Polarization of $\Lambda$ ($\bar{\Lambda}$) hyperons along the beam direction in Au+Au collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV

TL;DR

This study reports the first measurement of $\Lambda$ hyperon polarization along the beam direction in Au+Au collisions at 200 GeV, revealing a sine modulation pattern linked to vorticity, with results challenging existing hydrodynamic models.

Contribution

It provides the first experimental measurement of hyperon polarization along the beam axis in high-energy heavy-ion collisions, highlighting discrepancies with theoretical predictions.

Findings

Polarization exhibits sine modulation with emission angle.

Polarization increases with collision centrality.

Measured polarization is significantly smaller than model predictions.

Abstract

The $\Lambda$ ($\bar{\Lambda}$) hyperon polarization along the beam direction has been measured for the first time in Au+Au collisions at $\sqrt{s_{_{NN}}}$ = 200 GeV. The polarization dependence on the hyperons' emission angle relative to the second-order event plane exhibits a sine modulation, indicating a quadrupole pattern of the vorticity component along the beam direction. The polarization is found to increase in more peripheral collisions, and shows no strong transverse momentum ($p_T$) dependence at $p_T>1$ GeV/$c$. The magnitude of the signal is about five times smaller than those predicted by hydrodynamic and multiphase transport models; the observed phase of the emission angle dependence is also opposite to these model predictions. In contrast, blast-wave model calculations reproduce the modulation phase measured in the data and capture the centrality and transverse momentum dependence of the signal once the model is required to reproduce the azimuthal dependence of the Gaussian source radii measured via the Hanbury-Brown and Twiss intensity interferometry technique.

Equations

$$\frac{dN}{d\cos {\theta }^{\ast }}\propto 1+{\alpha }_H{P}_H\cos {\theta }^{\ast },$$

$${\Psi }_2^{\mathrm{obs}}={\tan }^{-1}\left(\sum _i{w}_i\sin (2{\varphi }_i)/\sum _i{w}_i\cos (2{\varphi }_i)\right),$$

$$P_z$$

$$⟨{\omega }_z\sin (2\varphi )⟩$$

$${\omega }_z$$

Full text

The properties of deconfined partonic matter, the quark-gluon plasma, have been explored in heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) J. Adams et al. (2005) (STAR Collaboration); K. Adcox et al. (2005) (PHENIX Collaboration); B. B. Back et al. (2005) (PHOBOS Collaboration); l. Arsene et al. (2005) (BRAHMS Collaboration) and the Large Hadron Collider K. Aamodt et al. (2011) (ALICE Collaboration); S. Chatrchyan et al. (2011) (CMS Collaboration); G. Aad et al. (2010) (ATLAS Collaboration). The matter created in non-central heavy-ion collisions should exhibit rotational motion in order to conserve the initial angular momentum carried by the two colliding nuclei. The direction of the angular momentum is perpendicular to the reaction plane, as defined by incoming beam and the impact parameter vector. It was predicted Z. T. Liang and X. N. Wang (2005); Voloshin (2004) that such a spinning motion of the matter would lead to a net spin polarization of particles produced in the collisions due to spin-orbit coupling. Hyperons are natural candidates to explore this phenomenon since in the parity violating weak decays of the hyperons the momentum vector of the decay baryon is highly correlated with the hyperon spin. In such decays the angular distribution of the daughter baryons is given by:

[TABLE]

where $\alpha_{H}$ is the hyperon decay parameter, $P_{H}$ is the hyperon polarization, and $\theta^{\ast}$ is the angle between the polarization vector and the direction of the daughter baryon momentum in the hyperon rest frame.

The Solenoidal Tracker at RHIC (STAR) Collaboration has observed positive polarizations of $\Lambda$ hyperons along the orbital angular momentum in Au+Au collisions for collision energies of $\sqrt{s_{{}_{NN}}}$ = 7.7 – 200 GeV L. Adamczyk et al. (2017); J. Adam et al. (2018). This polarization is evidence for the creation of the most vortical fluid ever observed, with vorticities of the order of $\omega\sim 10^{22}~{}s^{-1}$. These results open new opportunities for a better understanding of the dynamics and properties of the matter created in heavy-ion collisions.

