Polarization and Vorticity in the Quark Gluon Plasma
PPolarization and Vorticity in the Quark Gluon Plasma
Francesco Becattini and Michael A. Lisa, Dipartimento di Fisica e Astronomia, University of Florence, Florence, Italy,I-50019; email: becattini@fi.infn.it Department of Physics, The Ohio State University, Columbus, Ohio, USA43210; email: [email protected]. Xxx. Xxx. Xxx. YYYY. AA:1–30https://doi.org/10.1146/((please addarticle doi))Copyright c (cid:13)
YYYY by Annual Reviews.All rights reserved
Keywords polarization, quark gluon plasma, magnetic field, heavy ion collisions,hydrodynamics, vorticity
Abstract
The quark-gluon plasma produced by collisions between ultra-relativistic heavy nuclei is well described in the language of hydro-dynamics. Non-central collisions are characterized by very large an-gular momentum, which in a fluid system manifests as flow vorticity.This rotational structure can lead to a spin polarization of the hadronsthat eventually emerge from the plasma, providing experimental ac-cess to flow substructure at unprecedented detail. Recently, first ob-servations of Λ hyperon polarization along the direction of collisionalangular momentum have been reported. These measurements are inbroad agreement with hydrodynamic and transport-based calculationsand reveal that the QGP is the most vortical fluid ever observed. How-ever, there remain important tensions between theory and observationwhich might be fundamental in nature. In the relatively mature fieldof heavy ion physics, the discovery of global hyperon polarization andthree-dimensional simulations of the collision have opened an entirelynew direction of research. We discuss the current status of this rapidlydeveloping area and directions for future research. a r X i v : . [ nu c l - e x ] M a r ontents
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. QUARK GLUON PLASMA, HYDRODYNAMICS AND VORTICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1. Vorticity and polarization: overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2. Geometry of a nuclear collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. POLARIZATION IN RELATIVISTIC HEAVY ION COLLISIONS: THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1. Polarization in statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Hydrodynamic calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3. Effects of decays and rescattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4. Kinetic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5. Spin tensor and spin potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6. Contribution of the electro-magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144. POLARIZATION IN RELATIVISTIC HEAVY ION COLLISIONS: OBSERVATIONS . . . . . . . . . . . . . . . . . . . . . . 154.1. Measuring polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2. Global hyperon polarization - observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3. Global and local polarization at √ s NN = 200 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195. OPEN ISSUES AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.1. Local polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2. Λ − ¯Λ splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3. Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4. Future measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246. SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1. 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. Formore than two decades, a large community has systematically studied these collisions toextract insight about quantum chromodynamics (QCD) matter under extreme conditions.The resulting field of relativistic heavy ion physics is by now relatively mature. With theearly realization that the QGP in these collisions is a ”nearly perfect fluid, hydrodynamicshas 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 responseof the bulk medium to azimuthal (to the beam direction) anisotropies in the initial energydensity (6). Measured azimuthal correlations are well reproduced by modulations in theoutward-directed flow fields in the hydro simulations. However, despite the fact that heavyion collisions involve huge angular momentum densities (10 − (cid:126) over volumes ∼
250 fm ),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 couplingbetween rotational motion and quantum spin can lead, in the QGP, to polarization ofhadrons emitted from fluid cells, driven by the local vorticity of the cell. In 2017, thefirst 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 vorticalnature of the QGP. This line of investigation, only just now begun, represents one of thefew 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 igure 1 A heavy ion collision at relativistic energy is sketched, in the center of mass frame. The relevantgeometrical and physical quantities characterizing a collisions are shown in the left panel. TheQuark Gluon Plasma is formed out of the colliding nucleons of the nuclear overlapping region(right panel). The spectator deflection in the right panel is greatly exaggerated for clarity. in other physical systems and define geometrical conventions required for the heavy ioncase. We then discuss theoretical tools employed to model the complex rotational dynamicsof the plasma and the manifestation in particle polarization. In section 4, we discussexperimental measurements and observational systematics. We will see broad agreementbetween observation and theory, but tension in some important aspects. We conclude ourreview with open questions and an outlook.
2. QUARK GLUON PLASMA, HYDRODYNAMICS AND VORTICITY
That the QGP produced in collisions of nuclei at relativistic energies is, for a transient ofaround 10 − seconds, a nearly perfect fluid is based on the accumulated evidence collectedover a time span of more than ten years. The main fact is that this fluid breaks up intohadrons in a state very close to local thermodynamic equilibrium (8) at a temperature veryclose to the pseudo-critical QCD temperature of 160 MeV (9), (10).Local thermodynamic equilibrium implies that momentum spectra of produced hadronsare very well reproduced by the assumption of a local Bose-Einstein or Fermi-Dirac distri-bution function (for vanishing chemical potentials): f ( x, p ) = 1exp[ β · p ] ± β = (1 /T ) u ( x ) is the four-temperature vector including temperature and the four-velocity hydrodynamic field u ( x ). The formula 1 applies to the local fluid cell, and shouldbe integrated thereafter over the “freeze-out” 3D-hypersurface (see figure 2) defined as theboundary of local thermodynamic equilibrium, giving rise to what is well known in the fieldas “Cooper-Frye” formula (11). Indeed, this is analogous to the last-scattering surface inthe cosmological expansion where the background electro-magnetic radiation froze out.Apparently, local thermodynamic equilibrium is achieved - and a plasma at finite tem-perature is formed - at quite an early time in the process (see figure 2). This is confirmedby 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 themomentum spectra in the transverse plane perpendicular to the beam line (refer to fig. 1) • Polarization and Vorticity in the QGP 3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Au+Au, √ s NN = 200 GeV = 11.6 fm d N / d | — ω | | — ω | complete Σ fo time-like Σ fo space-like Σ fo Figure 2
