compat=1.1.0

# Long-time tails in the SYK chain from the effective field theory with a
large number of derivatives

Navid<EMAIL_ADDRESS>
__
_School of Nuclear Science and Technology, Lanzhou University,_
_222 South Tianshui Road, Lanzhou 730000, China_
__

We study the nonlinear energy diffusion through the SYK chain in the framework
of Schwinger-Keldysh effective field theory. We analytically construct the
interacting effective Lagrangian up to $40^{th}$ order in the derivative
expansion. According to this effective Lagrangian, we calculate the first
order loop correction of the energy density response function, the pole of
which is the dispersion relation of energy diffusion. As expected, we see that
the standard derivative expansion of that dispersion relation,
$\omega=-iD_{(1)}k^{2}-iD_{(2)}k^{4}+\mathcal{O}(k^{6})$, breaks down due to
the long-time tails. However, we find that the nonlinear contribution of order
$n$ to the self-energy is proportional to $\left(k^{2}\right)^{n+1/2}$. This
suggests to modify the dispersion relation by splitting it into two dispersion
relations and double the number of transport coefficients at any order as
$\omega=-ik^{2}\big{(}D_{(1,1)}\pm
iD_{(1,2)}\left(k^{2}\right)^{1/2}\big{)}-ik^{4}\big{(}D_{(2,1)}\pm
iD_{(2,2)}\left(k^{2}\right)^{1/2}\big{)}+\mathcal{O}(k^{6})$. We find that
the modified series, which include the effect of long-time tails, are
convergent. The radius of convergence is proportional to the ratio of thermal
conductivity to diffusion constant.

###### Contents

  1. 1 Introduction
  2. 2 Effective field theory around thermal equilibrium
    1. 2.1 Effective field theory of energy diffusion
    2. 2.2 Effective Lagrangian and perturbation strategy
  3. 3 The model
    1. 3.1 SYK model and hydrodynamics
    2. 3.2 SYK chain and classical EFT of energy diffusion
  4. 4 SYK Chain and EFT of nonlinear energy diffusion
    1. 4.1 KMS condition
      1. 4.1.1 quadratic action
      2. 4.1.2 cubic action
    2. 4.2 Effective action in terms of energy density
      1. 4.2.1 Quadratic action and the free propagators
      2. 4.2.2 Cubic action and coupling constants
  5. 5 Loop computations
    1. 5.1 Self-energy
    2. 5.2 Long time tails and breakdown of the derivative expansion
  6. 6 Modified dispersion relation
  7. 7 Discussion and outlook
  8. A Dimensions
  9. B Coefficients of $M_{2}$ term in the quadratic Lagrangian
  10. C Energy density in terms of reparametrization mode
  11. D KMS constraints
  12. E Coefficients of $\nabla\epsilon_{a}\partial_{t}^{\ell}(\nabla\epsilon_{a})\partial_{t}^{n-\ell-1}\mathcal{E}$ in the cubic action
  13. F Coefficients of $\partial_{t}\epsilon_{a}\partial_{t}^{\ell}\epsilon_{a}\partial_{t}^{n-\ell-1}\mathcal{E}$ in the cubic action
  14. G Loop computations
  15. H Series coefficients and convergence of series

##  1 Introduction

The classical picture of hydrodynamics, as a set of deterministic conservation
equations, can only describe the low-energy long-wavelength dissipation
processes, without considering fluctuations [1]. In order to include the
effect of fluctuations, one method is to place noise terms on the right side
of the conservation equations. Then fluctuation-dissipation theorem determines
the strength of the noise [2]. However, in this traditional “stochastic”
picture, the interaction between the noise fields is ignored. The recently
developed effective field theory (EFT) of hydrodynamics solves this problem by
considering the effects of non-Gaussian noise [3, 4, 5, 6, 7] 222See [8, 9,
10, 11, 12] for earlier works and also [13] for a review.. In this framework,
one can systematically derive the well known stochastic effects, such as long-
time tails [15].

The EFT of hydrodynamics has also some new predictions. An example is the
discovery of a positive contribution to the magneto-resistance in a $U(1)$
anomalous system [17]. Another interesting example is the prediction of
stochastic transport [16]. In two pioneering works, this EFT has also been
applied to quantum chaotic systems [18, 19]. The pole-skipping phenomenon as a
prediction of such EFT in Ref.[18] reveals the hydrodynamic origin of the
quantum chaos, at least in maximally chaotic systems [20]333See [22] for the
first observation of pole-skipping and the relation with hydrodynamics in
holography. See also [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43] and [44, 45] for the holographic and CFT
extensions, respectively. . In another direction, the EFT of hydrodynamics has
been recently applied to study the critical fluctuations near the QCD critical
point [47, 48] 444See also [49] for the application to the quasicrystals
systems and [50] to a system in a phase with a spontaneously broken U(1)
symmetry..

On the other hand, an important problem in classical hydrodynamics is to study
the large-order behavior of derivative expansion. In holographic systems, this
problem has been studied in real space and in momentum space [51][52, 53, 54].
In both cases it turns out that the large-order derivative expansion encodes
some information about the lowest non-hydrodynamic modes in the system. What
we want to do in this paper is to go one step further and include the effect
of fluctuations on the large-order derivative expansion within the framework
of EFT of hydrodynamics.

As is well known from stochastic [55, 56, 57, 58] and EFT [15] studies, the
derivative expansion breaks down due to the hydrodynamic fluctuations, and the
long-time tails. However, all these studies are based on the consideration of
the first order derivative expansion 555In the language of EFT, the above
works focus on the statistical fluctuations.. Paraphrasing the discussion in
the previous paragraph, we would like to determine how much information can be
extracted from the large-order derivative expansion in the presence of long-
time tails.

The EFT of hydrodynamics allows us to study the above issue, systematically.
We can construct EFT in the derivative expansion, and then compute the loop
corrections. In a general system, these two expansions should be truncated at
the same order. But in a large $N$ theory, due to suppression of coupling
constants in the $\frac{1}{N}$ expansion, the situation is different. One can
go through the derivative expansion while dropping out higher-than-one loops
[59]666We would like to thanks Luca Delacrétaz for pointing this out..
Therefore, , in a theory with a large number of degrees of freedom, we can
study the effect of one-loop interactions on the classical Green’s functions
at large-order in the derivative expansion.

The system that we choose to study is the low energy SYK chain with a limit of
$N\gg 1$. In the continuum limit of the chain, the effective action of the
system is found to be a local Schwarzian [60]. The system is not CFT and in
contrast to CFT in 1+1 dimension, it has a well-defined hydrodynamic regime
[61, 59]. In fact, energy is conserved in the system and diffuses through the
chain [60, 62]. Thus we can describe the low energy dynamics of the system by
the EFT of energy diffusion. In order to capture the effect of long-time tails
in this system, we assume that the diffusion constant is a function of energy
fluctuations. However, since the thermodynamic effective action of the system
includes quantum effects, we should use the quantum hydrodynamic EFT, which is
valid at finite $\hbar$ and to all order in derivatives [18]. Then by putting
two copies of the system on the two legs of closed time path (CTP) contour, we
construct the fluctuating EFT of hydrodynamics for the energy diffusion in the
chain. Although in the previous studies the EFT of energy diffusion was
constructed up to first [15] or second [16] order in derivatives, we
analytically construct it up to $40^{th}$ order in derivatives.

In the next step, we use the above EFT to compute the energy density response
function, whose pole is the diffusion dispersion relation. In the absence of
loop corrections, the “classical” dispersion relation is given in a derivative
expansion in momentum space,
${\mathfrak{w}}={\mathfrak{w}}({\mathfrak{q}}^{2})$. This series is convergent
in the entire range of momenta allowed by the EFT. When including the
interactions and loop effects, however, the derivative expansion breaks down.
Nonetheless, we show that one can still derive the diffusion pole in a power
series as ${\mathfrak{w}}={\mathfrak{w}}(|{\mathfrak{q}}|)$. Compared to the
classical dispersion relation, two new things would appear. First, the number
of transport coefficients in this series is twice of that in the classical
dispersion relation. In other words, the dispersion relation at order $n$ in
derivatives is characterized by $2n$ transport coefficients. Second, we find
that the dispersion relation series converges in momentum space. The radius of
convergence is given by the ratio of the thermal conductivity to the diffusion
constant. In the rest of the paper, we first briefly review the EFT of
hydrodynamics in § 2. Then in § 3 we discuss the relation between Schwarzian
theory and EFT of hydrodynamics. We then explain how to add fluctuations to
this picture. In § 4 we construct the EFT of fluctuating energy diffusion in
the SYK chain to $40^{th}$ order in the derivative expansion. § 5 is devoted
to compute the loop corrections of the energy density response function. We
then introduce the idea of modified dispersion relation and calculate the
corresponding series coefficients in § 6. Finally, in § 7 we end with review
of the results, mentioning possible applications and discussing some followup
directions.

##  2 Effective field theory around thermal equilibrium

The main quantity that we want to compute is the energy density response
function in a quantum chaotic system at finite temperature. We aim to do this
via computing the energy density correlation function

$\langle\mathcal{E}(t,x)\mathcal{E}(t^{\prime},x^{\prime})\rangle_{\beta}=\text{Tr}\big{(}\rho_{0}\,\mathcal{E}(t,x)\mathcal{E}(t^{\prime},x^{\prime})\big{)}$
(2.1)

where $\rho_{0}$ is the thermal density matrix. Note that the energy density
is a macroscopic dynamical variable. For this reason, we would like to compute
the above correlator in the framework of effective field theory (EFT). We will
construct the EFT we want according to ref. [13]. Below, we start from a
microscopic theory and briefly review the steps of ref. [13] to construct EFT
around thermal equilibrium.

The microscopic analogue of (2.1) can be written as
$\langle\psi(t,x)\psi(t^{\prime},x^{\prime})\rangle_{\beta}$ where $\psi$ is a
microscopic dynamical variable. Such an out of equilibrium observable
corresponds to insertion of microscopic fields $\psi_{1}$ and $\psi_{2}$ on
the two legs of a closed time path (CTP), namely on the Schwinger-Keldysh
contour 777Later on, we will elaborate on the connection between $\psi$ and
$\mathcal{E}$.. Then by coupling the fields to external sources $\phi_{1}$ and
$\phi_{2}$, one can compute the generating functional

$W[\phi_{1},\phi_{2}]=\,\int_{\rho_{0}}D\psi_{1}D\psi_{2}\,e^{iI_{0}[\psi_{1},\phi_{1}]-iI_{0}[\psi_{2},\phi_{2}]}\,,$
(2.2)

where $I_{0}[\psi_{1},\phi_{1}]$ is the microscopic action in the presence
external source $\phi$.

In the language of Wilsonian RG, one may integrate out the UV dynamical
variables, the so-called fast modes, in (2.2), to find an effective action
describing the low energy dynamics. Let us ideally assume that $\psi$ fields
can be separated into UV and IR variables, $\varphi$ and $\chi$, respectively.
Integrating out $\varphi$’s, (2.2) takes the following form:

$W[\phi_{1},\phi_{2}]=\,\int_{\ell_{\text{mic}}}D\chi_{1}D\chi_{2}\,e^{iI_{\text{EFT}}[\chi_{1},\chi_{2};\,\phi_{1},\phi_{2};\,\rho_{0}]}\,.$
(2.3)

Here, $I_{\text{EFT}}$ is the effective action of slow variables $\chi$ and is
valid at length scales larger than the microscopic scale $\ell_{\text{mic}}$,
or equivalently at energy scales much smaller than the UV cutoff
$1/\ell_{\text{mic}}$.

What are the slow modes $\chi$ in (2.3)? In order to answer the question, let
us limit the discussion to the situation in which the length (and time) scale
of perturbations in the thermal system is much larger than the microscopic
length (and time) of relaxation, i.e. $L\gg\ell$ (and $t_{L}\gg\tau$). In this
case, non-conserved quantities relax back to equilibrium very fast with the
rate of $\sim 1/\tau$. However, for conserved quantities, which cannot be
locally destroyed, the only way to relax to equilibrium is transport on scales
of order $L$ (and $t_{L}$). One concludes that at such macroscopic scales,
which denote the so-called IR limit in (2.3), the only relevant variables
$\chi$ are those associated with conserved quantities. The EFT of $\chi$ modes
is indeed hydrodynamics.

###  2.1 Effective field theory of energy diffusion

Let us consider the special case in which the only conserved quantity in the
system is energy. In this EFT of energy diffusion, the $\chi$ field in (2.3)
is associated with the energy conservation $\partial_{\mu}J^{\mu}=0$, with
$J^{\mu}$ being the energy current coupled to background gauge field
$A_{\mu}$. So, we should replace $\phi_{1,2}$ in (2.3) with $A_{s\mu}$,
$s=1,2$

$W[A_{1\mu},A_{2\mu}]=\,\int
D\chi_{1}D\chi_{2}\,e^{iI_{\text{EFT}}[\chi_{1},\chi_{2};\,A_{1\mu},A_{2\mu}]}\,.$
(2.4)

Then the conservation of $J^{\mu}_{1,2}$ translates to gauge invariance of
$W$:

$W[A_{1\mu},A_{2\mu}]=\,W[A_{1\mu}+\partial_{\mu}\lambda_{1},A_{2\mu}+\partial_{\mu}\lambda_{2}]\,.$
(2.5)

In addition, one can also show that the generating functional on CPT satisfies
the following properties:

$\displaystyle W[A_{\mu},A_{\mu}]=1$ (2.6) reflectivity: $\displaystyle
W^{*}[A_{1\mu},A_{2\mu}]=W[A_{2\mu},A_{1\mu}]$ (2.7) Cauchy-Schwarz ineq.:
$\displaystyle\mathfrak{Re}W[A_{1\mu},A_{2\mu}]\leq 0$ (2.8) KMS condition:
$\displaystyle
W[A_{1\mu}(x),A_{2\mu}(x)]=W[A_{1\mu}(-t,-\vec{x}),A_{2\mu}(-t-i\beta,-\vec{x})]$
(2.9)

Now one may ask in what sense $\epsilon$ field is associated with
$\partial_{\mu}J^{\mu}=0$? Let us denote that the above-mentioned $I_{EFT}$
should be such that the equations of motion of $\chi_{1,2}$ are equivalent to
conversations of $J^{\mu}_{1,2}$. This condition is fixed if $\chi_{1,2}$
always appear with external fields through the combinations

$B_{1\mu}=\,A_{1\mu}+\partial_{\mu}\chi_{1},\,\,\,\,\,B_{2\mu}=\,A_{2\mu}+\partial_{\mu}\chi_{2}\,.$
(2.10)

Therefore the $\chi_{1,2}$ are in the fact Stueckelberg fields associated with
the symmetry (2.5). The generating functional (2.4) then can be rewritten as

$W[A_{1\mu},A_{2\mu}]=\,\int
D\chi_{1}D\chi_{2}\,e^{iI_{\text{EFT}}[\,B_{1\mu},B_{2\mu}]}\,.$ (2.11)

Now the main question is how to construct $I_{\text{EFT}}$. The separation of
scales $L\gg\ell$ motivates to construct $I_{\text{EFT}}$ in a derivative
expansion with the expansion parameter
$\ell\partial_{\mu}\sim\frac{\ell}{L}\ll 1$. However, even writing down the
most general local derivative expansion of $I_{\text{EFT}}[B_{1\mu},B_{2\mu}]$
subject to conditions (2.5)-(2.9) is not enough 888Let us suppose $J^{\mu}$
current was related to an internal $U(1)$ symmetry. Then writing down the most
general local derivative expansion of $I_{\text{EFT}}[B_{1\mu},B_{2\mu}]$
would describe a super fluid phase with the spontaneously broken $U(1)$
symmetry [13].. One further symmetry has to be imposed. To understand it, it
is convenient to define

$\sigma=\,\frac{1}{2}(\chi_{1}+\chi_{2})\,,\,\,\,\,\,\,\,\,X^{a}=\chi_{1}-\chi_{2}\,.$
(2.12)

Calculating the equations of motion from local $I_{EFT}$, it turns out that
$\partial_{t}\sigma(t,\vec{x})$ is related to the energy density
$\mathfrak{h}+\mathcal{E}(t,\vec{x})$, and $X^{a}(t,\vec{x})$ corresponds to
the noise field999$\mathfrak{h}$ denotes the energy density in thermal
equilibrium.. This has two direct consequences:

  1. 1.

The derivative counting scheme should be such that $\partial_{t}\sigma\sim
X^{a}$ [15].

  2. 2.

Since only $\partial_{t}\sigma$ is physical, $\sigma$ must always appear with
at least one time derivative in the effective action. It is equivalent to say
that $I_{\text{EFT}}$ is required to be invariant under the so-called diagonal
shift symmetry [3]:

$\sigma(t,\vec{x})\rightarrow\sigma(t,\vec{x})+a(\vec{x})\,,\,\,\,\,\,\,\,X(t,\vec{x})\rightarrow
X^{a}(t,\vec{x})\,.$ (2.13)

###  2.2 Effective Lagrangian and perturbation strategy

So far, we have discussed all the elements that should be included when
constructing EFT of energy diffusion around thermal equilibrium. Considering
$I_{\text{EFT}}=\int dx^{d+1}\mathcal{L}_{\text{eff}}$, the most general
nonlinear Lagrangian satisfying the above-mentioned conditions, to second
order in $a$-fields, and in the absence of external sources, can be written as
[3, 13]

$\mathcal{L}_{\text{eff}}[\sigma,X_{a}]=\,-H\partial_{t}X_{a}-G_{i}\partial_{i}X_{a}+\,i\,\partial_{t}X_{a}M_{1}\partial_{t}X_{a}+\,i\,\partial_{i}X_{a}M_{2}\partial_{i}X_{a}+\mathcal{O}(X_{a}^{3})\,.$
(2.14)

In this equation, the coefficient functions $H$ and $G_{i}$ are (in general
nonlinear) functions of $\dot{\sigma(t,x)}$ and its partial derivatives.
Similarly, $M_{1,2}$ are differential operators constructed out of
$\dot{\sigma(t,x)}$ and differential operators $\partial_{i}$ and
$\partial_{t}$. At the quadratic level ($\mathcal{L}_{2}$), however, $M_{1,2}$
are purely differential operators constructed out of $\partial_{t}$ and
$\partial_{i}$ acting on the $X_{a}$ field sitting in their right side.

Another point with (2.14) is that the classical equation of motion of $X^{a}$
is simply the equation of conservation of energy:

$\partial_{t}H+\partial_{i}G_{i}=\,0\,.$ (2.15)

Thus $H$ and $G_{i}$ are classical parts of the energy density and energy
flux, respectively 101010By classical here, we mean non-noisy part of the
quantities.. In any particular system of interest, $H$ and $G_{i}$ should be
used as input data; then by applying KMS conditions to (2.14), one specifies
$M_{1}$ and $M_{2}$, as well.

Before introducing the system we are interested in in this work, let us recall
that our final goal is to compute one-loop corrections to (2.1) from (2.14),
to high orders in the derivative expansion. By doing this and obtaining the
corresponding energy diffusion pole, we will be able to study the large-order
behavior of derivative expansion in the presence of fluctuations.

Let us first mention our perturbation strategy for constructing Lagrangian
functions. We will include three expansions.

  1. 1.

expansion around equilibrium: The equilibrium state corresponds to
$\sigma(t,x)=t$ and $X_{a}(t,x)=0$. By considering the small deviations as

$\sigma(t,x)=t+\,\epsilon(t,x),\,\,\,\,\,\,\,\,X_{a}(t,x)=-\epsilon_{a}(t,x)\,,$
(2.16)

we construct our EFT to third order in the above out-of-equilibrium fields. We
will see that the cubic Lagrangian is sufficient to find the finite one-loop
correction to (2.1).

  2. 2.

noised field expansion: We limit our study to include the effect of thermal
fluctuations up to second order in $a-$fields.

  3. 3.

derivative expansion: In order to study the large-order behavior of the
hydrodynamic derivative expansion in the presence of fluctuations, we need to
construct $\mathcal{L}$ to the high-order derivatives. On the other hand, we
would like to find all terms in $\mathcal{L}$ analytically. By implementing a
systematic method, we will construct the Lagrangian to $40^{th}$ order in the
derivative expansion. Although due to the large size of analytic expressions,
we only explicitly display the Lagrangian terms to $20^{th}$ order, in the
appendices 111111It should be noted that, in principle, our method can be
implemented to any higher order in the derivative. However, we found that for
our current purposes, keeping terms to $40^{th}$ order is sufficient..

Before concluding this section, let us comment on the size of the spatial
derivative in our system. Since our expected EFT should describe a diffusion
process controlled by (2.15), we are interested in the disturbance of
$\omega\sim k^{2}$. This together with $\partial_{t}\sigma\sim X^{a}$ (see
below (2.12)) provide our derivative counting scheme in this work.

