Photon Self-energy in Magnetized Chiral Plasma from Kinetic Theory
aa r X i v : . [ h e p - ph ] F e b Photon Self-energy in Magnetized Chiral Plasma from Kinetic Theory
Han Gao, ∗ Zonglin Mo, † and Shu Lin ‡ School of Physics and Astronomy,Sun Yat-Sen University, Zhuhai 519082, China (Dated: February 20, 2020)
Abstract
We study the photon self-energy in magnetized chiral plasma by solving the response of electro-magnetic field perturbations in chiral kinetic theory with Landau level states. With lowest Landaulevel approximation and in collisionless limit, we find solutions for three particular perturbations:parallel electric field, static perpendicular electric and magnetic field, corresponding to chiral mag-netic wave, drift state and tilted state, from which we extract components of photon self-energy indifferent kinematics. We show no solution is possible for more general field perturbations. We arguethis is an artifact of the collisionless limit: while static solution corresponding to drift state andtilted state can be found, they cannot be realized dynamically without interaction between Landaulevels. We also discuss possible manifestation of side-jump effect due to both boost and rotation,with the latter due to the presence of background magnetic field. ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION There has been a long history of effort towards understanding of vacuum polariza-tion by electromagnetic fields. The full effective action of vacuum for arbitrary constantelectromagnetic field was established by Heisenberg and Euler [1], which predicted criticalelectric field in vacuum. It was later realized by Schwinger [2] that the critical electric fieldleads to pair production. On the other hand, while the magnetic field does not destabilizethe vacuum, it does modify vacuum properties: such as enhancing the pair production rate[3] and causing vacuum birefringence [4, 5].Recently there has been growing interests in the effect of magnetic field in chiralmedium in a variety of systems including quark-gluon plasma and Weyl semimetal etc. Themagnetic field in chiral medium is known to lead to novel anomalous transport such as chi-ral magnetic effect [6–9], chiral separation effect [10, 11] and chiral magnetic wave [12] etc.Furthermore, the presence of magnetic field also modifies existing transport coefficient likeconductivities [13–20] and viscosities [21, 22] nontrivially. In the regime of linear response,these transport phenomena are characterized by photon self-energy in magnetized chiralmedium. In the presence of magnetic field, the photon self-energy contains very rich struc-ture and is also very complicated in general. There have been many field theoretic attemptsto study the photon self-energy [23–26], see also studies on gluon self-energy [27, 28].A distinguishing feature of chiral fermion from classical particle is its spin, which isa genuine quantum quantity measured in ~ . A semi-classical expansion in ~ gives rise tothe chiral kinetic theory (CKT) [29–54]. It has been successfully applied to study transportphenomena of chiral medium in response to weak electromagnetic field [55–58], where eachpower of electromagnetic field contributes O ( ~ ). In the regime of strong magnetic field, adifferent expansion scheme is used giving rise to a chiral kinetic theory based on Landaulevel (LL) states [15, 19, 59]. The purpose of this paper is to apply this chiral kinetic theoryto study photon self-energy in magnetized chiral plasma as an alternative approach to theself-energy problem. For simplicity, we work in the strong magnetic field and collisionlesslimit. We will reproduce the field theoretic results to the accuracy of CKT approach andshow some new results for drift state.The paper is organized as follows: in Section II, we summarize generalities of photonself-energy; in Section III, we give a short review of chiral kinetic equations based on LLstates and analyze the structure of equations; in Section IV, we present solutions corre-sponding to three specific perturbations and discuss the physical implications of them. We2urther show it is not possible to obtain more solution for more general perturbations. Wewill argue it is an artifact of collisionless limit; in Section V, we summarize the results anddiscuss future directions.Throughout this paper, we primarily study chiral medium consisting of right-handedfermions with charge Q = | e | . Contribution of left-handed fermions will be added whencomparing with field theoretic results. For simplicity, we set e = 1 and reinstate it in theend. We use the following non-standard convention p µ = ( p , p , p , p ) for convenience.The magnetic field points in x direction. II. GENERALITIES OF PHOTON SELF-ENERGY IN MAGNETIZED MEDIUM
The photon self-energy in imaginary time formalism is defined by [60]Π µνE ( x, x ′ ) = h ˆ T (cid:0) J µE ( x ) J νE ( x ′ ) (cid:1) i = δJ µE ( x ) δA Eν ( x ′ ) , (1)where ˆ T denotes time ordering in Euclidean time. We can rewrite (1) in a simpler form inmomentum space: δJ µE ( q ) = Π µνE ( q ) δA Eν ( q ) . (2)Note that the Euclidean frequency q takes discrete values of Matsubara frequencies 2 πnT ,with T being temperature. It can be analytically continued to complex frequency plane.Taking q → i ( q + iǫ ), we obtain the more useful retarded photon self-energy δJ µ ( q ) = Π µνR ( q ) δA ν ( q ) , (3)with J E ( q ) → iJ ( q ) and A E ( q ) → iA ( q ). (3) expresses current as a response to externalelectromagnetic field perturbation, which can be studied in kinetic theory. We will mainlyuse the retarded self-energy in the paper.For parity breaking chiral