Improved quark coalescence model for spin alignment and polarization of hadrons
IImproved quark coalescence model for spin alignment and polarization of hadrons
Xin-Li Sheng, Qun Wang, and Xin-Nian Wang
2, 3 Peng Huanwu Center for Fundamental Theory and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan, Hubei 430079, China Nuclear Science Division, MS 70R0319, Lawrence Berkeley National Laboratory, Berkeley, California 94720
We propose an improved quark coalescence model for spin alignment of vector mesons and polar-ization of baryons by spin density matrix with phase space dependence. The spin density matrixis defined through Wigner functions. Within the model we propose an understanding of spin align-ments of vector mesons φ and K ∗ (including K ∗ ) in the static limit: a large positive deviation of ρ for φ mesons from 1/3 may come from the electric part of the vector φ field, while a negativedeviation of ρ for K ∗ may come from the electric part of vorticity tensor fields. Such a negativecontribution to ρ for K ∗ mesons, in comparison with the same contribution to ρ for φ mesonswhich is less important, is amplified by a factor of the mass ratio of strange to light quark timesthe ratio of (cid:10) p b (cid:11) on the wave function of K ∗ to φ ( p b is the relative momentum of two constituentquarks of K ∗ and φ ). These results should be tested by a detailed and comprehensive simulationof vorticity tensor fields and vector meson fields in heavy ion collisions. I. INTRODUCTION
The Barnett effect [1] and the Einstein-de Haas effect [2] are two well-known effects in materials to connect rotationand spin polarization which can be converted from one to another. Similar effects also exist in ultra-relativistic heavy-ion collisions (HIC), in which a huge orbital angular momentum (OAM) can be generated in the direction perpendicularto the reaction plane and is transferred to the hot and dense medium in the form of the global polarization of hadrons[3–8] (see, e.g. [9–12], for recent reviews). In microscopic scenarios the transfer of OAM to spin polarization of hadronsis through the spin-orbit coupling in particle scatterings [3, 8, 13, 14], while in macroscopic approaches it is throughthe spin-vorticity coupling in the fluid [15–22]. The global polarization can be measured through the the polarizationof hyperons such as Λ (including ¯Λ hereafter) since they have weak decay channels [3]. The STAR collaboration hasrecently measured a non-vanishing global polarization of Λ hyperons in Au+Au collisions at √ s NN = 7 . − GeV[23, 24].In principle vector mesons can also be polarized in heavy ion collisions, but the polarization of vector mesons cannotbe measured since they mainly decay through strong interaction. Instead, ρ , the 00-element of the vector meson’sspin density matrix, can be meaured through the angular distribution of its decay daughters [4, 25]. If ρ (cid:54) = 1 / , thedistribution is anisotropic and the spin of the vector meson is aligned to the spin quantization direction. In 2008, theSTAR collaboration measured ρ for the vector meson φ (1020) in Au+Au collisions at 200 GeV, but the result isconsistent to / , indicating no spin alignment within errors [26]. Recent preliminary data of STAR for the φ meson’s ρ (denoted as ρ φ hereafter) at lower energies show a significant positive deviation from / , which is beyondconventional understanding of the polarization [27]. In Ref. [28], some of us proposed that such a large positivedeviation of ρ φ from 1/3 may possibly be explained by the φ field. In such a proposal [28], a quark coalescencemodel is employed which is based on spin density operators in momentum space [25]. As the quark polarizationcomes mainly from vorticity and vector meson fields which are functions of space-time, the space dependence of thequark polarization in Ref. [28] is put in a phenomenological way. The purpose of this paper is to improve the quarkcoalescence model of Ref. [25] by defining and using spin density operators in phase space with the help of spinWigner functions. In such an improved quark coalescence model, the quark polarization as a function of space-timecan be treated in a rigorous and systematic way. So one can then naturally describe spin alignments of vector mesonssuch as φ and K ∗ (including K ∗ if not stated explicitly) as functions of space-time. It is expected to implement theimproved coalescence model in real time simulations and to provide insights in spin alignments of vector mesons.The paper is organized as follows. In Sect. II, we formulate the improved coalescence model through the spindensity matrix in phase space with coordinate dependence. In Sect. III, we give spin polarization of quarks in phasespace from vorticity and vector meson fields. In Sect. IV, we analyze global and local polarization of Λ (including ¯Λ if not stated explicitly) using the improved coalescence model. In Sect. V, using the improved coalescence modelwe formulate spin alignments of vector mesons φ and K ∗ . In Sect. VI, we solve the Klein-Gordon equation to givevector meson fields generated by point charge sources. Finally we make a summary of the results. Notations and conventions . We adopt the sign convention for the metric tensor g µν = (1 , − , − , − . A four-vector is represented by Greek indices, e.g, x µ or p µ with µ = 0 , , , . A three-vector is represented in a boldfaced a r X i v : . [ nu c l - t h ] S e p symbol, e.g., x or p . The components of a three-vector is represented by the Latin index, but we do not distinguishthe superscript and subscript, for example, we do not distinguish x i and x i with i = 1 , , . We use the shorthandnotation [ d p ] ≡ d p / (2 π ) . II. SPIN DENSITY MATRIX AND QUARK COALESCENCE MODEL IN PHASE SPACE
In Ref. [25], a quark coalescence model is constructed based on the spin density matrix in momentum representation.In order to describe space-time dependence of spin polarization, we need to formulate an improved coalescence modelthrough the spin density matrix in phase space with coordinate dependence. We work at the formation time t of ahadron, for simplicity of notation, throughout the paper we suppress the time dependence of all quantities unless itis necessary to show it explicitly.In momentum representation, the spin density operator for single particle states is defined as [25] ρ = 1Ω (cid:88) s (cid:90) [ d p ] w ( s, p ) | s, p (cid:105) (cid:104) s, p | , (2.1)where w ( s, p ) is the weight function corresponding to the particle state with spin s and momentum p , Ω is thespace volume, and the spin-momentum state | s, p (cid:105) is the direct product of the spin state and the momentum state, | s, p (cid:105) ≡ | s (cid:105) | p (cid:105) . The weight function is given by w ( s, p ) = (cid:104) s, p | ρ | s, p (cid:105) , (2.2)which satisfies the normalization condition Tr ρ = 1 equivalent to (cid:88) s (cid:90) [ d p ] w ( s, p ) = 1 . (2.3)The definition and convention of single particle states in non-relativistic quantum mechanics