Gradient resummation for nonlinear chiral transport: an insight from holography

Nonlinear transport phenomena induced by chiral anomaly are explored within a 4D field theory defined holographically as $U(1)_V\times U(1)_A$ Maxwell-Chern-Simons theory in Schwarzschild-$AdS_5$. In presence of weak constant background electromagnetic fields, the constitutive relations for vector and axial currents, resummed to all orders in the gradients of charge densities, are encoded in nine momenta-dependent transport coefficient functions (TCFs). These TCFs are first calculated analytically up to third order in gradient expansion, and then evaluated numerically beyond the hydrodynamic limit. Fourier transformed, the TCFs become memory functions. The memory function of the chiral magnetic effect (CME) is found to differ dramatically from the instantaneous response form of the original CME. Beyond hydrodynamic limit and when external magnetic field is larger than some critical value, the chiral magnetic wave (CMW) is discovered to possess a discrete spectrum of non-dissipative modes.

In this paper we continue exploring hydrodynamic regime of relativistic plasma with chiral asymmetries. We closely follow previous works [1,2] focusing on massless fermion plasma with two Maxwell gauge fields, U (1) V × U (1) A . Dynamics of hydrodynamic theories is governed by conservation equations (continuity equations) of the currents. As a result of chiral anomaly, which appears in relativistic QFTs with massless fermions, global U (1) A current coupled to external electromagnetic (e/m) fields is no longer conserved. The continuity equations turn into where J µ , J µ 5 are vector and axial currents and κ is an anomaly coefficient (κ = eN c /(24π 2 ) for SU (N c ) gauge theory with a massless Dirac fermion in fundamental representation and e is electric charge, which will be set to unit from now on). E and B are external vector electromagnetic fields.
The continuity equations could be regarded as time evolution equations for the charge densities ρ (ρ 5 ) sourced by three-current J ( J 5 ). However, these equations cannot be solved as an initial value problem without additional input, the currents J and J 5 . In hydrodynamics, the currents have to be expressed in terms of thermodynamical variables, such as the charge densities ρ and ρ 5 themselves, temperature T , and the external E and B fields if present. These are known as constitutive relations, which generically take the form The constitutive relations should be considered as "off-shell" relations, because they treat the charge density ρ (ρ 5 ) as independent of J ( J 5 ). Once (1) is imposed, the currents' constitutive relations (2) are put into "on-shell".
In addition to the charge current sector discussed above, one has to simultaneously consider energy-momentum conservation. In general, these two dynamical sectors are coupled.
However, in the discussion below, we will ignore back-reaction of the charge current sector on the energy-momentum conservation. This will be referred to as probe limit.
In the long wavelength limit, the constitutive relations are usually presented as a (truncated) gradient expansion. At any given order, the gradient expansion is fixed by thermodynamic considerations and symmetries, up to a finite number of transport coefficients (TCs).
The latter should be either computed from the underlying microscopic theory or deduced experimentally. Diffusion constant, DC conductivity or shear viscosity are examples of the lowest order TCs.
It is well known, however, that in relativistic theory truncation of the gradient expansion at any fixed order leads to serious conceptual problems such as violation of causality. Beyond conceptual issues, causality violation results in numerical instabilities rendering the entire framework unreliable. Causality is restored when all order gradient terms are included, in a way providing a UV completion to the "old" hydrodynamic effective theory. Below we will refer to such case as all order resummed hydrodynamics [3][4][5][6][7][8]. The first completion of the type was originally proposed by Müller, Israel, and Stewart [9][10][11][12] who introduced retardation effects in the constitutive relations for the currents. Formulation of [9][10][11][12] is the most popular scheme employed in practical simulations. Essentially, all order resummed hydrodynamics is equivalent to a non-local constitutive relation of the type (here we take the charge diffusion current as an example): whereD is the memory function of the diffusion function D(ω, q 2 ) [13], which is generally non-local both in time and space. Causality implies thatD(t) has no support for t < 0.
