Axial chiral vortical effect in a sphere with finite size effect

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Shu-Yun Yang, Ren-Hong Fang, De-Fu Hou and Hai-Cang Ren. Axial Chiral Vortical Effect in a Sphere with finite size effect[J]. Chinese Physics C. doi: 10.1088/1674-1137/acac6d
Shu-Yun Yang, Ren-Hong Fang, De-Fu Hou and Hai-Cang Ren. Axial Chiral Vortical Effect in a Sphere with finite size effect[J]. Chinese Physics C.  doi: 10.1088/1674-1137/acac6d shu
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Axial chiral vortical effect in a sphere with finite size effect

    Corresponding author: De-Fu Hou, houdf@mail.ccnu.edu.cn (Corresponding author)
    Corresponding author: Hai-Cang Ren, renhc@mail.ccnu.edu.cn (Corresponding author)
  • 1. Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS), Central China Normal University, Wuhan 430079, China
  • 2. Key Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao 266237, China
  • 3. Physics Department, The Rockefeller University, 1230 York Avenue, New York, NY 10021-6399, USA

Abstract: We investigate the axial vortical effect in a uniformly rotating sphere subject to finite size. We use the MIT boundary condition to limit the boundary of the sphere. For massless fermions inside the sphere, we obtain the exact axial vector current far from the boundary that matches the expression obtained in cylindrical coordinates in literature. On the spherical boundary, we find both the longitudinal and transverse (with respect to the rotation axis) components with magnitude depending on the colatitude angle. For massive fermions, we derive an expansion of the axial conductivity far from the boundary to all orders of mass, whose leading order term agrees with the mass correction reported in literature. We also obtain the leading order mass correction on the boundary, which is linear and stronger than the quadratic dependence far from the boundary. The qualitative implications on the phenomenology of heavy ion collisions are speculated.

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    I.   INTRODUCTION
    • Relativistic heavy ion collisions (RHICs) are utilized to produce quark-gluon plasmas (QGPs) at high temperature and nonzero baryon density. A typical (off-central) collision exposes the QGP thus generated under an ultra-strong magnetic field and endows it with a high angular momentum. A number of novel transport phenomena [115]have been proposed, including the axial-chiral-vortical-effect (ACVE). The ACVE refers to the axial vector current, i.e., the spin density of fermions in response to the global angular momentum, and it is expected to be detected via the polarization of Lambda post hadronization. The ACVE is also expected inside the core of a fast spinning neutron star [1618]. In this work, we shall focus on the theoretical aspect of the ACVE.

      In a thermal equilibrium ensemble, the ACVE is represented in terms of the global angular velocity ω by the formula

      JA=σω+,

      (1)

      where the coefficient σ is referred to as the axial vortical conductivity and the ellipsis represents higher power in ω. Based on a pioneer work by Son and Surowka [7, 19] and a supplemental work by Neiman and Oz [8], the axial vortical conductivity is restricted by thermodynamic laws to the following general form in the chiral limit,

      σ=μ2V+μ2A2π2+cT2,

      (2)

      where μV and μA are the vector and axial vector chemical potentials, respectively, and the coefficient c in front of the temperature square has to be determined by other means.

      Besides the hydrodynamic approach, Eq. (2) with c=1/6 was first derived by Vilenkin via the solution of a free Dirac equation in a rotating cylinder [2022], and the axial vortical conductivity for non-interacting fermions reads

      σ=μ2V+μ2A2π2+16T2.

      (3)

      The same expression was obtained by Landsteiner et al. via the Kubo formula to one-loop order [23]. There is also a large body of literature on the derivation of Eq. (3) from kinetic theory [24, 25] or holography [26, 27]. Beyond Eq. (3), the authors of [10, 14] discovered higher order corrections to the coefficient c in QED or QCD coupling. The authors of Ref. [28] determined the higher order terms in ω, i.e., the ellipsis in Eq. (1) for massless fermions, and ended up with a closed form of the axial-vector current, and the authors of Ref. [9] derived the leading order correction of the fermion mass. A recent calculation [29] of axial current for massless fermions in a general thermodynamic equilibrium with rotation and acceleration (within a formalism “far from the boundary,” that is, without enforcing boundary conditions) reproduces the known results for rotating equilibrium, such as those in Ref. [28], but it extends them to systems including acceleration.

      In this study, we explore the axial vortical effect in a finite sphere of radius R subject to the MIT boundary condition. Our motivation is twofold. First, a system rotating with constant angular velocity has to be finite in the direction transverse to the rotation axis, as restricted by the subluminal linear speed on the boundary. Second, a finite sphere serves as a better approximation to the shape of the QGP fireball in heavy ion collisions and the quark matter core of a neutron star than the infinitely long cylinder considered in literature. The MIT boundary condition effectively separates the deconfinement phase of the interior and the confinement phase outside. However, we could neither include the strong coupling underlying the near-perfect fluidity inside an actual QGP fireball nor describe its rapid expansion, especially in the early stage of its evolution. Far from the boundary, where the finite size effect can be ignored, we reproduce in spherical coordinates exactly the same form of the axial-vector current in the chiral limit derived in cylindrical coordinates [28]. We also carry out the fermion mass correction to all orders with the leading order matching the result in Ref. [9], which was derived with the Kubo formulation. The infinite series in powers of the mass correction indicates that the leading order correction for the mass of an s quark at the RIHC temperature is quite accurate. More importantly, we obtain an analytic approximation of the axial vector current on the spherical boundary with the aid of the asymptotic formula of the Bessel function of large argument and large order. For ω=ωˆz, we find that

      JA=(σˆz+σeρ)ω,

      (4)

      where eρ is the unit radial vector of the cylindrical coordinate systems (ρ,ϕ,z). For T1/R and the fermion mass MT, the axial vortical conductivity parallel to ω is

      σ={μ22π2+16T2M4π[μ+2Tln(1+eμT)]}cos2θ,

      (5)

      and that perpendicular to ω is

      σ={μ216π2+148T2M32π[μ+2Tln(1+eμT)]}sin2θ,

      (6)

      where θ is the polar angle with respect to the direction of the angular velocity. Note that we have to set μA=0 and μV=μ because the MIT boundary condition breaks the chiral symmetry even for massless fermions. To the best of our knowledge, the perpendicular component has never been reported in literature, and its existence may shed some light on the longitudinal (with respect to the beam direction) polarization in heavy ion collisions.

      The organization of the paper is as follows. In Sec. II, general properties of the axial vortical effect are discussed from symmetry perspectives. In Sec. III, we lay out the general formulation of the chiral magnetic effect in spherical coordinates with the MIT boundary condition. The axial vortical effects of massless and massive fermions are calculated in Sec. IV and Sec. V, respectively. Sec. VI concludes the paper with a qualitative speculation on the impact of the finite size effect for heavy ion collisions. Some technical details are deferred to Appendices. We also include two additional Appendices for self-containment, one for an alternative derivation of the closed end formula of the axial-current in cylindrical coordinates and the other one for the mass correction via the Kubo formula under dimensional regularization. Throughout the paper, we shall stay with the notation of Eqs. (5) and (6) by setting μA=0 and μμV. Furthermore, the size of the sphere is assumed to be sufficiently large in comparison with the length scale corresponding to the temperature or chemical potential for the boundary condition to be analytically soluble.

    II.   SYMMETRY CONSIDERATION
    • In this section, we explore the axial vortical effect from symmetry perspectives. The validity of the conclusion reached here is not limited to a free Dirac as considered in literature and the subsequent sections of this work.

      The axial vortical effect refers to the thermal average of the spatial component of the axial vector current density JA in the presence of a nonzero angular momentum. Taking the direction of the angular momentum as z-axis, we have

      JA(r)=Trϱ(μ,ω)JA(r)JA(r).

