arXiv:1704.04022v1 [nucl-th] 13 Apr 2017

Nuclear Physics A Nuclear Physics A 00 (2017) 1–8 www.elsevier.com/locate/procedia

arXiv:1704.04022v1 [nucl-th] 13 Apr 2017

XXVIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2017)

Global and local spin polarization in heavy ion collisions: a brief overview Qun Wang Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

Abstract We give a brief overview about recent developments in theories and experiments on the global and local spin polarization in heavy ion collisions. Keywords: global polarization, spin-orbital coupling, vorticity, angular momentum, heavy-ion collision

1. Introduction There is inherent correlation between rotation and polarization in materials as shown in the Barnett effect [1] and the Einstein-de Haas effect [2]. We expect that the same phenomena also exist in heavy ion collisions. Huge global angular momenta are generated in non-central heavy ion collisions at high energies [3–8]. How such huge global angular momenta are transferred to the hot and dense matter created in heavy ion collisions and how to measure them are two core questions in this field. There are some models to address the first question: the microscopic spin-orbital coupling model [3, 4, 8, 9], the statistical-hydro model [10–16] and the kinetic model with Wigner functions [17–20]. For the second question, it was proposed that the global angular momentum can lead to the local polarization of hadrons, which can be measured by the polarization of Λ hyperons and vector mesons [3, 4]. The global polarization is the net polarization of local ones in an event which is aligned in the direction of the reaction plane. Recently the STAR collaboration has measured the global polarization of Λ hyperons in the beam energy scan program [21, 22]. At all energies below 62.4 GeV, positive polarizations have been ¯ On average over all data, the gobal polarization for Λ and Λ ¯ are ΠΛ = (1.08 ± 0.15)% found for Λ and Λ. and ΠΛ¯ = (1.38 ± 0.30)%. This indicates that the matter created in ultra-relativistic heavy ion collisions is the most vortical fluid ever produced in the laboratory. In this note, we give a brief overview about recent developments in theories and experiments on the global and local spin polarization in heavy ion collisions.

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Qun Wang / Nuclear Physics A 00 (2017) 1–8

2. Theoretical models in particle polarization 2.1. Global orbital angluar momentum and local vorticity Let us consider two colliding nuclei with the beam momentum per nucleon pbeam ≡ pbeam ez (projectile) and −pbeam (target). The impact parameter b ≡ be x whose modulus is the transverse distance between the centers of the projectile and target nucleus points from the target to the projectile. The normal direction of the reaction plane or the direction of the global angular momentum is along bˆ × pˆ beam = −ey . The magnitude of the total orbital angular momentum Ly and the resulting longitudinal fluid shear can be estimated within the wounded nucleon model of particle production [3, 8]. The transverse distributions (integrated over y) of participant nucleons in each nucleus can be written as P,T dNpart

dx

=

ˆ

dydzρP,T A (x, y, z, b),

(1)

where ρP,T A denotes the number of participant nucleons in the projectile and target, respectively. One can use models to estimate ρP,T A such as the hard-sphere or Woods-Saxon model. Then we obtain  P ˆ P   dNpart dNpart  Ly = −pin dxx  − (2)  . dx dx 

The average collective longitudinal momentum per parton can be estimated as pz (x, b;

√

s) = p0

P T dNpart /dx − dNpart /dx

P /dx + dN T /dx dNpart part

,

(3)

