Polarization of $\Lambda$ and $\overline{\Lambda}$ hyperons along the beam direction in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV

The polarization of the $\Lambda$ and $\overline\Lambda$ hyperons along the beam ($z$) direction, $P_{\rm z}$, has been measured in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02TeV recorded with ALICE at the Large Hadron Collider (LHC). The main contribution to $P_{\rm z}$ comes from elliptic flow induced vorticity and can be characterized by the second Fourier sine coefficient $P_{\rm z,s2} = \langle P_{\rm z} \sin(2\varphi - 2 \Psi_{\rm 2}) \rangle$, where $\varphi$ is the hyperon azimuthal emission angle, and $\Psi_{\rm 2}$ is the elliptic flow plane angle. We report the measurement of $P_{\rm z,\,{\rm s2}}$ for different collision centralities, and in the 30-50% centrality interval as a function of the hyperon transverse momentum and rapidity. The $P_{\rm z,\,{\rm s2}}$ is positive similarly as measured by the STAR Collaboration in Au-Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV, with somewhat smaller amplitude in the semi-central collisions. This is the first experimental evidence of a non-zero hyperon $P_{\rm z}$ in Pb-Pb collisions at the LHC. The comparison of the measured $P_{\rm z,\,{\rm s2}}$ with the hydrodynamic model calculations shows sensitivity to the competing contributions from thermal and the recently found shear induced vorticity, as well as to whether the polarization is acquired at the quark-gluon plasma or the hadronic phase.

The system created in high-energy nuclear collisions behaves almost like an ideal fluid [1]. Its evolution is characterized by non-trivial velocity and vorticity fields, resulting in the polarization of the produced particles. In particular, the shear in the initial velocity distributions of the participants in off-center nuclear collisions leads to a non-zero vorticity component and a net particle polarization along the orbital momentum of the colliding nuclei, a phenomenon termed as global polarization [2][3][4]. Recent measurements at RHIC show a significant global polarization of Λ and Λ hyperons in Au-Au collisions at √ s NN = 7.7 -200 GeV with the polarization magnitude of a few to a fraction of a percent, monotonically decreasing with increasing √ s NN [5,6]. The global hyperon polarization measured by the ALICE Collaboration in Pb-Pb collisions at √ s NN = 2.76 and 5.02 TeV [7] was found to be at the per mil level, consistent with zero within experimental uncertainties. The ALICE measurements are also consistent with hydrodynamical model calculations for the LHC energies and empirical estimates based on the collision energy dependence of the directed flow due to the tilted source [4,8,9]. The decrease in the global polarization at midrapidity with collision energy is usually attributed to a decreasing role of the baryon stopping [10] in the initial velocity distributions.
In addition to the vorticity due to the orbital angular momentum of the entire system, other physics processes, such as anisotropic flow, jet energy deposition, and deviation from longitudinal boost invariance of the transverse velocity fields, generate vorticity [8,[11][12][13][14][15] along different directions depending on the location of the fluid elements in the created system. It was predicted that in non-central nucleus-nucleus collisions, the strong elliptic flow would generate a non-zero vorticity component along the beam axis (z) [8,12]. The vorticity and the corresponding polarization exhibits a quadrupole structure in the transverse plane. This polarization, characterized by the second harmonic sine component in the Fourier decomposition of the polarization along the beam axis (P z ) as a function of the particle azimuthal angle (ϕ) relative to the elliptic flow plane Ψ 2 , is evaluated as The sign of P z, s2 determines the phase of the P z modulation in azimuth relative to the elliptic flow plane.
