Global polarization effect and spin-orbit coupling in strong interaction
GGlobal polarization effect and spin-orbitcoupling in strong interaction
Jian-Hua Gao, Zuo-Tang Liang, Qun Wang and Xin-Nian Wang
Abstract
In non-central high energy heavy ion collisions the colliding system possesa huge orbital angular momentum in the direction opposite to the normal of thereaction plane. Due to the spin-orbit coupling in strong interaction, such huge orbitalangular momentum leads to the polarization of quarks and anti-quarks in the samedirection. This effect, known as the global polarization effect, has been recentlyobserved by STAR Collaboration at RHIC that confirms the theoretical predictionmade more than ten years ago. The discovery has attracted much attention on thestudy of spin effects in heavy ion collision. It opens a new window to study propertiesof QGP and a new direction in high energy heavy ion physics — Spin Physics inHeavy Ion Collisions. In this chapter, we review the original ideas and calculationsthat lead to the predictions. We emphasize the role played by spin-orbit coupling inhigh energy spin physics and discuss the new opportunities and challenges in thisconnection.
Jian-Hua GaoShandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment,Institute of Space Sciences, Shandong University, Weihai, Shandong 264209, China, e-mail: [email protected]
Zuo-tang LiangKey Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of Frontier andInterdisciplinary Science, Shandong University, Qingdao, Shandong 266237, China, e-mail: [email protected]
Qun WangDepartment of Modern Physics, University of Science and Technology of China, Hefei, Anhui230026, China, e-mail: [email protected]
Xin-Nian WangKey Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, CentralChina Normal University, Wuhan, 430079, China; and Nuclear Science Division, MS 70R0319,Lawrence Berkeley National Laboratory, Berkeley, California 94720, e-mail: [email protected] a r X i v : . [ nu c l - t h ] S e p Jian-Hua Gao, Zuo-Tang Liang, Qun Wang and Xin-Nian Wang
Recently, the global polarization effect (GPE) of Λ and ¯ Λ hyperons in heavy-ioncollisions (HIC) has been observed [1] by the STAR Collaboration at the RelativisticHeavy Ion Collider (RHIC) in Brookhaven National Laboratory (BNL). The dis-covery confirms the theoretical prediction [2] made more than ten years ago andhas attracted much attention on the study of spin effects in HIC. This opens a newwindow to study properties of QGP and a new direction in high energy heavy ionphysics — Spin Physics in HIC. New experiments along this line are being carriedout and/or planned. It is therefore timely to summarize the original ideas and theoret-ical calculations [2, 3, 4] that lead to the predictions and discuss new opportunitiesand challenges.Spin, as a fundamental degree of freedom of elementary particles, plays a veryimportant role in modern physics and often brings us surprises. There are many wellknown examples in the field of particle and nuclear physics. The anomalous magneticmoments of nucleons are usually regarded as one of the first clear signatures for theexistence of inner structure of nucleon. The explanation of these anomalous magneticmoments in 1960s was one of the great successes of the quark model that lead us tobelieve that it provides us the correct picture for hadron structure.High energy spin physics experiments started since 1970s. Soon after the be-ginning, a series of striking spin effects have been observed that were in strongcontradiction to the theoretical expectations at that time and been pushing the stud-ies move forward. The most famous ones might be classified as following.(i) Proton’s “spin crisis” : Measurements of spin dependent structure functions indeeply inelastic lepton-nucleon scatterings, started by E80 and E143 Collaborationsat SLAC [5, 6] and later on by the European Muon Collaboration (EMC) at CERN [7,8], seem to suggest that the contribution of the sum of spins of quarks and anti-quarksto proton spin is consistent with zero. This has triggered the so-called spin crisis ofthe proton and the intensive study on the spin structure of nucleon [9].(ii) Single spin left-right asymmetry (SSA): It has been observed [10, 11, 12, 13]that in inclusive hadron-hadron collisions with singly transversely polarized beamsor targets, the produced hadron has a large azimuthal angle dependence characterizedby the left-right asymmetry. The observed asymmetry can be as large as 40% but thetheoretical expectation at the quark level using pQCD at the leading order was closeto zero.(iii) Transverse hyperon polarization: It has been observed [14, 15, 16, 17, 18] thathyperons produced in unpolarized hadron-hadron and hadron-nucleus collisions aretransversely polarized with respect to the production plane. The observed polariza-tion can reach a magnitude as high as 40% but the leading order pQCD expectationwas again close to zero.(iv) Spin asymmetries in elastic pp -scattering: It has been observed [19, 20, 21, 18]that the azimuthal dependence, called the spin analyzing power, in scattering withsingle-transversely polarized proton and doubly polarized asymmetries are verysignificant, much larger than theoretical expectations available at that time. lobal polarization effect and spin-orbit coupling in strong interaction 3 Such striking spin effects came out often as such a shock to the field of stronginteraction physics that lead to the famous comment by Bjorken [22] in a QCDworkshop that “Polarization phenomena are often graveyards of fashionable theories....”. In last decades, the study on such spin effects lead to one of the most active fieldsin strong interaction or QCD physics.At the same time, high energy HIC physics has become the other active field instrong interaction physics in particular after the quark gluon plasma (QGP) has beendiscovered at RHIC [23, 24]. The study on properties of QGP in HIC is the core ofhigh energy HIC physics currently.We recall that RHIC is not only the first relativistic heavy ion collider in the worldbut also the first polarized high energy proton-proton collider. It is therefore naturalto ask whether we can do spin physics in HIC.Spin physics in HIC was however used to be regarded as difficult or impossiblebecause the polarization of the nucleon in a heavy nucleus is very small even if thenucleus is completely polarized. The breakthrough came out in 2005 when it wasrealized that [2] there is however a great advantage to study spin and/or angularmomentum effects in HIC, i.e., the reaction plane in a HIC can be determinedexperimentally by measuring flows and/or spectator nucleons and there exist a hugeorbital angular momentum for the participating system in a non-central HIC withrespect to the reaction plane! It provides a unique place in high energy reactions tostudy the mutual exchange of orbital angular momentum and the spin polarization.The discovery of GPE leads to an active field of Spin Physics in HIC [25].In this chapter, we review the original ideas and calculations [2] that lead to theprediction of GPE in HIC. We present a rough comparison to data available and anoutlook for future studies. The rest of the chapter is arranged as follows: In Sec. 2,we present the orbital angular momentum of the colliding system in non-central HICand the resulting gradient in momentum or rapidity distribution. In Sec. 3, we recallthe origin of spin-orbit coupling and famous example in electromagnetic and stronginteraction systems. In Sec. 4, we present calculations at the quark level and resultsfor the global quark polarization in HIC. In Sec. 5, we discuss the global hadronpolarization and finally a short summary and outlook is presented in Sec. 6.
We consider two colliding nuclei with the projectile of beam momentum per nucleon p in . For a non-central collision, there is a transverse separation between the centersof the two colliding nuclei. The impact parameter b is defined as the transversevector pointing from the target to the projectile. The reaction plane of a HIC isusually defined by b and p in and is illustrated in Fig. 1. The overlap parts, hereafterreferred as the colliding system, interact with each other and form the system denote Jian-Hua Gao, Zuo-Tang Liang, Qun Wang and Xin-Nian Wang by the red core in the middle while the other parts, denoted by the blues parts in thefigure, are just spectators and move apart in the original directions.
Fig. 1
Illustration diagram for the reaction plane in a non-central heavy ion collision. In contrastto high energy pp or e + e − collisions, the reaction plane in a high energy heavy collision can bedetermined experimentally. The geometry and the coordinate system are further specified in Fig. 2. The beamdirection of the colliding nuclei is taken as the z axis, as illustrated in the upper-leftpanel in the figure. The transverse separation is called the impact parameter b definedas the transverse distance of the projectile from the target nucleus and is taken as inthe x -direction. The normal of the reaction plane is given by, n ≡ p in × b /| p in × b | , (1)and is taken as the y -direction, where p in is the momentum per nucleon in theincident nucleus A .Usually in a high energy reaction such as a hadron-hadron, or lepton-hadron or e + e − annihilation, the size of the reaction region is typically less than 1fm. Thereaction plane in such collisions can be defined theoretically but can not be deter-mined experimentally. However, in a HIC, the reaction region is usually much largerand colliding parts give rise to a quark matter system with very high temperatureand high density and expand violently while the spectators just leave the region inthe original directions. Since the colliding system is not isotropic, the pressures indifferent directions are also different in different directions thus lead to a system thatexpands non-isotropically. In the transverse directions they behave like an ellipse asillustrated in the lower-right panel in Fig. 2. Such a non-isotropy is described by theelliptic flow v and the directed flow v that can be measured experimentally (seee.g. [26, 27]). Clearly, by measuring v , one can determine the reaction plane andfurther determine the direction of the plane by measuring the directed flow v .In experiments, the reaction plane in a HIC can not only be determined bymeasuring v and v but also determined by measuring the sidewards deflection lobal polarization effect and spin-orbit coupling in strong interaction 5 b A zy bz yz yx xx x R Fig. 2
Illustration of the geometry and coordinate system for the non-central HIC with impactparameter b . The global angular momentum of the produced matter is along the minus y direction,opposite to the reaction plane. This figure is taken from [2]. of the forward- and backward-going fragments and particles in the beamâĂŞbeamcounter detectors [1]. This is quite unique in different high energy reactions. Just as illustrated in Figs. 1 and 2, in a non-central HIC, there is a transverse separationbetween the overlapping parts of the two colliding nuclei in the same direction as theimpact parameter b . Hence the whole system that takes part in the reaction, i.e. thecolliding system carries a finite orbital angular momentum L y along the directionorthogonal to the reaction plane. We call L y the global orbital angular momentum.The magnitude of this global orbital angular momentum L y can be calculated by, L y = − p in ∫ x dx (cid:32) dN P part dx − dN T part dx (cid:33) , (2)where dN P , T part / dx is the transverse distributions (integrated over y and z ) of participantnucleons in each nucleus A along the x -direction, the superscript P or T denotesprojectile or target respectively. These transverse distributions are given by, dN P , T part dx = ∫ d y dz ρ P , TA ( x , y , z , b ) , (3)where ρ P , TA ( x , y , z , b ) is the number density of participant nucleons in nucleus A inthe coordinate system defined in Fig. 2. Jian-Hua Gao, Zuo-Tang Liang, Qun Wang and Xin-Nian Wang
The number density ρ P , TA ( x , y , z , b ) of participant nucleons in nucleus A can easilybe calculated if we take a hard sphere distribution of nucleons in the nucleus A . Inthis model, the overlapping area has a clear boundary and the participant nucleondensity is given by the overlapping area of two hard spheres, as illustrated in theupper-right panel of Fig. 2, i.e., ρ P , TA , HS ( x , y , z , b ) = f P , TA , HS ( x , y , z , b ) θ (cid:18) R A − (cid:113) ( x ± b / ) + y + z (cid:19) , (4)where f P , TA , HS ( x , y , z , b ) is the hard sphere nuclear distribution in A that is given by, f P , TA , HS ( x , y , z , b ) = A π R A θ (cid:18) R A − (cid:113) ( x ∓ b / ) + y + z (cid:19) , (5)where R A = . A / fm is the nuclear radius and A the atomic number.If we take the Woods-Saxon nuclear distribution, i.e., f P , TA , WS ( x , y , z , b ) = C (cid:32) + exp (cid:112) ( x ∓ b / ) + y + z − R A a (cid:33) − , (6)there is no clear boundary of the overlapping region and the participant nucleonnumber density is calculated using the Glauber model and is given by, ρ P , TA , WS ( x , y , z , b ) = f P , TWS ( x , y , z , b ) (cid:26) − exp (cid:104) − σ N N ∫ dz f T , PWS ( x , y , z , b ) (cid:105) (cid:27) , (7)where σ N N is the total cross section of nucleon-nucleon scatterings, C is thenormalization constant, C = A / π ∫ r dr (cid:16) + e ( r − R A )/ a (cid:17) − , (8)and a is the width parameter set to a = .
