Linear response theory for spin alignment of vector mesons in thermal media

We present a calculation of the spin alignment for unflavored vector mesons in thermalized quark-gluon plasma based on the Kubo formula in linear response theory. This is achieved by expanding the system to the first order of the coupling constant and the spatial gradient. The effect strongly relies on the vector meson's spectral functions which are determined by the interaction and medium properties. The spectral functions are calculated for the one-quark-loop self-energy with meson-quark interaction. The numerical results show that the correction to the spin alignment from the thermal shear tensor is of the order $10^{-4}\sim10^{-5}$ for the chosen values of quark-meson coupling constant, if the magnitude of thermal shear tensor is $10^{-2}$.


I. INTRODUCTION
Rotation and spin polarization are inherently connected and can be converted to each other as demonstrated in the Barnett effect [1] and Einstein-de Haas effect [2] in materials.The same phenomenon known as global polarization can also exist in peripheral heavy-ion collisions at high energies in which the huge orbital angular momentum is partially distributed into the strong interaction matter in the form of particles' spin polarization [3][4][5][6][7].The global polarization of hyperons has been observed in experiments [8,9] and been extensively studied in recent years [10][11][12][13][14][15].
Unlike the spin polarization of hyperons that can be measured through their weak decay, vector mesons can only decay by strong interaction which respects parity symmetry, which makes their spin polarization inaccessible in experiments.For spin-1 vector mesons, the only spin observables that can be measured are some elements of the spin density matrix ρ λ1λ2 with λ 1 and λ 2 denoting spin states along the spin quantization direction.One of them is ρ 00 that can be measured through the decay daughter's polar angle distribution in the rest frame of the vector meson.If ρ 00 is not 1/3, it means that the spin-0 state is not equally occupied among three spin states, which is called the spin alignment.The global spin alignment in heavy-ion collisions was first suggested by Liang and Wang [16].The global spin alignments of ϕ and K * 0 mesons were first measured by STAR collaboration in Au+Au collisions GeV in 2008 [17], but no signals were found.With the accumulation of experimental data, STAR Collaboration finally found a large spin alignment for ϕ mesons in Au+Au collision at lower energies but not for K * 0 [18].
Such a large spin alignment for ϕ mesons cannot be fully accounted by conventional mechanism [19][20][21][22][23].Some of us proposed that local fluctuations of vector fields in strong interaction may give a large deviation of ρ 00 from 1/3 for ϕ mesons [24].Such a prediction was made in a nonrelativistic quark coalescence model [19,25] that works for static or nearly static mesons in principle.Such a nonrelativistic quark coalescence model has been promoted to a relativistic version [26,27] based on quantum transport theory [28][29][30][31] with the help of covariant Wigner functions for massive particles [32][33][34][35][36][37][38][39] and matrix valued spin-dependent distributions [40,41].With fluctuation parameters of strong interaction fields extracted from transverse momentum-integrated data for ρ 00 as a function of the collision energy, the calculated transverse momentum dependence of ρ 00 agrees with STAR's data for ϕ mesons [18].The rapidity dependence of ρ 00 has also been predicted with same parameters before preliminary data of STAR was released: the main feature of the data can be described by the theoretical result [42].For recent reviews on the spin alignment of vector mesons, see, e.g., Refs.[43][44][45].
Recently, the contribution from the thermal shear tensor to the spin alignment of the vector meson has been calculated using the linear response theory [46] and kinetic theory [47].The authors of Ref. [46] argued that this contribution is quite large based on an estimate of the energy shift and width of the vector meson in medium without really calculating them.This work was inspired by Refs.[48][49][50][51][52] pointing out that there is a coupling between the spin polarization and the thermal shear tensor which can partially resolve the local polarization puzzle of Λ hyperons.
In this paper, we will calculate the spin alignment of vector mesons from the Kubo formula in linear response theory [53][54][55][56] in thermalized quark-gluon plasma (QGP).Vector mesons are assumed to be thermalized, and quarks and antiquarks are assumed to be unpolarized.The interaction is described by the vertex between vector meson and quark-antiquark [57][58][59][60].In Sec.II, we present two-point Green's functions of different kinds for vector mesons in the closed-time-path (CTP) formalism [28-31, 61, 62].In Sec.III, we give an introduce on spin density matrices for vector mesons from Wigner functions.In Sec.IV, we present the Dyson-Schwinger equation for retarded Green's functions.We give the expression for retarded self-energies of vector mesons including the contribution from one-quark-loop.In Sec.V, we give the general form of spectral functions in medium for vector mesons from retarded Green's functions.In Sec.VI, we use the Kubo formula in the linear response theory [53][54][55][56] to calculate the correction to the two-point Green's function proportional to the thermal shear tensor.From it we are able to calculate the correction to ρ 00 in Sec.VII.We adopt the hard-thermal-loop (HTL) [29,[63][64][65][66][67] and quasi-particle approximations [68] to calculate spectral functions.The HTL approximation provides a toy model to illustrate the physics inside this problem since we have analytical formula for spectra functions.Then we consider a more realistic quasi-particle approximation for spectral functions.Under a few approximations or assumptions, we obtain an analytical expression for the correction to ρ 00 , which depends on the width and energy shift from the self-energy.The numerical results for the tensor coefficients in the correction to ρ 00 are presented.The conclusion and discussion are given in Sec.VIII.
In this paper, we adopt following notational conventions: Greek letters denote components of four-vectors while lowercase Latin letters as subscripts denote components three-vectors.The four-momentum p µ is not necessarily on-shell unless we add an index 'on'.The summation of repeated indices is implied if not stated explicitly.The definition of two-point Green's functions G and Σ in this paper differs by a factor i = √ −1 from the usual one in quantum field theory, which are related by G = i G and Σ = i Σ.

