Evidence of spin-orbital angular momentum interactions in relativistic heavy-ion collisions

The first evidence of spin alignment of vector mesons ($K^{*0}$ and $\phi$) in heavy-ion collisions at the Large Hadron Collider (LHC) is reported. The spin density matrix element $\rho_{00}$ is measured at midrapidity ($|y|<$ 0.5) in Pb-Pb collisions at a center-of-mass energy ($\sqrt{s_{\rm NN}}$) of 2.76 TeV with the ALICE detector. $\rho_{00}$ values are found to be less than 1/3 (1/3 implies no spin alignment) at low transverse momentum ($p_{\rm T}<$ 2 GeV/$c$) for $K^{*0}$ and $\phi$ at a level of 3$\sigma$ and 2$\sigma$, respectively. No significant spin alignment is observed for the $K^0_S$ meson (spin = 0) in Pb-Pb collisions and for the vector mesons in $pp$ collisions. The measured spin alignment is unexpectedly large but qualitatively consistent with the expectation from models which attribute it to a polarization of quarks in the presence of angular momentum in heavy-ion collisions and a subsequent hadronization by the process of recombination.

The direction of the angular momentum in non-central heavy-ion collisions is perpendicular to the reaction plane (subtended by the beam axis and impact parameter) [12]. The spin-orbit interaction is expected to lead to spin alignment with respect to the reaction plane (RP). The reaction plane orientation cannot be measured directly, but is estimated from the final state distributions of particles. This experimentally measured plane is called the event plane (EP) [17]. The deviation of the EP with respect to the RP is corrected using the EP resolution (R) and observed ρ obs 00 [18], ρ 00 = 1 3 + ρ obs 00 − 1 3 There are specific qualitative predictions for the spin alignment effect [13]: (a) ρ 00 > 1/3 if the hadronization of a polarized parton proceeds via fragmentation and less than 1/3 for hadronization via recombination, (b) ρ 00 is expected to have a smaller deviation from 1/3 for both central (impact parameter 3 fm) and peripheral (impact parameter 11 fm) heavy-ion collisions, and a maximum deviation for mid-central collisions, where the angular momentum is also maximal, (c) the ρ 00 value is expected to have maximum deviation from 1/3 at low p T and reach the value of 1/3 at high p T in the recombination scenario, and (d) the effect is expected to be larger for K * 0 compared to φ due to their constituent quark composition. The initial large magnetic field might also affect the ρ 00 values [15]. This leads to ρ 00 > 1/3 for neutral and ρ 00 < 1/3 for charged vector mesons. Hence magnetic field and angular momentum could have opposite effects on electrically neutral K * 0 , φ . All of these features are probed for K * 0 and φ mesons in Pb-Pb collisions presented in this letter. As a cross check, a control measurement is carried out using pp collisions, which do not possess large initial angular momentum, and the same analysis is done in Pb-Pb collisions for K 0 S meson, which has zero spin. In addition, the measurements are carried out by randomizing the directions of the event (RndEP) and production planes (RndPP).
