Event shape engineering for inclusive spectra and elliptic flow in Pb-Pb collisions at $\sqrt{s_\rm{NN}}=2.76$ TeV

We report on results obtained with the Event Shape Engineering technique applied to Pb-Pb collisions at $\sqrt{s_\rm{NN}}=2.76$ TeV. By selecting events in the same centrality interval, but with very different average flow, different initial state conditions can be studied. We find the effect of the event-shape selection on the elliptic flow coefficient $v_2$ to be almost independent of transverse momentum $p_\rm{T}$, as expected if this effect is due to fluctuations in the initial geometry of the system. Charged hadron, pion, kaon, and proton transverse momentum distributions are found to be harder in events with higher-than-average elliptic flow, indicating an interplay between radial and elliptic flow.


Introduction
Results from Lattice Quantum Chromo-Dynamics [1,2] predict the existence of a plasma of deconfined quarks and gluons, known as the "Quark Gluon Plasma" (QGP). This state of matter can be produced in the laboratory by colliding heavy nuclei at relativistic energies [3,4,5]. The QGP was found to behave as a nearly perfect liquid and its properties can be described using relativistic hydrodynamics (for a recent review, see [6]). The current experimental heavy-ion programs at Brookhaven's Relativistic Heavy Ion Collider (RHIC) and at CERN's Large Hadron Collider (LHC) are aimed at a precise characterization of the QGP, in particular of its transport properties.
The system created in a heavy-ion collision expands and hence cools down, ultimately undergoing a phase transition to a hadron gas, which then decouples to the free-streaming particles detected in the experiments [6]. A precision study of the QGP properties requires a detailed understanding of this expansion process. If the initial geometry of the interaction region is not azimuthally symmetric, a hydrodynamic evolution of a nearly-ideal liquid (i.e. with a small value of the shear viscosity over entropy ratio η/s) gives rise to an azimuthally anisotropic distribution in momentum space for the produced particles.
This anisotropy can be characterized in terms of the Fourier coefficients v n of the particle azimuthal distribution [7]. The shape of the azimuthal distribution, and hence the values of these Fourier coefficients, depend on the initial conditions and on the expansion dynamics. The geometry of the initial state fluctuates event-by-event and measurements of the resulting v n fluctuations pose stringent constraints on initial state models. A quantitative understanding of the initial geometry of the produced system is therefore of primary importance [6]. A number of different experimental measurements and techniques have been proposed to disentangle the effects of the initial conditions from QGP transport, including measurements of correlations of different harmonics [8], event-by-event flow fluctuations [9,10,11,12] and studies in ultra-central collisions [13,14]. Recent results from pp and p-Pb collisions at the LHC, moreover, suggest that hydrodynamic models may be also applicable to small systems [15,16,17,18,19]. This further highlights the importance of studying Pb-Pb collisions with more differential probes, to investigate the interplay between the initial conditions and the evolution, in the system where the hydrodynamic models are expected to be most applicable.
One of the new tools for the study of the dynamics of heavy-ion collisions is the "Event Shape Engineering" (ESE) [20]. This technique is based on the observation that the event-by-event variation of the anisotropic flow coefficient (v n ) at fixed centrality is very large [12]. Hydrodynamic calculations show that the response of the system to the initial spatial anisotropy is essentially linear for the second and third harmonic, meaning that the final state v 2 (and v 3 ) are very well correlated with the second (and third) order eccentricities in the initial state for small values of η/s [7,21,22]. These observations suggest a possibility to select events in heavy-ion collisions based on the initial (geometrical) shape, providing new opportunities to study the dynamics of the system evolution and the role of the initial conditions. The ESE technique is proposed to study ensemble-averaged observables (such as v 2 and inclusive particle spectra) in a class of events corresponding to the same collision centrality, but different v n values. In this paper events are selected based on the magnitude of the second order reduced flow vector q 2 (see Sect. 3.1). The technique was recently applied to study correlations between different flow harmonics in the ATLAS experiment [23]. In this paper we present the results on elliptic flow and charged particle specta in Pb-Pb collisions at √ s NN = 2.76 TeV obtained with ESE technique. The events selected with the ESE technique are characterized by the measurement of v 2 , to quantify the effect of the selection on the global properties of the event. In order to search for a connection between elliptic and radial flow the effect of the ESE selection on the inclusive transverse momentum distribution of charged hadrons, pions, kaons and protons is then studied. The results are presented for primary charged particles, defined as all prompt particles produced in the collision including all decay products, except those from weak decays of light flavor hadrons and of muons. The differential measurement described in this work could provide important constraints to identify the correct model for initial conditions and for the determination

ALICE detector and data sample
The ALICE detector at the CERN LHC was designed to study mainly high-energy Pb-Pb collisions. It is composed of a central barrel (|η| 0.8 for full-length tracks), containing the main tracking and particle identification detectors, complemented by forward detectors for specific purposes (trigger, multiplicity measurement, centrality determination, muon tracking). A detailed description of the apparatus can be found in [24]. The main detectors used for the analysis presented in this paper are discussed below.
