Chiral vortical effect for vector fields

We consider photonic vortical effect, i.e. the difference of the flows of left- and right-handed photons along the vector of angular velocity in rotating photonic medium. Two alternative frameworks to evaluate the effect are considered, both of which have already been tried in the literature. First, the standard thermal fied theory and, alternatively, Hawking-radiation-type derivation. In our earlier attempt to compare the two approaches, we found a crucial factor of two difference. Here we revisit the problem, paying more attention to details of infrared regularizations. We find out that introduction of an infinitesimal mass of the vector field brings the two ways of evaluating the chiral vortical effect into agreement with each other. Some implications, both on the theoretical and phenomenological sides, are mentioned.


INTRODUCTION
We will consider thermodynamics of media whose constituents are massless particles of non-zero spin S. The best studied case is S = 1/2 and , as a starting point we quote some results obtained for spin 1/2 constituents. However, we are mainly interested in properties of photonic media, consisting of left-and right-handed photons. Since recently, the number of papers devoted to this case has also been growing, see in particular [1][2][3][4][5][6][7][8][9].
The chiral vortical effect was first evaluated by A. Vilenkin [10] who considered gas of non-interacting spin-1/2 fermions in a rotating coordinate system. It was demonstrated that there exists a current of the particle number flowing along the vector of the angular velocity Ω. Numerically, in case of a single right-handed Weyl spinor the current is given by: where µ R is the chemical potential, and T is the temperature. Nowadays, the literature on the chiral effects is huge and we cannot even briefly review the subject. Here we mention only the pioneering paper [11] which opened the chapter on theory of the chiral effects in the regime of strong coupling. It turns out that in the hydrodynamic approximation and in absence of dissipation one can derive chiral effects without exploiting the noninteracting gas approximation. Moreover, the magnitude of the chiral effects is determined uniquely in terms of the corresponding chiral anomaly of the fundamental theory underlying the phenomenological hydrodynamic approach, for review see the volume [12].
In particular, in case of the chiral vortical effect (1) the term proportional to the chemical potential squared is indeed related to the chiral anomaly. There is a simple substitution which allows to generate chiral hydrodynamic effects from the standard chiral anomaly [13]: where A α is the electromagnetic potential, e is the charge of the fundamental constituents and u α is the 4-velocity of an element of the medium. One can readily check that the substitution (2) does reproduce the µ 2 R term in the Eq. (1).
On the other hand, any field-theoretic interpretation of the T 2 term in the Eq. (1) had been missing until the recentl paper [14] which relates it to the gravitational chiral anomaly (for earlier attempts in the same direction see [15,16]). In case of massless spin 1/2 particles interacting with external gravitational field the anomaly reads: where R µνκλ is the Riemann tensor, R µνκλ = 1 2 ǫ µνρσ R κλ ρσ . To bridge (3) to the chiral vortical effect (1) one exploits the construction similar to the one introduced first to relate the Hawking radiation from a black hole to the field-theoretic anomalies [17]. Namely, one considers a space-time with a horizon. What is specific for the horizon is that there is a flow of particles from the horizon and absorption of the particles falling onto the horizon. One can say, there is a right-left asymmetry at the horizon. The rate of the particle production at the horizon can be fixed in terms of the anomalies of the field theory, or of the gravitational field on the horizon. In more detail, the relevant anomaly looks as:: where ǫ µα is the 2d antisymmetric tensor, c is the central charge and R is the Ricci scalar. On the other hand, far off from the horizon the flow of the particles can be compared to the thermal radiation. It was demonstrated [17] that the matching of the two expressions for the flow of the particles reproduces the Hawking, or Unruh temperature. The calculation is to be performed for each spherical wave separately and cumbersome technically. The paper in Ref. [14] evaluates the chiral vortical effect in a similar way. There is a significant simplification, however. Namely, the metric introduced is not a solution of the Einstein equations but rather imitates rotation of a fluid: At large distances z the function f (z) tends to unit, and the metric (5) reduces to that of the flat space in cylindrical coordinates: Integrating the r.h.s. of the Eq. (3) one evaluates the flow of particles at large z which is to be identified with the chiral thermal vortical effect, by the logic of the construction. The translation from the field theory to the thermal physics is achieved through the identification: where a horizon and Ω horizon are the gravitational acceleration and angular velocity on the horizon, while T and Ω are the flat-space values of the temperature and of angular velocity, respectively, as measured at large z. see eq. (5). It was demonstrated [14] that in this way one reproduces the T 2 term in the Eq. (1). Which is an amusing success of the modern ideas on the relation between gravitational acceleration and thermal physics.
In Ref. [8] it was suggested to extend the checks of the theory [14] by considering quantum particles of higher spin S. In particular, we concentrate on the S = 1 case, or photons. The corresponding gravitational anomaly was introduced in Ref. [18]. Using the machinery just described one can turn the knowledge of the gravitational anomaly into a prediction of the magntude of the chiral vortical effect for photons. Moreover this predition can be compared with the results of direct calculations by means of various techniques within the thermal field theory, see in particular [1][2][3][4][5][6][7].
There is a disagreement of a factor of two between the two ways of evaluating the chiral vortical effect for photons [8]. Here we revisit the problem of comparing various results for the photonic vortical effect. The main point we are emphasizing now is that both the evaluation of the gravitational anomaly and of the chiral vortical effect involve regulaization procedures. For our, pure theoretical purposes, we need identical regularizations on the both sides (gravitational and flat-space ones). Our overall conclusion here is that, in the sense indicated, there is no direct contradiction between the two ways of evaluating the chiral vortical effect for photons.