      (7)

      In terms of the field theoretic Hamiltonian H, conserved charge Q, and z-component of the angular momentum Jz, the density matrix at thermal equilibrium reads

      ϱ(μ,ω)=Z1exp(HμQωJzT),

      (8)

      where T is the temperature, μ is the chemical potential, ω is the angular velocity, and Z is the normalization constant such that Trϱ=1.

      Introducing the basic vector of cylindrical coordinates ˆz and

      eρ(ϕ)=ˆxcosϕ+ˆysinϕ,eϕ(ϕ)=ˆxsinϕ+ˆycosϕ,

      (9)

      the ensemble average (7) can be decomposed into its longitudinal component

      JzA(ρ,ϕ,z|μ,ω)=ˆzJA(ρ,ϕ,z|μ,ω),

      (10)

      and its transverse components

      J±A(ρ,ϕ,z|μ,ω)=e±(ϕ)JA(ρ,ϕ,z|μ,ω),

      (11)

      with

      e±(ϕ)=12(eρ±ieϕ),

      (12)

      where the dependence on the cylindrical coordinates, chemical potential, and angular velocity is explicitly indicated and will be suppressed in subsequent sections. We have

      JA(ρ,ϕ,z|μ,ω)=J+A(ρ,ϕ,z|μ,ω),

      (13)

      and consequently

      JρA(ρ,ϕ,z|μ,ω)=eρ(ϕ)JA(ρ,ϕ,z|μ,ω)=2ReJ+A(ρ,ϕ,z|μ,ω),JϕA(ρ,ϕ,z|μ,ω)=eϕ(ϕ)JA(ρ,ϕ,z|μ,ω)=2ImJ+A(ρ,ϕ,z|μ,ω).

      (14)

      Assuming that the Hamiltonian and the boundary condition are invariant under spatial rotation, spatial inversion, time reversal, and charge conjugation, we have

      R(α)ϱ(μ,ω)R(α)1=ϱ(μ,ω),

      (15)

      Pϱ(μ,ω)P1=ϱ(μ,ω),

      (16)

      Tϱ(μ,ω)T1=ϱ(μ,ω),

      (17)

      Cϱ(μ,ω)C1=ϱ(μ,ω),

      (18)

      where R(α) is a Hilbert space operator of a rotation about the z-axis by an angle α, and P, T, and Care Hilbert space operators for spatial inversion, time reversal, and charge conjugation, respectively. Together with the transformation laws of the axial vector current JA(r)

      R(α)JA(ρ,ϕ,z)R(α)1=D(α)JA(ρ,ϕα,z),

      (19)

      PJA(ρ,ϕ,z)P1=JA(ρ,ϕ+π,z),

      (20)

      TJA(ρ,ϕ,z)T1=JA(ρ,ϕ,z),

      (21)

      CJA(ρ,ϕ,z)C1=JA(ρ,ϕ,z),

      (22)

      it follows that

      JA(ρ,ϕ,z|μ,ω)=TrR(α)ϱ(μ,ω)JA(ρ,ϕ,z)R1(α)=D(α)JA(ρ,ϕα,z|μ,ω),

      (23)

      JA(ρ,ϕ,z|μ,ω)=TrPϱ(μ,ω)JA(ρ,ϕ,z)P1=JA(ρ,ϕ+π,z|μ,ω),

      (24)

      JA(ρ,ϕ,z|μ,ω)=TrTϱ(μ,ω)JA(ρ,ϕ,z)T1=JA(ρ,ϕ,z|μ,ω),

      (25)

      JA(ρ,ϕ,z|μ,ω)=TrCϱ(μ,ω)JA(ρ,ϕ,z)C1=JA(ρ,ϕ,z|μ,ω),

      (26)

      where D(α) is the dyadic notation of the 3×3 rotation matrix

      D(α)=(cosαsinα0sinαcosα0001).

      (27)

      Because of the relations

      es(ϕ)D(α)=es(ϕα),(s=±1)

      (28)

      and es(ϕ+π)=es(ϕ)s, Eqs. (23) and (24) imply that the longitudinal and transverse components of the axial current defined in Eq. (10) and (11) are independent of the azimuthal angle as expected, and the transverse component is odd in z, i.e.,

      JsA(ρ,ϕ,z|μ,ω)=JsA(ρ,ϕ,z|μ,ω).

      (29)

      Consequently, there cannot be a transverse axial vortical effect for an infinitely long cylinder because the axial vector current is independent of z. This, however, is not the case with a sphere as the z dependence cannot be ignored. The oddness with respect to z implies only zero transverse axial vector current on the equatorial plane of the sphere. Indeed, the subsequent sections show that the transverse component of the axial-vector current does exist on the spherical boundary for a free Dirac field and does vanish on the equatorial plane. The other two equations, Eqs. (25) and (26), imply that the thermal average of the axial-vector current is always odd with respect to the angular velocity and even with the chemical potential.

      Before concluding this section, we remark that some of the relations above can be readily generalized to a non-equilibrium density matrix with its time development dictated by the Liouville theorem. For instance, for a homogeneous and expanding system, as long as relations (15), (16), and (18) hold initially, they will hold always. Then, relation (29) and its implications discussed above remain valid always.

    III.   AXIAL VECTOR CURRENT IN SPHERICAL COORDINATES

      A.   Hamiltonian

    • The Hamiltonian for a Dirac fermion in a uniformly rotating system with angular velocity ω=ωez can be written as [28, 31]

      H=H0ωJzμ,

      (30)

      where H0=iα+βM is the free Hamiltonian, Jz=12Σ3i(xyyx) is the z-component of the total angular momentum, μ is the the chemical potential of the system, M is the mass of the Dirac fermion, and α=γ0γ, β=γ0. We work in the Dirac representation for gamma matrices γμ as follows:

      γ0=(1001),   γi=(0σiσi0).

      (31)

      The last two terms of Eq. (30) are included in the single particle Dirac Hamiltonian because it is Eq. (30), when being sandwiched between Dirac fields ψ,ψ

      H=d3rψ(r)Hψ(r),

      (32)

      to define the density operator e(HT) for the thermal average.

      In this section, we consider the eigenfunctions of the Hamiltonian in spherical coordinates. The eigenfunctions of the Hamiltonian satisfy

      Hψ=(EωJzμ)ψ,

      (33)

      where E is the eigen-energy of H0. The solutions of Eq. (33) can be chosen as the common eigenfunctions of these four commutative Hermitian operators: Hamiltonian H, square of total angular momentum J2, z-component of total angular momentum Jz, and parity operator P. We list the eigenfunctions in spherical coordinates as follows,

      ψj,l=j+12,m(r,θ,ϕ)=(f(r)Zj,j+12,m(θ,ϕ)ig(r)Zj,j12,m(θ,ϕ)),ψj,l=j12,m(r,θ,ϕ)=(f(r)Zj,j12,m(θ,ϕ)ig(r)Zj,j+12,m(θ,ϕ)),

      (34)

      where j, l, m denote the eigenvalues j(j+1),(1)l,m of J2,P,Jz, respectively; the spinor spherical harmonics Zj,j±12,m(θ,ϕ) are defined as

      Zj,j+12,m(θ,ϕ)=12(j+1)(jm+1Yj+12,m12(θ,ϕ)j+m+1Yj+12,m+12(θ,ϕ)),Zj,j12,m(θ,ϕ)=12j(j+mYj12,m12(θ,ϕ)jmYj12,m+12(θ,ϕ)),

      (35)

      and f(r), g(r) are the radial wave functions. Making use of the following relations,

      σˆrZj,j±12,m(θ,ϕ)=Zj,j12,m(θ,ϕ),σ[f(r)Zj,j±12,m(θ,ϕ)]=[f(r)+(1±j±12)f(r)r]Zj,j12,m(θ,ϕ),

      (36)

      one can obtain the following differential equation satisfied by f(r) in Eq. (34),

      r2f(r)+2rf(r)+[r2(E2M2)l(l+1)]f(r)=0,

      (37)

      which is the l-th order spherical Bessel equation. The radial function g(r) in Eq. (34) can be expressed by f(r),

      g(r)={1E+M[f(r)+l+1rf(r)],for l=j+12,1E+M[f(r)lrf(r)],for l=j12.

