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 demonstrate that the extension of the Zee-Babu model can generate not only the small neutrino masses but also the baryon number asymmetry in the universe. In particular, we show that the scale of the singlet scalar responsible for the leptogenesis
can be of order 1 TeV, that can be tested at the LHC and ILC. We also considered the possible minimal extension of this model to generate the dark matter.