There are three main sources of systematic uncertainties in the measurements of the angular distribution of vector meson decays. (a) Meson yield extraction: this contribution is estimated by varying the fit ranges for the yield extraction, the normalization range for the signal+background and background invariant mass distributions, the procedure to integrate the signal function to get the yields, and by leaving the width of the resonance peak free or keeping it fixed to the PDG value as discussed in Ref. [25,26]. The uncertainties for ρ 00 is at a level of 12(8)% at the lowest p T and decrease with p T to 4(3)% at the highest p T studied for the K * 0 (φ ). (b) Track selection: this contribution includes variations of the selection on the distance of closest approach to the collision vertex, the number of crossed pad rows in the TPC [23], the ratio of found clusters to the expected clusters, and the quality of the track fit. The systematic uncertainties for ρ 00 are 14(6)% at the lowest p T and about 11(5)% at the highest p T for K * 0 (φ ). (c) Particle identification: this is evaluated by varying the particle identification criteria related to the TPC and TOF detectors. The corresponding uncertainty is 5(3)% at the lowest p T and about 4(4.5)% at the highest p T studied for K * 0 (φ ). Systematic uncertainties due to different variations are considered as uncorrelated and the total systematic uncertainty on ρ 00 is obtained by adding all the contributions in quadrature. Several consistency checks are carried out and details can be found in the Supplemental   Material B. The final measurement is reported for the average yield of particles (K * 0 ) and anti-particles (K * 0 ) as results for K * 0 and K * 0 were consistent. Figure 2 shows the measured ρ 00 as a function of p T for K * 0 and φ mesons in pp collisions and Pb-Pb collisions, along with the measurements for K 0 S in Pb-Pb collisions. In mid-central (10-50%) Pb-Pb collisions, ρ 00 is below 1/3 at the lowest measured p T and increases to 1/3 within uncertainties for p T > 2 GeV/c. At low p T , the central value of ρ 00 is smaller for K * 0 than for φ , although the results are compatible within uncertainties. In pp collisions, ρ 00 is independent of p T and equal to 1/3 within uncertainties. For the spin zero hadron K 0 S , ρ 00 is consistent with 1/3 within uncertainties in Pb-Pb collisions. The results with random event plane directions are also compatible with no spin alignment for the studied p T range, except for the smallest p T bin, where ρ 00 less than 1/3 but still larger than for EP and PP measurements. The results for the random production plane (the momentum vector direction of each vector meson is randomized) are similar to RndEP measurements. These results indicate that a spin alignment is present at lower p T , which is a qualitatively consistent with predictions [13].  Table B.1 of Supplemental Material B). In the lowest p T range, ρ 00 shows maximum deviation from 1/3 for intermediate centrality and approaches 1/3 for both central and peripheral collisions. This centrality dependence is qualitatively consistent with the dependence of the initial angular momentum on impact parameter in heavy-ion collisions [4]. At higher p T , ρ 00 is consistent with 1/3 for all centrality classes. For the low-p T measurements in 10-30% (20-40% for φ meson w.r.t. PP) mid-central Pb-Pb collisions, the maximum deviations of ρ 00 from 1/3 with respect to the PP (EP) are 3.2 (2.6) σ and 2.1 (1.9) σ for K * 0 and φ mesons, respectively. The errors (σ ) are calculated by adding statistical and systematic uncertainties in quadrature.
The relation between the ρ 00 values with respect to different quantization axes can be expressed using Eq. 2 and calculating the corresponding factor R. This gives ∆ρ 00 (RndEP) = ∆ρ 00 (EP) × 1 4 (R = 0 for random plane) and ∆ρ 00 (PP) = ∆ρ 00 (EP) × 1+3v 2 4 (R = v 2 for production plane, where v 2 is the second Fourier coefficient of the azimuthal distribution of produced particles relative to the event plane angle).
