Chirality imbalance and chiral magnetic effect under a parallel electromagnetic field
aa r X i v : . [ h e p - ph ] J a n Chirality imbalance and chiral magnetic effect undera parallel electromagnetic field
Hayato Aoi and Katsuhiko Suzuki Department of Physics, Tokyo University of Science,Kagurazaka 1-3, Shinjuku, Tokyo 162-8601, Japan
Abstract
We study the time evolution of the chirality imbalance n and the chiral magneticeffect (CME) under the external parallel electromagnetic fields without assuming theartificial chiral asymmetric source. We adopt the time-dependent Sauter-type electricand constant magnetic field, and obtain analytical solutions of the Dirac equationfor a massive fermion. We use the point-split regularization to calculate the vacuumcontribution in the gauge invariant way. As a result, we find that n and CME currentincrease substantially as the electric field increases, and stay finite after the electricfield is switched off. The chirality imbalance and CME current are shown to consistof a dominant contribution, which is essentially proportional to relativistic velocity,and a small oscillating part. We find a simple analytical relation between n and thefermion pair-production rate from the vacuum. We also discuss dynamical origin ofthe chirality imbalance in detail. katsu [email protected] Introduction
Recently, roles of the chiral anomaly have attracted considerable theoretical and ex-perimental interests in various subjects of physics. The chiral (Adler-Bell-Jackiw)anomaly is violation of the (partial) axial-vector current conservation due to quan-tum effects[1, 2], and causes the CP-violating processes observed experimentally. Forthe last decade, macroscopic manifestations of the chiral anomaly are discussed in thecontext of hydrodynamic and transport phenomena in systems with chiral fermions,e.g. the quark-gluon plasma or the Dirac / Weyl semi-metals[3, 4, 5]. One of the im-portant effects induced by the anomaly is chiral magnetic effect (CME), which is thegeneration of ”non-dissipative” electric current along the direction of the magneticfield[6, 7, 8]; J = µ π B (1)where µ is the chiral chemical potential. The chiral chemical potential characterizes anasymmetry of the chirality of the system, and is conjugate to the chirality imbalance ofthe fermions, n , which is a difference of right-handed and left-handed fermion numberdensities, n ≡ n R − n L ≡ h ¯ ψγ γ ψ i .In the quark-gluon plasma produced in the heavy ion collisions, the interactionwith the non-trivial gluonic field would change quark chiralities and thus produce thechirality imbalance between right- and left-handed quarks[4]. With the strong magneticfield, eB ∼ m π , created by the heavy ion-collision, CME may produce asymmetry of thecharged particle distributions which can be measured experimentally[9]. On the otherhand, CME is an important topic in the condensed matter system[5], where the masslessDirac mode has been realized in the Dirac/Weyl semimetals[10, 11]. Experimentalresult for observing CME in such a system is reported in ref. [11].For various applications in QCD/condensed matter, the existence of the chiralityimbalance n and the chiral chemical potential µ is a priori assumed to study spe-cific transport phenomena. However, appearance of the initial chirality imbalance isstill under debate. For example, in the quark-gluon plasma, metastable local CP-violating domains may be generated by transitions of the non-perturbative gluonicconfigurations[7, 12, 13]. In ref. [11] for the semimetal system with the electromagnetic2eld, µ is estimated as µ = ~ v F (cid:18) e E · B τ (cid:19) , where τ is the relaxation time of thechirality imbalance. In our opinion, it is important to calculate the chirality imbalanceand the chiral magnetic effect within the field theoretical method without introducingadditional assumptions.On the other hand, it is also necessary to clarify use of eq. (1) in equilibrium.Although the CME formula eq. (1) is used for various applications, it has to be in-terpreted with care. It is pointed out that such a current is forbidden in equilibriumsystem[14, 15]. There are also some cautions from theoretical calculations[16, 17].It seems that the introduction of the chiral chemical potential implicitly assumes asystem out of equilibrium[18]. In order to clarify this issue, it is crucial to calculatetime-evolution of the chirality imbalance and the CME current within a specific modeland compare their characteristic time scales.In addition, the CME current for a massive fermion is to be studied carefully. It iswell-known that the anomaly relation receives a contribution from the mass-dependentterm: ∂ µ Z d x h ¯ ψγ µ γ ψ i = 2 im Z d x h ¯ ψγ ψ i + 2 απ Z d x E · B , (2)with α = e / π being the fine structure constant. To estimate the contribution fromthe mass, one should calculate a vacuum expectation value of the pseudoscalar density,which is also time-dependent. In particular, it is of interest to understand a relationbetween CME and the spontaneous breakdown of chiral symmetry. In ref. [19], theCME current is suppressed in the insulator phase, which