In practice, the memory function is typically modelled: Müller-Israel-Stewart formulation [9][10][11][12] models the memory functions with a simple exponential in time parametrised by a relaxation time.
The constitutive relations (2) are well known to receive contributions induced by the chiral anomaly. The most familiar example is the chiral magnetic effect (CME) [38][39][40]: a vector current is generated along an external magnetic field when a chiral imbalance between leftand right-handed fermions is present ( J ∼ ρ 5 B). Another important transport phenomenon induced by the chiral anomaly is the chiral separation effect (CSE) [41,42]: left and right charges get separated along an applied external magnetic field ( J 5 ∼ B). Combined, CME and CSE lead to a new gapless excitation called chiral magnetic wave (CMW) [43]. This is a propagating wave along the magnetic field. There is a vast literature on CME/CSE and other chiral anomaly-induced transport phenomena, which we cannot review here in full. We refer the reader to recent reviews [20,21,34,44,45] and references therein on the subject of chiral anomaly-induced transport phenomena.
Beyond naive CME/CSE, there are (infinitely) many additional effects induced or affected by chiral anomaly. Particularly, transport phenomena nonlinear in external fields were realised recently [46] to be of critical importance in having a self-consistent evolution of chiral plasma. This argument, together with the causality discussions mentioned earlier, would lead to the conclusion that the constitutive relations (2) should contain infinitely many "nonlinear" transport coefficients in order to guarantee applicability of the constitutive relations in a broad regime. Recently, this triggered strong interest in nonlinear chiral transport phenomena within chiral kinetic theory (CKT) [47][48][49][50]. Previous works on the subject of nonlinear chiral transport phenomena include [51] based on the notion of entropy current, and [52] based on the fluid-gravity correspondence [53].
The objective of present work is to explore all order gradient resummation for nonlinear transport effects induced by the chiral anomaly 1 , further extending the results of Refs. [1,2,58].
Just like in Refs. [1,2,58], our playground will be a holographic model, that is U (1) V × U (1) A Maxwell-Chern-Simons theory in Schwarzschild-AdS 5 [59,60] to be introduced in detail in Section III, for which we know to compute a zoo of transport coefficients exactly.
Hoping for some sort of universality, we could learn from this model both about general phenomena and relative strengths of the effects.
In our recent publication [58], we reviewed all different studies which were performed in [1,2,58]. Those studies and the present one are largely independent even though performed within the same holographic model. For brevity, we will not repeat this review here, but will make connection to these previous works whenever relevant.
Anomalous transport phenomenon is frequently discussed from the viewpoint of its dissipative nature and, equivalently, its contribution to entropy production [51,[61][62][63][64]. CME is well known to be non-dissipative [20,34,65]. What about the dissipative nature of other anomalous transport phenomena, say beyond CME? In [51] the transport coefficients that are odd in κ were identified as anomaly-induced and, based on space parity P arguments, are claimed to be non-dissipative. This is to distinguish from anomaly-induced corrections to normal transports, which appear to be even in κ. While the P-based arguments seem to work perfectly for the second order hydrodynamics [51], a more natural criterion of dissipation seems to be based on time-reversal symmetry T . T -odd transport coefficients describe dissipative currents, whereas T -even ones are non-dissipative [51]. The anomaly-induced phenomena explored below will involve terms both dissipative and not. 1 The asymptotic nature of the gradient expansion and problems related to resummation of the series have been a hot topic over the last few years, see recent works [54][55][56][57]. In our approach, however, we never attempt to actually sum the series and thus these discussions are of no relevance to our formalism.
In the next Section, we will review our results including connections to the previous works [1,2,58]. The following Sections present details of the calculations.

