Spin alignment of vector mesons from quark dynamics in a rotating medium

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Minghua Wei and Mei Huang. Spin alignment of vector mesons from quark dynamics in a rotating medium[J]. Chinese Physics C. doi: 10.1088/1674-1137/acf036
Minghua Wei and Mei Huang. Spin alignment of vector mesons from quark dynamics in a rotating medium[J]. Chinese Physics C.  doi: 10.1088/1674-1137/acf036 shu
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Received: 2023-05-19
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Spin alignment of vector mesons from quark dynamics in a rotating medium

  • 1. Institute of Modern Physics, Fudan University, Shanghai 200433, China
  • 2. School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China

Abstract: Vorticities in heavy-ion collisions (HICs) are supposed to induce spin alignment and polarization phenomena of quarks and mesons. In this work, we analyze the spin alignment of vector mesons ϕ and ρ induced by rotation from quark dynamics in the framework of the Nambu-Jona-Lasinio (NJL) model. The rotating angular velocity induces mass splitting of spin components for vector ϕ,ρ mesons Mϕ,ρ(Ω)Mϕ,ρ(Ω=0)szΩ. This behavior contributes to the spin alignment of vector mesons ϕ,ρ in an equilibrium medium and naturally explains the negative deviation of ρ001/3 for vector mesons. Incidentally, the positive deviation of ρ001/3 under the magnetic field can also be easily understood from quark dynamics.

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    I.   INTRODUCTION
    • Relativistic heavy-ion collision experiments provide a platform for studying quantum chromodynamics (QCD) matter under extreme conditions. It has been expected that quark gluon plasma (QGP) can be created through heavy ion collisions (HICs) [1]. In a specific case, a non-central collision, QGP is created with a large orbital angular momentum (OAM) at a range of 104105 [24]. Meanwhile, a strong magnetic field can reach the magnitude of 10m2π at the initial time of the HICs and evolves with time [57]. The magnitude of the magnetic field decays severely while the averaged angular velocity still maintains its magnitude along the axis, which is perpendicular to the reaction plane [2]. Therefore, the effects of a rotating medium will play significant roles in the QCD phase diagram, dilepton production rate, and spin polarization.

      Spin alignment of vector meson ϕ and K0 has been one of the intriguing topics in HICs. Experimental evidence suggests that spin density matrix element ρ00 has a remarkable deviation from 1/3 [8, 9]. ALICE collaboration has measured ρ00 for K0 and ϕ meson at sNN= 2.76 TeV in Pb-Pb collisions [9], and a negative deviation from 1/3 at lower transverse momentum has been reported. STAR collaboration has measured ρ00 between sNN=11.5 and 200 GeV in Au-Au collisions [8]. In this collision energy range, a positive deviation form 1/3 is reported for ρ00 of vector meson ϕ, which can be explained by a ϕ-meson field [10].

      In fact, spin alignment is a multifactorial phenomenon in heavy ion collisions. Theoretical researchers have developed a quark coalescence model for explaining the contribution of the vorticity and magnetic field [11, 12]. Furthermore, other physical mechanisms also contribute to the spin alignment, such as turbulent color fields[13], local vorticity [14], glasma fields [15], and gradient corrections in the vector-meson medium [16, 17].

      Both the magnetic field and vorticities are expected to contribute to the spin alignment. In Ref. [18], the effect of magnetic field on spin alignment was studied. In this work, we introduce the effects of global rotation on quark matter. Generally, the rotation has an inhibition effect on quark and anti-quark pairing [19], which is different from Magnetic Catalysis (MC) [2023]. As a consequence, chiral condensate will disappear as the angular velocity grows. Since the chiral condensate is an order parameter for chiral phase transition, its behaviors demonstrate that first order phase transition occurs at low temperature regions and crossover occurs in high temperature regions. The characteristics imply an analogy between the rotation and chemical potential [24, 25]. However, it is found that the rotation behaves beyond an effective chemical potential in dilepton production [26] and the rotation enhances the dilepton production rate and induces the ellipticity of lepton pairs.

      It is also worth emphasizing that previous studies have assumed an infinite size without boundary conditions [2, 24, 27]. The boundary conditions on a finite size system should be taken into account for a rotating system [2831], and inhomogeneous chiral condensation will be developed [32, 33]. For example, no-flux or MIT bag boundary conditions are applied in a spherical or cylindrical system [3436]. The choice of boundary conditions will influence the behavior of chiral condensate near the surface. In particular, when no-flux boundary condition is applied in a cylindrical system, chiral condensate will almost keep a constant in the inner part of the cylinder but get enhanced like a Gibbs phenomenon near the surface [28].

      In QCD phase diagram studies, it is also possible to consider inhomogeneous rotation with boundary conditions. Chiral condensate is enhanced near the area where the angular velocity changes severely, and this phenomenon is called the centrifugal effect [32]. Furthermore, chiral condensate reveals the dynamical mass of quarks. Besides the light quarks u and d, heavy flavor quarks, like s and c, are studied in electromagnetic and rotational fields [37]. Corresponding ϕ,D mesons are expected to be influenced by rotation as well. However, describing the freeze-out of heavy particles from the QGP is complicated [38]. Therefore, we only consider light-flavor vector meson modes excited and annihilated in a rotating medium.

