Flow dominance and factorization of transverse momentum correlations in Pb-Pb collisions at the LHC

We present the first measurement of the two-particle transverse momentum differential correlation function, $P_2\equiv\langle \Delta p_{\rm T} \Delta p_{\rm T} \rangle /\langle p_{\rm T} \rangle^2$, in Pb-Pb collisions at $\sqrt{s_{_{\rm NN}}} =$ 2.76 TeV. Results for $P_2$ are reported as a function of relative pseudorapidity ($\Delta \eta$) and azimuthal angle ($\Delta \varphi$) between two particles for different collision centralities. The $\Delta \phi$ dependence is found to be largely independent of $\Delta \eta$ for $|\Delta \eta| \geq$ 0.9. In 5% most central Pb-Pb collisions, the two-particle transverse momentum correlation function exhibits a clear double-hump structure around $\Delta \varphi = \pi$ (i.e., on the away side), which is not observed in number correlations in the same centrality range, and thus provides an indication of the dominance of triangular flow in this collision centrality. Fourier decompositions of $P_2$, studied as a function of collision centrality, show that correlations at $|\Delta \eta| \geq$ 0.9 can be well reproduced by a flow ansatz based on the notion that measured momentum correlations are strictly determined by the collective motion of the system.

Measurements of particle production and their correlations in heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) have provided very compelling evidence that the produced matter is characterized by extremely high temperatures and energy densities consistent with a deconfined, but strongly interacting quark-gluon plasma (sQGP).Evidence for the production of the sQGP is provided by observations of large suppression of particle production at momenta p T 3 GeV/c relative to that observed in pp collisions, strong suppression of away-side particles observed in two-particle number correlations, as well as by anisotropic flow studies (anisotropies in particle azimuthal distributions relative to the reaction plane defined by the beam axis and a line connecting the centers of colliding nuclei) [1][2][3][4][5][6][7][8][9][10][11].The comparison of measured flow coefficients, v n , with predictions from hydrodynamical models indicate that the sQGP has a vanishingly small shear viscosity over entropy density ratio [12].Furthermore, the observation of an approximate number of constituent quark scaling of flow coefficients in the 2 < p T < 4 GeV/c range, suggested as a signature of a deconfined medium [13], was reported by RHIC and LHC experiments [14,15].These results imply that the two-particle number correlations observed in the region of low p T (< 2 GeV/c), corresponding to the bulk of particle production, are largely determined by anisotropic flow.Such flow dominance is manifested, in particular, by an approximate factorization of the measured flow coefficients, ) , observed for pairs of particles at relative pseudorapidity ∆η > 0.8, in different transverse momentum bins up to p T ≈ 3 − 5 GeV/c [16].Two-particle transverse momentum correlations [17][18][19][20][21] provide additional insights into the dynamics of multiparticle production and can be used to further examine the flow dominance of two-particle correlation functions.One expects, in particular, that in the presence of anisotropic flow, the differential transverse momentum correlator ∆p T ∆p T should feature azimuthal Fourier decomposition coefficients calculable with a simple formula, hereafter called the flow ansatz, in terms of the regular and p T weighted flow coefficients [17].Such a simple relation, discussed in more details below, is not expected for particle production arising from processes not related to the common symmetry plane, known as non-flow, such as jets or resonance decays.An agreement between the Fourier coefficients of the ∆p T ∆p T correlator and those calculated with the flow ansatz should thus provide additional evidence of the dominance of collective flow effects.
