MMIT-CTP/4884
Chiral Vortical Effect for Bosons
Artur Avkhadiev ∗ and Andrey V. Sadofyev
2, 3, † Department of Physics, Brown University, Providence RI 02912, USA Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA ITEP, Bolshaya Cheremushkinskaya 25, Moscow, 117218, Russia
The thermal contribution to the chiral vortical effect is believed to be related to the axial anomalyin external gravitational fields. We use the universality of the spin-gravity interaction to extend thisidea to a wider set of phenomena. We consider the Kubo formula at weak coupling for the spincurrent of a vector field and derive a novel anomalous effect caused by the medium rotation: the chiralvortical effect for bosons. The effect consists in a spin current of vector bosons along the angularvelocity of the medium. We argue that it has the same anomalous nature as in the fermionic caseand show that this effect provides a mechanism for helicity transfer, from flow helicity to magnetichelicity.
INTRODUCTION
Recently, macroscopic manifestations of the axialanomaly have attracted significant attention in the liter-ature (for review, see [1, 2]). This triangle loop diagramviolates classical conservation of the axial charge in thepresence of electromagnetic (EM) fields (see e.g. [3]): ∂ µ J µA = 12 π E · B , (1)where J A is the axial current.In a chiral medium, it results in vector and axial cur-rents directed along the magnetic field or local angularvelocity: chiral effects. Moreover, the chiral vortical ef-fect (CVE) survives in the absence of EM fields while thetheory is non-anomalous in this limit, for instance J µA = (cid:18) µ V + µ A π + T (cid:19) ω µ , (2)where µ V ( A ) is a vector (axial) chemical potential, ω µ = (cid:15) µναβ u ν ∂ α u β is vorticity and u µ is the 4-velocity of thefluid element. There is an intensive discussion on the ori-gin of (2), see e.g. [4–13]. It is also known that this effectmay result in event-by-event contributions to parity andcharge parity violating observables in heavy ion collisions(see e.g. [2]).Despite the anomaly, it is possible to introduce a con-served generalization of the axial charge [14]. Includ-ing the contribution of the anomalous effects, the axialcharge is given by ∂ t ( N + H mh + H mfh + H fh ) = 0 (3)where H fh = (cid:82) (cid:16) µ V + µ A π + T (cid:17) ω d x is the flow he-licity, H mh = π (cid:82) A · Bd x is the magnetic helicity, H mfh = π (cid:82) µ V v · Bd x is the mixed magnetic/flowhelicity, and J = N is the difference between the num-ber of right and left particles. Note that the macroscopic contributions in (3) are of topological origin: these he-licities measure the linkage between field and flow lines.In this expression it may seem that the flow helicity H fh can be transferred into the chiral asymmetry. On theother hand, there is no known microscopic mechanismto support such a process. Indeed, the anomaly is un-modified by medium effects (at finite temperature anddensity). The issue persists in the other transition of theaxial charge between the macroscopic terms in (3).In this article, we concentrate on the latter issue in aneutral medium ( µ V ( A ) = 0), asking the following ques-tion: Is it possible to find a microscopic mechanism totransfer H fh to H mh ? If the answer is “yes,” then thereis at least an indirect way to generate chiral asymmetryfrom a helical flow via the anomaly caused by an inter-mediate generation of H mh . But first, it is instructiveto concentrate on the origin of the thermal contributionto (2), henceforth referred to as the tCVE. This effect isbelieved to be connected with the gravitational cousin ofthe axial anomaly (see e.g. [15]), consisting in an axialcharge non-conservation due to an external gravitationalfield. The coefficient in front of the gravitational anomalyis argued to be connected with the coefficient in the tCVEconductivity [8, 16]. Although this picture is widely ac-cepted, note that there is an ongoing discussion on otherpossible origins of this effect [12, 17, 18]. In particular,there are strong arguments in favor of the relation be-tween the tCVE and global anomalies in effective fieldtheory [12, 17].It is known that the gravitational anomaly is a moregeneral phenomenon taking place for chiral bosons as well[19–22]. Moreover, due to the features of gravitational in-teraction, this anomaly can appear in the axial currentconstructed on fields of arbitrary spin. Thus, followingthe conjectured connection of the gravitational anomalyand the tCVE, one expects to find other anomalous ef-fects for fields with s (cid:54) = .With this motivation, we begin our consideration ofthe relevant spin currents looking for effects analogousto the fermionic tCVE. We explore the novel anomaloustransport via the example of the chiral vortical effect for a r X i v : . [ h e p - t h ] A ug vector bosons (bCVE), concentrating on the direct andinstructive derivation relying on the Kubo formula in