The main tracking devices in the central barrel are the Inner Tracking System (ITS) and the Time Projection Chamber (TPC). They are immersed in a 0.5 T solenoidal field. The ITS is the detector closest to the interaction point. It is a six-layer silicon tracker with a very low material budget (∼ 7% of one radiation length X 0 ). The ITS provides information on the primary interaction vertex and is used to track particles close to the interaction point, with the first layer positioned at a radial distance of 3.9 cm from the interaction point and the sixth one at 43 cm. It can measure the transverse impact parameter (DCA xy ) of tracks with a resolution of about 300 (40) µm, for transverse momentum p T = 0.1 (4) GeV/c, allowing the contamination from secondary particles to be significantly reduced. The TPC [25] is a large-volume gas detector (external diameter 5 m) which measures up to 159 space points per track, providing excellent tracking performance and momentum resolution (σ p T /p T ∼ 6% at p T = 10 GeV/c) [26]. It is also used in this work to identify particles through the measurement of the specific energy loss, dE/dx. The dE/dx, computed as a truncated mean utilizing only 60% of the available samples, has a resolution of ∼ 5% in peripheral and ∼ 6.5% in central collisions [26]. At a radius of 3.7 m from the beam axis, the Time of Flight (TOF) detector measures the arrival time of particles with a total resolution of about 85 ps in Pb-Pb collisions, allowing a π/K (K/p) 2 σ separation up to p T = 3 (5) GeV/c. The ALICE reconstruction software performs tracking based either on the information from the TPC alone (TPC-only tracks) or on the combined information from the ITS and TPC (global tracks). The former have the advantage of an essentially flat azimuthal acceptance, and are used for v 2 and q 2 measurements. The latter provide better quality tracks (σ p T /p T ∼ 1.5% at p T = 10 GeV/c) [26], rejecting most of the secondary tracks. However, the acceptance and reconstruction efficiency of global tracks are not flat in azimuth and as a function of transverse momentum, mostly due to missing or inefficient regions of the ITS. These tracks are used for the p T distribution measurements. TPC-only tracks can be constrained to the primary vertex (reconstructed also using the ITS information) to provide better momentum resolution.
The data used for this analysis were collected in 2010, during the first Pb-Pb run at the LHC, at a centerof-mass energy per nucleon √ s NN = 2.76 TeV. The hadronic interaction rate was of the order of 100 Hz, low enough to avoid any space charge distortion effects in the TPC [27]. The trigger was provided by the 3 Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration V0 detector [28], a pair of forward scintillator hodoscopes placed on either side of the interaction region, covering the pseudorapidity regions 2.8 < η < 5.1 (V0A) and −3.7 < η < −1.7 (V0C). Events were requested to have a signal in both sides of the V0, selecting roughly 0-90% most central collisions [29]. The V0 measures a signal whose average amplitude is proportional to the multiplicity of charged particles. The V0 acceptance times detection efficiency is approximately 90% and flat as a function of the particle p T , with only a small reduction to about 85% for p T < 300 MeV/c. Events are further selected offline using the timing information from the V0 and from a set of two forward Zero Degree Calorimeters (ZDCs), in order to reject contamination from beam-induced backgrounds (see [29,30,31] for a detailed discussion). After all selections, the event sample used in the analysis consists of about 16 million events.

Analysis technique 3.1 Centrality and the event shape selection
The events which pass the basic selection described in Sect. 2 are divided in centrality classes based on the signal amplitude (proportional to the charged particle multiplicity) measured in the V0 detector, as described in [29]. Events in each centrality class are further subdivided into groups with different average elliptic event shapes based on the magnitude of the second order reduced flow vector q 2 [22] given as where M is the multiplicity and |Q Q Q 2 | = Q 2 2,x + Q 2 2,y is the magnitude of the second order flow vector.
In this paper, the flow vector Q Q Q 2 is calculated using the TPC or V0 detectors. In the TPC, tracks in the range 0.2 < p T < 20 GeV/c and |η| < 0.4 (to avoid an overlap with the η region used for the v 2 and p T distribution measurements) are used to measure where ϕ i is the azimuthal angle of the i-th particle and M is the number of tracks in an event.
In the forward rapidity region the V0 is used. This detector segmented into four rings, each consisting of 8 azimuthal sectors, the flow vector is hence calculated as where the sum runs over all 32 channels, ϕ i is the angle of the center of the sector containing channel i, w i is the amplitude measured in channel i and M is in this case the sum of the amplitudes measured in each channel.
The discriminating power of q 2 depends on the magnitude of elliptic flow as well as on the track multiplicity used in the q 2 calculation and on the performance of the detector, including the angular resolution or the linearity of the response to the charged particle multiplicity. The good resolution of the TPC and the large multiplicity at midrapidity are used to maximize the selectivity on q 2 . However, the ALICE central barrel acceptance enables only limited separation in pseudorapidity between the region used to calculate q 2 and the region used to calculate the observables (|∆η| = 0.1). This separation is introduced in order to suppress unwanted non-flow correlations, which typicaly involve only a few particles and are in general short-range. In order to further assess the contribution of non-flow correlations, the flow vector is also calculated using the V0 detectors. This leads to a separation of more than one unit in pseudorapidity between the two regions. In absence of correlations, the average length of Q Q Q 2 grows as √ M [22]: q 2 is introduced to remove this trivial part of the multiplicity dependence. In case of non-zero correlations (due to either collective flow or non-flow correlations), q 2 depends on multiplicity and on the strength of the flow as [32,22] where the parameter δ 2 accounts for non-flow correlations, and the angular brackets denote the average over all events.
