Global Fixed Point Potentials in the Abelian Higgs Model with $N$ flavors

Abstract: Existence of charged fixed points in the Abelian Higgs model with $N$ flavors in $d$ dimensions is studied using the functional renormalization group. We numerically solve the coupled fixed point equations for the scale dependent charge and the non-perturbative effective potential for the scalar field. We show that the $ε=4-d$ expansion, famously successful in theories with $O(N)$ symmetry, fails to produce reliable results when taking the $ε\rightarrow 1$ limit. By determining global fixed point potentials, it is shown that the critical flavor number at which charged fixed points appear, modifies significantly compared to the perturbative treatment. In $d=3$, signs of a richer fixed point structure with presumably multicritical fixed points are also found. Discussions include subtleties of the gauge fixing and the corresponding modified Ward-Takahashi identities, including the possibility of a nonzero dimensionless photon mass at the infrared fixed point.