54 fm.The calculations have been carried out in [2] and [4]. The obtained results areshown in Fig. 3. From the results shown in Fig. 3, we see that though there aresignificant differences between two nuclear geometry models the global orbitalangular momentum L y of the overlapped parts of two colliding nuclei is huge and isof the order of 10 at most impact parameters. How the global orbital angular momentum discussed above is transferred to the finalstate particles depends on the equation of state (EOS) of the dense matter. At low lobal polarization effect and spin-orbit coupling in strong interaction 7
Fig. 3
Global orbital angular momentum of the colliding system in the non-central HIC as afunction of the impact parameter obtained from the Woods-Saxon and hard-sphere distributions,respectively. This figure is taken from [4]. energies, the final state is expected to be the normal nuclear matter with an EOS ofrigid nuclei. In such cases, a rotating compound nucleus can be formed when thecolliding energy is comparable or smaller than the nuclear binding energy. The finitevalue of the global orbital angular momentum of the non-central collision at suchlow energies provides a useful tool for the study of the properties of super-deformednuclei under such rotation [28].At high colliding energies such as those at RHIC, the dense matter is expected tobe partonic with an EOS of QGP. Given such a soft EOS, the global orbital angularmomentum would probably not lead to the global rotation of the dense matter system.Instead, the global angular momentum could be distributed across the overlappedregion of nuclear scattering and is manifested in the shear of the longitudinal flowleading to a finite value of local vorticity density. Under such longitudinal fluid shear,a pair of scattering partons will on average carry a finite value of relative orbitalangular momentum that will be referred to as the local orbital angular momentumin the opposite direction to the reaction plane as defined in Eq. (1).By momentum conservation, the average initial collective longitudinal momen-tum at any given transverse position can be calculated as the total momentum differ-ence between participating projectile and target nucleons. Since the total multiplicityin HIC is proportional to the number of participant nucleons [29], we can make thesame assumption for the produced partons with a proportionality constant fixed at agiven center of mass energy √ s . How the global angular momentum is distributedto the longitudinal flow shear and the magnitude of the local relative orbital angu-lar momentum depends on the parton production mechanism and their longitudinalmomentum distributions. We consider two different scenarios: the Landau fireballand the Bjorken scaling model. Jian-Hua Gao, Zuo-Tang Liang, Qun Wang and Xin-Nian Wang
In the Landau fireball model, we assume that the produced partons thermalize quicklyand have a common longitudinal flow velocity at a given transverse position in theoverlapped region. The average collective longitudinal momentum per parton can bewritten as p z ( x , b , √ s ) = p R N ( x , b , √ s ) , (9)where p = √ s / c ( s ) is an energy dependent constant, √ s is the center of massenergy of a colliding nucleon pair, c ( s ) is the average number of partons producedper participating nucleon; and R N ( x , b , √ s ) is the ratio defined as, R N ( x , b , √ s ) = (cid:32) dN P part dx − dN T part dx (cid:33) (cid:46) (cid:32) dN P part dx + dN T part dx (cid:33) (10)It is clear that in the symmetric AA collision (where the beam and target nucleiare the same), the ratio R N ( x , b , √ s ) thus the distribution p z ( x , b , √ s ) is an oddfunction in both x and b and therefore vanishes at x = b =
0. In Fig. 4, p z ( x , b , √ s ) is plotted as a function of x at different impact parameters b . We seeclearly that p z ( x , b , √ s ) is a monotonically increasing function of x until the edge ofthe overlapped region | x ± b / | = R A beyond which it drops to zero (gradually forWoods-Saxon geometry).From p z ( x , b , √ s ) one can compute the transverse gradient of the average longi-tudinal collective momentum per parton dp z / dx which is an even function of x andvanishes at b =
0. One can then estimate the longitudinal momentum difference ∆ p z between two neighboring partons in QGP. On average, the relative orbital angularmomentum for two colliding partons separated by ∆ x in the transverse direction is l y ≡ −( ∆ x ) dp z dx . (11)With the hard sphere nuclear distribution, l y is proportional to dp dx ≡ p R A = √ s c ( s ) R A . (12)This provides a measure of order of magnitude of dp z / dx . In Au + Au collisionsat √ s =
200 GeV, the number of charged hadrons per participating nucleon is about15 [29]. Assuming the number of partons per (meson dominated) hadron is about 2,we have c ( s ) (cid:39)
45 (including neutral hadrons). Given R A = . dp / dx (cid:39) . l ≡ −( ∆ x ) dp / dx (cid:39) − . ∆ x = l y given by Eq. (11)for two neighboring partons separated by ∆ x = x for differentimpact parameter b for both Woods-Saxon and hard-sphere nuclear distributions.We see that l y is in general of the order of 1 and is comparable or larger than the spin lobal polarization effect and spin-orbit coupling in strong interaction 9 Fig. 4
The average longitudinal momentum distribution p z ( x , b , √ s ) in unit of p = √ s /[ c ( s )] as a function of x /( R A − b / ) for different values of b / R A with the hard sphere (upper panel) andWoods-Saxon (lower panel) nuclear distributions. This figure is taken from [4]. of a quark. It is expected that c ( s ) should depend logarithmically on the collidingenergy √ s , therefore l y should increases with growing √ s . In a three dimensional expanding system, there could be strong correlation betweenlongitudinal flow velocity and spatial coordinate of the fluid cell. The most simplifiedpicture is the Bjorken scaling scenario [30] in which the longitudinal flow velocityis identical to the spatial velocity η = log [( t + z )/( t − z )] . With such correlation, thelocal interaction and thermalization require that a parton only interacts with otherpartons in the same region of longitudinal momentum or rapidity Y . The width ofsuch region in rapidity is determined by the half-width of the thermal distribution f th ( Y , p T ) = exp [− p T cosh ( Y − η )/ T ] [31], which is approximately ∆ Y ≈ . (cid:104) p T (cid:105) ≈ T and T is the local temperature). The relevant measure of the localrelative orbital angular momentum between two interacting partons is, therefore, thedifference in parton rapidity distributions at transverse distance of the order of theaverage interaction range. Fig. 5
The average orbital angular momentum l y ≡ −( ∆ x ) dp z / dx of two neighboring partonsseparated by ∆ x = x /( R A − b / ) for differentvalues of the impact parameter b / R A with the hard-sphere (upper panel) and Woods-Saxon (lowerpanel) nuclear distributions. This figure is taken from [4]. The variation of the rapidity distributions with respect to the transverse coordinatecan be described by the normalized rapidity distribution f p ( Y , x ) at given x , f p ( Y , x , b , √ s ) = d NdxdY (cid:46) dNdx , (13)where d N / dxdY denotes the number density of particles produced with respect to x and Y and dN / dx ≡ ∫ dY d N / dxdY is the distribution of particles with respectto x . At a given x, the overall average value of the rapidity is given by, (cid:104) Y ( x , b , √ s )(cid:105) = ∫ Y dY f p ( Y , x , b , √ s ) . (14) (cid:104) Y ( x , b , √ s )(cid:105) just corresponds to p z ( x , b , √ s ) given by Eq. (9) discussed in theLandau fireball model. It measures the overall behavior of the rapidity distributionof partons at given transverse coordinate x . To further quantify such longitudinalfluid shear, one can calculate the average rapidity within an interval ∆ Y at a givenrapidity Y , i.e., lobal polarization effect and spin-orbit coupling in strong interaction 11 (cid:104) Y l ( Y , x , b , √ s )(cid:105) ≈ Y + ∆ Y
12 1 f p ∂ f p ∂ Y = Y + ∆ Y ∂ ln f p ∂ Y . (15)Here, we use the subscript l to denote that this is the average of Y in a localizedinterval [ Y − ∆ Y / , Y + ∆ Y / ] to differentiate it from the overall average (cid:104) Y ( x , b , √ s )(cid:105) given by Eq. (14). The average rapidity shear or the difference in average rapidityfor two partons separated by a unit of transverse distance ∆ x is then given by, ∂∂ x (cid:104) Y l ( Y , x , b , √ s )(cid:105) ≈ ∆ Y ∂ ln f p ∂ Y ∂ x . (16)The averaged longitudinal momentum is, (cid:104) p z (cid:105) ≈ p T sinh (cid:104) Y l (cid:105) ≈ p T (cid:32) sinh Y + cosh Y ∆ Y ∂ ln f p ∂ Y (cid:33) . (17)The corresponding local relative longitudinal momentum shear is given by, ∂ (cid:104) p z (cid:105) ∂ x ≈ p T cosh Y ∂ (cid:104) Y l (cid:105) ∂ x ≈ p T cosh Y ∆ Y ∂ ln f p ∂ Y ∂ x . (18)The corresponding local orbital angular momentum l y for two partons separatedby a transverse separation ∆ x at a given rapidity Y is (cid:104) l y ( Y )(cid:105) = − ∆ x ∆ (cid:104) p z (cid:105) = −( ∆ x ) ∂ (cid:104) p z (cid:105)/ ∂ x . We transform it into the co-moving frame or the center of massframe of the two partons and obtain, (cid:104) l ∗ y ( Y , x , b , √ s )(cid:105) = − ∆ x (cid:104) p ∗ z (cid:105) ≈ −( ∆ x ) p T ∆ Y ∂ ln f p ∂ Y ∂ x . (19)We see that they are all determined by a key quantity ξ p ( Y , x , b , √ s ) ≡ ∂ ln f p ( Y , x , b , √ s ) ∂ Y ∂ x , (20)that is determined by d N / dxdY . In terms of ξ p ( Y , x , b , √ s ) , we have, ∂ (cid:104) Y l (cid:105) ∂ x ≈ ∆ Y ξ p , (21) ∂ (cid:104) p z (cid:105) ∂ x ≈ ∆ Y ξ p p T cosh Y , (22) (cid:104) l ∗ y ( Y , x , b , √ s )(cid:105) ≈ − ∆ Y ξ p ( ∆ x ) p T . (23)The Y -dependence averaged over the transverse separation x is determined by theaverage value of ξ p ( Y , x , b , √ s ) defined by, (cid:104) ξ p (cid:105) = ∫ dx ξ p ( Y , x , b , √ s ) d NdxdY (cid:46) dNdY , (24)where dN / dY = ∫ dx ( d N / dxdY ) is the rapidity distribution of partons producedin a AA collision at the given impact parameter b . In the binary approximation, dNdY = N part dN pp dY . (25)To proceed with numerical calculations, one needs a dynamical model to estimatethe local rapidity distribution d N / dxdY of produced partons. For this purpose,two models, the HIJING Monte-Carlo model [32, 33] and the model proposed byBrodsky, Gunion and Kuhn (denoted as BGK model) [34], have been used [4, 35].We present the results [4, 35] obtained in the following respectively.(i) Results obtained using HIJING
In [4], the HIJING Monte Carlo model [32, 33] was used to calculate the hadronrapidity distributions at different transverse coordinate x and assume that partondistributions of the dense matter are proportional to the final hadron spectra. Weshow the results obtained in this way in [4] in the following.Shown in Fig. 6 is the average rapidity of particles in final state as a functionof the transverse coordinate x for different values of the impact parameter b . Wesee that, besides the edge effects, the distributions have exactly the same qualitativefeatures as given by the wounded nucleon model in Fig. 4. -1-0.500.51-10 -5 0 5 10 x (fm) < Y > b=0.2 R A b=0.6 R A b=1.0 R A b=1.4 R A Fig. 6
The average rapidity (cid:104) Y (cid:105) of the final state particles as a function of the transverse coordinate x from HIJING Mont Carlo simulations [32, 33] of non-central Au + Au collisions at √ s = In Fig. 7, we see the results of normalized rapidity distributions f p ( Y , x , b , √ s ) at different values of the transverse coordinate x . We see that at finite values of x , f p ( Y , x , b , √ s ) evidently peak at larger values of rapidity | Y | . The shift in the shapeof the rapidity distributions will provide the local longitudinal fluid shear or finiterelative orbital angular momentum for two interacting partons in the local co-moving lobal polarization effect and spin-orbit coupling in strong interaction 13 frame at any given rapidity Y . The fluid shear in the local co-moving frame at givenrapidity Y is finite and peaks at large value of rapidity | Y | ≈
2. It is also generallysmaller than the averaged fluid shear in the center of mass frame of two collidingnuclei in the Landau fireball model.
Fig. 7
The normalized rapidity distribution f p ( Y , x , b , √ s ) (in unit of 1/fm) of particles at differenttransverse position x from HIJING simulations of non-central Au + Au collisions at √ s = Shown in Fig. 8 is the average rapidity shear ∂ (cid:104) Y l (cid:105)/ ∂ x as a function of the rapid-ity Y at different values of the transverse coordinate x for ∆ Y =
1. As we can see,the average rapidity shear has a positive and finite value in the central rapidity re-gion. As given by Eq. (18), the corresponding local relative longitudinal momentumshear ∂ (cid:104) p z (cid:105)/ ∂ x is determined by this rapidity shear multiplied by p T cosh Y . With (cid:104) p T (cid:105) ≈ T ∼ . ∂ (cid:104) p z (cid:105)/ ∂ x ∼ .
