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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 [1–15]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 [16–18]. 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 andμA 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 [20–22], and the axial vortical conductivity for non-interacting fermions readsσ=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 thatJA=(σˆz+σ′eρ)ω,
(4) where
eρ is the unit radial vector of the cylindrical coordinate systems(ρ,ϕ,z) . ForT≫1/R and the fermion massM≪T , the axial vortical conductivity parallel toω isσ={μ22π2+16T2−M4π[μ+2Tln(1+e−μT)]}cos2θ,
(5) and that perpendicular to
ω isσ′={μ216π2+148T2−M32π[μ+2Tln(1+e−μT)]}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 andμV=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 andμ≡μV . 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. -
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 chargeQ , and z-component of the angular momentumJz , the density matrix at thermal equilibrium readsϱ(μ,ω)=Z−1exp(H−μQ−ωJzT),
(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 andeρ(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)=ˆz⋅JA(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)=1√2(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
J−A(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)P−1=ϱ(μ,ω),
(16) Tϱ(μ,ω)T−1=ϱ(μ,−ω),
(17) Cϱ(μ,ω)C−1=ϱ(−μ,ω),
(18) where
R(α) is a Hilbert space operator of a rotation about the z-axis by an angle α, andP ,T , andC are Hilbert space operators for spatial inversion, time reversal, and charge conjugation, respectively. Together with the transformation laws of the axial vector currentJA(r) R(α)JA(ρ,ϕ,z)R(α)−1=↔D(α)⋅JA(ρ,ϕ−α,z),
(19) PJA(ρ,ϕ,z)P−1=JA(ρ,ϕ+π,−z),
(20) TJA(ρ,ϕ,z)T−1=−JA(ρ,ϕ,z),
(21) CJA(ρ,ϕ,z)C−1=JA(ρ,ϕ,z),
(22) it follows that
JA(ρ,ϕ,z|μ,ω)=TrR(α)ϱ(μ,ω)JA(ρ,ϕ,z)R−1(α)=↔D(α)⋅JA(ρ,ϕ−α,z|μ,ω),
(23) JA(ρ,ϕ,z|μ,ω)=TrPϱ(μ,ω)JA(ρ,ϕ,z)P−1=JA(ρ,ϕ+π,−z|μ,ω),
(24) JA(ρ,ϕ,z|μ,ω)=TrTϱ(μ,ω)JA(ρ,ϕ,z)T−1=−JA(ρ,ϕ,z|μ,−ω),
(25) JA(ρ,ϕ,z|μ,ω)=TrCϱ(μ,ω)JA(ρ,ϕ,z)C−1=JA(ρ,ϕ,z|−μ,ω),
(26) where
↔D(α) is the dyadic notation of the3×3 rotation matrix↔D(α)=(cosα−sinα0sinαcosα0001).
(27) Because of the relations
es(ϕ)⋅↔D(α)=es(ϕ−α),(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 instance①, 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.
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The Hamiltonian for a Dirac fermion in a uniformly rotating system with angular velocity
ω=ωez can be written as [28, 31]H=H0−ωJz−μ,
(30) where
H0=−iα⋅∇+βM is the free Hamiltonian,Jz=12Σ3−i(x∂y−y∂x) 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, andα=γ0γ ,β=γ0 . We work in the Dirac representation for gamma matricesγμ as follows:γ0=(100−1), γi=(0σi−σi0).
(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=(E−ωJz−μ)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 momentumJ2 , z-component of total angular momentumJz , 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,j−12,m(θ,ϕ)),ψj,l=j−12,m(r,θ,ϕ)=(f(r)Zj,j−12,m(θ,ϕ)ig(r)Zj,j+12,m(θ,ϕ)),
(34) where
j, l, m denote the eigenvaluesj(j+1),(−1)l,m ofJ2,P,Jz , respectively; the spinor spherical harmonicsZj,j±12,m(θ,ϕ) are defined asZj,j+12,m(θ,ϕ)=1√2(j+1)(√j−m+1Yj+12,m−12(θ,ϕ)−√j+m+1Yj+12,m+12(θ,ϕ)),Zj,j−12,m(θ,ϕ)=1√2j(√j+mYj−12,m−12(θ,ϕ)√j−mYj−12,m+12(θ,ϕ)),
(35) and
f(r) ,g(r) are the radial wave functions. Making use of the following relations,σ⋅ˆrZj,j±12,m(θ,ϕ)=Zj,j∓12,m(θ,ϕ),σ⋅∇[f(r)Zj,j±12,m(θ,ϕ)]=[f′(r)+(1±j±12)f(r)r]Zj,j∓12,m(θ,ϕ),
(36) one can obtain the following differential equation satisfied by
f(r) in Eq. (34),r2f′′(r)+2rf′(r)+[r2(E2−M2)−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 byf(r) ,g(r)={1E+M[f′(r)+l+1rf(r)],for l=j+12,−1E+M[f′(r)−lrf(r)],for l=j−12.
