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Meson electro-magnetic form factors in an extended Nambu-Jona-Lasinio model including heavy quark flavors

  • Based on an extended NJL model including heavy quark flavors, we calculate the form factors of pseudo-scalar and vector mesons. After taking into account the vector-meson-dominance effect, which introduces a form factor correction to the quark vector coupling vertices, the form factors and electric radii of π+ and K+ pseudo-scalar mesons in the light flavor sector fit the experimental data well. The magnetic moments of the light vector mesons ρ+ and K*+ are comparable with other theoretical calculations. The form factors in the light-heavy flavor sector are presented to compare with future experiments or other theoretical calculations.
  • Quark gluon plasma (QGP) is created in high energy heavy ion collisions, constituting extremely hot and dense matter. An enormous magnetic field can be generated by high energy peripheral collisions [1-3]. One of the predictions in QGP is that positively and negatively charged particles seperate along the direction of the magnetic field, which is related to chiral magnetic effect (CME) [4-6]. Numerous efforts have been made to determine the CME in experiments [7-9]. However, due to background noise, no definite CME has been revealed to date. Numerous theoretical methods likewise investigated the CME, such as AdS/CFT [10, 11], hydrodynamics [12-14], finite temperature field theory [15-18], quantum kinetic theory [19], lattice method [20], et al.

    In this article, we study the CME in detail by determination of Landau levels. For the massive Dirac fermion system, several studies on CME addressed Landau levels. In Ref. [15], Fukushima et al. proposed four methods to derive the CME. One of these methods made use of Landau energy levels for the massive Dirac equation with chemical potential μ and chiral chemical potential μ5 in a homogeneous magnetic background B=Bez to construct the thermodynamic potential Ω. The macroscopic electric current jz along the z-axis can be obtained from the thermodynamic potential Ω. Another study on the CME addressing Landau levels is related to the second quantization of the Dirac field. In Ref. [21], the authors determined the Landau levels and corresponding Landau wavefunctions for the massive Dirac equation in a uniform magnetic field, likewise with chemical potential μ and chiral chemical potential μ5. Then, they second-quantized the Dirac field and expanded it by these solved Landau wavefunctions and creation/construction operators. The density operator ˆρ can then be determined from Hamiltonian ˆH and particle number operator ˆN of the system. Finally, they derived the macroscopic electric current jz along the z-axis through the trace of density operator ˆρ and electric current operator ˆjz, which is simply the CME equation.

    From the study on CME for massive Dirac fermions through Landau levels, we conclude that the contribution to CME arises uniquely from the lowest Landau level, while the contributions from higher Landau levels cancel each other. However, because of the mass m of the Dirac fermion, the physical picture of the CME for the massive Dirac fermion system is not as clear as in the massless fermion case, as the physical meaning of the chiral chemical potential μ5 for the massive fermion case is not entirely understood. To address this issue, we list the lowest Landau level as follows (we set the homogeneous magnetic background B=Bez along the z-axis and assume eB>0, which is also appropriate for following sections),

    ψ0λ(ky,kz;x)=c0λ(φ00F0λφ00)1Lei(yky+zkz), (λ=±1),

    (1)

    with energy E=λm2+k2z, where F0λ=(λm2+k2z+kz)/m, and φ0 is the zeroth harmonic oscillator wavefunction along the x-axis. To simplify the following discussions, we set ky=0. The z-component of the spin operator for the single particle is Sz=12diag(σ3,σ3), implying Szψ0λ=(+12)ψ0λ. When λ=+1, E=m2+k2z>0, then ψ0+ in Eq. (1) describes a particle with momentum kz and spin projection Sz=+12. When λ=1, E=m2+k2z<0, then ψ0 in Eq. (1) describes an antiparticle with momentum kz and spin projection Sz=12. Thus, in the homogeneous magnetic background B=Bez, we obtain a picture for the lowest Landau level (with ky=0): All particles spin along the (+z)-axis, while all antiparticles spin along the (z)-axis; however, the z-component momentum of particles and anti-particles can be along both the (+z)-axis or the(z)-axis. A net electric current is difficult to obtain along the magnetic field direction from the point of view of the lowest Landau level for the massive fermion case.

    In this article, we focus on a massless fermion (also referred to as the “chiral fermion”) system, where we show that it is easy to obtain a net electric current along the magnetic field direction, seen from the picture of the lowest Landau level. The chiral fermion field can be divided into two independent parts, namely the righthand and lefthand parts. First, we set up the notation. The electric charge of a fermion/antifermion is ±e. The chemical potential for righthand/lefthand fermions is μR/L, which can be employed to express the chiral and ordinary chemical potentials as μ5=(μRμL)/2 and μ=(μR+μL)/2, respectively. The chemical potential μ describes the imbalance of fermions and anti-fermions, while the chiral chemical potential μ5 describes the imbalance of righthand and lefthand chirality. Notably, the introduction of a chemical potential generally corresponds to a conserved quantity. The conserved quantity corresponding to the ordinary chemical potential μ is total electric charge of the system. However, due to chiral anomaly [22, 23], there is no conserved quantity corresponding to the chiral chemical potential μ5, which is crucial for the existence of CME [24].

    To study the CME in the chiral fermion system, first we show a succinct derivation of CME employing the Wigner function approach, which we can use to obtain the CME as a quantum effect of the first order in the expansion. Subsequently, we turn to determine the Landau levels for the chiral fermion system. Because chiral fermions are massless, the equations of righthand and lefthand parts of the chiral fermion field decouple with each other, which allows us to deal with righthand and lefthand fermion fields independently. Taking the righthand fermion field as an example, we first solve the energy eigenvalue equation of the righthand fermion field in an external uniform magnetic field and obtain a series of Landau levels. Then, we perform the second quantization for righthand fermion field, which can be expanded by complete wavefunctions of Landau levels. Finally, the CME can be derived through the ensemble average, explicitly indicating that the CME uniquely arises from the lowest Landau level. By analyzing the physical picture for the lowest Landau level, we conclude that all righthand (chirality is +1) fermions move along the positive z-direction, and all lefthand (chirality is -1) fermions move along the negative z-direction. This is the main result of this study. This result can qualitatively explain why a macroscopic electric current occurs along the direction of the magnetic field in a chiral fermion system, called the CME. We emphasize that the CME equation is derived by determining Landau levels, without the approximation of a weak magnetic field.

