  We study the superconductivity pairing mechanism in Sr2RuO4 in the limit of
small interaction by extending a renormalization group calculation developed by
Raghu et al. [Phys. Rev. B 81, 224505 (2010)] to include spin-orbit coupling
and multiband effects. We show these effects to be crucial to discriminate
between the possible order parameters. In contrast to previous results and
without the necessity of fine-tuning, we obtain pseudospin-triplet gaps of the
same order of magnitude on the two-dimensional gamma band and the
quasi-one-dimensional alpha and beta bands. The ratio of the gap amplitude on
the different bands varies continuously with the interaction parameter. The
favored pairing symmetry is shown to be chiral when gamma is slightly dominant
and helical when alpha and beta are slightly dominant.
