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Recent results on event-by-event fluctuations in ALICE at the LHC

Non-statistical event-by-event fluctuations in relativistic heavy-ion collisions have been proposed as a probe of the phase transition of hadronic matter to a deconfined phase of quarks and gluons, the so-called Quark-Gluon Plasma. In a thermodynamical picture of the strongly interacting system formed in heavy-ion collisions, the dynamical fluctuations of net-charge, fluctuations of the mean transverse momentum, mean multiplicity and balance functions are related to the fundamental properties of the system, hence they may reveal information about the QCD phase transition. In this article, recent results on event-by-event measurements of net-charge fluctuations, the measurement of the balance function and mean transverse momentum fluctuations are discussed.


Introduction
Heavy-ion collision experiments at the Relativistic Heavy-Ion Collision (RHIC) and the Large Hadron Collider (LHC) aim to study the phase transition of hadronic matter to the Quark-Gluon Plasma (QGP) phase and to characterize its properties. It is proposed that the study of eventby-event fluctuations and correlations are sensitive tools to look for evidence of a possible phase transition and its thermodynamic properties [1,2,3]. The measurement of fluctuations of particle multiplicities and their ratios, fluctuations of conserved charge (e.g. net-charge, net-proton), twoparticle correlations, mean transverse momentum fluctuations are common tools used in heavyion collision experiments to study the correlations and fluctuations originating from the system produced.
In this article, we discuss the recent results on event-by-event fluctuations measured at midrapidity with the ALICE [4] experiment at the LHC. The Time Projection Chamber (TPC) is used as the main tracking detector for these analyses and the Inner Tracking System (ITS) is used for vertex determination. The particle multiplicity per event measured with a set of scintillator arrays (VZERO) is used to determine the collision centrality. We have also compared our measurements with results obtained at lower collision energies from the RHIC. The article is organized as follows. First we discuss the results of net-charge fluctuations in Section 2. In Section 3, the results on the balance function are presented. In Section 4, the results of mean transverse momentum fluctuations are discussed. Finally in section 5, we summarize the findings and provide an outlook.

Net-charge fluctuations
It has been proposed that there is a significant reduction of net-charge fluctuations from the QGP medium to the final state hadronic medium due to the entropy conservation [1,2,5]. This can be argued in a very simple way as follows. In the QGP medium, quarks are the charge carriers, which have fractional electric charge, whereas the hadron gas has integral (unit) charge. So the variance of fluctuations of net-charge (δ Q 2 ) measured in the QGP medium will be substantially smaller than in the hadronic medium assuming that quark-quark correlation is negligible and that hadronization of gluons produces pairs of positive and negative charges hence doesn't contribute to the final state charge fluctuations. Thus the measurement of net-charge fluctuations can be regarded as a sensitive indicator of the formation of QGP in heavy-ion collisions [3]. However, there may be fluctuations of system size, which may affect the net-charge fluctuations. These volume fluctuations can be cancelled out by measuring the fluctuations of the ratio of positive (N + ) to negative (N − ) charges [3]. Theoretically, the ratio (R = N + /N − ) can be related to the D measure, which links the charged fluctuation per unit entropy as follows [6,7].
where δ R 2 = R 2 − R 2 , δ Q 2 is the variance of fluctuations of net-charge (Q = N + − N − ) and N ch is the total charge (N ch = N + − N − ) measured at the mid-rapidity. The value of D has been estimated from several theoretical works both for QGP and hadron gas (HG). According to Ref. [1], the value of D in a HG is approximately 4 times larger than in a QGP. Taking quark-quark interactions, lattice QCD calculations predict different values of D for different media. For an uncorrelated pion gas, D = 4 and after taking resonances into account, the value of D decreases to 3, for the QGP phase, D ≃ 1.0 − 1.5 [8]. The uncertainty in the D value for the QGP phase comes from the formulation of relating the entropy to the number of charged particles in the final state. Experimentally, D is estimated by measuring the dynamical fluctuations of the net-charge, ν (+−,dyn) , as follows [7,8].
where ν (+−,dyn) is given as ν (+−,dyn) measures the relative correlation strength of particle pairs and has been found robust against the detector efficiency [9]. Before correlating the measured ν (+−,dyn) , it needs to be corrected for global charge conservation (GCC) and finite net-charge effect [9]. So the corrected ν (+−,dyn) will be, N total is the number of total charged particles produced in the whole phase space which is estimated from experimental data. Therefore, taking this GCC correction, D can be written as, We have measured net-charge fluctuations by calculating ν (+−,dyn) and D as a function of collision centrality for minimum bias events in pp and Pb−Pb collisions at √ s NN = 2.76 TeV. The dynamical net-charge fluctuations, ν (+−,dyn) and ν corr (+−,dyn) for charged particles as a function of N part are shown in Figure 1 (a). The results for N ch ν corr (+−,dyn) as a function of N part for pp and Pb−Pb collisions are compared to results obtained from models, like HIJING [10] and PYTHIA [11] event generators, which are shown in Figure 1 The ν corr (+−,dyn) is measured taking two pseudorapidity intervals, i.e. ∆η = 1.0 (-0.5 < η < 0.5) and ∆η = 1.6 (-0.8 < η < 0.8). It can be seen from Figure 1 (b) that HIJING is unable to explain the Pb−Pb data. For a particular centrality range, the D value shows a decreasing trend with increasing pseudorapidity window. This suggests a strong dependence of dynamical charge fluctuations on the pseudorapidity window. In Ref. [12,13], this behavior is explained by considering the concept of diffusion in rapidity space. It is shown that there is a dissipation of the fluctuations originating from the initial stage of the collision during the evolution of the system. This nature of dissipation of the primordial fluctuation signal can be described by an error function of form erf(∆η/ 8σ f ).
In the error function, σ f represents the diffusion at freeze-out. We fit the data as a function of ∆η with that error function, which is shown in Figure 2 (a). From the fit, it can be seen that there is a decrease in the magnitude of fluctuations with increase in ∆η and it tends to saturate at ∆η = 2.3. In addition, the energy dependence of N ch ν corr (+−,dyn) and D measure as a function of √ s NN is given in Figure 2 (b). A monotonic decrease of the D value as a function of collision energy is observed. More importantly, the ALICE measurement for ∆η = 1.6 is closer to the QGP values than the other measurements at lower energies for ∆η = 1.0.