medium consisting of right handed fermions, Π µνR is ingeneral not symmetric in the Lorentz indices. Nevertheless, Π µνR is still constrained byanomalous Ward identity. To derive anomalous Ward identity in the regime of strongmagnetic field, we note that there is effective dimensional reduction from 3 + 1 D to 1 + 1 D .In this regime, the most interesting perturbations are time and longitudinal components ofphoton field δA ≡ a and δA ≡ a . For right handed current in the background magneticfield, the Ward identity is given by ∂ µ J µ = 1(2 π ) E B. (4)3ere E = ∂ a − ∂ a is electric field induced by perturbations. Fourier transforming (4)and doing variation with respect to a and a , we obtain the following constraints: q µ Π µ R = 1(2 π ) ( − q ) B,q µ Π µ R = 1(2 π ) q B. (5)Note that the anomalous Ward identity (5) involves all components of self-energy. We willuse chiral kinetic theory to study the response. III. CHIRAL KINETIC EQUATIONS WITH LANDAU LEVELS
The chiral kinetic equations with Landau level states are given by [19]∆ j + ∆ i j i = 0 ,p j + p i j i = 0 , ∆ j i + ∆ i j + 2 ǫ ijk p j j k = 0 , − p j i − p i j + 12 ǫ ijk ∆ j j k = 0 , (6)with ∆ µ = ∂ Xµ − ( F µν + f µν ) ∂∂p ν for µ = 0 , , ,
3. We use Greek letters for spacetime indicesand small Roman letters for spatial indices. F µν corresponds to background magnetic fieldwith the only nonvanishing components F = − F = − B . (6) is derived based on anexpansion in ~ , or equivalently in ∂ X . It is valid up to O ( ∂ X ) and to all order in B , whichimplicitly assumes the hierarchy of scales ∂ X ≪ p ∼ √ B . f µν = ∂ Xµ a ν − ∂ Xν a µ correspondsto perturbation of electromagnetic field, which is counted as O ( ∂ X ). Solving (6) we canobtain j µ and the momentum integral of j µ gives the induced current J µ J µ ( X ) = Z d pj µ ( X, p ) . (7)In the absence of perturbation, the background in the LLL approximation is given by j = j = 2(2 π ) δ ( p + p ) e − p T /B f ( | p | ) , j = j = 0 . (8)In equilibrium, the distribution function is given by Fermi-Dirac distribution f ± ( | p | ) = e ( | p |∓ µ ) /T +1 with the upper/lower sign for positively/negatively charged LLL states. HigherLL states are massive from the 1 + 1 D point of view with mass ∼ √ nB , thus their contri-bution are exponentially suppressed ∼ e −√ nB/T .Since (6) is valid to O ( ∂ X ), we seek solution of j µ order by order in gradient: δj µ = δj µ (0) + δj µ (1) + · · · , (9)4ith the subscript indicating order of gradient. We use δ to distinguish the induced j µ fromthe background one. Terms of O ( ∂ X ) are beyond the accuracy of (6). Substituting (9) into(6), we obtain to order O ( ∂ X ) and O ( ∂ X ) respectively D i δj i (0) = 0 ,D i δj + 2 ǫ ijk p j δj k (0) = 0 ,p δj + p i δj i (0) = 0 , − p δj i (0) − p i δj + 12 ǫ ijk D j δj k (0) = 0 , (10)and D i δj i (1) = S,D i δj + 2 ǫ ijk p j δj k (1) = V i ,p δj + p i δj i (1) = 0 , − p δj i (1) − p i δj + 12 ǫ ijk D j δj k (1) = V i , (11)with the right hand side defined as S = − (cid:16) δ ∆ j + δ ∆ i j i + ∂ δj + ∂ i δj i (0) (cid:17) ,V i = − (cid:16) δ ∆ j i + δ ∆ i j + ∂ δj i (0) + ∂ i δj (cid:17) ,V i = − (cid:18) ǫ ijk δ ∆ j j k + 12 ǫ ijk ∂ j δj k (0) (cid:19) . (12)We have defined D i = − ∂∂p j F ij and δ ∆ µ = − ∂∂p ν f µν . We also use the short-hand notation ∂ µ = ∂ Xµ . The structure of the equations are quite informative: (10) and (11) can beviewed as equations for δj µ (0) and δj µ (1) respectively. The only difference is that the formerare homogeneous and the latter are inhomogeneous. The homogeneous equations can’tuniquely determine δj µ (0) . The source of inhomogeneous equations involves perturbationsand the undetermined δj µ (0) . Nevertheless, (10) and (11) can still be solved thanks to theover-determinancy of the equations, which we will elaborate in the next section.Before closing this section, we verify that the first equation of (11) is consistentwith the anomalous Ward identity. We integrate the equation over four momentum andreorganize it as. Z d p (cid:16) ∂ µ δj µ (0) (cid:17) = − Z d p (cid:16) D i δj i (1) + δ ∆ j + δ ∆ j (cid:17) . (13)By our non-standard convention, we identify the left hand side (LHS) as ∂ µ J µ . For the righthand side (RHS), the first term becomes boundary terms upon integration over transverse5omentum Z d p T D i δj i (0) = Z d p T ∂∂p M Bǫ MN δj N (0) = 0 , (14)with the capital Roman letters run over indices in the plane perpendicular to the backgroundmagnetic field M, N = 1 ,
2. The second terms can be written explicitly as Z d p (cid:20)(cid:18) ∂∂p M f M + ∂∂p f (cid:19) j + (cid:18) ∂∂p f + ∂∂p M f M (cid:19) j (cid:21) . (15)The terms involving ∂∂p M vanish for the same reason as above. Using (8) and includingcontribution from both positively and negatively charged LLL states, we can combine theremaining terms as Z d p (cid:18) ∂∂p − ∂∂p (cid:19) f π ) exp ( − p T /B ) δ ( p + p ) ( f + ( | p | ) − f − ( | p | ))= E B (2 π ) (cid:18)Z ∞ dp (cid:18) − ∂∂p (cid:19) f + ( | p | ) − Z −∞ dp (cid:18) − ∂∂p (cid:19) f − ( | p | ) (cid:19) = E B (2 π ) ( f + (0) + f − (0)) = E B (2 π ) . (16)Therefore, we reproduce anomalous Ward identity. It also shows the anomalous Wardidentity is saturated by zeroth order solution in the regime of strong magnetic field. IV. SELF-ENERGY FROM SOLUTIONS TO KINETIC EQUATIONS
In this section, we present solutions to (10) and (11), which allow us to extractcomponents of self-energy in different kinematics, which in fact correspond to differentstates. After presenting three simple solutions, we will show no more solution is possible.We will argue this is an artifact of the collisionless limit.