Here ∆ρ 00 = ρ 00 -1/3. This is further confirmed using a toy model simulation with the PYTHIA 8.2 In the past, spin alignment measurements in e + e − [30][31][32], hadron-proton [33] and nucleon-nucleus collisions [34] were carried out to understand the role of spin in the dynamics of particle production, finding ρ 00 > 1/3 and off-diagonal elements close to zero with respect to the PP. For pp collisions at √ s = 13 TeV, we find ρ 00 ∼ 1/3 within the studied p T range (see Fig. 2). New preliminary results from RHIC have found deviations of ρ 00 from 1/3 indicating spin alignment for vector mesons at lower √ s NN [35,36]. The ρ 00 for φ mesons in mid-central Pb-Pb collisions at √ s NN = 2.76 TeV is less than 1/3 while the preliminary finding for mid-central Au-Au collisions at √ s NN = 200 GeV is ρ 00 greater than 1/3. The ρ 00 > 1/3 for φ mesons has been interpreted as evidence for a coherent φ meson field [37]. Similar conclusions cannot be easily applied to K * 0 as it consists of valence quarks of unequal mass (s andd), which makes it impossible to separate the effects of vorticity and due to electromangetic and mesonic fields. Significant polarization of Λ baryons (spin = 1/2) was reported at low RHIC energies. The polarization is found to decrease with increasing √ s NN [38,39]. At the LHC, the global polarization for Λ baryon is compatible with zero within uncertainties (P Λ (%) = 0.01 ± 0.06 ± 0.03) [40]. The spin alignment for vector mesons in heavy ion collisions could have contributions from angular momentum [12,13], electromagnetic fields [15] and mesonic fields [37]. While no quantitative theoretical calculation for vector meson polarization at LHC energies exists, the expected order of magnitude can be estimated and the measurements for vector mesons and hyperons can be related in a model dependent way. Considering only the angular momentum contribution and recombination as the process of hadronization [13], the ρ 00 of vector mesons are related to quark polarization as ρ 00 = 1−P q Pq 3+P q Pq where P q and Pq are quark and anti-quark polarization, respectively. Assuming P u = Pū = P d = Pd and P s = Ps, the measured p T integrated ρ 00 values for K * 0 and φ mesons in 10-50% Pb-Pb collisions could translate to light quark polarization of ∼0.8 and strange quark polarization of ∼0.2. Using a thermal and nonrelativistic approach as discussed in [41], vorticity (ω) and temperature (T ) are related to ρ 00 and vector meson polarization (P V ) as ρ 00 ≃ 1 3 1 − (ω/T ) 2 3 and P V ≃ (2ω/3T ), respectively. Also in this approach, the measured ρ 00 for K * 0 would correspond to K * 0 polarization of ∼0.6 and the ρ 00 for φ mesons would give φ meson polarization of ∼0.3.
In the recombination model, Λ polarization depends linearly on quark polarization whereas vector meson polarization depends quadratically on it. One would therefore expect the polarization for K * 0 to be of the same order or smaller than the one measured for the Λ at LHC [40], i.e. vanishing small (O( 0.01% )) rather than order 1. The large effect observed for the ρ 00 in mid-central Pb-Pb collisions at low p T is therefore puzzling. This result should stimulate further theoretical work in order to study which effects could make such a huge difference between Λ and K * 0 polarization. Possible reasons may include the transfer of the quark polarization to the hadrons (baryon vs. meson), details of the hadronization mechanism (recombination vs. fragmentation), re-scattering, regeneration, and possibly the lifetime and mass of the relevant hadron. Moreover, the vector mesons are predominantly directly produced whereas the hyperons have large contributions from resonance decays.
In conclusion, for the first time, evidence has been found for a significant spin alignment of vector mesons in heavy-ion collisions. The effect is strongest at low p T with respect to a vector perpendicular to the reaction plane and for mid-central (10-50%) collisions. These observations are qualitatively consistent with expectations from the effect of large initial angular momentum in non-central heavy-ion collisions, which leads to quark polarization via spin-orbit coupling, subsequently transferred to hadronic degrees of freedom by hadronization via recombination. However, the measured spin alignment is surprisingly large compared to the polarization measured for Λ hyperons where, in addition, a strong decrease in polarization with √ s NN is observed. In future measurements, the difference in the polarization of K * ± and K * o , due to their difference in magnetic moment, would be directly sensitive to the effect of the large initial magnetic field produced in heavy-ion collisions.

B Supplemental material B.1 Angular distribution of the decay products of the vector meson
The complete angular distribution of the decay products of the vector meson is expressed as, dN d cos θ * dϕ * ∝ [cos 2 θ * ρ 00 + sin 2 θ (ρ 11 + ρ −1−1 )/2 − sin 2θ (cos φ * Reρ 10 − sin φ * Imρ 10 )/ √ 2 The angle denoted here as θ * is that made by one of the decay daughters in the rest frame of the vector meson with respect to the quantization axis and ϕ * is the corresponding azimuthal angle. This expression is obtained by applying parity symmetry of QCD, the unit trace condition, and integrating over the azimuthal angle.

