Global quark spin correlations in relativistic heavy ion collisions

The observation of the vector meson's global spin alignment by the STAR Collaboration reveals that strong spin correlations may exist for quarks and antiquarks in relativistic heavy-ion collisions in the normal direction of the reaction plane. We propose a systematic method to describe such correlations in the quark matter. The correlations can be classified as local and long range types. We show in particular that the effective quark spin correlations contain the genuine spin correlations originated directly from the dynamical process as well as those induced by averaging over other degrees of freedom. We also show that such correlations can be studied by measuring the vector meson's spin density matrix and hyperon-hyperon and hyperon-anti-hyperon spin correlations. We present the relationships between these measurable quantities and spin correlations of quarks and antiquarks.


I. INTRODUCTION
The global hyperon polarization has been observed first by the STAR Collaboration at the Relativistic Heavy Ion Collider (RHIC) [1] and later in a series of subsequent experiments [2][3][4][5][6].
Recently, the STAR Collaboration published their measurements on vector mesons' global spin alignment [41].
It is clear that the spin correlation of quarks and antiquarks is an important property of QGP.It may contain important information on strong interaction and provide new clue to color confinement in quantum chromodynamics (QCD).So it is crucial to present a unified and description of spin correlations in quark matter and make connection with experimental observables.
The spin correlation in a system of spin-1/2 particles have been defined differently in text books.However, to study spin properties of QGP, it is convenient to give a definition in a way that one can study the correlation order by order.
Such a definition will facilitate the study and reveal the underlying dynamics.The connection between the spin correlation and experimental observables depend on hadronization mechanism and thus may be model dependent.
The purpose of this paper is to propose a systematic description of spin correlation of quarks and anti-quarks in heavy-ion collisions.We show that the spin correlation can be decomposed into the genuine and induced one and also into the local and long range one.We propose that the spin correlation at the quark level can be extracted from the vector meson's diagonal and offdiagonal elements of its spin density matrix together with the hyperon-anti-hyperon spin correlation.We present the relationships between the spin correlation at the quark level and measurable quantities in a simple quark recombination model.
The rest of the paper is organized as follows.In Sec.II, we propose the systematic way to describe quark spin correlation in the quark matter system and discuss its properties.In Sec.III A, we present the results of the vector meson's spin alignment and off-diagonal elements of the spin density matrix as functions of the quark spin polarization and correlation.In Sec.IV, we present the results for the hyperon polarization and hyperonhyperon or hyperon-anti-hyperon spin correlation.We also present numerical estimates of spin correlation parameters by fitting the existing data in Sec.VI.Finally, a short summary and an outlook are given in Sec.VII.

II. QUARK SPIN CORRELATIONS IN QUARK MATTER SYSTEM
We consider a quark matter system such as QGP consisting of quarks and anti-quarks.The spin properties of the system are described by the spin density matrix.For a single particle, we study the spin polarization while for two or more particles we can study not only the spin polarization but also the spin correlation.