The Λ and Λ polarization along the beam direction was measured by the STAR Collaboration in Au-Au collisions at √ s NN = 200 GeV [16] and compared with the hydrodynamic [12], transport (AMPT) [14,17], and Blast-Wave (BW) [8,16] model calculations. The measured P z, s2 was found to be about 5 times smaller in magnitude and of opposite sign compared to the hydrodynamic and AMPT model predictions. However, the BW model, tuned to spectra, elliptic flow, and azimuthally differential femtoscopic measurements, describes the magnitude and the sign of P z, s2 . Most model calculations estimate the particle polarization from the thermal vorticity [12,17] at the freeze-out surface assuming local thermodynamic equilibrium of the spin degrees of freedom. Unlike hydrodynamic and AMPT calculations, the BW model [8,16] accounts only for the kinematic vorticity associated with the velocity fields without contribution from the temperature gradients and acceleration. It was confirmed by other calculations that the kinematic vorticity alone describes the RHIC results much better than the thermal vorticity [18]. In addition, the chiral kinetic approach with AMPT initial conditions [15], accounting for the transverse vorticity fields due to deviation from longitudinal boost invariance, generates the correct sign for P z, s2 . The difference in the sign of P z, s2 between the experimental data and model calculations based on solely thermal vorticity has been a subject of intense investigations [14,15,[17][18][19].
Recently a possible explanation to the experimentally observed positive P z, s2 at RHIC was proposed based on the additional contribution to polarization from fluid shear [20,21]. The studies in Refs. [22,23] demonstrate that the fluid shear competes with thermal vorticity and contributes with an opposite phase to the azimuthal angle dependence of hyperon spin polarization. Under the assumptions of isothermal (at constant temperature) hadronization or that the hyperons inherit the spin polarization of the constituent strange quark, the effect of shear prevails over thermal vorticity and their combined effect qualitatively explains the experimentally observed azimuthal angle dependence of the hyperon spin polarization in Au-Au collisions at √ s NN = 200 GeV [22,23]. These studies indicate that the longitudinal polarization is very sensitive to the hydrodynamic gradients and the evolution of the spin degrees of freedom through different stages of the evolution of the system created in heavy-ion collisions. The measurement of P z, s2 in Pb-Pb collisions at √ s NN = 5.02 TeV and its comparison with measurements at RHIC as well as theoretical models can provide important insights into the fluid and spin dynamics in heavy-ion collisions.
In this Letter, we report the centrality, transverse momentum (p T ), and rapidity (y H ) dependences of P z, s2 for Λ (Λ) hyperons measured by the ALICE Collaboration in Pb-Pb collisions at √ s NN = 5.02 TeV and compare with the previous STAR measurements in Au-Au collisions at √ s NN = 200 GeV as well as with shear and vorticity based hydrodynamic model calculations for Pb-Pb collisions at √ s NN = 5.02 TeV.
As the spin of a particle cannot be measured directly, the parity violating weak decays of Λ → p + π − and Λ → p + π + in which the momentum of the daughter (anti)proton is correlated with the spin of the hyperon, are used to measure the polarization. The angular distribution of the (anti)proton in the hyperon rest frame is given by [24]: where P H is the polarization vector, α H is the hyperon decay parameter (α Λ = 0.750 ± 0.009, α Λ = −0.758 ± 0.01 [25]),p * p is the unit vector along the (anti)proton momentum in the hyperon rest frame, and θ * p is the angle between the (anti)proton momentum and the polarization vector in the hyperon rest frame. To measure the polarization component along the z direction, θ * p is considered as the polar angle of the (anti)proton momentum in the hyperon rest frame. The polarization P z can be estimated by averaging cos θ * p over all hyperons in all collisions [16] where p T , y H , and ϕ are the transverse momentum, rapidity, and azimuthal angle of the hyperon, respectively. The factor (cos θ * p ) 2 , which equals to 1/3 in the case of an ideal detector, is calculated directly from the data as a function of centrality, p T , and y H and serves as a correction for finite acceptance along the longitudinal direction.
