11footnotetext: These three authors contributed equally.

# Stochastic dynamics of chemotactic colonies with logistic growth

Riccardo Ben Alì Zinatia<EMAIL_ADDRESS>Sorbonne Université & CNRS,
Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005,
Paris, France  Charlie Ducluta<EMAIL_ADDRESS>Max-Planck-Institut für
Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany  Saeed
Mahdisoltania<EMAIL_ADDRESS>Rudolf Peierls Centre for
Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom  Max
Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen,
Germany  Andrea Gambassi<EMAIL_ADDRESS>SISSA — International School for
Advanced Studies & INFN, via Bonomea 265, I-34136 Trieste, Italy  Ramin
Golestanian<EMAIL_ADDRESS>Max Planck Institute for Dynamics and
Self-Organization (MPIDS), 37077 Göttingen, Germany  Rudolf Peierls Centre for
Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom

###### Abstract

The interplay between cellular growth and cell-cell signaling is essential for
the aggregation and proliferation of bacterial colonies, as well as for the
self-organization of cell tissues. To investigate this interplay, we focus
here on the collective properties of dividing chemotactic cell colonies by
studying their long-time and large-scale dynamics through a renormalization
group (RG) approach. The RG analysis reveals that a relevant but
unconventional chemotactic interaction – corresponding to a polarity-induced
mechanism – is generated by fluctuations at macroscopic scales, even when an
underlying mechanism is absent at the microscopic level. This emerges from the
interplay of the well-known Keller–Segel (KS) chemotactic nonlinearity and
cell birth and death processes. At one-loop order, we find no stable fixed
point of the RG flow equations. We discuss a connection between the dynamics
investigated here and the celebrated Kardar–Parisi–Zhang (KPZ) equation with
long-range correlated noise, which points at the existence of a strong-
coupling, nonperturbative fixed point.

##  I Introduction

The pervasiveness of emergent self-organization in living systems shows how
local interactions among individuals underpin global structures whose size can
significantly exceed the microscopic scale set by each unit. Chemotaxis,
namely the ability of cells and synthetic particles to direct their motion
according to the spatial gradient of some chemical signals, is one of the
primary mechanisms for such self-organization in many biological processes,
e.g. , immune response and cancer metastasis. Phenomenological models
describing chemotaxis as a directed motion guided by the chemical gradients
Keller and Segel (1970, 1971) and undergoing stochastic fluctuations Grima
(2005); Golestanian (2009); Sengupta  _et al._ (2009); Taktikos  _et al._
(2012) have proven useful in studying chemotactic aggregation (collapse) of
bacteria Brenner  _et al._ (1998); Chavanis and Sire (2004), the collective
behavior of synthetically active colloids and enzymes Golestanian (2012);
Thakur and Kapral (2012); Cohen and Golestanian (2014); Agudo-Canalejo  _et
al._ (2018); Agudo-Canalejo and Golestanian (2019), as well as the interplay
between birth and death processes among the living individuals of a colony
Gelimson and Golestanian (2015); Kruse  _et al._ (2005).

A thorough investigation of the emergent properties of active systems can
occasionally be achieved via a bottom-up approach and by coarse-graining the
microscopic dynamics of individual particles. Nevertheless, it is known that
such a direct coarse-graining may not necessarily expose all the interaction
terms that turn out to be relevant for describing the macroscopic dynamics of
the system. To obtain a sound macroscopic description of the possibly emerging
collective dynamics, one generally relies on statistical and field-theoretical
techniques, such as the renormalization group (RG). These dictate the
structure of the relevant interactions that are compatible with the symmetry
and conservation laws of the system under investigation Täuber (2014); Kardar
(2007). Such approaches have been successfully utilized in characterizing the
dynamical properties of a wide range of biological and synthetic systems Toner
and Tu (1995); Risler  _et al._ (2004); Caballero  _et al._ (2018); Cavagna
_et al._ (2019, 2021); Mahdisoltani  _et al._ (2021); Löffler  _et al._
(2021); Ro  _et al._ (2021).

In the present work, we investigate the collective behavior of chemotactic
particles undergoing birth and death processes by dynamic renormalization
group theory. We find that an unconventional chemotactic coupling, previously
studied in the context of polarity-induced chemotaxis Mahdisoltani  _et al._
(2021), is generated at mesoscopic scales upon coarse-graining the dynamics,
even when the underlying polarity mechanism is not explicitly present at the
single-particle level. We therefore extend the analysis presented in Ref.
Gelimson and Golestanian (2015), where the polarity-induced chemotactic term
was neglected, leading to a theory not closed under renormalization. The
extended model presents a critical point separating a homogeneous phase with
constant density from a collapsed phase characterized by aggregation and
nonuniform particle density. At that critical point, in the framework of a
perturbative dimensional expansion, we find that the RG flow exhibits a
runaway behavior, with no real-valued stable RG fixed points in the vicinity
of the upper critical dimension. However we highlight a connection between our
model and the celebrated Kardar–Parisi–Zhang (KPZ) equation with long-range
correlated noise Medina  _et al._ (1989), which suggests a possible
explanation for the runaway behavior in terms of the existence of a strong-
coupling fixed point which is beyond the reach of our analysis.

##  II Stochastic chemotactic model with growth

###  II.1 Stochastic Keller–Segel model

A corpuscular description of the dynamics of the mobile chemotactic particles
is obtained by considering $n$ particles moving in a $d$-dimensional space,
coupled according to the stochastic Langevin equation
$\dot{\mathbf{r}}_{i}=\mu_{1}\nabla\Phi(\mathbf{r}_{i},t)+\bm{\xi}_{i}\,,$
where $\mathbf{r}_{i}(t)$ describes the position of particle $i$, which
undergoes an overdamped motion. The trajectory of each particle is biased by
gradients of the chemical concentration field $\Phi$ according to the
Keller–Segel (KS) model Keller and Segel (1971); in addition, particles are
subject to Gaussian white noises $\bm{\xi}_{i}$ with vanishing mean and
correlations
$\langle\xi_{i,\alpha}(t)\xi_{j,\beta}(t^{\prime})\rangle=2D\delta_{ij}\delta_{\alpha\beta}\delta(t-t^{\prime})$,
where $D$ is the noise strength, $i,j=1,\dots,n$ are particle indices, and
$\alpha,\beta$ indicate Cartesian coordinates.

As we are interested in the large-scale behavior of the colony, it is
convenient to move from a microscopic description, to a coarser one involving
a smoothly-varying version $C(\mathbf{r},t)$ of the particle density
$\sum_{i=1}^{n}\delta^{d}(\mathbf{r}-\mathbf{r}_{i})$. To include the
important fluctuations in the dynamics of the chemotactic colony Grima (2005);
Newman and Grima (2004), we go beyond a mean-field description following the
Dean–Kawasaki approach Dean (1996); Kawasaki (1994), which yields a stochastic
field theory for the density distribution of chemotactic particles Chavanis
(2010); Mahdisoltani  _et al._ (2021). The resulting Langevin dynamics
describing the colony of chemotactic particles is given by Chavanis (2010)

$\displaystyle\partial_{t}C$
$\displaystyle=D\nabla^{2}C-\mu_{1}\nabla\cdot(C\nabla\Phi)+\nabla\cdot\bm{\xi}^{\rm
con}\,.$ (1)

This equation describes the stochastic evolution of the particle density with
diffusion coefficient $D$ and with KS chemotactic current $\mu_{1}C\nabla\Phi$
Keller and Segel (1971). The multiplicative noise field $\bm{\xi}^{\rm con}$,
which arises from the individual noises $\bm{\xi}_{i}$, has a vanishing mean
and correlations $\langle\xi^{\rm con}_{\alpha}(\mathbf{r},t)\xi^{\rm
con}_{\beta}(\mathbf{r}^{\prime},t^{\prime})\rangle=2DC(\mathbf{r},t)\delta_{\alpha\beta}\delta^{d}(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$.