are given in AppendixA.For the quark and antiquark with spin 1/2, the weight functions have the form w (q | s, p ) = 12 f q ( p ) [1 + sP q ( p )] ,w (¯q | s, p ) = 12 f ¯q ( p ) [1 + sP ¯q ( p )] , (2.4)where s = ± label two spin states with s z = ± / in the spin quantization direction z , and f q / ¯q ( p ) and P q / ¯q ( p ) denotethe distribution and polarization of the quark/antiquark respectively. Here the quark polarization is normalized to 1and given by P q ( p ) = w (q | + , p ) − w (q | − , p ) w (q | + , p ) + w (q | − , p ) . (2.5)The polarization for antiquark P ¯q ( p ) has the same form as above. We note that generally the weight functions (2.4)are × matrices in spin space. Throughout this paper we assume that they are diagonalized in the spin quantizationdirection.Now we generalize (2.1) by introducing the space variable into the density operator as ρ = (cid:88) s (cid:90) d x (cid:90) [ d p ] w ( s, x , p ) (cid:90) [ d q ] e − i q · x (cid:12)(cid:12)(cid:12) s, p + q (cid:69) (cid:68) s, p − q (cid:12)(cid:12)(cid:12) . (2.6)We see that the momenta of state bases differ by q with x being its conjugate position. The weight function w ( s, x , p ) is actually the Wigner function which can be obtained by projecting the above density operator onto two states withthe same spin and different momenta w ( s, x , p ) = (cid:90) [ d q ] e i q · x (cid:68) s, p + q | ρ | s, p − q (cid:69) . (2.7)By an integration over x for w ( s, x , p ) one can recover the weight function (2.2), therefore the normalization conditionfor w ( s, x , p ) reads (cid:88) s (cid:90) d x (cid:90) [ d p ] w ( s, x , p ) = 1 . (2.8)From above condition one can see that w ( s, x , p ) is dimensionless. For the quark and antiquark, with new weightfunctions w (q / ¯q | s, x , p ) we have similar formula to Eqs. (2.4,2.5) with the distribution f q / ¯q ( x , p ) and polarization P q / ¯q ( x , p ) as functions in phase space. A. Mesons
To describe the formation of mesons from a quark and an antiquark, we define the spin density operator for aquark-antiquark pair ρ q¯q = (cid:88) s ,s (cid:88) q , ¯q (cid:90) d x d x (cid:90) [ d p ][ d p ] (cid:90) [ d q ][ d q ] × w (q | s , x , p ) w (¯q | s , x , p ) e − i q · x e − i q · x × (cid:12)(cid:12)(cid:12) q , ¯q ; s , s ; p + q , p + q (cid:69) × (cid:68) q , ¯q ; s , s ; p − q , p − q (cid:12)(cid:12)(cid:12) , (2.9)where q = u,d,s and ¯q = ¯u , ¯d , ¯s denote the quark and antiquark respectively, the sum over quark and antiquarkflavors have been taken, the quark-antiquark state is the direct product of the quark state and the antiquark state (cid:12)(cid:12)(cid:12) q , ¯q ; s , s ; p + q , p + q (cid:69) = (cid:12)(cid:12)(cid:12) q , s , p + q (cid:69) (cid:12)(cid:12)(cid:12) ¯q , s , p + q (cid:69) = | q , ¯q (cid:105) | s , s (cid:105) (cid:12)(cid:12)(cid:12) p + q , p + q (cid:69) , (2.10)where | q , ¯q (cid:105) = | q (cid:105) | ¯q (cid:105) is the flavor state for the quark-antiquark pair, and s , s = ± / denote spins of the quarkand the antiquark in the quantization direction. All quantities with index ’1’ and ’2’ in (2.9) and (2.10) are those ofthe quark and antiquark respectively. The Wigner functions have similar forms to (2.4), w (q | s, x , p ) = 12 f q ( x , p ) [1 + sP q ( x , p )] ,w (¯q | s, x , p ) = 12 f ¯q ( x , p ) [1 + sP ¯q ( x , p )] . (2.11)The polarization P q / ¯q ( x , p ) can be determined from the Wigner function w (q / ¯q | s, x , p ) in a similar way to (2.5).Note that we do not include color wave functions for hadrons since they are totally decoupled from other parts ofwave functions. As we have mentioned, the spin Wigner functions in (2.11) are generally × matrices in spin space,but throughout the paper we assume that they are diagonalized in the spin quantization direction.To obtain spin density matrix elements of mesons, we put ρ q¯q between two meson states ρ M S z ,S z ( x , p ) = (cid:90) [ d q ] e i q · x (cid:68) M; S, S z ; p + q (cid:12)(cid:12)(cid:12) ρ q¯q (cid:12)(cid:12)(cid:12) M; S, S z ; p − q (cid:69) , (2.12)where M labels the flavor state of the meson, S and S z denote spin states which are the total spin and spin in aquantization direction (chosen to be + z or any direction) respectively, and p + q / and p − q / label two momentumstates. The details of the evaluation of (2.12) are given in Appendix B. The result is ρ M S z ,S z ( x , p ) = (cid:90) d x b [ d p b ][ d q b ] × exp ( − i q b · x b ) ϕ ∗ M (cid:16) p b + q b (cid:17) ϕ M (cid:16) p b − q b (cid:17) × (cid:88) s ,s w (cid:16) q (cid:12)(cid:12)(cid:12) s , x + x b , p p b (cid:17) w (cid:16) ¯q (cid:12)(cid:12)(cid:12) s , x − x b , p − p b (cid:17) × (cid:104) S, S z | s , s (cid:105) (cid:104) s , s | S, S z (cid:105) , (2.13) Figure 1: Quark positions and momenta inside a meson in its rest frame. where ϕ M is the meson wave function in relative momentum between the quark and the antiquark, and x b , p b and q b are relative position and momenta which are related to positions and momenta of the quark and the antiquark in(B4). Equation (2.13) is one of the main results in this paper.For convenience of notation, hereafter we use x = x + x b / , p = p / p b , x = x − x b / , and p = p / − p b ,see Fig. 1 for illustration. These relations can be obtained from (B4) by setting x a = x and p a = p .A simple choice of the meson wave function ϕ M ( k ) is the Gaussian distribution [29, 30] ϕ M ( k ) = (cid:18) √ πa M (cid:19) / exp (cid:18) − k a (cid:19) , (2.14)where a M is the momentum width parameter of the meson. If we use the above Gaussian form of the wave functionwe can complete the integral over q b in (2.13) to obtain the most simple form ρ M S z ,S z ( x , p ) = 1 π (cid:90) d x b d p b exp (cid:18) − p b a − a x b (cid:19) × (cid:88) s ,s w (q | s , x , p ) w (¯q | s , x , p ) × (cid:104) S, S z | s , s (cid:105) (cid:104) s , s | S, S z (cid:105) . (2.15)We see that the Gaussian wave packet form appears in the integral which depends on the relative position and relativemomentum between the quark and the antiquark.Now we apply (2.15) to the vector meson φ with S = 1 and S z = − , , . The diagonal elements of the spin densitymatrix for φ mesons are given in Eq. (B5). With spin Wigner functions (2.11), the normalization condition (2.8)reads (cid:90) d x (cid:90) [ d p ] f q / ¯q ( x , p ) = 1 . (2.16)Since we are concerned mainly with polarization functions that are small P q / ¯q ( x , p ) (cid:28) , without loss of generality,we can assume f q ( x , p ) = f q and f ¯q ( x , p ) = f ¯q are constants. Under these assumptions, with (B5) we obtain ¯ ρ φ = ρ φ ( x , p ) ρ φ ( x , p ) + ρ φ ( x , p ) + ρ φ − , − ( x , p ) ≈ − (cid:104) P s ( x , p ) P ¯s ( x , p ) (cid:105) φ , (2.17)where the average (cid:104)· · · (cid:105) M is taken on the meson wave packet (cid:104)· · · (cid:105) M ≡ π (cid:90) d x b d p b exp (cid:18) − p b a − a x b (cid:19) ( · · · ) . (2.18)If P s / ¯s are independent of positions, we can recover the result of Ref. [25]. In the remainder of this paper we willreuse ρ M00 to denote the normalized ¯ ρ M00 for simplicity of notation.In the same way, we can also obtain the normalized ρ for the vector meson K ∗ with the flavor content (d¯s) ρ K ∗ ≈ − (cid:104) P d ( x , p ) P ¯s ( x , p ) (cid:105) K ∗ . (2.19)The result for K ∗ with the flavor content (s¯d) can be obtained similarly. B. Baryons
