# Dynamical renormalization group for mode-coupling field theories
with solenoidal constraint

Andrea Cavagna  Istituto Sistemi Complessi, Consiglio Nazionale delle
Ricerche, UOS Sapienza, 00185 Rome, Italy  Dipartimento di Fisica, Università
Sapienza, 00185 Rome, Italy  Luca Di Carlo  Dipartimento di Fisica, Università
Sapienza, 00185 Rome, Italy  Istituto Sistemi Complessi, Consiglio Nazionale
delle Ricerche, UOS Sapienza, 00185 Rome, Italy  Irene Giardina  Dipartimento
di Fisica, Università Sapienza, 00185 Rome, Italy  Istituto Sistemi Complessi,
Consiglio Nazionale delle Ricerche, UOS Sapienza, 00185 Rome, Italy  INFN,
Unità di Roma 1, 00185 Rome, Italy  Tomas S. Grigera  Instituto de Física de
Líquidos y Sistemas Biológicos CONICET - Universidad Nacional de La Plata, La
Plata, Argentina  CCT CONICET La Plata, Consejo Nacional de Investigaciones
Científicas y Técnicas, Argentina  Departamento de Física, Facultad de
Ciencias Exactas, Universidad Nacional de La Plata, Argentina  Giulia Pisegna
Dipartimento di Fisica, Università Sapienza, 00185 Rome, Italy  Istituto
Sistemi Complessi, Consiglio Nazionale delle Ricerche, UOS Sapienza, 00185
Rome, Italy  Mattia Scandolo  Dipartimento di Fisica, Università Sapienza,
00185 Rome, Italy  Istituto Sistemi Complessi, Consiglio Nazionale delle
Ricerche, UOS Sapienza, 00185 Rome, Italy

###### Abstract

The recent inflow of empirical data about the collective behaviour of strongly
correlated biological systems has brought field theory and the renormalization
group into the biophysical arena. Experiments on bird flocks and insect swarms
show that social forces act on the particles’ velocity through the generator
of its rotations, namely the spin, indicating that mode-coupling field
theories are necessary to reproduce the correct dynamical behaviour.
Unfortunately, a theory for three coupled fields - density, velocity and spin
- has a prohibitive degree of intricacy. A simplifying path consists in
getting rid of density fluctuations by studying incompressible systems. This
requires imposing a solenoidal constraint on the primary field, an unsolved
problem even for equilibrium mode-coupling theories. Here, we perform an
equilibrium dynamic renormalization group analysis of a mode-coupling field
theory subject to a solenoidal constraint; using the classification of
Halperin and Hohenberg, we can dub this case as a solenoidal Model G. We
demonstrate that the constraint produces a new vertex that mixes static and
dynamical coupling constants, and that this vertex is essential to grant the
closure of the renormalization group structure and the consistency of dynamics
with statics. Interestingly, although the solenoidal constraint leads to a
modification of the static universality class, we find that it does not change
the dynamical universality class, a result that seems to represent an
exception to the general rule that dynamical universality classes are narrower
than static ones. Our results constitute a solid stepping stone in the
admittedly large chasm towards developing an off-equilibrium mode-coupling
theory of biological groups.

##  I Introduction

The phenomenon whereby systems with very different microscopic details have
similar critical behaviour, known as universality, finds an elegant
explanation in the context of the Renormalization Group (RG) Wilson (1971a,
b). As a system approaches a critical point, i.e. a second order phase
transition, it exhibits emergent collective phenomena on increasingly wider
length-scales, which leads to the presence of a diverging correlation length.
In this regime, a set of phenomenological scaling laws for near-critical
systems have been proposed Widom (1965); Kadanoff (1966) in order to describe
the critical behaviour through a set of critical exponents ruling the
divergence of thermodynamic quantities with respect to the correlation length.
In the RG framework, these scaling laws naturally arise as a consequence of
the diverging correlation length, thanks to the flow in the parameter space
generated by iterating an RG transformation many times. Through the concept of
attractive fixed points of the RG flow, it is possible to prove that systems
sharing only few general properties, as symmetries, dimensionality and range
of the interactions, develop the same large-scale behaviour near the critical
point, hence belonging to the same universality class. Moreover, the RG gives
a constructive method to compute the critical exponents Goldenfeld (1992),
making it one of the most powerful tools of the theory of critical phenomena,
broadly employed to characterise both static Pelissetto and Vicari (2002) and
dynamic Hohenberg and Halperin (1977) universality classes in equilibrium
systems.

When we shift our attention from equilibrium to non-equilibrium phenomena, the
study of emergent collective dynamics is of fundamental interest also in
biological systems, in which the response to external perturbations is more
efficient when individuals behave collectively Cresswell (1994); Roth  _et
al._ (2006). The existence of collective behaviour in biological systems is
strictly related to the presence of scale-free correlations among the system’s
components, as experimentally observed in bird flocks Cavagna  _et al._
(2010), sheep herds Ginelli  _et al._ (2015) and bacterial clusters Zhang  _et
al._ (2010), in strong similarity with emergent phenomena studied in condensed
matter, as superfluidity, superconductivity and ferromagnetism. The
experimental discovery of scale-free correlations provides some justification
to the theoretical investigation of collective behaviour in biological systems
using the tools developed for critical phenomena, including of course the
renormalization group.

The early efforts in the study of collective phenomena in active biological
systems focused on the ordered phase, namely in those groups exhibiting net
group motion, the most prominent example being represented by bird flocks
Toner and Tu (1995, 1998). More recently it has become experimentally evident
that also systems in their disordered phase, as natural swarms of insects,
display collective phenomena even in absence of collective order Attanasi  _et
al._ (2014a). Moreover, within natural swarms both static and dynamic scaling
laws have been found to hold Attanasi  _et al._ (2014a); Cavagna  _et al._
(2017). Since scaling laws are fundamental features of near-critical systems,
a renormalization group approach to swarming behaviour seems particularly
appealing.

Dynamic critical systems not only have a diverging correlation length, but
also a diverging relaxation time Cardy (1996). According to the dynamic
scaling hypothesis, the correlation length $\xi$ and the collective (i.e. zero
wavelength) relaxation time $\tau$ are linked by the relation
$\tau\sim\xi^{z}$, where $z$ is the dynamic critical exponent ruling the
dynamical relaxation of the system. Experiments show that the dynamic
behaviour of natural swarms is characterized by a critical exponent $z\sim 1$
Cavagna  _et al._ (2017), which is very far from the value $z\sim 2$ typical
of the simplest dissipative ferromagnetic models, generically represented by
Model A in the famous classification of Halperin and Hohenberg Hohenberg and
Halperin (1977); this is somewhat problematic, as the earliest theories for
active biological systems are essentially a mix of Model A ferromagnets and
Navier-Stokes equations Toner and Tu (1995). Finding what are the correct
dynamical equations that provide a value of the dynamical critical exponent
$z$ equal to the experimental value represents a very important step, not only
towards understanding collective behaviour in natural swarms (which will be
our focus here), but more broadly in the effort to prove that statistical
field theory is a predictive tool in biophysics.