II. SUMMARY OF THE RESULTS
The objective of [1,2,58] and of the present work is to systematically explore (2) under different approximations. Following [1,2,58], the charge densities are split into constant backgrounds and space-time dependent fluctuations where all the coefficients are scalar functionals of the derivative operator ∂ µ Taylor expanded, these coefficients contain information about infinitely many derivatives and associated TCs. Thanks to the linearisation, the constitutive relations could be conveniently expressed in Fourier space. Then the functionals of the derivatives are turned into functions of frequency and space momenta, (∂ t , ∇) → (−iω, i q), which we refer to as transport coefficients functions (TCFs) [6]. TCFs contain information about infinitely many derivatives and associated transport coefficients. In practice, they are not computed as a series resummation of order-by-order hydrodynamic expansion, and are in fact exact to all orders. TCFs go beyond the hydrodynamic low frequency/momentum limit and they contain collective effects of non-hydrodynamic modes. Fourier transformed back into real space, TCFs correspond to memory functions. Below we set πT = 1 for convenience. The dimensionful frequency and momentum should be πT ω and πT q.
In the hydrodynamic limit of ω, q 1, all the TCFs above were computed analytically in [1]. Here we only quote the results for the diffusion TCF D and and CME TCF σ χ The first term in (9) is the usual charge diffusion constant D 0 [66] while the second term corresponds to relaxation time. The first term in (10) is the usual CME conductivity. The following terms include relaxation effects.
Our goal here is to extend the work initiated in [1] and compute J (1)(1) and J Except for theσ aχH -term, all the rest of the terms in (11,12) have already appeared in our previous publication [58] at a fixed order in the gradient expansion. The present study generalises many of the TCs obtained at a fixed order gradient expansion into TCFs.
To the best of our knowledge, the TCF σχ is introduced here for the first time and will play a major role below. It is important to stress the difference between σχ and σ χ of [1,59,67]. Both TCFs generalise CME/CSE. Yet, while the latter is induced by variation of the magnetic field, the former is due to inhomogeneity of the charge densities ρ, ρ 5 . One might naively expect that both TCFs are equal. In fact they are not, as we demonstrate below.
Particularly, the first gradient corrections to CME/CSE, the relaxation time corrections, are different depending on if it is the magnetic field or the charge density that varies with time.  [47,48,58].σ aχH could be considered as an axial analogue of σ aχH . However, as will be clear later,σ aχH has an overall factor q 2 so that it will be non-vanishing starting from fourth order in the gradient expansion only. σ 1,2,3 andσ 3 are TCFs of the third order derivative operators (we remind the reader that the e/m fields are counted as of first order). σ 1 , σ 2 corresponds to rotor of Hall diffusion [58], and σ 3 ,σ 3 are rotors of anomalous chiral Hall effect [58].
The constitutive relations (11,12) could be re-written in a more compact way, where σ 1,2,3 andσ 3 constitute corrections to CME/CSE and, through spatial inhomogeneities of ρ, ρ 5 , influence the Ohmic conductivity. The diffusion constant D 0 was initially promoted into scalar TCF D in [13] and now becomes a tensor TCF, linearly depending on E and B because of the weak background field approximation adopted in the present work. It is important to emphasise that the tensorial structure emerges solely due to the anomaly, as clear from the coefficient κ in front.
These analytical results correspond to third order derivative expansion of J (1)(1) and . Beyond the hydrodynamic limit, the TCFs are computed numerically. The results are presented and discussed in subsection IV C. We observe that none of the TCFs survives at asymptotically large ω 5.
The TCF σχ enters the dispersion relation of CMW: The dispersion relation (25) is exact to all orders in q 2 . In the hydrodynamic limit, using (9,16), the dispersion relation can be solved analytically with the most comprehensive result reported in [58]. Yet, we have discovered a set of solutions with purely real ω. That is, for some (continuum set of) values of magnetic field B, there is a discrete density wave mode (ω B , q B ), which propagates without any dissipation. This is a quite intriguing result, which originates solely from the all order resummation procedure.
