VII.D Renormalization Group for the Coulomb Gas

To construct an RG for the Coulomb gas, note that the partition for the system, eq.(VII.59), involves two parameters, (K, y0), and has an implicit cutoff, a, related to the minimum separation between vortices. As discussed earlier, the distinction between regions inside and outside the core of a vortex are arbitrary. Increasing the core size to ba modifies not only the core energy, hence y0, but also the interaction parameter K. The latter is a consequence of change in the dielectric properties of the medium due to dipoles of separations between a and ba. The change in fugacity is obtained from eq.(VII.46) by changing a to ba as ỹ0(ba) = b y0(a) . (VII.68)