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We perform an analytical renormalization group (RG) study to address the roleof Coulomb repulsion, the competition between extended s-wave superconductingorder (s+-) and the spin-density wave (SDW) order and the angular dependence ofthe superconducting gap in multi-pocket models of Iron based superconductors.Previous analytic RG studies considered a toy model of one hole and oneelectron pocket. We consider more realistic models of two electron pockets andeither two or three hole pockets, and also incorporate the angular dependenceof interaction. We neglect for simplicity one of the two hole pockets centeredat k=0, which is less nested with electron pockets, i.e., consider 3-pocket and4-pocket models. In a toy 2-pocket model, SDW order always wins over s+- orderat perfect nesting, and s+- order only appears at a finite doping, and only ifRG flow extends long enough to overcome intra-pocket Coulomb repulsion. Formulti-pocket models, we find two new effects. First, there always exists anattractive component of the interaction in s+- channel, such that the systemnecessary becomes a superconductor once it overcomes the competition from theSDW state. Second, in 3-pocket case (but not in 4-pocket case), there aresituations 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 studiesby Thomale et al. [arXiv:1002.3599]

Renormalization group flow, competing phases, and the structure of superconducting gap in multi-band models of Iron based superconductors