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

Recent developments in chiral and spin polarization effects in heavy-ion collisions 1 corresp first-author gaojh@sdu.edu.cn gaojh@sdu.edu.cn gaojh@sdu.edu.cn 2 corresp glma@fudan.edu.cn glma@fudan.edu.cn glma@fudan.edu.cn 3 corresp shipu@ustc.edu.cn shipu@ustc.edu.cn shipu@ustc.edu.cn https://orcid.org/0000-0002-6784-7447 3 corresp qunwang@ustc.edu.cn qunwang@ustc.edu.cn qunwang@ustc.edu.cn https://orcid.org/0000-0003-4410-703X Shandong Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Institute of Space Sciences, Shandong University, Weihai 264209, China Shandong Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Institute of Space Sciences, Shandong University, Weihai 264209, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China 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 ρ 00 value of vector mesons. Relativistic heavy-ion collisions Chiral magnetic effect Chiral kinetic theory Spin polarization 1 Introduction 1 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 10 17-18 Gs [ 1 1 - 4 4 1-4 ] and are the strongest magnetic fields observed in nature. The QGP is also found to be the most vortical fluid [ 5 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 6 - 8 8 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: <math display="block" id="M1"><mrow><mi>j</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><msub><mi>μ</mi><mn>5</mn></msub><mi>B</mi><mn>,</mn></mrow></math> where μ 5 is the chiral chemical potential. The magnetic field can also generate a chiral current <math display="block" id="M2"><mrow><msub><mi>j</mi><mn>5</mn></msub><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mi>μ</mi><mi>B</mi><mn>,</mn></mrow></math> 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 μ 5 arises 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 9 ] and can be applied in quantum computing [ 10 10 ]. These phenomena are discussed in more detail in recent reviews [ 11 11 - 20 20 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 and <math id="M3"><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></math> hyperons has been measured in the STAR experiment, and the average angular velocity or vorticity was estimated as ω ∼10 22 s -1 [ 5 5 ]. The data on the global polarization of Λ and <math id="M4"><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></math> hyperons 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 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 Sec. 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 1 2.1 Theoretical progress 2 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 1 - 4 4 1-4 ]. Although the peak value of the magnetic field can be as large as a few times <math id="M5"><mrow><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></math> , where m π is the mass of a pion, it decays very rapidly in vacuum [ 7 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 4 , 22 22 - 24 24 22-24 ]. Magnetohydrodynamics (MHD) is widely used in astrophysics to investigate the coupling between a charged medium and a magnetic field. Ordinary MHD consists of the hydrodynamic conservation equations coupled with Maxwell’s equations. Very recently, several MHD studies [ 25 25 - 29 29 25-29 ] showed that the magnetic fields decay as 1/ τ in the infinite electrical conductivity limit, where τ is the proper time. The analytic solution for the anomalous MHD with the CME and chiral anomaly in a Bjorken flow has been derived [ 30 30 , 31 31 ]. The magnetic fields decay approximately as ∼1/ τ or e - / τ , where σ is the electrical conductivity. For numerical simulations of the ideal MHD, see Ref. [ 32 32 , 33 33 ]. Although the electrical conductivity of the QGP is not infinite, it is important to study the coupling between the electromagnetic field and QGP. The ratio of the magnetic density to the initial fluid density can reportedly be greater than 1 in event-by-event simulations of <math id="M6"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn><mtext> GeV</mtext></mrow></math> Au+Au collisions [ 3 3 ]; i.e., the magnetic field may also affect other physical quantities [ 25 25 , 29 29 , 34 34 ]. Further study of the relativistic MHD with a temperature-dependent electrical conductivity is needed. The ideal MHD framework has been extended to the second order using the self-consistent Grad moment expansion [ 35 35 , 36 36 ]. In this framework, the electromagnetic fields are coupled with normal dissipative terms, such as the shear viscous tensor and bulk viscous pressure. One can also use a similar approach based on the kinetic theory of massless fermions [ 37 37 ]. By comparing phenomenological studies and experimental data, one can constrain the lifetime of the magnetic field, t B ∼eq0.5-1 fm/c [ 38 38 ] or <math id="M7"><mrow><msub><mi>t</mi><mi>B</mi></msub><mo>=</mo><mi>A</mi><mo>/</mo><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> , where A =115±16 GeV⋅fm/ c [ 39 39 ]. It has been proposed that the magnetic field can be produced by a charged rotating fluid [ 39 39 ]. By using Maxwell’s equations in a charged fluid, ∂ μ F μν = j ν , and assuming the particle number density n is homogeneous or changes very slowly, the relationship between the vorticity and the magnetic field in the local rest frame, ω =(∇ 2 B )/ en , can be obtained. By introducing the average vorticity <math id="M8"><mover accent="true"><mi>ω</mi><mo>¯</mo></mover></math> , the average magnetic field per transverse area can be expressed as <math id="M9"><mrow><mover accent="true"><mi>B</mi><mo>¯</mo></mover><mo>≃</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mi>A</mi><mi>n</mi><mover accent="true"><mi>ω</mi><mo>¯</mo></mover></mrow></math> , where A is the transverse area of the vortex. In heavy-ion collisions, the evolution of both the vorticity and the charge density at freeze-out can be extracted from AMPT simulations. (The AMPT model is a very good tool for simulating the CME [ 40 40 - 44 44 40-44 ]; see Sec. 2.2.) For a transverse area of the vorticity of 16 π fm 2 , and assuming a centrality of 20–50% for Au+Au collisions at 10–200 GeV at the RHIC, the magnetic field and its evolution can be estimated. Then, the splitting in the polarization of Λ and <math id="M10"><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></math> hyperons can be estimated from the averaged magnetic field, <math id="M11"><mrow><msub><mi>P</mi><mi>Λ</mi></msub><mo>−</mo><msub><mi>P</mi><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></msub><mo>≃</mo><mn>2</mn><mo>|</mo><msub><mi>μ</mi><mi>Λ</mi></msub><mo>|</mo><mover accent="true"><mi>B</mi><mo>¯</mo></mover><mo>/</mo><msub><mi>T</mi><mrow><mtext>fo</mtext></mrow></msub></mrow></math> , where | μ Λ |=0.613 e /(2 M N ), M N =938 MeV, and T fo =155 MeV. The numerical results are close to the STAR data [ 5 5 ]. Another theoretical model is the anomalous-viscous fluid dynamics (AVFD) [ 45 45 - 47 47 45-47 ], which is the hydrodynamical realization of the CME in relativistic heavy-ion collisions. The latest event-by-event version of the AVFD includes the local charge conservation (LCC) effect [ 48 48 ] and the introduction of particlization, which may be the best way to quantify the LCC [ 49 49 ]. It also gives predictions for isobaric collisions [ 50 50 ]. For the event plane, AVFD predicts <math id="M12"><mrow><msubsup><mi>ζ</mi><mrow><mtext>isobar</mtext></mrow><mrow><mtext>EP</mtext></mrow></msubsup><mo>≡</mo><msub><mrow><mrow><mrow><msubsup><mi>γ</mi><mrow><mtext>Ru</mtext><mo>−</mo><mtext>Zr</mtext></mrow><mrow><mtext>OS</mtext><mo>−</mo><mtext>SS</mtext></mrow></msubsup></mrow><mo>|</mo></mrow></mrow><mrow><mtext>EP</mtext></mrow></msub><mo>/</mo><msub><mrow><mrow><mrow><msubsup><mi>δ</mi><mrow><mtext>Ru</mtext><mo>−</mo><mtext>Zr</mtext></mrow><mrow><mtext>OS</mtext><mo>−</mo><mtext>SS</mtext></mrow></msubsup></mrow><mo>|</mo></mrow></mrow><mrow><mtext>EP</mtext></mrow></msub><mo>≃</mo><mo>−</mo><mo stretchy="false">(</mo><mn>0.41</mn><mo>±</mo><mn>0.27</mn><mo stretchy="false">)</mo></mrow></math> , which yields 〈cos(2Ψ B -2Ψ EP )〉-0.46; for the reaction plane, the ratio is <math id="M13"><mrow><msubsup><mi>ζ</mi><mrow><mtext>isobar</mtext></mrow><mrow><mtext>RP</mtext></mrow></msubsup><mo>≡</mo><msub><mrow><mrow><mrow><msubsup><mi>γ</mi><mrow><mtext>Ru</mtext><mo>−</mo><mtext>Zr</mtext></mrow><mrow><mtext>OS</mtext><mo>−</mo><mtext>SS</mtext></mrow></msubsup></mrow><mo>|</mo></mrow></mrow><mrow><mtext>RP</mtext></mrow></msub><mo>/</mo><msub><mrow><mrow><mrow><msubsup><mi>δ</mi><mrow><mtext>Ru</mtext><mo>−</mo><mtext>Zr</mtext></mrow><mrow><mtext>OS</mtext><mo>−</mo><mtext>SS</mtext></mrow></msubsup></mrow><mo>|</mo></mrow></mrow><mrow><mtext>RP</mtext></mrow></msub><mo>≃</mo><mo>−</mo><mo stretchy="false">(</mo><mn>0.90</mn><mo>±</mo><mn>0.45</mn><mo stretchy="false">)</mo></mrow></math> , which yields <math id="M14"><mrow><mo><</mo><mtext>cos</mtext><mrow><mo>(</mo><mrow><mn>2</mn><msub><mi>Ψ</mi><mi>B</mi></msub><mo>−</mo><mn>2</mn><msub><mi>Ψ</mi><mrow><mi>E</mi><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>></mo><mo>≃</mo><mo>−</mo><mn>0.95</mn></mrow></math> . According to the calculations, these ratios are independent of the initial axial charge. As a natural extension to the CME, it is interesting to consider how large the mass correction is. This question can be addressed using a perturbation method in quantum kinetic theory, as described in Sect. 3. However, as noted in Ref. [ 51 51 ], the mass correction to the CME is related to another well-known phenomenon, Schwinger pair production. The operator equation for the CME with finite mass corrections is the axial Ward identity, <math display="block" id="M15"><mrow><msub><mo>∂</mo><mi>μ</mi></msub><msubsup><mi>j</mi><mn>5</mn><mi>μ</mi></msubsup><mo>=</mo><mn>2</mn><mi>i</mi><mi>m</mi><mover accent="true"><mi>ψ</mi><mo stretchy="true">¯</mo></mover><msup><mi>γ</mi><mn>5</mn></msup><mi>ψ</mi><mo>−</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mn>16</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>ϵ</mi><mrow><mi>μ</mi><mi>ν</mi><mi>α</mi><mi>β</mi></mrow></msup><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msub><mi>F</mi><mrow><mi>α</mi><mi>β</mi></mrow></msub><mn>.</mn></mrow></math> To simplify the discussion, one can assume that the electric and magnetic fields E and B , respectively, are constant and homogeneous in the z direction. In this case, the theory reduces to a (1+1)-dimensional problem; therefore, it can be computed using the worldline formalism. However, the original calculation by Schwinger [ 52 52 ] provides only the value of <math id="M16"><mrow><mrow><mo>〈</mo><mrow><mtext>out</mtext><mn>,0</mn><mrow><mo>|</mo><mrow><msub><mo>∂</mo><mi>μ</mi></msub><msubsup><mi>j</mi><mn>5</mn><mi>μ</mi></msubsup></mrow><mo>|</mo></mrow><mtext>in</mtext><mn>,0</mn></mrow><mo>〉</mo></mrow><mo>=</mo><mn>0</mn></mrow></math> , where |in, 0〉 and |out, 0〉 are the in and out states in vacuum, respectively. One needs to compute the exception values of all the operators in (3); i.e., one needs to compute <math id="M17"><mrow><mrow><mo>〈</mo><mrow><mtext>in</mtext><mn>,0</mn><mrow><mo>|</mo><mrow><msub><mo>∂</mo><mi>μ</mi></msub><msubsup><mi>j</mi><mn>5</mn><mi>μ</mi></msubsup></mrow><mo>|</mo></mrow><mtext>in</mtext><mn>,0</mn></mrow><mo>〉</mo></mrow></mrow></math> . The method of computing the Feynman propagator in the in-in state has been used to obtain the following result by a long and technical calculation [ 53 53 ]: <math display="block" id="M18"><mrow><msub><mo>∂</mo><mi>μ</mi></msub><msubsup><mi>j</mi><mn>5</mn><mi>μ</mi></msubsup><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup><mi>E</mi><mi>B</mi></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><mi>π</mi><msup><mi>m</mi><mn>2</mn></msup></mrow><mrow><mi>e</mi><mi>E</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> which reduces to the standard chiral anomaly, <math id="M19"><mrow><msub><mo>∂</mo><mi>μ</mi></msub><msubsup><mi>j</mi><mn>5</mn><mi>μ</mi></msubsup><mo>=</mo><msup><mi>e</mi><mn>2</mn></msup><mi>E</mi><mi>B</mi><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></math> , in the massless limit. It is also consistent with the physical scenario suggested in Ref. [ 51 51 ], which implies that the chirality production rate should be proportional to the Schwinger pair production rate. In addition, several other physical quantities can also be obtained by the real-time worldline formalism. The mass-corrected CME current is given by <math display="block" id="M20"><mrow><msup><mi>j</mi><mi>z</mi></msup><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup><mi>E</mi><mi>B</mi></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mi>coth</mi><mrow><mo>(</mo><mrow><mfrac><mi>B</mi><mi>E</mi></mfrac><mi>π</mi></mrow><mo>)</mo></mrow><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><mi>π</mi><msup><mi>m</mi><mn>2</mn></msup></mrow><mrow><mi>e</mi><mi>E</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where the factor coth( Bπ/E ) represents the sum over the Landau levels. There are also many studies of the anomaly and CME in the context of perturbation in quantum field theory [ 54 54 - 58 58 54-58 ]. The source of the CME, the chiral charge fluctuation [ 59 59 , 60 60 ], is another research topic and is important for determining the magnitude of the CME signal. 