Operator lifetime and the force-free electrodynamic limit of magnetised holographic plasma

Using the framework of higher-form global symmetries, we examine the regime of validity of force-free electrodynamics by evaluating the lifetime of the electric field operator, which is non-conserved due to screening effects. We focus on a holographic model which has the same global symmetry as that of low energy plasma and obtain the lifetime of (non-conserved) electric flux in a strong magnetic field regime. The lifetime is inversely correlated to the magnetic field strength and thus suppressed in the strong field regime.


I. INTRODUCTION
Hydrodynamics [1] is a well-established theoretical framework which universally describes the long wavelength, low frequency behaviour of interacting systems at finite temperature. Essentially, hydrodynamic theory is a description of conserved quantities and the manifestation of the corresponding symmetries in a system in thermal equilibrium. Theories with widely varying microscopics can have the same macroscopic hydrodynamic description. One possible explanation why such a universal description is possible is that all operators except conserved charges have parametrically short lifetimes compared to the scale of interest and, once the longest-lived nonconserved operator 1 has decayed away, the hydrodynamic description becomes viable (see Fig. 1). The hydrodynamic framework may be generalised to systems where the conserved charges are those of a higher-form symmetry [5] which counts the number density of extended objects. A recent exploration of this idea [6] (see also [7][8][9][10]) shows that the resulting hydrodynamics of a one-form U (1) charge reproduces the theory of magnetohydrodynamics (MHD) 2 . This should not come as a big surprise. MHD is, after all, a low energy effective theory of plasma where the (dynamical) electric field is screened -the one-form U(1) symmetry associated to electric flux is explicitly broken. This implies that in, for example, a plasma at zero magnetic field (where the Ohm's law j = σE is a good approximation) the electric field has a finite lifetime, with the lifetime of the electric field τ E = 1/σ. The conductivity σ can be computed from first principles. For instance, in quantum electrodynamics, it can be written as [14] 3 σ ∝ T e 2 log e −1 , (1.2) where e is the electromagnetic coupling. The lifetime of electric field τ ∼ 1/T , is then much shorter than the scale t 1/T (or ω/T 1 in Fourier space) where hydrodynamic behaviour is expected.
If, in this late-time limit, all other operators except energy density T tt and the momentum T ti have already decayed away, one can expect the hydrodynamic description of a plasma to be governed by ∂ µ T µν = 0 , ∂ µ J µν = 0 . (1. 3) The conserved currents T µν and J µν are expressed in terms of energy, momentum, magnetic flux J ti ≡ B i and their conjugates, organised order by order in the gradient expansion. This formulation of MHD only requires macroscopic consistency and does not require the introduction of the gauge field J = F = dA which, due to screening effect, is not a long-lived degree of freedom.
This brings us to the central question of the present paper: Is a hydrodynamic description of the form (1.3) applicable in the limit low temperature compared to magnetic flux density T 2 /|B| 1 ? This question is important if one wants to apply the MHD description to astrophysical plasmas where the magnetic field is many orders of magnitude larger than the scale set by the temperature.
If one were to naively extrapolate (1.1)-(1.2), the lifetime of the electric field appears to become arbitrarily long as the temperature decreases. However, there exists a macroscopic description of plasma in this regime that has been successfully applied. This theory is called force-free electrodynamics or FFE, and has been used extensively in astrophysical setups such as in the magnetosphere of black holes [15,16], neutron star [17] and solar corona [18] just to name a few. In its conventional form, this theory is applied to a system which is magnetically dominated (i.e. |B 2 | > |E 2 | or, covariantly F µν F µν > 0) and whose dynamics is governed by Here, the first relation implies that E · B = 0 while the second relation implies that the force j µ el F µν , with j µ el := ∇ ν F νµ via Maxwell's equations, acting on plasma vanishes (hence the name force-free electrodynamics). More details on the geometric and effective action view point of FFE can be found in e.g. [19,20] and [21][22][23][24]. One should emphasise that the system of equations in Eq.(1.4) is independent of the microscopic details of the cold plasma, which then strongly resembles hydrodynamic descriptions. In fact, it turns out that (1.4) can arise in a special limit where T |B| of a hydrodynamics description with one-form U (1) symmetry in (1.3), see [6,23,24] 4 .
