Global polarization of $\Xi$ and $\Omega$ hyperons in Au+Au collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV

Global polarization of $\Xi$ and $\Omega$ hyperons has been measured for the first time in Au+Au collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV. The measurements of the $\Xi^-$ and $\bar{\Xi}^+$ hyperon polarization have been performed by two independent methods, via analysis of the angular distribution of the daughter particles in the parity violating weak decay $\Xi\rightarrow\Lambda+\pi$, as well as by measuring the polarization of the daughter $\Lambda$-hyperon, polarized via polarization transfer from its parent. The polarization, obtained by combining the results from the two methods and averaged over $\Xi^-$ and $\bar{\Xi}^+$, is measured to be $\langle P_\Xi \rangle = 0.47\pm0.10~({\rm stat.})\pm0.23~({\rm syst.})\,\%$ for the collision centrality 20\%-80\%. The $\langle P_\Xi \rangle$ is found to be slightly larger than the inclusive $\Lambda$ polarization and in reasonable agreement with a multi-phase transport model (AMPT). The $\langle P_\Xi \rangle$ is found to follow the centrality dependence of the vorticity predicted in the model, increasing toward more peripheral collisions. The global polarization of $\Omega$, $\langle P_\Omega \rangle = 1.11\pm0.87~({\rm stat.})\pm1.97~({\rm syst.})\,\%$ was obtained by measuring the polarization of daughter $\Lambda$ in the decay $\Omega \rightarrow \Lambda + K$, assuming the polarization transfer factor $C_{\Omega\Lambda}=1$.

(Dated: December 29, 2020) Global polarization of Ξ and Ω hyperons has been measured for the first time in Au+Au collisions at √ s N N = 200 GeV. The measurements of the Ξ − andΞ + hyperon polarization have been performed by two independent methods, via analysis of the angular distribution of the daughter particles in the parity violating weak decay Ξ → Λ + π, as well as by measuring the polarization of the daughter Λ-hyperon, polarized via polarization transfer from its parent. The polarization, obtained by combining the results from the two methods and averaged over Ξ − and Ξ + , is measured to be PΞ = 0.47 ± 0.10 (stat.) ± 0.23 (syst.) % for the collision centrality 20%-80%. The PΞ is found to be slightly larger than the inclusive Λ polarization and in reasonable agreement with a multi-phase transport model (AMPT). The PΞ is found to follow the centrality dependence of the vorticity predicted in the model, increasing toward more peripheral collisions. The global polarization of Ω, PΩ = 1.11 ± 0.87 (stat.) ± 1.97 (syst.) % was obtained by measuring the polarization of daughter Λ in the decay Ω → Λ + K, assuming the polarization transfer factor CΩΛ = 1. The phenomenon of global polarization in heavy-ion collisions arises from the partial conversion of the orbital angular momentum of colliding nuclei into the spin angular momentum of the particles produced in the collision [1][2][3][4]. As a result, these particles become globally polarized along the direction of the initial orbital momentum of the nuclei. Global polarization was first observed by the STAR Collaboration in the beam energy scan Au+Au collisions [5] and was later confirmed, to better precision, in the analysis of the 200 GeV data with high statistics [6]. Assuming local thermal equilibrium, the polarization of the produced particles is determined by the local thermal vorticity of the fluid [3]. In the nonrelativistic limit (for hyperons m H T , where T is the temperature), the polarization of the particles is given by [7]: where s is the spin of the particle, s is the mean spin vector, and ω = 1 2 ∇×v is the local vorticity of the fluid velocity field. Averaged over the entire system volume, the vorticity direction should coincide with the direction of the system orbital momentum.