A. Parallel E field: chiral magnetic wave We begin with the case of parallel E field, which can be induced by either a ( t, x )or a ( t, x ). This case can be simplified by noting that the longitudinal motion of LL statesis classical. In the LLL approximation, parallel electric field only induces redistribution ofLLL states. Since we know LLL state satisfies homogeneous equation [19], we expect δj µ (0) proportional to LLL state and δj µ (1) = 0. It follows that (10) are satisfied automatically. Inorder for (11) to hold, we need to require the inhomogeneous terms to vanish: S = V i = V i = 0. It gives the following constraint on δj µ (0) f (cid:18) ∂∂p − ∂∂p (cid:19) j − ( ∂ + ∂ ) δj = 0 . (17)6n arriving at (17), we have used property of LLL state: δj = δj and assumed all ∂ M to vanish because the field perturbation f = ∂ a − ∂ a is independent on x T . (17) canbe solved easily in momentum space as δj = δj = 2(2 π ) exp ( − p T /B ) δ ( p + p ) f ′ ( p ) q q + iǫ + q a ,δj = δj = − π ) exp ( − p T /B ) δ ( p + p ) f ′ ( p ) q q + iǫ + q a , (18)for perturbations a ( t, x ) and a ( t, x ) respectively. Again f ( p ) can be f ± ( | p | ) for solu-tions corresponding to positively and negatively charged LLL states. We have made thesubstitution q → q + iǫ so that the solution corresponds to retarded response.Integrating the solution (18) over four momentum and using the following identity Z ∞ dp f ′ + ( | p | ) − Z −∞ dp f ′− ( | p | ) = − f + (0) − f − (0) = − , (19)we obtain the following retarded self-energy components from (3)Π ( q , q ) = Π ( q , q ) = − e B (2 π ) q q + iǫ + q , Π ( q , q ) = Π ( q , q ) = e B (2 π ) q q + iǫ + q . (20)We have reinstated powers of e in the above. Note that (20) is independent of temperatureand chemical potential. This is because the integration over p only picks up boundaryterms at p = 0, similar to (16). We can verify (20) indeed satisfies (5). The pole ofΠ µν gives dispersion relation q + q = 0 of collective excitation of the chiral medium. Itcorresponds to a wave propagating with speed of light in the x direction. In fact, this isnothing but chiral magnetic wave in the limit of strong magnetic field [12].It is also interesting to note Π µν is not symmetric with respect to Lorentz indices.The reason is that the state consisting of right-handed fermions are not parity invariant.Applying parity transformation on (20), we obtain components of self-energy for mediumconsisting of left-handed fermionsΠ ( q , q ) = − Π ( q , q ) = e B (2 π ) q q + iǫ − q , − Π ( q , q ) = Π ( q , q ) = e B (2 π ) q q + iǫ − q . (21)These results can also be obtained from explicit solution of chiral kinetic theory for left-handed medium, which we collect in Appendix A. Adding up contributions from bothleft-handed and right handed fermions, we obtain the following components of self-energy7or parity invariant stateΠ ( q , q ) = e B (2 π ) q ( q + iǫ ) − q , Π ( q , q ) = Π ( q , q ) = − e B (2 π ) q q ( q + iǫ ) − q , Π ( q , q ) = e B (2 π ) q ( q + iǫ ) − q . (22)These agree with field theoretic results in the LLL approximation up to an overall factorof e − q T /B [61]. The exponential factor at least is O ( ∂ X ), which lies beyond the accuracy ofour current chiral kinetic equation. B. Static perpendicular E field: drift state Next we consider the case of static perpendicular E induced by a ( x T ). We begin bysolving for δj and δj from the third and i = 3 component of the fourth equations of(11) to obtain δj δj = 1 p − p − p p M δj M (1) − p ǫ MN D M δj N (1) p p M δj M (1) + p ǫ MN D M δj N (1) . (23)We proceed with the following ansatz δj M (1) ∝ exp ( − p T /B ) δ ( p + p ) , (24)which converts (11) to the following equivalent equations Bδj M (1) p + p ) − p δj M (1) = − ǫ MN E N ∂∂p j − ǫ MN ∂ N δj , ǫ MN p M δj N (1) = − E M p M B j ,ǫ MN δj N (1) Bp + p + 2 ǫ MN δj N (1) p = E M ∂∂p j + ∂ M δj , (25)with E N = − ∂ N a being the perpendicular electric field perturbation. (25) adopts thefollowing solution δj = δj = a (cid:18) − δ ′ ( p + p ) − p B δ ( p + p ) (cid:19) exp ( − p T /B ) f ( p ) ,δj M (1) = ǫ MN E N B δ ( p + p ) exp ( − p T /B ) f ( p ) ,δj = δj = ǫ MN p M E N B δ ′ ( p + p ) exp ( − p T /B ) f ( p ) . (26)8ntegrating the solution over four momentum and using the following integrals Z ∞ dp p f + ( | p | ) − Z −∞ dp p f − ( | p | ) = π T µ , Z ∞ dp f + ( | p | ) − Z −∞ dp f − ( | p | ) = µ, we find the following components of self-energy after reinstating powers of e Π ( q M ) = Π ( q M ) = − e (2 π ) (cid:18) eB + π T µ (cid:19) , Π M ( q M ) = − e (2 π ) iǫ MN q N µ. (27)Note that δj and δj are odd function of p T , which vanishes upon integration over p T ,thus do not contribute to self-energy. Note that unlike the case of parallel electric field, thecase of perpendicular electric field gives rise to medium dependent self-energy components.In particular the medium dependent terms in Π and Π would not appear in the staticlimit q → E M and B . In the drift state, Hallcurrent is expected and is consistent with Π M above. Π and Π give deviation of chargedensity and current density of the drift state from those of the background. Interestinglythe deviation coincides with 00 component of the self-energy in chiral medium withoutbackground magnetic field [62].We can gain further insight of the drift state by “boosting” the equilibrium state.It is not difficult to see that boosting the equilibrium medium to a velocity − ǫ MN E N B , wehave then orthogonal electric field E M and background magnetic field B [64]. To describeit more quantitatively, we use the covariant form of the background solution j µ = 2(2 π ) δ ( p · ( u + b )) e − p T /B f ( p · u ) ( u + b ) µ , (28)which generalizes the solution in medium frame to arbitrary frame. We verify in appendixB that it indeed satisfies covariant chiral kinetic equations to the lowest order in gradient.Here u µ and b µ denote fluid velocity and magnetic direction and p T = − p + ( p · u ) − ( p · b ) . u µ and b µ are orthogonal to each other u · b = 0. In medium frame, we have u µ = (1 , , , b µ = (0 , , , δu M = ǫ MN E N B and δb = 0. It is easy tosee that δu M leads to δj M (1) . The remaining corrections are not from covariance and areonly present in δj and δj . At zeroth order δj and δj come from the fact that the9lectric field perturbation is not constant but x T dependent. In fact, a in (26) should beinterpreted as a ∼ E M /∂ M , thus the zeroth order correction characterizes redistributionof LL states in response to perturbation. Similar correction to zeroth order solution is alsopresent in chiral kinetic theory without background field [30]. At the first order δj and δj are entirely determined by δj M (1) from (23), which as we discussed above is not sensitiveto x T dependence of the perturbation. Note that δ ′ ( p + p ) f ( p ) = − δ ( p + p ) f ′ ( p ). Itis suggestive to interpret δj and δj as modification of distribution function f ( p ) → f ( p ) − ǫ MN p M E N B f ′ ( p ) = f ( p − ǫ MN p M E N B ), or p → p − ǫ MN p M E N B . This is analogous toside-jump effect in momentum in the absence of background field [35, 63]. Note that sinceour background solution is homogeneous in coordinate, possible jump in coordinate is notvisible from our comparison. We should not confuse the frame vector frequently used indescription of side-jump with the fluid velocity u µ . The latter is needed to define magneticfield in the background. C. Static perpendicular B field: tilted state We turn to the case of static perpendicular magnetic field induced by a ( x T ). Theanalysis is similar to the previous subsection. We will not spell out details. With the ansatz δj M (1) ∝ exp ( − p T /B ) δ ( p + p ), we obtain the following solution δj = δj = (cid:18) − p B δ ( p + p ) exp ( − p T /B ) f ( p ) (cid:19) a ,δj = δj = p M B ⊥ M B δ ′ ( p + p ) exp ( − p T /B ) f ( p ) ,δj M (1) = B ⊥ M B δ ( p + p ) exp ( − p T /B ) f ( p ) , (29)with B ⊥ M = − ǫ MN ∂ N a being the perpendicular magnetic field. It gives rise to the followingcomponents of self-energyΠ ( q M ) = Π ( q M ) = − e (2 π ) (cid:18) π T µ (cid:19) , Π M ( q M ) = − e (2 π ) iǫ MN q N µ. (30)Comparing (30) with the static limit of (20), which vanishes identically, we see the differenceis also medium dependent. The presence of B ⊥ can also be understood as tilt of thebackground. We can see Π M gives precisely chiral magnetic effect for right-handed fermionsdue to B ⊥ : J M = Π M a = − e (2 π ) ǫ MN ∂ N a µ = e (2 π ) µB M . (31)10 M also agrees with components of parity odd self-energy in the absence of backgroundfield [30], which is responsible for chiral magnetic effect.It is interesting to note that this particular components of self-energy actually givesrise to chiral plasma instability [62]. We can see some trace from the backreaction of theinduced chiral magnetic current to the electromagnetic field. The induced magnetic field∆ B can be found by solving the Maxwell equation( ∇ × ∆ B ) M = J M , (32)with the solution given by ∆ B i = e (2 π ) µa δ i . It enhances the original perturbation ofperpendicular magnetic field. The mechanism of enhancement seems independent of thebackground magnetic field. However, to answer the question dynamically, we need to knowthe self-energy away from the static limit.Let us again compare the tilted state with the equilibrium state rotated in such away that the background magnetic field coincides with that of the tilted state [65]. Wecan use the covariant form of background solution (28) with δb M = B ⊥ M /B and δu = 0. δb M gives precisely δj M (1) from the covariant factor ( u + b ) µ . The remaining differencebetween tilted state and equilibrium state is in δj and δj . The tilted state is not entirelyequivalent to rotated equilibrium state because of the x T dependence of perpendicularmagnetic field. The x T dependence leads to the the difference in δj and δj . Onthe contrary, δj and δj are fixed by δj M (1) through (23), thus are not sensitive to x T dependence of the perturbation. Using δ ′ ( p + p ) f ( p ) = − δ ( p + p ) f ′ ( p ), we interpret δj and δj as modification of distribution function: f ( p ) → f ( p ) − p M B M B f ′ ( p ) = f ( p − p M B M B ), or p → p − p M B M B . Like in the case of drift state, it is suggestive to interpretthe modification of distribution function as analog to side-jump effect in momentum in theabsence of background field [35, 63]. Since our background solution is homogeneous incoordinate, possible jump in coordinate is not visible. We stress that this is a new effectdue to the background field: conventional side-jump is manifested through boost, in ourcase side-jump can be manifested through both boost δu and rotation δb , as we see in bothdrift state and tilted state respectively. D. No more solutions
Finally, we look for solution for more general perturbations without using the ansatz.Note that S = V and ǫ MN V N = V M . We can eliminate redundant equations in (11) to11btain D M δj M (1) = 2 ǫ MN p M δj N (1) , (cid:18) ǫ MN D N + p M (cid:19) (cid:16) δj − δj (cid:17) + ( p + p ) δj M (1) = 0 , ǫ MN p M δj N (1) = − (cid:16) ( δ ∆ + δ ∆ ) j + ( ∂ + ∂ ) δj (cid:17) , ( p − p ) ǫ MN δj N (1) + (cid:18) D M + ǫ MN p N (cid:19) (cid:16) δj + δj (cid:17) = − (cid:16) δ ∆ M j + ∂ M δj (cid:17) . (33)To proceed, we define A M by pulling out a factor of e − p T /B from δj M (1) : δj M (1) = e − p T /B A M . (34) A M are functions of p and q . The solutions for static perpendicular electric and magneticfields correspond to A M being independent of p M . Plugging (34) into (33) and divide outcommon factor e − p T /B , we obtain ǫ MN ∂∂p N A M = 0 , (35a) − p − p (cid:18) p M − B ∂∂p M (cid:19) B ∂∂p K A K + ( p + p ) A M = 0 , (35b)2 ǫ MN p M A N = f δ ( p + p ) f ′ ( p ) − p K B ( f K + f K ) δ ( p + p ) − ( ∂ + ∂ ) δ ( p + p ) g ( p ) , (35c)( p − p ) ǫ MN A N + − p + p B ǫ MN ∂∂p N (cid:20)(cid:18) p K − B ∂∂p K (cid:19) A K (cid:21) = (cid:18) f M ∂∂p + f M ∂∂p (cid:19) ( δ ( p + p ) f ( p )) + ∂ M δ ( p + p ) g ( p ) . (35d)We have defined δj = δ ( p + p ) e − p T /B g ( p ). We already know δj µ (0) corresponds toredistribution of LLL states, so g has to be a function of p only. Below we will show this isnot possible except for the special cases presented in the above subsections. We first apply ǫ MN ∂∂p N to (35b) and use (35a) to arrive at p M ǫ MN ∂∂p N ∂∂p K A K = 0 . (36)We can also apply ǫ MN p N to (35b) and use (36) to obtain( p + p ) ǫ MN p M A N = 0 . (37)We then multiply (35d) by p M to obtain:( p − p ) p M ǫ MN A N + − p + p B ǫ MN p M ∂∂p N (cid:18) p K A K − B ∂∂p K A K (cid:19) = p M (cid:18) f M ∂∂p + f M ∂∂p (cid:19) ( δ ( p + p ) f ( p )) − p M ∂ M δ ( p + p ) g ( p ) . (38)12sing (37) to simplify the first term and using (36) to eliminate the second term in theround bracket on the LHS, we arrive at2 p ǫ MN p M A N + − Bp + p (cid:20) ǫ MN p M A N + ǫ MN p M p K ∂∂p N A K (cid:21) = p M (cid:18) f M ∂∂p + f M ∂∂p (cid:19) ( δ ( p + p ) f ( p )) − p M ∂ M δ ( p + p ) g ( p ) . (39)The second term in the square bracket can be further simplified using the following identity ǫ MN p K + ǫ NK p M + ǫ KM p N = 0 . (40)It follows that ǫ MN p M p K ∂∂p N A K = − (cid:0) ǫ NK p M + ǫ KM p N (cid:1) p M ∂∂p N A K = ǫ MN p M p K ∂∂p K A N , (41)where we have used (35a) and relabeled indices in the second equality. We can then rewritethe square bracket of (39) as (cid:20) ǫ MN p M A N + ǫ MN p M p K ∂∂p N A K (cid:21) = ǫ MN p M (cid:18) ∂∂p K (cid:19) A N = ǫ MN ∂∂p K p M A N . (42)With this, we arrive at the following simple form of (39) (cid:18) p + − Bp + p (cid:19) ǫ MN p M A N = p M (cid:18) f M ∂∂p + f M ∂∂p (cid:19) ( δ ( p + p ) f ( p )) − p M ∂ M δ ( p + p ) g ( p ) . (43)We can now plug (35c) into the above and compare coefficients of δ ( p + p ) and δ ′ ( p + p )to determine g ( p ). Note that the coefficients have to be matched separately rather thanusing δ ′ ( p + p ) = − δ ( p + p ) p + p , which involves dropping of boundary terms and is not alwaysjustified. We end up with two expressions for g : g = ˜ f f ′ ( p ) i ( q + q ) ,g = (2 p ˜ f − p M ˜ f M ) f ′ ( p ) − p B (cid:16) p M ˜ f M + 2 p M ˜ f M (cid:17) f ( p )2 i ( p ( q + q ) − p M q M ) , (44)with ˜ f µν = i ( q µ a ν − q ν a µ ). It is easy to verify that (44) include all three cases we discussedabove: in the case of parallel electric field, two expressions of (44) give the same result; inthe case of static perpendicular electric or magnetic field, only the second expression shouldbe used. In all three cases, g is independent of p T . This is a necessary condition for δj tobe a valid zeroth order solution as stressed above. However, any other field perturbationswould not allow for a p T independent g thus no more solution can be found.13he lack of nontrivial solution may sound odd. Indeed, it is actually an artifact of thecollisionless limit we work in. In the absence of interaction between LL states, the dynamicsof LL states is restricted to classical longitudinal motion. This can be induced by parallelelectric field leading to chiral magnetic wave. Perpendicular electric or magnetic field neces-sarily leads to quantum transition between LL states. However, this cannot occur withoutinteraction. The only possible solution is static ones in which no dynamics is involved. Inother words, although these static solutions can be found, solutions corresponding dynam-ical realization of these states is not possible in the absence of collision. The collisionlesslimit also lies behind the disagreement of the static limit of (20) and (27). We expect aconsistent limit will be reached by including collision. We leave it for future work. V. SUMMARY AND OUTLOOK
By using chiral kinetic theory with Landau level states, we studied photon self-energyin magnetized chiral plasma from response to electromagnetic field perturbations. In theregime of strong magnetic field, we studied the response of chiral plasma to three differentfield perturbations: parallel electric field, static perpendicular electric and static perpen-dicular magnetic fields. They give rise to components of self-energy in different kinematics.The three perturbations lead to chiral magnetic wave, drift state and tilted state respec-tively. From the case of chiral magnetic wave, we obtain self-energy components, whichare in agreement with field theoretic results up to the accuracy of the chiral kinetic theory.From the cases of drift state and tilted state, we obtain components of self-energy in thestatic case. We also compared the solutions of drift state and tilted state with boostedand rotated background solution respectively. The difference is understood from the spatialdependence of of the perturbations.We further showed no solution can be found in response to other more general per-turbations. We argued it is an artifact of the collisionless limit we work. Without collisions,quantum transition between LL states is not possible but only classical motion of LL stateis allowed. As a result, drift state and tilted state cannot be realized dynamically but canonly be found as static solutions. To study more general perturbations, it is crucial to in-troduce collision. It can be done by promoting photon as a dynamical field, which mediatesinteraction between LL states. It would also allow us to study self-energy of LL states andphoton in a systematic way. We hope to address these in the future.14
CKNOWLEDGMENTS
S.L. is grateful to Koichi Hattori, Defu Hou, Igor Shovkovy and Di-Lun Yang foruseful discussions. He also thanks Yukawa Institute of Theoretical Physics for hospitalityand the workshop “Quantum kinetic theories in magnetic and vortical fields” for providing astimulating environment during the final stage of this work. This work is in part supportedby NSFC under Grant Nos 11675274 and 11735007.