##  3 The model

So far, our discussion about EFT of hydrodynamics has been general, without
mentioning any special physical systems. When restricting the study to the
lower orders in the derivative expansion, it would be feasible to construct
$\mathcal{L}_{EFT}$ for the general system. For example, Ref.[15] constructs
the energy diffusion EFT in a general thermal system up to first order in the
derivative expansion. But at higher orders of the derivative, the calculation
will become more complicated because more possible terms may contribute to $H$
and $G_{i}$. Therefore, since our goal is to explore the large-order behavior
of the derivative expansion in this paper, we choose to use a specific system
with the well-known $H$ function.

Our system of interest is the SYK chain. The $(0+1)$ dimensional SYK model
[63, 64] and also the $(1+1)$ dimensional SYK chain [60] have been widely
studied in the literature [65]. However, in order to be clear about the
problem that we want to address, we need to recall some aspects of the SYK
model. In particular, it is well known that the infrared theory in the strong
coupling limit is described by hydrodynamics [66]. In § 3.1, we first briefly
review the model. Then following [18] and by using the language of Schwinger-
Keldysh EFT, we revisit the mentioned hydrodynamic picture and specify the $H$
function associated with the SYK model. Then in § 3.2, after a quick look at
the SYK chain model, we will review its connection with the EFT of
hydrodynamics and specify the corresponding $H$ and $G_{i}$ functions in
(2.14) [18].

Everything we will discuss in the following two subsections will be devoted to
classical aspects of EFT of hydrodynamics. But our ultimate goal is to couple
classical EFT with fluctuations, which is the subject of the next section.

###  3.1 SYK model and hydrodynamics

The SYK model consists of $N$ Majorana fermions
$\chi_{j}(\tau),\,j=1,2,\cdots,N$121212Here $\tau$ is the Euclidean time
coordinate. At finite temperature, we have $\tau\sim\tau+\beta$.  with a
random four-fermion interaction. The random coupling has zero mean
$\overline{J_{jklm}}=0$ and non-zero variance
$\frac{1}{3!}N^{3}\overline{J^{2}_{jklm}}=J^{2}$. The model is non-local in
space in the sense that the interaction is all-to-all. This model should be
actually considered as a (0 + 1)-dimensional quantum mechanical system.

The model is solvable in the large N limit. Specifically in the strong
coupling limit, $N\gg\beta J\gg 1$, the model shows up a conformal symmetry in
the infrared limit, that is the invariance under reparameterization
$\tau\rightarrow f(\tau)$. However, it turns out that the corresponding
conformal correlators, $G_{c}$, are only invariant under $SL(2,R)$ subgroup of
this conformal symmetry. In other words, the conformal symmetry is
spontaneously broken by the conformal solution $G_{c}$. As a result, the
infrared effective action of the model is zero when evaluated on fluctuations
that are reparameterizations of $G_{c}$.

Beyond the conformal limit, the picture above will no longer be the case. In
particular, due to explicitly breaking of the conformal symmetry, the
effective action becomes non-zero when evaluated on fluctuations of $\delta
G_{c}$, the reparameterizations of $G_{c}$. For an infinitesimal
reparameterization $\tau\rightarrow\tau+\epsilon(\tau)$, one finds [65]

$\frac{S}{N}=\frac{\alpha}{J}\int_{0}^{\beta}d\tau\frac{1}{2}\bigg{[}(\epsilon^{\prime\prime})^{2}-\lambda^{2}(\epsilon^{\prime})^{2}\bigg{]},\,\,\,\,\,\,\lambda=\frac{2\pi}{\beta}\,,$
(3.1)

with $\alpha$ being a numerical constant. Now let us take
$\mathcal{L}[\epsilon(t)]=C\big{(}(\epsilon^{\prime\prime})^{2}+\lambda^{2}(\epsilon^{\prime})^{2}\big{)}$
with the Lorentzian time coordinate $t$, and put two copies of it on the two
legs of CTP contour. The Schwinger-Keldysh action, with real time coordinate
$t$, then reads

$\mathcal{L}[\epsilon,\epsilon_{a}]=\,\mathcal{L}[\epsilon_{1}]-\mathcal{L}[\epsilon_{2}]=\,C\,\left(\lambda^{2}-\partial_{t}^{2}\right)\partial_{t}\epsilon\,\partial_{t}\epsilon_{a}\,.$
(3.2)

Considering (2.16), equation (3.2) is exactly the first term in the effective
Lagrangian of hydrodynamics (2.14), specific to a system in $(0+1)$ dimension
with

$H(\sigma)=C\left(\lambda^{2}-\partial^{2}_{t}\sigma\right)\partial_{t}\sigma,\,\,\,\,\,\,\,\,\,\,H_{0}=C\,\lambda^{2}\,.$
(3.3)

Since $H_{0}$ represents the energy density in thermal equilibrium,
$\sigma=t$, it is convenient to take $C=\frac{\mathfrak{h}}{\lambda^{2}}$,
where $\mathfrak{h}$ is the thermodynamic energy density. The classical
equation of motion is then simply the equation of conservation of energy, i.e.
$\partial_{t}H=0$. We conclude that the EFT describing dynamics of the
reparameterization mode $\epsilon$ in the SYK model is actually hydrodynamics
131313See [66] for another way of describing the hydrodynamic origin of
$\epsilon$..

We can extend the above-mentioned hydrodynamic action beyond the second order,
that is, the regime of nonlinear hydrodynamics. It can be done by generalizing
the action (3.1) to a finite reparameterization $\tau\rightarrow\tau+f(\tau)$.
One finds [65]

$\frac{S}{N}=-\frac{\alpha}{J}\int_{0}^{\beta}\,d\tau\,\text{Sch}(f(\tau),\tau),\,\,\,\,\,\,\,\text{Sch}(f(\tau),\tau)=\left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{\prime}-\frac{1}{2}\frac{f^{\prime\prime
2}}{f^{\prime 2}}$ (3.4)

with $f(\tau)=\tan\left(\frac{\pi\tau}{\beta}\right)$. We can have further
reparameterization on the Euclidean time circle,
$\tau\rightarrow\sigma(\tau)$. Then writing $\sigma(\tau)=\tau+\epsilon(\tau)$
and expanding $\text{Sch}(f(\sigma(\tau)),\tau)$ in (3.4), we get exactly the
two quadratic terms of (3.1).

In Lorentzian coordinate, $f(\tau)$ transforms to
$f(t)=\tanh\left(\frac{\pi\sigma(t)}{\beta}\right)$. Inverting this equation,
we define $u(t)$ as

$\frac{1}{2}e^{-\lambda\,\sigma(t)}=\,\frac{1}{2}\,\frac{1-f(t)}{1+f(t)}\equiv\frac{1}{2}u(t)\,.$
(3.5)

Since $\frac{1}{2}u$ is $SL(2,R)$ transformed of $f$, we immediately write:
$\text{Sch}(f(t),t)=\text{Sch}(\frac{1}{2}u(t),t)$. Thus in Lorentzian
coordinate, (3.4) takes the following form

$S=\,-i\int
dt\,\frac{\mathfrak{h}}{\lambda^{2}}\text{Sch}(u(t),t),\,\,\,\,\,\,\,\,\,u(t)=e^{-\lambda\,\sigma(t)}\,,$
(3.6)

where we have used
$\frac{1}{2}\frac{N\,\alpha}{J}=C=\frac{\mathfrak{h}}{\lambda^{2}}$.

We can repeat the discussion around (3.2) to find the Schwinger-Keldysh
analogue of (3.6). Doing so, we find that the reparameterization mode in the
SYK model, in general, is described by nonlinear hydrodynamics with the
corresponding $H$ function given by

$H=-\frac{\mathfrak{h}}{\lambda^{2}}\text{Sch}(u(t),t),\,\,\,\,\,\,\,\,\,u(t)=e^{-\lambda\,\sigma(t)}\,.$
(3.7)

Let us recall that equilibrium state corresponds to $\sigma(t)=t$.

It is worth nothing that this Schwarzian action goes actually beyond ordinary
long distance hydrodynamics. The reason is it includes modes whose time
variation rate is comparable to the temperature [67]. Correspondingly, the
framework of [3] can be used to any large-order in derivatives. Therefore, it
can be well applied to study the dynamics of the reparametrization mode in the
SYK model.

###  3.2 SYK chain and classical EFT of energy diffusion

The SYK chain describes a coupled array of SYK model sites. Each site contains
$N\gg 1$ Majorana fermions with four-fermion random coupling $J_{jklm,x}$.
Here $j=1,2,\cdots,N$ and $x=1,2,\cdots,M$ where $M$ is the number of sites.
Then there is further coupling between each pair of neighboring sites through
another four-fermion random coupling $J^{\prime}_{jklm,x}$. The random
couplings $J_{jklm,x}$ and $J^{\prime}_{jklm,x}$ are drawn independently and

$\overline{J_{jklm,x}}=\overline{J^{\prime}_{jklm,x}}=0,\,\,\,\,\,\,\,\,\,\,\frac{1}{3!}N^{3}\overline{J^{2}_{jklm,x}}=J_{0}^{2},\,\,\,\,\,\,N^{3}\overline{J^{\prime
2}_{jklm,x}}=J_{1}^{2}\,.$ (3.8)

One can show that the Schwinger-Dyson equations of the model reduce to exactly
the same form as those of a $(0+1)$-d SYK model with the coupling constant
$J=\sqrt{J_{0}^{2}+J_{1}^{2}}$.

As in the SYK model, the effective action of the SYK chain admits an
approximate reparametrization symmetry of time in the IR limit. For small
deformations of time $\tau\rightarrow\tau+\epsilon_{x}(\tau)$, the quadratic
effective action is found to be [60]

$\frac{S}{N}=\sum_{n,k}\,\frac{\alpha}{2J}\,\epsilon_{n,k}\bigg{(}\omega_{n}^{2}(\omega_{n}^{2}-\lambda^{2})+\,D\,k^{2}\,|\omega_{n}|(\omega_{n}^{2}-\lambda^{2})\bigg{)}\,\epsilon_{-n,-k}\,,$
(3.9)

where $\epsilon_{n,k}$ is the Fourier transform of $\epsilon_{x}(\tau)$ and
$D\sim J_{1}^{2}/J$ is the diffusion constant. The first term in (3.9) is
exactly the EFT of SYK model (3.1), if we remove the dependence on $k$.
Recalling (3.6), we find that in the limit of an infinite lattice,
$M\rightarrow\infty$, this term can be interpreted as a quadratic
approximation of the local Schwarzian action $\text{Sch}(u(t,x),t)$. What
about the second term in (3.9)? Let us consider two copies of the system
placing on the two legs of CTP contour. Then we find that

$\mathcal{L}[\epsilon,\epsilon_{a}]=\,\mathcal{L}[\epsilon_{1}]-\mathcal{L}[\epsilon_{2}]=\,C\,\bigg{(}\left(\lambda^{2}-\partial_{t}^{2}\right)\partial_{t}\epsilon\,\partial\epsilon_{a}+D\left(\lambda^{2}-\partial_{t}^{2}\right)\partial_{x}\partial_{t}\epsilon\,\partial_{x}\epsilon_{a}\bigg{)}\,.$
(3.10)

The above Lagrangian is actually the quadratic approximation to (2.14) in
$(1+1)$ dimension with141414In our setup, there is only one spatial direction
along the Chain. However, we use subscript $i$ to display the spatial
derivative so that we can discuss the generalization of things to higher
dimensions later.

$H=-\frac{\mathfrak{h}}{\lambda^{2}}\text{Sch}(u(t,x),t)\,\,\,\,\,\,\,\,\,\bigg{(}u(t,x)=e^{-\lambda\,\sigma(t,x)}\bigg{)},\,\,\,\,\,\,\,G_{i}=-D\,\partial_{i}H\,.$
(3.11)

Thus the second term in (3.10) denotes the diffusion of energy through the SYK
chain with $D$ being the diffusion constant.

In summary, given $H$ and $G_{i}$ by (3.11), the linearized version of
equation (2.15) represents the classical linear hydrodynamics of the diffusive
reparametrization mode in the SYK chain. In next section, we discuss the
nonlinear hydrodynamic description of the energy diffusion through the SYK
chain, taking into account the effects of fluctuations generated by going
beyond the quadratic order in (2.14).

##  4 SYK Chain and EFT of nonlinear energy diffusion

Let us recall that for the system to be specified in the framework of EFT of
hydro, the $H$ and $G_{i}$ associated with it need to be determined. As was
discussed earlier, our system of interest is the SYK chain with the $H$ and
$G_{i}$ functions given by (3.11). On the other hand, In order to include the
interaction between the hydrodynamic field $\epsilon$ and the fluctuation
field $\epsilon_{a}$, we have to take into account nonlinear terms in the
effective action. As mentioned before, we aim to construct a cubic interacting
Lagrangian. Considering $u=e^{-\lambda(t+\epsilon(t,x))}$, there are two
potential sources of nonlinearity in (3.11):

  1. 1.

$H$ itself is a nonlinear function of $\epsilon$; in order to construct the
cubic Lagrangian, we expand $H$ about its equilibrium value $\mathfrak{h}$ and
keep terms to second order:

$\begin{split}H&=\,-\frac{\mathfrak{h}}{\lambda^{2}}\,\text{Sch}(e^{-\lambda(t+\epsilon(t,x))},t)\\\
&=\,\,\,\frac{\mathfrak{h}}{2}+\,\mathfrak{h}\left(\partial_{t}\epsilon-\frac{1}{\lambda^{2}}\partial_{t}^{3}\epsilon\right)+\,\mathfrak{h}\left(\frac{1}{2}(\partial_{t}\epsilon)^{2}+\frac{3}{2\lambda^{2}}(\partial^{2}_{t}\epsilon)^{2}+\frac{1}{\lambda^{2}}\partial_{t}\epsilon\,\partial^{3}_{t}\epsilon\right)+\mathcal{O}(\epsilon^{3})\,.\end{split}$
(4.1)

Note that although we aim to perform our computations to high orders in the
derivative expansion, the above expression is by construction truncated at
$\partial_{t}^{2}$. Let us remind that as mentioned below (2.12),
$\partial_{t}\delta\sigma\sim\partial_{t}\epsilon\sim\mathcal{O}(\partial_{t}^{0})$.

  2. 2.

The diffusion coefficient $D$ can also be a nonlinear function of energy
fluctuations $\delta H=H-\frac{\mathfrak{h}}{2}$. We may write

$\begin{split}G_{i}=&\,-D(\delta H)\,\partial_{i}H\\\
=&\,-D_{0}\mathfrak{h}\left(\partial_{i}\partial_{t}\epsilon-\frac{1}{\lambda^{2}}\partial_{i}\partial_{t}^{3}\epsilon\right)-D_{1}\mathfrak{h}\left(\partial_{i}\partial_{t}\epsilon-\frac{1}{\lambda^{2}}\partial_{i}\partial_{t}^{3}\epsilon\right)\left(\partial_{t}\epsilon-\frac{1}{\lambda^{2}}\partial_{t}^{3}\epsilon\right)+\mathcal{O}(\epsilon^{3})\end{split}$
(4.2)

where we have used

$D(\delta H)=\,D_{0}+\,\frac{D_{1}}{\mathfrak{h}}\,\delta
H+\frac{D_{2}}{2\mathfrak{h}^{2}}(\delta H)^{2}+\cdots$ (4.3)

with $D_{n}=\,\mathfrak{h}^{n}\,\frac{\partial^{n}D}{\partial H^{n}}$. Note
that coefficients $D_{n}:n>1$ do not appear in the cubic Lagrangian. It is
also worth noting that in general, any of the $D_{n}$ coefficients can be a
differential operator, constructed out of $\partial_{t}$ and $\partial_{i}$.
However, in this work, we assume that these coefficients are constant. One
important consequence of this assumption is that the classical dispersion
relation of energy diffusion, namely the dispersion obtained from linear
hydrodynamic equations, will be exact in the derivative expansion and is given
by $\omega=D_{0}k^{2}$. In § 6 we will discuss the general case in which
$D_{0}$ itself has a derivative expansion. Finally, one of the problems we
will solve in this work is how this simple dispersion relation changes when
nonlinear effects are included.

Having specified the ingredients of the model, we should now compute $M_{1}$
and $M_{2}$ terms in (2.14). Since there are two $X_{a}$ fields, each of the
terms $M_{1}$ and $M_{2}$ is at least quadratic. So in order to construct a
cubic Lagrangian, we need to determine $M_{1}$ and $M_{2}$ up to first order
in $\epsilon$. Then we find that the appropriate ansatz in the most general
form is as follows

$\displaystyle M_{1}$ $\displaystyle=$
$\displaystyle\,\,\rho\,+\alpha_{1}\delta
H\,\alpha_{2}+\cdots=\,\rho+\,\alpha_{1}\mathfrak{h}\left(\partial_{t}\epsilon-\frac{1}{\lambda^{2}}\partial_{t}^{3}\epsilon\right)\alpha_{2}+\mathcal{O}(\epsilon^{2})\,,$
(4.4) $\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle\,\,\chi\,+\,\xi_{1}\delta
H\,\xi_{2}+\cdots\,=\,\chi+\,\xi_{1}\mathfrak{h}\left(\partial_{t}\epsilon-\frac{1}{\lambda^{2}}\partial_{t}^{3}\epsilon\right)\xi_{2}+\mathcal{O}(\epsilon^{2})\,.$
(4.5)

The coefficients in these two equations are all differential operators, each
of which acting on the first function on the right. Now the task is to apply
the KMS conditions to the effective action and specify these coefficients,
perturbatively in the derivative expansion, in terms of $D_{0}$, $D_{1}$,
$\beta=T^{-1}$ and $\mathfrak{h}$.

By defining
$\mathcal{F}\in\\{\rho(\partial_{t}),\,\alpha_{j}(\partial_{t})\,\cdots\\}$
and $\mathcal{G}\in\\{\chi(\partial_{t}),\,\xi_{j}(\partial_{t}),\cdots\\}$,
we want to find coefficients $\mathcal{F}^{(n)}$ in the following derivative
expansions:

$\mathcal{F}(\partial_{t})=\,\sum_{n=3}\mathcal{F}_{n}\times\left(\frac{\partial_{t}}{\lambda}\right)^{n-2},\,\,\,\,\,\,\,\,\,\,\,\mathcal{G}(\partial_{t})=\,\sum_{n=1}\mathcal{G}_{n}\times\left(\frac{\partial_{t}}{\lambda}\right)^{n-1}.$
(4.6)

In the following subsections we will find equations constraining
$\mathcal{F}_{n}$’s and $\mathcal{G}_{n}$’s.

###  4.1 KMS condition

The KMS condition (2.9) is actually a constraint equation on the generating
functional in the presence of external sources $A_{\mu}$. It is shown that a
sufficient condition for invariance of $W[A_{1\mu},A_{2\mu}]$ under (2.9) is
that $I_{EFT}[B_{1\mu},B_{2\mu}]$ be invariant under

$B_{1\mu}(x)\rightarrow B_{1\mu}(-t,-\vec{x}),\,\,\,\,\,B_{2\mu}(x)\rightarrow
B_{2\mu}(-t-i\beta,-\vec{x})$ (4.7)

which is the same as transformation of background $A_{\mu}$ field (see [3, 14]
for details). These constraints should be imposed to quadratic and cubic
action separately. Here, we do not derive the constraints and just follow
Ref.[3] and quote their results (see also [68] for an earlier derivation).

####  4.1.1 quadratic action

For a quadratic effective Lagrangian of the form

$\mathcal{L}^{(2)}_{\text{eff}}[B^{1},B^{2}]=\,K_{ra}^{\mu_{1}\mu_{2}}B_{a\mu_{1}}B_{r\mu_{2}}+\,\frac{i}{2}\,G_{rr}^{\mu_{1}\mu_{2}}B_{a\mu_{1}}B_{a\mu_{2}}\,,$
(4.8)

where $\mu_{1},\mu_{2}\in\\{t,i\\}$, it is found that the KMS constraint in
momentum space is given by

$i\,G_{rr}(\omega)=\,\frac{1}{2}\coth\left(\frac{\beta\omega}{2}\right)\bigg{(}K_{ra}(\omega)-K_{ar}(\omega)\bigg{)}\,.$
(4.9)

Here, $K_{ar}(\omega)=K_{ra}(-\omega)$ and for simplicity we have dropped the
superscripts $\mu_{1}$ and $\mu_{2}$. In the classical limit $\hbar\rightarrow
0$, the $\coth$ factor in the right side is simply replaced with
$(\beta\omega)^{-1}$. However, since quantum effects as well as the higher
order derivatives are important in our study, we continue to use the original
form (4.9).