may correspond to the chiralsymmetry breaking phase.For these purposes, we study time evolution of the chirality imbalance n and thechiral magnetic effect in the vacuum under the electromagnetic field solving the Diracequation analytically without initial chiral chemical potential[20]. We consider thevacuum state (zero temperature and zero fermion chemical potential) with externalparallel electromagnetic fields, which provide the chirality imbalance of the fermionnumber density due to the chiral anomaly. We adopt the time-dependent half-pulseelectric field (Sauter-type), and constant magnetic field in order to solve the Dirac equa-tion for a massive fermion analytically. To calculate the infinite vacuum contribution3n the gauge invariant way, we use the point-split regularization[21, 22] and calculatevacuum expectation values of the various bilinear fermion operators including n andCME. In addition, we expect production of fermion-antifermion pairs from the vacuumunder the electric field by the Schwinger mechanism[23, 24]. We systematically studyrelations between n , CME current and the pair-creation rate using the Bogoliubovtransformation. Our results are to be compared with the previous works obtained withthe Schwinger mechanism with constant electromagnetic field[18], the Wigner functionmethod with collinear electromagnetic fields[26], the Wigner function method withthe chiral chemical potential[27], cylindrical Dirac equation with the chiral chemicalpotential[28].This paper is organized as follows. In Sec. 2, we show analytical solutions of theDirac equation with the parallel electromagnetic fields. Using them we perform thecanonical quantization and define the vacuum state at t → −∞ in Sec. 3. We introducethe point-split regularization in Sec. 4 to calculate vacuum expectation values of thefermion operators in gauge invariant way. We present our numerical results for thetime evolution of the vacuum expectation values of the chirality imbalance and CMEin Sec. 5. In Sec. 6 we also discuss relations between the chirality imbalance and thefermion pair-production rate, and give a simple formula for CME current. Using them,we show how the chirality imbalance is dynamically generated in this model. Finally,Sec. 7 is devoted to the summary and discussion. We need an analytical solution of the Dirac equation under the constant magneticand the time-dependent electric field, which plays a key role in our work. The Diracequation for a fermion field ψ ( x ) with the mass m under an external electromagneticpotential, A µ ( x ), is given by [ i / D − m ] ψ ( x ) = 0 (3)4here we introduce the covariant derivative D µ = ∂ µ + ieA µ ( x ). Squared Dirac equationis given by / D Φ( x ) = − m Φ( x ) (4)Hereafter, we concentrate on finding solutions of the squared Dirac equation Φ( x ), fromwhich we can obtain ψ ( x ) by a suitable projection, ψ ( x ) = ( i / D + m )Φ( x ) . (5)We consider specific forms of the external electromagnetic field in this work toobtain analytical solutions; B = (0 , , − B ) , (6) E = (0 , , E/ cosh ( t/τ )) , (7)where parameters B and E are non-zero real constants, and τ >
0. The magnetic fieldis time-independent and uniform along the z -direction. On the other hand, the electricfield is spatially homogeneous but time-dependent with a pulse structure, which isknown as Sauter-type electric field[29]. The corresponding electromagnetic potential is A µ = (0 , , Bx, Eτ (tanh( t/τ ) + 1)) . (8)Note that the vector potential is finite even at t → ±∞ . If we adopted the constantelectric field[24, 22], the vector potential would diverge at t → ±∞ , A = E t .To understand roles of the electromagnetic field, it is convenient to introduce theso-called ”magnetic helicity” density[30] as h ( t ) ≡ V Z d x A · B . (9)Although the magnetic helicity is not gauge invariant in general, it is useful when wediscuss the topological structure of the gauge field. With eq. (8), the magnetic helicitydensity in our case is calculated as h ( t ) = − BEτ (tanh( t/τ ) + 1) . (10)At the initial state, t → −∞ , both electric field and the magnetic helicity h ( t ) arezero, and they increase as t increases. In the final state, t → ∞ , E vanishes rapidly,5hile the magnetic helicity h ( t ) is kept finite. This peculiar behavior of the magnetichelicity is due to the Sauter electric field, and appropriate to discuss the production ofthe chirality imbalance, as we will show later.With the chiral representation for the gamma matrices, the Dirac operator / D andits squared form / D are given by i / D = − ˆ c − i ˆ a − i ˆ a † ˆ c + ˆ c + − i ˆ a i ˆ a † − ˆ c − / D = ˆ c − ˆ c + + ˆ a ˆ a † c + ˆ c − + ˆ a † ˆ a c + ˆ c − + ˆ a ˆ a †
00 0 0 ˆ c − ˆ c + + ˆ a † ˆ a where we have defined the following operators,ˆ c + = ( − i∂ z + eEτ (tanh( t/τ ) + 1)) + i∂ t ˆ c − = ( − i∂ z + eEτ (tanh( t/τ ) + 1)) − i∂ t ˆ a = ( − i∂ y + eBx ) + ∂ x ˆ a † = ( − i∂ y + eBx ) − ∂ x . Because / D commute both ∂ y and ∂ z , the solution of the squared Dirac equation, Φ,can be written as a separable form, Φ( t, x ) = exp( ip y y + ip z z ) φ ( t, x ), with momentaof y and z directions being constants. For Φ( t, x ), we explicitly introduce the fourcomponent form as φ ( t, x ) = φ ( t, x ) φ ( t, x ) φ ( t, x ) φ ( t, x ) . (11)We then obtain a set of equations for φ i ( t, x ) ( i = 1 , , ,
