  We study the mechanism of the triplet superconductivity (TSC) in
Sr$_2$RuO$_4$ based on the multiorbital Hubbard model. The electronic states
are studied using the recently developed renormalization group method combined
with the constrained random-phase-approximation, called the RG+cRPA method.
Thanks to the vertex correction (VC) for the susceptibility, which is dropped
in the mean-field-level approximations, strong orbital and spin fluctuations at
$Q \approx(2\pi/3,2\pi/3)$ emerge in the quasi one-dimensional Fermi surfaces
(FSs) composed of $d_{xz}+d_{yz}$ orbitals. Due to the cooperation of both
fluctuations, we obtain the triplet superconductivity in the $E_u$
representation, in which the superconducting gap is given by the linear
combination of $(\Delta_x(k),\Delta_y(k))\sim (\sin 3k_x,\sin 3k_y)$. Very
similar results are obtained by applying the diagrammatic calculation called
the self-consistent VC method. Thus, the idea of "orbital+spin fluctuation
mediated TSC" is confirmed by both RG+cRPA method and the self-consistent VC
method. We also reveal that a substantial superconducting gap on the
$d_{xy}$-orbital FS is induced from the gaps on the quasi one-dimensional FSs,
in consequence of the large orbital-mixture due to the 4$d$ spin-orbit
interaction.
