# Renormalization group analysis of a self-organized critical system:
Intrinsic anisotropy vs random environment.

N. V. Antonov,1,2 P. I. Kakin,1 N. M. Lebedev2 and A. Yu. Luchin.1 1
Department of Physics, Saint Petersburg State University, 7/9 Universitetskaya
Naberezhnaya, Saint Petersburg 199034, Russia
2 N. N. Bogoliubov Laboratory of Theoretical Physics, Joint Institute for
Nuclear Research, Dubna 141980, Moscow Region, Russia<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study a self-organized critical system coupled to an isotropic random fluid
environment. The former is described by a strongly anisotropic continuous
(coarse-grained) model introduced by Hwa and Kardar [Phys. Rev. Lett. 62 1813
(1989); Phys. Rev. A 45 7002 (1992)]; the latter is described by the stirred
Navier–Stokes equation due to Forster, Nelson and Stephen [Phys. Rev. A 16 732
(1977)]. The full problem of two coupled stochastic equations is represented
as a field theoretic model, which is shown to be multiplicatively
renormalizable. The corresponding renormalization group equations possess a
semi-infinite curve of fixed points in the four-dimensional space of the model
parameters. The whole curve is infrared attractive for realistic values of
parameters; its endpoint corresponds to the purely isotropic regime where the
original Hwa-Kardar nonlinearity becomes irrelevant. There, one is left with a
simple advection of a passive scalar field by the external environment. The
main critical dimensions are calculated to the leading one-loop order (first
terms in the $\epsilon=4-d$ expansion); some of them are appear to be exact in
all orders. They remain the same along that curve, which makes it reasonable
to interpret it as a single universality class. However, the correction
exponents do vary along the curve. It is therefore not clear whether the curve
survives in all orders of the renormalization group expansion or shrinks to a
single point when the higher-order corrections are taken into account.

Keywords: self-organized criticality, renormalization group, nonequilibrium
behavior, disordered systems.

SB/.style = decoration=markings, mark=at position 0.6 with
[semithick]Stealth[black,width=3mm,length=3mm], mark=at position 0.8 with
[semithick]Bar[black,width=3mm,length=0mm];, line width=0.3mm,
postaction=decorate fermion’/.style = decoration= markings, mark=at position
0.7 with [xshift=2mm]Bar[black,width=3mm,length=0mm] , line width=0.3mm,
postaction=decorate fermion/.style = line width=0.3mm, postaction=decorate
boson’/.style = boson, decoration= markings, mark=at position 0.7 with
[xshift=2mm]Bar[black,width=3mm,length=0mm] , line width=0.3mm,
postaction=decorate antiboson’/.style = boson, decoration= markings, mark=at
position 0.2 with [xshift=0mm]Bar[black,width=3mm,length=0mm] , line
width=0.3mm, postaction=decorate bosonZerowidth’/.style = boson, decoration=
markings, mark=at position 0.05 with
[xshift=2mm]Bar[black,width=3mm,length=0mm] , line width=0.3mm,
postaction=decorate antifermion’/.style = decoration= markings, mark=at
position 0.2 with [xshift=0mm]Bar[black,width=3mm,length=0mm] , line
width=0.3mm, postaction=decorate Boson/.style = boson, line width=0.3mm,
postaction=decorate fermionZerowidth”/.style = decoration= markings, mark=at
position 0.05 with [xshift=2mm]Bar[black,width=3mm,length=0mm] , line
width=0.3mm, postaction=decorate fermionZerowidth”’/.style = decoration=
markings, mark=at position 0.05 with
[xshift=2.5mm]Bar[black,width=3mm,length=0mm] , line width=0.3mm,
postaction=decorate boson”/.style = boson, decoration=markings, mark=at
position 0.6 with [semithick]Stealth[black,width=3mm,length=3mm], mark=at
position 0.8 with [semithick]Bar[black,width=3mm,length=0mm];, line
width=0.3mm, postaction=decorate

##  1  Introduction

It has long been observed that numerous cooperative many-body systems achieve
a kind of critical state in the natural course of their intrinsic dynamics.
Such systems are termed to display self-organized criticality (SOC) [1] and
considered to stand in contrast with more common systems, nearly-equilibrium
[2] – [4] as well as strongly non-equilibrium [5, 6] ones, that need to be
“fine-tuned” to arrive at their critical points.

Systems with SOC are ubiquitous in Nature and are widely encountered beyond
the realm of physics in a narrow sense of the word [7] – [32]. Numerous
examples are provided by biological systems [17, 18], including neural systems
among them [19] – [24], social networks [25] – [29], and many others [31, 32].
That is why any further research of SOC remains actual in view of a large
constantly growing amount of examples and applications already known.

In original pioneering studies, SOC phenomena were described by certain
speculative or conjectural models, both discrete in time and space; see, e.g.
[7] – [9].

However, in the theory of much more common equilibrium critical behaviour, it
has long being realized that discrete models (like the Ising and Heisenberg
models of magnets) can be substituted with the continuous $\phi^{4}$ field
theory, as long as their critical behaviour is concerned [2] – [4]. Thus, one
can hope that the passage to effective continuous models of SOC also would not
“throw out the child along with the bathwater.”

Such kind of model for SOC was proposed by Hwa and Kardar in [41, 42]; it is a
continuous (coarse-grained) stochastic differential equation, designed on the
basis of simplicity, dimensionality, conservation and symmetry considerations.

Existing experience with nearly-critical systems has shown that they are
extremely sensitive to various kinds of external disturbances: inclusion of
impurities, gravity, effects of environment [33] – [37] and so on. In
practice, such systems can hardly be isolated from the influence of
surrounding medium, like turbulent motion in the atmosphere, in ocean or,
especially, in forest fires.

Moreover, it was observed that impressive macroscopic manifestations of
underlying microscopic dynamics of strongly interacting many-body systems
(such as formation of snow-flake structures, stochastic resonance etc.) may
result from such a rivalry between intrinsic dynamics and various external
disturbances that, in their turn, reveal themselves as a kind of effective
external friction [38] – [40].

In most common and ordinary situations, the original dynamics has some
symmetries (dimensional, translation, rotational or Galilean invariance),
violated by the initial or boundary conditions, specific geometry of an
experimental setup and other various external perturbations.

In this paper, we consider the situation which, in a sense, is opposite: we
study the strongly anisotropic Hwa-Kardar (HK) model [41, 42] of SOC coupled
to the isotropic random environment described by the stochastic Navier–Stokes
(NS) equation with a random stirring force due to Forster, Nelson, and Stephen
[43].

Some explanations are here in order. In the most intuitive and pictorial
interpretation, the HK model describes a box filled with sand, with an open
wall through which the sand is removed, and subjected to a “rain of grains”
from above. The system is supposed to achieve a SOC steady state. There, the
anisotropy of the external setup in the resulting dynamic model is conveyed
(“delegated”) to the effective differential equation and therefore can be
viewed as “intrinsic” or “local” one.

Thus, the original HK equation for the height field involves a distinguished
direction from the very beginning and, therefore, it is strongly anisotropic
due to its very formulation. Here, the anisotropy is termed “strong” in the
sense that one can introduce two independent canonical dimensions (length
scales): one for the chosen direction and the other for the orthogonal
subspace. The corresponding critical dimensions are also different; see, e.g.
[44, 45] for a more detailed discussion.

In the spirit of the theory of critical state, the model can be viewed as an
example of the universality class corresponding to a variety of physical
systems. Other possible representatives may imply the presence of local
anisotropy from the very beginning, due to a certain intrinsic microscopic
dynamics. This also justifies interest to the model for general spatial
dimension $d$.

In contrast to those basic dynamics, the external perturbation is represented
by an isotropic equilibrium random environment. For the latter, we employ the
stochastic NS equation with an external stirring force introduced and studied
in [43].

In an earlier work [44, 45], the velocity was modelled by the “rapid-change”
Kazantsev–Kraichnan velocity ensemble [46]. It was shown that such advection
“washes away” the nonlinearity of the original Hwa-Kardar equation [41, 42]
for the most realistic values of the model parameters [44, 45].

At the same time, in some asymptotic regimes, the coupling of anisotropic
dynamics with isotropic environment results in a kind of “dimensional
transmutation”111Not to be confused with the well-known mechanism of mass
generation in a massless field theory., i.e. a certain dimensionless parameter
acquires nontrivial canonical and critical dimensions; what is more, the very
notion of critical dimension should be revisited; see [44, 45] for detailed
explanation.

Thus, it is tempting to model the advection by more realistic velocity fields
governed by the stochastic dynamical NS equation rather than somewhat
artificial Gaussian ensemble.

The full-fledged HK+NS model of the two coupled stochastic equations for the
height and the velocity fields can be reformulated as a field theoretic model,
which is shown to be multiplicatively renormalizable.

This allows one to use the well-established techniques of the quantum field
theory to identify the possible types of critical behaviour, associated with
infrared (IR) attractive fixed points of the renormalization group (RG)
equations, to calculate critical dimensions of the fields and parameters, to
find their regions of stability and to discuss probable crossover phenomena.

It should be noted that the HK height field is “passive” in the sense that it
has no feedback on the velocity statistics. Thus, the field of competition
between various ingredients of the composite model lies in the dynamics of the
height field, while the velocity dynamics remains intact.

Despite the fact that the more symmetric perturbation appears in this sense
superior, the passive scalar dynamics can resist, and, in some cases, it can
retain its native strong anisotropy, or, in other cases, some kind of a
compromise is achieved: the resulting state is anisotropic, but no separate
dimensions in the two subspaces can be introduced.

The plan of the paper is the following. In Section 2, we give a detailed
description of the stochastic problem, while its field theoretic formulation
is presented in Section 3.

In Section 4, we demonstrate multiplicative renormalizability of the model by
using the dimensionality and symmetry considerations.

The practical calculations of the renormalization constants are performed in
the leading one-loop approximation; they are presented in Section 5.

The RG functions and the analysis of the RG flow attractors are given in
Section 6.

The full model involves four coupling constants. In their four-dimensional
space, there is a two-dimensional surface of trivial (Gaussian) fixed points,
IR attractive for $\epsilon<0$, where $\epsilon=4-d$ is the deviation of the
spatial dimension $d$ from its logarithmic value $d=4$.

There are also a few fixed points which are unstable for all $\epsilon$. Among
them there is a point in which the NS nonlinearity (and therefore the
advection) becomes irrelevant, and the pure HK model is restored. In this
regime, the ratios of certain kinematic coefficients acquire nontrivial
dimension due to the transmutation mechanism [44, 45], while the longitudinal
and transverse directions acquire independent dimensions, like in the original
HK model. However, in our model this interesting asymptotic regime cannot
materialize due to the instability of the corresponding fixed point.