There are two possible mechanisms that can explain a drop of the value of the
dynamical critical exponents in natural swarms compared to that of standard
ferromagnetic systems in the Model A class. The first possibility is activity,
namely the fact that individuals are self-propelled, which leads to a time-
dependent network of interactions between the particles’ velocities, thus
driving the system out of equilibrium. Activity gives rise to density
fluctuations, thus meaning that in addition to the velocity field
$\boldsymbol{v}\left(\boldsymbol{x},t\right)$, also the density field
$\rho\left(\boldsymbol{x},t\right)$ must be included in the hydrodynamic
description of the system Toner and Tu (1995, 1998). Broadly speaking, one can
say that the feedback between velocity and density fluctuations is responsible
for a more efficient transport of information within the system, which may
reflect into a lower value of $z$. Indeed, theoretical RG calculations in
presence of activity have been performed in the incompressible case, showing
that $z$ is indeed lower than its equilibrium value, leading the exponent from
$z\sim 2$ to $z\sim 1.7$ in $d=3$ dimensions Chen  _et al._ (2015); and yet
this is still far from the experimental value $z\sim 1$.

The second factor that may lower the value of $z$, and that has been shown to
be relevant in the description of both insect swarms and bird flocks, is the
presence of inertial dynamics, namely a coupling between the velocity field
$\boldsymbol{v}\left(\boldsymbol{x},t\right)$ and the generator of its
rotations, known as the spin $\boldsymbol{s}\left(\boldsymbol{x},t\right)$
Attanasi  _et al._ (2014b); Cavagna  _et al._ (2017). In equilibrium systems,
as for example planar ferromagnets, superfluids and quantum antiferromagnets
(respectively Model E, F and G in the classification of Hohenberg and Halperin
(1977)) this coupling (and the conservation law generated by it) radically
alters the mechanism of information propagation throughout the system and it
also lowers the dynamical critical exponent, which becomes $z=1.5$ in $d=3$
Cavagna  _et al._ (2019a, b). Even in this case, though, we are quite far from
the experimental value, but then it seems likely that a calculation that keeps
into account both the off-equilibrium active nature of swarms and the mode-
coupling nature of their interaction, may provide a value of the dynamical
critical exponent close to the experimental one, $z\sim 1$. The bad news is
that a full-fledged out of equilibrium field theory taking into account both
these features must contain a density field
$\rho\left(\boldsymbol{x},t\right)$, a velocity field
$\boldsymbol{v}\left(\boldsymbol{x},t\right)$ and a spin field
$\boldsymbol{s}\left(\boldsymbol{x},t\right)$, with a large number of
couplings between them and a near-prohibitive level of intricacy. Therefore,
to make theoretical progress, one needs some simplifying assumptions.

The first and most natural simplification is that of incompressibility.
Incompressibility in the RG study of the Navier-Stokes equations was first
used in the seminal paper by Forster, Nelson and Stephen Forster  _et al._
(1977), while it was used first in the context of dynamical field theories for
active matter in Chen  _et al._ (2015). This hypothesis seems quite reasonable
both in natural insect swarms and bird flocks, because experiments show that
density fluctuations are indeed very limited in these systems, and they do not
seem to be a crucial ingredient in their collective behaviour Attanasi  _et
al._ (2014a); Mora  _et al._ (2016). Incompressibility is obtained by
enforcing a solenoidal constraint on the velocity field, that is by adding to
the dynamics the condition, ${\boldsymbol{\nabla}}\cdot{\boldsymbol{v}}=0$.
This allows to neglect the (constant) density field, whose fluctuations are
suppressed thanks to the solenoidal constraint. Incompressible swarms in the
fully dissipative case (i.e. without velocity-spin coupling) have been studied
in Chen  _et al._ (2015), but we need to move to the mode-coupling case, of
velocity and spin, and here some new problems arise. In the case of a theory
with one single field (the velocity), the solenoidal constraint is simply
enforced by projecting the force acting on the velocity onto the direction
orthogonal to the wave vector $\boldsymbol{k}$ in Fourier space Forster  _et
al._ (1977); Chen  _et al._ (2015); in this way, the constraint is conserved
along the time trajectories. But in systems with mode-coupling dynamics
incompressibility cannot be achieved in this simple way, as the forces now act
on the spin, rather than on the velocity, and the spin is not subject to the
solenoidal constraint! Hence we are in a paradoxical situation: the assumption
that was supposed to radically simplify the calculation, reducing from three
to two the number of fields, creates itself a new challenges.

Instead of giving up the crucial simplification of incompressibility, we
choose to try and solve the problem of how to impose a solenoidal constraint
in a mode-coupling theory; and because this new problem has nothing to do with
the complications due to off-equilibrium dynamics, we tackle the problem in
the case of a zero-activity equilibrium system, namely a system living on a
fixed-network. Indeed, surprising as this may seem, the problem of how to
impose the solenoidal constraint on a mode-coupling theory has never been
investigated so far. Once we will have solved this problem, we will be in a
far better shape to push the calculation out of equilibrium, still retaining
the incompressibility condition. Notice that in absence of activity it does
not make much sense to insist in calling the primary field “velocity”. Hence,
we will use as our primary field ${\boldsymbol{\psi}}$, the field describing
the average direction of motion in the vanishing speed limit. The virtue of
this approach is that any future calculation performed in the active case must
have the present calculation as a limit in the zero-speed case.

To summarize, in the present work we will derive and study the coarse-grained
dynamic equations for a critical field ${\boldsymbol{\psi}}$ which is coupled
to the generator of its rotations, $\boldsymbol{s}$, in the presence of the
solenoidal constrain, ${\boldsymbol{\nabla}}\cdot{\boldsymbol{\psi}}=0$. We
shall refer to this theory as Solenoidal Model G (SMG), because in absence of
the solenoidal constraint this field theory is known in literature as Model G
Hohenberg and Halperin (1977): this classic model for equilibrium quantum
antiferromagnets will therefore be our stepping stone towards a theory for
off-equilibrium natural swarms. We will show that the solenoidal constraint
leads to the emergence of an additional non-linear term in the equation of
motion of the spin, as a consequence of the suppression of the
${\boldsymbol{\psi}}$ mode longitudinal to the wave-vector $\boldsymbol{k}$.
We will also prove that the equations we obtain are eigenstates of the
renormalization group, by calculating perturbative corrections up to one loop.
Moreover, the dynamic behaviour will be shown to be consistent with the static
behaviour of a ferromagnet with dipolar interactions Aharony and Fisher
(1973); Fisher and Aharony (1973). Finally, we will find as dynamic critical
exponent $z=\frac{d}{2}$, leading to the same dynamic universality class as
the unconstrained theory Cavagna  _et al._ (2019a); Hohenberg and Halperin
(1977). Therefore, while incompressibility affects the static universality
class, it does not modify the dynamic universality class, suggesting that the
critical behaviour of homogeneous systems is not affected by imposing
incompressibility.

In section II we will review the biological origin of the mode-coupling
dynamics and the reason for which incompressible constraint is imposed. In
section III the coarse-grained theory will be derived, starting form the
static Hamiltonian and the Poisson-bracket relations between the various
quantities. In section IV we will derive one-loop corrections to the couplings
of the theory and verify that the equations of motion are self-consistent.
Self-consistency is proven by showing that no new RG-relevant interaction is
generated and that the dynamic renormalization reproduces the correct static
behaviour. Finally, in section V we will derive the dynamical critical
exponent.