      (38)

      We list the solutions of f(r) and g(r) in Table 1, with k>0, Ek=k2+M2, and C the normalization factor.

      l=j±12E=EkE=Ek
      f(r)Cjl(kr)CEkMEk+Mjl(kr)
      g(r)CEkMEk+Mjl1(kr)Cjl1(kr)

      Table 1.  Solutions of f(r) and g(r).

      For a spherical volume of radius R, the quantization of the radial momentum k depends on the boundary condition. An approximate boundary condition for a fireball of QGP is derived from the MIT bag model [30] and reads

      iγˆrψj,l,m||r|=R=ψj,l,m||r|=R,

      (39)

      which requires that the solution of the Dirac equation on the boundary implements the eigenfunction of iγˆr of eigenvalue one. As γ5γˆr=γˆrγ5, the MIT boundary condition breaks the chiral symmetry even for massless fermions. In accordance with Eqs. (34) and (36), the radial wave function satisfies

      f(R)=±g(R),

      (40)

      for l=j1/2. For the solutions of the free Dirac equation in Table 1, the MIT boundary condition reads

      jj1/2(kR)=jj+1/2(kR)tanχ,

      (41)

      for the positive energy state of l=j1/2 and

      jj+1/2(kR)=jj1/2(kR)tanχ,

      (42)

      for the positive energy state of l=j+1/2, where

      tanχ=EkMEk+M.

      (43)

      The boundary conditions for the negative energy states are based on the charge conjugation, i.e.,

      ψcj,l,m=γ2ψj,l,m.

      (44)

      Employing the integration formula

      R0drrJ2ν(kr)=R22[J2ν(kR)+(1ν2k2R2)J2ν(kR)],

      (45)

      and the formulas of the derivative Jν(z) in terms of Jν(z) and Jν±1(z), the normalization constant in Table 1 is readily determined

      |C|2={2R3(sec2χ+csc2χ2jkRcotχ2j+2kRtanχ)1j2j1/2(kR),for l=j12,2R3(sec2χ+csc2χ+2jkRtanχ+2j+2kRcotχ)1j2j+1/2(kR),for l=j+12.

      (46)

      The boundary conditions Eqs. (41) and (42) can be solved approximately for kR1 and jkR with the aid of the asymptotic formula of the spherical Bessel function

      jl(x)1xsin(xlπ2)  as  xmax(1,l).

      (47)

      We find

      kRlπ2+χ=nπ,  (nZ)

      (48)

      for l=j±1/2. The summation of k can be converted to an integral

      k(...)=Rπ0dk(...),

      (49)

      and the normalization constant in Eq. (46) under both conditions of Eqs. (41) and (42) is simplified to

      |C|=2kR1tan2χ+1=kREk+MEk.

      (50)
    • B.   Quantized Dirac field

    • The quantized Dirac field can be expressed by the eigenfunctions of the Hamiltonian H as follows:

      ψ(r)=kjlm[akjlmukjlm(r)+bkjlmvkjlm(r)],

      (51)

      where akjlm and akjlm are the creation and annihilation operators of particles, whereas bkjlm and bkjlm are those of anti-particles. The explicit forms of ukjlm(r) and vkjlm(r) are

      ukjlm(r)=ψjlm(r),vkjlm(r)=γ2ψjlm(r).

      (52)

      We have

      Hukjlm=(Ekmωμ)ukjlm,Hvkjlm=(Ek+mωμ)vkjlm,

      (53)

      where Ek=k2+M2. The ensemble average (7) of akjlmakjlm and bkjlmbkjlm with the density operator (8) gives rise to the Fermi-Dirac distribution functions,

      akjlmakjlm=1eβ(Ekmωμ)+1,bkjlmbkjlm=1eβ(Ekmω+μ)+1,

      (54)

      and the thermal expectation values of akjlmbkjlm, bkjlmakjlm, akjlmbkjlm, and bkjlmakjlm are all zero.

      In the following, we calculate the axial vector current of the uniformly rotating system of Dirac fermions. The axial vector current is the ensemble average of the corresponding operator, i.e.,

      JA=ψΣψ=JvacA+kjlm(akjlmakjlmukjlmΣukjlmbkjlmbkjlmvkjlmΣvkjlm)=kjlm[1eβ(Ekmωμ)+1+1eβ(Ekmω+μ)+1]ukjlmΣukjlm,

      (55)

      where JvacA=kjlmvkjlmΣvkjlm=0 is the vacuum term, and the charge conjugation relation in Eq. (39) has been employed in the last step. It follows from the relation

      Zj,j12,m(θ,ϕ)=±i()m12Zj,j12,m(θ,ϕ),

      (56)

      that

      ukjlmΣukjlm=ukjlmΣukjlm,

      (57)

      and Eq. (55) becomes

      JA=kjlm[1eβ(Ekmωμ)+11eβ(Ek+mω+μ)+1]ukjlmΣukjlm.

      (58)

      Introducing the following ϕ independent functions

      ζjlm(θ)Zjlm(θ,ϕ)σ3Zjlm(θ),

      (59)

      and

      ηjlm(θ)Zjlm(θ,ϕ)σ+Zjlm(θ,ϕ),

      (60)

      we have

      uk,j,j12,m(r)Σ3uk,j,j12,m(r)=|C±|2[j2j12(kr)ζj,j12,m(θ)+j2j±12(kr)ζj,j±12,m(θ)tan2χk],

      (61)

      and

      uk,j,j12,m(r)Σ+uk,j,j12,m(r)=|C±|2[j2j12(kr)ηj,j12,m(θ)+j2j±12(kr)ηj,j±12,m(θ)tan2χk],

      (62)

      with |C±|2 given by the upper (lower) line of Eq. (46) In particular, the expression of ηjlm(θ) can be reduced to

      ηj,j12,m(θ)=12jeiϕYj12,m12(θ,ϕ)L+Yj12,m12(θ,ϕ),

      (63)

      and

      ηj,j+12,m(θ)=12(j+1)eiϕYj+12,m12(θ,ϕ)L+Yj+12,m12(θ,ϕ),

      (64)

      with

      L+=eiϕ(θ+icotθϕ).

      (65)

      It follows from the property Ylm(πθ,ϕ+π)= ()lYlm(θ,ϕ) that ηjlm(πθ)=ηjlm(θ) and thereby ηjlm(π2)=0.

      Before concluding this section, we point out an interesting property of the MIT boundary condition, which is not dictated by symmetries: the axial vector current vanishes along the equator of the fireball. Indeed, Eq. (40) implies that

      ukjlmΣ3ukjlm=f2(R)Θjm(θ),

      (66)

      with

      Θjm(θ)=Zj,j12,m(θ,ϕ)σ3Zj,j12,m(θ,ϕ)+Zj,j+12,m(θ,ϕ)σ3Zj,j+12,m(θ,ϕ).

      (67)

      Writing Θjm(θ) in terms of associated Legendre functions Pμl(cosθ) and using the explicit form of Pμl(0), we find that

      Θjm(π2)=0.

      (68)

      See Appendix A for details of the proof.

