Thermal Vorticity Research Articles

Revisiting the spin effects induced by thermal vorticity* *Supported in part by the National Natural Science Foundation of China (12175123, 11890710, 11890713), and the Major Program of Natural Science Foundation of Shandong Province, China (ZR2020ZD30)

We revisit the spin effects induced by thermal vorticity by calculating them directly from the spin-dependent distribution functions. For spin-1/2 particles, we provide the polarization up to the first order of thermal vorticity and compare it with the usual results calculated from the spin vector. For spin-1 particles, we show that all the non-diagonal elements vanish and there is no spin alignment up to the first order of thermal vortcity. We present the spin alignment at second-order contribution from thermal vorticity. We also show that the spin effects for both Dirac and vector particles receive an extra contribution when the spin direction is associated with the momentum of the particle.

Exact polarization of particles of any spin at global equilibrium

The polarization of the Λ particle offers the unique opportunity to study the hydrodynamic gradients in the Quark-Gluon Plasma formed in heavyion collisions. However, the theoretical formula commonly used to calculate polarization is only a linear order expansion in thermal vorticity and neglects higher-order corrections. Here, I present an exact calculation to all orders in (constant) thermal vorticity at global equilibrium, obtaining the analytic form of the spin density matrix and the polarization vector for massive particles of any spin. Finally, I extend these results to local equilibrium and assess their phenomenological impact by numerically calculating the polarization vector in a 3+1 hydrodynamic simulation.

Hyperon Polarization along the Beam Direction Relative to the Second and Third Harmonic Event Planes in Isobar Collisions at sqrt[s_{NN}]=200 GeV.

The polarization of Λ and Λ[over ¯] hyperons along the beam direction has been measured relative to the second and third harmonic event planes in isobar Ru+Ru and Zr+Zr collisions at sqrt[s_{NN}]=200 GeV. This is the first experimental evidence of the hyperon polarization by the triangular flow originating from the initial density fluctuations. The amplitudes of the sine modulation for the second and third harmonic results are comparable in magnitude, increase from central to peripheral collisions, and show a mild p_{T} dependence. The azimuthal angle dependence of the polarization follows the vorticity pattern expected due to elliptic and triangular anisotropic flow, and qualitatively disagrees with most hydrodynamic model calculations based on thermal vorticity and shear induced contributions. The model results based on one of existing implementations of the shear contribution lead to a correct azimuthal angle dependence, but predict centrality and p_{T} dependence that still disagree with experimental measurements. Thus, our results provide stringent constraints on the thermal vorticity and shear-induced contributions to hyperon polarization. Comparison to previous measurements at RHIC and the LHC for the second-order harmonic results shows little dependence on the collision system size and collision energy.

Suppression of spin polarization as an indicator of QCD critical point

We study the impact of the QCD critical point (CP) on the spin polarization of Lambda -hyperon generated by the thermal vorticity in viscous quark gluon plasma (QGP). The equations of the relativistic causal viscous hydrodynamics have been solved numerically in (3+1) dimensions to evaluate the thermal vorticity. The effects of the CP have been incorporated through the equation of state (EoS) and the scaling behavior of the transport coefficients. We observe a significant suppression of the spin polarization around mid-rapidity region induced by the CP and propose that such changed behaviour of the spin polarization can be used to detect the CP. To the best of our knowledge this is the first study on the effects of critical point on the spin polarization in QCD matter.

Impact of North Indian Atmospheric Diabatic Heating on Summer Precipitation in Central Asia