Thus, the most interesting attractor appears to be a semi-infinite curve of
nontrivial fixed points, where both the NS and the HK nonlinearities are
simultaneously relevant. The curve is IR attractive for $\epsilon>0$; its
endpoint corresponds to the regime where the HK nonlinearity becomes
irrelevant and one is left with the simple passive scalar advection. Although
this regime is anisotropic, no independent canonical dimensions can be
introduced for the chosen direction and the orthogonal subspace; the resulting
critical dimensions of the longitudinal and transverse subspaces are equal, in
contrast with the original HK model.

In Section 7, the corresponding critical dimensions are presented to the first
order of the expansion in $\epsilon=4-d$; some results appear to be exact to
all orders. It turns out that the dimensions of the basic fields and
parameters remain the same along the curve of the fixed points. Thus, it is
tempting to interpret the whole curve as a single universality class. On the
other hand, some correction exponents (eigenvalues of the stability matrix) do
vary along the curve, which opens the possibility to interpret it as a family
of classes parameterized by a point on the curve. The discussion of the
results and further perspectives are given in the concluding Section 8.

##  2  Description of the model

The HK stochastic equation introduced in [41, 42] is a semi-phenomenological
continuous model for the SOC behaviour in a coarse-grained “running” sandpile.
Being of a rather general importance, the model is intuitively best
illustrated by “sand in a box with an open top and an open side.” The system
is manifestly anisotropic: the new sand entering from above drives avalanches
that cause some sand to exit through the side. While the sandpile surface is
considered to be flat on average, it is getting rougher with time. The surface
tilt is specified by a constant unit vector ${\bf n}$: ${\bf x}={\bf
x_{\bot}}+{\bf n}x_{\parallel}$, $|{\bf n}|=1$, $({\bf x_{\bot}}{\bf n})=0$.
Here and below, $x=(t,{\bf x})$ represent the time-space coordinates.

More specifically, the HK equation describes the evolution of a scalar field
$h=h(t,{\bf x})$ that stands for the sandpile height deviation from its
average:

$\partial_{t}h=\nu_{\bot 0}\,\partial_{\bot}^{2}h+\nu_{\parallel
0}\,\partial_{\parallel}^{2}h-\partial_{\parallel}h^{2}/2+f.$ (1)

Here $\nu_{\bot 0}$ and $\nu_{\parallel 0}$ are kinetic coefficients; the
derivatives are

$\partial_{t}=\frac{\partial}{\partial{t}},\quad{\bf\partial}_{\perp}^{2}=({\bf\partial}_{{\perp}}{\bf\partial}_{{\perp}})=\frac{\partial}{\partial{x_{\perp
i}}}\frac{\partial}{\partial{x_{\perp i}}};$

summation over repeated tensor indices is implied here and throughout the
paper; index $i$ in $x_{\perp i}$ runs from 1 to $(d-1)$ with $d$ being the
dimension of surface, and $\partial_{\parallel}=({\bf n}\partial)$. The random
noise $f(x)$ has zero mean and prescribed Gaussian statistics:

$\langle
f(x)f(x^{\prime})\rangle_{f}=C_{0}\,\delta(t-t^{\prime})\,\delta^{(d)}({\bf
x}-{\bf x}^{\prime}),\quad$ (2)

where $C_{0}>0$ is a positive amplitude and brackets $\langle\dots\rangle_{f}$
denote averaging over the Gaussian statistics of the random noise $f$.

In this paper, we describe the environment motion by the NS stochastic
differential equation for an isotropic incompressible viscous fluid with an
external random stirring force [43]:

$\displaystyle\nabla_{t}v_{i}=\nu_{0}\partial^{2}v_{i}-\partial_{i}\wp+\eta_{i},$
(3)

where

$\displaystyle\nabla_{t}=\partial_{t}+({\bf v}{\bf\partial})$ (4)

is the Lagrangian (Galilean covariant) derivative, $\wp$ is the pressure and
$\nu_{0}$ is the kinematic viscosity coefficient. Due to incompressibility of
the fluid, the velocity field is transverse: $({\bf\partial}{\bf v})=0$, in
the sense that $({\bf k}{\bf v})=0$ in the momentum representation. Thus, the
pressure can be expressed in terms of ${\bf v}$ as
$\wp=-\partial^{-2}\partial_{i}\partial_{k}v_{i}v_{k}$. In a more explicit
notation,

$\displaystyle\wp(t,{\bf x})=-\int d{\bf x^{\prime}}\,\Delta({\bf x}-{\bf
x^{\prime}})\partial_{i}\partial_{k}\,v_{i}(t,{\bf x^{\prime}})v_{k}(t,{\bf
x^{\prime}}),$ (5)

where $\Delta({\bf x}-{\bf x^{\prime}})$ is the Green function for the Laplace
equation, $\partial^{2}\Delta({\bf x}-{\bf x^{\prime}})=\delta({\bf x}-{\bf
x^{\prime}})$.

Like the velocity field, the external random force per unit mass $\eta_{i}$
should also be taken to be transverse222There is no loss of generality: the
longitudinal (potential) part of the force would be canceled by the pressure
term.. In the spirit of the pioneering paper [43], we choose for $\eta_{i}$ a
Gaussian probability distribution with zero average and a given correlation
function

$\displaystyle\langle\eta_{i}(t,{\bf x})\eta_{j}(t^{\prime},{\bf
x}^{\prime})\rangle_{\eta}=D_{0}\,\delta(t-t^{\prime})\,D_{ij}({\bf x}-{\bf
x}^{\prime}),$ (6)

where the brackets $\langle\dots\rangle_{\eta}$ stand for the averaging over
the noise statistics, $D_{0}>0$ is a positive amplitude factor and

$\displaystyle D_{ij}({\bf x}-{\bf
x}^{\prime})=\,\int\frac{d\mathbf{k}}{(2\pi)^{d}}P_{ij}(\mathbf{k})\,\>\mathrm{exp}\,i(\mathbf{k}(\mathbf{x-x^{\prime}}))\,$
(7)

with the transverse projector $P_{ij}({\bf k})=\delta_{ij}-k_{i}k_{j}/k^{2}$
and the wave number $k\equiv|{\bf k}|$.

The temporal $\delta$ correlation in (6) is stipulated by the Galilean
symmetry, while the choice (7) is probably the simplest but a very
representative one in the following sense. Without the projector, it would
correspond to a uniform (that is, independent of the momentum and the
frequency) random stirring of the fluid at all scales ${\bf k}$. The presence
of the projector, caused by the incompressibility of the fluid, leads to
entanglement between different ${\bf k}$-modes and, simultaneously, introduces
non-locality. Indeed, in the coordinate representation one obtains:

$\displaystyle D_{ij}({\bf x}-{\bf
x}^{\prime})=P_{ij}({\bf\partial})\,\delta({\bf x}-{\bf
x^{\prime}})=\delta_{ij}\delta({\bf x}-{\bf
x^{\prime}})-\partial_{i}\partial_{j}\,{\Delta}({\bf x}-{\bf x^{\prime}}),$
(8)

with $\Delta({\bf x}-{\bf x^{\prime}})$ from (5).

It is easily seen that the correlation function (6), (7), (8) satisfies the
transversality condition $\partial_{i}\,D_{ij}=0$. Furthermore, (7) involves a
finite mode at ${\bf k}=0$; it can be interpreted as an overall macroscopic
random “shaking” of the fluid container as a whole; see footnote15 in [43].

The coupling of the fields $h$ and ${\bf v}$ is introduced by the “minimal”
substitution $\partial_{t}\to\nabla_{t}=\partial_{t}+({\bf v}{\bf\partial})$
in Eq. (1) with the Galilean covariant derivative $\nabla_{t}$ from Eq. (4).

Equations (1), (3) are studied on the entire $t$ axis; the retardation
condition is assumed; the asymptotic conditions for the fields $h$ and ${\bf
v}$ at $t\rightarrow-\infty$ are irrelevant due to the presence of the noises.
This completes formulation of the problem.

As we will show below, in the vicinity of $d=4$, the nonlinear terms in Eqs.
(1) and (3) simultaneously become logarithmic, that is, they are of the same
relevance in the sense of Wilson. This a posteriori justifies the choice of
the statistics of the random force $\eta_{i}$ that provides a full-blooded
diffusion-advection problem.

##  3 Field theoretic formulation of the model

The full stochastic problem (1)–(6) is equivalent (see, e.g. Sec. 5.3 in [4]
and references therein) to the field theoretic model with the doubled set of
fields $\Phi=\\{h,h^{\prime},{\bf v},{\bf v^{\prime}}\\}$ and the action
functional

$\displaystyle S(\Phi)$
$\displaystyle=C_{0}h^{\prime}h^{\prime}/2+h^{\prime}\\{-\nabla_{t}h+\nu_{\parallel
0}\partial_{\parallel}^{2}h+\nu_{\bot
0}\partial_{\bot}^{2}h-\partial_{\parallel}h^{2}/2\\}+$ (9)
$\displaystyle+D_{0}{\bf v^{\prime}}^{2}/2+{\bf v^{\prime}}\\{-\nabla_{t}{\bf
v}+\nu_{0}\partial^{2}{\bf v}\\}.$

For brevity, here and below in similar expressions, we tacitly imply the
needed integration over the arguments $x=\\{t,{\bf x}\\}$; in particular, the
first term in (9) stands for $C_{0}\int dt\int d{\bf x}\,h^{\prime}(t,{\bf
x})h^{\prime}(t,{\bf x})/2$. The pressure term is omitted owing to the
transversality of the auxiliary field ${\bf v}^{\prime}$ that acts as a kind
of transverse projector.

