BeyondPlanck XIV. Polarized foreground emission between 30 and 70GHz

We constrain polarized foreground emission between 30 and 70GHz with the Planck Low Frequency Instrument (LFI) and WMAP data within the framework of BeyondPlanck global Bayesian analysis. We combine for the first time full-resolution Planck LFI time-ordered data with low-resolution WMAP sky maps at 33, 40 and 61GHz. Spectral parameters are fit with a likelihood defined at the native resolution of each frequency channel. This analysis represents the first implementation of true multi-resolution component separation applied to CMB observations for both amplitude and spectral energy distribution (SED) parameters. For synchrotron emission, we approximate the SED as a power-law in frequency and find that the low signal-to-noise ratio of the data set strongly limits the number of free parameters that may be robustly constrained. We partition the sky into four large disjoint regions (High Latitude; Galactic Spur; Galactic Plane; and Galactic Center), each associated with its own power-law index. We find that the High Latitude region is prior-dominated, while the Galactic Center region is contaminated by residual instrumental systematics. The two remaining regions appear to be both signal-dominated and clean of systematics, and for these we derive spectral indices of $\beta_{\mathrm s}^{\mathrm{Spur}}=-3.15\pm0.07$ and $\beta_{\mathrm s}^{\mathrm{Plane}}=-3.12\pm0.06$. This agrees qualitatively with the WMAP-only polarization constraints presented by Dunkley et al. (2009), but contrasts with several temperature-based analyses. For thermal dust emission we assume a modified blackbody model and we fit the power-law index across the full sky. We find $\beta_{\mathrm{d}}=1.62\pm0.04$, which is slightly steeper than that previously reported from Planck HFI data, but still statistically consistent at a 2$\sigma$ confidence level.