    IV.   AXIAL CHIRAL VORTICAL EFFECT OF MASSLESS FERMIONS WITH FINITE-SIZE EFFECT

      A.   Axial vector current far from the boundary

    • For massless fermions, M=0 and Ek=k in Table 1. Far from the boundary, the main support of the axial vector current comes from the spherical Bessel function with l=O(1). Together with the condition T1/R and kT, we have kR1 for typical radial momentum and the approximation in the last paragraph of Sec. III.A becomes useful. Using the relations

      1ex+1=11ex+1.

      (69)

      the z-component of Eq. (55) reads

      JzA=R2πdkjlm[1eβ(kmωμ)+11eβ(k+mωμ)+1]ukjlmΣ3ukjlm,

      (70)

      where we have turned the summation over k to integral according to Eq. (49) and extended the integration domain to (,) via Eq. (69).

      The Taylor expansion of the axial current in Eq. (70) reads

      JzA=n=0Cnω2n+1,

      (71)

      where the coefficient

      CnR(2n+1)!πT2n+1dkf(2n+1)(kμT)jjm=jm2n+1×(uk,j,j12,mΣ3uk,j,j12,m+uk,j,j+12,mΣ3uk,j,j+12,m)=4(2n+1)!πT2n+1dkk2f(2n+1)(kμT)l=0j2l(kr)lm=l(m+12)2n+1|Ylm(θ,φ)|2,

      (72)

      with f(x)=1/(ex+1) the Fermi-Dirac distribution function and f(n)(x) its n-th derivative. In the second step of Eq. (72), we have substituted the explicit form of the wave function in Eq. (51) together with Eq. (35) for the spinor spherical harmonics. Applying the addition formula

      j0(k|rr|)=4πl=0j0(kr)j0(kr)lm=lYlm(θ,φ)Ylm(θ,φ),

      (73)

      for r=r, θ=θ, and ϕ=ϕ+ϵ, we find

      Cn=i()n(2n+1)!π2T(2n+1)d2n+1dϵ2n+1[eiϵ2ξdkkf(2n+1)(kμT)sin(kξ)]|ϵ=0,

      (74)

      where ξ2rsinθsinϵ2. After 2n times of integration by part with respect to k, we obtain that

      Cn=i(2n+1)!π2T(2n+1)d2n+1dϵ2n+1{eiϵ2ξ2n2dkkf(kμT)[2ncos(kξ)+kξsin(kξ)]}|ϵ=0.

      (75)

      Only the (2n+1)-th power of ξ inside the curly brackets contributes to Cn. Together with the integrals

      dxf(x)=1,dxx2f(x)=π23,

      (76)

      we have

      Cn=12π2[(n+1)(μ2+π23T2)ρ2n+112n(2n1)ρ2n2],

      (77)

      with ρ=rsinθ. Substituting into Eq. (71) and summing up the series, we end up with

      JzA=(16T2+μ22π2)ω(1ω2ρ2)2+ω324π21+3ω2ρ2(1ω2ρ2)3,

      (78)

      which is in agreement with the closed-end formula derived in cylindrical coordinates in Ref. [28]. An alternative derivation in cylindrical coordinates is presented in Appendix B. To the cubic order in ω, Eq. (78) yields the formula derived in Ref. [28]. As is shown in the step from Eq. (74) to Eq. (75), the key reason for having the closed form of the axial current Eq. (78) is that the density of states for massless fermions is proportional to an integer power of the energy Ek=k so that the integration by part terminates with a finite number of terms for arbitrary n. This is no longer the case for massive fermions.

      Equation (78) is plotted in Fig. 1, where we set ω=0.01T, a rough estimate of the vorticity of the QGP fireball created in RHICs. The pole at ωrsinθ=1 occurs where the linear speed of rotation reaches the speed of light and the linear speed beyond the pole becomes superluminal, which is not admissible. Therefore, the Hamiltonian in Eq. (30) applies only to a finite volume, which in the case of the sphere discussed in this section requires its radius below 1/ω. The axial vector current in Eq. (78) is thereby free from the pole within the sphere, but the finite size effect becomes significant. Unless the finite size effect falls to zero faster than a power series in r/R>ωr, its contribution will be of the same order of importance as that of the higher order terms of Eq. (78).

      Figure 1.  (color online) Ratio of axial vector current JzA over T3 of massless fermions far from the boundary in Eq. (78) with the angular velocity ω=0.01T as a function of ρT, where ρ and T are the radius and temperature, respectively. The black, blue, and red lines indicate μ/T = 0.5, 1.0, and 1.5, respectively. The inner panel is for ρT= 0–1.0.

      Regarding the transverse component far from the boundary, the typical contribution to the thermal average comes from jkR, and the sum over k in Eq. (55) and wave function normalization can be approximated by Eqs. (49) and (50), respectively. Following Eqs. (60), (63), (64), and (65), we obtain

      J+A=2π0dkk2j,mgm(k)[j2j12(kr)ηj,j12,m(θ)+j2j+12(kr)ηj,j+12,m(θ)]=2π0dkk2[l=0eiϕ2l+1mgm+12(k)[j2l(kr)Ylm(θ,ϕ)L+Ylm(θ,ϕ)]

      l=1eiϕ2l+1mgm+12(k)[j2l(kr)Ylm(θ,ϕ)L+Ylm(θ,ϕ)]]=2π0dkk2eiϕg12(k)j20(kr)Y00(θ,ϕ)L+Y00(θ,ϕ)=0,

      (79)

      where

      gm(k)1eβ(kmωμ)+1+1eβ(kmω+μ)+1.

      (80)

      The absence of the transverse components is expected because the finite size effect can be neglected in the bulk, and the spherical and cylindrical shapes of the volume make no difference there.

    • B.   Axial vector current on the boundary

    • Regarding the boundary of a QGP fireball, we have to distinguish the radial momentum k of the wave function ukjlm for l=j1/2 and l=j+1/2 because of the different quantization conditions in Eqs. (41) and (42). Based on Table 1 and Eq. (50), for M=0,

      uk,j,j±1/2,mΣ3uk,j,j±1/2,m=Θjm(θ)R3(2±2j+1kR).

      (81)

      The radial momentum k of Eq. (81) follows from Eqs. (41) and (42). The axial vector current on the boundary is obtained upon substitution of Eq. (81) into Eq. (55). An analytic expression of the boundary axial vector current can be derived for the linear order term of the Taylor expansion in ω, i.e., the chiral conductivity, at high temperature, i.e. T1/R. We have

      JzA=ωR3T[λ=±1,n,jf(λknjμT)22j+1knjR+λ=±1,n,jf(λk+njμT)2+2j+1k+njR]jm=jmΘjm(θ)+O(ω3),

      (82)

      where knj stands for the solutions of Eq. (41) ("" sign) or Eq. (42) ("+" sign). According to the definition in Eq. (67) and the explicit form of the spinor spherical harmonics in Eq. (35),

      jm=jmΘjm(θ)=2j+14π+ρj1/2(θ)ρj+1/2(θ),

      (83)

      where

      ρl(θ)22l+1lm=l(m)2|Ylm(θ,ϕ)|2=22l+1d2dϵ2lm=lYlm(θ,ϕ)Ylm(θ,ϕ+ϵ)|ϵ=0=12πd2dϵ2Pl(12sin2θsin2ϵ2)|ϵ=0=l(l+1)4πsin2θ,

      (84)

      and the addition formula of the spherical harmonics has been employed. Combining Eqs. (82), (83), and (84), we arrive at

      JzA=ω4πR3Tcos2θλ=±1,n,j[f(λknjμT)2j+122j+1knjR+f(λk+njμT)2j+122j+1k+njR].