theweakly coupled limit. In full similarity with [8, 17], thecorresponding conductivity is expressed as σ V = lim p k → (cid:15) ijk − i p k (cid:10) K i T j (cid:11) | ω =0 , (4)where K µ = (cid:15) µναβ A ν ∂ α A β is the Chern current [23].As in the fermionic case, the corresponding conduc-tivity appears as a resummation of a divergent series of3d “conductivities” indicating its relation to the globalanomalies [12, 17]. We further propose that there isa wider set of novel anomalous effects in spin currentsconstructed on higher spin fields. While the argu-ments on their relation with the gravitational anomalyare rather convincing in the holographic setup, furtherfield-theoretic study is required. However, it should bestressed that the bCVE and the tCVE have the same ori-gin, and that it is evident at the level of the derivation.When considering the bCVE for photons in a slowlyvarying helical motion of the medium, one finds a pro-cess generating H mh out of H fh . Therefore, the answerfor the question above is “yes.” The process of helic-ity transfer H fh → H mh extends the generalized axialcharge picture [14, 24, 25] to bosonic theories, where thehelical charge H mh could have a non-electromagnetic na-ture. We also argue that the generation of magnetic he-licity provides a microscopic mechanism to produce chiralasymmetry out of the helical motion of the medium.This article is organized as follows. First, we discussthe gravitational anomaly in K µ for vector bosons and itssimilarities with the fermionic case. We use this analogyas a guiding principle to motivate the study of polar-ization effects for vector bosons. Then we show that arotating bosonic system exhibits a novel contribution to K µ (bCVE), which is analogous to the tCVE in J µA . Westate that the similarity between gravitational anomalieshints at a possibly deeper connection between the twochiral vortical effects and argue that they have the sameorigin. Finally, we use bCVE to establish a microscopicmechanism for helicity transfer. GRAVITATIONAL ANOMALY
The gravitational anomaly for fermions is a conse-quence of the well-known triangle loop diagram in exter-nal gravitational fields [15]. Once it is taken into account,the divergence of the axial current reads ∇ µ J µ = − π (cid:15) µναβ R λρµν R ρλαβ (5)where J µ = ¯ ψγ µ γ ψ and ψ is the massless Dirac field. This result can be generalized to the case of other mass-less fields running in the triangle loop. Indeed, for vectorbosons such a diagram gives (see e.g. [20, 22]) ∇ µ K µ = 196 π (cid:15) µναβ R λρµν R ρλαβ , (6)where K µ = √− g (cid:15) µναβ A ν ∂ α A β . This anomaly is a par-ticular example of a wider set of phenomena which istied to the universality of the spin-gravity interaction(see e.g. [15, 22]). Note that despite the non-zero di-vergence ∇ µ K µ = F µν ˜ F µν , chirality is conserved formassless vector bosons and (cid:104)∇ µ K µ (cid:105) , na¨ıvely, vanishes inexternal gravitational fields.Some reservations should be made in the case of gaugebosons. The divergence ∂ µ K µ is a gauge invariant quan-tity, while the Chern current is not. However, a constantbulk current is always a sloppy concept: it is not theconstant bulk current that is measured, but rather thechange in the charge, tied to the gauge invariant diver-gence. In the following section, we derive perturbativelythe anomalous contribution to K µ constructed on a vec-tor boson field. When gauge bosons are involved, weassume that the observable is the change in the chargehappening, say, at the boundary of the system. Notethat the charge (cid:82) K d x is invariant under local gaugetransformations and gives ± K i should be understood only as a contributionto the 4-divergence. One may also reformulate that interms of the change in the angular momentum when po-larized photons leave the medium.The two anomalies (5) and (6) are fully analogous inthe sense of spin current. It is argued in [20] that ifone introduces an infinitesimal mass for fermions andvector bosons, which is always an eligible procedure inany consideration of the anomaly, then both currents areconnected with relativistic generalizations of one-particlePauli-Lubanski pseudovector. Further, we keep this lim-iting picture in mind when taking the massless limit inthe final results.It is worth mentioning that the origin of the tCVE isunder active discussion. In the hydrodynamic limit, thegravitational anomaly involves higher derivatives givingno contribution of the first order to the axial current.Some possible solutions [26] and alternative anomalousorigins [12, 17, 18] are suggested in the literature. Despitethis issue, there is a connection between the gravitationalanomaly coefficient and the tCVE conductivity [8] whichbecomes explicit in the holography [16]. Henceforth weemploy this picture, in particular, as a motivation. FIG. 1. The two-point function of the vector field spincurrent and the stress-energy tensor.
