# Scaling or multiscaling: Varieties of universality in a driven nonlinear
model

Sudip Mukherjee<EMAIL_ADDRESS><EMAIL_ADDRESS>Barasat
Government College, 10, KNC Road, Gupta Colony, Barasat, Kolkata 700124, West
Bengal, India  Condensed Matter Physics Division, Saha Institute of Nuclear
Physics, Calcutta 700064, West Bengal, India  Abhik Basu<EMAIL_ADDRESS><EMAIL_ADDRESS>Condensed Matter Physics Division, Saha Institute of
Nuclear Physics, Calcutta 700064, West Bengal, India

###### Abstract

Physical understanding of how the interplay between symmetries and nonlinear
effects can control the scaling and multiscaling properties in a coupled
driven system, such as magnetohydrodynamic turbulence or turbulent binary
fluid mixtures, remains elusive till the date. To address this generic issue,
we construct a conceptual nonlinear hydrodynamic model, parametrised jointly
by the nonlinear coefficients, and the spatial scaling of the variances of the
advecting stochastic velocity and the stochastic additive driving force,
respectively. By using a perturbative one-loop dynamic renormalisation group
method, we calculate the multiscaling exponents of the suitably defined equal-
time structure functions of the dynamical variable. We show that depending
upon the control parameters the model can display a variety of universal
scaling behaviours ranging from simple scaling to multiscaling.

##  I Introduction

In contrast to the conventional driven diffusive models ddlg , the
universality of the velocity fluctuations in the nonequilibrium steady state
(NESS) of forced three-dimensional ($3d$) homogeneous and isotropic fully
developed fluid turbulence are characterised by the multiscaling of equal-time
structure functions ${\mathcal{S}}_{n}^{v}(r)$ of the longitudinal component
of the velocity increments $\Delta v(r)=\hat{\bf r}\cdot[{\bf v}({\bf
x+r},t)-{\bf v}({\bf x},t)]$, where $\hat{\bf r}$ is the unit vector along
$\bf r$, the separation vector between the two space points:
${\mathcal{S}}_{n}^{v}(r)$ is defined as

${\mathcal{S}}_{n}^{v}(r)=\langle|\Delta v({\bf r})|^{n}\rangle,$ (1)

for all positive integer $n$; the separation $r$ belongs to the inertial
range: $L\gg r\gg\eta_{d}$, where $L$ and $\eta_{d}$, respectively, are the
integral scale or the forcing scale ($\sim$ system size) and small scale
dissipation scale, that depends on the fluid viscosity. Dimensional arguments
due to Kolmogorov (K41) k41  predicted that ${\mathcal{S}}_{n}^{v}(r)$ in
homogeneous and isotropic fully developed turbulence should be independent of
both $L$ and $\eta_{d}$ in the inertial regime $L\gg r\gg\eta_{d}$ and display
universal scaling, ${\mathcal{S}}_{n}^{v}(r)\sim
r^{\zeta_{n}^{v}},\,\zeta_{n}^{v}=n/3$ is positive, and has a linear
dependence on $n$, known as simple scaling. Later on, detailed experimental
and numerical studied have clearly shown the existence of corrections
nonlinear in $n$ to these scaling, making $\zeta_{n}^{v}$ depend nonlinearly
upon $n$. This is the essence of multiscaling frisch ; rahulrev . Closed
analytical forms for these nonlinear functions $\zeta_{n}^{v}$ are still
unknown. In spite of the extensive experimental and numerical results, a self-
consistent microscopic theory for multiscaling starting from the forced
Navier-Stokes equation is still lacking. Renormalisation group (RG) methods
designed to extract universal scaling exponents in critical phenomena and
critical dynamics zinn  as well as in nonequilibrium driven diffusive systems
ddlg , are so far less successful in calculating the universal multiscaling in
fully developed fluid turbulence; see Ref. rev1  for detailed expositions on
RG approaches to fluid turbulence.

Difficulties in the analytical progress of fluid turbulence studies have
inspired scientists to construct and study simpler models, amenable to
analytical methods, that might display qualitatively similar universal scaling
behaviour. One such well-known attempt is the studies of multiscaling in the
passive scalar turbulence model. In this model, a scalar $\theta$ (e.g., a
concentration field) is passively advected by a velocity field $\bf v$ without
affecting the dynamics of the flow field itself, i.e., the fluid motion
remains autonomous, independent of the passively advected $\theta$. The
generic form of the equation of motion of $\theta$ reads

$\frac{\partial\theta}{\partial t}+{\bf
v}\cdot{\bm{\nabla}}\theta=\nu_{\theta}\nabla^{2}\theta+f_{\theta},$ (2)

where, $\nu_{\theta}$ is a diffusivity and $f_{\theta}$ is an additive noise.
In the well-known Kraichnan model for passive scalar turbulence obu ; kraich ,
$\bf v$ is chosen to be zero-mean Gaussian-distributed with a variance that is
spatially long-ranged but temporally $\delta$-correlated, instead of being
solutions of the forced Navier-Stokes equation. This effectively reduces the
problem to a theory linear in $\theta$ for a given velocity, which in turn
makes the problem analytically tractable. Extensive studies on the passive
scalar turbulence adjhem ; kupi ; pagani  show universal scaling behaviour by
the appropriate structure functions of $\theta$:

${\mathcal{S}}_{2n}(r)\equiv\langle[\theta({\bf x+r})-\theta({\bf
x})]^{2n}\rangle\sim r^{\zeta_{n}}$ (3)

for positive integers $n$, where the (equal) time labels of $\theta$ are
suppressed; $r=|{\bf r}|$ in the inertial range. The scaling exponents
$\zeta_{2n}$ are found to be nonlinear functions of order $n$, similar to the
multiscaling in fluid turbulence. Unsurprisingly, the odd order structure
functions vanish identically, since the Kraichnan model depends linearly on
$\theta$. This remains true even when the passive scalar is advected by
velocity fields obtained from the isotropically Navier-Stokes equation,
instead of a Gaussian-distributed velocity sreeni1 .

In the Kraichnan model, the variance of the external (additive) stochastic
force is nonvanishing and concentrated only at the largest scales. The zero-
mean, incompressible, Gaussian-distributed velocity field ${\bf v}({\bf r},t)$
has a variance given by kupi ; adjhem

$\langle v_{i}({\bf q},t)v_{j}({\bf-q},0)\rangle=\frac{\tilde{D}P_{ij}({\bf
q})\delta(t)}{(q^{2}+1/L^{2})^{\tilde{\epsilon}}};$ (4)

see also Ref. others . Here, $\tilde{D}$ is a constant, $\bf q$ is a Fourier
wavevector and $P_{ij}=\delta_{ij}-q_{i}q_{j}/q^{2}$ is the transverse
projection operator. In the lowest order perturbative expansion in
$\tilde{\epsilon}$, the multiscaling exponents are given by

$\zeta_{n}=2n-\frac{n\tilde{\epsilon}(d+2n)}{d+2},$ (5)

where $d$ is the space dimension.

A major difference between homogeneous and isotropic fluid turbulence and the
passive scalar turbulence problem is the linearity of the latter problem in
$\theta$. Since the fully developed turbulence is a state where the advective
nonlinear term is relevant (in a RG sense), such that the Reynolds number
approaches infinity, it is generally believed that nonlinearity plays a
crucial role in the ensuing multiscaling. Due to the inherent linearity of the
problem, studies on the passive scalar turbulence models cannot shed much
light in this regard. In the absence of any general theoretical framework and
due to the considerable theoretical and technical challenges to study and
extract the scaling behaviour in fully developed turbulence in a systematic
manner starting from the forced Navier-Stokes equation, it is instructive to
construct simple driven models, which may be amenable to analytical
treatments. With this viewpoint, in the present work, we revisit the problem
of scaling and multiscaling in a simple conceptual nonlinear model. We
construct this model by generalising the well-known Kraichnan model for
passively advected substance concentration obu ; kraich  for a conserved
three-component spin vector field $\phi_{\alpha},\,\alpha=1,2,3$, driven by a
stochastic advecting velocity $\bf v$ and an additive force $f_{i}$. We
introduce nonlinear effects, by which we mean nonlinear interactions between
the components of $\phi_{\alpha}$, and their effects on the dynamics of
$\phi_{\alpha}$. The nonlinear effects are modeled by certain mode-coupling
terms of the form well-known in the dynamics of classical Heisenberg magnets
chaikin . The dynamics of $\phi_{\alpha}$ is controlled jointly by $y$ and
$\overline{y}$, spatial scaling exponents of the variances, respectively, of
$\bf v$ and $f_{i}$. By using Wilson momentum shell dynamic renormalisation
group (DRG) and within a one-loop perturbation approximation, we elucidate
scaling and multiscaling in these models. We show that the new mode coupling
term breaks the symmetry under a constant shift of $\phi_{\alpha}({\bf x},t)$,
present in the original passive scalar turbulence model. We argue that as a
result the appropriate structure functions that are invariant under same
transformations as the underlying equation of motion and can characterise the
universal scaling or multiscaling are of the form

${\cal\tilde{S}}_{2n}(r)\equiv\langle\left[\phi_{\alpha}({\bf
x+r})\phi_{\alpha}({\bf x})\right]^{n}\rangle\sim r^{\tilde{\zeta}_{n}}$ (6)

in the parameter regime where the coupling constant of the mode coupling term
is relevant (in a RG sense) that in turn is controlled by $y$ and
$\overline{y}$. By using a dynamic RG framework, we show that the relevant
structure functions ${\cal\tilde{S}}_{2n}(r)$ display only simple scaling but
no multiscaling. In the phase space region, where the mode coupling term is
irrelevant, the model effectively becomes linear in the long wavelength limit
and essentially reduces to the Kraichnan passive scalar model. This reveals
sensitive dependence of multiscaling properties (or, the lack of it) on the
nonlinear effects, a key qualitative outcome of the present study.