The data used in this analysis were collected by ALICE [26,27] at the LHC in 2018 for Pb-Pb collisions at √ s NN = 5.02 TeV. Two datasets, corresponding to positive and negative magnetic field polarities, are considered for this measurement. The centrality is determined using the sum of the charge deposited in the V0A (2.8 < η < 5.1) and the V0C (−3.7 < η < −1.7) scintillator arrays, denoted as the V0M centrality [28]. The event selection is based on the trigger criteria and quality of the event vertex reconstruction using the Time Projection Chamber (TPC) [29] and the Inner Tracking System (ITS) [30]. Events that pass central, semi-central, or minimum-bias trigger criteria with a z-component of the reconstructed event vertex (V z ) within ±10 cm are selected. To suppress the pile-up of multiple collisions in the TPC drift volume, events with a TPC multiplicity beyond 5 times the width of its distribution at any V0M centrality are rejected. A similar cut on the ITS centrality for the corresponding V0M centrality is applied to get rid of additional outliers in the sample. In total about 270M events are selected for the polarization measurement. The centrality dependence of P z, s2 is studied with 10% centrality intervals whereas the p T and y H dependence is studied in the semi-central (30-50%) collisions where elliptic flow is the largest.
The Λ and Λ hyperons are reconstructed inside the TPC using the decay topology of Λ → p + π − and Λ → p + π + (64% branching ratio) [31] as described in Refs. [27,32]. The daughter tracks are assigned the identity of a pion or a (anti)proton based on the charge and particle identification using the specific energy loss (dE/dx) measurement in the TPC. The tracks of the daughter pions and (anti)protons are selected within the pseudorapidity range of |η| < 0.8 inside the TPC. Topological cuts such as distance of closest approach (DCA) of the Λ and Λ candidates to the primary vertex (< 1.5 cm), DCA of the daughter tracks to the primary vertex (> 0.05 cm), DCA between the daughter tracks (< 0.5 cm), and cosine of the pointing angle which is the angle between the momentum direction of the hyperon and the direction from the primary vertex to the decay point (> 0.997) are used to reduce the combinatorial background contribution to the invariant mass spectrum. The Λ and Λ candidates having 1.103 < M inv < 1.129 GeV/c 2 with p T > 0.5 GeV/c and |y H | < 0.5 are considered in this measurement.
The event-plane method is used for the polarization measurement [33]. The second harmonic event plane is reconstructed using the TPC tracks, and signals in the V0A and V0C scintillators. The X 2 and Y 2 components of the second harmonic flow vector are given by where, in case of the TPC, w i = 1, ϕ i is the azimuthal angle of track i, and the sum runs over all the tracks used in the flow vector construction. In the case of V0A and V0C, which consist of four concentric rings with each ring divided into eight segments, ϕ i is the azimuthal angle of the centre of the i-th segment and w i is the measured signal proportional to the number of particles detected in that segment.   The TPC flow vectors are reconstructed using tracks in the positive (0.1 < η < 0.8) and negative (−0.8 < η < −0.1) pseudorapidity regions and transverse momentum within 0.2 < p T < 3.0 GeV/c. The flow vector for the V0A or the V0C is constructed by averaging the flow vectors of four rings using the energy deposited in each ring as a weight. Because of the imperfect detector acceptance, varying beam conditions, the averages X 2 and Y 2 might deviate from zero. To compensate these variations, flow vectors are re-centered [33] run-by-run as a function of the event centrality, event vertex position (V x , V y , V z ), and the time the event was taken during the run: The second harmonic event-plane angle (Ψ EP 2 ) is constructed from the re-centered flow vector components as The P z, s2 is experimentally measured using Ψ EP 2 via where ϕ is the azimuthal emission angle of the particle, and Ψ EP 2 is the reconstructed second harmonic event plane angle. The R(Ψ EP 2 ) is the event plane resolution correction [33]. The P z, s2 is estimated using the invariant mass method [16] by calculating Q = cos θ * p sin(2ϕ − 2Ψ EP 2 ) for all hyperon candidates as a function of the invariant mass and fitting it with the expression: where f S and f BG = 1 − f S are the signal and background fraction, respectively, of the Λ and Λ candidates estimated from the invariant mass yields. The constant Q S estimates the signal and Q BG (M inv ) estimates the possible contribution from the combinatorial background of Λ and Λ hyperons towards the measured polarization. By default, the results are obtained using the assumption of zero background contribution (Q BG (M inv ) = 0) to the hyperon polarization. The results obtained with the assumption of Q BG (M inv ) being a linear function of M inv are found to be consistent with the default case indicating the measured polarization is not sensitive to this assumption. Figure 1 shows an example of the fit to the invariant mass dependence of Q(M inv ) using Eq. 8 for Λ in the 30-40% centrality class. The P z, s2 is obtained from Q S after accounting for the finite detector acceptance ( (cos θ * p ) 2 ), event-plane resolution correction, and scaling it with the hyperon decay constant (α H ).