##  II Biophysical background

###  II.1 Dynamic scaling

The spatio-temporal statistical behavior of a collective system can be
described through the connected space-time correlation function of the
velocity, $C(\boldsymbol{r},t)$, which expresses how much the velocity
fluctuations of some individual in the system influence and are influenced by
the velocity fluctuations of another individual at distance $\boldsymbol{r}$
and after a time $t$ Cavagna  _et al._ (2018). For $t=0$ this is just the
static (i.e. equal times) correlation function, whose decay range defines the
correlation length, $\xi$. Conversely, for $r=0$ and $t\neq 0$, this is the
singe-particle auto-correlation function. In general, when both $r$ and $t$
are non-zero, the correlation function is quite a loaded concept, entailing
the fundamental relationship between relaxation in time and relaxation in
space. In the case of critical systems, such relationship acquires a
particularly illuminating form, which goes under the name of dynamic scaling
Ferrell  _et al._ (1967); Halperin and Hohenberg (1967). Dynamic scaling
states that the dynamic correlation function $C$, when expressed as a function
of wave-vector and frequency, takes the following simplified scaling form,

$C\left(\boldsymbol{k},\omega;\xi\right)=C_{0}\left(\boldsymbol{k};\xi\right)\,F\left(\frac{\omega}{\omega_{k}},k\xi\right)\\\
$ (1)

where $\xi$ is the correlation length and where the static correlation
function $C_{0}$ has in turns the scaling form,

$C_{0}(k,\xi)=k^{2-\eta}F_{0}\left(k\xi\right)$ (2)

and where the characteristic frequency at scale $k$ is given by,

$\omega_{k}=k^{z}\Omega\left(k\xi\right)$ (3)

In the relations above, $\Omega$, $F_{0}$ and $F$ are well-behaved scaling
functions, whose explicit form is inessential to capture the gists of dynamic
scaling Ferrell  _et al._ (1967); Halperin and Hohenberg (1967); $\eta$ is the
critical exponent for the static correlation function (normally called
anomalous dimension Binney  _et al._ (1992)). The fundamental meaning of
dynamic scaling is that in critical systems space and time do not scale
independently from each other, but they are linked by the dynamic critical
exponent $z$. The space-time correlation function has a very simple form, as
its whole spatio-temporal dependence goes through the product $k\xi$. When
$\boldsymbol{k}=0$, the collective relaxation time $\tau$ of the system is
linked to the correlation length $\xi$ through the relation,

$\tau\simeq\omega_{\boldsymbol{k}=0}^{-1}\sim\xi^{z}$ (4)

This phenomenon is known as critical slowing down and it represents a
consequence of the fact that the time $\tau$ needed to decorrelate a spatially
correlated region grows with the region’s size, namely with the correlation
length $\xi$, making the latter the only relevant scale also at a dynamic
level.

Notably, natural swarms of insects have been found to obey dynamic scaling
Cavagna  _et al._ (2017), with a dynamical critical exponent $z$ quite close
to $1$ ($z=1.2$, obtained by looking at the collapse of the correlation
functions Cavagna  _et al._ (2017), is the safest determination). On one hand,
because the validity of scaling laws is one of the hallmarks of critical
systems, the experimental evidence of static and dynamic scaling in natural
swarms Attanasi  _et al._ (2014a); Cavagna  _et al._ (2017) suggests that the
swarming phase can be theoretically described as a near-ordering phase of
classic ferromagnetic theories. On the other hand, the value $z\sim 1$ is
definitely anomalous, as the standard exponents in Model A class of
statistical systems Hohenberg and Halperin (1977) is $z\sim 2$, with small
corrections at two loops. This state of affairs suggest that a critical
dynamical universality class, different from Model A, must be found for
natural swarms. This new universality class will need to take into account the
two main features that affect the collective behaviour of these systems,
namely activity and mode-coupling dynamics. Let us illustrate the first factor
through the most venerated hydrodynamic theory describing active matter.

###  II.2 The hydrodynamic theory of Toner and Tu

In most collective biological systems, from bacterial clusters up to insect
swarms, bird flocks and animals herds, individuals are self-propelled thanks
to their ability to convert energy into systematic movement Schweitzer (2003).
The presence of metabolic processes, through which an energy supply from the
environment is provided, guarantees a constant speed. Activity means that the
interaction network over which the velocities of the individuals are
interacting with each other evolves in time, representing the principal non-
equilibrium feature of these systems.

The hydrodynamic theory developed by Toner and Tu Toner and Tu (1995, 1998),
based on the discrete model proposed by Vicsek and collaborators Vicsek  _et
al._ (1995), paved the way to a theoretical analysis of collective behaviours
in the presence of activity. Together with the velocity field
${\boldsymbol{v}}$, also the density number field $\rho$ is an hydrodynamic
variable of the system, since activity allows local density fluctuations. The
velocity field obeys to a dynamical behaviour that is a crossover between that
of a Landau-Ginzburg $O(n)$ model, reflecting the presence of an effective
alignment among the animals velocities, and a Navier-Stokes dynamics,
reflecting the fact that velocity and density are coupled as in a standard
fluid; the density evolves according to a continuity equation, since the total
number of individuals is assumed to be fixed. The essential terms of the Toner
and Tu (TT) theory are given by Ramaswamy (2010),

$\displaystyle\frac{\partial{\boldsymbol{v}}}{\partial
t}+\gamma_{v}\left({\boldsymbol{v}}\cdot{\boldsymbol{\nabla}}\right){\boldsymbol{v}}+\dots=-\Gamma\frac{\delta\mathcal{H}}{\delta{\boldsymbol{v}}}+{\boldsymbol{\nabla}}P+\boldsymbol{\theta}$
(5) $\displaystyle\frac{\partial\rho}{\partial
t}=-{\boldsymbol{\nabla}}\cdot\left(\rho{\boldsymbol{v}}\right)$ (6)

In Eq. (5), the r.h.s. encloses Model A dynamics Hohenberg and Halperin
(1977), with the gaussian random white noise $\boldsymbol{\theta}$ and the
hamiltonian force $-\frac{\delta\mathcal{H}}{\delta{\boldsymbol{v}}}$, to
which a pressure force is added contrasting an infinite compressibility. Here
the effective hamiltonian $\mathcal{H}$ takes the Landau-Ginzburg structure,
namely

$\mathcal{H}=\int
d^{d}x\left[\frac{1}{2}({\boldsymbol{\nabla}}{\boldsymbol{v}})^{2}+\frac{r_{0}}{2}{\boldsymbol{v}}\cdot{\boldsymbol{v}}+\frac{u_{0}}{4}\left({\boldsymbol{v}}\cdot{\boldsymbol{v}}\right)^{2}\right]$
(7)

Depending on the value of $r_{0}$, two phases can be identified: an ordered
phase for $r_{0}<r_{c}$ and a disordered phase for $r_{0}>r_{c}$, where
$r_{c}$ is the critical value of the parameter $r$. The main difference with
equilibrium dynamics is represented by the $\gamma_{v}$ term on the l.h.s. of
Eq. (5), which is the advection term111Since Galilean invariance is violated,
we cannot require $\gamma_{v}=1$. Moreover, the ellipsis in Eq. (5) denote
other two advection-like terms, involving one ${\boldsymbol{\nabla}}$ and two
${\boldsymbol{v}}$, allowed by symmetries typical of the Navier-Stokes
structure.