(38) We list the solutions of
f(r) andg(r) in Table 1, withk>0 ,Ek=√k2+M2 , and C the normalization factor.l=j±12 E=Ek E=−Ek f(r) Cjl(kr) −C√Ek−MEk+Mjl(kr) g(r) C√Ek−MEk+Mjl∓1(kr) Cjl∓1(kr) Table 1. Solutions of
f(r) andg(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
−iγ⋅ˆrψj,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
−iγ⋅ˆr of eigenvalue one. Asγ5γ⋅ˆr=−γ⋅ˆrγ5 , the MIT boundary condition breaks the chiral symmetry even for massless fermions. In accordance with Eqs. (34) and (36), the radial wave function satisfiesf(R)=±g(R),
(40) for
l=j∓1/2 . For the solutions of the free Dirac equation in Table 1②, the MIT boundary condition readsjj−1/2(kR)=jj+1/2(kR)tanχ,
(41) for the positive energy state of
l=j−1/2 andjj+1/2(kR)=−jj−1/2(kR)tanχ,
(42) for the positive energy state of
l=j+1/2 , wheretanχ=√Ek−MEk+M.
(43) The boundary conditions for the negative energy states are based on the charge conjugation, i.e.,
ψcj,l,m=γ2ψ∗j,l,m.
(44) Employing the integration formula
∫R0drrJ2ν(kr)=R22[J′2ν(kR)+(1−ν2k2R2)J2ν(kR)],
(45) and the formulas of the derivative
J′ν(z) in terms ofJν(z) andJν±1(z) , the normalization constant in Table 1 is readily determined|C|2={2R3(sec2χ+csc2χ−2jkRcotχ−2j+2kRtanχ)−1j−2j−1/2(kR),for l=j−12,2R3(sec2χ+csc2χ+2jkRtanχ+2j+2kRcotχ)−1j−2j+1/2(kR),for l=j+12.
(46) The boundary conditions Eqs. (41) and (42) can be solved approximately for
kR≫1 andj≪kR with the aid of the asymptotic formula of the spherical Bessel functionjl(x)≃1xsin(x−lπ2) as x≫max(1,l).
(47) We find
kR−lπ2+χ=nπ, (n∈Z)
(48) for
l=j±1/2 . The summation of k can be converted to an integral∑k(...)=Rπ∫∞0dk(...),
(49) and the normalization constant in Eq. (46) under both conditions of Eqs. (41) and (42) is simplified to
|C|=√2k√R√1tan2χ+1=k√R√Ek+MEk.
(50) -
The quantized Dirac field can be expressed by the eigenfunctions of the Hamiltonian H as follows:
ψ(r)=∑kjlm[akjlmukjlm(r)+b†kjlmvkjlm(r)],
(51) where
a†kjlm andakjlm are the creation and annihilation operators of particles, whereasb†kjlm andbkjlm are those of anti-particles. The explicit forms ofukjlm(r) andvkjlm(r) areukjlm(r)=ψjlm(r),vkjlm(r)=γ2ψ∗jlm(r).
(52) We have
Hukjlm=(Ek−mω−μ)ukjlm,Hvkjlm=(−Ek+mω−μ)vkjlm,
(53) where
Ek=√k2+M2 . The ensemble average (7) ofa†kjlmakjlm andb†kjlmbkjlm with the density operator (8) gives rise to the Fermi-Dirac distribution functions,⟨a†kjlmakjlm⟩=1eβ(Ek−mω−μ)+1,⟨b†kjlmbkjlm⟩=1eβ(Ek−mω+μ)+1,
(54) and the thermal expectation values of
a†kjlmb†kjlm ,b†kjlma†kjlm ,akjlmbkjlm , andbkjlmakjlm 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=⟨ψ†Σψ⟩=JvacA+∑kjlm(⟨a†kjlmakjlm⟩u†kjlmΣukjlm−⟨b†kjlmbkjlm⟩v†kjlmΣvkjlm)=∑kjlm[1eβ(Ek−mω−μ)+1+1eβ(Ek−mω+μ)+1]u†kjlmΣukjlm,
(55) where
JvacA=−∑kjlmv†kjlmΣvkjlm=0 is the vacuum term, and the charge conjugation relation in Eq. (39) has been employed in the last step. It follows from the relationZj,j∓12,m(θ,ϕ)=±i(−)m−12Z∗j,j∓12,m(θ,ϕ),
(56) that
u†kjlmΣukjlm=−u†kjl−mΣukjl−m,
(57) and Eq. (55) becomes
JA=∑kjlm[1eβ(Ek−mω−μ)+1−1eβ(Ek+mω+μ)+1]u†kjlmΣukjlm.
