Probing the Higgs boson-gluon coupling via the jet energy profile at $e^+e^-$ colliders

The effective coupling of the Higgs boson to a gluon pair is one of the most important parameters to test the Standard Model and search for the new physics beyond. In this paper, we propose several new observables based on the jet energy profile to extract the effective coupling. The statistical uncertainties of the effective coupling extracted by using new observables are derived and estimated based on the simulation at the future $e^+e^-$ collider for $250$ GeV center-of-mass energy and 5 ab$^{-1}$ integrated luminosity. We found that the statistical uncertainties of effective coupling via the optimized observable can reach about $1.6\%$ in the channels of a $Z$ boson decaying to lepton pairs and is reduced by $52\%$ compared to the relevant uncertainties in the conventional approach. These new observables potentially can be helpful for the measurement of effective coupling at future $e^+e^-$ colliders.