003 GeV/fm in the central rapidityregion of a non-central Au + Au collision at the RHIC energy given by the HIJINGsimulations, which is smaller than that from a Landau fireball model estimate.(ii) Results obtained using the BGK model
In a recent paper [35], a simple model [34] instead of HIJING [32, 33] was usedto repeat these calculations. Here, in this simple BGK model [34], the rapidity dis-tribution of produced hadrons is given by that in pp -collision, dN pp / dY , multipliedby the following Y linearly dependent factor, i.e., d Ndxd y dY = dN pp dY (cid:20) T PA ( x , y , b ) Y L + Y Y L + T TA ( x , y , b ) Y L − Y Y L (cid:21) , (26)where T P / TA is the thickness function for the projectile or target nucleus given by, T P , TA ( x , y , b ) = ∫ dz ρ P , TA ( x , y , z , b ) , (27) Y L ≈ ln (√ s / m N ) is the maximum of the rapidity of the produced hadron; dN pp / dY of hadrons produced in a pp -collision is taken as a modified Gaussian, Y ∂ < Y > / ∂ x b=1.0 R A x=0.0 fmx=-3.0 fmx=3.0 fm Fig. 8 (Color online) The average rapidity shear ∂ (cid:104) Y l (cid:105)/ ∂ x within a window ∆ Y = Y at different transverse position x from HIJING calculation of non-central Au + Au collisions at √ s =
200 GeV. This figure is taken from [4]. dN pp dY = a exp (− Y / a )/ (cid:113) + a cosh Y , (28)where a , a and a are parameters depending on the collision energy. They aredetermined by fitting the results obtained from PYTHIA8.2 [36] for pp collisions.A few examples obtained in [35] is given in Table 1. Table 1
The parameters a , a and a for the rapidity distribution dN pp / dY given by Eq. (28)determined from PYTHIA8.2 [36]. These numbers are taken from [35]. √ s (GeV) a a a
200 4.584 26.112 9 . × −
130 4.096 25.896 5 . × − . × −
39 3.420 18.779 6 . × −
27 3.421 13.555 2 . × − . × − One great advantage to take this simple model [34] is that we have analyticalexpressions for all the quantities need so the calculations are quite simplified so thatthe physical significance can be easily demonstrated. In Ref. [35], different resultsobtained using a hard sphere or Woods-Saxon nuclear distribution are given. In thefollowing, we show those obtained using a hard sphere distribution as an example.Those obtained using Woods-Saxon are similar.Shown in Fig. 9 are the contour plots for distributions of hadrons in the transverseplane with different rapidities. This provides us a very intuitive picture how particlesare distributed in the transverse plane at different rapidities. We see that at Y = x while at Y = − lobal polarization effect and spin-orbit coupling in strong interaction 15 to positive x and at Y = − x . But they are all symmetric or evenfunction of y . Fig. 9
Contour plots for distributions of hadrons obtained in BGK model [34] with a hard spherenuclear distribution in the transverse plane for non-central Au + Au collisions at √ s =
200 GeV at b = . R A and different rapidities. The number on the contour line denotes the value on the linenormalized by that at the origin. This figure is taken from [35]. We integrate over the transverse coordinates and obtain, d NdxdY = dN pp dY (cid:32) dN P part dx Y L + Y Y L + dN T part dx Y L − Y Y L (cid:33) , (29) dNdx = (cid:104) N pp (cid:105) (cid:32) dN P part dx + dN T part dx (cid:33) , (30)where (cid:104) N pp (cid:105) = ∫ dY ( dN pp / dY ) is the average total number of particles producedin the pp collision. The normalized rapidity distribution at given x is given by, f p ( Y , x , b , √ s ) = dN pp (cid:104) N pp (cid:105) dY (cid:20) + YY L R N ( x , b , √ s ) (cid:21) , (31)where the ratio R N ( x , b , √ s ) is defined by Eq. (10).The overall average value of Y at a given x is given by, (cid:104) Y ( x , b , √ s )(cid:105) = (cid:104) Y (cid:105) Y L R N ( x , b , √ s ) , (32)where (cid:104) Y (cid:105) = ∫ Y dY ( dN pp / dY )/(cid:104) N pp (cid:105) is the average value of Y in pp collision.Compare Eq. (30) with Eq. (9), we see that (cid:104) Y ( x , b , √ s )(cid:105) in this model has exactlythe same behavior as p z ( x , b , √ s ) in the Landau fireball model.Fig. 10 shows the average values of Y as functions of x plotted in the sameformat as that in Fig. 6. We see that, besides those in the edge regions where thecalculations need to be modified, the results exhibit the same qualitative featuresas those in Fig. 6, though the quantitative results show slight differences. Fig. 11shows the corresponding normalized distributions f p ( Y , x , b , √ s ) . The right panel is to compare with Fig. 7 where HIJING monte-Carlo model was used. We see inparticular a clear shift of the peak to positive Y for x > Y for x < − − − x(fm) − 〉 Y 〈 A b=0.3R A b=0.6R A b=0.9R A b=1.2R Fig. 10
The average rapidity (cid:104) Y (cid:105) of the final state particles as a function of the transverse coordinate x from BGK [34] with a hard sphere nuclear distribution in non-central Au + Au collisions at √ s =
200 GeV. This figure is taken from [35].
Fig. 11
The normalized distribution f p ( Y , x ) of hadrons in BGK model [34] with a hard spherenuclear distribution in the transverse plane for non-central Au + Au collisions at √ s =
200 GeVand b = . R A as a function of x at different rapidity Y (left panel), and as a function of Y atdifferent x (right panel). This figure is taken from [35]. To show the rapidity dependence of the local orbital angular momentum ormomentum shear, Ref. [35] also calculated (cid:104) ξ p (cid:105) defined in Eq. (20) as a function of Y at different energies. The obtained results are shown in Fig. 12. From this figure, wesee that the rapidity dependence of (cid:104) ξ p (cid:105) is quite weak except at the limiting regionwhen Y reaches its maximum. This represents the characteristics of the rapiditydependence of the microscopic local momentum shear and may also reflect therapidity dependence of the corresponding macroscopic observable effects. lobal polarization effect and spin-orbit coupling in strong interaction 17 Fig. 12
The averaged (cid:104) ξ p (cid:105) = (cid:104) ∂ ln f p / ∂ Y ∂ x (cid:105) as a function of rapidity Y of final state particlesin BGK model [34] with a hard sphere nuclear distribution for non-central Au + Au collisions atdifferent energies and impact parameter b = . R A . This figure is taken from [35]. The spin-orbit coupling is a well known effect in a quantum system. Here, we presenta short discussion of the origin and a brief review of related phenomena.
The spin-orbit coupling is an intrinsic property for a relativistic fermionic quantumsystem. This is derived explicitly from Dirac equation. A number of characteristicsof Dirac equation show that it describes particles of spin-1/2, and the spin and orbitalangular momentum couple to each other intrinsically even for free particles. Here,we recall a few of such characteristics in the following.First of all, it is well known that, even for a free Dirac particle, the Hamiltonian ˆ H does not commute with the orbital angular momentum ˆ L and the spin Σ separately,but commutes with the total angular momentum ˆ J = ˆ L + Σ /
2, i.e., [ ˆ H , ˆ L ] = − i α × ˆ p , [ ˆ H , Σ ] = i α × ˆ p , but [ ˆ H , ˆ J ] =
0. This shows clearly that spin and orbital angularmomentum couple to each other and transform from one to another in a relativisticfermionic quantum system, though the strength of the spin-orbit coupling can bedifferent for an electromagnetic or a strongly interacting system.Second, the magnetic momentum of a Dirac particle with electric charge e isobtained simply by replacing the classical expression M = e r × v / M = e r × α /
2. In an eigenstate | ψ (cid:105) of ˆ H , if we take the non-relativisticapproximation E ≈ m , we obtain immediately that [37], (cid:104) M (cid:105) ≈ e m (cid:104) ϕ |( ˆ L + σ )| ϕ (cid:105) , (33) where ϕ is the upper component of ψ . This is just the well known result for point-likespin-1/2 particles where the Landre factors are g L = g s = H , ˆ J , ˆ J z and the parity ˆ P with eigenvalues ( ε, j , m , P) , i.e., ψ ε jm P ( r , θ, φ, s ) = (cid:32) f ε l ( r ) Ω ljm ( θ, φ )(− ) ( l − l (cid:48) + ) g ε l (cid:48) ( r ) Ω l (cid:48) jm ( θ, φ ) (cid:33) , (34)where Ω ljm ( θ, φ ) is the 2 × f ε l ( r ) and g ε l (cid:48) ( r ) are the radial parts, j = l ± / = l (cid:48) ∓ / P = (− ) l . Inthe ground state ε = ε , j = / P = + , the magnetic moment is given by [37], (cid:104) ε , / , m , + | ˆ M | ε , / , m , + (cid:105) = µ q (cid:104) ξ ( m )| σ | ξ ( m )(cid:105) , (35)where µ q = − e ∫ r dr f ( r ) g ( r )/ ξ ( m ) is the eigenstate of σ z and is a Pauli spinor. Eq. 35 has exactlythe same form as that for a quark at rest. This explains why the static quark modelworks well in describing the magnetic moment of baryon although we know that thequark mass is small and the relativistic treatment has to be used.Third, we consider a Dirac particle moving in a magnetic field with potential A = ( φ, A ) . By replacing ˆ p with ˆ p − eA in the Dirac equation and taking the non-relativistic approximation, we obtain immediately,ˆ H nr = m ( ˆ p − e A ) − e φ − m d φ r dr ˆ L · σ , (36)where the spin-orbit coupling is obtained automatically. Intuitively, the spin-orbit coupling in systems under electromagnetic interactionshas a very clear physical picture and also leads to many well known effects. Themost famous textbook example might be the fine structure of atomic light spectra.Here, we consider the electron moving in the electromagnetic field induced by thehydrogen atom, we take the extra 1 / V ls ( r ) = − µ · B = e m σ · v × E = e m d φ r dr σ · L . (37)This is exactly the same as that in Eq. (36) derived from Dirac equation.The spin-orbit coupling plays also a very important in modern spintronics incondensed matter physics where spin transport in the electromagnetically interactingsystem is studied. There are also examples in electromagnetically interacting systems lobal polarization effect and spin-orbit coupling in strong interaction 19 where spin polarization (magnetization) and orbital angular momentum (rotation)are transferred from one to the other. Earlier examples may even be traced backto Einstein and deHaas [38] and Barnet [39]. It was known as the Einstein-deHaaseffect where the rotation is caused by magnetization and the Barnett effect that is thegyromagnetic effect where magnetization is caused by rotation. In systems under strong interactions, the spin-orbit coupling also leads to manydistinguished effects. One of such famous examples is the nuclear shell modeldeveloped by Mayer and Jensen [40, 41, 42] where the spin-orbit coupling plays acrucial role to produce the magic numbers of atomic nuclei.There is no such a clear intuitive picture for the spin-orbit interaction in systemsunder strong interactions as that for electromagnetic interactions so the strength cannot be derived explicitly. Usually in the covariant relativistic formalism, the spin-orbit coupling does appears explicitly. However, the role that it plays can be seenwhenever one separates spin and orbital angular momentum from each other. Besidesthe famous example in the nuclear shell model, another explicit example is the heavyquarkonium spectra where spin-orbit coupling has to be taken into account [43].Even more interesting is that, in the frontier of high energy spin physics, it seemsthat spin-orbit coupling plays a key role in understanding all the four classes ofstriking spin effects mentioned in Sec. 1 observed in experiments since 1970s. Thesimplest argument that orbital angular momentum contributes significantly to protonspin is that discussed in the first point in Sec. 3.1 where it has been shown that theorbital angular momentum for a Dirac particle is not a good quantum number. Henceeven if a quark is in the ground states in a central potential as given by Eq. (34)the average value of the orbital angular momentum is not zero. If we e.g. considera quark in the ground state in a spheric potential well with infinite depth such as inthe MIT bag model, the orbital angular momentum contributes ∼
35% to the totalangular momentum.Both phenomenological model [37, 44] and pQCD calculations [45] indicatethat orbital angular momentum of quarks in a polarized nucleon and the initial orfinal state interactions are responsible for SSA observed [10, 11, 12, 13] in inclusivehadron-hadron collisions. It has also been shown that transverse hyperon polarizationobserved [14, 15, 16, 17, 18] in unpolarized hadron-hadron collisions are closelyrelated to SSA thus has the same physical origins [46]. The spin analyzing powerobserved [19, 20, 21, 18] in elastic pp scattering is due to color magnetic interactionduring the scattering [47] thus originates also from the orbital angular momentumof the constituents in the polarized proton. The study of the role played by the orbitalangular momentum is one of the core issues currently in high energy spin physics.See recent reviews such as [48, 49, 9, 50, 51]. It has been shown [2] that due to spin-orbit interactions in a strongly interacting sys-tem such as QGP, the orbital angular momentum can be transferred to the polarizationof the constituents in the system such as the quarks and anti-quarks.