In the frequency-momentum $(\omega-{\bf k})$ representation, the bare
propagators for the model (9) are:

$\displaystyle\langle hh^{\prime}\rangle_{0}$ $\displaystyle=\langle
h^{\prime}h\rangle_{0}^{*}=\frac{1}{-i\omega+\epsilon({\bf k})},\quad\ \langle
h^{\prime}h^{\prime}\rangle_{0}=0,\quad\ \langle
hh\rangle_{0}=\frac{C_{0}}{\omega^{2}+\epsilon^{2}({\bf k})},$
$\displaystyle\langle v_{i}v^{\prime}_{j}\rangle_{0}$ $\displaystyle=\langle
v^{\prime}_{i}v_{j}\rangle_{0}^{*}=\frac{P_{ij}({\bf
k})}{-i\omega+\nu_{0}k^{2}},\quad\langle
v^{\prime}_{i}v^{\prime}_{j}\rangle_{0}=0,\,\,\,\,\,\,\langle
v_{i}v_{j}\rangle_{0}=\frac{D\,P_{ij}({\bf
k})}{\omega^{2}+{\nu_{0}}^{2}k^{4}},$ (10)

where $\epsilon({\bf k})=\nu_{\parallel 0}{k}_{\parallel}^{2}+\nu_{\perp
0}{\bf k}_{\perp}^{2}$ and $P_{ij}({\bf k})=\delta_{ij}-k_{i}k_{j}/k^{2}$. We
will use the following lines for the propagators when drawing Feynman’s
diagrams:

$\displaystyle\langle hh^{\prime}\rangle_{0}=\vbox{\hbox{ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=1.5cm of a] (b); \par\diagram* { (a) -- [fermion', thick] (b) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},$
$\displaystyle\quad\langle hh\rangle_{0}=\vbox{\hbox{ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=1.5cm of a] (b); \par\diagram* { (a) -- [fermion, thick] (b) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},\quad$
$\displaystyle\langle v_{i}v^{\prime}_{j}\rangle_{0}=\vbox{\hbox{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=1.5cm of a] (b); \par\diagram* { (a) -- [boson', thick] (b) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}},$
$\displaystyle\quad\langle v_{i}v_{j}\rangle_{0}=\vbox{\hbox{ \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=1.5cm of a] (b); \par\diagram* { (a) -- [boson, thick] (b) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
The three vertices $-h^{\prime}\partial_{\parallel}h^{2}/2$, $-h^{\prime}({\bf
v}{\bf\partial})h$ and $-{\bf v^{\prime}}({\bf v}{\bf\partial}){\bf v}$
correspond to the vertex factors $ik_{\parallel}^{h^{\prime}}$,
$ik_{j}^{h^{\prime}}$ and $i(k_{n}^{{\bf v^{\prime}}}\delta_{sj}+k_{s}^{{\bf
v^{\prime}}}\delta_{nj})$ respectively. The upper indices denote the field for
which this momentum is an argument (i.e., the momentum “flows” through the
corresponding leg of the vertex). The form of the vertex factors can be
explained by the fact that one can move the derivatives in all those vertices
onto the primed fields using the integration by parts and the transversality
of the field ${\bf v}$.

There are two coupling constants $g_{0}$ and $w_{0}$ (expansion parameters in
the ordinary perturbation theory): $g_{0}=C_{0}/\nu_{\bot
0}^{3/2}\nu_{\parallel 0}^{3/2}\sim{\Lambda}^{\epsilon},\quad
w_{0}=D_{0}/\nu_{0}^{3}\sim{\Lambda}^{\epsilon}$. These expressions follow
from the canonical dimensions analysis (see below) and define the typical
ultraviolet (UV) momentum scale $\Lambda$.

##  4 UV divergences and renormalization of the model

We study UV divergences in the Green’s functions of the model (9) using the
canonical dimensions analysis (see, e.g. [4], Sections 1.15 and 1.16). There
are two independent scales (the time scale $T$ and the length scale $L$), so
that the canonical dimension of any quantity $F$ is described by the frequency
dimension $d_{F}^{\omega}$ and the momentum dimension $d_{F}^{k}$:
$[F]\sim[T]^{-d_{F}^{\omega}}[L]^{-d_{F}^{k}}$ (see [4], Sections 1.17 and
5.14). Normalization conditions $d_{k_{i}}^{k}=-d_{x_{i}}^{k}=1,\
d_{k_{i}}^{\omega}=d_{x_{i}}^{\omega}=0,\ d_{\omega}^{k}=d_{t}^{k}=0,\
d_{\omega}^{\omega}=-d_{t}^{\omega}=1$ allow to calculate the canonical
dimensions of all the fields and the parameters entering the dimensionless
action functional (9). The total canonical dimension is defined as
$d_{F}=d_{F}^{k}+2d_{F}^{\omega}$; it should be used in RG analysis of dynamic
models instead of the momentum dimension (used in analysis of static models,
see [4], Section 5.14). Canonical dimensions for the model (9) are presented
in Table 1.

Table 1: Canonical dimensions in the model (9); $\epsilon=4-d$. $F$ | $h(x)$ | $h^{\prime}(x)$ | ${\bf v}(x)$ | ${\bf v^{\prime}}(x)$ |  $C_{0}$, $D_{0}$ |  $\nu_{0}$, $\nu_{\parallel 0}$, $\nu_{\bot 0}$ |  $g_{0}$, $w_{0}$ | $x_{10}$, $x_{20}$ |  $\mu$,$m$
---|---|---|---|---|---|---|---|---|---
$d^{\omega}_{F}$ | $1$ | $-1$ | $1$ | $-1$ | $3$ | $1$ | $0$ | $0$ | $0$
$d^{k}_{F}$ | $-1$ | $d+1$ | $-1$ | $d+1$ | $-d-2$ | $-2$ | $\epsilon$ | $0$ | $1$
$d_{F}$ | $1$ | $d-1$ | $1$ | $d-1$ | $\epsilon$ | $0$ | $\epsilon$ | $0$ | $1$

Here $m$ is the IR cut-off and $\mu$ is the renormalization mass (also
referred to as normalization point or reference momentum scale), the
additional parameters of the renormalized model; see below and [4].

As can be seen from Table 1, the three coefficients $\nu_{0}$, $\nu_{\parallel
0}$ and $\nu_{\bot 0}$ have identical dimensions. This means that their ratios
are completely dimensionless and may enter the expressions for the $\beta$
functions (see below). Thus, these ratios should be treated as additional
coupling constants, although they are not expansion parameters. One can choose
$x_{1,0}=\nu_{\parallel 0}/\nu_{0}$ and $x_{2,0}=\nu_{\perp 0}/\nu_{0}$.

All these coupling constants simultaneously become dimensionless (and the
respective interactions become marginal in the sense of Wilson) at $d=4$.
Thus, the model as a whole is logarithmic at $d=4$, and the role of the
expansion parameter in the RG perturbation theory is played by the single
variable $\epsilon=4-d$ that measures the deviation from logarithmicity.

To establish renormalizability of the model (9), let us calculate the UV
divergence indices of 1-irreducible Green’s functions for the model. The UV
divergence index $\delta_{\Gamma}$ for a Green’s function $\Gamma$ that
involves $N_{h}$ fields $h$, $N_{h^{\prime}}$ fields $h^{\prime}$, etc.,
coincides with the total canonical dimension $d_{\Gamma}$ of that function in
the frequency-momentum representation, taken at the logarithmic dimension
(see, e.g. Sec. 3.2 and Sec. 5.15 in [4] for detailed explanation):

$\displaystyle\delta_{\Gamma}$ $\displaystyle=$ $\displaystyle
d_{\Gamma}|_{d=4}=(d+2-d_{h}N_{h}-d_{h^{\prime}}N_{h^{\prime}}-d_{v}N_{{\bf
v}}-d_{{\bf v^{\prime}}}N_{{\bf v^{\prime}}})|_{d=4}=$ (11) $\displaystyle=$
$\displaystyle 6-N_{h}-3N_{h^{\prime}}-N_{{\bf v}}-3N_{{\bf v^{\prime}}}.$

Superficial UV divergences (those that must be eliminated by the
renormalization procedure) can only be present in the Green’s functions with
non-negative divergence index. However, the formal index $\delta_{\Gamma}$
should be adjusted: every field $h^{\prime}$ and ${\bf v^{\prime}}$ entering
an 1-irreducible Green’s function cause additional external momentum to appear
as an overall factor. It happens due to the form of the vertices that allows
spatial derivative to be moved onto those fields using integration by parts.
Thus, the real divergence index $\delta_{\Gamma}^{\prime}$ is given by the
expression:

$\delta_{\Gamma}^{\prime}=\delta_{\Gamma}-N_{h^{\prime}}-N_{{\bf
v^{\prime}}}=6-N_{h}-4N_{h^{\prime}}-N_{{\bf v}}-4N_{{\bf v^{\prime}}}.$ (12)

As a manifestation of causality, all the 1-irreducible Green’s functions with
$N_{h^{\prime}}=N_{{\bf v^{\prime}}}=0$ vanish and, therefore, do not require
counterterms; see, e.g. [4], Section 5.4.

Thus, we arrive at the following list of possible counterterms for the model
(9):

$\langle{\bf v}^{\prime}{\bf v}\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=2,\delta^{\prime}_{\Gamma}=1)$ with counterterm ${\bf
v}^{\prime}\partial^{2}{\bf v}$,
---
$\langle{\bf v}^{\prime}{\bf v}{\bf v}\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=1,\delta^{\prime}_{\Gamma}=0)$ with counterterm ${\bf
v}^{\prime}({\bf v}\partial){\bf v}$,
$\langle{\bf v}^{\prime}{\bf v}h\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=1,\delta^{\prime}_{\Gamma}=0)$ with counterterm $({\bf
v}^{\prime}{\bf v})\partial_{\parallel}h$,
$\langle{\bf v^{\prime}}hh\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=1,\delta^{\prime}_{\Gamma}=0)$ with counterterm $h({\bf
v^{\prime}}\partial)h$,
$\langle{\bf v^{\prime}}h\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=2,\delta^{\prime}_{\Gamma}=1)$ with counterterm $({\bf
v^{\prime}}\partial)h$,
$\langle h^{\prime}h\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=2,\delta^{\prime}_{\Gamma}=1)$ with counterterms
$h^{\prime}\partial_{\parallel}^{2}h$, $h^{\prime}\partial_{\bot}^{2}h$,
$\langle h^{\prime}hh\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=1,\delta^{\prime}_{\Gamma}=0)$ with counterterm
$h^{\prime}\partial_{\parallel}h^{2}$,
$\langle h^{\prime}h{\bf v}\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=1,\delta^{\prime}_{\Gamma}=0)$ with counterterm
$h^{\prime}({\bf v}\partial)h$,
$\langle h^{\prime}{\bf v}\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=2,\delta^{\prime}_{\Gamma}=1)$ with counterterm $({\bf
v}\partial)h^{\prime}$,
$\langle h^{\prime}{\bf v}{\bf v}\rangle_{\rm{1-ir}}$
$(\delta_{\Gamma}=1,\delta^{\prime}_{\Gamma}=0)$ with counterterm ${\bf
v}^{2}\partial_{\parallel}h^{\prime}$.

Here, the integration over $x=\\{t,{\bf x}\\}$ is implied. Thus, if the
difference between two counterterms is a total derivative, they should be
treated as the same item. Furthermore, the counterterms $({\bf
v}\partial)h^{\prime}$, $({\bf v^{\prime}}\partial)h$ and $h({\bf
v^{\prime}}\partial)h$ reduce to total derivatives due to the transversality
of the fields ${\bf v}$ and ${\bf v^{\prime}}$, vanish after the integration
over ${\bf x}$ and can be ignored.