    The rest of this article is organized as follows. In Sec. 2, we present a succinct derivation for the CME using Wigner function approach. In Sec. 3, we determine the Landau levels for the righthand fermion field. In Secs. 4 and 5, we perform the second quantization of the righthand fermion system and obtain CME through the ensemble average. In Sec. 6, we discuss the physical picture of the lowest Landau level. Finally, we summarize this study in Sec. 7. Some derivation details are presented in the appendixes.

    Throughout this article, we adopt natural units, where =c=kB=1. The convention for the metric tensor is gμν=diag(+1,1,1,1). The totally antisymmetric Levi-Civita tensor is ϵμνρσ with ϵ0123=+1, which is in agreement with Peskin [25], but not with Bjorken and Drell [26]. The Greek indices, μ,ν,ρ,σ, run over 0,1,2,3, or t,x,y,z, whereas Roman indices, i,j,k, run over 1,2,3 or x,y,z. We use the Heaviside-Lorentz convention for electromagnetism.

    We concisely derive the CME using the Wigner function approach for a chiral fermion system. Our starting point is the following covariant and gauge invariant Wigner function,

    Wαβ(x,p)=:1(2π)4d4yeipy¯Ψβ(x+y2)U(x+y2,xy2)×Ψα(xy2):,

    (2)

    where :: represents the ensemble average, Ψ(x) is the Dirac field operator for chiral fermions, α, β are Dirac spinor indices, and U(x+y/2,xy/2) is the gauge link of a straight line from (xy/2) to (x+y/2). This specific choice for the path in the gauge link in the definition of the Wigner function was first proposed in Ref. [27], where the authors argued that this type of gauge link can create the variable p in the Wigner function W(x,p) to represent the kinetic momentum, although in principle the path in the gauge link is arbitrary. The specific choice of the two points (x±y/2) in the integrand in Eq. (2) is based on the consideration of symmetry. We can also replace (x±y/2) by (x+sy) and (x(1s)y), where s is a real parameter [28].

    Suppose that the electromagnetic field Fμν is homogeneous in space and time, then from the dynamical equation satisfied by Ψ(x), one can obtain the dynamical equation for W(x,p) as follows,

    γKW(x,p)=0,

    (3)

    where Kμ=i2μ+pμ and μ=xμeFμννp. Because W(x,p) is a 4×4 matrix, we can decompose it into 16 independent Γ-matrices,

    W=14(F+iγ5P+γμVμ+γ5γμAμ+12σμνSμν).

    (4)

    The 16 coefficient functions F,P,Vμ,Aμ,Sμν are scalar, pseudoscalar, vector, pseudovector, and tensor, respectively, and they are all real functions because W=γ0Wγ0. Vector current and axial vector current can be expressed as the four-momentum integration of Vμ and Aμ,

    JμV(x)=d4pVμ,

    (5)

    JμA(x)=d4pAμ.

    (6)

    By multiplying Eq. (3) by γK from the lefthand side, we obtain the quadratic form of Eq. (3) as follows,

    (K2i2σμν[Kμ,Kν])W=0.

    (7)

    From Eq. (7), we can obtain two off mass-shell equations for Vμ and Aμ (see Appendix A for details),

    (p21422)Vμ=e˜FμνAν,

    (8)

    (p21422)Aμ=e˜FμνVν,

    (9)

    where ˜Fμν=12εμνρσFρσ. We explicitly showed the factor in Eqs. (8), (9). If we expand Vμ and Aμ order-by-order in as

    Vμ=Vμ(0)+Vμ(1)+2Vμ(2)+,

    (10)

    Aμ=Aμ(0)+Aμ(1)+2Aμ(2)+,

    (11)

    then, at order o(1) and o(), Eqs. (8), (9) become

    p2Vμ(0)=0,

    (12)

    p2Aμ(0)=0,

    (13)

    p2V(1)μ=e˜FμνAν(0),

    (14)

    p2A(1)μ=e˜FμνVν(0).

    (15)

    The zeroth order solutions Vμ(0) and Aμ(0) can be derived by directly calculating the Wigner function without the gauge link through the ensemble average in Eq. (2), which was already obtained by one of the authors and his collaborators [29]. The results for Vμ(0) and Aμ(0) are

    Vμ(0)=2(2π)3pμδ(p2)s[θ(p0)1eβ(p0μs)+1+θ(p0)1eβ(p0+μs)+1],

    (16)

    Aμ(0)=2(2π)3pμδ(p2)ss[θ(p0)1eβ(p0μs)+1+θ(p0)1eβ(p0+μs)+1],

    (17)

    where β=1/T is the inverse temperature of the system, μR/L is the chemical potential for righthand/lefthand fermions as mentioned in the introduction, and s=±1 corresponds to the chirality of righthand/lefthand fermions. The zeroth order solutions Vμ(0) and Aμ(0) satisfy Eqs. (12), (13), which indicates that they are both on shell. From Eqs. (14), (15) we directly obtain the first order solutions,

    Vμ(1)=2(2π)3e˜Fμνpνδ(p2)×ss[θ(p0)1eβ(p0μs)+1+θ(p0)1eβ(p0+μs)+1],

    (18)

    Aμ(1)=2(2π)3e˜Fμνpνδ(p2)×s[θ(p0)1eβ(p0μs)+1+θ(p0)1eβ(p0+μs)+1],

    (19)

    where we employ δ(p2)=δ(p2)/p2. Eqs. (18), (19) are the same as the second term in Eq. (3) of Ref. [30].

