Higgs boson $ZZ$ couplings in Higgs-strahlung at the ILC

We derive the fully differential cross section of the Higgs-strahlung process $f \bar{f} \to Z \to Z (\to f_Z \bar{f}_Z) X (\to f_X \bar{f}_X)$, where $f$, $f_Z$, and $f_X$ are arbitrary fermions and $X$ is a spin-zero particle with arbitrary couplings to $Z$ bosons and fermions. This process with $f = e$ and $X = h$ ($h$ denotes the Higgs boson) is planned to be measured at the ILC to put constraints on the couplings $g_1$, $g_2$, $g_3$ of the Higgs boson to $Z$ bosons. Using the obtained fully differential cross section, we define observables measurement of which yields the tightest constraints on the couplings. Explicit dependences of these observables on $g_i$ are derived.