√ where p0 = s/[2c(s)] denotes the maximum average longtitudinal momentum per parton. The average relative orbital angular momentum for two colliding partons separated by ∆x in the transverse direction is then ly ≡ −(∆x)2 d pz /dx. Note that ly is proportional to the local vorticity. As we all know that the strongly coupled quark gluon plasma (sQGP) can be well described by relativistic hydrodynamic models. So the sQGP can be treated as a fluid which is characterized by local quantities such as the momentum, energy and particle-number´ densities p(r), ǫ(r) and n(r), respectively. The total angluar momentum of a fluid can be written as L = d3 r r × p(r). The fluid velocity is defined by v(r) = p(r)/ǫ(r). In non-relativistic theory, the fluid vorticity is defined by ω = 12 ∇ × v(r). For a rigid-body rotation with a constant angular velocity ω, ¯ the velocity of a point on the rigid body is given by ¯ × r) = ω, ¯ i.e. for a rigid body in rotation the vorticity is v = ω ¯ × r. We can verify that ω = 12 ∇ × (ω identical´to the angular momentum. With the local vorticity, the total angluar momentum can be re-written as L = d3 r ǫ(r)[r2 ω − (ω · r)r]. We see that L is an integral of the moment of inertia density and the local vorticity. The time evolution of the local velocity and vorticity field can be simulated through the hydrodynamic model [23–25], the AMPT model [26, 27] or the HIJING model with a smearing technique [28]. 2.2. Spin-orbital coupling model We first consider a simple model for a spin-1/2 quark scattered in a static Yukawa potential V(r) = e−mD |r| /(4π|r|) with mD being the screening mass. The scattering amplitude is M(pi , λ′ → pf , λ) = Qu†λ (pf )V(q)uλ′ (pi ),

(4)

where V(q) = 1/(q2 +m2D ) is the Fourier transform of V(r) with q = pf −pi , Q denotes the coupling constant, and uλ′ (pi ) and uλ (pf ) are Dirac spinors of the quark before and after the scattering where (pi , λ′ ) and (pf , λ) are (4-momentum, spin) of the quark in the initial and final state, respectively. The spin-dependent cross section can be obtained 2 ˆ d3 pf 1 1X ′ M(p , λ → p , λ) σλ = (5) i f (2π)δ(Ef − Ei ), 2Ei vi 2 λ′ (2π)3 2Ef


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q where vi = |pi |/Ei and Ei = p2i + m2 . The polarized and total cross sections can thus be obtained by ∆σ = σ+ − σ− and σ = σ+ + σ− . In small angle scatterings, the corresponding differential cross sections are in the form d2 σ/d2 xT ∼ K0 (mD |xT |) and d2 ∆σ/d2 xT ∼ n · (xT × pi ), where xT is the impact parameter of the scattering in a small local cell [3]. We see that the polarized cross section is proportional to the spin-orbital coupling, n · (xT × pi ), where n is the spin quantization direction and L = xT × pi is the orbital angular momentum. The polarization of the particle for small angle scatterings in the non-relativistic limit can be cast into the form mD |pi | mD |pi | ∆ELS ∆σ (6) ∼ ∼ ∼ Π∼ σ Ei (Ei + m) E0 m2 which is proportional to the energy of the spin-ortibal coupling ∆ELS given by ∆ELS ∼ L · S

1 dV |pi | 1 · ∼ 2 (E0 m2D ) m2 r dr m mD

where E0 is an energy scale, L ∼ |pi |/mD is the angular momentum of the particle, r−1 dV/dr ∼ E0 m2D is the potential gradient divided by the typical range of the potential r ∼ 1/mD . One can elaborate the spin-orbital coupling model by considering a more realistic quark-quark scattering at a transverse distance of xT , whose polarized differential cross section is proportional to the spin-orbital coupling n · (xT × pi ), similar to the case of the static potential [8]. 2.3. Wigner function method For massive fermions, we can express its spin tensor density in terms of the Wigner function [19], D E y y 1 ¯ + ) lim Tr γ0 σαβ ψ(x − )ψ(x M αβ (x) = 2 y→0 2 2 ˆ h i 1 = d4 pTr γ0 σαβ W(x, p) . 2 Then we can define the spin tensor component of the Wigner function as i 1 h M αβ (x, p) ≡ Tr γ0 σαβ W(x, p) 2 i 1 h 0αβρ −ǫ Aρ + igα0 V β − igβ0 Tr(γα W) , = 2 If we take αβ = i j (spatial indices), we have a simple relation