    Now, we can calculate JA/V based on Eqs. (5) and (6). Because Vi(0),Ai(0) are odd functions of three-momentum p, the nonzero contribution to JiV/A arises uniquely from Vi(1) and Ai(1). We assume that only a uniform magnetic field exists B=Bez, i.e. F12=F21=B and ˜F03=˜F30=B (other components of Fμν, ˜Fμν are zero), which implies JxV/A=JyV/A=0. After integration over the z-components of Eqs. (18, 19) we have

    JzV=d4pVz(1)=eμ52π2B,

    (20)

    JzA=d4pAz(1)=eμ2π2B,

    (21)

    where μ5=(μRμL)/2 and μ=(μR+μL)/2. Eq. (20) indicates that if μ50, a current flows along the magnetic direction. Because appears in the coefficient of the magnetic field B, an enormous magnetic field is required to produce a macroscopic current, which may be realized in high energy heavy ion collisions. Thus far, we derived the CME in the chiral fermion system using the Wigner function approach, and we observe that the CME is a first order quantum effect in . In fact, the Wigner function approach is a quantum kinetic theory, which implies the presence of quantum effects of a multi-particle system, such as the CME.

    In this and the following sections, we derive the CME for a chiral fermion system by determining the Landau levels. The Lagrangian for a chiral fermion field is

    L=¯Ψ(x)iγDΨ(x),

    (22)

    with the covariant derivative Dμ=μ+ieAμ, and the electric charge ±e for particles/antiparticles. For a uniform magnetic field B=Bez along the z-axis, we choose the gauge potential as Aμ=(0,0,Bx,0). The equation of motion for the field Ψ(x) is

    iγDΨ(x)=0,

    (23)

    which can be written in the form of a Schrödinger equation,

    itΨ(t,x)=iαDΨ(t,x),

    (24)

    with D=+ieA, A=(0,Bx,0). In the chiral representation of Dirac γ-matrices, where γ5=diag(1,1), α=diag(σ,σ), we express Ψ in the form Ψ=(ΨTL,ΨTR)T. Then, Eq. (24) becomes

    it(ΨL(t,x)ΨR(t,x))=(iσDΨL(t,x)iσDΨR(t,x)),

    (25)

    which indicates that the two fields ΨL/R, which correspond to eigenvalues 1 of the matrix γ5, decouple with each other. The two fields ΨL/R are often referred to as lefthand/righthand fermion fields, respectively. Lefthand and righthand fermions are also referred to as chiral fermions.

    In the following, we focus on solving the eigenvalue equation for the righthand fermion field ΨR (similar results are obtained for the lefthand fermion field ΨL).

    To determine the Landau levels, we must solve the eigenvalue equation for the righthand fermion field as follows,

    iσDψR=EψR,

    (26)

    with D=(x,y+ieBx,z). The details for solving Eq. (26) are provided in Appendix B. We list the eigenfunctions and eigenvalues in the following : For n=0 Landau level, the wavefunction with energy E=kz is

    ψR0(ky,kz;x)=(φ0(ξ)0)1Lei(yky+zkz).

    (27)

    For n>0 Landau level, the wavefunction with energy E=λEn(kz) is

    ψRnλ(ky,kz;x)=cnλ(φn(ξ)iFnλφn1(ξ))1Lei(yky+zkz),

    (28)

    where λ=±1, En(kz)=2neB+k2z, Fnλ(kz)=[kzλEn(kz)]/2neB, normalized coefficient |cnλ|2=1/(1+F2nλ), and φn(ξ)=φn(eBxky/eB) is the n-th order wavefunction of a harmonic oscillator.

    For n>0 Landau levels, the wavefunctions with energies E=±En(kz) correspond to fermions and antifermions, respectively. For the lowest Landau level, the wavefunction with energy E=kz>0 corresponds to fermions, whereas that with energy E=kz<0 corresponds to antifermions. The wavefunctions of all Landau levels are orthonormal and complete. For the lefthand fermion field, the eigenfunctions of Landau levels are the same as the righthand case, but with the sign of the eigenvalues changed.

    In this section, we second-quantize the righthand fermion field ΨR(x), such that it becomes an operator and satisfies following anticommutative relations,

    {ΨR(x),ΨR(x)}=δ(3)(xx),{ΨR(x),ΨR(x)}=0.

    (29)

    Because all eigenfunctions for the Hamiltonian of the righthand fermion field are orthonormal and complete, we decompose the righthand fermion field operator ΨR(x) by these eigenfunctions as

    ΨR(x)=ky,kz[θ(kz)a0(ky,kz)ψR0(ky,kz;x)+θ(kz)b0(ky,kz)ψR0(ky,kz;x)]+n,ky,kz[an(ky,kz)ψRn+(ky,kz;x)+bn(ky,kz)ψRn(ky,kz;x)].

    (30)

    In contrast to the general Fourier decomposition for the second quantization, we place two theta functions θ(±kz) in front of a0(ky,kz) and b0(ky,kz) in the decomposition, which is very important for the subsequent second quantization procedure. From Eq. (29), we obtain following anticommutative relations,

    {θ(kz)a0(ky,kz),θ(kz)a0(ky,kz)}=θ(kz)δkykyδkzkz{θ(kz)b0(ky,kz),θ(kz)b0(ky,kz)}=θ(kz)δkykyδkzkz{an(ky,kz),an(ky,kz)}=δnnδkykyδkzkz{bn(ky,kz),bn(ky,kz)}=δnnδkykyδkzkz.