In Sec. 2, we have seen that in a non-central AA collision, there is a huge globalorbital angular momentum for the colliding system. Such a global angular momentumleads to the longitudinal fluid shear in the produced system of partons. A pair ofinteracting partons will have a finite value of relative orbital angular momentumalong the direction opposite to the normal of the reaction plane. We have also seenin Sec. 3 that spin-orbit coupling is an intrinsic property of a relativistic system. It isthus natural to ask whether the orbital angular momentum or momentum shear leadto the polarization of partons in the system.There is no field theoretical calculation that can be applied directly to answerthis question because usually the calculations are in the momentum space wherethe momentum shear with respect to x coordinate can not be taken into account.To achieve this, Ref. [2] took the approach by considering parton scattering withimpact parameter in the preferred direction and reach the positive conclusion. Wesummarize the studies of Refs. [2] and [4] in this section. To be explicit, we consider the scattering q ( p ) + q ( p ) → q ( p ) + q ( p ) of twoquarks with different flavors. The scattering matrix element in momentum space isgiven by, S f i = (cid:104) f | ˆ S | i (cid:105) = M f i ( q )( π ) δ ( p + p − p − p ) , (38)where p i = ( E i , p i ) is the four momentum of the quark, q = p − p = p − p isthe four momentum transfer and M f i ( q ) is the scattering amplitude in momentumspace. The incident momenta are taken as in z or − z direction and the transversemomentum is denoted as p T = p T = − p T . The differential cross section in themomentum space is given by, d σ = c qq F | S f i ( q )| TV d p ( π ) E d p ( π ) E , (39) lobal polarization effect and spin-orbit coupling in strong interaction 21 where T and V are interaction time and volume of the space, c qq = / F = (cid:113) ( p · p ) − m m is the flux factor. Here, just for clarity ofequations, we omit the spin indices and will pick them up later in the following.It can easily be verified that, S f i = ∫ d x T ∫ d q ⊥ ( π ) M f i ( q ) e − i ( q T + p T )· x T ( π ) δ ( p + p − p − p ) , (40)where we use x T to denote the impact parameter of the two scattering quarks todistinguish it from the impact parameter b = b e x of the two nuclei. By insertingEq. (40) into (39), we obtain, d σ = c qq F ∫ d x T ∫ d q ⊥ ( π ) d k ⊥ ( π ) e − i ( q T − k T )· x T M f i ( q ) Λ ( q ) M ∗ f i ( k ) Λ ( k ) , (41)where M f i ( q ) and M f i ( k ) are scattering amplitudes in momentum space with fourmomentum transfer q = ( q , q T , q z ) and k = ( k , k T , k z ) respectively; Λ ( q ) is akinematic factor obtained in carrying out the integration and is given by, Λ − ( q ) = ∫ δ ( p + p − p − p ) δ ( q T + p T ) d p E d p E = ( E + E ) p z , (42)where p z is the positive solution of (cid:113) q T + p z + m + (cid:113) q T + p z + m = E + E .Here, in obtaining Eq. (41), we have taken the symmetric form with exchange of q and k to guarantee the integrand of d x T to be positive definite.We pick up the spin indices and suppose that we are interested in the polarizationof quark q after the scattering. We therefore average over the spins of initial quarksand sum over the spin of quark q in the final state. In this case, we have, d σ λ d x T = c qq F (cid:213) λ ,λ ,λ ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T M( q ) Λ ( q ) M ∗ ( k ) Λ ( k ) . (43)We define, d ∆ σ d x T = d σ + d x T − d σ − d x T , (44) d σ d x T = d σ + d x T + d σ − d x T , (45)where λ = + or − denotes that the spin of q after the scattering is in the positiveor negative direction of the normal n of the reaction plane; d σ / d x T is just theunpolarized cross section at the fixed impact parameter.Suppose that the impact parameter x T has a given distribution f qq ( x T , b , Y , √ s ) ,we can calculate the polarization in the following way, (cid:104) ∆ σ (cid:105) = ∫ d x T f qq ( x T , b , Y , √ s ) d ∆ σ d x T , (46) (cid:104) σ (cid:105) = ∫ d x T f qq ( x T , b , Y , √ s ) d σ d x T , (47)and the polarization of the quark q after the scattering is given by, P q = (cid:104) ∆ σ (cid:105)/(cid:104) σ (cid:105) . (48)As discussed in Sec. 2, the average relative orbital angular momentum l of twoscattering quarks is in the opposite direction of the normal of the reaction planein non-central AA collisions. Since a given direction of l corresponds to a givendirection of x T , there should be a preferred direction of x T at a given direction of thenucleus-nucleus impact parameter b . The distribution f qq ( x T , b , Y , √ s ) of x T at given b depends on the collective longitudinal momentum distribution shown in Sec. 2.Clearly, it depends on the dynamics of QGP and that of AA collisions.To see the qualitative features of the physical consequences explicitly, Refs. [2, 4]took a simplified f qq ( x T , b , Y , √ s ) as an example, i.e., a uniform distribution of x T in the upper half x y -plane with x >
0, i.e., f qq ( x T , b , Y , √ s ) ∝ θ ( x ) , (49)so that (cid:104) ∆ σ (cid:105) ≈ ∫ ∞ dx ∫ ∞−∞ d y d ∆ σ d x T , (50) (cid:104) σ (cid:105) ≈ ∫ ∞ dx ∫ ∞−∞ d y d σ d x T . (51) To see the characteristics of the physical consequences clearly, in [2], we consideredfirst a quark scattering by a static potential. Here, it is envisaged that a quarkincident in z -direction and is scattered by an effective static potential induced byother constituents of QGP. In this case, we obtain, M f i ( q ) = ¯ u λ ( p + q ) A /( q ) u ( p ) , (52)where A ( q ) = ( A ( q ) , ) and A ( q ) = g /( q + µ D ) is the screened static potentialwith Debye screen mass µ D [52]. It follows that, M f i ( q )M ∗ f i ( k ) = A ( q ) A ( k ) ¯ u λ ( p + q )( ˜ p / + m q ) u λ ( p + k ) , (53)where ˜ p ≡ ( E , − p ) . We choose n as the quantization axis of spin and denote theeigenvalue by λ = ±
1. For small angle scattering, q T , k T ∼ µ D (cid:28) E , we obtain, lobal polarization effect and spin-orbit coupling in strong interaction 23 M f i ( q )M ∗ f i ( k ) ≈ E A ( q ) A ( k ) (cid:20) − i λ ( q T − k T ) · ( n × p ) E ( E + m q ) (cid:21) , (54)and the cross sections are given by, d σ d x T = g c T ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T ( q T + µ D )( k T + µ D ) , (55) d ∆ σ d x T = i g c T p ∫ d q T ( π ) d k T ( π ) ( n × p ) · ( k T − q T ) e i ( k T − q T )· x T ( q T + µ D )( k T + µ D ) . (56)where c T is the color factor. It is interesting to note that, under such approximation,these two parts of the cross section are related to each other, d ∆ σ d x T = p ( n × p ) · ∇ d σ d x T . (57)Completing the integrations over d q T and d k T by using the integration formulae, ∫ d q T ( π ) e i q T · x T q T + µ D = ∫ q T dq T π J ( q T x T ) q T + µ D = π K ( µ D x T ) , (58)we obtain from Eqs. (55) and (56) that [2], d σ d x T = α s c T K ( µ D x T ) , (59) d ∆ σ d x T = α s c T (cid:2) ( p × n ) · ˆ x T / p (cid:3) µ D K ( µ D x T ) K ( µ D x T ) . (60)where J and K are the Bessel and modified Bessel functions respectively and x T = | x T | . The unpolarized cross section just corresponds to d σ / d q T = πα s c T /( q T + µ D ) in the momentum space.It is evident from Eq. (60) that parton scattering polarizes quarks along thedirection opposite to the normal of the parton reaction plane determined by the impactparameter x T , i.e., along the direction of the relative orbital angular momentum. Thisis essentially the manifest of spin-orbit coupling in QCD. Ordinarily, the polarizedcross section along a fixed direction n vanishes when averaged over all possibledirection of the parton impact parameter x T . However, in non-central HIC the localrelative orbital angular momentum (cid:104) l y (cid:105) provides a preferred average reaction planefor parton collisions. This leads to a quark polarization opposite to the normal ofthe reaction plane of HIC. This conclusion should not depend on our perturbativetreatment of parton scattering as far as the effective interaction is mediated by thevector coupling in QCD.Averaging over the relative angle between parton x T and nuclear impact parameter b from − π / π / x T , one can obtain the global quark polarization, P q = − π µ D | p |/ E ( E + m q ) (61) via a single scattering for given E .If one takes the non-relativistic limit, E ∼ m q (cid:29) | p | , µ D , one obtains, P q ≈ − π µ D | p |/ m q . (62)One of the advantages in this limit is that one can check effects due to spin-orbit coupling explicitly. Here, the spin-orbit coupling is given by Eq. (36). Thecorresponding energy is roughly given by (cid:104) E ls (cid:105) ∼ (cid:104) l · s dV / r dr / m (cid:105) . Given theinteraction range is r ∼ / µ D , (cid:104) dV / r dr (cid:105) ∼ −(cid:104) V (cid:105) µ D ; (cid:104) l · s (cid:105) ∼ (cid:104) l (cid:105)/ ∼ | p |/ µ D . Thequark polarization is P q ∼ (cid:104) E ls (cid:105)/(cid:104) V (cid:105) . We obtain P q ∼ − µ D | p |/ m that is just theresult given by Eq. (62).If one takes the ultra-relativistic limit m q = | p | (cid:29) µ D , one expectsfrom Eq. (61) that P q ∼ − π µ D / E . However, given dp / dx = .
34 GeV/fm forsemi-peripheral ( b = R A ) collisions at RHIC, and an average range of interaction ∆ x − ∼ µ D ∼ . ∆ p z ∼ . µ D . In this case, one has to go beyond small angle approximation.We also note that the cross sections can be written in a general form as, d σ d x T = F ( x T , E ) , (63) d ∆ σ d x T = n · ( x T × p ) ∆ F ( x T , E ) , (64)where F ( x T , E ) and ∆ F ( x T , E ) are scalar functions of both x T ≡ | x T | and the c.m.energy E of the two quarks. We would like to emphasize that Eqs. (63) and (64)are in fact the most general forms of the two parts of the cross sections under parityconservation in the scattering process. The unpolarized part of the cross sectionshould be independent of any transverse direction thus can only take the form asgiven by Eq. (63), i.e. it depends only on the magnitude of x T but not on the direction.For the spin-dependent part, the only scalar that we can construct from the availablevectors is n · ( p × x T ) . Hence d ∆ σ / d x T can only take the form given by Eq. (64).We also note that, x T × p is nothing but the relative orbital angular momentumof the two-quark system, l = x T × p . Therefore, the polarized cross section takes itsmaximum when n is parallel or antiparallel to the relative orbital angular momen-tum, depending on whether ∆ F is positive or negative. This corresponds to quarkpolarization in the direction l or − l . The quark-quark scattering amplitude in a thermal medium can be calculated byusing the Hard Thermal Loop (HTL) resummed gluon propagator [53, 55], ∆ µν ( q ) = P µν T − q + Π T ( ξ ) + P µν L − q + Π L ( ξ ) + ( α − ) q µ q ν q , (65) lobal polarization effect and spin-orbit coupling in strong interaction 25 where q denotes the gluon four momentum and α is the gauge fixing parameter, x = ω / (cid:112) − ˜ q and ω = q · u , ˜ q = q − ω u , u is the fluid velocity of the local medium.The longitudinal and transverse projectors P µν T , L are defined by P µν L = q ˜ q ( ω q µ − q u µ )( ω q ν − q u ν ) , (66) P µν T = ˜ g µν − ˜ q µ ˜ q ν ˜ q , (67)where ˜ g µν = g µν − u µ u ν . Π L and Π T are the transverse and longitudinal self-energiesand are given by [53] Π L ( ξ ) = µ D (cid:20) − ξ (cid:18) + ξ − ξ (cid:19) + i π ξ (cid:21) ( − ξ ) , (68) Π T ( ξ ) = µ D (cid:20) ξ + ξ ( − ξ ) ln (cid:18) + ξ − ξ (cid:19) − i π ξ ( − ξ ) (cid:21) , (69)where the Debye screening mass is µ D = g ( N c + N f / ) T / M f i ( q ) in the momentum space can be expressed as, M f i ( q ) = ¯ u λ ( p ) γ µ u λ ( p ) ∆ µν ( q ) ¯ u λ ( p ) γ ν u λ ( p ) . (70)The product M f i ( q )M ∗ f i ( k ) can be converted to the following trace form, (cid:213) λ ,λ M f i ( q )M ∗ f i ( k ) = ∆ µν ( q ) ∆ αβ ∗ ( k ) Tr [ u λ ( p − k ) ¯ u λ ( p + q ) γ µ ( p / + m ) γ α ]× Tr [ u λ ( p − k ) ¯ u λ ( p − q ) γ ν ( p / + m ) γ β ] . (71)In calculations of transport coefficients such as jet energy loss parameter [54] andthermalization time [55] that generally involve cross sections weighted with trans-verse momentum transfer, the imaginary part of the HTL propagator in the magneticsector is enough to regularize the infrared behavior of the transport cross sections.However, in the calculation of quark polarization, the total parton scattering crosssection is involved. The contribution from the magnetic part of the interaction hastherefore infrared divergence that can only be regularized through the introductionof non-perturbative magnetic screening mass µ m ≈ . (cid:112) N c / g T [56].Since we have neglected the thermal momentum perpendicular to the longitudinalflow, the energy transfer ω = x = u = ( , , , ) . Thecorresponding HTL effective gluon propagator in Feynman gauge that contributesto the scattering amplitudes reduces to, ∆ µν ( q ) = g µν − u µ u ν q + µ m + u µ u ν q + µ D . (72)The spin-dependent part determines the polarization of the final state quark q via the scattering. The calculation is much involved. A detailed study is given in [4].We summarize part of the key results in the following.(i) Small angle approximation
We only consider light quarks and neglect their masses. Carrying out the tracesin Eq.(71), we can obtain the expression of the cross section with HTL gluonpropagators. The results are much more complicated than those as obtained inSec. 4.1.3 using a static potential model [2]. However, if we consider small transversemomentum transfer and use the small angle approximation, the results are still verysimple. In this case, with q z ∼ q T ≡ | q T | (cid:28) p , we obtain, d σ d x T = g c qq ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T × (cid:32) q T + µ m + q T + µ D (cid:33) (cid:32) k T + µ m + k T + µ D (cid:33) , (73) d ∆ σ d x T = − i g c qq p ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T [( k T − q T ) · ( p × n )]× (cid:32) q T + µ m + q T + µ D (cid:33) (cid:32) k T + µ m + k T + µ D (cid:33) . (74)We note that there exist the same relationship between the polarized and unpolarizedcross section as that as that given by Eq. (57) obtained in the case of static potentialmodel under the same small angle approximation. Completing the integration over d q T and d k T by using the formulae given by Eqs. (58), we obtain, d σ d x T = c qq α s [ K ( µ m x T ) + K ( µ D x T )] , (75) d ∆ σ d x T = c qq α s p [( p × n ) · ˆ x T ] [ K ( µ m x T ) + K ( µ D x T )]× (cid:2) µ m K ( µ m x T ) + µ D K ( µ D x T ) (cid:3) , (76)where ˆ x T = x T / x T is the unit vector of x T . We compare the above results withthose given by Eqs. (59) and (60) obtained in the screened static potential modelwhere one also made the small angle approximation. We see that the only differencebetween the two results is the additional contributions from magnetic gluons, whosecontributions are absent in the static potential model.(ii) Beyond small angle approximation