As mentioned in the Introduction, the TCFs could be Fourier transformed into memory functions, see an extensive discussion in e.g. [7,13]. The CME current with retardation effects is Via inverse Fourier transform, the CME/CSE memory function is (we focus on the case The memory functionσχ is displayed in figure 1. An important feature of this function is that it has no support at negative times, which is nothing but manifestation of causality. Another very interesting observation is that rather than having an instantaneous response picked at the origin, like in original CME, the actual response is significantly delayed and picked at a finite time of order temperature. This behaviour ofσχ is quite distinct from diffusion memory functionD(t) and shear viscosity memory functions computed previously in [7,13], which are picked at the origin. It is interesting to explore dependence of the TCFs on the chemical potentials. Of special interest is the case of zero background axial charge density,ρ 5 = 0, which is the most realistic scenario for any conceivable experiment. However, even in this case, µ 5 could be nonzero and would be proportional to E · B due to the chiral anomaly (1). Because of the linearisation approximation, the TCFs σ 1 ,σ 3 vanish in the limitρ 5 = 0. We expect them to be nonzero beyond the current approximation.
The generalised CME/CSE conductivity σχ in fact does not depend on either of the chemical potentials. At q = 0, for the remaining TCFs we discover some universal dependence:σ aχH vanishes; σ aχH , D H ,D H do not depend on the chemical potentials at all; σ 1 is linear in κ 2μμ 5 ; σ 3 is linear in κμ; similarly,σ 3 is linear in κμ 5 ; σ 2 has a normal component independent of the chemical potentials and anomaly induced correction which is linear in κ 2 (μ 2 +μ 2 5 ). All these features can be derived from the underlying equations (see Appendix A 2 for relevant ODEs).
Each TCF can be split into real part (even powers of frequency) and imaginary part (odd powers of frequency). Based on the time reversal criterion, we conclude that the real parts of σχ ,1,2 , D H ,D H and imaginary parts of σ aχH ,σ aχH ,σ 3 ,σ 3 are non-dissipative; all the rest do lead to dissipation of the currents. It is interesting to notice that there are points in the (ω, q) phase space, where some of the dissipative terms vanish. Particularly, this happens to Re[D] and Im [σχ].
The rest of this paper is structured as follows. In Section III, we present the holographic model. This section is identical to that of [58] and presented here solely for self-consistency.
Section IV contains the main part of the study: gradient resummation for nonlinear chiral transport. It is further split into four subsections. In subsection IV A, the constitutive relations (11,12) are derived from the dynamical components of the bulk anomalous Maxwell equations near the conformal boundary. In subsection IV B, the TCFs are analytically computed in the hydrodynamic limit. Subsection IV C numerically extends the results beyond this limit. Subsection IV D focuses on the CMW dispersion relation beyond hydrodynamic limit. Section V concludes our study. Appendix supplements calculational details for Section IV.
The holographic model is Maxwell-Chern-Simons theory in the Schwarzschild-AdS 5 . The bulk action is where and the counter-term action S c.t. is The gauge Chern-Simons terms (∼ κ) in the bulk action mimic the chiral anomaly of the boundary field theory. Note M N P QR is the Levi-Civita symbol with the convention rtxyz = +1, while the Levi-Civita tensor is M N P QR / √ −g. The counter-term action (30) is specified based on minimal subtraction, which excludes finite contribution to the boundary currents from the counter-term.
In the ingoing Eddington-Finkelstein coordinate, the Schwarzschild-AdS 5 is where f (r) = 1 − 1/r 4 . Thus, the Hawking temperature (identified as temperature of the boundary theory) is normalised to πT = 1. On the hypersurface Σ of constant r, the induced metric γ µν is It is convenient to split the bulk equations into dynamical and constraint components, dynamical equations : constraint equations : where The boundary currents are defined as which, in terms of the bulk fields, are where n M is the outpointing unit normal vector with respect to the slice Σ, and ∇ is compatible with the induced metric γ µν .
