Studies of azimuthal dihadron correlations in ultra-central PbPb collisions at sqrt(s[NN]) = 2.76 TeV

Azimuthal dihadron correlations of charged particles have been measured in PbPb collisions at sqrt(s[NN]) = 2.76 TeV by the CMS collaboration, using data from the 2011 LHC heavy-ion run. The data set includes a sample of ultra-central (0-0.2% centrality) PbPb events collected using a trigger based on total transverse energy in the hadron forward calorimeters and the total multiplicity of pixel clusters in the silicon pixel tracker. A total of about 1.8 million ultra-central events were recorded, corresponding to an integrated luminosity of 120 inverse microbarns. The observed correlations in ultra-central PbPb events are expected to be particularly sensitive to initial-state fluctuations. The single-particle anisotropy Fourier harmonics, from v[2] to v[6], are extracted as a function of particle transverse momentum. At higher transverse momentum, the v[2] harmonic becomes significantly smaller than the higher-order v[n] (n greater than or equal to 3). The pt-averaged v[2] and v[3] are found to be equal within 2%, while higher-order v[n] decrease as n increases. The breakdown of factorization of dihadron correlations into single-particle azimuthal anisotropies is observed. This effect is found to be most prominent in the ultra-central PbPb collisions, where the initial-state fluctuations play a dominant role. A comparison of the factorization data to hydrodynamic predictions with event-by-event fluctuating initial conditions is also presented.


Introduction
The azimuthal anisotropy of emitted charged particles is an important feature of the hot, dense medium produced in heavy-ion collisions. One of the main goals of studying the azimuthal anisotropies is to understand the collective properties of the medium and extract its transport coefficients, particularly the shear viscosity over entropy density ratio, η/s, using hydrodynamic models [1]. Earlier observations of strong azimuthal anisotropies in collisions of gold nuclei at nucleon-nucleon center-of-mass energies ( √ s NN ) up to 200 GeV at the Relativistic Heavy-Ion Collider (RHIC) indicated that a strongly coupled quark-gluon plasma is produced, which behaves as a nearly perfect liquid with a close-to-zero η/s value [2][3][4][5][6][7]. The azimuthal anisotropies have also been extensively measured at the Large Hadron Collider (LHC) over a wide kinematic range in PbPb collisions at √ s NN = 2.76 TeV [8][9][10][11][12][13][14][15][16][17].
In a non-central heavy-ion collision, the overlap region of the two colliding nuclei has a lenticular shape, and the interacting nucleons in this region are known as "participants." The "participant plane" is defined by the beam direction and the short axis of the participating nucleon distribution. Because of fluctuations that arise from the finite number of nucleons, the impact parameter vector typically does not coincide with the short axis of this lenticular region. Strong rescattering of the partons in the initial state may lead to local thermal equilibrium and the build-up of anisotropic pressure gradients, which drive a collective anisotropic expansion. The expansion is fastest along the largest pressure gradient, i.e., along the short axis of the lenticular region. Therefore, the eccentricity of initial-state collision geometry results in an anisotropic azimuthal distribution of the final-state hadrons. In general, the anisotropy can be characterized by the Fourier harmonic coefficient (v n ) in the azimuthal angle (φ) distribution of the hadron yield, dN/dφ ∝ 1 + 2 ∑ n v n cos[n(φ − Ψ n )], where Ψ n is the event-by-event azimuthal angle of the participant plane. As the participant plane is not a measurable quantity experimentally, it is often approximated by the "event plane", defined as the direction of maximum final-state particle density. The second-order Fourier component (v 2 ) is known as the "elliptic flow", and its event plane angle Ψ 2 approximately corresponds to the short axis direction of the lenticular region. Due to event-by-event fluctuations, higher-order deformations or eccentricities of the initial geometry can also be induced, which lead to higher-order Fourier harmonics (v n , n ≥ 3) in the final state with respect to their corresponding event plane angles, Ψ n [18][19][20][21][22][23][24]. For a given initial-state eccentricity, the finite η/s value of the system tends to reduce the azimuthal anisotropy observed for final-state particles. The higher-order Fourier harmonics are expected to be particularly sensitive to the shear viscosity of the expanding medium.
