Polarization and Vorticity in the Quark Gluon Plasma

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 $\Lambda$ hyperon polarization along the direction of collisional angular momentum have been reported. These measurements are in broad agreement with hydrodynamic and transport-based calculations and reveal that the QGP is the most vortical fluid ever observed. However, there remain important tensions between theory and observation which might be fundamental in nature. In the relatively mature field of heavy ion physics, the discovery of global hyperon polarization and three-dimensional simulations of the collision have opened an entirely new direction of research. We discuss the current status of this rapidly developing area and directions for future research.


INTRODUCTION
Collisions between heavy nuclei at ultra-relativistic energies create a Quark-Gluon Plasma (QGP) (1,2,3,4,5), characterized by colored partons as dynamic degrees of freedom. For more than two decades, a large community has systematically studied these collisions to extract insight about quantum chromodynamics (QCD) matter under extreme conditions. The resulting field of relativistic heavy ion physics is by now relatively mature. With the early realization that the QGP in these collisions is a "nearly perfect fluid, hydrodynamics has been the dominant theoretical framework in which to study the system.
Much of the evidence for the fluid nature of the QGP has been based on the response of the bulk medium to azimuthal (to the beam direction) anisotropies in the initial energy density (6). Measured azimuthal correlations are well reproduced by modulations in the outward-directed flow fields in the hydro simulations. However, despite the fact that heavy ion collisions involve huge angular momentum densities (10 3−4 over volumes ∼ 250 fm 3 ), relatively less focus has been placed on the consequences of this angular momentum.
In any fluid, angular momentum manifests as vorticity in the flow field. The coupling between rotational motion and quantum spin can lead, in the QGP, to polarization of hadrons emitted from fluid cells, driven by the local vorticity of the cell. In 2017, the first experimental observation of vorticity-driven polarization in heavy ions was reported (7). This has generated an intense theoretical activity and further experimental study. This manuscript reviews the tremendous progress and current understanding of the vortical nature of the QGP. This line of investigation, only just now begun, represents one of the few truly new directions in the soft sector of relativistic heavy ion physics for many years.
In the next section, we place these studies into a larger context of similar phenomena in other physical systems and define geometrical conventions required for the heavy ion case. We then discuss theoretical tools employed to model the complex rotational dynamics of the plasma and the manifestation in particle polarization. In section 4, we discuss experimental measurements and observational systematics. We will see broad agreement between observation and theory, but tension in some important aspects. We conclude our review with open questions and an outlook.

QUARK GLUON PLASMA, HYDRODYNAMICS AND VORTICITY
That the QGP produced in collisions of nuclei at relativistic energies is, for a transient of around 10 −22 seconds, a nearly perfect fluid is based on the accumulated evidence collected over a time span of more than ten years. The main fact is that this fluid breaks up into hadrons in a state very close to local thermodynamic equilibrium (8) at a temperature very close to the pseudo-critical QCD temperature of 160 MeV (9), (10). Local thermodynamic equilibrium implies that momentum spectra of produced hadrons are very well reproduced by the assumption of a local Bose-Einstein or Fermi-Dirac distribution function (for vanishing chemical potentials): f (x, p) = 1 exp[β · p] ± 1 1.
where β = (1/T )u(x) is the four-temperature vector including temperature and the fourvelocity hydrodynamic field u(x). The formula 1 applies to the local fluid cell, and should be integrated thereafter over the "freeze-out" 3D-hypersurface (see figure 2) defined as the boundary of local thermodynamic equilibrium, giving rise to what is well known in the field as "Cooper-Frye" formula (11). Indeed, this is analogous to the last-scattering surface in the cosmological expansion where the background electro-magnetic radiation froze out. Apparently, local thermodynamic equilibrium is achieved -and a plasma at finite temperature is formed -at quite an early time in the process (see figure 2). This is confirmed by the success of the hydrodynamic equations in determining the flow field u(x) in eq. 1. Particularly, the model is able to successfully account for the observed anisotropies of the momentum spectra in the transverse plane perpendicular to the beam line (refer to fig. 1) Left: A collision of two nuclei in space-time diagram. In the hydrodynamic model, local equilibrium is believed to occur on the hyperbola Σeq which is the initial 3D hypersurface of the thermalized QGP and to cease at the 3D hypersurface Σ F O Right: Distribution of the amplitude of thermal vorticity | | = | | µν µν | at the freeze-out hypersurface Σ F O , calculated with the ECHO-QGP code under the same conditions as in (12) with a freeze-out temperature of 130 MeV. The red dashed line is the contribution of the space-like part while the blue dashed line that of the time-like part.
as a function of the azimuthal angle. These anisotropies -encoded in the Fourier coefficients vn -have led to the conclusion that the viscosity of the QGP must be very small as compared to the entropy density, close to the conjectured universal lower bound of /4π (13).
Recently, the exploration of the QGP made a significant advance. The measurement of polarization of emitted hadrons made it clear that a new probe is accessible which may give a wealth of new and complementary information. In particular, in the hydrodynamic paradigm, while the momentum spectra provide direct information about the velocity and the temperature field, polarization is linked to the vorticity and more generally to the gradients of these fields (see Section 3). This is an interesting aspect. In ideal hydrodynamics, particle distribution function, such as eq. 1. is determined by intensive thermodynamic quantities of the local cell in the local rest frame, such as temperature and chemical potentials related to the various charges (baryon number, electric charge, etc). Likewise, assuming that spin degrees of freedom locally equilibrate, vorticity plays the role of a potential determining the "spin charge" distribution of particles, e.g. number of spin up versus spin down (see Section 3). Vorticity should be then considered a further intensive thermodynamic quantity needed to describe locally the fluid. In a sense, vorticity is an extra substructure of a hydrodynamic cell. This property makes polarization a very sensitive probe of the dy-namical process leading to the QGP formation and of its evolution. As has been mentioned in the Introduction, this field has only just begun and all the developments that polarization may lead to can be hardly envisioned for the present.