The spin polarization of hyperons along the orbital angular momentum of the entire system is referred to as the global polarization, meaning a net spin alignment along a globally defined direction. However, the vorticity and, consequently, the particle polarization may vary for different regions of the fluid due to anisotropic flow, energy deposits from jet quenching, density fluctuations, etc. The detailed structure of the vorticity fields may be complicated and the resulting particle polarization can depend on the particle transverse momentum and the azimuthal angle relative to the reaction plane, or even exhibit toroidal structures  Betz et al. (2007); Voloshin (2017); F. Becattini and Iu. Karpenko (2018); Pang et al. (2016).

Anisotropic flow, characterized by the Fourier coefficients of the particle azimuthal distribution in the transverse plane, has been extensively studied in heavy-ion collisions and was found to be well described by hydrodynamic calculations Voloshin et al. (2010); Heinz and Snellings (2013). Nontrivial velocity fields describing transverse anisotropic flow should lead to a vorticity component along the beam direction dependent on the azimuthal angle relative to the reaction plane Voloshin (2017); F. Becattini and Iu. Karpenko (2018). The observation of the large second-order coefficients, a.k.a. elliptic flow, in mid-central collisions indicates significantly stronger expansion in the reaction plane direction compared to that out-of-plane, which might lead to a quadrupole structure in the $z$-component of vorticity as illustrated in Fig. 1. Experimental measurements of such a component are the main goal of this analysis.

The beam direction component of the polarization arising from vorticity due to elliptic flow is expected to be more sensitive to later times from flow development in the system evolution D. Teaney and L. Yan (2011), unlike the global polarization that originates mostly from the initial velocity fields. It might also have different sensitivity to the relaxation time needed for the conversion of the vorticity into particle polarization. Therefore, it is of great interest to study the polarization along the beam direction for further understanding of the role of the vorticity in heavy-ion collisions and possibly to answer these questions. In this Letter, we report the beam direction component of polarization for $\Lambda$ and $\bar{\Lambda}$ hyperons in Au+Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV. The results are presented as functions of the collision centrality and hyperons’ transverse momentum ($p_{T}$).

The dataset for this analysis was collected in 2014 by the STAR detector during the period of Au+Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV. Charged-particle tracks were measured in the time projection chamber (TPC) M. Anderson et al. (2003), which covers the full azimuth and a pseudorapidity range of $-1<\eta<1$. The collision vertices were reconstructed using the measured charged-particle tracks. Events were selected to have the collision vertex position within 6 cm of the center of the TPC in the beam direction and within 2 cm in the radial direction with respect to the beam center. In addition, the difference between the vertex positions along the beam direction determined by the TPC and the vertex position detectors (VPD) W. J. Llope et al. (2014) located at forward and backward rapidities ($4.24<|\eta|<5.1$) was required to be less than 3 cm to suppress pileup events. These selection criteria yielded about one billion minimum bias events, where the minimum bias trigger required hits of both VPDs and the zero-degree calorimeters C. Adler, A. Denisov, E. Garcia, M. Murray, H. Strobele, and S. White (2001) located at $|\eta|>6.3$.

The collision centrality was determined from the measured multiplicity of charged particles within $|\eta|<0.5$ and a Monte-Carlo Glauber simulation L. Adamczyk et al. (2012) (STAR Collaboration). The second-order event plane ($\Psi_{2}$) as an experimental estimate of the reaction plane was determined by the charged-particle tracks within the transverse momentum range of 0.15<\mbox{p_{T}}<2 GeV/$c$ and $0.1<|\eta|<1$:

[TABLE]

where $\phi_{i}$ and $w_{i}$ are the azimuthal angle and $p_{T}$ of the $i^{\rm th}$ particle in the event. The resolution of the measured plane $\Psi_{2}^{\rm obs}$ defined as ${\rm Res}(\Psi_{2})=\langle\cos 2(\Psi_{2}^{\rm obs}-\Psi_{2})\rangle$ was estimated with the two-subevent method Poskanzer and Voloshin (1998), where the two subevents were taken from $0.1<|\eta|<1$. In mid-central collisions the event plane resolution peaks at $\sim$0.76.

