Directed flow of charged particles at mid-rapidity relative to the spectator plane in Pb-Pb collisions at $\sqrt{s_\rm{NN}}$=2.76 TeV

The directed flow of charged particles at mid-rapidity is measured in Pb-Pb collisions at $\sqrt{s_\rm{NN}}=2.76$ TeV relative to the collision symmetry plane defined by the spectator nucleons. A negative slope of the rapidity-odd directed flow component with approximately 3 times smaller magnitude than found at the highest RHIC energy is observed. This suggests a smaller longitudinal tilt of the initial system and disfavors the strong fireball rotation predicted for the LHC energies. The rapidity-even directed flow component is measured for the first time with spectators and found to be independent of pseudorapidity with a sign change at transverse momenta $p_{\rm T}$ between $1.2$ and $1.7$ GeV/$c$. Combined with the observation of a vanishing rapidity-even $p_{\rm T}$ shift along the spectator deflection this is strong evidence for dipole-like initial density fluctuations in the overlap zone of the nuclei. Similar trends in the rapidity-even directed flow and the estimate from two-particle correlations at mid-rapidity, which is larger by about a factor of 40, indicate a weak correlation between fluctuating participant and spectator symmetry planes. These observations open new possibilities for investigation of the initial conditions in heavy-ion collisions with spectator nucleons.