Vorticity and polarization: overview
While the QGP formed in heavy ion collisions is only a few times larger than a nucleus, "heavy" ions are utilized in order to form a bulk system, significantly larger than the confinement volume characteristic of a hadron. Otherwise stated, the system which is formed is much larger than the typical microscopic interaction scale, and such a separation of scales ("hydrodynamic limit") makes it possible to talk about a fluid and to use hydrodynamics as an effective tool to describe its evolution; in the hydro language, the Knudsen number is sufficiently small (14). Under such circumstances, the variation of the flow field in space and time can be slow enough to be dealt with as "macroscopic" motion of bulk matter and vorticity as well. As it will become clear later on in Section 3, the vortical structure is probed by the spin of hadrons that "freeze out" from local fluid cells in a state of local thermodynamic equilibrium, as has been discussed above. More specifically, the presence of a vortical motion (as well as an acceleration and a temperature gradient) entails a modification of 1 such that the distribution function becomes non-trivially dependent on the spin degrees of freedom.
That spin and vorticity are tightly related is not a new insight, and yet there are relatively few examples of physical systems which show the effect of the coupling between mechanical angular momentum of bulk matter and the quantum spin of particles that comprise (or emerge from) that matter.
Two seminal measurements were reported nearly simultaneously more than a century ago. Barnett (15) observed that an initially-unmagnetized steel cylinder would generate a magnetic field upon being rotated. In the same year, Einstein and de Haas (16) observed the complementary effect: a stationary unmagnetized ferromagnetic object will begin to rotate upon introduction of an external magnetic field. In both cases, the phenomenon is rooted in the conservation of total angular momentum on one hand and on equipartition of angular momentum, that is thermodynamic equilibrium, on the other. In the Barnett effect, the angular momentum which is imparted through a forced rotation gets partly distributed to the quantum spin of the constituents and, once thermodynamic equilibrium is reached, a stable magnetic field is generated as a consequence of the polarization of matter. In the Einstein-de Haas effect, the external magnetic field implies, at thermodynamic equilibrium, a polarization of matter, whence an angular momentum; if the magnetic field does not provide torque, the body should start spinning as to conserve the initial vanishing angular momentum. Indeed, a quantitative understanding of these phenomena was possible only a decade later, with the discovery of the electron spin and anomalous gyromagnetic ratio.
Another example is found in low-energy heavy ion reactions, in which a beam with kinetic energy of E kin ∼ 30 MeV per nucleon is incident on a stationary target. (In highenergy physics terms, √ sNN − 2mp ≈ 15 MeV, where mp is the proton mass.) This is the regime of quasi compound nucleus formation, in which the short-lived system is assumed to rotate as a whole, to first order. At "high" beam energies E kin 50 AMeV, projectile fragments are expected to experience positive deflection (c.f. section 2.2) due to collisional and bulk compression during the collision. At lower energies, collisions are Pauli-suppressed and attractive nuclear surface interactions are expected to produce an orbiting motion that leads to negative deflection. Disentangling the interplay between these physical mechanisms requires determination ofĴsys. This was achieved by correlating (17,18) the circular polarization of γ rays with forward fragment deflection angles. These measurements represent the first observation of "global polarization" in (nonrelativistic) heavy ion reactions.
In the above cases, the bulk mechanical motion is basically rigid-body rotation. 1 Only recently (19) has mechanically-induced spin polarization been observed in a fluid. Liquid Hg flowing through a channel experiences viscous forces along the channel walls, generating a local vorticity field whose strength and direction varies as a function of position. Hydrodynamic vorticity-spin coupling then produces a corresponding electron polarization field, which was measured using the inverse spin Hall effect (20). This experiment, where both the vorticity and the induced polarization are observable, is important to establish the phenomenon in fluids.
With respect to all above listed cases, polarization in relativistic heavy ions possesses two unique features. First of all, its measurement is not mediated by a magnetic field (like in the Barnett effect) but the mean spin of particles is directly observed; this is not possible in ordinary matter. Secondly, and maybe more importantly, the system at hand -QGP at very high energy -is almost neutral by charge conjugation, i.e. C-even. If it was precisely neutral, the observation of polarization by magnetization would be simply impossible because particles and antiparticles have opposite magnetic moment. In fact, as we will see, Λ andΛ in relativistic nuclear collisions at high energy have almost the same mean spin, which is an unmistakable signature of a thermal-mechanical driven polarization. If the electro-magnetic, or any other C-odd mean field, was responsible for this effect, the sign of the mean spin vector components would be opposite. Hence, altogether, while for non-relativistic matter (without anti-matter) it is impossible to resolve polarization by rotation and by magnetization -what lies at the very heart of the Barnett and Einsteinde Haas effects -in relativistic matter, because of the existence of antiparticles, they can be distinguished and QGP is the first relativistic system where the distinction has been observed.

Geometry of a nuclear collision
The left panel of figure 1 sketches the geometry of a heavy ion collision in its center of mass frame, prior to contact. Designating one nucleus the beam and the other the target 2 , the impact parameter b points from the center of the target to the center of the beam, perpendicular to the beam momentum p beam . The vectors b and p beam span the reaction plane, indicated by the grid. The total angular momentum of the collision Jsys = b × p beam .
The right panel sketches the situation after the collision. In the participant-spectator model (21) commonly used at high energies, a fireball at midrapidity is produced by the sudden and violent deposition of energy when "participant" nucleons overlap and collide. Meanwhile, projectile nucleons that do not overlap with oncoming nucleons in the target are considered "spectators" and continue with their forward motion essentially unchanged, later to undergo nuclear fragmentation.
However, this distinction is not so sharp in reality, as the forward "spectators" do receive a sideways repulsive impulse during the collision, as indicated by the deflected momentum arrows in the right panel of figure 1. The case shown in the figure is deflection to "positive" angles, to distinguish the case at much lower energy (e.g. 18) where attractive forces can produce negative deflection. In the parlance of relativistic collisions (22), the positive deflection corresponds to positive directed flow (v1) in the forward direction (v1 > 0 when y ≈ y beam ).
This deflection is important. While we are especially interested in the vortical structure of the fireball at midrapidity, we need to know the direction of the angular momentum, which must by symmetry give the average direction of vorticity. Forward detectors are used to estimateĴsys event-by-event, as discussed below.
A final note about coordinate systems and conventions. It is common to define a coordinate system in whichẑ ≡p beam andx ≡b. In this case,Ĵ = −ŷ; The azimuthal angle ofb aboutp beam in some coordinate system (say, the floor of the experimental facility) is often referred to as the reaction plane angle ΨRP . The aforementioned forward detectors use spectator fragment deflection to determine the event plane angle ΨEP,1. Standard techniques have been developed (22) to determine the event plane and the resolution with which it approximates ΨRP , i.e. the directionb.
Since the size and angular momentum of the QGP fireball depends on the overlap between the colliding nuclei, an estimate of the magnitude of the impact parameter is also important. Standard estimators (23), typically based on the charged particle multiplicity measured at midrapidity, quantify the "centrality" of each collision in terms of fraction of inelastic cross-section. Head-on (| b| = 0) and barely-glancing collisions are said to have centrality of 0% and 100% respectively.

POLARIZATION IN RELATIVISTIC HEAVY ION COLLISIONS: THEORY
The main purpose of the theoretical work is to calculate the amount of polarization of observable particles once the initial condition of the collision is known, that is the energy and the impact parameter of the two nuclei. The final outcome depends on the model of the collision (see Section 2) and on how the initial angular momentum may induce a global polarization of the particles.
The first calculation on global polarization in relativistic heavy ion collision was presented in ref. (24) based on a perturbative-QCD inspired model where colliding partons get polarized by means of a spin-orbit coupling. The amount of predicted polarization of Λ baryons was originally large (around 30%) and corrected thereafter by the same authors to be less than 4% (25). Besides the apparent large uncertainty, the main problem of the collisional approach -at the quark-gluon level -is the difficulty of reconciling it with the evidence of a strongly interacting QGP, which makes the kinetic approach dubious. Another problem is how to transfer the polarization at quark-gluon level to final hadrons, which requires a detailed hadronization model and more assumptions. This scenario, however, has been further developed and it will be addressed later in this Section.
About the time when the first measurement of global Λ polarization at RHIC appeared (26) setting an upper limit of few percent, the idea of a polarization related to hydrodynamic motion, and particularly vorticity, was put forward (27,28). If the QGP achieves and maintain local thermodynamic equilibrium until it decouples into freely streaming noninteracting hadrons and if this model -as discussed in the Introduction and in Section 2 -is very successful to describe the momentum spectra of particles, there is no apparent reason why it should not be applicable to the spin degrees of freedom as well. Hence, polarization must be derivable from the very fact that the system is at local thermodynamic equilibrium, whether in the plasma phase or in the hadron phase just before they freeze-out. This idea establishes a link between spin and vorticity (more precisely thermal vorticity as later described) and makes it possible to obtain quantitative predictions at hadronic level without the need of a mechanism to transfer polarization from partons to hadrons. The actual quantitative relation for a relativistic fluid was first worked out in global equilibrium (29), then at local equilibrium for spin 1/2 particles in ref. (30).
For a particle with spin 1/2 the mean spin vector is all is needed to describe polarization (this is not the case for spin greater than 1/2) and the relativistic formula was found to be, at the leading order (30): where p is the four-momentum of the particle, and nF = (1 + exp[β · p − µQ/T ] + 1) −1 is the Fermi-Dirac distribution with four-temperature β like in eq. 1 and with chemical potential µ coupled to a generic charge Q. The integration should be carried out over the freeze-out hypersurface (see fig. 2); in a sense, in the heavy ion jargon this can be called the "Cooper-Frye" formula for the spin. The key ingredient in equation 2 is the so-called thermal vorticity tensor (x), which reads: i.e. the anti-symmetric derivative of the four-temperature. This quantity is adimensional in natural units and it is the proper extension of the angular velocity over temperature ratio mentioned in the Introduction. Hence the spin depends, at the leading order, on the gradients of the temperature-velocity fields, unlike momentum spectra which depend, at the leading order, on the temperature-velocity field itself. Thereby, polarization can provide a complementary information about the hydrodynamic flow with respect to the spectra and their anisotropies. The formula 2 applies to anti-particles as well, so that in a charge-neutral fluid the spin vector is expected to be the same for particles and anti-particles, which is a remarkable feature as emphasized in Subsection 2.1. It is worth pointing out that formula 2 implies that a particle within a fluid in motionvat some space-time point x gets polarized according to (natural constants have been purposely restored): where γ = 1/ 1 − v 2 /c 2 and all three-vectors, including vorticity, acceleration and velocity, are observed in the particle rest frame. The decomposition 4 makes it clear what are the thermodynamic "forces" responsible for polarization: the last term corresponds to the acceleration-driven polarization, its expression is reminiscent of the Thomas precession and it is indeed tightly related to it (particle moving in an accelerated flow); the second term is the relativistic expression of polarization by vorticity; the first term is a polarization by combination of temperature gradient and hydrodynamic flow and is, to the best of our knowledge, a newly found effect.
First hydrodynamic calculations based on formula 2 predicted a global polarization of Λ baryons of a few percent at √ s NN = 200 GeV (31), hence compatible with the previous experimental limit. The new measurements with a larger statistics then confirmed that polarization value is of such order of magnitude. Formula 2 then became a benchmark for most phenomenological calculations of the polarization in heavy ion collisions.
We will now review in some detail the status of the theoretical understanding of the polarization in relativistic fluids and in nuclear collisions particularly.