We assume that the particles release chemicals at constant rate. Under the
assumption that the diffusion of chemicals is sufficiently fast compared to
the particles’ dynamics, the chemical field $\Phi$ instantaneously reaches the
steady-state profile with $\partial_{t}\Phi(\mathbf{r},t)\simeq 0$
corresponding to a given $C(\mathbf{r},t)$ and is governed by the screened
Poisson equation

$\displaystyle\left(-\nabla^{2}+\kappa^{2}\right)\Phi(\mathbf{r},t)=C(\mathbf{r},t)\,,$
(2)

where $\kappa^{-1}$ is an effective screening length determined by the
diffusive spread of the chemical signals before their degradation occurs.

###  II.2 Nonlinear birth and death processes

The combined effect of the cell birth and death processes and their long-range
chemotactic interactions are essential to understand the macroscopic
properties of growing cell colonies. As discussed in Ref. Gelimson and
Golestanian (2015), in order to account for birth and decay processes in the
coarse-grained description of self-chemotactic systems, we phenomenologically
extend the Langevin equation (1) by considering (i) the cell division process
$A\to 2A$ with rate $\beta_{\rm d}$, and (ii) the cell death/coagulation
process $2A\to A$ with rate $\beta_{\rm c}$. Utilizing a system-size expansion
van Kampen (1992); Gillespie (2000), one can encode these microscopic
processes into a Langevin description for the particle density within a system
of volume $\Omega$, which is valid in the limit of a large number of particles
and reads

$\displaystyle\left.\partial_{t}C\right|_{\rm growth}=\lambda
C(C_{0}-C)+\xi^{\rm nc}\,.$ (3)

Here, the subscript “growth” indicates that only the contributions from cell
growth and death processes are accounted for in this equation. We have
introduced the growth coefficient $\lambda=\beta_{\rm c}\Omega$, the carrying
capacity $C_{0}=(1+\beta_{\rm d}/\beta_{\rm c})/\Omega$, and the nonconserving
multiplicative noise $\xi^{\rm nc}$ with correlations

$\displaystyle\begin{split}\langle\xi^{\rm nc}(\bm{r},t)&\xi^{\rm
nc}(\bm{r}^{\prime},t^{\prime})\rangle=\\\ &\lambda
C(C_{0}-2/\Omega+C)\delta^{d}(\bm{r}-\bm{r}^{\prime})\delta(t-t^{\prime})\,.\end{split}$
(4)

Since we are interested in the dynamics in the asymptotic large-scale and
long-time regime, the specific form of the microscopic processes that led to
Eq. (3) is irrelevant, provided that $C_{0}$ is a stable steady state of the
growth process. It is worth mentioning that the Langevin description (3)
breaks down close to the absorbing state ($C_{0}\to 0$), as suggested by the
fact that, correspondingly, the noise correlations may become negative,
leading to a Langevin dynamics with an ill-defined imaginary noise. Close to
the absorbing state, one thus needs to make use of alternative descriptions of
the growth processes that are exact at low densities Benitez  _et al._ (2016);
Doering  _et al._ (2003). In the following, we focus on the case in which the
particle density is sufficiently large to be far from the absorbing state and
such that the description given by Eq. (4) is accurate.

###  II.3 Dynamics of density fluctuations

Provided the chemotactic colony is considered sufficiently far from the
absorbing state of the dynamics, we expand the particle density $C$ around the
constant carrying capacity $C_{0}$ as
$C(\mathbf{r},t)=C_{0}+\rho(\mathbf{r},t)$, where $\rho\ll C_{0}$, and
subsequently focus on the dynamics of the density-fluctuation field $\rho$.
The chemical concentration $\Phi$ may also be expanded around a spatially
uniform concentration $\Phi_{0}$ according to
$\Phi(\mathbf{r},t)=\Phi_{0}+\phi(\mathbf{r},t)$, where $\phi$ represents the
chemical fluctuations that induce chemotactic interactions among the particles
Chavanis and Sire (2007). As a result, Eq. (2) relating the chemical
concentration and particle density fields can be decomposed as
$(-\nabla^{2}+\kappa^{2})\Phi_{0}=C_{0}$ and
$(-\nabla^{2}+\kappa^{2})\phi\,=\rho(\mathbf{r},t)$, where the first equation
admits the uniform solution $\Phi_{0}=\kappa^{-2}C_{0}$. Accordingly, the
background chemical concentration $\Phi_{0}$ is maintained by the uniform part
of the particle density $C_{0}$, whereas the chemical fluctuation field
$\phi(\mathbf{r},t)$ is induced by the particle fluctuations,
$\rho(\mathbf{r},t)$. In the limit in which the chemical signals do not
degrade considerably within the system size and are therefore capable of
mediating interactions among distant particles Chavanis and Sire (2007);
Golestanian (2012), the chemical fluctuation $\phi$ is given by the Poisson-
like equation

$\displaystyle-\nabla^{2}\phi(\mathbf{r},t)=\rho(\mathbf{r},t).$ (5)

As we discuss below, focusing on particle and chemical fluctuations $\rho$ and
$\phi$ also renders the noise terms in the dynamics additive Gelimson and
Golestanian (2015); Mahdisoltani  _et al._ (2021).

###  II.4 Chemotactic model with growth

Let us consider the coarse-grained model of chemotaxis that also accounts for
birth and decay processes. Combining Eqs. (1) and (3), the stochastic Langevin
equation for the density fluctuations $\rho$ reads

$(\tau\partial_{t}-D\nabla^{2}+\sigma)\rho=-\lambda\rho^{2}-\mu_{1}\nabla(\rho\nabla\phi)-\mu_{2}\nabla^{2}(\nabla\phi)^{2}+\zeta.$
(6)

The linear term $\sigma=(\lambda-\mu_{1})C_{0}$ introduces a length scale
$\xi=\sqrt{D/\sigma}$ over which density fluctuations remain correlated. Such
correlation length diverges when the control parameter $\sigma$ approaches its
critical value $\sigma_{c}$. This critical point $\sigma_{c}$ demarks the
transition between a dilute, homogeneous phase for $\sigma>\sigma_{c}$, and a
dense, collapsed one for $\sigma<\sigma_{c}$ Mahdisoltani  _et al._ (2021). In
the absence of nonlinearities, the critical value $\sigma_{c,0}=0$ is achieved
when the linear part of the chemotactic aggregation term $\mu_{1}C_{0}$
balances the linear decay contribution $C_{0}\lambda$. In the presence of
nonlinearities, this critical value will be shifted, with a non-universal
fluctuation-induced shift that depends on microscopic details. Note also that
we have introduced the time-scale parameter $\tau$ since it eventually
acquires a non-trivial renormalization.

In addition to the KS chemotactic nonlinearity
$\mu_{1}\nabla\cdot(\rho\nabla\phi)$, in Eq. (6) we have also introduced a
second chemotactic nonlinearity $\mu_{2}\nabla^{2}(\nabla\phi)^{2}$; this
unconventional interaction is a relevant coupling in the macroscopic dynamics,
and, as it has been shown in Ref. Mahdisoltani  _et al._ (2021), it gives rise
to novel large-scale phenomena such as superdiffusion and non-Poissonian
number fluctuations in the absence of nonlinear growth processes. The
$\mu_{2}$ coupling may arise from the polarity effects in the microscopic
dynamics of chemotactic particles, and therefore this terms is referred to as
the polarity-induced chemotaxis Mahdisoltani  _et al._ (2021). As it will
become clear below, even if this term is not present in the single-particle
dynamics of Eq. (6), the polarity-induced coupling is effectively generated in
the macroscopic dynamics upon coarse-graining and it is as relevant (in the RG
sense) as the KS nonlinear term $\propto\mu_{1}$ and the nonlinear growth term
$\propto\lambda$. In this respect, the present analysis extends the model of
Ref. Gelimson and Golestanian (2015) by incorporating the effective polarity-
induced coupling in the chemotactic dynamics.