4) as follows,[ˆ c − ˆ c + + ˆ a ˆ a † + m ] φ ( t, x ) = 0 (12)[ˆ c + ˆ c − + ˆ a † ˆ a + m ] φ ( t, x ) = 0 (13)[ˆ c + ˆ c − + ˆ a ˆ a † + m ] φ ( t, x ) = 0 (14)[ˆ c − ˆ c + + ˆ a † ˆ a + m ] φ ( t, x ) = 0 . (15)6ote that the operators ˆ c + , ˆ c − include only t and ∂ t variables, whereas ˆ a, ˆ a † containonly x and ∂ x . Hence, these equations can be solved asΦ( x ) = exp( ip y y + ip z z ) f ( t ) g ( x ) f ( t ) g ( x ) f ( t ) g ( x ) f ( t ) g ( x ) (16)with eigenfunctions, f i ( t ) , g i ( x ) ( i = 1 , a ˆ a † g ( x ) = κg ( x ) (17)ˆ a † ˆ ag ( x ) = κg ( x ) (18)ˆ c − ˆ c + f ( t ) = − ( κ + m ) f ( t ) (19)ˆ c + ˆ c − f ( t ) = − ( κ + m ) f ( t ) . (20)We note that the eigenvalue κ is real and positive-semidefinite because the operatorsˆ a † ˆ a, ˆ a ˆ a † are Hermittian. x -dependent part The eigenfunction and the eignevalue of equations (17) and (18) are easily obtainedwith the standard technique for the harmonic oscillator.We find a solution with the normalized Hermite polynomial H n ( n = 0 , , , · · · ) g ( x ) = g n − ,p y ( x ) (21) g ( x ) = g n,p y ( x ) (22)with eigenvalues κ = 2 eBn and eigenfunctions g n,p y ( x ); g n,p y ( x ) = 1 √ n n ! ( | eB | π ) / H n ( η ) exp( − η /
2) (23) η ≡ p | eB | ( p y + | eB | x ) . where n denotes the Landau level. When n = 0, the normalizable solution of g ( x )does not exist, thus we define g − ,p y (˜ x ) = 0.7he eigenfunctions satisfy the orthonormal condition as Z dx g n,p y ( x ) g m,p y ( x ) = δ mn . (24)Moreover, the completeness identity also holds ∞ X n =0 g n,p y ( x ) g n,p y ( x ′ ) = δ ( x − x ′ ) . (25)Additionally, integration over p y also gives a relation Z dp y g n,p y ( x ) g m,p y ( x ) = | eB | δ mn (26)which guarantees the orthogonal condition for y in eq. (16). t -dependent part Next, we will solve equations for the time-dependent part. The operators ˆ c − ˆ c + , ˆ c + ˆ c − in eq. (19) and eq. (20) are written explicitly asˆ c + ˆ c − = ∂ t + ( p z + eEτ (tanh( t/τ ) + 1)) + ie E cosh ( t/τ ) (27)ˆ c − ˆ c + = ∂ t + ( p z + eEτ (tanh( t/τ ) + 1)) − ie E cosh ( t/τ ) . (28)which reduce to the hypergeometric differential equation for f ( x ). We obtain theeigenfunctions for f ( x ) in (19) and f ( x ) in (20) as follows˜ φ (+) n,p z ( t ) ≡ s ω (0) + p z ω (0) u − iτω (0)2 (1 − u ) iτω (1)2 F (cid:18) a, bc ; u ( t ) (cid:19) (29)˜ φ ( − ) n,p z ( t ) ≡ s ω (0) − p z ω (0) u iτω (0)2 (1 − u ) − iτω (1)2 × F (cid:18) − a, − b − c ; u ( t ) (cid:19) (30)where F (cid:0) a,bc ; u (cid:1) are Gauss’s hypergeometric function. The parameters a, b, c are givenby a = 1 − iτ ω n,p z (0)2 + iτ ω n,p z (1)2 + ieEτ (31a) b = − iτ ω n,p z (0)2 + iτ ω n,p z (1)2 − ieEτ (31b) c = 1 − iτ ω n,p z (0) , (31c)8here ω n,p z ( u ) = ( p z + 2 eEτ u ) + 2 | eB | n + m (32) u ( t ) = 12 (tanh( t/τ ) + 1) . (33)We find a simple relation, | ˜ φ (+) n,p z ( t ) | + | ˜ φ ( − ) n,p z ( t ) | = 1 , (34)which holds independent of t , and is useful for further calculations. We then obtain solutions of the squared Dirac equation, Φ n,p y ,p z ( x ), as follows:Φ n,p y ,p z = exp( ip y y + ip z z ) × g n − ,p y ( x ) { N (+)1 ˜ φ ∗ ( − ) n,p z ( t ) + N ( − )1 ˜ φ ∗ (+) n,p z ( t ) } g n,p y ( x ) { N ( − )2 ˜ φ ( − ) n,p z ( t ) + N (+)2 ˜ φ (+) n,p z ( t ) } g n − ,p y ( x ) { N ( − )3 ˜ φ ( − ) n,p z ( t ) + N (+)3 ˜ φ (+) n,p z ( t ) } g n,p y ( x ) { N (+)4 ˜ φ ∗ ( − ) n,p z ( t ) + N ( − )4 ˜ φ ∗ (+) n,p z ( t ) } (35)where N ( ± ) i are normalization constants. To construct the solutions of the Dirac equa-tion, we properly choose solutions in eq. (35), and extract the right/left-handed solu-tions by performing the suitable projection. Here, we choose 4-independent solutionsproportional to N ( ± )1 , N ( ± )4 in eq. (35), which satisfy the orthogonal and completenessrelations, as we will show later.First, we obtain the ”right-handed” solutions, operating iγ µ D µ + m to the 1st rowin eq. (35). ψ (+ , ˜ R ) p = N (+)1 exp( ip y y + ip z z ) × cos θ n · g n − ,p y ( x ) · ˜ φ ∗ ( − ) n,p z ( t )0 g n − ,p y ( x ) · ˜ φ (+) n,p z ( t ) i sin θ n · g n,p y ( x ) · ˜ φ ∗ ( − ) n,p z ( t ) (36) ψ ( − , ˜ R ) p = N ( − )1 exp( ip y y + ip z z ) × cos θ n · g n − ,p y ( x ) · ˜ φ ∗ (+) n,p z ( t )0 − g n − ,p y ( x ) · ˜ φ ( − ) n,p z ( t ) i sin θ n · g n,p y ( x ) · ˜ φ ∗ (+) n,p z ( t ) (37)9here θ n is defined by θ n = arctan( p | eB | nm ) . (38)Hereafter, we use the shorthand notation p = ( n, p y , p z ) for simplicity.Similarly, we obtain the ”left-handed” solutions operating iγ µ D µ + m to the 4throw in eq. (35). ψ (+ , ˜ L ) p = N (+)4 exp( ip y y + ip z z ) × i sin θ n · g n − ,p y ( x ) · ˜ φ ∗ ( − ) n,p z ( t ) g n,p y ( x ) · ˜ φ (+) n,p z ( t )0cos θ n · g n,p y ( x ) · ˜ φ ∗ ( − ) n,p z ( t ) (39) ψ ( − , ˜ L ) p = N ( − )4 exp( ip y y + ip z z ) × i sin θ n × g n − ,p y ( x ) · ˜ φ ∗ (+) n,p z ( t ) − g n,p y ( x ) · ˜ φ ( − ) n,p z ( t )0cos θ n · g n,p y ( x ) · ˜ φ ∗ (+) n,p z ( t ) (40)In massless limit, m →
