  We first examine the scaling argument for a renormalization-group (RG)
analysis applied to a system subject to the dimensional reduction in strong
magnetic fields, and discuss the fact that a four-Fermi operator of the
low-energy excitations is marginal irrespective of the strength of the coupling
constant in underlying theories. We then construct a scale-dependent effective
four-Fermi interaction as a result of screened photon exchanges at weak
coupling, and establish the RG method appropriately including the screening
effect, in which the RG evolution from ultraviolet to infrared scales is
separated into two stages by the screening-mass scale. Based on a precise
agreement between the dynamical mass gaps obtained from the solutions of the RG
and Schwinger-Dyson equations, we discuss an equivalence between these two
approaches. Focusing on QED and Nambu--Jona-Lasinio model, we clarify how the
properties of the interactions manifest themselves in the mass gap, and point
out an importance of respecting the intrinsic energy-scale dependences in
underlying theories for the determination of the mass gap. These studies are
expected to be useful for a diagnosis of the magnetic catalysis in QCD.