Abstract The impact of North Indian atmospheric diabatic heating variation on summer rainfall over Central Asia (CA) at an interannual scale during 1960–2019 was investigated from thermal adaptation and water vapor transportation perspective. The results showed that more precipitation in southeastern CA is associated with the southward subtropical westerly jet (SWJ), caused by the ascending motion and weakened water vapor output on the south side. When the SWJ moves southward, the high-level water vapor transportation on the south side changes from outward (−1.9 × 106 kg s−1) to inward (0.6 × 106 kg s−1), and the positive anomalous relative vorticity advections by the basic westerly winds produce ascending anomalies over southeastern CA. The position change in the SWJ was mainly related to atmospheric diabatic heating over northern India (NI). The thermal vorticity adaptation caused by a weakened heating rate over NI leads to an anomalous upper-level cyclone over southeastern CA, and the associated cold temperature advection eventually cools the upper troposphere of southeastern CA and reduces the temperature gradient at mid-to-high latitudes, leading to the southward SWJ. Thermal adaptation of the circulation and temperature anomaly over southeastern CA to the NI thermal forcing were also verified by numerical experiments. Both the abnormal ascending motions and the weakened outward water vapor associated with the southward SWJ, caused by the weakened heating rate over NI, lead to more summer rainfall in southeastern CA. The changes in diabatic heating over NI are closely related to Indian Ocean SST. When the Indian Ocean SST is warmer, the south Asian summer monsoon weakens, causing less precipitation and, thus, a weakened heating rate over NI. Significance Statement This study established that the northern Indian atmospheric diabatic heating anomalies associated with Indian Ocean SST variation play an important role in influencing precipitation in central Asia (CA). The weakening of the atmospheric diabatic heating over the NI would not only cause an abnormal cyclone and cooling over southern CA through thermal adaptation but also lead to southward subtropical westerly jet (SWJ), ascending motions, and decreased outward water vapor on the south side in southeastern CA, eventually resulting in more precipitation in southeastern CA. The results emphasize the influence of tropical SST and atmospheric heat sources on midlatitude climate and are important for understanding summer precipitation change in southeastern CA.

Exact spin polarization of massive and massless particles in relativistic fluids at global equilibrium

We present the exact form of the spin polarization vector and the spin density matrix of massive and massless free particles of any spin and helicity at general global equilibrium in a relativistic fluid with non-vanishing thermal vorticity, thus extending the known expression at the linear order. The exact form is obtained by means of the analytic continuation of the relativistic density operator to imaginary thermal vorticity and the resummation of the obtained series. The phenomenological implications for the polarization of the Lambda hyperon in relativistic heavy-ion collisions are addressed.

Spin polarization induced by magnetic field and the relativistic Barnett effect

First, I study the analogy between the magnetization of a material and the spin polarization of particles in a fluid. Using the relativistic version of the Barnett effect, i.e. the magnetization of a material induced by mechanical rotation, the spin polarization induced by thermal vorticity is obtained within a purely classical model, where spin is treated as an intrinsic magnetic moment and rotation is included as a non-inertial effect. I argue that since spin polarization induced by thermal vorticity can be obtained in a classical theory, it can not be dominated by quantum anomalies.Second, the spin polarization induced by magnetic field is obtained for a fluid at local thermal equilibrium using statistical quantum field theory. The obtained formula is valid beyond the weak field approximation and when contributions from the non-homogeneity of the magnetic field are small. The exact form of spin polarization is studied for a free Dirac field at global equilibrium, and, like magnetic susceptibility, it oscillates according to the de Haas - van Alphen effect.Finally, I briefly review how magnetic field contributes to the difference between the spin polarization of Λ and Λ¯ observed in heavy-ion collisions.

Colliding localized, lumpy holographic shocks with a granular nuclear structure

We apply a recent and simple technique which speeds up the calculation of localized collisions in holography to study more realistic models of the pre-hydrodynamic phase of heavy ion collisions using gauge/gravity duality. Our initial data reflects the lumpy nuclear structure of real heavy ions and our projectiles’ aspect ratio mimics the Lorentz contraction of nuclei during RHIC collisions. At the hydrodynamization time of the central region of the quark gluon plasma developed during the collision, we find that most of the system’s vorticity is located well outside the hydrodynamized part of the plasma. Only the relativistic corrections to the thermal vorticity within the hydrodynamized region are non-negligible. We compare the transverse flow shortly after the collision with previous results which did not use granular initial conditions and determine the proper energy density and fluid velocity in the hydrodynamized subregion of the plasma.

Λ and Λ¯ Freeze-Out Distributions and Global Polarizations in Au+Au Collisions

The gold–gold collisions at sNN=7.7 and 11.5 GeV are simulated within the PHSD transport model. In each collision event, the spectator nucleons are separated and the fluidization procedure for the participants is performed. The local velocities are determined in the Landau frame and the kinematic and thermal vorticity fields are evaluated. We analyze the thermodynamic properties of the cells where Λs and Λ¯s were born or had their last interaction. Such cells contribute to the formation of the observed global polarization of hyperons induced by the thermal vorticity of the medium. The Λ¯ polarization signal is found to be mainly determined by hot, dense, and highly vortical cells at the earlier stage of the collision, whereas the Λ polarization signal is accumulated over the longer time and includes cells with lower vorticity. The calculated global polarizations for both Λs and Λ¯s agree well with the experimental finding by the STAR collaboration at energy sNN=11.5 GeV. For collisions at sNN=7.7 GeV, we can reproduce the STAR data for Λ hyperons, but significantly underpredict the observed global polarization of Λ¯. Furthermore, we consider the centrality dependence of the hyperon polarization in collisions at 7.7 GeV. It increases with an increase of centrality, reaches a maximum at 65–75% and then starts decreasing rapidly for peripheral collisions.