2.2 Recent experimental results 2 The dipole charge separation due to the CME can be characterized by the first sine term a 1 in the Fourier series of the charged-particle azimuthal distribution: <math display="block" id="M21"><mrow><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><mi>ϕ</mi></mrow></mfrac><mo>∝</mo><mn>1</mn><mo>+</mo><mn>2</mn><mstyle displaystyle="true"><munder><mo>∑</mo><mi>n</mi></munder></mstyle><mo>{</mo><msub><mi>v</mi><mi>n</mi></msub><mi>cos</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo>−</mo><msub><mi>Ψ</mi><mrow><mtext>RP</mtext></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><mi>sin</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo>−</mo><msub><mi>Ψ</mi><mrow><mtext>RP</mtext></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>}</mo><mn>,</mn></mrow></math> where ϕ -Ψ RP represents the particle azimuthal angle with respect to the reaction plane angle Ψ RP in heavy-ion collisions (which is determined by the impact parameter and beam axis), and v n and a n are the coefficients of the P -even and P -odd Fourier terms, respectively. An azimuthal three-particle correlator [ 61 61 ], γ 112 , was proposed to explore the first coefficient, a 1 , of the P -odd Fourier terms characterizing the charge separation due to the CME; it is given by <math display="block" id="M22"><mrow><msub><mi>γ</mi><mrow><mn>112</mn></mrow></msub><mo>≡</mo><mrow><mo>〈</mo><mrow><mi>cos</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mi>α</mi></msub><mo>+</mo><msub><mi>ϕ</mi><mi>β</mi></msub><mo>−</mo><mn>2</mn><msub><mi>Ψ</mi><mrow><mtext>RP</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mo>〉</mo></mrow><mo>≃</mo><mrow><mo>〈</mo><mrow><mi>cos</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mi>α</mi></msub><mo>+</mo><msub><mi>ϕ</mi><mi>β</mi></msub><mo>−</mo><mn>2</mn><msub><mi>Ψ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo>〉</mo></mrow><mn>,</mn></mrow></math> where α and β denote particles with the same or opposite electric charge, respectively; angle brackets denote the averages over particles and events, and Ψ RP can be approximated by the azimuthal angle of the second-order event plane, Ψ 2 . Similarly, γ 123 , a charge-dependent correlator with respect to the azimuthal angle of the third-order event plane, Ψ 3 , is defined as <math display="block" id="M23"><mrow><msub><mi>γ</mi><mrow><mn>123</mn></mrow></msub><mo>≡</mo><mrow><mo>〈</mo><mrow><mi>cos</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mi>α</mi></msub><mo>+</mo><mn>2</mn><msub><mi>ϕ</mi><mi>β</mi></msub><mo>−</mo><mn>3</mn><msub><mi>Ψ</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow><mo>〉</mo></mrow></mrow></math> and can reflect the charge-dependent background effects unrelated to the CME, because Ψ 3 is not correlated with the magnetic field direction. Other types of azimuthal correlators γ ijk for charged particles can also be defined similarly for different purposes. Except in ambiguous cases, we use the shorthand notation γ to represent γ 112 hereafter. From the CME expectation, the difference between the opposite-sign and same-sign correlation for charged particles Δ γ = γ opp - γ same is expected to be proportional to B 2 and cos[2(Ψ B -Ψ 2 )] [ 1 1 , 42 42 , 64 64 ]: <math display="block" id="M24"><mrow><mi>Δ</mi><mi>γ</mi><mo>∝</mo><mrow><mo>〈</mo><mrow><msup><mi>B</mi><mn>2</mn></msup><mtext>c</mtext><mi>o</mi><mi>s</mi><mrow><mo>[</mo><mrow><mn>2</mn><mo stretchy="false">(</mo><msub><mi>Ψ</mi><mi>B</mi></msub><mo>−</mo><msub><mi>Ψ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow><mo>〉</mo></mrow><mn>.</mn></mrow></math> Because the magnitude of the magnetic field is proportional to the collision energy [ 2 2 , 65 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 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 (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 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 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 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 Fig. 3 shows the expected CME contribution to Δγ as a function of N part for Au+Au collisions at 200 GeV and U+U collisions at 193 GeV [ 66 66 ], which are multiplied by a factor N part / ≤ n to 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 N part larger 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 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, Δ γ 112 , Δ γ 123 , and Δ γ 132 , in Au+Au collisions at 200 GeV and U+U collisions at 193 GeV [ 66 66 ]. Their N part dependence is presented in the right panel of Fig. 3 Fig. 3 ; they are normalized by a factor of N part / v n for 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 N part larger than 300. The illustrations on the far right side of Fig. 3 Fig. 3 show that Δ γ 123 is not correlated with the magnetic field direction, in contrast to Δ γ 112 and Δ γ 132 ; 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 v 2 background and the CME signal: <math display="block" id="M25"><mtable columnalign="left"><mtr><mtd><mi>Δ</mi><mi>γ</mi><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>CME</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>Bkg</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><mi>Δ</mi><mi>γ</mi><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>ZDC</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>CME</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>ZDC</mtext></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>Bkg</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>ZDC</mtext></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where ψ TPC and ψ ZDC are 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 v 2 [ 67 67 ], one obtains <math display="block" id="M26"><mtable columnalign="left"><mtr><mtd><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>CME</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>CME</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>ZDC</mtext></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>Bkg</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>ZDC</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>Bkg</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where a =〈cos[2( ψ ZDC - ψ TPC )]〉 and can be obtained from the v 2 measurement: <math display="block" id="M27"><mrow><mi>a</mi><mo>=</mo><mfrac><mrow><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>ZDC</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> The CME signal relative to the inclusive Δ γ ( ψ TPC ) can be determined as <math display="block" id="M28"><mrow><msub><mi>f</mi><mrow><mtext>CME</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>CME</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>Δ</mi><mi>γ</mi><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mi>a</mi><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where A is defined as <math display="block" id="M29"><mrow><mi>A</mi><mo>=</mo><mfrac><mrow><mi>Δ</mi><mi>γ</mi><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>ZDC</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>Δ</mi><mi>γ</mi><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> It can be rewritten as <math display="block" id="M30"><mtable columnalign="left"><mtr><mtd><mi>A</mi><mo>=</mo><msub><mi>f</mi><mrow><mtext>CME</mtext></mrow></msub><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>a</mi><mo stretchy="false">)</mo><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>CME</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>a</mi><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>Bkg</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>Δ</mi><msub><mi>γ</mi><mrow><mtext>CME</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>ψ</mi><mrow><mtext>TPC</mtext></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mi>a</mi><mo>+</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>a</mi></mfrac><mo>−</mo><mi>a</mi></mrow><mo>)</mo></mrow><msub><mi>f</mi><mrow><mtext>CME</mtext></mrow></msub><mn>,</mn></mtd></mtr></mtable></math> which one can solve for f CME to obtain the last equality of (13). Note that A can also be measured experimentally. The left panel of Fig. 4 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 Δ γ ( ψ ZDC ) is consistently lower than Δ γ ( ψ TPC ), which indicates that A is less than unity. This result indicates that the flow background contributes less to Δ γ ( ψ ZDC ) than to Δ γ ( ψ TPC ). Applying the above method, the STAR Collaboration extracted the CME fraction f CME from 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 Fig. 4 . The combined result for the CME fraction for Au+Au at 200 GeV and U+U at 193 GeV is f CME =8±4±8% [ 66 66 ]. Note that the CMS Collaboration has systematically measured different types of correlators γ ijk in 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 68 , 69 69 ]. Many other methods of searching for the CME have been proposed, such as event shape engineering [ 69 69 ], the use of the H factor [ 71 71 ], and the invariant mass method [ 72 72 ]. See [ 11 11 , 73 73 ] for recent reviews. A new observable, the signed balance function, was proposed recently; it is defined as [ 74 74 ] <math display="block" id="M31"><mrow><msub><mi>B</mi><mrow><mi>P</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>S</mi><mi>y</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mrow><mo>+</mo><mo>−</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>S</mi><mi>y</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>N</mi><mrow><mo>+</mo><mo>+</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>S</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>N</mi><mo>+</mo></msub></mrow></mfrac></mrow></math> and <math display="block" id="M32"><mrow><msub><mi>B</mi><mrow><mi>N</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>S</mi><mi>y</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>N</mi><mrow><mo>−</mo><mo>+</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>S</mi><mi>y</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>N</mi><mo>–</mo></msub><mo stretchy="false">(</mo><msub><mi>S</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>N</mi><mo>−</mo></msub></mrow></mfrac><mn>.</mn></mrow></math> Note that S y =+1 if particle α is leading particle β (i.e., <math id="M33"><mrow><msubsup><mi>p</mi><mi>y</mi><mi>α</mi></msubsup><mo>></mo><msubsup><mi>p</mi><mi>y</mi><mi>β</mi></msubsup></mrow></math> ), and S y =-1 otherwise. N +- ( S y ) denotes the number of positive–negative charge pairs with sign S y in an event. N ++ ( S y ), N -+ ( S y ), and N –– ( S y ) 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 B P and B N : <math display="block" id="M34"><mrow><mi>δ</mi><msub><mi>B</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mo>±</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>B</mi><mrow><mi>P</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mo>±</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>B</mi><mrow><mi>N</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mo>±</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> and <math display="block" id="M35"><mrow><mi>Δ</mi><msub><mi>B</mi><mi>y</mi></msub><mo>=</mo><mi>δ</mi><msub><mi>B</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>δ</mi><msub><mi>B</mi><mi>y</mi></msub><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Note that Δ B x can 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, B P , y ( x ) and B N , y ( x ) measure the same quantity, in principle, and the distribution of Δ B y ( 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, B P , y and B N , y tend to differ; consequently, Δ B y has a wider distribution. By contrast, the distribution of Δ B x is 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 Δ B y to that of Δ B x as <math display="block" id="M36"><mrow><mi>r</mi><mo>=</mo><mfrac><mrow><msub><mi>σ</mi><mrow><mi>Δ</mi><msub><mi>B</mi><mi>y</mi></msub></mrow></msub></mrow><mrow><msub><mi>σ</mi><mrow><mi>Δ</mi><msub><mi>B</mi><mi>x</mi></msub></mrow></msub></mrow></mfrac></mrow></math> 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 ( r lab ) or in the rest frame ( r rest ) of the pair. One can take the ratio of these two cases: <math display="block" id="M37"><mrow><msub><mi>R</mi><mi>B</mi></msub><mo>≡</mo><mfrac><mrow><msub><mi>r</mi><mrow><mtext>rest</mtext></mrow></msub></mrow><mrow><msub><mi>r</mi><mrow><mtext>lab</mtext></mrow></msub></mrow></mfrac><mn>,</mn></mrow></math> where the subscript B indicates a balance function. It has been found that r lab , r rest , and R B are 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 74 ]. Figure 5 Figure 5 presents the recent STAR measurement of the centrality dependence of r lab , r rest , and R B from the signed balance functions in Au+Au collisions at 200 GeV [ 70 70 ]. In the 30–40% centrality bin, r rest , r lab , and R B are 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. Another new correlator, <math id="M38"><mrow><msub><mi>R</mi><mrow><msub><mi>Ψ</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>Δ</mi><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mrow><msub><mi>Ψ</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>Δ</mi><mi>S</mi><mo stretchy="false">)</mo><mo>/</mo><msubsup><mi>C</mi><mrow><msub><mi>Ψ</mi><mn>2</mn></msub></mrow><mo>⊥</mo></msubsup><mo stretchy="false">(</mo><mi>Δ</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></math> , has been designed to measure the magnitude of the CME-induced charge separation parallel to the B > field [ 76 76 , 77 77 ], where <math id="M39"><mrow><msub><mi>C</mi><mrow><msub><mi>Ψ</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>Δ</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></math> and <math id="M40"><mrow><msubsup><mi>C</mi><mrow><msub><mi>Ψ</mi><mn>2</mn></msub></mrow><mo>⊥</mo></msubsup><mo stretchy="false">(</mo><mi>Δ</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></math> are used to quantify the charge separation Δ S , i.e., the difference between the averaged 〈sin (Δ φ 2 )〉 values of negatively and positively charged particles emitted near the estimated Ψ 2 plane, Δ φ 2 = ϕ - Ψ 2 . Because the CME can result in charge separation along the B field, one expects a concave <math id="M41"><mrow><msub><mi>R</mi><mrow><msub><mi>Ψ</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>Δ</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></math> distribution having a width that reflects the magnitude a 1 of the CME-induced charge separation. Because the reaction plane angle Ψ 2 is very weakly correlated with the B field in small collision systems [ 78 78 ], the corresponding measurement in small systems can provide a baseline for the background contributions. The STAR Collaboration recently released the results for the R correlator [ 75 75 ]. Figure 6 Figure 6 compares the <math id="M42"><mrow><msub><mi>R</mi><mrow><msub><mi>Ψ</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>Δ</mi><msup><mi>S</mi><mo>″</mo></msup><mo stretchy="false">)</mo></mrow></math> correlators at 〈 N ch 〉 20 for p+Au, d+Au, and Au+Au collisions at 200 GeV. The convex or flat distributions shown for p(d)+Au collisions are consistent with the approximately random B field relative to Ψ 2 expected in small collisions. By contrast, the Au+Au measurement relative to Ψ 2 shows a concave distribution, which is consistent with CME-driven charge separation according to the AVFD [ 76 76 ] and AMPT models [ 43 43 , 44 44 ]. 