No anomalous spin fluctuation in optimally doped and overdoped iron-based oxypnictide superconductors LaFeAsO$_{1-x}$F$_x$ probed by muon spin relaxation

Zero field and longitudinal field muon spin relaxation measurements have been performed in optimally doped and overdoped superconductors LaFeAsO$_{1-x}$F$_x$ in order to investigate the magnetic fluctuation over a wide range of temperature and longitudinal field. We have observed no sign of magnetic fluctuation against temperature in the muons time window (10$^{-5}sim$ 10$^{-9}$s). Considering the current results and other results, i.e., spin fluctuation observed by neutron scattering, pseudogap-like behaviors by NMR and photoemission spectroscopy, it is suggested that not only the spin fluctuation but also the multiband character with several different orbital contributions at the Fermi surface may play an important role in the superconducting pairing mechanism of LaFeAsO$_{1-x}$F$_x$.