Now we present the complete results for the cross-section in impact parame-ter space using HTL gluon propagators without small angle approximation. Theunpolarized and polarized cross section can be expressed as, lobal polarization effect and spin-orbit coupling in strong interaction 27 d σ d x T = g c qq
16 ˆ s ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T f ( q , k ) Λ ( q ) Λ ( k ) , (77) d ∆ σ d x T = i g c qq s ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T ∆ f ( q , k ) Λ ( q ) Λ ( k ) , (78)where ˆ s is the c.m. energy squared of the quark-quark system, f ( q , k ) and ∆ f ( q , k ) are given by, f ( q , k ) = (cid:213) a , b A ab ( q , k )( q + µ a )( k + µ b ) , (79) ∆ f ( q , k ) = ( p × n ) · (cid:213) ab ∆ A ab ( q , k )( q + µ a )( k + µ b ) , (80)where the subscript a or b denotes m or D representing the magnetic or electric partand the sum runs over all possibilities of ( a , b ) . A ab are Lorentz scalar functions of ( q , k ) given by, A mm ( k , q ) = ˆ s [ ˆ s − ( q + k ) ] + ( q · k ) , (81) A DD ( q , k ) = ( ˆ s − q − k )[ ˆ s − ( q + k ) ] + ( q · k ) , (82) A mD ( q , k ) = A Dm ( k , q ) = ˆ s [ ˆ s − k − ( q + k ) ] + ( k − k · q ) + k q ˆ s ( q + k ) , (83) ∆ A ab ( q , k ) is a vector in the momentum space and can be written as, ∆ A ab ( q , k ) = ∆ g ( q ) ab ( q , k ) q T − ∆ g ( k ) ab ( q , k ) k T , (84)where ∆ g ( q ) ab ( q , k ) and ∆ g ( k ) ab ( q , k ) are Lorentz scalar functions given by, ∆ g ( q ) mm ( q , k ) = ∆ g ( k ) mm ( k , q ) = ˆ s ( ˆ s − q · k ) − ( ˆ s + q + k − q · k ) k , (85) ∆ g ( q ) DD ( q , k ) = ∆ g ( k ) DD ( k , q ) = ( ˆ s − q − k − q · k )( ˆ s − k ) , (86) ∆ g ( q ) mD ( q , k ) = ∆ g ( k ) Dm ( k , q ) = ˆ s ( ˆ s − k − q · k ) − ( k − q · k − q k ˆ s ) k , (87) ∆ g ( k ) mD ( q , k ) = ∆ g ( q ) Dm ( k , q ) = ˆ s ( ˆ s + q − k − q · k ) + ( q − q · k − q k ˆ s ) q , (88)We note that A ab ( q , k ) = A ab ( k , q ) , ∆ A ab ( q , k ) = − ∆ A ab ( k , q ) so that f ( q , k ) = f ( k , q ) and ∆ f ( q , k ) = − ∆ f ( k , q ) , i.e., they are symmetric or anti-symmetric w.r.t.the two variables respectively. Hence, the integration result in Eq. (77) is real whilethat in Eq. (78) is pure imaginary so that the cross section is real.We also note that f ( q , k ) and ∆ g ( q / k ) αβ ( q , k ) are all functions of Lorentz invariantsˆ s , q , k and q · k . Furthermore A ab ( k , q ) = (cid:205) n = − g ( n ) ab ( ˆ s , q , k )( q T · k T ) n , and ∆ g ( q / k ) ab ( k , q ) = (cid:205) n = , ∆ g ( q / k , n ) ab ( ˆ s , q , k )( q T · k T ) n . The angular parts of the inte-grations in Eqs. (77) and (78) can be carried out. For this purpose, we note that, e.g., for any scalar function f s of ( ˆ s , q , k ) , we have, ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T f s ( ˆ s , q , k ) = ∫ dq T π dk T π J ( q T x T ) J ( k T x T ) f s ( ˆ s , q , k ) ≡ F ( ) ( x T , ˆ s ) , (89) ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T ( q T · k T ) f s ( ˆ s , q , k ) = ∫ dq T π dk T π q T k T J (cid:48) ( q T x T ) J (cid:48) ( k T x T ) f s ( ˆ s , q , k ) ≡ F ( ) ( x T , ˆ s ) , (90) ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T g s ( ˆ s , q , k ) q T − i ˆ x T G ( ) ( x T , ˆ s ) , G ( ) ( x T , ˆ s ) = ∫ dq T π dk T π J (cid:48) ( q T x T ) J ( k T x T ) g s ( ˆ s , q , k ) , (91) ∫ d q T ( π ) d k T ( π ) e i ( k T − q T )· x T g s ( ˆ s , q , k )( q T · k T ) q T = − i ˆ x T G ( ) ( x T , ˆ s ) , G ( ) ( x T , ˆ s ) = ∫ dq T π dk T π q T k T J (cid:48)(cid:48) ( q T x T ) J (cid:48) ( k T x T ) g s ( ˆ s , q , k ) . (92)Hence, we see clearly that, d σ d x T = g c qq
16 ˆ s (cid:213) a , b F ab ( x T , ˆ s ) , (93) d ∆ σ d x T = g c qq s ( p × n ) · ˆ x T (cid:213) a , b ∆ F ab ( x T , ˆ s ) . (94)The scalar functions F ab ( x T , ˆ s ) and ∆ F ab ( x T , ˆ s ) are rather involved. However, ifwe take the simple form of f qq ( x T , Y , b , √ s ) given by Eq. (49) and calculate σ and ∆ σ using Eqs. (50) and (51), we may first carry out the integration over x T . In thiscase we obtain, (cid:104) σ (cid:105) = g c qq
32 ˆ s ∫ q T ≤ p d q T ( π ) f ( q , q ) Λ ( q ) , (95) (cid:104) ∆ σ (cid:105) = − g c qq s ∫ E − E dq y π ∫ √ E − q y − √ E − q y dq x π ∫ √ E − q y − √ E − q y dk x π ∆ f ( q x , q y ; k x , q y )( k x − q x ) Λ ( q ) Λ ( k ) . (96)These equations can be further simplified to the form suitable for carrying outnumerical calculations. Details are given in Ref. [4] where cases are also studied.Here, we present only the result of the quark polarization P q as function of c,m,energy of the quark-quark system √ ˆ s / T in Fig. 13. lobal polarization effect and spin-orbit coupling in strong interaction 29 Fig. 13
Quark polarization - P q as a function of √ ˆ s / T for different α s ’s obtained in quark-quarkscattering with a hard thermal loop propagator. This figure is taken from [4]. From Fig. 13, we see that the quark polarization changes drastically with √ ˆ s / T .It increases to some maximum values and then decreases with the growing energy,approaching the result of small angle approximation in the high-energy limit. Thisstructure is caused by the interpolation between the high-energy and low-energybehavior dominated by the magnetic part of the interaction in the weak couplinglimit α s <
1. Therefore, the position of the maxima in √ ˆ s should approximatelyscale with the magnetic mass µ m . Although approximations and/or models have to be used in the calculations presentedabove, the physical picture and consequence are very clear. It is confident that afterthe scattering of two constituents in QGP, the orbital angular momentum will betransferred partly to the polarization of quarks and anti-quarks in the system due tospin-orbit coupling in QCD. Such a polarization is very different from those thatwe meet usually in high energy physics such as the longitudinal or the transversepolarization. The longitudinal polarization refers to the helicity or the polarizationin the direction of the momentum, whereas the transverse polarization refers todirections perpendicular to the momentum, either in the production plane or alongthe normal of the production plane. These directions are all defined by the momentumof the individual particle and are in general different for different particles in the samecollision event. In contrast, the polarization discussed here refers to the normal ofthe reaction plane. It is a fixed direction for one collision event and is independent ofany particular hadron in the final state. Hence, in Ref. [2], this polarization was givena new name — the global polarization, and the QGP was referred to the globallypolarized QGP in non-central HIC. We illustrate this in Fig. 14.The following three points should be addressed in this connection.
Fig. 14
Illustration of the global quark polarization effect in non-central heavy ion collisions. (i) The results presented above are mainly a summary of those obtained in theoriginal papers [2, 4] where the global orbital angular momentum for the collidingsystem in HIC was first pointed out and the GPE were first predicted. These resultsare for a single quark-quark scattering. In a realistic HIC where QGP is created, suchquark-quark scatterings may take place for a few times before they hadronize intohadrons. The calculations presented above or in [2, 4] provide the theoretical basisfor GPE. They do not provide final results of global quark polarizations.(ii) The numerical results on quark polarization presented above are based onthe approximation by taking the simple form of f qq ( x T , Y , b , √ s ) given by Eq. (49).They provide a practical guidance for the magnitude of the quark polarization butcan not give us the relationship between the polarization and the local orbital angularmomentum. Further studies along this line are necessary. In practice, to describe theevolution of the global quark polarization, one can invoke a dynamical model ofQGP evolution or effectively a dynamical model for f qq ( x T , Y , b , √ s ) .(iii) If we consider QGP as a fluid, the momentum shear distribution discussedin Sec. 2 implies a non-vanishing vorticity ω = ( / )∇ × v . The spin-orbit couplingcan be replaced by spin-vortical coupling. This provides a good opportunity tostudy spin-vortical effects in strongly interacting system and has attracted muchattention [59, 60, 61, 62, 63, 64, 65, 66, 67]. See chapter ... in this series. The global polarization in heavy ion collisions arises from scattering processes ofpartons or hadrons with spin-orbit couplings. In a 2-to-2 particle scattering at a fixedimpact parameter, one can calculate the polarized cross section arising from thespin-orbit coupling. In a thermal medium, however, momenta of incident particlesare randomly distributed and particles participating in the scattering are located atdifferent space-time points. In order to obtain observables we have to take an en- lobal polarization effect and spin-orbit coupling in strong interaction 31 semble average over random momenta of incident particles and treat scatterings atdifferent space-time points properly. To this end, we propose a microscopic modelfor the polarization from the first principle through the spin-orbit coupling in particlescatterings in a thermal medium with a shear flow [68]. It is based on scatteringsof particles as wave packets, an effective method to deal with particle scatteringsat specified impact parameters. The polarization is then the consequence of particlecollisions in a non-equilibrium state of spins. The spin-vorticity coupling naturallyemerges from the spin-orbit one encoded in polarized scattering amplitudes of col-lisional integrals when one assumes local equilibrium in momentum but not in spin.As an illustrative example, we have calculated the quark polarization rate per unitvolume from all 2-to-2 parton (quark or gluon) scatterings in a locally thermalizedquark-gluon plasma. It can be shown that the polarization rate for anti-quarks is thesame as that for quarks because they are connected by the charge conjugate transfor-mation. This is consistent with the fact that the rotation does not distinguish particlesand antiparticles. The spin-orbit coupling is hidden in the polarized scattering am-plitude at specified impact parameters. We can show that the polarization rate perunit volume is proportional to the vorticity as the result of particle scatterings. Thuswe build up a non-equilibrium model for the global polarization.
We aim to derive the spin polarization rate in a thermal medium with a shear flowfrom particle scatterings through spin-orbit couplings. Before we do it in the nextsection, let us first look at the collision rate of spin-zero particles. It is easy togeneralize it to the spin polarization rate for spin-1/2 particles
Fig. 15
A collision or scattering in the Lab frame (left) and center-of-mass frame (right).
In the center of mass frame (CMS) of the incident particle A and B , the collisionrate (the number of collisions per unit time) per unit volume is given by R AB → = n A n B | v A − v B | σ d p A ( π ) d p B ( π ) f A ( x A , p A ) f B ( x B , p B )| v A − v B | ∆ σ, (97) where v A = | p A |/ E A and v B = −| p B |/ E B are the velocity of A and B respectivelywith p A = − p B , f A and f B are the phase space distributions for A and B respectively,and ∆ σ denotes the infinitesimal element of the cross section which is given by ∆ σ = C AB d x A d x B δ ( ∆ t ) δ ( ∆ x L ) d p ( π ) E d p ( π ) E ( E A )( E B ) K , (98)where we assumed that the scattering takes place at the same time and the samelongitudinal position in the CMS (these conditions are represented by two deltafunctions), the constant C AB makes ∆ σ have the correct dimension whose definitionwill be given later, and K is given by K = ( E A )( E B )| out (cid:104) p p | φ A ( x A , p A ) φ B ( x B , p B )(cid:105) in | , (99)with ( i = A , B ) | φ i ( x i , p i )(cid:105) in = ∫ d k i ( π ) (cid:112) E i , k φ i ( k i − p i ) e − i k i · x i | k i (cid:105) in , (100)being the wave packets for incident particles. If incoming particles are described bytwo plane waves, there is no initial angular momentum. This is why we should usewave packets for incoming particles. Normally one can choose a Gaussian form forthe wave packet amplitude, φ i ( k i − p i ) = ( π ) / α / i exp (cid:34) − ( k i − p i ) α i (cid:35) , (101)where α i denote the width of the wave packet. For simplicity, we use plane waves torepresent outgoing particles.Now we consider the scattering process in Fig. 16. The incoming particles arelocated at x A and x B . We can use new variables X = ( x A + x B )/ and y = x A − x B to replace x A and x B . We then define C AB ≡ ∫ d X = t X Ω int , where t X and Ω int arethe local time and space volume for the interaction. The local collision rate from Eq.(97) can be written as d N AB → dX = ( π ) ∫ d p A ( π ) E A d p B ( π ) E B d p ( π ) E d p ( π ) E ×| v A − v B | G G ∫ d k A d k B d k (cid:48) A d k (cid:48) B × φ A ( k A − p A ) φ B ( k B − p B ) φ ∗ A ( k (cid:48) A − p A ) φ ∗ B ( k (cid:48) B − p B )× δ ( ) ( k (cid:48) A + k (cid:48) B − p − p ) δ ( ) ( k A + k B − p − p )×M ({ k A , k B } → { p , p }) M ∗ (cid:0) { k (cid:48) A , k (cid:48) B } → { p , p } (cid:1) × ∫ d b f A (cid:16) X + y T , p A (cid:17) f B (cid:16) X − y T , p B (cid:17) exp (cid:2) i ( k (cid:48) A − k A ) · b (cid:3) , (102) lobal polarization effect and spin-orbit coupling in strong interaction 33 where N AB → is the number of collisions and G i ( i = , ) denote the distributionfactors which depends on the particle types in the final state. We have G i = G i = ± f i ( p i ) for bosons (upper sign) and fermions(lower sign). Fig. 16
Scattering of two particles in the center of mass frame.