We calculate the multiscaling exponents $\tilde{\zeta}_{2n}$ that depend
linearly (in case of simple scaling) or nonlinearly (in case of multiscaling)
on $n$ and are parametrised by $y$ and $\overline{y}$. We also show that in
the inertial range, ${\cal\tilde{S}}_{2n}(r)$ explicitly depends on $L$ that
ultimately leads to multiscaling (or lack thereof). We establish the crucial
role played by both the advecting velocity and the additive noise in the
dynamical equation for $\phi_{\alpha}$. We, in particular, show how the
spatial scaling of the variance of the additive stochastic force affects
$\zeta_{2n}$. Our calculational framework directly extends the standard DRG
calculations for scaling in driven diffusive models ddlg . The rest of this
article is organised as follows: In Sec. II, for our purposes we introduce two
similar models, Model I and Model II that differ in the variance of the
Gaussian distributed $\bf v$. We define the structure functions, that we use
to characterise the multiscaling, in Sec. III. Next, in Sec. IV, we calculate
the exact scaling behaviour in the linear limit of the model. Then in Sec.
V.1, we perform the renormalisation group analysis for the relevant model
parameters of Model I. In Sec. V.2.2, we analyse the scaling in the linear
model from the perspective of OPE. Next, in Sec. V.2 we calculate the scaling
and multiscaling exponents in Model I. We demonstrate that the nonlinear mode
coupling terms can make the multiscaling observed without these nonlinearities
(i.e., in the usual passive scalar model) disappear; instead, simple scaling
of a particular type is predicted to be displayed. In Sec. VI we summarise and
conclude. We provide some technical details and briefly discuss Model II in
Appendix for the interested reader.

##  II Nonlinear hydrodynamic model

Let $\phi_{\alpha}({\bf x},t)$ be a three-component “spin-vector” that is
being advected by a velocity field $v_{i}({\bf x},t)$. Here, $\alpha=1,2,3$
and $i=1,..,d$, where $d$ is the dimensionality of the physical space. The
spins span an order parameter space, distinct from the physical coordinate
space; spin index $\alpha$ have no relation with the Cartesian component index
$i$. For notational consistency, we reserve Greek letters to denote the spin
components; Roman letters are used to denote the Cartesian components. The
equation of motion for $\phi_{\alpha}({\bf x},t)$ reads

$\frac{\partial\phi_{\alpha}({\bf x},t)}{\partial
t}+g\epsilon_{\alpha\beta\gamma}\phi_{\beta}\nabla^{2}\phi_{\gamma}+\lambda{\bf
v}\cdot{\bm{\nabla}}\phi_{\alpha}=\nu\nabla^{2}\phi_{\alpha}+\overline{\eta}_{\alpha}.$
(7)

Here, $\epsilon_{\alpha\beta\gamma}$ is totally antisymmetric three-
dimensional tensor, $g$ and $\lambda$ are two coupling constants and can be of
either sign, and $\nu$ is a diffusion constant nu-comm . Evidently, through
the nonlinear mode-coupling term
$g\epsilon_{\alpha\beta\gamma}\phi_{\beta}\nabla^{2}\phi_{\gamma}$ different
components of $\phi_{\alpha}$ interact. Notice that the term
$\epsilon_{\alpha\beta\gamma}\phi_{\beta}\nabla^{2}\phi_{\gamma}=\epsilon_{\alpha\beta\gamma}{\bm{\nabla}}\cdot(\phi_{\beta}{\bm{\nabla}}\phi_{\gamma})$,
since $({\bm{\nabla}}{\bm{\phi}})\times({\bm{\nabla}}{\bm{\phi}})=0$
identically. Stochastic function $\overline{\eta}_{\alpha}$ is assumed to be
Gaussian distributed with zero mean and a variance given by (in the Fourier
space)

$\langle\overline{\eta}_{\alpha}({\bf q},\omega)\overline{\eta}_{\beta}(-{\bf
q},-\omega)\rangle=2\overline{D}|q|^{-\overline{y}},$ (8)

where $\overline{D}>0$ fixes the amplitude; ${\bf q}$ and $\omega$ are Fourier
wavevector and frequency, respectively. We are generally interested in
$\overline{y}>0$, corresponding to a spatially long-ranged noise in (7). When
$\lambda=0$, Eq. (7) reduces to that of the well-known Model E equation for
the classical Heisenberg model in its paramagnetic phase (i.e., above the
transition temperature) halpin ; u-term . The $\lambda{\bf
v}\cdot{\bm{\nabla}}\phi_{\alpha}$ describes advection of the spins by the
velocity $v_{i}$.

For a realistic 3D system, the incompressible velocity $v_{i}$ solves the
generalised Navier-Stokes equation that includes the stresses coming from the
spin fluctuations. Such velocity field itself, when strongly forced externally
at large spatial scales, displays nontrivial multiscaling or intermittency.
There are driven coupled system, where such an intermittent flow field advects
another field like the magnetic field in 3D MHD turbulence, or the
concentration field in a turbulent binary mixture. Any theoretical analysis of
the scaling or multiscaling of the second field is greatly complicated by the
multiscaling of the advecting velocity field. In the spirit of the Kraichnan
passive scalar model and in order to keep the ensuing calculation simple and
analytically tractable, instead of solving the velocity field from the Navier-
Stokes equation, it is assumed as an “input” as a zero-mean Gaussian
distributed field with a given variance that is spatially long-ranged. Thus
the velocity field in this description is autonomous, and has a variance
$D^{v}_{ij}({\bf q},\omega)$ in the Fourier space:

$\langle v_{i}({\bf q},\omega)v_{j}({\bf-q},-\omega)\rangle=D_{ij}^{v}({\bf
q},\omega).$ (9)

Equivalently, in the time domain

$\langle v_{i}({\bf q},t)v_{j}({\bf-q},0)\rangle=D_{ij}^{v}({\bf q},t).$ (10)

In particular, we assume

$D_{ij}^{v}({\bf q},t)=AP_{ij}({\bf q})q^{-y}{\rm exp}(-\tilde{\Gamma}(q)t),$
(11)

where, $A>0$ is a constant that sets the amplitude of the variance
$D_{ij}^{v}$, and the exponent $y>0$; $P_{ij}({\bf
q})\equiv\delta_{ij}-q_{i}q_{j}/q^{2}$ is the transverse projection operator.
Parameter $\tilde{\Gamma}(q)>0$ controls the temporal decay of the time-
dependent velocity correlator and parametrises (11). We consider two specific
choices for $\tilde{\Gamma}(q)$ tirtha :

(i) Model I: $\tilde{\Gamma}(q)=\Gamma$, a constant (independent of $q$)
together with $\Gamma\rightarrow\infty$, along with the condition $A$ scales
with $\Gamma$, i.e., $A/\Gamma=D_{1}>0$ a const.: relaxation of the velocity
modes are independent of wavevector $\bf q$. In that limit, (11) reduces to
being temporally $\delta$-correlated:

$D_{ij}^{v}({\bf q},t)=D_{1}P_{ij}({\bf q})q^{-y}\delta(t),$ (12)

Such a flow can exist when there is a (large) friction (e.g., through a porous
medium) and a large external stirring force applied on the fluid, so that the
friction and the force may balance and produce a nonequilibrium steady state.
This in turn produces a flow field correlated as in (12). The flow is self-
similar, which is evident from (12); it is however, not intermittent, since it
is Gaussian-distributed, in contrast to turbulent velocity fields that satisfy
the forced Navier-Stokes equation frisch .

(ii) Model II: $\tilde{\Gamma}=\eta q^{2}$. This is equivalent to $\bf v$
satisfying the linearised Navier-Stokes equation:

$\frac{\partial v_{i}}{\partial
t}=\eta\nabla^{2}v_{i}-{\nabla_{i}P}+\tilde{f}_{i},$ (13)

with $\langle v_{i}\rangle=0$; $P$ is the pressure and $\eta$ the kinematic
viscosity. Also, ${\bf v}$ is assumed to be incompressible, i.e.,
${\bm{\nabla}}\cdot{\bf v}=0$. We use this to eliminate $P$ from (13), and get

$\frac{\partial v_{i}}{\partial t}=\eta_{v}\nabla^{2}v_{i}+P_{ij}g_{j}.$ (14)

Here, $\eta_{v}$ is a fluid viscosity. Function $\tilde{f}_{i}$ is a zero-mean
Gaussian distributed stochastic force with a variance

$\langle\tilde{f}_{i}({\bf q},t)\tilde{f}_{j}({-\bf
q},0)\rangle=\tilde{D}_{1}|q|^{-y}\delta(t)\delta_{ij},$ (15)

where subscripts $i,j$ refer to Cartesian coordinates; $\tilde{D}_{1}>0$. This
yields

$D^{v}_{ij}({\bf q},\omega)=\frac{2\tilde{D}_{1}|q|^{-y}P_{ij}({\bf
q})}{\omega^{2}+\eta^{2}q^{4}}.$ (16)

We are again interested in the limit of large $\tilde{\Gamma}$, which in this
case implies a large $\eta_{v}$, with $\tilde{D}_{1}/\eta_{v}=D_{\eta}$ is
finite. This is the so-called Stokesian limit of the Navier-Stokes equation.

