  To understand the amazing variety of the superconducting states of Fe-based
superconductors, we analyze the multiorbital Hubbard models for LaFeAsO and
LiFeAs going beyond the random-phase approximation (RPA), by calculating the
vertex correction (VC) and self-energy correction. Due to the spin+orbital mode
coupling described by the VC, both orbital and spin fluctuations mutually
develop, consistently with the experimental phase diagram with the orbital and
magnetic orders. Due to both fluctuations, the s-wave gap function with
sign-reversal ($s_{\pm}$-wave), without sign-reversal ($s_{++}$-wave), and
nodal s-wave states are obtained, compatible with the experimental wide variety
of the gap structure. Thus, the present theory provides a microscopic
explanation of the normal and superconducting phase diagram based on the
realistic Hubbard model.

Superconductivity;Strongly Correlated Electrons