The radial gauge V r = A r = 0 will be assumed throughout this work. As a result, in order to determine the boundary currents (38) it is sufficient to solve dynamical equations (33) only, leaving the constraints aside. Indeed, the constraint equations (34) give rise to continuity equations (1) Practically, it is more instructive to relate the currents (38) to the coefficients of near boundary asymptotic expansion of the bulk gauge fields. Near r = ∞, where A possible constant term for A µ in (40) has been set to zero, in accordance with the fact that no axial external fields is assumed to be present in the current study. V µ is the gauge potential of external electromagnetic fields E and B, Dynamical equations (33) are sufficient to derive (40,41), where the near-boundary data µ have to be determined by completely solving (33) from the horizon to the boundary. The currents (38) become As the remainder of this section, we outline the strategy for deriving the constitutive relations for J µ and J µ 5 . To this end, we turn on finite vector/axial charge densities for the dual field theory, which are also exposed to external electromagnetic fields. Holographically, the charge densities and external fields are encoded in asymptotic behaviors of the bulk gauge fields. In the bulk, we will solve the dynamical equations (33) assuming the charge densities and external fields as given, but without specifying them explicitly.
Following [13] we start with the most general static and homogeneous profiles for the bulk gauge fields satisfying the dynamical equations (33), where V µ , ρ, ρ 5 are all constants for the moment. Regularity at r = 1 has been used to fix one integration constant for each V i and A i . As explained below (41), the constant term in A µ is set to zero. Through (43), the boundary currents are Hence, ρ and ρ 5 are identified as the vector/axial charge densities.
Next, following the idea of fluid/gravity correspondence [53], we promote V µ , ρ, ρ 5 into arbitrary functions of the boundary coordinates As a result, (44) ceases to solve the dynamical equations (33). To have them satisfied, suitable corrections in V µ and A µ have to be introduced: where V µ , A µ will be determined by solving (33). Appropriate boundary conditions are classified into three types. First, V µ and A µ are regular over the domain r ∈ [1, ∞). Second, at the conformal boundary r = ∞, we require which amounts to fixing external gauge potentials to be V µ and zero (for the axial fields).
Additional integration constants will be fixed by the Landau frame convention for the currents, The Landau frame convention corresponds to a residual gauge fixing for the bulk fields.
The vector/axial chemical potentials are defined as Generically, µ, µ 5 are nonlinear functionals of densities and external fields.
In terms of V µ and A µ , the dynamical equations (33) are In subsequent two sections we will present solutions to (51)(52)(53)(54) under two approximation schemes discussed in the Introduction.

IV. NONLINEAR CHIRAL TRANSPORT AND GRADIENT RESUMMATION
In this section, we focus on all order gradient resummation. While the background E and B fields are assumed here as time independent and space homogeneous, the charge densities are allowed to weakly fluctuate around some constant values. In this way we will account for the derivatives of the charge densities only. Following a more general ansatz (4,5), two expansion parameters are introduced: The corrections V µ and A µ are first expanded in powers of , and then each order in is further expanded in powers of α: The boundary currents are expanded accordingly where J t , J t 5 are fixed by the Landau frame convention (49). J (0)(1) , J This section is split into four subsections. The first one IV A is devoted to derivation of the constitutive relations (11,12). In the following subsections IV B and IV C, the TCFs in (11,12) are evaluated, first analytically in the hydrodynamic limit, and then numerically for arbitrary momenta. The last subsection IV D is about non-dissipative modes in the CMW dispersion relations. where were derived in [1] and are presented in Appendix-A 1.
Following the formalism introduced in [5,6], the corrections V where S i ,S i , V i andV i are functionals of the boundary derivative operator ∂ µ and functions of the radial coordinate r. They also depend on the constant valuesμ andμ 5 of the chemical potentials. Fourier transforming δρ and δρ 5 turns all the derivatives into momenta. Thus, in momentum space, these decomposition coefficients become functions of the radial coordinate, frequency ω and spatial momentum squared q 2 : which satisfy partially decoupled non-homogeneous ODEs listed in Appendix-A 2. The decomposition functions S i ,S i , V i andV i are nothing else but elements of the inverse Green function matrix for the system of the ODE's.