(7)

(8)

1 i jk k ǫ A (x, p), (9) 2 where ǫi jk is 3-dimensional anti-symmetric tensor. We see that one can treat the axial vector component as the spin pseudo-vector phase space density. So the polarization (or spin) pseudo-vector density (with a factor 1/2) is [19] ˆ 1 µ d4 pA µ (x, p) (10) Π (x) ≈ 2 M i j (x, p) =

at the non-relativistic limit. To match the Pauli-Lubanski pseudo-vector, we should add a E p /m factor as, ˆ 1 µ Π (x) ≈ d4 pE p A µ (x, p). (11) 2m The axial component of the Wigner function can be solved in a perturbation method, whose zeroth and first order solution are Aα(0) = m θ(p0 )nα (p, n) − θ(−p0 )nα (−p, −n) δ(p2 − m2 )A, Aα(1) (x, p) =

1 ˜ ασ dV δ(p2 − m2 ) − ~Ω , pσ δ(p2 − m2 ) − Q~F˜ αλ pλ V 2 2 d(βp0 ) p − m2

(12)

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where V = f+ + f− and A = f+ − f− with the phase space distribution f s for the spin state s = ± being defined by f s (x, p) =

2 θ(p0 ) fFD (p0 − µ s ) + θ(−p0 ) fFD (−p0 + µ s ) , (2π)3

(13)

where p0 ≡ pµ uµ with uµ being the fluid velocity, fFD is the Fermi-Dirac distribution function, and µ s is the chemical potential corresponding to the spin state s. In Eq. (12), the 4-vector of the spin quantization direction is given by ! n·p (n · p)p µ µ ν n (p, n) = Λ ν (−v p )n (0, n) = , (14) ,n+ m m(m + E p ) where Λµν (−v p ) is the Lorentz transformation with v p = p/E p and nν (0, n) = (0, n) is the spin quantization direction in the rest frame of the fermion. We note that the polarization pseudo-vector density at the zeroth order is vanishing if µ s does not depend on the spin s. The polarization density at the first order is obtained by integration over 4-momentum for Aα(1) (x, p), Πα(1)

≈

( ˆ i d3 p h eβ(E p −µ) 1 α α ~β E ω + QB p 2m (2π)3 [eβ(E p −µ) + 1]2 ) β(E +µ) i h e p , + E p ωα − QBα β(E +µ) [e p + 1]2

(15)

where Q > 0 is the fermion’s electric charge. The momentum spectra of the polarization pseudo-vector at the freezout hypersurface can be obtained ˆ dΠα (p) 1 ~ Ep 3 β ≈ dΣλ pλ 2m (2π)3 d p ˜ ασ pσ ± QF˜ ασ uσ f ± (x, p) 1 − f ± (x, p) , (16) × Ω FD FD

± where fFD are Dermi-Dirac distribution functions for fermions (+) and anti-fermions (−), respectively, and ˜ ξη = 1 ǫ ξηνσ Ωνσ Σλ denotes the freezeout hypersurface. In Eqs. (15,16), we have used F˜ ρλ = 21 ǫ ρλµν Fµν , Ω 2 with Ωνσ = 12 (∂ν uσ − ∂σ uν ), where ǫ µνσβ and ǫµνσβ are anti-symmetric tensors with ǫ µνσβ = 1(−1) and ǫµνσβ = −1(1) for even (odd) permutations of indices 0123, so we have ǫ 0123 = −ǫ0123 = 1. Instead of Ωνσ , ˜ ξη , Fµν and F˜ ρλ , we will also use the vorticity vector ωρ = 1 ǫ ρσαβ uσ ∂α uβ , the electric field E µ = F µν uν , Ω 2 and the magnetic field Bµ = 21 ǫ µνλρ uν Fλρ .

