Absence of cross-confinement for dynamically generated multi-Chern–Simons theories

We show that when the induced parity breaking part of the effective action for the low-momentum region of U(1)×⋯×U(1) Maxwell gauge field theory with massive fermions in 3 dimensions is coupled to a φ4 scalar field theory, it is not possible to eliminate the screening of the long-range Coulomb interactions and get external charges confined in the broken Higgs phase. This result is valid for non-zero temperature as well.