The goal of the heavy-ion program at the Large Hadron Collider (LHC) is to explore the properties of deconfined quark-gluon matter.Anisotropic transverse flow is sensitive to the early times of the collision, when the deconfined state of quarks and gluons is expected to dominate the collision dynamics (see reviews [1,2,3] and references therein).A positive (in-plane) elliptic flow was first observed at the Alternating Gradient Synchrotron (AGS) [4,5].A much stronger flow was then measured at the Super Proton Synchrotron (SPS) [6], Relativistic Heavy Ion Collider (RHIC) [7,8,9] and recently at the LHC [10,11,12].Elliptic flow at RHIC and the LHC is reproduced by hydrodynamic model calculations with a low value of the ratio of shear viscosity to entropy density [13,14,15,16].Despite the success of hydrodynamics in describing the equilibrium phase of matter produced in a relativistic heavy-ion collision, there are still large theoretical uncertainties in determining its initial conditions.Significant triangular flow measured recently at RHIC [17,18] and LHC [19,12] energies has demonstrated [20,21] that initial energy fluctuations play an important role in the development of the final momentum-space anisotropy in the distribution of produced particles.
The collision geometry is illustrated in Fig. 1 which depicts (a) the reaction plane and (b) the transverse to the beam plane views of the system produced in the overlap (participant) zone, as well as the projectile and target spectators.Figure 1(a) shows the projectile and target spectators assuming their deflection away from the collision (z) axis along the impact parameter direction (x-axis) in directions opposite to each other.An alternative possibility is discussed in [22] where it is predicted that low momentum projectile and target spectators are deflected towards the center of the colliding system.
The directed flow is characterized by the first harmonic coefficient v 1 in a Fourier decomposition of the particle azimuthal distribution with respect to one of the collision symmetry planes, Ψ, which are illustrated in Fig. 1(b) and discussed in details below Here η = − ln [tan (θ /2)], p T , θ and φ are the particle pseudo-rapidity, transverse momentum, polar and azimuthal angles, respectively.The brackets " ... " indicate an average over measured particles in all recorded events.
For a non-fluctuating nuclear matter distribution, the directed flow in the participant zone develops along the impact parameter direction.The collision symmetry requires the directed flow to be an antisymmetric function of pseudo-rapidity, v odd 1 (η) = −v odd 1 (−η).As illustrated in Fig. 1(b), due to eventby-event fluctuations in the initial energy density of the collision, the participant plane angle (Ψ (1) PP ) defined by the dipole asymmetry of the initial energy density [23,24] and that of projectile (Ψ p SP ) and target (Ψ t SP ) spectators, in which the flow develops, are different from the geometrical reaction plane angle Ψ RP (coincides with the x-axis).As a consequence, the directed flow can develop [23,24,25,26] a rapidity-symmetric component, v even 1 (η) = v even 1 (−η), which does not vanish at mid-rapidity.The slope of v odd 1 as a function of rapidity at the AGS [5,27] and SPS [28,29] energies is mainly driven by the difference between baryon and meson production and the shadowing by the nuclear remnants.At higher (RHIC) energies a zig-zag structure (multiple zero crossing as a function of rapidity) of v odd 1 outside of the nuclear fragmentation regions was predicted as a signature of the deconfined phase transition [30,31].However, the RHIC measurements [32,33,34,35] did not reveal such a structure.The magnitude of the directed flow depends on the amount of baryon stopping in the nuclear overlap zone [36] and these two can be related via realistic model calculations, which makes it an important experimental probe of the initial conditions in a heavy-ion collision.The set of initial conditions assumed in model calculations of v odd 1 at relativistic energies ranges from incomplete baryon stopping [36] with a positive space-momentum correlation to full nucleon stopping with a tilted [31,37] or rotating [38] source of matter produced in the overlap zone of the nuclei.Model calculations generally agree on the negative sign of the v odd 1 slope as a function of pseudorapidity measured at RHIC [32,33,34,35], while expectations differ for the LHC energies.In comparison to the measurement at the highest RHIC energy, the model predictions for v odd 1 at the LHC vary from the same slope but with smaller magnitude [37] to an opposite (positive) slope with significantly larger magnitude [39,38].
The v even 1 estimated from the two-particle azimuthal correlations at mid-rapidity for RHIC [40] (also discussed in [24]) and LHC [41,12,19] energies is in approximate agreement with ideal hydrodynamic model calculations [25,26] for dipole-like [23] energy fluctuations in the overlap zone of the nuclei.Interpretation of the two-particle correlations is complicated due to a possibly large bias from correlations unrelated to the initial geometry (non-flow) and due to the model dependence of the correction procedure for effects of momentum conservation [26].The directed flow measured relative to the spectator deflection is free from such biases and provides a cleaner probe of the initial conditions in a heavy-ion collision.It also allows for a study of the main features of the dipole-like energy fluctuations such as a vanishing transverse momentum shift of the created system along the direction of the spectator deflection.Directed flow and its fluctuations also play an important role in understanding effects due to the strong magnetic field in heavy-ion collisions [23] and interpretation of the observed charge separation relative to the reaction plane [42] in terms of the chiral magnetic effect [43].
In this Letter, we report on the measurement of the charged particle directed flow relative to the deflection of spectator neutrons in Pb-Pb collisions at √ s NN = 2.76 TeV.A sample of about 13 million minimumbias trigger [10] Pb-Pb collisions at √ s NN = 2.76 TeV in the 5-80% centrality range was analyzed.For the most central (0-5%) collisions, the small number of spectators does not allow for a reliable reconstruction of their deflection.Standard ALICE event selection criteria [10] were applied in the analysis.The amplitude measured by the two forward scintillator arrays (VZERO) [44] was used to determine the collision centrality.Charged particles reconstructed in the Time Projection Chamber (TPC) [45] with transverse momentum p T > 0.15 GeV/c and pseudorapidity |η| < 0.8 were selected for the analysis.
The event-by-event deflection of the projectile and target neutron spectators is reconstructed with a pair of Zero Degree Calorimeters (ZDC) [46].Each ZDC has a 2 × 2 segmentation in the transverse plane and is installed on each side, 114 meters from the interaction point covering the |η| > 8.78 (beam rapidity) region.A typical energy measured by both ZDCs for 30-40% centrality class is about 100 TeV [47].The spectator deflection in the transverse plane was quantified with a pair of two-dimensional vectors where "p" ("t") denotes the ZDC on the η > 0 (η < 0) side of the interaction point, E i is the measured signal and n i = (x i , y i ) are the coordinates of the i-th ZDC segment.An asymmetry of 0.1% [48] in energy calibration of two ZDCs as well as an absolute energy scale uncertainty cancel out in Eq. ( 2).An event-by-event correction (recentering) [3] of the Q t,p vectors for their event averages (Q t,p → Q t,p − Q t,p ) is applied as a function of collision centrality to compensate for the run-dependent variation of the LHC beam crossing position and the spread of the collision vertices with respect to its nominal position.Experimental values of the Q t,p event averages for 30-40% centrality class are Q p x(y) ≈ 2.0 (−1.5) mm and Q t x(y) ≈ −1.1 (0.01) mm.The directed flow is then determined with the scalar product method [3,49] from the average of correlation of Q t,p -vector components and that of a unit vector u(p T , η) ≡ (u x , u y ) = (cos φ , sin φ ) defined for charged particles The v odd 1 and v even 1 components of the directed flow relative to the spectator plane (Ψ = Ψ SP in Eq. ( 1)) are calculated from the equations and Equation ( 4) defines the sign of v odd 1 according to the same convention as used at RHIC [32,33] and implies a positive directed flow (or deflection along the positive x-axis direction in Fig. 1(a)) of the projectile spectators (η > 0).