    (31)

    The two theta functions θ(±kz) are always attached to the lowest Landau level operators, such as a0,a0,b0,b0. The Hamiltonian and total particle number of the righthand fermion system are

    H=d3xΨR(x)iσDΨR(x)=ky,kz[kzθ(kz)a0(ky,kz)a0(ky,kz)+(kz)θ(kz)b0(ky,kz)b0(ky,kz)]+n,ky,kzEn(kz)[an(ky,kz)×an(ky,kz)+bn(ky,kz)bn(ky,kz)],

    (32)

    N=d3xΨR(x)ΨR(x)=ky,kz[θ(kz)a0(ky,kz)a0(ky,kz)θ(kz)b0(ky,kz)b0(ky,kz)]×n,ky,kz[an(ky,kz)an(ky,kz)bn(ky,kz)bn(ky,kz)],

    (33)

    where we omitted the infinite vacuum term. This can be renormalized in the physics calculation and does not affect our result on the CME coefficient. Evidently, θ(kz)a0(ky,kz)a0(ky,kz) and an(ky,kz)an(ky,kz) are the occupied number operators of particles for different Landau levels, and θ(kz)b0(ky,kz)b0(ky,kz) and bn(ky,kz)bn(ky,kz) are occupied number operators of antiparticles for different Landau levels. Notably, without introduction of the two theta functions θ(±kz) in front of a0(ky,kz) and b0(ky,kz) in the decomposition of ΨR(x), the second quantization procedure could not be performed successfully. This is different from the massive case [21], where the authors determined the Landau levels and corresponding wavefunctions for the massive Dirac equation in a uniform magnetic field with chemical potential μ and chiral chemical potential μ5. The wave functions for the massive case are in a four-component Dirac form, and the θ function is not needed for the second quantization.

    Supposing that the system of the righthand ferimons within an external uniform magnetic field B=Bez is in equilibrium with a reservior with temperature T and chemical potential μR, then the density operator ˆρ for this righthand fermion system is

    ˆρ=1Zeβ(HμRN),

    (34)

    where β=1/T is the inverse temperature, and Z is the grand canonical partition function,

    Z=Treβ(HμRN).

    (35)

    The expectation value of an operator ˆF in the equilibrium state can be calculated as

    :ˆF:=Tr(ˆρˆF).

    (36)

    In the Appendix C, we calculated the expectation values of occupied number operators as

    :θ(kz)a0(ky,kz)a0(ky,kz):=θ(kz)eβ(kzμR)+1:θ(kz)b0(ky,kz)b0(ky,kz):=θ(kz)eβ(kz+μR)+1:an(ky,kz)an(ky,kz):=1eβ[En(kz)μR]+1:bn(ky,kz)bn(ky,kz):=1eβ[En(kz)+μR]+1.

    (37)

    The macroscopic electric current for the righthand fermion system is

    JR=:ΨR(x)σΨR(x):.

    (38)

    According to the rotational invariance of this system along the z-axis, JxR=JyR=0. In the following, we calculate JzR. Using Eq. (30), we see that

    JzR=:ΨR(x)σ3ΨR(x):=ky,kz(:θ(kz)a0(ky,kz)a0(ky,kz):+:θ(kz)b0(ky,kz)b0(ky,kz):)×ψR0(ky,kz;x)σ3ψR0(ky,kz;x)+n,ky,kz:an(ky,kz)an(ky,kz):ψRn+(ky,kz;x)σ3ψRn+(ky,kz;x)+n,ky,kz:bn(ky,kz)bn(ky,kz):ψRn(ky,kz;x)σ3ψRn(ky,kz;x)=ky,kz(θ(kz)eβ(kzμR)+1θ(kz)eβ(kz+μR)+1)ψR0(ky,kz;x)σ3ψR0(ky,kz;x)+n,ky,kz1eβ[En(kz)μR]+1ψRn+(ky,kz;x)σ3ψRn+(ky,kz;x)n,ky,kz1eβ[En(kz)+μR]+1ψRn(ky,kz;x)σ3ψRn(ky,kz;x).

    (39)

    First, we sum over ky for ψR0(ky,kz;x)σ3ψR0(ky,kz;x) and ψRnλ(ky,kz;x)σ3ψRnλ(ky,kz;x) in Eq. (39), the results are

    kyψR0(ky,kz;x)σ3ψR0(ky,kz;x)=1L2ky(φ0(ξ),0)(1001)(φ0(ξ)0)=12πLdky[φ0(eBxky/eB)]2=eB2πL,

    (40)

    and

    kyψRnλ(ky,kz;x)σ3ψRnλ(ky,kz;x)=12πLdkyc2nλ(kz)([φn(ξ)]22neB[φn1(ξ)]2[kz+λEn(kz)]2)=eB2πLc2nλ(kz)(12neB[kz+λEn(kz)]2)=eB2πLc2nλ(kz)[2c2nλ(kz)]=eB2πLλkzEn(kz).

    (41)

    Second, we sum over kz in the third equal sign of Eq. (39),

    JzR=kz(θ(kz)eβ(kzμR)+1θ(kz)eβ(kz+μR)+1)eB2πL+n,kz(1eβ[En(kz)μR]+1+1eβ[En(kz)+μR]+1)eB2πLkzEn(kz)=eB4π2dkz(θ(kz)eβ(kzμR)+1θ(kz)eβ(kz+μR)+1)+0=eB4π2μR.

    (42)

    Combining Eq. (42) and JxR=JyR=0 yields

    JR=eμR4π2B.

    (43)

    From the calculation above, we see that only the lowest Landau level contributes to Eq. (43). A similar calculation for the lefthand fermion system shows that

    JL=eμL4π2B.

    (44)

    We can also obtain Eq. (44) from Eq. (43) under space inversion: JRJL, μRμL, BB. If the system is composed of righthand and lefthand fermions, then the vector current JV and axial current JA are

    JV=JR+JL=eμ52π2B,

    (45)

    JA=JRJL=eμ2π2B,

    (46)

    where μ5=(μRμL)/2 is the chiral chemical potential and μ=(μR+μL)/2. Thus far, we derived the CME in the chiral fermion system by determining Landau levels. We emphasize that Eqs. (45), (46) are valid for any strength of magnetic field, in contrast to the weak magnetic field approximation through Wigner function approach in Sec. 2.

    We discuss the physical picture of the lowest Landau level. The wavefunction and energy of the lowest Landau level (n=0) for the righthand fermion field is

    ψR0(ky,kz;x)=(φ00)1Lei(yky+zkz),  E=kz.