The corresponding $G_{rr}$ and $K_{ra}$ functions associated with our system,
in the absence of external sources, can be written as

$\displaystyle K_{ra}^{\mu_{1}\mu_{2}}$ $\displaystyle=$
$\displaystyle\,\mathfrak{h}\,\left(1+\frac{\omega^{2}}{\lambda^{2}}\right)\,\delta^{\mu_{1}}_{t}\delta^{\mu_{2}}_{t}+\,(-D_{0}\,\mathfrak{h})\,(-i\omega)\,\left(1+\frac{\omega^{2}}{\lambda^{2}}\right)\,\delta^{\mu_{1}}_{i}\delta^{\mu_{2}}_{i}\,,$
(4.10) $\displaystyle G_{rr}^{\mu_{1}\mu_{2}}$ $\displaystyle=$
$\displaystyle\,2\,\rho_{-\omega}\,\delta^{\mu_{1}}_{t}\delta^{\mu_{2}}_{t}+\,2\,\chi_{-\omega}\,\delta^{\mu_{1}}_{i}\delta^{\mu_{2}}_{i}\,.$
(4.11)

Note that we take the Fourier components of $B_{\mu_{1}}$ and $B_{\mu_{2}}$ as
$B_{\mu_{1}}^{\omega}$ and $B_{\mu_{2}}^{-\omega}$, respectively. Plugging
above $G_{rr}$ and $K_{ra}$ into (4.9) and expanding it in the derivative
expansion, we can specify $\rho_{n}$ and $\chi_{n}$ coefficients, order by
order in the derivative expansion. We have computed these coefficients to
$40^{th}$ order in derivatives. The $\chi_{n}$ coefficients with $n\leq 20$
have been given in Appendix B. It is worth noting that $\rho$ vanishes to all
order in the derivative expansion. The reason is $K^{tt}_{ra}$ in (4.10), is
an even function of $\omega$. So the right side of (4.9) vanishes, causing
$G^{tt}_{rr}$ to be identically zero.

Putting the above-mentioned parts together, the quadratic Lagrangian takes the
following form:

$\mathcal{L}^{(2)}_{\text{eff}}=\mathfrak{h}\,\partial_{t}\epsilon_{a}\,\bigg{(}1-\frac{\partial_{t}^{2}}{\lambda^{2}}\bigg{)}\partial_{t}\epsilon-\,D_{0}\,\mathfrak{h}\,\partial_{i}\epsilon_{a}\,\bigg{(}1-\frac{\partial_{t}^{2}}{\lambda^{2}}\bigg{)}\partial_{i}\partial_{t}\epsilon+\,i\frac{D_{0}\,\mathfrak{h}}{\beta}\partial_{i}\epsilon_{a}\sum_{n=1}\chi_{n}\left(\frac{\partial_{t}}{\lambda}\right)^{n-1}\partial_{i}\epsilon_{a}\,.$
(4.12)

See Appendix B for the $\chi_{n}$ coefficients.

####  4.1.2 cubic action

For a cubic effective Lagrangian of the form:

$\mathcal{L}^{(3)}_{\text{eff}}[B^{1},B^{2}]=\,\,\frac{1}{2}\,G_{raa}^{\mu_{1}\mu_{2}\mu_{3}}\,B_{a\mu_{1}}B_{r\mu_{2}}B_{r\mu_{3}}+\,\frac{1}{2}\,G_{rra}^{\mu_{1}\mu_{2}\mu_{3}}\,B_{a\mu_{1}}B_{a\mu_{2}}B_{r\mu_{3}}\,,$
(4.13)

the KMS constraint in momentum space is given by [3, 15, 68]

$2\,G_{rra}=-\big{(}N_{1}+N_{2}\big{)}G^{*}_{aar}+N_{1}\,G_{ara}+N_{2}\,G_{raa}\,,\,\,\,\,\,\,\,\,\,N_{i}=\coth\left(\frac{\beta\,\omega_{i}}{2}\right)\,.$
(4.14)

Note that any of the above $G$’s is actually a function of $\omega_{1}$,
$\omega_{2}$ and $\omega_{3}$, where $\omega_{1}+\omega_{2}+\omega_{3}=0$. And
again, for simplicity, we have dropped the superscripts $\mu_{1}$, $\mu_{2}$
and $\mu_{3}$.

The corresponding $G_{raa}$ and $G_{rra}$ functions associated with our
system, in the absence of external sources, can be written as

$\begin{split}G_{raa}^{\mu_{1}\mu_{2}\mu_{3}}=&\,-\mathfrak{h}\,(-i\omega_{2})\bigg{[}D_{0}+D_{1}+3D_{0}\prod_{23}+(D_{0}-D_{1})\big{(}\prod_{33}+\prod_{22}\big{)}+D_{1}\prod_{22}\prod_{33}\bigg{]}\,\delta_{i}^{\mu_{1}}\delta_{i}^{\mu_{2}}\delta_{0}^{\mu_{3}}\\\
&\,\,\,\,+\frac{\mathfrak{h}}{2}\,\left[1+3\prod_{23}+\,2\prod_{22}\right]\delta_{0}^{\mu_{1}}\delta_{0}^{\mu_{2}}\delta_{0}^{\mu_{3}}\,+\,\big{(}2\leftrightarrow
3\big{)}\,,\end{split}$ (4.15)
$G_{rra}^{\mu_{1}\mu_{2}\mu_{3}}=\,\left(1+\prod_{3}\right)\,\bigg{[}\alpha_{2}^{\omega_{2},k_{2}}\alpha_{1}^{\omega_{3},k_{3}}\delta_{0}^{\mu_{1}}\delta_{0}^{\mu_{2}}\delta_{0}^{\mu_{3}}+\,\xi_{2}^{\omega_{2},k_{2}}\xi_{1}^{\omega_{3},k_{3}}\delta_{i}^{\mu_{1}}\delta_{i}^{\mu_{2}}\delta_{0}^{\mu_{3}}\bigg{]}+\,(1\leftrightarrow
2)\,.$ (4.16)

In the above expressions: $\omega_{1}+\omega_{2}+\omega_{3}=0$ and we have
also defined

$\prod_{j_{1}j_{2}\cdots}=\,\left(\frac{-i\,\omega_{j_{1}}}{\lambda}\right)\left(\frac{-i\,\omega_{j_{2}}}{\lambda}\right)\cdots\,.$
(4.17)

Substituting (4.15) and (4.16) into (4.14) and expanding the two sides in the
derivative expansion, we find equations between the expansion coefficients of
$\alpha_{1}$ and $\alpha_{2}$, defined in (4.6), and also equations between of
expansion coefficients of $\xi_{1}$ and $\xi_{2}$. Two comments about these
equations are as follows.

  1. 1.

Using KMS condition, it is clear that $\alpha_{1}$ and $\alpha_{2}$ are
related to the first line of (4.15) while $\xi_{1}$ and $\xi_{2}$ can be found
from the first expression in the second line of (4.15).

  2. 2.

For concreteness, let us consider $\xi_{1}$ and $\xi_{2}$ at $n^{th}$ order.
The corresponding expansion coefficients are $\xi^{(n)}_{1}$ and
$\xi^{(n)}_{2}$ (see (4.6)). It is clear that these coefficients always appear
in pairs as

$\xi^{(j)}_{1}\xi^{(k)}_{2}\equiv\,T_{j,k}\,.$ (4.18)

We find that at $n^{th}$ order in the derivative expansion, there are
$\left[\frac{n-1}{2}\right]+1$ equations between $n$ independent $T_{j,k}$
coefficients 151515It can be understood as follows. Equation (4.14) is
symmetric with respect to $\omega_{1}$ and $\omega_{2}$. While the symmetry in
the right side is obvious, in the left side it is due to (4.16). Thus at any
order in the derivative expansion, after imposing
$\omega_{1}+\omega_{2}+\omega_{3}=0$, equation (4.14) becomes a symmetric
polynomial of $\omega_{1}$ and $\omega_{2}$, giving a set of equations. The
number of independent equations then is equal to the number of independent
coefficients in the expansion of $(\omega_{1}+\omega_{2})^{n}$, i.e.
$\left[\frac{n-1}{2}\right]+1$. These equations describe all KMS constraints
of order $n$. On the other hand there are $n$ independent variables:
$T_{n,1},T_{n-1,2},\cdots,T_{1,n}$.. Therefore, KMS constraints seem
insufficient to specify all the coefficients of $T_{j,k}$ appearing in the
Lagrangian function. However, after solving the existing equations for an
appropriate set of $\left[\frac{n-1}{2}\right]+1$ coefficients 161616By
linearly combining the equations, we have found that the most appropriate set
of $T_{j,k}$’s at $n^{th}$ order is
$\\{T_{n,0},T_{n-2,2},T_{n-1,4},T_{n-1,6},\cdots,T_{n-1,2k-2},\cdots\\}$. As
it is seen, for $k>2$, the $k^{th}$ appropriate $T_{j,k}$ follows from a
general form: $T_{n-1,2k-2}$. Finally, in any order $n$, we choose the first
$\left[\frac{n-1}{2}\right]+1$ elements of the set as the appropriate
coefficients to solve from the KMS equations., and applying the solutions to
the Lagrangian, it turns out that all
$n-\left(\left[\frac{n-1}{2}\right]+1\right)$ unspecified $T_{j,k}$
coefficients appear as the coefficients of total derivative terms in the cubic
Lagrangian $\mathcal{L}^{(3)}_{\text{eff}}$. This observation shows that we
can actually start with an ansatz which is more constrained than (4.5) and
(4.4). However, we continue to use (4.5) and (4.4).

We have derived the equations between $T_{j,k}$ coefficients to $40^{th}$
order in the derivative expansion. In Appendix D , we display the
corresponding equations only for the first $10$ orders. We have also defined
$R_{j,k}=\alpha_{1}^{j}\alpha_{2}^{k}$ and calculated the equations between
$R_{j,k}$ coefficients to the $40^{th}$ order. The equations can be displayed
in a similar way as in Appendix D; however, we will not show them in the
paper.

Imposing all the above-mentioned KMS constraints to (4.13), we have
constructed the KMS invariant cubic Lagrangian
$\mathcal{L}_{\text{eff}}^{(3)}[\epsilon,\epsilon_{a}]$ to $40^{th}$ order in
the derivative expansion. However, as discussed in the Introduction, our goal
is to compute the energy density correlator defined by (2.1). For this reason,
we find that it is more appropriate to rewrite the effective Lagrangian in
terms of $\mathcal{E}$ and $\epsilon_{a}$, say
$\mathcal{L}_{\text{eff}}[\mathcal{E},\epsilon_{a}]$. Therefore, in order to
reduce complexity, we do not explicitly show the lengthy structure of
$\mathcal{L}_{\text{eff}}^{(3)}[\epsilon,\epsilon_{a}]$. Instead, in the next
section, we will discuss how to exchange $\epsilon$ with $\mathcal{E}$ and
then explicitly display
$\mathcal{L}^{(2)}_{\text{eff}}[\mathcal{E},\epsilon_{a}]$ nd
$\mathcal{L}^{(3)}_{\text{eff}}[\mathcal{E},\epsilon_{a}]$.

###  4.2 Effective action in terms of energy density

The off-shell hydrodynamic energy density current $J_{E}^{\mu}\equiv
J^{\mu}_{r}=(\mathcal{E},J_{i})$ is defined as $J^{\mu}_{E}=\,\delta
I_{\text{EFT}}/\delta A_{a\mu}$ [13]. Thus the energy density is given by

$\mathcal{E}=\,\frac{\delta I_{\text{EFT}}[B_{r\mu},B_{a\mu}]}{\delta
A_{a0}}=\,\frac{\delta
I_{\text{EFT}}[B_{r\mu},B_{a\mu}]}{\delta\partial_{t}\epsilon_{a}}\,.$ (4.19)

In the second equality we have used $B_{a0}=A_{a0}+\partial_{t}\epsilon_{a}$.
Calculating (4.19), we find 171717Note that $\partial_{t}\epsilon$ and
$\partial_{t}\epsilon_{a}$ are dimensionless quantities, while $\mathcal{E}$
and $\mathfrak{h}$ have dimension of energy density.

$\mathcal{E}=\,\mathfrak{h}\left(\partial_{t}\epsilon-\frac{\partial^{3}_{t}\epsilon}{\lambda^{2}}\right)+\frac{\mathfrak{h}}{2}\left((\partial_{t}\epsilon)^{2}+\frac{3}{\lambda^{2}}(\partial^{2}_{t}\epsilon)^{2}+\frac{2}{\lambda^{2}}\partial^{3}_{t}\epsilon\partial_{t}\epsilon\right)+\mathcal{O}(\epsilon^{3})\,.$
(4.20)

Note that this expression is exact in the derivative expansion. However,
$\partial_{t}\epsilon$ in terms of $\mathcal{E}$ can only be found
perturbatively in derivatives. For our later computations we need to go
through the derivative expansion up to $40^{th}$ order. We have done this and
the result related to the first $20$ orders are given in Appendix C.

The above discussion makes the statement below (2.12) more clear. In the
linear regime and to first order in derivatives, (4.20) takes the following
form:

$\mathcal{E}=\,\mathfrak{h}\,\partial_{t}\epsilon\,.$ (4.21)

Comparing this with $\mathcal{E}=c\,T\,\frac{\delta T}{T}$, we infer that
$\partial_{t}\epsilon$ in our EFT, actually acts as a temperate disturbance in
the system. Here, $c$ is the heat capacity. In addition, $\mathfrak{h}$ is
specified as $\mathfrak{h}=T\,c$.

####  4.2.1 Quadratic action and the free propagators

Substituting (C.2) into
$\mathcal{L}_{\text{eff}}^{(2)}[\epsilon,\epsilon_{a}]$, we find
$\mathcal{L}_{\text{eff}}^{(2)}[\mathcal{E},\epsilon_{a}]$ as

$\mathcal{L}_{\text{eff}}^{(2)}=\,\,-\epsilon_{a}\,\partial_{t}\mathcal{E}+D_{0}\,\epsilon_{a}\,\partial_{i}^{2}\mathcal{E}\,+\,i\frac{D_{0}\,\mathfrak{h}}{\beta}\partial_{i}\epsilon_{a}\sum_{n=1}^{40}\chi_{n}\left(\frac{\partial_{t}}{\lambda}\right)^{n-1}\partial_{i}\epsilon_{a}\,.$
(4.22)

From the above quadratic Lagrangian we find the free propagators as 181818Note
that in the original path integral, action appears as $\exp(iS)$. After
integrating out the UV fields, this factor is conventionally written as
$\exp(-S_{\text{EFT}})$. Actually this is $S_{\text{EFT}}$ from which, we
derive propagators and vertices. On the other hand, our EFT was given in terms
of $\mathcal{L}_{\text{eff}}$. Then it is clear that $S_{\text{EFT}}\equiv
I_{\text{EFT}}=-i\int\mathcal{L}_{\text{eff}}$ ; the $-i$ factor, indeed,
compensates the minus sign in $\exp(-S_{\text{EFT}})$ and gives the original
$i$ factor of $\exp(iS)$. Thus in order to do further computations, we have to
firstly multiply $\mathcal{L}_{\text{eff}}$ with the factor $-i$. Doing so,
the quadratic effective action can be written as
$S_{\text{EFT}}=\frac{1}{2}\int_{\omega,\boldsymbol{k}}\,\phi^{A\,*}_{\omega,\boldsymbol{k}}\,P_{AB}(\omega,\boldsymbol{k})\,\phi^{B}_{\omega,\boldsymbol{k}}$,
with $\phi_{A}=(\varphi_{a},\varepsilon)$. Then the free propagators are
simply given by $G^{(0)}_{AB}=(P^{-1})_{AB}$.

$G^{0}_{\mathcal{E}\mathcal{E}}=\frac{2\,D_{0}\,T\,\mathfrak{h}\,k^{2}}{\omega^{2}+(D_{0}\,k^{2})^{2}}\,\sum_{n=1}\chi_{n}\bigg{(}\frac{-i\,\omega}{\lambda}\bigg{)}^{n-1},\,\,\,\,\,\,G^{0}_{\mathcal{E}\epsilon_{a}}=\frac{1}{\omega+iD_{0}\,k^{2}},\,\,\,\,G^{0}_{\epsilon_{a}\mathcal{E}}=\frac{-1}{\omega-
iD_{0}\,k^{2}}\,.$ (4.23)

It should be mentioned that at $\lambda\rightarrow\infty$ our quadratic
Lagrangian simplifies to exactly that of Refs.[15, 16]. In this case, the
above propagators are exactly the same as the corresponding propagators in the
the mentioned references 191919However, as mentioned in [16],
$G^{0}_{\mathcal{E}\epsilon_{a}}$ and $G^{0}_{\epsilon_{a}\mathcal{E}}$
computed in [15] are different from (4.23) with an overall sign..

In order to compute $G^{R}_{\mathcal{E}\mathcal{E}}$, we use the Kramers-
Kronig relation (combined with KMS)202020We follow the convention of Refs.[3,
13] to define $G^{R}(\omega,k)=G_{ra}(\omega,k)=\frac{-i\delta^{2}{W}}{\delta
A_{a0}(\omega,k)\delta A_{r0}(-\omega,-k)}$. Then equation (4.24) matches with
the equation (2.31) in [13].

$G^{0}_{\mathcal{E}\mathcal{E}}=\coth\left(\frac{\beta\omega}{2}\right)\text{Im}\,G^{R(0)}_{\mathcal{E}\mathcal{E}},$
(4.24)

we then find

$G^{R(0)}_{\mathcal{E}\mathcal{E}}=\,\frac{i\,\mathfrak{h}\,D_{0}\,k^{2}\,\left(1+\frac{\omega^{2}}{\lambda^{2}}\right)}{\omega+\,i\,D_{0}k^{2}}\,,$
(4.25)

which agrees with [18] and [60] 212121Let us denote that this relation is
exact in the sense that it continues to hold to all orders in derivative. We
would like to thank Hong Liu for pointing this out..

At this point we can specify the value of $\mathfrak{h}$ in terms of physical
quantities in the system. Using (4.25), we compute the thermal conductivity:

$\kappa=\lim_{\omega\to 0}\lim_{k\to
0}\frac{\omega\beta}{k^{2}}\text{Im}G^{R(0)}_{\mathcal{E}\mathcal{E}}=\,\mathfrak{h}\,\beta\,D_{0}\,\,\,\,\,\,\,\,\,\xrightarrow[]{}\,\,\,\,\,\,\,\,\,\boxed{\kappa=\,D_{0}\,c}$
(4.26)

where we have used $\mathfrak{h}=T\,c$ found below (4.21) 222222Note that
$G_{i}$ in (4.2) can be rewritten in terms of temperature fluctuations:
$G_{i}=-\kappa(\delta T)\partial_{i}T$ with $\kappa(\delta
T)=\kappa+\kappa_{1}\frac{\delta T}{T}+\cdots$. In terms of our $D_{0}$ and
$D_{1}$, one can easily show that $\kappa=D_{0}\,c$ and
$\kappa_{1}=(D_{0}+D_{1})c$.. What we need to do in the following is to
compute the loop correction to (4.23) in order to find the one-loop corrected
version of (4.25). Doing so, we will be able to study the effect of nonlinear
fluctuations on the diffusion pole.

####  4.2.2 Cubic action and coupling constants

As in the quadratic Lagrangian, we substitute (C.2) into
$\mathcal{L}_{\text{eff}}^{(3)}[\epsilon,\epsilon_{a}]$ to find the cubic
Lagrangian in terms of $\mathcal{E}$ and $\epsilon_{a}$, namely
$\mathcal{L}_{\text{eff}}^{(3)}[\mathcal{E},\epsilon_{a}]$ 232323Let us remind
that as mentioned earlier, we do not display
$\mathcal{L}_{3}[\epsilon,\epsilon_{a}]$ in the paper. . The result is
formally written as

$\begin{split}\mathcal{L}_{\text{eff}}^{(3)}=\,\frac{\lambda_{1}}{2}\,\partial_{i}^{2}\epsilon_{a}\,\mathcal{E}^{2}+&i\,c\,T^{2}\,\lambda_{2}\,\sum_{n=1}\sum_{\ell=0}^{n-1}\frac{g^{(n)}_{\ell,n-\ell-1}}{\lambda^{n-1}}\,\partial_{i}\epsilon_{a}\partial_{t}^{\ell}(\partial_{i}\epsilon_{a})\partial_{t}^{n-\ell-1}\mathcal{E}\\\
&+i\,T^{3}\,\lambda_{3}\sum_{n=3}\sum_{\ell=1}^{n-1}\frac{h^{(n)}_{\ell,n-\ell-1}}{\lambda^{n-2}}\,\partial_{t}\epsilon_{a}\partial_{t}^{\ell}\epsilon_{a}\partial_{t}^{n-\ell-1}\mathcal{E}\,,\end{split}$
(4.27)

with the three distinct coupling constants

$\lambda_{1}=\frac{D_{1}}{T\,c},\,\,\,\,\,\,\,\,\,\lambda_{2}=\frac{D_{0}+D_{1}}{T\,c}\,,\,\,\,\,\,\,\,\,\,\,\lambda_{3}=\frac{6}{T^{2}\lambda}\,.$
(4.28)

It should be noted that $\lambda_{1}$ and $\lambda_{2}$ couplings come from
$M_{2}$ term in (2.14), while, $\lambda_{3}$ coupling comes from $M_{1}$ term
in that equation. At first order in the derivative expansion, our
$\lambda_{1}$ and $\lambda_{2}$ terms are exactly $\lambda$ and
$\tilde{\lambda}$ terms of equation (B.1) in Ref.[15]. Since this Reference
does not exceed the first order in the derivative expansion, our $\lambda_{3}$
term has no analogues there. Note that in (4.27), the $\lambda_{3}$ term
starts to contribute from the third order of the derivative expansion.

