  The effects of $\omega$ meson self-coupling (OMSC) on the properties of
finite nuclei and neutron stars are investigated within the framework of
effective field theory based relativistic mean-field (ERMF) model which
includes the contributions from all possible mixed interactions between the
scalar-isoscalar ($\sigma$), vector-isoscalar ($\omega$) and vector-isovector
($\rho$) mesons upto the quartic order. For a realistic investigation, several
parameter sets corresponding to different values of OMSC are generated by
adjusting the remaining parameters of the ERMF model to fit the properties of
the finite nuclei. Though, all these parameter sets give equally good fit to
the properties of the finite nuclei, only moderate values of OMSC are favored
from the "naturalness" point of view. The equation of state for the symmetric
nuclear and pure neutron matters resulting from the parameter sets with the
moderate values of OMSC are in close agreement with the ones obtained within
the Dirac-Brueckner-Hartree-Fock approximation. For such parameter sets the
limiting mass for the neutron stars composed of $\beta$-stable matter is $\sim
1.9M_\odot$. It is found that the direct Urca process can occur in the neutron
stars with "canonical" mass of $1.4M_\odot$ only for the moderate and higher
values of OMSC. Some other interesting properties for the neutron stars are
also discussed.
