  We perform an analytical renormalization group (RG) study to address the role
of Coulomb repulsion, the competition between extended s-wave superconducting
order (s+-) and the spin-density wave (SDW) order and the angular dependence of
the superconducting gap in multi-pocket models of Iron based superconductors.
Previous analytic RG studies considered a toy model of one hole and one
electron pocket. We consider more realistic models of two electron pockets and
either two or three hole pockets, and also incorporate the angular dependence
of interaction. We neglect for simplicity one of the two hole pockets centered
at k=0, which is less nested with electron pockets, i.e., consider 3-pocket and
4-pocket models. In a toy 2-pocket model, SDW order always wins over s+- order
at perfect nesting, and s+- order only appears at a finite doping, and only if
RG flow extends long enough to overcome intra-pocket Coulomb repulsion. For
multi-pocket models, we find two new effects. First, there always exists an
attractive component of the interaction in s+- channel, such that the system
necessary becomes a superconductor once it overcomes the competition from the
SDW state. Second, in 3-pocket case (but not in 4-pocket case), there are
situations when s+- order wins over SDW order even for perfect nesting,
suggesting that SDW order is not a necessary pre-condition for the s+- order.
Our results are in good agreement with recent numerical functional RG studies
by Thomale et al. [arXiv:1002.3599]