In this subsection we will derive the spin density matrix for baryons in phase space. The starting point is the spindensity operator for three quarks. The spin, flavor and momentum part of the wave function for three quarks is thedirect product of that for each single quark, | q , q , q ; s , s , s ; p , p , p (cid:105) ≡ | q , s , p (cid:105) | q , s , p (cid:105) | q , s , p (cid:105) = | q , q , q ; s , s , s (cid:105) | p , p , p (cid:105) , (2.20)where s , , = ± / denote spins in the quantization direction and q , , = u , d , s denote the spin states in the z-direction and quark flavors respectively. The second equality implies that the spin and flavor part of the wave functionfor three quarks is independent of the momentum part. The spin density operator for three quarks has the form ρ qqq = (cid:88) s ,s ,s (cid:88) q , q , q (cid:90) (cid:89) i =1 d x i (cid:89) i =1 [ d p i ] (cid:89) i =1 [ d q i ] × (cid:89) i =1 w (q i | s i , x i , p i ) e − i q i · x i × (cid:12)(cid:12)(cid:12) q , q , q ; s , s , s ; p + q , p + q , p + q (cid:69) × (cid:68) q , q , q ; s , s , s ; p − q , p − q , p − q (cid:12)(cid:12)(cid:12) . (2.21)The spin density matrix element for baryons with spin S is given by putting ρ qqq between two baryon states ρ B S z ,S z ( x , p ) = (cid:90) [ d q ] e i q · x (cid:68) B; S, S z ; p + q (cid:12)(cid:12)(cid:12) ρ qqq (cid:12)(cid:12)(cid:12) B; S, S z ; p − q (cid:69) . (2.22)For ground state (spin-1/2 octet and spin-3/2 decuplet) baryons, the spin-flavor part of the wave function isdecoupled from the momentum or spatial part, but for excited states of baryons, they are generally entangled. Inthis paper we only consider ground state baryons so the momentum or spatial part of the baryon wave function isdisentangled from the spin-flavor part. Using the Gaussian form of the baryon momentum wave function, we obtain ρ B S z ,S z ( x , p ) = 1 π (cid:90) d x b d x c d p b d p c × exp (cid:18) − p b a − p c a − a x b − a x c (cid:19) × (cid:88) s ,s ,s (cid:88) q , q , q × w (q | s , x , p ) w (q | s , x , p ) w (q | s , x , p ) × (cid:104) B ; S, S z | q , q , q ; s , s , s (cid:105)× (cid:104) q , q , q ; s , s , s | B ; S, S z (cid:105) , (2.23)where p i and x i ( i = 1 , , ) are expressed in terms of Jacobi variables p j and x j ( j = a, b, c ) defined in Eq. (C3) and(C6) respectively and finally by setting x a = x and p a = p , see Fig. 2 for illustration of positions of three quarksinside a baryon. The detailed derivation of (2.23) is given in Appendix C. We see that the wave packet form of thebaryon emerges as a function of relative coordinates and relative momenta of three quarks. Figure 2: Positions of three quarks inside a baryon. The momenta conjugate to Jocobi cooridinates x a = x , x b and x c are p a , p b and p c respectively, see Eq. (C3) and (C6). As an example, we can apply (2.23) to the octet baryon Λ with its SU(6) spin-flavor wave function. The spin-flavorwave function of Λ tells that its spin in the quantization direction is carried by the s-quark while spins of u- andd-quark cancel. Similar to mesons, we also assume the polarization is small, P q / ¯q ( x , p ) (cid:28) and f q ( x , p ) = f q and f ¯q ( x , p ) = f ¯q are constants. The result for the diagonal element of the spin density matrix ρ Λ++ ≡ ρ Λ , is then ρ Λ++ ( x , p ) = 124 π f u f d (cid:90) d x b d x c d p b d p c × exp (cid:18) − p b a − p c a − a x b − a x c (cid:19) × { w (s | + , x , p ) [2 − P u ( x , p ) P d ( x , p ) − P u ( x , p ) P d ( x , p )]+ w (s | + , x , p ) [2 − P u ( x , p ) P d ( x , p ) − P u ( x , p ) P d ( x , p )]+ w (s | + , x , p ) [2 − P u ( x , p ) P d ( x , p ) − P u ( x , p ) P d ( x , p )] } . (2.24)Another diagonal element ρ Λ −− ≡ ρ Λ − , − can be obtained from ρ Λ++ ≡ ρ Λ , by flipping the s-quark’s spin, i.e. w (s | + , x i , p i ) → w (s |− , x i , p i ) with i = 1 , , . Finally we can read out the polarization of Λ from spin densitymatrix elements P Λ ( x , p ) = ρ Λ++ ( x , p ) − ρ Λ −− ( x , p ) ρ Λ++ ( x , p ) + ρ Λ −− ( x , p ) ≈ (cid:104) P s ( x , p ) + P s ( x , p ) + P s ( x , p ) (cid:105) Λ , (2.25)where the average (cid:104) O ( x i , p i ) (cid:105) B with i = 1 , , are taken on the wave packet function of baryons (cid:104) O ( x i , p i ) (cid:105) B ≡ π (cid:90) d x b d x c d p b d p c O ( x i , p i ) × exp (cid:18) − p b a − p c a − a x b − a x c (cid:19) . (2.26)Note that the integral in the average is normalized to 1, i.e. (cid:104) (cid:105) = 1 . III. SPIN POLARIZATION OF QUARKS
In the last section we have constructed an improved quark coalescence model in phase space. The model is basedon the spin density operator for quarks with spin dependent Wigner functions as weights, from which one can obtainspin density matrix elements in phase space for mesons and baryons. Once spin polarization functions for quarks inphase space (or equivalently spin Wigner functions) are known, one can calculate a vector meson’s spin alignmentand a hyperon’s polarization.There are different sources of spin polarization for massive fermions: vorticity fields, electromagnetic fields, andmean fields of vector mesons. The first two sources, vorticity and electromagnetic fields, have been extensively studiedin quantum kinetic approach through Wigner functions [18, 19, 31–35]. The polarization effect by vector meson fieldswas first proposed in Ref. [36] in the study of Λ polarization. It was generalized to the spin alignment of vectormesons in Ref. [28]. For each kind of field, one can distinguish the electric and magnetic part. It is believed that thecontribution from electromagnetic fields is negligible [28, 36]. Therefore in the remainder of this paper we considervorticity and vector meson fields as main sources of spin polarization.The spin polarization distribution in phase space for quarks (upper sign) and antiquarks (lower sign) is in the form[25, 28] P µ ± ( x, p ) = 12 m (cid:18) ˜ ω µν th ± g V E p T ˜ F µνV (cid:19) p ν [1 − f F D ( E p ∓ µ )] , (3.1)where p µ = ( E p , ± p ) are on-shell momenta of quarks and antiquarks with E p = (cid:113) p + m , ˜ ω µν th = (cid:15) µνσρ ω th σρ is thedual of the thermal vorticity tensor defined by ω th σρ = [ ∂ σ ( βu ρ ) − ∂ ρ ( βu σ )] with β ≡ /T being the temperatureinverse (note that there is a sign difference in the definition of ω th σρ from Ref. [15]), ˜ F µνV = (cid:15) µνσρ F Vσρ is the dual ofthe field strength tensor of vector mesons, and f F D is the Fermi-Dirac distribution. The electric and magnetic partof vector meson fields as three-vectors are defined as E iV = E Vi = F i V and B iV = B Vi = − (cid:15) ijk F jkV respectively with i, j, k = x, y, z . In a similar way, one can define the three-vector of thermal vorticity as ω i = ω i = ˜ ω i which is themagnetic part of the thermal vorticity tensor, while the electric part of the thermal vorticity tensor is ε i = ε i = ω i .Written explicitly in three-vector forms, they are ω = 12 ∇ × ( β u ) , ε = −