Charged-particle tracks reconstructed with the TPC were selected to have good quality by requiring the following conditions. The number of hit points used in the track reconstruction was required to be larger than 15. The ratio of the number of hit points used to the maximum possible number of TPC space points for that trajectory was required to be larger than 0.52. Tracks within 0.15<\mbox{p_{T}}<10 GeV/$c$ and $|\eta|<1$ that passed through the track selections above were used to reconstruct $\Lambda$ hyperons. In order to reconstruct $\Lambda$ and $\bar{\Lambda}$, the decay channels of \mbox{\Lambda}\rightarrow p+\pi^{-} and \mbox{\bar{\Lambda}}\rightarrow\bar{p}+\pi^{+}, corresponding to (63.9$\pm$0.5)% of all decays C. Patrignani et al. (2016) (Particle Data Group), were utilized. The ionization energy loss $dE/dx$ in the TPC and the time of flight information of the particles from the time-of-flight detector W. J. Llope (2012) were used to select daughter pions and protons. Cuts on decay topology, such as a distance of the closest approach (DCA) between the trajectory of $\Lambda$ ($\bar{\Lambda}$) candidates and the primary vertex, DCA between the two daughters, and decay length of $\Lambda$ ($\bar{\Lambda}$) candidates were applied to reduce the combinatoric background. Additional details about the $\Lambda$ ($\bar{\Lambda}$) reconstruction can be found in Ref. J. Adam et al. (2018).

The longitudinal component of the polarization can be measured by projecting the polarization onto the beam direction:

[TABLE]

where $\theta_{p}^{\ast}$ is the polar angle of the daughter proton in the $\Lambda$ ($\bar{\Lambda}$) rest frame and $\langle\rangle$ represents an average over $\Lambda$ ($\bar{\Lambda}$) candidates in an event and then an average over all events. The decay parameter $\alpha_{H}$ is set to be $\alpha_{\Lambda}=-\alpha_{\bar{\Lambda}}=0.642\pm 0.013$ C. Patrignani et al. (2016) (Particle Data Group); not . If the detector has perfect acceptance and efficiency, $\langle\cos^{2}\theta_{p}^{\ast}\rangle$ leads to $1/3$. In this study $\langle\cos^{2}\theta_{p}^{\ast}\rangle$ was extracted from the data in order to account for pseudorapidity dependent detector acceptance effects. This term was found to be close to $1/3$ for all centralities but showed a systematic decrease for lower track $p_{T}$. To extract the signal $\langle\cos\theta_{p}^{\ast}\rangle$, two techniques were used: the event plane method and the invariant mass method as described in Ref. J. Adam et al. (2018). In the event plane method, $\langle\cos\theta_{p}^{\ast}\rangle$ was measured as a function of azimuthal angle of \mbox{\Lambda}(\mbox{\bar{\Lambda}}) relative to $\Psi_{2}$. The average polarization along the beam direction is expected to be zero due to symmetry. Effects due to detector acceptance and inefficiencies are removed by subtracting the azimuthal average of $\langle\cos\theta_{p}^{\ast}\rangle$ from each azimuthal bin $i$ of $\Lambda$ azimuthal angle: \mbox{\langle\cos\theta_{p}^{\ast}\rangle}^{\rm sub}_{i}=\mbox{\langle\cos\theta_{p}^{\ast}\rangle}_{i}-\sum_{i}^{\rm n_{bin}}\mbox{\langle\cos\theta_{p}^{\ast}\rangle}_{i}/{\rm n_{bin}}.

Figure 2 shows \mbox{\langle\cos\theta_{p}^{\ast}\rangle}^{\rm sub} of $\Lambda$ and $\bar{\Lambda}$ as a function of azimuthal angle relative to $\Psi_{2}$ for the 20%–60% centrality bin. The solid lines indicate the fit results to the function $p_{0}+2p_{1}\sin(2\phi-2\Psi_{2})$, where $p_{0}$ and $p_{1}$ are fit parameters. The data are consistent with a sine structure for both $\Lambda$ and $\bar{\Lambda}$ as expected from the elliptic flow. In the invariant mass method, the second-order Fourier sine coefficient of $P_{z}$, $p_{1}=\langle P_{z}\sin(2\phi-2\Psi_{2})\rangle$, was measured as a function of the invariant mass. Following the same procedure as described in Ref. J. Adam et al. (2018), the sine coefficient was directly extracted. The extracted coefficient in both methods was divided by Res($\Psi_{2}$) to account for the finite event plane resolution. The invariant mass method was used to calculate the sine coefficient of $P_{z}$ and the event plane method was used to cross-check and provide an estimate of the systematic uncertainty.

