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We study the off-shell production of the Higgs boson at the LHC to probe Higgs physics at higher energy scales utilizing the process $g g rightarrow h^{*} rightarrow ZZ$. We focus on the energy scale dependence of the off-shell Higgs propagation, and of the top quark Yukawa coupling, $y_t (Q^2)$. Extending our recent study in arXiv:1710.02149, we first discuss threshold effects in the Higgs propagator due to the existence of new states, such as a gauge singlet scalar portal, and a possible continuum of states in a conformal limit, both of which would be difficult to discover in other traditional searches. We then examine the modification of $y_t (Q^2)$ from its Standard Model (SM) prediction in terms of the renormalization group running of the top Yukawa, which could be significant in the presence of large flat extra-dimensions. Finally, we explore possible strongly coupled new physics in the top-Higgs sector that can lead to the appearance of a non-local $Q^2$-dependent form factor in the effective top-Higgs vertex. We find that considerable deviations compared to the SM prediction in the invariant mass distribution of the $Z$-boson pair can be conceivable, and may be probed at a $2sigma$-level at the high-luminosity 14 TeV HL-LHC for a new physics scale up to $mathcal{O}(1 {~rm TeV})$, and at the upgraded 27 TeV HE-LHC for a scale up to $mathcal{O}(3 {~rm TeV})$. For a few favorable scenarios, $5sigma$-level observation may be possible at the HE-LHC for a scale of about $mathcal{O}(1 {~rm TeV})$.
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