II. TWO-POINT GREEN'S FUNCTIONS
In this section we will give an introduction to two-point Green's functions for vector mesons on the CTP as shown in Fig. 1.The CTP formalism is a field-theory based method for many-body systems in off-equilibrium as well in equilibrium [28-31, 61, 62].When it is used for systems in equilibrium, it is actually the real time formalism of the thermal (finite temperature and density) field theory [56,69].Wigner functions can be obtained from two-point Green's functions and are related to spin density matrices, which will be addressed in the next section.We refer the readers to Section 2.2 of Ref. [70] for a very brief introduction to two-point Green's functions on the CTP.
The Lagrangian density for unflavored vector mesons with spin-1 and mass m V reads where A µ (x) is the real vector field for the meson, is the field strength tensor, and j µ is the source coupled to A µ (x).
The two-point Green's function on the CTP is defined as where ⟨• • • ⟩ denotes the ensemble average and T C denotes time order operator on the CTP contour.Depending on whether the field A µ lives on the positive or negative time branch, we have four components From the constraint G µν F + G µν F = G µν < + G µν > , only three of them are independent.In the so-called physical representation [28,71,72], three independent two-point Green's functions are where the subscripts "A" and "R" denote the advanced and retarded Green's function respectively.The two-point Green's functions in Eqs.(3)(4) can be used to express any two-point functions defined on the CTP contour such as the self energy Σ µν (x 1 , x 2 ).When dealing with the vacuum contributions to G µν R,A , the last equalities in the first and second line of Eq. ( 4) do not exactly hold since a singular term ∼ δ(t 1 − t 2 ) is missing.

