We derive the collision term in the Boltzmann equation using the equation of motion for the Wigner function of massive spin-1/2 particles. To next-to-lowest order in $\hbar$ it contains a nonlocal contribution, which is responsible for the conversion of orbital into spin angular momentum. In a proper choice of pseudo-gauge the antisymmetric part of the energy-momentum tensor arises solely from this nonlocal contribution. We show that the collision term vanishes only in global equilibrium and that the spin potential is then equal to the thermal vorticity. In the nonrelativistic limit, the equations of motion for the energy-momentum and spin tensors reduce to the well-known form for hydrodynamics for micropolar fluids.
Newly introduced equilibrium Wigner functions for particles with spin one-half are used in the semi-classical kinetic equations to study a possible relation between thermal vorticity and spin polarization. It is shown that in global equilibrium both the thermal-vorticity and spin polarization tensors are constant but not necessarily equal. In the case of local equilibrium, we define a procedure leading to hydrodynamic equations with spin. We introduce such equations for the de~Groot, van~Leeuwen, and van~Weert (GLW) formalism as well as for the canonical scheme (these two frameworks differ by …
We give a theoretical perspective on the vorticity and spin polarization in heavy-ion collisions. We discuss the recent progress in spin hydrodynamics and spin kinetic theory. We also discuss other effects caused by vorticity including the chiral vortical effect and rotation-induced phase transitions.
We study the local structure of the vorticity field and the $Λ$ polarization in Au+Au collisions in the energy range $\sqrt{s_{\mathrm{NN}}}=7.7$--$200$ GeV and Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2760$ GeV using A Multi-Phase Transport (AMPT) model. We focus on the vorticity field arising from the non-uniform expansion of the fireball, which gives the circular structure of the transverse vorticity $\boldsymbolω_{\perp}=(ω_{x},ω_{y})$ around the $z$ direction as well as the quadrupole pattern of the longitudinal vorticity $ω_{z}$ in the transverse plane. As a consequence, the three com…
We review studies of vortical motion and the resulting global polarization of $Λ$ and $\barΛ$ hyperons in heavy-ion collisions, in particular, within 3FD model. 3FD predictions for the global midrapidity polarization in the FAIR-NICA energy range are presented. The 3FD simulations indicate that energy dependence of the observed global polarization of hyperons in the midrapidity region is a consequence of the decrease of the vorticity in the central region with the collision energy rise because of pushing out the vorticity field into the fragmentation regions. At high collision energies this pu…
We study helicity polarization through the (3+1) dimensional relativistic viscous hydrodynamic models at $\sqrt{s_{NN}}=200$GeV Au+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 the low-energy collisions, the the fluid vorticity as part of thermal vorticity may play the crucial r…
The system created in ultrarelativistic nuclear collisions is known to behave as an almost ideal liquid. In non-central collisions, due to the large orbital momentum, such a system might be the fluid with the highest vorticity ever created under laboratory conditions. Particles emerging from such a highly vorticous fluid are expected to be globally polarized with their spins on average pointing along the system angular momentum. Vorticity-induced polarization is the same for particles and antiparticles, but the intense magnetic field generated in these collisions may lead to the splitting in p…
It has been proposed that the $Λ$ and $\overlineΛ$ polarizations observed in heavy-ion collisions are due to the interaction between quark spin and thermal vorticity. In this work we report on a computation of the relaxation time required for this alignment to occur at finite temperature and baryon chemical potential, considering quarks with a finite mass. The calculation is performed after modelling the interaction by means of an effective vertex which couples the thermal gluons and quarks within the vortical medium. We show that the effect of the quark mass is to reduce the relaxation time a…
Heavy ion collisions generate strong fluid vorticty in the produced hot quark-gluon matter which could in turn induce measurable spin polarization of hadrons. We review recent progress on the vorticity formation and spin polarization in heavy ion collisions with transport models. We present an introduction to the fluid vorticity in non-relativistic and relativistic hydrodynamics and address various properties of the vorticity formed in heavy ion collisions. We discuss the spin polarization in a vortical fluid using the Wigner function formalism in which we derive the freeze-out formula for the…
The polarization of $Λ$ hyperons from relativistic flow vorticity is studied in peripheral heavy ion reactions at FAIR and NICA energies, just above the threshold of the transition to the Quark-Gluon Plasma. Previous calculations at higher energies with larger initial angular momentum, predicted significant $Λ$ polarization based on the classical vorticity term in the polarization, while relativistic modifications decreased the polarization and changed its structure in the momentum space. At the lower energies studied here, we see the same effect namely that the relativistic modifications decr…
We study the behavior of four types of vorticities in non-central Au+Au collisions at energies $\sqrt{S_{NN}}$ = 5--200 GeV using a microscopic transport model PACIAE. The results show that the vorticities decay faster at lower energy and smaller impact parameter. The non-monotonic dependence of the initial vorticities on the collision energies is reconfirmed and the turning point is 10-15 GeV for different vorticities. The global $Λ$ polarization at energies $\sqrt{S_{NN}}$ = 7.7--200 GeV is reproduced by introducing an energy density freeze-out criterion.