The systematic uncertainties were estimated by variation of the topological cuts ($<2\%$), comparing the results from two methods for signal extraction ($5\%$) as mentioned above, using different subevents ($-1<\eta<-0.5$ and $0.5<\eta<1$) for $\Psi_{2}$ determination ($<11\%$), and estimates of the possible background contribution to the signal ($4.3\%$). The numbers are for mid-central collisions. Also the uncertainty from the decay parameter is accounted for (2% for $\Lambda$ and 9.6% for $\bar{\Lambda}$, see Ref. J. Adam et al. (2018) for the detail). We further studied the effect of a possible self-correlation between the particles used for the $\Lambda$ ($\bar{\Lambda}$) reconstruction and the event plane by explicitly removing the daughter particles from the event plane calculation in Eq. (2). There was no significant difference between the results. The $\Lambda$ and $\bar{\Lambda}$ reconstruction efficiencies were estimated using GEANT Brun et al. (1987) simulations of the STAR detector M. Anderson et al. (2003). The correction is found to lower mean values of the $P_{z}$ sine coefficient by $\sim$10% in peripheral collisions and increases up to $\sim$50% in central collisions, although the variations are within statistical uncertainties. No significant difference was observed between $\Lambda$ and $\bar{\Lambda}$ as expected. Therefore, results from both samples were combined to reduce statistical uncertainties.

Figure 3 presents the centrality dependence of the second Fourier sine coefficient $\langle P_{z}\sin(2\phi-2\Psi_{2})\rangle$. The increase of the signal with decreasing centrality is likely due to increasing elliptic flow contributions in peripheral collisions. We note that, unlike elliptic flow, the polarization does disappear in the most central collisions, where the elliptic flow is still significant due to initial density fluctuations. Because of large uncertainties in peripheral collisions, it is not clear whether the signal continues to increase or levels off. The results are compared to a multiphase transport (AMPT) model Xia et al. (2018) as shown with the dotted line. The AMPT model predicts the opposite phase of the modulations and overestimates the magnitude. The blast-wave model study is discussed later.

Since the elliptic flow also depends on $p_{T}$ as well as on the centrality, the polarization may have $p_{T}$ dependence. Figure 4 shows the sine coefficients of $P_{z}$ as a function of the hyperon transverse momentum. No significant $p_{T}$ dependence is observed for \mbox{p_{T}}>1 GeV/$c$, and the statistical precision of the single data point for \mbox{p_{T}}<1 GeV/$c$ is not enough to allow for definitive conclusions about the low $p_{T}$ dependence. In the hydrodynamic model calculation F. Becattini and Iu. Karpenko (2018), the sine coefficient of $P_{z}$ increases in magnitude with $p_{T}$ but shows the opposite sign to the data.

As shown in Figs. 3 and 4, the hydrodynamic and AMPT models predict the opposite sign in the sine coefficient of the polarization and their magnitudes differ from the data roughly by a factor of 5. The reason of this sign difference is under discussion in the community. However, the sign change may be due to the relation between azimuthal anisotropy and spatial anisotropy at freeze-out Voloshin (2017). There could be contributions from the kinematic vorticity originating from the elliptic flow as well as from the temporal gradient of temperatures at the time of hadronization F. Becattini and Iu. Karpenko (2018). A recent calculation using the chiral kinetic approach predicts the same sign as the data Sun and Ko (2019). The model accounts for the transverse component of the vorticity, resulting in axial charge currents. Note that both the hydrodynamic and transport models calculate local vorticity at freeze-out and convert it to the polarization assuming local thermal equilibrium of the spin degrees of freedom, while the chiral kinetic approach takes into account nonequilibrium effects but does not consider a contribution from the temperature gradient which is a main source of $P_{z}$ in the hydrodynamic model.