    (47)

    Setting ky=0 in Eq. (47), we calculate the Hamiltonian, particle number, z-component of momentum, and z-component of the spin angular momentum of the righthand fermion system for the lowest Landau level as follows,

    H=kz[kzθ(kz)a0(0,kz)a0(0,kz)+(kz)θ(kz)b0(0,kz)b0(0,kz)],N=kz[θ(kz)a0(0,kz)a0(0,kz)+(1)θ(kz)b0(0,kz)b0(0,kz)],Pz=kz[kzθ(kz)a0(0,kz)a0(0,kz)+(kz)θ(kz)b0(0,kz)b0(0,kz)],Sz=kz[12θ(kz)a0(0,kz)a0(0,kz)+(12)θ(kz)b0(0,kz)b0(0,kz)],

    (48)

    where the definitions of Pz and Sz are

    Pz=id3xΨR(x)zΨR(x),Sz=12d3xΨR(x)σ3ΨR(x).

    (49)

    Thus, we have a picture for the lowest Landau level: The operator θ(kz)a0(0,kz) produces a particle with charge e, energy kz>0, z-component of momentum kz>0, and z-component of spin angular momentum +12 (helicity h=+1); The operator θ(kz)b0(0,kz) produces a particle with charge e, energy kz>0, z-component of momentum kz>0, and z-component of spin angular momentum 12 (helicity h=1). This picture indicates that all righthand fermions/antifermions move along the (+z)-axis, with righthand fermions spinning along the (+z)-axis and righthand antifermions spinning along the z-axis. If μR>0, which indicates that there are more righthand fermions than righthand anti-fermions, a net electric current will move along the (+z)-axis, which is referred to as the CME for the righthand fermion system.

    The analogous analysis can be applied to lefthand fermions. The picture of the lowest Landau level for a lefthand fermion is: All lefthand fermions/antifermions move along the (z)-axis, with left fermions spinning along the (+z)-axis and lefthand antifermions spinning along the (z)-axis. If μL>0, which indicates that there is more lefthand fermions than lefthand anti-fermions, a net electric current will move along the (z)-axis, which is referred to as the CME for the lefthand fermion system.

    Because the total electric current JV of the chiral fermion system is the summation of the electric current JR of the righthand fermion system and the electric current JL of the lefthand fermion system, whether JV moves along the (+z)-axis will only depend on the sign of (μRμL). Thus, the CME for the chiral fermion system is described microscopically.

    CME arises from the lowest Landau level both for the massive Dirac fermion system and the chiral fermion system. For the massive case, the physical picture of how the lowest Landau level contributes to CME is not extensively clear. When the Landau levels are determined for the chiral fermion system in a uniform magnetic field, by performing the second quantization for the chiral fermion field, expanding the field operator by an eigenfunction of Landau levels, and calculating the ensemble average of the vector current operator, we natrually obtain the equation for the CME. Notably, no approximations were made for the strength of magnetic field in the calculation. Further, we introduced two theta functions θ(±kz) in front of a0(ky,kz) and b0(ky,kz) in the decomposition of ΨR(x), which is crucial for the successful performance of the subsequent procedure of second quantization. When we carefully analyze the lowest Landau level, we find that all righthand (chirality is +1) fermions move along the positive z-direction, and all lefthand (chirality is -1) fermions move along the negative z-direction. Thus, the CME is described microscopically within this picture of the lowest Landau level.

    We are grateful to Hai-Cang Ren and Xin-Li Sheng for valuable discussions. R.-H. F. thanks for the hospitality of Institute of Frontier and Interdisciplinary Science at Shandong University (Qingdao) where he is currently visiting.

    The quadratic form for the equation of motion of the Wigner function W(x,p) is

    (K2i2σμν[Kμ,Kν])W=0.

    (A1)

    Using K2=p2142+ip and [Kμ,Kν]=ieFμν, Eq. (A1) becomes

    (p2142+ip12eFμνσμν)W=0.

    (A2)

    W and σμν satisfy W=γ0Wγ0 and σμν=γ0σμνγ0. Employing the Hermi conjugation and subsequently multiplying γ0 to both sides of Eq. (A2) yields

    (p2142ip)W12eFμνWσμν=0.

    (A3)

    Eq. (A2) minus Eq. (A3) yields the Vlasov equation for W,

    ipW14eFμν[σμν,W]=0.

    (A4)

    Eq. (A2) added to Eq. (A3) yields the off mass-shell equation for W,

    (p2142)W14eFμν{σμν,W}=0.

    (A5)

    To calculate [σμν,W] and {σμν,W} in Eqs. (A4) (A5), we list the following useful identities,

    [σμν,1]=0,[σμν,iγ5]=0,[σμν,γρ]=2igρ[μγν],[σμν,γ5γρ]=2igρ[μγ5γν],[σμν,σρσ]=2igμ[ρσσ]ν2igν[ρσσ]μ,

    (A6)

    {σμν,1}=2σμν,{σμν,iγ5}=ϵμνρσσρσ,{σμν,γρ}=2ϵμνρσγ5γσ,{σμν,γ5γρ}=2ϵμνρσγσ,{σμν,σρσ}=2gμ[ρgσ]ν+2iϵμνρσγ5.

    (A7)

    Then, all matrices appearing in Eqs. (A4) (A5) are the 16 independent Γ-matrices, whose coefficients must be zero. These coefficient equations are the Vlasov equations and the off mass-shell equations for F,P,Vμ,Aμ,Sμν. The Vlasov equations are

    pF=0,pP=0,pVμ=eFμνVν,pAμ=eFμνAν,pQμν=eFρ [μQν]ρ,

    (A8)

    and the off mass-shell equations are

    (p2142)F=12eFμνQμν,(p2142)P=12e˜FμνQμν,(p2142)Vμ=e˜FμνAν,(p2142)Aμ=e˜FμνVν,(p2142)Qμν=e(FμνF˜FμνP),

    (A9)

    where ˜Fμν=12εμνρσFρσ.

    We solve following eigenvalue equation in detail,

    iσDψR(x)=EψR(x),

    (B1)

    with D=(x,y+ieBx,z). Because the operator iσD is commutative with ˆpy=iy,ˆpz=iz, we can choose ψR as the commom eigenstate of iσD, ˆpy and ˆpz as follows

    ψR(x,y,z)=(ϕ1(x)ϕ2(x))1Lei(yky+zkz),

    (B2)

    where L is the length of the system in y- and z- directions. The explicit form of σD is

    σD=(zx+iy+eBxxiyeBxz).