The counterterm $\langle h^{\prime}\rangle_{\rm{1-ir}}$ renormalizes the
average of the random noise mean $\langle f\rangle$ directly connected to the
average $\langle h\rangle$ by the averaging of Eq. (1). It is the deviations
from the averages that are of interest here, so that all those averages can be
simultaneously treated as constants and can be taken to be equal to zero which
allows one to ignore their renormalization. This treatment is similar to the
way in which the shift of the critical temperature is ignored within the RG
analysis of phase transitions in field-theoretic models; see, e.g. sections
1.20 and 1.21 in [4].

The remaining list of possible counterterms can be further pared down.

The passivity of the field $h$ (which broadly means that the dynamic of the
field ${\bf v}$ is not affected by the field $h$) in field-theoretic terms
translates to the vanishing of the full Green’s functions with
$N_{h^{\prime}}>0$, $N_{h}=0$, $N_{{\bf v}}+N_{{\bf v^{\prime}}}>0$ and the
1-irreducible Green’s functions with $N_{h}>0$, $N_{h^{\prime}}=0$, $N_{{\bf
v}}+N_{{\bf v^{\prime}}}>0$: no needed diagrams can be constructed. Thus, the
counterterms $({\bf v}^{\prime}{\bf v})\partial_{\parallel}h$ and $h({\bf
v^{\prime}}\partial)h$ should be omitted.

Furthermore, the action functional (9) is invariant with respect to the
Galilean transformation

$\displaystyle{\bf v}(t,{\bf x})\to{\bf v}(t,{\bf x}+{\bf u}\,t)-{\bf
u},\qquad$ $\displaystyle{\bf v^{\prime}}(t,{\bf x})\to{\bf v^{\prime}}(t,{\bf
x}+{\bf u}\,t),$ (13) $\displaystyle h(t,{\bf x})\to h(t,{\bf x}+{\bf
u}\,t),\quad$ $\displaystyle h^{\prime}(t,{\bf x})\to h^{\prime}(t,{\bf
x}+{\bf u}\,t)$

with a constant vector ${\bf u}$. Since the derivative (4) is covariant with
respect to this symmetry, it can appear in the counterterms only as a single
unit. For example, the counterterm $h^{\prime}\partial_{t}h$ is forbidden by
the real divergence index (due to the form of the vertices, where $h^{\prime}$
always stands under a spatial derivative). Although its Galilean partner
$h^{\prime}({\bf v}\partial)h$ is allowed by the real index of divergence, it
cannot appear by itself. Thus, the both structures are forbidden. This
symmetry also excludes the counterterms ${\bf v}^{\prime}({\bf v}\partial){\bf
v}$ and ${\bf v}^{2}\partial_{\parallel}h^{\prime}$.

The Hwa-Kardar equation (1) has its own Galilean-type symmetry $h(t,{\bf
x})\to h(t,{\bf x}+u{\bf n}\,t)-u{\bf n}$, where $u=const$, that is broken in
the full-scale model (9). However, it leaves its trace in some relations for
the Feynman diagrams without the fields ${\bf v}$ and ${\bf v^{\prime}}$. For
example, that symmetry, were it not violated, would forbid the counterterm
$h^{\prime}\partial_{\parallel}h^{2}$. Thus, in the diagrams for the Green’s
function $\langle h^{\prime}hh\rangle_{\rm{1-ir}}$ without the fields ${\bf
v}$ and ${\bf v^{\prime}}$ the UV divergences must cancel each other, as can
be confirmed by the practical calculation.

As a result, we arrive at the conclusion that there are only four
counterterms, namely, $h^{\prime}\partial_{\parallel}^{2}h$,
$h^{\prime}\partial_{\bot}^{2}h$, $h^{\prime}\partial_{\parallel}h^{2}$ and
$({\bf v^{\prime}}\partial^{2}{\bf v})$ related to the 1-irreducible functions
$\langle h^{\prime}h\rangle_{{\rm{1-ir}}}$, $\langle
h^{\prime}hh\rangle_{{\rm{1-ir}}}$, and $\langle{\bf v^{\prime}}{\bf
v}\rangle_{{\rm{1-ir}}}$, respectively. This means that the model (9) is
renormalizable and its renormalized action reads

$\displaystyle S_{R}(\Phi)$
$\displaystyle=g\mu^{\epsilon}\nu_{\bot}^{3/2}\nu_{\parallel}^{3/2}h^{\prime}h^{\prime}/2+h^{\prime}\\{-\nabla_{t}h+Z_{1}\nu_{\parallel}\partial_{\parallel}^{2}h+Z_{2}\nu_{\bot}\partial_{\bot}^{2}h-Z_{4}\partial_{\parallel}h^{2}/2\\}+$
(14) $\displaystyle+w\mu^{\epsilon}\nu^{3}{\bf v^{\prime}}^{2}/2+{\bf
v^{\prime}}\\{-\nabla_{t}{\bf v}+Z_{3}\nu\partial^{2}{\bf v}\\}.$

Here the fields were substituted with their renormalized counterparts without
a change of notation (the fields ${\bf v}$ and ${\bf v^{\prime}}$ are not
renormalized altogether while scalar fields were substituted as $h\to Z_{h}h,$
$h^{\prime}\to Z_{h^{\prime}}h^{\prime}$).

The coupling constants and other parameters are related to their counterparts
as follows:

$\displaystyle g_{0}=Z_{g}g\mu^{\epsilon},\quad
w_{0}=Z_{w}w\mu^{\epsilon},\quad x_{1,0}=Z_{x_{1}}x_{1},\quad
x_{2,0}=Z_{x_{2}}x_{2},$ $\displaystyle\nu_{0}=\nu Z_{\nu},\quad\nu_{\parallel
0}=\nu_{\parallel}Z_{\nu_{\parallel}},\quad\nu_{\perp
0}=\nu_{\perp}Z_{\nu_{\perp}}.$ (15)

Here $Z_{i}$ are the renormalization constants; they can be expressed in terms
of the constants $Z_{1}$, $Z_{2}$, $Z_{3}$, and $Z_{4}$:

$\displaystyle Z_{h}=Z^{-1}_{h^{\prime}}=Z_{4},$ $\displaystyle
Z_{g}=Z_{1}^{-3/2}Z_{2}^{-3/2}Z_{4}^{2},\quad Z_{w}=Z_{3}^{-3},\quad
Z_{x_{1}}=Z_{1}Z_{3}^{-1},\quad Z_{x_{2}}=Z_{2}Z_{3}^{-1},$ $\displaystyle
Z_{\nu_{\parallel}}=Z_{1},\quad Z_{\nu_{\perp}}=Z_{2},\quad Z_{\nu}=Z_{3}$
(16)

(we note that $Z_{{\bf v}}=Z_{{\bf v^{\prime}}}=1$).

##  5 One-loop calculations

In this section, we present the calculation of the renormalization constants
$Z_{1}$–$Z_{4}$ in the one-loop approximation, that is, in the leading order
in the couplings $g$ and $w$.

Let us briefly discuss the choice of the IR regularization. In the model (6),
the IR regularization in the diagrams is provided by the external frequencies
and momenta. However, in the simple way of calculation we adopt here, the
diagrams are calculated at zero external frequencies, while the integrands are
expanded in the external momenta to the desired order. Then, to ensure the IR
convergence, it is necessary to introduce the cut-off of integration momenta
at a certain value $k=m>0$ in expressions like (7). For the correlation
function (2) this implies the replacement

$\delta^{(d)}({\bf x}-{\bf
x}^{\prime})\to\int_{k>m}\frac{d\mathbf{k}}{(2\pi)^{d}}\>\mathrm{exp}\,i(\mathbf{k}(\mathbf{x-x^{\prime}})).\quad$

According to the general statement, the renormalization constants in the MS
scheme do not depend on the specific choice of the IR regularization; see,
e.g. Section 3.19 in [4]. Thus, the results we obtain here for the
renormalization constants, RG functions, fixed points and critical dimensions
are valid for all $m\geq 0$.

The one-loop approximation for the 1-irreducible Green’s functions $\langle
hh^{\prime}\rangle_{\rm 1-ir}$ and $\langle{{\bf v}}{{\bf
v}}^{\prime}\rangle_{\rm 1-ir}$ is:

$\displaystyle\langle hh^{\prime}\rangle_{\rm 1-ir}$ $\displaystyle=$
$\displaystyle{\rm
i}\omega-\nu_{\parallel}p_{\parallel}^{2}Z_{1}-\nu_{\bot}p_{\bot}^{2}Z_{2}+$
(17) $\displaystyle\vbox{\hbox{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=0.8cm of a] (b); \vertex[right=of b] (c); \vertex[right=0.8cm of
c] (e); \par\diagram* { (a) -- [fermion, thick] (b), (c) -- [fermion', half
right] (b), (b) -- [fermion, half right] (c), (c) -- [fermionZerowidth''] (e),
}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\ +\vbox{\hbox{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=0.8cm of a] (b); \vertex[right=of b] (c); \vertex[right=0.8cm of
c] (e); \par\diagram* { (a) -- [fermion] (b), (c) -- [fermion', half right]
(b), (b) -- [->, boson, half right, thick] (c), (c) -- [fermionZerowidth'']
(e), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\ \ ,$ $\langle{{\bf
v}}{{\bf v}}^{\prime}\rangle_{\rm 1-ir}={\rm i}\omega-\nu p^{2}Z_{3}\
+\vbox{\hbox{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=0.8cm of a] (b); \vertex[right=of b] (c); \vertex[right=0.8cm of
c] (e); \par\diagram* { (a) -- [Boson] (b),(c) -- [boson', half right] (b),
(b) -- [Boson, half right, looseness=1.5] (c), (c) -- [bosonZerowidth'] (e),
}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\ \ .$ (18)

Here we used diagrammatic technique introduced in Section 3.

The first diagram in Eq. (17) and the diagram in Eq. (18) are calculated in
the standard manner; see Appendix A in [50] (calculation of diagram $D_{2}$)
and Appendix in [47] (calculation of diagram $A_{9}$). The results are

$\displaystyle\vbox{\hbox{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=0.8cm of a] (b); \vertex[right=of b] (c); \vertex[right=0.8cm of
c] (e); \par\diagram* { (a) -- [fermion, thick] (b), (c) -- [fermion', half
right] (b), (b) -- [fermion, half right] (c), (c) -- [fermionZerowidth''] (e),
}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\frac{S_{d}}{(2\pi)^{d}}\nu_{\parallel}\nu_{\perp}^{\frac{4-d}{2}}p_{\parallel}^{2}\frac{1-d}{4d}\frac{g}{\epsilon}\,\Bigg{|}_{d=4}=-\frac{1}{8\pi^{2}}\nu_{\parallel}p_{\parallel}^{2}\frac{3}{16}\frac{g}{\epsilon};$
(19) $\displaystyle\vbox{\hbox{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=0.8cm of a] (b); \vertex[right=of b] (c); \vertex[right=0.8cm of
c] (e); \par\diagram* { (a) -- [Boson] (b),(c) -- [boson', half right] (b),
(b) -- [Boson, half right, looseness=1.5] (c), (c) -- [bosonZerowidth'] (e),
}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=-\frac{S_{d}}{(2\pi)^{d}}\nu
p^{2}\frac{(d-1)}{4(d+2)}\frac{w}{\epsilon}\,\Bigg{|}_{d=4}=-\frac{1}{8\pi^{2}}\nu
p^{2}\frac{1}{8}\frac{w}{\epsilon}.$ (20)

Here $S_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the area of the unit sphere in the
$d$-dimensional space.