The existence of FFE is usually justified by saying that the cold plasma is, on one hand, dense enough to screen the electric field (1.4a) but, on the other hand, dilute enough so that force-free condition (1.4b) is applicable. This statement can be made more precise in the light of relations between the equations of FFE and hydrodynamics. Thus, we propose a criterion for testing the validity of FFE using the lifetime of non-conserved operators -FFE, or equivalently, hydrodynamic description of cold plasma in the T |B| limit, is valid when the lifetime of all non-conserved operators is parametrically shorter than the time scale of interest. A key advantage of this approach 4 Recasting of force-free electrodynamics in the hydrodynamic language also allows the systematic gradient expansions [23,25]. This could serve to classify correction to FFE in order to account for phenomena such as pulsar radio emission where E · B = 0.
is that the operator lifetime can be, in principle, computed explicitly from microscopic description and therefore allows one to find the 'cutoff' scale where FFE description should break down.
Computing the operator lifetime from microscopic description is, however, not always an easy task. In fact, we are not aware of a genuine computation directly from quantum electrodynamics (in the sense of [14]) when both T and B are turned on. To simplify the computations, we shall demonstrate the validity of FFE in the strongly interacting magnetised plasma with a holographic dual as proposed in [26,27] where the one-form U (1) global symmetry is taken into account via a two-form gauge field in the gravity dual. This provides two key advantages. First, the computation of correlation functions boils down to solving simple linearised differential equations (see e.g. [28]).
Second, there is strong evidences that charge neutral operators, apart from energy and momentum, have a parametrically short lifetime in this class of theories 5 . Therefore, we shall focus on nonconserved operators in the electromagnetic sector of the theory: the electric flux operators, whose lifetime can be extracted via two-point correlation function as in (1.1). This will provide strong evidence for the validity of FFE limit in a strongly interacting holographic plasma.
On the technical side, the computations presented in this note show that there are no quasinormal modes present in the vicinity of the hydrodynamic regime ω/T 1 (and ω/ |B| 1).
The pole in the electric flux correlation function in this regime then implies that the operator has a parametrically long lifetime which could interfere with the hydrodynamic modes. The presence of such long-lived mode can be determined analytically in the usual hydrodynamic regime of ω/T 1 for a large class of theories. It is usually difficult to go beyond this regime towards the limit ω/T ∼ 1, ω/ |B| 1. Such computation can, however, be done analytically in the simple model of [26] thanks to the presence of the BTZ×R 2 bulk geometry in the deep IR [32]. We should also note that the treatment of a (long-lived) non-hydrodynamic modes has been extensively used to determine the breakdown of hydrodynamic descriptions in the context of QFTs with holographic duals, see e.g. [33][34][35][36] .
The remainder of this paper is organised as the follows. In section II, we summarise the procedure involved in the computation of the two-point correlation function in the holographic dual to one-form global symmetry. In section II A, we outline the method for exploring the existence of decaying modes in the vicinity of the usual hydrodynamic limit ω/T 1 at T / B| 1. Due to the simplicity of the bulk geometry, we are able to further extend the analysis to arbitrary value of 5 To be more precise, it has been shown in N = 4 supersymmetric Yang-Mills theory, which constitutes the matter sector of the holographic model [26,27], that there is no long-lived mode besides hydrodynamic modes at any T = 0 and |B| = 0 [29]. A similar conclusion was reached for the same theory in the charge neutral sector at finite non-dynamical magnetic field [30,31].
ω/T with ω/ |B| 1 and T / |B| 1. This is described in section II B. Further open questions and future directions are discussed in section III.

II. THE HOLOGRAPHIC MODEL
A simple holographic dual to a strongly interacting field theory of matter charged under dynamical U (1) electromagnetism (that is, the dynamical plasma described by low energy MHD) and formulated in the language of higher-form symmetry was constructed in [26,27]. We present a brief review here for completeness. The five-dimensional bulk theory is comprised of Einstein gravity coupled to a two-form bulk gauge field, B µν , and a negative cosmological constant, where H = dB and B ab is the bulk 2-form gauge field, Λ is the UV-cutoff, n a is the unit normal to the boundary, and S bnd denotes the Gibbons-Hawking and gravitational counter term. Roughly speaking, the two bulk fields G ab and B ab , asymptote to g µν and b µν respectively, which then source the currents, T µν and J µν .