##  III Structure functions

In both the problems of fully developed turbulence and passive scalar
turbulence, the forms (1) and (3), respectively, of the structure functions
are guided by the symmetry of the underlying equations of motion themselves.
In the same spirit, we note that the only symmetry of $\phi_{\alpha}$ that
keeps Eqs. (7) and (11) invariant is the rotation in the order parameter
space. This prompted us to define the structure function

${\cal\tilde{S}}_{2n}(r)\equiv\langle\left[\phi_{\alpha}({\bf
x+r})\phi_{\alpha}({\bf x})\right]^{n}\rangle\sim r^{\overline{\zeta}_{n}}.$
(17)

Here, $n$ is any positive integer. Rotational invariance ensures that the any
odd order analogues of the structure functions (17) should vanish. Thus only
even order structure functions are considered. On the other hand, in a
subspace of the full parameter space where the coupling constant $g=0$, i.e.,
the mode coupling term (7) vanishes, the reduced system has a higher symmetry:
In addition to the rotational invariance, it is also invariant under a
constant shift $\phi_{\alpha}\rightarrow\phi_{\alpha}+\,const.$. This allowed
us to define yet another class of structure functions ${\cal S}_{2n}(r)$ as

${\cal S}_{2n}(r)\equiv\langle[\phi_{\alpha}({\bf x+r})-\phi_{\alpha}({\bf
x})]^{2n}\rangle\sim r^{\zeta_{n}}.$ (18)

These are same as those used in the standard passive scalar turbulence problem
as given in (3) above tirtha . Here too, the odd order analogues of ${\cal
S}_{2n}(r)$ vanish.

##  IV Scaling in the linear theory

Before embarking on investigating the scaling properties of the full nonlinear
model, it is instructive to consider the same for the linearised version of
the model, i.e., with $g=0=\lambda$. Thus the dynamics of $\phi_{\alpha}$ is
decoupled from $v_{i}$ and just follows the linear equation

$\frac{\partial\phi_{\alpha}}{\partial
t}=\nu\nabla^{2}\phi_{\alpha}+\overline{\eta}_{\alpha}.$ (19)

In this section we closely follow the discussions in Ref. tirtha . By using
(19), we obtain in the Fourier space

$\langle\phi_{\alpha}({\bf k},\omega)\phi_{\beta}(-{\bf
k},-\omega)=\frac{2\overline{D}k^{-\overline{y}}\delta_{\alpha\beta}}{\omega^{2}+\nu^{2}k^{4}}.$
(20)

Linearity of the dynamics of $\phi_{\alpha}$ ensures that all order
correlation and structure functions can be calculated exactly.

Following Ref. tirtha , we write

$\displaystyle{\cal\tilde{S}}_{2}(r)\equiv\langle\phi_{\alpha}({\bf
x+r},t)\phi_{\alpha}({\bf x},t)\rangle$ (21) $\displaystyle=$
$\displaystyle\int_{2\pi/L}^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}\frac{\overline{D}\delta_{\alpha\alpha}}{q^{2+\overline{y}}}\exp(i{\bf
q\cdot r}).$

For $\overline{y}+2<d$, (21) is insensitive to the lower limit, which may be
brought to zero (i.e., $L\rightarrow\infty$) without any problem. This gives

${\cal\tilde{S}}_{2}\sim r^{\overline{y}+2-d}.$ (22)

The scaling of the higher order structure functions ${\cal\tilde{S}}_{2n}(r)$
can be readily found by noting that due to the linearity of the dynamics
$\phi_{\alpha}$ is Gaussian distributed. This gives

${\cal\tilde{S}}_{2n}(r)\sim r^{n(\overline{y}+2-d)}.$ (23)

This includes the case $\overline{y}=-2$, i.e., a conserved noise in (7).
Equation (23) shows that ${\cal\tilde{S}}_{2n}(r)$ is a decaying function of
$r$. On the other hand, for $\overline{y}+2>d$, (21) is dominated by the lower
limit, giving

${\cal\tilde{S}}_{2}(r)=\int_{2\pi/L}^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}\frac{D_{1}\delta_{\alpha\alpha}}{q^{2+\overline{y}}}\sim
L^{\overline{y}+2-d}$ (24)

in the asymptotic limit $L\gg r$. Thus, ${\cal\tilde{S}}_{2}(r)$ is
independent of $r$, a growing function of $L$ and in fact diverges as
$L\rightarrow\infty$ with $L\gg r$. Similarly, in the linearised theory using
the logic outlined above, and by employing Gaussianisation (exact for a linear
dynamical system) we find

${\cal\tilde{S}}_{2n}(r)\sim L^{n(\overline{y}+2-d)}$ (25)

independent of $r$ and diverges with $L$, since $\overline{y}+2>d$.

In the same way,

$\displaystyle{\cal S}_{2}(r)\equiv 2\langle\phi_{\alpha}({\bf
x},t)\phi_{\alpha}({\bf x},t)\rangle-2\langle\phi_{\alpha}({\bf
x},t)\phi_{\alpha}(0,t)\rangle$ (26) $\displaystyle=$ $\displaystyle
2\int_{2\pi/L}^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}\frac{\overline{D}\delta_{\alpha\alpha}}{q^{2+\overline{y}}}\left[1-\exp(i{\bf
q\cdot r})\right].$

This gives, as in Ref. tirtha , for $\overline{y}<d$,

${\cal S}_{2}(r)\sim r^{2-d+\overline{y}},\,{\cal S}_{2n}(r)\sim
r^{n(2-d+\overline{y})},$ (27)

where as for $\overline{y}>d$

${\cal S}_{2}(r)\sim r^{2}L^{\overline{y}-d},\,{\cal S}_{2n}(r)\sim
r^{2n}L^{n(\overline{y}-d)}.$ (28)

Unsurprisingly, in the linear model ${\cal S}_{2n}(r)$ shows only simple
scaling tirtha .

We will see below that complex scaling behaviours by ${\cal\tilde{S}}_{2n}(r)$
and ${\cal S}_{2n}(r)$ emerge, when the nonlinear effects are taken into
account. We address this systematically in the next Section.

##  V Nonlinear effects in Model I

When the nonlinear terms are considered, $\phi_{\alpha}({\bf x},t)$ can no
longer be solved exactly. This necessitates perturbative approaches. However,
naïve perturbative expansions produce diverging corrections. These
difficulties can be systematically handled within a renormalisation group
calculation framework.

The dynamic RG procedure is well-documented in the literature drg . We give
here a brief outline of the method for the convenience of the readers. This
starts by writing down the dynamic generating functional

${\cal Z}=\int{\cal D}\phi_{\alpha}{\cal D}\hat{\phi}_{\alpha}{\cal
D}v_{i}\exp[-S_{act}],$ (29)

where the action functional $S_{act}$ is given by

$\displaystyle{S}_{act}$ $\displaystyle=$ $\displaystyle-\int
d^{d}qd\omega[\overline{D}q^{-\overline{y}}\hat{\phi}_{\alpha}({\bf
q},\omega)\hat{\phi}_{\alpha}({\bf-q},-\omega)+\hat{\phi}_{\alpha}({\bf-q},-\omega)[-i\omega\phi_{\alpha}({\bf
q},\omega)+i\lambda q_{m}\sum_{{\bf q}_{1},\Omega}v_{m}({\bf
q}_{1},\Omega)\phi_{\alpha}({\bf q-q}_{1},\omega-\Omega)$ (30)
$\displaystyle+$ $\displaystyle\nu q^{2}\phi_{\alpha}({\bf q},\omega)+{\bf
q\cdot q}_{1}\phi_{\beta}({\bf q-q}_{1},\omega-\Omega)\phi_{\gamma}({\bf
q}_{1},\Omega)]-v_{i}({\bf
q},\omega)[D_{ij}^{v}(q,\omega)]^{-1}v_{j}({\bf-q},-\omega)/2].$

Here, $\hat{\phi}_{\alpha}({\bf x},t)$ is the usual response field janssen .

