Recent developments in chiral and spin polarization effects in heavy-ion collisions

Jian-Hua Gao · Guo-Liang Ma · Shi Pu· Qun Wang

Abstract We give a brief overview of recent theoretical and experimental results on the chiral magnetic effect and spin polarization effect in heavy-ion collisions.We present updated experimental results for the chiral magnetic effect and related phenomena.The time evolution of the magnetic fields in different models is discussed. The newly developed quantum kinetic theory for massive fermions is reviewed. We present theoretical and experimental results for the polarization of Λ hyperons and the ρ00 value of vector mesons.

Keywords Relativistic heavy-ion collisions · Chiral magnetic effect · Chiral kinetic theory · Spin polarization

1 Introduction

In relativistic heavy-ion collisions, two charged nuclei collide to produce a hot, dense state of matter known as a quark–gluon plasma(QGP).Very high magnetic fields and orbital angular momenta (OAMs) are generated in these collisions. The magnetic fields are on the order of 1017-18Gs [1–4] and are the strongest magnetic fields observed in nature. The QGP is also found to be the most vortical fluid [5], where a huge OAM is transferred to the fluid in the form of vorticity fields.These novel phenomena open a new window for the study of the QGP in heavy-ion collisions. These novel phenomena are quantum in nature and are usually negligible in classical fluids.

The chiral magnetic effect (CME) and chiral separation effect (CSE) [6–8] are two quantum effects on the magnetic field in a chiral fermion system.In the CME,a charge current is induced along the magnetic field:

where μ is the charge chemical potential. These two quantum effects are related to the chiral anomaly,which is absent from classical theories. The chiral chemical potential represents a chirality imbalance.The nonzero μ5arises from topological fluctuations in quantum chromodynamics(QCD), which are related to the local violation of parity and charge parity. Therefore, the observation of the CME in heavy-ion collisions implies local parity and charge parity violation. The collective modes associated with the CME and CSE are called the chiral magnetic wave(CMW). In addition, the CME has been observed in condensed matter [9] and can be applied in quantum computing[10].These phenomena are discussed in more detail in recent reviews [11–20] and references therein.

In non-central heavy-ion collisions, part of the very large OAM of colliding nuclei is converted to a vorticity field in the fluid.The vorticity can be regarded as the local OAM of the fluid and can polarize particles with spins through spin–orbit coupling. The global spin polarization of particles is along the direction of the global OAM of colliding nuclei (or the direction perpendicular to the reaction plane). The global polarization of Λ andhyperons has been measured in the STAR experiment,and the average angular velocity or vorticity was estimated as ω ～1022s-1[5]. The data on the global polarization of Λ andhyperons can be well described by various phenomenological models.

The STAR Collaboration has also measured the local spin polarization effect in Au+Au collisions at 200 GeV [21]. The data show a decreasing trend in the global spin polarization from the in-plane to the out-of-plane direction. In addition, the STAR Collaboration measured the azimuthal angle dependence of the spin polarization along the beam direction. These experimental results are still difficult to understand in terms of phenomenological models.

This paper is structured as follows. In Sect. 2.1, we review recent progress in theoretical studies of the CME and related effects in heavy-ion collisions.In Sect. 2.2,we present experimental results on the CME and CMW. In Sect. 3,we give a brief overview of recent developments in the quantum kinetic theory for massive fermions. In Sect. 4.1, we discuss the decomposition of the OAM and spin from the total angular momentum as well as the spin hydrodynamics. In Sect. 4.2, we discuss experimental results on the global and local polarization effects and spin alignments of vector mesons. We summarize the paper in Sect. 5.

2 Chiral magnetic and related effects in heavy-ion collisions

2.1 Theoretical progress

To study the magnetic-field-related effects in heavy-ion collisions,we need to know the magnetic field as a function of time. The electromagnetic field can be estimated using the Lienard–Wiechert potential [1–4]. Although the peak value of the magnetic field can be as large as a few times m2π, where mπis the mass of a pion, it decays very rapidly in vacuum [7]. Such a magnetic field in vacuum could not provide sufficient time to generate the CME and other chiral transport effects; thus, intermediate effects must be present [4, 22–24].