A. The spin density matrix
In a spin-1/2 particle system, the spin state of a particle is described by the spin density matrix that can be expanded in terms of the complete set of the 2 × 2 hermitian matrices {I, σi }, i.e., where P qi = ⟨σ i ⟩ = Tr[ρ (q) σi ] with i = x, y, z is the i-th component of the quark polarization vector P q = (P qx , P qy , P qz ), σi denotes Pauli matrices and Trρ (q) = 1 is normalized to one.The symbol I denotes the unit 2×2 matrix, and in the following of this paper, we simply write it as 1.Also, we use the convention that a sum over repeated indices is implicit through out the paper.
For two particle state in the system, we denote the spin density matrix by ρ (12) .Conventionally, one expends ρ (12) in terms of the complete set of hermitian matrices where t is called the spin correlation of particles 1 and 2.Here as well as in the following of this paper, we take the following scheme of notations: the superscript of the spin density matrix ρ or the spin correlation t denotes the type of particle where for hadrons we simply use the symbol while for quarks or anti-quarks we put it in a bracket; the subscript of them denotes the indices of matrix elements or spatial components such as mm ′ or ij.For polarization vectors, we simply use double subscripts to specify particle type and spatial component respectively.
There is however a shortcoming in the definition of the spin correlation through t (12) ij in Eq. ( 2).In case of no spin correlation between particles 1 and 2, we should have ρ(12) = ρ(1) ⊗ ρ(2) and then t (12) ij = P 1i P 2j that is nonvanishing.The situation is the same for spin correlations of three or more particles if they are defined in a similar way.This is in particular inconvenient if we study the spin correlations order by order.
To overcome such a shortcoming, we propose to expend ρ(12) in the following way, where the spin correlation is described by c = 0 if there is no spin correlation between particles 1 and 2. In the same way, we expand the spin density matrix for a system of three or four particles as The polarizations and spin correlations can be extracted by taking expectation values of a direct product of Pauli matrices on spin density matrices.The results are For a four-particle system, according to Eq. ( 5), if the system do not have any spin correlations, i.e., the spin density matrix of the system is the direct product of spin density matrices of single particle, we have If there are only two-particle spin correlations, we have If there are only two-particle and three-particle spin correlations, we have c In this way, we can include spin correlations order by order.
We note that if we define the spin correlation for two spin-1/2 particles h 1 h 2 in the conventional way, i.e, where n stands for the spin quantization direction n, is the fraction of the particle pair in the spin state |m 1 m 2 ⟩.We then obtain the relationship between c nn and c (12) ij defined above as c nn = c (12)  nn + P 1n P 2n .

B. With other degrees of freedom
We suppose particles in the system have other degrees of freedom that are denoted in general by α.We consider here a very simple case that the polarization and spin correlations have α-dependence so that spin density matrices are given by Now suppose we have a system (12) composed of 1 and 2. We assume that the system is at the state |α 12 ⟩ so the probability to find particle 1 and 2 at α 1 and α 2 respectively is determined by the amplitude ⟨α 1 , α 2 |α 12 ⟩.We obtain the effective spin density matrix for the system (12) at α 12 as ρ(12) (α 12 ) = ⟨α 12 |ρ (12) where the average of ρ(1) (α 1 ) and that of P 1i (α 1 ) weighted by the wave function ⟨α 1 , α 2 |α 12 ⟩ squared are given by ρ(1) (α 12 ) = ⟨ρ (1) Here as well as in the following of this paper we use ⟨• • • ⟩ to denote such an average on the state of the system.In the case of c( 12) ij (α 12 ), we have, We see that c( 12) ij (α 12 ) is not simply the average of c We see clearly that c(12;0) To distinguish them from each other, we propose to call c (12) ij (α 1 , α 2 ) the genuine spin correlation but the corresponding c( 12) ij (α 12 ) the effective spin correlation and c(12;0) ij (α 12 ) the induced spin correlation.To be consistent, we will also call Pqi (α 12 ) the effective and P qi (α q ) the genuine polarization.
Also we suggest to distinguish the induced spin correlations into local and long range correlations depending on whether they are short or long ranged in the α-space.An example that leads to such induced spin correlations was given in Refs.[51][52][53][54][55].The spin correlation between s and s was shown to be strong and local in phase space due to strong interaction exchanged ϕ-meson field.
Similarly, for a three-particle system (123), the α-dependent spin density matrix reads If the system (123) is in the state |α 123 ⟩, the effective spin density matrix are given by where the effective polarizations such as P1i (α 123 ) and effective two-particle correlations such as c( 12) ij (α 123 ) have similar expressions as those in the two-particle system given by Eqs. ( 18) and ( 19), and the effective three-particle correlation c(123) ijk (α 123 ) is given by ijk (α 1 , α 2 , α 3 ) = 0, the spin density matrix of the system in Eq. ( 21) is the direct product of single-particle spin density matrices.In this case we have a similar result for the induced two-particle spin correlations to Eq. ( 20), and the induced three-particle spin correlation becomes ijk (α 1 , α 2 , α 3 ) = 0 but there are two-particle spin correlations, the induced three-particle spin correlation has the form We see that if the system only has two-particle spin correlations but has α-dependence, three-particle spin correlations are not vanishing due to averages over α in a given region and/or given α-dependent weight.We call c(123;1) Eq. ( 25) the first order induced spin correlation and c(123;0) ijk (α 123 ) in Eq. ( 24) the zeroth order induced spin correlation.