By redefining charge $w/8\pi^{2}\rightarrow w$ and putting result (20) in Eq.
(18), one arrives at one-loop approximation for renormalization constant
$Z_{3}$:

$\displaystyle Z_{3}=1-\frac{1}{\epsilon}\frac{w}{8}.$ (21)

To find $Z_{1}$ and $Z_{2}$ we have to calculate the second diagram in Eq.
(17):

$\displaystyle\vbox{\hbox{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=0.8cm of a] (b); \vertex[right=of b] (c); \vertex[right=0.8cm of
c] (e); \vertex[below=0.4cm of b, black] (b2) {$a$}; \vertex[below=0.3cm of c,
black] (c2) {$b$}; \par\diagram* { \par(a) -- [fermion, momentum={$p$}] (b),
(b) -- [antifermion', half left, reversed momentum'={[arrow
shorten=0.25]$\omega$}, momentum={[arrow shorten=0.35]$p-k$}] (c), (b) --
[Boson, half right, looseness=1.5, momentum'={[arrow shorten=0.35]$k$}] (c),
(c) -- [fermionZerowidth'', momentum={$p$}] (e), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\,=\int_{k>m}\frac{d\omega}{2\pi}\frac{d^{d}{\bf
k}}{(2\pi)^{d}}\frac{D_{0}P_{ab}({\bf
k})\,ip_{a}i(p-k)_{b}}{(\omega^{2}+\nu^{2}_{0}k^{4})(-i\omega+\epsilon(p-k))}.$
(22)

By integrating over the frequency $w$ (using residues) and dropping the factor
$k_{b}$ from the numerator, we arrive at the following expression:

$\displaystyle-\frac{D_{0}}{2\nu_{0}}\int_{k>m}\frac{d^{d-1}{\bf
k_{\perp}}dk_{\parallel}}{(2\pi)^{d}}\frac{P_{ab}({\bf
k})p_{a}p_{b}}{k^{2}((\nu_{0}+\nu_{\parallel
0})k^{2}_{\parallel}+(\nu_{0}+\nu_{\perp 0})k^{2}_{\perp})}.$ (23)

The numerator $P_{ab}({\bf k})p_{a}p_{b}$ can be decomposed into the sum

$\displaystyle\left(p^{2}k^{2}-(p_{\parallel}k_{\parallel})^{2}-(p_{\perp}k_{\perp})^{2}\right)/k^{2},$
(24)

so well-known relation for $d$-dimensional integral

$\displaystyle\int d^{d}{\bf
k}\frac{k_{i}k_{j}}{k^{2}}f(k)=\frac{\delta_{ij}}{d}\int d^{d}{\bf k}f(k),$
(25)

($f(k)$ here is an arbitrary function that depends only on $k$) allows us to
transform integral (23) into the following one:

$\displaystyle-\frac{D_{0}}{2\nu_{0}}\int_{k>m}\frac{d^{d-1}{\bf
k_{\perp}}dk_{\parallel}}{(2\pi)^{d}}\frac{\left[p^{2}k^{2}-p^{2}_{\parallel}k^{2}_{\parallel}-\frac{p^{2}_{\perp}k^{2}_{\perp}}{d-1}\right]}{k^{4}((\nu_{0}+\nu_{\parallel
0})k^{2}_{\parallel}+(\nu_{0}+\nu_{\perp 0})k^{2}_{\perp})}=$
$\displaystyle-\frac{D_{0}}{2\nu_{0}}\int_{k>m}\frac{d^{d-1}{\bf
k_{\perp}}dk_{\parallel}}{(2\pi)^{d}}\frac{k^{2}_{\parallel}p^{2}_{\perp}+k^{2}_{\perp}\left(p^{2}-\frac{p^{2}_{\perp}}{d-1}\right)}{k^{4}((\nu_{0}+\nu_{\parallel
0})k^{2}_{\parallel}+(\nu_{0}+\nu_{\perp 0})k^{2}_{\perp})}.$ (26)

Integration over $k_{\parallel}$ by residue produces

$\displaystyle\frac{-D_{0}}{16\nu_{0}\left(\sqrt{\nu_{0}+\nu_{\parallel
0}}+\sqrt{\nu_{0}+\nu_{\perp 0}}\right)^{2}}\int_{k>m}\frac{d^{d-1}{\bf
k_{\perp}}}{(2\pi)^{d-1}}\frac{1}{\left|k_{\perp}\right|^{3}}\cdot$
$\displaystyle\left[p^{2}_{\perp}+\left(p^{2}-\frac{p^{2}_{\perp}}{d-1}\right)\left(1+2\sqrt{\frac{\nu_{0}+\nu_{\parallel
0}}{\nu_{0}+\nu_{\perp 0}}}\right)\right].$ (27)

Reducing integral over momenta to the dimensionless scalar integral as follows

$\displaystyle\int_{k>m}\frac{d^{d}{\bf
k}}{k^{d+\epsilon}}=\frac{S_{d}}{\epsilon}\frac{1}{m^{\epsilon}},$ (28)

we arrive at the final result for (22):

$\displaystyle=$
$\displaystyle-\frac{1}{8\pi^{2}}\cdot\frac{w\nu^{2}}{2\epsilon\left(\sqrt{\nu+\nu_{\parallel}}+\sqrt{\nu+\nu_{\perp}}\right)^{2}}\cdot$
(29)
$\displaystyle\left[p^{2}_{\parallel}\left(1+2\sqrt{\frac{\nu+\nu_{\parallel}}{\nu+\nu_{\perp}}}\right)+\frac{p^{2}_{\perp}}{3}\left(5+4\sqrt{\frac{\nu+\nu_{\parallel}}{\nu+\nu_{\perp}}}\right)\right].$

Note that we put $d=4$ and substituted all quantities with the first order of
their decomposition into their renormalized counterparts (in case of MS
scheme, this implies that renormalization constants were substituted with a
unity).

Let us rewrite this expression using several new notations; firstly, let $w$
again stand for $w/8\pi^{2}$. Secondly, charges $x_{1}$, $x_{2}$ should be
used instead of ratios $\nu_{\parallel}/\nu$ and $\nu_{\perp}/\nu$. Finally,
let us introduce the functions $f_{1}(x_{1},x_{2})$ and $f_{2}(x_{1},x_{2})$:

$\displaystyle
f_{1}(x_{1},x_{2})=\frac{1}{2x_{1}\left(\sqrt{1+x_{1}}+\sqrt{1+x_{2}}\right)^{2}}\left(1+2\sqrt{\frac{1+x_{1}}{1+x_{2}}}\right);$
(30) $\displaystyle
f_{2}(x_{1},x_{2})=\frac{1}{6x_{2}\left(\sqrt{1+x_{1}}+\sqrt{1+x_{2}}\right)^{2}}\left(5+4\sqrt{\frac{1+x_{1}}{1+x_{2}}}\right).$
(31)

Then we arrive at the following expression:

$\displaystyle\vbox{\hbox{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a);
\vertex[right=0.8cm of a] (b); \vertex[right=of b] (c); \vertex[right=0.8cm of
c] (e); \par\diagram* { (a) -- [fermion] (b), (c) -- [fermion', half right]
(b), (b) -- [->, boson, half right, thick] (c), (c) -- [fermionZerowidth'']
(e), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=-\frac{w}{\epsilon}\,\left[\nu_{\parallel}p^{2}_{\parallel}f_{1}(x_{1},x_{2})+\nu_{\perp}p^{2}_{\perp}f_{2}(x_{1},x_{2})\right].$
(32)

Equations (17), (21) combined with Eq. (32) give us one-loop results for
renormalization constants $Z_{1}$ and $Z_{2}$:

$\displaystyle
Z_{1}=1-\frac{1}{\epsilon}\left[g\frac{3}{16}+w\,f_{1}(x_{1},x_{2})\right],$
(33) $\displaystyle Z_{2}=1-\frac{1}{\epsilon}w\,f_{2}(x_{1},x_{2}).$ (34)

From now on, we change the notation as $g/8\pi^{2}\rightarrow g$.

Finally, consider the one-loop approximation for the 1-irreducible Green’s
function $\langle h^{\prime}hh\rangle_{\rm 1-ir}$:

$\displaystyle\langle
h^{\prime}hh\rangle_{1-ir}=\raisebox{-4.30554pt}[0.0pt][0.0pt]{
\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(a1) ;
\vertex[below=0.4cm of a1] (a2); \vertex[below left=0.4cm of a2] (a5);
\vertex[below right=0.4cm of a2] (a6); \diagram{ (a1) --[fermionZerowidth''']
(a2), (a2) --[fermion] (a5), (a2) --[fermion](a6), };
\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(a1) ;
\vertex[below=0.4cm of a1] (a2); \vertex[below right= 1.5cm of a2] (a3);
\vertex[below left= 1.5cm of a2] (a4); \vertex[below left=0.4cm of a4] (a5);
\vertex[below right=0.4cm of a3] (a6); \diagram{ (a1) --[fermionZerowidth''']
(a2), (a2) --[fermion'] (a3), (a2) --[fermion'](a4), (a5) --[fermion] (a4),
(a3) --[fermion](a4), (a3) --[fermion](a6), }; \par
\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(a1) ;
\vertex[below=0.4cm of a1] (a2); \vertex[below right= 1.5cm of a2] (a3);
\vertex[below left= 1.5cm of a2] (a4); \vertex[below left=0.4cm of a4] (a5);
\vertex[below right=0.4cm of a3] (a6); \diagram{ (a1) --[fermionZerowidth''']
(a2), (a2) --[fermion] (a3), (a2) --[fermion'](a4), (a4) --[fermion] (a5),
(a4) --[fermion'](a3), (a3) --[fermion](a6), }; \par
\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{ }\par\feynman \vertex(a1) ;
\vertex[below=0.4cm of a1] (a2); \vertex[below right= 1.5cm of a2] (a3);
\vertex[below left= 1.5cm of a2] (a4); \vertex[below left=0.4cm of a4] (a5);
\vertex[below right=0.4cm of a3] (a6); \diagram{ (a1) --[fermionZerowidth''']
(a2), (a2) --[fermion] (a3), (a2) --[fermion'](a4), (a4) --[fermion] (a5),
(a4) --[fermion'](a3), (a3) --[fermion](a6), }; \par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$
$\displaystyle+\raisebox{-17.22217pt}[48.36958pt][0.0pt]{ \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(a1) ;
\vertex[below=0.4cm of a1] (a2); \vertex[below right= 1.5cm of a2] (a3);
\vertex[below left= 1.5cm of a2] (a4); \vertex[below left=0.4cm of a4] (a5);
\vertex[below right=0.4cm of a3] (a6); \diagram{ (a1) --[fermionZerowidth''']
(a2), (a2) --[fermion'] (a3), (a2) --[fermion'](a4), (a5) --[fermion] (a4),
(a3) --[Boson](a4), (a3) --[fermion](a6), }; \par
\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(a1) ;
\vertex[below=0.4cm of a1] (a2); \vertex[below right= 1.5cm of a2] (a3);
\vertex[below left= 1.5cm of a2] (a4); \vertex[below left=0.4cm of a4] (a5);
\vertex[below right=0.4cm of a3] (a6); \diagram{ (a1) --[fermionZerowidth''']
(a2), (a2) --[fermion'] (a3), (a2) --[Boson](a4), (a5) --[fermion] (a4), (a3)
--[fermion'](a4), (a3) --[fermion](a6), }; \par
\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{ }\par\feynman \vertex(a1) ;
\vertex[below=0.4cm of a1] (a2); \vertex[below right= 1.5cm of a2] (a3);
\vertex[below left= 1.5cm of a2] (a4); \vertex[below left=0.4cm of a4] (a5);
\vertex[below right=0.4cm of a3] (a6); \diagram{ (a1) --[fermionZerowidth''']
(a2), (a2) --[Boson] (a3), (a2) --[fermion'](a4), (a4) --[fermion] (a5), (a4)
--[fermion'](a3), (a3) --[fermion](a6), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(35)

The UV divergences in the first three diagrams cancel each other out due to
Galilean symmetry of the Hwa-Kardar equation (1): $h(t,{\bf x})\to h(t,{\bf
x}+u{\bf n}\,t)-u{\bf n}$, where $u=$const. If one considered possible
counterterms to the action functional that corresponds to the “pure” Hwa-
Kardar equation without turbulent mixing, this symmetry would forbid the
counterterm $h^{\prime}\partial_{\parallel}h^{2}$. In practice, this fact
translates to cancellation of the UV divergences.

The divergent part of the three remaining diagrams (the diagrams in the second
line of Eq. (35)) vanish due to incompressibility of velocity field ${\bf v}$.
Indeed, without loss of the generality the routing of the external momenta can
be chosen such that corresponding integrands would appear to be proportional
to $P_{ab}({\bf k})\,k_{b}=0$, where ${\bf k}$ is the momentum flowing through
a propagator $\langle{\bf v\,v}\rangle_{0}$, so each of the integrals would
vanish as a whole.

As a result, the renormalization constant $Z_{4}$ appears trivial in the one-
loop approximation: $Z_{4}=1$. It is tempting to assume that the symmetry and
the incompressibility would keep this relation exact in all orders of
perturbation theory. However, while the symmetry guarantees that all diagrams
without the velocity field ${\bf v}$ cancel each other in all orders, the same
cannot be said about the incompressibility. Thus, the found value of $Z_{4}$
cannot be claimed to be exact.

The one-loop results for renormalization constants are:

$\displaystyle
Z_{1}=1-\frac{1}{\epsilon}\left[g\frac{3}{16}+w\,f_{1}(x_{1},x_{2})\right],$
$\displaystyle Z_{2}=1-\frac{1}{\epsilon}w\,f_{2}(x_{1},x_{2}),$
$\displaystyle Z_{3}=1-\frac{1}{\epsilon}\frac{w}{8},$ $\displaystyle
Z_{4}=1,$ (36)

with the functions $f_{1,2}$ defined in (30) and (31) and the higher-order
corrections involving higher powers in $w$ and $g$.

##  6 RG functions and attractors of the RG equations

Let us define renormalization functions (anomalous dimensions $\gamma$ and
$\beta$ functions):

$\displaystyle\gamma_{Q}=\widetilde{\cal D}_{\mu}\ln Z_{Q},$
$\displaystyle\beta_{r}=\widetilde{\cal D}_{\mu}r.$ (37)

Here $Q$ is a given quantity with renormalization constant $Z_{Q}$ and $r$ is
a coupling constant. Differential operator $\widetilde{\cal D}_{\mu}$

$\widetilde{\cal
D}_{\mu}=\mu\partial_{\mu}|_{\\{g_{0},w_{0},x_{10},x_{20},\nu_{0}\\}}$ (38)

emerges from the relation $\widetilde{\cal D}_{\mu}F=0$ for a physical
quantity $F$ that encapsulates the fact that $F$ cannot depend on
renormalization mass $\mu$ (which is not an observable).

Anomalous dimensions for the model (9) are

$\displaystyle\gamma_{h}=-\gamma_{h^{\prime}}=\gamma_{4},\quad\gamma_{{\bf
v}}=\gamma_{{\bf v^{\prime}}}=0,$
$\displaystyle\gamma_{g}=-3\gamma_{1}/2-3\gamma_{2}/2+2\gamma_{4},\quad\gamma_{w}=-3\gamma_{3},\quad\gamma_{x_{1}}=\gamma_{1}-\gamma_{3},\quad\gamma_{x_{2}}=\gamma_{2}-\gamma_{3},$
$\displaystyle\gamma_{\nu_{\parallel}}=\gamma_{1},\quad\gamma_{\nu_{\perp}}=\gamma_{2},\quad\gamma_{\nu}=\gamma_{3},$
(39)

while one-loop approximation for $\gamma_{1}$–$\gamma_{4}$ is

$\displaystyle\gamma_{1}=g\frac{3}{16}+w\,f_{1}(x_{1},x_{2}),\quad\gamma_{2}=w\,f_{2}(x_{1},x_{2}),\quad\gamma_{3}=\frac{w}{8},\quad\gamma_{4}=0.$
(40)

The $\beta$ functions for the coupling constants $g$, $w$, $x_{1}$, and
$x_{2}$ read

$\displaystyle\beta_{g}=-g\left[\epsilon+\gamma_{g}\right],\quad\beta_{w}=-w\left[\epsilon+\gamma_{w}\right],\quad\beta_{x_{1}}=-x_{1}\gamma_{x_{1}},\quad\beta_{x_{2}}=-x_{2}\gamma_{x_{2}}.$
(41)

In the one-loop approximation, the $\beta$ functions have the forms:

$\displaystyle\beta_{g}=$
$\displaystyle-g\left[\epsilon-\frac{9}{32}g-\frac{3}{2}w\,\left(f_{1}(x_{1},x_{2})+f_{2}(x_{1},x_{2})\right)\right]$
(42) $\beta_{w}=-w\left[\epsilon-\frac{3}{8}w\right],$ (43)
$\beta_{x_{1}}=-x_{1}\left[\frac{3}{16}g-\frac{1}{8}w+w\,f_{1}(x_{1},x_{2})\right],$
(44) $\beta_{x_{2}}=-x_{2}\left[-\frac{1}{8}w+w\,f_{2}(x_{1},x_{2})\right].$
(45)

The functions (41) satisfy the exact identity

$\beta_{g}=-g\left[\frac{3}{2}\frac{\beta_{x_{1}}}{x_{1}}+\frac{3}{2}\frac{\beta_{x_{2}}}{x_{2}}-\frac{\beta_{w}}{w}+2\gamma_{4}\right]$
(46)

that follows from the definitions (37) and the relations (39).

In the one-loop approximation, one has $\gamma_{4}=0$, so that the relation
(46) becomes a linear dependence between the $\beta$ functions. As we will see
below, this makes the most interesting attractor of the RG equations to be a
curve of fixed points rather than a single point.

Universality classes of the IR behaviour are defined by the critical exponents
and the coordinates of the RG equation fixed points. The latter are provided
by the zeroes of the $\beta$ functions system: $\beta(g^{*})=0$; see, e.g. [4]
(Section 1.42). Furthermore, to correspond to IR range behavior, a fixed point
must be IR attractive, i.e. the eigenvalues $\lambda_{i}$ of matrix
$\Omega_{rn}=\\{\partial\beta_{g_{r}}/\partial g_{n}\\}|_{g^{*}}$ must have
strictly positive real parts; see, e.g. [4] (Section 1.42).

Taking into account that charges $x_{1}$, $x_{2}$ are positive and charge $g$
is non-negative, we find the following fixed points:

FP1 – a two-dimensional surface of Gaussian (trivial) fixed points:

$\displaystyle g^{*}=0;\quad w^{*}=0;\quad x_{1}^{*}\neq 0;\quad x_{2}^{*}\neq
0;$ $\displaystyle\lambda_{i}=\\{0,0,-\epsilon,-\epsilon\\}.$ (47)

FP2 – a curve of fixed points, parametrized by one of the coordinates (e.g.
$\\{g^{*}(x^{*}_{2}),\ w^{*},\ x^{*}_{1}(x^{*}_{2}),\ x^{*}_{2}\\}$) and
determined by the following expressions:

$\displaystyle
g^{*}=\frac{16\epsilon}{9}\left(1-8f_{1}(x_{1}^{*},x_{2}^{*})\right);\quad
w^{*}=\frac{8}{3}\epsilon;\quad f_{2}(x_{1}^{*},x_{2}^{*})=\frac{1}{8};$
$\displaystyle\lambda_{i}=\\{0,\epsilon,\lambda_{3},\lambda_{4}\\}.$ (48)

The curve is “semi-infinite” as two charges span finite segments of values
while charge $x_{1}$ runs through $\left(+\infty,(\sqrt{13}-1)/2\right]$ (see
Fig.1).

Figure 1: Curve of fixed points FP2 in three-dimensional space of coordinates
$\\{g^{*}(x^{*}_{2}),\ x^{*}_{1}(x^{*}_{2}),\ x^{*}_{2}\\}$ (we recall that
$w^{*}=8\epsilon/3$).

The eigenvalues $\lambda_{3}$ and $\lambda_{4}$ change along the curve FP2 and
can also be parametrized, see Fig.2. Both the eigenvalues are non-negative for
the relevant values of $x_{2}^{*}$ and positive values of $\epsilon$.

Figure 2: Eigenvalues $\lambda_{3}$ and $\lambda_{4}$ parameterized by
$x_{2}^{*}$.