    (B3)

    Inserting Eq. (B2) (B3) into Eq. (B1), we obtain the group of differential equations for ϕ1(x) and ϕ2(x) as

    i(kzE)ϕ1+(x+kyeBx)ϕ2=0,

    (B4)

    (xky+eBx)ϕ1i(kz+E)ϕ2=0.

    (B5)

    From Eq. (B5), we can express ϕ2 by ϕ1, then Eq. (B4) becomes

    2xϕ1+(E2+eBk2ze2B2(xkyeB)2)ϕ1=0,

    (B6)

    which is a typical harmonic oscillator equation. Defining a dimensionless variable ξ=eB(xky/eB), and ϕ1(x)=φ(ξ), then (B6) becomes

    d2φdξ2+(E2k2zeB+1ξ2)φ=0.

    (B7)

    With the boundary condition φ0 as ξ±, we must set

    E2k2zeB+1=2n+1,

    (B8)

    with n=0,1,2,. Thus, energy E can only assume the following discrete values,

    E=±En(kz)±2neB+k2z,

    (B9)

    where we define En(kz)=2neB+k2z. The corresponding normalized solution for equation (B6) is

    ϕ1(x)=φn(ξ)=Nneξ2/2Hn(ξ),

    (B10)

    where Nn=(eB)14π14(2nn!)12, and Hn(ξ)=(1)neξ2dndξneξ2. For energy E=λEn(kz) (λ=±1), we can obtain ϕ2 as

    ϕ2(x)=eB(ξ+ξ)φn(ξ)i(kz+E)=i[kzλEn(kz)]2neBφn1(ξ),

    (B11)

    where we used (ξ+ξ)φn(ξ)=2nφn1(ξ). Defining Fnλ(kz)= [kzλEn(kz)]/2neB, the eigenfunction with eigenvalue E=λEn(kz) becomes

    ψRnλ(ky,kz;x)=(φn(ξ)iFnλ(kz)φn1(ξ))1Lei(yky+zkz).

    (B12)

    This is very subtle when n=0 in Eq. (B11). When n=0,E=kz, the first equal sign of Eq. (B11) indicates ϕ2=0 due to (ξ+ξ)φ0(ξ)=0. Then, the corresponding eigenfunction becomes

    ψR0(ky,kz;x)=(φ0(ξ)0)1Lei(yky+zkz).

    (B13)

    When n=0,E=kz, the denominator of the first equal sign of Eq. (B11) becomes zero, in which case we must directly deal with Eqs. (B4) (B5). In this case, Eqs. (B4) (B5) become

    2ikzϕ1+(x+kyeBx)ϕ2=0,

    (B14)

    (xky+eBx)ϕ1=0.

    (B15)

    Eq. (B15) gives ϕ1(x)exp[12eBx2+xky], then Eq. (B14) becomes

    2ikzexp(12eBx2+xky)+(x+kyeBx)ϕ2=0.

    (B16)

    When x±, Eq. (B16) tends to

    (xeBx)ϕ2=0,

    (B17)

    whose solution is ϕ2exp(12eBx2), which is divergent as x±. Thus, there is no physical solution when n=0,E=kz.

    Thus far, we obtain the eigenfunctions and eigenvalues of the Hamiltonian of the righthand fermion field as follows:

    For n=0 Landau level, the wavefunction with energy E=kz is

    ψR0(ky,kz;x)=(φ00)1Lei(yky+zkz).

    (B18)

    For n>0 Landau level, the wavefunction with energy E=λEn(kz) is

    ψRnλ(ky,kz;x)=cnλ(φniFnλφn1)1Lei(yky+zkz),

    (B19)

    where λ=±1, En(kz)=2neB+k2z, Fnλ(kz)=[kzλEn(kz)]/2neB, |cnλ|2=1/(1+F2nλ).

    We calculate the expectation values of particle number operators. From the expression of the Hamiltonian and the total particle number operator in Eqs. (32, 33), we easily obtain following commutative relations,

    [N,θ(kz)a0(ky,kz)]=θ(kz)a0(ky,kz)[N,θ(kz)b0(ky,kz)]=θ(kz)b0(ky,kz)[N,an(ky,kz)]=an(ky,kz)[N,bn(ky,kz)]=bn(ky,kz),

    (C1)

    [H,θ(kz)a0(ky,kz)]=kzθ(kz)a0(ky,kz)[H,θ(kz)b0(ky,kz)]=(kz)θ(kz)b0(ky,kz)[H,an(ky,kz)]=En(kz)an(ky,kz)[H,bn(ky,kz)]=En(kz)bn(ky,kz),

    (C2)

    where we employ [AB,C]=A{B,C}{A,C}B. Defining

    θ(kz)a0(ky,kz;β)=eβ(HμRN)θ(kz)a0(ky,kz)eβ(HμRN)θ(kz)b0(ky,kz;β)=eβ(HμRN)θ(kz)b0(ky,kz)eβ(HμRN)

    an(ky,kz;β)=eβ(HμRN)an(ky,kz)eβ(HμRN)bn(ky,kz;β)=eβ(HμRN)bn(ky,kz)eβ(HμRN).

    (C3)

    For θ(kz)a0(ky,kz;β), we obtain

    β[θ(kz)a0(ky,kz;β)]=[HμRN,θ(kz)a0(ky,kz;β)]=eβ(HμRN)[HμRN,θ(kz)a0(ky,kz)]eβ(HμRN)=eβ(HμRN)[(kzμR)θ(kz)a0(ky,kz)]eβ(HμRN)=(kzμR)[θ(kz)a0(ky,kz;β)],

    (C4)

    with the boundary condition θ(kz)a0(ky,kz;0)=θ(kz)a0(ky,kz), which implies

    θ(kz)a0(ky,kz;β)=θ(kz)a0(ky,kz)eβ(kzμR).