CHIRAL VORTICAL EFFECT IN THE EQUILIBRIUM
Massless spin-1/2 particles There are different ways of evaluating the chiral vortical effect in the one-loop approximation. The most straightforward way is to find energy levels, evaluate the current for each mode and weight the results with the Fermi or Bose (whichever is relevant) distribution of the levels. In particular, this was the strategy adopted first in the pioneering work [10]. In this section we review briefly evaluation of the photonic chiral effect in the equilibrium. In the next section we comment on the applications of the Kubo relations.
We begin with quoting the results obtained first in [10] in the form most suitable for generalizaions to higherspin cases. For the sake of normalization, let us remind the reader that the Fermi distribution for a single Weyl fermion is written as: The degree of freedom with a positive energy, describes a fermion polarized along its momentum. The second, subtraction term in the r.h.s. of Eq. (8) is introduced to ensure vanishing of the density of the energy at T, µ R = 0. Alternatively, the Fermi distribution (8) can be written in terms of states with positive energies, which unifies particles and anti-particles: where we keep the overall factor of 2 in the r.h.s. to emphasize that there are two degenerate levels for each energy. Now, the statistically averaged matrix element of the current can be represented in the following form [10,14] convenient for interpretation: where J N is written, as usual for the case of a single righthanded Weyl fermion, and we restore a non-vanishing chemical potential, µ R = 0. Moreover, the first term in the parentheses represents contribution of particles while the second term refers to the anti-particles. Upon integration over the energy, Eq. (10) reduces to the Eq. (1).
The interpretation of the Eq. (10) is straightforward. Indeed, by introducing the rotation we remove the twofold degeneracy of all the levels and get two levels split by the energy ∆E = Ω. Indeed, in the equilibrium one introduces an effective interaction: whereˆ M is the angular momentum operator for spin S=1/2. Note that such an interpretation assumes that the quantization is performed in the cylindrical coordinates (while the Eq. (8) can be derived, say, in the Cartesian coordinates as well). Therefore the energy levels now correspond to the states which have a definite projection of the momentum on z-axis, p z and projection of the angular momentum, L z .