KUBO FORMULA
This section is focused on the derivation of the vorticalconductivity for vector bosons in the limit of a weakly in-teracting medium: a gas of vector bosons. The operatorsin the corresponding Kubo formula (4) read K µ = (cid:15) µναβ A ν ∂ α A β T µν = F µλ F νλ − g µν (cid:18) F − m A (cid:19) . (7)We remind the reader that σ V is defined by the equilib-rium behavior of the corresponding retarded two-pointfunction at zero frequency and in the small momentumlimit (for details see [8]): G i, jR ( ω, p ) | ω =0 = i(cid:15) ijk p k σ V + O ( p ) . (8)The two-point function at zero frequency is given by theEuclidean Green’s function G i, jR = − i G i, j = (cid:10) K i T j (cid:11) .The relevant leading diagram is given in Fig.1.To avoid ambiguity, we begin by considering a massivevector field. In the calculation below, we employ theProca formalism, taking the massless limit at the end ofthe procedure. According to the Matsubara technique,the two-point function reads G µ, α (0 , (cid:126)p ) = β − (cid:88) n (cid:90) d q (2 π ) (cid:15) µνρσ p σ q ρ (cid:0) q α δ ν + q δ αν (cid:1) (( p − q ) + m ) ( q + m )where p = 0 and q is the bosonic thermal frequencyrunning in the loop: q = ω n = 2 πiT n . For brevity,we also omit the terms that result in zero contributionson symmetry grounds. It is worth mentioning that thesame diagram with photon lines is gauge independent.The issues originate in treating K µ by itself, not thisparticular contribution.For massless fermions, the effect of interest is knownto be related to the ζ -function resummation of conduc-tivities in 3d Euclidean theories. This result provides anargument towards another origin of the tCVE based on global anomalies [12, 17]. We find it instructive to par-tially generalize this statement. The two-point functionabove is clearly divergent in UV and it has to be regu-larized. The derivation is simplified in the dimensionalregularization where it is sufficient to relate some mo-mentum integrals [27]. Then, expanding in powers of p and taking the m → G i, j (0 , (cid:126)p ) = − β − (cid:88) n (cid:15) ikl p l (cid:90) d q (2 π ) δ jk ω n + q k q j ( ω n − (cid:126)q ) . (9)Finally, in similarity with the fermionic case, the bCVEconductivity reads σ V = 12 T ∞ (cid:88) n = −∞ | n | = T . (10)Note that the factor of in the bCVE conductivity isdue to the ζ -function regularization of a formally diver-gent sum (see [17]). In the case of finite mass, the result ismodified and tends to zero in the limit of infinite mass-to-temperature ratio; in what follows we assume the mass-less limit. A similar feature is present in the case of thechiral separation effect: a magnetic-field-driven contribu-tion to the fermionic axial current (see [28]). Finally, forthe resulting spin current of massless vector bosons wehave (cid:126)K = T (cid:126) Ω , (11)where (cid:126) Ω is the local angular velocity of the medium. Westress that the coefficient in bCVE is tied to the gravi-tational anomaly for bosons as much as in the fermioniccase [8, 16, 26] and that this perspective may be extendedto higher spin effects.Notably, one may arrive at the same result by follow-ing the one-point function calculation. This procedure isanalogous to the axial current derivation in the fermioniccase [4], and we omit it here.