The generating functional takes the form, and diffeomorphism invariance and gauge symmetry lead to the following equations, H = db is the three-form field strength of the two-form external source. The equilibrium solution of this holographic model is a domain wall interpolating between an asymptotic AdS 5 geometry in the UV (r → ∞ in our convention), and BT Z × R 2 in the near-horizon IR (r = r h ). It is described by the following metric and gauge field Modulo the subtleties due to the mixed boundary conditions, this is nothing but the hodge dual of the magnetised black brane solution of [32]. The radial coordinate is chosen such that r → ∞ corresponds to the usual asymptotic AdS 5 with in the r → ∞ limit. The BT Z × R 2 solution near the horizon can be written as The temperature is set by the horizon radius via 4πT = r 2 h |f (r h )| = 6r h /L 2 . We set L = 1 for simplicity. Note also that B is related to the z−component of the 'physical' magnetic field B which differs by a prefactor L or the 2-form gauge field coupling in the bulk (e.g. if one were to define the action with S ∼ (1/g 2 )H 2 ). We will keep using B to emphasise its holographic origin but there is no harm in thinking of it as simply B.
One interesting feature of this model is that the leading divergence of B µν in the Fefferman-Graham expansion is logarithmic. Thus, the definition of the source b µν requires mixed boundary Requiring the source b µν to be independent of the UV cutoff fixes the form of the 'coupling constant' 1/κ(Λ) which turns out to be logarithmically running. This is a common feature for fields with this type of near-boundary behaviour where the counterterm also plays the role of the double-trace deformation [37,38], see also [26,27] for a discussion in the present context. Mapping J µν in to a more familiar dynamical field strength via J µν = 1 2 µνρσ F ρσ , one can see that the double-trace deformation plays a role similar to the Maxwell term for the dynamical gauge field in the dual QFT with 1/κ(Λ) as a (logarithmically running) electromagnetic coupling.
The finite part of 1/κ(Λ) plays a crucial role in this setup. While the finite counterterm in the ordinary bulk Maxwell theory simply results in a contact term in the correlation function, the mixed boundary condition for B ab implies the existence of the purely decaying mode ω = −i/τ E that can interfere with the gapless hydrodynamic excitation. This is nothing but the life-time of the electric flux operator Q E ∼ dS ij J ij which appears in the following correlation function [27,33] Note that, due to the anisotropy introduced by finite equilibrium magnetic field, the value of τ E depends on which direction of the electric field in consideration. The limit where τ E is small, but finite, compared to the length scale of interest (set by temperature or magnetic flux density) is of particular interest as it allows one to extract τ E analytically, via a matching procedure that we outline below. As argued in the introduction, the lifetime of the electric flux determines the validity of MHD and FFE description.
A. Linearised solutions in ω/T 1 limit and matching procedure In this section, we outline the computation required to obtain the relaxation time of the electric field. We focus on the hydrodynamic regime where ω/T 1, and the low temperature limit 6 T / |B| 1. This allows us to solve the bulk equation of motion analytically via a matching method similar to that was employed in [29] (see also [33] for a recent review). We consider the decay rate of the electric field both along and perpendicular to the equilibrium magnetic field denoted by E = J xy and E ⊥ = J xz , J yz respectively.
Before proceeding, let us summarise the matching procedure for the ω/T 1 expansion. It involves separating the bulk into three suitably defined pieces: inner region, intermediate region and outer region. The inner region is a suitably defined region close to the horizon while the outer region is defined to be the range of r such that ω/r 1 so that one can drop terms quadratic in (ω/r) 2 , which includes the near boundary region. The integration constants of the solution in the outer region are determined by matching the form of inner region solution for intermediate value of r that connect the two regions together. In our case, this is the region of r close to r h but This intermediate region defined above is also consistent with the outer region assumption where ω/r 1 and thus we are able to match the two solution together. Note that, while this procedure is applicable to any bulk solution with event horizon, the limit ω/T 1 is crucial.
We now present the key equations and resulting lifetime of the electric flux.

Perturbation parallel to equilibrium magnetic field
As the magnetic field in equilibrium points along the z−direction, we are interested in E = 1 2 ε zxy J xy . The corresponding bulk perturbation is δB xy which decouples from the metric perturbation in the zero wave vector limit. The bulk equation of motion can be written as where (...) denotes a derivative w.r.t. the radial coordinate r. The inner region solution for δB xy , where we substitute the BT Z × R 2 solution for f, V, W , with the ingoing boundary condition can be written as The outer region solution can be obtained by considering the solution at linear order in ω/r and one obtains, where φ(r) is a function regular everywhere in the bulk defined as Note that, for the source to be independent of the UV cutoff, one requires κ(Λ) −1 = finite term − log Λ. This is the logarithmically running coupling usually found in a double-trace deformed theory and resembles the running of electromagnetic coupling as pointed out in [26,27,33].