The observed non-zero negative correlations Q t x Q p x and Q t y Q p y [50] indicate deflection of the projectile and target spectators in opposite directions.These correlations are sensitive to a combination of the spectator's directed flow relative to the reaction plane Ψ RP and an additional contribution due to flow of spectators along fluctuating Ψ p SP and Ψ t SP directions (see Fig. 1(b)).The two contributions are not separable within the current experimental technique and should be both considered for theoretical interpretation of the results obtained with Eqs.(3)- (5).Given that a typical transverse deflection of spectators x Q p y and Q t y Q p x in orthogonal directions, which can be non-zero only due to residual detector effects are measured [50] to be less than 5% of the correlations in the aligned directions.The extracted difference between the correlations Q t x Q p x and Q t y Q p y for mid-central collisions is about 10-20% [50], which is mainly due to a different offset of the beam spot from the center of the ZDCs in-plane and perpendicular to the LHC accelerator ring.This asymmetry is the dominant source of systematics in this measurement.The corresponding systematic uncertainty is evaluated from the spread of results calculated with different Q t,p -vector components according to Eq. ( 3) and estimated to be below 20%.The results obtained with Eq. (3) were compared with calculations using the event plane method [3] and are consistent within the statistical precision of the measurement.The variation of the results obtained for the nominal ±10 cm range of the collision vertex along the beam direction from the center of the ALICE detector and for the range reduced to ±7 cm are within 5%.The results with opposite polarity of the magnetic field of the ALICE detector are consistent within 5%.Variation of the results with the collision centrality estimated with the TPC, VZERO, and Silicon Pixel Detectors [46] is less than 5%.Altering the selection criteria for the tracks reconstructed with the TPC resulted in a 3-5% variation of the directed flow results.The systematic error evaluated for each of the sources listed above were added in quadrature to obtain the total systematic uncertainty of the measurement.in Pb-Pb collisions compared to the STAR data [33] for Au-Au collisions at √ s NN = 200 (62.4)GeV downscaled with a factor 0.37 (0.12).
Figure 2(a) shows the charged particle directed flow as a function of pseudorapidity measured in Pb-Pb collisions at √ s NN = 2.76 TeV for 10-20%, 30-40%, and 10-60% centrality classes.The v odd 1 (η) component exhibits a negative slope as a function of pseudorapidity.The v even 1 (η) component is found to be negative and independent of pseudo-rapidity within the statistical and systematic uncertainties of the measurement.The STAR data [33] for v odd 1 in Au-Au collisions at √ s NN = 200 (62) GeV in Fig. 2(c) are downscaled with a factor 0.37 (0.12) which is the value of the ratio of v odd 1 (η) slope at the LHC to that at RHIC energy.Compared to the measurement at the highest RHIC energy presented in Fig. 2(c), v odd 1 (η) has the same sign of the slope and a factor of three smaller magnitude.This is in contrast to the positive slope of v odd 1 (η) expected from the model calculations [39,38] with stronger rotation of the participant zone at the LHC than at RHIC.A smaller value of v odd 1 at the LHC is consistent with the model prediction [37] where a smaller tilt of the participant zone in x-z plane (see Fig. 1(a)) is predicted for the LHC compared to RHIC energies.The ratio of 0.37 (0.12) of v odd 1 slope at the LHC to that in Au-Au collisions at √ s NN = 200 (62) GeV indicates a strong violation of the beam rapidity scaling discussed in [35] by a factor 1.82 (4.55).3) by introducing a p T / p T weight in front of u x and u y .The observed non-zero p x odd / p T shift has a smaller magnitude than v odd 1 , while p x even / p T is zero in a measured pseudorapidity region.A vanishing p x even is consistent with the dipole-like event-by-event fluctuations of the initial energy density in a system with zero net transverse momentum.Disappearance of p x at η ≈ 0 indicates that particles produced at mid-rapidity are not involved in balancing the transverse momentum carried away by spectators. 1 in Pb-Pb collisions compared with the STAR data [33] for Au-Au collisions at √ s NN = 200 (62.4)GeV downscaled with a factor 0.37 (0.12).odd components an average in the |η| < 0.8 range was calculated by taking values at negative η with an opposite sign.Both, v odd 1 and v even 1 have weak centrality dependence.The p x even component is zero at all centralities, while p x odd / p T is a steeper function of centrality than v odd 1 .This suggest that v odd 1 has two contributions.A first contribution has a similar origin as v even 1 due to asymmetric dipole-like initial energy fluctuations.A second contribution grows almost linear from central to peripheral collisions and represents an effect of side-ward collective motion of particles at non-zero rapidity due to expansion of the initially tilted source with p x balancing the transverse momentum of the particles produced at opposite rapidity and in very forward (spectator) regions.The magnitude of v odd 1 measured at the LHC is significantly smaller than at RHIC (see Fig. 3(c)), while the centrality dependence is very similar at the different energies.The p T dependence of v even 1 {Ψ SP } is similar to that of v even 1 {Ψ PP } estimated from the Fourier fits of the two-particle correlations [41,12,19], while the magnitude of v even 1 {Ψ SP } is smaller by a factor of forty [26,51].The latter can be interpreted as a weak (but non-zero) correlation, cos Ψ PP − Ψ SP 1, between the orientation of the participant and spectator collision symmetry planes.
According to hydrodynamic model calculations [52,23,26] the particles with low transverse momentum should flow in the direction opposite to the largest magnitude of the density gradient.This, together with the negative even and odd v 1 {Ψ SP } components measured for particles at mid-rapidity with low transverse momentum (p T 1.2 GeV/c) allows one in fact to determine if spectators deflect away from or towards the center of the system.A detailed theoretical calculation of the correlation between fluctuations in the spectator positions and energy density in the participant zone such as in [22] is required to provide a definitive answer on this question.
In summary, the v odd 1 and v even 1 components of charged particle directed flow at mid-rapidity, |η| < 0.8, are measured relative to the spectator plane for Pb-Pb collisions at √ s NN = 2.76 TeV.The v odd 1 has a negative slope as a function of pseudo-rapidity with a magnitude about three times smaller than at the highest RHIC collision energy.This suggests a smaller tilt of the medium created in the participant zone at the LHC, with insufficient rotation to alter the slope of v odd 1 (η) as predicted in [39,38].As a function of transverse momentum, v odd 1 and v even 1 cross zero at p T ∼ 1.2 − 1.7 GeV/c for semi-central collisions.Disappearance of p x for particles produced close to zero rapidity suggest that they do not play a role in balancing the transverse momentum kick of spectators.The shape of v even 1 (p T ) and a vanishing p x even is consistent with dipole-like fluctuations of the initial energy density in the participant zone.A similar shape but with about forty times larger magnitude was observed for an estimate of v even 1 (p T ) relative to the participant plane from the Fourier fits of the two-particle correlation [41,12].This indicates that fluctuating participant and spectator collision symmetry planes are weakly correlated which is an important experimental input for modeling a not so well constrained initial conditions of a heave-ion collision.Future studies of the directed flow at mid-rapidity using identified particles and extension of the v 1 measurements to forward rapidities should provide a stronger constraint on the effects of initial density fluctuations in the formation of directed flow.