0, the solutions of eq. (36) are exact eigenspinors of the chiralityoperator γ with the eigenvalue +1, while eq. (39) is the chirality eigenstate with theeigenvalue −
1. Note that ψ ( ± , ˜ R )0 ,p y ,p z = 0 because of g − ,p y = 0 and sin θ = 0.These solutions of the Dirac equation form the complete orthonormal basis. Bychoosing the normalization constants, N (+)1 = N ( − )1 = N (+)4 = N ( − )4 = 1, the orthonor-mal relations are given by Z d x [ ψ ( u ′ ,s ′ ) p ′ ( x )] † [ ψ ( u,s ) p ( x )= (2 π ) δ uu ′ δ ss ′ δ nn ′ δ ( p y − p ′ y ) δ ( p z − p ′ z );which holds except for ψ ( ± , ˜ R )0 ,p y ,p z . Moreover, one can show the completeness relation foreqs. (36,37,39,40) X p X s = ˜ R, ˜ L X u = ± [ ψ ( u,s ) p ( t, x )] α [ ψ † ( u,s ) p ( t, x ′ )] β = (2 π ) δ αβ δ (3) ( x − x ′ ) , which guarantees the validity of our construction from eq. (35).10 Quantization and vacuum expectation values ofcurrents
To construct the quantum field theory with the external EM field, we first introducethe fermionic field operators from eqs. (36), (37), (39), (40);ˆ ψ ( x ) = ∞ X n =0 Z dp y √ π Z dp z √ π × X s = ˜ R, ˜ L (ˆ b s, p ψ (+) s, p ( x ) + ˆ d † s, − p ψ ( − ) s, p ( x )) (41)where ˆ b † s,p , ˆ d † s,p (ˆ b s,p , ˆ d s,p ) are interpreted as creation (annihilation) operators of the par-ticles and anti-particles. These operators obey the anti-commutation relations, { ˆ b s,p , ˆ b † s ′ ,p ′ } = { ˆ d s,p , ˆ d † s ′ ,p ′ } = δ ss ′ δ nn ′ δ ( p y − p ′ y ) δ ( p z − p ′ z )which is equivalent to the anti-commutation relations for the field operators, { ˆ ψ α ( t, x ) , ˆ ψ † β ( t, x ′ ) } = δ (3) ( x − x ′ ) δ αβ . In order to describe the fermion field under the time-dependent EM field, we adoptthe Heisenberg picture in the following calculations, and define the vacuum state | i at t → −∞ , ˆ b s, p | i = 0 , ˆ d s, p | i = 0 (for all s, p ) , h | i = 1 . (42)We obtain asymptotic behavior of the eigenfunctions ˜ φ ( ± ) n,p z ( t ) at t → −∞ as˜ φ (+) n,p z ( t ) ∝ exp( − iω n,p z (0) t ) ( t → −∞ ) (43a)˜ φ ( − ) n,p z ( t ) ∝ exp(+ iω n,p z (0) t ) ( t → −∞ ) (43b)Apparently, the eigenfunction ˜ φ (+) n,p z ( t ) ( ˜ φ ( − ) n,p z ( t )) at t → −∞ coincides a positive (neg-ative) energy solution of the free Dirac fermion.By using the quantized fields, the classical current, j (Γ; x ) = ¯ ψ ( x )Γ ψ ( x ), is replacedby the current operatorˆ j (Γ; x ) = 12 [ ˆ¯ ψ ( x ) , Γ ˆ ψ ( x )]= 12 [ ˆ¯ ψ α ( x )Γ αβ ˆ ψ β ( x ) − Γ αβ ˆ ψ β ( x ) ˆ¯ ψ α ( x )]11here Γ are products of γ matrices, i.e. Γ = (1 , iγ , γ µ , γ γ µ ). We can calculate thevacuum expectation value (VEV) of the corresponding current as follows h ¯ ψ ( x )Γ ψ ( x ) i = h | ˆ j (Γ; x ) | i (44a)= ∞ X n =0 Z dp y π Z dp z π S p ( x ; Γ) (44b)where we define S p ( x ; Γ) as S p ( x ; Γ) ≡ X s = ˜ R, ˜ L [ ¯ ψ ( − ,s ) p ( x )Γ ψ ( − ,s ) p ( x ) − ¯ ψ (+ ,s ) p ( x )Γ ψ (+ ,s ) p ( x )] (45)Using eqs. (36),(37),(39),(40) we find S p ( x ; Γ) for various Γ as, S p ( x ; γ γ ) = [ g n − ,p y − g n,p y ][ | ˜ φ (+) n,p z | − | ˜ φ ( − ) n,p z | ] (46) S p ( x ; γ ) = [ g n − ,p y + g n,p y ][ | ˜ φ (+) n,p z | − | ˜ φ ( − ) n,p z | ] (47) S p ( x ; iγ ) = 2[ g n − ,p y − g n,p y ] cos θ n Im[ ˜ φ (+) n,p z φ ( − ) n,p z ] (48) S p ( x ; γ γ ) = 2[ g n − ,p y + g n,p y ] cos θ n Im[ ˜ φ (+) n,p z φ ( − ) n,p z ] (49) S p ( x ; iγ γ ) = 2[ g n − ,p y − g n,p y ] cos θ n Re[ ˜ φ (+) n,p z φ ( − ) n,p z ] (50) S p ( x ; 1) = 2[ g n − ,p y + g n,p y ] cos θ n Re[ ˜ φ (+) n,p z φ ( − ) n,p z ] , (51)For further calculations, we shall integrate the right hand side over p y with payingattention to g − ,p y = 0, namely, Z dp y [ g n − ,p y ( x ) − g n,p y ( x )] = −| eB | δ n (52) Z dp y [ g n − ,p y ( x ) + g n,p y ( x )] = | eB | α n (53)where α n are defined by α n = ( n = 02 if n = 1 , , , · · · . (54) The VEVs of the current derived in the previous section diverge when we integrate over p z , thus we need some sort of the regularization. Because we could obtain these VEVs12s a result of the subtle cancellation of the divergent integrals, use of the gauge invariantregularization is certainly important. Here, we use the point-split regularization[21, 22],which is known as the gauge invariant regularization scheme.The regularization method in the p z integral essentially introduces the non-localityin the z space. We replace the local current operator, ¯ ψ ( x )Γ ψ ( x ), by the integral ofthe non-local current as follows¯ ψ ( z )Γ ψ ( z ) = Z dz ′ ¯ ψ ( z ′ ) δ ( z − z ′ )Γ ψ ( z ) (55)= lim ε → Z dz ′ ¯ ψ ( z ′ ) h ε ( z − z ′ )Γ ψ ( z ) (56)where h ε ( z − z ′ ) ≡ √ πε exp( − ( z − z ′ ) ε ) (57)which is reduced to the delta function δ ( z − z ′ ) as ε → U ( z ′ , z ) = P exp[ ie Z zz ′ d ˜ x µ A µ (˜ x )] (58)where the choice of the integral path P is arbitrary. We may choose a straight linewhich connects z ′ and z for the path function U ( z ′ , z ).