For the outer and inner region solutions to match, we consider both solutions in the intermediate region where we can write the inner solution as The matching condition δB inner xy = δB outer xy in this region prompts yield the following algebraic relations between the boundary quantities b xy , J xy : (2.16) Solving these equations at vanishing source b xy = 0 yields the spectrum of the form ω = −i/τ E where τ E is the lifetime of the electric flux parallel to the equilibrium magnetic field. This is the first key result that we advertised earlier, namely φ(r h ) and e −2 r T 2 /B 1 so that ωτ E ∼ ω/T 1. The second limit is essential as the matching procedure assumes that ω/T 1 and the solution outside this regime has to be discarded. Taking these factors into account, one concludes that the temperature dependence of the electric flux is different from the high temperature T / √ B 1 limit where τ E ∼ 1/T ( see Fig 2). Naively taking the limit T → 0 in (2.17) will result in the vanishing lifetime of the electric flux in contrast to the result in (1.2). However, one has to carefully remove the limit ω/T 1 in order to access the lower temperature limit ω/T ∼ 1, ω/ √ B 1.

Perturbation perpendicular to equilibrium magnetic field
Unlike the previous case, the perturbation δB xz that corresponds to E ⊥ = 1 2 yzx J zx is coupled to the metric perturbation. This is manifest in the equations of motion The solution for δB xz , δG tx in the outer region can be found by using the property of the background geometry combined with the Wronskian method as in [33]. To be more precise, one first notes that the time-independent solution of the magnetised black brane can be written in a total derivative form, which implies the existence of two radially conserved currents.
where we write the equilibrium ansatz for the gauge field as B = h(r) dt ∧ dz with gauge choice h(r h ) = 0, which, together with the horizon regularity, sets Q 1 = 0. The relation between h(r) and the 3-form field strength is More details on obtaining these radially conserved quantities can be found in e.g. [41]. With this ansatz, we can compare (2.18) and (2.19) and find that one of the solutions of (2.18) when ω/r → 0 One can use the Wronskian method to find find a pair of solution of (2.18) that are linearly independent to {Φ 1 , Ψ 1 }. These solutions are As a result, the outer region solution can be written as where J xz := (r 2 f e −W δB xz + BδG t x ) is an integration constant of (2.18) at ω = 0. One can substitute the BT Z × R 2 ansatz into the solution in (2.22) to check that Φ 1 , Ψ 1,2 are finite at r = r h while Φ 2 is singular. It is convenient to separate out the singular part of Φ 2 in the following One can also check that J xz is identical to the one-point function δJ xz via the definition (2.8).
Note also that the ratio c 2 /J xz is finite due to the cancellation of the logarithmic divergence of 1/κ(Λ) and that of the near boundary solution of h(r), obtained via (2.19c).
Let us also pointed out another way to organise the equations of motion for δB xz . It turns out that (2.18) can be combined into a single equation of motion that reduces to a total derivative form at ω = 0. Following the procedure in e.g. [42] and some manipulation, we find Substituting the form of c 1 , c 2 in terms of δJ xz , we can write the relations in a form similar to δJ xy , namely In the case of vanishing sources, we can write c 2 and the relaxation time of the electric field perpendicular to the equilibrium magnetic field is In contrast to the result at e −2 r 1 and zero equilibrium magnetic field in [27,33], the lifetime at strong magnetic field B/T 2 has a very different form. To see this, it is useful to examined that the combinations that enter the lifetime as follows with proportionality constants given by some numbers of order O(1). In the limit of large electromagnetic coupling 1/κ(Λ) small ω/ |B| regime. Next, we further extend the regime of validity to that of ω/ |B| 1 but for arbitrary ω/T . The purpose of the latter is to show that τ E ∝ T / |B| without relying on the ω/T 1 limit.

Zero temperature
A simple argument for the non-existence of such a slowly decaying mode, is the presence of Lorentz symmetry at zero temperature on the AdS 3 submanifold in the deep infrared. On the other hand, one can also show this, using matching methods similar to those in [39,44,45].