2.4. Statistical-hydro model The polarization of a partical in a locally rotating fluid can be described by the statistical-hydro model. The derivation of relativistic hydrodynamics in quantum statistical theory was proposed in late 1970s [11] and early 1980s [10] and further developed by several authors [12–16]. With the density operator, h one ican calculate the energy-momentum tensor and current as functions of space-time, T µν (x) = Tr ρˆ Tˆ µν (x) ≡ D E h i D E Tˆ µν (x) and jµ (x) = Tr ρˆ ˆjµ (x) ≡ ˆjµ (x) . One can employ the principle of maximum entropy to derive the density operator at local equilibrium. We then use Lagrange multiplier to maximize the entropy under the condition of fixed T µν (x) and jµ (x), ˆ nhD E i dΣµ Tˆ µν (x) − T µν (x) βν (x) S = Tr (ρˆ ln ρ) ˆ + Σ(τ) hD E i o − ˆjµ (x) − jµ (x) ζ(x) , (17)


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where Σµ = Σnµ is the space like hypersurface with nµ being the time-like vector, βν = βuν with uν being the fluid velocity. in which leads to ρˆ LE at local equilibrium (LE), "ˆ # 1 (18) dΣµ −T µν βν + ζ ˆjµ . ρˆ LE = exp Z Σ(τ) D E Given nµ , one can determine the local equilibrium value of βα and ζ by nµ Tˆ µν (x) LE = nµ T µν (x) and D E nµ ˆjµ (x) = nµ jµ (x). LE The global equilibrium of the fluid can be found by imposing the stationary condition under which the operator does not depend on a particular choice of space-like hypersurface Σ, so we have ´ density ´ µ µ ˆ ˆ ˆ µ ≡ −Tˆ µν βν + ζ ˆjµ , or in another form dΣ Φ = dΣ Φ , where Φ µ µ Σ1 Σ2 ˛ ˆ µ ˆ ˆ µ = 0, dΣµ Φ = d4 x∂µ Φ (19) Σ1 +Σ2 +ΣT

V

where ΣT is the transverse surface to Σ1 and Σ2 . The above equation leads to ˆµ ∂µ Φ

1 − Tˆ µν (∂µ βν + ∂ν βµ ) + (∂µ ζ) ˆjµ = 0. 2

=

So we obtain the stationary conditions ∂µ βν + ∂ν βµ = 0, ∂µ ζ = 0,

(20)

where the former condition is called the Killing condition whose solution is in the form βµ = βuµ + ̟µν xν , where ̟µν = − 21 (∂µ βν − ∂ν βµ ). So we obtain the density operator at global equilibrium # " 1 1 ρˆ GE = exp −βuν Pˆ ν + Jˆνρ ̟νρ + ζ Qˆ , (21) Z 2 ´ ´ ´ where Pˆ ν = Σ dΣµ Tˆ µν , Jˆνρ = Σ dΣµ (xν Tˆ µρ − xρ Tˆ µν ) and Qˆ = Σ dΣµ ˆjµ . We can also add the spin tensor to the angular momentum tensor density Sˆ µ;νρ : ˆ νρ ˆ dΣµ (xν Tˆ µρ − xρ Tˆ µν + Sˆ µ;νρ ) J = Σ

=

νρ JˆOAM + JˆSνρ .

(22)

1 ˆS ˆ The spin tensor JˆSνρ gives the Pauli-Lubanski pseudo-vector (spin vector), Sˆ µ = − 2m Jνρ Pσ , which satisfies µ ˆν µˆ µˆ ˆ ˆ ˆ [S , P ] = 0, S Pµ = 0 and S S µ = −S (S + 1) with S is spin quantum number of the particle. The expectation value of spin vector is given by S µ = Tr(ρˆ GE Sˆ µ ). Then the polarization is obtained by Πµ = S µ /S . Since the spin pseudo-vector Sˆ µ invloves the momentum operator, we need to know a particle’s momentum to evaluate its polarization. In general, this requires the knowledge of the Wigner function, which allows to express the mean values of operators as integrals over space-time and 4-momentum. The mean spin pseudo-vector of a spin-1/2 particle with 4-momentum pµ , produced at xµ on particlization hypersurface, at the leading order in the thermal vorticity reads [14, 29]