Based on the collision rate for spin-zero particles in the above section, we nowconsider spin-1/2 particles. We assume that particle distributions are independent ofspin states, so the spin dependence comes only from scatterings of particles carryingthe spin degree of freedom. In this section we will distinguish quantities in the CMSfrom those in the lab frame, we will put an index c for a CMS quantity.If the system has reached local equilibrium in momentum, we can make anexpansion of f A f B in y c , T = ( , b c ) , and thus, f A (cid:16) X c + y c , T , p c , A (cid:17) f B (cid:16) X c − y c , T , p c , B (cid:17) = f A ( X , p A ) f B ( X , p B ) + y µ c , T [ Λ − ] νµ ∂ ( β u ρ ) ∂ X ν × (cid:20) p ρ A f B ( X , p B ) df A ( X , p A ) d ( β u · p A ) − p ρ B f A ( X , p A ) df B ( X , p B ) d ( β u · p B ) (cid:21) , (103)where we have used the defination of the Lorentz transformation matrix ∂ X ν / ∂ X µ c = [ Λ − ] νµ = Λ νµ , and the scalar invariance f A ( X , p A ) = f A (cid:0) X c , p c , A (cid:1) and f B ( X , p B ) = f B (cid:0) X c , p c , B (cid:1) . From Eq. (103) we see that the local vorticity ∂ ( β u ρ )/ ∂ X ν shows up.We look closely at the term y µ c , T [ ∂ ( β u c ,ρ )/ ∂ X µ c ] p ρ c , A , y µ c , T p ρ c , A ∂ ( β u ρ ) ∂ X µ c = y [ µ c , T p ρ ] c , A (cid:20) ∂ ( β u c ,ρ ) ∂ X µ c − ∂ ( β u c ,µ ) ∂ X ρ c (cid:21) + y { µ c , T p ρ } c , A (cid:20) ∂ ( β u c ,ρ ) ∂ X µ c + ∂ ( β u c ,µ ) ∂ X ρ c (cid:21) = − y [ µ c , T p ρ ] c , A (cid:36) ( c ) µρ + y { µ c , T p ρ } c , A (cid:20) ∂ ( β u c ,ρ ) ∂ X µ c + ∂ ( β u c ,µ ) ∂ X ρ c (cid:21) = − L µρ ( c ) (cid:36) ( c ) µρ + y { µ c , T p ρ } c , A (cid:20) ∂ ( β u c ,ρ ) ∂ X µ c + ∂ ( β u c ,µ ) ∂ X ρ c (cid:21) , (104)where [ µρ ] and { µρ } denote the anti-symmetrization and symmetrization of two in-dices respectively, L µρ ( c ) ≡ y [ µ c , T p ρ ] c , A is the OAM tensor, and ω ( c ) µρ ≡ −( / )[ ∂ X c µ ( β u c ,ρ )− ∂ X c ρ ( β u c ,µ )] is the thermal vorticity. We see that the coupling term of the OAM andvorticity appear in Eq. (103). The second term in last line of Eq. (104) is related tothe Killing condition required by the thermal equilibrium of the spin.Now we consider the scattering process A + B → + s A , s B , s and s ( s i = ± / i = A , B , ,
2) respectively. For simplicity, we sum over s A , s B , s , and leave s open.Defining the direction of the reaction plane in the CMS as n c = ˆ b c × ˆ p c , A , we have,from Eq. (102), the polarization rate of particle 2 per unit time and unit volume is d P AB → ( X ) dX = − ( π ) ∫ d p A ( π ) E A d p B ( π ) E B d p c , ( π ) E c , d p c , ( π ) E c , ×| v c , A − v c , B | ∫ d k c , A d k c , B d k (cid:48) c , A d k (cid:48) c , B × φ A ( k c , A − p c , A ) φ B ( k c , B − p c , B ) φ ∗ A ( k (cid:48) c , A − p c , A ) φ ∗ B ( k (cid:48) c , B − p c , B )× δ ( ) ( k (cid:48) c , A + k (cid:48) c , B − p c , − p c , ) δ ( ) ( k c , A + k c , B − p c , − p c , )× ∫ d b c exp (cid:2) i ( k (cid:48) c , A − k c , A ) · b c (cid:3) b c , j [ Λ − ] ν j ∂ ( β u ρ ) ∂ X ν × (cid:2) p ρ A − p ρ B (cid:3) f A ( X , p A ) f B ( X , p B ) ∆ I AB → M n c , (105)where P AB → denotes the polarization vector. In the derivation of Eq. (105), wehave used Boltzmann distributions for f A ( X , p A ) f B ( X , p B ) with G G =
1. Thequantity ∆ I AB → M is defined as ∆ I AB → M = (cid:213) s A , s B , s , s (cid:213) color s M (cid:0) { s A , k c , A ; s B , k c , B } → { s , p c , ; s , p c , } (cid:1) ×M ∗ (cid:16) { s A , k (cid:48) c , A ; s B , k (cid:48) c , B } → { s , p c , ; s , p c , } (cid:17) . (106)Since we consider the polarization of quarks, there are seven processes involvedas shwon in Fig. 17. Evaluate all these diagrams will give more than 5000 terms.However, all these terms are spin-orbit coupling ones [2, 4] that have four types ofstructures: ( n × p ) · ˆ k A , ( n × p ) · ˆ k (cid:48) A , ( n × ˆ k A ) · ˆ k (cid:48) A and ( p × ˆ k A ) · ˆ k (cid:48) A . lobal polarization effect and spin-orbit coupling in strong interaction 35 Fig. 17
Tree level Feynman diagrams of all 2-to-2 parton scatterings. The final states contain atleast one quark. Here a and b denote the quark flavor, s i = ± / i = A , B , ,
2) denote the spinstates, k i ( i = A , B , ,
2) denote the momenta, q , q , q , q denote the momenta in propagators.The processes for antiquark are similar. Finally the polarization rate of quarks per unit time and unit volume in Eq. (105) canbe put into a compact form d P q ( X ) dX = π ( π ) ∂ ( β u ρ ) ∂ X ν (cid:213) A , B , ∫ d p A ( π ) E A d p B ( π ) E B | v c , A − v c , B |×[ Λ − ] ν j e c , i (cid:15) ikh ˆ p hc , A f A ( X , p A ) f B ( X , p B ) (cid:0) p ρ A − p ρ B (cid:1) Θ jk ( p c , A )≡ ∂ ( β u ρ ) ∂ X ν W ρν , (107)where the tensor W ρν , defined in the last line, contains 64 components, and each ofits component a is 16 dimensional integration.This is a major challenge in the numerical calculation. To handle this high dimen-sion integration, we split the integration into two parts: a 10-dimension (10D) inte-gration over ( p c , , p c , , k Tc , A , k (cid:48) Tc , A ) and a 6-dimension (6D) integration over ( p A , p B ) . We first carry out the 10D integration by ZMCintegral-3.0, a Monte Carlo integra-tion package that we have newly developed and runs on multi-GPUs [69]. Then wesave this 10D result Θ jk ( p c , A ) as a function of p c , A (and p c , B = − p c , A ). Finallywe perform the 6D integration using the pre-calculated 10D integral. The main pa-rameters are set to following values: the quark mass m q = . u , d , s , ¯ u , ¯ d , ¯ s ), the gluon mass m g = m g = m D = . t and u channel to regulate the possible divergence, the width α = .
28 GeV of the Gaussianwave packet, and the temperature T = . W ρν as W ρν = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) W e z − W e y − W e z W e x W e y − W e x (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , (108)or in a compact form W ρν = W (cid:15) ρν j e j . (109)Therefore Eq. (107) becomes d P q ( X ) dX = (cid:15) j ρν ∂ ( β u ρ ) ∂ X ν W e j = (cid:15) jkl ω kl W e j = W ∇ X × ( β u ) , (110)where ω ρν = −( / )[ ∂ X ρ ( β u ν ) − ∂ X ν ( β u ρ )] . We have constructed a microscopic model for the global polarization from particlescatterings in a many body system. The core of the idea is the scattering of particlesas wave packets so that the orbital angular momentum is present in the initial state ofthe scattering which can be converted to the spin polarization of final state particles.As an illustrative example, we have calculated the quark/antiquark polarization in aQGP. The quarks and gluons are assumed to obey the Boltzmann distribution whichsimplifies the heavy numerical calculation. There is no essential difficulty to treatquarks and gluons as fermions and bosons respectively.To simplify the calculation, we also assume that the quark distributions are thesame for all flavors and spin states. As a consequence, the inverse process is absentthat one polarized quark is scattered by a parton to two final state partons as wavepackets. So the relaxation of the spin polarization cannot be described without inverseprocesses and spin dependent distributions. We will extend our model by includingthe inverse process in the future. In Ref. [70], local and nonlocal collision termsin the Boltzmann equation for massive spin-1/2 particles in the Wigner functionapproach [71] have been derived for spin dependent distributions. The equilibrationof spin degrees of freedom can be fully described by such a spin Boltzmann equation.Nonlocal collision terms are found to be responsible for the conversion of orbital lobal polarization effect and spin-orbit coupling in strong interaction 37 g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q - × - - × - × - × - b ( fm ) W x ( G e V ) g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q × - × - × - b ( fm ) W y ( G e V ) g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q - × - - × - - × - b ( fm ) W z ( G e V ) g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q - × - - × - × - × - b ( fm ) W x ( G e V ) g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q - × - - × - - × - b ( fm ) W y ( G e V ) g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q - × - - × - - × - × - × - b ( fm ) W z ( G e V ) g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q - × - - × - - × - × - × - × - b ( fm ) W x ( G e V ) g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q - × - × - × - × - b ( fm ) W y ( G e V ) g + q → g + q q ( q ) + q → q ( q ) + qg + g → q + q - × - × - × - × - × - b ( fm ) W z ( G e V ) Fig. 18
Numerical results for components of W ρν . Here b is the cut-off of the impact parameterin the CMS of the scattering.8 Jian-Hua Gao, Zuo-Tang Liang, Qun Wang and Xin-Nian Wang into spin angular momentum. It can be shown that collision terms vanish in globalequilibrium and that the spin potential is equal to the thermal vorticity. Such aBoltzmann equation can be applied to parton collisions in quark matter. The global polarization of quarks and anti-quarks in QGP produced in non-centralHIC has different direct consequences. The most obvious and measurable effects isthe global polarization of hadrons produced after the hadronization of QGP. In [2],the global polarization of produced hyperons has been given. The spin alignment ofvector mesons has been calculated in [3].It is clear that the global hadron polarization depends not only on the global quarkpolarization but also on the hadronization mechanism. In the following, we discussthe results obtained in quark combination and fragmentation respectively.