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 82 , 83 83 ]. It can ultimately be transformed into a charge-asymmetry-dependent elliptic flow of pions by collective expansion [ 84 84 ]. Therefore, the elliptic flow of positively and negatively charged pions is given by <math display="block" id="M44"><mrow><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msup><mi>π</mi><mo>±</mo></msup><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>v</mi><mn>2</mn><mrow><mtext>b</mtext><mi>a</mi><mi>s</mi><mi>e</mi></mrow></msubsup><mo>∓</mo><mfrac><mrow><mi>r</mi><mo stretchy="false">(</mo><msup><mi>π</mi><mo>±</mo></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><msub><mi>A</mi><mrow><mtext>ch</mtext></mrow></msub><mn>,</mn></mrow></math> where <math display="block" id="M45"><mrow><msub><mi>A</mi><mrow><mtext>ch</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi>N</mi><mo>+</mo></msup><mo>−</mo><msup><mi>N</mi><mo>−</mo></msup></mrow><mrow><msup><mi>N</mi><mo>+</mo></msup><mo>+</mo><msup><mi>N</mi><mo>−</mo></msup></mrow></mfrac></mrow></math> is the charge asymmetry. The elliptic flow difference between positive and negative pions, [Δ v 2 = v 2 ( π - )- v 2 ( π + )], can then be fitted using the relation Δ v 2 = rA ch +Δ v 2 (0). The left panel of Fig. 7 Fig. 7 presents the results for Au+Au collisions at 200 GeV (30–40%) measured in the STAR experiment [ 79 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 v 3 can provide a reference from the background in comparison with that from elliptic flow v 2 . 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 Δ v n ( A ch ) owing to non-flow 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 85 ] <math display="block" id="M46"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>d</mi><mi>n</mi></msub><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>π</mi><mo>±</mo></msup><mi>h</mi><mo>}</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><msub><mi>d</mi><mi>n</mi></msub><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>π</mi><mo>±</mo></msup><msup><mi>h</mi><mo>+</mo></msup><mo>}</mo><mo>+</mo><msub><mi>d</mi><mi>n</mi></msub><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>π</mi><mo>±</mo></msup><msup><mi>h</mi><mo>−</mo></msup><mo>}</mo></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mfrac><mrow><msub><mi>d</mi><mi>n</mi></msub><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>π</mi><mo>±</mo></msup><msup><mi>h</mi><mo>+</mo></msup><mo>}</mo><mo>−</mo><msub><mi>d</mi><mi>n</mi></msub><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>π</mi><mo>±</mo></msup><msup><mi>h</mi><mo>−</mo></msup><mo>}</mo></mrow><mn>2</mn></mfrac><msub><mi>A</mi><mrow><mtext>ch</mtext></mrow></msub><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The second term on the right-hand side of (24) is proportional to A ch and 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 d n {2; pairs} from like-sign pairs and opposite-sign pairs differ. Therefore, the second term on the right-hand 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 A ch term 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 <math id="M47"><mrow><msubsup><mi>v</mi><mi>n</mi><mi>π</mi></msubsup><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>h</mi><mo>+</mo></msup><mo>}</mo></mrow></math> and <math id="M48"><mrow><msubsup><mi>v</mi><mi>n</mi><mi>π</mi></msubsup><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>h</mi><mo>−</mo></msup><mo>}</mo></mrow></math> , and then take an average: <math display="block" id="M49"><mrow><msubsup><mover accent="true"><mi>v</mi><mo>¯</mo></mover><mi>n</mi><mi>π</mi></msubsup><mo>≡</mo><mfrac><mrow><msubsup><mi>v</mi><mi>n</mi><mi>π</mi></msubsup><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>h</mi><mo>+</mo></msup><mo>}</mo><mo>+</mo><msubsup><mi>v</mi><mi>n</mi><mi>π</mi></msubsup><mo>{</mo><mn>2</mn><mo>;</mo><msup><mi>h</mi><mo>−</mo></msup><mo>}</mo></mrow><mn>2</mn></mfrac><mn>.</mn></mrow></math> Previous STAR results showed significant negative slopes for v 3 [ 86 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 Fig. 7 , the normalized v 3 slopes are consistent with positive values (1.5 σ above zero for 20–60% centrality), which are similar to the normalized v 2 slopes 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 Δ v n ( A ch ) owing to competition among different pion sources and the large multiplicity dilution of π + ( π - ) at positive (negative) A ch [ 80 80 , 81 81 ]. Therefore, more detailed studies are needed to search for the CMW in the future. 3 Quantum kinetic theory 1 In recent years, chiral kinetic theory (CKT) has been developed significantly [ 87 87 - 102 102 87-102 ] to describe various chiral effects in heavy-ion collisions. Numerical simulations based on the CKT have been developed [ 103 103 - 110 110 103-110 ]. However, with the discovery of global polarization at relatively low energies [ 5 5 , 111 111 ], 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. [ 112 112 - 114 114 112-114 ] are based on early works on the covariant Wigner functions [ 115 115 - 117 117 115-117 ]. The covariant Wigner function W ( x , p ) for the Dirac fermion is defined as a two-point function: <math display="block" id="M50"><mrow><msub><mi>W</mi><mrow><mi>α</mi><mi>β</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>4</mn></msup><mi>y</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>p</mi><mo>⋅</mo><mi>y</mi></mrow></msup><mrow><mo>〈</mo><mrow><msub><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>β</mi></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mi>y</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mi>U</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mi>y</mi><mn>2</mn></mfrac><mn>,</mn><mi>x</mi><mo>−</mo><mfrac><mi>y</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><msub><mi>ψ</mi><mi>α</mi></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mfrac><mi>y</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mo>〉</mo></mrow><mn>,</mn></mrow></math> where U denotes the gauge link along the straight line between x-y /2 and x + y /2. The Wigner function can generally be expanded as <math display="block" id="M51"><mrow><mi>W</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mrow><mo>[</mo><mrow><mi mathvariant="script">F</mi><mo>+</mo><mi>i</mi><msup><mi>γ</mi><mn>5</mn></msup><mi mathvariant="script">P</mi><mo>+</mo><msup><mi>γ</mi><mi>μ</mi></msup><msub><mi mathvariant="script">V</mi><mi>μ</mi></msub><mo>+</mo><msup><mi>γ</mi><mn>5</mn></msup><msup><mi>γ</mi><mi>μ</mi></msup><msub><mi mathvariant="script">A</mi><mi>μ</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>σ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mi mathvariant="script">S</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mrow><mo>]</mo></mrow><mtext> </mtext><mn>,</mn></mrow></math> where σ μν =( i /2)[ γ μ , γ ν ]. One can choose the scalar function 𝒡 and axial vector function 𝒜 μ as independent variables [ 112 112 ]; they are related to F and A μ as <math display="block" id="M52"><mrow><mi mathvariant="script">F</mi><mo>=</mo><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mi mathvariant="script">F</mi><mo>+</mo><mfrac><mi>ℏ</mi><mi>m</mi></mfrac><msub><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msup><mi>p</mi><mi>μ</mi></msup><msup><mi mathvariant="script">A</mi><mi>ν</mi></msup><msup><mi>δ</mi><mo>'</mo></msup><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M53"><mrow><msub><mi mathvariant="script">A</mi><mi>μ</mi></msub><mo>=</mo><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><msub><mi mathvariant="script">A</mi><mi>μ</mi></msub><mo>+</mo><mfrac><mi>ℏ</mi><mi>m</mi></mfrac><msup><mi>p</mi><mi>ν</mi></msup><msub><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mi mathvariant="script">F</mi><msup><mi>δ</mi><mo>'</mo></msup><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> The kinetic equations for F and 𝒜 μ are given by <math display="block" id="M54"><mtable columnalign="left"><mtr><mtd><mi>p</mi><mo>⋅</mo><mo>∇</mo><mrow><mo>[</mo><mrow><mi mathvariant="script">F</mi><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mi>ℏ</mi><mi>m</mi></mfrac><msub><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msup><mi>p</mi><mi>μ</mi></msup><msup><mi mathvariant="script">A</mi><mi>ν</mi></msup><msup><mi>δ</mi><mo>'</mo></msup><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mi>ℏ</mi><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mo>∂</mo><mi>λ</mi><mi>x</mi></msubsup><msub><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo stretchy="false">)</mo><msubsup><mo>∂</mo><mi>p</mi><mi>λ</mi></msubsup><mrow><mo>[</mo><mrow><msup><mi>p</mi><mi>μ</mi></msup><msup><mi mathvariant="script">A</mi><mi>ν</mi></msup><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow/></mtd></mtr></mtable></math> and <math display="block" id="M55"><mtable columnalign="left"><mtr><mtd><mi>p</mi><mo>⋅</mo><mo>∇</mo><mrow><mo>[</mo><mrow><msub><mi mathvariant="script">A</mi><mi>μ</mi></msub><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mi>ℏ</mi><mi>m</mi></mfrac><msup><mi>p</mi><mi>ν</mi></msup><msub><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mi mathvariant="script">F</mi><msup><mi>δ</mi><mo>'</mo></msup><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mrow><mo>[</mo><mrow><msup><mi mathvariant="script">A</mi><mi>ν</mi></msup><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mi>ℏ</mi><mi>m</mi></mfrac><msub><mi>p</mi><mi>λ</mi></msub><msup><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>ν</mi><mi>λ</mi></mrow></msup><mi mathvariant="script">F</mi><msup><mi>δ</mi><mo>'</mo></msup><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mi>ℏ</mi><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mo>∂</mo><mi>λ</mi><mi>x</mi></msubsup><msub><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo stretchy="false">)</mo><msubsup><mo>∂</mo><mi>p</mi><mi>λ</mi></msubsup><mrow><mo>[</mo><mrow><msup><mi>p</mi><mi>ν</mi></msup><mi mathvariant="script">F</mi><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> together with the subsidiary condition p⋅𝒜δ ( p 2 - m 2 )=0, where <math id="M56"><mrow><msup><mo>∇</mo><mi>μ</mi></msup><mo>=</mo><msubsup><mo>∂</mo><mi>x</mi><mi>μ</mi></msubsup><mo>−</mo><msup><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msubsup><mo>∂</mo><mi>ν</mi><mi>p</mi></msubsup></mrow></math> , and <math id="M57"><mrow><msub><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>=</mo><msub><mi>є</mi><mrow><mi>μ</mi><mi>ν</mi><mi>ρ</mi><mi>σ</mi></mrow></msub><msup><mi>F</mi><mrow><mi>ρ</mi><mi>σ</mi></mrow></msup><mo>/</mo><mn>2</mn></mrow></math> . The integrated kinetic equation in three-vector form can be obtained by integrating the zero component of the momentum: <math display="block" id="M58"><mrow><mrow><mo>(</mo><mrow><msub><mo>∇</mo><mi>t</mi></msub><mo>+</mo><mstyle mathvariant="bold" mathsize="normal"><mi>v</mi></mstyle><mo>⋅</mo><mo>∇</mo></mrow><mo>)</mo></mrow><mi mathvariant="script">F</mi><mo>=</mo><mo>−</mo><mfrac><mi>ℏ</mi><mrow><mn>2</mn><mi>m</mi><msub><mi>E</mi><mi>p</mi></msub></mrow></mfrac><mrow><mo>[</mo><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mi>E</mi><mo>×</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>v</mi><mo>⋅</mo><mo>∇</mo><mo>+</mo><msub><mi>E</mi><mi>p</mi></msub><msub><mrow><mover accent="true"><mo>∇</mo><mo stretchy="true">←</mo></mover></mrow><mi>x</mi></msub><mo>⋅</mo><msub><mo>∇</mo><mi>p</mi></msub><mo stretchy="false">)</mo></mrow></mrow><mo>−</mo><mrow><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>⋅</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mtext>v</mtext><mo>⋅</mo><mo>∇</mo><mo>+</mo><msub><mi>E</mi><mi>p</mi></msub><msub><mrow><mover accent="true"><mo>∇</mo><mo stretchy="true">←</mo></mover></mrow><mi>x</mi></msub><mo>⋅</mo><msub><mo>∇</mo><mi>p</mi></msub><mtext>v</mtext></mrow><mo>]</mo></mrow><mo>⋅</mo><mover accent="true"><mi mathvariant="script">A</mi><mo stretchy="true">→</mo></mover><mn>,</mn></mrow></math> <math