      (85)

      To evaluate the summation over k and j under the condition T1/R or μ1/R, we notice that kR1and the wave functions of the large j become important because of the centrifugal force. The asymptotic formula Eq. (47) is no longer sufficient to serve the purpose, and one has to switch to the Debye formula [32] for the Bessel function of a large argument and large order,

      Jν(νsecβ)2νπtanβcos(νtanβνβπ4),  ν1,

      (86)

      which implies

      jl(kR)=π2kRJl+12(kR)=1(l+12)secβtanβcos[(l+12)(tanββ)π4],

      (87)

      for a spherical Bessel function. Then, the MIT boundary conditions in Eqs. (41) and (42) become

      1jsecβtanβcos[j(tanββ)π4]=±1(j+1)secβtanβcos[(j+1)(tanββ)π4],

      (88)

      with

      jsecβ=(j+1)secβ=kR.

      (89)

      The large j serves as the guideline to sort the order of approximation. Eq. (89) gives rise to the leading order relation between β and β

      β=β1jcotβ.

      (90)

      Substituting Eq. (90) to the RHS of Eq. (88) and dropping the terms beyond the order of 1/j, the boundary condition is reduced to

      cos[j(tanββ)π4]=±cos[j(tanββ)π4β],

      (91)

      with the solutions

      j(tanββ)β2=(n+14)π,

      (92)

      for the upper sign and

      j(tanββ)β2=(n+34)π,

      (93)

      for the lower sign, where n is a positive integer. Together with the relation between β and the radial momentum k in Eq. (89), we have [33]

      δn=RπsinβδkRπ1(jkR)2δk,

      (94)

      to the leading order of a large j for both signs in Eq. (88). Converting the summation over k and j in Eq. (85) to integrals, we obtain the leading order axial vector current on the boundary

      JzAω4π2R2Tcos2θ0dkj1(jkR)2(j1jkR+j1+jkR)λ=±1f(λkμT)ω4π2Tcos2θ0dkk2λ=±1f(λkμT)10du1u2(u1u+u1+u)=ω2π2Tcos2θ0dkk2λ=±1f(λkμT)10duu1u2=ω2π2Tcos2θ0dkk2λ=±1f(λkμT)=(16T2ω+μ22π2ω)cos2θ.

      (95)

      Therefore, the longitudinal axial vortical conductivity vanishes along the equator, which is consistent with the general statement according to Eq. (68) and matches the axial vortical conductivity far from the boundary at the poles (θ=0,π).

      To the linear order in ω, the transverse component of the axial vector current JA is obtained by replacing Θjm(θ) of the formula for the longitudinal component Eq. (82) by

      ηj,j12,m(θ)+ηj,j+12,m(θ)=12jeiϕYj12,m12(θ,ϕ)L+Yj12,m12(θ,ϕ)12(j+1)eiϕYj+12,m12(θ,ϕ)L+Yj+12,m12(θ,ϕ),

      (96)

      i.e.

      J+A=ωR3T[λ=±1,n,jf(λknjμT)22j+1knjR+λ=±1,n,jf(λk+njμT)2+2j+1k+njR]×jm=jm[12jeiϕYj12,m12(θ,ϕ)L+Yj12,m12(θ,ϕ)12(j+1)eiϕYj+12,m12(θ,ϕ)L+Yj+12,m12(θ,ϕ)].

      (97)

      The summation over m can be carried out in a similar manner to that in Eq. (84), and we find, with m=m+12 and l=j±12, that

      m(m+12)Ylm(θ,ϕ)L+Ylm(θ,ϕ)=ilim(θ,ϕ)(θ,ϕ)ei2ϕϕei2ϕL+mYlm(θ,ϕ)Ylm(θ,ϕ)=i2l+14πlim(θ,ϕ)(θ,ϕ)ei2ϕϕei2ϕ(θ+icotθϕ).Pl[cosθcosθ+sinθsinθcos(ϕϕ)]=l(l+1)(2l+1)16πsin2θ,

      (98)

      where the derivative formula of the Legendre polynomial

      Pl(1)=l(l+1)2

      (99)

      is employed. Approximating the sum over n and j by integrals of Eqs. (97) and (98), we obtain the transverse component of the axial vector current

      J+A=ω32π2R2sin2θ0dkj1(jkR)2(j1jkR+j1+jkR)λ=±1f(λkμT)=ω32π2sin2θ0dkk2λ=±1f(λkμT)10du1u2(u1u+u1+u)=116π2(π23T2+μ2)ωsin2θ,

      (100)

      which is of the same order of magnitude as the longitudinal component. Restoring the cylindrical coordinates via

      cosθ=zρ2+z2sinθ=ρρ2+z2

      (101)

      we have

      J+A=18π2(π23T2+μ2)ρzρ2+z2

      (102)

      which is independent of the azimuthal angle and odd in z and is consistent with the symmetry argument in Sec. II.

    V.   AXIAL CHIRAL VORTICAL EFFECT OF MASSIVE FERMIONS WITH FINITE-SIZE EFFECT

      A.   Mass correction of axial vector current far from the boundary

    • For massive fermions, the same approximation of the MIT boundary condition applied to massless fermions reduces the axial vector current JA in Eq. (55) far from the boundary to

      JA=R2πdkjlm[1eβ(Ekmωμ)+11eβ(Ek+mωμ)+1]ukjlmΣukjlm,

      (103)

      with Ek=k2+M2. As dkk2=dEkEkE2kM2, the density of states is no longer an integer power of the energy Ek, and a closed-end formula such as Eq. (78) does not exist. We shall stay with the linear response of JzA to ω in what follows and calculate the axial vertical conductivity. It is straightforward to verify that the combination

      uk,j,j1/2,mΣuk,j,j1/2,m+uk,j,j+1/2,mΣuk,j,j+1/2,m,

      (104)

      with the radial wave functions in Table 1 and the normalization constant Eq. (50) at a given k, is independent of the mass M and thereby takes the same massless form. For the longitudinal component, the spinor spherical harmonics part can be reduced the same way as in Sec. IV.A, and Eq. (103) becomes, to the order ω,

      JzA=4ωπT0dkk2[f(EμT)+f(E+μT)]l=0j2l(kr)lm=l(m+12)|Ylm(θ,φ)|2.

      (105)

      Using the relation of Eq. (73), Eq. (105) becomes

      JzA=12π2T2ωM/Tλ=±1λx(λx)2(MT)2eλxμ/T(eλxμ/T+1)2dx,

      (106)

      where we have transformed the integration variable from k to x=E/T with E=k2+M2. The integral Eq. (106) can be converted to a contour integral by the observation that

      M/Tλ=±1λx(λx)2(MT)2eλxμ/T(eλxμ/T+1)2dx=Re[+i0++i0+zz2(MT)2ezμ/T(ezμ/T+1)2dz]=Re[I+I],

      (107)

      where the first two terms of the Taylor expansion of z2M2T2 in the powers of M are included in I, i.e.,

      Re[I]=Re[+i0++i0+[z2a22]ezμ/T(ezμ/T+1)2dz]=[[(x+μT)2a22]ex(ex+1)2dx]=π23+μ2T2a22,

      (108)

      with a=MT. Then, the integrand of I vanishes sufficiently fast at infinity so that the integration path can be closed from infinity on the upper or lower z-plane and the integral equals the sum of residues at the poles of the distribution function within the contour. Closing the path from the upper plane, we have the poles

      z=μT+(2n+1)iπivn,

      (109)

      within the contour, i.e., n=0,1,2,.... Consequently,

      I=+i0++i0+[zz2a2z2+a22]ezμ/T(ezμ/T+1)2dz,=2Re[πn=0vn((1+a2v2n)12+(1+a2v2n)122)],

      (110)

      Combining Eqs. (106), (107), and (108), we have

      JzA=σω,

      (111)

      with the axial vertical conductivity of massive fermions

      σ=16T2+μ22π2+T2π2Re[πn=0vn((1+a2v2n)12+(1+a2v2n)122)].