(58) Introducing the following ϕ independent functions
ζjlm(θ)≡Z†jlm(θ,ϕ)σ3Zjlm(θ),
(59) and
ηjlm(θ)≡Z†jlm(θ,ϕ)σ+Zjlm(θ,ϕ),
(60) we have
u†k,j,j∓12,m(r)Σ3uk,j,j∓12,m(r)=|C±|2[j2j∓12(kr)ζj,j∓12,m(θ)+j2j±12(kr)ζj,j±12,m(θ)tan2χk],
(61) and
u†k,j,j∓12,m(r)Σ+uk,j,j∓12,m(r)=|C±|2[j2j∓12(kr)ηj,j∓12,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 ofηjlm(θ) can be reduced toηj,j−12,m(θ)=12je−iϕY∗j−12,m−12(θ,ϕ)L+Yj−12,m−12(θ,ϕ),
(63) and
ηj,j+12,m(θ)=−12(j+1)e−iϕY∗j+12,m−12(θ,ϕ)L+Yj+12,m−12(θ,ϕ),
(64) with
L+=eiϕ(∂∂θ+icotθ∂∂ϕ).
(65) It follows from the property
Ylm′(π−θ,ϕ+π)= (−)lYlm′(θ,ϕ) thatηjlm(π−θ)=−ηjlm(θ) and therebyηjlm(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
u†kjlmΣ3ukjlm=f2(R)Θjm(θ),
(66) with
Θjm(θ)=Z†j,j−12,m(θ,ϕ)σ3Zj,j−12,m(θ,ϕ)+Z†j,j+12,m(θ,ϕ)σ3Zj,j+12,m(θ,ϕ).
(67) Writing
Θjm(θ) in terms of associated Legendre functionsPμl(cosθ) and using the explicit form ofPμl(0) , we find thatΘjm(π2)=0.
(68) See Appendix A for details of the proof.
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For massless fermions,
M=0 andEk=k in Table 1. Far from the boundary, the main support of the axial vector current comes from the spherical Bessel function withl=O(1) . Together with the conditionT≫1/R andk∼T , we havekR≫1 for typical radial momentum and the approximation in the last paragraph of Sec. III.A becomes useful. Using the relations1ex+1=1−1e−x+1.
(69) the z-component of Eq. (55) reads
JzA=R2π∫∞−∞dk∑jlm[1eβ(k−mω−μ)+1−1eβ(k+mω−μ)+1]u†kjlmΣ3ukjlm,
(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
Cn≡−R(2n+1)!πT2n+1∫∞−∞dkf(2n+1)(k−μT)∑jj∑m=−jm2n+1×(u†k,j,j−12,mΣ3uk,j,j−12,m+u†k,j,j+12,mΣ3uk,j,j+12,m)=−4(2n+1)!πT2n+1∫∞−∞dkk2f(2n+1)(k−μT)∞∑l=0j2l(kr)l∑m′=−l(m′+12)2n+1|Ylm′(θ,φ)|2,
(72) with
f(x)=1/(ex+1) the Fermi-Dirac distribution function andf(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 formulaj0(k|r−r′|)=4π∞∑l=0j0(kr)j0(kr′)l∑m′=−lY∗lm′(θ,φ)Ylm′(θ′,φ′),
(73) for
r′=r ,θ′=θ , andϕ′=ϕ+ϵ , we findCn=i(−)n(2n+1)!π2T(2n+1)d2n+1dϵ2n+1[eiϵ2ξ∫∞−∞dkkf(2n+1)(k−μT)sin(kξ)]|ϵ=0,
(74) where
ξ≡2rsinθsinϵ2 . After2n times of integration by part with respect to k, we obtain thatCn=i(2n+1)!π2T(2n+1)d2n+1dϵ2n+1{eiϵ2ξ2n−2∫∞−∞dkkf′(k−μT)[−2ncos(kξ)+kξsin(kξ)]}|ϵ=0.
(75) Only the
(2n+1) -th power of ξ inside the curly brackets contributes toCn . Together with the integrals∫∞−∞dxf′(x)=−1,∫∞−∞dxx2f′(x)=π23,
(76) we have
Cn=12π2[(n+1)(μ2+π23T2)ρ2n+112n(2n−1)ρ2n−2],
(77) with
ρ=rsinθ . Substituting into Eq. (71) and summing up the series, we end up withJzA=(16T2+μ22π2)ω(1−ω2ρ2)2+ω324π21+3ω2ρ2(1−ω2ρ2)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 atωrsinθ=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 below1/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 inr/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 overT3 of massless fermions far from the boundary in Eq. (78) with the angular velocityω=0.01T as a function ofρT , where ρ and T are the radius and temperature, respectively. The black, blue, and red lines indicateμ/T = 0.5, 1.0, and 1.5, respectively. The inner panel is forρT = 0–1.0.Regarding the transverse component far from the boundary, the typical contribution to the thermal average comes from
j≪kR , 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 obtainJ+A=2π∫∞0dkk2∑j,mgm(k)[j2j−12(kr)ηj,j−12,m(θ)+j2j+12(kr)ηj,j+12,m(θ)]=2π∫∞0dkk2[∞∑l=0e−iϕ2l+1∑m′gm′+12(k)[j2l(kr)Y∗lm′(θ,ϕ)L+Ylm′(θ,ϕ)]
−∞∑l=1e−iϕ2l+1∑m′gm′+12(k)[j2l(kr)Y∗lm′(θ,ϕ)L+Ylm′(θ,ϕ)]]=2π∫∞0dkk2e−iϕg12(k)j20(kr)Y∗00(θ,ϕ)L+Y00(θ,ϕ)=0,
(79) where
gm(k)≡1eβ(k−mω−μ)+1+1eβ(k−mω+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.
