  In order to investigate various decay channels of the Higgs boson $h$ or the
hypothetical dilaton, we consider a neutral particle $X$ with zero spin and
arbitrary $CP$ parity. This particle can decay into two off-mass-shell $Z$
bosons ($Z_1^*$ and $Z_2^*$) decaying to identical fermion-antifermion pairs
($f \bar{f}$): $X \to Z_1^* Z_2^* \to f \bar{f} f \bar{f}$. We derive
analytical formulas for the fully differential width of this decay and for the
fully differential width of $h \to Z_1^* Z_2^* \to 4 \ell$ ($4 \ell$ stands for
$4 e$, $4 \mu$, or $2 e 2 \mu$). Integration of these formulas yields some
Standard Model histogram distributions of the decay $h \to Z_1^* Z_2^* \to 4
\ell$, which are compared with corresponding Monte Carlo simulated
distributions obtained by ATLAS and with ATLAS experimental data.