Chiral photonic current
The notion of chirality for photons is well known. Namely, the left-and right-handed polarized photons are chiral states since they correspond to a certain projection of the spin of the photon on its momentum, S p = ±1. The chiral current, therefore, can be defined as where A µ is the vector potential of the electromagnetic field, and we also reserved for a non-trivial determinant, g of the metric tensor. The current (12) is defined in such a way that the eigenvalues of the associated charge are indeed ±1. Note, however, that the current (12) is not gauge invariant, and we should be careful to associate observables only with a kind of gauge-invariant projections of K µ . In particular, a well-known example of such a gauge-invariant observable is the charge coresponding to the current (12): where H is the magnetic field. Then, under the gauge transformation, δ gauge A i = ∂ i Λ, the variation of the magnetic-helicity charge density is given by: Here ∇ · H = 0 by virtue of the equations of motion while the integral over d 3 x from the first term becomes a boundary term and can be neglected. This completes the proof that the Q magnetic helicity is gauge invariant. Note that in the momentum space definiton of the charge assumes q i ≡ 0, q 0 → 0 limiting procedure (where (q 0 , q i ) is the 4-momentum carried by the current).
Note also that the matrix element of the operator of the magnetic-helicity charge,Q magnetic helicity counts the difference between the numbers of the left-and righthanded photons n L,R : In other words, the normalization of the chiral currents for spins S = 1/2, 1 is similar. Now, to evaluate the photonic vortical effect we need to consider the spatial component, K of the current (12). Moreover, since we are interested in the physics of equilibrium we have to consider the static (or stationary) limit with no time dependence. In the momentum space, as is first emphasized by [16], we are interested in the limiting procedure, q 0 ≡ 0, q i → 0. Let us check that the current K = A 0 H + A × E is gauge invariant in this limit. Under the gauge transformation: As a result, we get the local term proportional to which vanishes because of the equations of motion, plus total derivatives which become boundary terms upon the integration over the volume, or time.
To summarize, the apparent gauge-dependence of the chiral photonic current does not imply, generally speaking, that the chiral photonic current in the equilibrium is gauge dependent. There is a reservation, however, that the problem considered is infrared sensitive. In particular, considering uniform rotation everywhere in the space is inconsistent with finiteness of the speed of light, for discussion see, for example, [6]. Thus, neglecting the boundary terms just discussed above might be in conflict with some other constraints on the behaviour of the fields on the boundaries.
It might worth emphasizing that the current (12) is not a Noether current and its conservation is not automatic: whereF µν = 1 2 √ −g ǫ µναβ F αβ However, on the mass shell, or for the electromagnetic plane waves the r.h.s. of Eq. (17) vanishes, and the current (12) is conserved for free photons. What is even more fascinating, one can introduce non-trivial dynamics through interaction of the photons with external gravitational fields, and the chiral photonic current is still (naively) conserved. We come back to discuss this point further later.