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Regarding the boundary of a QGP fireball, we have to distinguish the radial momentum k of the wave function
ukjlm forl=j−1/2 andl=j+1/2 because of the different quantization conditions in Eqs. (41) and (42). Based on Table 1 and Eq. (50), forM=0 ,u†k,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.
T≫1/R . We haveJzA=−ωR3T[∑λ=±1,n,jf′(λk−nj−μT)2−2j+1k−njR+∑λ=±1,n,jf′(λk+nj−μT)2+2j+1k+njR]j∑m=−jmΘjm(θ)+O(ω3),
(82) where
k∓nj 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),j∑m=−jmΘjm(θ)=2j+14π+ρj−1/2(θ)−ρj+1/2(θ),
(83) where
ρl(θ)≡22l+1l∑m′=−l(m′)2|Ylm′(θ,ϕ)|2=−22l+1d2dϵ2l∑m′=−lY∗lm′(θ,ϕ)Ylm′(θ,ϕ+ϵ)|ϵ=0=−12πd2dϵ2Pl(1−2sin2θsin2ϵ2)|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=−ω4πR3Tcos2θ∑λ=±1,n,j[f′(λk−nj−μT)2j+12−2j+1k−njR+f′(λk+nj−μT)2j+12−2j+1k+njR].
(85) To evaluate the summation over k and j under the condition
T≫1/R orμ≫1/R , we notice thatkR≫1 and 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 order③,Jν(νsecβ)≅√2νπtanβcos(νtanβ−νβ−π4), ν≫1,
(86) which implies
jl(kR)=√π2kRJl+12(kR)=1(l+12)√secβtanβcos[(l+12)(tanβ−β)−π4],
(87) for a spherical Bessel function. Then, the MIT boundary conditions in Eqs. (41) and (42) become
1j√secβtanβcos[j(tanβ−β)−π4]=±1(j+1)√secβ′tanβ′cos[(j+1)(tanβ′−β′)−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′=β−1jcotβ.
(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 tocos[j(tanβ−β)−π4]=±cos[j(tanβ−β)−π4−β],
(91) with the solutions
j(tanβ−β)−β2=(n+14)π,
(92) for the upper sign and
j(tanβ−β)−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=Rπsinβδk≃Rπ√1−(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
JzA≅−ω4π2R2Tcos2θ∫∞0dk∑j√1−(jkR)2(j1−jkR+j1+jkR)∑λ=±1f′(λk−μT)≅−ω4π2Tcos2θ∫∞0dkk2∑λ=±1f′(λk−μT)∫10du√1−u2(u1−u+u1+u)=−ω2π2Tcos2θ∫∞0dkk2∑λ=±1f′(λk−μT)∫10duu√1−u2=−ω2π2Tcos2θ∫∞0dkk2∑λ=±1f′(λk−μT)=(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 replacingΘjm(θ) of the formula for the longitudinal component Eq. (82) byηj,j−12,m(θ)+ηj,j+12,m(θ)=12je−iϕY∗j−12,m−12(θ,ϕ)L+Yj−12,m−12(θ,ϕ)−12(j+1)e−iϕY∗j+12,m−12(θ,ϕ)L+Yj+12,m−12(θ,ϕ),
(96) i.e.
J+A=−ωR3T[∑λ=±1,n,jf′(λk−nj−μT)2−2j+1k−njR+∑λ=±1,n,jf′(λk+nj−μT)2+2j+1k+njR]×j∑m=−jm[12je−iϕY∗j−12,m−12(θ,ϕ)L+Yj−12,m−12(θ,ϕ)−12(j+1)e−iϕY∗j+12,m−12(θ,ϕ)L+Yj+12,m−12(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 andl=j±12 , that∑m′(m′+12)Y∗lm′(θ,ϕ)L+Ylm′(θ,ϕ)=ilim(θ′,ϕ′)→(θ,ϕ)ei2ϕ′∂∂ϕ′e−i2ϕ′L+∑m′Y∗lm′(θ′,ϕ′)Ylm′(θ,ϕ)=i2l+14πlim(θ′,ϕ′)→(θ,ϕ)ei2ϕ′∂∂ϕ′e−i2ϕ′(∂∂θ+icotθ∂∂ϕ).Pl[cosθ′cosθ+sinθ′sinθcos(ϕ′−ϕ)]=−l(l+1)(2l+1)16πsin2θ,
(98) where the derivative formula of the Legendre polynomial
P′l(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=ω32π2R2sin2θ∫∞0dk∑j√1−(jkR)2(j1−jkR+j1+jkR)∑λ=±1f′(λk−μT)=ω32π2sin2θ∫∞0dkk2∑λ=±1f′(λk−μT)∫10du√1−u2(u1−u+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=z√ρ2+z2sinθ=ρ√ρ2+z2
(101) we have
J+A=18π2(π23T2+μ2)ρz√ρ2+z2
(102) which is independent of the azimuthal angle and odd in z and is consistent with the symmetry argument in Sec. II.