      External fields, such as magnetic and vortical fields, will induce the mass splitting of spin-1 vector mesons. In recent years, considerable attention has been paid to the magnetic field effect of vector mesons, such asρ, ϕ. For example, for charged ρ mesons, mass splitting behaves like:

      Mρ±(eB)=Mρ±(0)κeB,

      (1)

      where κ is a coefficient. For point-like particles, κ=1; when quark-antiquark loop effect is considered, κ1. Similarly, ρ meson under rotation also exhibits a mass splitting phenomenon [39]:

      Mρ(Ω)=Mρ(0)szΩ.

      (2)

      This relation indicates that vector mesons tend to occupy the sz=+1 state. In this study, a similar result can be obtained for vector meson ϕ.

      Recently, the mass splitting and spin alignment of vector meson ϕ was investigated under a magnetic field [18]. Further, another important contribution from the rotation on the spin alignment should also be considered. Therefore, in this work, we calculate the rotational contribution on matrix element ρ00.

      This paper is organized as follows. In Sec. II, quantum field theory in a rotating frame is introduced. Based on this, the quark propagator and the self-energy are modified by uniform rotation. Consequently, the masses of vector mesons are obtained by random phase approximation. In Sec. III, the numerical results of quark mass, meson mass, and spin alignment are presented. Finally, the conclusion and summary are provided in Sec. IV.

    II.   FORMALISM

      A.   Quantum field theory in a uniformly rotating frame

    • In quantum field theory, a Lorentz vector field Va(x) can be described by tetrad ea(x)=eμa(x)xμ in curved space-time. In this section, Latin letters, a,b=0,1,2,3, stand for Lorentz indices. And Greek letters, μ,ν= 0, 1, 2, 3, stand for coordinate indices. Parallel transport of Va(x) acts as follows:

      Va(x+dx)=Va(x)+ΓabμVbdxμ,

      (3)

      where Γabμ is called the spin connection. Metric compatibility ensures Γabμ=Γbaμ, and non-torsion condition provides a relation between the tetrad and the spin connection:

      Γabμ=12[eaν(μebννebμ)ebμ(μeaννeaμ)eaρebl(ρecσσecρ)ecμ].

      (4)

      For spinor field, corresponding spinor connection is given by Γμ=σabΓabμ, where σab is spinor representation of Lorentz group. In the co-moving frame of the QGP, free fermions are described by the modified Dirac equation:

      [iˉγμ(μ+Γμ)Mf]ψ=0,

      (5)

      where Mf is the mass of a fermion and ˉγμ=eμaγa satisfies {ˉγμ,ˉγν}=gμν.

      Particularly, in a uniformly rotating frame, the tetrad can be described by

      eaμ=δaμ+δaiδ0μvi,eμa=δμaδ0aδμivi,

      (6)

      where v=Ω×x gives the velocity in the lab frame. We use the capital Greek letter Ω to represent the magnitude of angular velocity. If we substitute Eq. (6) into Eq. (4), non-zero terms of spin connection will be expressed as follows:

      Γij0=12(ivjjvi),Γi0j=12(ivj+jvi),Γ0ij=12(ivj+jvi),Γ0i0=12(vjivj+vjjvi).

      (7)

      Finally, in a uniformly rotating frame, Dirac equation can be rewritten by [40]

      [iγaa+γ0Ω^JzMf]ψ=0.

      (8)

      Here, the z-axis is chosen as the direction of the rotation, and it is perpendicular to the reaction plane. The total angular momentum ^Jz can be expressed as ˆJz=ˆLz+ˆSz where ˆLz is the orbital angular momentum and ˆSz=12(ˆσz00ˆσz) contributes to the spin part.

      For solving modified Dirac equation Eq. (8), Ref. [19] has given the solutions in the cylindrical coordinates where a position in space-time is labeled by ˜r=(t,r,θ,z). We can write down a complete set of commuting operators: the Hamiltonian ˆH, the longitudinal momentum ˆkz, the square of transverse momentum ˆk2t, the total angular momentum ˆJz, and the helicity operator on transverse plan ˆht. The eigenstates for fermion and anti-fermion [19] are given by

      U=Ek+Mf4Ekeikzzeinθ(Jn(ktr)seiθJn+1(ktr)kzisktEk+MfJn(ktr)skz+iktEk+MfeiθJn+1(ktr)),

      (9)

      V=Ek+Mf4Ekeikzzeinθ(kzisktEk+MfJn(ktr)skziktEk+MfeiθJn+1(ktr)Jn(ktr)seiθJn+1(ktr)),

      (10)

      where Jn(ktr) is the n-th order Bessel function and nZ stands for the quantum number of angular momentum. In Eq. (9) and Eq. (10), kt and kz are the eigenvalues of transverse and longitudinal momentum, respectively, and s=±1 is the eigenvalue of the helicity operator ˆht. Besides, Ek is defined by Ekk2z+k2t+M2f and the energy is given by E=±Ek(n+1/2)Ω.