Finally, the noise term $\zeta=\xi^{\rm nc}+\nabla\cdot\bm{\xi}^{\rm con}$ in
Eq. (6) is Gaussian and combines an additive and a multiplicative part. As
long as the system evolves far from the absorbing state, the expansion
$C=C_{0}+\rho$ allows us to discard the multiplicative part of the noise
$\zeta$, as it is also RG irrelevant Mahdisoltani  _et al._ (2021). The
remaining additive noise has correlations given by

$\langle\zeta(\mathbf{r},t)\zeta(\mathbf{r}^{\prime},t^{\prime})\rangle=2(\mathcal{D}_{0}-\mathcal{D}_{2}\nabla^{2})\delta^{d}(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime}),$
(7)

where the conserving part $\mathcal{D}_{2}=DC_{0}$ comes from the stochastic
KS equation (1), whereas the nonconserving part $\mathcal{D}_{0}=\lambda
C_{0}^{2}$ comes from the expansion111Note that we have neglected the term
$2/\Omega$ compared to $C_{0}$ in Eq. (4), which is consistent with the
assumption of being far from an absorbing state and of considering large
populations, such that $\Omega\gg 1$. of the stochastic growth equation (3).

To conclude this section, let us analyse the symmetries of the stochastic
chemotaxis model with growth. The Langevin equation (6) possesses a trivial
$\phi$-shift symmetry and a more subtle pseudo-Galilean symmetry. The
$\phi$-shift symmetry is a direct consequence of the fact that the dynamics
considered here is only sensitive to the gradients of the chemical field
$\phi$, which makes the Langevin equation trivially invariant under the shift
$\phi\to\phi^{\prime}=\phi+\text{const}$. The pseudo-Galilean symmetry
expresses the invariance of the dynamics under the transformation Mahdisoltani
_et al._ (2021)

$\displaystyle\phi^{\prime}(\mathbf{r},t)$
$\displaystyle=\phi(\mathbf{r}+t(\mu_{1}-2\mu_{2})\mathbf{v}/\tau,t)-\mathbf{v}\cdot\mathbf{r}\,,$
(8a) $\displaystyle\rho^{\prime}(\mathbf{r},t)$
$\displaystyle=\rho(\mathbf{r}+t(\mu_{1}-2\mu_{2})\mathbf{v}/\tau,t)\,,$ (8b)

where $\mathbf{v}$ is an arbitrary $d$-dimensional boost vector. As we discuss
below, this symmetry enforces an identity between critical exponents.
Moreover, in the absence of the $\lambda\rho^{2}$ term, Eq. (6) reduces to the
number-conserving chemotaxis model considered in Ref. Mahdisoltani  _et al._
(2021). In that case, the pseudo-Galilean symmetry, accompanied by the non-
renormalization of the noise term, is such that the dynamical scaling
exponents can be determined exactly. In the present case of chemotactic
dynamics with logistic growth, namely Eq. (6), even if an exact exponent
identity holds as a consequence of the pseudo-Galilean transformation (8) (see
Eq. (13)), the noise term $\mathcal{D}_{0}$ acquires a non-trivial
renormalization due to the presence of $\lambda$. As a consequence, the
critical exponents are no longer known exactly, but we describe in the
following how they can be calculated by using RG techniques.

##  III Renormalization group analysis

###  III.1 Perturbative expansion

Transforming Eq. (6) in Fourier space222Here we use the following convention
$f(\mathbf{x},t)=\int_{\hat{k}}f(\mathbf{k},\omega)\mathrm{e}^{-\mathrm{i}\omega
t+\mathrm{i}\mathbf{k}\cdot\mathbf{x}}$ with shorthand notation
$\hat{k}\equiv(\mathbf{k},\omega)$ and
$\int_{\hat{k}}\equiv\int{\rm{d}}^{d}\mathbf{k}\,{\rm{d}}\omega/(2\pi)^{d+1}$.
we obtain

$\rho(\hat{k})=G_{0}(\hat{k})\zeta(\hat{k})+G_{0}(\hat{k})\int_{\hat{k}_{1}}V_{0}(\mathbf{k},\mathbf{k}_{1})\rho(\hat{k}_{1})\rho(\hat{k}-\hat{k}_{1})\,,$
(9)

where $G_{0}(\hat{k})\equiv(-\mathrm{i}\tau\omega+Dk^{2}+\sigma)^{-1}$ is the
bare propagator, while the nonlinearities $\mu_{1,2}$ and $\lambda$ enter via
the the vertex function $V_{0}(\mathbf{k},\mathbf{k}_{1})$, which upon
symmetrization $\mathbf{k}_{1}\leftrightarrow\mathbf{k}-\mathbf{k}_{1}$, reads

$\begin{split}V_{0}(\mathbf{k},\mathbf{k}_{1})=&\,-\lambda-\mu_{2}\,\frac{\mathbf{k}_{1}\cdot(\mathbf{k}-\mathbf{k}_{1})k^{2}}{(\mathbf{k}-\mathbf{k}_{1})^{2}k_{1}^{2}}+\\\
&+\frac{1}{2}\mu_{1}\left[\frac{\mathbf{k}\cdot\mathbf{k}_{1}}{k_{1}^{2}}+\frac{\mathbf{k}\cdot(\mathbf{k}-\mathbf{k}_{1})}{(\mathbf{k}-\mathbf{k}_{1})^{2}}\right]\,.\end{split}$
(10)

We then denote the bare correlation function (spectral function)
$N_{0}(\hat{k})=2\mathcal{D}_{0}|G_{0}(\hat{k})|^{2}$. Note that we have
discarded the conserving part of the noise correlations (7) proportional to
$\mathcal{D}_{2}$, as it is irrelevant in the RG sense. In view of setting the
perturbative expansion, it is convenient to introduce the following
diagrammatic representation Forster  _et al._ (1977); Medina  _et al._ (1989)

$G_{0}(\hat{k})=\leavevmode\hbox to15.63pt{\vbox
to3.2pt{\pgfpicture\makeatletter\hbox{\hskip 1.19998pt\lower-1.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.6532pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6532pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},~{}~{}~{}~{}V_{0}(\mathbf{k},\mathbf{k}_{1})=\leavevmode\hbox
to14.63pt{\vbox to14.63pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.31319pt\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{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{14.22638pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},~{}~{}~{}~{}N_{0}(\hat{k})=\leavevmode\hbox
to20.32pt{\vbox to4.4pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.2pt\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{ } {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.6532pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.6532pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
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{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
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}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{13.26364pt}{0.0pt}\pgfsys@invoke{
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}\pgfsys<EMAIL_ADDRESS>(11)

The integral equation (9) is then solved perturbatively in the vertex $V_{0}$:
the one-loop corrections to the bare quantities (11) are schematically given
by