The curve FP2 includes a special point where the HK nonlinearity becomes IR
irrelevant:

FP2a:

$\displaystyle g^{*}=0;\quad w^{*}=\frac{8}{3}\epsilon;\quad
x_{1}^{*}=x_{2}^{*}=\frac{1}{2}(\sqrt{13}-1);$
$\displaystyle\lambda_{i}=\left\\{0,\epsilon,\frac{47+\sqrt{13}}{162}\epsilon,\frac{13-\sqrt{13}}{18}\epsilon\right\\}.$
(49)

This endpoint corresponds to the marginal case where the nonlinearity of the
original HK equation becomes completely irrelevant making the dynamics
isotropic in the leading order of the IR asymptotic behaviour.

The existence of the curve of fixed points is a consequence of the exact
equation (46) which turns to a linear relation between the $\beta$ functions
in the one-loop approximation, where $\gamma_{4}=0$, see the last relation in
(40).

However, there are no obvious reasons for the relation $\gamma_{4}=0$ in (40)
to hold in all orders of perturbation theory. Thus, at the one-loop level, it
is impossible to determine whether we are dealing with a curve or a single
point “in disguise”, i.e. whether the curve FP2 would be reduced to a single
point, if one were to take into account all higher order corrections.

Nevertheless, it is indeed peculiar that the point FP2a lies on the curve FP2.
Fixed points similar to FP2a arise in most problems where stochastic system is
affected by environment motion; it is natural to expect appearance of such a
point. However, it now shares stability region with all the other points on
the curve FP2.

The entire curve is IR stable simultaneously (see Fig. 2) which means that
points with different coordinates and different values of eigenvalues are IR
attractive for the same values of parameter $\epsilon$. It is tempting to
explain away this odd occurrence as an artefact of one-loop approximation and
to expect the curve to shrink to its endpoint FP2a in higher order
approximations.

As we will see in the next Section, the critical dimensions of the fields are
the same for all points on the curve FP2, while the correction exponents,
determined by the eigenvalues, vary along the line. Regime of critical
behavior related to the point FP2a (in which the isotropic environment
dominates the dynamics) thus “extends” to the entire curve despite the fact
that other points on the curve has non-zero coordinate $g^{*}$, i.e. the non-
linearity of the Hwa-Kardar equation is relevant.

If the curve FP2 exists in all orders of perturbation theory, then coinciding
stability regions imply a loss of universality, i.e. that critical behavior of
the system depends on local parameters. Direct numerical simulations of the RG
flows333RG flow is defined by the system of ordinary differential equations
${\cal D}_{s}\overline{g}_{i}=\beta_{i}(\overline{g}_{j})$ for the invariant
(“running”) coupling constants $\overline{g}_{i}(s,g)$, where $g=\\{g_{i}\\}$
is the full set of coupling constants and $s=k/\mu$ features some momentum
variable $k$; see, e.g. Section 1.42 in [4]. (based on one-loop approximation
for $\beta$ functions) for positive $\epsilon$ show that each point on the
curve is indeed IR-attractive, see Fig. 3.

Figure 3: RG flows for positive $\epsilon$; arrows indicate movement towards
IR limit; (a) RG flow for $g^{*}=0$; the dot is the fixed point from FP2 with
$g^{*}=0$. (b) RG flow for $g^{*}=1,2$; the dot is the fixed point from FP2
with $g^{*}=1,2$.

We also considered marginal values of $x_{1}^{*}$ and $x_{2}^{*}$, where
$x_{1}^{*},x_{2}^{*}=0$ and $x_{1}^{*},x_{2}^{*}\rightarrow\infty$. To do that
one should pass to another set of charges, (e.g. $g$, $w$,
$\alpha_{1}=x_{1}^{-1}$, $\alpha_{2}=x_{2}^{-1}$), and investigate possible
fixed points. However, all of these points turned out to have nearly empty
stability regions defined, e.g. by contradictory inequalities like
$\epsilon>0$ and $\epsilon<0$. For example, the fixed point related to the
regime where only nonlinearity of HK equation is relevant with coordinates
$g^{*}=32\epsilon/9$, $w^{*}=0$, $\alpha_{1}^{*}=0$, arbitrary
$\alpha_{2}^{*}$, has eigenvalues $\\{0,-\epsilon,2\epsilon/3,\epsilon\\}.$
Such “saddle” points can be useful for general analysis of RG flows but they
do not correspond to possible critical behavior in the same way IR attractive
fixed points do. Thus, we do not investigate them further.

##  7 Critical dimensions

Critical behaviour of the Green’s functions is described by critical
dimensions (related to critical exponents) or powers in the scaling law in the
IR range. Critical dimensions are defined for IR relevant parameters, i.e. for
the parameters that are dilated in critical scaling as opposed to IR
irrelevant quantities (in our case, that includes renormalization mass $\mu$
and all four coupling constants) that stay fixed. Expression

$\Delta_{F}=d^{k}_{F}+\Delta_{\omega}d^{\omega}_{F}+\gamma_{F}^{*},$ (50)

gives a critical dimension $\Delta_{F}$ of a field $F$ (see, e.g. Section 2.1
in [48], Section 3.1 in [49] and Sections 5.16 and 6.7 in [4]). Here
$\gamma_{F}^{*}$ is anomalous dimension of the field taken at some IR
attractive fixed point. Frequency critical dimension $\Delta_{\omega}$ is
given by the expression

$\Delta_{\omega}=2-\gamma_{\nu}^{*}.$ (51)

For the fixed points FP1, all the critical dimensions are found exactly:

$\Delta_{h}=\Delta_{{\bf v}}=1,\quad\Delta_{h^{\prime}}=\Delta_{{\bf
v^{\prime}}}=d-1,\quad\Delta_{\omega}=2.$ (52)

For the curve of fixed points FP2 (including the endpoint FP2a), the critical
dimensions

$\Delta_{h}=\Delta_{{\bf
v}}=1-\epsilon/3,\quad\Delta_{h^{\prime}}=\Delta_{{\bf
v^{\prime}}}=d-1+\epsilon/3,\quad\Delta_{\omega}=2-\epsilon/3$ (53)

are also known perturbatively exactly in the sense that they have no higher-
order corrections in $\varepsilon$ (see discussion in the next section). As it
should be, the dimensions $\Delta_{{\bf v}}$, $\Delta_{{\bf v^{\prime}}}$ and
$\Delta_{\omega}$ for FP2a coincide with those derived in [43].

The fixed point $g^{*}=32\epsilon/9$, $w^{*}=0$, $\alpha_{1}^{*}=0$, arbitrary
$\alpha_{2}^{*}$ mentioned in Sec. 6 allows for a kind of dimensional
transmutation, in the sense that the ratio of the diffusivity coefficients
$\nu_{\parallel}$ and $\nu_{\perp}$ acquires nontrivial canonical and critical
dimensions. A new canonical symmetry arises that allows one to introduce two
independent momentum canonical dimensions $d^{\parallel}$ and $d^{\perp}$ for
the two subspaces. The scaling behavior corresponding to that fixed point is
the one where the coordinates $x_{\parallel}$ and $x_{\perp}$ are diluted in
the different way resulting in nontrivial critical dimension
$\Delta_{\parallel}$. However, in our model, that fixed point is never IR
attractive and this interesting scaling regime cannot realize for general
initial conditions. A thorough discussion of this dimensional transmutation
can be found in [44, 45].

##  8 Conclusion

We studied a semi-phenomenological, strongly anisotropic continuous model of a
system with self-organized criticality (1)–(2) proposed in [41] and subjected
to isotropic randomly moving environment modelled by the Navier-Stokes
stochastic differential equation for an incompressible viscous fluid (3)–(6)
inspired by [43].

We constructed the corresponding field theoretic model (9) and established its
renormalizability. The renormalization constants, the RG functions and the
coordinates of the IR attractors were calculated in the leading one-loop
approximation. The RG analysis showed that the IR behaviour of the model is
governed by the IR attractor, referred to as FP2 in Section 6, that is a curve
of fixed points rather than a single point. Moreover, the critical dimensions
of the basic quantities (the fields and the frequency) remain the same along
the curve. Thus, it is natural and tempting to interpret the whole curve as a
single universality class.

In this respect, our situation differs from those where the attractors of the
RG flows were given by lines [51, 52, 53] or surfaces [54]-[59] with the
critical exponents continuously varying along the attractive parts of the
manifolds. In our case, only the correction exponents (the eigenvalues of the
stability matrix $\Omega$) vary along the curve.

It should be stressed that the appearance of the curve of fixed points is a
consequence of the linear relation between the $\beta$ functions that results
from (46) for $\gamma_{4}=0$. The latter expression follows form the one-loop
result $Z_{4}=1$ in (36), which, in its turn, reflects the absence of a
counterterm $h^{\prime}\partial_{\parallel}h^{2}$. However, this counterterm
is not forbidden by the real index of divergence nor some evident symmetry, so
we can see no clear reason to expect that the relations $Z_{4}=1$ and
$\gamma_{4}=0$ are in fact exact and hold in all orders of perturbation
theory.

The two-loop calculations are expected to be rather difficult for our
complicated model. However, even without such a calculation it is clear that
there are two main possibilities.

First option: for some reasons, like a certain obscure hidden symmetry, the
last relation $Z_{4}=1$ in (36) and its consequence $\gamma_{4}=0$ in (40) are
exact identities. Then the IR attractor is indeed a curve. The critical
dimensions of the fields $h^{\prime}$ and $h$ remain the same along the curve
and are given exactly by (53). The exactness of those expressions follow from
(50) and (51). Indeed, the anomalous dimensions of the fields $h$ and
$h^{\prime}$ are equal (up to the sign) with the vanishing anomalous dimension
$\gamma_{4}$. From the second equality in (41) it follows that
$\gamma_{w}^{*}=-\varepsilon$ along the curve, while the exact relation
$\gamma_{\nu}=-\gamma_{w}/3$ delivers the exact value of the
$\gamma_{\nu}^{*}=\varepsilon/3$ for the whole curve FP2. As a result, in the
option consireded, only the correction exponents (eigenvalues of the matrix
$\Omega$) which are, in principle, observable quantities, vary along the curve
of fixed points.

Second option: the relation $\gamma_{4}=0$ in (40) is an artefact of the one-
loop approximation and is violated by the higher-order corrections. Then the
IR attractive curve will shrink to a single point satisfying the equation
$\gamma_{4}^{*}=0$ following from (46). From physics reasons, it is clear that
this point corresponds to the regime when the HK non-linearity becomes
irrelevant and the expressions (53) for the critical dimensions are in fact
exact. This is the point referred to as FP2a in Section 6.

In the future, it would be interesting to consider other types of the velocity
ensemble, especially the power-like correlation function of the stirring force
that describes turbulent motion [48, 49] or its combination with the simple
stirring (7) due to [43]. It is also desirable to go beyond the leading one-
loop approximation. This work is already in progress.