These models indicate that the contribution from the kinematic vorticity to $P_{z}$ is negligible or opposite in the sign to the naive expectation from the elliptic flow. In order to estimate the contribution from the kinematic vorticity we employed the blast-wave model (BW) Schnedermann et al. (1993); Adler et al. (2001); Retiere and Lisa (2004). Following Ref. Retiere and Lisa (2004) we parameterize the system velocity field at freeze-out with temperature ($T$) and transverse flow rapidity ($\rho$) defined as $\rho=\tilde{r}[\rho_{0}+\rho_{2}\cos(2\phi_{b})]$. Here $\rho_{0}$ and $\rho_{2}$ are the maximal radial expansion rapidity and its azimuthal modulation, $\tilde{r}$ is the relative distance to the edge of the source, and $\phi_{b}$ defines the direction of the local velocity as indicated in Fig. 1. The source shape, assumed to be elliptical in the transverse plane, is parameterized by the $R_{y}$ and $R_{x}$ radii. Boost invariance is assumed. Two fits to the data are performed: in one only spectra and elliptic flow of $\pi$, K, and p($\bar{p}$) are fit; the second fit Adams et al. (2005) also includes azimuthal-angle-dependence of the pion Gaussian source radii at freeze-out as measured via Hanbury-Brown and Twiss (HBT) intensity interferometry. The average longitudinal vorticity is calculated according to the following formula:

[TABLE]

where the integration is over the transverse cross-sectional area of the source, $u_{\mu}$ is a four-vector of the local flow velocity Retiere and Lisa (2004), $\phi_{s}$ is the azimuth of the production point (see Fig. 1 for the relation to $\phi_{b}$), $\alpha_{t}=p_{T}/T\,\sinh\rho$, $\beta_{t}=m_{T}/T\cosh\rho$; $I_{n}$ and $K_{1}$ are the modified Bessel functions. Assuming a local thermal equilibrium, the longitudinal component of the polarization is estimated as $P_{z}\approx\omega_{z}/(2T)$. The uncertainties shown for the BW model calculations corresponds to 1 $\sigma$ variation in the model parameters. See Ref. Dobrin et al. for more details.

The BW calculations are compared to the data in Figs. 3 and 4. From central to mid-central collisions both BW calculations show positive sine coefficients which are compatible in both sign and magnitude to the measurement, although the BW model is based on a very simple picture of the freeze-out condition. It was shown in Ref. Voloshin (2017) that the vorticity in the BW model has the effects of the velocity field anisotropy ($\rho_{2}/\rho_{0}$) and the spacial source anisotropy ($R_{y}/R_{x}$) contributing with opposite signs, which can explain a strong sensitivity of the BW model predictions in the peripheral collisions to the inclusions of the HBT radii.

We have presented the first measurements of the longitudinal component of the polarization for $\Lambda$ and $\bar{\Lambda}$ hyperons in Au+Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV. Finite signals of a quadrupole modulation of both $\Lambda$ and $\bar{\Lambda}$ polarization along the beam direction are observed and found to be qualitatively consistent with the expectation from the vorticity component along the beam direction due to the elliptic flow. The results exhibit a strong centrality dependence with increasing magnitude as the collision centrality becomes more peripheral. No significant $p_{T}$ dependence is observed above \mbox{p_{T}}>1 GeV/$c$. A drop-off of the signal is hinted at for \mbox{p_{T}}<1 GeV/$c$. The data were compared to calculations from hydrodynamic and AMPT models, both of which show the opposite phase of the modulation and overpredict the magnitude of the polarization. This might indicate incomplete thermal equilibration of the spin degrees of freedom for the beam direction component of the vorticity/polarization, as it develops later in time compared to the global polarization. On the other hand, the blast-wave model calculations are much closer to the data, even more so when the azimuthally sensitive HBT results along with the $p_{T}$ spectra and $v_{2}$ are included in the model fit. The blast-wave model predicts the correct phase of $P_{z}$ modulation and a similar $p_{T}$ dependence; the version with HBT radii included in the fit also reasonably describes the centrality dependence. These results together with the results of the global polarization may provide information on the relaxation time needed to convert the vorticity to particle polarization. Further theoretical and experimental studies are needed for better understanding.