12 [ ∂ t ( β u ) + ∇ ( βu )] . (3.2)We take xz plane as the reaction plane with one nucleus moving along + z direction at x = − b/ while the othernucleus moving along − z direction at x = b/ . The global OAM is along + y direction. Therefore we assume that thespin quantization direction is + y , and that the Wigner functions in (2.11) are diagonalized in + y direction. Then thepolarization distribution for q and ¯q along + y direction can be written as [28] P y q / ¯q ( x , p ) = 12 ω y ± m q ( ε × p ) y ± g V m q T B Vy + g V m q E p T ( E V × p ) y , (3.3)where g V is the coupling constant of quarks and antiquarks to vector meson fields, and we have taken the Boltzmannlimit − f F D ( E p ∓ µ ) (cid:39) . The last term of Eq. (3.3) is the spin-orbit term for quarks and antiquarks involving theelectric part of vector meson fields, the similar term is the key to the nuclear shell structure if applying to nucleonsin meson fields [37, 38]. For q = s and ¯q = ¯s , the vector meson field should be the φ field, i.e. V = φ . IV. GLOBAL AND LOCAL POLARIZATION OF Λ In this section we look at the polarization of Λ (including ¯Λ if not stated explicitly) in Eq. (2.25) with thepolarization of s and ¯s given in Eq. (3.3). In this case the vector meson field is the φ field, i.e. V = φ . By choosing + y as the spin quantization direction, the spin polarization of Λ and ¯Λ in phase space is now P y Λ / ¯Λ ( x , p ) ≈ (cid:68) P y s / ¯s ( x , p ) + P y s / ¯s ( x , p ) + P y s / ¯s ( x , p ) (cid:69) Λ / ¯Λ ≈ (cid:104) ω y ( x ) + ω y ( x ) + ω y ( x ) (cid:105) Λ / ¯Λ ± m s ˆ y · (cid:104) ε ( x ) × p + ε ( x ) × p + ε ( x ) × p (cid:105) Λ / ¯Λ ± g φ m s T (cid:10) B φy ( x ) + B φy ( x ) + B φy ( x ) (cid:11) Λ / ¯Λ + g φ m T ˆ y · (cid:104) E φ ( x ) × p + E φ ( x ) × p + E φ ( x ) × p (cid:105) Λ / ¯Λ . (4.1)where we have taken non-relativistic limit E p ≈ m s . We can take an average over a space volume at the formationtime of Λ . If all fields change slowly inside Λ , we can approximate O ( x i ) ≈ O ( x ) for i = 1 , , . Then we obtain (cid:68) P y Λ / ¯Λ ( x , p ) (cid:69) ≈ (cid:104) ω y ( x ) (cid:105) ± m s [ (cid:104) ε ( x ) (cid:105) × p ] y ± g φ m s (cid:10) β B φy ( x ) (cid:11) + g φ m [ (cid:104) β E φ ( x ) (cid:105) × p ] y , (4.2)where (cid:104)·(cid:105) represents the volume average at the formation time of Λ . Note that the spin-orbit term E φ × p has thesame sign for Λ and ¯Λ .For static Λ with p = 0 , the terms involving ε and E φ are vanishing [28], but for non-static Λ with non-vanishingmomenta, they are generally present. However, for the global spin polarization in the direction of + y (direction ofthe global OAM) with all Λ and ¯Λ in momentum spectra being included, these terms of ε and E φ are vanishing. Sothe global polarization for Λ and ¯Λ measured in STAR experiments [23, 24] comes mainly from ω y and B φy . Notethat the B φy term for ¯Λ has an opposite sign to Λ . This provides a possible explanation of the difference betweenmagnitudes of P y Λ and P y ¯Λ , similar to the scenario of Ref. [36]. The fact P y ¯Λ > P y Λ shown in experimental data indicates g φ (cid:10) β B φy ( x ) (cid:11) < .Recent STAR measurements [39] of the longitudinal spin polarization of Λ as functions show a positive sin (2 φ − ) behavior with φ and Ψ being the azimuthal angle of Λ and the second-order event plane respectively, while theoreticalresults of relativistic hydrodynamics model [40] and transport models [41–43] show an opposite sign. The simulationfrom chiral kinetic theory in Ref. [44] and results from a simple phenomenological model in Ref. [45] gives the correctsign as the data. The sign problem in local polarization may indicate the assumption of global equilibrium of spinmay not be justified, so the thermal vorticity may not be the right quantity for the spin chemical potential [46]. Theazimuthal angle dependence of P y Λ / ¯Λ has been measured by the STAR collaboration with the trend that P y Λ / ¯Λ in thereaction plane is larger than that out of the reaction plane. This phenomenon has not been well understood [46].The spin-orbit term may provide an additional contribution to the polarization along the beam direction P z Λ / ¯Λ inheavy ion collisions [39]. To this end, we split the whole space into four parts corresponding to four quadrants of thetransverse plane which we denote as ++ , − + , −− and + − respectively. Let us look at (cid:68) P z Λ / ¯Λ (cid:69) in the first and secondquadrant (cid:68) P z Λ / ¯Λ ( x , p ) (cid:69) ++ ∼ g φ m (cid:20)(cid:10) β E xφ (cid:11) ++ p T sin( φ p ) − (cid:68) β E yφ (cid:69) ++ p T cos( φ p ) (cid:21) , (cid:68) P z Λ / ¯Λ ( x , p ) (cid:69) − + ∼ g φ m (cid:20)(cid:10) β E xφ (cid:11) − + p T sin( φ p ) − (cid:68) β E yφ (cid:69) − + p T cos( φ p ) (cid:21) . (4.3)If (cid:104) β E φ (cid:105) is dominated by the x component in the first and second quadrant and if g φ (cid:68) β E xφ (cid:69) ++ = − g φ (cid:68) β E xφ (cid:69) − + > ,then we can obtain the patterns observed in experiments [39]: (cid:68) P y Λ / ¯Λ ( x , p ) (cid:69) ++ > and (cid:68) P y Λ / ¯Λ ( x , p ) (cid:69) − + < .Furthermore the spin-orbit term E φ × p in P y Λ / ¯Λ may also provide a possible additional contribution to the azimuthalangle dependence of the polarization along + y in heavy ion collisions [47], if there is a correlation between E φ and p ina certain region. In order to look at the relevant observable, we choose the region for taking average to be x > , y > corresponding to the first quadrant of the transverse plane in collisions, the average quantity is denoted as (cid:104) β E φ (cid:105) ++ which may not be vanishing (the average of β E φ over the full space should be vanishing). Then the azimuthal anglepart of P y Λ / ¯Λ in the first quadrant of the transverse plane is (cid:68) P y Λ / ¯Λ ( x , p ) (cid:69) ++ ∼ g φ m (cid:10) β E zφ (cid:11) ++ p T cos( φ p ) , (4.4)where φ p is the azimuthal angle relative to that of the reaction plane, and p T ≡ | p T | is the scalar transverse momentum.We see that the spin-orbit term may provide an additional contribution to the the azimuthal angle dependence of P y Λ / ¯Λ . V. SPIN ALIGNMENTS OF φ AND K ∗ We now investigate spin alignments of vector mesons φ and K ∗ . In the remainder of this paper, when we say K ∗ we imply to include K ∗ if there is no ambiguity.Let us first look at the spin alignment of φ . Substituting Eq. (3.3) for q = s and ¯q = ¯s into Eq. (2.17) and takingan average on a space volume, we obtain the spin density matrix element for φ mesons (cid:68) ρ φ ( x , p ) (cid:69) ≈ − (cid:104) P y s ( x , p ) P y ¯s ( x , p ) (cid:105) φ, Vol ≈ − (cid:10) ω y (cid:11) + 19 m s (cid:68) ( ε × p ) y ( ε × p ) y (cid:69) φ, Vol + g φ m (cid:68)(cid:0) β B φy (cid:1) (cid:69) − g φ m (cid:28) β E p E p ( E φ × p ) y ( E φ × p ) y (cid:29) φ, Vol , (5.1)where the spin quantization direction is chosen as + y , (cid:104)·(cid:105) denotes the volume average at the formation time of φ mesons, and we have put index ’Vol’ to distinguish the volume average from the average on the φ meson wavefunction if both averages are taken. In deriving Eq. (5.1) we have made approximations: (a) The size of the vectormeson is much smaller than the hydrodynamic scale, so we put x ≈ x ≈ x for vorticity fields and the φ fields;(b) We neglect correlation in the volume between different fields except between themselves [28], for example, nocorrelation between E φ and B φ , between ε and E φ , or between ω and B φ , etc.. We also neglect correlation in thevolume between different components of the same field, for example, between E xφ and E zφ or between ε z and ε x , etc..We now simplify terms involving ε and E φ in (5.1). The ε term is evaluated as (cid:68) ( ε × p ) y ( ε × p ) y (cid:69) φ, Vol ≈ (cid:10) ε z (cid:11) p x + 14 (cid:10) ε x (cid:11) p z − (cid:10) ε z (cid:11) (cid:10) p b,x (cid:11) φ − (cid:10) ε x (cid:11) (cid:10) p b,z (cid:11) φ = 14 (cid:10) ε z (cid:11) p x + 14 (cid:10) ε x (cid:11) p z − (cid:0)(cid:10) ε z (cid:11) + (cid:10) ε x (cid:11)(cid:1) (cid:10) p b (cid:11) φ , (5.2)and the E φ term is evaluated as (cid:28) β E p E p ( E φ × p ) y ( E φ × p ) y (cid:29) φ, Vol ≈ (cid:10) β E φ,z (cid:11) (cid:28) E p E p (cid:29) φ p x + 14 (cid:10) β E φ,x (cid:11) (cid:28) E p E p (cid:29) φ p z − (cid:10) β E φ,z (cid:11) (cid:42) p b,x E p E p (cid:43) φ − (cid:10) β E φ,x (cid:11) (cid:42) p b,z E p E p (cid:43) φ , (5.3)where we have used p , = p / ± p b and dropped terms with mixture of different fields or different components ofthe same field. Inserting (5.2) and (5.3) into (5.1) we obtain (cid:68) ρ φ ( x , p ) (cid:69) ≈ − (cid:10) ω y (cid:11) − m s (cid:0)(cid:10) ε z (cid:11) + (cid:10) ε x (cid:11)(cid:1) (cid:10) p b (cid:11) φ + g φ m (cid:68)(cid:0) β B φy (cid:1) (cid:69) + g φ m (cid:10) β E φ,z (cid:11) (cid:42) p b,x E p E p (cid:43) φ + (cid:10) β E φ,x (cid:11) (cid:42) p b,z E p E p (cid:43) φ + ρ p ( ε z ) p x + ρ p ( ε x ) p z − ρ p ( φ, E φ,z ) p x − ρ p ( φ, E φ,x ) p z , (5.4)where we have used following positive coefficients ρ p ( ε i ) = 136 m s (cid:10) ε i (cid:11) ,ρ p ( φ, E φ,i ) = g φ m (cid:10) β E φ,i (cid:11) (cid:28) E p E p (cid:29) φ , (5.5)with i = x, y, z . For nearly static φ mesons with | p | (cid:28) | p b | the terms proportional to p x and p z in (5.4) are very smalland can be neglected compared with the (cid:10) p b (cid:11) φ term, in this case we recover the result of Ref. [28] in nonrelativistic0limit with E p ≈ E p ≈ m s (cid:68) ρ φ ( x , p ≈ (cid:69) ≈ − (cid:10) ω y (cid:11) − m s (cid:0)(cid:10) ε z (cid:11) + (cid:10) ε x (cid:11)(cid:1) (cid:10) p b (cid:11) φ + g φ m (cid:68)(cid:0) β B φy (cid:1) (cid:69) + g φ m (cid:104)(cid:10) β E φ,z (cid:11) (cid:10) p b,x (cid:11) φ + (cid:10) β E φ,x (cid:11) (cid:10) p b,z (cid:11) φ (cid:105) . (5.6)In Eq. (5.4) and (5.6) there are averages of squares of relative momenta of two quarks on the wave function of φ mesons and there are also space volume averages of field squares.Equation (5.4) with (5.6) as its static limit is part of our main results in the paper. A few remarks are in orderabout Eq. (5.4): (a) All contributions appear independently as positive or negative quantities. This is the mainfeature of ρ for φ mesons. (b) The second term is from the vorticity vector (magnetic part of the vorticity tensor),while the third term is from the electric part of the vorticity tensor. Both terms are negative definite. (c) The fourthterm is from the magnetic part of the φ field, while the fifth term is from the electric part of the φ field. Both termsare positive definite. (d) The last line collects contributions proportional to momentum squares of the φ meson, wherethe contribution from the electric part of the vorticity tensor is always positive while that from the electric part ofthe φ field is always negative. (e) We have argued in Ref. [28] that the dominant contribution to ρ φ may possibly befrom the electric part of the φ field which is positive definite.Let us turn to the spin alignment of another vector meson K ∗ . Different from the φ meson with flavor content (s¯s) , K ∗ has flavor (d¯s) . Vector meson ( ρ or ω ) fields that can polarize light quarks are different from the φ fieldwhich mainly polarize s and ¯s . We will see that such a difference has significant consequences on ρ K ∗ . Following thesame procedure and taking the same approximations as in deriving (5.1), we obtain the spin density matrix elementfor K ∗ , a counterpart of Eq. (5.4), (cid:68) ρ K ∗ ( x , p ) (cid:69) ≈ − (cid:10) ω y (cid:11) − m s m d (cid:0)(cid:10) ε z (cid:11) + (cid:10) ε x (cid:11)(cid:1) (cid:10) p b (cid:11) K ∗ + g φ g V m s m d (cid:10) β B φy B Vy (cid:11) + g φ g V m s m d (cid:34)(cid:10) β E φz E Vz (cid:11) (cid:42) p b,x E d p E ¯s p (cid:43) K ∗ + (cid:10) β E φx E Vx (cid:11) (cid:42) p b,z E d p E ¯s p (cid:43) K ∗ (cid:35) + m s m d ρ p ( ε z ) p x + m s m d ρ p ( ε x ) p z − ρ p ( K ∗ , E φz E Vz ) p x − ρ p ( K ∗ , E φx E Vx ) p z , (5.7)where B φi and E φi with i = x, y, z are from the polarization of ¯s , while B Vi and E Vi are vector meson fields ( ρ or ω mesons) that polarize the d-quark, and ρ p ( K ∗ , E φi E Vi ) are defined as ρ p ( K ∗ , E φi E Vi ) = g φ g V m s m d (cid:68) β E φi E Vi (cid:69) (cid:42) E d p E ¯s p (cid:43) K ∗ . (5.8)In (5.7) we have shown terms of volume averages of different fields, (cid:10) β B φy B Vy (cid:11) and (cid:68) β E φi E Vi (cid:69) , for the purposeof illustration and comparison, since they should have been neglected in accordance with the approximation thatdifferent fields do not have large correlation in space as compared with the correlation between the same fields. Afterimplementing this approximation, we obtain (cid:68) ρ K ∗ ( x , p ) (cid:69) ≈ − (cid:10) ω y (cid:11) − m s m d (cid:0)(cid:10) ε z (cid:11) + (cid:10) ε x (cid:11)(cid:1) (cid:10) p b (cid:11) K ∗ + m s m d (cid:2) ρ p ( ε z ) p x + ρ p ( ε x ) p z (cid:3) . (5.9)We can see that the slope of ρ K ∗ with respect to p T is positive. For nearly static K ∗ mesons with | p | (cid:28) | p b | , theterms proportional to p x and p z in (5.4) are very small and can be neglected, then we have (cid:68) ρ K ∗ ( x , p ≈ (cid:69) ≈ − (cid:10) ω y (cid:11) − m s m d (cid:0)(cid:10) ε z (cid:11) + (cid:10) ε x (cid:11)(cid:1) (cid:10) p b (cid:11) K ∗ . (5.10)We see in (5.9) and (5.10) the absence of the contribution from vector meson fields. Therefore the spin alignment of K ∗ is dominated by the vorticity contribution which must be negative for nearly static K ∗ . This is the significant1 Figure 3: An example for the effects of vector meson fields on the spin density matrices of φ and K ∗ mesons in their