In the case when the multiplicity is measured via the signal amplitude in the V0 detector, the first term in Eq. 4 (unity) has to be substituted by e 2 i / e i 2 , where e i is the energy deposition of a single particle i. The fluctuations in e i lead to an increase in the flow vector length and reduce the corresponding event plane resolution. Large-q The q 2 distribution measured with the TPC (q TPC 2 ) and V0C (q V0C 2 ) is shown in Fig. 1 as a function of centrality, and in two narrow centrality classes, 0-1% and 30-31%. As can be seen, q 2 reaches values twice as large as the mean value, as expected in case of large initial state fluctuations [20]. The q V0C 2 is larger than q TPC 2 , as the former is measured in a larger pseudorapidity window (integrating a larger multiplicity) and is sensitive to the fluctuations in e i . Note also that the selectivity (discrimination power) of the two selection cuts is in principle different, due to the different detector resolution, and, in the case of V0C, smaller v 2 value at forward η, fluctuations in e i and large contribution of secondary particles.
In the present analysis, the effect of the event shape engineering on v 2 and p T distributions is studied. The average flow and particle spectra are measured in the pseudorapidity range 0.5 < |η| < 0.8 in order to avoid overlap with the region used to calculate q TPC 2 . The V0C selection is used to estimate the contribution of non-flow correlations to the event-shape selection, since it provides a large η gap. As a further crosscheck, the analysis was also repeated using the V0A detector. The results obtained with V0A and V0C show a qualitative agreement with a better selectivity when the V0C is used (mostly due to the larger multiplicity in the acceptance of this detector and to the η dependence of the elliptic flow). We therefore report the results for events selected using q TPC 2 and q V0C 2 in this paper.
Due to the limited statistics, the analysis has to be performed in relatively wide centrality classes (∼ 10%). The length of q 2 changes within such large centrality intervals (Fig. 1), and a cut at a fixed value of q 2 would introduce a dependence on the multiplicity that would obscure the effect of the event-shape selection. The q 2 selection is therefore evaluated in narrow (1%-wide) centrality classes. The results presented in the next sections are obtained in two event-shape classes, corresponding to the 10% of the events having the top (bottom) value of the q 2 (estimated in the narrow centrality classes). In the following, we refer to these two classes as "large−q 2 " (90-100%) and "small−q 2 " (0-10%) or, generically, as ESE-selected events. Conversely, we refer to the totality of data within a given centrality class as the "unbiased" sample.
The correlation between q TPC 2 and q V0C 2 is illustrated for events in the 30-31% centrality class in Fig. 2. The left (right) panel shows the distribution of q 2 measured with the TPC (V0C) for all events and for events in the large−q 2 and small−q 2 classes, selected with the V0C (TPC). The average q 2 changes by about 18% and 14% in the large−q 2 and small−q 2 samples respectively. In order to control the effect of fluctuations in a given detector the detailed comparison of the results obtained with q TPC 2 and q V0C 2 is 6 Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration crucial, as discussed in detail below. In order to disentangle the effect of the η gap and of the q 2 cut, the selection on q TPC 2 is also adjusted such that the average flow measured at mid-rapidity is similar to the one in the large−q 2 sample (Sect. 4).
The ESE becomes less selective in peripheral events regardless of the detector used to compute q 2 , due to the low multiplicity. This limits the present analysis to the 60% most central events.
Space charge distortion effects in the TPC, which accumulate over many events could, in principle, bias the q 2 selection. In order to check for this and other possible instrumental effects it was verified that the results are not sensitive to the instantaneous luminosity.

Elliptic flow measurement
The elliptic flow, v 2 , is measured in the pseudorapidity range 0.5 < |η| < 0.8 using the Scalar Product (SP) method [22], according to: where u u u 2,k =exp(i2ϕ k ) is the particle's unit flow vector, ϕ k is the azimuthal angle of the k-th particle of interest, Q Q Q 2 is the flow vector and M is the multiplicity. The full event is divided in two independent sub-events, labeled as A and B, covering two different pseudorapidity ranges, 0.5 < η < 0.8 and −0.8 < η < −0.5. The particle's unit flow vector u u u 2,k is evaluated in the sub-event A while the flow vector Q Q Q 2 and the multiplcity M in the sub-event B and vice-versa, ensuring a pseudorapidity gap of |∆η| > 1 between the particle of interest and the reference charged particles, which suppresses the non-flow contribution in the calculation of v 2 {SP}. A flat acceptance in azimuth is achieved in this analysis selecting TPC-only tracks, constrained to the primary vertex. Tracks are required to have at least 70 clusters and a χ 2 ≤ 4 per TPC cluster (two degrees of freedom). Tracks with a transverse distance of closest approach to the vertex (computed before constraining tracks to the primary vertex) DCA xy > 2.4 cm or a longitudinal distance of closest approach DCA z > 3.2 cm are rejected to reduce the contamination from secondary tracks. The effect of secondary particles is corrected applying the same analysis procedure to Monte Carlo events, simulated with the AMPT event generator [33] and propagated through a GEANT3 [34] model of the detector. The v 2 {SP} computed using reconstructed tracks is then compared with the one computed with generated primary particles, and the difference (< 5%) is used as a correction factor.