      Based on the solutions of the Dirac equation, the quark propagator can be defined as follows: S(˜r;˜r)=0|Tψ(˜r)ˉψ(˜r)|0. As a standard procedure, ψ(˜r) and ˉψ(˜r) are expanded by Eq. (9) and Eq. (10), and an explicit form can be obtained as follows:

      S(˜r;˜r)=1(2π)2ndk02πktdktdkzein(θθ)eik0(tt)+ikz(zz)[k0+(n+12)Ω]2k2tk2zM2f+iϵ×{[[k0+(n+12)Ω]γ0kzγ3+Mf][Jn(ktr)Jn(ktr)P++ei(θθ)Jn+1(ktr)Jn+1(ktr)P]iγ1kteiθJn+1(ktr)Jn(ktr)P+γ2kteiθJn(ktr)Jn+1(ktr)P}.

      (11)

      Here, we have simplified the expression by projection operators P±=12(1±iγ1γ2). In Appendix A, we will present another procedure for obtaining the quark propagator in a rotating medium.

      Nevertheless, Eq. (11) is the result under the infinite size approximation. A finite size version with a boundary condition can be found in Ref. [28]. The transverse momentum ktis discrete, and its integral is replaced by the summation of the series. In this study, we have to first calculate the spectral functions of vector mesons in a rotating medium. Therefore, we apply the infinite size approximation to avoid the tedious summation of the series. This compromise will violate the causality in the case of a large angular velocity. In this work, we study the phenomena and quantities at a fixed radius r=0.1GeV1 and in a range of angular velocity from Ω=0 GeV to Ω=1.2 GeV so that the velocity of a fixed point is smaller than the speed of light, i.e. Ωr<1.

    • B.   The 3-flavor NJL model

    • In order to study the microscopic properties of vector mesons, we use the Nambu-Jona-Lasinio(NJL) model to estimate the strong interaction[41]. The 3-flavor NJL model is required to investigate ϕ meson, which contains s quark [42]. The Lagrangian of the 3-flavor NJL model is given as follows:

      L3NJL=ˉψ[iˉγμ(μ+Γμ)mf]ψ+GS8a=0[(ˉψλaψ)2+(ˉψiγ5λaψ)2]GV8a=0[(ˉψγμλaψ)2+(ˉψiγμγ5λaψ)2]K[detˉψ(1+γ5)ψ+detˉψ(1γ5)ψ],

      (12)

      where K is the coupling constant of six-fermion interaction, Gs and GV are coupling constants of four-fermion interaction for scalar and vector channels, respectively. In the 3-flavor NJL model, ψ=(ψu,ψd,ψs) is a Dirac spinor which contains u, d and s quarks. Correspondingly, λa are the Gell-Man matrices and mf is the current quark mass with different flavors.

      The NJL model is an effective model which only contains quarks. The rotation affects the quark dynamics through the non-zero term of spinor connection Γμ. Furthermore, by applying the mean field approximation, the Lagrangian can be rewritten as

      LMF=f=u,d,sˉψf[iˉγμ(μ+Γμ)Mf]ψf2GSf=u,d,sσ2f+4Kσuσdσs

      (13)

      where σf is the chiral condensate σfˉψfψf for a specific flavor. And Mf is dynamical quark mass, which is given by

      Mfmf4GSσf+2Kffσf.

      (14)

      In a uniformly rotating medium, the spinor connection Γμ has been estimated in Sec. II.A. We assume the direction of rotation is parallel to the z-axis. By applying the standard procedure in finite-temperature field theory [43], the grand potential for quarks with specific flavors is shown as follows:

      Ωf(r)=Nc8π2Tndk2tdkz[Jn(ktr)2+Jn+1(ktr)2]×[Ek/T+ln(1+e(Ek(n+12)Ω)/T)+ln(1+e(Ek+(n+12)Ω)/T)].

      (15)

      Consequently, the total grand potential is

      Ωtot(r)=f=u,d,s(2GSσ2fΩf)+4Kσuσdσs.

      (16)

      Here, Nc=3 is the degeneracy of color, and T is the temperature of the medium. We can obtain the dynamical quark mass Mf and chiral condensate σf by solving the gap equations:

      Ωtotσf=0,2Ωtotσ2f>0.

      (17)

      In principle, if other parameters are fixed, the dynamical quark mass Mf(r) will be a function of radius r. Currently, most studies have assumed that Mf(r) is changed smoothly, i.e. Mf(r)/r0. Such an assumption is called local density approximation(LDA) [19, 24, 32]. In this paper, we use the LDA and choose a fixed radius r=0.1 GeV1. Then, Mf(Ω) will be evaluated numerically in Sec. II.A.

    • C.   Vector meson mass splitting under the rotation

    • In the NJL model, the vector meson ϕ is treated as a sˉs bound state or a resonance state, and it can be constructed by quark-antiquark scattering [41]. In the random phase approximation (RPA), the vector meson propagator can be obtained by summation of quark loops, and the one loop polarization function is given by

      Πμν(q)=id4˜rTrsfc[iγμS(0;˜r)iγνS(˜r;0)]eiq˜r.