Λ Polarization and Vortex Rings in Heavy-Ion Collisions at NICA Energies

We review recent studies of vortical motion and the resulting polarization of Λ hyperons in heavy-ion collisions at NICA energies, in particular, within the model of three-fluid dynamics (3FD). This includes predictions of the global Λ polarization and ring structures that appear in Au+Au collisions. The global Λ polarization in Au+Au collisions is calculated, including its rapidity and centrality dependence. The contributions of the thermal vorticity and meson-field term (proposed by Csernai, Kapusta, and Welle) to the global polarization are considered. The results are compared with data from recent STAR and HADES experiments. It is predicted that the polarization maximum is reached at sNN≈ 3 GeV if the measurements are performed with the same acceptance. It is demonstrated that a pair of vortex rings are formed, one at forward rapidities and another at backward rapidities, in ultra-central Au+Au collisions at sNN> 4 GeV. The vortex rings carry information about the early stage of the collision, in particular, the stopping of baryons. It is shown that these rings can be detected by measuring the ring observable RΛ, even in the midrapidity region at sNN= 5–20 GeV. At forward/backward rapidities, the RΛ signal is expected to be stronger. The possibility of observing the vortex-ring signal against the background of non-collective transverse polarization is discussed.

Shear induced polarization: collisional contributions

It has been realized that thermal shear plays a similar role as thermal vorticity in polarizing spin of particles in heavy ion collisions. We point out that shear has a funda- mental difference that it leads to particle redistribution in the medium. The redistribution gives rise to an additional contribution to spin polarization through the self-energy, which is parametrically the same order as the one considered so far in the literature. The self-energy contribution is in general gauge dependent. We introduce double gauge links stretching along the Schwinger-Keldysh contour to restore gauge invariance. We also generalize the straight path to adapt to the Schwinger-Keldysh contour. We find another contribution associated with the gauge link, which is also parametrically the same order. We illustrate the two contributions with a massive probe fermion in massless QED plasma with shear. A modest suppression of spin polarization is found from the combined contributions when the probe fermion has momentum much greater than the temperature.

Lambda polarization in 108Ag[formula omitted]Ag and 197Au[formula omitted]Au collisions around a few GeV

Within the framework of Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model, we extract the global spin polarization of Λ hyperon in 108Ag+108Ag and 197 Au+197Au collisions at sNN=2.42–62.4 GeV. We use two different approaches to calculate the Λ polarization Py: approach I is based on equilibrium assumption so that Py is determined by thermal vorticity and approach II assumes a proportional relation between Py and the system's angular momentum in Λ's rest frame. We find that both approaches can describe the experimental data at low energies around a few GeV but only approach I describes well also higher-energy data. This suggests that at higher energies the relativistic effect plays an important role. After taking into such effect properly, the relativity-improved approach II can describe the higher-energy data as well.

Measurement of global polarization of Λ hyperons in few-GeV heavy-ion collisions

The global polarization of Λ hyperons along the total orbital angular momentum of a relativistic heavy-ion collision is presented based on the high statistics data samples collected in Au+Au collisions at sNN=2.4 GeV and Ag+Ag at 2.55 GeV with the High-Acceptance Di-Electron Spectrometer (HADES) at GSI, Darmstadt. This is the first measurement below the strangeness production threshold in nucleon-nucleon collisions. Results are reported as a function of the collision centrality as well as a function of the hyperon's transverse momentum (pT) and rapidity (yCM) for the range of centrality 0–40%. We observe a strong centrality dependence of the polarization with an increasing signal towards peripheral collisions. For mid-central (20 – 40%) collisions the polarization magnitudes are 〈PΛ〉(%)=6.8±1.3(stat.)±2.1(syst.) for Au+Au and 〈PΛ〉(%)=6.2±0.4(stat.)±0.6(syst.) for Ag+Ag, which are the largest values observed so far. This observation thus provides a continuation of the increasing trend previously observed by STAR and contrasts expectations from recent theoretical calculations predicting a maximum in the region of collision energies about 3 GeV. The observed polarization is of a similar magnitude as predicted by 3D-fluid-dynamics and the UrQMD plus thermal vorticity model and significantly above results from the AMPT model.