The uncertainty on the tracking efficiency was assessed with different track samples and selections: using a set of hybrid tracks, built from a combination of global and TPC only tracks to obtain a uniform azimuthal acceptance [35], using TPC only tracks not constrained to the primary vertex, varying the minimum number of TPC clusters required in the analysis from 70 to 50 (Track reconstruction in Tab. 1 and 2) and weighting each track by the inverse of the (p T -dependent) efficiency (Tracking efficiency).
The procedure used to estimate the centrality percentiles leads to a ∼1% uncertainty in the definition of the centrality classes [29]. In order to propagate this uncertainty to the results presented in this paper, the measurement is repeated displacing the centrality percentile by 1%. For instance, the analysis in the 30-40% centrality class is repeated for the selection 30.3-40.4% (Centrality resolution). Moreover, tracks reconstructed at midrapidity (instead of the V0 signal) are used as the centrality estimator (Centrality estimator).
The correction for the effect of secondary particles mentioned above is strongly model dependent, therefore the difference between the v 2 estimated using generated AMPT particles and reconstructed tracks was used to estimate the corresponding systematic uncertainty, ∼ 3.5% (0.7%) at p T = 0.2 (1.5) GeV/c (Secondary particles).
Moreover, the following systematic checks were considered: the dependence on the magnetic field configuration was studied analyzing separately samples of events collected with different polarities of the 7 Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration  magnetic field (Magnetic field), analyzing positive and negative particles separately (charge) and analyzing samples of tracks produced at different vertex positions: −10 < z vtx < 0 cm and 0 < z vtx < 10 cm (Vertex). These effects are found to be not significant.
The systematic uncertainties in the v 2 measurements and in the ratios of v 2 in ESE-selected over unbiased events are summarized in Tab. 1 and 2. Only the checks and variations which are found to be statistically significant are considered in the systematic uncertainties [36]. Whenever the p T dependence of the uncertainty is not negligible, values for characteristic p T are given in the tables.

Transverse momentum distribution measurement
The measurement of the p T distributions uses global tracks, which provide good resolution on DCA xy (Sect. 2), and hence good separation of primary and secondary particles. The track selection requires at least 70 clusters in the TPC and at least 2 points in the ITS, out of which at least one must be in the first two layers, to improve the DCA xy resolution. A p T -dependent cut on the DCA xy , corresponding to 7 times the experimental resolution on DCA xy , is applied to reduce the contamination from secondary particles. Tracks with a χ 2 per point larger than 36 in the ITS and larger than 4 in the TPC are rejected. Finally, to further reduce the contamination from fake tracks, a consistency cut between the track parameters of TPC and global tracks was applied. For each reconstructed TPC track, the χ 2 -difference between the  Table 3: Summary of systematic errors for the ratio of p T distributions between large-q 2 and unbiased events. NS = not statistically significant. Table 4: Summary of systematic errors for the ratio of p T distributions between small-q 2 and unbiased events. NS = not statistically significant.
track parameters computed using only the TPC information constrained to the vertex and the associated global track is required to be less than 36 [37]. Charged tracks are studied in the pseudorapidity window 0.5 < |η| < 0.8, to avoid an overlap with the q TPC 2 calculation.
Particles are identified using the specific energy loss dE/dx in the TPC and their arrival time in the TOF. The technique is similar to the one presented in [15]. A track is identified as either a pion, a kaon or a proton based on the difference, in the detector resolution units, from the expected energy loss and/or time of flight nσ i PID (with i being the particle identity under study). Below p T = 0.5 GeV/c, only the TPC information is used (nσ i PID = nσ i TPC ). For larger p T , the TPC and TOF information is combined using a geometrical mean: Tracks are required to be within 3σ PID of the expected value to be identified as π ± , K ± or p (p). In the region where the 3σ PID identification bands of two species overlap, the identity corresponding to the smaller nσ PID is assigned. This technique gives a good track-by-track identification in the following p T ranges: 0.2 < p T < 4 GeV/c for π ± , 0.3 < p T < 3.2 GeV/c for K ± , 0.5 < p T < 4 GeV/c for p (p). The misidentification of tracks is below 4% for pions, 25% for kaons and 10% for protons in those ranges. Further discussion on the ALICE Particle Identification (PID) performance can be found in [26,38]. The results for identified particles are provided in the pseudorapidity range 0.5 < |η| < 0.8. However, in the case of the q V0C 2 selection the results were also studied at mid-rapidity |y| < 0.5. Results for positive and negative particles are consistent. In the following, "pions", "kaons" and "protons", as well as the symbols "π", "K" and "p", refer to the sum of particles and antiparticles.
9 Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration The results for the spectra in ESE-selected events are presented in terms of ratios between the distributions measured in the large−q 2 (small−q 2 ) samples and the unbiased sample. The unbiased spectra have already been reported in [37,38]. Most of the corrections (and uncertainties) cancel out in these ratios, allowing for a precise determination of the effect due to the event-shape selection, as discussed in detail below. The uncertainties can mostly arise due to effects that depend on the local track density, which are found to be small [39].
The systematic uncertainties are summarized in Tab. 3 and Tab. 4. As mentioned before, only the checks and variations which are found to be statistically significant are considered in the systematic uncertainties [36].