-
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 toJA=R2π∫∞−∞dk∑jlm[1eβ(Ek−mω−μ)+1−1eβ(Ek+mω−μ)+1]u†kjlmΣukjlm,
(103) with
Ek=√k2+M2 . Asdkk2=dEkEk√E2k−M2 , the density of states is no longer an integer power of the energyEk , and a closed-end formula such as Eq. (78) does not exist. We shall stay with the linear response ofJzA to ω in what follows and calculate the axial vertical conductivity. It is straightforward to verify that the combinationu†k,j,j−1/2,mΣuk,j,j−1/2,m+u†k,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=−4ωπT∫∞0dkk2[f′(E−μT)+f′(E+μT)]∞∑l=0j2l(kr)l∑m′=−l(m′+12)|Ylm′(θ,φ)|2.
(105) Using the relation of Eq. (73), Eq. (105) becomes
JzA=12π2T2ω∫∞M/T∑λ=±1λx√(λx)2−(MT)2eλx−μ/T(eλx−μ/T+1)2dx,
(106) where we have transformed the integration variable from k to
x=E/T withE=√k2+M2 . The integral Eq. (106) can be converted to a contour integral by the observation that∫∞M/T∑λ=±1λx√(λx)2−(MT)2eλx−μ/T(eλx−μ/T+1)2dx=Re[∫∞+i0+−∞+i0+z√z2−(MT)2ez−μ/T(ez−μ/T+1)2dz]=Re[I+I′],
(107) where the first two terms of the Taylor expansion of
√z2−M2T2 in the powers of M are included inI′ , i.e.,Re[I′]=Re[∫∞+i0+−∞+i0+[z2−a22]ez−μ/T(ez−μ/T+1)2dz]=[∫∞−∞[(x+μT)2−a22]ex(ex+1)2dx]=π23+μ2T2−a22,
(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 polesz=μT+(2n+1)iπ≡ivn,
(109) within the contour, i.e.,
n=0,1,2,... . Consequently,I=∫∞+i0+−∞+i0+[z√z2−a2−z2+a22]ez−μ/T(ez−μ/T+1)2dz,=2Re[π∞∑n=0vn((1+a2v2n)12+(1+a2v2n)−12−2)],
(110) Combining Eqs. (106), (107), and (108), we have
JzA=σω,
(111) with the axial vertical conductivity of massive fermions
σ=16T2+μ22π2+T2π2Re[π∞∑n=0vn((1+a2v2n)12+(1+a2v2n)−12−2)].
(112) The binomial expansions of the square roots in Eq. (112) enable us to write
σ=16T2+μ22π2−M24π2+T2∞∑r=2[(r−1)(2r−3)!!](−1)rr!2r−1(2π)−2rζ(2r−1,12+b2πi)a2r,
(113) where
ζ(...) denotes the Hurwitz zeta function defined byζ(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=μ22π2−M24π2+∫∞−iμ−iμdξ(√ξ2+M2+ξ2√ξ2+M2−2ξ)={0(μ<M)12π2μ√μ2−M2(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 limitT→0 for all k. The case withμ>M returns the massless result derived in Sec. IV.A forM=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 isJz(1)A . Here,Jz(0)A is the axial vector current atM=0 ,Jz(1)A is the axial vector current with onlyM2 correction (the result ofJz(1)A is also obtained in Ref. [9]), andJz(n)A is the result including mass correction up toM2n . Their concrete expressions areFigure 2. (color online) Ratio of axial vector current
Jz(n)A including mass correction up toM2n over massless currentJz(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 indicateμ/T =0, 0.5, 1.0, and 1.5, respectively. The dashed lines areJz(1)A/Jz(0)A , and the solid lines areJz(n)A/Jz(0)A .Jz(0)A=(T26+μ22π2)ω,Jz(1)A=(T26+μ22π2)ω−M24π2ω,
Jz(n)A=(T26+μ22π2)ω−M24π2ω+T2ωn∑r=2ArM2r.