In this Letter, we present the first measurements of the differential transverse momentum correlations in Pb-Pb collisions at √ s NN = 2.76 TeV in terms of the dimensionless correlator P 2 defined as where ∆p T,i = p T,i − p T , with p T = ρ 1 p T d p T / ρ 1 d p T , the inclusive average transverse momentum of particles observed in the p T,min ≤ p T ≤ p T,max range.The quantities ρ 1 and ρ 2 represent single and two-particle densities, respectively.For particle correlations induced strictly by anisotropic emission relative to the reaction plane, the Fourier coefficients of P 2 , v n [P 2 ], should be determined by regular and the p T weighted flow coefficients defined according to the following flow ansatz [17]: where v n and v p T n = ρ 1 v n (p T )p T d p T / ρ 1 d p T are the regular, and p T weighted coefficients, respectively [17,22].Thus, we shall compare the Fourier coefficients of the P 2 correlator to values expected from this ansatz based on coefficients v n and v p T n measured with traditional flow methods, e.g., the scalar product method [22].
This study is based on an analysis of a 14 × 10 6 events subset of a sample of minimum bias (MB) trigger events recorded with the ALICE detector during the LHC Run 1 in 2010.Detailed descriptions of the ALICE detector, its subsystems, and their respective performance have been reported in [23][24][25][26].For this study, the Inner Tracking System (ITS) and the Time Projection Chamber (TPC) were used to reconstruct charged-particle tracks, while the V0 detector and the Silicon Pixel Detector (SPD) formed the basis of the online minimum bias trigger used to acquire the data, as described in [5,6].
The ALICE solenoidal magnet was operated with a field of 0.5 T with both positive and negative polarities.Events included in this analysis were required to have a single reconstructed primary vertex within 10 cm of the nominal interaction point along the beam axis, hereafter taken to be the z-axis.The fraction of pile-up events in the analysis sample is found to be negligible after applying dedicated pile-up removal criteria [26].
Correlation functions reported in this article are based on charged-particle tracks measured in the pseudorapidity range |η| < 1.0 and with full azimuthal coverage 0 ≤ ϕ < 2π.The analysis was limited to particles produced with 0.2 < p T < 2.0 GeV/c corresponding largely to particles emerging from the bulk of the matter.Only tracks with a minimum of 70 reconstructed space points in the TPC, out of a maximum of 159, were included in the analysis.Contributions from photon conversions into e + e − pairs were suppressed based on an electron rejection criterion relying on the truncated average of the specific ionization energy loss dE/dx measured in the TPC.Tracks with dE/dx lying within 3σ dE/dx of the Bethe-Bloch parametrization of the dE/dx expectation value for electrons and at least 3σ dE/dx away from the relevant parameterizations for π, K, p were removed.In addition, suppression of the contamination from secondary particles originating from weak decays and from interaction of particles with the detector material was accomplished by imposing upper limits of 3.2 cm and 2.4 cm (RMS ∼ 0.36 cm) for the distance of closest approach (DCA) of a track to the reconstructed vertex in the longitudinal (DCA z ) and radial (DCA xy ) directions, respectively.These criteria lead to a reconstruction efficiency of about 80% for primary particles and contamination from secondaries of about 5% at p T = 1 GeV/c [27].No filters were used to suppress like-sign (LS) particle correlations resulting from Hanbury Brown and Twiss (HBT) effects, which produce a strong and narrow peak centered at ∆η, ∆ϕ = 0 in LS correlation functions.Corrections for single track losses were carried out using the weight technique described in [28] with weights calculated separately for positively and negatively charged tracks, positive and negative solenoidal magnetic fields, and with 40 vertex position bins in the fiducial range |z| ≤ 10 cm.Pair inefficiencies associated with track merging or crossing (e.g. two tracks being partly or entirely reconstructed as a single track) within the TPC were corrected for based on track charge and momentum ordering techniques [29].The P 2 correlators were measured separately for charge pair combinations ++, +−, and −− and were combined with equal weights to produce the charge independent correlation functions reported in this article.Systematic uncertainties were investigated by repeating the analysis for different operational and analysis conditions including two solenoidal magnetic field polarities, different event and track selection criteria, as well as different track reconstruction methods.Track selection criteria, most particularly the maximum value of distance of closest approach to the primary vertex, dominate systematic effects.The systematic uncertainties assigned to the measurements of v n coefficients are the quadratic sums of individual contributions and range from 4% in central 0 -10% collisions to 14% in peripheral 70 -80% collisions.
