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 + \( p\left(\overline{p}\right) \) 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.