2.2 Recent experimental results

The dipole charge separation due to the CME can be characterized by the first sine term a1in the Fourier series of the charged-particle azimuthal distribution:

where φ-ΨRPrepresents the particle azimuthal angle with respect to the reaction plane angle ΨRPin heavy-ion collisions (which is determined by the impact parameter and beam axis),and vnand anare the coefficients of the P-even and P-odd Fourier terms,respectively.An azimuthal threeparticle correlator [61], γ112, was proposed to explore the first coefficient,a1,of the P-odd Fourier terms characterizing the charge separation due to the CME;it is given by

where α and β denote particles with the same or opposite electric charge, respectively; angle brackets denote the averages over particles and events, and ΨRP can be approximated by the azimuthal angle of the second-order event plane, Ψ2. Similarly, γ123, a charge-dependent correlator with respect to the azimuthal angle of the thirdorder event plane, Ψ3, is defined as

Because the magnitude of the magnetic field is proportional to the collision energy [2, 65], the CME should produce large differences between the γ correlators at very different energies, such as the RHIC and LHC energies. However,Fig. 1 shows a very weak energy dependence of the γ correlator in a wide energy range from the RHIC energy to the LHC energy.

As Eq. (9) shows, the Δγ resulting from the CME is proportional to both the squared magnetic field and the correlation in the magnetic field direction with respect to the event plane. The left panel of Fig. 2 shows the simulation results of Δγ due to the CME as a function of the centrality for Pb+Pb collisions at 5.02 TeV and Xe+Xe collisions at 5.44 TeV.They are very different except in the most central collisions. However, the right panel of Fig. 2 shows the preliminary ALICE data for Xe+Xe collisions,which show a centrality dependence that is very similar to that for Pb+Pb collisions in Fig. 1. The observed weak independence of Δγ from the collision energy indicates that the dominant component of Δγ is most likely due to the background.

One may argue, however, that the CME signal also depends on the lifetime of the magnetic field. Because of the shorter magnetic field lifetime at higher collision energies, the signal is very weak at very high energies.Fortunately, experimentalists are searching for a possible signal by comparing Au+Au collisions and U+U collisions at the RHIC. The top-left panel of Fig. 3 shows the expected CME contribution to Δγ as a function of Npartfor Au+Au collisions at 200 GeV and U+U collisions at 193 GeV [66],which are multiplied by a factor Npart/∊nto scale the effect from the elliptic flow and transverse momentum conservation. The CME contributions for Au+Au and U+U collisions are different except in very peripheral collisions. The difference is sizable for Npartlarger than 150. In addition, the background contribution to Δγ is simulated using a hydrodynamics model with and without maximal LCC,as shown in the bottom-left panel of Fig. 3.For each case,the expected contribution is almost the same for Au+Au collisions at 200 GeV and U+U collisions at 193 GeV, although local charge conservation can significantly increase the magnitude of Δγ. To check these results, the STAR Collaboration has measured three correlators, Δγ112, Δγ123, and Δγ132, in Au+Au collisions at 200 GeV and U+U collisions at 193 GeV [66].Their Npartdependence is presented in the right panel of Fig. 3; they are normalized by a factor of Npart/vnfor the above reason.It is observed that the mixed harmonic correlations do not follow the background-only expectations. Differences between these correlators in Au+Au collisions and U+U collisions appear only for very central collisions with Npartlarger than 300. The illustrations on the far right side of Fig. 3 show that Δγ123is not correlated with the magnetic field direction, in contrast to Δγ112and Δγ132; it is hard to see whether this difference is due to the signal or the background.A detailed study of these correlators is needed to clarify their nature.

To extract the contribution of the CME to the observable Δγ, the Δγ measured in the STAR experiment was decomposed into the v2background and the CME signal:

where ψTPCand ψZDCare the event planes measured for mid-rapidity particles in the time projection chamber(TPC) and for spectator neutrons in the zero degree calorimeter (ZDC), respectively. Assuming that the CME is proportional to the magnetic field squared and the background is proportional to v2[67], one obtains