$\displaystyle G(\hat{k})$ $\displaystyle=~{}~{}\leavevmode\hbox
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{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.6532pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,+\,4~{}\leavevmode\hbox
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{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{4.8749pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{5.33488pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}{}{{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{33.32765pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{33.78764pt}{0.0pt}\pgfsys@lineto{35.56595pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}{}{}{}{}{{}}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{-5.89282pt}{11.89015pt}{-10.66978pt}{17.32298pt}{-10.66978pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{17.78297pt}{-10.66978pt}\pgfsys@curveto{23.6758pt}{-10.66978pt}{28.45276pt}{-5.89282pt}{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}{}{{}}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{5.17793pt}{10.83092pt}{9.60876pt}{15.47722pt}{10.42793pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}{}{{}}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@curveto{28.45276pt}{5.17793pt}{24.73503pt}{9.60876pt}{20.08873pt}{10.42793pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.98482}{0.17361}{-0.17361}{-0.98482}{20.08875pt}{10.42792pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{17.78297pt}{10.66978pt}\pgfsys@moveto{19.78297pt}{10.66978pt}\pgfsys@curveto{19.78297pt}{11.77437pt}{18.88756pt}{12.66978pt}{17.78297pt}{12.66978pt}\pgfsys@curveto{16.67839pt}{12.66978pt}{15.78297pt}{11.77437pt}{15.78297pt}{10.66978pt}\pgfsys@curveto{15.78297pt}{9.5652pt}{16.67839pt}{8.66978pt}{17.78297pt}{8.66978pt}\pgfsys@curveto{18.88756pt}{8.66978pt}{19.78297pt}{9.5652pt}{19.78297pt}{10.66978pt}\pgfsys@closepath\pgfsys@moveto{17.78297pt}{10.66978pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{17.78297pt}{10.66978pt}\pgfsys@moveto{19.78297pt}{10.66978pt}\pgfsys@curveto{19.78297pt}{11.77437pt}{18.88756pt}{12.66978pt}{17.78297pt}{12.66978pt}\pgfsys@curveto{16.67839pt}{12.66978pt}{15.78297pt}{11.77437pt}{15.78297pt}{10.66978pt}\pgfsys@curveto{15.78297pt}{9.5652pt}{16.67839pt}{8.66978pt}{17.78297pt}{8.66978pt}\pgfsys@curveto{18.88756pt}{8.66978pt}{19.78297pt}{9.5652pt}{19.78297pt}{10.66978pt}\pgfsys@closepath\pgfsys@moveto{17.78297pt}{10.66978pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},$ $\displaystyle
V(\mathbf{k},\mathbf{k}_{1})$ $\displaystyle=~{}~{}\leavevmode\hbox
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0.2pt\lower-7.31319pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{14.22638pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,+\,4~{}\leavevmode\hbox
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where the prefactors in front of each one-loop contribution accounts for all
the possible noise contractions (the corresponding explicit expressions are
given in Supplementary Material). The renormalization group procedure can then
be set by first integrating out fluctuations within the momentum shell
$\Lambda\mathrm{e}^{-\ell}\leq|\mathbf{k}|\leq\Lambda$, where $\Lambda$
represents the momentum cutoff (set, e.g. , by the inverse particle size).
This momentum integration is followed by a rescaling back to the original
cutoff $\Lambda$, a rescaling of the fields according to
$\rho\to\rho^{\prime}=\mathrm{e}^{\ell\chi}\rho$,
$\eta\to\eta^{\prime}=\mathrm{e}^{-\ell\tilde{\chi}}\eta$, and a temporal
rescaling $t\to t^{\prime}=\mathrm{e}^{\ell z}t$. The roughness exponent
$\chi$, the dynamical exponent $z$, and the exponent $\tilde{\chi}$,
characterize the scaling behavior of the critical system in the macroscopic
limit. For instance, the long-time and large-scale particle density
correlations assume the scaling form Täuber (2014); Medina  _et al._ (1989)

$\langle\rho(\mathbf{x},t)\rho(\mathbf{x}^{\prime},t)\rangle\sim|\mathbf{x}^{\prime}-\mathbf{x}|^{2\chi}F\left(\frac{|t^{\prime}-t|}{|\mathbf{x}^{\prime}-\mathbf{x}|^{z}}\right)\,,$
(12)

where $F$ is a scaling function.

The exponent $\tilde{\chi}$ determines the scaling properties of the response
function $R$ of the system to an external force
$h(\mathbf{x}^{\prime},t^{\prime})$. Namely, by defining
$R(\mathbf{x},t,\mathbf{x}^{\prime},t^{\prime})=\delta\langle\rho(\mathbf{x},t)\rangle/\delta
h(\mathbf{x}^{\prime},t^{\prime})|_{h=0}$, we have
$R(\mathbf{x},t,\mathbf{x}^{\prime},t^{\prime})\sim|\mathbf{x}^{\prime}-\mathbf{x}|^{\chi+\tilde{\chi}}H(|t^{\prime}-t|/|\mathbf{x}^{\prime}-\mathbf{x}|^{z})$
in the asymptotic limit, with $H$ a scaling function. The three critical
exponents $z$, $\chi$ and $\tilde{\chi}$ are in principle independent;
however, the pseudo-Galilean symmetry (8) enforces the exact exponent identity
Mahdisoltani  _et al._ (2021)

$\displaystyle z+\chi=0\,.$ (13)

Futhermore, we note that for systems relaxing to an equilibrium configuration
(which is not the case here), the fluctuation-dissipation theorem provides a
relation between response function and correlation function Aron  _et al._
(2010), which implies the exponent identity $2\chi-z=\chi+\tilde{\chi}$. This
identity is broken333Note that the latter identity is accidentally recovered
at the fixed points of the dynamics when the nonlinear growth coupling is set
to zero ($\lambda=0$) – even though the polarity-induced coupling is itself a
nonequilibrium interaction and is not derivable from a free-energy functional
Mahdisoltani  _et al._ (2021). for the critical dynamics of Eq. (6).

We obtain the RG functions in the limit $\ell\to 0$ and through a dimensional
expansion in the parameter $\varepsilon=d_{c}-d\ll 1$ around the upper
critical dimension $d_{c}=6$. On defining the rescaled variables
$\bar{\lambda}^{2}=\lambda^{2}K_{d}\Lambda^{-\varepsilon}\mathcal{D}_{0}/(D^{3}\tau)$
and
$\bar{\mu}^{2}_{1,2}=\mu^{2}_{1,2}K_{d}\Lambda^{-\varepsilon}\mathcal{D}_{0}/(D^{3}\tau)$,
where $K_{d}\equiv S_{d}/(2\pi)^{d}$ with $S_{d}$ the area of a unit sphere in
$d$ dimensions, we obtain the following RG flow equations

$\displaystyle\frac{\partial_{\ell}\mathcal{D}_{0}}{\mathcal{D}_{0}}=\,$
$\displaystyle d+z+2\tilde{\chi}+\bar{\lambda}^{2}\,,$ (14a)
$\displaystyle\begin{split}\frac{\partial_{\ell}D}{D}=\,&d+z+\chi+\tilde{\chi}-2-\frac{25}{48}\bar{\mu}_{1}^{2}-\frac{15}{8}\bar{\mu}_{1}\bar{\mu}_{2}\\\
&-\frac{1}{16}\bar{\mu}_{1}\bar{\lambda}+\frac{53}{24}\bar{\mu}_{2}\bar{\lambda}+\frac{2}{3}\bar{\lambda}^{2}\,,\end{split}$
(14b) $\displaystyle\partial_{\ell}\tau=\,$
$\displaystyle(d+\chi+\tilde{\chi})\tau-\frac{5}{6}\bar{\mu}_{1}\bar{\lambda}+\bar{\lambda}^{2}\,,$
(14c)
$\displaystyle\begin{split}\partial_{\ell}{\bar{\mu}}_{1}=\,&\left(d+z+2\chi+\tilde{\chi}\right){\bar{\mu}}_{1}-\frac{7}{8}\bar{\mu}_{1}^{2}\bar{\lambda}+\frac{11}{6}\bar{\mu}_{1}\bar{\mu}_{2}\bar{\lambda}\\\
&+\frac{1}{2}\bar{\mu}_{2}^{2}\bar{\lambda}+\bar{\mu}_{1}\bar{\lambda}^{2}-2\bar{\mu}_{2}\bar{\lambda}^{2}\,,\end{split}$
(14d)
$\displaystyle\begin{split}\partial_{\ell}\bar{\mu}_{2}=\,&\left(d+z+2\chi+\tilde{\chi}\right){\bar{\mu}}_{2}-\frac{1}{48}\bar{\mu}_{1}^{2}\bar{\lambda}\\\
&+\frac{1}{12}\bar{\mu}_{1}\bar{\mu}_{2}\bar{\lambda}+\frac{1}{4}\bar{\mu}_{2}^{2}\bar{\lambda}\,,\end{split}$
(14e)
$\displaystyle\begin{split}\partial_{\ell}{\bar{\lambda}}=\,&(d+z+2\chi+\tilde{\chi})\bar{\lambda}+\frac{35}{16}\bar{\mu}_{1}^{2}\bar{\lambda}+\frac{5}{4}\bar{\mu}_{1}\bar{\mu}_{2}\bar{\lambda}\\\
&+\frac{5}{12}\bar{\mu}_{2}^{2}\bar{\lambda}-6\bar{\mu}_{1}\bar{\lambda}^{2}-\frac{4}{3}\bar{\mu}_{2}\bar{\lambda}^{2}+4\bar{\lambda}^{3}\,.\end{split}$
(14f)

