This study investigates the origin of superconductivity in the iron-based superconductor LaFeAsO$_{1-x}$F$_x$. The researchers constructed a minimal model that includes all five Fe d bands and applied the random-phase approximation (RPA) to analyze the superconducting mechanism. They concluded that the multiple spin fluctuation modes arising from the nesting across the disconnected Fermi surfaces lead to an extended s-wave pairing, while d-wave pairing is also a possible candidate. LaFeAsO belongs to the family of quaternary oxypnictides, and superconductivity was first observed in LaFePO with a critical temperature $ T_c \approx 3K $, which was increased to $ T_c \approx 7K $ by F doping. Recently, superconductivity was discovered in LaFeAsO$_{1-x}$F$_x$ with $ T_c \sim 26K $ when $ x \approx 0.11 $. The high $ T_c $ suggests unconventional superconductivity, supported by experimental evidence such as specific heat measurements showing a $ \sqrt{H} $ behavior and point-contact conductance measurements indicating a zero-bias peak, suggesting a sign change in the gap function. Theoretical studies show that the band structure of LaFeAsO has five Fermi surface sheets, which contradicts experimental results for undoped LaFeAsO. However, dynamical mean-field theory indicates that electron correlations enhance the crystal field splitting, leading to a band-semiconducting behavior consistent with experiments. Local spin-density calculations suggest the system is near the boundary between magnetic and nonmagnetic states, with a tendency toward ferromagnetism and antiferromagnetism. The electron-phonon coupling is too weak to explain the high $ T_c $. The researchers constructed a microscopic electronic model for LaFeAsO$_{1-x}$F$_x$, which includes all five Fe d orbitals. They applied RPA to solve the Eliashberg equation and found that the Fermi surface consists of multiple pockets, leading to multiple spin-fluctuation modes. The spin fluctuations dominate over orbital fluctuations when $ U > U' $, and the spin susceptibility matrix shows peaks around $ (\pi, 0) $, $ (0, \pi) $, and a ridge-like structure from $ (\pi, \pi/2) $ to $ (\pi/2, \pi) $, reflecting the Fermi surface nesting. The gap function is primarily s-wave but changes sign between the Fermi surfaces of bands 3 and 4, across the nesting vector $ (\pi, 0) $, $ (0, \pi) $. This sign change is similar to those in models studied by Bulut et al. and the authors. The gap nodes intersect the $ \beta $ Fermi surface due to spin fluctuations from $ \beta_{1}-\beta_{This study investigates the origin of superconductivity in the iron-based superconductor LaFeAsO$_{1-x}$F$_x$. The researchers constructed a minimal model that includes all five Fe d bands and applied the random-phase approximation (RPA) to analyze the superconducting mechanism. They concluded that the multiple spin fluctuation modes arising from the nesting across the disconnected Fermi surfaces lead to an extended s-wave pairing, while d-wave pairing is also a possible candidate. LaFeAsO belongs to the family of quaternary oxypnictides, and superconductivity was first observed in LaFePO with a critical temperature $ T_c \approx 3K $, which was increased to $ T_c \approx 7K $ by F doping. Recently, superconductivity was discovered in LaFeAsO$_{1-x}$F$_x$ with $ T_c \sim 26K $ when $ x \approx 0.11 $. The high $ T_c $ suggests unconventional superconductivity, supported by experimental evidence such as specific heat measurements showing a $ \sqrt{H} $ behavior and point-contact conductance measurements indicating a zero-bias peak, suggesting a sign change in the gap function. Theoretical studies show that the band structure of LaFeAsO has five Fermi surface sheets, which contradicts experimental results for undoped LaFeAsO. However, dynamical mean-field theory indicates that electron correlations enhance the crystal field splitting, leading to a band-semiconducting behavior consistent with experiments. Local spin-density calculations suggest the system is near the boundary between magnetic and nonmagnetic states, with a tendency toward ferromagnetism and antiferromagnetism. The electron-phonon coupling is too weak to explain the high $ T_c $. The researchers constructed a microscopic electronic model for LaFeAsO$_{1-x}$F$_x$, which includes all five Fe d orbitals. They applied RPA to solve the Eliashberg equation and found that the Fermi surface consists of multiple pockets, leading to multiple spin-fluctuation modes. The spin fluctuations dominate over orbital fluctuations when $ U > U' $, and the spin susceptibility matrix shows peaks around $ (\pi, 0) $, $ (0, \pi) $, and a ridge-like structure from $ (\pi, \pi/2) $ to $ (\pi/2, \pi) $, reflecting the Fermi surface nesting. The gap function is primarily s-wave but changes sign between the Fermi surfaces of bands 3 and 4, across the nesting vector $ (\pi, 0) $, $ (0, \pi) $. This sign change is similar to those in models studied by Bulut et al. and the authors. The gap nodes intersect the $ \beta $ Fermi surface due to spin fluctuations from $ \beta_{1}-\beta_{