  The experimental discoveries of fractional quantum anomalous Hall effects
under zero magnetic fields in both transition metal dichalcogenide and
pentalayer graphene moir\'e superlattices have aroused significant research
interest. In this work, we theoretically study the fractional quantum anomalous
Hall states (also known as fractional Chern insulator states) in pentalayer
graphene moir\'e superlattice. Starting from the highest energy scale
($\sim\!2\,$eV) of the continuum model, we first construct a renormalized
low-energy model that applies to a lower cutoff $\sim\!0.15\,$eV using
renormalization group approach. Then, we study the ground states of the
renormalized low-energy model at filling 1 under Hartree-Fock approximation in
the presence of tunable but self-consistently screened displacement field $D$
with several experimentally relevant background dielectric constant
$\epsilon_r$. Two competing Hartree-Fock states are obtained at filling 1,
which give rise to two types of topologically distinct isolated flat bands with
Chern number 1 and 0, respectively. We continue to explore the interacting
ground states of the two types of isolated flat bands at hole dopings of 1/3,
2/5, 3/5, and 2/3 (corresponding electron fillings of 2/3, 3/5, 2/5, and 1/3
with respect to charge neutrality). Setting $\epsilon_r=5$, our
exact-diagonalization calculations suggest that the system stays in fractional
Chern insulator (FCI) state at 2/3 electron filling when
$0.9\,\textrm{V/nm}\leq\!D\!\leq 0.92\,\textrm{V/nm}$; while no robust FCI
state is obtained at 1/3 electron filling. We have also obtained
composite-fermion type FCI ground states at 3/5 electron filling within
$0.9\,\textrm{V/nm}\leq\! D \!\leq\!0.95\,\textrm{V/nm}$ and $\epsilon_r=5$.
These numerical results are quantitatively consistent with experimental
observations.
