  The Cosmic Microwave Background (CMB) primordial B-modes signal is predicted
to be much lower than the polarized Galactic emission (foregrounds) in any
region of the sky pointing to the need for sophisticated component separation
methods. Among them, the blind Needlet-ILC (NILC) has great relevance given our
current poor knowledge of the B-modes foregrounds. However the expected level
of spatial variability of the foreground spectral properties complicates the
NILC subtraction of the Galactic contamination. In order to reach the ambitious
targets of future CMB experiments, we therefore propose a novel extension of
the NILC approach, the Multi-Clustering NILC (MC-NILC), which performs NILC
variance minimization on separate regions of the sky (clusters) properly chosen
to have similar spectral properties of the B-modes foregrounds emission.
Clusters are identified thresholding the ratio of B-modes maps at two separate
frequencies which is used as tracer of the spatial distribution of the spectral
indices of the Galactic emission in B modes. We consider ratios either of
simulated foregrounds-only B modes (ideal case) or of cleaned templates of
Galactic emission obtained from realistic simulations. In this work we present
an application of MC-NILC to the future LiteBIRD satellite, which targets the
observation of both reionization and recombination peaks of the primordial
B-modes angular power spectrum with a total error on the tensor-to-scalar ratio
$\delta r < 0.001$. We show that MC-NILC provides a CMB solution with residual
foregrounds and noise contamination that is significantly reduced with respect
to NILC and lower than the primordial signal targeted by LiteBIRD at all
angular scales for the ideal case and at the reionization peak for a realistic
ratio. Thus, MC-NILC will represent a powerful method to mitigate B-modes
foregrounds for future CMB polarization experiments.
