  Recently, the cross-correlation between $E$- and $B$-mode polarization of the
cosmic microwave background (CMB), which is well explained by cosmic
birefringence with rotation angle $\beta\approx 0.3$ deg, has been found in CMB
polarization data. We carefully investigate the possibility of explaining the
observed $EB$ correlation by the primordial chiral gravitational waves (CGWs),
which can be generated in the parity-violating theories in the primordial
Universe. We found that the CGWs scenario does not work due to the
overproduction of the $BB$ auto-correlation which far exceeds the observed one
by SPTPol and POLARBEAR.