Since its development, the TT theory has been able to explain many fundamental
aspects of collective behaviours in active systems, from the existence of an
ordered phase also in $2$ dimensions, to the presence of linear sound modes
Toner and Tu (1995); Tu  _et al._ (1998). The success of this remarkable
theoretical effort is due to the minimal description given by the TT equations
of motion, which tie together activity and effective alignment. However,
experimental evidence has shown how this theory is not able to fully describe
some aspects of biological systems’, in particular information propagation in
flocks Attanasi  _et al._ (2014b), and dynamical correlation functions in
swarms Cavagna  _et al._ (2017). These discrepancies arise as consequence of
the structure of Eq. (5), which is essentially that of an overdamped Langevin
equation for the velocity,

$D_{t}{\boldsymbol{v}}=-\partial_{\boldsymbol{v}}\mathcal{H}$ (8)

Moreover, RG analysis of the near-ordering dynamics of the TT theory in the
incompressible case predictss a dynamical critical exponent $z\simeq 1.7$ for
$d=3$ Chen  _et al._ (2015), significantly different from the value $z\sim 1$
observed in natural swarms.

###  II.3 Restoring inertia: the mode-coupling theory

The first hints about the necessity of restoring an underdamped dynamics came
from experiments on bird flocks. In order to explain the information
propagation dispersion law within turning flocks, the generator of the
rotations of the velocity vector has to be conserved during the turn Attanasi
_et al._ (2014b). In analogy with quantum mechanics, this generator of the
rotations in the internal space of the velocity, which must not be confused
with the generator of spatial rotations (the standard angular momentum), takes
the name of spin.

A second piece of empirical evidence that inertial dynamics was missing from
the original Vicsek model and Toner-Tu theory came from the calculation of the
dynamical correlation functions in real swarms Cavagna  _et al._ (2017). The
overdamped dynamics for the velocity typical of the Vicsek model and TT theory
generates a classic exponential decay of the dynamical correlation functions;
however, in natural swarms this is not the case, and the correlation function
displays a non-exponential form typical of underdamped inertial systems. This
feature too can be included at the theoretical level by incorporating in the
dynamics the generator of the rotations of the velocity.

The discrete model describing this kind of spin-velocity mode-coupling
behaviour is known as the Inertial Spin Model (ISM) Cavagna  _et al._ (2015),
and it is characterized by an underdamped structure for the equations of
motion; where the alignment force exerted on the focal particle by its
neighbours does not act directly on the velocity, but it is mediated by the
generator of rotations, thus restoring an inertial behaviour.

Adding a conservation law leads to a modification of the hydrodynamic
behaviour of a system, which now must include the dynamics of the spin density
and the effects of a conserved total spin on the other variables. In the
present case, the conservation of the spin leads a mode-coupling between the
order parameter and the spin density field. This structure is not new to
physical systems: this is what happens in the case of planar magnets,
superfluid helium and isotropic anti-ferromagnets (Models E, F and G in
Hohenberg and Halperin (1977) respectively). The equations of motion for these
models, in terms of the order parameter ${\boldsymbol{\psi}}$ and the spin
$\boldsymbol{s}$, take the following form for a $3$ dimensional order
parameter (Model G)

$\displaystyle\frac{\partial{\boldsymbol{\psi}}}{\partial t}$
$\displaystyle=-\Gamma\frac{\delta\mathcal{H}}{\delta{\boldsymbol{\psi}}}+g\,{\boldsymbol{\psi}}\times\frac{\delta\mathcal{H}}{\delta\boldsymbol{s}}+\boldsymbol{\theta}$
(9) $\displaystyle\frac{\partial\boldsymbol{s}}{\partial t}$
$\displaystyle=\lambda\nabla^{2}\frac{\delta\mathcal{H}}{\delta\boldsymbol{s}}+g\,{\boldsymbol{\psi}}\times\frac{\delta\mathcal{H}}{\delta{\boldsymbol{\psi}}}+\boldsymbol{\zeta}$
(10)

where $\boldsymbol{\theta}$ and $\boldsymbol{\zeta}$ are gaussian random white
noises, while the Hamiltonian has the usual Landau-Ginzburg form for
${\boldsymbol{\psi}}$, plus a Gaussian non-interacting kinetic term in the
spin,

$\mathcal{H}=\int
d^{d}x\left[\frac{1}{2}\left({\boldsymbol{\nabla}}{\boldsymbol{\psi}}\right)^{2}+\frac{r_{0}}{2}{\boldsymbol{\psi}}\cdot{\boldsymbol{\psi}}+\frac{u_{0}}{4}\left({\boldsymbol{\psi}}\cdot{\boldsymbol{\psi}}\right)^{2}+\frac{\boldsymbol{s}\cdot\boldsymbol{s}}{2}\right]$
(11)

In Eq. (9) and (10) the two terms proportional to $g$, namely
$\partial_{t}\psi\sim\delta_{s}\mathcal{H}$ and
$\partial_{t}s\sim\delta_{\psi}\mathcal{H}$, represent the Hamiltonian
conservative dynamics enforced by the Poisson-braket relation

$\left\\{s_{\alpha},\psi_{\beta}\right\\}=g_{0}\,\epsilon_{\alpha\beta\gamma}\psi_{\gamma}$
(12)

where $\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita antisymmetric symbol,
meaning that $\boldsymbol{s}$ is rotating ${\boldsymbol{\psi}}$. Moreover, the
equation of motion of $\boldsymbol{s}$ can be written in the form of a
continuity equation, reflecting the fact that due to the rotational symmetry,
the total spin $\boldsymbol{S}(t)=\int d^{d}x\ \boldsymbol{s}(x,t)$ is
conserved222In the biophysical context, the spin is not strictly conserved.
Thus, the presence of a dissipative term
$\partial_{t}\boldsymbol{s}\sim-\eta\delta_{s}\mathcal{H}$ is allowed in Eq.
(10). In this system a crossover between a conservative and a dissipative
dynamics has been observed Cavagna  _et al._ (2019a, b), in which the former
regulates the behaviour of finite-size systems as swarms. As we will see later
on, the solenoidal constraint will violate the spin conservation, but does not
generate a dissipative behaviour, thus allowing us to work at $\eta=0$.

###  II.4 Incompressible flow and fixed network assumptions

In order to join the two ingredients described in the last two sections,
namely activity and inertia (i.e. mode-coupling dynamics) into one single
dynamical field theory, one needs to write equations for three coupled fields
- density, velocity and spin - with several non-linear couplings among them,
giving rise to a diagrammatic RG proliferation that it is impossible to keep
under control without any simplifying assumption. One such simplification is
that of incompressibility. The huge advantage of working under the hypothesis
of incompressibility is that the density field drops out the theoretical
description Forster  _et al._ (1977); Chen  _et al._ (2015), thus reducing the
number of fields that have to be studied from three to two, hence dramatically
simplifying the theoretical investigation. One may worry that
incompressibility introduces some non-local interactions in the system, then
potentially changing the critical exponents with respect to the compressible
case. However, numerical simulations of the standard compressible theory have
found the dynamic scaling exponent to be in perfect agreement with theoretical
RG results for incompressible theory Cavagna  _et al._ (2021), thus indicating
that the effects of activity on the dynamical critical exponent are
independent from whether incompressibility is enforced or not. This is very
useful indeed, as it means that incompressibility can be used as a simplifying
tool, without changing critical dynamics.