      (112)

      The binomial expansions of the square roots in Eq. (112) enable us to write

      σ=16T2+μ22π2M24π2+T2r=2[(r1)(2r3)!!](1)rr!2r1(2π)2rζ(2r1,12+b2πi)a2r,

      (113)

      where ζ(...) denotes the Hurwitz zeta function defined by

      ζ(s,b)=n=01(n+b)s.

      (114)

      Away from the branch points of the square roots in the summands, the infinite series Eq. (110) converges uniformly with respect to a, and thereby, the radius of convergence of the power series Eq. (113) corresponds to the absolute value of the closest branch point to the origin of the complex a-plane, i.e.,π2+(μ/T)2. This can also be inferred from the asymptotic behavior of the expansion coefficients of Eq. (113). We also obtain Eq. (113) in Appendix B by using a cylindrical coordinate system and in Appendix C by the Kubo formula via a thermal diagram, which shows that this result, derived by different methods, is robust. In particular, the thermal diagram requires UV regularization, but the result is independent of regularization schemes.

      At zero temperature, the summation over n in Eq. (112) can be converted to an integral, and we obtain that

      σ=μ22π2M24π2+iμiμdξ(ξ2+M2+ξ2ξ2+M22ξ)={0(μ<M)12π2μμ2M2(μ>M).

      (115)

      The zero σ for μ<M is obvious from Eq. (105), where the derivative of the distribution function vanishes exponentially in the limit T0 for all k. The case with μ>M returns the massless result derived in Sec. IV.A for M=0.

      The axial vector current with mass correction is plotted in Fig. 2, where the solid line is Jz(n)A, and the dashed line is Jz(1)A. Here, Jz(0)A is the axial vector current at M=0, Jz(1)A is the axial vector current with only M2 correction (the result of Jz(1)A is also obtained in Ref. [9]), and Jz(n)A is the result including mass correction up to M2n. Their concrete expressions are

      Figure 2.  (color online) Ratio of axial vector current Jz(n)A including mass correction up to M2n over massless current Jz(0)A (Jz(n)A/Jz(0)A) as a function of the product of M over T (M/T), where the black, blue, green, and red lines indicate μ/T=0, 0.5, 1.0, and 1.5, respectively. The dashed lines are Jz(1)A/Jz(0)A, and the solid lines are Jz(n)A/Jz(0)A.

      Jz(0)A=(T26+μ22π2)ω,Jz(1)A=(T26+μ22π2)ωM24π2ω,

      Jz(n)A=(T26+μ22π2)ωM24π2ω+T2ωnr=2ArM2r.

      (116)

      We can see clearly that Jz(n)A/Jz(0)A decreases with M/T. This is because the presence of mass generally inhibits the fluidity, thus suppressing the vortical conductivity. While the presence of chemical potential slows down this inhibition, when we fix M/T, Jz(n)A/Jz(0)A and Jz(1)A/Jz(0)A increase with increasing μ/T.

      An s quark is taken as an example. We set M=150 MeV, μ/T=1.0, and n=2000 and list the numerical values of the mass correction in Table 2.

      T/MeVJz(1)A/Jz(0)AJz(n)A/Jz(0)A(T2ωnr=2ArM2r)/Jz(0)A
      1000.7377540.7419614.20713×103
      1500.8834460.8842858.38337×104
      2000.9344390.9347042.65816×104
      2500.9580410.958151.08964×104

      Table 2.  Mass correction of axial current when M=150 MeV, μ/T=1.0, n=2000.

      Far from the boundary, the mass correction for the s quark is modest for the selected temperature and chemical potential and is dominated by the leading order O(M2) correction. On the boundary, the leading order mass correction is O(M), as shown below. The mass suppression for the s quark is thereby much stronger there.

      For the transverse component of the axial vector current of massive fermions far from the boundary, all we need is to replace k in gm(k) of Eq. (79) with Ek, and the result remains zero, the same as in the massless case.

    • B.   Mass correction of axial vector current on the boundary

    • An analytical result can also be obtained for the leading order mass correction on the spherical boundary under the same approximation of Sec. IV.B, i.e., T1/R. For massive fermions, it follows from Eq. (46) that Eq. (81) is replaced by

      uk,j,j±1/2,mΣ3uk,j,j±1/2,m=Θjm(θ)2R3b(b±jkR),

      (117)

      with b=Ek/E2kM2, where we have substituted Eq. (43) for the trigonometric functions in the normalization constant Eq. (46) and made the approximation 2j+22jin the last term inside the parentheses for a large j. The conversion from the sum of the radial momentum into an integral proceeds in the same way as for the massless case in Sec. IV.B, and we obtain the following form of the axial vector current to the orderO(ω)

      JzA=ω4π2Tcos2θ0dkk2λ=±1f(λEkμT)1b10du1u2(ubu+ub+u).

      (118)

      The integration over u can be carried out readily

      1b10du1u2(ubu+ub+u)=22b21tan11b21=2πMEk+O(M2E2k).

      (119)

      Consequently, the leading order mass correction is O(M), stronger than O(M2) for the mass correction far from the boundary. Substituting Eq. (119) into Eq. (118) and setting Ek=k, we find

      JzA=Jz(0)A+Jz(1)A+...,

      (120)

      where the first term, Jz(0)A, is the axial-vector current of massless fermions given by Eq. (95), and the leading order mass correction reads

      Jz(1)A=Mω4πTcos2θ0dkk[f(kμT)+f(k+μT)]=Mω4π[μ+2Tln(1+eμT)]cos2θ,

      (121)

      which is an even function of μ. Adding Eqs. (95) and (121), we have the longitudinal axial vector current on the boundary up to the leading order mass correction.

      Jz(B)A={T26+μ22π2M4π[μ+2Tln(1+eμT)]}ωcos2θ,

      (122)

      where Jz(B)Ais the axial vector current with only leading order mass correction on the boundary. We can clearly observe that the mass correction is stronger on the boundary than that far from the boundary. The coefficient of ω of Eq. (116) gives rise to the axial vortical conductivity on the boundary, Eq. (5), presented in the introduction. AsT0,

      1Tf(λEkμT)δ(λEkμ).

      (123)

      With the aid of the integral Eq. (119), together with the definition of b, we obtain a closed-end formula of the axial vortical conductivivity to all orders of mass on the boundary

      σ={0(μ<M)12π2μμ2M2(1Mμ2M2tan1μ2M2M)cos2θ(μ>M).

      (124)

      in parallel to Eq. (115) in the bulk.

      It is straightforward to extend the above analysis to the transverse component. Starting with Eq. (62) and going through the gymnastics from Eq. (118) to Eq. (116) with cos2θ replaced by 18sin2θ, we find the transverse axial vector current on the boundary up to the leading order of mass correction, i.e.,

      J+(B)A={T248+μ216π2M24π[μ+2Tln(1+eμT)]}ωsin2θ.

      (125)

      At zero temperature, we have

      J+(B)A={0(μ<M)116π2μμ2M2(1Mμ2M2tan1μ2M2M)ωsin2θ(μ>M).

      (126)

      This is valid up to all orders of the mass M.

    VI.   CONCLUDING REMARKS
    • Our study can be summarized as follows. We started with a general discussion of the axial vortical effect from symmetry perspectives and investigated the axial vortical effect of a free Dirac field in a finite sphere rotating with a given angular velocity ω. For massless fermions far from the boundary, we were able to reproduce the closed-end formula derived within a cylinder in literature. On the boundary, the axial vector current displays both longitudinal and transverse components with respect to the rotation axis, and the magnitude of each component depends on the colatitude angle of the spherical coordinates. For massive fermions, we obtained the mass correction of the chiral conductivity far from and on the boundary. In the former case, we expanded the chiral conductivity to all orders of mass with the leading order correction in agreement with what was reported in literature. In the latter case, we found that the leading order mass correction was stronger than that of the former, O(M) versus O(M2). To the best of our knowledge, the axial vortical effect on the boundary, especially the emergence of the transverse component, has not been explored in literature.