display="block" id="M59"><mrow><mrow><mo>(</mo><mrow><msub><mo>∇</mo><mi>t</mi></msub><mo>+</mo><mstyle mathvariant="bold" mathsize="normal"><mi>v</mi></mstyle><mo>⋅</mo><mo>∇</mo></mrow><mo>)</mo></mrow><mover accent="true"><mi mathvariant="script">A</mi><mo stretchy="true">→</mo></mover><mo>=</mo><mi>B</mi><mo>×</mo><mover accent="true"><mi mathvariant="script">A</mi><mo stretchy="true">→</mo></mover><mo>−</mo><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo>⋅</mo><mover accent="true"><mi mathvariant="script">A</mi><mo stretchy="true">→</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mi>ℏ</mi><mtext> </mtext></mrow><mrow><mn>2</mn><mi>m</mi><msub><mi>E</mi><mi>p</mi></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mi>E</mi><mo>×</mo><mtext>v</mtext><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mtext>v</mtext><mo>⋅</mo><mo>∇</mo><mo>+</mo><msub><mi>E</mi><mi>p</mi></msub><msub><mrow><mover accent="true"><mo>∇</mo><mo stretchy="true">←</mo></mover></mrow><mi>x</mi></msub><mo>⋅</mo><msub><mo>∇</mo><mi>p</mi></msub><mo stretchy="false">)</mo><mi mathvariant="script">F</mi><mn>,</mn></mrow></math> where v = p / E p , <math id="M60"><mrow><msub><mi>E</mi><mi>p</mi></msub><mo>=</mo><msqrt><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mi>m</mi><mn>2</mn></msup></mrow></msqrt></mrow></math> , ▽ t =∂ t + E ·▽ p , ▽=▽ x + B ×▽ p , and the left arrow over ∇ x denotes that it acts only on the electromagnetic fields on its left. When this method is used, various spin effects such as the chiral anomaly, CSE, quantum magnetization effect, and global polarization effect emerge naturally. It can be shown that the Dirac sea or vacuum contribution originating from the anti-commutation relations between antiparticle field operators in the Wigner function without a normal-order operator plays a crucial role in generating the chiral anomaly in quantum kinetic theory for both massive and massless fermions [ 118 118 ]. The coefficient of the chiral anomaly derived in this way is universal and is independent of the phase space distribution function at zero momentum. Weickgenannt et al. [ 113 113 , 114 114 ] 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 S μν , 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 moment <math id="M61"><mrow><msub><mover accent="true"><mo>∑</mo><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mrow></math> as independent variables of O ( ℏ ): <math display="block" id="M62"><mrow><mi mathvariant="script">F</mi><mo>=</mo><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mi>m</mi><mi>V</mi><mo>−</mo><mfrac><mi>ℏ</mi><mn>2</mn></mfrac><mi>m</mi><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msup><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msup><mi>δ</mi><mo>'</mo></msup><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M63"><mrow><msub><mi mathvariant="script">S</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>=</mo><mi>δ</mi><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mi>m</mi><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>−</mo><mi>ℏ</mi><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mi>V</mi><msup><mi>δ</mi><mo>'</mo></msup><mrow><mo>(</mo><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where δ′ ( x )≡d δ ( x )/d x , V = V (0) + ħV (1) , and <math id="M64"><mrow><msup><mover accent="true"><mo>∑</mo><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><msubsup><mo>∑</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mi>A</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><mi>ℏ</mi><msubsup><mo>∑</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> . In addition, V (0) and A (0) are scalar functions of the phase space and can be expressed as particle number distributions [ 113 113 , 114 114 ]. The kinetic equations for these functions are given by <math display="block" id="M65"><mtable columnalign="left"><mtr><mtd><mn>0</mn><mo>=</mo><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mrow><mo>[</mo><mrow><mi>p</mi><mo>⋅</mo><mo>∇</mo><mi>V</mi><mo>+</mo><mfrac><mi>ℏ</mi><mn>4</mn></mfrac><mo stretchy="false">(</mo><msubsup><mo>∂</mo><mi>x</mi><mi>α</mi></msubsup><msup><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo stretchy="false">)</mo><msubsup><mo>∂</mo><mi>α</mi><mi>p</mi></msubsup><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mi>ℏ</mi><mn>2</mn></mfrac><msup><mi>δ</mi><mo>'</mo></msup><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mi>F</mi><mrow><mi>α</mi><mi>β</mi></mrow></msup><mtext> </mtext><mi>p</mi><mo>⋅</mo><mo>∇</mo><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>α</mi><mi>β</mi></mrow></msub></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>p</mi><mo>⋅</mo><mo>∇</mo><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>−</mo><msubsup><mi>F</mi><mrow><mtext> </mtext><mo stretchy="false">[</mo><mi>μ</mi></mrow><mi>α</mi></msubsup><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>ν</mi><mo stretchy="false">]</mo><mi>α</mi></mrow></msub><mo>+</mo><mfrac><mi>ℏ</mi><mn>2</mn></mfrac><mo stretchy="false">(</mo><msubsup><mo>∂</mo><mi>x</mi><mi>α</mi></msubsup><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo stretchy="false">)</mo><msubsup><mo>∂</mo><mi>p</mi><mi>α</mi></msubsup><mi>V</mi><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>ℏ</mi><msup><mi>δ</mi><mo>'</mo></msup><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mtext> </mtext><mi>p</mi><mo>⋅</mo><mo>∇</mo><mi>V</mi><mn>.</mn></mtd></mtr></mtable></math> together with one constraint equation, <math id="M66"><mrow><msup><mi>p</mi><mi>ν</mi></msup><msub><mover accent="true"><mo>∑</mo><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mi>ℏ</mi><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msubsup><mo>∇</mo><mi>μ</mi><mi>V</mi></msubsup><mo>/</mo><mn>2</mn></mrow></math> . In this method, the kinetic equations and the components of the Wigner function are invariant under the transformation <math display="block" id="M67"><mtable columnalign="left"><mtr><mtd><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>→</mo><msub><mover accent="true"><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mo stretchy="true">^</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>=</mo><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mi>δ</mi><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mtext> </mtext><mn>,</mn></mtd></mtr><mtr><mtd><mi>V</mi><mo>→</mo><mover accent="true"><mi>V</mi><mo stretchy="true">^</mo></mover><mo>=</mo><mi>V</mi><mo>−</mo><mfrac><mi>ℏ</mi><mn>2</mn></mfrac><msup><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mi>δ</mi><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mtext> </mtext><mn>,</mn></mtd></mtr><mtr><mtd><mrow/></mtd></mtr></mtable></math> or the transformation <math display="block" id="M68"><mtable columnalign="left"><mtr><mtd><mi>V</mi><mo>→</mo><mover accent="true"><mi>V</mi><mo stretchy="true">^</mo></mover><mo>=</mo><mi>V</mi><mo>+</mo><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mi>δ</mi><mi>V</mi><mtext> </mtext><mn>,</mn></mtd></mtr><mtr><mtd><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>→</mo><msub><mover accent="true"><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mo stretchy="true">^</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>=</mo><msub><mover accent="true"><mi>Σ</mi><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>−</mo><mi>ℏ</mi><msub><mi>F</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mi>δ</mi><mi>V</mi><mtext> </mtext><mn>,</mn></mtd></mtr></mtable></math> where δV and <math id="M69"><mrow><mi>δ</mi><msub><mover accent="true"><mo>∑</mo><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mrow></math> are arbitrary functions of x and p that are nonsingular on the mass shell. When these transformations are used, the derivative terms of the delta function can be omitted from the kinetic equations, greatly reducing the results. Agreement with the CKT has been obtained by replacing the dipole-moment tensor <math id="M70"><mrow><msub><mover accent="true"><mo>∑</mo><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mrow></math> with its counterpart in the massless case. The kinetic equations yield a special single-particle distribution in global equilibrium with rigid rotation. When this distribution is used, analogs of various spin effects appear, such as the CME and CSE, for massive fermions. A smooth connection between the kinetic theory of massive spin-1/2 particles and the CKT was found recently [ 119 119 ] by introducing a frame-dependent decomposition of the dipole-moment tensor <math id="M71"><mrow><msub><mover accent="true"><mo>∑</mo><mo>¯</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mrow></math> . All the components of the covariant Wigner function can be derived from V μ and A μ for massive fermions in the Schwinger–Keldysh formalism [ 120 120 ]. 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 [ 121 121 ]. A covariant kinetic equation for massive fermions in curved spacetime and an external electromagnetic field can also be derived. Liu et al. [ 122 122 ] 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 and global equilibrium has also been studied. Integration of the covariant Wigner function over p 0 yields the equal-time Wigner function [ 123 123 - 125 125 123-125 ]. 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 [ 126 126 , 127 127 ]. The kinetic equation can also be derived from the equal-time Wigner functions. The equal-time Wigner function W ( x , p ) can be obtained from the covariant one W ( x , p ): <math display="block" id="M72"><mrow><mi mathvariant="script">W</mi><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>d</mi><msub><mi>p</mi><mn>0</mn></msub><mi>W</mi><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><msup><mi>γ</mi><mn>0</mn></msup><mn>.</mn></mrow></math> It can be decomposed into the following components: <math display="block" id="M73"><mtable columnalign="left"><mtr><mtd><mi mathvariant="script">W</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mrow><mo>[</mo><mrow><msub><mi>f</mi><mn>0</mn></msub><mo>+</mo><msub><mi>γ</mi><mn>5</mn></msub><msub><mi>f</mi><mn>1</mn></msub><mo>−</mo><mi>i</mi><msub><mi>γ</mi><mn>0</mn></msub><msub><mi>γ</mi><mn>5</mn></msub><msub><mi>f</mi><mn>2</mn></msub><mo>+</mo><msub><mi>γ</mi><mn>0</mn></msub><msub><mi>f</mi><mn>3</mn></msub><mo>+</mo><msub><mi>γ</mi><mn>5</mn></msub><msub><mi>γ</mi><mn>0</mn></msub><mi>γ</mi><mo>⋅</mo><msub><mi>g</mi><mn>0</mn></msub></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>+</mo><msub><mi>γ</mi><mn>0</mn></msub><mi>γ</mi><mo>⋅</mo><msub><mi>g</mi><mn>1</mn></msub><mo>−</mo><mi>i</mi><mi>γ</mi><mo>⋅</mo><msub><mi>g</mi><mn>2</mn></msub><mo>−</mo><msub><mi>γ</mi><mn>5</mn></msub><mi>γ</mi><mo>⋅</mo><msub><mi>g</mi><mn>3</mn></msub></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> The fermion number density f 0 and spin current g 0 are chosen to be independent components; their kinetic equations are derived as [ 128 128 ] <math display="block" id="M74"><mrow><mrow><mo>(</mo><mrow><msub><mo>∇</mo><mi>t</mi></msub><mo>±</mo><mfrac><mi>p</mi><mrow><msub><mi>E</mi><mi>p</mi></msub></mrow></mfrac><mo>⋅</mo><mo>∇</mo></mrow><mo>)</mo></mrow><msubsup><mi>f</mi><mn>0</mn><mo>±</mo></msubsup><mo>=</mo><mfrac><mrow><mi>ℏ</mi><mi>E</mi></mrow><mrow><mn>2</mn><msubsup><mi>E</mi><mi>p</mi><mn>2</mn></msubsup></mrow></mfrac><mo>⋅</mo><mo>∇</mo><mo>×</mo><msubsup><mi>g</mi><mn>0</mn><mo>±</mo></msubsup><mo>∓</mo><mfrac><mi>ℏ</mi><mrow><mn>2</mn><msubsup><mi>E</mi><mi>p</mi><mn>3</mn></msubsup></mrow></mfrac><mi>B</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>p</mi><mo>⋅</mo><mo>∇</mo><mo stretchy="false">)</mo><msubsup><mi>g</mi><mn>0</mn><mo>±</mo></msubsup><mo>+</mo><mfrac><mrow><mi>ℏ</mi><mi>B</mi><mo>×</mo><mi>p</mi></mrow><mrow><msubsup><mi>E</mi><mi>p</mi><mn>4</mn></msubsup></mrow></mfrac><mo>⋅</mo><mi>E</mi><mo>×</mo><msubsup><mi>g</mi><mn>0</mn><mo>±</mo></msubsup><mn>,</mn></mrow></math> <math display="block" id="M75"><mtable columnalign="left"><mtr><mtd><mrow><mo>(</mo><mrow><msub><mo>∇</mo><mi>t</mi></msub><mo>±</mo><mfrac><mi>p</mi><mrow><msub><mi>E</mi><mi>p</mi></msub></mrow></mfrac><mo>⋅</mo><mo>∇</mo></mrow><mo>)</mo></mrow><msubsup><mtext>g</mtext><mn>0</mn><mo>±</mo></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><msubsup><mi>E</mi><mi>p</mi><mn>2</mn></msubsup></mrow></mfrac><mrow><mo>[</mo><mrow><mtext>p</mtext><mo>×</mo><mrow><mo>(</mo><mrow><mi>E</mi><mo>×</mo><msubsup><mtext>g</mtext><mn>0</mn><mo>±</mo></msubsup></mrow><mo>)</mo></mrow><mo>∓</mo><msub><mi>E</mi><mi>p</mi></msub><mi>B</mi><mo>×</mo><msubsup><mtext>g</mtext><mn>0</mn><mo>±</mo></msubsup></mrow><mo>]</mo></mrow><mo>∓</mo><mi>ℏ</mi><mrow><mo>(</mo><mrow><mfrac><mi>B</mi><mrow><mn>2</mn><msubsup><mi>E</mi><mi>p</mi><mn>3</mn></msubsup></mrow></mfrac><mo>±</mo><mfrac><mrow><mi>E</mi><mo>×</mo><mi>p</mi></mrow><mrow><mn>2</mn><msubsup><mi>E</mi><mi>p</mi><mn>4</mn></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><mi>p</mi><mo>⋅</mo><mo>∇</mo><msubsup><mi>f</mi><mn>0</mn><mo>±</mo></msubsup></mtd></mtr><mtr><mtd><mo>∓</mo><mi>ℏ</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>⋅</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>E</mi><mo>×</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><msubsup><mi>E</mi><mi>p</mi><mn>5</mn></msubsup></mrow></mfrac><mo>±</mo><mfrac><mrow><mi>p</mi><mo>×</mo><mo stretchy="false">(</mo><mi>B</mi><mo>×</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msubsup><mi>E</mi><mi>p</mi><mn>4</mn></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><msubsup><mi>f</mi><mn>0</mn><mo>±</mo></msubsup><mn>.