In Appendices E and F, we have shown $g^{(n)}_{\ell,n-\ell-1}$’s and
$h^{(n)}_{\ell,n-\ell-1}$’s for the first $20$ orders of the derivative
expansion. Although as any other part of the computations, we have computed
these coefficients to $40^{th}$ order in the derivative expansion.

##  5 Loop computations

In order to compute the one-loop renormalized Green’s function, we use the
standard textbook method:

$\langle\mathcal{E}\mathcal{E}\rangle=\,\langle\mathcal{E}\mathcal{E}\rangle_{0}+i\langle\mathcal{E}S_{\text{int}}\mathcal{E}\rangle-\frac{1}{2}\langle\mathcal{E}S_{\text{int}}^{2}\mathcal{E}\rangle+\cdots\,.$
(5.1)

Then following [15], we parameterize the corrections to the Green’s function
as a numerator $C(\omega,\textbf{k})$ and a self-energy
$\Sigma(\omega,\textbf{k})$. The latter corresponds to diagrams that resum,
shown by the gray blobs below, while the former corresponds to those that do
not so, shown by shaded blobs in the following. Diagrammatically, we write

$\begin{split}G_{\mathcal{E}\epsilon_{a}}(p)(-C(p))G_{\epsilon_{a}\mathcal{E}}(p)=&\,\feynmandiagram[small,horizontal=atod]{a--[small]b};+\,\feynmandiagram[small,horizontal=atod]{a--[small]b--[scalar]c[blob]d--[scalar]e--[small]f,};\,,\\\
G_{\mathcal{E}\epsilon_{a}}(p)(-\Sigma(p))G_{\mathcal{E}\mathcal{E}}(p)=&\,\left[\feynmandiagram[small,horizontal=atod]{a--[small]b--[scalar]c[blob,{/tikz/fill=gray}]d};+\feynmandiagram[small,horizontal=atod]{a--[small]b--[scalar]c[blob,{/tikz/fill=gray}]d};\feynmandiagram[small,horizontal=atod]{a--[small]b--[scalar]c[blob,{/tikz/fill=gray}]d};+\cdots\right]\feynmandiagram[small,horizontal=atod]{a--[small]b};\\\
=&\,\frac{\feynmandiagram[small,horizontal=atob]{a--[small]b--[scalar]c[blob,{/tikz/fill=gray}]d--[small]e};\,}{1-\feynmandiagram[small,horizontal=atod]{a--[small]b--[scalar]c[blob,{/tikz/fill=gray}]d};}\,.\end{split}$
(5.2)

Then the Green’s functions can be written as

$G_{\mathcal{E}\mathcal{E}}(p)=\frac{\feynmandiagram[small,horizontal=atod]{a--[small]b};+\,\feynmandiagram[small,horizontal=atod]{a--[small]b--[scalar]c[blob]d--[scalar]e--[small]f,};}{1-\left(\feynmandiagram[small,horizontal=atod]{a--[small]b--[scalar]c[blob,{/tikz/fill=gray}]d};+\feynmandiagram[small,horizontal=atod]{a[blob,{/tikz/fill=gray}]b--[scalar]c--[small]d};\right)}\\\
=\,\frac{G^{0}_{\mathcal{E}\epsilon_{a}}(p)(-C(p))G^{0}_{\epsilon_{a}\mathcal{E}}(p)}{1+\left(G^{0}_{\mathcal{E}\epsilon_{a}}\Sigma(p)-\Sigma^{*}(p)G^{0}_{\epsilon_{a}\mathcal{E}}(p)\right)}\,.$
(5.3)

By using the free propagators given in (4.23), we simply find

$G_{\mathcal{E}\mathcal{E}}(\omega,\textbf{k})=\,\frac{C(\omega,\textbf{k})}{\omega^{2}+D_{0}^{2}\textbf{k}^{4}+2\,\omega\,\text{Re}\,\Sigma(\omega,\textbf{k})+2\,D_{0}\,\textbf{k}^{2}\,\text{Im}\,\Sigma(\omega,\textbf{k})}$
(5.4)

where

$C(\omega,\textbf{k})=2\,T^{2}\,\kappa\,\textbf{k}^{2}\,\sum_{n=1}^{40}\chi_{n}\bigg{(}\frac{-i\,\omega}{\lambda}\bigg{)}^{n-1}+\text{loop
corrections}\,.$ (5.5)

Using Kramers-Kronig relations, we can also parameterize the one-loop retarded
Green’s function as the following:

$G^{R}_{\mathcal{E}\mathcal{E}}(\omega,\textbf{k})=\,\frac{i\bigg{(}(\kappa+\delta\kappa)\big{(}1+\frac{\omega^{2}}{\lambda^{2}}\big{)}-\kappa\,\frac{2\delta\lambda}{\lambda}\,\frac{\omega^{2}}{\lambda^{2}}\bigg{)}T\,\textbf{k}^{2}}{\omega+i\,D_{0}\,\textbf{k}^{2}+\Sigma(\omega,\textbf{k})}\,.$
(5.6)

In the expression above, our focus will be on the denominator. We want to
compute the self-energy, $\Sigma$, perturbatively, in the derivative
expansion. Then we will be able to see the influence of fluctuations on the
diffusion pole in the derivative expansion.

###  5.1 Self-energy

The detailed calculation of $\Sigma$ is given in the Appendix G. We find the
cutoff independent part of the self-energy as

$\Sigma(\omega,k)=\frac{cT^{2}}{4D_{0}^{2}}\,k^{2}\,\lambda_{1}\,\left[k^{2}-\frac{2i\omega}{D_{0}}\right]^{-1/2}\left(\sigma_{1}(\omega,k)+\sigma_{2}(\omega,k)+\sigma_{3}(\omega,k)+\cdots\right)\,,$
(5.7)

where $\sigma_{j}$ function denotes the $j^{th}$ order correction in the
derivative expansion. We have found that

$\begin{split}\sigma_{1}(\omega,k)=&\,\omega\,\lambda_{1}-(\omega+iD_{0}k^{2})\lambda_{2}\,,\\\
\sigma_{2n}(\omega,k)=&\,0\,,\\\
\sigma_{2n+1}(\omega,k)=&\,(\omega+iD_{0}k^{2})\,\bigg{[}F_{2n}(\omega,iD_{0}k^{2})\,\lambda_{1}+G_{2n}(\omega,iD_{0}k^{2})\,\lambda_{2}\bigg{]}\,,\end{split}$
(5.8)

where $F_{2n}$ and $G_{2n}$ are polynomials of $\omega$ and $iD_{0}k^{2}$ (see
Appendix G). We have also defined $r=D_{1}/D_{0}$. Here, some important points
should be mentioned:

  * •

The corrections at any even order in the derivative expansion vanish.

  * •

The $\lambda_{3}$-coupling in (4.27) does not contribute to $\Sigma$ at any
order in the derivative expansion. The reason is that the corresponding
Feynman diagrams are fully divergent after the renormalization, without any
finite parts.

  * •

Except for the first order in derivatives, at any other odd order, the
correction vanishes at the classical diffusion pole, namely at
$\omega=-iD_{0}k^{2}$.

###  5.2 Long time tails and breakdown of the derivative expansion

The standard derivative expansion in our diffusion model is based on the
assumption that $\omega\sim k^{2}$. On the other hand, our loop computations
show that242424Note that the absence of even orders contributions to $\Sigma$
is just due to the special form of the Schwarzian.

$\begin{split}\text{at}\,\,1^{st}\,\text{order in
derivative}:&\,\,\,\Sigma(\omega,k)\sim|\omega|^{3/2}\sim(k^{2})^{3/2}\,,\\\
\text{at}\,\,3^{rd}\,\text{order in
derivative}:&\,\,\,\Sigma(\omega,k)\sim|\omega|^{3/2}\sim(k^{2})^{7/2}\,,\\\
\cdots\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&\\\
\text{at}\,\,(n_{\text{odd}})^{th}\,\text{order in
derivative}:&\,\,\,\Sigma(\omega,k)\sim|\omega|^{(2n_{\text{odd}}+1)/2}\sim(k^{2})^{n_{\text{odd}}+1/2}\,.\\\
\end{split}$ (5.9)

Obviously, these contributions are not consistent with our derivative counting
scheme; the self-energy cannot be expressed by the natural powers of $\omega$
and $k^{2}$, but it shows some non-analyticities.

The appearance of non-analytic contributions of the form $|\omega|^{n/2}$ is
the result of nonlinear interactions in $\mathcal{L}_{\text{EFT}}$. It is easy
to see that the presence of $|\omega|^{n/2}$ in
$G^{R}_{\mathcal{E}\mathcal{E}}(\omega,\textbf{k}\rightarrow 0)$ leads
$G^{R}_{\mathcal{E}\mathcal{E}}(t,x)$ to show the power-law relaxation $\sim
t^{-(n+2)/2}$. This non-exponential damping of the response function is
obviously similar to the long-time tail effect in stochastic hydrodynamics
[55, 15].

The nonlinear interactions in our model have another consequence; the
breakdown of derivative expansion. In order to understand why this is the
case, we only need to consider (5.9) at first order. In the hydrodynamic
limit, $|\omega|^{3/2}$ is smaller than the first order contribution
$\mathcal{O}(\omega)$, but is larger than the second order contribution
$\mathcal{O}(\omega^{2})$. It simply indicates that the derivative expansion
in $1+1$ dimensional hydrodynamics breaks down beyond the first order
252525See [57] for the discussion about the breakdown of derivative expansion
in $(2+1)$ and $(3+1)$ dimensional cases. 262626It has recently been
discovered that the breakdown of hydrodynamics may also occur in certain
condensed matter systems. For instance, the breakdown of diffusion at the edge
of dirty Quantum Hall systems [69] and breakdown of hydrodynamics in a fracton
fluids [70]. . Note that this statement is independent of the assumptions
based on which, we have studied the SYK chain in this work.

##  6 Modified dispersion relation

It should be recalled that in our model, the linear hydro dispersion relation
is exact to all order in the derivative expansion, i.e. $\omega=-iD_{0}k^{2}$.
In a more general case, we could consider the following dispersion
relation272727The truncation we choose to use is just to simplify the
calculation. By performing tedious calculations, it can be proved that even if
we use the non-truncated dispersion relation, (5.9) will be satisfied.

$\omega=-iD_{(1)}k^{2}-iD_{(2)}k^{4}-iD_{(3)}k^{6}+\,\mathcal{O}(k^{8})=\,-i\sum_{n=1}D_{(n)}k^{2n}\,.$
(6.1)

The interesting point is that in any order in derivatives, the corresponding
linear and nonlinear contributions, coming from (6.1) and (5.9), are two
consecutive powers of $(k^{2})^{1/2}$. For instance, at $3^{rd}$ order, the
linear contribution $\sim(k^{2})^{3}$ and the nonlinear contribution
$\sim(k^{2})^{7/2}$. This suggests that:

Although the expansion (6.1) fails to work due to nonlinear effects given in
(5.9), one can still represents $\omega$ as a power series in $(k^{2})^{1/2}$.
We find it convenient to write the series as follows

$\begin{split}\omega=&\,-i\sum_{n=1}\,k^{2n}\,\big{(}D_{(n,1)}\pm\,iD_{(n,2)}(k^{2})^{1/2}\big{)}\\\
=&\,-i\,k^{2}\,\underbrace{\big{(}D_{(1,1)}\pm\,iD_{(1,2)}(k^{2})^{1/2}\big{)}}_{1^{{st}}\,\text{order}}-i\,k^{4}\,\underbrace{\big{(}D_{(2,1)}\pm\,iD_{(2,2)}(k^{2})^{1/2}\big{)}}_{2^{nd}\,\text{order}}\,+\,\mathcal{O}(k^{6})\,.\end{split}$
(6.2)

It is important to note that by “$n^{th}$ order” in the right side of (6.2),
we mean the contribution to the dispersion relation that comes from the
nonlinear Lagrangian at order $n$ in the derivative expansion. From now on we
will refer to (6.2) as modified dispersion relation. Let us emphasize that the
above expansion is not actually a derivative expansion. The derivative
expansion is meaningful only at the Lagrangian level. Our results show that at
the level of dispersion relation, the derivative expansion should be replaced
by (6.2). The modified dispersion relation has some properties:

  1. 1.

The $\pm$ in (6.2) indicates that, due to the nonlinear effects, the classical
diffusion pole is split into two poles with opposite real parts (see Fig.2
below). This is consistent with the same result in the first order of the
derivative expansion found in Ref.[15].

  2. 2.

At any order in the derivative expansion, the modified dispersion relation is
determined by twice the transport coefficients of the classical dispersion
relation.

  3. 3.

As we will show below, it turns out that that $D_{(1,1)}=D_{(1)}$. On the
other hand, in our model $D_{(n)}=0,n>1$. It would be interesting to see if
there is a general relation between $D_{(n,1)}$ and $D_{(n)}$ in the extended
model.

It is convenient to use dimensionless momentum and frequency at this time. To
this end, we rearrange the series as

${\mathfrak{w}}=\,-i\sum_{n=1}{\mathfrak{q}}^{2n}\,\bigg{(}\tilde{D}_{(n,1)}(r,s)\,\pm\,i\tilde{D}_{(n,2)}(r,s)\,({\mathfrak{q}}^{2})^{1/2}\bigg{)}$
(6.3)

with

${\mathfrak{w}}=\,\frac{D_{0}\,\omega}{\kappa^{2}},\,\,\,\,\,\,\,\,\,\,\,{\mathfrak{q}}=\frac{D_{0}\,k}{\kappa},\,\,\,\,\,\,\,\,\,\,\,\tilde{D}_{(n,j)}=\,\frac{\kappa^{2n-3+j}}{D_{0}^{2n-2+j}}\,D_{(n,j)}:\,j=1,2\,.$
(6.4)

The parameters $r$ and $s$ are defined as

$r=\frac{D_{1}}{D_{0}},\,\,\,\,\,\,s=\,\frac{\kappa^{2}}{D_{0}\,T}\,.$ (6.5)

For concreteness, we discuss the convergence of (6.2) for $r=\frac{1}{2}$ and
$s=250$. The corresponding first forty coefficients have been given in
Appendix H. For now, let us limit the discussion to real values of momentum.
Then (6.3) takes the following simple form

${\mathfrak{w}}=\,-i\sum_{n=1}^{\mathcal{M}}{\mathfrak{q}}^{2n}\,\bigg{(}\tilde{D}_{(n,1)}(r,s)\,\pm\,i\tilde{D}_{(n,2)}(r,s)\,|{\mathfrak{q}}|\bigg{)}\equiv\,-i\sum_{n=1}^{2\mathcal{M}}c_{n}|{\mathfrak{q}}|^{n}\,.$
(6.6)

Figure 1: Illustration of coefficients of the series (6.6). In our current
case, $\mathcal{M}=40$, so the modified series is determined by $80$ transport
coefficients.

In Fig.1 we have displayed the logarithm of the absolute value of the
coefficients $c_{n}$. For the mentioned value of the parameters $r$ and $s$,
the slope of the plot at large $n$ tends to $4.720$. So the series
approximately converges to

${\mathfrak{q}}_{c}=\lim_{n\rightarrow\infty}\left(\ln|c_{n+1}|-\ln|c_{n}|\right)^{-1}=\,\lim_{n\rightarrow\infty}\left(\ln\bigg{|}\frac{c_{n+1}}{c_{n}}\bigg{|}\right)^{-1}\approx
0.211\,.$ (6.7)

Let us denote that the value of ${\mathfrak{q}}_{c}$ only depends on the
choice of $r=D_{1}/D_{0}$. In fact, as can be seen in Appendix H, at large
$n$, all coefficients are proportional to $s^{2}$ resulting in $c_{n+1}/c_{n}$
independent of $s$. Then the only place where the effect of $s$ may appear is
in the low-order terms in Fig.1.

In summary, we find that for a given value of $r$, the series (6.3) converges
to a finite value. Using (6.4), the dimensionfull radius of convergence is
given by

$k_{c}=\,{\mathfrak{q}}_{c}\,\frac{\kappa}{D_{0}}=\,\\#\frac{\kappa}{D_{0}}\,,$
(6.8)

where the numerical factor depends on the value of $r$.

Figure 2: Splitting of the classical diffusion mode (represented by large
colored dots) into two mode represented by the small dots. Each group of
colored path-like points, starting with dark red and ending with purple, shows
the change of a particular mode when ${\mathfrak{q}}$ discretely increases
from $0.1$ to $0.2$. We have considered 10 regular steps by increments of
$\Delta{\mathfrak{q}}=0.01$.

Before ending this section let us mention two points regarding the dispersion
relations.

  * •

As mentioned earlier, one of the effects of nonlinear fluctuations is to split
the classical diffusion pole into two poles with non-vanishing real parts. It
can be seen from Fig.2 that the real parts of these modes are opposite to each
other.

  * •

It would be interesting to study these modes at complex momenta to find the
reason behind the finite radius of convergence of the modified series.
Following [53], it should be determined whether there are any mode collisions
that limit convergence 282828See also [52, 54, 71, 38, 76, 40, 79, 81, 78, 73,
74, 75, 77, 72, 80, 82, 83, 84, 85] for recent studies on various aspects of
mode collision and convergence radius of hydrodynamics.. We leave more
investigations on this issue for future work.

##  7 Discussion and outlook

In this paper we investigated the effect of hydrodynamic fluctuations in the
SYK chain. By constructing the interacting EFT of energy diffusion up to
$40^{th}$ order in derivatives, as given by (4.22) and (4.27), we computed the
one-loop self-energy to the same order in derivatives (see (5.7)). The special
form of derivative corrections in the self-energy led us to the following
results:

  1. 1.

The dispersion relation of the energy diffusion does no longer follow the
derivative expansion, namely the classical relation
${\mathfrak{w}}={\mathfrak{w}}({\mathfrak{q}}^{2})$. However, the dispersion
relation can be still represented as a power series, but in terms of of
$|{\mathfrak{q}}|$, say ${\mathfrak{w}}={\mathfrak{w}}(|{\mathfrak{q}}|)$. We
referred to it as the modified (or quantum) dispersion relation.

  2. 2.

Up to any order in the derivative expansion, the number of transport
coefficients in the modified dispersion relation is twice of that in the
classical dispersion relation.

  3. 3.

The modified dispersion relation is convergent in the momentum space. The
radius of convergent, $k_{c}$, is proportional to the ratio of the thermal
conductivity to the diffusion constant. Since $\kappa=D_{0}\,c$, one
immediately concludes that $k_{c}\sim c$, that is, it is proportional to the
number of degrees of freedom. Therefore, in the large $N$ limit, $k_{c}$ has a
large value.

As the first generalization of our results, one might think of including the
two-loop corrections along with derivative orders larger than $40$. Although
the latter is a technical problem to obtain better convergence, the former one
is physically important; it is actually equivalent to considering
$\frac{1}{N^{2}}$ corrections to the self-energy.

To be more realistic, a larger EFT cut-off value should be considered to make
room for the inclusion of non-hydrodynamic modes. In the simplest case, only
the lowest quasi-normal mode can be included. The corresponding EFT on CTP
contour then will have two additional dynamical variables, associated with
this quasinormal mode. In this setting, the influence of nonlinear
fluctuations on diffusion and non-hydro mode should be studied. Then it would
be interesting to see what kind of singularity limits the convergence of the
modified dispersion relation.

More realistically, one can think of a 3+1 dimensional model. One way is to
consider that the Schwarzian (3.7) depends on three spatial dimensions.
Repeating all the steps taken in the paper, one would arrive at the analogue
of (5.7) in 3+1 dimension as follows

$\tilde{\Sigma}(\omega,k)=\frac{cT^{2}}{4D_{0}^{2}}\,k^{2}\,\lambda_{1}\,\left[k^{2}-\frac{2i\omega}{D_{0}}\right]^{1/2}\left(\tilde{\sigma}_{1}(\omega,k)+\tilde{\sigma}_{2}(\omega,k)+\tilde{\sigma}_{3}(\omega,k)+\cdots\right)\,.$

As it is seen, at the $n^{th}$ order in derivatives,
$\tilde{\Sigma}_{(n)}\sim(k^{2})^{n+1/2}$. It can be concluded that for the
3+1 dimensional model, the expansion (6.2) should be changed to allow four
transport coefficients at any $n$; i.e. $D_{(n,j)}:j=1,2,3,4$. However, it
must be carefully considered; it turns out the coefficients of $(k^{2})^{n}$
and $(k^{2})^{n+1/2}$ will get contributions from two different orders of $n$:

$\begin{split}\text{coefficient of
}(k^{2})^{n}\,\,=&\,D_{(n-1,3)}+D_{(n,1)}\,,\\\ \text{coefficient of
}(k^{2})^{n+1/2}=&\,D_{(n-1,4)}+D_{(n,2)}\,.\end{split}$

Because of the above pairing between the coefficients $D_{(n,j)}$, one needs
to use effectively $2n+4$ coefficients. Here, $4$ corresponds to $D_{(1,1)}$,
$D_{(1,2)}$, $D_{(n,3)}$ and $D_{(n,4)}$; they are not paired with any other
coefficients at order $n$. We leave more investigation on 3+1 dimensional case
to a future work.