## Acknowledgments

The authors are indebted to Abhik Basu for valuable discussion and for
bringing our attention to the paper [53]. One of the authors (N.V.A.) is
indebted to M. I. Vyazovsky for enlightening discussions.

The work was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS” (P. I. K., project number 22-1-3-33-1) and by
Ministry of Science and Higher Education of the Russian Federation, agreement
№075–15–2022–287 (P. I. K. and A. Yu. L.).

## References

## References

  * [1] Bak P 1999 How Nature Works: the Science of Self-Organized Criticality (Copernicus)
  * [2] Amit D J 1984 Field Theory, Renormalization Group, and Critical Phenomena 2nd edition (World Scientific, Singapore)
  * [3] Zinn-Justin J 1989 Quantum Field Theory and Critical Phenomena (Clarendon Press: Oxford)
  * [4] Vasiliev A N 2004 The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Chapman & Hall/CRC, Boca Raton)
  * [5] Hinrichsen H 2000 Non-Equilibrium Critical Phenomena and Phase Transitions into Absorbing States Advances in Physics 49 815-958
  * [6] Henkel M, Hinrichsen H, Lübeck S and Pleimling M 2008 Non-Equilibrium Phase Transitions vol 1 (Springer, Dordrecht)
  * [7] Bak P, Tang C and Wiesenfeld K 1987 Self-Organized Criticality: An Explanation of the $1/f$ noise Phys. Rev. Lett. 59 381
  * [8] Tang C and Bak P 1988 Critical exponents and scaling relations for self-organized critical phenomena Phys. Rev. Lett. 60 2347
  * [9] Bak P and Sneppen K 1993 Punctuated equilibrium and criticality in a simple model of evolution Phys. Rev. Lett. 71 4083
  * [10] Jensen H J 1998 Self-Organized Criticality: Emergent Complex behavior in Physical and Biological Systems (Cambridge University Press, Cambridge)
  * [11] Turcotte D L 1999 Self-organized criticality Rep. Prog. Phys. 62 1377
  * [12] Pruessner G 2012 Self-Organized Criticality: Theory, Models and Characterisation (Cambridge University Press)
  * [13] Watkins N W, Pruessner G, Chapman S C, Crosby N B and Jensen H J 2016 25 years of self-organized criticality: Concepts and controversies Space Sci. Rev. 198, 3
  * [14] Muñoz M A 2018 Colloquium: Criticality and dynamical scaling in living systems Rev. Mod. Phys. 90 031001
  * [15] Markovic D and Gros C 2014 Power laws and self-organized criticality in theory and nature Phys. Rep. 536 41
  * [16] Aschwanden M J 2013 Self-Organized Criticality Systems (Open Academic Press, Berlin, Warsaw)
  * [17] Ellis G F R and Kopel J 2019 The dynamical emergence of biology from physics: branching causation via biomolecules Front. Physiol. 9 1966
  * [18] Mora T and Bialek W 2011 Are biological systems poised at criticality? J. Stat. Phys. 144 268
  * [19] Hesse J and Gross T 2014 Self-organized criticality as a fundamental property of neural systems Front. Syst. Neurosci. 8 166
  * [20] Pasquale V, Massobrio P, Bologna L L, Chiappalone M and Martinoia S 2008 Self-organization and neuronal avalanches in networks of dissociated cortical neurons Neuroscience 153 1354
  * [21] Orlandi J G, Soriano J, Alvarez-Lacalle E, Teller S and Casademunt J 2013 Noise focusing and the emergence of coherent activity in neuronal cultures Nat. Phys. 9 582
  * [22] Timme N M, Marshall N J, Bennett N, Ripp M, Lautzenhiser E and Beggs J M 2016 Criticality maximizes complexity in neural tissue Front. Physiol. 7 425
  * [23] Kossio F Y K, Goedeke S, van den Akker B, Ibarz B, Memmesheimer R-M 2018 Growing Critical: Self-Organized Criticality in a Developing Neural System Phys. Rev. Lett. 121 058301
  * [24] Levina A, Herrmann J M and Geisel T 2007 Dynamical synapses causing self-organized criticality in neural networks Nat. Phys. 3 857
  * [25] Tadić B, Mitrović Dankulov M and Melnik R 2017 Self-organised criticality and emergent hyperbolic networks: blueprint for complexity in social dynamics Phys. Rev. E 96 032307
  * [26] Tadić B 2019 Self-organised criticality and emergent hyperbolic networks: blueprint for complexity in social dynamics European Journal of Physics 40 024002
  * [27] Tadić B, Gligorijevic V, Mitrovicć M and Suvakov M 2013 Co-evolutionary mechanisms of emotional bursts in online social dynamics and networks Entropy 15 5084
  * [28] Suvakov M and Tadić B 2014 Collective emotion dynamics in chats with agents, moderators and Bots Condens. Matter Phys. 17 33801
  * [29] Holovatch Yu, Mrygold O, Szell M and Thurner S 2017 Math Meets Myths: Quantitative Approaches to Ancient Narratives ed R Kenna (Springer Int. Publishing)
  * [30] Kou G, Zhao Y, Peng Y and Shi Y 2012 Multi-level opinion dynamics under bounded confidence PLoS One 7 e43507
  * [31] Torres-Rojo J M and Bahena-González R 2018 Scale invariant behavior of cropping area losses Agricultural Systems 165 33
  * [32] Tonello L, Giacobbi L, Pettenon A, Scuotto A, Cocchi M, Gabrielli F and Cappello G 2018 Crisis Behavior in Autism Spectrum Disorders: A Self-Organized Criticality Approach Complexity 5128157
  * [33] Onuki A and Kawasaki K 1980 Critical phenomena of classical fluids under flow. I: Mean field approximation Progr. Theor. Phys. 63 122
  * [34] Aronowitz A and Nelson D R 1984 Turbulence in phase-separating binary mixtures Phys. Rev. A 29 2012
  * [35] Satten G and Ronis D 1986 Critical phenomena in randomly stirred fluids: Correlation functions, equation of motion, and crossover behavior Phys. Rev. A 33 3415
  * [36] Nandy M K and Bhattacharjee J K 1998 Renormalization-group analysis for the infrared properties of a randomly stirred binary fluid J. Phys. A Math. Gen. 31 2621
  * [37] Antonov N V, Hnatich M and Honkonen J 2006 Effects of mixing and stirring on the critical behavior J. Phys. A: Math. Gen. 39 7867
  * [38] Benzi R, Parisi G, Sutera A and Vulpiani A 1983 A theory of stochastic resonance in climate changes SIAM J. Appl. Math. 43 565
  * [39] Feigenbaum M J, Procaccia I and Davidovich B 2001 Dynamics of Finger Formation in Laplacian Growth Without Surface Tension J. Stat. Phys. 103 973–1007
  * [40] Giorgio Parisi–Facts–2021. NobelPrize.org. Nobel Prize Outreach AB 2021. Mon. 11 Oct. 2021. Available online: https://www.nobelprize.org/prizes/physics/2021/parisi/facts/ (accessed on 24 November 2022 ).
  * [41] Hwa T and Kardar M 1989 Dissipative transport in open systems: An investigation of self-organized criticality Phys. Rev. Lett. 62(16) 1813
  * [42] Hwa T and Kardar M 1992 Avalanches, hydrodynamics and great events in models of sandpiles Phys. Rev. A 45 7002
  * [43] Forster D, Nelson D R and Stephen M J 1977 Large-distance and long-time properties of a randomly stirred fluid Phys. Rev. A 16 732
  * [44] Antonov N V, Gulitskiy N M, Kakin P I and Kochnev G E 2020 Effects of turbulent environment on self-organized critical behavior: isotropy vs. Anisotropy Universe 6 145
  * [45] Antonov N V, Gulitskiy N M, Kakin P I and Semeikin M N 2022 Dimensional transmutation and nonconventional scaling behaviour in a model of self-organized criticality
  * [46] Falkovich G, Gawȩdzki K and Vergassola M 2001 Particles and fields in fluid turbulence Rev. Mod. Phys. 73 913
  * [47] Antonov N V, Kakin P I and Lebedev N M 2019 The Kardar–Parisi–Zhang model of a random kinetic growth: effects of a randomly moving medium J. Phys. A: Math. Theor. 52 505002
  * [48] Adzhemyan L T, Antonov N V and Vasil’ev A N 1999 The Field Theoretic Renormalization Group in Fully Developed Turbulence (Gordon & Breach: London, UK)
  * [49] Adzhemyan L T, Antonov N V and Vasil’ev A N 1996 Quantum field renormalization group in the theory of fully developed turbulence Phys.-Usp. 39 1193 (Translated from the Russian: 1996 Usp. Fiz. Nauk 166 1257)
  * [50] Antonov N V, Gulitskiy N M, Kakin P I and Serov V D 2021 Effects of turbulent environment and random noise on self-organized critical behavior: Universality versus nonuniversality Phys. Rev. E 103 042106
  * [51] Antonov N V and Kapustin A S 2010 Effects of turbulent mixing on critical behaviour in the presence of compressibility: Renormalization group analysis of two models J. Phys. A: Math. Theor. 43 405001
  * [52] Antonov N V and Kapustin A S 2012 Critical behaviour of the randomly stirred dynamical Potts model: Novel universality class and effects of compressibility J. Phys. A: Math. Theor. 45 505001
  * [53] Haldar A and Basu A 2022 Disorders can induce continuously varying universal scaling in driven systems Phys. Rev. E 105 034104
  * [54] Kohmoto M, den Nijs M and Kadanoff L P 1981 Hamiltonian studies of the $d=2$ Ashkin-Teller model Phys. Rev. B 24 5229
  * [55] Antonov N V and Vasil’ev A N 1995 The quantum-field renormalization group in the problem of a growing phase boundary JETP 81 485 (Translated from the Russian: ZhETF 108 885)
  * [56] Antonov N V 1997 The renormalization group in the problem of turbulent convection of a passive scalar impurity with nonlinear diffusion JETP 85 898–906 (Translated from the Russian: 1997 ZhETF 112 1649)
  * [57] Antonov N V and Kakin P I 2017 Scaling in erosion of landscapes: Renormalization group analysis of a model with infinitely many couplings Theor. Math. Phys. 190(2) 193-203
  * [58] Antonov N V and Kakin P I 2017 Scaling in erosion of landscapes: Renormalization group analysis of a model with turbulent mixing J. Phys. A: Math. Theor. 50 085002
  * [59] Antonov N V, Babakin A A and Kakin P I 2022 Strongly Nonlinear Diffusion in Turbulent Environment: A Problem with Infinitely Many Couplings Universe 8 121