Local and global polarization of Λ hyperons across RHIC-BES energies: The roles of spin hall effect, initial condition, and baryon diffusion

We perform a systematic study on the local and global spin polarization of $\mathrm{\ensuremath{\Lambda}}$ and $\overline{\mathrm{\ensuremath{\Lambda}}}$ hyperons in relativistic heavy-ion collisions at beam energy scan energies via the ($3+1$)-dimensional CLVisc hydrodynamics model with a multiphase transport (AMPT) and simulating many accelerated strongly interacting hadron (SMASH) initial conditions. Following the quantum kinetic theory, we decompose the polarization vector as the parts induced by thermal vorticity, shear tensor and the spin Hall effect (SHE). We find that the polarization induced by the SHE and the total polarization strongly depends on the initial conditions. At $7.7\phantom{\rule{0.28em}{0ex}}\mathrm{GeV}$, the SHE gives a sizable contribution and even flips the sign of the local polarization along the beam direction for the AMPT initial condition, which is not observed for the SMASH initial condition. Meanwhile, the local polarization along the out-of-plane direction induced by the SHE with the AMPT initial condition does not always increase with decreasing collision energies. Next, we find that the polarization along the beam direction is sensitive to the baryon diffusion coefficient, but the local polarization along the out-of-plane direction is not. Our results for the global polarization of $\mathrm{\ensuremath{\Lambda}}$ and $\overline{\mathrm{\ensuremath{\Lambda}}}$ agree well with the data of the STAR Collaboration. Interestingly, the global polarization of $\overline{\mathrm{\ensuremath{\Lambda}}}$ is not always larger than that of $\mathrm{\ensuremath{\Lambda}}$ due to various competing effects. Our findings are helpful for understanding the polarization phenomenon and the detailed structure of quark-gluon plasma in relativistic heavy-ion collisions.

Effect of thermal shear on longitudinal spin polarization in a thermal model

By including the recently introduced thermal shear term that contributes to the spin-polarization vector at local equilibrium, we determine longitudinal polarization of $\mathrm{\ensuremath{\Lambda}}$ hyperons emitted from a hot and rotating hadronic medium using the thermal model with single freeze-out. In our analysis, we consider the RHIC top energies and use the model parameters which were determined in the earlier analyses of particle spectra and elliptic flow. We confirm that, unlike the previous calculations done by using only the thermal vorticity, the thermal shear term alone leads to the correct sign of the quadrupole structure of the longitudinal component of the polarization three-vector measured in experiments. However, we find almost complete cancellation between thermal shear and vorticity terms, which eventually leads to disagreement with the data. To clarify the role played by velocity and temperature gradient terms, we present a systematic analysis of different contributions to the longitudinal polarization. Finally, replacing the mass of $\mathrm{\ensuremath{\Lambda}}$ hyperon with its constituent strange quark mass in the polarization formula we achieve agreement with the data.

Hydrodynamic helicity polarization in relativistic heavy ion collisions

We study helicity polarization through the $(3+1)$-dimensional relativistic viscous hydrodynamic models at $\sqrt{{s}_{NN}}=200$ GeV $\text{Au}+\text{Au}$ collisions. Similar to the local spin polarization, we consider the helicity polarization beyond global equilibrium and investigate the contributions induced by thermal vorticity, shear viscous tensor, and the fluid acceleration. We find that the local helicity polarization induced by thermal vorticity dominates over other contributions. It also implies that, in low-energy collisions, the fluid vorticity as part of thermal vorticity may play the crucial role in the total helicity polarization. Such a finding could be useful for probing the local strength of vorticity in rotational quark gluon plasmas by measuring helicity polarization. Our simulation confirms the strict space reversal symmetry, whereas we also compare our numerical results with approximated relations derived from ideal Bjorken flow. Our studies also provide a baseline for the future investigation on local parity violation through the correlations of helicity polarization.

Pseudogauge dependence of the spin polarization and of the axial vortical effect

The mean spin polarization vector of spin 1/2 particles in a relativistic fluid at local thermal equilibrium (LTE) in different pseudogauges (PGs), i.e., with different choices for the decomposition of orbital and spin angular momentum, is obtained. The spin polarization obtained in the canonical PG differs from the one obtained in the Belinfante PG. It is found that this difference can not be written by replacing the thermal vorticity with the spin potential in the usual polarization formula. Other PG choices affect the contribution of thermal shear to the spin polarization. Therefore, the choice of a PG affects the predictions for the polarization measured in heavy-ion collisions. In general, it is shown that the Wigner function of a noninteracting Dirac field at LTE is affected by the PG transformations. Explicit expressions of the mean axial current are also calculated and it is found that even the axial vortical effect conductivity depends on the PG.