III. SPIN DENSITY MATRIX FOR VECTOR MESONS
We take a simple case that quarks and anti-quarks in the system combine with each other to form hadrons.We use this as an illustrating example to show the relationship between the quark-quark spin correlations and the polarization of hadrons as well as other measurable quantities.
In this section, we consider the combination process q 1 + q2 → V and present the results for the spin density matrix of the vector meson V .We use M to denote the transition matrix for a q 1 q2 to combine together to form V in the combination process so that the spin density matrix of V is given by Using this, we will calculate elements of ρV in various cases in this section.

A. With only spin degree of freedom
If we only consider the spin degree of freedom, the matrix element of ρV is given by where j = 1 is the spin of V .Hereafter, we will use shorthand notations for spin states of quark-antiquark system in case of no ambiguity.
The transition matrix element ⟨jm| M|m n ⟩ can be further written as where ⟨m n |jm⟩ is the well-known Clebsch-Gordan coefficient.The space rotation invariance demands that j = j ′ and m = m ′ and that jm| M|jm be independent of m.We therefore obtain that where N V is a constant that can be absorbed into the normalization constant.We insert ρ(q1 q2) by Eq. (3) into Eq.( 29) and obtain the element of the vector meson's spin density matrix where + P q1i P q2i is the normalization constant.From Eqs. (30)(31)(32)(33), we see clearly that we have contributions from quark-anti-quark spin correlations in all elements of the spin density matrix of the vector meson.

B. With other degrees of freedom
If there are other degrees of freedom, we have where similar to m n , we use α n to denote (α 1 , α 2 ); and we consider only the case discussed in Sec.II B, i.e. we consider only the α-dependence of ρ(q1 q2) but neglect its off-diagonal elements with respect to α.
The transition matrix element ⟨jm, If we consider only the case where all j, m and α are conserved, we obtain where the rotation invariance of M leads to that ⟨jm, α V | M|jm, α V ⟩ is independent of m.In this case, the matrix element ρ V mm ′ (α V ) is obtained as where that in general depends on m, m ′ , m n and m ′ n through the function Note that ψ does not depend on m and m n if the wave function is factorized, i.e, In this case, we have ψ( We have a similar formula to Eq. ( 16) for ρ(q1 q2) (α V ), i.e., where the effective spin density of q 1 and the effective spin correlation are given by and similar for ρ(q2) (α V ) and Pq2i (α V ).We see that the above results are similar to Eqs. ( 18) and (19).Eq. ( 41) just corresponds to the case discussed in Sec.II B. We note that the factorization in Eq. ( 40) is true in non-relativistic quark models.In relativistic quantum systems, spin is not an independent degree of freedom so the wave function is not factorizable as in Eq. ( 40).In the following of this paper, we limit ourselves to the factorizable case and leave the general case for future studies.
Using Eqs. ( 37) and ( 42), we can obtain the results for ρ V mm ′ (α V ) similar to ρ V mm ′ in Eqs.(30)(31)(32)(33).The results can be obtained from Eqs. (30)(31)(32)(33) by the replacement of spin polarization and correlation quantities, , where Pq1i (α V ) and c(q1 q2) ij (α V ) are defined in Eq. (43).By applying Eq. ( 19), the above results can be put in similar forms as those given by Eqs.(30)(31)(32)(33) but with the average taken over each numerator and each denominator separately weighted by |⟨α n |α V ⟩| 2 .Here we show the result of ρ V 00 corresponding to Eq. ( 30), In practice, in particular in experiments, we often study ρV (α V ) averaged over α V in a given kinematic region.In this case, we obtain e.g for ρ V 00 , 1 + ⟨⟨P q1x P q2x + P q1y P q2y − P q1z P q2z + c (q1 q2) where S denotes the kinematic region of α V or the sub-system over that we average.We emphasized [48] in particular that the average now is two folded.For example, for c where f V (α V ) is the α V distribution of V .We see that for the genuine quark spin correlation c (q1 q2) ij (α q1 , α q2 ) and polarizations P q1 (α q1 ) and P q2 (α q2 ) of q 1 and q2 , we first average over (α q1 , α q2 ) inside the vector meson V .In this step, we obtain ⟨c It is clear that in this step only local quark-anti-quark spin correlations contribute.In the second step, we average over V with different α V .For both the genuine and induced correlations, we average the results obtained in the first step at different α V weighted by the α V distribution f V (α V ) of V .We never consider a q 1 and q2 separated by a large distance in the α-space.Hence we do not have any contribution from long range correlation.This implies that by studying vector meson spin alignment and off-diagonal elements of the spin density matrix, we study only local quark-anti-quark spin correlations.