</mn></mtd></mtr></mtable></math> 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 of ℏ and without collision terms for particle scattering. In addition to these works based on Wigner functions, Li and Yee [ 129 129 ] 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 g 4 log(1/ g ): <math display="block" id="M76"><mrow><mfrac><mrow><mo>∂</mo><mi>f</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mfrac><mrow><msubsup><mi>m</mi><mi>D</mi><mn>2</mn></msubsup><msup><mi>g</mi><mn>2</mn></msup><mi>log</mi><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mi>π</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>p</mi><msub><mi>E</mi><mi>p</mi></msub></mrow></mfrac><msub><mi>Γ</mi><mi>f</mi></msub><mtext> </mtext><mn>,</mn><mfrac><mrow><mo>∂</mo><mstyle mathvariant="bold" mathsize="normal"><mi>S</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>C</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mfrac><mrow><msubsup><mi>m</mi><mi>D</mi><mn>2</mn></msubsup><msup><mi>g</mi><mn>2</mn></msup><mi>log</mi><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mi>π</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>p</mi><msub><mi>E</mi><mi>p</mi></msub></mrow></mfrac><mtext> </mtext><msub><mi>Γ</mi><mi>S</mi></msub><mtext> </mtext><mn>,</mn></mrow></math> where <math id="M77"><mrow><msub><mi>C</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>N</mi><mi>c</mi><mn>2</mn></msubsup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><mn>2</mn><msub><mi>N</mi><mi>c</mi></msub></mrow></math> , and Γ f and Γ S are diffusion-like differential operators in momentum space that contain up to the second-order derivatives of the momentum. The explicit forms of Γ f and Γ S are given in Eq. (4.70) of Ref. [ 129 129 ]. 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. [ 119 119 , 130 130 ]. 4 Spin polarization effects 1 4.1 Theoretical progress 2 In early works [ 131 131 , 132 132 ], 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 [ 133 133 ]. Becattini and his collaborators [ 134 134 - 136 136 134-136 ] performed a systematic study of statistical models of relativistic spinning particles. In addition, recent review articles are cited in Refs. [ 137 137 - 139 139 137-139 ]. The initial global OAM is estimated to be as large as J 0 ∼10 5 ℏ in 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 [ 140 140 - 142 142 140-142 ], UrQMD [ 143 143 ], and hydrodynamic models [ 144 144 - 149 149 144-149 ]. The vorticity is found to be more strongly suppressed at higher collision energies. One can find further discussion in Refs. [ 150 150 - 152 152 150-152 ] 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 [ 136 136 ] and the Wigner functions [ 153 153 ]: <math display="block" id="M78"><mrow><msup><mi>S</mi><mi>μ</mi></msup><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>8</mn><mi>m</mi></mrow></mfrac><mfrac><mrow><msup><mi>ϵ</mi><mrow><mi>μ</mi><mi>ν</mi><mi>ρ</mi><mi>σ</mi></mrow></msup><msub><mi>p</mi><mi>ν</mi></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>λ</mi></msub><msup><mi>p</mi><mi>λ</mi></msup><msup><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>⋅</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>f</mi><mrow><mtext>FD</mtext></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>f</mi><mrow><mtext>FD</mtext></mrow></msub><mo stretchy="false">)</mo><msubsup><mi>ω</mi><mrow><mi>ρ</mi><mi>σ</mi></mrow><mrow><mtext>th</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>λ</mi></msub><msup><mi>p</mi><mi>λ</mi></msup><msub><mi>f</mi><mrow><mtext>FD</mtext></mrow></msub></mrow></mfrac><mn>,</mn></mrow></math> where ∑ λ denotes the freeze-out hypersurface, S μ is the Pauli–Lubanski pseudo-vector, m is the fermion mass, f FD ≡ f FD ( x , p ) is the Fermi–Dirac distribution. In addition, <math display="block" id="M79"><mrow><msubsup><mi>ω</mi><mrow><mi>ρ</mi><mi>σ</mi></mrow><mrow><mtext>th</mtext></mrow></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><msub><mo>∂</mo><mi>σ</mi></msub><msub><mi>β</mi><mi>ρ</mi></msub><mo>−</mo><msub><mo>∂</mo><mi>ρ</mi></msub><msub><mi>β</mi><mi>σ</mi></msub></mrow><mo>)</mo></mrow></mrow></math> is the thermal vorticity, where u μ is the fluid velocity, and β ρ = u ρ / T is the temperature four-vector. Equation (44) has been widely used in hydrodynamics models and transport models to calculate the spin polarization of hyperons. In quantum field theory, the decomposition of the total angular momentum into the orbital and spin components for massive particles with spin is not unique. Different decompositions are related by a pseudo-gauge transformation. Although the energy-momentum tensor <math id="M80"><mrow><msup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></mrow></math> and spin tensor <math id="M81"><mrow><msup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msup></mrow></math> depend on the decomposition, the total energy-momentum and total angular momentum are invariant under the pseudo-gauge transformation. A special pseudo-gauge transformation is the transformation of the canonical definitions of <math id="M82"><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> and <math id="M83"><mrow><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> into the Belinfante <math id="M84"><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> , which absorbs <math id="M85"><mrow><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> ; i.e., <math id="M86"><mrow><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><mn>0</mn></mrow></math> . Thus, the total angular momentum consists only of the orbital component. The Belinfante energy-momentum tensor is symmetric with respect to its two indices: <math id="M87"><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>ν</mi><mi>μ</mi></mrow></msubsup></mrow></math> . The thermal properties of a quantum field system can be described by the density operator <math id="M88"><mover accent="true"><mi>ρ</mi><mo>^</mo></mover></math> . The density operator in local equilibrium can be determined from the maximal entropy principle [ 154 154 - 156 156 154-156 ] under given densities of conserved currents on the space-like hypersurface ∑ . This is done by maximizing <math id="M89"><mrow><mi>S</mi><mo>=</mo><mo>−</mo><mtext>Tr</mtext><mo stretchy="false">(</mo><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mi>log</mi><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mo stretchy="false">)</mo></mrow></math> under the conditions <math display="block" id="M90"><mtable columnalign="left"><mtr><mtd><msub><mi>n</mi><mi>μ</mi></msub><mtext>Tr</mtext><mrow><mo>(</mo><mrow><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><msup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>n</mi><mi>μ</mi></msub><msup><mi>T</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mn>,</mn></mtd></mtr><mtr><mtd><msub><mi>n</mi><mi>μ</mi></msub><mtext>Tr</mtext><mrow><mo>(</mo><mrow><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>n</mi><mi>μ</mi></msub><msup><mi>j</mi><mi>μ</mi></msup><mn>,</mn></mtd></mtr><mtr><mtd><mrow/></mtd></mtr></mtable></math> where n μ is the normal vector of ∑ . The constraint on the total angular momentum density can generally be included as well: <math display="block" id="M91"><mtable columnalign="left"><mtr><mtd><msub><mi>n</mi><mi>λ</mi></msub><mtext>Tr</mtext><mrow><mo>(</mo><mrow><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><msup><mover accent="true"><mi mathvariant="script">J</mi><mo>^</mo></mover><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>n</mi><mi>λ</mi></msub><mtext>Tr</mtext><mrow><mo>[</mo><mrow><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>μ</mi></msup><msup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mrow><mi>λ</mi><mi>ν</mi></mrow></msup><mo>−</mo><msup><mi>x</mi><mi>ν</mi></msup><msup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mrow><mi>λ</mi><mi>μ</mi></mrow></msup><mo>+</mo><msup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>n</mi><mi>λ</mi></msub><msup><mi mathvariant="script">J</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msup><mn>.</mn></mtd></mtr></mtable></math> For the Belinfante energy-momentum tensor, however, the above condition is automatically satisfied by using (46). In this case, the density operator at local equilibrium has the form <math display="block" id="M92"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>Z</mi></mfrac><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>Σ</mi></msub></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>μ</mi></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mi>β</mi><mi>ν</mi></msub><mo>−</mo><mi>ζ</mi><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where β ν and ζ are the Lagrange multiplier functions of space-time; the former is the temperature four-vector, and the latter is the ratio of the chemical potential to the temperature [ 156 156 ]. In general, <math id="M93"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> in (48) depends on the time τ as ∑ μ ( τ ). One can rewrite the exponent of <math id="M94"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> as <math display="block" id="M95"><mtable columnalign="left"><mtr><mtd><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mrow><mi>Σ</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>μ</mi></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mi>β</mi><mi>ν</mi></msub><mo>−</mo><mi>ζ</mi><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mrow><mi>Σ</mi><mo stretchy="false">(</mo><msub><mi>τ</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></msub></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>μ</mi></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mi>β</mi><mi>ν</mi></msub><mo>−</mo><mi>ζ</mi><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>Θ</mi></msub></mrow></mstyle><mi>d</mi><mi>Θ</mi><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mo>∂</mo><mi>μ</mi></msub><msub><mi>β</mi><mi>ν</mi></msub><mo>−</mo><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup><msub><mo>∂</mo><mi>μ</mi></msub><mi>ζ</mi></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> where Θ is the space-time volume bounded by ∑( τ ), ∑( τ 0 ), and the time-like boundary connecting ∑( τ ) and ∑( τ 0 ). Here we used <math display="block" id="M96"><mrow><msub><mo>∂</mo><mi>μ</mi></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mi>β</mi><mi>ν</mi></msub><mo>−</mo><mi>ζ</mi><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow><mo>)</mo></mrow><mo>=</mo><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>B</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mo>∂</mo><mi>μ</mi></msub><msub><mi>β</mi><mi>ν</mi></msub><mo>−</mo><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup><msub><mo>∂</mo><mi>μ</mi></msub><mi>ζ</mi><mn>,</mn></mrow></math> where we used the conservation of energy-momentum and current. To make <math id="M97"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> independent of the choice of ∑ μ ( τ ), i.e., to obtain the global equilibrium density matrix <math id="M98"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>GE</mtext></mrow></msub></mrow></math> , the following conditions must be met: <math display="block" id="M99"><mtable columnalign="left"><mtr><mtd><msub><mo>∂</mo><mi>μ</mi></msub><msub><mi>β</mi><mi>ν</mi></msub><mo>+</mo><msub><mo>∂</mo><mi>ν</mi></msub><msub><mi>β</mi><mi>μ</mi></msub><mo>=</mo><mn>0,</mn></mtd></mtr><mtr><mtd><msub><mo>∂</mo><mi>μ</mi></msub><mi>ζ</mi><mo>=</mo><mn>0,</mn></mtd></mtr></mtable></math> according to which ζ is constant and β μ is a Killing vector: <math display="block" id="M100"><mrow><msup><mi>β</mi><mi>μ</mi></msup><mo>=</mo><msubsup><mi>ω</mi><mrow><mtext>th</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mi>x</mi><mi>ν</mi></msub><mo>+</mo><msup><mi>b</mi><mi>μ</mi></msup><mn>,</mn></mrow></math> where <math id="M101"><mrow><msubsup><mi>ω</mi><mrow><mtext>th</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> is a constant antisymmetric tensor given by (45), and b μ is a constant vector. The local equilibrium density operator in (48) can be expressed in terms of the canonical energy-momentum and spin tensors: <math display="block" id="M102"><mtable columnalign="left"><mtr><mtd><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>Z</mi></mfrac><mi>exp</mi><mrow><mo>{</mo><mrow><mo>−</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>Σ</mi></msub></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>μ</mi></msub><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mi>β</mi><mi>ν</mi></msub><mo>−</mo><mi>ζ</mi><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow></mrow></mrow></mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mi>ω</mi><mrow><mi>λ</mi><mi>ν</mi></mrow><mrow><mtext>th</mtext></mrow></msubsup><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>μ</mi><mo>;</mo><mi>λ</mi><mi>ν</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><msub><mo>∂</mo><mi>λ</mi></msub><msub><mi>β</mi><mi>ν</mi></msub><mo>+</mo><msub><mo>∂</mo><mi>ν</mi></msub><msub><mi>β</mi><mi>λ</mi></msub></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>ν</mi><mo>;</mo><mi>μ</mi><mi>λ</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> If we substitute (51) and (52) in the above expression for <math id="M103"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> , we obtain <math id="M104"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>GE</mtext></mrow></msub></mrow></math> : <math display="block" id="M105"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>GE</mtext></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>Z</mi></mfrac><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><msub><mi>b</mi><mi>μ</mi></msub><msup><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>μ</mi></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mi>ω</mi><mrow><mi>λ</mi><mi>ν</mi></mrow><mrow><mtext>th</mtext></mrow></msubsup><msup><mover accent="true"><mi>J</mi><mo>^</mo></mover><mrow><mi>λ</mi><mi>ν</mi></mrow></msup><mo>+</mo><mi>ζ</mi><mover accent="true"><mi>Q</mi><mo>^</mo></mover></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where <math id="M106"><mrow><msup><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow></math> , <math id="M107"><mrow><msup><mover accent="true"><mi>J</mi><mo>^</mo></mover><mrow><mi>λ</mi><mi>ν</mi></mrow></msup></mrow></math> , and <math id="M108"><mover accent="true"><mi>Q</mi><mo>^</mo></mover></math> are given by <math display="block" id="M109"><mtable columnalign="left"><mtr><mtd><msup><mover accent="true"><mi>P</mi><mo>^</mo></mover><mi>ν</mi></msup><mo>=</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>Σ</mi></msub></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>μ</mi></msub><msup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mn>,</mn></mtd></mtr><mtr><mtd><msup><mover accent="true"><mi>J</mi><mo>^</mo></mover><mrow><mi>λ</mi><mi>ν</mi></mrow></msup><mo>=</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>Σ</mi></msub></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>μ</mi></msub><msup><mi mathvariant="script">J</mi><mrow><mi>μ</mi><mo>;</mo><mi>λ</mi><mi>ν</mi></mrow></msup><mn>,</mn></mtd></mtr><mtr><mtd><mover accent="true"><mi>Q</mi><mo>^</mo></mover><mo>=</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>Σ</mi></msub></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>μ</mi></msub><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup><mn>.