Fig. 1 :
Fig. 1: (color online) Sketch of non-central heavy-ion collision.See text for description of the figure.

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is tiny compared to the ZDC detector position |z ZDC | = 114 m along the beam direction, one can make a rough estimate of the corresponding transverse momentum carried by an individual spectator: p spec T ≈ √ s NN d spec /z ZDC .The measured d spec is about 0.67 (0.92) mm [50] for 5-10% (30-40%) centrality class which yields p spec T ∼ 16 (22) MeV/c.Correlations between Q t

Figure 2 (
Figure 2(b) shows the relative momentum shift p x / p T ≡ p T cos (φ − Ψ SP ) / p T , along the spectator symmetry plane as a function of pseudorapidity.It is obtained based on Eq. (3) by introducing a p T / p T weight in front of u x and u y .The observed non-zero p x odd / p T shift has a smaller magni-

Figures 3 (Fig. 3 :
Figures 3(a) and 3(b) present the charged particle v 1 and p x / p T versus collision centrality.In case of

Figure 4 (odd 1 and v even 1 components change sign around p T ∼ 1 . 2 − 1 . 7
Figure 4(a) presents results for the charged particle directed flow as a function of transverse momentum.The v odd 1 and v even 1 components change sign around p T ∼ 1.2 − 1.7 GeV/c.The observed zero crossing of v even 1 (p T ) is expected for the dipole-like energy fluctuations when momentum of the low p T particles is balanced by that of the high p T particles [23, 24, 25, 26].Compared to the measurements at the highest RHIC energy, in Fig. 4(b), v odd 1 shows a similar trend including the sign change around p T ∼ 1.5 GeV/c in central collisions and negative value at all p T for peripheral collisions.

Fig. 4 :
Fig. 4: (color online) v 1 versus transverse momentum in Pb-Pb collisions at √ s NN = 2.76 TeV.The statistical (systematic) uncertainties are indicated by the error bars (shaded bands).Lines (to guide the eye) represent fits with the third order polynomial.(a) v 1 for 5-80% centrality range.(b) v odd1 in Pb-Pb collisions compared with the STAR data[33] for Au-Au collisions at √ s NN = 200 GeV downscaled with a factor 0.37.