The hot and dense matter generated in heavy-ion collisions contains intricate vortical structure in which the local fluid vorticity can be very large. Such vorticity can polarize the spin of the produced particles. We study the event-by-event generation of the so-called thermal vorticity in Au + Au collisions at energy region $\sqrt{s}=7.7-200$ GeV and calculate its time evolution, spatial distribution, etc., in a multiphase transport (AMPT) model. We then compute the spin polarization of the $Λ$ and $\barΛ$ hyperons as a function of $\sqrt{s}$, transverse momentum $p_T$, rapidity, and azimuth…
We critically compare thermodynamic and kinetic approaches, that have been recently used to study relations between the spin polarization and fluid vorticity in systems consisting of spin-one-half particles. The thermodynamic approach refers to general properties of global thermal equilibrium with a rigid-like rotation and demonstrates that the spin-polarization and thermal-vorticity tensors are equal. On the other hand, the kinetic approach uses the concept of the Wigner function and its semi-classical expansion. In most of the works done so far, the Wigner functions satisfy kinetic equations…
Polarization of $Λ$ hyperons and their antiparticles is calculated in a 3+1 dimensional viscous hydrodynamic model with initial state from UrQMD hadron/string cascade. We find that, along with recent results from STAR, the mean polarization at fixed centrality decreases as a function of collision energy from 1.5% at $\sqrt{s_{\rm NN}}=7.7$ GeV to 0.2% at $\sqrt{s_{\rm NN}}=200$ GeV. We explore the effects which lead to such collision energy dependence, feed-down corrections and a difference between $Λ$ and $\barΛ$.
We discuss the information that can be deduced from a measurement of particle (hyperon or vector meson) polarization in ultrarelativistic nuclear collisions. We describe the sensitivity of polarization to initial conditions, hydrodynamic evolution and mean free path, and find that the polarization observable is sensitive to all details and stages of the system's evolution. We suggest that an experimental investigation covering production plane and reaction plane polarizations, as well as the polarization of jet-associated particles in the plane defined by the jet and particle direction, can he…
We use a thermal model with single freeze-out to determine longitudinal polarization of $Λ$ hyperons emitted from a hot and rotating hadronic medium. We consider the top RHIC energies and use the model parameters determined in the previous analyses of particle spectra and elliptic flow. Using a direct connection between the spin polarization tensor and thermal vorticity, we reproduce earlier results which indicate a quadrupole structure of the longitudinal component of the polarization three-vector with an opposite sign compared to that found in the experiment. We further use only the spatial …
With a Yang-Mills field, stratified shear flow initial state and a high resolution (3+1)D Particle-in-Cell Relativistic (PICR) hydrodynamic model, we calculate the $Λ$ polarization for peripheral Au+Au collisions at RHIC energy of $\sqrt{S_{NN}}=200$ GeV. The obtained longitudinal polarization in our model agrees with the experimental signature and the quadrupole structure on transverse momentum plane. It is found that the relativistic correction (2nd term), arising from expansion and from the time component of the thermal vorticity, plays a crucial role in our results. This term is changing t…
The rotation of polarization occurs for light interacting with chiral materials. It requires the light states with opposite chiralities interact differently with the materials. We demonstrate analogous rotation of polarization also exists for chiral fermions interacting with quantum electrodynamics plasma with vorticity using chiral kinetic theory. We find that the rotation of polarization is perpendicular both to vorticity and fermion momentum. The effect also exists for chiral fermions in quantum chromodynamics plasma with vorticity. It could lead to generation of a vector current when the p…
We study the spin polarization generated by the hydrodynamic gradients. In addition to the widely studied thermal vorticity effects, we identify an undiscovered contribution from the fluid shear. This shear-induced polarization (SIP) can be viewed as the fluid analog of strain-induced polarization observed in elastic and nematic materials. We obtain the explicit expression for SIP using the quantum kinetic equation and linear response theory. Based on a realistic hydrodynamic model, we compute the differential spin polarization along both the beam direction $\hat{z}$ and the out-plane directio…