Πµ (x, p) = −

1 [1 − fFD (x, p)]ǫ µνσρ pν ̟σρ , 8m

(23)

where fFD (x, p) is the Fermi-Dirac distribution function. The mean polarization of the particle with 4momentum pµ over the particlization hypersurface is given by ´ dΣρ pρ fFD (x, p)Πµ (x, p) µ ´ Π (p) = . (24) dΣρ pρ fFD (x, p) Note that at a constant temperature, Eqs. (23,24) are consistent to Eq. (16) [19].

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3. Experimental measurements of global polarization The global polarization can be measured by the Λ hyperon’s weak decay into a proton and a negatively charged pion. Due to its nature of weak interaction, the proton is emitted preferentially along the direction of the Λ’s spin in the Λ’s rest frame, so the parity is broken in the decay process. In this sense, we say that Λ is self-analyzing since we can determine the Λ’s polarization by measuring the daughter proton’s momentum [30]. The solid angle distribution for the daughter proton in the Λ’s rest frame is given by dN dΩ∗

=

1 1 (1 + αH ΠΛ cos θ∗ ) , 1 + αH pˆ ∗p · ΠΛ = 4π 4π

where pˆ ∗p is the direction of the daughter proton’s momentum in the Λ’s rest frame, ΠΛ is the Λ’s polarization vector with its modulus ΠΛ < 1, θ∗ is the angle between the momentum of the daughter proton’s and that of Λ, and αH = 0.642 ± 0.013 is the Λ’s decay constant measured in experiments. The Λ’s polarization can be determined by an event average of the proton’s momentum direction in the Λ’s rest frame, ΠΛ =

3 hcos θ∗ iev . αH

(25)

We assume the beam direction is along ez , pˆ beam = (0, 0, 1), and the direction of the impact parameter is bˆ = (cos ψRP , sin ψRP , 0) where ψRP is the azimuthal angle of the reaction plane. The global polarization L is along bˆ × pˆ beam = (sin ψRP , − cos ψRP , 0). The direction of the daughter proton’s momentum in the Λ’s rest frame is assumed to be pˆ ∗p = (sin θp∗ cos φ∗p , sin θp∗ sin φ∗p , cos θp∗ ). If ΠΛ is in the direction of the global polarization L, we have ˆ Λ = sin θp∗ sin(ψRP − φ∗p ). cos θ∗ = pˆ ∗p · Π (26) We can obtain the proton’s distribution in φ∗p after an integration over θp∗ , dN dφ∗p

π

=

ˆ

=

1 1 + αH ΠΛ sin(ψRP − φ∗p ). 8 8

dθp∗ sin θp∗

0

dN dΩ∗ (27)

Then we obtain ΠΛ by taking an event average of sin(ψRP − φ∗p ) [22], ΠΛ = −

E 8 D sin(φ∗p − ψRP ) . ev παH

(28)

The above equation is similar to that used in directed flow measurements [31–33], which allows us to use the corresponding anisotropic flow measurement technique [34, 35]. The reaction plane angle in Eq. (28) is estimated by calculating the angle of the first order event plane, so we need to correct the final results by the reaction plane resolution R(1) EP . Then we can rewrite Eq. (28) in terms of the first-order event plane angle (1) [22], and its resolution R Ψ(1) EP EP ΠΛ = −

8 παH R(1) EP

D

E sin φ∗p − Ψ(1) . EP ev

(29)