The momentum shell dynamic RG consists of tracing over the short wavelength
Fourier modes of $\phi_{\alpha}({\bf x},t),\,\hat{\phi}_{\alpha}({\bf x},t)$
and ${\bf v}({\bf x},t)$, followed by rescaling of lengths and time. We follow
the standard procedure of initially restricting the wavevectors to lie in a
$d$-dimensional Brillouin zone: $q\leq\Lambda$, where $\Lambda$ is an ultra-
violet cutoff, which should be of the order of the inverse lattice spacing
$a_{0}$. However, its precise value has no effect on the theory that we
develop. The fields ${\phi_{\alpha}}({\bf x},t),\,\hat{\phi}_{\alpha}({\bf
x},t)$ and ${\bf v}({\bf x},t)$ are then separated into high and low wave
vector parts: $\phi_{\alpha}({\bf x},t)=\phi_{\alpha}^{>}({\bf
x},t)+\phi_{\alpha}^{<}({\bf x},t),\,\hat{\phi}_{\alpha}({\bf
x},t)=\hat{\phi}_{\alpha}^{>}({\bf x},t)+\hat{\phi}_{\alpha}^{<}({\bf x},t)$
and ${\bf v}({\bf x},t)={\bf v}^{>}({\bf x},t)+{\bf v}^{<}({\bf x},t)$, where
$\phi_{\alpha}^{>}({\bf x},t),\,\hat{\phi}_{\alpha}^{>}({\bf x},t)$ and ${\bf
v}^{>}({\bf x},t)$ all have support in the large wave vector (short
wavelength) range $\Lambda/b<|q|<\Lambda$, while $\phi_{\alpha}^{<}({\bf
x},t),\,\hat{\phi}_{\alpha}^{<}({\bf x},t)$ and ${\bf v}^{<}({\bf x},t)$ have
support in the small wave vector (long wavelength) range
$|{q}|<\Lambda/b,\,b>1$. In the next step, we integrate over
$\phi_{\alpha}^{>}({\bf x},t),\,\hat{\phi}_{\alpha}^{>}({\bf x},t)$ and ${\bf
v}^{>}({\bf x},t)$, which can be done perturbatively in the anhamornic
couplings in (30). The perturbation theory that is constructed here is
represented by the Feynman diagrams, with the order of perturbation theory
being given by the number of loops in the Feynman diagrams that we consider
here. After this perturbative step, we are to rescale lengths with ${\bf
x}=b{\bf x^{\prime}},\,b>1$, so as to restore the UV cutoff to $\Lambda$
again, and time with $t=t^{\prime}b^{z}$, where $z$ is the dynamic exponent.
This is then followed by rescaling of $\phi_{\alpha}^{<}({\bf
x},t),\,\hat{\phi}_{\alpha}^{<}({\bf x},t)$ and ${\bf v}^{<}({\bf x},t)$. In
particular, we rescale $\phi_{\alpha}^{<}({\bf x},t)$ by $b^{\chi}$, where the
scaling exponent $\chi$ is used to produce a fixed point, and charactrises the
spatial scaling of the auto-correlations of $\phi_{\alpha}({\bf x},t)$.

###  V.1 Renormalisation of the model parameters

The RG procedure allows us to obtain the fluctuation-corrected model
parameters. First we consider $\overline{y}=-2$. One can formally construct a
one-loop diagram for $\overline{D}$ that comes from the mode-coupling term.
However, these one-loop fluctuation corrections to $\nu$ and $\overline{D}$
that come from the mode coupling nonlinear term are not infra-red divergent
and hence are not relevant. Only the advective nonlinearity contributes to the
relevant corrections to $\nu$ and $\overline{D}$ tirtha . Unsurprisingly, our
results for $\overline{y}=-2$ are identical to those in Ref. tirtha . We
revisit those results here for the sake of completeness. The relevant one-loop
Feynman graphs for $\nu$ are shown in Appendix A, some of which come from the
mode-coupling nonlinearity.

$\displaystyle\nu^{<}$ $\displaystyle=$
$\displaystyle\nu\left[1+\frac{\lambda^{2}D_{1}}{\nu}\left(1-\frac{1}{d}\right)k_{d}\frac{\Lambda^{d-y}}{d-y}\left(1-\frac{1}{b^{d-y}}\right)\right],$
(31) $\displaystyle\overline{D}^{<}$ $\displaystyle=$
$\displaystyle\overline{D}\left[1+\frac{\lambda^{2}D_{1}}{\nu}\left(1-\frac{1}{d}\right)k_{d}\frac{\Lambda^{d-y}}{d-y}\left(1-\frac{1}{b^{d-y}}\right)\right],$
(32)

where $k_{d}=S_{d}/(2\pi)^{d}$ with $S_{d}$ being the hypersurface area of a
$d$-dimensional sphere of unit radius. Writing $b=\exp(\delta l)\approx
1+\delta l$, we obtain the flow equations

$\displaystyle\frac{d\nu}{dl}$ $\displaystyle=$
$\displaystyle\nu\left[z-2+\frac{\lambda^{2}D}{\nu}\left(1-\frac{1}{d}\right)k_{d}\Lambda^{d-y}\right],$
(33) $\displaystyle\frac{d\overline{D}}{dl}$ $\displaystyle=$
$\displaystyle\overline{D}\left[z-2\chi-d-2+\frac{\lambda^{2}D}{\nu}\left(1-\frac{1}{d}\right)k_{d}\Lambda^{d-y}\right].$

Solution of the above flow equations yield $z=2$ and $\chi=-d/2$ tirtha .
Here, $\chi$ is the scaling dimension of the field $\phi_{\alpha}({\bf x},t)$.

We now focus on the case with $\overline{y}>-2$. With this, notice that there
are no relevant (in a RG sense) corrections to $\overline{D}$. This is because
the corrections to $\overline{D}$ are all ${\cal O}(q^{2})$, hence cannot
generate a singular (in the IR limit) contribution that may correct
$\overline{D}q^{-\overline{y}}$. This gives, as in Ref. tirtha ,

$\displaystyle\frac{d\overline{D}}{dl}$ $\displaystyle=$
$\displaystyle\overline{D}\left[z-2\chi-d+\overline{y}\right],$ (35)

for all values of $\overline{y}$. Thus

$\chi=(\overline{y}+z-d)/2$ (36)

is an exact exponent identity. This in the linear theory reduces to
$\chi=(\overline{y}+2-d)/2$ with $z=2$.

There are relevant one-loop fluctuation corrections to $\nu$, however. The
fluctuation-corrected diffusivity $\nu^{<}$ is given by

$\displaystyle\nu^{<}$ $\displaystyle=$
$\displaystyle\nu[1+\frac{\lambda^{2}D_{1}}{\nu d}(d-1)k_{d}\int dqq^{-y+d-1}$
(37) $\displaystyle+$
$\displaystyle\frac{g^{2}\overline{D}\overline{y}}{2\nu^{3}d}k_{d}\int
dqq^{-\overline{y}+d-3}]$

see Appendix A for detailed calculations. Equation (37) allows us to define
two effective coupling constants:
$g_{1}=k_{d}\frac{g^{2}\overline{D}\overline{y}}{\nu^{3}}\Lambda^{d-\overline{y}-2}$,
$g_{2}=k_{d}\frac{\lambda^{2}D_{1}}{\nu}\Lambda^{d-y}$.There are no
fluctuation corrections to $g,\,\lambda,\,\overline{D},\,D_{1}$. That $g$ does
not renormalise is a consequence of the rotational invariance of the order
parameter space. There are actually one-loop graphical corrections to
$\lambda$; these however produce subleading corrections (see Ref. tirtha ).

Rescaling of the space, time and fields allow us to convert (37) into a
differential RG flow equation. We get

$\frac{d\nu}{dl}=\nu\left[z-2+\frac{g_{1}}{2d}+g_{2}\frac{d-1}{d}\right].\\\ $
(38)

The RG flow equations for the coupling constants $g_{1}$ and $g_{2}$ are as
follows:

$\displaystyle\frac{dg_{1}}{dl}$ $\displaystyle=$ $\displaystyle
g_{1}\left[\overline{y}-d+2-\frac{3g_{1}}{2d}-3g_{2}\frac{d-1}{d}\right],$
(39) $\displaystyle\frac{dg_{2}}{dl}$ $\displaystyle=$ $\displaystyle
g_{2}\left[y-d-\frac{g_{1}}{2d}-g_{2}\frac{d-1}{d}\right].$ (40)

It is convenient to define two “small parameters”
$\delta\equiv\overline{y}-d+2$ and $\epsilon\equiv y-d$. Clearly, $\delta$ and
$\epsilon$, respectively, are the scaling dimensions of $g_{1}$ and $g_{2}$.
In fact, the one-loop perturbation theory that we set up below is effectively
an expansion in $\epsilon$ and $\delta$ up to the linear order in these two
parameters.

The RG fixed points $(g^{*}_{1},\,g^{*}_{2})$ are the solutions of
$dg_{1}/dl=0=dg_{2}/dl$. Flow Eqs. (39) and (40) four sets of fixed points
(FP):

(i) FP1: $(g^{*}_{1},\,g^{*}_{2})=(0,\,0)$: this is linearly stable for
$\overline{y}<d-2$, i.e., $\delta<0$ and $y<d$, i.e., $\epsilon<0$. At this
fixed point, unsurprisingly, the dynamic exponent $z=2$, same as that in the
linear theory.

(ii) FP2: $(g^{*}_{1},\,g^{*}_{2})=(\frac{2d}{3}\delta,\,0)$: this is linearly
stable for $\delta/3>\epsilon$. At this FP,

$z=2-(\overline{y}-d+2)/3=2-\frac{\delta}{3}.$ (41)

(iii) FP3: $(g^{*}_{1},\,g^{*}_{2})=(0,\,\frac{d}{d-1}\epsilon)$: this is
linearly stable for $\epsilon>\delta/3$. At this fixed point,
$z=2-(y-d)=2-\epsilon$. Since $y>d$, i.e., $\epsilon>0$, at this FP, we have
$z<2$.

(iv) Nontrivial FP: If $y-d=(\overline{y}-d+2)/3>0$, i.e., $\epsilon=\delta/3$
then

$\frac{g^{*}_{1}}{2}+g^{*}_{2}(d-1)=d(y-d)=(\overline{y}-d+2)\frac{d}{3}.$
(42)

However, $g^{*}_{2}$ and $g^{*}_{1}$ cannot however be separately evaluated
foot2 . By using (38), we conclude that

$z=2-\frac{g^{*}_{1}}{d}-g^{*}_{2}\frac{d-1}{d}=2-(y-d)<2.$ (43)

We find that

$y-d=\left(\overline{y}-d+2\right)/3,\;\;{\rm i.e.},\epsilon=\frac{\delta}{3}$
(44)

is the separatrix between the two fixed points in the $g_{1}-g_{2}$ plane. We
note that the systems with parameters satisfying $y-d>(\overline{y}-d+2)/3>0$
flow to the linearly stable fixed point

$g^{*}_{1}=0,\,g^{*}_{2}=\frac{d}{d-1}(y-d).$ (45)

This gives

$z=2-(y-d)=2-\epsilon<2.$ (46)

However, all systems with $2(\overline{y}-d+2)/3>y-d>0$ flow to the linearly
stable fixed point

$g^{*}_{1}=\frac{2d}{3}(\overline{y}-d+2),\,g^{*}_{2}=0,\,.$ (47)

This gives $z=2-(\overline{y}-d+2)/3=2-\delta/3<2$.