The quark-gluon plasma produced by collisions between ultra-relativistic heavy nuclei is well described in the language of hydrodynamics. Non-central collisions are characterized by very large angular momentum, which in a fluid system manifests as flow vorticity. This rotational structure can lead to a spin polarization of the hadrons that eventually emerge from the plasma, providing experimental access to flow substructure at unprecedented detail. Recently, first observations of $Λ$ hyperon polarization along the direction of collisional angular momentum have been reported. These measurements…
Experiments at the Relativistic Heavy Ion Collider (RHIC) have measured the net polarization of $Λ$ and $\barΛ$ hyperons and attributed it to a coupling between their spin and the vorticity of the fluid created in heavy ion collisions, but how the spin comes to equilibrium with vorticity is an open problem. Recently we found that vorticity fluctuations and helicity flip of strange quarks in quark-gluon plasma through perturbative QCD processes resulted in equilibration times far too long to be relevant. Here we consider the Nambu--Jona-Lasinio model with the inclusion of the six-quark Kobayash…
We study the effect of vorticity present in heavy ion collisions (HICs) on the temperature evolution of hot quark-gluon plasma in the presence of spin-vorticity coupling. The initial global rotation entails a nontrivial dependence of the longitudinal flow velocity on the transverse coordinates and also develops a transverse velocity component that depends upon the longitudinal coordinate. Both of these velocities lead to a 2+1 dimensional expansion of the fireball. It is observed that with finite vorticity and spin-polarization the fireball cools faster as compared to the case without vorticit…
Initial-state control of vorticity--shear competition in longitudinal $Λ$ polarization in Pb--Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.36\,\mathrm{TeV}$
School of Mathematics and Physics, China University of Geosciences (Wuhan)（中国地质大学（武汉）数学与物理学院）
We study the structure of vorticity and hydrodynamic helicity fields in peripheral heavy-ion collisions using the kinetic Quark-Gluon String Model. The angular momentum conservation within this model holds with a good accuracy. We observe the formation of specific toroidal structures of vorticity field (vortex sheets). Their existence is mirrored in the polarization of hyperons of the percent order.
By including the recently introduced thermal shear term that contributes to the spin polarization vector at local equilibrium, we determine longitudinal polarization of $Λ$ 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 qua…
A power expansion scheme is set up to determine the Wigner function that satisfies the quantum kinetic equation for spin-1/2 charged fermions in a background electromagnetic field. Vector and axial-vector current induced by magnetic field and vorticity are obtained simultaneously from the Wigner function. The chiral magnetic and vortical effect and chiral anomaly are shown as natural consequences of the quantum kinetic equation. The axial-vector current induced by vorticity is argued to lead to a local polarization effect along the vorticity direction in heavy-ion collisions.
Vorticity of matter generated in noncentral heavy-ion collisions at energies of the Nuclotron-based Ion Collider fAcility (NICA) at the Joint Institute for Nuclear Research (JINR) in Dubna is studied. Simulations are performed within the model of the three-fluid dynamics (3FD) which reproduces the major part of bulk observables at these energies. Comparison with earlier calculations is done. Qualitative pattern of the vorticity evolution is analyzed. It is demonstrated that the vorticity is mainly located at the border between participants and spectators. In particular, this implies that the r…
A formalism for describing an all-sky map of the polarization of the cosmic microwave background is presented. The polarization pattern on the sky can be decomposed into two geometrically distinct components. One of these components is not coupled to density inhomogeneities. A non-zero amplitude for this component of polarization can only be caused by tensor or vector metric perturbations. This allows unambiguous identification of long-wavelength gravity waves or large-scale vortical flows at the time of last scattering.
