Fixed point structure of the Abelian Higgs model

The order of the superconducting phase transition is analyzed via the functional renormalization group approach. For the first time, we derive fully analytic expressions for the $\beta$ functions of the charge and the self-coupling in the Abelian Higgs model with one complex scalar field in $d=3$ dimensions that support the existence of two charged fixed points: an infrared (IR) stable fixed point describing a second-order phase transition and a tritical fixed point controlling the region of the parameter space that is attracted by the former one. It is found that the region separating first- and second-order transitions can be uniquely characterized by the Ginzburg-Landau parameter $\kappa$, and the system undergoes a second order transition, only if $\kappa >{\kappa }_c\approx 0.62\mathrm{/}\sqrt{2}$.