  Here we have developed a FLEX+DMFT formalism, where the symmetry properties
of the system are incorporated by constructing a SO(4) generalization of the
conventional fluctuation-exchange approximation (FLEX) coupled
self-consistently to the dynamical mean-field theory (DMFT). Along with this
line, we emphasize that the SO(4) symmetry is the lowest group-symmetry that
enables us to investigate superconductivity and antiferromagnetism on an equal
footing. We have imposed this by decomposing the electron operator into
auxiliary fermionic and slave-boson constituents that respect SU(2)$_{\rm
spin}\otimes$SU(2)$_{\eta{\rm spin}}$. This is used not in a mean-field
treatment as in the usual slave-boson formalisms, but instead in the DMFT
impurity solver with an SU(2)$_{\rm spin}\otimes$SU(2)$_{\eta{\rm spin}}$
hybridization function to incorporate the FLEX-generated bath information into
DMFT iterations. While there have been attempts such as the doublon-less SU(2)
slave-boson formalism, the present "full-SU(2)" slave-boson formalism is
expected to provide a new platform for addressing the underlying physics for
various quantum orders, which compete with each other and can coexist.