For all hyperons belong to the J P = ( / ) + baryon octet except Σ , the polarizationcan be measured via the angular distribution of decay products in the correspondingweak decay. Such decay process is often called “spin self analyzing parity violatingweak decay". Because of this, hyperon polarizations are widely studied in the fieldof high energy spin physics.(i) Hyperon polarization in the quark combination
Different aspects of experimental data suggest that hadronization of QGP proceedsvia combination of quarks and/or anti-quarks. This mechanism is phrased as “quarkre-combination”, or “quark coalescence” or simply as “quark combination”. Wesimply refer it as “the quark combination mechanism” and use it to calculate thehyperon polarization in the following.In the quark combination mechanism, it is envisaged that quarks and anti-quarksevolve into constituent quarks and anti-quarks and combine with each other to formhadrons. We choose the minus direction of the normal of the reaction plane − n asthe quantization axis. The spin density matrix of quark or anti-quark is given by,ˆ ρ q = (cid:18) + P q
00 1 − P q (cid:19) . (111)We do not consider the correlation between the polarizations of different quarksand/or anti-quarks hence the spin density matrix for a q q q is given by,ˆ ρ q q q = ˆ ρ q ⊗ ˆ ρ q ⊗ ˆ ρ q . (112)Suppose a hyperon H is produced via the combination of q q q , we obtain, lobal polarization effect and spin-orbit coupling in strong interaction 39 ρ H ( m (cid:48) , m ) = (cid:205) m i , m (cid:48) i ρ q q q ( m (cid:48) i , m i )(cid:104) j H , m (cid:48) | m (cid:48) , m (cid:48) , m (cid:48) (cid:105)(cid:104) m , m , m | j H , m (cid:105) (cid:205) m , m i , m (cid:48) i ρ q q q ( m (cid:48) i , m i )(cid:104) j H , m | m (cid:48) , m (cid:48) , m (cid:48) (cid:105)(cid:104) m , m , m | j H , m (cid:105) , (113)where | j H , m (cid:105) is the spin wave function of H in the constituent quark model, and (cid:104) j H , m | m , m , m (cid:105) is the Clebsh-Gordon coefficient. The polarization of H is, P H = ρ H ( / , / ) − ρ H (− / , − / ) . (114)Since ˆ ρ q is diagonal so is ˆ ρ q q q , i.e. ρ q q q ( m (cid:48) i , m i ) = Π i ( + ˜ P q i ) δ m i , m (cid:48) i / P q i ≡ sign ( m i ) P q i , Eq. (113) reduces to, ρ H ( m (cid:48) , m ) = (cid:205) m i Π j ( + ˜ P q j )(cid:104) j H , m (cid:48) | m , m , m (cid:105)(cid:104) m , m , m | j H , m (cid:105) (cid:205) m , m i Π j ( + ˜ P q j )|(cid:104) j H , m | m , m , m (cid:105)| . (115)The remaining calculations are straight forward and we list the results in table 2. Itis also obvious that if P u = P d = P s ≡ P q , we obtain P H = P q for all hyperons. Table 2
Polarization of hyperons directly produced in the quark combination or fragmentationmechanism. The results for fragmentation are for the leading hadrons only where n s and f s infragmentation are the strange quark abundances relative to up or down quarks in QGP and quarkfragmentation, respectively. These results are taken from [2].hyperon Λ Σ + Σ Σ − Ξ Ξ − combination P s P u − P s ( P u + P d )− P s P d − P s P s − P u P s − P d fragmentation n s P s n s + f s f s P u − n s P s ( f s + n s ) f s ( P u + P d )− n s P s ( f s + n s ) f s P d − n s P s ( f s + n s ) n s P s − f s P u ( n s + f s ) n s P s − f s P d ( n s + f s ) (ii) Hyperon polarization in the quark fragmentation
In the high p T region, hadron production is dominated by the quark fragmentationmechanism, described by quark fragmentation functions defined via the quark-quarkcorrelator such as, D ( z ) = (cid:213) S h ∫ d ξ − π e − i ξ − p + h / z Tr γ + (cid:104) |L( , + ∞) ψ ( )| p h , S h , X (cid:105)× (cid:104) p h , S h , X | ¯ ψ ( ξ )L( ξ, + ∞)| (cid:105) , (116)which is the number density of hadron h produced in the fragmentation process q → h + X ; z = p + h / p + is the momentum fraction of quark q carried by hadron h ,where p and p h denote the momenta of q and h respectively. Here the light conecoordinate is used and the superscript + denotes the + component. L is the gaugelink that originates from the multiple gluon scattering and guarantees the gaugeinvariance. The polarization transfer is described by G ( z ) = ∫ d ξ − π e − i ξ − p + h / z Tr γ γ + (cid:104) | ψ ( )| p h , + , X (cid:105)(cid:104) p h , + , X | ¯ ψ ( ξ )| (cid:105) , (117) H T ( z ) = ∫ d ξ − π e − i ξ − p + h / z Tr γ T γ + (cid:104) | ψ ( )| p h , + T , X (cid:105)(cid:104) p h , + T , X | ¯ ψ ( ξ )| (cid:105) , (118)for the longitudinal and transverse polarization respectively; the + or + T in | p h , S h , X (cid:105) represents that the spin of h is in the S hz = + / S hT = + / γ or γ T = γ · n T introducesthe dependence on the spin of the fragmenting quark q .Fragmentation functions are best studied in e + e − annihilations. They can not becalculated using pQCD so currently we have to rely on parameterizations or models.There are still not much data available yet. For longitudinal polarization, we havedata from LEP at CERN for Λ polarization [72, 73]. A recent parameterizationof G can be found in [78]. For the transversely polarized case, little data and noparameterization of H T is available. ALEPHOPAL
Fig. 19
Longitudinal polarization of Λ in e + e − → Λ + X as described by using a parameterizationof G L ( z ) . The data points are from experiments at LEP [72, 73]. This figure is taken from [78]. To get a feeling of the z -dependence of the spin transfer in quark fragmentations,we show the fit obtained in [78] to the LEP data in Fig. 19. We see that, although theaccuracy is still need to be improved, it is definite that there is a strong z -dependenceof G and the spin transfer G / D is usually significantly smaller than unity. Thisimplies that the hyperon polarization obtained in the fragmentation mechanismshould be much smaller than that obtained in the combination case.In [2], a model estimation was made for the polarization of the leading hyperonproduced in the fragmentation of a polarized quark. It was assumed that two unpolar-ized quarks are created in the fragmentation and they combine with the polarized q to form the leading hyperon. In this case, we obtain the results as given in table 2. Wesee if n s = f s the result from fragmentation is just 1 / lobal polarization effect and spin-orbit coupling in strong interaction 41 Vector meson spin alignment can also be measured via angular distribution of decayproducts in the strong two body decay V → + Vector meson alignment in the quark combination
Similar to q q q , we do not consider the correlation between polarizations ofquarks and anti-quarks, and obtain the spin density matrix for a q ¯ q -system as,ˆ ρ q ¯ q = ˆ ρ q ⊗ ˆ ρ ¯ q . (119)The spin density matrix for a vector meson V produced via the combination of q ¯ q is given by, ρ Vm (cid:48) m = (cid:205) m i , m (cid:48) i ρ q ¯ q ( m (cid:48) i , m i )(cid:104) j V , m (cid:48) | m (cid:48) , m (cid:48) (cid:105)(cid:104) m , m | j V , m (cid:105) (cid:205) m , m i , m (cid:48) i ρ q ¯ q ( m (cid:48) i , m i )(cid:104) j V , m | m (cid:48) , m (cid:48) (cid:105)(cid:104) m , m | j V , m (cid:105) , (120)where | j V , m (cid:105) is the spin wave function of V in the constituent quark model. Fordiagonal ˆ ρ q and ˆ ρ ¯ q , we have, ρ Vm (cid:48) m = (cid:205) m i ( + ˜ P q )( + ˜ P ¯ q )(cid:104) j V , m (cid:48) | m , m (cid:105)(cid:104) m , m | j V , m (cid:105) (cid:205) m , m i ( + ˜ P q )( + ˜ P ¯ q )|(cid:104) j V , m | m , m (cid:105)| , (121)The spin alignment is described by ρ V and is obtained as [3], ρ V = − P q P ¯ q + P q P ¯ q . (122)From Eq. (122), we see clearly that the global vector meson spin alignment ρ V obtained in quark combination should be less than 1 /
3. We also see that in contrastto the hyperon polarization P H , ρ V is a quadratic effect of P q .(ii) Vector meson spin alignment in the quark fragmentation
To define the fragmentation functions for spin-1 hadrons in q → V + X , one usuallydecomposes the 3 × ρ in terms of the 3 × Σ i and Σ ij = ( Σ i Σ j + Σ j Σ i ) − δ ij , i.e., ρ = ( + S i Σ i + T ij Σ ij ) , (123)where the spin polarization tensor T ij = Tr ( ρ Σ ij ) and is parameterized as, T = (cid:169)(cid:173)(cid:171) − S LL + S xxTT S xyTT S xLT S xyTT − S LL − S xxTT S yLT S xLT S yLT S LL (cid:170)(cid:174)(cid:172) . (124) The spin alignment ρ is directly related to S LL by ρ = ( − S LL )/ S LL = (cid:104) Σ z (cid:105)/ − S LL -dependence is given by, D LL ( z ) = (cid:213) λ (− ) λ + ∫ d ξ − π e − i ξ − p + Tr γ + (cid:104) | ψ ( )| p h λ X (cid:105)(cid:104) p h λ X | ¯ ψ ( ξ )| (cid:105) , (125)where λ = ± , D LL ( z ) in fact does not depends on the spin of the fragmenting quark q .There are data available on the vector meson spin alignment from experiments atLEP [74, 75, 76]. A parameterization of D LL ( z ) is given in [79, 80] and the fit tothe data is shown in Fig. 20. OPAL
Fig. 20
Spin alignment of K ∗ in e + e − → K ∗ + X as described by using a parameterization of D LL ( z ) . The data points are from experiments at LEP [74, 75]. This figure is taken from [79]. From Fig. 20, we see clearly that, in contrast to quark combination mechanism, ρ obtained in fragmentation is larger than 1 /
3. This indicates that the spin of ¯ q produced in the fragmentation q → h + X has larger probability to be in the oppositedirection as q . For the leading meson, a parameterization of P ¯ q = − β P q (where β ∼ .
5) for the anti-quark ¯ q produced in the fragmentation process and combinewith the fragmenting quark to form the vector meson was obtained [77] to fit thedata [74, 75]. Ref. [3] also made an estimation for such leading vector mesons infragmentation based on the this empirical relation and obtained that, ρ V = ( + β P q )/( − β P q ) . (126)We see that the spin alignment ρ V obtained this way is indeed larger than 1 / lobal polarization effect and spin-orbit coupling in strong interaction 43 It is clear that final state hadrons in a high energy reaction usually contain thecontributions from decays of heavier resonances in particular those from strongand electromagnetic decays. To compare with the data, we need to take such decaycontributions into account.The decay contributions have influences both on the momentum distribution andon the polarization of final hadrons. Such influences have been discussed repeatedlyin literature calculating hyperon polarizations in high energy reactions (see e.g. [81,82, 83, 84] and recently in HIC [85, 86]). For hadrons consisting of light flavorsof quarks, we usually consider only the production of J P = ( / ) + octet and J P = ( / ) + decuplet baryons, and J P = − pseudo-scalar and J P = − vector mesons. Inthis case, there is no decay contribution to vector mesons. We only need to considerthose to hyperons and most of them are just two body decay H j → H i + M where H j and H i are two hyperons and M is a pseudo-scalar meson. We limit our discussionsto this process in the following.To be explicit, we consider the fragmentation mechanism and study decay contri-butions to fragmentation functions. For quark combination, we need only to replacethe fragmentation function by the corresponding distribution function and z by thecorresponding variable. We start with the unpolarized case and the contribution from H j → H i + M to the unpolarized fragmentation function of H i is given by, D ij ( z i , p Ti ) = Br ( H i , H j ) ∫ dz j d p T j K ji ( z i , p Ti ; z j , p T j ) D j ( z j , p T j ) , (127)where Br ( H i , H j ) is the decay branch ratio. K ji ( z i , p Ti ; z j , p T j ) is a kernel functionrepresenting the probability for a H j with ( z j , p T j ) to decay into a H i with ( z i , p Ti ) . Itis just the normalized distribution of H i from H j → H i + M and should be determinedby the dynamics of the decay process. However, in the unpolarized case, for two bodydecay, it is determined completely by the energy momentum conservation.From energy conservation, we obtain that, in the rest frame of H j , E ∗ i = ( M j + M i − M m )/ M j ≡ E ∗ , (128) | p ∗ | = λ / ( M j , M i , M m )/ M j ≡ p ∗ , (129)where the λ -function is λ ( x , y , z ) = x + y + z − x y − y z − zx . We see thatthe magnitude of p ∗ is completely fixed. Furthermore, because there is no specifieddirection in the initial state, the decay product should be distributed isotropically.Hence, in the Lorentz invariant form, the distribution of H i from H j → H i + M isgiven by, E i d Nd p i = M j πλ / ( M j , M i , M m ) δ (cid:16) ( p j − p i ) − M m (cid:17) . (130)By replacing variables p with z and p T , we obtain the kernel function K ji as, K ji ( z i , p Ti ; z j , p T j ) = d Ndz i d p Ti = z j M j πλ / ( M j , M i , M m ) δ (cid:18) ( p T j z j − p Ti z i ) + ( M j z j − M i z i ) − ∆ M − M m z i z j (cid:19) , (131)where ∆ M ≡ M j − M i is the mass difference between the two hyperons.In practice, we often use the following approximation. We note that the Lorentztransformation of the four-momentum of H i from the rest frame of H j to the labora-tory frame is given by, E i = ( E j E ∗ i + p j · p ∗ i )/ M j , (132) p i = p ∗ i + p j · p ∗ i + ( E j − M j ) E ∗ i M j ( E j − M j ) p j , (133)We take the average over the distribution of p ∗ i at given p j , and obtain, (cid:104) p i (cid:105) = p j ξ ij , ξ ij = ( M j + M i − M m )/ M j . (134)In the case that ∆ M (cid:28) M j ∼ M i and p ∗ (cid:28) | p i | , one can simply neglect thedistribution and take, p i ≈ (cid:104) p i (cid:105) = p j ξ ij so that z i ≈ z j ξ ij , p Ti ≈ p T j ξ ij and, K ij ( z i , p Ti ; z j , p T j ) ≈ δ ( z j − z i / ξ ij ) δ ( p T j − p Ti / ξ ij ) , (135) D ij ( z i , p Ti ) ≈ Br ( H i , H j ) D j ( z i / ξ ij , p Ti / ξ ij ) . (136)In the polarized case, we need also to consider the polarization transfer t ijD . Ingeneral, in the rest frame of H j , t ijD may depend on the momentum p ∗ i of H i . Bytransforming it to the Lab frame, we should obtain a result depending on ( z i , p Ti ) and ( z j , p T j ) and it is different for the longitudinal and transverse polarization. Thisis much involved. In practice, we often take the approximation by neglecting themomentum dependence and calculate t ijD in the rest frame of H j . In this case it isthe same for the longitudinal and transverse polarization. E.g., for the longitudinalpolarized case, we have, G ij L ( z i , p Ti ) = Br ( H i , H j ) t ijD ∫ dz j d p T j K ij ( z i , p Ti ; z j , p T j ) G j L ( z j , p T j ) . (137)Under the approximation given by Eq. (135), we have, G ij L ( z i , p Ti ) ≈ Br ( H i , H j ) t ijD G j L ( z i / ξ ij , p Ti / ξ ij ) , (138)For parity conserving decays, the polarization transfer factor t ijD can easily becalculated from angular momentum conservation. The results are given in table 3.For the weak decay Ξ → Λ π , t D = ( + γ )/ γ is a decay parameter that canbe found in Review of Particle Properties (see e.g. [87]). lobal polarization effect and spin-orbit coupling in strong interaction 45 Table 3
The decay spin transfer factor in parity conserving tow body decay H j → H i + M . Thefirst column specifies the spin and parity J P of hadrons. H j → H i + M relative orbital angular momentum t i jD = P H i / P H j / + → / + + − l = − / / − → / + + − l = / + → / + + − l = / − → / + + − l = − / If we taken only J P = ( / ) + hyperons into account and use spin counting forrelative production weights, we obtain P f inal Λ = P direct Λ [ + λ ( + γ )]/ ( + λ ) , (139)where λ is the strangeness suppression factor for s -quarks. This leads to a reductionfactor between 0 .