Following from Eq. 1, all particles, as well as antiparticles of the same spin should have the same polarization. A difference could arise from effects of the initial magnetic field [7], from the fact that different particles are produced at different times or regions as the system freezes out [8], or through meson-baryon interactions [9]. Therefore, to establish the global nature of the polarization, it is important to measure the polarization for different particles, and if possible, particles of different spins. In order to study the possible contribution from the initial magnetic field, the polarization measurement with particles of different magnetic moment would provide additional information. Thus far, only Λ andΛ polarizations have been measured [5,6,10], and they differ by a couple of standard deviations at most, with available statistics.
In this paper we present the first measurements of the global polarization of spin s = 1/2 Ξ − andΞ + hyperons, as well as spin s = 3/2 Ω hyperons in Au+Au collisions at √ s NN =200 GeV.
Hyperon weak decays present the most straightforward possibility for measuring the polarization of the produced particles [11]. In parity-violating weak decays the daughter particle distribution in the rest frame of the hyperon directly depends on the hyperon polarization: where α H is the hyperon decay parameter, P * H is the hyperon polarization, andp * B is the unit vector in the direction of the daughter baryon momentum, both in the parent rest frame denoted by an asterisk.
Ξ − (Ξ + ) hyperon decay happens in two steps: Ξ − → Λ + π − with subsequent decay Λ → p + π − . If Ξ − is polarized, its polarization is partially transferred to the daughter Λ. Both steps in such a cascade decay are parity violating and thus can be used for an independent measurement of the polarization of Ξ − (Ξ + ).
The polarization of the daughter baryon in a weak decay of a spin 1/2 hyperon is described by the Lee-Yang formula [12][13][14] in terms of the three parameters α (parity violating part), β (violation of the time reversal symmetry), and γ (satisfying α 2 +β 2 +γ 2 = 1). For a particular case of Ξ → Λ + π decay it reads: wherep * Λ is the unit vector of the Λ momentum in the Ξ rest frame. Averaging over the angular distribution of the Λ in the rest frame of the Ξ given by Eq. 2 yields Using the measured value for the γ Ξ parameter [14,15], the polarization transfer coefficient for Ξ − to Λ decay is: The polarization of the daughter baryon in a two particle decay of spin 3/2 hyperon, Ω → Λ + K, is also described by three parameters α Ω , β Ω , and γ Ω [16]. The decay parameter α Ω , determines the angular distribution of Λ in the Ω rest frame and is measured to be small [15]: α Ω = 0.0157±0.0021; this makes it practically impossible to measure the Ω polarization via analysis of the daughter Λ angular distribution. The polarization transfer in this case is determined by the γ Ω parameter via [16][17][18]: The time-reversal violation parameter β Ω is expected to be small. This combined with the constraint that α 2 + β 2 +γ 2 = 1 limits the unmeasured parameter to γ Ω ≈ ±1, resulting in a polarization transfer Our analysis is based on the data of Au+Au collisions at √ s N N = 200 GeV collected in 2010, 2011, 2014, and 2016 by the STAR detector. Charged-particle tracks were measured in the time projection chamber (TPC) [19], which covers the full azimuth and a pseudorapidity range of |η| < 1. The collision vertices were reconstructed using the measured charged-particle tracks and were required to be within 30 cm relative to the TPC center in the beam direction for the 2010 and 2011 datasets to ensure a good acceptance of reconstructed tracks. The narrower vertex selection to be within 6 cm was applied in the 2014 and 2016 data due to online trigger requirement for the Heavy Flavor Tracker installed prior to 2014 data taking. The vertex in the radial direction relative to the beam center was also required to be within 2 cm to reject background from collisions with beam pipe. Additionally, the difference in the vertex positions along the beam direction from the vertex position detectors (VPD) [20] located at forward and backward pseudorapidities (4.24 < |η| < 5.1) was required to be less than 3 cm to suppress pileup events in