Cosmological birefringence due to CPT-even Chern-Simons-like term with Kalb-Ramond and scalar fields

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 sizable rotation polarization angle in the cosmic microwave background radiation polarization.