      (18)

      Here, Trsfc stands for the trace in the spinor, flavor, and color spaces. Since ϕ meson is purely constituted by s quark, S(0;˜r) stands for the s quark propagator with the dynamical mass Ms given by mean field approximation. The polarization function of ϕ meson is supposed to be modified in a rotating medium.

      Currently, we focus on the mesons which remain at rest in the rotating frame, i.e. q=0. In this case, polarization vectors are given by

      ϵμ1=12(0,1,i,0),ϵμ2=12(0,1,i,0),bμ=(0,0,0,1)

      (19)

      where ϵμ1 and ϵμ2 are the right and left-hand polarization vectors, respectively. The longitudinal polarization vector is parallel to the direction of rotation. Correspondingly, the projection operators are given by

      Pμν1=ϵμ1ϵν1,(sz=1 for ϕ meson ),Pμν2=ϵμ2ϵν2,(sz=+1 for ϕ meson ),Lμν=bμbν,(sz=0 for ϕ meson ).

      (20)

      As we have assumed q=0, nonzero elements of the polarization functions can be read in the following matrix:

      Πμνϕ=(00000Π11Π1200Π21Π220000Π33).

      (21)

      The explicit expressions of matrix elements are shown in Ref. [26]. This tensor can be decomposed by projection operators in Eq. (20) as follows:

      Πμνϕ=A1,1Pμν1+A11Pμν2+A00Lμν,

      (22)

      Similarly, the vector meson propagator can be decomposed as follows:

      Dμνϕ(q)=D1,1(q)Pμν1+D11(q)Pμν2+D00(q)Lμν.

      (23)

      Here, coefficients Dλ are obtained by RPA summation and expressed by:

      Dλλ(q)=4GV1+4GVAλλ,

      (24)

      where

      A1,1=(Π11iΠ12),(sz=1 for ϕ meson ),A11=Π11iΠ12,(sz=+1 for ϕ meson ),A00=Π33,(sz=0 for ϕ meson ).

      (25)

      Then, we can obtain the corresponding spectrum functions for different spin components, taking the following form:

      ξλλ(ω)1πImDλλ(ω)=(4GV)2ImAλλ(ω)π{[1+4GVReAλλ(ω)]2+[4GVImAλλ(ω)]2},

      (26)

      where ω is the energy of the meson, and we set the momentum to q=0.

    • D.   spin alignment of vector meson ϕ

    • Heavy-ion collisions will create an ensemble of particles under extreme conditions. Particularly, we consider vector meson ϕ with spin-1. For a fixed direction, a normalized spin state of a ϕ meson is labeled by |λ with λ=1,0,1. Spin density operator ρ is defined by

      ρ=λλρλλ|λλ|,

      (27)

      Here, ρλλ comprises a 3×3 spin density matrix as follows:

      ρλλ=(ρ11ρ10ρ1,1ρ01ρ00ρ1,1ρ1,1ρ1,0ρ1,1),

      (28)

      An ensemble of ϕ mesons, for which spin information is described by ρλλ, will decay to

      ϕK++K.

      (29)

      In this process, daughter particles will have an angular distribution [44, 45]:

      dNdΩ=34π{cos2θρ00+sin2θ(ρ11+ρ11)/2sin2θ(cosϕReρ10sinϕImρ10)/2+sin2θ(cosϕReρ10+sinϕImρ10)/2sin2θ[cos(2ϕ)Reρ11sin(2ϕ)Imρ11]}.

      (30)

      From experimental data of θ-distribution, ρ00 can be obtained as a coefficient of angular distribution. Since experiments can measure the value of ρλλ, theoretical studies should explain the results of spin alignment and evaluate ρλλ qualitatively. Magnetic field and vorticity are two factors considered in many models. The approaches are generalized as follows:

      B,Ω(quarks) influence ρλλ(ϕ) determine dNdΩ(K+,K).

      (31)

      Theoretical studies are interested in the first step of Eq. (31). In a real heavy-ion collision process, the evolution is complicated, and the history will influence ρλλ after the freeze out. One possible method is the quark coalescence model. However, it is unable to take the medium effects into account in the quark coalescence model. As we mentioned in previous sections, the quark and meson masses will be modified in a rotating medium.

      In this investigation, we aim at a uniformly rotating medium, and the created ϕ mesons are in global equilibrium. In this case, the particle number density ¯ρλλ(q) can be expressed by

      ¯ρλλ(q)=dω2ωeω/T1ξλλ(ω,q),

      (32)

      where ξλλ(ω,k) is the spectral function given in Eq. (26). At the one loop level, the spectral function in a rotating medium is calculated in Ref. [26], and we present it in Appendix B for convenience. Generally, the spectral function is the imaginary part of the full propagator:

      ξλλ(q)1πImDλλ(q).

      (33)

      For a general nonzero momentum q, the matrix ξλλ(ω,q) and ¯ρλλ(q) may have nonzero off-diagonal elements. Particularly, if the quantization direction coincides with the rotation axis and the momentum q is parallel/antiparallel to the rotation axis, or q=0, ξλλ(ω,q) and ¯ρλλ(q) will become diagonal:

      ¯ρλλ(q)=(¯ρ11000¯ρ00000¯ρ1,1).