IV. GLOBAL HYPERON POLARIZATION
For hyperon formation from three quarks, q 1 + q 2 + q 3 → H, similar to the vector meson discussed in Sec.III A, the spin density matrix for the hyperon is given by The calculation of the hyperon's polarization is also similar to the vector meson which we will present in this section.

A. With only spin degrees of freedom
In this case, the matrix element of ρ H mm ′ is written as where j = 1/2 is the spin of the hyperon H.For the quark spin state, we still omit j i in |j i m n ⟩ since they are all 1/2 and use the shorthand notation |m n ⟩ to stand for |m 1 , m 2 , m 3 ⟩.Similar to Eq. ( 28), the transition matrix element ⟨jm| M|m n ⟩ is given by We use again the space rotation invariance that demands that j = j ′ and m = m ′ and that ⟨jm| M|jm⟩ is independent of m.We therefore obtain that Note in particular that the spin density matrix in Eq. ( 50) for the hyperon has the same form as that in Eq. ( 29) for the vector meson.
By inserting Eq. ( 4) into Eq.( 50), we obtain the spin density matrix ρH and the hyperon polarization and C H = Trρ H is the normalization constant.The result for Λ is given by From Eqs. (51-53), we see in particular that the Λ polarization is not simply equal to that of the s-quark when quark spin correlations are considered.This is because to produce a spin-up or spin-down Λ, we need not only a spin-up or spin-down s-quark but also a spin zero ud-di-quark.If the spin of s and those of u and d are correlated, the probability to have a spin zero ud-di-quark in the case that s is spin-up can be different from that in the case that s is spin-down.This leads to influences on the final polarization of Λ.
For other J P = (1/2) + hyperons, we obtain where δρ H112 , C H112 , δρ Σ 0 and C Σ 0 are given by where H 112 denotes J P = (1/2) + hyperon with quark flavor q 1 q 1 q 2 and we have used c .We see again that there are contributions from spin correlations in hyperons' polarizations.

