Bose-Einstein Correlations and Collective Flow in Proton-Proton Collisions at √ 𝑠 = 13 TeV: A Tensorial Framework via the Haddad-Λ Formalism

This article introduces a novel geometric framework for interpreting Bose–Einstein correlations (BEC) in high-energy proton–proton collisions. Traditionally used to probe the femtoscopic structure of particle-emitting sources, BEC measurements provide access to source radii and coherence parameters—but are often interpreted through purely statistical or hydrodynamic models. In this work, the author proposes a new tensor—the Haddad–Λ tensor—that captures the interplay between quantum phase coherence, space-time curvature, and entropic flow within the emission region. By applying this tensor to BEC observables such as the homogeneity radius R_{\mathrm{inv}} and the correlation strength \lambda, the article reinterprets femtoscopic results as geometric and information-theoretic projections of an underlying quantum system. The study uses experimental data from CMS and ATLAS at \sqrt{s} = 13\,\mathrm{TeV}, and validates the tensor model through detailed Monte Carlo simulations, Lévy and Gaussian fits, and direct mapping of experimental parameters to tensor eigenvalues. The results suggest that even in small systems, such as high-multiplicity pp collisions, collective behavior and geometric coherence can emerge and be described by a unified tensorial formalism. Bridging ideas from quantum gravity, information geometry, and collider physics, this article lays the foundation for a deeper theoretical understanding of QCD collectivity—and opens new paths for incorporating curvature, coherence, and entropy into the analysis of femtoscopic data.