The correction for the resolution of the second order event planes reconstructed in the TPC, V0A, and V0C detectors are estimated using the three-subevent method with the set of [TPC, V0A, V0C] and [V0, TPC-left (−0.8 < η < −0.1), TPC-right (0.1 < η < 0.8)] event planes [33]. For mid-central collisions, the event-plane resolution correction peaks at ∼ 0.88 for the TPC and ∼ 0.84 for the combined V0A and V0C detectors. The results obtained using the event planes reconstructed in the TPC and V0 detectors are found to be consistent with each other and are combined to reduce the statistical uncertainty considering the correlations between the event planes reconstructed in two detectors. The P z, s2 measured for Λ and Λ hyperons are consistent with each other as expected for the polarization due to the elliptic flow induced vorticity and combined to calculate the average hyperon polarization along the beam direction. A large fraction of the measured Λ and Λ hyperons originate from the decay of heavier resonances. In Ref. [34] it was shown that under the assumption of similar vorticity induced polarization for all final-state particles, the effect of feed-down is small, of the order of 15%. Similar to the previous STAR measurement [16], this measurement is not corrected for this effect.
The systematic uncertainties of this measurement are evaluated by varying the criteria for the selection of the events, hyperon daughters and topology of the decay, assumptions on the possible contributions from the Λ and Λ background towards the measured polarization, the p T dependent reconstruction efficiency, and comparing results obtained with different magnetic field orientations. The efficiency is estimated from a Monte Carlo event generator HIJING [35] by transporting the generated particles through GEANT3 [36] simulated detector response and performing track reconstruction in the ALICE reconstruction framework. The effect of the efficiency dependence on the hyperon transverse momentum is found to be negligible. The differences between the results estimated with the default and varied parameters, if found statistically significant from the Barlow criterion [37], are considered as a source of systematic uncertainty. The Barlow criterion is applied for each interval of centrality, p T , and y H for which the final polarization results are presented. If the Barlow criterion passes for more than 25% of the total intervals, the contribution of that particular systematic source is included in the measurement uncertainty. The contributions from the different sources are added in quadrature to estimate the total systematic uncertainty.
The centrality, p T , and y H dependences of P z, s2 in Pb-Pb collisions at √ s NN = 5.02 TeV are shown in Figs. 2, 3, and 4. The P z, s2 decreases towards more central collisions, similar to the elliptic flow. For centralities larger than 60%, the large uncertainties prevent a firm conclusion on its centrality dependence. The P z, s2 also shows an increase with p T up to p T ≈ 2.0 GeV/c in the 30-50% centrality interval. For higher p T (p T > 2.0 GeV/c), the P z, s2 is consistent with being constant but the uncertainty in the measurement does not allow for a strong conclusion. The ALICE results are compared with the STAR measurements in Au-Au collisions at √ s NN = 200 GeV [16] in Figs. 2 and 3. As the STAR results were obtained with α H = 0.642 whereas the ALICE measurement uses updated values α H = 0.750 (Λ) and −0.758 (Λ), the STAR results are rescaled with a factor 0.856 for a proper comparison. Figure 2 indicates that the hyperon polarization in Pb-Pb collisions at √ s NN = 5.02 TeV is similar in magnitude for the central collisions with somewhat smaller value in the semi-central collisions compared to the top RHIC energy. The latter seems to originate at lower transverse momenta (p T < 2.0 GeV/c), where P z, s2 at the LHC is smaller than that at the top RHIC energy in semi-central collisions as shown in Fig. 3. The P z, s2 does not exhibit a significant dependence on rapidity as shown in Fig. 4.  [38] for Λ and strange quark for Pb-Pb collisions at √ s NN = 5.02 TeV using the approach described in Ref. [23] are shown by dash-dotted lines.   [38] for Λ and strange quark for Pb-Pb collisions at √ s NN = 5.02 TeV in the 30-50% centrality interval using the approach described in Ref. [23] are shown by dash-dotted lines.