The first-order event plane angle is estimated experimentally by measuring the sidewards deflection of the forward- and backward-going fragments and particles in the STAR’s BBC detectors. The STAR’s recent measurements for the global Λ polarization at all collisional energies in the Beam Energy Scan (BES) program are shown in Fig. 1. At each energy, a positive polarization at the level of ¯ Taking all data at different energies into account, the global polarization (1.1−3.6)σ is observed for Λ and Λ. ¯ are ΠΛ = (1.08±0.15)% and ΠΛ¯ = (1.38±0.30)% respectively. It seems that the Λ’s ¯ polarization for Λ and Λ is larger than the Λ’s. Such a difference may possibly due to the contributions from their magnetic moments with an opposite sign. But this difference is indistinguishable within the range of errors. So the magnetic


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Fig. 1. STAR results for the global Λ polarization.

fields extracted from the data are consistent to zero within error bars. Another source of difference may possibly be due to more Pauli blocking effect for fermions than anti-fermions in lower collisional energies where fermions have non-vanishing chemical potentials [19, 36]. But still such a difference is too small to be observed within error bars. The global polarization decreases with the collisional energy since the Bjorken scaling works better at higher energies than lower energies [27, 37]. The fluid vorticity can be estimated from the data by the hydro-statistical model ω ≈ T (ΠΛ + ΠΛ¯ ), where T is the temperature of the fluid at the moment of particle freezeout. The polarization data averaged over collisional energies imply that the vorticity is about (9 ± 1) × 1021 s−1 . This is much larger than any other fluids that exist in the universe. Then the sQGP created in heavy ion collisions is not only the hottest, least viscous, but also the most vortical fluid that is ever produced in the laboratory. Acknowledgment. QW thanks M. Lisa and F. Becattini for helpful discussions. QW is supported in part by the Major State Basic Research Development Program (MSBRD) in China under the Grant No. 2015CB856902 and 2014CB845402 and by the National Natural Science Foundation of China (NSFC) under the Grant No. 11535012. References [1] S. Barnett, Gyromagnetic and Electron-Inertia Effects, Rev. Mod. Rev. 7 (2) (1935) 129. doi:10.1103/RevModPhys.7.129. [2] A. Einstein, W. de Haas, Experimenteller Nachweis der Ampereschen Molekularstroeme, Deutsche Physikalische Gesellschaft, Verhandlungen 17 (1915) 152. [3] Z.-T. Liang, X.-N. Wang, Globally polarized quark-gluon plasma in non-central A+A collisions, Phys. arXiv:nucl-th/0410079, Rev. Lett. 94 (2005) 102301, [Erratum: Phys. Rev. Lett.96,039901(2006)]. doi:10.1103/PhysRevLett.94.102301,10.1103/PhysRevLett.96.039901. [4] Z.-T. Liang, X.-N. Wang, Spin alignment of vector mesons in non-central A+A collisions, Phys. Lett. B629 (2005) 20–26. arXiv:nucl-th/0411101, doi:10.1016/j.physletb.2005.09.060. [5] S. A. Voloshin, Polarized secondary particles in unpolarized high energy hadron-hadron collisions?arXiv:nucl-th/0410089. [6] B. Betz, M. Gyulassy, G. Torrieri, Polarization probes of vorticity in heavy ion collisions, Phys. Rev. C76 (2007) 044901. arXiv:0708.0035, doi:10.1103/PhysRevC.76.044901. [7] F. Becattini, F. Piccinini, J. Rizzo, Angular momentum conservation in heavy ion collisions at very high energy, Phys. Rev. C77 (2008) 024906. arXiv:0711.1253, doi:10.1103/PhysRevC.77.024906. [8] J.-H. Gao, S.-W. Chen, W.-t. Deng, Z.-T. Liang, Q. Wang, X.-N. Wang, Global quark polarization in non-central A+A collisions, Phys. Rev. C77 (2008) 044902. arXiv:0710.2943, doi:10.1103/PhysRevC.77.044902. [9] S.-w. Chen, J. Deng, J.-h. Gao, Q. Wang, A General derivation of differential cross-section in quark-quark scatterings at fixed impact parameter, Front. Phys. China 4 (2009) 509–516. arXiv:0801.2296, doi:10.1007/s11467-009-0064-0 .

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