AZIMUTHAL CORRELATIONS AND MIXED HIGHER ORDER FLOW HARMONICS FROM CMS AT THE LHC ∗

Two-particle correlations measurements of v n ( n = 2 –4) in 8.16 TeV p Pb collisions, and event-by-event correlations of diﬀerent v n are measured us-ing symmetric cumulants in 13 TeV pp , 5.02 and 8.16 TeV p Pb and 5.02 TeV PbPb collisions at the LHC. These new results give important insights to the origin of collectivity observed in small collision systems. Additionally, using the scalar product method and the method of two-particle correlations, the mixed higher order ﬂow harmonics and extracted nonlinear response coeﬃcients of charged particles are measured for the ﬁrst time as a function of p T and centrality in 2.76 and 5.02 TeV PbPb collisions. The obtained results are compared with diﬀerent theoretical predictions.


Introduction
Initial geometry of colliding nuclei overlapping region and its fluctuations lead to space anisotropy.As a consequence, there is a different pressure gradient in different directions in created medium -strongly-coupled quark-gluon plasma (QGP).Therefore, there are preferential directions of emission of particles created in the collision.Particle distribution over azimuthal angle can be described by the Fourier decomposition [1] where coefficients v n present the magnitude of azimuthal anisotropy, while the event plane angle (Ψ n ) stands for the direction of the maximum final-state particle density.Since Fourier harmonics v n are driven by initial conditions and medium transport properties, they can be used for studying the properties of created matter in heavy-ion collision.Furthermore, a correlation observed between the event plane angles of different order [2] points out that higher order harmonics can be measured with respect to the direction of multiple lower order event planes, which can provide additional constraints on understanding the properties of QGP [3].Also event-by-event anti-correlation of flow amplitudes between v 2 and v 3 is shown that is much more sensitive to the initial conditions, while the correlation between v 2 and v 4 is sensitive to both initial conditions and medium properties, in PbPb collisions [4].Since the collectivity is also observed in pp and pPb collisions [5][6][7][8][9], this new observable can offer a possibility for better understanding the origin of collectivity in small systems.

Analysis techniques
For studying correlations between flow harmonics, two different methods have been used: mixed order harmonics, applied on PbPb data at √ s N N = 2.76 and 5.02 TeV, and symmetric cumulant applied on pp at √ s = 13 TeV, pPb at √ s N N = 5.02 and 8.16 TeV, and PbPb collisions at √ s N N = 5.02 TeV.All the data are taken with the CMS detector [10].

Mixed order harmonics
Mixed harmonics measurement is based on scalar product method (SP) [11] and follow the procedure introduced here [3].For second and third harmonic, flow coefficients are linearly proportional to the initial eccentricity, i.e. v n ∝ n [12].For n ≥ 4, on the other hand, there are nonlinear terms coming from lower order harmonics [3].In standard (SP) approach, v n with respect of its own event plane, Ψ n , would be where V n = v n exp(inΨ n ) is the n th flow harmonic complex coefficient.But if one measures v 5 , for example, with respect to the direction of 2 nd , Ψ 2 , and 3 rd , Ψ 3 , harmonic v 5 in (2) becomes A contribution from v 2 and v 3 in v 5 can be evaluated via nonlinear response coefficient Measurements of mixed order flow harmonics, v 4 with respect to the Ψ 2 (v 4 {Ψ 22 }), v 5 with respect to the Ψ 2 and Ψ 3 (v 5 {Ψ 23 }), v 6 with respect to the Ψ 2 (v 6 {Ψ 222 }) and with respect to the Ψ 3 (v 6 {Ψ 33 }), v 7 with respect to the Ψ 2 and Ψ 3 (v 7 {Ψ 223 }), and corresponding nonlinear response coefficients χ 422 , χ 523 , χ 6222 , χ 633 and χ 7223 are presented in this contribution.