III. WIGNER FUNCTIONS AND SPIN DENSITY MATRICES
In this section, we will introduce how one can obtain spin density matrices for vector mesons from Wigner functions.We refer the readers to some recent reviews [45,73] for details of the topic.
The second quantization of the vector field is in the form where p µ on = (E V p , p) is the on-shell momentum of the vector meson, E V p = |p| 2 + m 2 V is the vector meson's energy, λ denotes the spin state, a(λ, p) and a † (λ, p) are annihilation and creation operators respectively, and ϵ µ (λ, p) ≡ ϵ µ (λ, p on ) represents the polarization vector obeying the following relations where ∆ µν (p) = g µν − p µ p ν /p 2 is the projector perpendicular to p µ .One can check that the quantum field A µ defined in Eq. ( 5) is Hermitian, A µ = A µ † .The Wigner function can be defined from ] by taking a Fourier transform with respect to the relative position y Inserting the quantized field (5) into the definition of the Wigner function (7), we obtain where the superscript "(0)" denotes the leading order contribution in ℏ or gradient expansion, and the MVSD [40,41] at the leading order for the vector meson is defined as Note that f (0) λ1λ2 (x, p) is actually the (unnormalized) spin density matrix ρ λ1λ2 , which can be decomposed into the scalar, polarization (P i ) and tensor polarization (T ij ) parts as [45,73] where i, j = 1, 2, 3, Tr(f λλ , and Σ i and Σ ij are 3 × 3 traceless matrices defined as Let us define an integrated or on-shell Wigner function It is easy to check that the second equality holds for the leading order Wigner function G (0)< µν (x, p) given by Eq. ( 8).But we assume that it hold at any order.One can check that W µν (x, p on ) is always transverse to the on-shell momentum, p on µ W µν (x, p on ) = 0.The on-shell Wigner function can be decomposed into the scalar (S), polorization (W [µν] ) and tensor polarization (T µν ) parts as [45,73] where each part is defined as With Eq. ( 13) one can show that both W [µν] and T µν are traceless, g µν W [µν] = g µν T µν = 0. Inserting Eq. ( 10) into Eq.( 12), we have We see that W [µν] is related to P i while T µν is related to T ij .We can extract f 00 ∝ ρ 00 by projecting onto W µν in Eq. ( 12) as In (16), ϵ µ (0, p) is the polarization vector along the spin quantization direction.With the first line of Eq. ( 15) and Eq. ( 17), we obtain The above formula relates the Wigner function to ρ 00 , which we will use to calculate the correction to ρ 00 in Sect.VII.

IV. DYSON-SCHWINGER EQUATION ON CTP
In this section we will give an introduction to the Dyson-Schwinger equation (DSE) on the CTP which incorporates retarded and advanced self-energies to be used for spectral functions in the next section.
We start from the integral form of the Dyson-Schwinger equation (DSE) on the CTP for the vector meson [27,74] where , C denotes the integral on the CTP contour, G µν (0) and G µν are the bare and full propagator respectively, and Σ ρσ is the self-energy.In Eq. ( 19) we have suppressed the index 'CTP' in two-point functions G µν (0) , G µν and Σ ρσ .Contracting (G µν (0) ) −1 on both sides of Eq. ( 19) and writing the DSE in the matrix form, we obtain where the integral over x ′ 2 is an normal one (not on the CTP).Under a unitary transformation, Eq.( 20) can be put into the physical representation where we used the shorthand notation , and the spatial inhomogeneity of the system, as required by the Kubo formula, is induced by a perturbation.One can obtain the Dyson-Schwinger equation for retarded and advanced Green's functions in momentum space (propagators) The free retarded and advanced propagators are given by One can check that G ρν (0)A/R (p) satisfies Eq. ( 22) neglecting the last term in the right-hand-side.The coupling between the vector meson and quark-antiquark in QGP or the qqV vertex is assumed to be g V Bψ q γ µ ψ q A µ [57][58][59][60].Here B denotes the Bethe-Salpeter wave function and can be parameterized as [75,76] where p−p ′ and p ′ are momenta of the quark and anti-quark respectively.We see that the wave function only depends on the relative momentum.We can assume that only when the distance between the quark and anti-quark is zero can they form a meson, thus we have 1/σ → 0 and B = 1.Then the vector meson's self-energy to the lowest order of the coupling constant g V from the quark one-loop is shown in Fig. (2).Applying Eq. ( 4), we can construct retarded and advanced self-energies as where is the two-point Green's function of quarks on the CTP.We have included a negative sign for the quark loop in Eq. (25).Under the assumption that the system is homogeneous in position space, we obtain self-energies in momentum space The retarded self-energy is our starting point for derivation of spectral functions for vector mesons.