rest frame.There is large correlation between vector meson fields acting on s and ¯s in the φ meson but almost no correlation between vectormeson fields acting on d and ¯s in K ∗ . Due to the short distance nature of vector meson fields, the dominant contribution tothe fields at the position of a constituent quark of φ or K ∗ is from the quark of its nearest neighbor. The relative momentumof the quark and antiquark inside the meson is shown as p (intead of p b in the text). difference from the spin alignment of φ mesons which may possibly be dominated by φ fields whose contribution ispositive definite for nearly static φ mesons. Another feature of ρ K ∗ in (5.9) and (5.10) is that the contribution fromthe electric part of the vorticity tensor is amplified by a factor ( m s /m d ) (cid:16)(cid:10) p b (cid:11) K ∗ / (cid:10) p b (cid:11) φ (cid:17) compared with ρ φ . Notethat the ratio (cid:10) p b (cid:11) K ∗ / (cid:10) p b (cid:11) φ is about . ∼ . in the quark model. This may provide a sizable magnitude of thenegative contribution to ρ K ∗ as shown in ALICE experiments [48].We note that the above arguments are only valid for primary K ∗ . The life time of K ∗ is much shorter andthe interaction of K ∗ with the surrounding matter is much stronger than the φ meson. This may bring othercontributions to ρ K ∗ from the interaction of K ∗ with medium. A caveat is that the above arguments are basedon the approximation that different fields do not have large correlation in space as compared with the correlationbetween the same fields. This seems to work for ρ φ since there are squares of the same vector meson field. Howeverit is not the case for ρ K ∗ that all terms of vector meson fields are mixture of differenct fields which are thought to beequally small. In this case, in order to justify the approximation, we may need to evaluate these terms and comparetheir magnitudes with the negative comtribution from vorticity tensor fields. This is beyond the scope of this paperand will be studied in the future.To summarize, in the picture of the coalescence model, we propose that a large positive contribution to the spinmatrix element ρ φ should be from the φ field [28]. This is due to the correlation between the φ field that polarizesthe s-quark and that polarizes ¯s , see Fig. 3 for illustration. However this is not the case for ρ K ∗ : the φ field thatpolarizes ¯s does not correlate much with vector meson fields ( ρ or ω mesons) that polarize the d-quark, the formeris from other strange quarks not belonging to K ∗ , while the latter come from other light quarks surrounding d, seeFig. 3. Therefore ρ K ∗ is dominated by the contribution from vorticity fields which is negative definite for static K ∗ .Such a negative contribution from vorticity fields in ρ K ∗ is amplified relative to ρ φ by the mass ratio of strange tolight quark and by the ratio of (cid:10) p b (cid:11) on K ∗ ’s to φ ’s wave function. VI. SOLVING VECTOR MESON FIELDS GENERATED BY SOURCES
In this section we solve the mean vector field which satisfies the Klein-Gordon equation [36] ∂ µ F µνV + m V V ν = g V J ν , (6.1)2where F µνV ≡ ∂ µ V ν − ∂ ν V µ is the field strength tensor, the source of the field J µ is the current density associatedwith a conserved quantum number, m V is the vector meson mass, and g V is the coupling constant. We can write V µ and J µ explicitly as V µ = ( φ, A ) and J µ = ( ρ, j ) . We can also define the electric and magnetic part of F µνV asthree-vectors as in Sect. III. If m V is very large compared with the derivative term, we can just neglect latter in Eq.(6.1). In this case V µ can be approximately proportional to the current density [36], V µ ≈ ( g V /m V ) J µ , known asthe current-field identity [49, 50] in the vector dominance model [51, 52].We can use the Green’s function method to solve the Klein-Gordon equation (6.1) as to solve Maxwell’s equationsin Ref. [53]. The only difference is the presence of the vector meson mass which brings a little more complexity. Weconsider a point charge Q located at the original point at t = 0 moves with velocity v in + z direction. The charge Q corresponds to a quantum number such as the baryon number for quarks or the strangeness number for s and ¯s .Finally we obtain the electric and magnetic parts of vector meson fields as E xV ( t, x ) = g V Q γv (1 + m V ∆)4 π ∆ x e − m V ∆ , E yV ( t, x ) = g V Q γv (1 + m V ∆)4 π ∆ y e − m V ∆ , E zV ( t, x ) = g V Q γv (1 + m V ∆)4 π ∆ ( z − vt ) e − m V ∆ , B xV ( t, x ) = − g V Q γv (1 + m V ∆)4 π ∆ y e − m V ∆ , B yV ( t, x ) = g V Q γv (1 + m V ∆)4 π ∆ x e − m V ∆ , B zV ( t, x ) = 0 , (6.2)where ∆ = (cid:112) x + y + γ ( vt − z ) with γ = 1 / √ − v being the Lorentz contraction factor. We see that anexponential decay factor e − m V ∆ appears in vector meson fields produced by a point charge, which reflects the finitedistance nature of vector meson fields. Such a factor is absent in electromagnetic fields produced by electric charges[53, 54]. The detailed derivation of (6.2) is given in Appendix D.If we can determine the strangeness current, we can apply Eq. (6.2) to obtain the φ field with Q being thestrangeness number. Due to the short distance nature of the vector meson field, the φ field that can polarize theconstituent s-quark in a φ meson is dominated by the field produced by its constituent partner ¯s in the same φ mesonwhich is in its nearest neighborhood in space, and vice versa. VII. SUMMARY
We have constructed an improved quark coalecence model based on the spin density matrix in phase space withcoordinate dependence. The spin density matrices for mesons and ground state baryons depend on spin Wignerfunctions of quark systems. The quark spin polarization functions in phase space are encoded in spin Wigner functions.The spin polarization of baryons can be obtained from spin density matrices for hadrons. As an example we obtainthe spin polarization of Λ which is determined by that of strange quarks. The spin polarization of quarks comesmainly from vorticity tensor fields and vector meson fields. We discussed a possible role that the electric part of thevector meson field may play in understanding experimental observations in local polarization of Λ . Most importantlywe propose an understanding of spin alignments of vector mesons φ and K ∗ (including K ∗ ) in the static limit: alarge positive deviation of ρ for φ mesons from 1/3 may come from the electric part of the vector φ field, while alarge magnitude of negative deviation of ρ for K ∗ may come from the electric part of vorticity tensor fields. Such alarge negative contribution to ρ for K ∗ , in contrast to the same contribution to that for φ which is less important,may be due to a large mass ratio of strange quarks to light quarks. These results should be tested by a detailed andcomprehensive simulation of vorticity tensor fields and vector meson fields in heavy ion collisions. Acknowledgments