Photonic vortical effect
After these preliminary remarks we are set to consider a direct, a la Vilenkin evaluation of the chiral vortical effect for photons. There are no general reasons to believe that such a calculation is less reliable than the showcase [10,14] of spin-1/2 massless fermions. Probably, the naive expectation would be that the final result for the photons is very similar to (1), with some obvious changes (that is, increasing the splitting, due to the rotation, between the levels by a factor of two, due to the spin of the photon, and replacing the Fermi distribution by the Bose distribution). The hard work of quantizing photons in the cylindrical coordinates, finding the levels and the corresponding wave functions, evaluating the magnetichelicity current on the modes has been done in Ref. [6], with the following result: Almost everything looks like what we expected to find naively. Except for the bald-faced overall factor of 2/3. Thus, in the rest of this section we will look for a convincing interpretation of this factor. For further discussion of the relation between the cases of massless and nearly massless photons see also Appendix We introduced the current K µ , with an idea that it provides us with a unique definition of the chirality of the photon. However, it is actually well known that there exist various currents which in case of massless particles measure their spin or chirality, see [18], and references therein.
The best known spin-related 4-vector has been introduced by Pauli and Lubanski [19]. This vector, Γ µ is defined in the most general terms of the generators of the Poincare group: where P ν и M κλ are the generators of the coordinate shifts and of the Lorentz rotations, respectivrly. For the one-partilce state of a massless field, p 2 = 0, of spin S we have: where λ is the helicity λ = ±S. The helicity plays a crucial role because it determines the energy splitting induced by the rotation. Indeed, we already quoted the expression (11) for the effective interaction introduced in the statistical physics to describe the equilibrium. Once we quantize the projection of the angular momentum onto the vector Ω the energy induced by the effective interaction becomes equal to: where λ is the helicity. In view of this, it is useful to introduce a current j µ helicity , such that the corresponding charge is equal to the helicity of the one-particle states. Explicit expression for the j µ helicity is: where S νρλ is the spin part of the density of the angular momenum density in the Lagrangian formaism: where (Σ ρλ ) is the represntation of the generators of the Lorentz rotations on the fields φ a : (Σ ρλ ) ab = i(g ρa g λb − g ρb g λa ).
The crucial point is that for one-particle massless states the values of K µ and of j µ helicity are proportinal to each other but are not identical. In particular for a 4-vector field: while in case of the Dirac field the matrix elements of the currents j µ helicity and of K µ coincide with each other. Apparently, the factors of 2/3 in Eqs (10) and (25) are of pure geometric origin and related to each other. But at the moment we are not aware of any clear derivation of such a relation. Note also that these states do not coincide with the propagating states. To construct propagator in terms of the equilibrium states one needs to derive reexpansion of one complete set of functions over the other set. A similar recent analysis can be found in Ref. [20].
Coming back to our main problem of evaluating the photonic vortical effect, our next step is the introduction of the infrared regularization by a finite photon mass m γ = 0.

Chiral anomaly and infrared regularization
Introduction of m γ = 0 is a logical step, within our approach. Indeed, we are going to compare predictions for the photonic vortical effect obtained in two different ways, namely, in terms of the gravitational chiral anomaly for the K µ current and and in terms of the statistically averaged matrix element of the same current. As we remind the reader next, a finite photon mass is introduced to regularize in the infrared the gravitational anomaly, [18]. Therefore, we are invited to consider the statistical-theory approach at a finite photon mass as well.
To substantiate the point, let us reiterate basic steps of derivation of the photonic gravitational anomaly [18]. As we already mentioned, see Eq. (17), there is no conservation of the K µ current off-mass shell. However, for electromagnetic waves, or on-mass shell E · H = 0 and the current is conserved. It is only natural then that, upon inclusion of interaction of photons with external gravitational field, we expect covariant conservation, However this, "naive" expectation is to be checked against possibility of existence of an anomaly. To uncover the anomaly, one considers the matrix element of transition of the K µ current into two gravitons, in the annihilation channel. If the gravitons are on the mass shell the matrix element is defined in terms of a single form factor f (q 2 ): where q µ is the 4-momentum, carried in by the K µ current. The next step is to use dispersion relations to evaluate f (q 2 ). The imaginary part, imf (q 2 ) is given by tree graphs and, naively, it respects all the symmetries of the problem. However, a direct calculation of the imaginary part fails because one of the propagators of intermediate particles has a pole which-for all the particles being massless-falls onto the physical region of integration. To regularize the calculation one introduces then an infinitesimal photon mass. As a result: where v is the velocity of the intermediate photons in the c.o.m. system. Finally, the real part of f (q 2 ) corresponding to (27) is given by: (28) whereR µνκδ = (1/2)ǫ κδρσ R µν ρσ . Eq. (28) is nothing else but the photonic gravitational anomaly.
Following then Ref. [14] we conclude that the photonic vortical current is predicted to be: or four times larger than that for massless spin-1/2 particles [8].
Turn now to the statistical-theory approach. Introdcution of a finite photon mass simplifies the evaluation of the chiral vortical effect greatly. The reason is that, for massive photons, we recover the factorization property which makes Eq. (10) to look so simple. Namely, we start with non-interacting gas of massive photons in absence of the rotation. Each level is degenerate three times since projection of the spin of the massive photon is now S z = ±1, 0. Account for the rotation splits the levels so that the energies now are ǫ + Ω, ǫ, ǫ − Ω. These energy differences are readily calculable in the rest frame of the massive photon and are invariant under the boosts along the z-axis. In case of massless photons there is no rest frame for photons and this makes the calculation much more involved, for further comments see, in particular, [9].
Thus, for massive photons the vortical current is gven by: in agreement with the prediction (29). This coincidence of the results obtained in case of massive photons within the thermal field theory and via the gravitational anomaly is our main result in these notes. In view of this, we will check it against calculations of the vortical effect by means of the Kubo relations.