V. SPECTRAL FUNCTIONS FOR VECTOR MESONS
In this section, we will derive spectral functions for vector mesons from the retarded self-energy.We use the CTP formalism in grand-canonical equilibrium which is also called the real time formalism of the thermal field theory.The vacuum and thermal equilbrium contributions are incorporated in the same framework.We assume that quarks and antiquarks are unpolarized and their distributions are the Fermi-Dirac distribution (A2).
Evaluating the retarded self-energy in Eq. ( 26) using the quark propagators in Appendix A, we obtain where I µν 1 and I µν 2 are medium parts while I µν vac is the vacuum part.The derivation of I µν 1 , I µν 2 and I µν vac are presented in Appendix B. From Eq. (B10) we have where Σ ⊥ (p) denotes the transverse part of Σ ij R (p).Using above relations, we can greatly simplify the result of full propagators.
From Eq. (A1), one can see the only difference between the retarded and advanced propagators is the sign of the small positive number ϵ, so the retarded and advanced propagators or self-energies are complex conjugate to each other, Σ µν A = −Σ µν * R (note that there is an i factor in the definition of the self-energy).It can be checked that Σ µν R is transverse to p µ as required by the current conservation.We note that the vacuum contribution and its real part is divergent and can be renormalized [68].The imaginary part of the vacuum contribution corresponds to the pair production or annihilation processes.Inserting Eq.( 27) into Eq.( 22) and introducing we obtain From the definition of I µν 1,2 , we find that they are written in terms of projectors related to three momentum p.Therefore, we assume G µν R has the same structure with Σ µν R and can be written as where A, B, C, D are functions of p and are not independent since G µν ≶ are transverse to p µ .By solving where Σ ⊥ = −iΣ ⊥ .Other two functions B and D can be expressed in terms of A and will be discussed later.We can also define where n B (p 0 ) = 1/(e βp0−βµ V − 1) is the Bose-Einstein distribution with the inverse temperature β ≡ 1/T and the vecctor meson's chemical potential µ V (µ V = 0 for the unflavored meson).Note that there is an i factor in the definition of the propagator without tilde.From Eq. ( 33), we find the real part of A, B, C, D have no contributions to the spectral function, and the imaginary part of A, B, D have following constraints from p µ G µν < = 0, Inserting Eq. ( 31) into Eq.( 33), one can obtain or equivalently In Eqs.(35,36), we defined as the transverse and longitudinal projector respectively with p µ = (0, p), and ρ T,L are spectral functions in the transverse and longitudinal directions given by where Σ ⊥ and Σ 00 are from Σ µν R : is the sign of p 0 , and ε is an infinitesimal positive number.One can check in Eq. ( 37) that p µ ∆ µν T = p µ ∆ µν L = 0.In Eq. ( 38), one can verify that the real parts of Σ ⊥ and Σ 00 contribute to the mass correction while the imaginary parts of Σ ⊥ and Σ 00 determines the width or lifetime of the quasi-particle mode.For free vector mesons, the spectral functions are ρ for the free vector meson following Eq.( 35) and Im G µν R (p) for the free vector meson following Eq.(36).

VI. KUBO FORMULA IN LINEAR RESPONSE THEORY
In this section we use the Kubo formula in linear response theory to calculate the non-equilibrium correction to G µν < (p).The Kubo formula has been derived in Zubarev's approach to non-equilibrium density operator [55,77,78].
According to the Kubo formula, the linear response of the expectation value of an operator Ô to the perturbation ∂ µ β ν has the form where Ô(x) ≡ Tr ρ Ô(x) and Ô(x) LE ≡ Tr ρLE Ô(x) with ρ and ρLE being the non-equilibrium and local equilibrium density operator respectively [78], β µ (x) ≡ u µ (x)/T (x) with u µ (x) and T (x) being the local velocity and temperature respectively, K µ is the momentum roughly equals to π/L with L being the length of the system, and is the energy-momentum tensor for the vector field.Detailed derivation of Eq. ( 39) is given in Ref. [78].Now we set Ô(x) to be the operator corresponding to which gives G µν < = Ĝµν < (x, p) .In Eqs. ( 40) and ( 41) we explicitly show the "hat" on the field operator Âµ which we have suppressed in Sec.II and III just to emphasize their operator's nature in the Kubo formula (39).When inserting Eqs. ( 40) and ( 41) into Eq.( 39), the vector field Âµ can be approximated as the free field at the leading order in space-time gradient, since ∂ µ β ν (x) is already of the next-to-leading order.
Substituting Ĝµν < (x, p) in (41) into Eq.(39), one obtains the next-to-leading order term of G µν < as where is the Bose-Einstein distribution defined after Eq. ( 33), and ρ L,T are given in Eq. (38).Note that integral ranges for p 0 1,2 are different from Ref. [46].The tensor I µνγλ ab (p 1 , p 2 ) can be expressed in terms of projectors ∆ µν L,T as Then we integrate Eq. ( 42) over p 0 from 0 to +∞ to exclude the contribution from anti-particles.As we have mentioned above, the limit K µ → 0 should be taken in the last step, thus the integral of Eq. ( 42) can be simplified as where ξ γλ = ∂ (γ β λ) denotes the thermal shear tensor.
The spin alignment coupled with the thermal shear tensor is given by where G < µν (x, p) is given in Eq. ( 35) while δG < µν (x, p) is given in (44), and L µν (p on ) is defined in Eq. (16).The above formula is the starting point for us to evaluate the correction to ρ 00 from the shear stress tensor in the next section.
We should note about the difference between the average taken in G < µν (x, p) given by Eq. (35) [as well as other avarages in Sec.II] and the one taken in Ĝµν < (x, p) in Eq. (42).The local equilibrium average is implied for the former, while the non-equilibrium average is implied for the latter.For notational simplicity, we do not put "LE" index to local equilibrium averages in this paper except in the Kubo formula Eqs.(39) and (42).