The CME signal relative to the inclusive Δγ（ψTPC） can be determined as

which one can solve for fCMEto obtain the last equality of(13).Note that A can also be measured experimentally.The left panel of Fig. 4 shows the STAR preliminary data on the centrality dependence of Δγ with respect to the ZDC and TPC event planes in Au+Au collisions at 200 GeV and U+U collisions at 193 GeV. We see that Δγ（ψZDC） is consistently lower than Δγ（ψTPC）,which indicates that A is less than unity. This result indicates that the flow background contributes less to Δγ（ψZDC） than to Δγ（ψTPC）.Applying the above method, the STAR Collaboration extracted the CME fraction fCMEfrom different datasets for Au+Au collisions at 200 GeV and U+U collisions at 193 GeV; their results are summarized in the right panel of Fig. 4. The combined result for the CME fraction for Au+Au at 200 GeV and U+U at 193 GeV is fCME=8±4±8% [66].Note that the CMS Collaboration has systematically measured different types of correlators γijkin p+Pb collisions at 5.02 and 8.16 TeV and Pb+Pb collisions at 5.02 TeV, which provide constraints on the upper limit of the CME fraction, i.e., 13% for p+Pb and 7% for Pb+Pb collisions at the 95% confidence level[68, 69].

Many other methods of searching for the CME have been proposed, such as event shape engineering [69], the use of the H factor [71], and the invariant mass method[72]. See [11, 73] for recent reviews. A new observable,the signed balance function, was proposed recently; it is defined as [74]

Note that Sy=+1 if particle α is leading particle β (i.e.,pαy＞pβy), and Sy=-1 otherwise. N+-（Sy） denotes the number of positive–negative charge pairs with sign Syin an event. N++（Sy）, N-+（Sy）, and N--（Sy） have similar definitions. N+（-）is the number of positive (negative) charge pairs in an event. Here, the x-axis is along the reaction plane,the z-axis is along the beam direction,and the y-axis is perpendicular to both the x- and z-axes. One can calculate an event using the event difference between BPand BN:

Note that ΔBxcan be defined similarly. When the CME is absent, for a positive–negative charge pair, the probability of the positive particle leading the negative one is equal to the probability of the opposite case.Thus,BP，y（x）and BN，y（x）measure the same quantity, in principle, and the distribution of ΔBy（x）is subject only to statistical fluctuation.When the CME is present, the two probabilities become unbalanced within an event; thus, more pairs of particles of one charge type lead the other type. Therefore, for each event,BP，yand BN，ytend to differ;consequently,ΔByhas a wider distribution. By contrast, the distribution of ΔBxis not broadened, as there is no charge separation in the x direction. To cancel out the statistical fluctuation, one can define the ratio of the width of ΔByto that of ΔBxas

to characterize the magnitude of the CME, because r will be greater than unity or unity with or without the CME,respectively. That is, the strength of the CME will be positively correlated with the deviation of r from unity.The ratio r can be calculated either in the laboratory frame(rlab)or in the rest frame(rrest)of the pair.One can take the ratio of these two cases:

where the subscript B indicates a balance function. It has been found that rlab, rrest, and RBare sensitive not only to the strength of the CME, but also to the elliptic flow of primordial pions and ρ resonances, and even to the global spin alignment of the resonances [74]. Figure 5 presents the recent STAR measurement of the centrality dependence of rlab, rrest, and RBfrom the signed balance functions in Au+Au collisions at 200 GeV [70]. In the 30–40% centrality bin, rrest, rlab, and RBare all larger than the AVFD result without the CME and larger than unity, which supports the presence of the CME. However, none of the AVFD results for CMEs of different strengths can describe all three of these observables simultaneously. Therefore,more experimental and theoretical studies are needed.

In addition to the CME, a gapless CMW could be formed by the interplay between the CME and CSE. The CMW results in an electric quadrupole moment in the initial coordinate space of the QGP [82, 83]. It can ultimately be transformed into a charge-asymmetry-dependent elliptic flow of pions by collective expansion [84]. Therefore, the elliptic flow of positively and negatively charged pions is given by

is the charge asymmetry. The elliptic flow difference between positive and negative pions,[Δv2=v2（π-）-v2（π+）], can then be fitted using the relation Δv2=rAch+Δv2（0）. The left panel of Fig. 7 presents the results for Au+Au collisions at 200 GeV(30–40%) measured in the STAR experiment [79], where the slope parameter r is expected to reflect the strength of the CMW. In addition, because the third order of the event plane is not correlated with the magnetic field direction,measuring the slope parameter r from the triangular flow v3can provide a reference from the background in comparison with that from elliptic flow v2. However, care should be taken, because flow measurements by the Q-cumulant method using all the charged particles as a reference can introduce a trivial linear term to Δvn（Ach） owing to nonflow correlations.When all the charged hadrons are used as reference particles,as is typically done in data analysis,the two-particle cumulant can be rewritten as [85]