Polarization in statistical mechanics
The calculation of spin at global or local thermodynamic equilibrium requires a quantum framework, spin being inherently a quantum observable. The most appropriate framework is thus quantum statistical mechanics, and since we are dealing with a relativistic fluid, in a relativistic setting. However, many quantitative features can be found out starting from the simplest non-relativistic case.
As a simple illustrative case, consider a rotating ideal gas with angular velocity ω within a cylindrical vessel of radius R. At equilibrium, the statistical operator 3 reads (32): Since the particles are free, both the Hamiltonian and angular momentum operator are the sum of individual single-particle operators and the density operator can be factorized. Since the total angular momentum includes both orbital and spin part, that is Ji = Li + Si for each particle i, the spin density matrix for a particle with momentum p turns out to be: implying a mean spin vector of particles: where the last expression is the leading order for small ratios ω/T . Equation 6 also implies that the so-called alignment Θ00 for spin 1 particles is quadratic in ω/T at the leading order, which puts a severe limitation to its observability in relativistic heavy ion collisions (see also subsection 5.3).
In the more general, relativistic case, the equilibrium operator 5 is replaced by (33): where b is a constant time-like four-vector and is the thermal vorticity which, at global thermodynamic equilibrium ought to be constant; P and J are the four-momentum and angular momentum-boost operators. It is important to point out that thermal vorticity includes both vorticity and acceleration besides the gradient of the temperature. For instance, at global equilibrium, it turns out (34): where u is the four-velocity, A the four-acceleration and ω the vorticity four-vector. The entanglement of vorticity and acceleration is a typical signature of relativity, much like that of electric and magnetic field in the electromagnetic tensor Fµν . An intermediate step towards formula 2 is the free single-particle quantum relativistic calculation. In this case, for a single particle, the operator 8 leads to the spin density matrix (35): where D S () stands for the (2S + 1)-dimensional representation of the group SL(2,C) universal covering of the Lorentz group, ΣS are the (2S + 1) × (2S + 1) matrices representing the Lorentz generators, [p] is the so-called standard Lorentz transformation which takes the unit time vectort into the direction of the four-momentum p (36). The spin density matrix in eq. 10 implies a mean spin four-vector, for sufficiently low values of the thermal vorticity: which is a direct relativistic extension of the formula 7 (37).
For a system of many particles, just like those emerging from a nuclear collisions, one would take the formula 11 and average it over the different particle-emitting spots, i.e. over the hydrodynamic cells of the freeze-out hypersurface. The result is the formula 2 except the factor (1 − nF ). Indeed, the latter is the typical signature of Fermi statistics and it naturally comes out in a quantum-field theoretical calculation. Indeed, this was the approach taken into the original calculation at local thermodynamic equilibrium in ref. (30) where the density operator is the extension of equation 8: where βν (x) is the four-temperature function (dependent on space and time), ζ(x) is the ratio between chemical potential and temperature and T , j are the stress-energy tensor and current operators respectively. The integration should be done over the freeze-out 3D-hypersurface (see figure 2) which is supposedly the boundary of local thermodynamic equilibrium. Indeed, calculating the mean spin vector from the density operator 12 is not straightforward and some key assumptions are needed to get to the formula 2. The most important is the usual hydrodynamic limit: microscopic lengths should be much smaller than the hydrodynamic scale, that is β(x) should be a slowly varying function. The second main assumption used in the original calculation (30) was an ansatz for the covariant Wigner function at global equilibrium with acceleration and rotation, that is with the density operator 8. In spite of these assumptions, there are good reasons to believe that the exact formula at the leading order in thermal vorticity in a quantum field theory calculation would precisely be equation. 2. Indeed, the same formula was found with a different approach, based on the expansion of the Wigner equation (38) and, furthermore, it is the only possible linear expression in yielding the correct single-particle 11. and non-relativistic limit. For instance, a term proportional to µν pν , even if orthogonal to p, would not yield correct limiting cases. What is still unknown is the exact global equilibrium formula at all orders in thermal vorticity including quantum statistics.
While the local equilibrium calculation of the spin density matrix and related quantities at leading order seems to be established at the most fundamental level of quantum field theory, some questions remain to be addressed. It is not known how large are the higher order terms in thermal vorticity at local equilibrium, nor we have an exact solution at global equilibrium with the density operator 8 including quantum field effects, namely quantum statistics. Very little is known about the the dissipative, non local-equilibrium terms, and their magnitude. Recently, a phenomenological approach to spin dissipation has been taken (39) generalizing a familiar classical method to constrain constitutive equations in dissipative hydrodynamics, based on the positivity of entropy current divergence (40). It remains to be understood whether such a method includes all possible quantum terms in the entropy current and if it agrees with the most fundamental quantum approach to dissipation, based on Zubarev non-equilibrium density operator (41). Another very recent study (42) studied the possible dissipative terms of the spin tensor in the relaxation time approximation.