Precise extraction of η/s from the anisotropy data is crucial for investigating the transport properties of the hot and dense medium created in heavy-ion collisions in detail [1]. This effort is, however, complicated by large uncertainties in our understanding of the initial-state conditions of heavy-ion collisions, especially in terms of event-by-event fluctuations. Different initial-state models predict different values of eccentricity and its fluctuations, leading to large uncertainties on the extracted η/s values. In order to better constrain the initial-state condition, it was suggested [25] that in ultra-central heavy-ion collisions (e.g., top 1% most central collisions), the initial collision geometry is predominantly generated by fluctuations such that various orders of eccentricities predicted by different models tend to converge. Here, collision centrality is defined as the fraction of the total inelastic PbPb cross section, with 0% denoting the most central collisions. Therefore, studies of azimuthal anisotropy in ultra-central heavy-ion collisions can help to reduce the systematic uncertainties of initial-state modeling in extracting the η/s value of the system, although quantitative comparison to theoretical calculations is beyond the scope of this paper.
Furthermore, since the event plane angle, Ψ n , is determined by the final-state particles, selecting particles from different ranges of transverse momentum (p T ) may lead to different estimates of event plane angles. Also due to the effect of initial-state fluctuations, it was recently predicted by hydrodynamic models that a p T -dependence of the event plane angle will be induced, which could be one of the sources responsible for the breakdown of factorization in extracting v n harmonics from dihadron correlations [26,27]. As mentioned already, the ultracentral heavy-ion events are dominated by the initial-state eccentricity fluctuations. Thus, they provide an ideal testing ground for the effect of a p T -dependent event plane angle.
This paper presents the measurement of azimuthal anisotropy harmonics, from v 2 to v 6 , extracted using long-range (large |∆η|) dihadron correlations as a function of p T from 0.3 to 8.0 GeV/c in the top 0.2% most central PbPb collisions at a center-of-mass energy per nucleon pair ( √ s NN ) of 2.76 TeV. Here, ∆η is the difference in pseudorapidity η =− ln[tan(θ/2)] between the two particles, where the polar angle θ is defined relative to the beam axis. The p T -averaged v n values for 0.3 < p T < 3.0 GeV/c are also derived up to n = 7. Factorization of the Fourier coefficients from dihadron correlations into a product of single-particle azimuthal anisotropies is investigated. This study of factorization is quantitatively compared to hydrodynamic predictions with different models of initial-state fluctuations and η/s values for two centrality classes.

Experimental Setup
The data used in this analysis correspond to an integrated luminosity of 120 µb −1 and were recorded with the CMS detector during the 2011 PbPb LHC running period at √ s NN = 2.76 TeV.
A detailed description of the CMS detector can be found in Ref. [28]. The CMS uses a righthanded coordinate system, with the origin at the nominal interaction point, the x axis pointing to the centre of the LHC, the y axis pointing up (perpendicular to the LHC plane), and the z axis along the anticlockwise-beam direction. The polar angle θ is measured from the positive z axis and the azimuthal angle (φ) is measured in the x-y plane. The central feature of the apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the field volume are the silicon pixel and strip trackers, the crystal electromagnetic calorimeter, and the brass/scintillator hadron calorimeter. In PbPb collisions, trajectories of charged particles with p T > 0.2 GeV/c are reconstructed in the tracker covering the pseudorapidity region |η| < 2.5, with a track momentum resolution of about 1% at p T = 100 GeV/c. In addition, CMS has extensive forward calorimetry, in particular two steel/quartz-fiber Cherenkov hadron forward (HF) calorimeters, which cover the pseudorapidity range 2.9 < |η| < 5.2. The HF calorimeters are segmented into towers, each of which is a two-dimensional cell with a granularity of 0.5 units in η and 0.349 rad in φ. The zero-degree calorimeters (ZDC) are tungsten/quartz Cherenkov calorimeters located at ±140 mm from the interaction point [29]. They are designed to measure the energy of photons and spectator neutrons emitted from heavy ion collisions. Each ZDC calorimeter has electromagnetic and hadronic sections with an active area of ±40 mm in x and ±50 mm in y. When the LHC beam crossing angle is 0 degree, this corresponds to an η acceptance that starts at η = 8.3 and is 100% by η = 8.9 for √ s NN = 2.76 TeV.
For one neutron, the ZDCs have an energy resolution of 20%. Since each neutron interacts independently, the resolution improves as the square root of the number of neutrons.