The second term on the right-hand side of (24) is proportional to Achand is opposite in sign for π+and π-. These characteristics result directly in a trivial contribution to the CMW-sensitive slope parameter. Because non-flow correlations are always present in experimental data, the flow coefficients dn｛2;pairs｝ from like-sign pairs and oppositesign pairs differ. Therefore, the second term on the righthand side of (24) is always finite as a trivial term, and it should be removed to detect the possible CMW signal. To eliminate the trivial linear Achterm in practice,one can use hadrons of a single charge sign instead of all the charged hadrons as reference particles. One may use positive and negative particles separately as reference particles to obtain vπn｛2;h+｝ and vπn｛2;h-｝ and then take an average:

Previous STAR results showed significant negative slopes for v3[86], which were thought to support the presence of the CMW. By applying the new method of removing the trivial term,the slope r can be corrected to some extent.As shown in the right-hand plot in Fig. 7, the normalized v3slopes are consistent with positive values(1.5σ above zero for 20–60%centrality),which are similar to the normalized v2slopes in terms of the relative magnitudes.In addition,it has been found that the non-flow correlations give rise to additional background in the slope of Δvn（Ach） owing to competition among different pion sources and the large multiplicity dilution of π+(π-) at positive (negative)Ach[80,81].Therefore,more detailed studies are needed to search for the CMW in the future.

3 Quantum kinetic theory

In recent years, chiral kinetic theory (CKT) has been developed significantly[87–102]to describe various chiral effects in heavy-ion collisions. Numerical simulations based on the CKT have been developed [103–110]. However,with the discovery of global polarization at relatively low energies [5, 111], it is necessary to go beyond the chiral limit and develop a more general and practical quantum kinetic theory to describe spin effects for massive fermions.In this brief review,we consider only some of the works that were reported at the 2019 Quark Matter conference.

Most of these works are based on the Wigner functions but use slightly different realizations.The methods used in Refs. [112–114]are based on early works on the covariant Wigner functions [115–117]. The covariant Wigner function W(x, p)for the Dirac fermion is defined as a two-point function:

Weickgenannt et al. [113, 114] also derived the kinetic theory for massive spin-1/2 particles in the covariant Wigner function formalism. They chose the scalar and tensor components, F and Sμν, as the basis from which the other components can be derived;these components are related to the distribution function V and tensor distribution or dipole momentμνas independent variables of O（）:

All the components of the covariant Wigner function can be derived from Vμand Aμfor massive fermions in the Schwinger–Keldysh formalism[120].The CKT for the massless case can be recovered from the formula for massive fermions. The kinetic equation has been generalized to include collision terms in the Schwinger–Keldysh formalism [121].

A covariant kinetic equation for massive fermions in curved space-time and an external electromagnetic field can also be derived.Liu et al. [122]use a general covariant Wigner function formalism not only under the U(1) gauge and local Lorentz transformation but also under diffeomorphism,which is compatible with general relativity.The spin polarization in the presence of Riemann curvature and an electromagnetic field in both local equilibrium and global equilibrium has also been studied.

Integration of the covariant Wigner function over p0yields the equal-time Wigner function [123–125]. Compared to the covariant form, the equal-time form loses obvious Lorentz covariance, but it is more convenient for time evolution problems such as pair production in a strong electric field [126, 127]. The kinetic equation can also be derived from the equal-time Wigner functions. The equaltime Wigner function W（x，p） can be obtained from the covariant one W(x, p):

where the superscripts + and - indicate the particle and antiparticle,respectively.The small mass expansion can be performed.The mass correction changes only the structure of the chiral kinetic equation in terms of the effective collision terms, which is only a first-order quantum correction to the CME.

All the works mentioned above are valid only up to the first order ofand without collision terms for particle scattering. In addition to these works based on Wigner functions, Li and Yee [129] derived the kinetic equations for the spin-density matrix in the Schwinger–Keldysh formalism.Specifically,they formulated collision terms for fermions from their interactions with the QGP medium.However,they consider only the spatial homogeneity limit;thus, the collisions are local and do not involve the vorticity. The time evolution equations are derived for the particle number distribution f（p，t） and the spin polarization density S（p，t） in the leading log order of g4log（1/g）:

where C2（F）=（N2c-1）/2Nc, and Γfand ΓSare diffusion-like differential operators in momentum space that contain up to the second-order derivatives of the momentum. The explicit forms of Γfand ΓSare given in Eq. (4.70) of Ref. [129].