Left: A collision of two nuclei in space-time diagram. In the hydrodynamic model, localequilibrium is believed to occur on the hyperbola Σ eq which is the initial 3D hypersurface of thethermalized QGP and to cease at the 3D hypersurface Σ FO Right: Distribution of the amplitude of thermal vorticity | (cid:36) | = | (cid:112) | (cid:36) µν (cid:36) µν | at the freeze-outhypersurface Σ FO , 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 thespace-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 coeffi-cients v n - have led to the conclusion that the viscosity of the QGP must be very small ascompared to the entropy density, close to the conjectured universal lower bound of (cid:126) / π (13).Recently, the exploration of the QGP made a significant advance. The measurementof polarization of emitted hadrons made it clear that a new probe is accessible which maygive a wealth of new and complementary information. In particular, in the hydrodynamicparadigm, while the momentum spectra provide direct information about the velocity andthe temperature field, polarization is linked to the vorticity and more generally to the gra-dients 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 thermodynamicquantities of the local cell in the local rest frame, such as temperature and chemical poten-tials related to the various charges (baryon number, electric charge, etc). Likewise, assumingthat spin degrees of freedom locally equilibrate, vorticity plays the role of a potential deter-mining 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 thermodynamicquantity needed to describe locally the fluid. In a sense, vorticity is an extra substructureof a hydrodynamic cell. This property makes polarization a very sensitive probe of the dy- amical process leading to the QGP formation and of its evolution. As has been mentionedin the Introduction, this field has only just begun and all the developments that polarizationmay lead to can be hardly envisioned for the present. 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 con-finement volume characteristic of a hadron. Otherwise stated, the system which is formedis 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 hydrodynamicsas an effective tool to describe its evolution; in the hydro language, the Knudsen numberis sufficiently small (14). Under such circumstances, the variation of the flow field in spaceand time can be slow enough to be dealt with as ”macroscopic” motion of bulk matterand vorticity as well. As it will become clear later on in Section 3, the vortical structureis probed by the spin of hadrons that ”freeze out” from local fluid cells in a state of localthermodynamic equilibrium, as has been discussed above. More specifically, the presence ofa vortical motion (as well as an acceleration and a temperature gradient) entails a modifi-cation of 1 such that the distribution function becomes non-trivially dependent on the spindegrees of freedom.That spin and vorticity are tightly related is not a new insight, and yet there arerelatively few examples of physical systems which show the effect of the coupling betweenmechanical angular momentum of bulk matter and the quantum spin of particles thatcomprise (or emerge from) that matter.Two seminal measurements were reported nearly simultaneously more than a centuryago. Barnett (15) observed that an initially-unmagnetized steel cylinder would generate amagnetic field upon being rotated. In the same year, Einstein and de Haas (16) observedthe complementary effect: a stationary unmagnetized ferromagnetic object will begin torotate upon introduction of an external magnetic field. In both cases, the phenomenon isrooted in the conservation of total angular momentum on one hand and on equipartition ofangular momentum, that is thermodynamic equilibrium, on the other. In the Barnett effect,the angular momentum which is imparted through a forced rotation gets partly distributedto 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 theEinstein-de Haas effect, the external magnetic field implies, at thermodynamic equilibrium,a polarization of matter, whence an angular momentum; if the magnetic field does notprovide torque, the body should start spinning as to conserve the initial vanishing angularmomentum. Indeed, a quantitative understanding of these phenomena was possible only adecade 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 withkinetic energy of E kin ∼
30 MeV per nucleon is incident on a stationary target. (In high-energy physics terms, √ s NN − m p ≈
15 MeV, where m p is the proton mass.) Thisis the regime of quasi compound nucleus formation, in which the short-lived system isassumed to rotate as a whole, to first order. At ”high” beam energies E kin (cid:38)
50 AMeV,projectile fragments are expected to experience positive deflection (c.f. section 2.2) dueto collisional and bulk compression during the collision. At lower energies, collisions arePauli-suppressed and attractive nuclear surface interactions are expected to produce an • Polarization and Vorticity in the QGP 5 rbiting motion that leads to negative deflection. Disentangling the interplay between thesephysical mechanisms requires determination of ˆ J sys . This was achieved by correlating (17,18) the circular polarization of γ rays with forward fragment deflection angles. Thesemeasurements 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. 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 polarizationfield, which was measured using the inverse spin Hall effect (20). This experiment, whereboth the vorticity and the induced polarization are observable, is important to establishthe phenomenon in fluids.With respect to all above listed cases, polarization in relativistic heavy ions possessestwo 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 notpossible 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 itwas precisely neutral, the observation of polarization by magnetization would be simplyimpossible because particles and antiparticles have opposite magnetic moment. In fact, aswe will see, Λ and ¯Λ in relativistic nuclear collisions at high energy have almost the samemean 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, thesign of the mean spin vector components would be opposite. Hence, altogether, whilefor non-relativistic matter (without anti-matter) it is impossible to resolve polarization byrotation and by magnetization - what lies at the very heart of the Barnett and Einstein-de Haas effects - in relativistic matter, because of the existence of antiparticles, they canbe distinguished and QGP is the first relativistic system where the distinction has beenobserved.
The left panel of figure 1 sketches the geometry of a heavy ion collision in its center ofmass frame, prior to contact. Designating one nucleus the beam and the other the target ,the impact parameter (cid:126)b points from the center of the target to the center of the beam,perpendicular to the beam momentum (cid:126)p beam . The vectors (cid:126)b and (cid:126)p beam span the reactionplane, indicated by the grid. The total angular momentum of the collision (cid:126)J sys = (cid:126)b × (cid:126)p beam .The right panel sketches the situation after the collision. In the participant-spectatormodel (21) commonly used at high energies, a fireball at midrapidity is produced by thesudden and violent deposition of energy when ”participant” nucleons overlap and collide.Meanwhile, projectile nucleons that do not overlap with oncoming nucleons in the targetare considered ”spectators” and continue with their forward motion essentially unchanged,later to undergo nuclear fragmentation. Low-energy compound nuclei have surface vibrations and breathing, but generally do not featureinternal fluid flow structure. This initial designation is of course arbitrary, but the convention must be kept consistently. Inthe age of collider-mode nuclear physics, confusion is not uncommon and leads to sign errors. owever, this distinction is not so sharp in reality, as the forward ”spectators” do receivea sideways repulsive impulse during the collision, as indicated by the deflected momentumarrows 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 canproduce negative deflection. In the parlance of relativistic collisions (22), the positivedeflection corresponds to positive directed flow ( v ) in the forward direction ( v > y ≈ y beam ).This deflection is important. While we are especially interested in the vortical structureof 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 usedto estimate ˆ J sys event-by-event, as discussed below.A final note about coordinate systems and conventions. It is common to define acoordinate system in which ˆ z ≡ ˆ p beam and ˆ x ≡ ˆ b . In this case, ˆ J = − ˆ y ; The azimuthalangle of ˆ b about ˆ p beam in some coordinate system (say, the floor of the experimental facility)is often referred to as the reaction plane angle Ψ RP . The aforementioned forward detectorsuse spectator fragment deflection to determine the event plane angle Ψ EP , . Standardtechniques have been developed (22) to determine the event plane and the resolution withwhich it approximates Ψ RP , i.e. the direction ˆ b .Since the size and angular momentum of the QGP fireball depends on the overlapbetween the colliding nuclei, an estimate of the magnitude of the impact parameter is alsoimportant. Standard estimators (23), typically based on the charged particle multiplicitymeasured at midrapidity, quantify the ”centrality” of each collision in terms of fraction ofinelastic cross-section. Head-on ( | (cid:126)b | = 0) and barely-glancing collisions are said to havecentrality of 0% and 100% respectively.