Selections of Events and Tracks
Minimum bias PbPb events were triggered by coincident signals from both ends of the detector in either the beam scintillator counters (BSC) at 3.23 < |η| < 4.65 or in the HF calorimeters. Events due to noise, cosmic rays, out-of-time triggers, and beam backgrounds were suppressed by requiring a coincidence of the minimum bias trigger with bunches colliding in the interaction region. The trigger has an efficiency of (97 ± 3)% for hadronic inelastic PbPb collisions. In total, about 2% of all minimum bias PbPb events were recorded.
To maximize the event sample for very central PbPb collisions, a dedicated online trigger on the 0-0.2% ultra-central events was implemented by simultaneously requiring the HF transverse energy (E T ) sum to be greater than 3260 GeV and the pixel cluster multiplicity to be greater than 51400 (which approximately corresponds to 9500 charged particles over 5 units of pseudorapidity). The selected events correspond to the 0.2% most central collisions of the total PbPb inelastic cross section. The correlation between the HF E T sum and pixel cluster multiplicity for minimum bias PbPb collisions at √ s NN = 2.76 TeV is shown in Fig. 1. The dashed lines indicate the selections used for the 0-0.2% centrality range. This fractional cross section is determined relative to the standard 0-2.5% centrality selection in PbPb collisions at CMS by selecting on the total energy deposited in the HF calorimeters [8]. The inefficiencies of the minimum bias trigger and event selection for very peripheral events are properly accounted. In a similar way, the 0-0.02% centrality range is also determined by requiring the HF E T sum greater than 3393 GeV and pixel cluster multiplicity greater than 53450 (a subset of 0-0.2% ultra-central events). With this trigger, the ultra-central PbPb event sample is enhanced by a factor of about 40 compared to the minimum bias sample. For purposes of systematic comparisons, other PbPb centrality ranges, corresponding to 40-50%, 0-10%, 2.5-5.0%, 0-2.5% and 0-1%, are studied based on the HF E T sum selection using the minimum bias sample. As a cross-check, the 0-1% centrality range is also studied using combined HF E T sum and pixel cluster multiplicity, similar to the centrality selection of 0-0.2% ultra-central events.
Centrality selections of ultra-central events are investigated in Monte Carlo (MC) simulations using the AMPT [30] heavy-ion event generator, which provides a realistic modeling of the initial-state fluctuations of participating nucleons. The generated particles are propagated through the full GEANT4 [31] simulation of the CMS detector. The equivalent centrality requirements on the HF E T sum and pixel cluster multiplicity are applied in order to evaluate the selected ranges of impact parameter and number of participating nucleons, N Part , for various centrality ranges. A summary of the mean and RMS values of N Part distributions for selected events of each very central PbPb centrality range can be found in Table 1. As one can see, there is only a moderate increase of average N Part value for events that are more central than 0-1% centrality, although the RMS value still decreases significantly for more central selections. Standard offline event selections [8] are also applied by requiring energy deposits in at least three towers in each of the HF calorimeters, with at least 3 GeV of energy in each tower, and the presence of a reconstructed primary vertex containing at least two tracks. The reconstructed primary vertex is required to be located within ±15 cm of the average interaction region along the beam axis and within a radius of 0.02 cm in the transverse plane. These criteria further reduce the background from single-beam interactions (e.g., beam-gas and beam-halo), cosmic muons, and ultra peripheral collisions that lead to the electromagnetic breakup of one or both Pb nuclei [32]. These criteria are most relevant for selecting very peripheral PbPb events but have little effect (< 0.01%) on the events studied in this paper.
During the 2011 PbPb run, there was a probability of about 10 −3 to have two collisions recorded in a single beam crossing (pileup events). This probability is even higher for ultra-central triggered events, which sample the tails of the HF E T sum and pixel cluster multiplicity distributions. If a large HF E T sum or pixel cluster multiplicity event is due to two mid-central collisions instead of a single ultra-central collision, more spectator neutrons will be released, resulting in a large signal in the ZDC. To select cleaner single-collision PbPb events, the correlation of energy sum signals between ZDC and HF detectors is studied. Events with large signals in both ZDC and HF are identified as pileup events (about 0.1% of all events), and thus rejected.