    (C5)

    Similarly, we obtain

    θ(kz)b0(ky,kz;β)=θ(kz)b0(ky,kz)eβ(kz+μR)an(ky,kz;β)=an(ky,kz)eβ[En(kz)μR]bn(ky,kz;β)=bn(ky,kz)eβ[En(kz)+μR].

    (C6)

    We calculate the expectation value of :θ(kz)a0(ky,kz)a0(ky,kz):. We see that

    :θ(kz)a0(ky,kz)a0(ky,kz):=Tr[ρθ(kz)a0(ky,kz)a0(ky,kz)]=1ZTr(θ(kz)a0(ky,kz;β)eβ(HμRN)a0(ky,kz))=1ZTr(θ(kz)a0(ky,kz)a0(ky,kz;β)eβ(HμRN))=:θ(kz)a0(ky,kz)a0(ky,kz;β):=:θ(kz)a0(ky,kz)a0(ky,kz):eβ(kzμR)=θ(kz)eβ(kzμR):θ(kz)a0(ky,kz)a0(ky,kz):eβ(kzμR),

    (C7)

    thus, we obtain

    θ(kz)a0(ky,kz)a0(ky,kz)=θ(kz)eβ(kzμR)+1.

    (C8)

    Similar calculations obtain

    θ(kz)b0(ky,kz)b0(ky,kz)=θ(kz)eβ(kz+μR)+1an(ky,kz)an(ky,kz)=1eβ[En(kz)μR]+1bn(ky,kz)bn(ky,kz)=1eβ[En(kz)+μR]+1.