      (34)

      Therefore, we divide the spectral function into two parts: a delta function part and a continuum part, i.e.

      ξλλ(ω,q)=δ(ω2M2ϕ,λ)+ξλλ(ω,q),

      (35)

      where ξλλ(ω,q) is the continuum part of spectral function and Mϕ,λ is the vector meson mass for different spin components. Correspondingly, the particle number density ˉρλλ(q) takes the form of:

      ˉρλλ(q)=1exp(Mϕ,λ/T)1+dω2ωξλλ(ω,q)exp(ω/T)1.

      (36)

      In experiment measurement, spin alignment is evaluated by matrix element ρ00 which can be expressed by:

      ρ00(q)¯ρ00(q)λ=0,±1¯ρλλ(q).

      (37)

      In the non-rotating case, ϕ mesons with different spin components have the same mass Mϕ(sz=0,±1), which leads to ρ00=1/3. In a finite angular velocity, the mass of ϕ meson with sz=+1 component decreases linearly; thus, ϕ mesons tend to occupy the sz=+1 state. Consequently, spin polarization and alignment can be obtained in our formalism.

      In this section, Eqs. (32)−(37) are momentum-dependent. However, in this investigation, we evaluate the spectral functions and spin alignment with q=0, which means vector mesons are staying at rest in the rotating frame. In general, Ref. [18] has revealed the spin density matrix for arbitrary measuring direction, which is characterized by Euler angles (α,β,γ). The explicit form is:

      ¯ρλλ(0;α,β,γ)=λ1,λ2Rλλ1(α,β,γ)¯ρλ1λ2(0)R1λ2λ(α,β,γ).

      (38)

      Here, Rλλ(α,β,γ) is the spin-1 representation of the rotation. As a result, the spin alignment only depends on Euler angles β, and the explicit form is:

      ρ00(0;α,β,γ)=¯ρ00(0)cos2β+¯ρ11(0)sin2βˉρ00(0)+ˉρ11(0)+ˉρ1,1(0).

      (39)

      In Sec. III.C, numerical results will be presented for β=0, and the following formula is used:

      ρΩ00(0)ˉρ00(0)ˉρ00(0)+ˉρ11(0)+ˉρ1,1(0).

      (40)
    III.   NUMERICAL RESULT
    • Since a rotating system has broken the Lorentz symmetry, it is not necessary to use Pauli-Villas regularization. In fact, the Bessel function will not reflect the oscillation behavior in the cut-off energy scale Λ. Therefore, we choose a soft cut-off function:

      fΛ=Λ210Λ210+p210,

      (41)

      where Λ=620.411 MeV is the energy scale for cut-off, and the corresponding coupling constants are: GS=1.710Λ2, GV=0.671.710Λ2 and K=120351000Λ5. In this case, mπ=140 MeV and mϕ=1020 MeV in vacuum. Current quark masses are: mu=md=5.5 MeV, ms=135.433 MeV.

    • A.   Chiral condensates and dynamical quark masses

    • As mentioned in previous sections, chiral condensate will be suppressed under rotation. In Fig. 1 and Fig. 2, chiral condensates are demonstrated as a function of angular velocity Ω at temperature T=10MeV and T= 150 MeV, respectively. Correspondingly, dynamical quark masses are also demonstrated as a function of angular velocity Ω in Fig. 3 and Fig. 4. Chiral symmetry will be restored as angular velocity grows. In Fig. 1 and Fig. 3, at an almost zero temperature, T=10 MeV, rotation induces a first order transition at angular velocity Ωc= 0.713 GeV. Both Mu and Md stay at a constant mass when the angular velocity is below Ωc. From Eq. (14), we know that dynamical quark masses are determined by the chiral condensates σu, σd and σs. In fact, σu=σd and drops at Ωc while σs still varies smoothly. Consequently, Ms has a small jump at Ωc and then decrease smoothly. The behavior of Mu,d,s as a function of angular velocity, Ω is very similar to the case of finite quark chemical potential [46, 47], but the first order phase transition will occur at μc0.33 GeV at finite density.

      Figure 1.  (color online) Chiral condensates as functions of angular velocity at temperature T=10 MeV. Red lines stand for σs and the blue lines stand for σu and σd.

      Figure 2.  (color online) Chiral condensates as functions of angular velocity at temperature T=150 MeV. Red lines stand for σs and the blue lines stand for σu and σd.

      Figure 3.  (color online) Dynamical quark masses as functions of angular velocity at temperature T=10 MeV. Red lines stand for mass of s quark and the blue lines stand for light quark u and d.

      Figure 4.  (color online) Dynamical quark masses as functions of angular velocity at temperature T=150 MeV. Red lines stand for mass of s quark and the blue lines stand for light quark u and d.

      At a higher temperature, T=150 MeV, Fig. 4 reveals that the chiral phase transition will be a crossover, which occurs around Ωc0.4 GeV. The phase transition takes place in a smaller angular velocity. Furthermore, It is noticed that the mass decreases slowly before the phase transition. Above all, the rotational effect on the dynamical quark mass is similar to that of the chemical potential μq.

