Running Flavor Number and Asymptotic Freedom in the Normal Phase of QED

In the normal phase (where no dynamical fermion mass generation occurs) of the D-dimensional quantum electrodynamics with $N_f$ flavors of fermions, we derive an integral equation which should be satisfied by (the inverse of) the wave function renormalization of the fermion in the Landau gauge. For this we use the inverse Landau-Khalatnikov transformation connecting the nonlocal gauge with the Landau gauge. This leads to a similar equation for the running flavor number in the framework of the $1/N_f$ resumed Schwinger-Dyson equation. Solving the equation analytically and numerically, we study the infrared behavior and the critical exponent of the 3-dimensional QED (QED$_3$). This confirms that the flavor number in QED$_3$ runs according to the $\beta$ function which is consistent with the asymptotic freedom as that in 4-dimensional QCD.