See Fig. 1 for a schematic RG flow diagram in the $g_{1}-g_{2}$ plane.

We now obtain the spatial scaling exponent $\chi$. We note that the condition
of the nonrenormalisation of $\overline{D}$ (or equivalently the flow equation
(35)) gives

$\chi=(z+\overline{y}-d)/2,$ (48)

an exact relation that holds at the fixed points discussed above. Since the
value of $z$ depends upon the stable fixed point, so does the value of $\chi$.
For instance, when $g^{*}_{1}=0,\,g^{*}_{2}>0$, we get by using $z=2-(y-d)$

$\chi=(2-y+\overline{y})/2.$ (49)

Here, $\chi$ can be made positive by tuning $y$ and $\overline{y}$. Assume
$y-d\equiv\epsilon>0$ and $\overline{y}-d+2\equiv\delta/2>0$, with
$\epsilon>\delta/3$. The last condition ensures that $g^{*}_{2}>0$ and
$g^{*}_{1}=0$ at the FP. This gives

$\chi=\frac{1}{2}(2-y+\overline{y})=\frac{1}{2}(\delta-\epsilon)>0.$ (50)

Thus, $\delta/3<\epsilon<\delta$ is the necessary condition for $\chi>0$. On
the other hand, when $g^{*}_{2}=0,\,g^{*}_{1}>0$, together with
$z=(\overline{y}-d+2)/3$, we get

$\chi=\frac{1}{2}(z+\overline{y}-d)=\frac{1}{2}(2-\frac{\delta}{3}+\delta-2)=\frac{2\delta}{3}>0$
(51)

along with $\delta/3>\epsilon$. We find that the fluctuation corrections to
the values of the scaling exponents $z$ and $\chi$ in the linear theory are
linear in $\epsilon$ or $\delta$. We further note that at the stable fixed
point FP2, $g_{1}^{*}\sim{\cal O}(\delta),\,g_{2}^{*}=0$, whereas at FP3,
$g_{1}^{*}=0,\,g_{2}^{*}\sim{\cal O}(\epsilon)$, establishing the fact that
the one-loop perturbation theory here is in effect an an expansion in
$\epsilon$ and $\delta$. See Fig. 2 for a schematic phase diagram in the
$\delta-\epsilon$ plane.

Figure 1: RG flow diagram in the $g_{2}-g_{1}$ plane with $\overline{y}>-2$
and $y>0$. FP2 and FP3 are the RG fixed points obtained above. The inclined
broken line is the separatrix $y-d=(\overline{y}-d+2)/3$ for given
$y,\overline{y},d$. Figure 2: Phase diagram of Model I in the
$\delta-\epsilon$ plane. The mode-coupling term is relevant but the advective
nonlinearity is irrelevant (in a RG sense) with $\chi>0$ in the renormalised
theory in the lower purple triangle. In the intermediate light blue triangle,
the mode coupling term is irrelevant, but the advective nonlinearity is now
relevant with $\chi>0$ in the renormalised theory. In the light green upper
triangle, $\chi<0$ (see text).

###  V.2 Multiscaling

With the knowledge of the renormalised parameters, and the associated scaling
exponents, we now focus on the scaling properties of the structure functions
defined above.

####  V.2.1 Case I: $g_{1}^{*}>0,\,g_{2}^{*}=0$

We consider multiscaling corresponding to the fixed point
$g_{1}^{*}=\frac{2d}{3}\left(\overline{y}-d+2\right)>0$ and $g_{2}^{*}=0$. At
this fixed point, the velocity field decouples from $\phi_{\alpha}$ in the
long wavelength limit. In this case, there is no invariance under a constant
shift of all $\phi_{\alpha}({\bf x})$; only the rotation in the space of
$\phi_{\alpha}({\bf x})$ keeps the equation of motion invariant. We are
interested in the scaling properties of the structure functions $\tilde{\cal
S}_{2n}(r)$ defined in (17) here.

On dimension ground, we can write

$\tilde{\cal S}_{2n}(r)\sim r^{2n\chi}\tilde{\theta}_{n}(r/\tilde{L}),$ (52)

where, $\tilde{\theta}_{n}$ is a dimensionless scaling function of
$r/\tilde{L}$; length scale $\tilde{L}$ is yet to be specified. Whether
$\tilde{L}$ is a “large-scale” (e.g., the integral scale $L$ in a fully
developed turbulence), or a “small scale” (e.g., the dissipation scale
$\eta_{d}$) is still to be decided. Regardless of the interpretation of
$\tilde{L}$, we assume

$\tilde{\theta}_{n}\left(\frac{r}{\tilde{L}}\right)\sim\left(\frac{\tilde{L}}{r}\right)^{\tilde{\Delta}_{2n}}$
(53)

in the asymptotic scaling limit adjhem ; tirtha . If $\tilde{\Delta}_{2n}=0$,
then scaling function $\theta_{n}$ approaches a constant in the asymptotic
limit, and the scaling of $\tilde{\cal S}_{2n}(r)$ is completely given by
$r^{2n\chi}$. On the other hand, a non-zero $\tilde{\Delta}_{n}$ implies
correction to the scaling given by $r^{2n\chi}$; in fact, a nonlinear
dependence of $\Delta_{n}$ on $n$ should give rise to multiscaling. Notice
that if we compare (53) with the corresponding result (28) in the linear
theory, we can identify $\tilde{L}$ with $L$, a large scale. We shall see
below that inclusion of nonlinear effects once again identifies $\tilde{L}$
with $L$; see also Ref. tirtha .

Calculations of $\tilde{\cal S}_{2n}(r)$ involve averaging over spatially non-
local quantities with respect to the action functional (30). We use the idea
of the operator product expansion (OPE) adjhem ; cardy ; book  to write

$\phi_{\alpha}^{n}({\bf x+r})\phi_{\alpha}^{n}({\bf
x})\sim\sum_{m}\tilde{C}_{m}(r)\tilde{\cal O}_{m}({\bf x}),$ (54)

in the asymptotic limit of $r/\tilde{L}\ll 1$. Here, $\tilde{C}_{m}(r)$ is a
function of the separation $r$ with a power law dependence in the scaling
limit, and $\tilde{\cal O}_{m}({\bf x})$ is an “operator” that depends upon
$\phi_{\alpha}({\bf x})$. Since the rhs of the OPE in (54) must be invariant
under the rotation of $\phi_{\alpha}({\bf x})$, so should the rhs of (54) be.
The composite operators that can appear on the rhs of (54) are of the form

$\tilde{\cal O}_{m}({\bf x})=\left(\phi_{\alpha}({\bf x})\phi_{\alpha}({\bf
x})\right)^{m}.$ (55)

Evidently, these are invariant under rotation.

####  V.2.2 OPE for the linear problem

The idea of OPE in the problem remains true even in the linear limit, for
which the fluctuation corrections to all the parameters immediately disappear,
and exponents $z$ and $\chi$ are given exactly by (36). We now reconsider the
exactly known scaling of $\tilde{\cal S}_{2n}(r)$ in the linear case in the
context of the OPE discussed above and examine the consistency of the latter.
Our aim is to obtain (24) and (25) by using the prescription of OPE as
elucidated above: according to that we should have

$[\phi_{\alpha}({\bf x}+{\bf r})\phi_{\alpha}({\bf x})]\sim\tilde{\cal
C}_{0}(r){\bf I}+\tilde{\cal C}_{2}(r)[\phi_{\alpha}({\bf x})]^{2},$ (56)

such that

${\cal\tilde{S}}_{2}(r)\sim{\cal\tilde{C}}_{0}(r){\bf
I}+{\cal\tilde{C}}_{2}(r)\langle[\phi_{\alpha}({\bf x})]^{2}\rangle.$ (57)

Here, $\bf I$ is the identity operator. Now in the linear theory for
$\overline{y}+2<d$, or $\chi<0$, the left had side of (56) scales as
$b^{2\chi}$, when all spatial coordinates are scaled by $b$. Each term on the
rhs of (56) on dimensional ground must scale the same way under spatial
rescaling as the lhs. This consideration gives

$\tilde{\cal C}_{0}(r)\sim r^{2\chi}.$ (58)

On the other hand,

$\langle[\phi_{\alpha}({\bf x})]^{2}\rangle\sim a_{0}^{2\chi}$ (59)

where $a_{0}$ is the small scale cut-off. This gives on the dimensional ground

$\tilde{\cal C}_{2}(r)\sim r^{0}.$ (60)

Therefore, in the scaling limit of $r\gg a_{0}$, the first term on the rhs of
(56) dominates. This gives

$\tilde{\cal S}_{2}(r)\sim r^{2\chi}\sim r^{-(2-d+\overline{y})}.$ (61)

Using the linearity of the $\phi_{\alpha}$-dynamics then,

$\tilde{\cal S}_{2n}(r)\sim r^{2\chi}\sim r^{-n(2-d+\overline{y})},$ (62)

in agreement with (23).