</mn></mtd></mtr></mtable></math> If we work with the canonical energy-momentum and spin tensors, we have to impose the constraint in (47), which also constrains the spin tensor: <math display="block" id="M110"><mrow><msub><mi>n</mi><mi>λ</mi></msub><mtext>Tr</mtext><mrow><mo>(</mo><mrow><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><msub><mi>n</mi><mi>λ</mi></msub><msubsup><mi>S</mi><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mn>.</mn></mrow></math> This introduces a spin chemical potential, the antisymmetric tensor Ω λν , as a Lagrange multiplier into the equation for <math id="M111"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> : <math display="block" id="M112"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>Z</mi></mfrac><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mi>Σ</mi></msub></mrow></mstyle><mi>d</mi><msub><mi>Σ</mi><mi>μ</mi></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><msub><mi>β</mi><mi>ν</mi></msub><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>Ω</mi><mrow><mi>λ</mi><mi>ν</mi></mrow></msub><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>μ</mi><mo>;</mo><mi>λ</mi><mi>ν</mi></mrow></msubsup><mo>−</mo><mi>ζ</mi><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>.</mn></mrow></math> Comparing (57) with (53), we see that <math id="M113"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> is the same for the Belinfante and canonical tensors under the following conditions: (a) the field β μ is the same in both cases; (b) <math id="M114"><mrow><msub><mi>Ω</mi><mrow><mi>λ</mi><mi>ν</mi></mrow></msub><mo>=</mo><msubsup><mi>ω</mi><mrow><mi>λ</mi><mi>ν</mi></mrow><mrow><mtext>th</mtext></mrow></msubsup></mrow></math> ; and (c) <math id="M115"><mrow><mrow><mo>(</mo><mrow><msub><mo>∂</mo><mi>λ</mi></msub><msub><mi>β</mi><mi>ν</mi></msub><mo>+</mo><msub><mo>∂</mo><mi>ν</mi></msub><msub><mi>β</mi><mi>λ</mi></msub></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mi>C</mi><mrow><mi>ν</mi><mo>;</mo><mi>μ</mi><mi>λ</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math> . We can see that if <math id="M116"><mrow><msub><mi>Ω</mi><mrow><mi>λ</mi><mi>ν</mi></mrow></msub><mo>=</mo><msubsup><mi>ω</mi><mrow><mi>λ</mi><mi>ν</mi></mrow><mrow><mtext>th</mtext></mrow></msubsup></mrow></math> and ∂ λ β ν +∂ ν β λ =0, then <math id="M117"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> for the canonical tensors becomes the same as <math id="M118"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>GE</mtext></mrow></msub></mrow></math> for the Belinfante tensor in (54). Therefore, one can conclude that the density operator in global equilibrium is pseudo-gauge-invariant, but in local equilibrium, it depends on the pseudo-gauge [ 157 157 ]. If we choose the Belinfante tensors or <math id="M119"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> in (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 <math id="M120"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> in (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 [ 157 157 ]. Similar problems involving the decomposition of the total angular momentum of quarks and gluons in the proton spin (see Ref. [ 158 158 ] 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 [ 159 159 ] discusses the difference between the Belinfante and canonical forms. For a chiral medium, the decomposition can be found in Ref. [ 160 160 ]. If we choose a specific pseudo-gauge, for example, the canonical form that has the spin tensor, the spin chemical potential Ω λν must be introduced into <math id="M121"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> , as shown in (57). Therefore, we have another conservation equation in addition to the conservation of energy-momentum and charge: <math display="block" id="M122"><mrow><msub><mo>∂</mo><mi>μ</mi></msub><msup><mi>j</mi><mi>μ</mi></msup><mo>=</mo><mn>0,</mn><mtext> </mtext><mtext> </mtext><msub><mo>∂</mo><mi>μ</mi></msub><msup><mi>T</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><mn>0,</mn><mtext> </mtext><mtext> </mtext><msub><mo>∂</mo><mi>λ</mi></msub><msup><mi>S</mi><mrow><mi>λ</mi><mn>,</mn><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><msup><mi>T</mi><mrow><mi>ν</mi><mi>μ</mi></mrow></msup><mo>−</mo><msup><mi>T</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mn>,</mn></mrow></math> where the energy-momentum tensor, charge current, and spin tensor can be obtained through <math id="M123"><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub></mrow></math> : <math display="block" id="M124"><mtable columnalign="left"><mtr><mtd><msup><mi>T</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>=</mo><mtext>tr</mtext><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub><msup><mover accent="true"><mi>T</mi><mo>^</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr><mtr><mtd><msup><mi>j</mi><mi>μ</mi></msup><mo>=</mo><mtext>tr</mtext><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub><msup><mover accent="true"><mi>j</mi><mo>^</mo></mover><mi>μ</mi></msup></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr><mtr><mtd><msup><mi>S</mi><mrow><mi>μ</mi><mn>,</mn><mi>ν</mi><mi>λ</mi></mrow></msup><mo>=</mo><mtext>tr</mtext><mrow><mo>(</mo><mrow><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mrow><mtext>LE</mtext></mrow></msub><msup><mover accent="true"><mi>S</mi><mo>^</mo></mover><mrow><mi>μ</mi><mn>,</mn><mi>ν</mi><mi>λ</mi></mrow></msup></mrow><mo>)</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> All the above quantities are functions of β μ , ζ , and Ω λν . 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 [ 35 35 , 157 157 , 161 161 - 165 165 161-165 ]. Another approach using Lagrangian techniques can also provide some information about the local equilibrium [ 166 166 - 169 169 166-169 ]. 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 [ 136 136 ]: <math display="block" id="M125"><mtable columnalign="left"><mtr><mtd><msubsup><mi>f</mi><mrow><mi>r</mi><mi>s</mi></mrow><mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><msub><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mi>X</mi><mo>+</mo></msup><msub><mi>u</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><msubsup><mi>f</mi><mrow><mi>r</mi><mi>s</mi></mrow><mo>−</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><msub><mover accent="true"><mi>v</mi><mo>¯</mo></mover><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mi>X</mi><mo>−</mo></msup><msub><mi>v</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where r , s =±1; u r and v r are Dirac spinors normalized by <math id="M126"><mrow><msub><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mi>u</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>m</mi><msub><mi>δ</mi><mrow><mi>r</mi><mi>s</mi></mrow></msub></mrow></math> and <math id="M127"><mrow><msub><mover accent="true"><mi>v</mi><mo>¯</mo></mover><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mi>v</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><mi>m</mi><msub><mi>δ</mi><mrow><mi>r</mi><mi>s</mi></mrow></msub></mrow></math> , respectively; and X ± are 44 matrices defined as <math display="block" id="M128"><mrow><msup><mi>X</mi><mo>±</mo></msup><mo>=</mo><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><msub><mi>β</mi><mi>μ</mi></msub><msup><mi>p</mi><mi>μ</mi></msup><mo>±</mo><mi>ζ</mi><mo>±</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>Ω</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><msup><mi>σ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> The Wigner functions in equilibrium for fermions and antifermions are given by <math display="block" id="M129"><mtable columnalign="left"><mtr><mtd><msubsup><mi>W</mi><mrow><mtext>eq</mtext></mrow><mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>r</mi><mn>,</mn><mi>s</mi></mrow></munder></mstyle><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>p</mi><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mi>δ</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mi>u</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mover accent="true"><mi>u</mi><mo>¯</mo></mover><mi>s</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msubsup><mi>f</mi><mrow><mi>r</mi><mi>s</mi></mrow><mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>p</mi><mi>θ</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mi>δ</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>p</mi><mi>μ</mi></msub><msup><mi>γ</mi><mi>μ</mi></msup><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo><msup><mi>X</mi><mo>+</mo></msup><mo stretchy="false">(</mo><msub><mi>p</mi><mi>μ</mi></msub><msup><mi>γ</mi><mi>μ</mi></msup><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><msubsup><mi>W</mi><mrow><mtext>eq</mtext></mrow><mo>−</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>r</mi><mn>,</mn><mi>s</mi></mrow></munder></mstyle><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>p</mi><mi>θ</mi><mo stretchy="false">(</mo><mo>−</mo><msub><mi>p</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mi>δ</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mi>v</mi><mi>r</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mover accent="true"><mi>v</mi><mo>¯</mo></mover><mi>s</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msubsup><mi>f</mi><mrow><mi>r</mi><mi>s</mi></mrow><mo>−</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>p</mi><mi>θ</mi><mo stretchy="false">(</mo><mo>−</mo><msub><mi>p</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mi>δ</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>p</mi><mi>μ</mi></msub><msup><mi>γ</mi><mi>μ</mi></msup><mo>−</mo><mi>m</mi><mo stretchy="false">)</mo><msup><mi>X</mi><mo>−</mo></msup><mo stretchy="false">(</mo><msub><mi>p</mi><mi>μ</mi></msub><msup><mi>γ</mi><mi>μ</mi></msup><mo>−</mo><mi>m</mi><mo stretchy="false">)</mo><mn>.</mn></mtd></mtr></mtable></math> The total Wigner function is the sum of them: <math display="block" id="M130"><mrow><msub><mi>W</mi><mrow><mtext>eq</mtext></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>W</mi><mrow><mtext>eq</mtext></mrow><mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>W</mi><mrow><mtext>eq</mtext></mrow><mo>−</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><mi>k</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> The components of W eq ( x , k ) are given in (27) and can be extracted by taking the traces of Γ i W eq ( x , k ) with Γ i =1, γ 5 , γ μ , γ 5 γ μ , μν . The energy-momentum and spin tensors proposed by de Groot, van Leeuwen, and van Weert (GLW) [ 170 170 ] can be extracted from the Wigner function [ 163 163 ]: <math display="block" id="M131"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>T</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mn>1</mn><mi>m</mi></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi><msup><mi>k</mi><mi>μ</mi></msup><msup><mi>k</mi><mi>ν</mi></msup><mi mathvariant="script">F</mi><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>S</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi><mtext>Tr</mtext><mrow><mo>[</mo><mrow><mrow><mo>{</mo><mrow><msup><mi>σ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mn>,</mn><msup><mi>γ</mi><mi>λ</mi></msup></mrow><mo>}</mo></mrow><mi>W</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>i</mi></mrow><mi>m</mi></mfrac><mrow><mo>(</mo><mrow><msup><mi>γ</mi><mrow><mo stretchy="false">[</mo><mi>μ</mi></mrow></msup><msup><mi>k</mi><mrow><mi>ν</mi><mo stretchy="false">]</mo></mrow></msup><msup><mi>γ</mi><mi>λ</mi></msup><mo>−</mo><msup><mi>γ</mi><mi>λ</mi></msup><msup><mi>γ</mi><mrow><mo stretchy="false">[</mo><mi>μ</mi></mrow></msup><msup><mi>k</mi><mrow><mi>ν</mi><mo stretchy="false">]</mo></mrow></msup></mrow><mo>)</mo></mrow><mi>W</mi></mrow><mo>]</mo></mrow><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> In equilibrium, we use W = W eq in (63) and obtain <math id="M132"><mrow><msubsup><mi>T</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><msubsup><mi>T</mi><mrow><mtext>eq</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> and <math id="M133"><mrow><msubsup><mi>S</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><msubsup><mi>S</mi><mrow><mtext>eq</mtext></mrow><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> . Because the GLW energy-momentum tensor is symmetric, the GLW spin tensor should be conserved separately. Thus, the conservation laws read <math display="block" id="M134"><mrow><msub><mo>∂</mo><mi>μ</mi></msub><msup><mi>j</mi><mi>μ</mi></msup><mo>=</mo><mn>0,</mn><mtext> </mtext><mtext> </mtext><msub><mo>∂</mo><mi>μ</mi></msub><msubsup><mi>T</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><mn>0,</mn><mtext> </mtext><mtext> </mtext><msub><mo>∂</mo><mi>λ</mi></msub><msubsup><mi>S</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><mn>0.