    • B.   Mass spectra of ϕ and ρ meson under rotation

    • After we obtain the dynamical quark masses in different temperatures and angular velocities, we can apply this result in Eq. (18) and obtain corresponding meson masses.

      At zero temperature and low temperatures, similar to ρ meson, ϕ meson mass with different spin components sz=0,±1 also show mass splitting effect with Mϕ(Ω)=Mϕ(0)szΩ.

      At T=150 MeV and μ=0 MeV, Fig. 5 shows ϕ meson mass with different spin components sz=0,±1 as a function of angular velocity. The mass of sz=0 component for ϕ meson almost remains unchanged with the angular velocity. The mass of sz=1 component of ϕ meson grows almost linearly with the angular velocity. It implies that ϕ meson will be less likely to stay at the sz=1 state. In contrast, the sz=+1 component ϕ meson mass decreases almost linearly with the angular velocity. sz=+1 component will be a preferred state under the rotation. Above all, the nearly linear mass splitting behavior of ϕ meson at T=150 MeV can be summarized in the following expression:

      Figure 5.  (color online) ϕ meson mass as a function of angular velocity at temperature T=150 MeV and μ=0 MeV.

      Mϕ(Ω)Mϕ(0)szΩ.

      (42)

      In Fig. 6, we reveal the deviation of ϕ meson mass from Eq. (42) at T=150 MeV. From Eq. (42), we will know that: Mϕ(Ω)+szΩMϕ(0), and the Fig. 6 has shown the difference between "" and "=". The deviation is caused by the inhibition of chiral condensate. If we expand Mϕ(Ω) to order Ω2, the deviation can be revealed as follows:

      Figure 6.  (color online) The deviation of ϕ meson mass as a function of angular velocity at temperature T=150 MeV.

      Mϕ(Ω,sz=+1)=0.951.00Ω0.54Ω2,Mϕ(Ω,sz=0)=0.95+0.01Ω0.31Ω2,Mϕ(Ω,sz=1)=0.95+1.00Ω0.54Ω2.

      (43)

      Eq. (43) shows the deviation of sz=0,±1 components, caused by quark mass descending in finite temperature and angular velocity. ϕ meson is an almost pure sˉs state [41], so its mass is obviously influenced by the quark mass. It is seen that the deviation of the sz=±1 component states are larger than that of the sz=0 component state.

      Figure 7 and Fig. 8 reveal rotational effect on the spectra function ξ(ω) for ϕ meson and ρ meson with different spin components as a function of the frequency ω respectively. The blue lines stand for spectral functions at Ω=0 without rotation, while the red, orange, and gray dashed lines stand for spectral functions under finite rotation for Ω=0.1,0.2, and 0.3 GeV, respectively. In the zero rotation case, ϕ mesons with different spin states shared the same spectral function which is constituted by a delta function part and a continuum part. The location of the delta function indicates the pole mass.

      Figure 7.  (color online) The spectral function ξ(ω) for ϕ meson with different spin component as a function of the frequency ω under different angular velocities Ω=0, 0.1, 0.2, 0.3 GeV at T=150 MeV and μ=0.

      Figure 8.  (color online) The spectral function ξ(ω) for ρ meson with different spin component as a function of the frequency ω under under different angular velocities Ω=0, 0.1, 0.2, 0.3 GeV at T=150 MeV and μ=0.

      For the sz=0 component of ϕ meson, the rotational effect is less remarkable than in the other two cases. So, the scale has been amplified, and we only present the ξ(ω) in the range of energy ω from 0.85 GeV to 1.20 GeV. It is found that spectral functions are shifted to the left slightly. And the peaks of the continuum parts are enhanced significantly.

      For sz=+1 component of ϕ meson, the rotational effect will shift the spectral function to the left side, and the rotation will change the height or the shape of the continuum part of the spectral function as well. For the sz=1 component, the spectral function is shifted to the right side correspondingly.

      A similar analysis can be applied for the vector meson ρ, since we have assumed Mu=Md. We can obtain the spectral function by substituting Ms for Mu/Md. For ϕ mesons, a bound state is labeled by mass Mϕ , which is very close to the 2Ms. However, ρ mesons are dissociated at temperature T=150 MeV. So, in Fig. 8, a spectral function only has a continuum part and appears as a single peak. The top panel in Fig. 8 shows the spectral functions of ρ mesons with spin components sz=0. Different colored lines stand for different strengths of angular velocities ranging from Ω=0, 0.1, 0.2 and 0.3 GeV. Rotational effects are reflected in two aspects: the heights of the peaks are suppressed, and the widths are broadened by the angular velocities. It can be understood that mesons tend to be less bounded in a rotating medium. The location of the peaks is almost unchanged in the case of sz=0. However, in the case of sz=+1, the locations of resonance peaks are shifted to the left side by rotation. Similarly, in the case of sz=1, mass spectra are shifted to the right side by rotation. Above all, on the shape of spectral functions, the rotation effects are similar.

    • C.   Spin alignment of vector meson ϕ and ρ

    • In Fig. 9, we show the deviation of spin alignment ρ00 from 1/3 for ϕ meson as a function of angular velocity at a finite temperature T = 150 MeV. In the case of rotation, ρ00 is always smaller than 1/3, and the deviation will become more significant as the angular velocity grows. Furthermore, resonance states will contribute less to spin alignment and the deviation between the bounded state and the total result is negligible. In Fig. 10, we compare the spin alignment ρ00 from 1/3 for ϕ meson with ρ meson, and it is found the difference is quite small.