The systematic uncertainty related to the tracking is estimated varying the track selection cuts. Instead of the standard TPC cluster cut, at least 120 (out of 159) pad-rows hits in the TPC and a fraction of shared clusters in the TPC < 0.4, are required (Track reconstruction in Tab. 3 and 4).
The possible effect of a track-density-dependent efficiency (which would influence in a different way events with the large− and small−q 2 selection) is investigated using simulations based on the AMPT event generator [33] and a parametric event generator tuned to reproduce the ALICE spectra and v 2 measurements [39]. This effect leads to an uncorrelated systematic error of about 0.2% and a normalization error of 0.4% (Tracking efficiency).
The uncertainty on the centrality is estimated varying the definitions of centrality classes by 1% and using tracks as the centrality estimator. These checks lead to an uncorrelated uncertainty of about 0.1% and 0.35%, respectively and a normalization uncertainty below 1% in the ratios of spectra (Centrality resolution and Centrality estimator).
The systematic effect related to the particle identification is studied performing several variations to the PID approach described above. The nσ PID cut is varied between 2 and 4. Alternatively, if a track is consistent with more than one particle assignment within the nσ PID cut, double counting is allowed. As compared to the standard strategy where only the identity closest to the measured nσ PID is selected, this approach leads to a slightly larger contamination from misidentified tracks, but also to a larger efficiency. Finally, an exclusive nσ PID strategy was used, which drastically reduces misidentification: a particle is only accepted if it is compatible with only one mass hypothesis at 3σ PID . As a further cross-check, a Bayesian approach [26] was also considered. This method allows for better control of contamination at high p T . Overall, the uncertainty related to the particle identification strategy is less than 0.1% (PID).
The effect of secondary particles depends on the p T distribution of weakly-decaying primary particles, and could be different for the large− and small−q 2 samples. This effect is estimated to be at most ∼ 0.1% for protons with the TPC ESE selection and negligible in all other cases (Secondary particles).
Possible effects related to the magnetic field and to the charge state are addressed studying separately events collected with different magnet polarities (Magnetic field) and different charges (Charge), as in the case of the v 2 {SP} measurement. Particles produced at different longitudinal position cross a different portion of the detector, with different reconstruction efficiency. The samples of events produced with a negative (−10 < z vtx < 0 cm) and positive (0 < z vtx < 10 cm) longitudinal vertex coordinate with respect to the nominal interaction point were studied separately (Vertex).

Charged particle elliptic flow
The event-shape selection is studied in Fig. 3     (small−q 2 ) selection and the unbiased sample. Selecting the 10% highest (lowest) q TPC 2 samples leads to a change of 30-50% in the v 2 {SP} measured, depending on centrality. The change is smaller (∼ 10-25%) in the case of q V0C 2 -based selection, as compared to the q TPC 2 case. As already indirectly inferred from the difference between 2 nd and 4 th order flow cumulants v 2 {2} and v 2 {4} in [12], the elliptic flow response of the system to geometry fluctuations is almost independent of p T . For all centralities, the change observed in Fig. 4 depends indeed weakly on p T , up to at least 4-5 GeV/c. This indicates that a cut on q 2 selects a global property of the event, likely related to the initial shape in the overlap region. The only exception to the previous observation is the 0-5% centrality class, where for the q TPC 2 selection an increasing trend with p T is observed. In this centrality class the mean value of v 2 is small, due to the almost isotropic shape in the initial state. Moreover, relative flow fluctuations are large in central collisions, with a p T dependence similar to the one shown in Fig. 4 [12]. The analysis of the p T spectra presented in Section 4.2 gives additional insight into the trend observed in Figure 4.
For p T 4 − 5 GeV/c, the ratio ESE-selected/unbiased v 2 {SP} increases for the large−q 2 selection. This trend is more pronounced for the q TPC 2 selection and for the most central and the most peripheral classes. A fit with a constant over the full p T range yields χ 2 per degree of freedom values in the range 2-6 (depending on centrality) for the q TPC 2 selection and < 2 for the q V0C 2 selection. Fitting the ranges p T < 5 GeV/c and p T > 5 GeV/c with two different constants indicates an increase for the large−q 2 selection of order 5% and 10% for the q V0C 2 and q TPC 2 selections, respectively. This difference could be due to a small non-flow-induced bias. At high p T the v 2 is believed to be determined by the path-length dependence of parton energy loss [12].
The difference between the q TPC 2 and q V0C 2 can be due to the different selectivity (see Sect. 3.1), but also to a different contribution of non-flow correlations between the q 2 and the v 2 measurements. Replacing the q TPC 2 selection with the q V0C 2 one changes both non-flow and selectivity at the same time. To disentangle these two contributions, the selectivity of the q TPC 2 selection was artificially reduced. This is achieved  either relaxing the selection itself or rejecting a random fraction of tracks for the computation of q TPC 2 , while still selecting 10% of the events. It is found that selecting the class 65-100% for the large−q 2 sample (0-55% for the small−q 2 sample) with q TPC 2 , or alternatively rejecting 70% of the TPC tracks, leads to an average variation of the v 2 {SP} in the range 0.2 < p T < 4 GeV/c comparable to the one obtained with the standard 10% q V0C 2 selection. The results are shown in Fig. 5 for the centrality class 30-40%. Not only is it possible to find a cut which leads to the same average variation in v 2 {SP}, but the p T dependence is very similar in both cases. Rejecting randomly 70% of the tracks changes the selectivity of q TPC 2 without affecting non-flow correlations between the q TPC 2 selection and v 2 {SP} measurement (as the η gap is not varied). Also in this case, it is found that the effect of the q 2 selection does not depend on p T . A similar result, with the same value of the relaxed cut or fraction of rejected tracks, is found for the centrality interval 10-50%. Moreover, as it will be discussed in the next section, the same relaxed selections lead to the same effect on the p T distributions.