3. POLARIZATION IN RELATIVISTIC HEAVY ION COLLISIONS: THEORY
The main purpose of the theoretical work is to calculate the amount of polarization ofobservable particles once the initial condition of the collision is known, that is the energyand the impact parameter of the two nuclei. The final outcome depends on the model ofthe collision (see Section 2) and on how the initial angular momentum may induce a globalpolarization of the particles.The first calculation on global polarization in relativistic heavy ion collision was pre-sented in ref. (24) based on a perturbative-QCD inspired model where colliding partonsget polarized by means of a spin-orbit coupling. The amount of predicted polarization ofΛ baryons was originally large (around 30%) and corrected thereafter by the same authorsto be less than 4% (25). Besides the apparent large uncertainty, the main problem of thecollisional approach - at the quark-gluon level - is the difficulty of reconciling it with the ev-idence of a strongly interacting QGP, which makes the kinetic approach dubious. Anotherproblem is how to transfer the polarization at quark-gluon level to final hadrons, whichrequires 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 hydrodynamicmotion, and particularly vorticity, was put forward (27, 28). If the QGP achieves andmaintain local thermodynamic equilibrium until it decouples into freely streaming non-interacting hadrons and if this model - as discussed in the Introduction and in Section 2 • Polarization and Vorticity in the QGP 7 is very successful to describe the momentum spectra of particles, there is no apparentreason 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 thermodynamicequilibrium, 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 aslater described) and makes it possible to obtain quantitative predictions at hadronic levelwithout the need of a mechanism to transfer polarization from partons to hadrons. Theactual 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): S µ ( p ) = − m (cid:15) µρστ p τ (cid:82) d Σ λ p λ n F (1 − n F ) (cid:36) ρσ (cid:82) d Σ λ p λ n F p is the four-momentum of the particle, and n F = (1 + exp[ β · p − µQ/T ] + 1) − is the Fermi-Dirac distribution with four-temperature β like in eq. 1 and with chemicalpotential µ coupled to a generic charge Q . The integration should be carried out over thefreeze-out hypersurface (see fig. 2); in a sense, in the heavy ion jargon this can be calledthe ”Cooper-Frye” formula for the spin. The key ingredient in equation 2 is the so-called thermal vorticity tensor (cid:36) ( x ), which reads: (cid:36) µν = −
12 ( ∂ µ β ν − ∂ ν β µ ) 3.i.e. the anti-symmetric derivative of the four-temperature. This quantity is adimensionalin natural units and it is the proper extension of the angular velocity over temperatureratio 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 theleading order, on the temperature-velocity field itself. Thereby, polarization can provide acomplementary information about the hydrodynamic flow with respect to the spectra andtheir anisotropies. The formula 2 applies to anti-particles as well, so that in a charge-neutralfluid the spin vector is expected to be the same for particles and anti-particles, which is aremarkable feature as emphasized in Subsection 2.1. It is worth pointing out that formula 2implies that a particle within a fluid in motionvat some space-time point x gets polarizedaccording to (natural constants have been purposely restored): S ∗ ( x, p ) ∝ (cid:126) KT γ v × ∇ T + (cid:126) KT γ ( ω − ( ω · v ) v /c ) + (cid:126) KT γ A × v /c γ = 1 / (cid:112) − v /c and all three-vectors, including vorticity, acceleration and velocity,are observed in the particle rest frame. The decomposition 4 makes it clear what arethe thermodynamic ”forces” responsible for polarization: the last term corresponds to theacceleration-driven polarization, its expression is reminiscent of the Thomas precession andit is indeed tightly related to it (particle moving in an accelerated flow); the second termis the relativistic expression of polarization by vorticity; the first term is a polarizationby combination of temperature gradient and hydrodynamic flow and is, to the best of ourknowledge, a newly found effect. irst hydrodynamic calculations based on formula 2 predicted a global polarization ofΛ baryons of a few percent at √ s NN = 200 GeV (31), hence compatible with the previousexperimental limit. The new measurements with a larger statistics then confirmed thatpolarization value is of such order of magnitude. Formula 2 then became a benchmark formost phenomenological calculations of the polarization in heavy ion collisions.We will now review in some detail the status of the theoretical understanding of thepolarization in relativistic fluids and in nuclear collisions particularly. The calculation of spin at global or local thermodynamic equilibrium requires a quantumframework, spin being inherently a quantum observable. The most appropriate frameworkis thus quantum statistical mechanics, and since we are dealing with a relativistic fluid, ina relativistic setting. However, many quantitative features can be found out starting fromthe simplest non-relativistic case.As a simple illustrative case, consider a rotating ideal gas with angular velocity ω withina cylindrical vessel of radius R . At equilibrium, the statistical operator reads (32): (cid:98) ρ = 1 Z exp (cid:34) − (cid:98) HT + ω · (cid:98) J T (cid:35) (cid:98) J i = (cid:98) L i + (cid:98) S i foreach particle i , the spin density matrix for a particle with momentum p turns out to be:Θ( p ) rs ≡ (cid:104) p, s | (cid:98) ρ i | p, r (cid:105) = (cid:104) p, s | exp[ ω · (cid:98) S i /T ] | p, r (cid:105) (cid:80) St = − S (cid:104) p, t | exp[ ω · (cid:98) S i /T ] | p, t (cid:105) = δ rs exp[ − sω/T ] (cid:80) St = − S exp[ − tω/T ] 6.implying a mean spin vector of particles: S = ˆ ω ∂∂ ( ω/T ) sinh[( S + 1 / ω/T ]sinh[ ω/ T ] (cid:39) S ( S + 1)3 ω T ω/T . Equation 6 also impliesthat the so-called alignment Θ for spin 1 particles is quadratic in ω/T at the leadingorder, 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): (cid:98) ρ = 1 Z exp (cid:20) − b µ (cid:98) P µ + 12 (cid:36) µν (cid:98) J µν (cid:21) b is a constant time-like four-vector and (cid:36) is the thermal vorticity which, at globalthermodynamic equilibrium ought to be constant; (cid:98) P and (cid:98) J are the four-momentum andangular momentum-boost operators. It is important to point out that thermal vorticity in-cludes both vorticity and acceleration besides the gradient of the temperature. For instance,at global equilibrium, it turns out (34): (cid:36) µν = 12 ε µνρσ T ω ρ u σ + 1 T ( A µ u ν − A ν u µ ) 9. We will denote quantum operators with an upper wide hat throughout. • Polarization and Vorticity in the QGP 9 here u is the four-velocity, A the four-acceleration and ω the vorticity four-vector. Theentanglement of vorticity and acceleration is a typical signature of relativity, much like thatof electric and magnetic field in the electromagnetic tensor F µν .An intermediate step towards formula 2 is the free single-particle quantum relativisticcalculation. In this case, for a single particle, the operator 8 leads to the spin density matrix(35): Θ( p ) = D S ([ p ] − exp[(1 / (cid:36) : Σ S ][ p ]) + D S ([ p ] † exp[(1 / (cid:36) : Σ † S ][ p ] − † )tr(exp[(1 / (cid:36) : Σ S ] + exp[(1 / (cid:36) : Σ † S ]) , D S () stands for the (2 S + 1)-dimensional representation of the group SL(2,C) uni-versal covering of the Lorentz group, Σ S are the (2 S + 1) × (2 S + 1) matrices representingthe Lorentz generators, [ p ] is the so-called standard Lorentz transformation which takes theunit time vector ˆ t into the direction of the four-momentum p (36). The spin density matrixin eq. 10 implies a mean spin four-vector, for sufficiently low values of the thermal vorticity: S µ ( p ) = − m S ( S + 1)3 (cid:15) µαβν (cid:36) αβ p ν , − n F ). Indeed, the latter is the typical signature of Fermi statisticsand it naturally comes out in a quantum-field theoretical calculation. Indeed, this was theapproach taken into the original calculation at local thermodynamic equilibrium in ref. (30)where the density operator is the extension of equation 8: (cid:98) ρ = 1 Z exp (cid:20) − (cid:90) Σ FO dΣ n µ (cid:16) (cid:98) T µν ( x ) β ν ( x ) − ζ ( x ) (cid:98) j µ ( x ) (cid:17)(cid:21) β ν ( x ) is the four-temperature function (dependent on space and time), ζ ( x ) is theratio between chemical potential and temperature and (cid:98) T , (cid:98) j are the stress-energy tensorand current operators respectively. The integration should be done over the freeze-out3D-hypersurface (see figure 2) which is supposedly the boundary of local thermodynamicequilibrium. Indeed, calculating the mean spin vector from the density operator 12 isnot straightforward and some key assumptions are needed to get to the formula 2. Themost important is the usual hydrodynamic limit: microscopic lengths should be muchsmaller 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 thecovariant Wigner function at global equilibrium with acceleration and rotation, that is withthe density operator 8. In spite of these assumptions, there are good reasons to believethat the exact formula at the leading order in thermal vorticity in a quantum field theorycalculation would precisely be equation. 2. Indeed, the same formula was found with adifferent approach, based on the (cid:126) expansion of the Wigner equation (38) and, furthermore,it is the only possible linear expression in (cid:36) yielding the correct single-particle 11. andnon-relativistic limit. For instance, a term proportional to (cid:36) µν p ν , even if orthogonal to p ,would not yield correct limiting cases. What is still unknown is the exact global equilibriumformula at all orders in thermal vorticity including quantum statistics.
