Renormalization-group analysis of the generalized sine-Gordon model and of the Coulomb gas for $d>~3$ dimensions

Renormalization-group (RG) flow equations have been derived for the generalized sine-Gordon model (GSGM) and the Coulomb gas (CG) in $d>~3$ of dimensions by means of the Wegner-Houghton method, and by way of the real-space RG approach. The UV scaling laws determined by the leading-order terms of the flow equations are in qualitative agreement for all dimensions $d>~3,$ independent of the dimensionality, and in sharp contrast to the special case $d=2.\mathrm{}$ For the 4-dimensional GSGM it is demonstrated explicitly (by numerical calculations) that the blocked potential tends to a constant effective potential in the infrared limit, satisfying the requirements of periodicity and convexity. The comparison of the RG flows for the three-dimensional GSGM, the CG, and the vortex-loop gas reveals a significant dependence on the renormalization schemes and the approximations used.