Primordial non-Gaussianity with $\mu$-type and $y$-type spectral
  distortions: exploiting Cosmic Microwave Background polarization and dealing
  with secondary sources
  Cross-correlations between Cosmic Microwave Background (CMB) temperature and
$y$-spectral distortions anisotropies have been previously proposed as a way to
measure the local bispectrum parameter $f_{\rm NL}^{\rm loc.}$ in a range of
scales inaccessible to either CMB ($T$, $E$) bispectra or $T$-$\mu$
correlations. This is useful e.g. to test scale dependence of primordial
non-Gaussianity. Unfortunately, the primordial $y$-T signal is strongly
contaminated by the late-time correlation between the Integrated Sachs Wolfe
and Sunyaev-Zel'dovich (SZ) effects. Moreover, SZ itself generates a large
noise contribution in the $y$-parameter map. We consider two original ways to
address these issues. In order to remove the bias due to the SZ-CMB temperature
coupling, while also adding new signal, we include in the analysis the
cross-correlation between $y$-distortions and CMB {\em polarization}. In order
to reduce the noise, we propose to clean the $y$-map by subtracting a SZ
template, reconstructed via cross-correlation with external tracers (CMB and
galaxy-lensing signals). We combine this SZ template subtraction with the
previously adopted solution of directly masking detected clusters. Our final
forecasts show that, using $y$-distortions, a PRISM-like survey can achieve
$1\sigma(f_{\rm NL}^\text{loc.}) = 300$, while an ideal experiment will achieve
$1\sigma(f_{\rm NL}^\text{loc.}) = 130$, with improvements of a factor $\sim 3$
from adding the $y$-$E$ signal, and a further $20-30 \%$ from template
cleaning. These forecasts are much worse than current $f_{\rm NL}^\text{loc.}$
boundaries from {\em Planck}, but we stress again that they refer to completely
different scales.