Figure 1 (a) presents the correlator P 2 measured as a function of ∆η and ∆ϕ in the 5% most central Pb-Pb collisions.The central range around ∆η ∼ 0 and ∆ϕ ∼ 0(rad) is left under-corrected by the weight correction procedure mainly due to track merging effects.It is thus not considered in this analysis. 1The correlator P 2 features a prominent near-side ridge centered at ∆ϕ = 0, extending across the full pseudorapidity range of the measurement.It also features two distinct away-side humps at |∆ϕ −π| ≈ 60 • separated by a weak dip centered at ∆ϕ = π and also extending across the full pseudorapidity range of the acceptance.Such an away-side correlation feature, which indicates the presence of a strong third harmonic, was previously reported in ultra-central (0-2%) Pb-Pb collisions at the LHC [16,31,32] η ∆ as well as in central Au-Au collisions at RHIC but for the latter case only after the subtraction of a correlated component whose shape was exclusively attributed to elliptic flow [33][34][35].
To further study the azimuthal angle dependence of transverse momentum correlations, projections of the measured P 2 correlation function are fitted with an unconstrained 6-th order Fourier decomposition in ∆ϕ according to F(∆ϕ) = b 0 + 2 ∑ 6 n=1 b n cos(n∆ϕ), as illustrated in Fig. 1 (b).We verified that higher-order contributions, with n > 6, do not significantly improve the fits for |∆η| ≥ 0.9.Coefficients b 5 , b 6 feature large relative errors and are thus not reported in this paper.The double-hump at |∆ϕ − π| ≈ 60 • implies the presence of a strong third harmonic, v 3 , in the Fourier decompositions of the correlation functions.The large v 3 likely originates from fluctuations in the initial density profile of colliding nuclei [36].
The flow coefficients obtained from two-particle transverse momentum correlations, v n [P 2 ], calculated according to v n = b n /(b 0 + 1), are plotted in Fig. 2 as a function of centrality for central (0 -5%) up to peripheral collisions (70 -80%).The v n [P 2 ] coefficients exhibit a collision centrality dependence qualitatively similar to that of regular flow coefficients obtained from standard flow measurement methods [22].In addition, they feature a hierarchy such that v 2 > v 3 > v 4 at all centralities except in the 5% most central Pb-Pb collisions where the third is slightly larger than the second harmonic, thereby explaining the presence of the away-side double-hump seen in Fig. 1.This is at variance with the dependence of the regular flow coefficients which, even in the centrality range 0-5%, exhibit the basic hierarchy v 2 > v 3 > v 4 .The observed higher value of v 3 [P 2 ] relative to v 2 [P 2 ] implies that v 3 should rise faster with increasing p T than v 2 , in agreement with explicit measurements of the flow coefficients dependence on p T [37].