¯ ψ ( x )Γ ψ ( x ) → Z ∞−∞ dz ′ ¯ ψ ( z ′ ) h ε ( z − z ′ ) U ( z ′ , z )Γ ψ ( z ) . (59)Carrying out the z ′ integration, we obtain the regularized VEV of the current, h ¯ ψ ( x )Γ ψ ( x ) i = ∞ X n =0 Z dp y π Z dp z π × X s = ˜ R, ˜ L exp( − ε [ p z + eA z ( t )] ) S p ( x ; Γ) (60)The regularization factor, exp( − ε [ p z + eA z ( t )] ), is now inserted into the integrand forthe VEVs. Because the parameter ε has the dimension of (mass) − , we introduce thecutoff parameter Λ ≡ /ε with the dimension of Λ being (mass) .13e arrive at final expressions for the regularized VEVs of the currents: . h ¯ ψγ γ ψ i = eB π Z dp z π f Λ ( p z )[ | ˜ φ (+)0 ,p z ( t ) | − | ˜ φ ( − )0 ,p z ( t ) | ] (61) h ¯ ψiγ γ ψ i = 2 eB π Z dp z π f Λ ( p z ) · Re[ ˜ φ (+)0 ,p z ( t ) φ ( − )0 ,p z ( t )] (62) h ¯ ψiγ ψ i = − eB π Z dp z π f Λ ( p z ) · Im[ ˜ φ (+)0 ,p z ( t ) φ ( − )0 ,p z ( t )] (63) h ¯ ψγ ψ i = | eB | π Z dp z π ∞ X n =0 α n f Λ ( p z )[ | ˜ φ (+) n,p z ( t ) | − | ˜ φ ( − ) n,p z ( t ) | ] (64) h ¯ ψψ i = − | eB | π Z dp z π ∞ X n =0 α n f Λ ( p z ) cos θ n × Re[ ˜ φ (+) n,p z ( t ) ˜ φ ( − ) n,p z ( t )] (65) h ¯ ψiγ γ ψ i = 2 | eB | π Z dp z π ∞ X n =0 α n f Λ ( p z ) cos θ n × Im[ ˜ φ (+) n,p z ( t ) ˜ φ ( − ) n,p z ( t )] (66)where f Λ ( p z ) = exp( − [ p z + eA z ( t )] / Λ ).It is clear that only the lowest Landau level (LLL), n = 0, contributes to the chiralityimbalance eq. (61), psedoscalar density eq. (63), and the tensor density eq. (62). On theother hand, we need sum up contributions from all possible Landau levels for the vectorcurrent eq. (64) as well as the scalar density (65). This is different from calculationswith the chiral chemical potential [27, 28], where the vector current is given by thecontribution from only the lowest Landau level. In our case, when there is no electricfield ( t = −∞ ), | ˜ φ (+) n,p z | and | ˜ φ ( − ) n,p z | in (64) show the same momentum distribution foreach n , and a cancellation gives null vector current. After the electric field is turned on( t ≥ | ˜ φ (+) n,p z | and | ˜ φ ( − ) n,p z | with the regularizatinfunction become different in eqs.(29,30), and the resulting vector current is finite foreach n , although contributions from higher Landau levels are small. Our results areconsistent with ref. [22].We also find that the VEVs for all other Γs vanish. In particular, the spin expec-tation value of the z component vanishes, h ¯ ψγ γ ψ i ∼ h S z i = 0. Thus, there is nomagnetization of the vacuum due to the chiral anomaly. This is in contrast with theresult for the tensor matrix element, h ¯ ψσ ψ i 6 = 0, in eq. (62), which is non-zero. Thisdifference may come from roles of antifermions for these matrix elemets, i.e. h ¯ ψγ γ ψ i z -component of the electric current, J z = e h ¯ ψγ ψ i , in view of the chiral magnetic effect. From eqs. (61) and (64), we find asimple relation between J z and the chirality imbalance n in the LLL approximation( n = 0) as J z ≃ e | eB | π Z dp z π f Λ ( p z )[ | ˜ φ (+)0 ,p z ( t ) | − | ˜ φ ( − )0 ,p z ( t ) | ]= e | eB | eB n = sgn( eB ) e n (67)Eq. (67) tells us that J z is essentially proportional to n in the limit of the strongmagnetic field, where the use of the LLL approximation can be justified. The resultagrees with one obtained in the previous work[8], although the existence of the chiralchemical potential is assumed in ref. [8]. Here, we recover eq. (67) by considering themassive fermion under the external EM fields, without assuming the chirality asym-metric source. We are also interested in the modification of the conservation law for the axial-vectorcurrent[1]. Neglecting the surface term from the current divergence, the chiral anomalyrelation is given by ∂ t Z d x h ¯ ψγ γ ψ i = 2 im Z d x h ¯ ψγ ψ i + 2 απ Z d x E · B . (68)Here, the second term of RHS is just an input of the model calculation in our case.On the other hand, we have already calculated the LHS and the 1st term of RHSindividually. Thus, we can check a consistency of our calculations with the point-splitregularization.For the LHS of the chiral anomaly, we simply calculate the time-derivative of thechirality imbalance. If we used eq. (46) for the chirality imbalance without invoking15he momentum regularization, the time derivative would yield, ∂ t h ¯ ψγ γ ψ i = 2 im · g n − ,p y ( x ) − g n,p y ( x )] cos θ n Im[ ˜ φ (+) n,p z ( t ) φ ( − ) n,p z ( t )]= 2 im h ¯ ψγ ψ i . (69)This is just a classical conservation law for the axial-vector current.However, the gauge invariant regularization provides an additional time-dependentfactor coming from exp( − [ p z + eA z ( t )] / Λ ), in the integrand of the chirality imbalance.Using the regularized result, eq. (61), we obtain a modified conservation law as ∂ t h ¯ ψγ γ ψ i = 2 im h ¯ ψγ ψ i + 2 απ E z ( t ) B F Λ ( t ) (70)where F Λ ( t ) = Z ∞−∞ dp z p z + eA z ( t )Λ z f Λ ( p z )[ | ˜ φ (+)0 ,p z ( t ) | − | ˜ φ ∗ ( − )0 ,p z ( t ) | ] . This is the axial-vector current conservation law in our framework. If the momentumcutoff is large enough, Λ ≫ m , we obtain a simple relation lim Λ →∞ F Λ ( t ) = 1, which isexplicitly shown in Appendix, and thus reproduce the correct anomaly relation eq. (68). In this section, we show numerical results for the time evolution of the chirality im-balance and CME in the vacuum. Here, we have three independent