(116) We can see clearly that
Jz(n)A/Jz(0)A decreases withM/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 fixM/T ,Jz(n)A/Jz(0)A andJz(1)A/Jz(0)A increase with increasingμ/T .An s quark is taken as an example. We set
M=150 MeV,μ/T=1.0 , andn=2000 and list the numerical values of the mass correction in Table 2.T/MeV Jz(1)A/Jz(0)A Jz(n)A/Jz(0)A (T2ωn∑r=2ArM2r)/Jz(0)A 100 0.737754 0.741961 4.20713×10−3 150 0.883446 0.884285 8.38337×10−4 200 0.934439 0.934704 2.65816×10−4 250 0.958041 0.95815 1.08964×10−4 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 isO(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) withEk , and the result remains zero, the same as in the massless case. -
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.,
T≫1/R . For massive fermions, it follows from Eq. (46) that Eq. (81) is replaced byu†k,j,j±1/2,mΣ3uk,j,j±1/2,m=Θjm(θ)2R3b(b±jkR),
(117) with
b=Ek/√E2k−M2 , where we have substituted Eq. (43) for the trigonometric functions in the normalization constant Eq. (46) and made the approximation2j+2≃2j in 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=−ω4π2Tcos2θ∫∞0dkk2∑λ=±1f′(λEk−μT)1b∫10du√1−u2(ub−u+ub+u).
(118) The integration over u can be carried out readily
1b∫10du√1−u2(ub−u+ub+u)=2−2√b2−1tan−11√b2−1=2−πMEk+O(M2E2k).
(119) Consequently, the leading order mass correction is
O(M) , stronger thanO(M2) for the mass correction far from the boundary. Substituting Eq. (119) into Eq. (118) and settingEk=k , we findJzA=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 readsJz(1)A=Mω4πTcos2θ∫∞0dkk[f′(k−μT)+f′(k+μT)]=−Mω4π[μ+2Tln(1+e−μT)]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+μ22π2−M4π[μ+2Tln(1+e−μT)]}ωcos2θ,
(122) where
Jz(B)A is 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. AsT→0 ,1Tf′(λEk−μT)→δ(λEk−μ).
(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)12π2μ√μ2−M2(1−M√μ2−M2tan−1√μ2−M2M)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 by18sin2θ , we find the transverse axial vector current on the boundary up to the leading order of mass correction, i.e.,J+(B)A={T248+μ216π2−M24π[μ+2Tln(1+e−μT)]}ωsin2θ.
(125) At zero temperature, we have
J+(B)A={0(μ<M)116π2μ√μ2−M2(1−M√μ2−M2tan−1√μ2−M2M)ωsin2θ(μ>M).
(126) This is valid up to all orders of the mass M.
-
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) versusO(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).To observe the latter effect clearly, we assume that the beam is along
ˆy and rotate the coordinate system by90∘ around the x-axis, i.e.,y=−z′=−rcosθ′ ,z=y′=rsinθ′sinϕ′ , andx=x′=rsinθ′cosϕ′ , with r being the radial coordinate. In terms of the polar angle,θ′ , and azimuthal angleϕ′ associated with the primed coordinates, the longitudinal component in Eqs. (4) and (5) takes the formJA⋅ˆz′=−bsin2θ′sinϕ′
(127) with
b={μ216π2+148T2−M32π[μ+2Tln(1+e−μT)]}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, withϕ′ being the angle of the transverse momentum with respect to the reaction plan andθ′ being related to the pseudorapidity viaη=−lntanθ′2 .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.
-
We thank Ren-Da Dong and Xin-Li Sheng for fruitful discussions.
-
To prove Eq. (68), we substitute the explicit form of
Z†j,l,m(θ,ϕ) into Eq. (67), i.e.,Θjm(θ)=12j[(j+m)|Yj−12,m−12(θ,ϕ)|2−(j−m)|Yj−12,m+12(θ,ϕ)|2]+12(j+1)[(j−m+1)|Yj+12,m−12(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 withm>0 . Settingj=l+1/2 andm=μ+1/2 and using the expression of spherical harmonics in terms of the associated Legendre function, we haveΘjm(θ)=14π(l+μ+1)(l−μ)!(l+μ)![(l+μ+1)2Pμl(cosθ)2−Pμ+1l(cosθ)2+(l−μ+1)2Pμl+1(cosθ)2−Pμ+1l+1(cosθ)2],
with
μ⩾0 . It follows from the generating function of Legendre polynomials1√1−2zt+t2=∞∑l=0tlPl(z),
and the definition
Pμl(z)=(−)μ(1−z2)μ2dμPl(z)dzμ,
that
(−)μ(2μ−1)!!(1−z2)μ2tμ(1−2zt+t2)−12−μ=∞∑l=μtlPμl(z).
Setting
z=0 and comparing the coefficients oftl on both sides, we obtain that [34]Pμl(0)=2μ√πΓ(l−μ2+1)Γ(−l−μ+12).
It is straightforward to verify that
Pμ+1l+1(0)=−(l+μ+1)Pμl(0),
and
Pμ+1l(0)=(l−μ+1)Pμl+1(0).
Equation (68) is thereby proved.