These checks demonstrate that the selectivity of the cut is the main reason for the difference between the TPC and V0C selections. Due to the large η gap, the non-flow contribution is expected to be negligible in the case of the q V0C 2 selection. The agreement observed in Fig. 5 indicates that, in the centrality classes 10-50%, this is also the case for the q TPC 2 selection in the range p T < 5 GeV/c, a transverse momentum region dominated by hydrodynamic effects [38]. It is worth noticing that the ATLAS Collaboration measured a modification of the elliptic flow of ∼ 35%, nearly independent of p T up to ∼ 12 GeV/c in the 20-30% centrality class, while measuring v 2 and q 2 with a pseudorapidity gap of 0.7 units [23]. The increasing trend in the centrality class 0-5% is also observed in [23] 1 .
To study the centrality and the q 2 dependence of v 2     the average change for each centrality class fitting the ratios in the range 0.2 < p T < 4 GeV/c with a constant 2 . The centrality dependence of the average change in the large−q 2 and small−q 2 selection is reported in Fig. 6. The trend obtained with the q TPC 2 and q V0C 2 selections is very similar, except for the most central class 0-5%, where the average is influenced by the non-flat trend seen in Fig. 4. This once again reinforces the conclusion that the non-flow contamination is small also in the TPC selection case for the bulk of particles. The relative importance of non-flow changes with centrality. A large non-flow bias would therefore introduce a centrality dependence in the relative trend between the q TPC 2 and q V0C 2 selections, which is not observed. The dependence of the v 2 {SP} variation on q TPC 2 and q V0C 2 is shown for the centrality classes 5-10%, 30-40% and 50-60% in Fig. 7. The left panel shows the absolute q 2 Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration values on the x axis, while the right panel depicts the self-normalized values, defined as the average q 2 value in ESE-selected events over the average q 2 values for all events in a given centrality class. The V0C selection spans a larger range but the TPC is more selective, as is clearly seen from the different slope of the TPC and V0C curves. In both cases the average q 2 reaches values twice as large compared to those in the unbiased sample, (Fig. 7, right).
In summary, the observations reported in this section indicate that the ESE selects a global property of the collisions, as suggested by the flat modification in the v 2 as a function of p T . The q TPC 2 leads to a change twice as large than the corresponding q V0C 2 selection. The difference between the two seems to be mostly due to the different discriminating power rather than to non-flow effects.

Transverse momentum distributions
In order to study the interplay between the initial configuration of the system and the dynamics of the expansion of the fireball, the effect of the ESE selection on the single particle p T distribution is reported in Fig. 8, for the q TPC 2 and q V0C 2 selections. As discussed in Sect. 3.1 the reduced flow vector is calculated in the TPC detector in the pseudorapidity range |η| < 0.4. In order to avoid overlap between the q TPC 2 and p T distribution measurements, only the region 0.5 < |η| < 0.8 is used to measure the p T distributions. This ensures at least 0.1 units of pseudorapidity separation between the q 2 and spectra measurements, thus suppressing the effect of short-range correlations. For consistency with the TPC analysis, the same pseudorapidity range is used in the case of the V0C selection. In the q V0C 2 case, it is also possible to study the spectra at mid-rapidity |η| < 0.8 without any overlap with the q 2 measurement. The results agree within uncertainty with those in 0.5 < |η| < 0.8.
The spectra in the large−q 2 sample are harder than those in the small−q 2 one. The ratio to the unbiased spectra reaches a maximum around p T = 4 GeV/c, and then stays approximately constant within large uncertainties.
The effect of the selection is more pronounced in semi-central events (∼ 30-50%), and decreases both towards more central and more peripheral collisions. This can be understood as due to the fact that the q 2 15 Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration    spans a larger dynamic range in semi-central collisions ( Fig. 1 and Fig. 7). In the most peripheral centrality class studied in this paper (50-60%) the effect of the TPC-based selection is still very pronounced, while the q V0C 2 selection is less effective. This may indicate a small contamination from non-flow effects in the most peripheral bin, consistent with observations discussed for the v 2 {SP} measurement in Sect. 4.1. In the most central class (0-5%) the modification of the spectrum is very small. This suggests that the trend observed in the same centrality class in Fig. 4     rather than non-flow contributions.
As in the previous section, we disentangle the effect of non-flow and q 2 selectivity either relaxing the q TPC 2 selection or randomly rejecting a fraction of the tracks. The relaxed cut and the fraction of rejected tracks tuned to reproduce the v 2 variation in 0.2 < p T < 4 GeV/c in Sect. 4.1 are used. Figure 9 shows that these selections yield results compatible with the standard q V0C Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration  same relaxed cuts or fraction of rejected tracks) is found for all centralities up to ∼ 50%, after which non-flow effects seem to become relevant.
