  We present an improved and extended analysis of the cross-correlation between
the map of the Cosmic Microwave Background (CMB) lensing potential derived from
the \emph{Planck} mission data and the high-redshift galaxies detected by the
\emph{Herschel} Astrophysical Terahertz Large Area Survey (H-ATLAS) in the
photometric redshift range $z_{\rm ph} \ge 1.5$. We compare the results based
on the 2013 and 2015 \textit{Planck} datasets, and investigate the impact of
different selections of the H-ATLAS galaxy samples. Significant improvements
over our previous analysis have been achieved thanks to the higher
signal-to-noise ratio of the new CMB lensing map recently released by the
\textit{Planck} collaboration. The effective galaxy bias parameter, $b$, for
the full galaxy sample, derived from a joint analysis of the cross-power
spectrum and of the galaxy auto-power spectrum is found to be $b =
3.54^{+0.15}_{-0.14}$. Furthermore, a first tomographic analysis of the
cross-correlation signal is implemented, by splitting the galaxy sample into
two redshift intervals: $1.5 \le z_{\rm ph} < 2.1$ and $z_{\rm ph}\ge 2.1$. A
statistically significant signal was found for both bins, indicating a
substantial increase with redshift of the bias parameter: $b=2.89\pm0.23$ for
the lower and $b=4.75^{+0.24}_{-0.25}$ for the higher redshift bin.
Consistently with our previous analysis we find that the amplitude of the cross
correlation signal is a factor of $1.45^{+0.14}_{-0.13}$ higher than expected
from the standard $\Lambda$CDM model for the assumed redshift distribution. The
robustness of our results against possible systematic effects has been
extensively discussed although the tension is mitigated by passing from 4 to
3$\sigma$.