Hydrodynamic calculations
The main goal of hydrodynamic calculations is to provide the key input to the formula 2, that is the thermal vorticity at the freeze-out hypersurface. In principle, the thermal vorticity field depends on the assumed initial conditions of the hydrodynamic calculations, on the equation of state, on the hydrodynamic constitutive equations and on the freeze-out conditions. Nevertheless, different hydrodynamic calculations have provided similar results, which is reassuring regarding the robustness of theoretical computations of polarization.
It is important to stress that polarization studies demand a 3+1D hydrodynamic simulation. This is a crucial requirement because the components of the thermal vorticity driving the projection of the mean spin vector along the total angular momentum involve the gradients of the longitudinal flow velocity, which are neglected by 2+1D codes.
A common feature of all calculations is the fact that the values of thermal vorticity are, on the average, sufficiently less than 1 so as to justify a linear approximation in the relation between mean spin vector and thermal vorticity (see e.g. eq. 7); this is shown in the histogram in figure 2. Nevertheless, a role of quadratic corrections cannot be excluded and it is yet to be studied.
The codes that have been used so far to calculate polarization based on formula 2 are few: 1. A 3+1D Particle in Cell simulation of ideal relativistic hydrodynamics (43). All published calculations of polarization assume peculiar initial conditions for heavy ion collisions, implying a non-vanishing initial vorticity. 2. A 3+1D code implementing relativistic dissipative hydrodynamics, ECHO-QGP (44) with initial conditions adjusted to reproduce the directed flow as a function of rapidity (45). 3. A 3+1D code implementing relativistic dissipative hydrodynamics, vHLLE (46) with initial state determined by means of a pre-stage of nucleonic collisions, and including a post-hadronization rescattering stage, all adjusted to reproduce the basic hadronic observables in relativistic heavy ion collisions, that is (pseudo)rapidity and transverse momentum distributions and elliptic flow coefficients. 4. A 3+1D code implementing relativistic dissipative hydrodynamics, CLVisc (47) with initial conditions provided by another transport-based simulation package AMPT (48).
Furthermore, many calculations of polarization in literature are based on the coarse-graining of the output provided by the transport-based simulation code AMPT (48) to obtain the thermal vorticity field in eq. 2; we will refer to these calculations as transport-hybrid.
Overall, while the global polarization is in excellent quantitative agreement with the hydrodynamic calculations based on 2, the azimuthal dependence of the polarization along angular momentum and the sign of the longitudinal component disagree with the data (c.f. sections 4.3 and 4.3). These issues will be discussed later in Section 5.

Effects of decays and rescattering
Most of the calculations presented in literature involve the primary Λ, i.e. those which are emitted from the freeze-out hypersurface. However, they are just a fraction of the measured Λ's, about 25% at √ s NN = 200 GeV according to statistical hadronization model estimates (49), while most of them are decay products of higher lying states, such as Σ 0 , Σ * , Ξ etc. Those states are expected to be polarized as well, according to the formula 2 with the suitable spin-dependent coefficient (see e.g. eq. 11), hence with the same momentum pattern as for the primary Λ's. The secondary Λ from decays of polarized particles turns out to be polarized in turn and its polarization vector depends on the properties of the interaction responsible for the decay (strong, electromagnetic, weak) and on the polarization of the decaying particle. The formula for the global polarization inherited by the Λ's in several decay channels was obtained in ref. (37) and its effect studied in (50). While single channels involve a sizeable correction to the primary polarization, the overall effect is small, of the order of 10% or so. This result was confirmed by more detailed studies where the polarization transfer in 2-body decays producing a Λ hyperon was determined as a function of momentum (51,35). Surprisingly, the combination of relative production rates of different hyperons, their decay branching ratios and the coefficients of the polarization transfer produce an accidental cancellation of the contribution of secondary Λ's polarization so that the dependence of polarization as a function of momentum is almost the same as predicted for primary Λ's alone (51,35).
While the contribution of secondary decays is under control, little is known about the effect of post-hadronization secondary hadronic scattering after the hydrodynamic motion ceases. In general, one would naturally expect an overall dilution of the primary polarization. However, it has been speculated (52) that final-state hadronic rescattering could generate some polarization and a model was put forward in ref. (53) showing that initially unpolarized hyperons in pA collisions can become polarized because of secondary interactions. However, the same model applied to AA yields a secondary polarization consistent with zero (54).

Kinetic models
If, for some reason, spin degrees of freedom relax more slowly than momentum, local thermodynamic equilibrium is not possibly a good approximation and the calculation of polarization becomes more complicated. A possible substitute theoretical approach is kinetic theory. However, as has been mentioned, near the pseudo-critical temperature, the QGP is a strongly interacting system for which a kinetic description is dubious because the thermal wavelength of partons is comparable to their mean free path; particles interact so strongly that they are not free for most of their time. Notwithstanding, one may hope that kinetic theory provides a good approximation for the spin degrees of freedom if the spin-orbit coupling is weak. Recent estimates of the spin-flip rate in perturbative QCD imply, though, indicate a too large equilibration time (55) so that non-perturbative effects appear to be essential.
A formulation of relativistic kinetic theory with spin dates back to De Groot and collaborators (56), and it has been the subject of intense studies over the past few years. While the development of a relativistic kinetic theory of massless fermions was motivated by the search of the Chiral Magnetic Effect (57,58), the corresponding theory for massive fermions is mostly motivated by the observation of polarization. The goal of the relativistic kinetic theory of fermions is the study of the evolution of the covariant Wigner function, which extends the notion of the phase space distribution function of relativistic Boltzmann equation. For free particles this reads: where Ψ is the Dirac field, A, B are spinorial indices and : denotes normal ordering; this definition should be changed to make it gauge invariant in quantum electrodynamics. Most recent studies aimed at a formulation of the covariant Wigner function kinetic equations in a background electromagnetic field (59,60,61,62,63) at some order in . A different approach was taken in ref. (64), where the polarization rate was obtained including the spin degrees of freedom in the collisional rate of the relativistic Boltzmann equation.
Kinetic theory with spin is in a theoretical development stage and has not yet produced stable numerical estimates of polarization in heavy ion collisions. However, important steps toward this goal have been recently made. In ref. (65) an estimate of the evolution equation of the spin density matrix in perturbative QCD has been obtained. Computing tools are also being developed for the numerical solution of relativistic kinetic equations (66).
A sensitive issue of this approach is how to transfer the calculated polarization of partons to the hadrons, which is not relevant for the hydrodynamic-statistical model, see the discussion at the beginning of this Section. More generally, there is a gap between the perturbative, collisional quark-gluon stage and the hadronic final state which is highly nontrivial and needs to be bridged.

Spin tensor and spin potential
A very interesting theoretical issue concerned with the description of spin effects in relativistic fluids, is the possible physical separation between orbital and spin angular momentum. A similar discussion has been going on for several years in hadronic physics in connection with the proton spin studies (67). A comprehensive introduction and discussion of the subject is beyond the scope of this work, we refer the reader to the specialized literature.
In Quantum Field Theory, the angular momentum current has in general two contributions: a so-called orbital part involving the stress-energy tensor and a spin part involving a rank three tensor S λ,µν called spin tensor. However, this separation seems to be unphysical and one can make a transformation of the stress-energy and the spin tensor so as to make the current all orbital, obtaining the so-called Belinfante stress-energy tensor, with the total angular momentum unchanged. This transformation is called pseudo-gauge transformation (68) and it looks much like a gauge transformation in gauge field theories where the stress-energy and the spin tensor play the role of gauge potentials, while the total energy-momentum P µ and angular momentum-boost J µν are gauge-invariant. The question is whether an observation of a polarization in the QGP breaks pseudo-gauge invariance, making it possible to single out a specific spin tensor. This would be obviously a breakthrough with remarkable consequences, as it would have an impact on fundamental physics, such as relativistic gravity theories.
Indeed, the first derivation of the formula 2 made use of a specific spin tensor and this has led to some confusion, even in the original paper (30). In fact, it was later observed (69) that the resulting expression of the polarization is the same regardless of the spin tensor used, among the most common choices. It has recently become clear that the definition of spin density matrix and of the spin vector (70,71) in Quantum Field Theory do not indeed require any angular momentum or spin operator, just the density operator and creationdestruction operators (35); so, their expressions are completely independent of the spin tensor. In fact, the mean value of the polarization may depend on the spin tensor, insofar as the density operator does. At global thermodynamic equilibrium, the density operator 8 is manifestly independent of the spin tensor because only the total angular momentum appears, but in the case of local thermodynamic equilibrium, the density operator 12 is not invariant under a pseudo-gauge transformation (72). Then, in principle, one might be able to distinguish between two spin tensors by measuring the polarization. Of course, this is a principle statement because, in practice, there are many uncertainties limiting the accuracy of the theoretical predictions (e.g. the hydrodynamic initial conditions) and it is not clear yet to what extent the measurements could solve the issue.
The inclusion of the spin tensor in relativistic hydrodynamics has been explored in some detail by W. Florkowski et al. in a series of papers (73),(70) and a first hydrodynamic calculation of polarization presented in a simplified boost-invariant scenario (74). As far as the heavy ion phenomenology is concerned, a general comment is in order for the spin tensor scenario: an extended version of relativistic hydrodynamics requires six additional fields (the anti-symmetric spin potential Ωµν ) which in turn need six additional initial and boundary conditions, which are completely unknown in nuclear collisions. Polarization measurements could then be used to adjust them, but this would strongly reduce the probing power of polarization in all other regards.

