  Inspired by the picture portraying the KNO scaling violation as an extension
of the geometrical scaling violation, the current study proposes a
phenomenological model for multi-particle production in hadron collisions based
on the geometrical approach and using the $U$-Matrix unitarization scheme of
the scattering amplitude. The model has been fine-tuned and all parameters have
been derived from optimal fits to various hadronic multiplicity distributions
data in $p + p(\bar{p})$ collisions across a broad range of energies. The
results have revealed that our model furnishes a reasonable description of
diverse multiplicity distributions at various energies. Besides, they have
demonstrated a pronounced violation of the geometrical scaling, which
eventually resulted in a significant violation of the KNO scaling. The study
has also analyzed the higher-order moments of the multiplicity distribution. We
have observed an unexpected overestimation of the fluctuations and correlations
between final state particles with increasing energy, particularly above LHC
energy. It is claimed that this overestimation is due to statistical
fluctuations embedded in the $U$-matrix scheme. The findings of this study have
shed light on the key role of the $U$-matrix scheme in the impact of collision
geometry on multi-particle production processes at high energy.