By taking the logarithm of the definition of $\bar{\mu}_{1,2}$ and
$\bar{\lambda}$, differentiating w.r.t. $\ell$, and taking advantage of Eq.
(14), we obtain the following closed flow equations for arbitrary $\chi$,
$\tilde{\chi}$ and $z$,

$\displaystyle\begin{split}\partial_{\ell}\bar{\lambda}=\,&\frac{1}{2}\bar{\lambda}\varepsilon+3\bar{\lambda}^{3}-\frac{527}{96}\bar{\mu}_{1}\bar{\lambda}^{2}-\frac{223}{48}\bar{\mu}_{2}\bar{\lambda}^{2}\\\
&+\frac{95}{32}\bar{\mu}_{1}^{2}\bar{\lambda}+\frac{5}{12}\bar{\mu}_{2}^{2}\bar{\lambda}+\frac{65}{16}\bar{\mu}_{1}\bar{\mu}_{2}\lambda\,,\end{split}$
(15a)
$\displaystyle\begin{split}\partial_{\ell}\bar{\mu}_{1}=\,&\frac{1}{2}\bar{\mu}_{1}\varepsilon+\frac{25}{32}\bar{\mu}_{1}^{3}-\frac{35}{96}\bar{\mu}_{1}^{2}\bar{\lambda}+\frac{45}{16}\bar{\mu}_{1}^{2}\bar{\mu}_{2}\\\
&-\frac{71}{48}\bar{\mu}_{2}\bar{\mu}_{1}\bar{\lambda}+\frac{1}{2}\bar{\mu}_{2}^{2}\bar{\lambda}-2\bar{\mu}_{2}\bar{\lambda}^{2}\,,\end{split}$
(15b)
$\displaystyle\begin{split}\partial_{\ell}\bar{\mu}_{2}=\,&\frac{1}{2}\bar{\mu}_{2}\varepsilon-\frac{49}{16}\bar{\mu}_{2}^{2}\bar{\lambda}+\frac{45}{16}\bar{\mu}_{1}\bar{\mu}_{2}^{2}-\bar{\lambda}^{2}\bar{\mu}_{2}\\\
&+\frac{19}{32}\bar{\mu}_{1}\bar{\mu}_{2}\bar{\lambda}+\frac{25}{32}\bar{\mu}_{1}^{2}\bar{\mu}_{2}-\frac{1}{48}\bar{\mu}_{1}^{2}\bar{\lambda}\,.\end{split}$
(15c)

These one-loop RG equations are the main technical result of this paper.
Consistently with the proximity to the Gaussian fixed point, we choose the
values of $\chi$, $\tilde{\chi}$ and $z$ such that $D$, $\mathcal{D}_{0}$ and
$\tau$ do not flow under RG, i.e. , we impose
$\partial_{\ell}D=\partial_{\ell}\mathcal{D}_{0}=\partial_{\ell}\tau=0$. This
choice is such that the scaling of the correlation length $\xi$, which is
controlled by the ratio $D/\sigma$, is determined solely by the scaling of
$\sigma$. Likewise, the density fluctuations in the renormalized system, which
are controlled by the ratio $\mathcal{D}_{0}/(D\tau)$, remain unchanged at the
critical point and their scaling can therefore be determined directly from
$\chi$, $\tilde{\chi}$ and $z$. Adopting this choice, we find

$\displaystyle z$
$\displaystyle=2\\!+\\!\frac{25}{48}\bar{\mu}_{1}^{2}\\!+\\!\frac{15}{8}\bar{\mu}_{1}\bar{\mu}_{2}\\!-\\!\frac{37}{48}\bar{\mu}_{1}\bar{\lambda}\\!-\\!\frac{53}{24}\bar{\mu}_{2}\bar{\lambda}\\!+\\!\frac{1}{3}\bar{\lambda}^{2}\,,$
(16a) $\displaystyle\tilde{\chi}$
$\displaystyle=-\frac{1}{2}(z+d+\bar{\lambda}^{2})\,,$ (16b)
$\displaystyle\chi$
$\displaystyle=-d-\tilde{\chi}-\bar{\lambda}^{2}+\frac{5}{6}\bar{\mu}_{1}\bar{\lambda}\,.$
(16c)

Lastly, the critical state of the system can be reached by fine-tuning
$\sigma$ to its critical value $\sigma_{c}$. This is a fluctuation-induced and
non-universal quantity, the value of which we report in the Supplementary
Material. It is customary to introduce the reduced control parameter
$r\equiv\sigma-\sigma_{c}$ such that the critical state of the system is
reached for $r\to 0$. The corresponding RG flow then reads

$\frac{\partial_{\ell}r}{r}=d+z+\chi+\tilde{\chi}-\frac{7}{2}\bar{\mu}_{1}\bar{\lambda}+\frac{1}{3}\bar{\mu}_{2}\bar{\lambda}+4\bar{\lambda}^{2}\,.$
(17)

###  III.2 Fixed points

The system of Eqs. (15) admits the Gaussian fixed point
$\bar{\mu}_{1}^{\star}=\bar{\mu}_{2}^{\star}=\bar{\lambda}^{\star}=0$ as a
solution, which is unstable below the upper critical dimension $d<d_{c}=6$ and
stable above it. In the parameter space, we then focus on the plane
$(\bar{\mu}_{1},\bar{\mu}_{2},\bar{\lambda}^{\star}=0)$, for which Eqs. (15)
reduce to a closed system in the variables $(\bar{\mu}_{1},\bar{\mu}_{2})$.
The RG flow on this plane is governed by the stable lines of fixed points
defined by the conics

$\bar{\mu}_{2}^{\star}=-\frac{8\varepsilon}{45\bar{\mu}_{1}^{\star}}-\frac{5\bar{\mu}_{1}^{\star}}{18}\hskip
10.0pt\text{and}\hskip 10.0pt\bar{\lambda}^{\star}=0\,.$ (18)

The asymptotic lines corresponding to these hyperbolas separate regions of the
reduced couplings plane where the RG flow runs away to infinity, shown by the
blue arrows in the top panel of Fig. 1, from the basin of attraction of the
fixed points that are shaded in red in the same figure. (See Ref. Mahdisoltani
_et al._ (2021) for a detailed analysis of the flow equations of the
chemotactic model without the nonlinear growth term.)

As discussed in Ref. Mahdisoltani  _et al._ (2021), the runaway behavior of
the one-loop RG flows in the $\lambda=0$ plane could hint to several different
scenarios, and thus it calls for further investigations. In particular, it may
be an artifact of the one-loop approximation used here. On the other hand, the
runaway solution could signal a first-order phase transition, or,
alternatively, it may suggest that the critical behavior of the dynamics is
ruled by a strong-coupling fixed point that is not within the reach of the
perturbative RG calculations. The latter possibility is supported by a
connection between the chemotactic Langevin dynamics (6) and the KPZ equation
Kardar  _et al._ (1986), which describes the fluctuations of a growing
interface and is known to exhibit a nonperturbative behavior Wiese (1998);
Kloss  _et al._ (2014). We further discuss this connection below.

Figure 1: (Top) One-loop RG flows in the parameter space. The red hyperbolas
in the $\lambda=0$ plane correspond to the fixed-point solutions in Eq. (18).
The blue arrows show four instances of RG flow lines with $\lambda\neq 0$, all
appearing to exhibit runaway behavior. (Bottom) Projected RG flow on the plane
$\mu_{2}=-2\mu_{1}$ (gray plane in the top panel). Note the crossover behavior
of the trajectories starting from $\lambda\neq 0$, which first approach the
line of fixed points at $\lambda=0$ before diverging.

