Cross-correlation between CMB B-modes and Faraday Rotation as a Novel Probe of Primordial Magnetic Fields

We investigate the cross-bispectrum between the cosmic microwave background (CMB) B-mode polarization and two Faraday rotation (FR) angles as a new probe of primordial magnetic fields (PMFs). Assuming Gaussian PMFs with both non-helical and helical components, we derive the Bαα bispectrum and decompose it into parity-even and parity-odd contributions. The mixed non-helical--helical contribution appears in the parity-even sector, whereas the purely non-helical and purely helical contributions contribute to the parity-odd sector. This parity dependence provides sensitivity to PMF helicity that is absent in the FR power spectrum alone. We then forecast the detectability of the cross-bispectrum for SO-like and LiteBIRD-like CMB experiments, and find that future observations can probe PMFs with amplitudes at the sub-nG level. The Bαα bispectrum is dominated by the helicity-sensitive parity-even sector and provides a promising avenue for constraining both the amplitude and helicity of PMFs.