      Figure 9.  (color online) Spin alignment ρ00 for vector meson ϕ as a function of angular velocity at temperature T=150 MeV.

      Figure 10.  (color online) Spin alignment ρ00 for resonance states of vector meson ρ and ϕ as a function of angular velocity at temperature T=150 MeV.

      Here, we mention the result from the quark coalescence model [11]:

      ρϕ00(Ω)=1319(βΩ)2,

      (44)

      where β=1/T is the inverse of temperature. Eq. (44) is only valid in the vicinity of Ω=0 GeV, and the value of the coefficient is 19β2=4.94 GeV2 for T=0.15 GeV. As a comparison, we fit our result with polynomial functions in a range of angular velocities from Ω=0 GeV to Ω=0.15 GeV. The numerical results give:

      ρϕ00(Ω)=135.10Ω2+39.62Ω4.

      (45)

      Here, the dimensions of the angular velocity and the coefficients are omitted. In the vicinity of Ω=0 GeV, the absolute value of the coefficient of Ω2 is larger than 19β2=4.94 GeV2.

      Compared with the spin alignment under an external magnetic field and rotation, the deviation ρ001/3 is positive under the magnetic field, while it is negative in the presence of rotation. It is natural to understand from quark dynamics that the spin of a particle tends to align along the direction of angular momentum due to the spin-orbital coupling. For sz=+1 component, the vector ϕ,ρ mesons masses are suppressed in the rotating medium. As a consequence, vector mesons are more likely to occupy the sz=+1 state and less likely to occupy sz=0 state. So, ρ001/3 is negative in the rotating medium. On the contrary, ρ001/3 of ϕ meson is positive under the magnetic field [18]. Actually, vector meson masses in the magnetic field are charge dependent. The ϕ meson is a neutral particle; its property under the magnetic field can be extended from the result of neutral ρ0 meson mass spectra under the magnetic field [48]. Under the magnetic field, neutral ϕ meson with sz=±1 will have a larger mass than ϕ meson with sz=0. So, ϕ mesons are more likely to occupy the sz=0 state in the presence of the magnetic field, which naturally explains why ρ001/3 for ϕ meson is positive under the magnetic field.

      Similarly, our theoretical method can be applied to other species of vector mesons, such as ρ and K0. The difference is the dynamical mass of constituent quarks u, d, and s in the rotating medium. In Eq. (18), only one species of quark propagator exists in the one loop polarization function. Since we have applied the assumption Mu=Md, spin alignment of ρ meson is demonstrated in Fig. 10. In the rotating medium with a temperature of 150 MeV, ρ mesons are resonance states. So, we compare it with the resonance states of ϕ mesons in Fig. 10. The tendency of the deviation ρ001/3 is still close to the quadratic polynomial.

      It is worth reminding that those results are calculated for vector mesons that stay at rest in a rotating medium, i.e., at q=0. In the q0 case, the calculation will be more complicated, and it is still a puzzle to switch the physical quantities in the rotating frame into the counterparts in the lab frame. Above all, the contribution from the rotating medium is significant, although the spin alignment of vector mesons is affected by a combination of many factors.

    IV.   CONCLUSION AND DISCUSSION
    • In this study, we investigated the spin alignment of vector mesons ϕ and ρ induced by rotation. By applying a three flavor NJL model with a vector interaction channel, we obtain the dynamical quark mass under rotation. The curves of Mf(Ω) is similar to Mf(μ). For the s quark, the first order phase transition occurs at a critical angular velocity Ωc, and Ms decreases smoothly after the phase transition, which is similar to the quark mass behavior at a finite chemical potential.

      After substituting the dynamical quark mass, the mass spectra of vector mesons can be obtained through the quark-antiquark polarization function. The rotating angular velocity induces mass splitting of spin components for vector ϕ,ρ mesons Mϕ,ρ(Ω)Mϕ,ρ(Ω=0)szΩ. This behavior contributes to the spin alignment of vector mesons ϕ,ρ in an equilibrium medium. In a rotating medium, ρ00 of vector mesons has a negative deviation from 1/3 , indicating a spin alignment phenomenon, which can be easily understood from quark dynamics that the spin of a particle tends to align along the direction of angular momentum due to spin-orbital coupling. For the sz=+1 component, the vector mesons masses are suppressed in the rotating medium. As a consequence, vector mesons are more likely to occupy the sz=+1 state and less likely to occupy the sz=0 state. Therefore, ρ001/3 is negative in the rotating medium.

      On the contrary, the deviation ρ001/3 is positive under the magnetic field, which can also be easily understood from quark dynamics. Under the magnetic field, the sz=±1 components of ϕ,ρ meson will have a larger mass than that of the sz=0 component of ϕ,ρ mesons. Therefore, ϕ,ρ mesons are more likely to occupy the sz=0 state in the presence of the magnetic field, which naturally explains the positive ρ001/3 for ϕ meson under the magnetic field.