Kubo-type relation in case of massive photons
We are interested to evaluate the coefficient σ V entering the definition of the vortical current J 5µ : where u µ is the 4-velocity of an element of the fluid. The Kubo-type relation fixes the coefficient σ V in terms of the correlator between the spatial components of the current K i and the T 0j component of the energy-momentum tensor [1,2,15]: where p k is the momentum brought in by the current K i , where A µ is now the field describing massive photons. The propagator of the massive vector field in the momentum space is given by: In case of massless photons, the correlator (32) was calculated in Refs. [1,2], with the result quoted above, σ V (S = 1) = T 2 /6. We are calculating now the change in σ V (S = 1) due to m γ = 0. It turns out that in the limit m γ ≪ T there is a finite jump in the value of σ V (S = 1) which stems from the cancellation of the factor m −2 γ in the propagators (34) and of the factor m 2 γ in the component T 0j , see Eq. (33). The final result is: This result is in full agreement with the expectations (29) based on the gravitational anomaly and with the evaluation (30) of the vortical effect within the statistical approach for massive photons.

Vector meson chirality and baryon polarization
One may ask whether the chirality of massive vector particles has any phenomenological implications. The answer is provided by the relation of axial charge to the average polarization of baryons which is the way to implement the quark-hadron duality in this problem [35] which may be realized in both kinetic [36] and hydrodynamic [37] calculations. There is a natural explanation of the fact theΛ polarization is larger than that of Λ as the same (C-even) chiral charge is distributed between the smaller number of particles. For quantitative description of the effect It is mandatory [38] to take into account the axial charge carried by K * mesons. Therefore, their chirality is implicitly present here. The role of the numerical factors studied here depends on the assumptions on the distribution of chirality or axial charge between baryons and mesons and remains to be studied.
It is interesting whether meson chirality can affect the measured tensor polarization of vector mesons [39]. One should stress that contrary to baryon polarization it is P-even quantity and may emerge due to the product of quark polarizations as well as due to their spin correlations [40]: where enter the parallel and orthogonal to quantization axis components of tensor C containng contributions of average quark polarizations and (boldfaced) correlations. The relative smallness of first term is implied by its relation to squared baryon polarization and squared vorticity [41] so that the terms probing the entanglement of quark spins may play the dominant role.
At the same time, the role of squared vorticity may be overtaken by square (and higher even powers) of magnetic field [42,43]. The emerging longitudinal polarization is related [44] to the conductivity in magnetic field [45,46] and supports its growth.
Let us finally note that vector (related to chirality) and tensor polarizations are mixed in the positivity constraints and invariants of density matrix [47,48] providing another possible direction of experimental investigations.