The reconstruction of the primary event vertex and the trajectories of charged particles in PbPb collisions is based on signals in the silicon pixel and strip detectors and described in detail in Ref. [8]. From studies based on PbPb events simulated using HYDJET [33] (version 1.8), the combined geometrical acceptance and reconstruction efficiency of the primary tracks is about 70% at p T ∼ 1 GeV/c and |η| < 1.0 for the 0-0.2% central PbPb events but drops to about 50% for p T ∼ 0.3 GeV/c. The fraction of misidentified tracks is kept at the level of < 5% over most of the p T (p T > 0.5 GeV/c) and η (|η| < 1.6) ranges. It increases up to about 20% for very low p T (p T < 0.5 GeV/c) particles in the forward (|η| ≈ 2) region.

Analysis procedure
Following the same procedure of dihadron correlation analysis as in Refs. [9,[34][35][36][37], the signal and background distributions of particle pairs are first constructed. Any charged particle associated with the primary vertex and in the range |η| < 2.4 can be used as a "trigger" particle. A variety of bins of trigger particle transverse momentum, denoted by p trig T , are considered. In a single event, there can be more than one trigger particle and their total multiplicity is denoted by N trig . Within each event, every trigger particle is then paired with all of the remaining particles (again within |η| < 2.4). Just as for the trigger particles, these associated particles are also binned in transverse momentum (p assoc T ).
The signal distribution, S(∆η, ∆φ), is the per-trigger-particle yield of pairs found in the same event, where N same is the number of such pairs within a (∆η,∆φ) bin, and ∆φ and ∆η are the differences in azimuthal angle φ and pseudorapidity η between the two particles. The background distribution, B(∆η, ∆φ), is found using a mixed-event technique, wherein trigger particles from one event are combined (mixed) with all of the associated particles from a different event. In the analysis, associated particles from 10 randomly chosen events with a small z vtx range (±0.5 cm) near the z vtx of the event with trigger particles are used. The result is given by where N mix denotes the number of mixed-event pairs. This background distribution represents the expected correlation function assuming independent particle emission, but taking into account effects of the finite acceptance.
The two-dimensional (2D) differential yield of associated particles per trigger particle is given by where N pair is the total number of hadron pairs. The value of the background distribution at ∆η = 0 and ∆φ = 0, B(0, 0), represents the mixed-event associated yield for both particles of the pair going in approximately the same direction and thus having full pair acceptance (with a bin width of 0.3 in ∆η and π/16 in ∆φ). Therefore, the ratio B(0, 0)/B(∆η, ∆φ) accounts for the pair-acceptance effects. The correlation function described in Eq. (3) is calculated in 0.5 cm wide bins of the z vtx along the beam direction and then averaged over the range |z vtx | < 15 cm.
To extract the azimuthal anisotropy harmonics, v n , the one-dimensional (1D) azimuthal dihadron correlation function as a function of ∆φ, averaged over |∆η| > 2 (to avoid the shortrange correlations from jets and resonance decays), can be decomposed into a Fourier series given by Here, V n∆ are the Fourier coefficients from dihadron correlations, and N assoc represents the total number of hadron pairs per trigger particle for a given |∆η| range and (p trig T , p assoc T ) bin.
In Refs. [9,[35][36][37], a fit to the azimuthal correlation function by a Fourier series was used to extract the V n∆ coefficients. In this paper, a slightly different approach is applied. The V n∆ values are directly calculated as the average value of cos(n∆φ) of all particle pairs for |∆η| > 2 (to avoid the short-range correlations from jets and resonance decays): Here, denotes averaging over all particles in each event and over all the events. The subscripts S and B correspond to the average over signal and background pairs. With an ideal detector, cos(n∆φ) S equals to V n∆ by definition. The cos(n∆φ) B term is subtracted in order to remove the effects of detector non-uniformity. The advantage of the present approach is that the extracted Fourier harmonics will not be affected by the finite bin widths of the histogram in ∆η and ∆φ. This is particularly important for very-high-order harmonics (V n∆ is extracted up to n = 7 in this analysis) that are sensitive to the finer variations of the correlation functions.
It was thought [9,14,16] that, for correlations purely driven by the hydrodynamic flow, V n∆ can be factorized into a product of single-particle Fourier harmonics, v n (p trig T ), for trigger particles and v n (p assoc T ), for associated particles: The single-particle azimuthal anisotropy harmonics can then be extracted as a function of p T as follows: where a fixed p ref T range is chosen for the "reference particles". However, as pointed out in Refs. [26,27], due to fluctuating initial-state geometry, the factorization of V n∆ could also break down for flow-only correlations. Direct tests of the factorization relation for V n∆ in Eq. (6) are carried out in this paper, as will be discussed in Section 5.3. These tests may provide new insights into the initial-state density fluctuations of the expanding hot medium.
