  We study the importance of precise modelling of the photometric redshift
error distributions on the estimation of parameters from cross correlation
measurements and present a working example of the scattering matrix formalism
to correct for the redshift bin mismatch of objects in tomographic cross
correlation analysis. We measured the angular galaxy auto-power spectrum and
cross-power spectrum in four tomographic bins with redshift intervals $z =
[0.0,0.3,0.45,0.6,0.8]$ from the cross correlation of Planck Cosmic Microwave
Background lensing potential and photometric galaxy catalogue from the Dark
Energy Spectroscopic Instrument Legacy Imaging Survey Data Release 8. We
estimated galaxy linear bias and amplitude of cross correlation using maximum
likelihood estimation to put constraints on the $\sigma_{8}$ parameter. We show
that the modified Lorentzian function used to fit the photometric redshift
error distribution performs well only near the peaks of the distribution. We
adopt a sum of Gaussians model to capture the broad tails of the error
distribution. Our sum of Gaussians model yields $\sim 2-5\,\sigma$ smaller
values of cross correlation amplitude compared to the $\Lambda$CDM
expectations. We compute the $\sigma_{8}$ parameter after correcting for the
redshift bin mismatch of objects following the scattering matrix approach. The
$\sigma_{8}$ parameter becomes consistent with $\Lambda$CDM model in the last
tomographic bin but shows $\sim 1-3\,\sigma$ tension in other redshift bins.