As discussed in Section III, the boundary conditions for the decomposition coefficients in (63)(64)(65)(66) are Additional integration constants will be fixed by the Landau frame convention (49).

C. Beyond the hydrodynamic limit: numerical results
In this section, we present our results for the TCFs in (11,12) for finite frequency/momentum via solving the ODEs (A13-A28) numerically. Pseudo-spectral collation method is employed, which essentially converts the continuous boundary value problem of linear ODEs into that of discrete linear algebra. For more details on the numerical method, we recommend the references [68][69][70]. Thanks to the symmetry relations (78), we plot the TCFs σ aχH ,σ aχH , σ 1,2 for κμ κμ 5 only without loss of generality. For D H and σ 3 , this constraint is abandoned so thatD H andσ 3 could be extracted from D H and σ 3 via the exchangeμ ↔μ 5 .
First, consider TCF σχ, which generalises the original CME (CSE) and measures the response to inhomogeneity of charge density ρ (ρ 5 ). Note σχ does not depend on the vector/axial chemical potentials at all, as can be seen from the relevant ODEs (A14,A15,A18).
In Figure 2 we show the 3D plot of σχ. The plots in Figure 3 are 2D slices of Figure 2 when either ω = 0 or q = 0. While σχ is different from the chiral magnetic conductivity σ χ of [1], it has roughly the same dependence on frequency/momentum as σ χ as is clear from these plots. Namely, σχ shows a relatively weak dependence on q 2 while its dependence on ω is more profound: damped oscillations towards asymptotic regime around ω 5 where σχ vanishes essentially. As will be clear later, this damped oscillating behavior is also observed in all other TCFs. This phenomenon can be related to quasi-normal modes in the presence of background fields, but here we are not pursuing this connection any further. When q = 0 we computed the inverse Fourier transform of σχ, that is the memory functionσχ(t) of (27), as displayed in Figure 1.  Thus, we will mainly focus on D H . σ aχH is the anomalous chiral Hall TCF andσ aχH is its axial analogue. Since V 4 = q 2 V 5 andV 4 = q 2V 5 (see (A31)), from the ODE (A25) it is obvious thatσ aχH has an overall q 2 factor, so we will plotσ aχH /q 2 in order to see non-trivial behavior.

D. CMW dispersion relation to all orders: non-dissipative modes
The TCF σχ enters the dispersion relation of CMW: (97)  The dispersion relation (97) is exact to all orders in q 2 , provided κB 1. General solutions of this equation are complex and cannot be studied with our present results. This is because σχ(ω, q 2 ) and D(ω, q 2 ) have been computed for real values of ω only. We believe that beyond the hydrodynamic limit, equation (25) has infinitely many gapped modes. Exploring this point in general would require going into complex ω plane for the TCFs, which is beyond the scope of the present work. Yet, quite intriguingly, there is a set of purely real non-dissipative solutions to (97). In order to find these solutions we have devised the following procedure.
First, the equation is split into real and imaginary parts (assuming q parallel to B): For a fixed value of κB, say κB = 0.33, the functions φ I and φ R are shown in Figure 21 (left) as contour plots in (ω, q 2 ) space (the function D(ω, q 2 ) is taken from [13] ). The dashed (blue) and solid (red) curves stand for φ I and φ R respectively. The numbers indicated on the curves correspond to the values of these functions along the curves. Our interest is when both functions vanish simultaneously, that is a crossing point of φ I = 0 and φ R = 0 curves.
