model
with a new scalar field ($S$) coupled to the Standard Model Higgs doublet ($H$)
in a form $ |S|^2 |H|^2$. In the process $p p \rightarrow h^* \rightarrow ZZ$,
the dominant momentum-dependent part of the one-loop scalar singlet
corrections, especially above the new threshold at $2m_S$, leads to a
measurable deviation in the differential distribution of the $Z$-pair invariant
mass, in accordance with the quadratic divergence cancellation to the Higgs
mass. We find that it is conceivable to probe such new physics at the $5\sigma$
level at the high-luminosity LHC, improving further with the upgraded $27$ TeV
LHC, without requiring the precise measurement of the Higgs boson total width.
The discovery of such a Higgs portal could also have important implications for
thermal dark matter as well as for electroweak baryogenesis.