For chemotactic dynamics with logistic growth, our one-loop analysis shows
that no stable real-valued RG fixed point exists in the space
$(\bar{\mu}_{1},\bar{\mu}_{2},\bar{\lambda}\neq 0)$ and in the vicinity of the
upper critical dimension (see the Supplementary Material for a discussion of
possible fixed points at lower dimensions). A linear stability analysis which
considers a perturbation around the fixed-point solutions (18) gives the
eigenvalues $\uptheta_{1,2,3}$ of the associated stability matrix as

$\displaystyle\uptheta_{1}=-\varepsilon\,,\quad\uptheta_{2}=0\,,\quad\uptheta_{3}=\frac{16-220\bar{\mu}_{1}^{2}+2275\bar{\mu}_{1}^{4}}{1215\bar{\mu}_{1}^{2}}\varepsilon\,.$
(19)

The eigenvalue $\uptheta_{1}$ is negative for $\varepsilon>0$ and indicates
that the line of fixed points is attractive in the plane $\lambda=0$, as
already discussed above. The eigenvalue $\uptheta_{2}=0$ is the marginal
direction along the line of fixed points. The eigenvalue $\uptheta_{3}$ is
always positive for $\varepsilon>0$ and with $\bar{\mu}_{1}\in\mathbb{R}$, and
it indicates that the perturbations in a direction orthogonal to the plane
$\lambda=0$ are unstable.

The analysis presented above therefore shows the absence of stable
perturbative fixed point for the chemotactic system with growth. This
contrasts with the findings of Ref. Gelimson and Golestanian (2015) where a
stable fixed point was reported for the dynamics of dividing chemotactic cells
without the polarity-induced coupling. Importantly, even if not present at the
microscopic level, the polarity-induced chemotactic term proportional to
$\mu_{2}$ is immediately generated by the presence of the growth term
$\propto\lambda$ (see Eqs. (15)). This observation highlights the fundamental
role played by the polarity-induced mechanism in describing the macroscopic
properties of dividing chemotactic colonies.

In passing, we mention that along the line
$(\bar{\mu}_{1}^{\star}=0,\bar{\mu}_{2}^{\star}=0,\bar{\lambda})$, we find two
imaginary RG fixed points with coordinates
$\bar{\lambda}^{\star}=\pm\mathrm{i}\sqrt{\varepsilon/6}$. These correctly
describe the dynamics of the Yang–Lee edge singularity Yang and Lee (1952);
Lee and Yang (1952) for which, at one-loop order, we recover the known
critical exponents $z=2-\varepsilon/18$ and $\chi=-2+5\varepsilon/9$ Fisher
(1978); Breuer and Janssen (1981).

As a final remark, we discuss the relation between Eq. (6) and the KPZ
dynamics Kardar  _et al._ (1986). Consider the roughening of an interface with
height profile $h(\mathbf{r},t)$ and subject to a long-ranged correlated noise
$\tilde{\eta}$. The corresponding dynamics reads Medina  _et al._ (1989)

$\displaystyle\partial_{t}h=D\nabla^{2}h+g(\nabla h)^{2}+\tilde{\eta}\,,$ (20)

where the noise $\tilde{\eta}$ has correlations
$\langle\tilde{\eta}(\mathbf{k},\omega)\tilde{\eta}(\mathbf{k}_{1},\omega_{1})\rangle=2\mathcal{D}_{\rm
LR}k^{-2\theta}\delta^{d}(\mathbf{k}+\mathbf{k}_{1})\delta(\omega+\omega_{1})$,
with $\theta$ characterizing the long-range nature of its spatial correlations
(the original short-ranged KPZ equation is recovered for $\theta=0$). By
taking the Laplacian of Eq. (20), one then arrives at

$\displaystyle\partial_{t}\nabla^{2}h=D\nabla^{2}(\nabla^{2}h)+g\nabla^{2}(\nabla
h)^{2}+\eta\,,$ (21)

where the noise $\eta$ is now characterized by
$\langle\eta(\mathbf{k},\omega)\eta(\mathbf{k}_{1},\omega_{1})\rangle=2\mathcal{D}_{\rm
LR}k^{4-2\theta}\delta^{d}(\mathbf{k}+\mathbf{k}_{1})\delta(\omega+\omega_{1})\,.$
In the absence of the nonlinearities $\mu_{1}$ and $\lambda$, the Langevin
equation (6), which now describes the dynamics of chemotactic particles with
only the polarity-induced mechanism, can be mapped to the KPZ model (21) with
long-range correlated noise, by setting $\theta=2$ and identifying
$\phi\leftrightarrow h$ and $\mu_{2}\leftrightarrow g$. The pseudo-Galilean
symmetry (8) of the original chemotactic model then corresponds to the
Galilean symmetry of the KPZ equation. Note that in this case, Eq. (16) yields
the critical exponents $z=2$, $\chi=-2-\varepsilon/2$ and
$\tilde{\chi}=-4+\varepsilon/2$, which are consistent with the fact that these
exponents should not acquire correction at any order of the perturbation
theory with $\varepsilon$-expansion (rather than a “fixed-$d$” scheme) Wiese
(1998); Lässig (1995).

##  IV Conclusion

Starting from a microscopic model of chemotactic particles undergoing birth
and death processes, we have derived a coarse-grained, field-theoretical
description for the dynamics of density fluctuations in a growing self-
chemotactic colony (see Eq. (6)). In this Langevin description, we identify a
pseudo-Galilean symmetry, which is also present in the case of non-dividing
chemotactic particles Mahdisoltani  _et al._ (2021); this symmetry is
reminiscent of the Galilean symmetry of the Kardar–Parisi–Zhang equation
Kardar  _et al._ (1986). We then use perturbative RG techniques to investigate
the scaling behavior of the dynamics at its critical point. Crucially, our
analysis reveals that the interplay of the fluctuations effects in the
presence of the Keller–Segel chemotactic interactions and the logistic growth
nonlinearity, generate the so-called polarity-induced chemotactic coupling
($\propto\mu_{2}$ in Eq. (6)) at macroscopic scales. Although the microscopic
origin of this unconventional coupling may be attributed to particle polarity
effects in chemotactic dynamics Mahdisoltani  _et al._ (2021), its generation
along the RG flow in the presence of growth processes shows that it needs to
be incorporated in models of dividing chemotactic particles, even if the cells
have an isotropic gradient-sensing mechanism with no polarity effects at the
microscopic level.

We show that including the polarity-induced coupling changes the critical
picture of the system compared to the analysis of Ref. Gelimson and
Golestanian (2015). In particular, the chemotactic lines of fixed points that
exist in the absence of the growth coupling $\lambda$ Mahdisoltani  _et al._
(2021) are unstable in the direction of $\lambda$, and we report the absence
of any stable fixed point of the one-loop perturbative RG flow near the upper
critical dimension $d_{c}=6$ (see the Supplemental Material for the discussion
of a stable fixed point in $d=2$). Importantly, even in the absence of a
stable fixed point, the behavior of large but finite systems can still be
dictated by the presence of the line of fixed points in the plane $\lambda=0$,
as illustrated by the crossover behavior in Fig. 1.

Determining whether a stable perturbative fixed point would be found at higher
order in the perturbative expansion, or whether the absence of a stable
perturbative fixed point is the signature of a strong-coupling critical point
that would govern the critical behavior, requires further investigation, e.g.,
using nonperturbative RG techniques Canet  _et al._ (2011); Duclut and
Delamotte (2017a, b); Dupuis  _et al._ (2021).

###### Acknowledgements.

R.B.A.Z. and C.D. thank Davide Squizzato for illuminating discussions. RBAZ
acknowledges the support from the French ANR through the project NeqFluids
(grant ANR-18-CE92-0019). S.M. acknowledges support from the University of
Oxford through the Clarendon Fund and St John’s College Kendrew Scholarship,
and the Rudolf Peierls Centre for Theoretical Physics. A.G. acknowledges
support from the MIUR PRIN Project “Coarse-grained description for
nonequilibrium systems and transport phenomena (CO-NEST),” No. 201798CZL. R.G.
acknowledges support from the Max Planck School Matter to Life and the
MaxSynBio Consortium which are jointly funded by the Federal Ministry of
Education and Research (BMBF) of Germany and the Max Planck Society.