A second remark about incompressibility is in order. In the standard
compressible theory, the presence of density fluctuations in the system, as a
consequence of activity, makes the phase transition of Vicsek-like models a
first-order (i.e. discontinuous) transition Grégoire and Chaté (2004). In
incompressible systems, however, the absence of density fluctuations makes the
phase transitions always second-order (continuous) Chen  _et al._ (2015), thus
scaling behaviour is observed at all sizes and all the complications about
phase separation are avoided. Of course, this is only reasonable because in
real natural swarms of insects (as in bird flocks), density fluctuations are
negligible and play no determinant role in ruling the collective dynamics of
the system Attanasi  _et al._ (2014a, c). Hence, incompressibility is not only
a simplifying hypothesis, but also a sound theoretical description of actual
empirical data.

Thanks to the continuity equation (6), the condition of incompressibility
reduces to imposing a solenoidal constraint on the velocity field,

${\boldsymbol{\nabla}}\cdot{\boldsymbol{v}}=0$ (13)

As we shall see, the introduction of a solenoidal constraint on the primary
field within a mode-coupling dynamics is far from trivial. The only
encouraging thing is that the complications arising are completely unrelated
to activity, so that it seems sound to first find a conceptually consistent
way to impose the solenoidal constraint on a theory at equilibrium, and then
to use that results to make progress off-equilibrium in the future. Therefore,
in this work activity will be neglected by assuming the adjacency network to
be fixed in time. Since in the microscopic description of Vicsek-like systems
each individual has a fixed speed $v_{0}$, the fixed-network approximation can
be formally seen as a limit where $v_{0}$ vanishes, which is equivalent to
freeze the position of each particle. While in this limit the velocity field
acquires a singular behaviour, since no particle is actually moving, the
coarse-grained direction of motion ${\boldsymbol{\psi}}$, defined by the
relation

${\boldsymbol{v}}\left(\boldsymbol{x},t\right)=v_{0}\,{\boldsymbol{\psi}}\left(\boldsymbol{x},t\right)$
(14)

still has a smooth behaviour when $v_{0}\to 0$, and therefore represents the
ideal candidate to be the order parameter in the fixed network approximation.
Since the solenoidal constraint given by Eq. (13) holds also for
${\boldsymbol{\psi}}$ at any finite value of $v_{0}$, it must hold also in the
fixed network approximation.

##  III The solenoidal mode-coupling theory

The first problem we have is that the presence of the solenoidal constraint,

${\boldsymbol{\nabla}}\cdot{\boldsymbol{\psi}}=0$ (15)

forces the order parameter to have the same dimensionality as the space in
which the theory is defined. Thus, since in the following sections we will
need to perform an RG expansion near $d=4$, we must work with the
generalization of Model G to arbitrary dimensions, known as the Sasvari-
Schwabl-Szepfalusy (SSS) model Sasvári  _et al._ (1975); Sasvári and
Szépfalusy (1977).333 Given that we will be calculating the properties of the
Solenoidal Sasvari-Schwabl-Szepfalusy model, strictly speaking we should use
the nomenclature “SSSS model”; however, chiefly for aesthetic reasons, we
prefer not to do that, and we will rather stick to Solenoidal Model G (SMG)
even in generic dimension $d$. In this model, the spin $\boldsymbol{s}$ is an
anti-symmetric $d\times d$ matrix, with one independent component for each
possible plane around which a rotation can be performed. The Poisson-bracket
relation between $\boldsymbol{s}$ and $\psi$ becomes,

$\left\\{s_{\alpha\beta}\left(\boldsymbol{x}\right),\psi_{\gamma}\left(\boldsymbol{x}^{\prime}\right)\right\\}=2g_{0}\,\mathbb{I}_{\alpha\beta\gamma\rho}\psi_{\rho}\left(\boldsymbol{x}\right)\delta^{\left(d\right)}\left(\boldsymbol{x}-\boldsymbol{x}^{\prime}\right)$
(16)

where repeated greek-letter indices are intended to be summed, if not
otherwise indicated, and the tensor $\mathbb{I}$ is the identity tensor in the
space of $d\times d$ anti-symmetric matrices,444 The factor $\frac{1}{2}$ in
the definition of $\mathbb{I}$ arises as a consequence of the fact that, when
$\boldsymbol{s}$ is represented as an anti-symmetric matrix, each independent
component appears twice. given by

$\mathbb{I}_{\alpha\beta\gamma\rho}=\frac{1}{2}\left(\delta_{\alpha\gamma}\delta_{\beta\rho}-\delta_{\alpha\rho}\delta_{\beta\gamma}\right)$
(17)

The static properties of the SSS model are fully determined by its effective
Hamiltonian, which reflects the fact that ${\boldsymbol{\psi}}$ is a critical
field with $O\left(d\right)$ symmetry group and local interactions, having the
static critical properties of the Landau-Ginzburg universality class, while
$\boldsymbol{s}$ is a non-critical massive field. The effective Hamiltonian of
the system takes the following natural generalization

$\mathcal{H}=\int
d^{d}x\left[\frac{1}{2}\left(\partial_{\alpha}\psi_{\beta}\right)\left(\partial_{\alpha}\psi_{\beta}\right)+\frac{r_{0}}{2}\psi_{\alpha}\psi_{\alpha}+\frac{u_{0}}{4}\left(\psi_{\alpha}\psi_{\alpha}\right)^{2}+\frac{s_{\alpha\beta}s_{\alpha\beta}}{4}\right]$
(18)

The equations of motion of the critical SSS model can be derived starting from
the Poisson-bracket relation (16) and the effective hamiltonian (18) by
following a procedure coming from the works of Mori et al. Mori  _et al._
(1974) and Zwanzig Zwanzig (1961) (see Ma and Mazenko (1975) and Frey and
Schwabl (1994)). These equations are given by

$\displaystyle\frac{\partial\psi_{\alpha}}{\partial t}$
$\displaystyle=-\Gamma_{0}\frac{\delta\mathcal{H}}{\delta\psi_{\alpha}}+g_{0}\frac{\delta\mathcal{H}}{\delta
s_{\alpha\beta}}\psi_{\beta}+\theta_{\alpha}$ (19)
$\displaystyle\frac{\partial s_{\alpha\beta}}{\partial t}$
$\displaystyle=\lambda_{0}\nabla^{2}\frac{\delta\mathcal{H}}{\delta
s_{\alpha\beta}}+2g_{0}\mathbb{I}_{\alpha\beta\gamma\nu}\,\psi_{\gamma}\frac{\delta\mathcal{H}}{\delta\psi_{\nu}}+\zeta_{\alpha\beta}$
(20)

where $\theta$ and $\zeta$ are two gaussian white noises, with variance

$\displaystyle\langle\theta_{\alpha}\left(\boldsymbol{x},t\right)\theta_{\beta}\left(\boldsymbol{x}^{\prime},t^{\prime}\right)\rangle$
$\displaystyle=2\Gamma_{0}\delta_{\alpha\beta}\delta^{(d)}\left(\boldsymbol{x}-\boldsymbol{x}^{\prime}\right)\delta\left(t-t^{\prime}\right)$
(21)
$\displaystyle\langle\zeta_{\alpha\beta}\left(\boldsymbol{x},t\right)\zeta_{\gamma\nu}\left(\boldsymbol{x}^{\prime},t^{\prime}\right)\rangle$
$\displaystyle=-4\lambda_{0}\mathbb{I}_{\alpha\beta\gamma\nu}\nabla^{2}\delta^{(d)}\left(\boldsymbol{x}-\boldsymbol{x}^{\prime}\right)\delta\left(t-t^{\prime}\right)$
(22)

The $\lambda_{0}\nabla^{2}$ terms in the stochastic parts of the dynamics of
$\boldsymbol{s}$ ensure that the spin is conserved.