Considering all aforementioned points, one may think of studying the same
problem in the case of Quark Gluon Plasma (QGP). The hydrodynamic fluctuations
in QGP have been studied previously on the static [56] and also on the
evolutionary [86] background. In order to extend the calculations of this
paper to the QGP case, two important points should be considered. First, the
dynamics of energy density in our model is similar to that of charge density
or transverse momentum density in QGP, which is a diffusion process. In QGP,
however, there are also sound modes. They actually propagate and diffuse into
the medium. Second, QCD is not a large-N theory. Therefore, to be applicable
to QGP, our setting should be extended to include relativistic hydrodynamic
fluctuations and also to go beyond one-loop computations.

Another direction to explore this problem is holography. In the pioneering
work of Ref.[87], the one-loop calculation of Einstein’s gravity in AdS space
has been proven to reproduce the long-time tail effect in boundary quantum
field theory at a finite temperature. Another interesting calculation can be
found in [89], where the leading derivative contribution of the self energy is
directly computed from gravity. But one can do it in yet another way. The idea
might be to use the holographic prescription of the Schwinger-Keldysh contour
[88] 292929See [90, 91] for other holographic prescriptions. and compute the
boundary nonlinear effective action from gravity to higher orders in
derivatives303030According to the prescription of [88], the “quadratic”
effective action for the boundary theory has been computed in several
different situations [92, 93, 94].. The next step will be simple, calculating
the loop correction of the energy density response function in the boundary.

Lastly, it would be interesting to continue the loop computations in this
paper and specify all parameters introduced in (5.6). Doing so, the full one-
loop corrected response function will be determined. This response function
has an important characteristic. It is actually related to a maximally chaotic
quantum system [19]. Then it would be very interesting to find the pole-
skipping phenomenon in this response function and extract the chaos point from
it [18]. It is well-known that the chaos point might be located out of the
regime validity of hydrodynamics [54, 71]. However, the fact that we have
determined $\mathcal{L}_{\text{eff}}$ to high orders in the derivative
expansion allows to probe the chaos point. Then the main question to be
answered will be how the Lyapunov exponent and butterfly velocity are affected
by hydrodynamic fluctuations.

## Acknowledgment

We are grateful to Ali Davody, Luca Delacrétaz, Paolo Glorioso, Sean Hartnoll,
Matthias Kaminski, Pavel Kovtun, Hong Liu and Misha Stephanov for valuable
discussions and comments. We would also like to thank Armin Ghazi and Omid
Tavakol for discussing the different aspects of EFT of hydrodynamics. This
work was supported by grant number 561119208 “Double First Class” start-up
funding of Lanzhou University, China.

##  Appendix A Dimensions

The mass dimension of quantities involved in our system is as follows:

$\begin{split}[H]=\,[\mathfrak{h}]=\,[\mathcal{E}]=&\,2\\\
[\omega]=\,[k]=\,[c]=&\,1\\\ [\kappa]=&\,0\\\
[\epsilon]=\,[\epsilon_{a}]=\,[D_{0}]=\,[D_{1}]=&\,-1\\\
[\lambda_{1}]=\,[\lambda_{2}]=\,[\lambda_{3}]=&\,-3\,.\end{split}$ (A.1)

##  Appendix B Coefficients of $M_{2}$ term in the quadratic Lagrangian

$\begin{split}\chi_{1}=1,\,\,\,\,\chi_{3}=\frac{1}{12}\left(-\beta^{2}\lambda^{2}-12\right),\,\,\,\,\,\chi_{5}=-\frac{1}{720}\beta^{2}\lambda^{2}\left(\beta^{2}\lambda^{2}-60\right),\,\,\,\,\,\,\,\,\,\chi_{7}=-\frac{\beta^{4}\lambda^{4}\left(\beta^{2}\lambda^{2}-42\right)}{30240},\,\,\,\,\,\,\,\chi_{9}=\\-\frac{\beta^{6}\lambda^{6}\left(\beta^{2}\lambda^{2}-40\right)}{1209600}\\\
\chi_{11}=-\frac{\beta^{8}\lambda^{8}\left(5\beta^{2}\lambda^{2}-198\right)}{239500800},\,\,\,\,\,\,\,\,\,\,\chi_{13}=-\frac{\beta^{10}\lambda^{10}\left(691\beta^{2}\lambda^{2}-27300\right)}{1307674368000},\,\,\,\,\,\,\chi_{15}=-\frac{\beta^{12}\lambda^{12}\left(35\beta^{2}\lambda^{2}-1382\right)}{2615348736000}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\
\chi_{17}=-\frac{\beta^{14}\lambda^{14}\left(3617\beta^{2}\lambda^{2}-142800\right)}{10670622842880000}\,\,\,\,\,\,\,\,\,\,\,\chi_{19}-\frac{\beta^{16}\lambda^{16}\left(219335\beta^{2}\lambda^{2}-8659098\right)}{25545471085854720000}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{split}$
(B.1)

##  Appendix C Energy density in terms of reparametrization mode

In order to find the invert equation (4.20), we take an anstaz for
$\partial_{t}\epsilon$ of the form

$\partial_{t}\epsilon=\,\sum_{n=0}\frac{1}{\mathfrak{h}}e_{n}\frac{\partial_{t}^{n}\mathcal{E}}{\lambda^{n}}+\,\sum_{n,m=0}\frac{1}{\mathfrak{h}^{2}}e_{n,m}\frac{\partial_{t}^{n}\mathcal{E}\partial_{t}^{m}\mathcal{E}}{\lambda^{n+m}}\,.$
(C.1)

Substituting this into (4.20), we can then read the coefficients $e_{n}$ and
$e_{n,m}$, order by order in the derivative expansion. Here is the result
related the first $20$ orders:

$\begin{split}\partial_{t}\epsilon&=\,\frac{1}{\mathfrak{h}}\left[\mathcal{E}+\frac{\partial^{2}_{t}{\mathcal{E}}}{\lambda^{2}}+\frac{\partial^{4}_{t}{\mathcal{E}}}{\lambda^{4}}+\cdots+\frac{\partial^{20}_{t}{\mathcal{E}}}{\lambda^{20}}+\mathcal{O}\left(\frac{1}{\mathfrak{h}}\frac{\partial_{t}^{21}\mathcal{E}}{\lambda^{21}}\right)\right]-\frac{1}{\mathfrak{h}^{2}}\bigg{[}\frac{\mathcal{E}^{2}}{2}+\frac{1}{\lambda^{2}}\left(\frac{5}{2}(\partial_{t}\mathcal{E})^{2}+3\,\mathcal{E}\partial^{2}_{t}\mathcal{E}\right)+\frac{1}{\lambda^{4}}\left(\frac{19}{2}(\partial^{2}_{t}\mathcal{E})^{2}+14\,\partial\mathcal{E}\partial^{3}_{t}\mathcal{E}+5\,\mathcal{E}\partial^{4}_{t}\mathcal{E}\right)\\\
&+\frac{1}{\lambda^{6}}\left(\frac{69}{2}(\partial^{3}_{t}\mathcal{E})^{2}+55\,\partial_{t}^{2}\mathcal{E}\partial^{4}_{t}\mathcal{E}+27\,\partial_{t}\mathcal{E}\partial^{5}_{t}\mathcal{E},+7\,\mathcal{E}\partial^{6}_{t}\mathcal{E}\right)\\\
&+\frac{1}{\lambda^{8}}\left(\frac{251}{2}(\partial^{4}_{t}\mathcal{E})^{2}+209\,\partial_{t}^{3}\mathcal{E}\partial^{5}_{t}\mathcal{E}+119\,\partial^{2}_{t}\mathcal{E}\partial^{6}_{t}\mathcal{E}\,+44\,\partial_{t}\mathcal{E}\partial^{7}_{t}\mathcal{E}+9\,\mathcal{E}\partial^{8}_{t}\mathcal{E}\right)\\\
&+\frac{1}{\lambda^{10}}\left(\frac{923}{2}(\partial^{5}_{t}\mathcal{E})^{2}+791\,\partial_{t}^{4}\mathcal{E}\partial^{6}_{t}\mathcal{E}+494\,\partial^{3}_{t}\mathcal{E}\partial^{7}_{t}\mathcal{E}+219\,\partial^{2}_{t}\mathcal{E}\partial^{8}_{t}\mathcal{E}+65\,\partial_{t}\mathcal{E}\partial^{9}_{t}\mathcal{E}+11\,\mathcal{E}\partial^{10}_{t}\mathcal{E}\right)\\\
&+\frac{1}{\lambda^{12}}\bigg{(}\frac{3431}{2}(\partial^{6}_{t}\mathcal{E})^{2}+3002\,\partial_{t}^{5}\mathcal{E}\partial^{7}_{t}\mathcal{E}+2001\,\partial^{4}_{t}\mathcal{E}\partial^{8}_{t}\mathcal{E}+1000\,\partial^{3}_{t}\mathcal{E}\partial^{9}_{t}\mathcal{E}+363\,\partial^{2}_{t}\mathcal{E}\partial^{10}_{t}\mathcal{E}+90\,\partial_{t}\mathcal{E}\partial^{11}_{t}\mathcal{E}+\,13\,\mathcal{E}\partial^{12}_{t}\mathcal{E}\bigg{)}\\\
&+\frac{1}{\lambda^{14}}\bigg{(}\frac{12869}{2}(\partial^{7}_{t}\mathcal{E})^{2}+11439\,\partial_{t}^{6}\mathcal{E}\partial^{8}_{t}\mathcal{E}+8007\,\partial^{5}_{t}\mathcal{E}\partial^{9}_{t}\mathcal{E}+4367\,\partial^{4}_{t}\mathcal{E}\partial^{10}_{t}\mathcal{E}+1819\,\partial^{3}_{t}\mathcal{E}\partial^{11}_{t}\mathcal{E}+559\,\partial^{2}_{t}\mathcal{E}\partial^{12}_{t}\mathcal{E}+\,119\,\partial_{t}\mathcal{E}\partial^{13}_{t}\mathcal{E}+\,15\,\mathcal{E}\partial^{14}_{t}\mathcal{E}\bigg{)}\\\
&+\frac{1}{\lambda^{16}}\bigg{(}\frac{48619}{2}(\partial^{8}_{t}\mathcal{E})^{2}+43757\,\partial_{t}^{7}\mathcal{E}\partial^{9}_{t}\mathcal{E}+31823\,\partial^{6}_{t}\mathcal{E}\partial^{10}_{t}\mathcal{E}+18563\,\partial^{5}_{t}\mathcal{E}\partial^{11}_{t}\mathcal{E}+8567\,\partial^{4}_{t}\mathcal{E}\partial^{12}_{t}\mathcal{E}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+3059\,\partial^{3}_{t}\mathcal{E}\partial^{13}_{t}\mathcal{E}+\,815\,\partial^{2}_{t}\mathcal{E}\partial^{14}_{t}\mathcal{E}+\,152\,\partial_{t}\mathcal{E}\partial^{15}_{t}\mathcal{E}+\,17\,\mathcal{E}\partial^{16}_{t}\mathcal{E}\bigg{)}\\\
&+\frac{1}{\lambda^{18}}\bigg{(}\frac{184755}{2}(\partial^{9}_{t}\mathcal{E})^{2}+167959\,\partial_{t}^{8}\mathcal{E}\partial^{10}_{t}\mathcal{E}+125969\,\partial^{7}_{t}\mathcal{E}\partial^{11}_{t}\mathcal{E}+77519\,\partial^{6}_{t}\mathcal{E}\partial^{12}_{t}\mathcal{E}+38759\,\partial^{5}_{t}\mathcal{E}\partial^{13}_{t}\mathcal{E}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+15503\,\partial^{4}_{t}\mathcal{E}\partial^{14}_{t}\mathcal{E}+\,4844\,\partial^{3}_{t}\mathcal{E}\partial^{15}_{t}\mathcal{E}+\,1139\,\partial_{t}^{2}\mathcal{E}\partial^{16}_{t}\mathcal{E}+\,189\,\partial_{t}\mathcal{E}\partial^{17}_{t}\mathcal{E}+\,19\,\mathcal{E}\partial^{18}_{t}\mathcal{E}\bigg{)}\\\
&+\frac{1}{\lambda^{20}}\bigg{(}\frac{705431}{2}(\partial^{10}_{t}\mathcal{E})^{2}+646645\,\partial_{t}^{9}\mathcal{E}\partial^{11}_{t}\mathcal{E}+497419\,\partial^{8}_{t}\mathcal{E}\partial^{12}_{t}\mathcal{E}+319769\,\partial^{7}_{t}\mathcal{E}\partial^{13}_{t}\mathcal{E}+170543\,\partial^{6}_{t}\mathcal{E}\partial^{14}_{t}\mathcal{E}\\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+74612\,\partial^{5}_{t}\mathcal{E}\partial^{15}_{t}\mathcal{E}+\,26333\,\partial^{4}_{t}\mathcal{E}\partial^{16}_{t}\mathcal{E}+\,7314\,\partial_{t}^{3}\mathcal{E}\partial^{17}_{t}\mathcal{E}+\,1539\,\partial^{2}_{t}\mathcal{E}\partial^{18}_{t}\mathcal{E}+230\,\partial_{t}\mathcal{E}\partial^{19}_{t}\mathcal{E}+\,21\,\mathcal{E}\partial^{20}_{t}\mathcal{E}\bigg{)}\bigg{]}+\mathcal{O}\left(\frac{1}{\mathfrak{h}^{2}}\frac{\partial_{t}^{21}\mathcal{E}^{2}}{\lambda^{21}}\right)\,.\end{split}$
(C.2)

##  Appendix D KMS constraints

In the following table, we display the set of KMS constraints for the first
ten orders in the derivative expansion.

$n$ | KMS constraints at order $n$ of the derivative expansion
---|---
$1$ | $\frac{D_{0}+D_{1}}{2\beta}-T_{0,0}=0$
$2$ | $T_{0,1}-2T_{1,0}=0$
$3$ | $-12\beta\left(2T_{0,0}-T_{0,2}+T_{1,1}-2T_{2,0}\right)+\frac{1}{2}\beta^{2}D_{0}\lambda^{2}+\frac{1}{2}D_{1}\left(\beta^{2}\lambda^{2}+36\right)=0$
| $12\beta
T_{0,2}+\frac{1}{2}D_{0}\left(\beta^{2}\lambda^{2}-12\right)+\frac{1}{2}D_{1}\left(\beta^{2}\lambda^{2}+12\right)=0$
$4$ | $T_{0,1}-T_{0,3}-2T_{1,0}+T_{1,2}-T_{2,1}+2T_{3,0}=0$
| $2T_{1,2}-3T_{0,3}=0$
$5$ | $\beta\left(\frac{1}{2}\beta D_{0}\lambda^{2}\left(\beta^{2}\lambda^{2}-60\right)-720\left(T_{0,2}-T_{0,4}-T_{1,1}+T_{1,3}+2T_{2,0}-T_{2,2}+T_{3,1}-2T_{4,0}\right)\right)+\frac{1}{2}D_{1}\left(\beta^{4}\lambda^{4}-120\beta^{2}\lambda^{2}-720\right)=0$
| $\beta\left(\frac{1}{2}\beta
D_{0}\lambda^{2}\left(\beta^{2}\lambda^{2}-45\right)-180\left(2T_{0,2}-4T_{0,4}+3T_{1,3}-2T_{2,2}\right)\right)+\frac{1}{2}D_{1}\left(\beta^{4}\lambda^{4}-90\beta^{2}\lambda^{2}-360\right)=0$
| $720T_{0,4}+\frac{1}{2}\beta
D_{0}\lambda^{2}\left(\beta^{2}\lambda^{2}+60\right)+\frac{1}{2}\beta
D_{1}\lambda^{2}\left(\beta^{2}\lambda^{2}-60\right)=0$
$6$ | $T_{0,3}-T_{0,5}-T_{1,2}+T_{1,4}+T_{2,1}-T_{2,3}-2T_{3,0}+T_{3,2}-T_{4,1}+2T_{5,0}=0$
| $3T_{0,3}-5T_{0,5}-2T_{1,2}+4T_{1,4}-3T_{2,3}+2T_{3,2}=0$
| $5T_{0,5}-2T_{1,4}=0$
$7$ | $\frac{\left(-T_{0,4}+T_{0,6}+T_{1,3}-T_{1,5}-T_{2,2}+T_{2,4}+T_{3,1}-T_{3,3}-2T_{4,0}+T_{4,2}-T_{5,1}+2T_{6,0}\right)}{\lambda^{6}}+\frac{\left(\beta D_{1}\left(\beta^{4}\lambda^{4}-84\beta^{2}\lambda^{2}+2520\right)+\beta^{3}D_{0}\lambda^{2}\left(\beta^{2}\lambda^{2}-42\right)\right)}{60480\lambda^{4}}=0$
|
$\frac{\left(-5040\left(4T_{0,4}-6T_{0,6}-3T_{1,3}+5T_{1,5}+2T_{2,2}-4T_{2,4}+3T_{3,3}-2T_{4,2}\right)+\frac{1}{2}\beta
D_{1}\lambda^{2}\left(\beta^{4}\lambda^{4}-70\beta^{2}\lambda^{2}+1680\right)+\frac{1}{2}\beta^{3}D_{0}\lambda^{4}\left(\beta^{2}\lambda^{2}-35\right)\right)}{\lambda^{6}}=0$
| $-10080\left(2T_{0,4}-9T_{0,6}+5T_{1,5}-2T_{2,4}\right)+\frac{1}{2}\beta
D_{1}\lambda^{2}\left(3\beta^{4}\lambda^{4}-154\beta^{2}\lambda^{2}+1680\right)+\frac{1}{2}\beta^{3}D_{0}\lambda^{4}\left(3\beta^{2}\lambda^{2}-56\right)=0$
|
$30240T_{0,6}+\frac{1}{2}\beta^{3}D_{0}\lambda^{4}\left(\beta^{2}\lambda^{2}+42\right)+\frac{1}{2}\beta^{3}D_{1}\lambda^{4}\left(\beta^{2}\lambda^{2}-42\right)=0$
$8$ | $T_{0,5}-T_{0,7}-T_{1,4}+T_{1,6}+T_{2,3}-T_{2,5}-T_{3,2}+T_{3,4}+T_{4,1}-T_{4,3}-2T_{5,0}+T_{5,2}-T_{6,1}+2T_{7,0}=0$
|
$5T_{0,5}-7T_{0,7}-4T_{1,4}+6T_{1,6}+3T_{2,3}-5T_{2,5}-2T_{3,2}+4T_{3,4}-3T_{4,3}+2T_{5,2}=0$
| $5T_{0,5}-14T_{0,7}-2T_{1,4}+9T_{1,6}-5T_{2,5}+2T_{3,4}=0$
| $2T_{1,6}-7T_{0,7}=0$
$9$ | $\frac{\left(-T_{0,6}+T_{0,8}+T_{1,5}-T_{1,7}-T_{2,4}+T_{2,6}+T_{3,3}-T_{3,5}-T_{4,2}+T_{4,4}+T_{5,1}-T_{5,3}-2T_{6,0}+T_{6,2}-T_{7,1}+2T_{8,0}\right)}{\lambda^{8}}+\frac{\left(\beta^{5}D_{0}\lambda^{2}\left(\beta^{2}\lambda^{2}-40\right)+\beta^{3}D_{1}\left(\beta^{4}\lambda^{4}-80\beta^{2}\lambda^{2}+1680\right)\right)}{2419200\lambda^{4}}=0$
|
$\frac{\left(-151200\left(6T_{0,6}-8T_{0,8}-5T_{1,5}+7T_{1,7}+4T_{2,4}-6T_{2,6}-3T_{3,3}+5T_{3,5}+2T_{4,2}-4T_{4,4}+3T_{5,3}-2T_{6,2}\right)+\frac{1}{2}\beta^{5}D_{0}\lambda^{6}\left(\beta^{2}\lambda^{2}-35\right)+\frac{1}{2}\beta^{3}D_{1}\lambda^{4}\left(\beta^{4}\lambda^{4}-70\beta^{2}\lambda^{2}+1260\right)\right)}{\lambda^{8}}=0$
|
$\frac{\left(-60480\left(9T_{0,6}-20T_{0,8}-5T_{1,5}+14T_{1,7}+2T_{2,4}-9T_{2,6}+5T_{3,5}-2T_{4,4}\right)+\frac{1}{2}\beta^{5}D_{0}\lambda^{6}\left(\beta^{2}\lambda^{2}-26\right)+\frac{1}{2}\beta^{3}D_{1}\lambda^{4}\left(\beta^{4}\lambda^{4}-58\beta^{2}\lambda^{2}+756\right)\right)}{\lambda^{8}}=0$
|
$-151200\left(2T_{0,6}-16T_{0,8}+7T_{1,7}-2T_{2,6}\right)+\frac{1}{2}\beta^{5}D_{0}\lambda^{6}\left(2\beta^{2}\lambda^{2}-15\right)+\beta^{3}D_{1}\lambda^{4}\left(\beta^{4}\lambda^{4}-45\beta^{2}\lambda^{2}+210\right)=0$
|
$1209600T_{0,8}+\frac{1}{2}\beta^{5}D_{0}\lambda^{6}\left(\beta^{2}\lambda^{2}+40\right)+\frac{1}{2}\beta^{5}D_{1}\lambda^{6}\left(\beta^{2}\lambda^{2}-40\right)=0$
$10$ | $-T_{0,7}+T_{0,9}+T_{1,6}-T_{1,8}-T_{2,5}+T_{2,7}+T_{3,4}-T_{3,6}-T_{4,3}+T_{4,5}+T_{5,2}-T_{5,4}-T_{6,1}+T_{6,3}+2T_{7,0}-T_{7,2}+T_{8,1}-2T_{9,0}=0$
|
$7T_{0,7}-9T_{0,9}-6T_{1,6}+8T_{1,8}+5T_{2,5}-7T_{2,7}-4T_{3,4}+6T_{3,6}+3T_{4,3}-5T_{4,5}-2T_{5,2}+4T_{5,4}-3T_{6,3}+2T_{7,2}=0$
|
$-14T_{0,7}+27T_{0,9}+9T_{1,6}-20T_{1,8}-5T_{2,5}+14T_{2,7}+2T_{3,4}-9T_{3,6}+5T_{4,5}-2T_{5,4}=0$
| $7T_{0,7}-30T_{0,9}-2T_{1,6}+16T_{1,8}-7T_{2,7}+2T_{3,6}=0$
| $9T_{0,9}-2T_{1,8}=0$

