Topological analysis of scale-invariant spatial fluctuations in ultrarelativistic heavy-ion collisions
The QGP-to-hadronic matter phase transition and QCD critical point in heavy-ion collisions can be identified by studying spatial fluctuations among final-state particles using intermittency analysis. First CMC-based intermittency analysis in the two-dimensional angular ($\eta$, $\varphi$) phase space, using EPOS as the background model is presented. Critical fluctuation signals are extremely weak, constituting only a few percent of the total event sample and are severely diluted by the overwhelming non-critical background, rendering traditional intermittency analyses insufficient for reliable signal extraction. To extract the weak critical signal, we employ a two-stage topological machine learning framework combining Topological Data Analysis (TDA) with deep learning. In the first stage, particle events are represented as two-dimensional point clouds and a Delaunay-based sub-level set filtration is constructed to extract Betti curves as multiscale topological invariants, corrected for multiplicity bias via azimuthal randomisation and classified by two complementary architectures, a TopoPointNet (TPN) and Boosted Decision Trees (BDT). Since event-level classification alone is insufficient to restore the critical scaling, a second stage applies a particle-level density filter, explicitly stripping away the diffuse thermal background and isolating the densely packed critical clusters. The two stage pipeline successfully restores the power-law scaling of the normalized factorial moments, enabling accurate recovery of the intermittency index in ($\eta$, $\varphi$) space and establishing topological machine learning as a robust data driven tool for probing the QCD critical point and the phase structure of strongly interacting matter in heavy-ion collisions at LHC energies.