VII. SPIN ALIGNMENT CORRECTION FROM SHEAR TENSOR
In this section, we will calculate the spin alignment correction from the shear tensor.To this end, we adopt two approximations to evaluate self-energies and spectral functions of unflavored vector mesons: the HTL and quasiparticle approximation.

A. HTL approximation
Under the HTL approximation, the external momentum of the vector meson's self-energy is of order g V T which is called "soft" while the quark loop momentum is of order T which is called "hard" [29,[63][64][65][66][67].This condition is not satisfied for the real vector meson in the thermal environment at RHIC and LHC with p 0 > m V ≫ T .The reason that we still consider the HTL approximation is that the self-energies in this approximation is analytical and the calculation of the spin density matrix is transparent.In other words, we treat the HTL approximation as a toy model to show the underlying physics.
We can consider massless quarks for simplicity.Note that the vacuum term is not included since the imaginary part of the vacuum term corresponds to the process that one particle decomposes into two on-shell quarks, i.e. p 0 > k 0 , which is beyond the HTL approximation.The vacuum contribution is proportional to p 2 ∆ µν as required by the Ward identity, which is of order g 4 V T 2 since p ∼ g V T .The self-energy in the HTL approximation reads where m 2 T = g 2 V T 2 /9 denotes the thermal mass.The real and imaginary parts of Σ 00 and Σ ⊥ can be obtained as where θ(x) is the Heaviside step function.We see that the imaginary parts are non-vanishing only in space-like region of p.Under the HTL approximation, one can get the inequality p 0 ∼ m T ∼ g V T ≪ m V , which provides a natural power counting in α ≡ m T /m V .We also assume p 2 0 < |p| 2 , so there is no pole contribution.Then the spectral function ρ L/T in (38) can be approximated as for p 0 ≪ m V .Using Eq. ( 48) in Eq. ( 45), we can get the leading order term of δρ 00 where ∆ µν L,T are projectors defined in Eq. ( 37).In Eq. ( 49), the polarization vector can be approximated as ϵ µ (0, p on ) = (0, 0, 1, 0) + O(α) with |p| ≪ E p ≈ m V , where we choose y direction as the spin quantization direction.So ∆ µν (p on ) can be approximated as g µν − g µ0 g ν0 .The leading order I µνγλ ab,(0) is given by which is O(m 2 V ).Finally we can estimate where ξ ≡ |ξ γλ | is the magnitude of the thermal shear tensor.If we set the parameters' values as g V = 1, T =150 MeV, m V =1020 MeV, the coupling between the spin alignment and the shear tensor is about α 2 ∼ O(10 −2 ).If we further use ξ ∼ 0.01, then we obtain δρ (0) 00 ∼ O(10 −4 ), which is much smaller than the contribution from the coalescence model via strong force fields [26].