      While the values of the above results are mainly theoretical and cannot describe quantitatively the ACVE of a strongly interacting and expanding fireball of QGP, some qualitative speculations on the finite size effect in heavy ion collisions remain instructive. The quadrupole factor cos2θ in Eq. (5) would suppress the global polarization (z-component of Eq. (4)) and the perpendicular component in Eq. (4) would contribute to the polarization in the reaction plane shown in Fig. 3, e.g., the longitudinal polarization (the polarization along the beam).

      Figure 3.  x and the beamline define the reaction plane.

      To observe the latter effect clearly, we assume that the beam is along ˆy and rotate the coordinate system by 90 around the x-axis, i.e., y=z=rcosθ, z=y=rsinθsinϕ, and x=x=rsinθcosϕ, with r being the radial coordinate. In terms of the polar angle, θ, and azimuthal angle ϕ associated with the primed coordinates, the longitudinal component in Eqs. (4) and (5) takes the form

      JAˆz=bsin2θsinϕ

      (127)

      with b={μ216π2+148T2M32π[μ+2Tln(1+eμT)]}ω. As the fragmenthadrons, e.g., Λ hyperons originating from the boundary layer, are more likely flying in the radial direction, Eq. (127) maps out the longitudinal polarization profile of these hadrons, withϕbeing the angle of the transverse momentum with respect to the reaction plan and θbeing related to the pseudorapidity via η=lntanθ2.

      More investigations are required for the finite size effect discovered in this work to be practical with respect to the phenomenology of heavy ion collisions. These include exploring the ACVE with the solution of the Dirac equation in an expanding sphere and/or incorporating the anisotropic ACVE conductivity in Eq. (4) in hydrodynamic models. We hope to report the progress along this line in the near future.

    ACKNOWLEDGMENTS
    • We thank Ren-Da Dong and Xin-Li Sheng for fruitful discussions.

    APPENDIX A: AXIAL VECTOR CURRENT ALONG THE EQUATOR
    • To prove Eq. (68), we substitute the explicit form of Zj,l,m(θ,ϕ) into Eq. (67), i.e.,

      Θjm(θ)=12j[(j+m)|Yj12,m12(θ,ϕ)|2(jm)|Yj12,m+12(θ,ϕ)|2]+12(j+1)[(jm+1)|Yj+12,m12(θ,ϕ)|2(j+m+1)|Yj+12,m+12(θ,ϕ)|2].

      As Θjm(θ) is odd in m, we only need to consider the case with m>0. Setting j=l+1/2 and m=μ+1/2 and using the expression of spherical harmonics in terms of the associated Legendre function, we have

      Θjm(θ)=14π(l+μ+1)(lμ)!(l+μ)![(l+μ+1)2Pμl(cosθ)2Pμ+1l(cosθ)2+(lμ+1)2Pμl+1(cosθ)2Pμ+1l+1(cosθ)2],

      with μ0. It follows from the generating function of Legendre polynomials

      112zt+t2=l=0tlPl(z),

      and the definition

      Pμl(z)=()μ(1z2)μ2dμPl(z)dzμ,

      that

      ()μ(2μ1)!!(1z2)μ2tμ(12zt+t2)12μ=l=μtlPμl(z).

      Setting z=0 and comparing the coefficients of tl on both sides, we obtain that [34]

      Pμl(0)=2μπΓ(lμ2+1)Γ(lμ+12).

      It is straightforward to verify that

      Pμ+1l+1(0)=(l+μ+1)Pμl(0),

      and

      Pμ+1l(0)=(lμ+1)Pμl+1(0).

      Equation (68) is thereby proved.

    APPENDIX B: AXIAL VECTOR CURRENT IN CYLINDRICAL COORDINATE SYSTEM
    • In this appendix, we first solve the free Dirac equation in a cylindrical coordinate system and then calculate the axial vector current of the system of massive Dirac fermions, which uniformly rotates with angular velocity ω=ωez along the z-axis. We consider only the axial vector current far from the boundary and thereby ignore the finite size effect.

    • 1.   Solution of the free Dirac equation in cylindrical coordinate system

    • We work in the chiral representation of gamma matrices as adopted in Ref. [35],

      γ0=(0110),  γi=(0σiσi0),  γ5=(1001),

      with σi(i=x, y, z) being the three Pauli matrices. The equation of motion for the free Dirac field Ψ(t, r) can be written as

      itΨ(t,r)=ˆHΨ(t,r),

      with the Hamiltonian ˆH=iγ0γ+γ0M and the Dirac fermion mass M. Suppose that Ψ(t,r) is an energy eigenstate with eigenvalue E, i.e., Ψ(t,r)=eiEtψ(r); then, Eq. (B2) becomes

      ˆHψ(r)=Eψ(r),

      which is the energy eigenvalue equation of the Hamiltonian. It can be proved that these four Hermitian operators, ˆH,ˆpz,ˆJz,Σˆp, are commutative with each other, where Σ=diag(σ,σ), ˆp=i, ˆJ=r׈p+12Σ, and ˆpz,ˆJz are the z-components of ˆp and ˆJ , respectively. In the following, we will calculate the common eigenstates of these four operators in a cylindrical coordinate system. We set ψ=(ψ1,ψ2)T, where ψ1,ψ2 are both two-component spinors; then, Eq. (B3) can be replaced by the following two equations,

      (2+E2M2)ψ1=0,

      ψ2=1M(Eiσ)ψ1.

      In a cylindrical coordinate system, the form of 2 is

      2=2r2+1rr+1r22ϕ2+2z2.

      Now, we solve ψ1 from Eq. (B4). ψ1 can be chosen as

      ψ1=(f(r)ei(j12)ϕg(r)ei(j+12)ϕ)eizpz,

      which is the common eigenstate of ˆpz and iϕ+12σz with eigenvalues pz and j. Plugging Eq. (B7) into Eq. (B4) gives

      [d2dr2+1rddr+(E2M2p2z(j12)2r2)]f(r)=0,

      [d2dr2+1rddr+(E2M2p2z(j+12)2r2)]g(r)=0,

      which are the Bessel equations of order (j12). The boundary conditions of ψ1 at r=0 and r= require that E2>M2+p2z. We can introduce a transverse momentum α=E2M2p2z; then, the eigen-energy becomes E=λM2+p2z+α2, with λ=±1 corresponding to the positive and negative modes. Now, one can obtain ψ1 as

      ψ1=(Jj12(αr)ei(j12)ϕAJj+12(αr)ei(j+12)ϕ)eizpz,

      where A is a constant to be determined. As ψ is also an eigenstate of iΣ, then

      iΣψ=sϵψ

      where ϵ=α2+p2z is the magnitude of the total momentum and s=±1 correspond to the two opposite helicities. From Eq. (B11), one can obtain iσψ1=sϵψ1, which leads to A=iα(sϵpz) and

      ψ2=1M(E+sϵ)ψ1.

      Finally, we obtain the eigenfunctions and corresponding eigen-energy as follows,

      Ψ(λ)ϵpzjs(t,r,ϕ,z)=14πXeitλX+izpz((Xλsϵ)(ϵ+spz)Jj12(αr)ei(j12)ϕis(Xλsϵ)(ϵspz)Jj+12(αr)ei(j+12)ϕλ(X+λsϵ)(ϵ+spz)Jj12(αr)ei(j12)ϕiλs(X+λsϵ)(ϵspz)Jj+12(αr)ei(j+12)ϕ),

      E(λ)ϵpzjs=λM2+ϵ2,

      where X=M2+ϵ2, and λ=±1 correspond to the positive and negative modes. All eigenfunctions are orthonormal,

      dVΨ(λ)ϵpzjsΨ(λ)ϵpzjs=δλλδjjδssδ(ϵϵ)δ(pzpz).