The authors thank Xian-Gai Deng, Xu-Guang Huang, Guo-Liang Ma, Yu-Gang Ma, Jia-Lun Ping, Ai-Hong Tang,Xiao-Liang Xia and Hao-Jie Xu for helpful discussions. X.-L.S. and Q.W. are supported in part by the NationalNatural Science Foundation of China (NSFC) under Grant No. 11535012, 11890713, and by the Strategic PriorityResearch Program of Chinese Academy of Sciences under Grant No. XDB34030102. X.-N.W. is supported in part byNational Natural Science Foundation of China (NSFC) under Grant No. 11935007, 11861131009 and 11890714.3
Appendix A: Single particle state in coordinate and momentum space
In this appendix, we give definition and convention for single particle states in coordinate and momentum repre-sentation in non-relativistic quantum mechanics.A position eigenstate is denoted as | x (cid:105) and satisfies following orthogonality and completeness conditions (cid:104) x (cid:48) | x (cid:105) = δ (3) ( x (cid:48) − x ) , (cid:90) d x | x (cid:105) (cid:104) x | . (A1)The normalization of the state | x (cid:105) is then (cid:104) x | x (cid:105) = δ (3) ( x − x ) = (cid:90) [ d p ] = 1Ω (cid:88) p , (A2)where Ω is the space volume and [ d p ] ≡ d p / (2 π ) .A momentum eigenstate is denoted as | p (cid:105) and satisfies following orthogonality and completeness conditions (cid:104) p (cid:48) | p (cid:105) = (2 π ) δ (3) ( p − p (cid:48) ) , (cid:90) [ d p ] | p (cid:105) (cid:104) p | . (A3)The normalization of | p (cid:105) is then (cid:104) p | p (cid:105) = (2 π ) δ (3) ( p − p ) = Ω . (A4)From Eq. (A1) and (A3) we can define the inner product (cid:104) x | p (cid:105) as (cid:104) x | p (cid:105) = e i p · x . (A5)With the above relation we can check δ (3) ( x − x (cid:48) ) = (cid:104) x (cid:48) | x (cid:105) = (cid:90) [ d p ] (cid:104) x (cid:48) | p (cid:105) (cid:104) p | x (cid:105) = (cid:90) [ d p ] e i p · ( x (cid:48) − x ) , (A6)where we have inserted the completeness relation (A3). We can express | x (cid:105) in terms of | p (cid:105) and vice versa, | x (cid:105) = (cid:90) [ d p ] | p (cid:105) (cid:104) p | x (cid:105) = (cid:90) [ d p ] e − i p · x | p (cid:105) , | p (cid:105) = (cid:90) d x | x (cid:105) (cid:104) x | p (cid:105) = (cid:90) d x e i p · x | x (cid:105) . (A7) Appendix B: Derivation of density matrix elements for mesons
In this Appendix, we evaluate (2.12), the spin density matrix element on two meson states, ρ M S z ,S z ( x , p ) = (cid:90) [ d q ] e i q · x (cid:90) d x d x (cid:90) [ d p ][ d p ][ d q ][ d q ] × e − i q · x e − i q · x × (cid:68) M; p + q (cid:12)(cid:12)(cid:12) p + q , p + q (cid:69) (cid:68) p − q , p − q (cid:12)(cid:12)(cid:12) M; p − q (cid:69) × (cid:88) s ,s w (q | s , x , p ) w (¯q | s , x , p ) × (cid:104) S, S z | s , s (cid:105) (cid:104) s , s | S, S z (cid:105) , (B1)4where (cid:104) S, S z | s , s (cid:105) denotes the Clebsch-Gordan coefficient for spin states, (cid:80) q , ¯q |(cid:104) q , ¯q | M (cid:105)| = 1 with | M (cid:105) beingthe flavor part of the meson’s wave function, (cid:104) q , ¯q | M (cid:105) denotes the Clebsch-Gordan coefficient for the flavor state(here we have used the fact that the flavor part is decoupled from its spin part in a meson’s wave function), and theamplitudes between the meson’s and quark-antiquark’s momentum states are (M | q , ¯q) = (cid:68) M; p + q (cid:12)(cid:12)(cid:12) p + q , p + q (cid:69) = (2 π ) δ (3) (cid:18) p + p − p + q + q − q (cid:19) × ϕ ∗ M (cid:18) p − p q − q (cid:19) , (q , ¯q | M) = (cid:68) p − q , p − q (cid:12)(cid:12)(cid:12) M; p − q (cid:69) = (2 π ) δ (3) (cid:18) p + p − p − q + q − q (cid:19) × ϕ M (cid:18) p − p − q − q (cid:19) , (B2)where the meson wave function in relative momentum of two quarks is normalized as (cid:82) [ d k ] | ϕ M ( k ) | = 1 with ϕ M ( k ) being related to the wave function in relative position, ϕ M ( k ) = (cid:82) d y e − i k · y ϕ M ( y ) . Here we have used the samesymbol ϕ M to denote the meson wave function in coordinate and momentum without ambiguity.Equation (B1) can be simplified as ρ M S z ,S z ( x , p ) = (cid:90) d x a d x b (cid:90) [ d p b ][ d q a ][ d q b ] × exp ( − i q b · x b ) exp [ − i q a · ( x a − x )] × ϕ ∗ M (cid:16) p b + q b (cid:17) ϕ M (cid:16) p b − q b (cid:17) × (cid:88) s ,s w (cid:16) q (cid:12)(cid:12)(cid:12) s , x a + x b , p p b (cid:17) w (cid:16) ¯q (cid:12)(cid:12)(cid:12) s , x a − x b , p − p b (cid:17) × (cid:104) S, S z | s , s (cid:105) (cid:104) s , s | S, S z (cid:105) , (B3)where we have used p a = p + p , p b = 12 ( p − p ) , q a = q + q , q b = 12 ( q − q ) , x a = 12 ( x + x ) , x b = x − x . (B4)Note that q a and q b are conjugate momenta of x a and x b respectively. Completing integrals in (B3) over q a and x a ,we obtain Eq. (2.13).Using the Gaussian form of the meson wave function in Eq. (2.14), we can further simplify Eq. (2.13) to obtainthe most simple form in Eq. (2.15) for the spin matrix elements. Applying Eq. (2.15) to the vector meson φ with5 S = 1 and S z = − , , , we obtain diagonal elements of the spin density matrix for φ , ρ φ ( x , p ) = 12 π (cid:90) d x b d p b exp (cid:32) − p b a φ − a φ x b (cid:33) × (cid:104) w (cid:16) s (cid:12)(cid:12)(cid:12) + , x + x b , p p b (cid:17) w (cid:16) ¯s (cid:12)(cid:12)(cid:12) − , x − x b , p − p b (cid:17) + w (cid:16) s (cid:12)(cid:12)(cid:12) − , x + x b , p p b (cid:17) w (cid:16) ¯s (cid:12)(cid:12)(cid:12) + , x − x b , p − p b (cid:17)(cid:105) ,ρ φ ( x , p ) = 1 π (cid:90) d x b d p b exp (cid:32) − p b a φ − a φ x b (cid:33) × w (cid:16) s (cid:12)(cid:12)(cid:12) + , x + x b , p p b (cid:17) w (cid:16) ¯s (cid:12)(cid:12)(cid:12) + , x − x b , p − p b (cid:17) ,ρ φ − , − ( x , p ) = 1 π (cid:90) d x b d p b exp (cid:32) − p b a φ − a φ x b (cid:33) × w (cid:16) s (cid:12)(cid:12)(cid:12) − , x + x b , p p b (cid:17) w (cid:16) ¯s (cid:12)(cid:12)(cid:12) − , x − x b , p − p b (cid:17) . (B5) Appendix C: Derivation of density matrix elements for baryons
In this appendix we will evaluate Eq. (2.22) for ground state baryons to give Eq. (2.23). The spatial or momentumparts of wave functions for these baryons are independent of spin-flavor parts. Inserting (2.21) into (2.22) we obtain ρ B S z ,S z ( x , p ) = (cid:90) [ d q ] e i q · x (cid:90) (cid:89) i =1 d x i (cid:89) i =1 [ d p i ] (cid:89) i =1 [ d q i ] × exp [ − i ( q · x + q · x + q · x )] × (cid:68) B; p + q (cid:12)(cid:12)(cid:12) p + q , p + q , p + q (cid:69) × (cid:68) p − q , p − q , p − q (cid:12)(cid:12)(cid:12) B; p − q (cid:69) × (cid:88) s ,s ,s (cid:88) q , q , q (cid:89) i =1 w (q i | s i , x i , p i ) × (cid:104) B ; S, S z | q , q , q ; s , s , s (cid:105)× (cid:104) q , q , q ; s , s , s | B ; S, S z (cid:105) . (C1)The amplitudes between momentum states of the baryon and three quarks are given by (q , q , q | B) = (cid:68) q; p − q , p − q , p − q (cid:12)(cid:12)(cid:12) B; p − q (cid:69) = (2 π ) δ (3) (cid:18) p a − p − q a − q (cid:19) ϕ B (cid:16) p b − q b , p c − q c (cid:17) , (B | q , q , q) = (cid:68) B; p + q (cid:12)(cid:12)(cid:12) q; p + q , p + q , p + q (cid:69) = (2 π ) δ (3) (cid:18) p a − p + q a − q (cid:19) ϕ ∗ B (cid:16) p b + q b , p c + q c (cid:17) , (C2)6where ϕ B ( k b , k c ) is wave function of the baryon in the momentum representation to defined in (C5) and (C7), andwe have used momenta in Jacobi form p a = p + p + p , p b = 13 ( p + p − p ) , p c = 12 ( p − p ) , q a = q + q + q , q b = 13 ( q + q − q ) , q c = 12 ( q − q ) . (C3)To obtain the amplitudes (C2), we have inserted the completeness relation (cid:90) (cid:89) i =1 d x i | x i (cid:105) (cid:104) x i | = 1 , (C4)between the baryon and three-quarks state. We have also used (cid:104) x , x , x | B; p (cid:105) = exp ( i p · x a ) ϕ B ( x b , x c ) , (C5)where ϕ B ( x b , x c ) is the spatial wave function of the baryon depending on relative distance x b and x c of Jacobicoordinates defined as x a = 13 ( x + x + x ) , x b = 12 ( x + x ) − x , x c = x − x . (C6)The momentum state ϕ B ( k b , k c ) in (C2) can be obtained from ϕ B ( x b , x c ) by Fourier transformation ϕ B ( k b , k c ) = (cid:90) d x b d x c exp ( − i k b · x b − i k c · x c ) ϕ B ( x b , x c ) , (C7)where k b and k c are conjugate momenta to x b and x c respectively. Note that we have used for simplicty of notation thesame symbol ϕ B for the wave function in both coordinate and momentum representation. We assume normalizationconditions for ϕ B ( x b , x c ) and ϕ B ( k b , k c ) as (cid:90) d x b d x c | ϕ B ( x b , x c ) | = 1 , (cid:90) [ d k b ][ d k c ] | ϕ B ( k b , k c ) | = 1 . (C8)7We insert (C2) into (C1) and complete integrals over q , x a and p a , then we obtain ρ B S z ,S z ( x , p ) = (cid:90) (cid:89) i = b,c d x i (cid:89) i = b,c [ d p i ] (cid:89) i = b,c [ d q i ] × exp [ − i ( q b · x b + q c · x c )] × ϕ B (cid:16) p b − q b , p c − q c (cid:17) ϕ ∗ B (cid:16) p b + q b , p c + q c (cid:17) × (cid:88) s ,s ,s (cid:88) q , q , q × w (cid:18) q (cid:12)(cid:12)(cid:12)(cid:12) s , x + 13 x b + 12 x c , p + 12 p b + p c (cid:19) × w (cid:18) q (cid:12)(cid:12)(cid:12)(cid:12) s , x + 13 x b − x c , p + 12 p b − p c (cid:19) × w (cid:18) q (cid:12)(cid:12)(cid:12)(cid:12) s , x − x b , p − p b (cid:19) × (cid:104) B ; S, S z | q , q , q ; s , s , s (cid:105)× (cid:104) q , q , q ; s , s , s | B ; S, S z (cid:105) . (C9)The above equation is another main result in this paper. Now we assume that the baryon’s momentum wave functionhas the Gaussian form [29, 30] ϕ B ( k b , k c ) = (cid:90) d x b d x c exp ( − i k b · x b − i k c · x c ) ϕ B ( x b , x c )= (2 √ π ) (cid:18) a B1 a B2 (cid:19) / exp (cid:18) − k b a − k c a (cid:19) , (C10)where a B1 and a B2 are two width parameters in the Gaussian wave function of the baryon. One can verify thenormalization condition (C8) holds for the above form of ϕ B ( k b , k c ) . Substituting (C10) into (C9), we can completeintegrals over q b and q c to arrive at Eq. (2.23). Appendix D: Solving Klein-Gordon equation for vector meson fields
In this appendix, we will solve the Klein-Gordon equation (6.1) for vector meson fields using the Green’s functionmethod [53].In terms of V µ = ( φ, A ) and J µ = ( ρ, j ) , the Klein-Gordon equation (6.1) can be put in a three-vector form ∂ φ − ∂ t ( ∂ t φ + ∇ · A ) + m φ = gρ,∂ A + ∇ ( ∂ t φ + ∇ · A ) + m A = g j . (D1)For simple notations, in this appendix we suppress the index ’V’ of following quantities: m ≡ m V , g ≡ g V , E ≡ E V ,and B = B V . The electric and magnetic vector meson fields are given by E = − ∂ t A − ∇ φ, B = ∇ × A . (D2)From the equations for φ and A we derive the following equation for E and B , ( ∂ + m ) E = − g ( ∂ t j + ∇ ρ ) , ( ∂ + m ) B = g ∇ × j . (D3)We can solve Eq. (D3) by taking Fourier transformation ˜ f ( ω, k ) = (cid:90) dtd x exp( iωt − i k · x ) f ( t, x ) ,f ( t, x ) = (cid:90) d k (2 π ) exp( − iωt + i k · x ) ˜ f ( ω, k ) , (D4)8where f can be E , B , ρ , and j . Then in momentum representation Eq. (D3) becomes ( − ω + k + m ) E ( ω, k ) = igω j ( ω, k ) − ig k ρ ( ω, k ) , ( − ω + k + m ) B ( ω, k ) = ig k × j ( ω, k ) , (D5)where we have suppressed tildes on all variables in momentum representation for simple notations. The solutions havethe form E ( ω, k ) = − ig ω j ( ω, k ) − k ρ ( ω, k ) ω − k − m , B ( ω, k ) = − ig k × j ( ω, k ) ω − k − m . (D6)The solutions in space-time can be obtained from their momentum forms by Fourier transformation E ( t, x ) = g∂ t (cid:90) dωd k (2 π ) exp( − iωt + i k · x ) j ( ω, k ) ω − k − m + g ∇ (cid:90) dωd k (2 π ) exp( − iωt + i k · x ) ρ ( ω, k ) ω − k − m , B ( t, x ) = − g ∇ × (cid:90) dωd k (2 π ) exp( − iωt + i k · x ) j ( ω, k ) ω − k − m . (D7)We consider a point charge located at the original point at t = 0 and moves with velocity v in + z direction. Thenthe charge and current density are in the forms in space-time and momentum, ρ ( t, x ) = Qδ ( x ) δ ( y ) δ ( z − vt ) , j ( t, x ) = Qv e z δ ( x ) δ ( y ) δ ( z − vt ) ,ρ ( ω, k ) = 2 πQδ ( ω − k z v ) , j ( ω, k ) = 2 πQv e z δ ( ω − k z v ) = v e z ρ ( ω, k ) . (D8)We evaluate the integral of ρ ( ω, k ) in (D7) I = (cid:90) dωd k (2 π ) exp( − iωt + i k · x ) ρ ( ω, k ) ω − k − m = − Q (cid:90) d k (2 π ) exp [ − ik z ( vt − z ) + i k T · x T ] 1 k z /γ + k T + m = − Q (cid:90) dk T dθdk z (2 π ) exp [ − ik z ( vt − z ) + ik T x T cos θ ] k T k z /γ + k T + m , (D9)where γ = 1 / √ − v , k T ≡ | k T | , x T ≡ | x T | , k z ≡ k z , and we have used cylindrical coordinates in the last step. Wethen use the formula for the Bessel function, πJ ( x ) = (cid:82) π dθ exp( i x cos θ ) , and complete the k z integral by contourintegral around the poles at k z = ± iγ (cid:112) k T + m , where ± depends on the sign of vt − z . The result is I = − Q (2 π ) (cid:90) dk T dk z exp [ − ik z ( vt − z )] × γ k T J ( k T x T ) (cid:16) k z + iγ (cid:112) k T + m (cid:17) (cid:16) k z − iγ (cid:112) k T + m (cid:17) = − Qγ π (cid:82) dk T exp (cid:104) − γ ( vt − z ) (cid:112) k T + m (cid:105) k T J ( k T x T ) √ k T + m , vt − z > − Qγ π (cid:82) dk T exp (cid:104) γ ( vt − z ) (cid:112) k T + m (cid:105) k T J ( k T x T ) √ k T + m , vt − z < (D10)The integral over k T can also be worked out by the formula (cid:90) ∞ dx e − a √ x + m xJ ( bx ) √ x + m = m (cid:90) ∞ dy e − amy J (cid:16) bm (cid:112) y − (cid:17) = 1 √ a + b exp (cid:16) − m (cid:112) a + b (cid:17) . (D11)9Finally we obtain I = − Qγ π ∆ e − m ∆ , (D12)with ∆ = (cid:112) x + y + γ ( vt − z ) . In the same way we can also obtain I = (cid:90) dωd k (2 π ) exp( − iωt + i k · x ) j ( ω, k ) ω − k − m = − v e z Qγ π ∆ e − m ∆ . (D13)Inserting Eq. (D12) and (D13) into (D7), we obtain Eq. (6.2). [1] S. Barnett, Rev. Mod. Phys. , 129 (1935).[2] A. Einstein and W. de Haas, Deutsche Physikalische Gesellschaft, Verhandlungen , 152 (1915).[3] Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. , 102301 (2005), [Erratum: Phys. Rev. Lett.96,039901(2006)], nucl-th/0410079.[4] Z.-T. Liang and X.-N. Wang, Phys. Lett. B629 , 20 (2005), nucl-th/0411101.[5] S. A. Voloshin (2004), nucl-th/0410089.[6] B. Betz, M. Gyulassy, and G. Torrieri, Phys. Rev.
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