###  III.1 Static critical behaviour of the solenoidal theory

We will now review, for the benefit of the reader, the classic results of the
universality class of dipolar ferromagnets Aharony and Fisher (1973); Fisher
and Aharony (1973), since the static behaviour of the ${\boldsymbol{\psi}}$
field in SMG belongs to this class. From now on, if not explicitly indicated,
we shall work with the Fourier transformed fields, defined by the relation

$\psi_{\alpha}\left(\boldsymbol{x}\right)=\int_{\boldsymbol{k}}\text{e}^{\text{i}\boldsymbol{k}\cdot\boldsymbol{x}}\psi_{\alpha}\left(\boldsymbol{k}\right)$
(23)

where we introduced the notation
$\int\limits_{\boldsymbol{k}}=\int\limits_{\mathinner{\\!\left\lvert\boldsymbol{k}\right\rvert}<\Lambda}\frac{d^{d}k}{\left(2\pi\right)^{d}}$.

In Eq. (23) the cutoff $\Lambda$ is the value of the wave-vector above which
fluctuations have no physical meaning. Since a field theory, in the context of
biophysics, is obtained by coarse-graining a discrete model, fluctuations
cannot occure on distances shorter than the microscopic length $a$, therefore
meaning that the cutoff can be taken to be $\Lambda\sim a^{-1}$. In Fourier
space the solenoidal constraint reads

$k_{\alpha}\psi_{\alpha}\left(\boldsymbol{k}\right)=0$ (24)

An equivalent way in which this can be formulated is by requiring that

$P_{\alpha\beta}\left(\boldsymbol{k}\right)\psi_{\beta}\left(\boldsymbol{k}\right)=\psi_{\alpha}\left(\boldsymbol{k}\right)$
(25)

where we have introduced the transverse projection operator

$P_{\alpha\beta}\left(\boldsymbol{k}\right)=\delta_{\alpha\beta}-\frac{k_{\alpha}k_{\beta}}{k^{2}}$
(26)

The static behaviour of a field characterized by the Hamiltonian of the SSS
model, given in Eq. (18), to which a solenoidal constraint is applied, is
described by the static universality class of isotropic dipolar ferromagnets
Aharony and Fisher (1973); Bruce and Aharony (1974). The universality class of
dipolar ferromagnets describes the more generic class of critical theories
described by an Hamiltonian with a structure as that of Eq. (18), in which the
$\psi_{\parallel}\left(\boldsymbol{k}\right)$ mode is taken to be non-
critical.

Table 1: Values of $2\nu$ and $\eta$ up to order $\varepsilon^{2}$ Bruce and Aharony (1974) |  | Landau-Ginzburg | Dipolar Ferromagnets
---|---|---|---
| Mean-field |  $\varepsilon$-expansion | $\varepsilon=1$ |  $\varepsilon$-expansion | $\varepsilon=1$
$2\nu$ | $1$ | $1+\frac{1}{4}\varepsilon+\frac{1}{8}\varepsilon^{2}$ | $1.375$ | $1+\frac{9}{34}\varepsilon+\frac{7013}{58956}\varepsilon^{2}$ | $1.384$
$\eta$ | $0$ | $\frac{1}{48}\varepsilon^{2}$ | $0.0208$ | $\frac{20}{867}\varepsilon^{2}$ | $0.0231$

The renormalization group analysis of isotropic dipolar ferromagnets Aharony
and Fisher (1973) shows that the non-criticality of the
$\psi_{\parallel}\left(\boldsymbol{k}\right)$ mode leads to its full
suppression in the long-wavelength limit, meaning that the RG stable fixed
point describes a solenoidal-constrained theory. To order $\varepsilon=4-d$,
the RG recursive relation for the ferromagnetic coupling constant is Aharony
and Fisher (1973),

$u_{l+1}=b^{\varepsilon}u_{l}\left[1-\frac{17}{2}{u}_{l}\ln{b}\right]$ (27)

In dimension $d<4$ the stable fixed point $u^{*}$ ruling the critical
behaviour of dipolar ferromagnets is given by Aharony and Fisher (1973)

$u^{*}=\frac{2}{17}\varepsilon$ (28)

(for the sake of simplicity we have set to 1 the volume of the $d$-dimensional
unit sphere). The fact that at the stable fixed point the
$\psi_{\parallel}\left(\boldsymbol{k}\right)$ mode is suppressed Aharony and
Fisher (1973); Bruce and Aharony (1974) means that the solenoidal theory is
robust with respect to weak violations of the constraint, i.e. small non-
critical fluctuations of the $\psi_{\parallel}\left(\boldsymbol{k}\right)$
mode. Even though the change of static universality class is very interesting
at the theoretical level, it must be said that the new critical exponents are
so close to those of the Landau-Ginzburg universality class that the
difference is often experimentally difficult to observe Bruce and Aharony
(1974). A two-loop estimation of the critical exponents $2\nu$, ruling the
behaviour of the mass $r$ while the transition is approached, and $\eta$,
modifying the spatial dependence of the correlation function, are reported in
Table 1. Here it is clear the little difference between the scaling behaviour
of dipolar ferromagnets class and Landau-Ginzburg class.

###  III.2 Dynamics of the solenoidal theory

In this crucial section, the dynamical equations of the mode-coupling theory
subject to a solenoidal constraints will be derived. We will use the Poisson
relations, the effective Hamiltonian of the SSS model, and the classic Mori-
Zwanzig formalism Zwanzig (1961); Mori  _et al._ (1974).

