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Simultaneous determination of the cosmic birefringence and miscalibrated polarization angles II: Including cross frequency spectra

We develop a strategy to determine the cosmic birefringence and miscalibrated polarization angles simultaneously using the observed $EB$ polarization power spectra of the cosmic microwave background and the Galactic foreground emission. We extend the methodology of Y. Minami et al. (Prog. Theor. Exp. Phys. $\mathbf{2019}$, 083E02, 2019), which was developed for auto frequency power spectra, by including cross frequency spectra. By fitting one global birefringence angle and independent miscalibration angles at different frequency bands, we determine both angles with significantly smaller uncertainties (by more than a factor of two) compared to the auto spectra.