New conformal field theories by gauging electric and magnetic currents

Old folklore says that there is no non-trivial renormalization group fixed point with $U(1)$ gauge symmetry in four dimensions, but it can be circumvented by the existence of magnetic monopoles. We propose to construct (potentially infinitely many) novel types field theories gauging $U(1) \times U(1)$ global electrically and magnetically. If construction consistent, resulting theory will most likely possess a for coupling constant conformal without $U(1)\times infrared upon tunings other relevant deformations. CP-violating, $\theta$-parameter accompanied.