10 Becattini and Lisa hile the local equilibrium calculation of the spin density matrix and related quantitiesat leading order seems to be established at the most fundamental level of quantum fieldtheory, some questions remain to be addressed. It is not known how large are the higherorder terms in thermal vorticity at local equilibrium, nor we have an exact solution atglobal equilibrium with the density operator 8 including quantum field effects, namelyquantum statistics. Very little is known about the the dissipative, non local-equilibriumterms, and their magnitude. Recently, a phenomenological approach to spin dissipation hasbeen taken (39) generalizing a familiar classical method to constrain constitutive equationsin dissipative hydrodynamics, based on the positivity of entropy current divergence (40).It remains to be understood whether such a method includes all possible quantum termsin the entropy current and if it agrees with the most fundamental quantum approach todissipation, based on Zubarev non-equilibrium density operator (41). Another very recentstudy (42) studied the possible dissipative terms of the spin tensor in the relaxation timeapproximation.
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 vor-ticity field depends on the assumed initial conditions of the hydrodynamic calculations, onthe equation of state, on the hydrodynamic constitutive equations and on the freeze-outconditions. 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 sim-ulation. This is a crucial requirement because the components of the thermal vorticitydriving the projection of the mean spin vector along the total angular momentum involvethe 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 vorticityare, on the average, sufficiently less than 1 so as to justify a linear approximation in therelation between mean spin vector and thermal vorticity (see e.g. eq. 7); this is shown inthe histogram in figure 2. Nevertheless, a role of quadratic corrections cannot be excludedand it is yet to be studied.The codes that have been used so far to calculate polarization based on formula 2 arefew:1. A 3+1D
Particle in Cell simulation of ideal relativistic hydrodynamics (43). Allpublished calculations of polarization assume peculiar initial conditions for heavy ioncollisions, 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) withinitial state determined by means of a pre-stage of nucleonic collisions, and includinga post-hadronization rescattering stage, all adjusted to reproduce the basic hadronicobservables in relativistic heavy ion collisions, that is (pseudo)rapidity and transversemomentum distributions and elliptic flow coefficients.4. A 3+1D code implementing relativistic dissipative hydrodynamics, CLVisc (47) withinitial conditions provided by another transport-based simulation package AMPT • Polarization and Vorticity in the QGP 11
Most of the calculations presented in literature involve the primary
Λ, i.e. those whichare emitted from the freeze-out hypersurface. However, they are just a fraction of themeasured Λ’s, about 25% at √ s NN = 200 GeV according to statistical hadronization modelestimates (49), while most of them are decay products of higher lying states, such as Σ , Σ ∗ ,Ξ etc. Those states are expected to be polarized as well, according to the formula 2 withthe suitable spin-dependent coefficient (see e.g. eq. 11), hence with the same momentumpattern as for the primary Λ’s. The secondary Λ from decays of polarized particles turnsout to be polarized in turn and its polarization vector depends on the properties of theinteraction responsible for the decay (strong, electromagnetic, weak) and on the polarizationof the decaying particle. The formula for the global polarization inherited by the Λ’s inseveral decay channels was obtained in ref. (37) and its effect studied in (50). While singlechannels 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 wherethe polarization transfer in 2-body decays producing a Λ hyperon was determined as afunction of momentum (51, 35). Surprisingly, the combination of relative production ratesof different hyperons, their decay branching ratios and the coefficients of the polarizationtransfer produce an accidental cancellation of the contribution of secondary Λ’s polarizationso that the dependence of polarization as a function of momentum is almost the same aspredicted for primary Λ’s alone (51, 35).While the contribution of secondary decays is under control, little is known about theeffect of post-hadronization secondary hadronic scattering after the hydrodynamic motionceases. In general, one would naturally expect an overall dilution of the primary polar-ization. However, it has been speculated (52) that final-state hadronic rescattering couldgenerate some polarization and a model was put forward in ref. (53) showing that initiallyunpolarized hyperons in pA collisions can become polarized because of secondary interac-tions. However, the same model applied to AA yields a secondary polarization consistentwith zero (54). If, for some reason, spin degrees of freedom relax more slowly than momentum, local ther-modynamic equilibrium is not possibly a good approximation and the calculation of po-larization becomes more complicated. A possible substitute theoretical approach is kinetictheory. However, as has been mentioned, near the pseudo-critical temperature, the QGP isa strongly interacting system for which a kinetic description is dubious because the thermalwavelength of partons is comparable to their mean free path; particles interact so strongly
12 Becattini and Lisa hat they are not free for most of their time. Notwithstanding, one may hope that kinetictheory provides a good approximation for the spin degrees of freedom if the spin-orbit cou-pling 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 beessential.A formulation of relativistic kinetic theory with spin dates back to De Groot and col-laborators (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 motivatedby the search of the Chiral Magnetic Effect (57, 58), the corresponding theory for massivefermions is mostly motivated by the observation of polarization. The goal of the relativistickinetic 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 Boltzmannequation. For free particles this reads: (cid:99) W ( x, k ) AB = − π ) (cid:90) d y e − ik · y : Ψ A ( x − y/ B ( x + y/
2) : 13.where Ψ is the Dirac field,
A, B are spinorial indices and : denotes normal ordering; thisdefinition should be changed to make it gauge invariant in quantum electrodynamics. Mostrecent studies aimed at a formulation of the covariant Wigner function kinetic equationsin a background electromagnetic field (59, 60, 61, 62, 63) at some order in (cid:126) . A differentapproach was taken in ref. (64), where the polarization rate was obtained including the spindegrees 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 producedstable numerical estimates of polarization in heavy ion collisions. However, important stepstoward this goal have been recently made. In ref. (65) an estimate of the evolution equationof the spin density matrix in perturbative QCD has been obtained. Computing tools arealso 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 par-tons to the hadrons, which is not relevant for the hydrodynamic-statistical model, see thediscussion at the beginning of this Section. More generally, there is a gap between theperturbative, collisional quark-gluon stage and the hadronic final state which is highly non-trivial and needs to be bridged.