When calculating cos(n∆φ) , each pair is weighted by the product of correction factors for the two particles. These factors are the inverse of an efficiency that is a function of each particle's pseudorapidity and transverse momentum, where A(η, p T ) is the geometrical acceptance, E(η, p T ) is the reconstruction efficiency, and F(η, p T ) is the fraction of misidentified tracks. The effect of this weighting factor only changes the overall scale of dihadron correlation functions, and has almost no effect on cos(n∆φ) . However, the misidentified tracks may have different v n values from those of correctly reconstructed tracks. Therefore, the effects of misidentified tracks are investigated and corrected using the same procedure as done in Ref. [8]. The v n values for the true charged tracks (v true n ) can be expressed as a combination of v n for all the observed tracks (v obs n ) and for misidentified tracks (v mis An empirical correction for the misidentified track v n based on the simulation studies is found to be independent of track selections or the fraction of misidentified tracks. The correction is where v n is the yield-weighted average over the p T range from 0.3 to 3.0 GeV/c, folding in the efficiency-corrected spectra. The estimated values of the correction factor, f , as well as its uncertainty, are summarized in Table 2 for different v n . Table 2: The factor, f , for estimating the v n values of misidentified tracks, as well as its uncertainty, for various orders of Fourier harmonics.
The systematic uncertainties due to misidentified tracks, which are most important at low p T where the misidentified track rate is high, are reflected in the uncertainty of the f factor in Table 2. At low p T , the systematic uncertainty from this source is 1.4% for v 2 and 5-8% for v 3 to v 6 . By varying the z-coordinate of vertex binning in the mixed-event background, the results of the v n values vary by at most 2-8% for v 2 to v 6 , respectively. Systematic uncertainties due to the tracking efficiency correction are estimated to be about 0.5%. By varying the requirements on the ZDC sum energy used for pileup rejection, the results are stable within less than 1%. The various sources of systematic uncertainties are added in quadrature to obtain the final uncertainties shown as the shaded color bands for results in Section 5.

Single-particle azimuthal anisotropy, v n
Results of azimuthal anisotropy harmonics, from v 2 to v 6 , as a function of p T in 0-0.2% central PbPb collisions at √ s NN = 2.76 TeV, are shown in Fig. 2 (left). The v n values are extracted from long-range (|∆η| > 2) dihadron correlations using Eq. (5), and by assuming factorization in Eq. (7). The p ref T range is chosen to be 1-3 GeV/c. The error bars correspond to statistical uncertainties, while the shaded color bands indicate the systematic uncertainties. As the collisions are extremely central, the eccentricities, n , are mostly driven by event-by-event participant fluctuations and are of similar sizes within a few % for all orders. Consequently, the magnitudes of v 2 and v 3 are observed to be comparable (within 2% averaged over p T as will be shown in Fig. 4), which is not the case for non-central collisions. Different v n harmonics have very different dependencies on p T . At low p T (p T < 1 GeV/c), the v 2 harmonic has the biggest magnitude compared to other higher-order harmonics. It becomes smaller than v 3 at p T ≈ 1 GeV/c, and even smaller than v 5 for p T > 3 GeV/c. This intriguing p T dependence can be compared quantitatively to hydrodynamics calculations with fluctuating initial conditions, and it provides important constraints on theoretical models. For a given value of p T , the magnitude of v n for n ≥ 3 decreases monotonically with n, as will be shown later.
If a system created in an ultra-relativistic heavy-ion collision behaves according to ideal hydrodynamics, the Fourier harmonics, v n , are expected to follow a p T dependence that has a power-law, p n T , functional form in the low-p T region [38,39]. Hence, the scaling ratio, v 1/n n /v 1/2 2 , will be largely independent of p T , as was seen by the ATLAS collaboration for not very central events [16]. In Fig. 2 (right), the v 1/n n /v 1/2 2 ratios are shown as a function of p T for n = 3-6  an increase as a function of p T . This trend is consistent to what was observed by the ATLAS collaboration for very central events (e.g., 0-1% centrality) [16].