B. Vector mesons as resonances
Now we consider the realistic case that vector mesons are resonances so that the coalescence and dissociation processes can happen.In this case, we have p 0 > E k and (p − k) 2 > 0. The small imaginary numbers in the quark loop integral become ±i(E k ± p 0 )ϵ ∝ iϵ in J ± (p; n 1 , n 2 ) in Eq. (B4).Therefore, the vector meson's self-energies read where Note that the vacuum contributions to real parts of self-energies are canceled by renormalization.When evaluating imaginary parts, we note that J 0 (n 1 , n 2 ) is real and ImJ + is non-zero in the region p 2 + 2p 0 E k < 2|k||p|, which cannot be satisfied under the quasi-particle approximation with p 0 > |p| and E k > |k|.

So the imaginary parts come from ImJ
where µ q is the chemical potential of quarks, m q is the quark mass, and E max/min is We see that imaginary parts exist only when p 2 > 4m 2 q .Then the imaginary parts of self-energies read Note that vacuum contributions are included in imaginary parts of self-energies, which correspond to pair production and annihilation (dissociation and combination) processes involving on-shell particles in the initial and final states (the meson, quark and antiquark are all on-shell).

Quasi-particle approximation
We take the quasi-particle approximation (QPA) for the vector meson that g V is not very large and the self-energies are assumed to be small compared with m 2 V .In this case, the spectral functions in Eq. ( 38) have narrow peaks around E V p .In the region near p 0 = E V p , we can approximate the self-energies as their on-shell values, i.e., Σ 00 (p) ≈ Σ 00 (p on ) and Σ ⊥ (p) ≈ Σ ⊥ (p on ).Then spectral functions for transverse/longitudinal modes can be approximated as where Γ T /L are widths and ∆E T /L are energy shifts for transverse/longitudinal modes approximated as We see in Eq. ( 56) that ρ pole T /L (p) denote pole contributions while ρ cut T /L (p) denote cut contributions.We plot widths and energy shifts in Fig. 3 as functions of |p| at g V = 1, 2. We choose two sets of values for the strange quark chemical potential and temperature corresponding to the freezeout conditions at √ s N N ≈20 and 200 GeV in heavy-ion collisions [79,80]: µ s ≈ µ B /3 ≈ 64.5 MeV and T ≈ 155.7 MeV (black) and µ s ≈ µ B /3 ≈ 7.4 MeV and T ≈ 158.4 MeV (red).Other parameters are set to g V = 1, m V = 1.02GeV, and m s = 419 MeV.We can check ρ pole T /L (p) = 0 for these values of parameters since 4m 2 q − p 2 < 0 at the corrected mass-shell Re Σ 00 (p on ) for transverse and longitudinal modes respectively.One can see in Fig. 3 that the width and energy shift are almost independent of freezeout conditions at the collision energy 20 and 200 GeV.
We find that the Γ T /L and ∆E T /L are much smaller than m V , which allows us to introduce the following power counting scheme where we have introduced ϵ as a small power counting parameter.Since Γ T and Γ L are positive definite, we expect that their difference is a second-order contribution ∆Γ/E . On the other hand, such a cancellation may not happen for ∆E T and ∆E L , because they may have different signs.Therefore (∆E T − ∆E L )/E V p ≲ O(ϵ) could be a first-order contribution.According to hydrodynamic simulation of the strong interaction matter in heavy-ion collisions, the thermal shear tensor ξ ≡ |ξ γλ | is a small quantity of O(10 −2 ), which can be treated as another power counting parameter.With Eq. ( 56) for spectral functions, one can prove that the term with the p 0 integral of δG µν < (x, p) in the denominator of the right-hand-side of Eq. ( 45) is of the order ξE V p /Γ T /L ∼ ξ/ϵ, while the term with the p 0 integral of G µν < (x, p) is O(1).In order for the linear response theory to work, one has to require ξ/ϵ ≪ 1.
It is clear that the integrands in Eq. ( 45) are suppressed by spectral functions in the region of p µ far from the mass-shell.Therefore we can make an approximation by expanding p 0 in the integrands around the on-shell energy E V p in powers of δp 0 = p 0 − E V p except spectral functions.To the first order in δp 0 , the p 0 integral of G µν < (p) gives while the p 0 integral of δG µν < (p) from the linear response to the shear tensor gives Detailed calculations for the integrand in Eq. ( 60) are given in Appendix C. The integrals over p 0 in Eqs. ( 59) and ( 60) can be completed and the results are listed in Appendix D. Then δρ 00 (p) is calculated by substituting Eqs. ( 59) and ( 60) into Eq.( 45).Up to linear order in ϵ or ξ, the result reads in Eq. ( 63) for the transverse (left) and parallel (right) configurations in which the momentum is transverse and parallel to the spin quantization direction z respectively.The results under the quasi-particle approximation (QPA) using Eq. ( 61) are shown for comparison.
where the dimensionless coefficients are defined as Noting that ρ 00 could deviate from 1/3 due to a nonzero C 0 independent of the shear tensor.Such a deviation arises from the possible difference between spectral functions for transverse and longitudinal modes [81].In the power counting scheme, we can check that C 0 ∼ O(ϵ) and other coefficients C i with i = 1, 2, T, L are all O(1).The numerical results show that C 0 ∼ O(10 −3 ) and other coefficients The dominant term that is proportional to the shear tensor is the C 1 term, which is controlled by 1/Γ T − 1/Γ L for the current values of g V .