      Based on a dynamical quark model, we calculated the spin alignment ρ00 for the transverse momentum dependent case. Furthermore, we can apply it for vector meson K0 in the future. In this series of studies, we have studied the vector meson ϕ and ρ. In this case, the quark loops only contain s and ˉs quarks, and light quarks are assumed to have the same value of mass, i.e. Mu=Md. For vector meson K0, the mass difference of quarks is expected to explain the different measurements between vector meson ϕ and K0 in the experiment. Our study is an attempt at this target in a rotating medium.

    ACKNOWLEDGMENTS
    • We are grateful for the helpful discussions with Li Yan, Anping Huang, Xinli Sheng, and Kun Xu.

    APPENDIX A: QUARK PROPAGATOR IN A ROTATING MEDIUM
    • To obtain the quark propagator in a rotating and dense medium, we adopt the method from Vladimir A. Miransky and Igor A. Shovkovy [23]. This derivation has considered the chemical potential μ and the rotation term Ω^Jz. According to an alternative definition, the quark propagator is given by

      S(˜r,˜r)=i˜r|[(it+μ+Ω^Jz)γ0πγMf]1|˜r,

      where π is the canonical momentum and γ is the Dirac matrix. Their expressions depend on the coordinates of the position ˜r. At this moment, we treat them as abstract operators, and the propagator can be rewritten as

      S(˜r,˜r)=i˜r|[(it+μ+ΩˆJz)γ0πγ+Mf]×[(it+μ+ΩˆJz)γ0πγ+Mf]1[(it+μ+Ω^Jz)γ0πγMf]1|˜r=i˜r|[(t+μ+ω^Jz)γ0πγ+Mf]×[(it+μ+ΩˆJz)2π2M2f]1|˜r

      Owing to the translation invariance in the t and z directions, we can perform the Fourier transformation on the quark propagator as follows:

      S(E,kz;r,r)=dtdzeiE(tt)ikz(zz)S(˜r,˜r).

      Here, r=(r,θ) is the position in cylindrical coordinates, and E is the energy. The propagator can be expressed as follows:

      S(E,kz;r,r)=i[(E+μ+ΩˆJz(r))γ0πrγkzγ3+Mf]r|[(E+μ+ΩˆJz)2(kz)2π2M2f]1|r

      where πr and ˆJz(r) are the canonical momentum and the angular momentum operator in cylindrical coordinate space, respectively. According to Ref. [19], the operators π2 and ˆLz commute with each other and have a common eigenstate |nkt, in which the explicit form in coordinate space is given by Eqs. (9) and (10). Here, we present several useful equations:

      rnkt=einθJn(ktr)ˆLznkt=nnktπnkt=ktnkt,

      where ˆLz is the operator of orbital angular momentum. As a result, the right hand side of Eq. (49) can be evaluated as follows:

      r|[(E+μ+ΩˆJz)2(kz)2π2M2f]1|r=nktdktr|[(E+μ+ΩˆJz)2(kz)2π2M2f]1|nktnktr=nktdktr|[(E+μ+ΩˆLz)2+2(E+μ)ΩSz+Ω2(Sz)2k2zπ2M2f]1|nktnktr=n+0ktdktJn(ktr)Jn(ktr)ein(θθ)[(E+μ+Ωn)2+2(E+μ)ΩSz+Ω214k2zk2tM2f]1,

      (51)

      where Sz=i2γ1γ2 is the spin angular momentum term. We have inserted the completeness condition in the second line of Eq. (51). Now, it is easy to obtain Eq. (11) using projection operator P±=12(1±iγ1γ2). By inserting I4=P+P+, the summation in Eq. (51) can be replaced by:

      nJn(ktr)Jn(ktr)ein(θθ)P++Jn+1(ktr)Jn+1(ktr)ei(n+1)(θθ)P[E+μ+(n+12)Ω]2k2zk2tM2f.

      Finally, we can calculate Eq. (49) by acting the operators ˆJz(r)γ0, πrr and kzγ3 on Eq. (52).

    APPENDIX B: SPECTRAL FUNCTIONS
    • The explicit form and the derivation of spectral functions can be found in Refs. [26, 39]. In our previous work, we have utilized the properties of Bessel functions, i.e. Jn(0)=0 for n0 and J0(0)=1. As a consequence, the infinite summation is reduced to finite terms. Then, Eq. (18) can be evaluated at zero and finite temperature. To save space in this manuscript, we merely present one component of polarization functions. For example, the imaginary part of the 00-component is [26]:

      ImΠ00(ω,q)=π2NfNcη=±1d3p(2π)31EpEk{[EpEk+pk+M2f][f(EpμηΩ2)+f(Ep+μηΩ2)]×[δ(ω+EpEk)δ(ωEp+Ek)]+[EpEkpkM2f]δ(ωEpEk)×[1f(EpμηΩ2)f(Ep+μηΩ2)]}.

      where k=p+q and Ek=k2+M2f. Here, Nf and Nc are the flavor number and color number in the quark loop, and f(x) is Fermi-Dirac distribution functions with a finite temperature T. Other components can be obtained similarly.

Reference (48)

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