Symmetric cumulant
The symmetric cumulant method is based on four-particle correlation calculation with cumulants, first time introduced by the ALICE Collaboration in [4].As a first step, one should build 2-and 4-particle correlator where . . .denotes averaging over all particles and events of interest.More details about standard cumulant technique and generic code, which this analysis framework is based on, can be found in [13].Symmetric cumulant for n th and m th flow coefficient is then defined as and it is a measure of correlations between flow amplitudes of the harmonics.In order to obtain results that are independent of magnitude of v n and v m , one can normalise SC(n, m) with v 2 n v 2 m .In this work, v 2 n and v 2 m are extracts from two-particle correlations technique, which is described in [6].SC measurement has been done as a function of multiplicity (N offline trk ) in 0.3 < p T < 3.0 GeV/c transverse momentum region.

Results
Mixed higher order harmonics, v 4 {Ψ 22 }, v 5 {Ψ 23 }, v 6 {Ψ 222 }, v 6 {Ψ 33 } and v 7 {Ψ 223 } as a function of centrality are measured at two energies, 2.76 and 5.02 TeV [14].The results are integrated over pseudorapidity and transverse momentum intervals η < 2.4 and 0.3 < p T < 3.0 GeV/c, respectively.They are presented in Fig. 1 and compared with hydrodynamic calculations from [3].All mixed harmonics show a weak energy dependence and all except v 6 {Ψ 33 } show a strong centrality dependence.Corresponding nonlinear response coefficients, χ 422 , χ 523 , χ 6222 , χ 633 and χ 7223 , as a function of centrality, with |η| < 2.4 in the 0.3 < p T < 3.0 GeV/c range [14], are presented in Figs. 2 and 3.In the former, the results are compared with predictions from AMPT and hydrodynamics with a deformed symmetric Gaussian density profile with the initial condition using η/s = 0.08 in Ref. [3], and from iEBE-VISHNU hydrodynamics with Glauber and KLN initial conditions using the same η/s [16].Predictions from AMPT are favoured by the data.While in Fig. 3 the same results are compared with predictions from AMPT in Ref. [3] and from iEBE-VISHNU hydrodynamics with KLN initial condition using η/s = 0, 0.08 and 0.2 [16].
The large difference between hydrodynamic predictions with different viscosities indicates that our results can provide constraints on the value of viscosity at freeze-out [3,16].However, based only on these comparisons, it is not unambiguously clear which viscosity and initial conditions are the best choices.Figure 4 shows symmetric cumulants, normalised by flow magnitudes, of second and third flow harmonics and second and forth flow harmonics for 0.3 < p T < 3.0 GeV/c region, as a function of multiplicity, in pp at 13 TeV, pPb at 5.02 and 8.16 TeV and PbPb collisions at 5.02 TeV [15].Results for PbPb are in agreement with results published by ALICE [4].Similar results for high multiplicity pPb and PbPb suggest that initial-state fluctuations are the same in those two systems.This conclusion may also be applicable to pp collisions, but statistical uncertainties are too large to make stronger claim.At the same time, clear ordering in normalised SC(2, 4) results of different systems may suggest different medium properties if one assumes hydrodinamical origin of collectivity in small systems.

Summary
Results of mixed higher-order harmonics and nonlinear response coefficients show sensitivity to the initial conditions and medium properties close to the freeze-out stage.Comparison with different theoretical predictions is shown and AMPT model is favoured by data.A quantitative agreement between pPb and PbPb collisions of normalised symmetric cumulant for the second and third harmonics suggest similar initial conditions in those two systems.High multiplicity pp collisions may follow the similar trend but statistical uncertainties are too large to make stronger claim.At the same time, different results for normalised correlations of the second and fourth harmonic may suggest different medium properties if one assumes hydrodynamical origin of flow.

Fig. 4 .v 2 3 and v 2 2 v 2 4
Fig. 4. The SC for the second and third harmonic (left) and the second and forth harmonic (right) normalised by v 2 2 v 2 3 and v 2 2 v 2 4 from dihardon correlations respectively.The p T range for considered tracks is 0.3 < p T < 3.0 GeV/c and the results are shown as a function of N offline trk in pp at √ s = 13 TeV, pPb at √ s N N = 5.02 and 8.16 TeV, and PbPb at √ s N N = 5.02 TeV [15].