##  Appendix E Coefficients of
$\nabla\epsilon_{a}\partial_{t}^{\ell}(\nabla\epsilon_{a})\partial_{t}^{n-\ell-1}\mathcal{E}$
in the cubic action

$(n)$ | non-vanishing $g^{(n)}_{\ell,n-\ell-1}$ to $20^{th}$ order $\left(r=\frac{D_{1}}{D_{0}},\,\,\,\,\lambda=\frac{2\pi}{\beta},\,\,\,\,\,\beta=T^{-1}\right)$
---|---
$(1)$ | $g^{(1)}_{0,0}=1$
$(3)$ | $g^{(3)}_{0,2}=\frac{1}{2r+2},\,\,g^{(3)}_{2,0}=-\frac{r\left(\beta^{2}\lambda^{2}+12\right)+\beta^{2}\lambda^{2}-12}{12(r+1)}$
$(5)$ | $g^{(5)}_{0,4}=\frac{8-\beta^{2}\lambda^{2}}{16r+16},\,\,\,\,\,\,g^{(5)}_{2,2}=\frac{5\beta^{2}\lambda^{2}+24}{24r+24},\,\,\,\,\,\,\,\,g^{(5)}_{4,0}=-\frac{\beta^{2}\lambda^{2}\left(r\left(\beta^{2}\lambda^{2}-60\right)+\beta^{2}\lambda^{2}+60\right)}{720(r+1)}$
$(7)$ | $g^{(7)}_{0,6}=\frac{\beta^{4}\lambda^{4}-30\beta^{2}\lambda^{2}+240}{480(r+1)},\,\,\,\,\,\,\,g^{(7)}_{2,4}=\frac{-\beta^{4}\lambda^{4}+20\beta^{2}\lambda^{2}+96}{96r+96}$
|
$g^{(7)}_{4,2}=\frac{\beta^{2}\lambda^{2}(-120+11\beta^{2}\lambda^{2})}{1440(1+r)},\,\,\,\,\,g^{(7)}_{6,0}=-\frac{\beta^{4}\lambda^{4}\left(r\left(\beta^{2}\lambda^{2}-42\right)+\beta^{2}\lambda^{2}+42\right)}{30240(r+1)}$
$(9)$ | $g^{(9)}_{0,8}=\frac{-17\beta^{6}\lambda^{6}+168\beta^{4}\lambda^{4}-5040\beta^{2}\lambda^{2}+40320}{80640(r+1)}$
|
$g^{(9)}_{2,6}=\frac{\beta^{6}\lambda^{6}-10\beta^{4}\lambda^{4}+200\beta^{2}\lambda^{2}+960}{960(r+1)},\,\,\,g^{(9)}_{4,4}=-\frac{\beta^{2}\lambda^{2}\left(5\beta^{4}\lambda^{4}-44\beta^{2}\lambda^{2}+480\right)}{5760(r+1)}$
|
$g^{(9)}_{6,2}=\frac{\beta^{4}\lambda^{4}\left(17\beta^{2}\lambda^{2}-84\right)}{60480(r+1)},\,\,\,\,g^{(9)}_{8,0}=-\frac{\beta^{6}\lambda^{6}\left(r\left(\beta^{2}\lambda^{2}-40\right)+\beta^{2}\lambda^{2}+40\right)}{1209600(r+1)}$
$(11)$ | $g^{(11)}_{0,10}=\frac{31\beta^{8}\lambda^{8}-170\beta^{6}\lambda^{6}+1680\beta^{4}\lambda^{4}-50400\beta^{2}\lambda^{2}+403200}{806400(r+1)},\,\,g^{(11)}_{2,8}=\frac{-51\beta^{8}\lambda^{8}+280\beta^{6}\lambda^{6}-2800\beta^{4}\lambda^{4}+56000\beta^{2}\lambda^{2}+268800}{268800(r+1)}$
|
$g^{(11)}_{4,6}=\frac{\beta^{2}\lambda^{2}\left(9\beta^{6}\lambda^{6}-50\beta^{4}\lambda^{4}+440\beta^{2}\lambda^{2}-4800\right)}{57600(r+1)},\,\,\,g^{(11)}_{6,4}=-\frac{\beta^{4}\lambda^{4}\left(63\beta^{4}\lambda^{4}-340\beta^{2}\lambda^{2}+1680\right)}{1209600(r+1)}$
|
$g^{(11)}_{8,2}=\frac{\beta^{6}\lambda^{6}\left(23\beta^{2}\lambda^{2}-80\right)}{2419200(r+1)},\,\,\,\,g^{(11)}_{10,0}-\frac{\beta^{8}\lambda^{8}\left(r\left(5\beta^{2}\lambda^{2}-198\right)+5\beta^{2}\lambda^{2}+198\right)}{239500800(r+1)}$
$(13)$ | $g^{(13)}_{0,12}=\frac{-3455\beta^{10}\lambda^{10}+12276\beta^{8}\lambda^{8}-67320\beta^{6}\lambda^{6}+665280\beta^{4}\lambda^{4}-19958400\beta^{2}\lambda^{2}+159667200}{319334400(r+1)}$
|
$g^{(13)}_{2,10}=\frac{775\beta^{10}\lambda^{10}-2754\beta^{8}\lambda^{8}+15120\beta^{6}\lambda^{6}-151200\beta^{4}\lambda^{4}+3024000\beta^{2}\lambda^{2}+14515200}{14515200(r+1)},\,\,\,g^{(13)}_{10,2}=\frac{\beta^{8}\lambda^{8}\left(145\beta^{2}\lambda^{2}-396\right)}{479001600(r+1)}$
|
$g^{(13)}_{6,6}=\frac{\beta^{4}\lambda^{4}\left(35\beta^{6}\lambda^{6}-126\beta^{4}\lambda^{4}+680\beta^{2}\lambda^{2}-3360\right)}{2419200(r+1)}$
|
$g^{(13)}_{4,8}=-\frac{\beta^{2}\lambda^{2}\left(425\beta^{8}\lambda^{8}-1512\beta^{6}\lambda^{6}+8400\beta^{4}\lambda^{4}-73920\beta^{2}\lambda^{2}+806400\right)}{9676800(r+1)},\,\,\,\,g^{(13)}_{8,4}=-\frac{\beta^{6}\lambda^{6}\left(25\beta^{4}\lambda^{4}-92\beta^{2}\lambda^{2}+320\right)}{9676800(r+1)}$
|
$g^{(13)}_{12,0}=-\frac{\beta^{10}\lambda^{10}\left(r\left(691\beta^{2}\lambda^{2}-27300\right)+691\beta^{2}\lambda^{2}+27300\right)}{1307674368000(r+1)}$
$(15)$ | $g^{(15)}_{0,14}=\frac{3773551\beta^{12}\lambda^{12}-9432150\beta^{10}\lambda^{10}+33513480\beta^{8}\lambda^{8}-183783600\beta^{6}\lambda^{6}+1816214400\beta^{4}\lambda^{4}-54486432000\beta^{2}\lambda^{2}+435891456000}{871782912000(r+1)}$
|
$g^{(15)}_{2,12}=\frac{-477481\beta^{12}\lambda^{12}+1193500\beta^{10}\lambda^{10}-4241160\beta^{8}\lambda^{8}+23284800\beta^{6}\lambda^{6}-232848000\beta^{4}\lambda^{4}+4656960000\beta^{2}\lambda^{2}+22353408000}{22353408000(r+1)}$
|
$g^{(15)}_{4,10}=\frac{\beta^{2}\lambda^{2}\left(21421\beta^{10}\lambda^{10}-53550\beta^{8}\lambda^{8}+190512\beta^{6}\lambda^{6}-1058400\beta^{4}\lambda^{4}+9313920\beta^{2}\lambda^{2}-101606400\right)}{1219276800(r+1)}$
|
$g^{(15)}_{6,8}=-\frac{\beta^{4}\lambda^{4}\left(11747\beta^{8}\lambda^{8}-29400\beta^{6}\lambda^{6}+105840\beta^{4}\lambda^{4}-571200\beta^{2}\lambda^{2}+2822400\right)}{2032128000(r+1)}$
|
$g^{(15)}_{8,6}=\frac{\beta^{6}\lambda^{6}\left(691\beta^{6}\lambda^{6}-1750\beta^{4}\lambda^{4}+6440\beta^{2}\lambda^{2}-22400\right)}{677376000(r+1)}$
|
$g^{(15)}_{10,4}=-\frac{\beta^{8}\lambda^{8}\left(7601\beta^{4}\lambda^{4}-20300\beta^{2}\lambda^{2}+55440\right)}{67060224000(r+1)}$
|
$g^{(15)}_{12,2}=\frac{\beta^{10}\lambda^{10}\left(691\beta^{2}\lambda^{2}-1560\right)}{74724249600(\alpha+1)}$
|
$g^{(15)}_{14,0}=-\frac{\beta^{12}\lambda^{12}\left(r\left(35\beta^{2}\lambda^{2}-1382\right)+35\beta^{2}\lambda^{2}+1382\right)}{2615348736000(r+1)}$
$(n)$ | non-vanishing $g^{(n)}_{\ell,n-\ell-1}$ to $20^{th}$ order
---|---
$(17)$ | $g^{(17)}_{0,16}=\frac{-32534915\beta^{14}\lambda^{14}+60376816\beta^{12}\lambda^{12}-150914400\beta^{10}\lambda^{10}+536215680\beta^{8}\lambda^{8}-2940537600\beta^{6}\lambda^{6}+29059430400\beta^{4}\lambda^{4}-871782912000\beta^{2}\lambda^{2}+6974263296000}{13948526592000(r+1)}$
|
$g^{(17)}_{2,14}=\frac{6689725\beta^{14}\lambda^{14}-12414506\beta^{12}\lambda^{12}+31031000\beta^{10}\lambda^{10}-110270160\beta^{8}\lambda^{8}+605404800\beta^{6}\lambda^{6}-6054048000\beta^{4}\lambda^{4}+121080960000\beta^{2}\lambda^{2}+581188608000}{581188608000(r+1)}$
|
$g^{(17)}_{4,12}=-\frac{\beta^{2}\lambda^{2}\left(507885\beta^{12}\lambda^{12}-942524\beta^{10}\lambda^{10}+2356200\beta^{8}\lambda^{8}-8382528\beta^{6}\lambda^{6}+46569600\beta^{4}\lambda^{4}-409812480\beta^{2}\lambda^{2}+4470681600\right)}{53648179200(r+1)}$
|
$g^{(17)}_{6,10}=\frac{\beta^{4}\lambda^{4}\left(37975\beta^{10}\lambda^{10}-70482\beta^{8}\lambda^{8}+176400\beta^{6}\lambda^{6}-635040\beta^{4}\lambda^{4}+3427200\beta^{2}\lambda^{2}-16934400\right)}{12192768000(r+1)}$
|
$g^{(17)}_{8,8}=-\frac{\beta^{6}\lambda^{6}\left(2975\beta^{8}\lambda^{8}-5528\beta^{6}\lambda^{6}+14000\beta^{4}\lambda^{4}-51520\beta^{2}\lambda^{2}+179200\right)}{5419008000(r+1)}$
|
$g^{(17)}_{10,6}=\frac{\beta^{8}\lambda^{8}\left(8085\beta^{6}\lambda^{6}-15202\beta^{4}\lambda^{4}+40600\beta^{2}\lambda^{2}-110880\right)}{134120448000(r+1)}$
|
$g^{(17)}_{12,4}=-\frac{\beta^{10}\lambda^{10}\left(1365\beta^{4}\lambda^{4}-2764\beta^{2}\lambda^{2}+6240\right)}{298896998400(r+1)}$
|
$g^{(17)}_{14,2}=\frac{\beta^{12}\lambda^{12}\left(1435\beta^{2}\lambda^{2}-2764\right)}{5230697472000(r+1)}$
|
$g^{(17)}_{16,0}=-\frac{\beta^{14}\lambda^{14}\left(r\left(3617\beta^{2}\lambda^{2}-142800\right)+3617\beta^{2}\lambda^{2}+142800\right)}{10670622842880000(r+1)}$
$(19)$ | $g^{(19)}_{0,18}=\frac{1}{7113748561920000(r+1)}[11582686547\beta^{16}\lambda^{16}-16592806650\beta^{14}\lambda^{14}+30792176160\beta^{12}\lambda^{12}-76966344000\beta^{10}\lambda^{10}$
|
$+273469996800\beta^{8}\lambda^{8}-1499674176000\beta^{6}\lambda^{6}+14820309504000\beta^{4}\lambda^{4}-444609285120000\beta^{2}\lambda^{2}+3556874280960000]$
|
$g^{(19)}_{2,16}=\frac{1}{418455797760000(r+1)}[-3362251073\beta^{16}\lambda^{16}+4816602000\beta^{14}\lambda^{14}-8938444320\beta^{12}\lambda^{12}+22342320000\beta^{10}\lambda^{10}$
|
$-79394515200\beta^{8}\lambda^{8}+435891456000\beta^{6}\lambda^{6}-4358914560000\beta^{4}\lambda^{4}+87178291200000\beta^{2}\lambda^{2}+418455797760000]$
|
$g^{(19)}_{4,14}=\frac{\beta^{2}\lambda^{2}\left(138267059\beta^{14}\lambda^{14}-198075150\beta^{12}\lambda^{12}+367584360\beta^{10}\lambda^{10}-918918000\beta^{8}\lambda^{8}+3269185920\beta^{6}\lambda^{6}-18162144000\beta^{4}\lambda^{4}+159826867200\beta^{2}\lambda^{2}-1743565824000\right)}{20922789888000(r+1)}$
|
$g^{(19)}_{6,12}=-\frac{\beta^{4}\lambda^{4}\left(17495429\beta^{12}\lambda^{12}-25063500\beta^{10}\lambda^{10}+46518120\beta^{8}\lambda^{8}-116424000\beta^{6}\lambda^{6}+419126400\beta^{4}\lambda^{4}-2261952000\beta^{2}\lambda^{2}+11176704000\right)}{8047226880000(r+1)}$
|
$g^{(19)}_{8,10}=\frac{\beta^{6}\lambda^{6}\left(112127\beta^{10}\lambda^{10}-160650\beta^{8}\lambda^{8}+298512\beta^{6}\lambda^{6}-756000\beta^{4}\lambda^{4}+2782080\beta^{2}\lambda^{2}-9676800\right)}{292626432000(r+1)}$
|
$g^{(19)}_{10,8}=-\frac{\beta^{8}\lambda^{8}\left(676379\beta^{8}\lambda^{8}-970200\beta^{6}\lambda^{6}+1824240\beta^{4}\lambda^{4}-4872000\beta^{2}\lambda^{2}+13305600\right)}{16094453760000(r+1)}$
|
$g^{(19)}_{12,6}=\frac{\beta^{10}\lambda^{10}\left(47021\beta^{6}\lambda^{6}-68250\beta^{4}\lambda^{4}+138200\beta^{2}\lambda^{2}-312000\right)}{14944849920000(r+1)}$
|
$g^{(19)}_{14,4}=-\frac{\beta^{12}\lambda^{12}\left(3617\beta^{4}\lambda^{4}-5740\beta^{2}\lambda^{2}+11056\right)}{20922789888000(\alpha+1)}$
|
$g^{(19)}_{16,2}=\frac{\beta^{14}\lambda^{14}\left(169999\beta^{2}\lambda^{2}-285600\right)}{21341245685760000(r+1)}$
|
$g^{(19)}_{18,0}=-\frac{\beta^{16}\lambda^{16}\left(r\left(219335\beta^{2}\lambda^{2}-8659098\right)+219335\beta^{2}\lambda^{2}+8659098\right)}{25545471085854720000(r+1)}$

##  Appendix F Coefficients of
$\partial_{t}\epsilon_{a}\partial_{t}^{\ell}\epsilon_{a}\partial_{t}^{n-\ell-1}\mathcal{E}$
in the cubic action