A very interesting theoretical issue concerned with the description of spin effects in relativis-tic 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 connectionwith the proton spin studies (67). A comprehensive introduction and discussion of thesubject 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 contribu-tions: a so-called orbital part involving the stress-energy tensor and a spin part involvinga rank three tensor S λ,µν called spin tensor . However, this separation seems to be un-physical and one can make a transformation of the stress-energy and the spin tensor soas 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-gaugetransformation (68) and it looks much like a gauge transformation in gauge field theorieswhere the stress-energy and the spin tensor play the role of gauge potentials, while the • Polarization and Vorticity in the QGP 13 otal energy-momentum P µ and angular momentum-boost J µν are gauge-invariant. Thequestion is whether an observation of a polarization in the QGP breaks pseudo-gauge in-variance, making it possible to single out a specific spin tensor. This would be obviouslya breakthrough with remarkable consequences, as it would have an impact on fundamentalphysics, such as relativistic gravity theories.Indeed, the first derivation of the formula 2 made use of a specific spin tensor and thishas 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 tensorused, among the most common choices. It has recently become clear that the definition ofspin density matrix and of the spin vector (70, 71) in Quantum Field Theory do not indeedrequire any angular momentum or spin operator, just the density operator and creation-destruction operators (35); so, their expressions are completely independent of the spintensor. In fact, the mean value of the polarization may depend on the spin tensor, insofaras the density operator does. At global thermodynamic equilibrium, the density operator 8is manifestly independent of the spin tensor because only the total angular momentumappears, but in the case of local thermodynamic equilibrium, the density operator 12 is notinvariant under a pseudo-gauge transformation (72). Then, in principle, one might be ableto distinguish between two spin tensors by measuring the polarization. Of course, this is aprinciple statement because, in practice, there are many uncertainties limiting the accuracyof the theoretical predictions (e.g. the hydrodynamic initial conditions) and it is not clearyet to what extent the measurements could solve the issue.The inclusion of the spin tensor in relativistic hydrodynamics has been explored insome detail by W. Florkowski et al. in a series of papers (73),(70) and a first hydrodynamiccalculation of polarization presented in a simplified boost-invariant scenario (74). As far asthe heavy ion phenomenology is concerned, a general comment is in order for the spin tensorscenario: an extended version of relativistic hydrodynamics requires six additional fields (theanti-symmetric spin potential Ω µν ) which in turn need six additional initial and boundaryconditions, which are completely unknown in nuclear collisions. Polarization measurementscould then be used to adjust them, but this would strongly reduce the probing power ofpolarization in all other regards. As has been mentioned, an important feature of the statistical-thermodynamic approachis 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 magnitudeand sign for Λ and ¯Λ polarization (see figure 3 later in this work). Indeed, for a fluidwith some charge current, a difference in the polarization of particle and anti-particle isencoded in the Fermi-Dirac distributions in eq. 2 in that the e.g. baryon chemical potentialis larger at lower energy, favouring the ¯Λ’s polarization through the factor n F (1 − n F ) inthe numerator (38). However, the known values of baryon chemical potential/temperatureratios at the relevant collision energies imply a much smaller difference in the polarizationthan observed.A possible source of particle-antiparticle polarization splitting is the electro-magneticfield, which would lead - at local equilibrium - to a modification of the formula 2 withthermal vorticity (cid:36) µν replaced by (37): (cid:36) ρσ → (cid:36) ρσ + µS F ρσ
14 Becattini and Lisa ith µ the particle magnetic moment. Indeed, in peripheral heavy ion collisions a largeelectro-magnetic field is present at the collision time which may steer the spin vector of Λand ¯Λ and lead to a splitting of polarization, their magnetic moments being opposite.Therefore, the polarization splitting might be taken advantage of to determine themagnitude of the electro-magnetic field at the freeze-out (or earlier if the relaxation timeis not small) (37) or its lifetime (75). Pinning down the electro-magnetic field would be avery important achievement in the search of local parity violation in relativistic heavy-ioncollisions (76) through the so-called Chiral Magnetic Effect (57, 58). However, alternativeexplanations of the splitting have been proposed and this feature needs to be exploredexperimentally and theoretically. We will return to this in section 5.2.
4. POLARIZATION IN RELATIVISTIC HEAVY ION COLLISIONS:OBSERVATIONS
As of this writing, there is only a handful of measurements of spin polarization in relativisticheavy ion collisions. These measurements require excellent tracking and vertex resolutionin the region of interest (typically midrapidity); large coverage and good particle identi-fication to measure decay products; high statistics to measure relatively small correlationsignals; and a suite of detectors to correlate forward-rapidity momentum anisotropies withmidrapidity decay topologies. Several such experiments exist today, and more will soon becommissioned. The initial measurements described here will eventually be part of a fullerset of mapped systematics.
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 veryclean 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 involveslarge tracking systems (e.g. 78). A particle’s polarization may be determined by the topologyof its decay into charged particles, if the angular distribution of daughters’ momenta isrelated to the spin direction of the parent.For weak parity-violating hyperon decays with spin and parity
12 + →
12 + + 0 − , thedaughter baryon is emitted preferentially in the direction of the polarization vector ( P ∗ H )of the parent, as (79)d N dΩ ∗ = 14 π (1 + α H P ∗ H · ˆp ∗ D ) = 14 π (1 + α H cos ξ ∗ ) , ˆp ∗ D is a unit vector pointing in the direction of the daughter baryon momentum, and ξ ∗ is the angle between the ˆp ∗ D and the polarization direction. Here and throughout, anasterisk ( ∗ ) denotes a quantity as measured in the rest frame of the decaying parent. Thedecay parameter α H depends on the hyperon species (80).The general task to extract polarization from experimental data is to identify a potentialdirection, say ˆn (specific examples discussed below). The ensemble-averaged projection ofthe daughter baryon’s momentum along ˆn gives the projection of P : (cid:104) ˆp ∗ D · ˆn (cid:105) = α H P ∗ H · ˆn . • Polarization and Vorticity in the QGP 15 irst measurements (26, 7, 81, 82, 83) of polarization in relativistic heavy ion collisionshave used Λ → p + π − (Λ → p + π + ) decays. The decay parameter for an antiparti-cle is expected and observed (80, 84) to be of equal magnitude and opposite sign of thecorresponding particle within measurement uncertainties. Polarization of other hadronic species may also also be measured, in principle. Thereduced efficiency associated with identifying two displaced vertices, as well as the reducedyield of doubly strange baryons makes using Ξ − ( α Ξ − → Λ+ π − = − . ( α Ξ → Λ+ π = − . α Ω 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, whichhas been extensively discussed in this work. For particles with spin > /
2, a full descriptionof the polarization state requires more quantities; in practice, one should quote the full spindensity matrix Θ rs ( p ) (see Section 3). Particularly, for spin 1 particles, a quantity inde-pendent of the mean spin vector related to the polarization state is the so-called alignment (86): A = Θ ( p ) − /
3A randomly-oriented ensemble would have Θ = , hence vanishing A ; a value Θ (cid:54) = indicates spin alignment, though by symmetry it is impossible to distinguish the sign in (cid:68) (cid:126)S (cid:69) (cid:107) ˆ n . The 2-particle decay topology of a vector meson is related to the alignmentaccording to (87): d N d cos ξ ∗ = 34 (cid:2) − Θ + (3Θ −
1) cos ξ ∗ (cid:3) , ξ ∗ is defined as in equation 15. At local thermodynamic equilibriu, − ρ is quadraticin thermal vorticity to first order, as mentioned in section 3.Thus far, the first measurements of global spin alignment of vector mesons in heavy ioncollisions are difficult to understand in a consistent picture. We discuss these in subsection5.3, and focus here on hyperon polarization. By symmetry, the average vorticity of the QGP fireball must point in the direction of thefireball’s angular momentum (cid:126)J
QGP , and on average (cid:126)J
QGP (cid:107) (cid:126)J sys (c.f. figure 1). Similarly,even without appealing to a connection to vorticity, when averaged over all particles, sym-metry demands an average (over all emitted particles) polarization aligned with ˆ J sys . Inthe current context, the ”global polarization” of a subset of particles refers to the use ofˆ n = ˆ J sys in equation 16.As discussed in section 2.2, the momentum-space anisotropy of particle emission isused (22) to extract an event plane angle Ψ EP, which approximates the reaction planewith some finite resolution. Standard methods have been developed (22) to correct for theeffects of this resolution on measured asymmetries in the emission pattern about the beam Until very recently, the accepted world average value has been (80) α Λ = 0 . ± . α Λ = 0 . ± . ± . σ . Although the source of this large discrepancy not entirely clear, in itsonline 2019 update, the Particle Data Group adopted this new value. Therefore, we have decidedto scale all reported polarizations to reflect the BESIII value.