Using the AMPT + MUSIC+UrQMD hybrid model, we study the global and local spin polarizations of $Λ$ 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 $Λ$ hyperons due to the thermal vorticity decrease by a fa…
Local Spin Polarization in Anisotropic Gubser Flow: Suppression Mechanism and Formulation Dependence
The University of Tokyo（东京大学）；University of Science and Technology of China（中国科学技术大学）；Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）
Vorticity generated in heavy-ion collisions at energy of $\sqrt{s_{NN}}=$ 39 GeV is studied. Simulations are performed within a model of the three-fluid dynamics. A peculiar structure consisting of two vortex rings is found: one ring in the target fragmentation region and another one in the projectile fragmentation region. These rings are also formed in central collisions. The matter rotation is opposite in this two rings. These vortex rings are already formed at the early stage of the collision together with primordial fragmentation regions. The average vorticity, responsible for the global p…
We derive the Boltzmann equation and the collision kernel for massive spin-1/2 particles, using the Wigner-function formalism and employing an expansion in powers of $\hbar$. The phase space is enlarged to include a variable related to the spin degrees of freedom. This allows to reduce the transport equations of the independent components of the Wigner function to one scalar equation. To next-to-leading order in $\hbar$, we find that the collision kernel contains both local and nonlocal terms. We show that off-shell contributions cancel in the Boltzmann equation. Our framework can be used to s…
We compute the fermion spin distribution in the vortical fluid created in off-central high-energy heavy-ion collisions. We employ the event-by-event (3+1)D viscous hydrodynamic model. The spin polarization density is proportional to the local fluid vorticity in quantum kinetic theory. As a result of strong collectivity, the spatial distribution of the local vorticity on the freeze-out hyper-surface strongly correlates to the rapidity and azimuthal angle distribution of fermion spins. We investigate the sensitivity of the local polarization to the initial fluid velocity in the hydrodynamic mode…
We study the energy dependence of global polarization of $Λ$ hyperons in peripheral $Au-Au$ collisions. We combine the calculation of vorticity and strange chemical potential in the framework of kinetic Quark-Gluon String Model with the anomalous mechanism related to axial vortical effect. The earlier found effect of helicity separation implies the quadrupole structure of longitudinal vorticity. We pay special attention to the temperature dependent contribution related to gravitational anomaly and found that the preliminary RHIC data are compatible with its suppression discovered earlier in la…
We study local spin polarization in the relativistic hydrodynamic model. Generalizing the Wigner functions previously obtained from chiral kinetic theory by Y. Hidaka et al. [Phys. Rev. D 97, 016004 (2018)] to the massive case, we present the possible contributions up to the order of $\hbar$ from thermal vorticity, shear viscous tensor, other terms associated with the temperature and chemical-potential gradients, and electromagnetic fields to the local spin polarization. We then implement the (3+1)-dimensional viscous hydrodynamic model to study the spin polarizations from these sources with a…
We compute the relaxation time for quark/antiquark spin and thermal vorticity alignment in a quark-gluon plasma at finite temperature and quark chemical potential. We model the interaction of quark/antiquark spin with thermal vorticity as driven by a phenomenological modification of the elementary quark interaction with gluons. We find that in a scenario where the angular velocity of the quark-gluon plasma produced in a peripheral heavy-ion collision is small, quarks/antiquarks take a long time to align their spin with the vorticity. However, when the angular velocity created in the reaction i…
In hydrodynamic approach to relativistic heavy ion collisions, hadrons with nonzero spin, produced out of the hydrodynamic medium, can acquire polarization via spin-vorticity thermodynamic coupling mechanism. The hydrodynamical quantity steering the polarization is the thermal vorticity, that is minus the antisymmetric part of the gradient of four-temperature field. Based on this mechanism there have been several calculations in hydrodynamic and non-hydrodynamic models for non-central heavy ion collisions in the RHIC Beam Energy Scan energy range, showing that the amount of polarization of pro…
The operator-product expansion(OPE) could be employed to obtain the lowest-order, nonlocal quark scalar condensate component of gluon vacuum polarization. In particular, nonlocal quark scalar condensate can be calculated by solving Dyson-Schwinger Equation(DSE) of QCD. Then, field-theoretic aspects of the gluon vacuum polarization and nonperturbative gluon propagator will be considered in the Landau gauge of the Lorentz gauge fixing. The gluon propagator we obtained is finite in the infrared domain where the single gluon mass $m_g$ can be determined. Our results of the ratio $m_{g}/Λ_{QCD}$ th…