Contribution of the electro-magnetic field
As has been mentioned, an important feature of the statistical-thermodynamic approach is that polarization is independent of the charge of the particles for a charge-neutral fluid. This has been confirmed by the measurements, which essentially find the same magnitude and sign for Λ andΛ polarization (see figure 3 later in this work). Indeed, for a fluid with some charge current, a difference in the polarization of particle and anti-particle is encoded in the Fermi-Dirac distributions in eq. 2 in that the e.g. baryon chemical potential is larger at lower energy, favouring theΛ's polarization through the factor nF (1 − nF ) in the numerator (38). However, the known values of baryon chemical potential/temperature ratios at the relevant collision energies imply a much smaller difference in the polarization than observed. A possible source of particle-antiparticle polarization splitting is the electro-magnetic field, which would lead -at local equilibrium -to a modification of the formula 2 with thermal vorticity µν replaced by (37): with µ the particle magnetic moment. Indeed, in peripheral heavy ion collisions a large electro-magnetic field is present at the collision time which may steer the spin vector of Λ andΛ and lead to a splitting of polarization, their magnetic moments being opposite. Therefore, the polarization splitting might be taken advantage of to determine the magnitude of the electro-magnetic field at the freeze-out (or earlier if the relaxation time is not small) (37) or its lifetime (75). Pinning down the electro-magnetic field would be a very important achievement in the search of local parity violation in relativistic heavy-ion collisions (76) through the so-called Chiral Magnetic Effect (57,58). However, alternative explanations of the splitting have been proposed and this feature needs to be explored experimentally and theoretically. We will return to this in section 5.2.

POLARIZATION IN RELATIVISTIC HEAVY ION COLLISIONS: OBSERVATIONS
As of this writing, there is only a handful of measurements of spin polarization in relativistic heavy ion collisions. These measurements require excellent tracking and vertex resolution in the region of interest (typically midrapidity); large coverage and good particle identification to measure decay products; high statistics to measure relatively small correlation signals; and a suite of detectors to correlate forward-rapidity momentum anisotropies with midrapidity decay topologies. Several such experiments exist today, and more will soon be commissioned. The initial measurements described here will eventually be part of a fuller set of mapped systematics.

Measuring polarization
If spin is locally equilibrated, as we have discussed, all hadrons with spin will be polarized. However, while polarimeters (77) may directly detect the polarization of particles in very clean environments, their use is infeasible in a final state involving thousands of hadrons. Recording the debris from the midrapidity region in a heavy ion collision usually involves large tracking systems (e.g. 78). A particle's polarization may be determined by the topology of its decay into charged particles, if the angular distribution of daughters' momenta is related to the spin direction of the parent.
For weak parity-violating hyperon decays with spin and parity 1 2 + → 1 2 + + 0 − , the daughter baryon is emitted preferentially in the direction of the polarization vector (P * H ) of the parent, as (79) wherep * D is a unit vector pointing in the direction of the daughter baryon momentum, and ξ * is the angle between thep * D and the polarization direction. Here and throughout, an asterisk ( * ) denotes a quantity as measured in the rest frame of the decaying parent. The decay parameter αH depends on the hyperon species (80).
The general task to extract polarization from experimental data is to identify a potential direction, sayn (specific examples discussed below). The ensemble-averaged projection of the daughter baryon's momentum alongn gives the projection of P: First measurements (26,7,81,82,83) of polarization in relativistic heavy ion collisions have used Λ → p + π − (Λ → p + π + ) decays. The decay parameter for an antiparticle is expected and observed (80,84) to be of equal magnitude and opposite sign of the corresponding particle within measurement uncertainties. 4 Polarization of other hadronic species may also also be measured, in principle. The reduced efficiency associated with identifying two displaced vertices, as well as the reduced yield of doubly strange baryons makes using Ξ − (α Ξ − →Λ+π − = −0.458) more difficult. The neutral decay of Ξ 0 (α Ξ 0 →Λ+π 0 = −0.406) is more difficult still. Relatively low production rates (85) and very small αΩ values (80) strongly disfavor the use of triply-strange Ω baryons.
For spin-1/2 particles, polarization is entirely described by the mean spin vector, which has been extensively discussed in this work. For particles with spin > 1/2, a full description of the polarization state requires more quantities; in practice, one should quote the full spin density matrix Θrs(p) (see Section 3). Particularly, for spin 1 particles, a quantity independent of the mean spin vector related to the polarization state is the so-called alignment indicates spin alignment, though by symmetry it is impossible to distinguish the sign in S n. The 2-particle decay topology of a vector meson is related to the alignment according to (87): where ξ * is defined as in equation 15. At local thermodynamic equilibriu, 1 3 −ρ00 is quadratic in thermal vorticity to first order, as mentioned in section 3.
Thus far, the first measurements of global spin alignment of vector mesons in heavy ion collisions are difficult to understand in a consistent picture. We discuss these in subsection 5.3, and focus here on hyperon polarization.