-
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. -
We work in the chiral representation of gamma matrices as adopted in Ref. [35],
γ0=(0110), γi=(0σi−σi0), γ5=(−1001),
with
σi(i=x, y, z) being the three Pauli matrices. The equation of motion for the free Dirac fieldΨ(t, r) can be written asi∂∂tΨ(t,r)=ˆHΨ(t,r),
with the Hamiltonian
ˆH=−iγ0γ⋅∇+γ0M and the Dirac fermion mass M. Suppose thatΨ(t,r) is an energy eigenstate with eigenvalue E, i.e.,Ψ(t,r)=e−iEtψ(r) ; then, Eq. (B2) becomesˆHψ(r)=Eψ(r),
which is the energy eigenvalue equation of the Hamiltonian. It can be proved that these four Hermitian operators,
ˆH,ˆpz,ˆJz,Σ⋅ˆp , are commutative with each other, whereΣ=diag(σ,σ) ,ˆp=−i∇ ,ˆJ=r׈p+12Σ , andˆpz,ˆJz are the z-components ofˆp andˆJ , respectively. In the following, we will calculate the common eigenstates of these four operators in a cylindrical coordinate system. We setψ=(ψ1,ψ2)T , whereψ1,ψ2 are both two-component spinors; then, Eq. (B3) can be replaced by the following two equations,(∇2+E2−M2)ψ1=0,
ψ2=1M(E−iσ⋅∇)ψ1.
In a cylindrical coordinate system, the form of
∇2 is∇2=∂2∂r2+1r∂∂r+1r2∂2∂ϕ2+∂2∂z2.
Now, we solve
ψ1 from Eq. (B4).ψ1 can be chosen asψ1=(f(r)ei(j−12)ϕg(r)ei(j+12)ϕ)eizpz,
which is the common eigenstate of
ˆpz and−i∂ϕ+12σz with eigenvaluespz and j. Plugging Eq. (B7) into Eq. (B4) gives[d2dr2+1rddr+(E2−M2−p2z−(j−12)2r2)]f(r)=0,
[d2dr2+1rddr+(E2−M2−p2z−(j+12)2r2)]g(r)=0,
which are the Bessel equations of order
(j∓12) . The boundary conditions ofψ1 atr=0 andr=∞ require thatE2>M2+p2z . We can introduce a transverse momentumα=√E2−M2−p2z ; then, the eigen-energy becomesE=λ√M2+p2z+α2 , withλ=±1 corresponding to the positive and negative modes. Now, one can obtainψ1 asψ1=(Jj−12(αr)ei(j−12)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−iΣ⋅∇ψ=sϵψ
where
ϵ=√α2+p2z is the magnitude of the total momentum ands=±1 correspond to the two opposite helicities. From Eq. (B11), one can obtain−iσ⋅∇ψ1=sϵψ1 , which leads toA=iα(sϵ−pz) andψ2=1M(E+sϵ)ψ1.
Finally, we obtain the eigenfunctions and corresponding eigen-energy as follows,
Ψ(λ)ϵpzjs(t,r,ϕ,z)=14π√Xe−itλ√X+izpz(√(X−λsϵ)(ϵ+spz)Jj−12(αr)ei(j−12)ϕis√(X−λsϵ)(ϵ−spz)Jj+12(αr)ei(j+12)ϕλ√(X+λsϵ)(ϵ+spz)Jj−12(αr)ei(j−12)ϕiλs√(X+λsϵ)(ϵ−spz)Jj+12(αr)ei(j+12)ϕ),
E(λ)ϵpzjs=λ√M2+ϵ2,
where
X=√M2+ϵ2 , andλ=±1 correspond to the positive and negative modes. All eigenfunctions are orthonormal,∫dVΨ(λ′)†ϵ′p′zj′s′Ψ(λ)ϵpzjs=δλ′λδj′jδs′sδ(ϵ′−ϵ)δ(p′z−pz).
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The Dirac equation in a uniformly rotating system with angular velocity
ω=ωez can be written as [28, 31]i∂∂tΨ(t,r)=(−iγ0γ⋅∇+γ0M−ωˆJz)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 velocityω=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 currentJμA of this system. According to the rotational symmetry along the z-axis of the system, we can obtainJxA=JyA=0 . Due to the absence of axial chemical potentialμ5 in our formalism,J0A vanishes [24]. The unique non-zero component isJzA . From the approach of statistical mechanics used in Refs. [20, 22], one can obtainJzA=∑λ,j,s∫∞0dϵ∫ϵ−ϵdpzλeβ[√M2+ϵ2−λ(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 functionJn(x) withn∈N ,[Jn(x)]2=∞∑i=0(−1)i(2n+2i)!i![(n+i)!]2(2n+i)!22n+2ix2n+2i,
Equation (B17) becomes
JzA=T3π2∞∑N=0ρ2N2N+1∞∑n=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 , andc=M/T .CN,n ,IN(α,c) are defined asCN,n=N∑j=0(−1)N−j(N−j)!(N+j)!(1+δj,0)[(j+12)2n+1−(j−12)2n+1],
IN(α,c)=∫∞0dyy2N+2(1e√y2+c2−α+1−1e√y2+c2+α+1).