Note that the quantum kinetic equation for massive fermions is very different from the chiral kinetic equation for massless fermions. The kinetic equation for massive fermions includes four coupled independent functions;one of them gives the particle distribution function, and the other three give the spin polarization vector. By contrast,the kinetic equation for massless fermions contains only one distribution function. The reason is that the spin direction coincides with the momentum of the fermion and is not a dynamical quantity at the chiral limit;however,for the massive case, the spin does not coincide with the momentum and becomes dynamical.It was thought that the chiral kinetic equation could be obtained from the quantum kinetic equation for massive fermions by naively taking the massless limit; however, this limit was found to be more important than it appears. The most recent works on this problem can be found in Refs. [119, 130].

4 Spin polarization effects

4.1 Theoretical progress

In early works [131, 132], Liang and Wang proposed that particles can be polarized as a result of the global OAM in non-central heavy-ion collisions.The formation of the vorticity in heavy-ion collisions was subsequently studied [133]. Becattini and his collaborators [134–136]performed a systematic study of statistical models of relativistic spinning particles. In addition, recent review articles are cited in Refs. [137–139].

The initial global OAM is estimated to be as large as J0～105in 200 GeV Au+Au collisions with the impact parameter b=10 fm. Consequently, the QGP is found to be the most vortical fluid ever observed in nature. The vorticity properties can be studied using the AMPT and HIJING [140–142], UrQMD [143], and hydrodynamic models [144–149]. The vorticity is found to be more strongly suppressed at higher collision energies. One can find further discussion in Refs. [150–152] and reference therein.

The spin polarization per particle for spin-1/2 fermions with momentum p at freeze-out can be derived using the statistical model for relativistic spinning particles [136]and the Wigner functions [153]:

The thermal properties of a quantum field system can be described by the density operator ^ρ.The density operator in local equilibrium can be determined from the maximal entropy principle [154–156] under given densities of conserved currents on the space-like hypersurface Σ. This is done by maximizing S=-Tr（^ρ log^ρ） under the conditions

If we choose the Belinfante tensors or ^ρLEin (48) or(53),the spin relaxation time is microscopically small,and the value of the spin potential agrees with the thermal vorticity almost immediately. If we choose the canonical tensors or ^ρLEin(57)with the spin chemical potential, the spin density slowly reaches global equilibrium, just as a conserved charge density or energy density does, and finally, the spin chemical potential should converge to the thermal vorticity [157].

Similar problems involving the decomposition of the total angular momentum of quarks and gluons in the proton spin(see Ref.[158]for a review)have been widely debated in the QCD community for years. Two well-known decomposition methods are that of Jaffe and Manohar and that of Ji, which are close to the canonical and Belinfante tensor forms,respectively.A review article [159]discusses the difference between the Belinfante and canonical forms.For a chiral medium, the decomposition can be found in Ref. [160].

If we choose a specific pseudo-gauge, for example, the canonical form that has the spin tensor, the spin chemical potential Ωλνmust be introduced into ^ρLE, as shown in(57). Therefore, we have another conservation equation in addition to the conservation of energy–momentum and charge:

where the energy–momentum tensor, charge current, and spin tensor can be obtained through ^ρLE:

All the above quantities are functions of βμ, ζ, and Ωλν.There are a total of 11 independent equations and variables.These equations make up a full set of hydrodynamic equations including the spin degrees of freedom. There have been a few attempts in spin hydrodynamics[37, 157, 161–165]. Another approach using Lagrangian techniques can also provide some information about the local equilibrium [166–169].

Spin degrees of freedom can be introduced into the phase space distribution for spin-1/2 fermions by generalizing the scalar function to a 2×2 Hermitian matrix [136]:

where r，s=±1;urand vrare Dirac spinors normalized by（p）us（p）=2mδrsand（p）vs（p）=-2mδrs, respectively; and X±are 4×4 matrices defined as