Numerical results
In this subsection we will numerically calculate spectral functions and δρ 00 using Eqs.( 38) and ( 55).We will compare numerical results with the QPA results using Eq. ( 61).The parameters are set to the same values as in Subsection VII B 1. We can express δρ 00 (p) as where C µν are dimensionless constants.The numerical results for δρ (ξ=0) 00 (p) are shown in Fig. 4. The QPA results using Eq. ( 61) are shown for comparison.We choose two configurations for the mometum direction with respect to the spin quantization one: transverse or parallel configuration.The analytic results using Eq. ( 61) are also shown for comparison.The results of the configuration with an arbitrary angle are between these two limits.We see that the magnitude of δρ (ξ=0) 00 (p) is about 10 −3 for the values of parameters we choose.
The numerical results for the tensor coefficient C µν (p) are shown in Figs. 5 and 6 for transverse and parallel configurations respectively.The QPA results using Eq. ( 61) are shown for comparison.We see that the magnitude of C µν (p) is about 10 −2 ∼ 10 −3 for the values of parameters we choose, which is consistent with the result of Ref. [47] in the order of magnitude.

VIII. DISCUSSION AND CONCLUSION
We study thermal medium effects for the spin alignment of vector mesons from the meson-quark interaction in the thermalized QGP, in which quarks, antiquarks and vector mesons are assumed to be thermalized.Quarks and antiquarks are also assumed to be unpolarized.We calculate the retarded self-energy of the vector meson from the quark loop.The spectral function can be obtained from the retarded two-point Green's function including the contribution of the retarded self-energy.Other types of two-point Green's functions with interaction can all be expressed in spectral functions.Then we calculate the linear response of the two-point Green's function to the thermal shear tensor using the Kubo formula, which provides a correction to the Green's function.Such an effect is caused by interaction.63) for parallel directions of the spin quantization and momentum.The spin quantization is chosen to be in the z direction.The results under the quasi-particle approximation (QPA) using Eq. ( 61) are shown for comparison.
Finally the correction to ρ 00 can be expressed in terms of spectral functions through one-loop self-energies.In order to obtain an analytical formula for the correction to ρ 00 , we take the quasi-particle approximation: (a) the energy shifts and widths from real and imaginary parts of self-energies are much smaller than energies of vector mesons; (b) the difference between widths for transverse and longitudinal modes is much smaller than widths themselves.This approximation is supported by numerical results with the parameters we have chosen.Under this approximation we derive an analytical formula for the correction to ρ 00 to the linear order in the expansion parameter in terms of energy shifts and widths.The numerical results show that dimensionless coefficients of the thermal shear tensor are presented as functions of scalar momentum with the magnitude of O(10 −2 ∼ 10 −3 ) for the chosen values of quark-meson coupling constant.The magnitude of the contribution from the thermal shear tensor to ρ 00 is then O(10 −4 ∼ 10 −5 ) if the thermal shear tensor is O(10 −2 ).
Our results are based the one-loop self-energy with meson-quark interaction in the QGP.One can also consider other interactions, such as ρππ or ϕKK couplings, in the nuclear matter [81][82][83].