Appendix A: Chiral kinetic equations for left handed fermions
The chiral kinetic equation for left handed fermions can be derived from the equationof motion for the corresponding Wigner function W : (cid:18)
12 ∆ µ − ip µ (cid:19) ¯ σ µ W = 0 , (cid:18)
12 ∆ µ + ip µ (cid:19) W ¯ σ µ = 0 . (A1)The difference with counterpart of right handed fermions is σ µ → ¯ σ µ . We then decomposethe Wigner function into components j µ as W = j j i σ i . (A2)This decomposition keeps the integral representation of current (7) the same for left handedfermions. The chiral kinetic equations for components follow immediately from (A1)∆ j − ∆ i j i = 0 ,p j − p i j i = 0 , ∆ j i − ∆ i j − ǫ ijk p j j k , − p j i + p i j − ǫ ijk ∆ j j k = 0 . (A3)They are obtainable from the counterpart of right handed fermions by the replacement∆ i → − ∆ i and p i → − p i . It follows that the contributions of left handed fermions andright handed fermions to self-energy are related by the replacement q i → − q i , which agreeswith what we used in the text. 15 ppendix B: Covariance of the background solution In this appendix, we show the covariance of the background solution (28). We firstwrite down the covariant chiral kinetic equations∆ µ j µ = 0 , (B1a) − ∆ µ j ν + ∆ n j µ + 2 ǫ µνρσ p ρ j σ = 0 , (B1b) p µ j µ = 0 , (B1c) p µ j ν − p ν j µ + 12 ǫ µνρσ ∆ ρ j σ = 0 , (B1d)with ∆ µ = ∂ µ − ∂∂p ν ( F µν + f µν ). To the lowest order in gradient, we have ∆ µ = − ∂∂p ν F µν = ∂∂p ν Bǫ µνρσ b ρ u σ . Here u µ and b µ are unit vector corresponding to fluid velocity and magneticfield direction with u · b = 0. B ≡ p F µν F µν . The solution (8) corresponds to u µ = (1 , , , b µ = (0 , , , ǫ αβµν and using the identity ǫ αβµν ǫ µνρσ = − (cid:16) δ αρ δ βσ − δ ασ δ βρ (cid:17) . (B2)Below we show at the lowest order in gradient (B1a), (B1c) and (B1d) are indeed satisfiedby the following covariant solution j µ ∼ ( u + b ) µ δ ( p · ( u + b )) e − p T /B f ( p · u ) , (B3)with p T = − p +( p · u ) − ( p · b ) . We first see (B1a) and (B1c) are satisfied by anti-symmetryof indices and on-shell condition:∆ µ j µ ∼ ∂∂p ν Bǫ µνρσ b ρ u σ ( u + b ) µ = 0 ,p µ j µ ∼ p · ( u + b ) δ ( p · ( u + b )) = 0 . (B4)(B1d) requires some work: p µ j ν − p ν j µ + 12 ǫ µνρσ ∆ ρ j σ ∼ ( p µ ( u + b ) ν − p ν ( u + b ) µ ) δ ( p · (( u + b )) e − p T /B f ( p · u )+ 12 ǫ µνρσ ∂∂p λ Bǫ ρσαβ b α u β ( u + b ) σ δ ( p · ( u + b )) e − p T /B f ( p · u ) . (B5)The second term can be simplified by noting that ∂∂p λ can pull out p λ , u λ and b λ . The lasttwo cases always vanish when contracting with ǫ ρσαβ b α u β . Keeping only the p λ contributionand using the following identity ǫ µνρσ ǫ ρσαβ = − (cid:16) δ µλ δ να δ σβ + δ µα δ νβ δ σλ + δ µβ δ νλ δ σα − δ µλ δ νβ δ σα − δ µβ δ να δ σλ − δ µα δ νλ δ σβ (cid:17) , (B6)16e obtain from (B5) ∼ p · ( u + b ) ( b µ u ν − b ν u µ ) δ ( p · ( u + b )) e − p T /B f ( p · u ) , (B7)which vanishes by the on-shell condition. [1] W. Heisenberg and H. Euler, Z. Phys. , no. 11-12, 714 (1936) doi:10.1007/BF01343663,10.1007/978-3-642-70078-1 9 [physics/0605038].[2] J. S. Schwinger, Phys. Rev. , 664 (1951). doi:10.1103/PhysRev.82.664[3] G. V. Dunne, In *Shifman, M. (ed.) et al.: From fields to strings, vol. 1* 445-522doi:10.1142/9789812775344 0014 [hep-th/0406216].[4] K. Hattori and K. Itakura, Annals Phys. , 23 (2013) doi:10.1016/j.aop.2012.11.010[arXiv:1209.2663 [hep-ph]].[5] K. Hattori and K. Itakura, Annals Phys. , 58 (2013) doi:10.1016/j.aop.2013.03.016[arXiv:1212.1897 [hep-ph]].[6] A. Vilenkin, Phys. Rev. D , 3080 (1980). doi:10.1103/PhysRevD.22.3080[7] D. Kharzeev, Phys. Lett. B , 260 (2006) doi:10.1016/j.physletb.2005.11.075 [hep-ph/0406125].