The comparison between the ALICE results and the P z, s2 values estimated from the fluid shear and thermal vorticity in a hydrodynamic model following the scheme used in Ref. [23] is shown in Figs. 2, 3, and 4. The 3+1 D hydrodynamical model MUSIC [39,40] with AMPT initial conditions [41,42], tuned to describe the dN ch /dη [43], p T spectra [44], and v 2 (p T ) of pions, kaons, and protons [32,45] [38] for Λ and strange quark for Pb-Pb collisions at √ s NN = 5.02 in the 30-50% centrality interval using the approach described in Ref. [23] are shown by dash-dotted lines.
the polarization is calculated for the Λ and Λ at the freeze-out using hyperon mass for the mass of the spin carrier, the effect of thermal vorticity dominates over the shear induced polarization and total P z, s2 shows a negative sign. However, considering the constituent strange quark as the spin carrier and the hyperons inheriting the spin polarization of the strange quark at hadronization, the effect of fluid shear prevails over thermal vorticity and generates the correct sign for resulting P z, s2 as shown in Figs. 2, 3, and 4. In both cases, the effect of hadronic scatterings on the hyperon spin polarization is not considered. Note that theoretical models, including the spin degrees of freedom consistently through all the stages of heavy-ion collisions are not yet well developed [46,47]. Comparison of the experimental results with the two scenarios discussed here provides a qualitative idea about the possible consequences of the different assumptions used to estimate hyperon polarization from the velocity and temperature gradients generated by hydrodynamic or transport models [23].
In summary, the polarization of Λ and Λ hyperons along the beam direction has been measured in Pb-Pb collisions at √ s NN = 5.02 TeV using the ALICE detector at the LHC. The polarization exhibits a clear second harmonic sine modulation as expected due to elliptic flow. This is the first experimental evidence of a significant z-component of the hyperon polarization due to elliptic flow induced vorticity at the LHC. The P z, s2 measured in Pb-Pb collisions at √ s NN = 5.02 TeV is of similar magnitude as the one measured by the STAR Collaboration in Au-Au collisions at √ s NN = 200 GeV. No significant dependence of P z, s2 on the rapidity is observed. The sign of the P z, s2 is positive at both RHIC and the LHC and in disagreement with hydrodynamic and AMPT models estimations accounting only for the thermal vorticity. The introduction of shear induced polarization [22,23] along with additional assumptions on the hadronization temperature or mass of the spin carrier reproduces the experimentally observed positive P z, s2 at RHIC and the LHC energies. These studies indicate that longitudinal polarization is sensitive to the hydrodynamic gradients as well as the dynamics of the spin degrees of freedom through the different stages of the evolution of the system created in heavy-ion collisions. For a quantitative data to model comparison, a detailed theoretical understanding of the quark spin polarization in the quarkgluon plasma (QGP), spin transfer at the hadronization, and the effect of hadronic scattering on the spin polarization are required. The upcoming Run 3 at the LHC will provide much larger data samples for more differential and precision measurements of local and global hyperon polarization and provide further constraints to the models aiming to explain the vorticity and the particle polarization in heavy-ion collisions.

Acknowledgements
The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The