33 and 0 .
44 for λ = Σ ± or Ξ where decay influences are negligible. The novel predictions [2, 3] on GPE attracted immediate attention, both experi-mentally and theoretically. A new preprint [88] only three days after the first predic-tion [2] attempted to extend the idea to other reactions. Experimentalists in the STARCollaboration had started measurements shortly after the publication of theoreticalpredictions [2, 3], both on the global Λ hyperon polarization and on spin alignmentsof K ∗ and φ [89, 90, 91, 92, 93, 94, 95]. Studies on both aspects have advantagesand disadvantages. Hyperon polarization is a linear effect where the polarization fordirectly produced Λ is equal to that of quarks. The spin alignment of vector mesonis a quadratic effect proportional to the square of the quark polarization. Hence themagnitude of the latter should be much smaller than that of the former. However,to measure the polarization of hyperon, one has to determine the direction of thenormal of the reaction plane, which is not needed for measurements of vector mesonspin alignments. Also the contamination effects due to decay contributions to vectormesons are negligible but not for Λ hyperons.Although there were some promising indications, the results obtained in the earlymeasurements [94, 95] by the STAR Collaboration were consistent with zero withinlarge errors. STAR measurements continued during the beam energy scan (BES)experiments and positive results were obtained in lower energy region with improvedaccuracies [1]. The obtained value averaged over energy is 1 . ± . ± .
11 per centand 1 . ± . ± .
13 per cent for Λ and ¯ Λ respectively. With much higher statistics, the STAR Collaboration has repeated measurements [96] in Au-Au collisions at200AGeV and obtained positive result of P Λ ∼ − .
003 with much higher accuracies.To compare with experiments at this stage, we start with the following roughestimations: (i) From both Figs. 8 and 12 obtained using HIJING and BGK respec-tively, we obtain at Y ∼ ∆ p ∼ . ∆ x ∼ T ∼
140 MeV, ∆ p / T ∼ . P q is unfortunatelyin the small and rapidly changing region. Nevertheless, the order of magnitude isin the same range of STAR data [96]. (ii) If we take ω ∼ ∂ u z / ∂ x , u z ∼ (cid:104) p z (cid:105)/ p T ,we obtain, ω ∼ ∆ Y cosh Y ξ p /
12 from Eq. (22). By using the results for (cid:104) ξ p (cid:105) shownin Fig. 8 or Fig. 12 and T ∼
140 MeV, we obtain P q ∼ − .
003 at √ s = δ u ∼ | p |/ m q , δ x ∼ / µ D , sothat ω ∼ δ u / δ x ∼ µ D | p |/ m q , and quark polarization is P q ∼ πω / m q . If we takean effective quark mass m q ∼
200 MeV at the hadronization, this is clearly also ofthe same order of magnitude as ω / T .Such rough estimations are rather encouraging. We continue with more realisticestimations. We note that quark polarization is given by Eqs. (46-48) and d σ and d ∆ σ take the general form given by Eqs. (63) and (64). Before we construct a dynamicalmodel for f qq ( x T , b , Y , √ s ) , we present the following qualitative discussion.It is clear that at b = f qq ( x T , , Y , √ s ) should be independent of the direction of x T . The ˆ x T -dependent term should given by ˆ x T · b . We take the linearly dependentterm into account and have, f qq ( x T , b , Y , √ s ) = f qq ( x T , , Y , √ s ) + f qq ( x T , b , Y , √ s ) ˆ x T · b , (140)We insert Eq. (140) into (63) and (64) and obtain immediately that P q ∝ (cid:104) l ∗ y (cid:105) , i.e., P q = α ( b , Y , √ s )(cid:104) l ∗ y ( b , Y , √ s )(cid:105) . (141)We insert the result of (cid:104) l ∗ y (cid:105) given by Eq. (23) into (141), average over the impactparameter b and obtain, P q = − κ ( Y , √ s )(cid:104) p T (cid:105)(cid:104) ξ p (cid:105) . (142)where κ = α ( ∆ x ) ∆ Y /
24. The proportional coefficient α in Eq. (141) hence also (cid:60) inEq. (142) are very involved. They are determined by the dynamics in QGP formationand evolution. Averaged over b , κ can still be dependent of Y and √ s . In [35], thesimplest choice, i.e., κ is taken as a constant independent of √ s at Y =
0, was firstconsidered and obtained the energy dependence of P q shown in Fig. 21(a). Takingan energy dependent κ , Ref. [35] made a better fit to the data [1, 96] available asshown in Fig. 21(b).Fig. 22 shows the rapidity dependence of the polarization at different energiesobtained in [35] in the different cases. The Y -dependence of κ was obtained by assum-ing that the dependence is mediated by the chemical potential. The Y -dependence of (cid:104) p T (cid:105) was taken empirically [35]. See [35] for details. lobal polarization effect and spin-orbit coupling in strong interaction 47 (GeV) NN s − [ % ] H P =6.4 κ HS =8.4 κ Woods Saxon L (a) (GeV) NN s − [ % ] H P /200) NN s =2.14/(0.05+ κ HS /200) NN s =2.8/(0.05+ κ Woods Saxon (b)
Fig. 21
Energy dependence of the global polarization of Λ obtained by taking κ as a constant oran energy dependent form. The data points are taken from [1, 96]. This figure is taken from [35]. Y [ % ] H P
200 GeV62.4GeV39 GeV27 GeV19.6GeV14.5GeV11.5GeV9.2 GeV7.7 GeV (a) Y [ % ] H P
200 GeV62.4GeV39 GeV27 GeV19.6GeV14.5GeV11.5GeV9.2 GeV7.7 GeV (b) Y [ % ] H P
200 GeV62.4GeV39 GeV27 GeV19.6GeV14.5GeV11.5GeV9.2 GeV (c) Y [ % ] H P
200 GeV62.4GeV39 GeV27 GeV19.6GeV14.5GeV11.5GeV9.2 GeV (d)
Fig. 22
Rapidity dependence of the global polarization of Λ obtained in four different cases (a)neither κ nor (cid:104) p T (cid:105) depends on Y ; (b) κ depends on Y but (cid:104) p T (cid:105) does not; (c) κ does not but (cid:104) p T (cid:105) depends on Y ; (d) both κ and (cid:104) p T (cid:105) depend on Y . This figure is taken from [35]. To summarize, high energy HIC is usually non-central thus the colliding system andthe produced partonic system QGP carries a huge global orbital angular momentumas large as 10 (cid:126) in Au-Au collisions at RHIC energies. Due to the spin-orbit couplingin QCD, such huge orbital angular momentum can be transferred to quarks and anti-quarks thus leads to a globally polarized QGP. The global polarization of quarks and anti-quarks manifest itself as the global polarization of hadrons such as hyperonsand vector mesons produced in HIC.The early theoretical prediction [2] and discovery by the STAR Collaboration [1]open a new window to study properties of QGP and a new direction in high energyheavy ion physics. Similar measurements have been carried in other experimentssuch as those by ALICE Collaboration at the Large Hadron Collider (LHC) in Pb-Pb collisions [97]. Other efforts have also been made on measurements of vectormeson spin alignments [98, 99]. The STAR Collaboration has just finished majordetector upgrades and started the beam energy scan at phase II (BES II). Thesuccessful detector upgrade with improved inner time projection chamber (iTPC)and event plane detector (EPD) will be crucial to the measurements of global hadronpolarizations. The STAR BES II will provide an excellent opportunity to study GPEin HIC and we expect new results with higher accuracies in next years.The experimental efforts in turns further inspire theoretical studies. The rapidprogresses and continuous studies along this line lead to a very active researchdirection – the Spin Physics in HIC in the field of high energy nuclear physics.Among the most active aspects, we have in particular the following.(i) GPE phenomenology
This includes different model approaches [59, 100, 60, 101, 102, 103, 104, 105,106, 107, 108, 109, 110, 111, 35] to numerical calculations of GPE in HIC and itsdependences on different kinematic variables. The model approaches are basicallydivided into two categories, i.e., microscopic approaches based on the spin-orbit(or spin-vorticity) coupling and hydrodynamic approaches based on equilibriumassumptions. The various dependences of GPE are studied on kinematic variablesdescribing (a) the initial state such as energy, centrality (impact parameter), differentincident nuclei even pA collisions etc; (b) the produced hadron such as transversemomentum, rapidity, azimuthal angle, different types of hyperons and/or vectormesons; (c) other related measurable effects such as longitudinal polarization, theinterplay with other effects and so on. Short summaries can e.g. be found in plenarytalks given at recent Quark Matter conferences [112, 113].(ii) Spin-vortical effects in strong interacting system
If we can treat QGP as a vortical ideal fluid consisting of quarks and anti-quarks,the global polarization of hadrons is directly related to the vorticity of the system [85].The fluid vorticity may be estimated from the data [1] on GPE of Λ hyperon using therelation given in the hydro-dynamic model, and it leads to a vorticity ω ≈ ( ± ) × s − . This far surpasses the vorticity of all other known fluids. It was thereforeconcluded that QGP created in HIC is the most vortical fluid in nature observed yet.GPE in HIC therefore provides a very special place to study spin-vortical effects instrong interaction and attracts many studies [59, 60, 61, 62, 63, 64, 65, 66, 67]. Seechapter ... for discussions in this aspect.(iii) Spin-magnetic effects in HIC
Because of the huge orbital angular momentum, there exists also a very strongmagnetic field for the colliding system in HIC. In Au-Au collision at RHIC, it canreach at least instantaneously the order 10 − Tesla. Such a strong magnetic lobal polarization effect and spin-orbit coupling in strong interaction 49 field can manifest itself in different aspects and lead to different measurable effects.The most frequently discussed currently are the following three aspects.(a) The fine structure of GPE of different hadrons. The spin-orbit coupling inQCD predicts e.g. the same polarization of quarks and anti-quarks thus also the samefor hyperons and anti-hyperons. The strong magnetic field can lead to differencesbetween the polarization of quarks and that of anti-quarks thus lead to difference inthe polarization of hyperons and anti-hyperons. Indeed, the STAR data in Ref. [1]suggests such a fine-structure pattern, and if errorbars are ignored, would indicate B ∼ T. However, much smaller uncertainties– available with the new BES-IIdata– will be needed to resolve the issue. Also the magnetic field may lead to differentbehavior of vector meson spin alignment [117].(b) Chiral magnetic effect. In Ref. [114, 115], a novel electromagnetic spin effect– the chiral magnetic effect was proposed. It was argued that such effects havedeep connection to P and CP violation. Clearly, if they exist, strong magnetic filedin HIC provides good opportunity to detect such effects [116]. This has attractedmuch attention both experimentally and theoretically. See a number of reviews suchas [118, 119, 120, 121], plenary talks at QM2019 by Xu-guang Huang and Jin-fengLiao [122, 123] and chapter ... in this series.(c) Spin-electromagnetic effects in ultra-peripheral collisions (UPC) in HIC. Froma field theoretical point of view, the electromagnetic coupling for a HIC is enhancedby a factor Z (number of protons in the nucleus). Hence, many electromagneticeffects become visible in UPC with nuclei of large A . This provides a good place tostudy the spin-electromagnetic effects and develop the theoretical methodology inparticular those developed in studying nucleon structure in the small- x region. Seee.g. the plenary talk at QM2019 by Zhangbu Xu [124] for a brief summary.(iv) Spin transport theory in relativistic quantum system
Theoretically, a very challenging task is to derive GPE, describe the spin transport,calculate the polarization and other related spin effects directly from QCD. This israther involved since, to describe orbital angular momentum or vorticity of thesystem, not only momentum but also space coordinate are needed. It seems thatquantum kinetic theory based on the Wigner function formalism [125, 126, 127, 128]is very promising thus has attracted much attention recently. Many progresses havebeen made. Besides others, the local polarization effect has been first derived [129]and a disentanglement theorem [131] in the massless case has proposed. It has nowextended to massive case [133, 134, 135] and has been shown that different spineffects can indeed be derived. See chapter ... for more discussions in this aspect.
Acknowledgements
We thank in particular many collaborators for excellent collaborations on this subject.This work was supported in part by the National Natural Science Foundation of China(Nos. 11890713, 11675092, 11535012, 11935007, 11861131009 and 11890714),and by the Director, Office of Energy Research, Office of High Energy and Nuclear
Physics, Division of Nuclear Physics, of the U.S. Department of Energy under grantNo. DE-AC02-05CH11231, the National Science Foundation (NSF) under grant No.ACI-1550228 within the framework of the JETSCAPE Collaboration.
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