  In the context of the standard model of particles, the weak interaction of
cosmic microwave background (CMB) and cosmic neutrino background (C$\nu$B), can
generate non-vanishing TB and EB power spectra in the order of one loop forward
scattering, in the presence of scalar perturbation, which is in contrast with
the standard scenario cosmology. Comparing our results with the current
experimental data may provide, significant information about the nature of
C$\nu$B, including CMB-C$\nu$B forward scattering for TB, TE, and EB power
spectra. To this end, different cases were studied, including Majorana C$\nu$B
and Dirac C$\nu$B. On the other hand, it was shown that the mean opacity due to
cosmic neutrino background could behave as an anisotropic birefringent medium
and change the linear polarization rotation angle. Considering the
contributions from neutrino and anti-neutrino forward scattering with CMB
photons (in the case of Dirac neutrino), we introduce relative neutrino and
anti-neutrino density asymmetry ($\delta_\nu=\frac{\Delta
n_\nu}{n_\nu}=\frac{n_\nu-n_{\bar{\nu}}}{n_\nu}$). Then, using the cosmic
birefringence angle reported by the Planck data release
$\beta=0.30^\circ\pm0.11^\circ$ ($68\%C.L.$), some constraints can be put on
$\delta_\nu$. Also, the value of cosmic birefringence due to Majorana C$\nu$B
medium is estimated at about $\beta|_\nu\simeq0.2$ rad. In this respect, since
Majorana neutrino and anti-neutrino are exactly the same, both CB contributions
will be added together. However, this value is at least two orders larger than
the cosmic birefringence angle reported by the Planck data release,
$\beta=0.30^\circ\pm0.11^\circ$ ($68\%C.L.$).