Because the field ${\boldsymbol{\psi}}$ is subject to the constraint,

$k_{\alpha}\psi_{\alpha}\left(\boldsymbol{k}\right)=0$ (29)

the most natural representation in which one would like to derive the
equations of motion is,

${\boldsymbol{\psi}}\left(\boldsymbol{k}\right)=\psi_{\parallel}\left(\boldsymbol{k}\right)\boldsymbol{\hat{e}}^{\parallel}\left(\boldsymbol{k}\right)+\sum_{i=1}^{d-1}\psi_{\perp,i}\left(\boldsymbol{k}\right)\boldsymbol{\hat{e}}^{\perp,i}\left(\boldsymbol{k}\right)$
(30)

where
$\boldsymbol{\hat{e}}^{\parallel}=\boldsymbol{k}/\mathinner{\\!\left\lvert\boldsymbol{k}\right\rvert}$
is the unitary vector identifying the direction of $\boldsymbol{k}$, while the
$\boldsymbol{\hat{e}}^{\perp,i}$ are orthogonal unitary vectors spanning the
space perpendicular to $\boldsymbol{k}$, in such a way that
$\boldsymbol{\hat{e}}^{\parallel}\cdot\boldsymbol{\hat{e}}^{\perp,i}=0$ and
$\boldsymbol{\hat{e}}^{\perp,i}\cdot\boldsymbol{\hat{e}}^{\perp,j}=\delta_{ij}$;
indeed, within this set of coordinates, the constraint simply reads,

$\psi_{\parallel}\left(\boldsymbol{k}\right)=0$ (31)

leaving only the $(d-1)$ independent modes
$\psi_{\perp,i}\left(\boldsymbol{k}\right)$ to take care of. The advantage of
this notation is that we can formulate the constrained field theory in terms
of the $(d-1)$ independent transverse modes and not in terms of a constrained
set of $d$ cartesian coordinates, $\psi_{\alpha}$, to which it is not clear
how to apply the Mori-Zwanzig procedure. However, the explicit form of both
the effective hamiltonian $\mathcal{H}$ and of the Poisson-brackets are given
in terms of the cartesian coordinates, $\psi_{\alpha}$, while their form in
terms of the $\psi_{\perp,i}$ would be extremely cumbersome. What we shall do,
then, will be to first obtain the equation of motions for the $\psi_{\perp,i}$
and then to go back to the standard $\psi_{\alpha}$ fields by using the chain
rule. In doing that, something new will pop out in the spin equation.

The dynamic behaviour of the constrained variables $\psi_{\alpha}$ is given,
in terms of the independent variables $\psi_{\perp}$, by

$\frac{\partial\psi_{\alpha}}{\partial
t}\left(\boldsymbol{k},t\right)=\sum_{i=1}^{d-1}\hat{e}_{\alpha}^{\perp,i}\left(\boldsymbol{k}\right)\frac{\partial\psi_{\perp,i}}{\partial
t}\left(\boldsymbol{k},t\right)$ (32)

Following the Mori-Zwanzig procedure, the equations of motion for the $(d-1)$
independent fields $\psi_{\perp,i}$ and $\boldsymbol{s}$ take the following
form

$\displaystyle\frac{\partial\psi_{\perp,i}}{\partial
t}\left(\boldsymbol{k},t\right)=-\Gamma_{0}\frac{\delta\mathcal{H}}{\delta\psi_{\perp,i}\left(-\boldsymbol{k}\right)}+\frac{1}{2}\int_{\boldsymbol{q}}\left\\{s_{\gamma\nu}\left(-\boldsymbol{q}\right),\psi_{\perp,i}\left(\boldsymbol{k}\right)\right\\}\frac{\delta\mathcal{H}}{\delta
s_{\gamma\nu}\left(-\boldsymbol{q}\right)}+\hat{e}_{\beta}^{\perp,i}\left(\boldsymbol{k}\right)\tilde{\theta}_{\beta}$
(33) $\displaystyle\frac{\partial s_{\alpha\beta}}{\partial
t}\left(\boldsymbol{k},t\right)=-k^{2}\Lambda_{\alpha\beta\gamma\nu}\left(\lambda_{0},\boldsymbol{k}\right)\frac{\delta\mathcal{H}}{\delta
s_{\gamma\nu}\left(-\boldsymbol{k}\right)}-\sum_{i=1}^{d-1}\int_{\boldsymbol{q}}\left\\{s_{\alpha\beta}\left(\boldsymbol{k}\right),\psi_{\perp,i}\left(-\boldsymbol{q}\right)\right\\}\frac{\delta\mathcal{H}}{\delta\psi_{\perp,i}\left(-\boldsymbol{q}\right)}+\zeta_{\alpha\beta}$
(34)

where the $t$ dependence is always understood even when not made explicit for
reasons of space. In Eqs. (33) and (34) $\mathcal{H}$ is the Hamiltonian given
by Eq. (18) in which $\psi_{\parallel}$ is set to $0$;
$\boldsymbol{\tilde{\theta}}$, $\boldsymbol{\zeta}$ are white gaussian noises
with variance respectively given by $2\Gamma_{0}\delta_{\alpha\beta}$ and
$4k^{2}\Lambda_{\alpha\beta\gamma\nu}$. The tensor
$\Lambda_{\alpha\beta\gamma\nu}=\Lambda_{\alpha\beta\gamma\nu}\left(\lambda_{0};\boldsymbol{k}\right)$
is function of the diffusive coefficient $\lambda_{0}$ and potentially also of
the wave-vector $\boldsymbol{k}$; in the field theory without constraint it
takes the form
$\Lambda_{\alpha\beta\gamma\nu}=\lambda_{0}\mathbb{I}_{\alpha\beta\gamma\nu}$;
however, since the solenoidal constraint generates an anisotropy in Fourier
space for the order parameter, we expect that anisotropic effects can affect
also the spin dynamics; therefore, we generalize
$\Lambda_{\alpha\beta\gamma\nu}$ by allowing $\lambda_{0}$ to take different
values for the longitudinal and transverse components, namely by taking,

$\Lambda_{\alpha\beta\gamma\nu}\left(\lambda_{0}^{\perp},\lambda_{0}^{\parallel};\boldsymbol{k}\right)=\lambda_{0}^{\perp}\mathbb{P}_{\alpha\beta\gamma\nu}\left(\boldsymbol{k}\right)+\lambda_{0}^{\parallel}\left[\mathbb{I}_{\alpha\beta\gamma\nu}-\mathbb{P}_{\alpha\beta\gamma\nu}\left(\boldsymbol{k}\right)\right]$
(35)

where $\mathbb{P}$ is the generalization of the projection operator $P$
defined in Eq. (26) acting on the space of 2-indices antisymmetric tensors,
and takes the following form

$\mathbb{P}_{\alpha\beta\gamma\nu}\left(\boldsymbol{k}\right)=\mathbb{I}_{\alpha\beta\gamma\nu}-\mathbb{I}_{\alpha\beta\sigma\tau}P_{\sigma\gamma}\left(\boldsymbol{k}\right)P_{\tau\nu}\left(\boldsymbol{k}\right)$
(36)

In order to find the equations of motion of $\psi_{\alpha}$ and
$\boldsymbol{s}_{\alpha\beta}$, we will proceed by making explicit the terms
in Eqs. (33) and (34) exploiting the chain rule between the fields
$\psi_{\perp}$ and $\psi_{\alpha}$. The chain rule applied to the variations
of the Hamiltonian with respect to $\psi_{\perp,i}$ reads,