$(n)$ | non-vanishing $h^{(n)}_{\ell,n-\ell-1}$ to $20^{th}$ order
---|---
$(3)$ | $h^{(3)}_{1,1}=1$
$(5)$ | $h^{(5)}_{1,3}=\frac{1}{72}\left(72-\beta^{2}\lambda^{2}\right),\,\,\,\,\,\,\,h^{(5)}_{3,1}=-\frac{1}{36}\beta^{2}\lambda^{2}$
$(7)$ | $h^{(7)}_{1,5}=\frac{\beta^{4}\lambda^{4}-30\beta^{2}\lambda^{2}+2160}{2160},\,\,\,\,\,h^{(7)}_{3,3}=-\frac{1}{432}\beta^{2}\lambda^{2}\left(\beta^{2}\lambda^{2}+12\right),\,\,\,\,\,\,h^{(7)}_{5,1}=\frac{\beta^{4}\lambda^{4}}{2160}$
$(9)$ | $h^{(9)}_{1,7}=\frac{-17\beta^{6}\lambda^{6}+168\beta^{4}\lambda^{4}-5040\beta^{2}\lambda^{2}+362880}{362880},\,\,\,h^{(9)}_{3,5}=\frac{\beta^{2}\lambda^{2}\left(\beta^{4}\lambda^{4}-10\beta^{2}\lambda^{2}-120\right)}{4320}$
|
$h^{(9)}_{5,3}=-\frac{\beta^{4}\lambda^{4}\left(5\beta^{2}\lambda^{2}-12\right)}{25920},\,\,\,\,h^{(9)}_{7,1}=\frac{\beta^{6}\lambda^{6}}{30240}$
$(11)$ | $h^{(11)}_{1,9}=\frac{31\beta^{8}\lambda^{8}-170\beta^{6}\lambda^{6}+1680\beta^{4}\lambda^{4}-50400\beta^{2}\lambda^{2}+3628800}{3628800}$
|
$h^{(11)}_{3,7}=-\frac{\beta^{2}\lambda^{2}\left(51\beta^{6}\lambda^{6}-280\beta^{4}\lambda^{4}+2800\beta^{2}\lambda^{2}+33600\right)}{1209600},\,\,\,\,h^{(11)}_{5,5}=\frac{\beta^{4}\lambda^{4}\left(9\beta^{4}\lambda^{4}-50\beta^{2}\lambda^{2}+120\right)}{259200}$
|
$h^{(11)}_{7,3}=\frac{120\beta^{6}\lambda^{6}-42\beta^{8}\lambda^{8}}{3628800},\,\,\,\,h^{(11)}_{9,1}=\frac{\beta^{8}\lambda^{8}}{725760}$
$(13)$ | $h^{(13)}_{1,11}=\frac{-3455\beta^{10}\lambda^{10}+12276\beta^{8}\lambda^{8}-67320\beta^{6}\lambda^{6}+665280\beta^{4}\lambda^{4}-19958400\beta^{2}\lambda^{2}+1437004800}{1437004800}$
|
$h^{(13)}_{3,9}=\frac{17050\beta^{10}\lambda^{10}-60588\beta^{8}\lambda^{8}+332640\beta^{6}\lambda^{6}-3326400\beta^{4}\lambda^{4}-39916800\beta^{2}\lambda^{2}}{1437004800}$
|
$h^{(13)}_{5,7}=-\frac{\beta^{4}\lambda^{4}\left(425\beta^{6}\lambda^{6}-1512\beta^{4}\lambda^{4}+8400\beta^{2}\lambda^{2}-20160\right)}{43545600},\,\,\,\,h^{(13)}_{7,5}=\frac{\beta^{6}\lambda^{6}\left(35\beta^{4}\lambda^{4}-126\beta^{2}\lambda^{2}+360\right)}{10886400}$
|
$h^{(13)}_{9,3}=\frac{1980\beta^{8}\lambda^{8}-825\beta^{10}\lambda^{10}}{1437004800},\,\,\,\,h^{(13)}_{11,1}=\frac{\beta^{10}\lambda^{10}}{20528640}$
$(15)$ | $h^{(15)}_{1,13}=\frac{3773551\beta^{12}\lambda^{12}-9432150\beta^{10}\lambda^{10}+33513480\beta^{8}\lambda^{8}-183783600\beta^{6}\lambda^{6}+1816214400\beta^{4}\lambda^{4}-54486432000\beta^{2}\lambda^{2}+3923023104000}{3923023104000}$
|
$h^{(15)}_{3,11}=\frac{-18621759\beta^{12}\lambda^{12}+46546500\beta^{10}\lambda^{10}-165405240\beta^{8}\lambda^{8}+908107200\beta^{6}\lambda^{6}-9081072000\beta^{4}\lambda^{4}-108972864000\beta^{2}\lambda^{2}}{3923023104000}$
|
$h^{(15)}_{5,9}=\frac{\beta^{4}\lambda^{4}\left(21421\beta^{8}\lambda^{8}-53550\beta^{6}\lambda^{6}+190512\beta^{4}\lambda^{4}-1058400\beta^{2}\lambda^{2}+2540160\right)}{5486745600}$
|
$h^{(15)}_{7,7}=-\frac{\beta^{6}\lambda^{6}\left(11747\beta^{6}\lambda^{6}-29400\beta^{4}\lambda^{4}+105840\beta^{2}\lambda^{2}-302400\right)}{9144576000}$
|
$h^{(15)}_{9,5}=\frac{\beta^{8}\lambda^{8}\left(691\beta^{4}\lambda^{4}-1750\beta^{2}\lambda^{2}+4200\right)}{3048192000}$
|
$h^{(15)}_{11,3}=\frac{\beta^{10}\lambda^{10}\left(14700-7601\beta^{2}\lambda^{2}\right)}{301771008000}$
| $h^{(15)}_{13,1}=\frac{691\beta^{12}\lambda^{12}}{435891456000}$
$(17)$ | $h^{(17)}_{1,15}=-\frac{929569\beta^{14}\lambda^{14}}{1793381990400}+\frac{3773551\beta^{12}\lambda^{12}}{3923023104000}-\frac{691\beta^{10}\lambda^{10}}{287400960}+\frac{31\beta^{8}\lambda^{8}}{3628800}-\frac{17\beta^{6}\lambda^{6}}{362880}+\frac{\beta^{4}\lambda^{4}}{2160}-\frac{\beta^{2}\lambda^{2}}{72}+1$
|
$h^{(17)}_{3,13}=\frac{5461\beta^{14}\lambda^{14}}{2134978560}-\frac{477481\beta^{12}\lambda^{12}}{100590336000}+\frac{31\beta^{10}\lambda^{10}}{2612736}-\frac{17\beta^{8}\lambda^{8}}{403200}+\frac{\beta^{6}\lambda^{6}}{4320}-\frac{\beta^{4}\lambda^{4}}{432}-\frac{\beta^{2}\lambda^{2}}{36}$
|
$h^{(17)}_{5,11}=\frac{\beta^{4}\lambda^{4}\left(-507885\beta^{10}\lambda^{10}+942524\beta^{8}\lambda^{8}-2356200\beta^{6}\lambda^{6}+8382528\beta^{4}\lambda^{4}-46569600\beta^{2}\lambda^{2}+111767040\right)}{241416806400}$
|
$h^{(17)}_{7,9}=\frac{\beta^{6}\lambda^{6}\left(37975\beta^{8}\lambda^{8}-70482\beta^{6}\lambda^{6}+176400\beta^{4}\lambda^{4}-635040\beta^{2}\lambda^{2}+1814400\right)}{54867456000}$
|
$h^{(17)}_{9,7}=-\frac{\beta^{8}\lambda^{8}\left(2975\beta^{6}\lambda^{6}-5528\beta^{4}\lambda^{4}+14000\beta^{2}\lambda^{2}-33600\right)}{24385536000}$
|
$h^{(17)}_{11,5}=\frac{\beta^{10}\lambda^{10}\left(8085\beta^{4}\lambda^{4}-15202\beta^{2}\lambda^{2}+29400\right)}{603542016000}$
|
$h^{(17)}_{13,3}=\frac{\beta^{12}\lambda^{12}\left(24876-15925\beta^{2}\lambda^{2}\right)}{15692092416000}$
| $h^{(17)}_{15,1}=\frac{\beta^{14}\lambda^{14}}{20379340800}$
$(n)$ | non-vanishing $h^{(n)}_{\ell,n-\ell-1}$ to $20^{th}$ order
---|---
$(19)$ | $h^{(19)}_{1,17}=\frac{11582686547\beta^{16}\lambda^{16}}{32011868528640000}-\frac{929569\beta^{14}\lambda^{14}}{1793381990400}+\frac{3773551\beta^{12}\lambda^{12}}{3923023104000}-\frac{691\beta^{10}\lambda^{10}}{287400960}+\frac{31\beta^{8}\lambda^{8}}{3628800}$
|
$-\frac{17\beta^{6}\lambda^{6}}{362880}+\frac{\beta^{4}\lambda^{4}}{2160}-\frac{\beta^{2}\lambda^{2}}{72}+1$
|
$h^{(19)}_{3,15}=-\frac{3362251073\beta^{16}\lambda^{16}}{1883051089920000}+\frac{5461\beta^{14}\lambda^{14}}{2134978560}-\frac{477481\beta^{12}\lambda^{12}}{100590336000}+\frac{31\beta^{10}\lambda^{10}}{2612736}-\frac{17\beta^{8}\lambda^{8}}{403200}+\frac{\beta^{6}\lambda^{6}}{4320}-\frac{\beta^{4}\lambda^{4}}{432}-\frac{\beta^{2}\lambda^{2}}{36}$
|
$h^{(19)}_{5,13}=\frac{1}{94152554496000}[\beta^{4}\lambda^{4}(138267059\beta^{12}\lambda^{12}-198075150\beta^{10}\lambda^{10}+367584360\beta^{8}\lambda^{8}$
|
$-918918000\beta^{6}\lambda^{6}+3269185920\beta^{4}\lambda^{4}-18162144000\beta^{2}\lambda^{2}+43589145600)-198075150\beta^{10}\lambda^{10}$
|
$+367584360\beta^{8}\lambda^{8}-918918000\beta^{6}\lambda^{6}+3269185920\beta^{4}\lambda^{4}-18162144000\beta^{2}\lambda^{2}+43589145600]$
|
$h^{(19)}_{7,11}=\frac{\beta^{6}\lambda^{6}\left(-17495429\beta^{10}\lambda^{10}+25063500\beta^{8}\lambda^{8}-46518120\beta^{6}\lambda^{6}+116424000\beta^{4}\lambda^{4}-419126400\beta^{2}\lambda^{2}+1197504000\right)}{36212520960000}$
|
$h^{(19)}_{9,9}=\frac{\beta^{8}\lambda^{8}\left(112127\beta^{8}\lambda^{8}-160650\beta^{6}\lambda^{6}+298512\beta^{4}\lambda^{4}-756000\beta^{2}\lambda^{2}+1814400\right)}{1316818944000}$
|
$h^{(19)}_{11,7}=\frac{\beta^{10}\lambda^{10}\left(-676379\beta^{6}\lambda^{6}+970200\beta^{4}\lambda^{4}-1824240\beta^{2}\lambda^{2}+3528000\right)}{72425041920000}$
|
$h^{(19)}_{13,5}=\frac{\beta^{12}\lambda^{12}\left(329147\beta^{4}\lambda^{4}-477750\beta^{2}\lambda^{2}+746280\right)}{470762772480000}$
|
$h^{(19)}_{15,3}=\frac{\beta^{14}\lambda^{14}\left(4620-3617\beta^{2}\lambda^{2}\right)}{94152554496000}$
| $h^{(19)}_{17,1}=\frac{3617\beta^{16}\lambda^{16}}{2462451425280000}$

##  Appendix G Loop computations

The numerator $C(p)$ is given by the following diagram with $p=(\omega,k)$

$\begin{split}&G_{\mathcal{E}\epsilon_{a}}(p)(-C(p))G_{\epsilon_{a}\mathcal{E}}(p)=\\\
&\feynmandiagram[scale=0.5,transformshape][inline=(a.base),horizontal=atob]{a--[thick]c--[thick]b};+\scalebox{0.65}{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) ;
\vertex[right=of a] (a1) ; \vertex[right=of a1] (a2); \vertex[above=of a2]
(a3); \vertex[ right=of a2] (a4); \vertex[ right=of a4] (a5); \diagram* { (a)
-- [very thick](a1)-- [boson,very thick] (a2) -- [out=140, in=180, loop, min
distance=1.cm,very thick] (a3)--[out=0, in=40, loop, min distance=1.cm,very
thick](a2) , (a2) -- [boson,very thick] (a4)-- [very thick](a5), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) ;
\vertex[right=of a] (a1) ; \vertex[right=of a1] (a2); \vertex[above right=of
a2] (a3); \vertex[below right=of a2] (a4); \vertex[above right=of a4] (a5);
\vertex[ right=of a5] (a6); \vertex[ right=of a6] (a7); \diagram* { (a) -- [
very thick](a1)-- [boson,very thick] (a2) -- [quarter left, very thick]
(a3)--[quarter left, very thick](a5)--[quarter left, very thick](a4)--[quarter
left, very thick](a2) , (a5) -- [boson,very thick] (a6)-- [very thick](a7), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\left[\,\scalebox{0.5}{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) ;
\vertex[right=of a] (a1) ; \vertex[right=of a1] (a2); \vertex[above right=of
a2] (a3); \vertex[below right=of a2] (a4); \vertex[above right=of a4] (a5);
\vertex[ right=of a5] (a6); \vertex[ right=of a6] (a7); \diagram* { (a)
--[very thick](a1)-- [boson,very thick] (a2) -- [quarter left, very thick]
(a3)--[quarter left,boson,very thick](a5)--[quarter left, very
thick](a4)--[quarter left, very thick](a2) , (a5) -- [boson,very thick] (a6)--
[very thick](a7), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+\text{c.c}\right]\,\,.\end{split}$

Since we are not interested in computing $C(p)$ we do not write the
corresponding loop integrals. The self-energy $\Sigma(p)$ is what we want to
compute. Diagrammatically:

$\begin{split}&G_{\mathcal{E}\epsilon_{a}}(p)(-\Sigma(p))G_{\mathcal{E}\mathcal{E}}(p)=\\\
&\scalebox{0.65}{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) ;
\vertex[right=of a] (a1) ; \vertex[right=of a1] (a2); \vertex[above=of a2]
(a3); \vertex[ right=of a2] (a4); \vertex[ right=of a4] (a5); \diagram* { (a)
-- [very thick](a1)-- [boson,very thick] (a2) -- [out=140, in=180, loop, min
distance=1.cm,very thick] (a3)--[out=0, in=40, loop, min distance=1.cm,very
thick](a2) , (a2) -- [very thick] (a4)-- [very thick](a5), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) ;
\vertex[right=of a] (a1) ; \vertex[right=of a1] (a2); \vertex[above right=of
a2] (a3); \vertex[below right=of a2] (a4); \vertex[above right=of a4] (a5);
\vertex[ right=of a5] (a6); \vertex[ right=of a6] (a7); \diagram* { (a)
--[very thick](a1)-- [boson,very thick] (a2) -- [quarter left, very thick]
(a3)--[boson,quarter left,very thick](a5)--[quarter left, very
thick](a4)--[quarter left,very thick](a2) , (a5) -- [very thick] (a6)-- [very
thick](a7), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) ;
\vertex[right=of a] (a1) ; \vertex[right=of a1] (a2); \vertex[above right=of
a2] (a3); \vertex[below right=of a2] (a4); \vertex[above right=of a4] (a5);
\vertex[ right=of a5] (a6); \vertex[ right=of a6] (a7); \diagram* { (a)
--[very thick](a1)-- [boson,very thick] (a2) -- [quarter left, very thick]
(a3)--[quarter left,boson,very thick](a5)--[quarter left,boson,very
thick](a4)--[quarter left, very thick](a2) , (a5) -- [very thick](a6)-- [very
thick](a7), };\,. \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\end{split}$ (G.1)

which leads to

$\begin{split}\Sigma(p)=&\,\text{quartic}+\,\lambda_{1}^{2}k^{2}\int_{p^{\prime}}k^{\prime
2}G_{\epsilon_{a}\mathcal{E}}(p^{\prime})G_{\mathcal{E}\mathcal{E}}(p+p^{\prime})-\,icT^{2}\lambda_{1}k^{2}\int_{p^{\prime}}\,\tilde{\lambda}_{2}\,(k^{\prime
2}+kk^{\prime})G_{\epsilon_{a}\mathcal{E}}(p^{\prime})G_{\mathcal{E}\epsilon_{a}}(p+p^{\prime})\\\
&\,-\,iT^{3}\lambda_{1}k^{2}\int_{p^{\prime}}\,\tilde{\lambda}_{3}\,(\omega^{\prime
2}+\omega\omega^{\prime})G_{\epsilon_{a}\mathcal{E}}(p^{\prime})\,G_{\mathcal{E}\epsilon_{a}}(p+p^{\prime})\end{split}$
(G.2)

In the above expression

$\begin{split}\tilde{\lambda}_{2}=&\frac{1}{2}\lambda_{2}\,\sum_{n=1}\sum_{\ell=0}^{n-1}\frac{g^{(n)}_{\ell,n-\ell-1}}{\lambda^{n-1}}(-i)^{n-1}\left[\omega^{\prime\ell}+(\omega+\omega^{\prime})^{\ell}\right]\omega^{n-\ell-1}\,,\\\
\tilde{\lambda}_{3}=&\frac{1}{2}\lambda_{3}\,\sum_{n=3}\sum_{\ell=1}^{n-1}\frac{h^{(n)}_{\ell,n-\ell-1}}{\lambda^{n-2}}(-i)^{n-1}\left[\omega^{\prime\ell}+(\omega+\omega^{\prime})^{\ell}\right]\omega^{n-\ell-1}\,,\end{split}$
(G.3)

with $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ given in (4.28). Let us
recall that $h$ and $g$ coefficients have been given in Appendix . In eq.
(G.2), “quartic” denotes the contribution coming from the quartic Lagrangian.
We have found that this term does not have any cutoff independent part.

From (G.3), we see that at first order in derivatives, namely $n=1$,
$\tilde{\lambda}_{2}=\lambda_{2}$ and $\tilde{\lambda}_{3}=0$. This is exactly
the case considered in [15]. In this case, the frequency integrals are UV
finite and one can easily perform the momentum integrals by considering a hard
cutoff momentum. Beyond $n=1$, however, the frequency integrals are UV
divergent, as can be seen by substituting (G.3) into (G.2). In order to
regularize them, we follow the ideas developed in [14] and use the following
regularization for the three integrals in (G.2), respectively:

$G_{\mathcal{E}\mathcal{E}}(\omega^{\prime},k^{\prime})\rightarrow\left(\frac{\Lambda}{\Lambda+i\omega}\right)^{n-1}G_{\mathcal{E}\mathcal{E}}(\omega^{\prime},k^{\prime}),\,\,\,\,\,\,\,\,\,\tilde{\lambda}_{2,3}\rightarrow\left(\frac{\Lambda}{\Lambda-i\omega}\right)^{n-1}\tilde{\lambda}_{2,3}$
(G.4)

where $\Lambda$ is a UV energy cutoff and $n$ denotes the order of derivative,
which is actually $40$ in our case. Note that to perform the integral in the
$\lambda_{1}^{2}$ term, we close the contour along the semicircle in the lower
half of the complex $\omega$-plane, while for the next two integrals in (G.2),
we close the contour along the upper semicircle in the complex plane. These
choices are consistent with the regularizations introduced in (G.4). After
performing the frequency integrals, we consider a hard cutoff momentum
$\Lambda^{\prime}$ and evaluate the momentum integrals.

Another point that should be noted is the inclusion of ghosts. As mentioned in
[3], some anti-commuting ghosts variables and BRST symmetry are necessary to
protect the unitarity condition (2.6) at the loop level. However, it was shown
that they do not play any role in low energy dynamics [14, 13].