Appendix A: Quark propagators The propagators for unpolarized quarks at the leading order are given by where ϵ is a small positive number, m q = m q is the quark mass, and are Fermi-Dirac distributions for quarks/antiquarks as functions of the energy E k = |k| 2 + m 2 q and chemical potentials µ q/q .Appendix B: Retarded self-energy We evaluate the retarded self-energy in Eq. ( 26) using quark propagators in (A1).The result reads where I µν 1 , I µν 2 and I µν vac are defined as where k µ on = (E k , k) is an on-shell momentum for the quark or anti-quark.The tensors I µν 1 and I µν 2 can be expressed in special functions J ± (p; n 1 , n 2 ) and J 0 (n 1 , n 2 ) defined as We now evaluate each element of I µν 1 separately.The result for I 00 1 is In evaluating I 0i , we decompose the vector k into the component parallel and perpendicular to p as k = p(k • p) + k T with k T • p = 0.The integral over the component perpendicular to p vanishes.The result for I 0i 1 is To evaluate I ij , we notice that it is symmetric in i and j, so we can decomposes it into components proportional to pi pj and δ ij − pi pj using Then we obtain the result for I ij as In Eqs.(B5), (B6) and (B8) we have express the result of I µν 1 in terms of special functions J ± (p; n 1 , n 2 ) and J 0 (n 1 , n 2 ) in Eq. (B4).
The derivation of I µν 2 is similar and straightforward.Here we just list the result as follows Using the results for elements of I µν 1 and I µν 2 , we obtain the elements of 2I µν 1 + I µν 2 in Eq. (B1), )] −4p 0 p 2 [J + (p; 0, 1) − J − (p; 0, 1)] +4|p| 2 [J + (p; −1, 3) + J − (p; −1, 3)] . (B10) The vacuum contribution I µν vac for p 0 > 0 can be evaluated by dimensional regularization as where ϵ = 4 − d (d is an arbitrary space-time dimension in regularization) and γ ≈ 0.5772.Here the term proportional to delta functions in I µν vac is vanishing.The real part of I µν vac can be canceled by introducing a renormalization term with the condition ReI µν vac (p 2 = m 2 V ) = 0.The imaginary part I µν vac is nonzero when p 2 > 2m q and contributes to the spectral density.This corresponds to pair production or annihilation processes.
Appendix C: Expansion of part of integrand in Eq. ( 44) at mass-shell In this appendix, we will expand p 0 around E V p in powers of δp 0 for the integrand in the second term of Eq. ( 44) except spectral functions.The integrand can be written as  43) at p 0 = E V p .We can express p µ = p µ on + δp 0 g µ0 , where δp 0 = p 0 − E V p .We note that ∆ µν T (p) does not depend on p 0 , but ∆ µν L (p) = ∆ µν (p) − ∆ µν T (p) depends on p 0 through ∆ µν .Then to the first order in δp 0 they can be expanded as The function ∂n B (p 0 )/∂p 0 is expanded to the first order in δp 0 as To the first order in δp 0 , the integrand is expanded as

Figure 1 .
Figure 1.Illustration of the closed-time-path upon which the non-equilibrium quantum field theory is built.

Figure 3 .
Figure 3.The width Γ (a,b) and energy shift ∆E (c,d) for transverse (solid lines) and longitudinal (dashed lines) modes as functions of |p| at gV = 1 (a,c) and gV = 2 (b,d).Two sets of values are chosen for the s-quark chemical potential and temperature corresponding to the freezeout conditions at √ sNN ≈ 20 GeV and 200 GeV: µs = 64.5 MeV, T = 155.7 MeV (black) and µs = 7.4 MeV and T = 158.4MeV (red).

Figure 5 .
Figure 5.The numerical results for C µν in Eq. (63) for the transverse configuration in which the momentum is perpendicular to the spin quantization direction z.The results under the quasi-particle approximation (QPA) using Eq.(61) are shown for comparison.

Figure 6 .
Figure 6.The numerical results for C µν in Eq. (63) for parallel directions of the spin quantization and momentum.The spin quantization is chosen to be in the z direction.The results under the quasi-particle approximation (QPA) using Eq.(61) are shown for comparison.