16 Becattini and Lisa
10 (GeV) NN s0246 P ( % ) STARALICE
L L =0.75 L a scaled using L and L Average of hydrodynamicsparton cascade (AMPT)hadron cascade (UrQMD) 3-fluid dynamicschiral kinetic
Figure 3
Left: The vectors and angles involved in an analysis of hyperon polarization along the angularmomentum of the collision are shown. In the lab coordinate system (not shown), the azimuthalangle of ˆ b is defined to be Ψ RP . Thus, the angle between the projection of ˆ p ∗ D and ˆ b is φ ∗ D − Ψ RP .The minus sign on the angle indicated arises from the fact that azimuthal angles are measuredcounterclockwise about the beam axis.Right: The energy dependence of Λ and Λ global polarization at mid-rapidity from mid-centralAu+Au (20-50%) or Pb+Pb (15-50%) collisions. Data (7, 26, 81, 83) are compared to polarizationsimulations of viscous hydrodynamics (50); partonic transport (88); hadronic transport (89);chiral-kinetic transport plus coalescence (90); and a three-fluid hydro model applicable at lowerenergies (91). Experimental data points have been corrected for the recent change in α Λ , asdiscussed in section 4.1. For (50) and (88), the values shown represent both primary andfeed-down hyperons (c.f 37). See text for details. axis, so it is convenient to rewrite equation 16 as (26) P H, ˆ J = 3 α H (cid:68) ˆp ∗ D · (cid:16) ˆb × ˆp beam (cid:17)(cid:69) = 3 α H (cid:104) cos ξ ∗ (cid:105) = − α H (cid:104) sin ( φ ∗ D − Ψ RP ) sin θ ∗ D (cid:105) . φ ∗ D and θ ∗ D are the angles between the daughter momentum and ˆb and ˆp beam , respec-tively, and in the last step, a trigonometric relationship between the angles is used. Theseangles are shown in figure 3.Integrating over polar angle θ ∗ P H, ˆ J = − πα H (cid:104) sin ( φ ∗ D − Ψ RP ) (cid:105) = − πα H R (1)EP (cid:104) sin ( φ ∗ D − Ψ EP , ) (cid:105) , R (1)EP (22). See reference (92) for a discussion of significant experimental challenges to perform the averagein equation 18 Detectors in which Λs are reconstructed usually do not measure the charged daughters at veryforward angles at collider energies. Corrections (26, 93) on order ∼
3% (94) are applied in order toaccount for this. • Polarization and Vorticity in the QGP 17 he resolution with which ˆ J sys is measured is critical. Polarization affects daughteranisotropies only at the few percent level, and statistical uncertainties can dominate exper-imental results. Using equation 19. the statistical uncertainty on polarization goes as δP H, ˆ J ∼ (cid:16) R (1)EP · √ N H (cid:17) − , N H is the total number of hyperons analyzed in the dataset. This dependence isgenerically true for any measurement that involves correlation with the first order eventplane or ˆ J sys . Increasing the resolution by a factor of two (95) thus decreases the requiredduration of an experimental campaign four-fold.Figure 3 shows the world dataset of P Λ , ˆ J and P Λ , ˆ J as a function of collision energyfor semi-peripheral collisions. As discussed in section 4.1, the recent change in acceptedvalue for α Λ requires a rescaling of the published experimental values. From a maximumof ∼ .
5% at √ s NN = 7 . with energy. At LHCenergies, they vanish within experimental uncertainties.Strikingly, all available hydrodynamic and transport calculations reproduce the obser-vations in sign, magnitude and energy dependence, as discussed in section 3.2. This isnontrivial; since they all use formula 2 it means that they all predict a very similar ther-mal vorticity field. These models have been to some extent ”tuned” to reproduce earlierobservations such as anisotropic flow (6), which is sensitive to the bulk motion of fluid cellsfrom which particles emerge; it is thus satisfying that they produce similar and correctpredictions for this more sensitive observable.Clearly, | (cid:126)J sys | increases with increasing √ s NN , and transport calculations (96, 97) pre-dict that about 20% of this angular momentum is transferred to the QGP fireball. Whilesome early calculations (25) predicted an increased polarization at high collision energies,a strongly decreasing trend is produced by most hydrodynamic (50, 98) and transport-hybrid codes in which the thermal vorticity field is obtained through a coarse-grainingprocedure (96, 99, 100, 101, 102).Driving mechanisms may include increased temperature (103) at increased √ s NN ; in-creases 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); anincreased fluid moment of inertia due to increased mass-energy (96); reduced longitudinalfluctuations and boost-invariance at high energy (104).In addition to the overall energy dependence, the data in figure 3 suggests a fine splittingbetween particles and antiparticles at low √ s NN . While statistically not significant at anygiven energy, very important physical effects are predicted to manifest in P Λ , ˆ J > P Λ , ˆ J , aswe discuss in section 5.2.Even as we note possible differences between the polarizations of Λ and Λ, it is clear thatto good approximation they are the same, even at the lowest energies, suggesting similaraverage vorticity of the cells from which they arise. This is remarkable, in light of the factthat the directed flow of these particles diverge strongly (105) as the energy is reducedbelow √ s NN ≈
20 GeV, even taking opposite signs at midrapidity. In the hydrodynamicparadigm, directed flow (22), essentially the sidewards push of forward-going particles (c.f.figure 1) reflects the anisotropy of the bulk fluid velocity about the ˆ J axis at a large scale. While eye-catching, the value of P Λ , ˆ J = (7 . ± . σ abovethe general systematics and is marginally significant.
18 Becattini and Lisa
10 20 30 40 50 60 70 80Centrality (%)00.20.40.60.8 ) / L , J + P L , J ( P =0.75 L a scaled using ( % ) æ ) EP , Y - H f s i n ( z P Æ =0.75 L a scaled using Figure 4
The centrality dependence of hyperon (average of Λ and Λ) polarization in 200 GeV Au+Aucollisions. As in figure 3, published data have been rescaled to reflect the new accepted value of α Λ . Cartoons at the bottom of each panel roughly sketch the geometry of the overlap region for agiven centrality. Left panel: Global polarization (81). Right panel: Second-order oscillationamplitude of the longitudinal polarization (82). Meanwhile, global polarization reflects rotational flow structure about ˆ J at a more localscale. There may be a coupling in a hydrodynamic picture (106, 103, 12). Whether there isa tension here is unclear, though a three-fluid hydrodynamic code is able to approximatelyreproduce proton and antiproton flow (107) and Λ polarization (91). √ s NN = 200 GeV
Systematic studies of the dependence of P ˆ J,H in Au+Au collisions have so far only beenpossible at √ s NN = 200 GeV (81). Statistics are poor at low energies, while at higherenergies, the signal itself vanishes. More detailed measurements can provide stringent chal-lenges to theoretical models and may provide new insight. In √ s NN = 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 | (cid:126)b | ≈ −
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 phenomenondriven by bulk mechanical angular momentum; this is in agreement with transport-hybridcalculations (96).The ”global” polarization– i.e. integrated over all particles at midrapidity– is non-zeroand aligned with an event-specific direction. Momentum-differential (”local”) polariza- This is in strong contrast to the well-known phenomenon (109, 110) in p+p and p+A collisions,in which Λ (but not Λ, for unclear reason) hyperons emitted at very forward angles are polarizedalong their production plane, spanned by (cid:126)p Λ × (cid:126)p beam . This effect is rapidity-odd, vanishing atmidrapidity. In principle, convolution of the production-plane polarization with finite directed • Polarization and Vorticity in the QGP 19
EP,1 Y - L f ( % ) J H , P STAR Collaboration Preliminary Data =0.75 L a scaled using Figure 5
Left: Preliminary results (108) from the STAR collaboration for the global polarization of Λ andΛ as a function of hyperon emission angle relative to the event plane, for mid-central Au+Aucollisions at √ s NN = 200 GeV. As in figure 3, published data have been rescaled to reflect thenew accepted value of α Λ . Right: Hydrodynamic calculations (12) of P ˆ J in the transversemomentum plane, for the same colliding system. tion structures, in the local equilibrium picture, are more sensitive to the thermal vorticityvariations as a function of space and time, convoluted with flow-driven space-momentumcorrelations. First measurements (81) report P ˆ J, Λ / Λ to be independent of transverse mo-mentum for p T (cid:46) | η | < P ˆ J, Λ&Λ is significantly stronger for particles emitted perpendicular to ˆ J sys ( | φ Λ − Ψ RP | = π/
2) than for ˆ p Λ (cid:107) ˆ J . Indeed, P ˆ J may vanish for hyperons emitted out ofthe reaction plane. This stands in contradiction to rather robust predictions of hydrody-namic (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 oppo-site dependence. If the STAR results are confirmed in a final analysis, this represents anontrivial challenge to the theory.By symmetry, polarization components perpendicular to ˆ J sys must vanish, when av-eraging over all momenta. Locally in momentum space, however, these components areallowed to be non vanishing. Particularly, there can be non-vanishing values oscillatingas a function of the azimuthal emission angle φ H over the transverse plane with a typicalquadrupolar pattern. Hydrodynamic (12) and transport calculations (100) predict the signand the magnitude of these oscillations. Here, ˆ n = ˆ p beam in equation 16 so ξ ∗ D = θ ∗ D , thepolar 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 thosewe discuss here (31).