STAR collaboration at RHIC has recently measured the polarization of $Λ$ hyperons in non-central heavy ion collisions in the RHIC Beam Energy Scan (BES) program. The magnitude of the polarization was found to decrease from few percents at the lowest BES energies to $\approx0.2$% at the top RHIC energy. The polarization signal has been reproduced in different hydrodynamic calculations assuming a thermodynamic spin-vorticity coupling mechanism at the Cooper-Frye hypersurface. In this work an extension of our existing calculations of the $Λ$ polarization in the RHIC BES program to the top RHIC an…
We present a quantitative study of vorticity formation in peripheral ultrarelativistic heavy ion collisions at sqrt(s)NN = 200 GeV by using the ECHO-QGP numerical code, implementing relativistic dissipative hydrodynamics in the causal Israel-Stewart framework in 3+1 dimensions with an initial Bjorken flow profile. We consider and discuss different definitions of vorticity which are relevant in relativistic hydrodynamics. After demonstrating the excellent capabilities of our code, which proves to be able to reproduce Gubser flow up to 8 fm/c, we show that, with the initial conditions needed to …
Based on the three-fluid model, the global $Λ$ polarization in Au+Au collisions at 2.4 $\leq\sqrt{s_{NN}}\leq$ 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 $\sqrt{s_{NN}}\approx$ 3 GeV, if the measurements are performed with the same acceptance. The value of the polarization is …
Fermions become polarized in a vortical fluid due to spin-vorticity coupling. The spin polarization density is proportional to the local fluid vorticity at the next-to-leading order of a gradient expansion in a quantum kinetic theory. Spin correlations of two $Λ$-hyperons can therefore reveal the vortical structure of the dense matter in high-energy heavy-ion collisions. We employ a (3+1)D viscous hydrodynamic model with event-by-event fluctuating initial conditions from A MultiPhase Transport (AMPT) model to calculate the vorticity distributions and $Λ$ spin correlations. The azimuthal correl…
We present a calculation of the global polarization of Lambda hyperons in relativistic Au-Au collisions at RHIC Beam Energy Scan range sqrt{s}_NN = 7.7 - 200 GeV with a 3+1 dimensional cascade + viscous hydro + cascade model, UrQMD+vHLLE. Within this model, the mean polarization of Lambda in the out-of-plane direction is predicted to decrease rapidly with collision energy from a top value of about 2% at the lowest energy examined. We explore the connection between the polarization signal and thermal vorticity and estimate the feed-down contribution to Lambda polarization due to the decay of hi…
We propose an improved quark coalescence model for spin alignment of vector mesons and polarization of baryons by spin density matrix with phase space dependence. The spin density matrix is defined through Wigner functions. Within the model we propose an understanding of spin alignments of vector mesons $ϕ$ and $K^{*0}$ (including $\bar{K}^{*0}$) in the static limit: a large positive deviation of $ρ_{00}$ for $ϕ$ mesons from 1/3 may come from the electric part of the vector $ϕ$ field, while a negative deviation of $ρ_{00}$ for $K^{*0}$ may come from the electric part of vorticity tensor fields…
We calculated the azimuthal angle dependence of the local spin polarization of hyperons in 200 GeV Au+Au and 2.76 TeV Pb+Pb collisions in the framework of the (3+1)D viscous hydrodynamic model CLVisc with AMPT initial conditions encoding initial orbital angular momenta. We find that the azimuthal angle dependence of the hyperon polarization strongly depends on the choice of the spin chemical potential $Ω_{μν}$. With $Ω_{μν}$ chosen to be proportional to the temperature vorticity, our simulation shows coincidental results with the recent measurements at RHIC.
We study the relaxation time required for the alignment between the spin of a finite-mass quark/antiquark and the thermal vorticity, at finite temperature and baryon chemical potential, in the context of relativistic heavy-ion collisions. The relaxation time is computed as the inverse of the total reaction rate that in turn is obtained from the imaginary part of the quark/antiquark self-energy. We model the interaction between spin and thermal vorticity within the medium by means of a vertex coupling quarks and thermal gluons that, for a uniform temperature, is proportional to the global angul…
The global spin polarization of hadrons in heavy ion collisions has been measured in STAR (the Solenoidal Tracker At Relativistic heavy ion collider) experiments, which opens up a new window in the study of the hottest, least viscous and most vortical fluid that has ever been produced in the laboratory. We present a different approach to spin polarization from conventional ones: a topological realization of spin polarization through quantum vortex formation in collisions of Bose-Einstein condensates (BEC). This approach is based on the observation that the vortex is a topological excitation in…