B. With other degrees of freedom
If there are other degrees of freedom besides spin, the matrix elements of ρ H mm ′ (α H ) is given by The transition matrix element ⟨jm, α H |M|m n , α n ⟩ can be further written as In the case that j, m and α are conserved, we obtain so that the matrix element ρ H mm ′ (α H ) is given by where ρ(q1q2q3) (α H ) in general depends on m, m n , m ′ , m ′ n and is given by where ψ(m n , m; α n , α H ) is defined as If the wave function is factorizable, ⟨m n , α n |jm, α H ⟩ = ⟨α n |α H ⟩⟨m n |jm⟩, we have ψ(m n , m; α n , α H ) = ⟨α n |α H ⟩. So Eq. ( 64) is simplified as Inserting Eq. ( 66) into Eq.( 63) we can calculate the hyperon polarization and results take exactly the same form as Eqs.(51)(52)(53)(54)(55)(56)(57)(58)(59) except that all quantities are replaced by effective ones, e.g.
ij and so on.For example, the polarization of Λ is in the form, By applying Eqs. ( 16) and ( 23), we can rewrite Eq. ( 67) as where the averages are taken with the weight |⟨α n |α Λ ⟩| 2 .We see that the situation is similar to vector mesons.If all the genuine quark correlations c (q1q2) ij = 0 and c (q1q2q3) ijk = 0, we still have induced correlations, Similar to the vector meson's spin alignment, we often run into P Λ (α Λ ) averaged over α Λ in a given kinematic region.Then we will have an additional average over the distribution f Λ (α Λ ).It is also obvious that we have only local quark-quark spin correlations in this case.
Together with the results obtained in Sec.III, we see that by studying spin polarizations of one hadron we always study only local quark spin correlations.

V. GLOBAL SPIN CORRELATIONS OF HYPERONS
The calculations can be extended in a straightforward manner to hyperon-hyperon and hyperon-anti-hyperon spin correlations.In this section, we take hyperon-antihyperon as an example to show the calculation and results.
For a spin-1/2 hyperon pair H 1 H2 , the spin correlation is usually defined in the conventional way as given by Eq. ( 12), i.e., where f H1 H2 m H 1 m H2 = ⟨m H1 m H2 |ρ H1 H2 |m H1 m H2 ⟩ and m H1 , m H2 = ± denoting the spin states parallel or anti-parallel to n-direction respectively.We simply adopt this definition and calculate its relationship to those quantities defined at the quark level using quark combination mechanism.In the calculation, the most convenient way is to rotate the Cartesian system so that n-direction is z-direction in the new system.We choose this case as an example to do the calculations and denote c H1 H2 nn in this case by c H1 H2 zz in the following of this paper.Now the task is to compute the spin density matrix element ρ H1 H2 Similar to spin density operators for vector mesons and hyperons given by Eqs. ( 26) and ( 47), ρH1 H2 is related to that of the six body system q 1 q 2 q 3 q4 q5 q6 by where we simply used '1 • • • 6' to label q 1 q 2 q 3 q4 q5 q6 .The matrix element of ρH1 H2 is given by The complete expansion of ρ(1 5) ⊗ ρ(6) + 14 exchange terms ] ijkl σ1i ⊗ σ2j ⊗ σ3k ⊗ σ4l ⊗ ρ(5) ⊗ ρ(6) + 14 exchange terms ] In the following of this section, we take Λ Λ as an example to show the calculation of the hyperon-anti-hyperon spin correlation.For simplicity, we only consider two-particle spin correlations and set all others with more than two particles as zero.As before, we consider two cases, the one with only spin degree of freedom and the one with other degrees of freedom denoted by α.The calculations can be extended to other hyperons and/or include spin correlations of more than two particles in a straightforward way.