20 Becattini and Lisa
EP,2 Y - L f - - - · æ * p q c o s Æ Figure 6 (cid:10) cos θ ∗ p (cid:11) for 20-60% centrality Au+Au collisions at √ s NN = 200 GeV, as a function of hyperonemission angle relative to the event plane (82). Small detector effects (see footnote 6) andevent-plane resolution effects have not been corrected for, in this figure. A sinusoidal curve isdrawn to guide the eye. Right: Hydrodynamic calculations (12) of P ˆ z in the transversemomentum plane, for the same colliding system. predict a negative sign of the longitudinal component of the polarization vector in thefirst quadrant of the p T plane rotating counterclockwise from the reaction plane. One suchcalculation 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 morestrikingly, the sign of the predicted oscillation is opposite that seen in the data, reminiscentof the discrepancy in figure 5.Understanding and resolving the tension in figures 5 and 6 is among the most pressingopen issues in this area. This is further discussed in Section 5.
5. OPEN ISSUES AND OUTLOOK
Above, we have presented the theoretical framework (mostly hydrodynamics) in whichto calculate the vorticity of the QGP; the theoretical connection between the vorticityand the polarization of hadrons emitted from the plasma, based on local thermodynamicequilibrium of hadrons and their generalized distribution function; and the measurementsof this polarization with Λ hyperons. Overall, the hydrodynamic and statistical equilibriumparadigm predicted first experimental observations of global polarization strikingly well.However, qualitative discrepancies between theory and experiment may indicate thatsome fundamental feature of the dynamics itself (encoded hydro or transport) is misunder-stood or unaccounted for. Alternatively, we may misunderstand the interface (“Cooper-Frye” and thermal vorticity) between hydrodynamics or its coarse-grained approximationand the polarization observable. Clearly, the existing data demands more theoretical workand 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 demandexperimental tests. These will involve new detectors, future facilities, and new analysistechniques.Finally, two topics deserve separate attention. One is the possibility that polarizationsof Λ and Λ are different. The other concerns the spin alignment of vector mesons. • Polarization and Vorticity in the QGP 21 .1. Local polarization
The discrepancies between hydrodynamic calculations and the polarization pattern in mo-mentum space have been presented in subsection 4.3 and they have been the subject ofinvestigations 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 de-pendence polarization as the primary, if all the primary are polarized according to thehydrodynamic predictions. The other simple explanation is a polarization change in post-hadronization 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 shouldproduce an amplification in some selected momentum regions. The available hadronictransport codes do not include the spin degrees of freedom mostly because the helicity-dependent scattering amplitudes are unknown and, even resorting to educated guesses, itis a formidable computational task to include them into Monte-Carlo codes.Within the hydrodynamic paradigm, there are more options yet to be explored. Thefirst is concerned with the formula 2, which is first-order in thermal vorticity. Indeed, ther-mal vorticity is moderately smaller than 1 (see figure 2) and the exact formula at all ordersis not known yet, so a sizeable role of higher order corrections cannot be ruled out for thepresent.Since polarization is steered by thermal vorticity, it is possible that the thermal vorticityfield is different from the predictions obtained with the presently used initial hydrodynamicconditions, 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 specificinitial 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 forthe stress-energy tensor, which are not included in the local thermodynamic equilibriumassumption, are sizeable. As has been mentioned in Section 3, the theory of dissipationand 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 itis pseudo-gauge dependent (see subsection 3.5).Furthermore, it has been considered that other kinds of vorticity, instead of thermal vor-ticity 3, enter in the polarization definition. In ref. (102) it has been shown that the rightsign of the longitudinal polarization is retrieved if the thermal vorticity is replaced by the atensor proportional to the T-vorticity (12) whereas in ref. (115) the agreement was restoredby replacing the thermal vorticity with its double projection perpendicular to the velocityfield. 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 beenobtained for the polarization of quarks within a chiral kinetic model; the question remainson 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 spinpotential to describe its hydrodynamic evolution, with six additional degrees of free-dom and six additional hydrodynamic equations (see subsection 3.5).
22 Becattini and Lisa f course, both should be able to explain why the global polarization is in very goodagreement with local equilibrium with thermal vorticity. Finally, we should always considerthe 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 P Λ ,J at the lower collision energies where polarizationitself is large. A possible interpretation of such a splitting is the presence of a large electro-magnetic field and one could use the observed difference to extract the value of the magneticfield in the rest frame of the particles, as discussed in subsection 3.6.To first approximation, the ˆ J -component of the vorticity is determined by the sumof P Λ ,J and P J, Λ , and the magnetic field by their difference (37). However, feed-downcorrections can be important, and should be accounted for (37). For example, in the absenceof feed-down, a finite B -field would produce P Λ ,J > P Λ ,J , and B = 0 would result in nodifference 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” withopposite sign, i.e. P Λ ,J < P Λ ,J (37). Applying formula 2 (with the substitution 14) to thedata in figure 3, and accounting for feeddown effects (37), results (117) in an estimate of B = (6 ± × T when averaging over results from 10 GeV < √ s NN <
40 GeV. Such anaverage is hardly justified, but it nevertheless provides a valuable estimate of the magnitudesof the magnetic field– and the measurement uncertainty– that may be associated with thedata. 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. Whilethey may decay well before freeze-out, a highly conductive QGP itself can significantlyextend the lifetime of the initially large field (118) and vorticity certainly helps in thisrespect (119). At low √ s NN , field lifetimes may be longer (120) and QGP evolution timeshorter. Relativistic magneto-hydrodynamics it the standard tool to study the evolutionof the electro-magnetic field in a plasma and there have been major advances recently(121, 122). Neglecting feed-down corrections, the similarity of P Λ ,J and P Λ ,J places (123)an upper limit on the magnetic field at freezeout of about 10 T at top RHIC energy.Transport calculations may provide more insight, with respect to local thermodynamicequilibrium. Simplified calculations estimate that the expected field could be on order10 − T, and the energy dependence of the splitting would resemble that seen in thedata (119). A more sophisticated calculation (75) with partonic transport argues that thedifference between P Λ ,J and P Λ ,J may be reasonably attributed to the accumulated effectof an evolving magnetic field; interestingly, in the absence of a magnetic field, P Λ ,J < P Λ ,J .A firm statement on the existence of a long-lived (several fm/c) magnetic field on thescale of 10 T would have tremendous implications for the Chiral Magnetic Effect (57).However, several effects have been postulated, which may complicate the interpretation ofthe splitting. Especially at low collision energies where baryon stopping is significant, Λand Λ may originate from different regions within the fireball (89), their final polarizationsthus reflecting differently-weighted averages over vorticity. Han et al argue that the quark-antiquark vector potential in the presence of the net quark flux at these energies maygenerate a splitting largely indistinguishable that expected from a magnetic field. Thevector meson field may play an equivalent role in the hadronic sector; however, existingcalculations (124, 125) reproduce the splitting only by adjusting by hand the unknown • Polarization and Vorticity in the QGP 23 ign, magnitude and energy dependence of the effect.In principle, disentangling these effects could require a full three-dimensional magne-tohydrodynamic calculation which includes appropriate vector potentials, conserved non-trivial baryon currents and QGP conductivity, potentially followed by a hadronic cascadewith spin-transfer collisional dynamics. Hopefully, however, sophisticated but achievablecalculations, in conjunction with targeted measurements, can lead to reasonable estimatesof the individual contributions of these important effects.