Global hyperon polarization -observation
By symmetry, the average vorticity of the QGP fireball must point in the direction of the fireball's angular momentum JQGP, and on average JQGP Jsys (c.f. figure 1). Similarly, even without appealing to a connection to vorticity, when averaged over all particles, symmetry demands an average (over all emitted particles) polarization aligned withĴsys. In the current context, the "global polarization" of a subset of particles refers to the use of n =Ĵsys in equation 16.
As discussed in section 2.2, the momentum-space anisotropy of particle emission is used (22) to extract an event plane angle ΨEP,1 which approximates the reaction plane with some finite resolution. Standard methods have been developed (22) to correct for the effects of this resolution on measured asymmetries in the emission pattern about the beam  Left: The vectors and angles involved in an analysis of hyperon polarization along the angular momentum of the collision are shown. In the lab coordinate system (not shown), the azimuthal angle ofb is defined to be Ψ RP . Thus, the angle between the projection ofp * D andb is φ * D − Ψ RP . The minus sign on the angle indicated arises from the fact that azimuthal angles are measured counterclockwise about the beam axis. Right: The energy dependence of Λ and Λ global polarization at mid-rapidity from mid-central Au+Au (20-50%) or Pb+Pb (15-50%) collisions. Data (7,26,81,83) are compared to polarization simulations of viscous hydrodynamics (50); partonic transport (88); hadronic transport (89); chiral-kinetic transport plus coalescence (90); and a three-fluid hydro model applicable at lower energies (91). Experimental data points have been corrected for the recent change in α Λ , as discussed in section 4.1. For (50) and (88), the values shown represent both primary and feed-down hyperons (c.f 37). See text for details. axis, so it is convenient to rewrite equation 16 as 5 (26) 18.
Here, φ * D and θ * D are the angles between the daughter momentum andb andp beam , respectively, and in the last step, a trigonometric relationship between the angles is used. These angles are shown in figure 3.
Integrating 6 over polar angle θ * where in the final step, the experimentally-determined event plane angle replaces the reaction plane angle, accounting for the resolution with a calculable correction factor R EP (22). 5 See reference (92) for a discussion of significant experimental challenges to perform the average in equation 18 6 Detectors in which Λs are reconstructed usually do not measure the charged daughters at very forward angles at collider energies. Corrections (26,93) on order ∼ 3% (94) are applied in order to account for this.
The resolution with whichĴsys is measured is critical. Polarization affects daughter anisotropies only at the few percent level, and statistical uncertainties can dominate experimental results. Using equation 19. the statistical uncertainty on polarization goes as where NH is the total number of hyperons analyzed in the dataset. This dependence is generically true for any measurement that involves correlation with the first order event plane orĴsys. Increasing the resolution by a factor of two (95) thus decreases the required duration of an experimental campaign four-fold. Figure 3 shows the world dataset of P Λ,Ĵ and P Λ,Ĵ as a function of collision energy for semi-peripheral collisions. As discussed in section 4.1, the recent change in accepted value for αΛ requires a rescaling of the published experimental values. From a maximum of ∼ 1.5% at √ sNN = 7.7 GeV, the polarizations fall smoothly 7 with energy. At LHC energies, they vanish within experimental uncertainties. Strikingly, all available hydrodynamic and transport calculations reproduce the observations in sign, magnitude and energy dependence, as discussed in section 3.2. This is nontrivial; since they all use formula 2 it means that they all predict a very similar thermal vorticity field. These models have been to some extent "tuned" to reproduce earlier observations such as anisotropic flow (6), which is sensitive to the bulk motion of fluid cells from which particles emerge; it is thus satisfying that they produce similar and correct predictions for this more sensitive observable.
Clearly, | Jsys| increases with increasing √ sNN , and transport calculations (96,97) predict that about 20% of this angular momentum is transferred to the QGP fireball. While some early calculations (25) predicted an increased polarization at high collision energies, a strongly decreasing trend is produced by most hydrodynamic (50,98) and transporthybrid codes in which the thermal vorticity field is obtained through a coarse-graining procedure (96,99,100,101,102). Driving mechanisms may include increased temperature (103) at increased √ sNN ; increases in evolution timescale (50,90); vorticity migrating to forward rapidity (28,96,91), perhaps due to reduced baryon stopping / increased transparency at high energy (50); an increased fluid moment of inertia due to increased mass-energy (96); reduced longitudinal fluctuations and boost-invariance at high energy (104).
In addition to the overall energy dependence, the data in figure 3 suggests a fine splitting between particles and antiparticles at low √ sNN . While statistically not significant at any given energy, very important physical effects are predicted to manifest in P Λ,Ĵ > P Λ,Ĵ , as we discuss in section 5.2. Even as we note possible differences between the polarizations of Λ and Λ, it is clear that to good approximation they are the same, even at the lowest energies, suggesting similar average vorticity of the cells from which they arise. This is remarkable, in light of the fact that the directed flow of these particles diverge strongly (105) as the energy is reduced below √ sNN ≈ 20 GeV, even taking opposite signs at midrapidity. In the hydrodynamic paradigm, directed flow (22), essentially the sidewards push of forward-going particles (c.f. figure 1) reflects the anisotropy of the bulk fluid velocity about theĴ axis at a large scale.
7 While eye-catching, the value of P Λ,Ĵ = (7.4 ± 3.1)% at the lowest energy is less than 2σ above the general systematics and is marginally significant.  Meanwhile, global polarization reflects rotational flow structure aboutĴ at a more local scale. There may be a coupling in a hydrodynamic picture (106,103,12). Whether there is a tension here is unclear, though a three-fluid hydrodynamic code is able to approximately reproduce proton and antiproton flow (107) and Λ polarization (91).  (81). Statistics are poor at low energies, while at higher energies, the signal itself vanishes. More detailed measurements can provide stringent challenges to theoretical models and may provide new insight. In √ sNN = 200 GeV collisions, polarizations of Λ and Λ are identical within uncertainties, so here we discuss their average. In figure 3, global hyperon polarization was shown for collisions with centrality of 20-50% (c.f. section 2.2), corresponding to | b| ≈ 7−11 fm. Figure 4 shows the centrality dependence. Both the global polarization and the oscillation of the longitudinal local polarization (c.f. section 4.3) increase monotonically with impact parameter, as expected for a phenomenon driven by bulk mechanical angular momentum; this is in agreement with transport-hybrid calculations (96).

Global and local polarization at
The "global" polarization-i.e. integrated over all particles at midrapidity-is non-zero and aligned with an event-specific direction. 8 Momentum-differential ("local") polariza-  tion structures, in the local equilibrium picture, are more sensitive to the thermal vorticity variations as a function of space and time, convoluted with flow-driven space-momentum correlations. First measurements (81) report PĴ ,Λ/Λ to be independent of transverse momentum for pT 2 GeV/c, in agreement with hydrodynamic predictions (12,111) when realistic initial conditions are used. It was also seen (81) to be independent of pseudorapidity, though only a limited range, |η| < 1 could be explored. As we discuss in section 5, several theories suggest there is much to be learned at forward rapidity.
A recurring theme in heavy ion physics has been that azimuthal dependencies often present surprises and the opportunity for new physical insight. The same may well be true for polarization. Figure 5 shows preliminary data from the STAR collaboration (108) suggesting that PĴ ,Λ&Λ is significantly stronger for particles emitted perpendicular toĴsys (|φΛ − ΨRP| = π/2) than forpΛ Ĵ . Indeed, PĴ may vanish for hyperons emitted out of the reaction plane. This stands in contradiction to rather robust predictions of hydrodynamic (31,12,50,112,98) and coarse-grained transport (96,100,101,102) calculations, one of which is shown on the right panel of the figure, which predict precisely the opposite dependence. If the STAR results are confirmed in a final analysis, this represents a nontrivial challenge to the theory. By symmetry, polarization components perpendicular toĴsys must vanish, when averaging over all momenta. Locally in momentum space, however, these components are allowed to be non vanishing. Particularly, there can be non-vanishing values oscillating as a function of the azimuthal emission angle φH over the transverse plane with a typical quadrupolar pattern. Hydrodynamic (12) and transport calculations (100) predict the sign and the magnitude of these oscillations. Here,n =p beam in equation 16 so ξ * D = θ * D , the polar angle of the daughter in the hyperon frame; c.f. figure 3.
Hydrodynamic (12,50,112,111) and transport-hybrid (96, 100, 101, 102) calculations flow (22) could produce a global effect. However, in practice, this effect is much smaller than those we discuss here (31).  predict a negative sign of the longitudinal component of the polarization vector in the first quadrant of the pT plane rotating counterclockwise from the reaction plane. One such calculation is shown in the right panel of figure 6, while the corresponding measurement (82) is on the left. The magnitude of the effect is significantly larger in the model, but more strikingly, the sign of the predicted oscillation is opposite that seen in the data, reminiscent of the discrepancy in figure 5.
Understanding and resolving the tension in figures 5 and 6 is among the most pressing open issues in this area. This is further discussed in Section 5.