However, it is still an open question whether the QGP reaches a local equilibrium of the spin degrees of freedom.A microscopic model of the spin polarization generated by spin–orbit coupling in particle collisions has been proposed [171].Without assuming a local equilibrium of spins,it uses an effective method of wave packets to handle particle scattering for specified impact parameters. The spin–vorticity coupling naturally emerges from the spin–orbit coupling encoded in the polarized scattering amplitudes of collisional integrals when a local equilibrium of the momentum is assumed.First,the collision rate is calculated:

where vA=|pA|/EAand vB=-|pB|/EBare the longitudinal velocities, with pA=-pBin the center-of-mass frame of colliding particles;fAand fBare the phase space distributions of the incident particles A and B,respectively;and Δσ denotes the infinitesimal element of the cross section.After incorporating the components of the wave packets, the matrix elements of 2-to-2 scattering,the spin projection,and the proper Lorentz transformations, one obtains the polarization production rate per unit volume:

where PAB→12denotes the polarization vector of particle 2;φAand φBare the wave packets for A and B, respectively;nc= ^bc× ^pc，Ais the direction of the reaction plane in the center-of-mass frame, with ^bcbeing the unit impact parameter vector; fAand fBare the distributions at the coordinates Xc+yc，T/2 and Xc-yc，T/2, respectively; and M denotes the amplitude of the 2-to-2 scattering with all the spins and momenta specified. All the momenta are defined in the center-of-mass frame and indicated by the index c.For more details on(68),see Ref. [171].The wave packets ensure that the colliding particles have a nonvanishing initial angular momentum; the matrix elements encode the collision probability, and the spin projection and Lorentz transformation provide a consistent treatment of particle scattering in the thermal bath frame. One can apply (68) to the quark–gluon system. Then, the quark polarization rate per unit volume with all the 2-to-2 parton scatterings in a locally thermalized QGP in momentum is

which is proportional to the thermal vorticity. Here, the tensor Wρνcontains 64 components, each of which involves a 16-dimensional integration. The numerical calculation of Wρνis challenging owing to the very large number of scattering amplitudes and high-dimensional collision integrals. To tackle this problem, a new Monte Carlo integration algorithm, ZMCintegral, which can handle 16-dimensional integration,has been developed for use on multiple GPUs [172, 173]. The most recent application of this algorithm is the solution of the Boltzmann equations of a partonic system [174].The numerical result shows that Wρνhas an antisymmetric form:

where W is approximately constant. Note that W depends on the cutoff of the impact parameter, as shown in Fig. 8.Finally,the polarization rate per unit volume for one quark flavor can be expressed compactly:

This is a good example of how spin–vorticity coupling emerges naturally from particle scattering.

Note that the result of Ref. [171] or Eq. (69) does not include the back reaction to contain the growing polarization with increasing vorticity in the absence of a cutoff.A systematic derivation of the spin polarization from the vorticity through non-local collisions with back reactions has been performed [175]by expanding the collision terms in the Planck constant for massive fermions [121].

4.2 Experimental results

The global polarization of Λ hyperons can be measured through their weak decays. The angular distribution of the daughter baryons is [5, 111]

where αHis the decay constant of the hyperon, PHis the polarization of the hyperon(a fraction),and θ*is the angle between the momentum of the daughter baryon and the polarization direction in the hyperon rest frame. The experimental results for the global polarization of Λ andhyperons measured by the STAR Collaboration[5,111]are shown in the left panel of Fig. 9. PH（Λ） and PH（）decrease with the collision energy.From the data,the fluid vorticity can be estimated as ω ≃kBT（PΛ+P）/[176].It also appears that PH（）PH（Λ）, although the difference is not significant; this result may suggest a possible contribution from the magnetic field or other effects[177, 178]. The data can be described by various phenomenological models using (44) [105, 142, 143,179–181].

The right panel of Fig. 9 shows the recent STAR Collaboration measurement [21] of the longitudinal polarization as a function of the azimuthal angle φ relative to the second-order event plane Ψ2. The longitudinal spin polarization data show positive sin（2φ-2Ψ2） behavior,whereas the theoretical results of the relativistic hydrodynamics model [182] and transport models [143, 180, 183]show the opposite sign along the beam line direction. A simulation using CKT [110]and the results obtained using a simple phenomenological model [184] give the correct sign as the data. The azimuthal angle dependence of the spin polarization in the direction of the global OAM has been measured by the STAR Collaboration. Some phenomenological models do not well describe the STAR data for the azimuthal angle dependence of the global polarization.

The sign problem of the local polarization requires further investigations. It may indicate a need for new frameworks to describe the spin dynamics, such as the quantum spin kinetic theory for massive particles (see Sect. 3) or the spin hydrodynamics (see Sect. 4.1).