As discussed in Sect. 4.1, we conclude that the effect of non-flow is small and that the main factor driving these observations is the average v 2 at mid-rapidity.
The modification on the spectra of identified π, K, and p is reported in Fig. 10 and Fig. 11 for different centrality classes. The same pattern measured in the case of non-identified hadrons is observed. Moreover, a clear mass ordering is seen: the modification is more pronounced for heavier particles. Conversely, the spectra in the small−q 2 sample are softer. In the case of the V0C selection the analysis was Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration also repeated in the region |y| < 0.5, yielding consistent results.
These observations suggest that the spectra in the large−q 2 (small−q 2 ) sample are affected by a larger (smaller) radial flow push. This hypothesis was tested with a blast-wave [40] study. A ratio of two blast-wave functions was used to fit the spectra ratios shown in Fig. 10 and Fig. 11. The parameters were initially fixed to the values from [38], where they were tuned to describe the inclusive spectra of pions, kaons and protons. Then, the β T parameter of the numerator function was allowed to change (while keeping the overall integral of the function constant). The fit was performed as in [38] in the transverse momentum ranges 0.5-1 GeV/c, 0.2-1.5 GeV/c, 0.3-3 GeV/c for π, K, p, respectively. The agreement with the data is good, also outside the range used to determine the parameters, up to p T ∼ 3GeV/c. The fits yield the following result for the difference ∆ β T between the β T parameter of the numerator and denominator function: ∆ β T = (0.41 ± 0.03)% (large−q 2 ) and ∆ β T = (−0.22 ± 0.03)% (small−q 2 ) for the centrality class 30-40%, as shown in Fig. 12.

Discussion
In this paper the first application of the Event Shape Engineering (ESE) [20] to the analysis of ALICE data was presented.
The results on the v 2 {SP} measurement suggest that the ESE technique selects a global property of the collision, likely related to the eccentricity in the initial state. The measurement of p T spectra indicates that events with larger eccentricity show an increased radial flow. A correlation between elliptic and radial flow could be introduced either at the initial stage, due to the specific fluctuation patterns in the energy deposition, or during the hydrodynamic evolution of the system, due to an interplay of bulk and shear viscosity [7].
A Glauber Monte Carlo simulation was performed to estimate the possible correlation between the initial eccentricity and azimuthally-averaged pressure gradients. In the model, the multiplicity of charged particles in the acceptance of the V0 detector, used to determine the centrality classes, is computed following [29]. A "number of ancestors" N ancestors is derived from the number of participant nucleons (N part ) and binary collisions (N coll ) as Each ancestor is assumed to produce particles following a negative binomial distribution with parameters taken from [29].
The participant density, defined following [41,42,9,43] as N part /S, is used as a proxy for the magnitude of the pressure gradients. The average cross-sectional area S and participant eccentricity ε are computed as where The unprimed coordinates are given in the fixed laboratory coordinate frame. Primed coordinates, x ′ and y ′ , are calculated in the so-called participant coordinate system, rotated with respect to the laboratory coordinate frame such that the minor symmetry axis of the participant nucleon distribution coincides   with the x ′ direction. The normalization of the area is chosen such that for a Gaussian distribution the average density coincides with N part /S.
Two narrow centrality classes, selected based on the simulated charged particle multiplicity, roughly corresponding to 0-2% (central) and 30-32% (semi-central), are studied in Fig. 13. The observed correlation between the density and the participant eccentricity is reminiscent of the correlation between radial flow and event shape measured in this paper. The average density in events with the 10% largest ε is about 1% (7%) larger than in events with the smallest ε for central (semi-central) collisions, qualitatively consistent with what is observed in Fig. 10 and Fig. 11, where the effect of the ESE selection is much stronger for semi-central collisions. This reinforces our conclusion that ESE is an effective tool to select the initial shape and density, thereby opening the possibility of further studies.
A quantitative comparison would require a full hydrodynamical calculation. The correlation can in fact be modified by the transport in the hydrodynamic phase. In particular, it was shown [7,44]   are reduced as compared to the ideal hydrodynamics case. At the same time, shear viscosity increases the radial flow. In principle, bulk viscosity reduces the radial flow, reducing the correlation observed in this paper, but the latter effect was estimated to be negligible [44]. Therefore, the measurement we present in this paper is sensitive to the interplay of initial conditions and transport coefficients in the hydrodynamic phase. As such, it poses stringent constraints on hydrodynamic calculations, and it could allow the extraction of the value of average shear viscosity at the LHC.
A study of the relation of the fluctuation in the initial size to the spectra was performed in [45,46] with a full hydrodynamic simulation. It was shown that the event-by-event fluctuations in the Glauber initial conditions lead to fluctuations in the initial size of the system that reflect in fluctuations of the radial flow and hence p T . It is found that the relative p T fluctuations computed with Glauber initial conditions overestimate the data, indicating a strong sensitivity of event-by-event measurements on the initial conditions model. It is also shown that the p T fluctuations are not sensitive to the shear viscosity. The study in [45,46] (fluctuations in p T ), however, does not address the relation between the elliptic and radial flow. It may be expected that the present measurement will also be sensitive to the transport coefficient of the medium.