In contrast, for $\overline{y}+2>d$ or $\chi>0$ the second term in the right
hand side of (56) dominates. This may be argued as follows. On dimensional
ground, we still have $\tilde{\cal C}_{2}(r)\sim r^{2\chi}$. On the other
hand, with $\chi>0$

$\langle[\phi_{\alpha}({\bf x})]^{2}\rangle\sim L^{\overline{y}+2-d},$ (63)

an $L$-dependence identical to that in (24), giving $\tilde{\cal C}_{2}(r)\sim
r^{0}$, again using dimensional analysis. Therefore, in the scaling limit
$r\ll L$, the second term on the rhs of (56) dominates over the first term. We
thus conclude from Eq. (63) that

$\tilde{\cal S}_{2}(r)\sim L^{\overline{y}+2-d},$ (64)

unsurprisingly same as (24). This may be extended to higher order structure
($n>1$) easily. For instance for $n=2$, the most dominant operator that
contributes to $\tilde{\cal S}_{4}(r)$ in the scaling limit is
$(\phi_{\alpha}({\bf x}))^{4}$. It is easy to see $\langle(\phi_{\alpha}({\bf
x}))^{4}\rangle\sim L^{2(\overline{y}-d)}$, giving $\tilde{\cal C}_{4}(r)\sim
r^{4}$. Putting together everything then,

$\tilde{\cal S}_{4}(r)\sim L^{2\overline{y}+4-2d},$ (65)

which is same as that obtained by direct calculations above; see (25) above.
This evidently lends credence to our analysis even when $\lambda\neq 0$. In
general, we conclude

$\tilde{\cal S}_{n}(r)\sim L^{n(\overline{y}+2-d)}.$ (66)

We note that only when $\chi>0$, the structure functions can depend on $L$.
Having re-established the exactly known scaling exponents of $\tilde{\cal
S}_{2n}(r)$ in the linear limit by the arguments of OPE, we now analyse the
nonlinear cases below.

####  V.2.3 OPE for Model I

We now use the idea of OPE to calculate the scaling of
${\cal\tilde{S}}_{2n}(r)$ in the nonlinear case: We start from the general
expansion of OPE

${\cal\tilde{S}}_{2n}(r)\sim\sum_{m}\tilde{C}_{m}(r)\langle{\cal\tilde{O}}({\bf
x})\rangle.$ (67)

Thus, in order to have a one-loop renormalised theory for multiscaling we must
now find out how the naïve scaling of ${\cal\tilde{O}}_{m}({\bf x})$ changes
due to fluctuations. We continue to consider the case with $\chi>0$ for which
composite operators are expected to contribute to the scaling properties of
the structure functions. We are required to find out which composite operator
on the rhs of (67) makes the most dominant contribution in the asymptotic
limit $L\gg r$. Naïve expansion of the lhs of (67) suggests that $\tilde{\cal
O}_{m}({\bf x})$ with $m=n$ should make the desired most dominant
contribution. Assuming that $\tilde{\cal O}_{n}({\bf x})$ do not generate any
higher order relevant operators, we must now calculate the one loop
renormalisation of ${\cal\tilde{O}}_{n}({\bf x})$ for arbitrary $n$ and then
find the scaling forms for the renormalised composite operators; see Appendix
B for the relevant one-loop Feynman diagrams contributing to the
renormalisation of ${\cal\tilde{O}}_{m}({\bf x})$; see also Appendix C for
some related technical discussions (including higher order operators). We find

$\langle\tilde{\cal O}^{<}_{m}({\bf x})\rangle=\langle\tilde{\cal O}_{m}({\bf
x})\rangle\left[1+\delta m^{\prime}\right],$ (68)

where, $\delta m^{\prime}$ is the fluctuation correction.

We now consider the the one-loop correction for $\tilde{\cal O}_{n}({\bf x})$.
The one-loop diagram is shown in Fig. 4 in Appendix B.

Evaluation of this diagram is shown in Appendix. We find

$\tilde{\cal O}^{<}_{n}({\bf x})=\tilde{\cal O}_{n}({\bf
x})\left[1+2\frac{D_{1}g^{2}}{2\nu^{3}}\int_{\Lambda/b}^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}\frac{|q|^{-y}}{q^{2}}\right].$
(69)

This results into the RG flow equation for $\langle\tilde{\cal O}^{<}_{n}({\bf
x})\rangle$

$\frac{d\langle\tilde{\cal O}_{n}\rangle}{dl}=\langle\tilde{\cal
O}_{n}\rangle\left[2n\chi+\tilde{\delta}_{n}\right],$ (70)

where $\tilde{\delta}_{n}=\frac{nD_{1}g^{2}}{\nu^{3}}=ng_{1}$ is linear in
$n$. This implies

$\langle\tilde{\cal O}_{n}({\bf x})\rangle\sim
L^{2n\chi+\tilde{\delta}_{n}}\sim L^{\tilde{\Delta}n}.$ (71)

This gives for the scaling behaviour of the structure functions $\tilde{\cal
S}_{2n}(r)$ as

$\tilde{\cal S}_{2n}(r)\sim r^{2n\chi}f_{2n}\left(\frac{L}{r}\right)\sim
r^{2n\chi}\left(\frac{L}{r}\right)^{\tilde{\Delta}_{n}}\sim
r^{-\tilde{\delta}_{n}}L^{\tilde{\Delta}_{n}},$ (72)

where $\tilde{\Delta}_{n}$ or $\tilde{\delta}_{n}$ should be evaluated at the
RG fixed point with $g_{1}=g_{1}^{*}$. In this linear theory, $\tilde{\cal
S}_{2n}(r)$ is independent of $r$ in the asymptotic limit $r\ll L$, but depend
on $L$ with exponents that vary linearly with $n$. Nonlinear effects change
this picture. Structure functions $\tilde{\cal S}_{2n}(r)$ now do depend on
$r$, but with negative exponents, i.e., they decrease with $r$:

$\tilde{\zeta}_{n}=-\tilde{\delta}_{n}=-ng_{1}^{*}=n\frac{2d}{3}\delta.$ (73)

On the other hand, their dependence on $L$ are now more sensitive; the
exponents that characterise their dependence on $L$ are positive and bigger
than their values in the linear theory. We notice that the scaling exponents
$\tilde{\zeta}_{n}$ are ${\cal O}(\delta)$ at the lowest order in the
perturbation theory, consistent with the structure of the perturbation theory
with $\delta$ and $\epsilon$ as “small expansion parameters”.

####  V.2.4 Case II: $g_{1}^{*}=0,\,g_{2}^{*}>0$

We now consider the part of the phase space in the $y-\overline{y}$ plane
where $g^{*}_{2}=d(y-d)/(d-1)>0$ and $g^{*}_{1}=0$. In this part of the phase
space, the mode coupling term is irrelevant (in a RG sense), and thus there is
an emergent symmetry under $\phi_{\alpha}({\bf
x},t)\rightarrow\phi_{\alpha}({\bf x},t)+{\rm const.}$ in the long wavelength
limit. The effective model for each of $\phi_{\alpha}$ is in fact identical to
that used for the passive scalar turbulence problem tirtha . In this case, it
is meaningful to consider the scaling properties of the structure functions
${\cal S}_{2n}(r)$ defined above. We revisit the calculation for the scaling
properties of ${\cal S}_{2n}(r)$ for the sake of completeness below. On purely
dimensional ground, we expect

${\cal S}_{2n}(r)=r^{2n\chi}\theta_{n}(r/\tilde{L}),$ (74)

where $\theta_{n}$ is a dimensionless scaling function of $r/\tilde{L}$,
$\tilde{L}$ is a length scale that is yet to be specified. As before,
independent of the interpretation of $\tilde{L}$, we write

$\theta_{n}\left(\frac{r}{\tilde{L}}\right)\sim\left(\frac{\tilde{L}}{r}\right)^{\Delta_{2n}}$
(75)

in the asymptotic scaling regime tirtha . For $\Delta_{2n}=0$, the scaling
function $\theta_{n}$ approaches a constant in the asymptotic limit, and
${\cal S}_{2n}(r)\sim r^{2n\chi}$. In contrast, a non-zero $\Delta_{n}$
implies correction to the scaling given by $r^{2n\chi}$.

We briefly revisit the results and the discussions in Ref. tirtha . We express
the structure functions in terms of local composite operators

$[\phi_{\alpha}({\bf x+r}-\phi_{\alpha}({\bf
x})]^{2n}\sim\sum_{m}C_{m}(r){\cal O}_{m}({\bf x}).$ (76)

Using (76),

${\cal S}_{2n}(r)\sim C_{n}(r)\langle{\cal O}_{n}({\bf x})\rangle.$ (77)

In the asymptotic limit of $L/r\rightarrow\infty$, we have

${\cal S}_{2n}(r)\sim r^{2n\chi}\left(\frac{L}{r}\right)^{\Delta_{n}}.$ (78)

As explained in Ref. tirtha , the most dominant operator is

${\cal O}_{n}({\bf x})=(\partial_{i}\phi_{\alpha}({\bf
x})\partial_{i}\phi_{\alpha}({\bf x})^{n}.$ (79)

As discussed in Ref. tirtha  in details, there are one-loop corrections to
${\cal O}_{n}({\bf x})$ with the fluctuation-corrected composite operator

${\cal O}_{n}^{<}({\bf x})={\cal O}_{n}({\bf x})[1+\delta n],$ (80)

where

$\delta n=\frac{\lambda_{1}^{2}Dn(d-1)(d+2n)}{\nu d(d+2)},$ (81)

giving for the renormalised composite operator

$\langle{\cal O}_{n}({\bf x})\rangle\sim L^{2n(\chi-1)+\delta n}.$ (82)

This implies

${\cal S}_{2n}(r)\sim r^{2n-\delta n}\sim r^{\zeta_{n}},$ (83)

where

$\zeta_{n}=n(\overline{y}-y)+\frac{(y-d)n(d+2n)}{d+2}.$ (84)

Thus, $\zeta_{n}$ depends nonlinearly on $n$, a hallmark of multiscaling.
Structure functions ${\cal S}_{2n}(r)$ grows with $r$ in the renormalised
theory, albeit slower than in the linear theory. Contrast this with the
results for Case I (i.e., with $g_{1}^{*}>0,\,g_{2}^{*}=0$). In that case, the
relevant structure functions ${\cal\tilde{S}}_{2n}(r)$ decay with $r$ in the
renormalised theory. Here too, $\delta_{n}\sim{\cal O}(\epsilon)$.