    (C9)
    [1] Nambu Y, Jona-Lasinio G. Phys. Rev., 1961, 122: 345
    [2] Nambu Y, Jona-Lasinio G. Phys. Rev., 1961, 124: 246
    [3] Klevansky S. Rev. Mod. Phys., 1992, 64: 649
    [4] Klimt S, Lutz M F, Vogl U, Weise W. Nucl. Phys. A, 1990, 516: 429
    [5] Vogl U, Lutz M F, Klimt S, Weise W. Nucl. Phys. A, 1990, 516: 469
    [6] Vogl U, Weise W. Prog. Part. Nucl. Phys., 1991, 27: 195
    [7] Shifman M A, Voloshin M. Sov. J. Nucl. Phys., 1987, 45: 292
    [8] Politzer H D, Wise M B. Phys. Lett. B, 1988, 208: 504
    [9] Isgur N, Wise M B. Phys. Lett. B, 1989, 232: 113
    [10] Eichten E, Hill B. R. Phys. Lett. B, 1990, 234: 511
    [11] Georgi H. Phys. Lett. B, 1990, 240: 447
    [12] Grinstein B. Nucl. Phys. B, 1990, 339: 253
    [13] Falk A F, Georgi H, Grinstein B, Wise M B. Nucl. Phys. B, 1990, 343: 1
    [14] Mannel T, Roberts W, Ryzak Z. Nucl. Phys. B, 1992, 368: 204
    [15] Ebert D, Feldmann T, Friedrich R, Reinhardt H. Nucl. Phys. B, 1995, 434: 619
    [16] GUO X Y, CHEN X L, DENG W Z. Chin. Phys. C, 2013, 37: 033102
    [17] DENG H B, CHEN X L, DENG W Z. Chin. Phys. C, 2014, 38: 013103
    [18] Amendolia S, Badelek B, Batignani G et al. Phys. Lett. B, 1984, 146: 116
    [19] Amendolia S, Batignani G, Beck G et al. Phys. Lett. B, 1986, 178: 435
    [20] Amendolia S et al. (NA7 collaboration). Nucl. Phys. B, 1986, 277: 168
    [21] Bernard V, Meissner U G. Phys. Rev. Lett., 1988, 61: 2296
    [22] Lutz M F, Weise W. Nucl. Phys. A, 1990, 518: 156
    [23] Schulze H. J. Phys. G, 1994, 20: 531
    [24] Roberts C D. Nucl. Phys. A, 1996, 605: 475
    [25] Chung P, Coester F, Polyzou W. Phys. Lett. B, 1988, 205: 545
    [26] Ito H, Buck W, Gross F. Phys. Lett. B, 1990, 248: 28
    [27] Woloshyn R, Kobos A. Phys. Rev. D, 1986, 33: 222
    [28] Nemoto Y et al. (RBC collaboration). Nucl. Phys. Proc. Suppl., 2004, 129: 299
    [29] Efremov A, Radyushkin A. Phys. Lett. B, 1980, 94: 245
    [30] Lepage G P, Brodsky S J. Phys. Lett. B, 1979, 87: 359
    [31] Rothstein I Z. Phys. Rev. D, 2004, 70: 054024
    [32] Bakulev A, Passek-Kumericki K, Schroers W, Stefanis N. Phys. Rev. D, 2004, 70: 033014
    [33] Bakker B L, Choi H M, Ji C R. Phys. Rev. D, 2002, 65: 116001
    [34] Cardarelli F, Grach I, Narodetsky I, Salme G, Simula S. Phys. Lett. B, 1995, 349: 393
    [35] de Melo J, Frederico T. Phys. Rev. C, 1997, 55: 2043
    [36] Bhagwat M, Maris P. Phys. Rev. C, 2008, 77: 025203
    [37] Lee F X, Moerschbacher S, Wilcox W. Phys. Rev. D, 2008, 78: 094502
    [38] Hedditch J, Kamleh W, Lasscock B et al. Phys. Rev. D, 2007, 75: 094504
    [39] Rubinstein H, Solomon S, Wittlich T. Nucl. Phys. B, 1995, 457: 577
    [40] Badalian A, Simonov Y. Phys. Rev. D, 2013, 87: 074012
    [41] Aliev T, Ozpineci A, Savci M. Phys. Lett. B, 2009, 678: 470
    [42] Gomez-Rocha M, Schweiger W. Phys. Rev. D, 2012, 86: 053010
    [43] Brodsky S J, Hiller J R. Phys. Rev. D, 1992, 46: 2141
    [44] Arnold R, Carlson C E, Gross F. Phys. Rev. C, 1981, 23: 363
  • [1] Nambu Y, Jona-Lasinio G. Phys. Rev., 1961, 122: 345
    [2] Nambu Y, Jona-Lasinio G. Phys. Rev., 1961, 124: 246
    [3] Klevansky S. Rev. Mod. Phys., 1992, 64: 649
    [4] Klimt S, Lutz M F, Vogl U, Weise W. Nucl. Phys. A, 1990, 516: 429
    [5] Vogl U, Lutz M F, Klimt S, Weise W. Nucl. Phys. A, 1990, 516: 469
    [6] Vogl U, Weise W. Prog. Part. Nucl. Phys., 1991, 27: 195
    [7] Shifman M A, Voloshin M. Sov. J. Nucl. Phys., 1987, 45: 292
    [8] Politzer H D, Wise M B. Phys. Lett. B, 1988, 208: 504
    [9] Isgur N, Wise M B. Phys. Lett. B, 1989, 232: 113
    [10] Eichten E, Hill B. R. Phys. Lett. B, 1990, 234: 511
    [11] Georgi H. Phys. Lett. B, 1990, 240: 447
    [12] Grinstein B. Nucl. Phys. B, 1990, 339: 253
    [13] Falk A F, Georgi H, Grinstein B, Wise M B. Nucl. Phys. B, 1990, 343: 1
    [14] Mannel T, Roberts W, Ryzak Z. Nucl. Phys. B, 1992, 368: 204
    [15] Ebert D, Feldmann T, Friedrich R, Reinhardt H. Nucl. Phys. B, 1995, 434: 619
    [16] GUO X Y, CHEN X L, DENG W Z. Chin. Phys. C, 2013, 37: 033102
    [17] DENG H B, CHEN X L, DENG W Z. Chin. Phys. C, 2014, 38: 013103
    [18] Amendolia S, Badelek B, Batignani G et al. Phys. Lett. B, 1984, 146: 116
    [19] Amendolia S, Batignani G, Beck G et al. Phys. Lett. B, 1986, 178: 435
    [20] Amendolia S et al. (NA7 collaboration). Nucl. Phys. B, 1986, 277: 168
    [21] Bernard V, Meissner U G. Phys. Rev. Lett., 1988, 61: 2296
    [22] Lutz M F, Weise W. Nucl. Phys. A, 1990, 518: 156
    [23] Schulze H. J. Phys. G, 1994, 20: 531
    [24] Roberts C D. Nucl. Phys. A, 1996, 605: 475
    [25] Chung P, Coester F, Polyzou W. Phys. Lett. B, 1988, 205: 545
    [26] Ito H, Buck W, Gross F. Phys. Lett. B, 1990, 248: 28
    [27] Woloshyn R, Kobos A. Phys. Rev. D, 1986, 33: 222
    [28] Nemoto Y et al. (RBC collaboration). Nucl. Phys. Proc. Suppl., 2004, 129: 299
    [29] Efremov A, Radyushkin A. Phys. Lett. B, 1980, 94: 245
    [30] Lepage G P, Brodsky S J. Phys. Lett. B, 1979, 87: 359
    [31] Rothstein I Z. Phys. Rev. D, 2004, 70: 054024
    [32] Bakulev A, Passek-Kumericki K, Schroers W, Stefanis N. Phys. Rev. D, 2004, 70: 033014
    [33] Bakker B L, Choi H M, Ji C R. Phys. Rev. D, 2002, 65: 116001
    [34] Cardarelli F, Grach I, Narodetsky I, Salme G, Simula S. Phys. Lett. B, 1995, 349: 393
    [35] de Melo J, Frederico T. Phys. Rev. C, 1997, 55: 2043
    [36] Bhagwat M, Maris P. Phys. Rev. C, 2008, 77: 025203
    [37] Lee F X, Moerschbacher S, Wilcox W. Phys. Rev. D, 2008, 78: 094502
    [38] Hedditch J, Kamleh W, Lasscock B et al. Phys. Rev. D, 2007, 75: 094504
    [39] Rubinstein H, Solomon S, Wittlich T. Nucl. Phys. B, 1995, 457: 577
    [40] Badalian A, Simonov Y. Phys. Rev. D, 2013, 87: 074012
    [41] Aliev T, Ozpineci A, Savci M. Phys. Lett. B, 2009, 678: 470
    [42] Gomez-Rocha M, Schweiger W. Phys. Rev. D, 2012, 86: 053010
    [43] Brodsky S J, Hiller J R. Phys. Rev. D, 1992, 46: 2141
    [44] Arnold R, Carlson C E, Gross F. Phys. Rev. C, 1981, 23: 363
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Get Citation
LUAN Yi-Long, CHEN Xiao-Lin and DENG Wei-Zhen. Meson electro-magnetic form factors in an extended Nambu-Jona-Lasinio model including heavy quark flavors[J]. Chinese Physics C, 2015, 39(11): 113103. doi: 10.1088/1674-1137/39/11/113103
LUAN Yi-Long, CHEN Xiao-Lin and DENG Wei-Zhen. Meson electro-magnetic form factors in an extended Nambu-Jona-Lasinio model including heavy quark flavors[J]. Chinese Physics C, 2015, 39(11): 113103.  doi: 10.1088/1674-1137/39/11/113103 shu
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Meson electro-magnetic form factors in an extended Nambu-Jona-Lasinio model including heavy quark flavors

    Corresponding author: DENG Wei-Zhen,
  • 1. School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China

Abstract: Based on an extended NJL model including heavy quark flavors, we calculate the form factors of pseudo-scalar and vector mesons. After taking into account the vector-meson-dominance effect, which introduces a form factor correction to the quark vector coupling vertices, the form factors and electric radii of π+ and K+ pseudo-scalar mesons in the light flavor sector fit the experimental data well. The magnetic moments of the light vector mesons ρ+ and K*+ are comparable with other theoretical calculations. The form factors in the light-heavy flavor sector are presented to compare with future experiments or other theoretical calculations.

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