We investigate the generation of large-scale magnetic fields due to the breaking of the conformal invariance in the electromagnetic field through the $CPT$-even dimension-six Chern-Simons-like effective interaction with a fermion current by taking ac
count of the dynamical Kalb-Ramond and scalar fields in inflationary cosmology. It is explicitly demonstrated that the magnetic fields on 1Mpc scale with the field strength of $sim 10^{-9}$G at the present time can be induced.
We study the CPT-even dimension-six Chern-Simons-like term by including dynamical Kalb-Ramond and scalar fields to examine the cosmological birefringence. We show that the combined effect of neutrino current and Kalb-Ramond field could induce a sizab
le rotation polarization angle in the cosmic microwave background radiation polarization.