Such crossing is clearly seen in the region ω < 0.5 and q 2 < 0.5. We denote this point by (ω B , q B ). This is a discrete density wave mode propagating in the medium without any dissipation.  in (11,12) correspond to all order resummation of gradients of the the charge density fluctuations parameterised by TCFs, first computed analytically in the hydrodynamic limit (section IV B) and then numerically for large frequency/momentum (section IV C). A common feature of all TCFs in (11,12) is that they depend weakly on spatial momentum but display pronounced dependence on frequency in the form of damped oscillations vanishing asymptotically at ω 5.
Most of our results are presented in Summary section. Among new results worth highlighting is the CME memory function computationσχ(t − t ). The memory function is found to differ dramatically from a delta-function form of instantaneous response. In fact,σχ(t−t ) vanishes at t = t and the CME response gets built only after a finite amount of time of order temperature.
Another result we find of interest is related to CMW dispersion relation, which for the first time was considered to all orders in momentum q. Beyond the perturbative hydrodynamic limit, we found a continuum set of discrete density wave modes, which can propagate in the medium without any dissipation. While the original CMW dissipates and that could be one of the problems for its detection, the new modes that we discover should be long lived and have some potential experimental signature 2 . It is important to remember that our calculation of the CMW dispersion relation is done for a weak magnetic field only.
One can obviously question the validity of the results beyond this approximation. Both TCFs σχ and D that enter the CMW dispersion relation are functions of E and B. In our previous work [58], we initiated this study, still in perturbative in E and B regions, but a full non-perturbative analysis will be reported elsewhere [71].
We have found a wealth of non-linear phenomena all induced entirely by the chiral anomaly. An important next step in deriving a full chiral MHD would be to abandon the probe limit adopted in this paper and include the dynamics of a neutral flow as well.
This will bring into the picture additional effects such as thermoelectric conductivities, nor-mal Hall current, the chiral vortical effect [72,73], and some nonlinear effects discussed in [47]. We plan to address these in the future.
Appendix A: Supplement for section IV 1. Review of the relevant results of [1] In this Appendix we summarise the relevant results from [1] at the orders O ( 0 α 1 ) and At the order O( 1 α 0 ), the corrections are (A5) g 3 and g 4 satisfy coupled ordinary differential equations (ODEs), 0 = r 2 ∂ 2 r g 3 + 3r∂ r g 3 − q 2 ∂ r g 4 , which were solved both analytically in the hydro limit (ω, q 1) and numerically for generic values of ω, q. In the hydro limit, having introduced the expansion parameter λ, ω → λω, q → λq, Up to O(λ 2 ), the results for g 3 , g 4 are g (0) 3 + g The relevant boundary currents are presented in Section II.
2. ODEs and the constraints for the decomposition coefficients in (63)(64)(65)(66) We first collect the ODEs satisfied by the decomposition coefficients in (63)(64)(65)(66) and then derive some constraint relations obeyed by these coefficients. Plugging (63)(64)(65)(66) into (51)(52)(53)(54) and performing Fourier transform ∂ µ → (−iω, i q), we obtained ODEs for the decomposition coefficients S i ,S i , V i ,V i . These ODEs can be grouped into partially decoupled sub-sectors: The remaining decomposition coefficients satisfy the same ODEs as above. More specifically, the sub-sector {S 3 , S 3 ,V 7 , V 7 ,V 8 , V 8 ,V 9 , V 9 } satisfies the same equations as the sub- obeis the same equations as sub-sector (ii): In what follows, we explore some "mirror symmetry relations" among these decomposition coefficients, which are useful in simplifying the expressions for currents' constitutive relations at the order O( 1 α 1 ). First, notice that since these two sub-sectors satisfy identical system of ODEs and have the same boundary conditions. Following this reasoning, The "equal sign" in (A29,A30) should be understood in the specific order as shown therein.
These relations also help to reduce the number of the ODEs to be solved.

Perturbative solutions
Here, we summarize the pertubative solutions of (A13-A28) in the hydrodynamic limit where C is the Catalan constant. It is straightforward to read off the boundary data v i and v i from the solutions presented above. The resultant hydrodynamic expansion of v i andv i are summarised in (85-96).