Balance function
The measurement of charge dependent two-particle correlations via the so-called balance function is argued to probe the charge creation time and the collective motion of the system [14,15]. The general definition of the balance function is given as follows [16].
where C ab (P 2 , P 1 ) = N ab (P 2 , P 1 )/N b (P 1 ) is the distribution of the pairs of particles, of type a and b, are in momentum space P 1 and P 2 , respectively, normalized to the numbers of particles b. Here particles a and b refer to all positive and negative particles, respectively. If the correlation is measured in terms of pseudorapidity difference ∆η = |η b − η a |, the balance function is given as, Each term of Eq. 3.2 is corrected for detector acceptance and tracking efficiency [16]. Similarly, Eq. 3.2 can be written for azimuthal angle (∆φ = |φ b − φ a |). ALICE has measured the widths of balance function which are characterized by ∆η and ∆φ studied in pseudorapidity and azimuthal angle, respectively. For example, ∆η is calculated as follows.
where B +− (∆η i ) is the balance function value for each bin ∆η i with the sum running over all bins k. The ALICE results of balance function in Pb−Pb collisions at √ s NN = 2.76 TeV as a function of centrality percentile are shown in Figure 3 (a). The data are compared with the Monte Carlo simulation results of HIJING [10] and AMPT [17]. It is found that the widths of the balance function measured for ∆η and ∆φ decrease while moving from peripheral to central collisions.
The results could not be reproduced by HIJING, whereas AMPT tuned to describe the v 2 values reported by ALICE [18] can qualitatively reproduce the centrality dependence of ∆η but fails to describe the ∆φ values. The relative decrease of width of the balance function in the relative pseudorapdity and azimuthal angle from peripheral to central collisions as a function of N part is shown in Figure 3 (b). The ALICE points are compared with the results obtained from highest SPS and RHIC energies. The correlation in relative pseudorapidity ( ∆η / ∆η peripheral ) and in relative azimuthal angle ( ∆φ / ∆φ peripheral ) shows a decrease of ≈ (9.5 ± 2.0(stat) ± 2.5(sys)) % and ≈ (14.0 ± 1.3(stat) ± 1.9(sys))% respectively, for the most central collisions compared to RHIC results. The latter can be understood as additional increase in radial flow between central and peripheral collisions at the LHC compared to RHIC.

Mean p T fluctuations
The measurement of event-by-event fluctuations of mean p T ( p T ) is proposed as an excellent tool to characterize the thermodynamics and collectivity of the system produced in heavy-ion collisions [1,2,3]. Mean p T fluctuations measured in pp collisions also serve as a model independent baseline to figure out the non-trivial fluctuations in A-A collisions. The event-by-event mean transverse momentum is measured by the mean value of M EbE (p T ) k of the transverse momenta p T,i of the N acc , k accepted charged particles in event k as follows.
The event-by-event fluctuations of M EbE (p T ) k have both dynamical and statistical contributions. The dynamical component is measured by the two-particle correlator C m = ∆p T,i , ∆p T,j and is defined as follows [19].
where n ev,m is the number of events in a given multiplicity class m, N pairs k is the number of pairs constructed out of N k number of particles in an event and equal to 0.5N k (N k − 1). M(p T ) m is the average p T of all tracks in all events of multiplicity class m, which is defined as follows.

Summary and outlook
In summary, the measurements of dynamical net-charge fluctuations are presented as a function of centrality and compared with theoretical expectations for HRG and QGP. That data are compared with previously measured results by different experiments at different collision energies. Moreover, the ν corr (+−,dyn) are measured as a function of the pseudorapidity windows. The measurement of the width of balance function is found to decrease from peripheral to central collisions. We also presented the mean p T fluctuation results for pp and Pb−Pb collisions. A decrease of the fluctuations with increasing multiplicity is observed in pp collisions. The mean p T fluctuation results for Pb−Pb and Au−Au collision data at √ s NN = 2.76 TeV and 0.2 TeV deviate from a power-law fit to the pp data at the same centrality. In future, ALICE will extend its fluctuation studies to other analyses, like, multiplicity fluctuations, higher moments and cumulants of conserved charge fluctuations. ALICE has the plan to extend the balance function studies to pp and p−Pb collisions both for charged particles and identified particles.