parameters of themodel, magnitudes of the electric and magnetic fields, and the fermion mass, whichare expressed in units of the electron mass m e = 0 . m e , which is much larger than thefermion mass scale.In our study, we calculate the VEVs of the vacuum under parallel constant magneticand the time-dependent Sauter Electric fields, whose magnitudes can be fixed indepen-dently. However, as far as we understand, the chirality imbalance is well studied byconsidering the magnetic helicity density h defined in eq. (9). As we have discussed inthe previous section, our calculation is fully consistent to the chiral anomaly relation.16n the massless limit, it is simplified as ∂ t Z d x h ¯ ψγ γ ψ i ≃ απ Z E · B = − ∂ t (cid:18) απ Z d x A · B (cid:19) (71)since we consider the time-independent magnetic field. Integrand of the RHS is justmagnetic helicity h ( t ). Hence, with our EM field, the chirality imbalance becomes n ( t = ∞ ) = − απ h ( t = ∞ ) = e BEτ /π , (72)which is true only for massless fermions. Nevertheless, it is convenient to express thechirality imbalance (and CME) in unit of the magnetic helicity, e BEτ /π .In Fig. 1, we first show the chirality imbalance n as a function t with a shape ofthe Sauter electric field by the dash-dotted curve. In the massless case (solid curve), n increases by the electric field, and approaches a finite value, e BEτ /π , at t → ∞ evenafter E field diminished. On the other hand, in the case of the finite fermion mass, thechirality imbalance consists of both a constant part and an oscillating part at t → ∞ .When the mass is comparable with the magnitude of the electric field, m ∼ ( eE ),the chirality imbalance is largely suppressed as depicted by the dotted curve. Thus,we find that the average chirality imbalance is almost zero, if m > ( eE ). We willrelate these results with the fermion pair production from the vacuum in view of theSchwinger mechanism[24].We also examine effects of the magnetic field on the chirality imbalance. If weincrease the strength of the magnetic field, the magnitude of the chirality imbalanceis also increased which is just proportional to the magnetic helicity. However, thetime-dependence of n is never changed as expected.We then show the vector current along the z -direction in Fig. 2 which could beunderstood as the chiral magnetic effect. Again, the vector current is shown in unitsof the magnetic helicity density. In the case of the massless limit, the vector currentdepicted by the solid curve consists of dominant constant part and tiny oscillating part,which is somewhat different from the behavior of the chirality imbalance n . This isbecause n is solely determined by the lowest Landau level contribution, while thevector current gets contributions from higher Landau levels in eq. (64). The average17 - Figure 1: The time evolution of chirality imbalance n τ /π . eE/m e = 4 . , τ m e =0 . , eB/m e = 8 . m > ( eE ), which is similarwith the chirality imbalance.From Fig. 2, for t/τ ≫ j z ≃ e BEτπ = α π B (8 Eτ ) . (73)The form of eq.(73) is the same as eq. (1) if we substitute 8 Eτ for µ . This crudeidentification is justified only if t is large enough compared with the time scale τ of theelectric field in eq. (7). - - Figure 2: The time evolution of vector current density j z τ m e = 0 . , eB/m e = 8 . t ∼
0. 18 - - - - Figure 3: The time evolution of pseudoscalar density. m e = 0 . , eE/m e =4 . , τ m e = 0 . , eB/m e = 8 . In order to understand appearance of the chirality imbalance from the vacuum, werelate it with the fermion pair-production[24, 25]. To do so, we try to find a relationbetween the ”in-state” vacuum at t → −∞ and the ”out-state” vacuum at t → ∞ .As discussed in eqs. (74,75), our original ”in state” vacuum at t → −∞ , | i , coincideswith the free particle vacuum (although B = 0). However, due to the Sauter-electricfield, the vacuum at t → ∞ , | i out, is not the same as the original vacuum | i .To proceed calculations, we need asymptotic forms of the eigenfunction ˜ φ (+) n,p z ( t ) at t → −∞ and at t → ∞ . For the ”in-state”, the eigenfunctions reduce to˜ φ (+) n,p z ( t ) ∝ exp( − iω n,p z (0) t ) ( t → −∞ ) (74)˜ φ ( − ) n,p z ( t ) ∝ exp(+ iω n,p z (0) t ) ( t → −∞ ) (75)which agree with positive/negative energy plane wave solutions with the energy ± ω (0)defined eq. (32). On the other hand, with the help of the connection formula for theGauss hypergeometric function, ”out-state” eigenfunctions are rewritten as˜ φ (+) n,p z ( t ) = α n,p z ˜ φ (+)out ,n,p z ( t ) − β ∗ n,p z ˜ φ ( − )out ,n,p z ( t ) (76)˜ φ ( − ) n,p z ( t ) = α ∗ n,p z ˜ φ ( − )out ,n,p z ( t ) + β n,p z ˜ φ (+)out ,n,p z ( t ) (77)19here ˜ φ (+)out ,n,p z ( t ) ≡ s ω (1) + [ p z + 2 eEτ ]2 ω (1) u − iτω (0)2 (1 − u ) iτω (1)2 × F (cid:18) a, b a + b − c ; 1 − u (cid:19) (78)˜ φ ( − )out ,n,p z ( t ) ≡ s ω (1) − [ p z + 2 eEτ ]2 ω (1) u iτω (0)2 (1 − u ) − iτω (1)2 × F (cid:18) − a, − b c − a − b ; 1 − u (cid:19) . (79)and α n,p z = s ω (0) + p z ω (0) s ω (1) ω (1) + [ p z + 2 eEτ ] 2 iτ [ ω (0) + ω (1) − eEτ ] × Γ(1 − iτ ω (0))Γ( − iτ ω (1))Γ( − iτω (0)2 − iτω (1)2 − ieEτ )Γ( − iτω (0)2 − iτω (1)2 + ieEτ ) (80) β n,p z = s ω (0) + p z ω (0) s ω (1) ω (1) − [ p z + 2 eEτ ] 2 iτ [ ω (0) − ω (1) − eEτ ] × Γ(1 + iτ ω (0))Γ( − iτ ω (1))Γ( iτω (0)2 − iτω (1)2 + ieEτ )Γ( iτω (0)2 − iτω (1)2 − ieEτ ) . (81)˜ φ (+)out ,n,p z ( t ) and ˜ φ ( − )out ,n,p z ( t ) are further simplified as˜ φ (+)out ,n,p z ( t ) ∝ exp( − iω (1) t ) ( t → ∞ ) (82)˜ φ ( − )out ,n,p z ( t ) ∝ exp( iω (1) t ) ( t → ∞ ) .. (83)which are the free fermion wave functions with the energy ω (1). From these functions,we can construct the Bogoliubov transformation between ”in-state” and ”out-state”[24,25]. We already introduced the annihilation operators and the vacuum for the ”in-state”. ˆ b s, p | i = 0 , ˆ d s, p | i = 0 (for all s, p ) , h | i = 1 . (84)Similarly, we define the ”out-state” vacuum with operators ˆ b (out) s, p , ˆ d (out) s, p ,ˆ b (out) s, p | i out = 0 , ˆ d (out) s, p | i out = 0 (for all s, p ) , h | i = 1 . (85)20here operators ˆ b (out) s, p , ˆ d (out) s, p are introduced as coefficients of ˜ φ (+)out ,n,p z ( t ) and ˜ φ ( − )out ,n,p z ( t )in the standard way. Thus, these operators are subject to the transformation, ˆ b (out) p , s ˆ d (out) † p , s ! = (cid:18) α n,p z β n,p z − β ∗ n,p z α ∗ n,p z (cid:19) (cid:18) ˆ b p , s ˆ d † p , s (cid:19) (86)where the Bogoliubov coefficients satisfy the unitary condition | α n,p z | + | β n,p z | = 1.The expectation value of the number operator at t = ∞ between the original vacuumbecomes h | ˆ b (out) † p , s ˆ b (out) p , s | i = | β n,p z | (87)which is understood as the probability to find a fermion produced by the electricfield with the momentum n, p z at t = ∞ [24, 25, 31]. It is well-known that | β n,p z | issignificant only if the electric field is larger than the fermion mass square, eE > m ,which means spontaneous creation of fermion pairs from the vacuum under the strongelectric field. Thus, we naively expect the chirality imbalance may emerge for eE ≫ m .Using these results, one can express the VEVs of the vacuum at t = ∞ in termsof the Bogoliubov coefficients. For example, the chirality imbalance n at t = ∞ iscalculated as n | t = ∞ = eB π Z dp z π f Λ ( p z ) (cid:20) − | β ,p z | p z + 2 eEτω (1) − mω (1) Re[ α ,p z β ,p z e − iω (1) t ] (cid:21) (88)where the regularization function at t = ∞ is f Λ ( p z ) = exp[ − ε ( p z + 2 eEτ ) / Λ ] (89)The first term is independent of time, and simply proportional to | β ,p z | which is theprobability to find a produced particle in the lowest Landau level with p z . On the otherhand, the second term is proportional to the mass, and is somehow interpreted as the”interference” term.At first sight, n is simply determined by the magnitude of | β ,p z | . However,existence of the chirality imbalance strongly depends on details of the integration over p z in eq. (88), which is sensitive to a parameter τ , the time scale of the electric field in21q. (7). We will discuss how the non-zero n appears in some detail. In the masslesslimit, the first term of eq. (88), which we call n (0)5 , becomes n (0)5 = eB π Z dp z π f Λ ( p z ) " − | β ,p z | p z + 2 eEτ p ( p z + 2 eEτ ) + m → eB π Z dp z π f Λ ( p z ) (cid:2) − p z + 2 eEτ ] | β ,p z | (cid:3) . (90)In the presence of the uniform magnetic field, all the fermions move along the z -direction, and the spin of the fermions in the lowest Landau level, which can contributeto | β ,p z | , is parallel to the z-direction. Hence, the fermions with positive canonical mo-menta, p z + 2 eEτ >
0, carry the right-handed chirality, while those with p z + 2 eEτ < n (0)5 would vanish, becauseof a cancellation between contributions from p z > p z < p z distribution of the pair-production probability β ,p z . However, non-zero electric field induces an asymmetry between momentum distributions of right- andleft-handed fermions in both the sign function sgn[ p z + 2 eEτ ] and the regularizationfunction f Λ , which indeed generates the chirality imbalance in this model. - - - - - Figure 4: p z distribution of the pair-production probability | β | and the signfunctionof p z + 2 eEτ /ω (1).To study n (0)5 in the case of the finite mass, we show | β ,p z | and ( p z + 2 eEτ ) /ω (1)in Fig. 4, where ( p z + 2 eEτ ) /ω (1) is no longer the sign function. The pair-creationprobability | β ,p z | peaked at p z = − eEτ , whereas ( p z + 2 eEτ ) /ω (1) changes its signat p z = − eEτ . Hence, if τ is very small ( ∼ p z is negligibledue to a cancellation, and thus the resulting chirality imbalance almost vanishes.For completeness, we show explicit τ dependence of the results. We first show thechirality imbalance as a function of τ m e in Fig. 5 for several values of eE . If eE < m ,22igure 5: τ dependence of n (0)5 for several values of eE .the chirality imbalance is almost zero, because the production of the fermion pairs isforbidden. The large electric field simply gives the larger chirality imbalance. However,if the time scale τ is quite small, τ ≪ /m e , situation becomes different. In Fig. 6, weshow n (0)5 for several values of τ m e . We find that, even if the magnitude of the electricfield is large enough, n (0)5 is very small for τ m e < .