##  VI Summary and outlook

In this work, we have studied the issue of multiscaling in stochastically
driven nonlinear dynamical models. To that end, we have considered a simple,
conceptual model in the form of the conserved dynamics of classical Heisenberg
spins above $T_{c}$ that included a mode-coupling contribution, advected by a
stochastic velocity field that is assumed to be Gaussian-distributed and
independent of the spin dynamics, and an additive noise. We define appropriate
structure functions ${\cal\tilde{S}}_{2n}(r)$ that should capture the
universal scaling when the mode coupling term is relevant in a RG sense. When
it is irrelevant, a different set of structure functions ${\cal S}_{2n}(r)$
that are essentially same as those used for the passive scalar turbulence
problem is useful for this purpose. We show that when the velocity and
additive noise correlations are sufficiently long-ranged, as measured by the
spatial scaling of the their variances, the structure functions start to
depend on the system size $L$. The precise $L$-dependences of these two sets
of the structure functions are, however, quite different. More interestingly,
${\cal\tilde{S}}_{2n}(r)$ do not show any multiscaling, i.e., the associated
scaling exponents ${\tilde{\zeta}}_{2n}$ are linear functions of $n$. In
contrast, ${\cal S}_{2n}$ show genuine multiscaling identical to those found
in Refs. adjhem ; kupi ; tirtha . We also show in Appendix D that these
results are rather insensitive to the specific choice for the large damping
limit of the the flow field equation, so long as the flow field remains
autonomous. On the other hand, nonlinear effects can generally be very crucial
in controlling the nature of multiscaling. In fact, in this particular case,
the nonlinear mode coupling term, when relevant, can lead to simple scaling
instead of multiscaling.

Our calculational scheme is technically challenging. Due to the mode coupling
term, composite operators of the spin variables of a particular order are now
connected to all higher order composite operators. We have argued in Appendix
C that due to the conservation law nature of the spin dynamics, the
perturbatively generated higher order composite operators from a composite
operator of a given order should be subleading due to the appearance of
additional gradient operators.

The quantitative accuracy of our results are limited by the low order of the
perturbation theory. Nonetheless, we still expect that the qualitative
features of our results will hold even in a more sophisticated perturbation
theory, or in numerical solutions of the equations of motion. It will indeed
be interesting to test our results by numerically solving the governing
equations of motion. In the more realistic problem multiscaling in a system
with coupled variables, e.g., $3d$ magnetohydrodynamic turbulence or binary
fluid turbulence, both the velocity field and the second dynamical field,
e.g., the magnetic field or the concentration field display non-trivial
multiscaling. While our results cannot be obviously carried over to these
problems due to the simplified and admittedly artificial nature of our model
equation and the significant technical challenges involved in $3d$
magnetohydrodynamic or binary fluid turbulence, we can certainly conclude with
a good degree of confidence that the presence of the different nonlinear terms
can affect the detailed form of the multiscaling exponents in a nontrivial
manner abmhd ; binfluid . In this work, we have been concerned with the
multiscaling (or scaling) of the (appropriately defined) equal-time structure
functions. More recently, the concepts of scaling and multiscaling have been
extended to their dynamic analogues as well dyn-mult . It will be interesting
to study the effects of the nonlinear term in the model equation studied here
on possible dynamic multiscaling. Recent studies have indicated that the
anomalous scaling in the Kraichnan model actually resembles more like that for
the Burgers equation for pressureless turbulence sreeni2 , instead of the form
predicted by perturbation theories. Similar studies could be undertaken for
the present nonlinear model to find out the validity of the perturbation
theory results discussed in the work. We hope in future our studies here will
provide new impetus to future analytical work to unearth the physics of
multiscaling in forced hydrodynamic turbulence and its analogous systems.

##  Appendix A One-loop corrections to $\nu$ for $\overline{y}>-2$

We start with the bare propagators and correlators in the action (30), which
are given by

$\displaystyle\langle\hat{\phi}_{\alpha}({\bf
q},\omega)\hat{\phi}_{\beta}(-{\bf q},-\omega)\rangle$ $\displaystyle=$
$\displaystyle 0$ (85) $\displaystyle\langle\hat{\phi}_{\alpha}({\bf
q},\omega)\phi_{\beta}(-{\bf q},-\omega)\rangle$ $\displaystyle=$
$\displaystyle\frac{\delta_{\alpha\beta}}{i\omega+\nu q^{2}}$
$\displaystyle\langle\hat{\phi}_{\alpha}(-{\bf q},-\omega)\phi_{\beta}({\bf
q},\omega)\rangle$ $\displaystyle=$
$\displaystyle\frac{\delta_{\alpha\beta}}{-i\omega+\nu q^{2}}$
$\displaystyle\langle\phi_{\alpha}({\bf q},\omega)\phi_{\beta}(-{\bf
q},-\omega)\rangle$ $\displaystyle=$
$\displaystyle\frac{2\delta_{\alpha\beta}\overline{D}q^{-\overline{y}}}{\omega^{2}+\nu^{2}q^{4}}.$

Figure 3: One-loop Feynman graphs that correct $\nu$. Diagrams (a-d) have
their origin in the mode-coupling term, where as diagram (e) comes from the
advective nonlinear term.

The relevant Feynman diagrams are shown in Fig. 3. We discuss the diagrams
which originate from the mode-coupling vertex in detail below.

(i) Diagrams (3)(a) and (b) combine to yield (each has symmetry factor 2)

$\displaystyle
g^{2}\hat{\phi}_{\alpha}\phi_{\delta}\epsilon_{\alpha\beta\gamma}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{d\Omega}{2\pi}\frac{2\overline{D}q^{-\overline{y}}}{\Omega^{2}+\nu^{2}q^{4}}\frac{1}{i\Omega+\nu({\bf
k-q})^{2}}\left(\epsilon_{\gamma\delta\mu}\delta_{\beta\mu}q^{2}({\bf
k-q})^{2}+\epsilon_{\beta\delta\mu}\delta_{\gamma\mu}q^{4}\right)$
$\displaystyle=g^{2}\hat{\phi}_{\alpha}\phi_{\delta}\epsilon_{\alpha\beta\gamma}\frac{\overline{D}}{2\nu^{2}}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{q^{-\overline{y}}}{q^{2}+({\bf
k-q})^{2}}\left(\epsilon_{\gamma\delta\mu}\delta_{\beta\mu}q^{2}({\bf
k-q})^{2}+\epsilon_{\beta\delta\mu}\delta_{\gamma\mu}q^{4}\right).$ (86)

We symmetrise this to obtain

$\displaystyle
g^{2}\hat{\phi}_{\alpha}\phi_{\delta}\epsilon_{\alpha\beta\gamma}\epsilon_{\gamma\delta\beta}\frac{\overline{D}}{4\nu^{2}}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{({\bf
k-q})^{2}-q^{2}}{q^{2}+({\bf k-q})^{2}}\left[q^{-\overline{y}}-|{\bf
k-q}|^{-\overline{y}}\right]$
$\displaystyle=g^{2}\hat{\phi}_{\alpha}\phi_{\delta}\epsilon_{\alpha\beta\gamma}\epsilon_{\gamma\delta\beta}\frac{\overline{D}}{4\nu^{2}}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{\bf
k\cdot
q}{q^{2}}\left[q^{-\overline{y}}-q^{-\overline{y}}\left(1+\overline{y}\frac{\bf
k\cdot q}{q^{2}}\right)\right]$
$\displaystyle=-g^{2}\hat{\phi}_{\alpha}\phi_{\delta}\left(\delta_{\alpha\delta}\delta_{\beta\beta}-\delta_{\alpha\beta}\delta_{\beta\delta}\right)\frac{\overline{D}}{4\nu^{2}}k_{d}k^{2}\int
dqq^{d-\overline{y}-3}=-\delta_{\alpha\delta}\frac{\overline{y}\overline{D}}{2d\nu^{2}}k_{d}k^{2}\int
dqq^{d-\overline{y}-3}\hat{\phi}_{\alpha}\phi_{\delta}.$ (87)

(ii) Diagrams (3)(c) and (d) combine to yield (each has a symmetry factor 2)

$\displaystyle\hat{\phi}_{\alpha}\phi_{\mu}g^{2}\epsilon_{\alpha\beta\gamma}k^{2}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{d\Omega}{2\pi}\frac{2\overline{D}|q|^{-\overline{y}}}{\Omega^{2}+\nu^{2}q^{4}}\frac{1}{i\Omega+\nu({\bf
k-q})^{2}}\left(\epsilon_{\gamma\delta\mu}\delta_{\beta\mu}({\bf
k-q})^{2}+\epsilon_{\beta\delta\mu}\delta_{\delta\mu}q^{2}\right)$
$\displaystyle=\hat{\phi}_{\alpha}\phi_{\mu}g^{2}\left(\epsilon_{\alpha\beta\gamma}\epsilon_{\gamma\delta\mu}\delta_{\beta\delta}+\epsilon_{\alpha\gamma\beta}\epsilon_{\gamma\delta\mu}\delta_{\beta\delta}\right)\frac{\overline{D}k^{2}}{2\nu^{2}}\int\frac{d^{d}}{(2\pi)^{d}}\frac{1}{q^{\overline{y}+2}}=0.$
(88)

The contribution from diagram 3(e) reads

$-\hat{\phi}_{\alpha}\phi_{\alpha}\lambda^{2}D_{1}\frac{d-1}{d}k_{d}\int
dqq^{d-1-y}.$ (89)

##  Appendix B Composite operators

Figure 4: One-loop Feynman graphs for the composite operator $\tilde{\cal
O}_{n}({\bf x})$. In both these diagrams, a small thick circle together with
the pair of small vertical parallel lines represent a composite operator, that
is expanded to form the one-loop diagrams in the lowest order perturbation
theory (see text).