</mn></mrow></math> The canonical forms of the energy-momentum and spin tensors can be obtained from the Dirac Lagrangian [ 113 113 , 117 117 ]: <math display="block" id="M135"><mtable columnalign="left"><mtr><mtd><msubsup><mi>T</mi><mi>C</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi><msup><mi>k</mi><mi>ν</mi></msup><msup><mi mathvariant="script">V</mi><mi>μ</mi></msup><mn>,</mn></mtd></mtr><mtr><mtd><msubsup><mi>S</mi><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi><mtext>Tr</mtext><mrow><mo>[</mo><mrow><mrow><mo>{</mo><mrow><msup><mi>σ</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mn>,</mn><msup><mi>γ</mi><mi>λ</mi></msup></mrow><mo>}</mo></mrow><mi>W</mi></mrow><mo>]</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>ϵ</mi><mrow><mi>λ</mi><mi>μ</mi><mi>ν</mi><mi>ρ</mi></mrow></msup><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi><msub><mi mathvariant="script">A</mi><mi>ρ</mi></msub><mn>.</mn></mtd></mtr></mtable></math> One can verify that the canonical tensor forms are related to the GLW forms by the pseudo-gauge transformation [ 163 163 ]. The explicit relation between the two tensor forms can be written as <math display="block" id="M136"><mtable columnalign="left"><mtr><mtd><msubsup><mi>T</mi><mi>C</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><msubsup><mi>T</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>−</mo><mfrac><mi>ℏ</mi><mrow><mn>2</mn><mi>m</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi><msup><mi>k</mi><mi>ν</mi></msup><msub><mo>∂</mo><mi>λ</mi></msub><msubsup><mi>S</mi><mrow><mtext>eq</mtext></mrow><mrow><mi>λ</mi><mi>μ</mi></mrow></msubsup><mo>=</mo><msubsup><mi>T</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>−</mo><msub><mo>∂</mo><mi>λ</mi></msub><msubsup><mi>S</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>ν</mi><mo>;</mo><mi>λ</mi><mi>μ</mi></mrow></msubsup><mn>,</mn></mtd></mtr><mtr><mtd><msubsup><mi>S</mi><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>=</mo><msubsup><mi>S</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup><mo>+</mo><msubsup><mi>S</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>μ</mi><mo>;</mo><mi>ν</mi><mi>λ</mi></mrow></msubsup><mo>+</mo><msubsup><mi>S</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>ν</mi><mo>;</mo><mi>λ</mi><mi>μ</mi></mrow></msubsup><mn>.</mn></mtd></mtr></mtable></math> One can verify that the conservation laws for <math id="M137"><mrow><msubsup><mi>T</mi><mi>C</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> and <math id="M138"><mrow><msubsup><mi>S</mi><mi>C</mi><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> follow those for <math id="M139"><mrow><msubsup><mi>T</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> and <math id="M140"><mrow><msubsup><mi>S</mi><mrow><mtext>GLW</mtext></mrow><mrow><mi>λ</mi><mo>;</mo><mi>μ</mi><mi>ν</mi></mrow></msubsup></mrow></math> in (64). 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 [ 171 171 ]. 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: <math display="block" id="M141"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>R</mi><mrow><mi>A</mi><mi>B</mi><mo>→</mo><mn>12</mn></mrow></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msub><mi>n</mi><mi>A</mi></msub><msub><mi>n</mi><mi>B</mi></msub><mo>|</mo><msub><mi>v</mi><mi>A</mi></msub><mo>−</mo><msub><mi>v</mi><mi>B</mi></msub><mo>|</mo><mi>σ</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><msub><mi>p</mi><mi>A</mi></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><msub><mi>p</mi><mi>B</mi></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><msub><mi>f</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>A</mi></msub><mn>,</mn><msub><mi>p</mi><mi>A</mi></msub><mo stretchy="false">)</mo><msub><mi>f</mi><mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>B</mi></msub><mn>,</mn><msub><mi>p</mi><mi>B</mi></msub><mo stretchy="false">)</mo><mo>|</mo><msub><mi>v</mi><mi>A</mi></msub><mo>−</mo><msub><mi>v</mi><mi>B</mi></msub><mo>|</mo><mi>Δ</mi><mi>σ</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where v A =| p A |/ E A and v B =-| p B |/ E B are the longitudinal velocities, with p A =- p B in the center-of-mass frame of colliding particles; f A and f B are the phase space distributions of the incident particles A and B , respectively; and Δ 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: <math display="block" id="M142"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msup><mi>d</mi><mn>4</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mrow><mi>A</mi><mi>B</mi><mo>→</mo><mn>12</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msup><mi>X</mi><mn>4</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><msub><mi>p</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mn>2</mn><msub><mi>E</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub></mrow></mfrac><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><msub><mi>p</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mn>2</mn><msub><mi>E</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub></mrow></mfrac><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><msub><mi>p</mi><mrow><mi>c</mi><mn>,1</mn></mrow></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mn>2</mn><msub><mi>E</mi><mrow><mi>c</mi><mn>,1</mn></mrow></msub></mrow></mfrac><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><msub><mi>p</mi><mrow><mi>c</mi><mn>,2</mn></mrow></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mn>2</mn><msub><mi>E</mi><mrow><mi>c</mi><mn>,2</mn></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mo>×</mo><mo>|</mo><msub><mi>v</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo>−</mo><msub><mi>v</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub><mo>|</mo><msub><mi>G</mi><mn>1</mn></msub><msub><mi>G</mi><mn>2</mn></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>3</mn></msup><msub><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><msup><mi>d</mi><mn>3</mn></msup><msub><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub><msup><mi>d</mi><mn>3</mn></msup><msubsup><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow><mo>'</mo></msubsup><msup><mi>d</mi><mn>3</mn></msup><msubsup><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow><mo>'</mo></msubsup></mtd></mtr><mtr><mtd><mo>×</mo><msub><mi>ϕ</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo stretchy="false">)</mo><msub><mi>ϕ</mi><mi>B</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub><mo stretchy="false">)</mo><msubsup><mi>ϕ</mi><mi>A</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow><mo>'</mo></msubsup><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo stretchy="false">)</mo><msubsup><mi>ϕ</mi><mi>B</mi><mo>*</mo></msubsup><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow><mo>'</mo></msubsup><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><msup><mi>δ</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><msubsup><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow><mo>'</mo></msubsup><mo>+</mo><msubsup><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow><mo>'</mo></msubsup><mo>−</mo><msub><mi>p</mi><mrow><mi>c</mi><mn>,1</mn></mrow></msub><mo>−</mo><msub><mi>p</mi><mrow><mi>c</mi><mn>,2</mn></mrow></msub><mo stretchy="false">)</mo><msup><mi>δ</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><msub><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo>+</mo><msub><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub><mo>−</mo><msub><mi>p</mi><mrow><mi>c</mi><mn>,1</mn></mrow></msub><mo>−</mo><msub><mi>p</mi><mrow><mi>c</mi><mn>,2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msup><mi>d</mi><mn>2</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mi>c</mi></msub><msub><mi>f</mi><mi>A</mi></msub><mrow><mo>(</mo><mrow><msub><mi>X</mi><mi>c</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>y</mi><mrow><mi>c</mi><mn>,</mn><mi>T</mi></mrow></msub></mrow><mn>2</mn></mfrac><mn>,</mn><msub><mi>p</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub></mrow><mo>)</mo></mrow><msub><mi>f</mi><mi>B</mi></msub><mrow><mo>(</mo><mrow><msub><mi>X</mi><mi>c</mi></msub><mo>−</mo><mfrac><mrow><msub><mi>y</mi><mrow><mi>c</mi><mn>,</mn><mi>T</mi></mrow></msub></mrow><mn>2</mn></mfrac><mn>,</mn><msub><mi>p</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mi>exp</mi><mrow><mo>[</mo><mrow><mi>i</mi><mo stretchy="false">(</mo><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow><mo>'</mo></msubsup><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>k</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>b</mi></mstyle><mi>c</mi></msub></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>s</mi><mi>A</mi></msub><mn>,</mn><msub><mi>s</mi><mi>B</mi></msub><mn>,</mn><msub><mi>s</mi><mn>1</mn></msub><mn>,</mn><msub><mi>s</mi><mn>2</mn></msub></mrow></munder></mstyle><mn>2</mn><msub><mi>s</mi><mn>2</mn></msub><msub><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mi>c</mi></msub><mi mathvariant="script">M</mi><mrow><mo>(</mo><mrow><mo>{</mo><msub><mi>s</mi><mi>A</mi></msub><mn>,</mn><msub><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo>;</mo><msub><mi>s</mi><mi>B</mi></msub><mn>,</mn><msub><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub><mo>}</mo><mo>→</mo><mo>{</mo><msub><mi>s</mi><mn>1</mn></msub><mn>,</mn><msub><mi>p</mi><mrow><mi>c</mi><mn>,1</mn></mrow></msub><mo>;</mo><msub><mi>s</mi><mn>2</mn></msub><mn>,</mn><msub><mi>p</mi><mrow><mi>c</mi><mn>,2</mn></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><msup><mi mathvariant="script">M</mi><mo>*</mo></msup><mrow><mo>(</mo><mrow><mo>{</mo><msub><mi>s</mi><mi>A</mi></msub><mn>,</mn><msubsup><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow><mo>'</mo></msubsup><mo>;</mo><msub><mi>s</mi><mi>B</mi></msub><mn>,</mn><msubsup><mi>k</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow><mo>'</mo></msubsup><mo>}</mo><mo>→</mo><mo>{</mo><msub><mi>s</mi><mn>1</mn></msub><mn>,</mn><msub><mi>p</mi><mrow><mi>c</mi><mn>,1</mn></mrow></msub><mo>;</mo><msub><mi>s</mi><mn>2</mn></msub><mn>,</mn><msub><mi>p</mi><mrow><mi>c</mi><mn>,2</mn></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> where P AB →12 denotes the polarization vector of particle 2; ϕ A and ϕ B are the wave packets for A and B , respectively; <math id="M143"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle><mi>c</mi></msub><mo>=</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mover accent="true"><mi>b</mi><mo>^</mo></mover></mstyle><mi>c</mi></msub><mo>×</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mover accent="true"><mi>p</mi><mo>^</mo></mover></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub></mrow></math> is the direction of the reaction plane in the center-of-mass frame, with <math id="M144"><mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mover accent="true"><mi>b</mi><mo>^</mo></mover></mstyle><mi>c</mi></msub></mrow></math> being the unit impact parameter vector; f A and f B are the distributions at the coordinates X c + y c , T /2 and X c - y c , 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. [ 171 171 ]. The wave packets ensure that the colliding particles have a non-vanishing 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 <math display="block" id="M145"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msup><mi>d</mi><mn>4</mn></msup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>P</mi></mstyle><mi>q</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msup><mi>X</mi><mn>4</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mi>π</mi><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mo stretchy="false">(</mo><mi>β</mi><msub><mi>u</mi><mi>ρ</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mi>X</mi><mi>ν</mi></msup></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>A</mi><mn>,</mn><mi>B</mi><mn>,1</mn></mrow></munder></mstyle><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><msub><mi>p</mi><mi>A</mi></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mn>2</mn><msub><mi>E</mi><mi>A</mi></msub></mrow></mfrac><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><msub><mi>p</mi><mi>B</mi></msub></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup><mn>2</mn><msub><mi>E</mi><mi>B</mi></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mo>×</mo><mo>|</mo><msub><mi>v</mi><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo>−</mo><msub><mi>v</mi><mrow><mi>c</mi><mn>,</mn><mi>B</mi></mrow></msub><mo>|</mo><mo stretchy="false">[</mo><msup><mi>Λ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mo