In a recent series of theoretical studies [47,48,49], it was suggested to use the Principal Component Analysis to study flow fluctuations. It was argued that most of the current methods to study flow do not fully capture the complexity of the initial state. Indeed, the PCA studies revealed the presence of sub-leading flow components (arising from radial geometry excitations), which break the factorization of flow harmonics [47,48], In particular, in [49] it is argued that the sub-leading component of v 2 reflects a non linear mixing with radial flow, which could address the same physics as reported in this paper.
A study of the relation of the fluctuation in the initial size to the spectra was performed in [45,46]   with a full hydrodynamic simulation. It was shown that the event-by-event fluctuations in the Glauber initial conditions lead to fluctuations in the initial size of the system that reflect in fluctuations of the radial flow and hence p T . It is found that the relative p T fluctuations computed with Glauber initial conditions overestimate the data, indicating a strong sensitivity of event-by-event measurements on the initial conditions model. It is also shown that the p T fluctuations are not sensitive to the shear viscosity. The study in [45,46] (fluctuations in p T ), however, does not address the relation between the elliptic and radial flow. It may be expected that the present measurement will also be sensitive to the transport coefficient of the medium.
In a recent series of theoretical studies [47,48,49], it was suggested to use the Principal Component Analysis to study flow fluctuations. It was argued that most of the current methods to study flow do not fully capture the complexity of the initial state. Indeed, the PCA studies revealed the presence of sub-leading flow components (arising from radial geometry excitations), which break the factorization of flow harmonics [47,48]. In particular, in [49] it is argued that the sub-leading component of v 2 reflects a non linear mixing with radial flow, which could be related to the same underlying physics phenomena reported in this paper.
To further understand the observed effect, we studied it in AMPT, a model known to reproduce many of the flow observables measured at the LHC [33]. This model is based on HIJING [50] to describe the initial conditions and on the Zhang's parton cascade [51] to describe the partonic evolution. The string melting configuration, described in [52], is used. To assess the impact of the detector resolution on the q 2 selection, the simulated AMPT events were transported through the ALICE apparatus using the GEANT [34] transport model. The q 2 was computed using either the reconstructed Monte Carlo tracks (q rec 2 ) or the generated primary particles in the same kinematic range (q gen 2 ). The elliptic flow and the transverse momentum distribution are calculated using generated Monte Carlo particles. Since the charged particle multiplicity distribution is different in AMPT and data, the q 2 selection is calibrated in the model as a function of multiplicity. The results are shown in Fig. 14 for the charged hadrons elliptic flow and in Fig. 15 for the transverse momentum distribution of charged hadrons. Using either q rec 2 or q gen 2 does not introduce any significant difference on the effect of the selection. This indicates that detector resolution effects are negligible for the q TPC 2 selection. The V0 detectors, on the other hand, have a coarser azimuthal resolution and are sensitive to fluctuations in the energy deposition of incident 22 Event shape engineering in Pb-Pb collisions at √ s NN = 2.76 TeV ALICE Collaboration particles. However, the study with the relaxed TPC selection discussed in Sect. 4 demonstrates that the properties of the ESE selected events are mostly determined by the average v 2 {SP} value. It is therefore advised that in any comparison of this data to theoretical models the selection in the model is tuned as to reproduce the average change in v 2 {SP} at mid-rapidity.
The p T dependence of the elliptic flow observed in data is not reproduced in AMPT (top panel). This model reproduces however the magnitude of the modification as well as the flatness of the ratio as a function of p T .
The effect of the ESE selection on the p T distribution of charged particles is well reproduced by AMPT below p T = 2 GeV/c, as shown in Fig. 15. However, the magnitude of the effect at intermediate p T (2 < p T < 6 GeV/c) is underestimated in AMPT. As previously observed for the v 2 measurement, a good agreement is observed between the selection based on q gen 2 and q rec 2 .

Conclusions
In summary, the first application of the Event Shape Engineering (ESE) technique to Pb-Pb collisions data measured by ALICE at √ s NN = 2.76 TeV has been presented.
The elliptic flow at mid-rapidity is observed to increase as a function of the q 2 calculated in the central or forward rapidity regions. The modification of the v 2 coefficient as a function of p T is nearly flat below p T = 4 GeV/c, suggesting that this technique allows the selection of a global property of the collision, likely related with the geometry of the participant nucleons in the initial state. In the region above p T > 5 GeV/c a small increase is observed within the large statistical uncertainties, possibly due to a small non-flow contamination. In this transverse momentum range the elliptic flow is believed to be driven by the different path-length traversed in-and out-of-plane by high-p T partons in the deconfined medium, rather than by the hydrodynamic evolution of the system.
The p T -distributions of unidentified hadrons in the p T region (0 < p T < 5 GeV/c) are harder (softer) in event with large−q 2 (small−q 2 ) values.
Identified pions, kaons and protons show a similar behavior with a clear mass ordering in the ratio between the large−q 2 and the unbiased spectra, thus suggesting this effect to be due to a stronger radial flow in such events. Glauber Monte Carlo calculations reveal a correlation between the transverse participant density and the participant eccentricity which could be the origin of this effect. This indicates that at least part of the correlation is generated in the initial state. However, these measurements are also sensitive to the transport coefficients in the hydrodynamic evolution. A quantitative comparison would require a full hydrodynamic calculation and may provide stringent constraints both on shear and bulk viscosity.