Interpreting map-based E/B spectral properties of CMB foregrounds

Map-space E/B decompositions of linear polarization are attractive for foreground and CMB analyses because they separate parity families: B-family patterns directly contaminate primordial tensor searches, while E-family patterns trace coherent Galactic structures. However, the E/B transform is not fully local and can induce apparent spectral complexity even when the underlying sky is spectrally simple in P=Q+iU. We quantify this effect for synchrotron emission using complex log–Taylor and moment expansions for P, its spin-preserving projections PE and PB, and its more standard scalar projections E and B. We relate the coefficients of these expansions to physical mechanisms such as line-of-sight mixing, synchrotron ageing, and Faraday effects. Using simple sky models, we show how PE and PB reorganize the spectral behaviour of P into parity families with a clear geometric meaning. They retain interpretable amplitudes and angles and satisfy the closure relation P=PE+PB, which extends to all moment orders. By contrast, scalar quantities such as |E| and |B| show larger induced variability of effective spectral parameters and enhanced spectral complexity, while E + iB lacks interpretable polarization amplitude and angle. Finally, we present simple CMB-oriented applications: three-frequency diagnostics to test whether the sky is better described by a power law in P or by separate effective laws in (PE, PB), and idealized ILC and masking examples showing that the preferred map-space field depends on where foreground simplicity and residual contamination reside. This framework provides practical diagnostics for choosing foreground modelling, cleaning, and masking strategies in Galactic and CMB B-mode analyses.