Global Λ polarization in heavy-ion collisions at energies 2.4–7.7 GeV: Effect of meson-field interaction

Based on the three-fluid model, the global $\mathrm{\ensuremath{\Lambda}}$ polarization in $\mathrm{Au}+\mathrm{Au}$ collisions at 2.4 $\ensuremath{\le}\sqrt{{s}_{NN}}\ensuremath{\le}$ 7.7 GeV is calculated, including its rapidity and centrality dependence. Contributions from the thermal vorticity and meson-field term (proposed by Csernai, Kapusta, and Welle) to the global polarization are considered. The results are compared with data from recent and ongoing STAR and HADES experiments. It is predicted that the polarization maximum is reached at $\sqrt{{s}_{NN}}\ensuremath{\approx}3$ GeV, if the measurements are performed with the same acceptance. The value of the polarization is very sensitive to interplay of the aforementioned contributions. In particular, the thermal vorticity results in quite strong increase of the polarization from the midrapidity to forward/backward rapidities, while the meson-field contribution considerably flattens the rapidity dependence. The polarization turns out to be very sensitive to details of the equation of state. While collision dynamics become less equilibrium with decreasing collision energy, the present approach to polarization is based on the assumption of thermal equilibrium. It is found that equilibrium is achieved at the freeze-out stage, but this equilibration takes longer at moderately relativistic energies.

Evolution of Λ polarization in the hadronic phase of heavy-ion collisions

Using the ampt$+$music$+$urqmd hybrid model, we study the global and local spin polarizations of $\mathrm{\ensuremath{\Lambda}}$ hyperons as functions of the freeze-out temperature of the spin degree of freedom in the hadronic phase of Au$+$Au collisions at $\sqrt{{s}_{NN}}=19.6$ GeV. Including contributions from both the thermal vorticity and thermal shear of the hadronic matter, we find that with the spin freeze-out temperature dropping from the hadronization temperature of 160 MeV to 110 MeV at the kinetic freeze-out, both the global and local spin polarizations of $\mathrm{\ensuremath{\Lambda}}$ hyperons due to the thermal vorticity decrease by a factor of 2, while those due to the thermal shear decrease quickly and become negligibly small at 140 MeV. Our results thus suggest that it is important to take into account the evolution of the spin degree of freedom in the hadronic stage of relativistic heavy-ion collisions when theoretically predicted global and local $\mathrm{\ensuremath{\Lambda}}$ spin polarizations are compared with the experimental measurements.

Rise and fall of Λ and Λ¯ global polarization in semi-central heavy-ion collisions at HADES, NICA and RHIC energies from the core-corona model

We compute the $\mathrm{\ensuremath{\Lambda}}$ and $\overline{\mathrm{\ensuremath{\Lambda}}}$ global polarizations in semicentral heavy-ion collisions using the core-corona model where the source of $\mathrm{\ensuremath{\Lambda}}\mathrm{s}$ and $\overline{\mathrm{\ensuremath{\Lambda}}}\mathrm{s}$ is taken as consisting of a high-density core and a less dense corona. We show that the overall properties of the polarization excitation functions can be linked to the relative abundance of $\mathrm{\ensuremath{\Lambda}}\mathrm{s}$ coming from the core versus those coming from the corona. For low collision energies, the former are more abundant whereas for higher energies the latter become more abundant. The main consequence of this reversing of the relative abundance is that both polarizations peak at collision energies $\sqrt{{s}_{NN}}\ensuremath{\lesssim}10$ GeV. The exact positions and heights of these peaks depend not only on this reversal of relative abundances, but also on the centrality class, which is directly related to the quark gluon plasma volume and lifetime, as well as on the relative abundances of $\mathrm{\ensuremath{\Lambda}}\mathrm{s}$ and $\overline{\mathrm{\ensuremath{\Lambda}}}\mathrm{s}$ in the core and corona regions. The intrinsic polarizations are computed from a field-theoretical approach that links the alignment of the strange quark spin with the thermal vorticity and modeling the quark gluon plasma volume and lifetime using a Bjorken expansion scenario. We predict that the $\mathrm{\ensuremath{\Lambda}}$ and $\overline{\mathrm{\ensuremath{\Lambda}}}$ global polarizations should peak at the energy range accessible to NICA and HADES.