##  Appendix A Details of the renormalization group analysis

In this section, following standard renormalization group (RG) methods Täuber
(2014), we compute the one-loop corrections to the bare quantities in Eq.
(11), which were outlined in the main text. Adopting the following notation
$\int_{\hat{q}}^{>}\equiv\int_{-\infty}^{\infty}\mathrm{d}\omega\int_{\Lambda/b\leq|\bm{q}|\leq\Lambda}\mathrm{d}^{d}\bm{q}$,
where $\Lambda$ and $b$ were introduced in the main text, we have

$\displaystyle G_{1}(\hat{k})$ $\displaystyle=\leavevmode\hbox to35.97pt{\vbox
to23.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-10.86978pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{35.56595pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}{}{}{}{}{{}}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{-5.89282pt}{11.89015pt}{-10.66978pt}{17.32298pt}{-10.66978pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.32298pt}{-10.66978pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
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} {}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}{}{}{}{}{{}}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@curveto{7.11319pt}{5.17793pt}{10.83092pt}{9.60876pt}{15.47722pt}{10.42793pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.98482}{0.17361}{-0.17361}{0.98482}{15.4772pt}{10.42792pt}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=\int^{>}_{\hat{q}}N_{0}(\hat{k}/2+\hat{q})G_{0}(\hat{k}/2-\hat{q})V_{0}(\mathbf{k},\mathbf{k}/2+\mathbf{q})V_{0}(\mathbf{k},\mathbf{k}/2-\mathbf{q})\,,$
(22a) $\displaystyle V_{1}^{\rm(a)}(\hat{k},\hat{k}/2+\hat{k}_{1})$
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(22b) $\displaystyle V_{1}^{\rm(b)}(\hat{k},\hat{k}/2+\hat{k}_{1})$
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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$\displaystyle=\int^{>}_{\hat{q}}N_{0}(\hat{k}/2+\hat{q})G_{0}(\hat{k}/2-\hat{q})G_{0}(\hat{k}_{1}-\hat{q})V_{0}(\mathbf{k},\mathbf{k}/2+\mathbf{q})V_{0}(\mathbf{k}/2-\mathbf{q},\mathbf{k}_{1}-\mathbf{q})V_{0}(\mathbf{k}_{1}-\mathbf{q},\mathbf{k}/2+\mathbf{k}_{1})\,,$
(22c) $\displaystyle V_{1}^{\rm(c)}(\hat{k},\hat{k}/2+\hat{k}_{1})$
$\displaystyle=\leavevmode\hbox to32.84pt{\vbox
to29.42pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-14.71074pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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$\displaystyle=\int^{>}_{\hat{q}}N_{0}(\hat{k}_{1}-\hat{q})G_{0}(\hat{k}/2+\hat{q})G_{0}(\hat{k}/2-\hat{q})V_{0}(\mathbf{k},\mathbf{k}/2+\mathbf{q})V_{0}(\mathbf{k}/2+\mathbf{q},\mathbf{k}/2+\mathbf{k}_{1})V_{0}(\mathbf{k}/2-\mathbf{q},\mathbf{k}/2-\mathbf{k}_{1})\,,$
(22d) $\displaystyle N_{1}(\hat{k})$ $\displaystyle=\leavevmode\hbox
to35.97pt{\vbox to25.74pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-12.86978pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@lineto{35.56595pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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$\displaystyle=\int^{>}_{\hat{q}}N_{0}(\hat{k}/2+\hat{q})N_{0}(\hat{k}/2-\hat{q})V_{0}(\mathbf{k},\mathbf{k}/2+\mathbf{q})V_{0}(-\mathbf{k},-\mathbf{k}/2-\mathbf{q})\,.$
(22e)

In the expressions above, $V_{0}$ is defined in Eq. (10) in the main text, and
$G_{0}$ and $N_{0}$ are given below Eq. (9) and below Eq. (10), respectively.
Loop integrals are first computed over the frequencies using the residue
theorem and then reducing the $d$-dimensional integral over the internal
momentum $\mathbf{q}$ to a one-dimensional integral over its modulus
$|\mathbf{q}|=q$. The integration over the internal momentum $q$ is then
facilitated by performing an expansion in series of $q\to\infty$ and by
keeping only the ultraviolet leading-order singularity. This procedure
simplifies the computation while allowing the determination of the
$1/\varepsilon$ poles (where $\varepsilon=d_{c}-d$ with $d_{c}=6$ is the upper
critical dimension) encoding the critical scaling of the theory. The
integration over the modulus $q$ itself is then obtained as
$\int^{>}_{q}f(q)=f(\Lambda)\Lambda\delta\ell+\mathcal{O}(\delta\ell^{2})$.

In order to properly define the one-loop corrections $\delta\mu_{1,2}$ to the
non-linear terms $\mu_{1,2}$, it is convenient to evaluate
$V_{1}=V_{1}^{\rm(a)}+V_{1}^{\rm(b)}+V_{1}^{\rm(c)}$ at external momenta
$\mathbf{k}$ and $\mathbf{k}_{1}$ such that $|\mathbf{k}_{1}|=|\mathbf{k}|=k$
and $\mathbf{k}\cdot\mathbf{k}_{1}=k^{2}\cos\theta$. This determines the
vertex function $V_{1}\equiv V_{1,\theta}$ solely in terms of $\theta$. Here,
we report the one-loop corrections to the various coefficients in the model
(6); being interested in the RG flow near the critical point, all the
corrections listed below are evaluated at $\sigma=0$ except for Eq. (23a) for
$\sigma$ itself, in terms of which we determine the critical point given by
Eq. (24):