B.2 Consistency checks
In order to verify the measurements, several consistency checks are carried out. Specifically, the yields of vector mesons are summed over cosθ * bins for each p T interval to obtain the p T distributions; these are found to be consistent within the statistical uncertainties with the published p T distributions in Pb-Pb collisions [25,26]. Similarly a closure test (comparison between generated and reconstructed angular distribution) is carried out for the Monte Carlo (MC) data which is used to obtain the reconstruction efficiencies for the mesons. Two different event generators are used to determine the reconstruction efficiency and the results are consistent. The effect of the shape of the p T distributions in the MC simulations is studied in detail and the impact on the ρ 00 measurement is found to be small. The dependence of the reconstruction efficiency for a cos θ * range on the azimuthal angle of vector meson (φ V ) relative to the event plane angle (Ψ) is also studied. The reconstruction efficiencies obtained in a cos θ * range by integrating over φ V − Ψ are similar to the efficiency obtained by averaging over the φ V − Ψ bins. Data samples with two different magnetic field polarities in the experiment are separately analyzed and the cos θ * distributions are found to be consistent. In addition, the analysis is performed separately for positive (0 < y < 0.5) and negative (-0.5 < y < 0) rapidity and also for K * 0 versus K * 0 ; the different samples are also consistent.

B.3 Analytical relation between EP and PP
The relation between measured ρ 00 with respect to two different frames of references is where frame A is obtained by rotating frame B by angle ψ. Averaging over angle ψ gives, The transition from the EP to PP is obtained by taking into account the elliptic flow of the vector meson which leads to cos 2ψ = 1 2π

B.4 Toy model simulation to understand the relation between EP, PP and RndEP
The spin density matrix element ρ 00 for vector mesons is measured with respect to EP, PP and RndEP. The measured ρ 00 values for K * 0 at 0.8 < p T < 1.2 GeV/c with respect to different planes in heavy-ion collisions has the following ordering ρ 00 (EP) < ρ 00 (PP) < ρ 00 (RndEP). To understand this ordering a toy model simulation is carried out by using PYTHIA (version 8.2) event generator [29] which has no azimuthal anisotropy and spin alignment.
For this study we have taken event plane angle as zero, which corresponds impact parameter along x-axis. In order to find the relations between different frames, v 2 (0.15 ± 0.06, value expected for hadrons with mass similar to K * 0 in Pb-Pb collisions at √ s NN = 2.76 TeV [22]) is introduced to K * 0 by appropriate rotation of its momentum in azimuthal plane. The modified angle in azimuthal plane is calculated by solving the following equation where φ 0 is azimuthal angle of a K * 0 in absence of v 2 and for a given value of v 2 , φ 0 transforms to φ . Then spin alignment (ρ 00 = 0.125, same as measured in data) is introduced with respect to event plane by rotating the momentum of decay daughters in K * 0 rest frame by solving Here θ * 0 is the angle made by the decay daughter of K * 0 with the quantization axis in absence of spin alignment. θ * 0 transforms to θ * to introduce a given input value of ρ 00 . In this study we assume that the φ * is remain fixed during the rotation. With these modifications, calculations as in the experimental data is carried out. The results are shown in Fig. B.1 for two cases, with and without v 2 . The result corresponding to event plane is correctly retrieved in the model. The model results for v 2 = 0, are same for production plane and random event plane. However with v 2 = 0.15, the ρ 00 (PP) value is lower and closer to data for PP. The toy model reproduces the hierarchy observed in the ρ 00 values for various planes as observed in data. The physical picture is that spin alignment with respect to the event plane is coupled to that in the production plane through the elliptic flow of the system. The ρ 00 (RndEP) is lower than 1/3 as the quantization axis is always perpendicular to the beam axis, resulting in a residual effect. If the quantization axis is random in 3 dimension, then the residual effect is not present and the ρ 00 value is consistent with 1/3.