[8] D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A , 67 (2007)doi:10.1016/j.nuclphysa.2007.10.001 [arXiv:0706.1026 [hep-ph]].[9] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D , 074033 (2008)doi:10.1103/PhysRevD.78.074033 [arXiv:0808.3382 [hep-ph]].[10] M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D , 045011 (2005)doi:10.1103/PhysRevD.72.045011 [hep-ph/0505072].[11] D. T. Son and A. R. Zhitnitsky, Phys. Rev. D , 074018 (2004)doi:10.1103/PhysRevD.70.074018 [hep-ph/0405216].[12] D. E. Kharzeev and H. U. Yee, Phys. Rev. D , 085007 (2011)doi:10.1103/PhysRevD.83.085007 [arXiv:1012.6026 [hep-th]].[13] D. T. Son and B. Z. Spivak, Phys. Rev. B , 104412 (2013) doi:10.1103/PhysRevB.88.104412[arXiv:1206.1627 [cond-mat.mes-hall]].[14] K. Hattori and D. Satow, Phys. Rev. D , no. 11, 114032 (2016)doi:10.1103/PhysRevD.94.114032 [arXiv:1610.06818 [hep-ph]].[15] K. Hattori, S. Li, D. Satow and H. U. Yee, Phys. Rev. D , no. 7, 076008 (2017)doi:10.1103/PhysRevD.95.076008 [arXiv:1610.06839 [hep-ph]].[16] K. Fukushima and Y. Hidaka, Phys. Rev. Lett. , no. 16, 162301 (2018)doi:10.1103/PhysRevLett.120.162301 [arXiv:1711.01472 [hep-ph]].[17] K. Fukushima and Y. Hidaka, arXiv:1906.02683 [hep-ph].
18] W. Li, S. Lin and J. Mei, Phys. Rev. D , no. 11, 114014 (2018)doi:10.1103/PhysRevD.98.114014 [arXiv:1809.02178 [hep-th]].[19] S. Lin and L. Yang, arXiv:1909.11514 [nucl-th].[20] N. Y. Astrakhantsev, V. V. Braguta, M. D’Elia, A. Y. Kotov, A. A. Nikolaev and F. Sanfilippo,arXiv:1910.08516 [hep-lat].[21] R. Critelli, S. I. Finazzo, M. Zaniboni and J. Noronha, Phys. Rev. D , no. 6, 066006 (2014)doi:10.1103/PhysRevD.90.066006 [arXiv:1406.6019 [hep-th]].[22] S. Li and H. U. Yee, Phys. Rev. D , no. 5, 056024 (2018) doi:10.1103/PhysRevD.97.056024[arXiv:1707.00795 [hep-ph]].[23] U. H. Danielsson and D. Grasso, Phys. Rev. D , 2533 (1995) doi:10.1103/PhysRevD.52.2533[hep-ph/9503459].[24] J. Chao, L. Yu and M. Huang, Phys. Rev. D , no. 4, 045033 (2014) Erratum: [Phys. Rev.D , no. 2, 029903 (2015)] doi:10.1103/PhysRevD.90.045033, 10.1103/PhysRevD.91.029903[arXiv:1403.0442 [hep-th]].[25] K. Fukushima, K. Hattori, H. U. Yee and Y. Yin, Phys. Rev. D , no. 7, 074028 (2016)doi:10.1103/PhysRevD.93.074028 [arXiv:1512.03689 [hep-ph]].[26] J. Chao and M. Huang, arXiv:1609.04966 [hep-ph].[27] K. Hattori and D. Satow, Phys. Rev. D , no. 1, 014023 (2018)doi:10.1103/PhysRevD.97.014023 [arXiv:1704.03191 [hep-ph]].[28] B. Singh, S. Mazumder and H. Mishra, arXiv:2002.04922 [hep-ph].[29] D. T. Son and N. Yamamoto, Phys. Rev. Lett. , 181602 (2012)doi:10.1103/PhysRevLett.109.181602 [arXiv:1203.2697 [cond-mat.mes-hall]].[30] D. T. Son and N. Yamamoto, Phys. Rev. D , no. 8, 085016 (2013)doi:10.1103/PhysRevD.87.085016 [arXiv:1210.8158 [hep-th]].[31] M. A. Stephanov and Y. Yin, Phys. Rev. Lett. , 162001 (2012)doi:10.1103/PhysRevLett.109.162001 [arXiv:1207.0747 [hep-th]].[32] J. H. Gao, Z. T. Liang, S. Pu, Q. Wang and X. N. Wang, Phys. Rev. Lett. , 232301 (2012)doi:10.1103/PhysRevLett.109.232301 [arXiv:1203.0725 [hep-ph]].[33] S. Pu, J. h. Gao and Q. Wang, Phys. Rev. D , 094017 (2011)doi:10.1103/PhysRevD.83.094017 [arXiv:1008.2418 [nucl-th]].[34] J. W. Chen, S. Pu, Q. Wang and X. N. Wang, Phys. Rev. Lett. , no. 26, 262301 (2013)doi:10.1103/PhysRevLett.110.262301 [arXiv:1210.8312 [hep-th]].[35] Y. Hidaka, S. Pu and D. L. Yang, Phys. Rev. D , no. 9, 091901 (2017)doi:10.1103/PhysRevD.95.091901 [arXiv:1612.04630 [hep-th]].[36] C. Manuel and J. M. Torres-Rincon, Phys. Rev. D , no. 9, 096002 (2014)doi:10.1103/PhysRevD.89.096002 [arXiv:1312.1158 [hep-ph]].[37] C. Manuel and J. M. Torres-Rincon, Phys. Rev. D , no. 7, 076007 (2014)doi:10.1103/PhysRevD.90.076007 [arXiv:1404.6409 [hep-ph]].
