Scaling of charged particles produced in ultra-relativistic nuclear collisions

The Large Hadron Collider (LHC) serves as an experimental platform to test theoretical predictions of the Standard Model. The Standard Model is a framework for particle physics that describes, in detail, all elementary particles and their interactions through electromagnetic, strong, and weak forces, which are three of the four fundamental forces in nature. A Large Ion Collider Experiment (ALICE) is one of the major experiments at the LHC at CERN, Geneva. It focuses on collecting data from high-energy heavy-ion collisions, aiming to recreate and study a new state of nuclear matter, known as Quark-Gluon Plasma (QGP) which is believed to have existed at the beginning of the universe. Thus, investigating relativistic heavy-ion collisions with ALICE is similar to probing the early universe. QGP is a state of matter where the constituent partons (quarks and gluons) are free particles rather than being confined within hadrons where they are glued to each other through the strong force. These interactions between fundamental particles that form hadrons is explained by the theory of strong interactions, known as Quantum Chromodynamics (QCD). By analyzing heavy-ion collision data, researchers can investigate formation of QGP and its properties, such as temperature, energy density, viscosity etc..

Confirmed experimental signals of QGP formation have been observed in the data coming from the heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) at BNL, USA and at the LHC. Experimental results from these colliders, combined with theoretical advances from various frameworks have greatly expanded our understanding of the properties of hot QCD matter. The QGP produced in these collisions behaves as a strongly coupled liquid, resembling a near-ideal Fermi liquid with the lowest specific viscosity ever measured, indicating almost no frictional resistance. Studying this newly formed matter at range of temperatures is essential to accurately characterize its properties and firmly establish the QCD theory. To achieve this goal, the LHC is designed to reach 5.5~A~TeV for Pb-Pb collisions, allowing higher energy densities and temperatures conducive for the formation of QGP. It provides an opportunity to understand not only the properties of the QGP but also the mechanism of hadronization that occurs as the QGP cools and transitions from state of free quarks and gluons to hadronic matter. Direct observation of QGP is challenging due to its short lifetime (about 10 fm/c), as it rapidly thermalises and expands, forming hadron gas. Indirect methods are therefore used to identify the formation of the QGP. In addition to heavy-ion collisions, LHC also collides smaller systems such as proton-proton (\textit{pp}) and proton-lead (\textit{p-Pb}).

Numerous observables that can serve as evidence to qualify and quantify the creation of QGP have been proposed which can serve as a test for QCD theory. Among these fluctuations in physical measurables from the heavy-ion collisions serve as an important signal of QGP formation. Since fluctuations are inextricably linked to particle correlations, studying fluctuations provides valuable insight into the mechanisms behind the multiparticle production in high-energy collisions and other properties, such as the initial energy density, the nature of particle interactions, and the dynamics of the system as it evolves, etc.. Specifically fluctuations in multiplicity distributions are critical for determining the properties of the QGP. Multiplicity fluctuations are sensitive to the presence of critical point in the QCD phase diagram. At the critical endpoint (CEP), where the correlation length diverges, the systems undergoing phase transition exhibits large fluctuations in various observables, such as particle multiplicity, transverse momentum, temperature, pion to kaon ratio, etc.. At CEP the system becomes scale-invariant and self-similar, like the critical opalescence in quantum electrodynamic (QED) systems and leads to significant local density fluctuations. So studying these changes is a good way to understand the type of phase transition, find the critical point on the phase diagram, grasp how particles are produced in heavy-ion collisions, and therefore learn about how the matter created in those collisions evolves over time.

The transition from QGP to a hadronic state may preserve, or "freeze in," fractal fluctuations. Thus one can also probe QGP formation and dynamics through the analysis of fractal properties in the distribution of hadrons produced in high-energy collisions and can serve as a potential signal of the QCD phase transition. Fractal structures are manifested as intermittent fluctuations in kinematic phase space that exhibit power-law scaling, a characteristic signature of self-similar cascades and a hallmark of fractal systems. Observation of multifractality--where the scaling behaviour varies across different scales--in the phase space indicates complex, non-equilibrium dynamics, further reinforcing the connection between fractal patterns and the underlying physics of the QCD phase diagram. Thus observation of self-similarity, scaling behaviour, and intermittency in final-state particle distributions suggest a deep connection between QGP phenomena and critical behaviour associated with phase transitions. 

One of the effective methods for examining fluctuations in the system created in heavy-ion collisions is to carry event-by-event study of the obsevables. This approach involves measuring a particular observable for each event and then studying fluctuations in the observable across the ensemble of events. Intermittency analysis that examines patterns  in the particle configurations on event-by-event basis can reveal correlations in the multiparticle production in heavy-ion collisions. Intermittency, defined as an increase in the normalized factorial moments (NFMs) with an increase in phase space resolution, is a key indicator of scale-invariant local number density fluctuations.  The NFMs of the multiplicity distributions have the beauty that these contain, in integrated form the correlations of particles in the system and filter out statistical effects. These moments of the distributions of the produced particles gave first successful explanation of a high multiplicity spike event recorded by the JACEE (Japanese-American Collaborative Emulsion Experiment) experiment. Intermittency analysis performed for the low-energy systems, in the early 1990s, were constrained due to low bin multiplicities and low resolution of the detectors. With heavy-ion collisions at ultra-relativistic energies in recent colliders such as RHIC and LHC, a large number of particles per event are produced, allowing for detailed event-by-event analysis. This gives the opportunity to investigate the scaling behaviour of NFM and allowing to analyze multiplicity fluctuations at high bin resolution. 
Of the many observables, the study of multiplicity distributions and their fluctuations is critical for determining the properties of the QGP. 

The research work embodied in this thesis pertains to the study of local-multiplicity fluctuations in the charged particles that are produced in Xe-Xe collision at $\sqrt{s_{\rm{NN}}}$~=~5.44~TeV, recorded with the ALICE detector at LHC. NFM of the charged particle multiplicity configurations are determined and are studied for their scaling behaviour in the contours of intermittency, to understand the multiparticle production processes and the dynamics of the system produced during collisions. The scaling behaviour of these moments and a scaling exponent ($\nu$) are studied for various kinematic acceptance regions. Centrality and transverse momentum dependence of the observables is also studied. A study of fractal dimension $D_{\rm{q}}$ on the order of the moments $q$ is also performed. The analysis is also performed for the generated charged particles in the event samples obtained using the String Melting version of the A Multi Phase Transport (AMPT) model.