HELICITY TRANSFER
Considering the bCVE for photons in the hydrody-namic limit, we are interested in the magnetic helicitychange. As mentioned above, the Chern current is agauge-dependent quantity requiring careful treatment.However, a gauge transformation cannot influence thedivergence ∂ µ K µ , which, in turn, defines the helicitychange (cid:82) d x ∂ µ K µ ∼ ∂ t H mh .One expects that multiple photons of the same po-larization result in a helical “condensate” equivalent toa non-zero H mf in the considered region of the space.Thus, in a na¨ıve approximation, the current (11) gener-ates regions of opposite helical charges at the boundaries.This idea could be extended to a local form with the rel-ativistic version of (11), given by K µ = T ω µ . (12)Relying on the vector boson condensation and turning onthe hydrodynamic perturbations, we find (cid:104) E · B (cid:105) = ∂ µ T ω µ . (13)Consequently, a flow with ∂∂t H fh (cid:54) = 0 is expected to gen-erate H mh [29].This process is of particular interest since it provides amechanism to generate microscopic asymmetry througha change in macroscopic motion of the medium via anintermediate generation of (cid:104) E · B (cid:105) . It is instructive to notethe connection of the discussion above with the tCVErenormalization (see [17, 30]). There, it is pointed outthat in the presence of dynamical gauge fields, the tCVEreads J µ = T (cid:18) e π (cid:19) ω µ . (14)The coupling e is explicitly restored to make the relationof the second term to the axial anomaly evident. This re-sult supports the proposed mechanism of helicity transferto chirality and provides an example of an interesting in-terplay between the bosonic anomalous spin current andits fermionic counterpart. Indeed, the condensate of po-larized photons results in an additional anomalous contri-bution to the axial current. Note that the correspondingcoefficient is fixed by two multiples coming from differentanomalous phenomena. Other chiral effects may also bemodified, see discussions in [18, 31–35].In the theory of magnetohydrodynamics (MHD), it iswell known that the dynamics of a magnetic fluid consid-erably depends on topological properties of the field andflow configuration. Particularly, in the ideal MHD limit,magnetic field lines are frozen in the medium volume el-ement and the magnetic helicity is conserved. In otherwords, the electric field is completely screened and thereis no way to change H mh . When the ideal limit of infi-nite conductivity is relaxed, the helicity is changing dueto reconnections. However, this process is slower thanmagnetic energy dissipation and the helicity constrainsthe dynamics of the system. The bCVE current con-nects the two helicities in a plasma of light bosons andmay result in a considerable modification of the MHDevolution. The photon properties are modified in the medium,including the Debye mass screening by the Coulomb in-teraction. In the case of gluons or other non-Abelian vec-tors, this issue is additionally complicated by a possiblegeneration of a magnetic mass. These effects depend onthe coupling and disappear in the non-interacting limit.In this note we restrict ourselves to the leading order con-tributions and leave a detailed analysis of possible modi-fications of the conductivity by the interaction for futureconsiderations. Note, however, that the connection ofchiral effects with the anomaly results in an additionalrobustness, which is also supported in the strongly cou-pled limit by holographic considerations. In the case ofthe bCVE, one may rely on a bottom-up construction fora strongly interacting theory with no fermions. The keyproperties of such a model involving the relevant gravita-tional anomaly remain the same as in [16], and the axial(spin) current gains the bCVE contribution connectedwith its anomaly. DISCUSSIONS
In this note, we propose a set of novel anomalous effectsconsisting in spin currents of various fields along vortic-ity. We argue that these effects have a common origin fordifferent spin values and can be thought of as a general-ization of the tCVE. We explicitly derive the anomalouscontribution for the case of bCVE in the weakly coupledlimit using the Kubo formula.We further discuss the connection between the bCVEand the corresponding anomaly (6). While the relationof the tCVE with the gravitational and global anomaliesis under discussion, the arguments based on the holo-graphic picture [16] are rather convincing. The gravita-tional anomaly takes place for fields of any spin (see e.g.[15, 22]). We argue that the universality of spin-gravityinteraction provides a strong argument in favor of theexistence of other chiral effects in the spin currents for s > s >
Note added –
Soon after we released this work, thefirst study of the bCVE in the context of chiral kinetictheory appeared in [41].
ACKNOWLEDGMENTS
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