Other choices of p ref T ranges are also studied in order to examine the assumption of factorization made for extracting v n . As an example, Fig. 3 shows the comparison of v n as a function of p T for 1 < p ref T < 3 GeV/c and 0.5 < p ref T < 1.0 GeV/c. The v n values extracted with two choices of p ref T ranges are consistent within statistical uncertainties for n > 2 over the entire p T range. However, a significant discrepancy is observed for v 2 at higher p T , e.g., up to about 40% for p T ∼ 4 GeV/c, while the low p T region shows a good agreement between the two p ref T ranges. A detailed study of factorization breakdown for Eq. (6) as well as its physical implication is    Fig. 4 as a function of n up to n = 7 (the v 7 value as a function of p T is not presented in Fig. 2 due to limited statistical precision). The 0-0.2% ultra-central events are compared to several other very central PbPb centrality ranges including 2.5-5.0%, 0-2.5%, 0-1% and 0-0.02%. As mentioned earlier, results for 0-1% centrality are compared with both the HF E T sum selection (not shown) and HF E T sum plus pixel cluster multiplicity (NPixel) selection as a systematic check. The two methods of centrality selection yield consistent v n results within statistical uncertainties. Therefore, only results from HF E T sum plus pixel cluster multiplicity centrality selection are shown in Fig. 4. Beyond the 2.5-5.0% centrality range, the v n values are still decreasing toward more central collisions, especially for v 2 . Going from 0-0.2% to 0-0.02% centrality, v n shows almost no change, indicating events do not become significantly more central by requiring larger HF E T sum and pixel cluster multiplicity, especially in terms of eccentricities. This is consistent with the studies using the AMPT model. The v n values remain finite up to n = 6 within the statistical precision of our data. Beyond n = 6, v n becomes consistent with zero. The magnitude of v 2 and v 3 are very similar, while the v n become progressively smaller for n ≥ 4. This is qualitatively in agreement with expectations from hydrodynamic calculations [38].

Correlation Functions
Dihadron correlation functions are also constructed using Eq. (3) in order to check the consistency of extracting V n∆ using Eq. (5) with the fit method to the correlation function by a Fourier series in Eq. (4). a significant local minimum (at ∆φ ∼ π along ∆η) is present. On the near side (∆φ ∼ 0) of the correlation function, a long-range structure extending over the entire ∆η region is present. The observed features of the correlation function are similar to what was seen previously at CMS in other centrality ranges of PbPb collisions [9,35], although the dip on the away side is not seen in non-central PbPb collisions. This may indicate that the contribution of higher-order Fourier components (e.g., v 3 ) is more relevant for very central events.
Averaging over ∆η, the 1D ∆φ dihadron correlation function, for 1 < p trig T < 3 GeV/c and 1 < p assoc T < 3 GeV/c in 0-0.2% central PbPb collisions at √ s NN = 2.76 TeV, is shown in Fig. 5 (right). The range of |∆η| < 2 is excluded from the average to avoid non-flow effects from other source of correlations, such as jet fragmentation. The dashed curves represent different V n∆ components and are constructed from the v n values extracted in Section 5.1 by assuming factorization. The solid curve is the sum of all V n∆ components, which is in good agreement with the measured dihadron correlation function.

Factorization breakdown and p T dependence of event plane angle
The breakdown of factorization observed in Fig. 3 could be caused by non-flow effects that contribute to the dihadron correlation function at large ∆η, e.g., back-to-back jet correlations. However, in hydrodynamics, it has been recently suggested that one possible source of factorization breakdown is related to the initial-state eccentricity fluctuations [26,27]. The event plane angle, Ψ n , as determined by final-state particles, could be dependent on the particle p T event-by-event, instead of a unique angle for the entire event (which is the case for a nonfluctuating smooth initial condition). Because of this effect, the factorization of V n∆ extracted from dihadron correlations could be broken, even if hydrodynamic flow is the only source of correlations. The breakdown effect can be explored more quantitatively in the following analysis.
A ratio for testing factorization defined as has been proposed as a direct measurement of p T -dependent event plane angle fluctuations [27].
Here, the V n∆ coefficients are calculated by pairing particles within the same p T interval (denominator) or from different p T intervals (numerator). If V n∆ factorizes, this ratio will be equal to unity. With the presence of a p T -dependent event plane angle, it has been shown that the ratio, r n , is equivalent to where Ψ n (p trig T ) and Ψ n (p assoc T ) represent the event plane angles determined for trigger and associated particles from two p T intervals [26,27]. One can see from Eq. (11) that r n is in general less than unity if event plane angle Ψ n depends on p T .