We calculate the one-loop corrections to the composite operators $\tilde{\cal
O}_{n}({\bf x})$. The only relevant contribution comes from the model coupling
vertex in (30), since the advective vertex is ${\cal O}(q)$. The one-loop
diagram in Fig. (4) may be constructed by contracting any two of the
$\phi_{\alpha}$-fields in ${\cal\tilde{O}}_{n}$ with the two trilinear
anhamornic terms, each of the form
$(\nabla_{m}\hat{\phi}_{\alpha})\epsilon_{\alpha\beta\gamma}\phi_{\beta}\nabla_{m}\phi_{\gamma}$.
There are two distinct ways these contractions can be done:

(i) The two $\phi$-fields in ${\cal\tilde{O}}_{n}$ which are to be contracted
with the anhamornic terms have the same indices. The contribution reads

$n\frac{\overline{D}g^{2}}{\nu^{3}}\phi_{\alpha}\phi_{\alpha}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{1}{q^{\overline{y}+2}}.$
(90)

(ii) The two $\phi$-fields in ${\cal\tilde{O}}_{n}$ which are to be contracted
with the anhamornic terms have different indices. The corresponding
contribution reads

$\displaystyle
2n(n-1)\phi_{\beta}\phi_{\lambda}g^{2}\epsilon_{\alpha\beta\gamma}\epsilon{\mu\lambda\nu}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{d\Omega}{2\pi}\frac{2\overline{D}q^{4-\overline{y}}\delta_{\gamma\nu}}{(\Omega^{2}+\nu^{2}q^{4})^{2}}=n(n-1)\overline{D}\phi_{\beta}\phi_{\lambda}g^{2}\epsilon_{\alpha\beta\gamma}\epsilon{\mu\lambda\gamma}\int\frac{d^{d}q}{(2\pi)^{d}}\frac{1}{q^{\overline{y}+2}}$
(91) $\displaystyle=$ $\displaystyle
n(n-1)\overline{D}\phi_{\beta}\phi_{\lambda}g^{2}[\delta_{\alpha\mu}\delta_{\beta\lambda}-\delta_{\alpha\lambda}\delta_{\beta\mu}]\phi_{\beta\lambda}=0.$

Thus at the one-loop order, the fluctuation corrections to
${\cal\tilde{O}}_{n}$ are linear in $n$.

##  Appendix C Higher order composite operators

Due to the mode-coupling anharmonic term in (30), a composite operator of the
form ${\tilde{O}}_{m}({\bf x})=(\phi_{\alpha}({\bf x})\phi_{\alpha}({\bf
x}))^{m}$ for some positive integer $m$ can perturbatively generate a
composite operator ${\tilde{O}}_{p}({\bf x})$, where $p>m$. To the lowest
order perturbation expansion, $p=m+1$; see Fig. 5.

Figure 5: Generation of ${\cal\tilde{O}}_{3}({\bf x})$ from
${\cal\tilde{O}}_{2}({\bf x})$ in the lowest order perturbation theory.

This is distinct from the traditional passive scalar problem, where due to the
linearity of the equation of motion, there is no possibility of generating
higher order composite operators. This unambiguously allowed one to determine
the operator that made the largest contribution to the multiscaling of the
structure functions in the passive scalar problem. This holds true in the
present problem in the subspace where the renormalised coupling of the
nonlinear mode coupling term vanishes. In the other region of the phase space,
where this coupling is relevant, this is generally not true. We however note
that

(i) $\langle{\cal\tilde{O}}_{3}({\bf x})\rangle$ vanishes due to the
rotational invariance in the order parameter space. Thus, in the lowest order
perturbative expansion $\langle{\cal\tilde{O}}_{2}({\bf x})\rangle$ does not
connect to any higher order composite operators.

(ii) At higher order perturbation expansions, ${\cal\tilde{O}}_{2}({\bf x})$
can however generate higher order composite operators. For instance, in the
second order in $g$ (the coupling constant for the mode coupling
nonlinearity), ${\cal\tilde{O}}_{2}({\bf x})$ can generate
${\cal\tilde{O}}_{4}({\bf x})$; see Fig. 6.

Figure 6: Generation of ${\cal\tilde{O}}_{4}({\bf x})$ from
${\cal\tilde{O}}_{2}({\bf x})$ in the second order perturbation expansion.

Due to the particular form of the mode-coupling nonlinear term, it is however
clear that the fourth order composite operators are generated as well; this,
however, has the form $\phi_{\alpha}({\bf x})(\nabla^{2}\phi_{\beta}({\bf
x}))\phi_{\gamma}({\bf x})\nabla^{2}\phi_{\delta}({\bf x})$, which should be
subleading to $\phi_{\alpha}({\bf x})\phi_{\beta}({\bf x})\phi_{\gamma}({\bf
x})\phi_{\delta}({\bf x})$ for any $\alpha,\beta,\gamma,\delta$. We thus
expect this issue of the generation of the higher order composite operators is
unimportant in the asymptotic long wavevelength limit. Nonetheless, this issue
continues to persist in the realistic problems of hydrodynamic turbulence and
its analogue systems. We do not discuss it here any further.

##  Appendix D Multiscaling in Model II

Let us now consider the multiscaling in Model II, where we set
$\tilde{\Gamma}(q)=\eta q^{2}$. We work in the large $\eta$ limit, in which
the velocity dynamics effectively reduces to the Stokes’ equation. In this
limit, the Galilean invariance is restored. We focus on $\overline{y}<-2$. We
closely follow the calculational scheme outlined for Model I above; see also
Ref. tirtha . Model II is Galilean invariant and as a result, $\lambda$ does
not renormalise at any order in the perturbation theory. Furthermore, there
are no renormalisations to $g,\overline{D}$ and $D_{\eta}$ as well, for
reasons identical to those in Model I. It is clearly only the diagrams that
originate from the advective vertex in (30) are going to be affected by the
choice $\tilde{\Gamma}(q)=\eta q^{2}$. In other words, only the flow equation
for $\nu$ is affected. Noting that the infrared divergence those Feynman
graphs, for the same choice of $y$, are now more than what they were with
$\tilde{\Gamma}(q)=\Gamma$. In fact, it is easy to see read these
contributions from the corresponding contributions in Model I by replacing $y$
by $y+2$. This, after performing simple and standard procedure, gives

$\frac{d\tilde{g}_{2}}{dl}=\tilde{g}_{2}\left[y+2-d-\frac{g_{1}}{d}-2\tilde{g}_{2}\frac{d-1}{d}\right],$
(92)

where $\tilde{g}_{2}\equiv
k_{d}\frac{\lambda^{2}D_{\eta}}{\nu^{2}}\Lambda^{d-y-2}$. The flow equation
for the other dimensionless coupling constant $g_{1}$ is still given by (39)
with $g_{2}$ replaced by $\tilde{g}_{2}$. The fixed points can be directly
read off the fixed points obtained for Model I: (i) (0,0), (ii)
$(\frac{2d}{3}(\overline{y}+2-d),\,0)$, (iii) $(0,\,\frac{d}{2(d-1)}(y+2-d))$,
and (iv)
$2\frac{g^{*}_{1}}{4}+\tilde{g}^{*}_{2}(d-1)=d(y+2-d)=2(\overline{y}-d+2)\frac{d}{3}$.

We focus on the fixed point $g_{1}^{*}>0,\,\tilde{g}_{2}^{*}=0$. We note that
the condition of the nonrenormalisation of $\overline{D}$ (or equivalently the
flow equation (35)) still holds giving $\chi=\delta/3$, as in Model I together
with $2\delta/3>\tilde{\epsilon}\equiv y+2-d$. At this fixed point, the mode
coupling term is relevant and, as in Model I, the structure functions
${\cal\tilde{S}}_{2n}(r)$ characterise the universality in the inertial range.
Unsurprisingly, ${\cal\tilde{S}}_{2n}(r)$ show only scaling identical to Model
I. At the other fixed point $g_{1}^{*}=0,\tilde{g}_{2}^{*}>0$, the mode
coupling term is irrelevant (in a RG sense) and the structure functions ${\cal
S}_{2n}(r)$ are the ones which describe the universal multiscaling with
exponents identical to those in Ref. tirtha .

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