stretchy="false">]</mo><mrow><mtext> </mtext><mi>j</mi></mrow><mi>ν</mi></msubsup><msub><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>i</mi></mrow></msub><msub><mi>ϵ</mi><mrow><mi>i</mi><mi>k</mi><mi>h</mi></mrow></msub><msubsup><mstyle mathvariant="bold" mathsize="normal"><mover accent="true"><mi>p</mi><mo>^</mo></mover></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow><mi>h</mi></msubsup></mtd></mtr><mtr><mtd><mo>×</mo><msub><mi>f</mi><mi>A</mi></msub><mrow><mo>(</mo><mrow><mi>X</mi><mn>,</mn><msub><mi>p</mi><mi>A</mi></msub></mrow><mo>)</mo></mrow><msub><mi>f</mi><mi>B</mi></msub><mrow><mo>(</mo><mrow><mi>X</mi><mn>,</mn><msub><mi>p</mi><mi>B</mi></msub></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msubsup><mi>p</mi><mi>A</mi><mi>ρ</mi></msubsup><mo>−</mo><msubsup><mi>p</mi><mi>B</mi><mi>ρ</mi></msubsup></mrow><mo>)</mo></mrow><msub><mi>Θ</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mrow><mi>c</mi><mn>,</mn><mi>A</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>≡</mo><mfrac><mrow><mo>∂</mo><mo stretchy="false">(</mo><mi>β</mi><msub><mi>u</mi><mi>ρ</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mi>X</mi><mi>ν</mi></msup></mrow></mfrac><msup><mstyle mathvariant="bold" mathsize="normal"><mi>W</mi></mstyle><mrow><mi>ρ</mi><mi>ν</mi></mrow></msup><mn>,</mn></mtd></mtr></mtable></math> which is proportional to the thermal vorticity. Here the tensor W ρν contains 64 components, each of which involves a 16-dimensional integration. The numerical calculation of W ρν is 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 [ 172 172 , 173 173 ]. The most recent application of this algorithm is the solution of the Boltzmann equations of a partonic system [ 174 174 ]. The numerical result shows that W ρν has an antisymmetric form: <math display="block" id="M146"><mrow><msup><mstyle mathvariant="bold" mathsize="normal"><mi>W</mi></mstyle><mrow><mi>ρ</mi><mi>ν</mi></mrow></msup><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mi>W</mi><msub><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi></mstyle><mi>z</mi></msub></mrow></mtd><mtd><mrow><mo>−</mo><mi>W</mi><msub><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi></mstyle><mi>y</mi></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><mo>−</mo><mi>W</mi><msub><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi></mstyle><mi>z</mi></msub></mrow></mtd><mtd><mn>0</mn></mtd><mtd><mrow><mi>W</mi><msub><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi></mstyle><mi>x</mi></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow><mi>W</mi><msub><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi></mstyle><mi>y</mi></msub></mrow></mtd><mtd><mrow><mo>−</mo><mi>W</mi><msub><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi></mstyle><mi>x</mi></msub></mrow></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where W is approximately constant. Note that W depends on the cutoff of the impact parameter, as shown in Fig. 8 Fig. 8 . Finally, the polarization rate per unit volume for one quark flavor can be expressed compactly: <math display="block" id="M147"><mrow><mfrac><mrow><msup><mi>d</mi><mn>4</mn></msup><msub><mi>P</mi><mi>q</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msup><mi>X</mi><mn>4</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn><mi>W</mi><msub><mo>∇</mo><mi>X</mi></msub><mo>×</mo><mrow><mo>(</mo><mrow><mfrac><mi>u</mi><mi>T</mi></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> This is a good example of how spin–vorticity coupling emerges naturally from particle scattering. Note that the result of Ref. [ 171 171 ] 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 [ 175 175 ] by expanding the collision terms in the Planck constant for massive fermions [ 121 121 ]. 4.2 Experimental results 2 The global polarization of Λ hyperons can be measured through their weak decays. The angular distribution of the daughter baryons is [ 5 5 , 111 111 ] <math display="block" id="M148"><mrow><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><mi>cos</mi><msup><mi>θ</mi><mo>*</mo></msup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>α</mi><mi>H</mi></msub><msub><mi>P</mi><mi>H</mi></msub><mi>cos</mi><msup><mi>θ</mi><mo>*</mo></msup></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where α H is the decay constant of the hyperon, P H is 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 and <math id="M149"><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></math> hyperons measured by the STAR Collaboration [ 5 5 , 111 111 ] are shown in the left panel of Fig. 9 Fig. 9 . P H (Λ) and <math id="M150"><mrow><msub><mi>P</mi><mi>H</mi></msub><mo stretchy="false">(</mo><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow></math> decrease with the collision energy. From the data, the fluid vorticity can be estimated as <math id="M151"><mrow><mi>ω</mi><mo>≃</mo><msub><mi>k</mi><mi>B</mi></msub><mi>T</mi><mo stretchy="false">(</mo><msub><mi>P</mi><mi>Λ</mi></msub><mo>+</mo><msub><mi>P</mi><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></msub><mo stretchy="false">)</mo><mo>/</mo><mi>ℏ</mi></mrow></math> [ 176 176 ]. It also appears that <math id="M152"><mrow><msub><mi>P</mi><mi>H</mi></msub><mo stretchy="false">(</mo><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>≳</mo><msub><mi>P</mi><mi>H</mi></msub><mo stretchy="false">(</mo><mi>Λ</mi><mo stretchy="false">)</mo></mrow></math> , although the difference is not significant; this result may suggest a possible contribution from the magnetic field or other effects [ 177 177 , 178 178 ]. The data can be described by various phenomenological models using (44) [ 105 105 , 142 142 , 143 143 , 179 179 - 181 181 179-181 ]. The right panel of Fig. 9 Fig. 9 shows the recent STAR Collaboration measurement [ 21 21 ] of the longitudinal polarization as a function of the azimuthal angle ϕ relative to the second-order event plane Ψ 2 . The longitudinal spin polarization data show positive sin(2 ϕ -2Ψ 2 ) behavior, whereas the theoretical results of the relativistic hydrodynamics model [ 182 182 ] and transport models [ 143 143 , 180 180 , 183 183 ] show the opposite sign along the beam line direction. A simulation using CKT [ 110 110 ] and the results obtained using a simple phenomenological model [ 184 184 ] 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 Sec. 3) or the spin hydrodynamics (see Sec. 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. [ 185 185 , 186 186 ] concluded that feed-down effects decreased by approximately 10for 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. [ 148 148 , 149 149 ] tested four different types of vorticity: the kinematic vorticity, relativistic extension of the non-relativistic 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 33 Hermitian spin-density matrix [ 187 187 , 188 188 ]. The 00-component of the spin-density matrix enters the angular distribution of its decay daughter as <math display="block" id="M154"><mrow><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><mi>cos</mi><msub><msup><mi>θ</mi><mo>*</mo></msup><mrow/></msub></mrow></mfrac><mo>∝</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>ρ</mi><mrow><mn>00</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><msub><mi>ρ</mi><mrow><mn>00</mn></mrow></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>cos</mi></mrow></mstyle><mn>2</mn></msup><msup><mi>θ</mi><mo>*</mo></msup><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where θ * is the angle between the decay daughter and the spin quantization direction in the vector meson’s rest frame. Thus, the ρ 00 value of the vector meson can be measured using the angular distribution of its decay daughter. The vector meson ( K ∗0 and ϕ ) spin alignments have been measured by the ALICE Collaboration [ 189 189 ]. The ρ 00 value is consistent with 1/3 for both K *0 and ϕ mesons in p+p collisions. In Pb+Pb collisions, the ρ 00 value of K *0 is approximately 1/3 at high p T and less than 1/3 at low p T . Theoretical calculations using the statistical-hydro model [ 176 176 ] and quark coalescence model [ 190 190 ] give <math display="block" id="M155"><mrow><msubsup><mi>ρ</mi><mrow><mn>00</mn></mrow><mi>ϕ</mi></msubsup><mo>≈</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>−</mo><mfrac><mn>4</mn><mn>9</mn></mfrac><msub><mi>P</mi><mi>Λ</mi></msub><msub><mi>P</mi><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></msub><mo>≲</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mn>.</mn></mrow></math> Using the STAR data, P Λ ≈(1.08±0.15±0.11)% and <math id="M156"><mrow><msub><mi>P</mi><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></msub><mo>≈</mo><mo stretchy="false">(</mo><mn>1.38</mn><mo>±</mo><mn>0.30</mn><mo>±</mo><mn>0.13</mn><mo stretchy="false">)</mo><mi>%</mi></mrow></math> [ 5 5 , 111 111 ], one can estimate <math id="M157"><mrow><mo stretchy="false">(</mo><mn>4</mn><mo>/</mo><mn>9</mn><mo stretchy="false">)</mo><msub><mi>P</mi><mi>Λ</mi></msub><msub><mi>P</mi><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></msub><mo>≈</mo><mn>6.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup></mrow></math> , which may suggest that <math id="M158"><mrow><msubsup><mi>ρ</mi><mrow><mn>00</mn></mrow><mi>ϕ</mi></msubsup></mrow></math> cannot be significantly larger than 1/3 and thus contradict the STAR preliminary data for <math id="M159"><mrow><msubsup><mi>ρ</mi><mrow><mn>00</mn></mrow><mi>ϕ</mi></msubsup></mrow></math> . To solve this puzzle, it is proposed that a strangeness current can exist in heavy-ion collisions and give rise to a non-vanishing mean ϕ field [ 191 191 ]. Like the magnetic field, the magnetic part of the ϕ field can also polarize s and <math id="M160"><mover accent="true"><mi>s</mi><mo>¯</mo></mover></math> through their magnetic moments; this polarization contributes to the polarization of Λ and <math id="M161"><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></math> , whereas the contribution from the electric part of the ϕ field is absent and therefore is not constrained by the polarization of Λ and <math id="M162"><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></math> . However, the electric part of the ϕ field contributes significantly to <math id="M163"><mrow><msubsup><mi>ρ</mi><mrow><mn>00</mn></mrow><mi>ϕ</mi></msubsup></mrow></math> , which is positive definite [ 191 191 ]: <math display="block" id="M164"><mrow><msubsup><mi>ρ</mi><mrow><mn>00</mn></mrow><mi>ϕ</mi></msubsup><mo>≈</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mfrac><mrow><msubsup><mi>g</mi><mi>ϕ</mi><mn>2</mn></msubsup></mrow><mrow><mn>27</mn><msubsup><mi>m</mi><mtext>s</mtext><mn>4</mn></msubsup><msubsup><mi>T</mi><mrow><mtext>eff</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><msub><mrow><mrow><mo>〈</mo><mrow><msup><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mn>2</mn></msup></mrow><mo>〉</mo></mrow></mrow><mi>ϕ</mi></msub><mrow><mo>〈</mo><mrow><msubsup><mi>E</mi><mrow><mi>ϕ</mi><mn>,</mn><mi>z</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>E</mi><mrow><mi>ϕ</mi><mn>,</mn><mi>x</mi></mrow><mn>2</mn></msubsup></mrow><mo>〉</mo></mrow><mn>,</mn></mrow></math> where E ϕ is the electric part of the ϕ field, m s is the strange quark mass, 〈 p 2 〉 ϕ is the mean value of p 2 for s or <math id="M165"><mover accent="true"><mi>s</mi><mo>¯</mo></mover></math> in the ϕ meson wave function, g ϕ is the coupling constant of the s quark to the ϕ meson in the quark–meson model, and T eff is the average temperature of the fireball. The contribution originates from the spin–orbit coupling term that polarizes s and <math id="M166"><mover accent="true"><mi>s</mi><mo>¯</mo></mover></math> . Thus, one can see that ρ 00 for the ϕ meson is a good tool for analyzing the ϕ mean field even if it may strongly fluctuate in space-time [ 191 191 ]. The theoretical prediction of <math id="M167"><mrow><msubsup><mi>ρ</mi><mrow><mn>00</mn></mrow><mi>ϕ</mi></msubsup></mrow></math> as a function of the collision energy is shown in Fig. 10 Fig. 10 . However, this theory does not work for another vector meson, K *0 , for several reasons. First, because of the unequal masses of <math id="M170"><mover accent="true"><mi>s</mi><mo>¯</mo></mover></math> and 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 *0 with the surrounding matter is much stronger than that of the ϕ meson. The above points are supported by preliminary data from ALICE [ 189 189 ]. 5 Summary 1 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 non-flow correlations for the CMW give rise to additional backgrounds in the slope of Δ v 2 ( A ch ). 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 and <math id="M171"><mover accent="true"><mi>Λ</mi><mo>¯</mo></mover></math> hyperons, 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 <math id="M172"><mrow><msubsup><mi>ρ</mi><mrow><mn>00</mn></mrow><mi>ϕ</mi></msubsup></mrow></math> may indicate the existence of a mean ϕ field. References J. Bloczynski , X.-G. Huang , X. 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What can we learn from global spin alignment of ϕ meson in heavy-ion collisions?, in preparation 10.1007/s41365-020-00801-x This work was supported in part by the National Natural Science Foundation of China (No. 11890713, 11890714, 11835002, 11961131011, 11421505, 11535012 and 11890713), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Nos. XDB34030202 and XDB34030102). 2020-05-14 2020-07-13 2020-07-13 2020-08-30 2020-09-01 2021-04-10 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2020 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-020-00801-x http://dx.doi.org/10.1007/s41365-020-00801-x . 2020 Nuclear Science and Techniques 9 31