In this paper, the proposed factorization ratio, r n , is studied as a function of p For the second Fourier harmonics (Fig. 6), the r 2 ratio significantly deviates from one as the collisions become more central. For any centrality, the effect gets larger with an increase of the difference between p trig T and p assoc T values. To explicitly emphasize this observation, p trig T − p assoc T , instead of p assoc T , is used as the horizontal axis of figures 6-8. The deviation reaches up to 20% for the lowest p assoc T bins in the ultra-central 0-0.2% events for 2.5 < p trig T < 3.0. This is expected as event-by-event initial-state geometry fluctuations play a more dominant role as the collisions become more central. Calculations from viscous hydrodynamics in Ref. [27] are compared to data for 0-10% and 40-50% centralities with MC Glauber initial condition model [40,41] and η/s = 0.08 (dashed lines), and MC-KLN initial condition model [42] and η/s = 0.2 (solid lines). The qualitative trend of hydrodynamic calculations is the same as what is observed in the data. The observed r 2 values are found to be more consistent with the MC-KLN model and an η/s value of 0.2. However, future theoretical studies, particularly with comparison to the precision ultra-central collisions data presented in this paper, are still needed to achieve better constraints on the initial-state models and the η/s value of the system.
For higher-order harmonics (n = 3, 4), shown in Fig. 7 and Fig. 8, the factorization is fulfilled over a wider range of p trig T , p assoc T , and centrality ranges than for v 2 . The factorization only breaks by about 5% at large values of p trig T − p assoc T , i.e., greater than 1 GeV/c. Due to large statistical uncertainties, r 5 is not included in this result. Again, the qualitative trend of the data is described by hydrodynamics for 0-10% centrality, while no conclusion can be drawn for 40-50% centrality based on the present statistical precision of the data.

Conclusion
In summary, azimuthal dihadron correlations were studied for PbPb collisions at √ s NN = 2.76 TeV using the CMS detector at the LHC. Assuming factorization, these two-particle correlations were used to extract the single-particle anisotropy harmonics, v n , as a function of p T from 0.3 to 8.0 GeV/c. The data set includes a sample of ultra-central (0-0.2% centrality) PbPb events collected using a trigger based on total transverse energy in the hadron forward calorimeters and the total multiplicity of pixel clusters in the silicon pixel tracker. In the context of hydrodynamic models, anisotropies in such ultra-central heavy-ion collisions arise predominantly from initialstate eccentricity fluctuations. The magnitude of the flow harmonics decreases from v 3 to v 6 . As a function of p T , these four harmonics all display a common maximum around p T = 3.5 GeV/c. Although the v 2 harmonic exceeds the others at low p T , it falls below v 3 around p T = 1 GeV/c and reaches its maximum around p T = 2.5 GeV/c.
The p T -averaged v n for 0.3 < p T < 3.0 GeV/c were also derived up to n = 7, and results for 0-0.2% collisions were compared to those for other slightly less central ranges. Between the 2.5-5.0% and 0-0.2% centrality ranges, all v n harmonics decrease. The decrease is largest for  v 2 , reaching up to 45%. Only small variations of v n are observed for events that are even more central than 0-0.2% (e.g., 0-0.02%). For the most central collisions, the p T -averaged v 2 and v 3 are found to be comparable within 2%, while higher-order v n decrease as n increases.
Detailed studies indicate that factorization of dihadron correlations into single-particle azimuthal anisotropies does not hold precisely. The observed breakdown of factorization increases up to about 20% as the p T difference between the two particles becomes larger in ultra-central PbPb events. This behavior is expected in hydrodynamic models, in which a p Tdependent event plane angle is induced by initial-state fluctuations. The factorization data for the 0-10% and 40-50% centrality ranges were compared to viscous hydrodynamic calculations with different models of initial-state fluctuations and different η/s values. Future quantitative theoretical comparisons to the high-precision data of ultra-central PbPb collisions presented by the CMS collaboration in this paper can provide a new stringent test of hydrodynamic models, particularly for constraining the initial-state density fluctuations and the η/s value.

Acknowledgment
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWF and FWF (Austria); FNRS and