38] Y. Wu, D. Hou and H. c. Ren, Phys. Rev. D , no. 9, 096015 (2017)doi:10.1103/PhysRevD.96.096015 [arXiv:1601.06520 [hep-ph]].[39] N. Mueller and R. Venugopalan, Phys. Rev. D , no. 1, 016023 (2017)doi:10.1103/PhysRevD.96.016023 [arXiv:1702.01233 [hep-ph]].[40] N. Mueller and R. Venugopalan, Phys. Rev. D , no. 5, 051901 (2018)doi:10.1103/PhysRevD.97.051901 [arXiv:1701.03331 [hep-ph]].[41] A. Huang, S. Shi, Y. Jiang, J. Liao and P. Zhuang, Phys. Rev. D , no. 3, 036010 (2018)doi:10.1103/PhysRevD.98.036010 [arXiv:1801.03640 [hep-th]].[42] J. H. Gao, Z. T. Liang, Q. Wang and X. N. Wang, Phys. Rev. D , no. 3, 036019 (2018)doi:10.1103/PhysRevD.98.036019 [arXiv:1802.06216 [hep-ph]].[43] S. Carignano, C. Manuel and J. M. Torres-Rincon, Phys. Rev. D , no. 7, 076005 (2018)doi:10.1103/PhysRevD.98.076005 [arXiv:1806.01684 [hep-ph]].[44] S. Lin and A. Shukla, JHEP , 060 (2019) doi:10.1007/JHEP06(2019)060 [arXiv:1901.01528[hep-ph]].[45] S. Carignano, C. Manuel and J. M. Torres-Rincon, arXiv:1908.00561 [hep-ph].[46] Y. C. Liu, L. L. Gao, K. Mameda and X. G. Huang, Phys. Rev. D , no. 8, 085014 (2019)doi:10.1103/PhysRevD.99.085014 [arXiv:1812.10127 [hep-th]].[47] ¨O. F. Dayi and E. Kilinarslan, Phys. Rev. D , no. 8, 081701 (2018)doi:10.1103/PhysRevD.98.081701 [arXiv:1807.05912 [hep-th]].[48] N. Weickgenannt, X. L. Sheng, E. Speranza, Q. Wang and D. H. Rischke, Phys. Rev. D ,no. 5, 056018 (2019) doi:10.1103/PhysRevD.100.056018 [arXiv:1902.06513 [hep-ph]].[49] J. H. Gao and Z. T. Liang, Phys. Rev. D , no. 5, 056021 (2019)doi:10.1103/PhysRevD.100.056021 [arXiv:1902.06510 [hep-ph]].[50] K. Hattori, Y. Hidaka and D. L. Yang, Phys. Rev. D , no. 9, 096011 (2019)doi:10.1103/PhysRevD.100.096011 [arXiv:1903.01653 [hep-ph]].[51] Z. Wang, X. Guo, S. Shi and P. Zhuang, Phys. Rev. D , no. 1, 014015 (2019)doi:10.1103/PhysRevD.100.014015 [arXiv:1903.03461 [hep-ph]].[52] J. H. Gao, Z. T. Liang and Q. Wang, arXiv:1910.11060 [hep-ph].[53] Y. C. Liu, K. Mameda and X. G. Huang, arXiv:2002.03753 [hep-ph].[54] D. L. Yang, K. Hattori and Y. Hidaka, arXiv:2002.02612 [hep-ph].[55] E. V. Gorbar, I. A. Shovkovy, S. Vilchinskii, I. Rudenok, A. Boyarsky and O. Ruchayskiy, Phys.Rev. D , no. 10, 105028 (2016) doi:10.1103/PhysRevD.93.105028 [arXiv:1603.03442 [hep-th]].[56] J. W. Chen, T. Ishii, S. Pu and N. Yamamoto, Phys. Rev. D , no. 12, 125023 (2016)doi:10.1103/PhysRevD.93.125023 [arXiv:1603.03620 [hep-th]].[57] Y. Hidaka, S. Pu and D. L. Yang, Phys. Rev. D , no. 1, 016004 (2018)doi:10.1103/PhysRevD.97.016004 [arXiv:1710.00278 [hep-th]].[58] N. Abbasi, F. Taghinavaz and O. Tavakol, JHEP , 051 (2019)doi:10.1007/JHEP03(2019)051 [arXiv:1811.05532 [hep-th]].
59] X. L. Sheng, R. H. Fang, Q. Wang and D. H. Rischke, Phys. Rev. D , no. 5, 056004 (2019)doi:10.1103/PhysRevD.99.056004 [arXiv:1812.01146 [hep-ph]].[60] M. L. Bellac, “Thermal Field Theory,” doi:10.1017/CBO9780511721700[61] K. Fukushima, Phys. Rev. D , 111501 (2011) doi:10.1103/PhysRevD.83.111501[arXiv:1103.4430 [hep-ph]].[62] Y. Akamatsu and N. Yamamoto, Phys. Rev. Lett. , 052002 (2013)doi:10.1103/PhysRevLett.111.052002 [arXiv:1302.2125 [nucl-th]].[63] J. Y. Chen, D. T. Son, M. A. Stephanov, H. U. Yee and Y. Yin, Phys. Rev. Lett. , no. 18,182302 (2014) doi:10.1103/PhysRevLett.113.182302 [arXiv:1404.5963 [hep-th]].[64] The magnetic field in boosted frame is √ B − E , whose deviation from background is negligibleto linear order in E M .[65] To linear order in the perturbation, the magnitude of the magnetic field p B + B ⊥ = B doesnot change. Only the direction changes.doesnot change. Only the direction changes.