DISCUSSION AND CONCLUSIONS
As is noticed in Ref. [8], knowing the gravitational chiral anomaly and following the logic of Ref. [14] one can predict the value of the chiral vortical effect for any spin S of the massless constituents: where σ V is defined in Eq. (31). Note that there are no free parameters in this prediction. Alternatively, one can calculate σ V within the framework of the thermal field theory and, in this sense, test the Eq. (38). First, and with great success, the prediction (38) was tested in the original paper [14] in case of massless spin-1/2 constituents. This case is remarkable for the fact that σ V (S = 1/2) was evaluated in a few ways within the statistical approach, and the value σ V (S = 1/2) = T 2 /(12) is well established and non-controversial.
Proceeding to the case S = 1 [1-9] we notice σ V (S = 1) obtained in the literature [1,2] on the basis of the Kubo relation differs from (38) by a factor of 2. Moreover, these results themselves are not without controversy, see [1][2][3][4][5][6][7][8][9] which is not easy to resolve. Finally, consideration of the limit of large spin S apparently brings Eq (38) to a qualitative disagreement with the thermal field theory.
It is on this background that we have to appreciate significance of the new observation that for a massive vector field there is full agreement of results for σ V (S = 1, m γ = 0) obtained within the field-theoretic and statistical approaches.
As is mentioned above, the main argument against the duality between the statistical and anomalybased approaches is an apparent conflict between the predictions for the chiral vortical effect obtained within the two frameworks in the limit of large spin S. The finding that in case of the vector field introduction of a finite mass m γ = 0 brings the consequences from the Kubo-type relation and from the anomaly into agreement with each other does not settle by itself the issue of the violation of the duality for large spin S. However, increasing spin S generically makes the theory more and more dependent on details of the infrared regularization [8]. In particular, the rotational vacuum becomes unstable at spin S ≥ 3/2. This infrared instability does not affect directly the derivation [26] of the graitational anomaly. However, beginning with S = 3/2 one has to assume that the infrared issues are settled somehow without changing prediction (38) which, in the language of the thermal field theory, is expected to be saturated by contribution of high energies of order temperature.
The results for the chiral vortical effect obtained at m γ = 0 make the validity of this extra assumption more questionable. Indeed, we have demonstrated that introduction of m γ ≪ T results in a finite jump in the vaue of σ V (S = 1). Explicit evaluation of σ V in case of S = 3/2 becomes a crucial step to be made. (For a recent discussion of the theory of massless charged spin-3/2 particles see [49]).
We conclude this section with a remark on possible phenomenological implications of the evaluation of σ V (S = 1). The point is that in case of superfluidity the chiral vortical efect can be manifested through polarization, or spin of heavy particles (for details and references see [50,51]). In particular, in case of superfluidity the average value of the vortical current < J 5 > is equal to the spin density carried by the cores of the vortices, which, in turn, is equal to the spin density carried by heavy particles < σ heavy > [52,53]: In view of the low viscosity of the quark-gluon plasma such a relation might work well in case of heavy-ion collisions [53]. Usually the relation (39) is used in case of spin-1/2 constituents and applied to hyperons, as heavy particles. Since σ V (S = 1) = 4σ V (S = 1/2) one can speculate that the contribution of heavy mesons is not less important than the contribution of hyperons.

ACKNOWLEGMENTS
We are grateful to A. I. Vainshtein and P. G. Mitkin for discussions and critical remarks. The work was partially supported by the grant RFBR18-02-40056.

APPENDIX
In the main body of the paper we considered the cases of strictly massless photons and of photons with infinitesimally small mass. The predictions for the photonic vortical effect differ by a finite factor of 2/3, see Eq. (18). Nevertheless, it is rather obvious that the two results are absolutely consistent with each other. To appreciate this, we should be more careful to formulate the question to be answered.
Let us start with no rotation. Then, depending on whether m γ is strictly zero or small we have energy densities which differ from each other by a factor of 2/3: If we switch on the effect of the rotation, the number of levels is not changed. Moreover the distribution between the levels with L z = +1, 0 , −1 apparently is the same for massless photons and photons with infinitesimal mass (i.e. m γ ≪ T ). For this reason the factor of 2/3 which reflects the difference in the total number of levels goes through to the final answer for the chiral vortical effect. Thus, the factor of 2/3 in Eq. (18) reflects so to say renomalization of the total amount of thermal energy stored in the system. On the technical side, we argued that the simple, "factorized" form of distribution of levels is valid for massive particles and is not valid for strictly massless partciles. This is in accord with the theoretical expectations. The difference vetween massive and strictly massless cases goes back to the fact that for massive particles the 4-momentum vector is ortogonal to the Pauli-Lubanski vector while for strictly massless cases the two vectors are parallel to each other, for a related discussion see [9].