A. With spin degree of freedom
As in previous sections, we insert the completeness identity mn |m n ⟩⟨m n | = 1 into Eq.(72), and obtain where we have suppressed j H1 = j H2 = 1/2 for H 1 = Λ and H2 = Λ.The transition matrix element can be written as where we have assumed a factorization form for ⟨m H1 m H2 | M|m H1 m H2 ⟩ with M = MH M H , so that the transition matrix contributes only to the normalization constant and has no effect on the spin part.Then we obtain We note that in production processes of H 1 and H2 , q 1 + q 2 + q 3 → H 1 and q4 + q5 + q6 → H2 , the Clebsch-Gordan coefficient ⟨m H1 m H2 |m n ⟩ is just the product of ⟨m H1 |m 1 m 2 m 3 ⟩ and ⟨m H2 |m 4 m 5 m 6 ⟩.When all two-particle spin correlations are considered, the result for the spin correlation of Λ Λ is iz P ui + c (us) iz P ūi + c (ūs) iz P ui + c (us) iz P ūi + c (ūs) where the normalization constant C Λ Λ is given by where C Λ is given by Eq. ( 53) and C Λ is obtained from C Λ with the replacement of everything by that of the corresponding anti-quark.We compare the results given by Eqs.(77-78) with those given by Eqs.(51)(52)(53) for Λ-polarization.We note that we need to put all spin correlations of more than two quarks and/or anti-quarks and products of two particle spin correlations as zero since we consider only two particle spin correlations.In this way, we obtain, From Eq. (79), we see clearly that the spin correlation of Λ and Λ comes from those of quarks and anti-quarks.We also see clearly that c Λ Λ zz = P Λz P Λz if only quark-quark and anti-quark-anti-quark spin correlations are considered.

B. With other degrees of freedom
It is clear that in this case the six-quark (anti-quark) spin density matrix ρ(1
Then we obtain a similar formula for ρ H1 H2

H2
(α H1 , α H2 ) to Eq. (76).By assuming a factorization condition similar to Eq. ( 75) for H 1 and H2 and that for the spin and α parts of wave functions, we obtain where the effective density matrix is given by We see the difference between Eq. (76) for the case with only the spin degree of freedom and Eq. ( 80) is the replacement We emphasize the average in Eq. ( 81) can be carried out inside H 1 and H2 for quarks and antiquarks successively.This is different from the case for vector meson discussed in Sec.III B where the average inside V is carried out for the quark and antiquark simultaneously.More precisely, we have Pq l (α H1 ) = αn P q l (α q l )|⟨α n |α H1 ⟩| 2 = ⟨P q l (α q l )⟩ H1 , (82) We obtain two-particle spin correlations as We see that c(q1q2) because ⟨P q1i (α q1 )P q2j (α q2 )⟩ H1 H2 = ⟨P q1i (α q1 )⟩ H1 ⟨P q2j (α q2 )⟩ H2 . (88) Here we neglect the overlap of H 1 and H2 in α-space.In this case, we have no contributions from the induced spin correlation between the quark and anti-quark, i.e., c(q1 q2;0) ij (α H1 , α H2 ) = 0. Also, because α qi is inside H 1 while α qj is inside H2 , we do not have contributions from local spin correlations between quarks and antiquarks.This can be seen more clearly if we assume all genuine two-particle correlations vanish.In this case, the average of each term in Eqs.(81) can be separated into a product two factors, one is inside H 1 for quarks and the other is inside H2 for antiquarks.This shows explicitly that we have contributions from local quark-quark and antiquark-antiquark correlations but no contribution from local quark-antiquark correlations.Hence in spin correlations between the hyperon and anti-hyperon, there is no contribution from local correlations between the quark and antiquark.Now we compute the spin correlation of Λ Λ with above formula.With only two-particle spin correlations, the result is just those given by Eqs.(77-79) with the replacement of all quantities by the corresponding effective ones, i.e.
We emphasize in particular that all quantities for quarks and/or anti-quarks on the r.h.s. of Eq. (89) are effective ones and are functions of α Λ and/or α Λ.More precisely, Pqi and Pqi are functions of α Λ and α Λ respectively, while c(q1 q2) ij is a function of (α Λ , α Λ).These results are valid in the case that we neglect the overlap of the wave function of Λ with that of Λ.
We now discuss a simple case where both quark polarizations and quark spin correlations are small so that we can neglect the last two terms in Eq. ( 89) compared with the first two.In this case, when further averaged over α Λ and α Λ in a given kinematic region, we obtain where ⟨c Compared to those inside a hadron, we call this long range correlation.We see that ⟨c (ss;0) zz ⟩ is the contribution from the induced spin correlation while ⟨c (ss) zz ⟩ is from the genuine quark spin correlation of ss.
Since we do not consider the overlap of the wave functions of the hyperon and that of the anti-hyperon, the results obtained above can be extended directly to hyperon-hyperon spin correlations.In particular, those given by Eqs.(89-90) can be extended to ΛΛ spin correlations if we neglect the overlap of the wave function of the two Λ's.In this case, we need only to replace Pq (α Λ ) and Pq (α Λ) by Pq (α Λ1 ) and Pq (α Λ2 ) respectively in order to obtain c ΛΛ zz (α Λ1 , α Λ2 ).
In the simple case considered above for ⟨c Λ Λ zz ⟩ we obtain similar result for c ΛΛ zz as We see that in this case the spin correlation between Λ's measures the spin correlation between s-quarks.
To compare with the results obtained in Sec.III, we see clearly that by studying the spin alignment of ϕ meson, we study the spin correlations between s and s inside the vector meson.In contrast, by studying Λ Λ and ΛΛ spin correlations, we study the spin correlations between ss or ss in the whole QGP system [13,54,55].The former is in general short ranged while the latter includes long range contributions.The strength of such correlations is determined by the dynamics of the system and is an important direction for future study.