OPEN ISSUES AND OUTLOOK
Above, we have presented the theoretical framework (mostly hydrodynamics) in which to calculate the vorticity of the QGP; the theoretical connection between the vorticity and the polarization of hadrons emitted from the plasma, based on local thermodynamic equilibrium of hadrons and their generalized distribution function; and the measurements of this polarization with Λ hyperons. Overall, the hydrodynamic and statistical equilibrium paradigm predicted first experimental observations of global polarization strikingly well.
However, qualitative discrepancies between theory and experiment may indicate that some fundamental feature of the dynamics itself (encoded hydro or transport) is misunderstood or unaccounted for. Alternatively, we may misunderstand the interface ("Cooper-Frye" and thermal vorticity) between hydrodynamics or its coarse-grained approximation and the polarization observable. Clearly, the existing data demands more theoretical work and a report of the recent and ongoing work is the subject of the next subsection.
On the other side, there are many important theoretical predictions which demand experimental tests. These will involve new detectors, future facilities, and new analysis techniques.
Finally, two topics deserve separate attention. One is the possibility that polarizations of Λ and Λ are different. The other concerns the spin alignment of vector mesons.

Local polarization
The discrepancies between hydrodynamic calculations and the polarization pattern in momentum space have been presented in subsection 4.3 and they have been the subject of investigations over the past year.
The simplest explanation of them being an effect of secondary decays (see subsection 3.3) has been ruled out (51), (35); the secondary Λ's have almost the same momentum dependence polarization as the primary, if all the primary are polarized according to the hydrodynamic predictions. The other simple explanation is a polarization change in posthadronization rescattering, which is not taken into account in simulation codes; however, this seems to be very unlikely, see discussion in subsection 3.3, especially because it should produce an amplification in some selected momentum regions. The available hadronic transport codes do not include the spin degrees of freedom mostly because the helicitydependent scattering amplitudes are unknown and, even resorting to educated guesses, it is a formidable computational task to include them into Monte-Carlo codes.
Within the hydrodynamic paradigm, there are more options yet to be explored. The first is concerned with the formula 2, which is first-order in thermal vorticity. Indeed, thermal vorticity is moderately smaller than 1 (see figure 2) and the exact formula at all orders is not known yet, so a sizeable role of higher order corrections cannot be ruled out for the present. Since polarization is steered by thermal vorticity, it is possible that the thermal vorticity field is different from the predictions obtained with the presently used initial hydrodynamic conditions, tuned to reproduce a set of observables in momentum space. Recently, ref. (113) obtained the right sign of the longitudinal polarization at √ s NN = 200 GeV with specific initial conditions (114), while the same model predicts the "wrong" sign at lower energy √ s NN = 8 GeV (112). Another possibility is that spin dissipative corrections, analogous to viscous corrections for the stress-energy tensor, which are not included in the local thermodynamic equilibrium assumption, are sizeable. As has been mentioned in Section 3, the theory of dissipation and spin in hydrodynamic framework has recently drawn the attention of several authors; as yet, it is not clear whether such an approach includes relevant quantum terms and if it is pseudo-gauge dependent (see subsection 3.5). Furthermore, it has been considered that other kinds of vorticity, instead of thermal vorticity 3, enter in the polarization definition. In ref. (102) it has been shown that the right sign of the longitudinal polarization is retrieved if the thermal vorticity is replaced by the a tensor proportional to the T-vorticity (12) whereas in ref. (115) the agreement was restored by replacing the thermal vorticity with its double projection perpendicular to the velocity field. So far, these observations are not borne out by fundamental theoretical justifications. Finally, it should be mentioned that in ref. (116) the correct polarization patterns have been obtained for the polarization of quarks within a chiral kinetic model; the question remains on the effect of hadronization.
If all of the above ideas will fail to describe the data, two scenarios may be envisioned: • Spin does not locally equilibrate and it has to be described within a kinetic approach; c.f. section 3.4; • Spin equilibrates locally, but pseudo-gauge invariance is broken and one needs a spin potential to describe its hydrodynamic evolution, with six additional degrees of freedom and six additional hydrodynamic equations (see subsection 3.5).
Of course, both should be able to explain why the global polarization is in very good agreement with local equilibrium with thermal vorticity. Finally, we should always consider the possibility of a thus far unsuspected important ingredient.

Λ −Λ splitting
As we have discussed, while the difference is statistically insignificant at any given energy, P Λ,J is systematically larger than PΛ,J at the lower collision energies where polarization itself is large. A possible interpretation of such a splitting is the presence of a large electromagnetic field and one could use the observed difference to extract the value of the magnetic field in the rest frame of the particles, as discussed in subsection 3.6.
To first approximation, theĴ-component of the vorticity is determined by the sum of PΛ,J and P J,Λ , and the magnetic field by their difference (37). However, feed-down corrections can be important, and should be accounted for (37). For example, in the absence of feed-down, a finite B-field would produce P Λ,J > PΛ,J , and B = 0 would result in no difference in the polarizations. However, if B = 0, feed-down effects at low collision energies (where there are significant chemical potentials at freezeout) can generate a "splitting" with opposite sign, i.e. P Λ,J < PΛ,J (37). Applying formula 2 (with the substitution 14) to the data in figure 3, and accounting for feeddown effects (37), results (117) in an estimate of B = (6 ± 6) × 10 13 T when averaging over results from 10 GeV < √ sNN < 40 GeV. Such an average is hardly justified, but it nevertheless provides a valuable estimate of the magnitudes of the magnetic field-and the measurement uncertainty-that may be associated with the data. In the equilibrium paradigm, this is the magnetic field at freezeout. Theoretically, fields of this magnitude are present in the first instants of a heavy ion collision. While they may decay well before freeze-out, a highly conductive QGP itself can significantly extend the lifetime of the initially large field (118) and vorticity certainly helps in this respect (119). At low √ sNN , field lifetimes may be longer (120) and QGP evolution time shorter. Relativistic magneto-hydrodynamics it the standard tool to study the evolution of the electro-magnetic field in a plasma and there have been major advances recently (121,122). Neglecting feed-down corrections, the similarity of PΛ,J and P Λ,J places (123) an upper limit on the magnetic field at freezeout of about 10 12 T at top RHIC energy. Transport calculations may provide more insight, with respect to local thermodynamic equilibrium. Simplified calculations estimate that the expected field could be on order 10 12 − 10 13 T, and the energy dependence of the splitting would resemble that seen in the data (119). A more sophisticated calculation (75) with partonic transport argues that the difference between PΛ,J and P Λ,J may be reasonably attributed to the accumulated effect of an evolving magnetic field; interestingly, in the absence of a magnetic field, P Λ,J < PΛ,J .
A firm statement on the existence of a long-lived (several fm/c) magnetic field on the scale of 10 13 T would have tremendous implications for the Chiral Magnetic Effect (57). However, several effects have been postulated, which may complicate the interpretation of the splitting. Especially at low collision energies where baryon stopping is significant, Λ and Λ may originate from different regions within the fireball (89), their final polarizations thus reflecting differently-weighted averages over vorticity. Han et al argue that the quarkantiquark vector potential in the presence of the net quark flux at these energies may generate a splitting largely indistinguishable that expected from a magnetic field. The vector meson field may play an equivalent role in the hadronic sector; however, existing calculations (124,125) reproduce the splitting only by adjusting by hand the unknown sign, magnitude and energy dependence of the effect.
In principle, disentangling these effects could require a full three-dimensional magnetohydrodynamic calculation which includes appropriate vector potentials, conserved nontrivial baryon currents and QGP conductivity, potentially followed by a hadronic cascade with spin-transfer collisional dynamics. Hopefully, however, sophisticated but achievable calculations, in conjunction with targeted measurements, can lead to reasonable estimates of the individual contributions of these important effects.