01. This is because the small τ cannot provide enough asymmetry in the integrad of eq. (90).Figure 6: eE dependence of n (0)5 Similar argument holds for the chiral magnetic effect, the z -component of vectorcurrent at t = ∞ . We can write the CME current in terms of Bogoliubov coeffcientsas, h ¯ ψγ ψ i| t = ∞ = − | eB | π Z dp z π f Λ ( p z ) ∞ X n =0 α n (cid:20) − | β n,p z | p z + 2 eEτω (1) − √ m + 2 eBnω (1) Re[ α n,p z β n,p z e − iω (1) t ] , (91)which is similar with one of the chirality imbalance. The first term is independentof time and essentially given by a product of | β ,p z | and ( p z + 2 eEτ ) /ω (1), which is23nterpreted as the z -component of relativistic velocity of particles. Hence, this term isunderstood as a classical analogue of the electric current of the z -component carriedby the produced fermions. Note that the second oscillating term is non-zero even inthe massless limit. We have studied the chirality imbalance of the vacuum under the time-independentmagnetic field and the Sauter-type pulsed electric field. In particular, we have focusedon the time evolution of the chirality imbalance and the chiral magnetic effect from t = −∞ to t = ∞ . Solving the squared Dirac equation with the EM field, we haveconstructed the quantized fermion field and the vacuum at t = −∞ . Then, we havecalculated the vacuum expectation values of various fermion current operators includ-ing n and CME in terms of the point-split regularization. Use of the gauge invariantregularization method is important in our study, because the VEVs diverge by the mo-mentum integration. Subtle cancellation between positive and negative energy statesprovides non-zero contribution to CME. We note that calculated VEVs are finite at t = ∞ , and differ from the case with the constant electric and magnetic fields, wherethe several VEVs become infinite at t → ∞ [22]. In addition, we have found expressionsfor the VEVs of other bilinear operators, e.g. h ¯ ψγ γ ψ i = 0, whereas h ¯ ψσ ψ i 6 = 0.We have shown the time evolution of the chirality imbalance and CME current. Theresulting chirality imbalance is finite at t = ∞ where the Sauter electric field is alreadyturned off. We have also demonstrated that a part of the chirality imbalance consists ofthe time-oscillating contribution, which is proportional to the fermion mass. The CMEcurrent also consists of the dominant time-independent part and the oscillating part,which is similar with the chirality imbalance. We have also discussed a connectionbetween the fermion pair-creation by the electric field and the chirality imbalance.As we have obtained in eqs.(90,91), there are simple physical interpretations for thegeneration of n and CME.The magnitudes of n and CME in this model are essentially determined by thefollowing conditions;1. enough magnitude of the electric field which is much larger than the fermion mass24cale.2. enough asymmetric p z distribution of the produced fermions (in the integrand ofthe chirality imbalance).Asymmetries of the pair-production rate between p z > p z < Acknowledgments
We acknowledge Prof. A. Suzuki for useful and critical comments. We also thankall the members of the quark/hadron group in Tokyo University of Science for usefulconversations. K.S. thanks Dr. S. Saitoh for contributions on early stage of this work.25 ppendix : Anomaly relation
We will provide a proof of the anomaly equation, eq.(70), ∂ t h ¯ ψγ γ ψ i = 2 im h ¯ ψγ ψ i + 2 απ E z ( t ) B F Λ ( t ) (92)where F Λ ( t ) = Z ∞−∞ dp z p z + eA z ( t )Λ z exp( − ( p z + eA ( t )) Λ )[ | ˜ φ (+)0 ,p z ( t ) | − | ˜ φ ∗ ( − )0 ,p z ( t ) | ] → → ∞ )Using the solutions of the Dirac equation for the lowest Landau level, ˜ φ ,p z , we firstrewrite the integrand of F Λ ( t ) as | ˜ φ (+)0 ,p z ( t ) | − | ˜ φ ∗ ( − )0 ,p z ( t ) | = p z + eA ( t ) p m + ( p z − eA ( t )) + G ( p z , t ) . (93)where the first term gives a finite contribution as p z → ∞ , while the second term, G ( p z , t ), is a rapidly decreasing function, lim | p z |→∞ | G ( p z , t ) | →
0. We decompose theintegrand of the first term using p z p m + p z = | p z | + | p z | H ( p z m ) , (94)where a function H ( p z , t ) satisfies lim p z →∞ H ( p z /m, t ) →
0. Thus, we rewrite the firstterm as Z ∞−∞ dp z p z + eA z ( t )Λ exp( − ( p z + eA ( t )) Λ ) p z + eA ( t ) p m + ( p z − eA ( t )) = 2Λ Z ∞ dp z | p z | exp( − p z Λ ) + 2Λ Z ∞ dp z | p z | H ( p z m ) exp( − p z Λ )=1 + 2 m Λ Z ∞ du H ( u ) exp( − m Λ u ) → → ∞ , because of the fact H ( p z m ) → p z → ∞ ).On the other hand, the second term of eq.(93) can be shown to vanish in thesimilar way. In our model with the Sauter electric field, G ( p z , t ), decreases rapidly for | p z | ≫ eEτ independent of time t , as shown in Fig. 4. Hence, in the limit Λ → ∞ ,i.e. Λ ≫ eEτ , the integal of the second term is independent of Λ, and proportional to( eEτ ) by the dimensional analysis. (If the fermion mass is comparable with ( eE ) / ,26he argument should be modified. However, the essential result is not changed forΛ ≫ ( eE ) / or m .) It leads to Z ∞−∞ dp z p z + eA z ( t )Λ z exp( − ( p z + eA ( t )) Λ ) G ( p z , t )= ( eEτ )Λ × (Λ-independent constant) → References [1] S.L. Adler, Phys. Rev. , 2426 (1969); J.S. Bell and R. Jackiw, Nuovo Cim.
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