    • 2.   Axial vector current of a uniformly rotating system of massive Dirac fermions

    • The Dirac equation in a uniformly rotating system with angular velocity ω=ωez can be written as [28, 31]

      itΨ(t,r)=(iγ0γ+γ0MωˆJz)Ψ(t,r).

      Compared with the free case in Sec. B.1, it can be observed that the eigenfunctions of Eq. (B16) are the same as the free case but with an energy shift ΔE=jω. Now, we consider a uniformly rotating system of massive Dirac fermions with angular velocity ω=ωez, where the interaction among fermions is ignored. This system is in equilibrium with a reservoir, which keeps a constant temperature T and constant chemical potential μ. In the following, we will calculate the axial vector current JμA of this system. According to the rotational symmetry along the z-axis of the system, we can obtain JxA=JyA=0. Due to the absence of axial chemical potential μ5 in our formalism, J0A vanishes [24]. The unique non-zero component is JzA. From the approach of statistical mechanics used in Refs. [20, 22], one can obtain

      JzA=λ,j,s0dϵϵϵdpzλeβ[M2+ϵ2λ(jω+μ)]+1Ψ(λ)ϵpzjsΣzΨ(λ)ϵpzjs,

      where the Fermi-Dirac distribution has been inserted, and β=1/T. Making use of the following series for Bessel function Jn(x) with nN,

      [Jn(x)]2=i=0(1)i(2n+2i)!i![(n+i)!]2(2n+i)!22n+2ix2n+2i,

      Equation (B17) becomes

      JzA=T3π2N=0ρ2N2N+1n=0CN,nΩ2n+1(2n+1)!d2n+1dα2n+1IN(α,c),

      where we have defined four dimensionless quantities, ρ=rT, Ω=ω/T, α=μ/T, and c=M/T. CN,n, IN(α,c) are defined as

      CN,n=Nj=0(1)Nj(Nj)!(N+j)!(1+δj,0)[(j+12)2n+1(j12)2n+1],

      IN(α,c)=0dyy2N+2(1ey2+c2α+11ey2+c2+α+1).

      The coefficient CN,n can also be expressed as follows:

      CN,n=1(2N)!(xddx)2n+1[xN+12(x1)2N]|x=1=22N2n1(2N)!(d2n+1dt2n+1sinh2N+1t)t=0,

      where we have used the variable transformation x=et in the second line. According to the Taylor expansion of sinht, one can readily show that CN,n=0 for n<N. In principle, one can calculate CN,n for any nN from Eq. (B22). For example, for n=N,N+1, one can obtain

      CN,N=12(2N+1),CN,N+1==124(2N+1)2(N+1)(2N+3).

      According to the calculation method in the appendixes of the recent articles by some of us [6, 36], the integral IN(α,c) in Eq. (B21) can be expanded at c=0 as follows,

      IN(α,c)=l=0(2l2N5)!!(2N+3)(2N5)!!(2l)!!c2lDN,l(α),

      with DN,l(α) expanded at α=0 as

      DN,l(α)=k=0(1)l+k+N(222+2N2l2k)(2l+2k2N2)!(2l2N4)!(2k+1)!ζ(2l+2k2N1)π2l+2k2N2α2k+1.

      Plugging Eqs. (B24) and (B25) into Eq. (B19), one can obtain the series expansion of JzA at ρ=0, Ω=0, α=0, c=0 or r=0, ω=0, M=0, μ=0 as follows:

      JzA=T3N=0ρ2N(2N5)!!(2N+1)(2N+3)n=NCN,nΩ2n+1(2n+1)!l=0(2l2N5)!!(2l)!!(2l2N4)!c2l×k=n(1)l+k+N(222+2N2l2k)(2l+2k2N2)!(2k2n)!ζ(2l+2k2N1)π2l+2k2Nα2k2n.

      If we only keep the linear term of Ω and set α=0 in Eq. (B26), then

      JzA=T2ωl=0(1)l(2222l)(l1)ζ(2l1)(2l3)!!(2l)!!c2lπ2l.

      For the massless fermion case, we can obtain an analytic expression for JzA,

      JzA=(T26+μ22π2)ω(1r2ω2)2+ω3(1+3r2ω2)24π2(1r2ω2)3,

      which is divergent as the speed-of-light surface is approached [28].

    APPENDIX C: KUBO FORMULA VIA DIMENSIONAL REGULARIZATION
    • The Kubo formula relates the axial vortical conductivity to the static Fourier component of the correlation between the axial vector current JiA and the stress tensor T0j via

      JiAT0j=2iϵijnknσ,

      in the limit k0. Ignoring the interactions, the LHS is represented by the one-loop thermal diagram in Fig. 4 [9, 14, 23, 37]. Calculating the thermal diagram with the Matsubara formulation, we have

      Figure 4.  One-loop correction to the vortical conductivity [23].

      JiAT0jiβvntr(qμγμ+M)γiγ5[(q+k)νγν+M][γ0(q+k2)j+γjq0](q2M2)2[(q+k)2M2]2=4iϵijnknβvnd3q(2π)313q2v2n(q2+v2n+M2)2=2iϵijnknπ2dqq21βvn13q2v2n(q2+v2n+M2)2,

      where the Matsubara frequency vn=(2n+1)πTiμ. The summation and integral in Eq. (C2) appear UV-divergent, and we apply the dimensional regularization by extending the spatial components of the loop momentum from 3-dimensional to D-dimensional, i.e.,

      d3q(2π)3dDq(2π)D.

      It is straightforward to obtain that

      σ=ωD(2π)Dvn[1DB(1+D2,1D2)(v2n+M2)D21B(D2,2D2)v2n(v2n+M2)D22]=πωDT(2π)DsinπD2n=0vD2n[(1+M2v2n)D21(2D)(1+M2v2n)D22]=σ|M=0+Δσ,

      where the D-dimensional solid angle ωD=2πD/Γ(D2), and B(x,y) is the beta function. The last line of Eq. (C4) separates σ into two terms, where σ|M=0 is the vortical conductivity of massless fermions, and Δσ is the mass correction of the vortical conductivity. Here,

      σ|M=0=πωDT(2π)DsinπD2n=0vD2n(D1),

      and

      Δσ=πωDT(2π)DsinπD2n=0vD2n[(1+M2v2n)D21(2D)(1+M2v2n)D22D+1].

      With the dimensionality D=3ϵ and taking the limit ϵ0, we find

      σ|M=0=ωD(D1)TD1(2π)D1sinπD2Reζ(2D,12iμ2πT)T26+μ22π2(ϵ0).

      Upon expanding Δσ in the power of M2, the leading term, the M2 term, is of the form 0× at D=3, and the limit has to be taken carefully. Let cD be the coefficient of M2; we have

      cD=πωDT(2π)DsinπD2(D21)(D3)Ren=0vD4n(Tπ)12(1ϵ)(ϵ)(πT)1(121)Reζ(1+ϵ,12iμ2πT)14π2.

      For a higher power of M2, however, the naive limit works. Taken together, we obtain the limit

      Δσ=M24π2+T2πn=0vn[(1+M2v2n)12+(1+M2v2n)122].

      Adding Eq. (C7) and Eq. (C9), we replicate Eq. (112). We have also verified that the same result emerges with the Pauli-Villars regularization.

Reference (37)

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