We next consider the possible role of non-flow correlations on the correlator P 2 by comparing, in Fig. 2, the v n [P 2 ] coefficients obtained in ranges 0.2 ≤ |∆η| ≤ 0.9 and 0.9 ≤ |∆η| ≤ 1.9 with values predicted by the flow ansatz, introduced above.In the range 0.9 ≤ |∆η| ≤ 1.9 (see Fig. 2(c)), one observes that the coefficients v n [P 2 ] are in a very good agreement, at all measured collision centralities, with expectations from the flow ansatz.This agreement provides additional evidence that two-particle correlations in this relative pseudorapidity range are predominantly determined by the collective nature of particle emission at low p T , which motivates the factorization hypothesis used to derive Eq. (2).It also suggests that awayside jets, that might be associated with the near-side peak, are significantly suppressed and contribute minimally to the away-side correlated yield in that η range.In contrast, in the range 0.2 ≤ |∆η| ≤ 0.9 (see Fig. 2(a)), the v n [P 2 ] coefficients exhibit a stronger and monotonic centrality evolution.In particular, the v n [P 2 ] deviate significantly from the flow ansatz for collision centralities larger than 40%, where one Using the same measurement technique, we further compare features of the P 2 correlation function to that of the number correlation function, R 2 , defined as Figure 3 presents the ∆η dependence of v n , n =2, 3 and 4, coefficients obtained from these correlation functions for the 5% most central collisions.In this centrality interval one finds that the hierarchies indeed hold for all measured ∆η.The dominance of v 3 [P 2 ] across all ∆η is likely a consequence of the third harmonic's (triangular flow) stronger dependence on p T relative to that of the second harmonic (elliptic flow).The v 2 , v 3 , and v 4 dependencies on ∆η reveal additional interesting features.In the case of the R 2 correlation, the coefficients v 2 and v 3 monotonically decrease over the entire ∆η range, whereas coefficients extracted from P 2 exhibit a more pronounced decrease for |∆η| ≤ 0.9.From |∆η| ∼ 1.0 to ∼ 2.0, the relative decrease of v 2 is about 5% for both correlators, and somewhat smaller for v 3 .These contrasting dependencies reflect the different shapes of the near-side peaks of the two correlation functions.The narrower shape of the near-side peak of the P 2 distribution suggests that the near-side peak of R 2 might involve two components, one of which is characterized by a vanishing ∆p T ∆p T for pairs with |∆η| ≤ 0.9.While the origin of this behavior is not fully understood, it offers the benefit of enabling the determination of flow coefficients with smaller non-flow effects using a narrower ∆η gap.
In summary, we presented the first measurements of the two-particle transverse momentum differential In the 5% most central Pb-Pb collisions, P 2 has a shape qualitatively different to that observed in measurements of the number density correlations, with a relatively narrow near-side peak near |∆η|, |∆ϕ| < 0.5, and a longitudinally broad and double-hump structure on the away-side.The double-hump structure in the 5% most central P 2 correlation indicates that this observable is more sensitive to the presence of a triangular flow component than the number correlations R 2 , and consequently provides an indication that triangular flow features a stronger dependence on p T than elliptic flow does.Comparison of the Fourier decompositions of the R 2 and P 2 correlators, calculated as a function of |∆η|, suggests that the v 2 , v 3 , and v 4 coefficients extracted from P 2 reach approximately constant values beyond |∆η| ∼ 0.9, while coefficients v 2 , v 3 obtained from R 2 decrease monotonically for increasing |∆η|.The observed agreement between the flow coefficients measured from P 2 correlations, at |∆η| > 0.9, and the values predicted from the flow ansatz provide new and independent support to the notion that the observed long range correlations are largely due to the initial collision geometry.These results may be used to further constrain particle production models.This agreement to the flow ansatz also provides further evidence for flow coefficient factorization in heavy-ion collisions.

Fig. 2 :
Fig. 2: (Color online) v n coefficients where n = 2, 3, 4 in the range (a) 0.2 ≤ |∆η| ≤ 0.9 and (c) 0.9 ≤ |∆η| ≤ 1.9 obtained from the P 2 correlation function.The coefficients are compared with the expectations from the flow ansatz calculated in their respective ∆η ranges in Pb-Pb collisions.Statistical errors are shown as vertical solid lines, whereas systematic errors are displayed as colored bands.Ratios of the v n coefficients and their corresponding flow ansatz values are shown in (b) and (d).The errors on the ratios are only statistical.

Fig. 3 :
Fig. 3: (Color online) v n coefficients, n = 2, 3, 4, obtained from (a) P 2 and (b) R 2 correlators, as a function of |∆η| in the 5% most central Pb -Pb collisions.Statistical errors are shown as vertical solid lines whereas systematic uncertainties are displayed as shaded bands.Panels (c -d): ratios of the v n , n = 2, 3, 4, by the corresponding values of v n measured at ∆η = 0.3.