which more than one heavy-ion collision occurred. These selection criteria yielded about 180 (350) million minimum bias (MB) events for the 2010 (2011) dataset, 1 billion MB events for the 2014 dataset, and 1.5 billion MB events for the 2016 dataset. The MB trigger requires hits of both VPDs and the zero-degree calorimeters (ZDCs) [21], which detect spectator neutrons in |η| > 6.3. The collision centrality was determined from the measured multiplicity of charged particles within |η| < 0.5 and a Monte-Carlo Glauber simulation [22,23]. The first-harmonic event plane angle Ψ 1 as an experimental estimate of the impact parameter direction was determined by measuring the neutron spectator deflection [24] in the ZDCs equipped with Shower Maximum Detectors (SMD) [25][26][27]. The event plane resolution [28] is largest (∼41%, the resolution is better if it is closer to 100%) at 30%-40% collision centrality for the 2014 and 2016 datasets, and is decreased by 4% for the 2010 and 2011 datasets [6].  [15]. Charged pions (kaons) and protons of the daughter particles were identified based on the ionization energy loss in the TPC gas, and the timing information measured by the Time-Of-Flight detector [29]. Reconstruction of Ξ − (Ξ + ), Ω − (Ω + ), and Λ (Λ) was performed using the KF Particle Finder package based on the Kalman Filter method initially developed for the CBM and ALICE experiments [30][31][32], which utilizes the quality of the track fit as well as the decay topology. Figure 1 shows the invariant mass distributions for reconstructed Ξ − (Ξ + ) and Ω − (Ω + ) for 20%-80% centrality. The purities for this centrality bin are higher than 90% for both species. The significance with the Kalman Filter method is found to be increased by ∼ 30% for Ξ compared to the traditional method for reconstruction of short-lived particles (e.g. see Refs. [6,33]). The hyperon candidates were also ensured not to share their decay products with other particles of interest.
The polarization along the initial angular momentum direction can be defined as [34]: where α H is the hyperon decay parameter and φ * B is the azimuthal angle of the daughter baryon in the parent hyperon rest frame. The azimuthal angle of the first-order event plane is Ψ obs 1 , and Res(Ψ 1 ) is the resolution [28] with which it estimates the reaction plane.
The extraction of sin(Ψ obs 1 − φ * ) was performed in the same way as in our previous studies [5,6]. The decay parameters of Λ, Ξ − , and Ω − have been recently updated by the Particle Data Group [15] and the latest values are used in this analysis; α Λ = 0.732 ± 0.014, α Ξ = −0.401 ± 0.010, and α Ω = 0.0157 ± 0.0021. When comparing to earlier measurements, the previous results are rescaled by using the new values, i.e. α old /α new . In case of the Ξ and Ω hyperon polarization measurements via measurements of the daughter Λ polarization, the polarization transfer factors C ΞΛ(ΩΛ) from Eqs. 4 and 6 are used to obtain the parent polarization.
The largest systematic uncertainty (37%) was attributed to the variation of the results obtained with datasets taken in different years. Weighted average over different datasets was used as the final result, and all other systematic uncertainties were assessed based on the weighted average: by comparing different polarization signal extractions [6] (11%), by varying the mass window for particles of interest from 3σ to 2σ (15%), by varying the decay lengths of both parent and daughter hyperons (4%), and by considering uncertainties on the decay parameter α H (2%), where the numbers in parentheses represent the uncertainty for the Ξ polarization via the daughter Λ polarization measurement. A correction for non-uniform acceptance effects [34] was applied for the appropriate detector configuration for the given dataset. This correction, depending on particle species, was less than 2%. Due to a weak p T dependence on the global polarization [6], effects from the p T dependent efficiency of the hyperon reconstruction were found to be negligible. Figure 2 shows the collision energy dependence of the Λ hyperon global polarization measured earlier [5,6,10,34] together with the new results on Ξ and Ω global polarizations at √ s N N = 200 GeV. (Note that the statistical and systematic uncertainties for the Λ are smaller than the symbol size.) For both Ξ and Ω polarizations, the particle and antiparticle results are averaged to reduce the statistical uncertainty. Also to maximize the significance of the polarization signal, the results were integrated over the centrality range 20%-80%, transverse momentum p T > 0.5 GeV/c, and rapidity |y| < 1. Global polarization of Ξ − andΞ + measurements via daughter Λ polarization show positive values, with no significant difference between Ξ − andΞ + (P Ξ (%) = 0.77 ± 0.16 (stat.) ± 0.49 (syst.) and PΞ (%) = 0.49 ± 0.16 (stat.) ± 0.20 (syst.)). The average polarization value obtained by this method is P Ξ (%) = 0.63 ± 0.11 (stat.) ± 0.26 (syst.). The Ξ +Ξ polarization was also measured via analysis of the angular distribution of daughter Λ in Ξ rest frame. This result, P Ξ (%) = −0.07 ± 0.19 (stat.) ± 0.50 (syst.), has larger uncertainty in part due to a smaller value of α Ξ compared to α Λ , which leads to smaller sensitivity of the measurement. The weighted average of the two measurements is P Ξ (%) = 0.47 ± 0.10 (stat.) ± 0.23 (syst.), which is larger than the polarization of inclusive Λ +Λ measured at the same energy for 20%-80% centrality, P Λ (%) = 0.24 ± 0.03 ± 0.03 [6], although the difference is still not significant considering the statistical and systematic uncertainties of both measurements. Note that the above quoted values for the inclusive Λ have been rescaled by the new decay parameter as mentioned earlier.
Calculations [35] carried out with a multi-phase transport model (AMPT) can describe the particle species dependence in data at 200 GeV as well as the energy dependence for Λ. These calculations indicate that the lighter particles with higher spin could be more polarized by the vorticity [35]. The multi-strange particles might freeze out at earlier times, which may lead to larger polarization for Ξ and Ω compared to Λ [8]. The feed-down effect can also lead to a 15 ∼ 20% reduction of the primary Λ polarization [7,[36][37][38], while the Ξ has less contribution from the feed-down. All these effects can contribute to small differences in the measured polarizations between inclusive Λ and Ξ hyperons.
Global polarization of Ω − was also measured and is presented in Fig. 2 under the assumption of γ Ω = +1 and therefore C ΩΛ = 1, as discussed with respect to Eq. 6. The result has large uncertainty, P Ω (%) = 1.11 ± 0.87 (stat.) ± 1.97 (syst.) for 20%-80% centrality. Assumption of γ Ω = −1 (therefore C ΩΛ = −0.6) results in P Ω (%) = −0.67 ± 0.52 (stat.) ± 1.18 (syst.). Assuming the validity of the global polarization picture, the result favors γ Ω ≈ +1 instead of γ Ω ≈ −1, but the uncertainties are large and more precise measurements are needed to make a definitive statement. The centrality dependence of Ξ +Ξ polarization via the measurement of daughter Λ polarization is shown in Fig. 3, where the inclusive Λ polarization [6] is plotted for comparison. The hyperon polarization increases in more peripheral collisions as expected from the centrality dependence of the fluid vorticity [39,40]. The Ξ polarization looks larger than that of the inclusive Λ in peripheral collisions as already discussed in relation to Fig. 2, although the uncertainties preclude a more definite conclusion.
In summary, we have presented the first measurements of the global polarization for Ξ − (Ξ + ) hyperons in Au+Au collisions at √ s N N = 200 GeV. Our results of Ξ hyperon polarization, along with the previous measurements of Λ polarization, confirm the global polarization picture based on the system fluid vorticity. The average polarization of Ξ +Ξ seems to be larger than that of the inclusive Λ, which is qualitatively captured by the AMPT model. The measured polarization seems to exhibit a centrality dependence as expected from the impact parameter dependence of the vorticity. Global polarization of Ω − hyperons was, also for the first time, extracted via measurements of the polarization of the daughter Λ and presented with the assumption that γ Ω = +1. Fu-ture measurements with higher precision will shed light on the uncertainty of the decay parameter γ Ω , as well as experimental results on the global polarization of spin-3/2 particles, providing critical information about spin dynamics in heavy-ion collisions.