The coefficient
CN,n can also be expressed as follows:CN,n=1(2N)!(xddx)2n+1[x−N+12(x−1)2N]|x=1=22N−2n−1(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 ofsinht , one can readily show thatCN,n=0 forn<N . In principle, one can calculateCN,n for anyn⩾N from Eq. (B22). For example, forn=N,N+1 , one can obtainCN,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 atc=0 as follows,IN(α,c)=∞∑l=0(2l−2N−5)!!(2N+3)(−2N−5)!!(2l)!!c2lDN,l(α),
with
DN,l(α) expanded atα=0 asDN,l(α)=∞∑k=0(−1)l+k+N(2−22+2N−2l−2k)(2l+2k−2N−2)!(2l−2N−4)!(2k+1)!ζ(2l+2k−2N−1)π2l+2k−2N−2α2k+1.
Plugging Eqs. (B24) and (B25) into Eq. (B19), one can obtain the series expansion of
JzA atρ=0 ,Ω=0 ,α=0 ,c=0 orr=0 ,ω=0 ,M=0 ,μ=0 as follows:JzA=T3∞∑N=0ρ2N(−2N−5)!!(2N+1)(2N+3)∞∑n=NCN,nΩ2n+1(2n+1)!∞∑l=0(2l−2N−5)!!(2l)!!(2l−2N−4)!c2l×∞∑k=n(−1)l+k+N(2−22+2N−2l−2k)(2l+2k−2N−2)!(2k−2n)!ζ(2l+2k−2N−1)π2l+2k−2Nα2k−2n.
If we only keep the linear term of Ω and set
α=0 in Eq. (B26), thenJzA=T2ω∞∑l=0(−1)l(2−22−2l)(l−1)ζ(2l−1)(2l−3)!!(2l)!!c2lπ2l.
For the massless fermion case, we can obtain an analytic expression for
JzA ,JzA=(T26+μ22π2)ω(1−r2ω2)2+ω3(1+3r2ω2)24π2(1−r2ω2)3,
which is divergent as the speed-of-light surface is approached [28].
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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 tensorT0j viaJiAT0j=2iϵijnknσ,
in the limit
k→0 . 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 haveFigure 4. One-loop correction to the vortical conductivity [23].
JiAT0j≅iβ∑vntr(qμγμ+M)γiγ5[(q+k)νγν+M][γ0(q+k2)j+γjq0](q2−M2)2[(q+k)2−M2]2=4iϵijnknβ∑vn∫d3q(2π)313q2−v2n(q2+v2n+M2)2=2iϵijnknπ2∫dqq21β∑vn13q2−v2n(q2+v2n+M2)2,
where the Matsubara frequency
vn=(2n+1)πT−iμ . 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)3→∫dDq(2π)D.
It is straightforward to obtain that
σ=−ωD(2π)D∑vn[1DB(1+D2,1−D2)(v2n+M2)D2−1−B(D2,2−D2)v2n(v2n+M2)D2−2]=−πωDT(2π)DsinπD2∞∑n=0vD−2n[(1+M2v2n)D2−1−(2−D)(1+M2v2n)D2−2]=σ|M=0+Δσ,
where the D-dimensional solid angle
ωD=2πD/Γ(D2) , andB(x,y) is the beta function. The last line of Eq. (C4) separates σ into two terms, whereσ|M=0 is the vortical conductivity of massless fermions, andΔσ is the mass correction of the vortical conductivity. Here,σ|M=0=−πωDT(2π)DsinπD2∞∑n=0vD−2n(D−1),
and
Δσ=−πωDT(2π)DsinπD2∞∑n=0vD−2n[(1+M2v2n)D2−1−(2−D)(1+M2v2n)D2−2−D+1].
With the dimensionality
D=3−ϵ and taking the limitϵ→0 , we findσ|M=0=−ωD(D−1)TD−1(2π)D−1sinπD2Reζ(2−D,12−iμ2πT)→T26+μ22π2(ϵ→0).
Upon expanding
Δσ in the power ofM2 , the leading term, theM2 term, is of the form0×∞ atD=3 , and the limit has to be taken carefully. LetcD be the coefficient ofM2 ; we havecD=−πωDT(2π)DsinπD2(D2−1)(D−3)Re∞∑n=0vD−4n→−(−Tπ)12(1−ϵ)(−ϵ)(πT)−1(1−2−1)Reζ(1+ϵ,12−iμ2πT)→−14π2.
For a higher power of
M2 , however, the naive limit works. Taken together, we obtain the limitΔσ=−M24π2+T2π∞∑n=0vn[(1+M2v2n)12+(1+M2v2n)−12−2].
Adding Eq. (C7) and Eq. (C9), we replicate Eq. (112). We have also verified that the same result emerges with the Pauli-Villars regularization.
Axial chiral vortical effect in a sphere with finite size effect
- Received Date: 2022-11-13
- Available Online: 2023-03-15
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.