One possible effect is that of feed-down decays. The hyperons measured in experiments may be produced by decays of heavier resonance particles. However, the authors of Refs. [185, 186] concluded that feed-down effects decreased by approximately 10% for the Λ primordial spin polarization; thus, they do not solve the spin sign problem.

In addition, the sign problem of the local polarization may also indicate that the assumption of global or local equilibrium of the spin may not be justified, and thus, the thermal vorticity may not be the correct quantity for the spin chemical potential. Wu et al. [148, 149] tested four different types of vorticity: the kinematic vorticity, relativistic extension of the nonrelativistic vorticity,temperature vorticity, and thermal vorticity. They calculated the local polarization of hyperons corresponding to each type of vorticity. By using the （3+1）-dimensional hydrodynamic model with the AMPT initial conditions encoding the global OAM, they found a few remarkable differences between the results for different vorticities.First, although all four types of vorticity give the correct sign and magnitude of the polarization along the global OAM direction, only the temperature vorticity agrees with the STAR preliminary result in showing a decreasing trend of the azimuthal angle dependence. Second, only the temperature vorticity can simultaneously provide the correct sign and magnitude of the longitudinal polarization.This result suggests the possibility of spin–temperature vorticity coupling, analogous to magnetic moment–magnetic field coupling. It is also possible that the agreement may be a coincidental result of the main assumption that the spin vector depends on the temperature vorticity in the same way as it does on the thermal vorticity. Further investigation is needed to clarify why the results for the temperature vorticity are the most satisfactory.

The vector meson spin alignment is another recent research topic. The spin alignment of a vector meson is described by the 3×3 Hermitian spin-density matrix[187, 188]. The 00-component of the spin-density matrix enters the angular distribution of its decay daughter as

where θ*is the angle between the decay daughter and the spin quantization direction in the vector meson’s rest frame. Thus, the ρ00value of the vector meson can be measured using the angular distribution of its decay daughter. The vector meson (K*0and φ) spin alignments have been measured by the ALICE Collaboration [189].The ρ00value is consistent with 1/3 for both K*0and φ mesons in p+p collisions. In Pb+Pb collisions, the ρ00value of K*0is approximately 1/3 at high pTand less than 1/3 at low pT.

Theoretical calculations using the statistical-hydro model [176] and quark coalescence model [190] give

To solve this puzzle, it is proposed that a strangeness current can exist in heavy-ion collisions and give rise to a nonvanishing mean φ field [191]. Like the magnetic field,the magnetic part of the φ field can also polarize s andthrough their magnetic moments; this polarization contributes to the polarization of Λ and, whereas the contribution from the electric part of the φ field is absent and therefore is not constrained by the polarization of Λ andHowever, the electric part of the φ field contributes significantly to ρφ

00, which is positive definite [191]:

However, this theory does not work for another vector meson, K*0, for several reasons. First, because of the unequal masses ofand d, one cannot derive the same formula as that for φ mesons, in which the contributions from the vorticity and those from the electric and magnetic fields are decoupled. Second, the interaction of K*0with the surrounding matter is much stronger than that of the φ meson.The above points are supported by preliminary data from ALICE [189].

5 Summary

We gave a brief overview of recent theoretical developments and experimental results on the effects of chirality and vorticity on heavy-ion collisions. We focus on works reported at the Quark Matter conference of 2019.

We discussed the time evolution of magnetic fields in various models of the QGP, for example, the Lienard–Wiechert potential and MHD models.Macroscopic models such as the second-order dissipative MHD and AVFD models are applied in phenomenological studies.

It is still experimentally challenging to obtain a clear CME signal against the dominant backgrounds. The nonflow correlations for the CMW give rise to additional backgrounds in the slope of Δv2（Ach）. More detailed studies of the CME and CMW are needed.

The kinetic theory of massive fermions has been developed for the covariant and equal-time Wigner functions and the Schwinger–Keldysh formalism.The collision terms have been studied at the leading and next-to-leading order in an expansion of the Planck constant.

The decomposition of the total angular momentum into the OAM and spin has been debated for quite a long time.Another important issue that remains to be resolved is the relationship between the theoretical spin and the results of experimental measurements.

For the global polarization of Λ andhyperons,various phenomenological models give consistent descriptions of the experimental data.For the local polarization,the nature of the spin sign problem is still unclear. It has been proposed that a significant positive deviation of 1/3 for ρφ00may indicate the existence of a mean φ field.