$(2n+1)$ |  $F_{2n}$ and $G_{2n}$ functions defined in (5.8)
---|---
$(3)$ |  $\scriptstyle F_{2}=\frac{\omega\left(\beta^{2}\lambda^{2}+12\right)\left(\omega+iD_{0}k^{2}\right)}{48\lambda^{2}}$
|  $\scriptstyle
G_{2}=-\frac{D_{0}^{2}k^{4}\left(\beta^{2}\lambda^{2}+r\left(\beta^{2}\lambda^{2}+12\right)-12\right)-2iD_{0}k^{2}\omega\left(\beta^{2}\lambda^{2}+r\left(\beta^{2}\lambda^{2}+12\right)-12\right)+\omega^{2}\left(\beta^{2}\lambda^{2}+r\left(\beta^{2}\lambda^{2}+12\right)-36\right)}{48\lambda^{2}(r+1)}$
$(5)$ |  $\scriptstyle F_{4}=-\frac{\beta^{2}\omega\left(\beta^{2}\lambda^{2}-60\right)\left(\omega+iD_{0}k^{2}\right)\left(3D_{0}^{2}k^{4}-6iD_{0}k^{2}\omega+\omega^{2}\right)}{11520\lambda^{2}}$
|  $\scriptstyle
G_{4}=\frac{720\omega^{4}\left(\beta^{2}\lambda^{2}-8\right)-120\omega^{2}\left(5\beta^{2}\lambda^{2}+24\right)\left(D_{0}^{2}k^{4}-2iD_{0}k^{2}\omega+\omega^{2}\right)+\beta^{2}\lambda^{2}\left(D_{0}^{4}k^{8}-4iD_{0}^{3}k^{6}\omega+2D_{0}^{2}k^{4}\omega^{2}-12iD_{0}k^{2}\omega^{3}+\omega^{4}\right)\left(\beta^{2}\lambda^{2}+r\left(\beta^{2}\lambda^{2}-60\right)+60\right)}{11520\lambda^{4}(r+1)}$
$(7)$ |  $\scriptstyle F_{6}=\frac{\beta^{4}\omega\left(\beta^{2}\lambda^{2}-42\right)\left(\omega+iD_{0}k^{2}\right)\left(5D_{0}^{4}k^{8}-20iD_{0}^{3}k^{6}\omega-10D_{0}^{2}k^{4}\omega^{2}-20iD_{0}k^{2}\omega^{3}+\omega^{4}\right)}{1935360\lambda^{2}}$
|  $\scriptstyle
G_{6}=\frac{1}{1935360\lambda^{6}(r+1)}\bigg{[}4032\omega^{6}\left(\beta^{4}\lambda^{4}-30\beta^{2}\lambda^{2}+240\right)-5040\omega^{4}\left(\beta^{2}\lambda^{2}-24\right)\left(\beta^{2}\lambda^{2}+4\right)\left(D_{0}^{2}k^{4}-2iD_{0}k^{2}\omega+\omega^{2}\right)+84\beta^{2}\lambda^{2}\omega^{2}\left(11\beta^{2}\lambda^{2}-120\right)\left(D_{0}^{4}k^{8}-4iD_{0}^{3}k^{6}\omega+2D_{0}^{2}k^{4}\omega^{2}-12iD_{0}k^{2}\omega^{3}+\omega^{4}\right)$
|
$\scriptstyle-\beta^{4}\lambda^{4}\left(D_{0}^{2}k^{4}-2iD_{0}k^{2}\omega+\omega^{2}\right)\left(D_{0}^{4}k^{8}-4iD_{0}^{3}k^{6}\omega+10D_{0}^{2}k^{4}\omega^{2}-28iD_{0}k^{2}\omega^{3}+\omega^{4}\right)\left(\beta^{2}\lambda^{2}+r\left(\beta^{2}\lambda^{2}-42\right)+42\right)\bigg{]}$
$(9)$ |  $\scriptstyle F_{8}=-\frac{\beta^{6}\omega\left(\beta^{2}\lambda^{2}-40\right)\left(\omega+iD_{0}k^{2}\right)\left(7D_{0}^{6}k^{12}-42iD_{0}^{5}k^{10}\omega-49D_{0}^{4}k^{8}\omega^{2}-84iD_{0}^{3}k^{6}\omega^{3}-119D_{0}^{2}k^{4}\omega^{4}-42iD_{0}k^{2}\omega^{5}+\omega^{6}\right)}{309657600\lambda^{2}}$
|  $\scriptstyle
G_{8}=\frac{1}{309657600\lambda^{8}(r+1)}\bigg{[}3840\omega^{8}\left(17\beta^{6}\lambda^{6}-168\beta^{4}\lambda^{4}+5040\beta^{2}\lambda^{2}-40320\right)-80640\omega^{6}\left(\beta^{6}\lambda^{6}-10\beta^{4}\lambda^{4}+200\beta^{2}\lambda^{2}+960\right)\left(D_{0}^{2}k^{4}-2iD_{0}k^{2}\omega+\omega^{2}\right)$
|
$\scriptstyle+3360\beta^{2}\lambda^{2}\omega^{4}\left(5\beta^{4}\lambda^{4}-44\beta^{2}\lambda^{2}+480\right)\left(D_{0}^{4}k^{8}-4iD_{0}^{3}k^{6}\omega+2D_{0}^{2}k^{4}\omega^{2}-12iD_{0}k^{2}\omega^{3}+\omega^{4}\right)-80\beta^{4}\lambda^{4}\omega^{2}\left(17\beta^{2}\lambda^{2}-84\right)\left(D_{0}^{2}k^{4}-2iD_{0}k^{2}\omega+\omega^{2}\right)\left(D_{0}^{4}k^{8}-4iD_{0}^{3}k^{6}\omega+10D_{0}^{2}k^{4}\omega^{2}-28iD_{0}k^{2}\omega^{3}+\omega^{4}\right)$
|
$\scriptstyle+\beta^{6}\lambda^{6}\left(D_{0}^{8}k^{16}-8iD_{0}^{7}k^{14}\omega+4D_{0}^{6}k^{12}\omega^{2}-136iD_{0}^{5}k^{10}\omega^{3}-250D_{0}^{4}k^{8}\omega^{4}-56iD_{0}^{3}k^{6}\omega^{5}-252D_{0}^{2}k^{4}\omega^{6}-56iD_{0}k^{2}\omega^{7}+\omega^{8}\right)\times\left(\beta^{2}\lambda^{2}+r\left(\beta^{2}\lambda^{2}-40\right)+40\right)\bigg{]}$
$(11)$ |  $\scriptstyle F_{10}=\frac{\beta^{8}\omega\left(5\beta^{2}\lambda^{2}-198\right)\left(\omega+iD_{0}k^{2}\right)\left(3D_{0}^{2}k^{4}-6iD_{0}k^{2}\omega+\omega^{2}\right)\left(3D_{0}^{6}k^{12}-18iD_{0}^{5}k^{10}\omega-9D_{0}^{4}k^{8}\omega^{2}-84iD_{0}^{3}k^{6}\omega^{3}-75D_{0}^{2}k^{4}\omega^{4}-66iD_{0}k^{2}\omega^{5}+\omega^{6}\right)}{245248819200\lambda^{2}}$
|  $\scriptstyle
G_{10}=\frac{1}{245248819200\lambda^{10}(r+1)}\bigg{[}304128\omega^{10}\left(31\beta^{8}\lambda^{8}-170\beta^{6}\lambda^{6}+1680\beta^{4}\lambda^{4}-50400\beta^{2}\lambda^{2}+403200\right)+228096\omega^{8}\left(-51\beta^{8}\lambda^{8}+280\beta^{6}\lambda^{6}-2800\beta^{4}\lambda^{4}+56000\beta^{2}\lambda^{2}+268800\right)\left(D_{0}^{2}k^{4}-2iD_{0}k^{2}\omega+\omega^{2}\right)$
|
$\scriptstyle-3168\beta^{4}\lambda^{4}\omega^{4}\left(63\beta^{4}\lambda^{4}-340\beta^{2}\lambda^{2}+1680\right)\left(D_{0}^{2}k^{4}-2iD_{0}k^{2}\omega+\omega^{2}\right)\left(D_{0}^{4}k^{8}-4iD_{0}^{3}k^{6}\omega+10D_{0}^{2}k^{4}\omega^{2}-28iD_{0}k^{2}\omega^{3}+\omega^{4}\right)$
|
$\scriptstyle+266112\beta^{2}\lambda^{2}\omega^{6}\left(9\beta^{6}\lambda^{6}-50\beta^{4}\lambda^{4}+440\beta^{2}\lambda^{2}-4800\right)\left(D_{0}^{4}k^{8}-4iD_{0}^{3}k^{6}\omega+2D_{0}^{2}k^{4}\omega^{2}-12iD_{0}k^{2}\omega^{3}+\omega^{4}\right)+396\beta^{6}\lambda^{6}\omega^{2}\left(23\beta^{2}\lambda^{2}-80\right)$
|
$\scriptstyle\times\left(D_{0}^{8}k^{16}-8iD_{0}^{7}k^{14}\omega+4D_{0}^{6}k^{12}\omega^{2}-136iD_{0}^{5}k^{10}\omega^{3}-250D_{0}^{4}k^{8}\omega^{4}-56iD_{0}^{3}k^{6}\omega^{5}-252D_{0}^{2}k^{4}\omega^{6}-56iD_{0}k^{2}\omega^{7}+\omega^{8}\right)$
|
$\scriptstyle-\beta^{8}\lambda^{8}\left(D_{0}^{2}k^{4}-2iD_{0}k^{2}\omega+\omega^{2}\right)\left(5\beta^{2}\lambda^{2}+r\left(5\beta^{2}\lambda^{2}-198\right)+198\right)\left(D_{0}^{8}k^{16}-8iD_{0}^{7}k^{14}\omega+20D_{0}^{6}k^{12}\omega^{2}-232iD_{0}^{5}k^{10}\omega^{3}-346D_{0}^{4}k^{8}\omega^{4}-312iD_{0}^{3}k^{6}\omega^{5}-620D_{0}^{2}k^{4}\omega^{6}-88iD_{0}k^{2}\omega^{7}+\omega^{8}\right)\bigg{]}$
$(13)$ |  $\scriptstyle F_{12}=-\frac{\beta^{10}\omega\left(691\beta^{2}\lambda^{2}-27300\right)\left(\omega+iD_{0}k^{2}\right)\left(11D_{0}^{10}k^{20}-110iD_{0}^{9}k^{18}\omega-275D_{0}^{8}k^{16}\omega^{2}-440iD_{0}^{7}k^{14}\omega^{3}-2618D_{0}^{6}k^{12}\omega^{4}+2156iD_{0}^{5}k^{10}\omega^{5}-2574D_{0}^{4}k^{8}\omega^{6}+2376iD_{0}^{3}k^{6}\omega^{7}-1265D_{0}^{2}k^{4}\omega^{8}-110iD_{0}k^{2}\omega^{9}+\omega^{10}\right)}{5356234211328000\lambda^{2}}$
|  $\scriptstyle
G_{12}=\frac{1}{5356234211328000(r+1)\lambda^{12}}\big{[}16773120\left(\beta^{2}\lambda^{2}\left(3455\beta^{8}\lambda^{8}-12276\beta^{6}\lambda^{6}+67320\beta^{4}\lambda^{4}-665280\beta^{2}\lambda^{2}+19958400\right)-159667200\right)\omega^{12}$
|
$\scriptstyle-92252160\left(\beta^{2}\left(775\beta^{8}\lambda^{8}-2754\beta^{6}\lambda^{6}+15120\beta^{4}\lambda^{4}-151200\beta^{2}\lambda^{2}+3024000\right)\lambda^{2}+14515200\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)\omega^{10}$
|
$\scriptstyle+34594560\beta^{2}\lambda^{2}\left(425\beta^{8}\lambda^{8}-1512\beta^{6}\lambda^{6}+8400\beta^{4}\lambda^{4}-73920\beta^{2}\lambda^{2}+806400\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+2\omega^{2}D_{0}^{2}k^{4}-12i\omega^{3}D_{0}k^{2}+\omega^{4}\right)\omega^{8}$
|
$\scriptstyle-34594560\beta^{4}\lambda^{4}\left(35\beta^{6}\lambda^{6}-126\beta^{4}\lambda^{4}+680\beta^{2}\lambda^{2}-3360\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+10\omega^{2}D_{0}^{2}k^{4}-28i\omega^{3}D_{0}k^{2}+\omega^{4}\right)\omega^{6}$
|
$\scriptstyle+2162160\beta^{6}\lambda^{6}\left(25\beta^{4}\lambda^{4}-92\beta^{2}\lambda^{2}+320\right)\left(D_{0}^{8}k^{16}-8i\omega
D_{0}^{7}k^{14}+4\omega^{2}D_{0}^{6}k^{12}-136i\omega^{3}D_{0}^{5}k^{10}-250\omega^{4}D_{0}^{4}k^{8}-56i\omega^{5}D_{0}^{3}k^{6}-252\omega^{6}D_{0}^{2}k^{4}-56i\omega^{7}D_{0}k^{2}+\omega^{8}\right)\omega^{4}$
|
$\scriptstyle-10920\beta^{8}\lambda^{8}\left(145\beta^{2}\lambda^{2}-396\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)\left(D_{0}^{8}k^{16}-8i\omega
D_{0}^{7}k^{14}+20\omega^{2}D_{0}^{6}k^{12}-232i\omega^{3}D_{0}^{5}k^{10}-346\omega^{4}D_{0}^{4}k^{8}-312i\omega^{5}D_{0}^{3}k^{6}-620\omega^{6}D_{0}^{2}k^{4}-88i\omega^{7}D_{0}k^{2}+\omega^{8}\right)\omega^{2}$
|
$\scriptstyle+\beta^{10}\lambda^{10}\left(691\beta^{2}\lambda^{2}+r\left(691\beta^{2}\lambda^{2}-27300\right)+27300\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+2\omega^{2}D_{0}^{2}k^{4}-12i\omega^{3}D_{0}k^{2}+\omega^{4}\right)$
|  $\scriptstyle\times\left(D_{0}^{8}k^{16}-8i\omega
D_{0}^{7}k^{14}+36\omega^{2}D_{0}^{6}k^{12}-328i\omega^{3}D_{0}^{5}k^{10}-570\omega^{4}D_{0}^{4}k^{8}-56i\omega^{5}D_{0}^{3}k^{6}-476\omega^{6}D_{0}^{2}k^{4}-120i\omega^{7}D_{0}k^{2}+\omega^{8}\right)\big{]}$
$(2n+1)$ |  $F_{2n}$ and $G_{2n}$ functions defined in (5.8)
---|---
$(15)$ |  $\scriptstyle F_{14}=\frac{1}{42849873690624000\lambda^{2}}\big{[}\beta^{12}\omega\left(35\beta^{2}\lambda^{2}-1382\right)\left(\omega+iD_{0}k^{2}\right)\big{(}13D_{0}^{12}k^{24}-156iD_{0}^{11}k^{22}\omega-494D_{0}^{10}k^{20}\omega^{2}-780iD_{0}^{9}k^{18}\omega^{3}-7033D_{0}^{8}k^{16}\omega^{4}$
|
$\scriptstyle+10088iD_{0}^{7}k^{14}\omega^{5}-7124D_{0}^{6}k^{12}\omega^{6}+21736iD_{0}^{5}k^{10}\omega^{7}+715D_{0}^{4}k^{8}\omega^{8}+10868iD_{0}^{3}k^{6}\omega^{9}-2782D_{0}^{2}k^{4}\omega^{10}-156iD_{0}k^{2}\omega^{11}+\omega^{12}\big{)}\big{]}$
|  $\scriptstyle
G_{14}=\frac{1}{42849873690624000(r+1)\lambda^{14}}\big{[}49152\left(\beta^{4}\left(3773551\beta^{8}\lambda^{8}-9432150\beta^{6}\lambda^{6}+33513480\beta^{4}\lambda^{4}-183783600\beta^{2}\lambda^{2}+1816214400\right)\lambda^{4}-54486432000\beta^{2}\lambda^{2}+435891456000\right)\omega^{14}$
|
$\scriptstyle+479232\left(\beta^{4}\left(-477481\beta^{8}\lambda^{8}+1193500\beta^{6}\lambda^{6}-4241160\beta^{4}\lambda^{4}+23284800\beta^{2}\lambda^{2}-232848000\right)\lambda^{4}+4656960000\beta^{2}\lambda^{2}+22353408000\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)\omega^{12}$
|
$\scriptstyle+2196480\beta^{2}\lambda^{2}\left(\beta^{2}\lambda^{2}\left(21421\beta^{8}\lambda^{8}-53550\beta^{6}\lambda^{6}+190512\beta^{4}\lambda^{4}-1058400\beta^{2}\lambda^{2}+9313920\right)-101606400\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+2\omega^{2}D_{0}^{2}k^{4}-12i\omega^{3}D_{0}k^{2}+\omega^{4}\right)\omega^{10}$
|
$\scriptstyle-329472\beta^{4}\lambda^{4}\left(11747\beta^{8}\lambda^{8}-29400\beta^{6}\lambda^{6}+105840\beta^{4}\lambda^{4}-571200\beta^{2}\lambda^{2}+2822400\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+10\omega^{2}D_{0}^{2}k^{4}-28i\omega^{3}D_{0}k^{2}+\omega^{4}\right)\omega^{8}$
|
$\scriptstyle+247104\beta^{6}\lambda^{6}\left(691\beta^{6}\lambda^{6}-1750\beta^{4}\lambda^{4}+6440\beta^{2}\lambda^{2}-22400\right)\left(D_{0}^{8}k^{16}-8i\omega
D_{0}^{7}k^{14}+4\omega^{2}D_{0}^{6}k^{12}-136i\omega^{3}D_{0}^{5}k^{10}-250\omega^{4}D_{0}^{4}k^{8}-56i\omega^{5}D_{0}^{3}k^{6}-252\omega^{6}D_{0}^{2}k^{4}-56i\omega^{7}D_{0}k^{2}+\omega^{8}\right)\omega^{6}$
|
$\scriptstyle-624\beta^{8}\lambda^{8}\left(7601\beta^{4}\lambda^{4}-20300\beta^{2}\lambda^{2}+55440\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)\left(D_{0}^{8}k^{16}-8i\omega
D_{0}^{7}k^{14}+20\omega^{2}D_{0}^{6}k^{12}-232i\omega^{3}D_{0}^{5}k^{10}-346\omega^{4}D_{0}^{4}k^{8}-312i\omega^{5}D_{0}^{3}k^{6}-620\omega^{6}D_{0}^{2}k^{4}-88i\omega^{7}D_{0}k^{2}+\omega^{8}\right)\omega^{4}$
|
$\scriptstyle+140\beta^{10}\lambda^{10}\left(691\beta^{2}\lambda^{2}-1560\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+2\omega^{2}D_{0}^{2}k^{4}-12i\omega^{3}D_{0}k^{2}+\omega^{4}\right)\left(D_{0}^{8}k^{16}-8i\omega
D_{0}^{7}k^{14}+36\omega^{2}D_{0}^{6}k^{12}-328i\omega^{3}D_{0}^{5}k^{10}-570\omega^{4}D_{0}^{4}k^{8}-56i\omega^{5}D_{0}^{3}k^{6}-476\omega^{6}D_{0}^{2}k^{4}-120i\omega^{7}D_{0}k^{2}+\omega^{8}\right)\omega^{2}$
|
$\scriptstyle-\beta^{12}\lambda^{12}\left(35\beta^{2}\lambda^{2}+r\left(35\beta^{2}\lambda^{2}-1382\right)+1382\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)$
|  $\scriptstyle\times\left(D_{0}^{12}k^{24}-12i\omega
D_{0}^{11}k^{22}+30\omega^{2}D_{0}^{10}k^{20}-740i\omega^{3}D_{0}^{9}k^{18}-2449\omega^{4}D_{0}^{8}k^{16}-280i\omega^{5}D_{0}^{7}k^{14}-12636\omega^{6}D_{0}^{6}k^{12}+13720i\omega^{7}D_{0}^{5}k^{10}-9617\omega^{8}D_{0}^{4}k^{8}+13092i\omega^{9}D_{0}^{3}k^{6}-3554\omega^{10}D_{0}^{2}k^{4}-180i\omega^{11}D_{0}k^{2}+\omega^{12}\right)\big{]}$
$(17)$ |  $\scriptstyle F_{16}=-\frac{1}{699309938630983680000\lambda^{2}}\big{[}\beta^{14}\omega\left(3617\beta^{2}\lambda^{2}-142800\right)\left(\omega+iD_{0}k^{2}\right)\left(3D_{0}^{2}k^{4}-6iD_{0}k^{2}\omega+\omega^{2}\right)\left(5D_{0}^{4}k^{8}-20iD_{0}^{3}k^{6}\omega-10D_{0}^{2}k^{4}\omega^{2}-20iD_{0}k^{2}\omega^{3}+\omega^{4}\right)$
|
$\scriptstyle\times\left(D_{0}^{8}k^{16}-8iD_{0}^{7}k^{14}\omega+4D_{0}^{6}k^{12}\omega^{2}-136iD_{0}^{5}k^{10}\omega^{3}-186D_{0}^{4}k^{8}\omega^{4}-312iD_{0}^{3}k^{6}\omega^{5}-444D_{0}^{2}k^{4}\omega^{6}-184iD_{0}k^{2}\omega^{7}+\omega^{8}\right)$
|  $\scriptstyle
G_{16}=\frac{1}{699309938630983680000(r+1)\lambda^{16}}\big{[}50135040\left(\beta^{6}\left(32534915\beta^{8}\lambda^{8}-60376816\beta^{6}\lambda^{6}+150914400\beta^{4}\lambda^{4}-536215680\beta^{2}\lambda^{2}+2940537600\right)\lambda^{6}-29059430400\beta^{4}\lambda^{4}+871782912000\beta^{2}\lambda^{2}-6974263296000\right)\omega^{16}$
|
$\scriptstyle-300810240\left(\beta^{6}\left(6689725\beta^{8}\lambda^{8}-12414506\beta^{6}\lambda^{6}+31031000\beta^{4}\lambda^{4}-110270160\beta^{2}\lambda^{2}+605404800\right)\lambda^{6}-6054048000\beta^{4}\lambda^{4}+121080960000\beta^{2}\lambda^{2}+581188608000\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)\omega^{14}$
|
$\scriptstyle+814694400\beta^{2}\lambda^{2}\left(\beta^{4}\left(507885\beta^{8}\lambda^{8}-942524\beta^{6}\lambda^{6}+2356200\beta^{4}\lambda^{4}-8382528\beta^{2}\lambda^{2}+46569600\right)\lambda^{4}-409812480\beta^{2}\lambda^{2}+4470681600\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+2\omega^{2}D_{0}^{2}k^{4}-12i\omega^{3}D_{0}k^{2}+\omega^{4}\right)\omega^{12}$
|
$\scriptstyle-896163840\beta^{4}\lambda^{4}\left(\beta^{2}\lambda^{2}\left(37975\beta^{8}\lambda^{8}-70482\beta^{6}\lambda^{6}+176400\beta^{4}\lambda^{4}-635040\beta^{2}\lambda^{2}+3427200\right)-16934400\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+10\omega^{2}D_{0}^{2}k^{4}-28i\omega^{3}D_{0}k^{2}+\omega^{4}\right)\omega^{10}$
|
$\scriptstyle+504092160\beta^{6}\lambda^{6}\left(2975\beta^{8}\lambda^{8}-5528\beta^{6}\lambda^{6}+14000\beta^{4}\lambda^{4}-51520\beta^{2}\lambda^{2}+179200\right)\left(D_{0}^{8}k^{16}-8i\omega
D_{0}^{7}k^{14}+4\omega^{2}D_{0}^{6}k^{12}-136i\omega^{3}D_{0}^{5}k^{10}-250\omega^{4}D_{0}^{4}k^{8}-56i\omega^{5}D_{0}^{3}k^{6}-252\omega^{6}D_{0}^{2}k^{4}-56i\omega^{7}D_{0}k^{2}+\omega^{8}\right)\omega^{8}$
|
$\scriptstyle-5091840\beta^{8}\lambda^{8}\left(8085\beta^{6}\lambda^{6}-15202\beta^{4}\lambda^{4}+40600\beta^{2}\lambda^{2}-110880\right)\left(D_{0}^{2}k^{4}-2i\omega
D_{0}k^{2}+\omega^{2}\right)$
|  $\scriptstyle\times\left(D_{0}^{8}k^{16}-8i\omega
D_{0}^{7}k^{14}+20\omega^{2}D_{0}^{6}k^{12}-232i\omega^{3}D_{0}^{5}k^{10}-346\omega^{4}D_{0}^{4}k^{8}-312i\omega^{5}D_{0}^{3}k^{6}-620\omega^{6}D_{0}^{2}k^{4}-88i\omega^{7}D_{0}k^{2}+\omega^{8}\right)\omega^{6}$
|
$\scriptstyle+571200\beta^{10}\lambda^{10}\left(1365\beta^{4}\lambda^{4}-2764\beta^{2}\lambda^{2}+6240\right)\left(D_{0}^{4}k^{8}-4i\omega
D_{0}^{3}k^{6}+2\omega^{2}D_{0}^{2}k^{4}-12i\omega^{3}D_{0}k^{2}+\omega^{4}\right)$
|  $\scriptstyle\times\left(D_{0}^{8}k^{16}-8i\omega