To obtain this result, one first realises that the geometry of the magnetised black brane is that of an interpolation between IR AdS 3 × R 2 and UV AdS 5 . Roughly speaking, the IR geometry starts to becomes a good approximation as one starts to probe the scale below the magnetic field i.e. r ∼ |B|. The inner and outer regions are defined such that they start off from the IR and UV geometry respectively, and extend to cover the overlap region (see Figure 3). This is achievable when ω/ |B| 1.

FIG. 3:
A sketch of the bulk geometry at zero temperature. The inner region, whose solutions only depends on the ratio ω/r extended from the near horizon limit r → 0 to the one where ω/r ∼ ω/ |B| → 0 as we are working in the ω/ |B| 1 limit. The outer region is defined to be the region where the ω 2 /r 2 and higher power in ω/r is suppressed, which can be extended toward r |B| as long as the frequency is small.
For concreteness, let us demonstrate how this works in the E channel that involves the bulk field δB xy governed by Eq. (2.11). The solution can be written in the same form as (2.13) evaluated at zero temperature (i.e. r h = 0). It is worth noting that the singular behaviour near r/ √ B → 0 is different from that in earlier section. Instead, it can be written as where the integration constants can be related to source and response via (2.8). It is worth noting that the logarithmic divergence appears at order ω 2 . This is can be confirmed via Frobenius analysis in AdS 5 region (see e.g. [46]) and AdS 3 × R 2 region (see appendix B). The prefactor of the r −2 divergence is obtained by evaluating e 2V (r)−W (r) /f (r) at the horizon r → 0. Hereφ(r) is the integral in (2.13) subtracted by the r −2 divergent and logarithmic divergent pieces. The resulting integral evalutated from r = r 0 ∼ √ B of the overlapping region to the UV cutoff r = Λ is finite and its number is not extremely relevant for us as long as one keep e −2 r large. Next, we consider the inner region solution, which can be obtained by solving (2.11) in the AdS 3 × R 2 region. Upon imposing horizon regularity at r → 0, we find that the inner region solution is For these two branches of solutions to match, we extend the inner region solution to the regime where ζ = ω/r 1. We find that the 'near boundary' expansion takes the form δB inner xy = c H 1 + 1 2 γζ 2 + 1 2 ζ 2 log ζ + ... (2.33) Matching this solution to the outer region, we find that c 2 ∝ ω 2 unlike what happened in the previous section. Carrying on the matching procedure, we find that the polynomial governing the spectrum only depends on ω 2 and thus rules out the purely imaginary mode ω = −i/τ . The same argument can also be made for the E ⊥ channel involving δB xz . This is because, the part that is relevant to the matching procedure only depends on ζ 2 . See appendix B for more details on the form of δB xz in the AdS 3 × R 2 region.

T ω |B| limit
In this section, we show that the electric flux lifetime can also be obtained regime where ω/T 1 and ω/ |B| 1 while keeping |B|/T 1. The calculations closely resembles that of the zero temperature case except that the deep IR geometry is now BT Z × R 2 instead of AdS 3 × R 2 . Figure   4 illustrates this geometry where the AdS 5 joined with the BT Z × R 2 at the 'boundary' AdS 3 × R 2 of the IR geometry. We will only focus on the E fluctuations as it is the only channel that contains the decaying modes in the ω/T 1 regime. Similar computation for this type of geometry can also be found in [45].

FIG. 4:
A sketch of the bulk geometry at low temperature T |B|. The inner region, whose solutions only depends on the ratio ω/r extends from the near horizon limit r → r h √ B to the one where ω/r ∼ ω/ |B| 1, which corresponds to the near boundary region of BT Z × R 2 geometry, described by AdS 3 × R 2 . The outer region is defined to be the region where ω 2 /r 2 (and higher powers) is negligible and, therefore, can be extended toward r ∼ |B| in the ω/ |B| 1 limit.
The outer region solution, which extends from the UV AdS 5 to the intermediate AdS 3 × R 2 region has the same form as in (2.31). This is possible only in the limit where √ B T so that r/ √ B is always much greater than T / √ B ∼ r h / √ B in this region.
The inner region solution in the BT Z ×R 2 region can be expressed in terms of a hypergeometric function (upon imposing ingoing boundary condition) where w = ω/(2πT ) = ω/3r h . Extending this solution in the r r h limit (which is possible due to r h /r → 0 as we approach the limit ω/r → 0) yields the following expansion [47] δB inner indicates that the lifetime will become extremely short in the extremely strong magnetic field regime T / |B| 1 and cannot interfere with the low energy regime of ω/ |B| 1 where the FFE limit is thought to be applicable.