$\displaystyle\delta\sigma$
$\displaystyle=\left.G_{1}(\hat{k})\right|_{\hat{k}=\hat{0}}$
$\displaystyle=-\frac{K_{d}\Lambda^{2-\varepsilon}\mathcal{D}_{0}}{D^{2}\tau}\left[2\lambda^{2}-\left(\frac{3}{2}+\frac{1}{d}\right)\mu_{1}\lambda-\left(1-\frac{6}{d}\right)\mu_{2}\lambda\right]\delta\ell+\frac{K_{d}\Lambda^{-\varepsilon}\mathcal{D}_{0}\sigma}{D^{3}\tau}\left[4\lambda^{2}-\left(3+\frac{3}{d}\right)\mu_{1}\lambda-\left(2-\frac{14}{d}\right)\mu_{2}\lambda\right]\delta\ell\,,$
(23a) $\displaystyle\delta\tau$
$\displaystyle=\left.\mathrm{i}\partial_{\omega_{k}}G_{1}(\hat{k})\right|_{\hat{k}=0}$
$\displaystyle=K_{d}\Lambda^{-\varepsilon}\frac{\mathcal{D}_{0}}{D^{3}\tau}\tau\left[\lambda^{2}-\left(\frac{3}{4}+\frac{1}{2d}\right)\mu_{1}\lambda-\left(\frac{1}{2}-\frac{3}{d}\right)\mu_{2}\lambda\right]\delta\ell\,,$
(23b) $\displaystyle\delta D$
$\displaystyle=\left.-\partial_{k^{2}}G_{1}(\hat{k})\right|_{\hat{k}=\hat{0}}$
$\displaystyle=\frac{K_{d}\Lambda^{-\varepsilon}\mathcal{D}_{0}}{D^{3}\tau}D\left[\frac{\mu_{1}\lambda}{4}\left(\frac{9}{d}-\frac{6}{d+2}-1\right)+\mu_{2}\lambda\left(\frac{3}{2}-\frac{23}{2d}+\frac{21}{d+2}\right)+\lambda^{2}\frac{d-2}{d}+\frac{\mu_{1}^{2}}{4}\left(\frac{10}{d}-\frac{6}{d+2}-3\right)+\right.$
$\displaystyle\left.+\mu_{1}\mu_{2}\left(\frac{9}{d+2}-\frac{6}{d}-2\right)+\mu_{2}^{2}\left(\frac{6}{d}-1\right)\right]\delta\ell\,,$
(23c) $\displaystyle\delta\mu_{1}$
$\displaystyle=\delta\lambda-\frac{9}{16}V_{1,\theta=\pi}+\frac{25}{16}V_{1,\theta=\pi/2}$
$\displaystyle=-\frac{K_{d}\Lambda^{-\varepsilon}\mathcal{D}_{0}}{D^{3}\tau}\lambda\left[\lambda(2\mu_{2}-\mu_{1})+\mu_{1}^{2}\left(\frac{3}{4}-\frac{3}{2d}+\frac{3}{d+2}\right)-\mu_{1}\mu_{2}\left(1-\frac{16}{d}+\frac{28}{d+2}\right)-\mu_{2}^{2}\left(1+\frac{42}{d}-\frac{60}{d+2}\right)\right]\delta\ell\,,$
(23d) $\displaystyle\delta\mu_{2}$
$\displaystyle=\frac{5}{4}\delta\lambda+\frac{15}{32}V_{1,\theta=\pi}+\frac{25}{32}V_{1,\theta=\pi/2}$
$\displaystyle=\frac{K_{d}\Lambda^{-\varepsilon}\mathcal{D}_{0}}{D^{3}\tau}\lambda\left[\mu_{1}^{2}\left(\frac{1}{d}-\frac{3}{2(d+2)}\right)-\mu_{1}\mu_{2}\left(\frac{10}{d}-\frac{14}{d+2}\right)+\mu_{2}^{2}\left(\frac{24}{d}-\frac{30}{d+2}\right)\right]\delta\ell\,,$
(23e) $\displaystyle\delta\lambda$
$\displaystyle=-\left.V_{1}(\hat{k},\hat{k}/2+\hat{k}_{1})\right|_{\hat{k}=\hat{0}}$
$\displaystyle=\frac{K_{d}\Lambda^{-\varepsilon}\mathcal{D}_{0}}{D^{3}\tau}\lambda\left[4\lambda^{2}-\mu_{1}\lambda\left(5+\frac{6}{d}\right)-\mu_{2}\lambda\left(6-\frac{28}{d}\right)+\mu_{1}^{2}\left(\frac{3}{2}+\frac{15}{2d}-\frac{9}{2(d+2)}\right)+\mu_{1}\mu_{2}\left(4-\frac{48}{d}+\frac{42}{d+2}\right)\right.$
$\displaystyle\left.+\mu_{2}^{2}\left(2+\frac{58}{d}-\frac{90}{d+2}\right)\right]\delta\ell\,,$
(23f) $\displaystyle\delta\mathcal{D}_{0}$
$\displaystyle=\left.N_{1}(\hat{k})\right|_{\hat{k}=\hat{0}}$
$\displaystyle=\frac{K_{d}\Lambda^{-\varepsilon}\mathcal{D}_{0}}{D^{3}\tau}\mathcal{D}_{0}\lambda^{2}\delta\ell\,.$
(23g)

Equations (14) in the main text are finally obtained by evaluating Eq. (23) at
the upper critical dimension $d=d_{c}=6$ and after rescaling the variables as
described in the main text. Note that the fluctuation-induced shift of the
critical value $\sigma_{c}$ of the parameter $\sigma$ discussed in the main
text is obtained from Eq. (23a). In terms of the rescaled variables, it reads:

$\displaystyle\sigma_{c}=-\frac{\Lambda^{2}}{D}\left[2\bar{\lambda}^{2}-\left(\frac{3}{2}+\frac{1}{d}\right)\bar{\mu}_{1}\bar{\lambda}-\left(1-\frac{6}{d}\right)\bar{\mu}_{2}\bar{\lambda}\right]\,.$
(24)

##  Appendix B Extrapolated fixed points in spatial dimensions far from
$d_{c}$

Evaluating the corrections at $d=d_{c}$ is consistent with the rationale of
the $\varepsilon$-expansion, which is based on a perturbative dimensional
expansion valid close to $d_{c}$. For completeness, however, we have reported
the one-loop corrections in Eq. (23) with the $d$-dependent coefficients which
result from evaluating the corresponding loop integrals. Any extrapolation of
these coefficients to $d<d_{c}$ generally leads to uncontrolled, scheme-
dependent predictions typical of “fixed-$d$ RG” computations Frey and Täuber
(1994); Lässig (1995); Wiese (1998).

Nonetheless, it is informative to investigate the one-loop flows away from the
upper critical dimension. To this aim, it is convenient to first simplify the
one-loop RG flows by enforcing the exponent identity $z+\chi=0$ which follows
from the pseudo-Galilean symmetry discussed in the main text. In addition to
the line of fixed points in the plane $\lambda^{*}=0$ discussed in the main
text, which are unstable in the $\lambda$ direction, we find the following
five additional fixed points $(\mu_{1}^{*},\mu_{2}^{*},\lambda^{*})$ of the RG
flow equations:

  1. 1.

The Gaussian fixed point with $\mu_{1}^{*,0}=\mu_{2}^{*,0}=\lambda^{*,0}$
which is stable above the critical dimension $d_{\rm c}=6$ and unstable below.

  2. 2.

Two fixed points $\pm(\mu_{1}^{*,1},\mu_{2}^{*,1},\lambda^{*,1})$ with (here
and below we report the expressions of the dimensionless variables, but we
drop the bar on the corresponding symbols to lighten the notation)

$\displaystyle\lambda^{*,1}=\frac{5}{6}\mu_{1}^{*,1}=5\mu_{2}^{*,1}=\sqrt{\frac{5d(d-6)(d+2)}{2[7d^{2}-10d-12]}},$
(25)

which are real-valued for $d<\frac{5+\sqrt{109}}{7}\approx 2.21$ or $d\geq 6$.
These fixed points are only stable for $d<2$ and unstable otherwise.

  3. 3.

Two fixed points $\pm(\mu_{1}^{*,2},\mu_{2}^{*,2},\lambda^{*,2})$ with

$\displaystyle\lambda^{*,2}=\left[1+\frac{2}{d(d-8)}\right]\mu_{1}^{*,2}=2\left[1+\frac{2(1-2d)}{d(d-4)}\right]\mu_{2}^{*,2}=\sqrt{\frac{2(d-6)(d+2)(d^{2}-8d+2)^{2}}{5d^{6}-97d^{5}+606d^{4}-1358d^{3}+1320d^{2}-872d+96}}\,.$
(26)

These fixed points are real-valued for $0.13\lesssim d\lesssim 2.3$, and also
for $6\leq d\lesssim 7.54$ or $d\gtrsim 8.55$ above the upper critical
dimension. They are stable RG fixed points only for $2\leq d\lesssim 2.3$.

In Fig. 2 we summarize this analysis by plotting the three coordinates
$(\mu_{1}^{*},\mu_{2}^{*},\lambda^{*})$ of the fixed points discussed above as
functions of the spatial dimension $d$. At one-loop, our analysis reveals no
stable fixed point of the RG flow equations in $2.3\lesssim d<6$. A stable
fixed emerges at $d\simeq 2.3$ (orange solid line and green solid line) and
can be followed up to $d=0$. Determining whether this fixed point captures
correctly the physics in $d=1$ and 2 or whether it is an artefact of the one-
loop computation is beyond the scope of this work.

Figure 2: Coordinates $\mu_{1}^{*}$ (left), $\mu_{2}^{*}$ (middle) and
$\lambda^{*}$ (right) of the RG fixed points as functions of the spatial
dimension $d$. Solid lines refer to the stable fixed points and dashed lines
to unstable ones. The Gaussian fixed point is plotted in blue, the fixed point
$(\mu_{1}^{*,1},\mu_{2}^{*,1},\lambda^{*,1})$ given by Eq. (25) is shown in
green, and the fixed point $(\mu_{1}^{*,2},\mu_{2}^{*,2},\lambda^{*,2})$ given
by Eq. (26) is shown in red.

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