In peripheral collisions the anisotropy generated by the collective angular momentum impliesthat 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. Atlocal thermodynamic equilibrium, the equation 10 predicts an alignment which is quadraticin the thermal vorticity (29). Since thermal vorticity is less than 1 throughout (see figure2), 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, Θ (c.f. equation 17) is considerablydifferent than . For φ at LHC and K ∗ at both colliders would imply a vorticity atleast two orders of magnitude higher than calculations or expectations from the hyperonmeasurements. More surprisingly, Θ for the φ mesons at RHIC is greater than (126),something that cannot be understood in hydrodynamic or recombination model (128). Thisobservation may require fundamentally new physics mechanisms (129) for alignment thatapply at RHIC but not at LHC. Altogether, the situation with these preliminary spinalignment measurements is not sufficiently well understood to discuss in a review. As we have discussed, the first few positive observations of hyperon polarization at RHIChave generated tremendous theoretical activity. Much of this work has focused on the degreeto which models can reproduce the measurements, but a growing body of work points tothe ways in which new measurements can strictly test our understanding of QGP dynamicsand 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 collisionenergy is reduced. Exploring this trend for even lower energies may touch on several impor-tant questions: does a hydrodynamic description of the system break down at lower energydensity? What are the effects of increased viscosity (130, 50, 131, 132)? Can spin equilibraterapidly enough to justify a local thermodynamic equilibrium approximation – and if so, isthis due to hadronic mechanisms or to the QCD phase transition? Some hydrodynamicmodels tuned for low energies predict an uninterrupted continued rise (103, 112), regardlessof equation of state (133, 91), though initial state and thermalization assumptions mayaffect 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 andpolarization at low collision energy; how tightly coupled are the large-scale and smaller-scalerotational 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) signalsa phase transition in the equation of state. Alternatively, it may arise from a convolution
24 Becattini and Lisa f baryon stopping and quark coalescence (137, 105).Again regarding Λ and Λ, it has been suggested (89) that differences in polarizationare dominated by differences in the phase space from which these particles arise; suchdifferences are largest at low collision energy. Testing this hypothesis and that discussed inthe previous paragraph of course will require measurement of local polarization, that is asa function of momentum.Addressing these questions will require new measurements at √ s NN (cid:46)
10 GeV, withgood tracking, event plane resolution, and high statistics, especially given plummeting Λyields. These will be pursued at the future NICA (138, 139) and FAIR (140) facilities, aswell 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 theQGP fireball. However, calculations with a variety of models suggest a vortical structurethat evolves with rapidity.A geometric calculation (28) based on the BGK model of hadron production (143) andboost-invariance suggests that vorticity will increase with rapidity, and speculates that arapid 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 dependson √ s NN at RHIC energies, and that it is sensitive to important physical parameters of themodel itself. Numerical calculations with transport-hybrid codes (96, 99, 102) also indicatea forward migration of vorticity, especially as the collision energy increases (99). Finally,hydrodynamic models predict much higher vorticity in the beam fragmentation region atboth 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 providecoverage and tracking over a physically important region (144). If the event plane can bereconstructed, the LHCb experiment (147) could be used to explore the rapidity evolutionat the highest energies. Other polarization projections
Referring to equation 16, experiments have reported polarization projections along ˆ n = ˆ J and ˆ n = ˆ p beam . The geometry of the collision itself suggests other natural directions.For particles emitted at forward rapidity, symmetry permits an average polarizationprojection along ˆ n = ˆ b . In fact, ”vortex rings” or ”cyclones” are predicted (100, 146) atforward rapidity at RHIC (100) or NICA (97, 146) energies, as well as at midrapidity innon-symmetric systems (148). In this case, ˆ n (cid:107) (cid:126)p Λ × ˆ z .One of the first model studies of vorticity in heavy ion collisions predicted similar ring-like structures relative to jets. High-momentum partons formed in the initial stages of theheavy ion collision lose energy in the QGP fireball (149) and can locally perturb the flowfield (150). This may produce a cone or ring of vortical structure locally perpendicular tothe direction of the deposited momentum (28), ˆ n = ˆ p dep × ˆ p H , where the hyperon H hasacquired 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 character-istic or symmetry-breaking direction. However, the assumption is that the polarizations ofall particles emitted from a cell are aligned with the vorticity of that cell, and flow-inducedspace-momentum correlations (6) cause particles from the same cell to be emitted in thesame direction. Hence, if experimental complications can be overcome (92), spin-spin cor- • Polarization and Vorticity in the QGP 25 elations as a function of relative momentum (or angle) are a promising way to probe theturbulent vortical substructure of the QGP (104).
6. SUMMARY AND OUTLOOK
Polarization has opened an exciting new direction in relativistic heavy ion physics; one ofthe increasingly rare truly new developments in this rather mature field. Its measurementhas definitely proved that a new degree of freedom other than momentum is now availableto probe the QGP formation and dynamics. In the hydrodynamic model, unlike particlemomentum, 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, polar-ization can help to constrain the electro-magnetic field, which would be incredibly valuablefor the search of Chiral Magnetic Effect (57). The hydrodynamic model predicts, and themeasurements have shown, that polarization increases at low energy, and it will be furtherexplored in future low-energy heavy ion programs. At RHIC and LHC energies, flow sub-structure is already being probed in unprecedented detail, presenting theory with new andas yet unsolved challenges. Directions for future studies at these energies were discussed.There are several pressing issues to be solved which require considerable advances intheory and phenomenology. Indeed, at this time, after having played the leading role, theoryappears to have been surpassed by the experiments which have proved to be able to mea-sure 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 beavailable which will help to constrain or disprove theoretical models and assumptions; po-larization of different species (e.g. Σ and Ξ − ), spin-spin correlations (104); measurement ofpolarization 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, theinclusion of dissipative effects and the application of alternative methods such as kinetictheory as well as the development of a hydrodynamic with spin potential. Equally impor-tant is a major advance in phenomenology and numerical computation, with the inclusionof hadronic rescattering effects and the systematic study of polarization dependence on theinitial conditions.Since its experimental discovery a few years ago, there has been tremendous progressin the study of polarization in heavy ion collisions. Yet, at this early stage, the potentialof this new tool is still to be explored. It may well be that this direction of research yieldsnew 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 holdingsthat 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 ofthe 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.
26 Becattini and Lisa
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