• For the component of electric flux E ⊥ perpendicular to the magnetic field, we find that there is no pole in the vicinity of ω/T 1. The dependence of the lifetime on the renormalised electromagnetic coupling disappears as one approaches the strong magnetic field limit.
We also performed a consistency check at T → 0 to ensure that there are no modes in the deep IR limit of ω/ |B| 1. In this regime, the modes that indicate (potentially) long lifetime of E disappear from the low energy spectrum as anticipated.
These computations are basic checks on the validity of FFE description. In the holographic context, it would be interesting to check if all the accessible non-conserved operator truly have a parametrically short lifetime as well as confirming the low energy spectrum predicted by force-free electrodynamics (and its subsequent derivative corrections). Extraction of FFE effective action from gravity akin to [48][49][50] or the full constitutive relation as in [51][52][53] would be desirable as a definitive proof of FFE description in the dynamically magnetised black brane geometry. Last but not least, it would be very interesting to investigate operators lifetime in (weakly coupled) quantum electrodynamics at finite T and B to better understand FFE and its limitations in a system more directly connected to astrophysical plasma than the strongly coupled holographic model considered here.
to choose r h = 1, r 2 h f (r h ) = 1 (or equivalently T = 1/4π). It is also convenient to set V (r h ) = W (r h ) = 0 which results in the UV boundary metric of the form lim r→∞ ds 2 = r 2 −dt 2 + v(dx 2 + dy 2 ) + wdz 2 + dr 2 r 2 (A1) Upon rescaling of spatial coordinates {dx, dy, dz} → {dx/ √ v, dy/ √ v, dz/ √ w}, we recover the desired background solutions. Note also that the physical magnetic flux is related to the input parameter (that produced the metric in (A1)) by B physical = B input /v. A small caveat of this method is that one cannot find a smooth solution beyond B input √ 3/2 which corresponds to the 05. This is most likely an artifact of the presented numerical method as there exists a smooth solution in the zero temperature limit corresponding to the AdS 3 × R 2 geometry in the deep IR. We should also note that this is a sufficiently low energy temperature as the entropy becomes sufficiently close to s ∝ T obtained from BT Z × R 2 geometry (c.f. [26,32]). The background is generated for r from [1 + 10 −3 , 10 6 ] and varying the (numerical) cutoffs within this order of magnitude does not change the obtained numerical results.
Let us also remark on the the numerical value of the renormalised electromagnetic coupling e −2 r = log(Λ/r h ) + κ(Λ) −1 . This quantity strongly influences both the thermodynamics and low energy spectrum [26,27,33] of the model. In particular a small value of e −2 r would result in the speed of sound becoming imaginary [26]. Another way to see that this quantity should be large is to write it in terms of a renormalisation group independent scale M * that denotes the energy scale of a Landau pole [27] i.e. e −2 r ∼ log(M /T ) where M T . We take this to be the largest scale in the problem-much larger than the accessible value of √ B/T .
Numerical value of the integral for φ(r h ) in (2.17) is shown in Figure 5. For a larger temperature (when φ(r h ) ≈ O(1)), the lifetime can be sensibly approximated to be τ E ≈ 2π(T /B)e −2 r . As T / √ B decreases, the lifetime becomes shorter and, if we are to extrapolate the fitting function φ ∼ B T 2 log B T 2 to even lower temperature where e −2 r φ, it will escape the regime of the validity of small ω/T, ω/ √ B expansions. In this scenario, one shall conclude that there are no long-lived modes that can interfere with the low energy excitations. 2 , h except p [2] 1 , which can be set to zero without loss of generality [46], can be obtained recursively. The important piece of information here is the coefficient h = 1 which can be obtained by recursively solving the equation (B1). Another easy way to see this is to recast (B1) as the Bessel equation of order 1 as pointed out earlier. Then, using the fact that the Bessel functions K 1 (ζ) and I 1 (ζ) are two independent solutions of such equation and, for small ζ they admit the following asymptotic expansions (see e.g. §3.3 of [55]) will result in the series expansions of the solution in AdS 3 × R 2 region in (B2a).
A similar procedure can also be applied for E ⊥ using Eq.(2.25). Substituting δB xz = ζ 2 c(ζ), one finds that it obeys the Bessel equation of order 2 whose ζ 1 expansion only yields even power in ζ.