VI. NUMERICAL ESTIMATES
The global quark spin polarizations and correlations are determined by the QCD dynamics in heavy-ion collisions and can be calculated using QCD-based theoretical models.Having the relationships between measurable quantities at the hadron level and global spin properties at the quark level, we can also extract them from data available and make predictions for other measurable quantities.The available data are however still far from enough to make high precision predictions.In this section, we just present a rough estimate based on the data available [1][2][3][4][5][6]41].
We use Eqs.(69,44,90) and take approximately where all quark spin correlations are effective ones and are in general sums of genuine and induced contributions.We use these equations to extract ⟨P s ⟩ and c(ss) zz;ϕ from data available on ⟨P Λ ⟩ and ⟨ρ ϕ 00 ⟩ [1-6, 41], and make estimates of ⟨c Λ Λ zz ⟩.We take the following forms of ⟨P s ⟩ and c(ss) zz;ϕ as functions of

VII. SUMMARY AND OUTLOOK
The STAR measurements of the global spin alignment of vector mesons ρ ϕ 00 [41] indicate that there are strong global quark-antiquark spin correlations in relativistic heavy-ion collisions.
It opens a new window to study properties of QGP and reaction mechanisms of relativistic heavy ion collisions.
We propose a systematic way of describing quark and/or antiquark spin correlations in the QGP.We show that effective quark spin correlations contain contributions from genuine spin correlations from dynamics and induced spin correlations due to average over other degrees of freedom.We derive the relationships between these spin correlations at the quark level and those for hyperons and vector mesons that are measurable in experiments.We show in particular that the vector meson's spin density matrix elements, either diagonal or off-diagonal, are sensitive to local spin correlations between the quark and antiquark, while hyperon-anti-hyperon spin correlations are sensitive to long rang quark spin correlations.We present a rough estimate of spin correlations based on available data [1][2][3][4][5][6]41] to guide future measurements.
When another degree of freedom characterized by α is considered, we assume that the spin and α part of the wave function are factorizable.This is general true in the non-relativistic case.However, in the relativistic case, spin and other degree of freedom such as momentum are usually coupled in an intrinsic way so that such a factorization is impossible.In such cases, the calculation is more complicated but can be done, which we reserve for a future study.

FIG. 3 .
FIG. 3. (a) The effective global spin correlation c(ss) zz between s and s as a function of energy √ sNN obtained by fitting the data [1-6, 41] using Eqs.(93) and (94) described in the text.(b) Estimates of ⟨c Λ Λ zz ⟩ as functions of √ sNN in the two extreme cases described in the text.