Alignment
In peripheral collisions the anisotropy generated by the collective angular momentum implies that all particles with spin can, in principle, have a non-vanishing polarization. Particularly, for vector mesons, this implies a non-vanishing alignment, as discussed in subsection 4.1. At local thermodynamic equilibrium, the equation 10 predicts an alignment which is quadratic in the thermal vorticity (29). Since thermal vorticity is less than 1 throughout (see figure  2), and at freezeout it is on order 0.02, the expected resulting alignment is tiny.
In fact, preliminary results on the alignment have been reported for two vector mesons, K * and φ at RHIC (126) and LHC (127). In all cases, Θ00 (c.f. equation 17) is considerably different than 1 3 . For φ at LHC and K * at both colliders would imply a vorticity at least two orders of magnitude higher than calculations or expectations from the hyperon measurements. More surprisingly, Θ00 for the φ mesons at RHIC is greater than 1 3 (126), something that cannot be understood in hydrodynamic or recombination model (128). This observation may require fundamentally new physics mechanisms (129) for alignment that apply at RHIC but not at LHC. Altogether, the situation with these preliminary spin alignment measurements is not sufficiently well understood to discuss in a review.

Future measurements
As we have discussed, the first few positive observations of hyperon polarization at RHIC have generated tremendous theoretical activity. Much of this work has focused on the degree to which models can reproduce the measurements, but a growing body of work points to the ways in which new measurements can strictly test our understanding of QGP dynamics and may provide enhanced sensitivity to important physics. Lower energy collisions As seen in figure 3, both the observed and predicted polarization signals rise as the collision energy is reduced. Exploring this trend for even lower energies may touch on several important questions: does a hydrodynamic description of the system break down at lower energy density? What are the effects of increased viscosity (130,50,131,132)? Can spin equilibrate rapidly enough to justify a local thermodynamic equilibrium approximation -and if so, is this due to hadronic mechanisms or to the QCD phase transition? Some hydrodynamic models tuned for low energies predict an uninterrupted continued rise (103,112), regardless of equation of state (133,91), though initial state and thermalization assumptions may affect this behavior at the lowest energies, producing a non-monotonic behavior (134).
In section 4.2, we remarked on the possible tension between Λ and Λ directed flow and polarization at low collision energy; how tightly coupled are the large-scale and smaller-scale rotational structures in the flow fields probed by these particles? It has been suggested (107,135) that the diverging behavior of baryon and antibaryon directed flow (136, 105) signals a phase transition in the equation of state. Alternatively, it may arise from a convolution of baryon stopping and quark coalescence (137,105).
Again regarding Λ and Λ, it has been suggested (89) that differences in polarization are dominated by differences in the phase space from which these particles arise; such differences are largest at low collision energy. Testing this hypothesis and that discussed in the previous paragraph of course will require measurement of local polarization, that is as a function of momentum.
Addressing these questions will require new measurements at √ sNN 10 GeV, with good tracking, event plane resolution, and high statistics, especially given plummeting Λ yields. These will be pursued at the future NICA (138,139) and FAIR (140) facilities, as well as the STAR/RHIC fixed-target program (141) and the HADES/GSI experiment (142). Measurements at forward rapidity Thus far, polarization has been measured at midrapidity, to focus on the hottest part of the QGP fireball. However, calculations with a variety of models suggest a vortical structure that evolves with rapidity.
A geometric calculation (28) based on the BGK model of hadron production (143) and boost-invariance suggests that vorticity will increase with rapidity, and speculates that a rapid change in the evolution could signal a phase change at some critical density. A similar, more recent calculation (144) finds that the rapidity dependence of vorticity itself depends on √ sNN at RHIC energies, and that it is sensitive to important physical parameters of the model itself. Numerical calculations with transport-hybrid codes (96,99,102) also indicate a forward migration of vorticity, especially as the collision energy increases (99). Finally, hydrodynamic models predict much higher vorticity in the beam fragmentation region at both NICA (97,145) and RHIC (146,91) energies.
Exploring vorticity away from midrapidity in fixed-target experiments (discussed above) is relatively straightforward. At collider energies, the STAR forward upgrade will provide coverage and tracking over a physically important region (144). If the event plane can be reconstructed, the LHCb experiment (147) could be used to explore the rapidity evolution at the highest energies. Other polarization projections Referring to equation 16, experiments have reported polarization projections alongn =Ĵ andn =p beam . The geometry of the collision itself suggests other natural directions.
For particles emitted at forward rapidity, symmetry permits an average polarization projection alongn =b. In fact, "vortex rings" or "cyclones" are predicted (100,146) at forward rapidity at RHIC (100) or NICA (97,146) energies, as well as at midrapidity in non-symmetric systems (148). In this case,n pΛ ×ẑ.
One of the first model studies of vorticity in heavy ion collisions predicted similar ringlike structures relative to jets. High-momentum partons formed in the initial stages of the heavy ion collision lose energy in the QGP fireball (149) and can locally perturb the flow field (150). This may produce a cone or ring of vortical structure locally perpendicular to the direction of the deposited momentum (28),n =p dep ×pH , where the hyperon H has acquired an outward velocity from the radial flow (6) of the QGP.
Finally, the QGP depicted in figure 1 is likely to be characterized by turbulence (151,106,152), in which the vorticity of a fluid cell is not correlated with a global event characteristic or symmetry-breaking direction. However, the assumption is that the polarizations of all particles emitted from a cell are aligned with the vorticity of that cell, and flow-induced space-momentum correlations (6) cause particles from the same cell to be emitted in the same direction. Hence, if experimental complications can be overcome (92), spin-spin cor-relations as a function of relative momentum (or angle) are a promising way to probe the turbulent vortical substructure of the QGP (104).

SUMMARY AND OUTLOOK
Polarization has opened an exciting new direction in relativistic heavy ion physics; one of the increasingly rare truly new developments in this rather mature field. Its measurement has definitely proved that a new degree of freedom other than momentum is now available to probe the QGP formation and dynamics. In the hydrodynamic model, unlike particle momentum, polarization is primarily sensitive to the gradients of the hydro-thermal fields, and this appears to be a unique feature among the known observables. Moreover, polarization can help to constrain the electro-magnetic field, which would be incredibly valuable for the search of Chiral Magnetic Effect (57). The hydrodynamic model predicts, and the measurements have shown, that polarization increases at low energy, and it will be further explored in future low-energy heavy ion programs. At RHIC and LHC energies, flow substructure is already being probed in unprecedented detail, presenting theory with new and as yet unsolved challenges. Directions for future studies at these energies were discussed.
There are several pressing issues to be solved which require considerable advances in theory and phenomenology. Indeed, at this time, after having played the leading role, theory appears to have been surpassed by the experiments which have proved to be able to measure polarization as a function of many relevant variables in relativistic heavy ion collisions: azimuthal angle, rapidity, centrality, etc. In the near future, more measurements will be available which will help to constrain or disprove theoretical models and assumptions; polarization of different species (e.g. Σ 0 and Ξ − ), spin-spin correlations (104); measurement of polarization in different colliding systems (153). On the theory side, as has been mentioned, one expects improved formulae including more terms and corrections to the equation 2, the inclusion of dissipative effects and the application of alternative methods such as kinetic theory as well as the development of a hydrodynamic with spin potential. Equally important is a major advance in phenomenology and numerical computation, with the inclusion of hadronic rescattering effects and the systematic study of polarization dependence on the initial conditions.
Since its experimental discovery a few years ago, there has been tremendous progress in the study of polarization in heavy ion collisions. Yet, at this early stage, the potential of this new tool is still to be explored. It may well be that this direction of research yields new insights and major results in the study of the QCD matter with nuclear collisions.

DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS
We are greatly indebted to Gabriele Inghirami for his invaluable help in making some of the figures of this article. F. B. was partly supported by the INFN project SIM. M.A.L. supported by the U.S. Department of Energy.