$\frac{\delta\mathcal{H}}{\delta\psi_{\perp,i}\left(-\boldsymbol{k}\right)}=\int
d^{d}p\;\frac{\delta\psi_{\beta}\left(-\boldsymbol{p}\right)}{\delta\psi_{\perp,i}\left(-\boldsymbol{k}\right)}\frac{\delta\mathcal{H}}{\delta\psi_{\beta}\left(-\boldsymbol{p}\right)}=\hat{e}_{\beta}^{\perp,i}\left(\boldsymbol{k}\right)\frac{\delta\mathcal{H}}{\delta\psi_{\beta}\left(-\boldsymbol{k}\right)}$
(37)

while the Poisson-bracket relation between $\boldsymbol{s}$ and
$\psi_{\perp,i}$ can be written as

$\left\\{s_{\gamma\nu}\left(-\boldsymbol{q}\right),\psi_{\perp,i}\left(\boldsymbol{k}\right)\right\\}=\int
d^{d}p\left\\{s_{\gamma\nu}\left(-\boldsymbol{q}\right),\psi_{\beta}\left(\boldsymbol{p}\right)\right\\}\frac{\delta\psi_{\perp,i}\left(\boldsymbol{k}\right)}{\delta\psi_{\beta}\left(\boldsymbol{p}\right)}=\hat{e}_{\beta}^{\perp,i}\left(\boldsymbol{k}\right)\left\\{s_{\gamma\nu}\left(-\boldsymbol{q}\right),\psi_{\beta}\left(\boldsymbol{k}\right)\right\\}$
(38)

where in the last equality of both Eq. (37) and (38) we used the relations,

$\displaystyle\frac{\delta\psi_{\perp,i}\left(\boldsymbol{k}\right)}{\delta\psi_{\beta}\left(\boldsymbol{p}\right)}$
$\displaystyle=\hat{e}_{\beta}^{\perp,i}\left(\boldsymbol{k}\right)\delta^{\left(d\right)}\left(\boldsymbol{k}-\boldsymbol{p}\right)$
$\displaystyle\frac{\delta\psi_{\beta}\left(\boldsymbol{p}\right)}{\delta\psi_{\perp,i}\left(\boldsymbol{k}\right)}$
$\displaystyle=\hat{e}_{\beta}^{\perp,i}\left(\boldsymbol{k}\right)\delta^{\left(d\right)}\left(\boldsymbol{k}-\boldsymbol{p}\right)$
(39)

Thanks to Eq. (37) and (38) we can write the mode-coupling term of the
dynamics of the spin in the following way

$\begin{split}\sum_{i=1}^{d-1}\int_{\boldsymbol{q}}\left\\{s_{\alpha\beta}\left(\boldsymbol{k}\right),\psi_{\perp,i}\left(-\boldsymbol{q}\right)\right\\}\frac{\delta\mathcal{H}}{\delta\psi_{\perp,i}\left(-\boldsymbol{q}\right)}&=\int_{\boldsymbol{q}}\left\\{s_{\alpha\beta}\left(\boldsymbol{k}\right),\psi_{\nu}\left(-\boldsymbol{q}\right)\right\\}\left[\sum_{i=1}^{d-1}\hat{e}_{\nu}^{\perp,i}\left(\boldsymbol{q}\right)\hat{e}_{\rho}^{\perp,i}\left(\boldsymbol{q}\right)\right]\frac{\delta\mathcal{H}}{\delta\psi_{\rho}\left(-\boldsymbol{q}\right)}=\\\
&=\int_{\boldsymbol{q}}\left\\{s_{\alpha\beta}\left(\boldsymbol{k}\right),\psi_{\nu}\left(-\boldsymbol{q}\right)\right\\}P_{\nu\rho}\left(\boldsymbol{q}\right)\frac{\delta\mathcal{H}}{\delta\psi_{\rho}\left(-\boldsymbol{q}\right)}\end{split}$
(40)

where we used the following relation, which comes directly from the definition
of the unitary vectors $\boldsymbol{\hat{e}}^{\perp,i}$

$P_{\alpha\beta}\left(\boldsymbol{k}\right)=\sum_{i=1}^{d-1}\hat{e}^{\perp,i}_{\alpha}\left(\boldsymbol{k}\right)\hat{e}^{\perp,i}_{\beta}\left(\boldsymbol{k}\right)$
(41)

Thanks to the relations we found in Eq. (37), (38) and (40) we can write Eq.
(33) and (34) in the following form

$\frac{\partial\psi_{\perp,i}}{\partial
t}\left(\boldsymbol{k},t\right)=\hat{e}_{\beta}^{\perp,i}\left(\boldsymbol{k}\right)\left[-\Gamma_{0}\frac{\delta\mathcal{H}}{\delta\psi_{\beta}\left(-\boldsymbol{k}\right)}+\frac{1}{2}\int_{\boldsymbol{q}}\left\\{s_{\gamma\nu}\left(-\boldsymbol{q}\right),\psi_{\beta}\left(\boldsymbol{k}\right)\right\\}\frac{\delta\mathcal{H}}{\delta
s_{\gamma\nu}\left(-\boldsymbol{q}\right)}+\tilde{\theta}_{\beta}\right]$ (42)
$\frac{\partial s_{\alpha\beta}}{\partial
t}\left(\boldsymbol{k},t\right)=-k^{2}\Lambda_{\alpha\beta\gamma\nu}\frac{\delta\mathcal{H}}{\delta
s_{\gamma\nu}\left(-\boldsymbol{k}\right)}-\int_{\boldsymbol{q}}\left\\{s_{\alpha\beta}\left(\boldsymbol{k}\right),\psi_{\nu}\left(-\boldsymbol{q}\right)\right\\}P_{\nu\rho}\left(\boldsymbol{q}\right)\frac{\delta\mathcal{H}}{\delta\psi_{\rho}\left(-\boldsymbol{q}\right)}+\zeta_{\alpha\beta}$
(43)

By substituting Eq. (42) in Eq. (32) and by using the explicit Poisson-bracket
relation given in Eq. (16), the equations of motion for the standard Cartesian
components $\psi_{\alpha}$ and for $s_{\alpha\beta}$ can finally be written,

$\frac{\partial\psi_{\alpha}}{\partial
t}\left(\boldsymbol{k},t\right)=-\Gamma_{0}P_{\alpha\beta}\left(\boldsymbol{k}\right)\frac{\delta\mathcal{H}}{\delta\psi_{\beta}\left(-\boldsymbol{k}\right)}+g_{0}P_{\alpha\rho}\left(\boldsymbol{k}\right)\mathbb{I}_{\rho\beta\gamma\nu}\int_{\boldsymbol{q}}\psi_{\beta}\left(\boldsymbol{k}-\boldsymbol{q}\right)\frac{\delta\mathcal{H}}{\delta
s_{\gamma\nu}\left(-\boldsymbol{q}\right)}+\theta_{\alpha}$ (44)

$\frac{\partial s_{\alpha\beta}}{\partial
t}\left(\boldsymbol{k},t\right)=-k^{2}\Lambda_{\alpha\beta\gamma\nu}\left(\boldsymbol{k}\right)\frac{\delta\mathcal{H}}{\delta
s_{\gamma\nu}\left(-\boldsymbol{k}\right)}+2\,g_{0}\mathbb{I}_{\alpha\beta\gamma\nu}\int_{\boldsymbol{q}}\psi_{\gamma}\left(\boldsymbol{k}-\boldsymbol{q}\right)P_{\nu\rho}\left(\boldsymbol{q}\right)\frac{\delta\mathcal{H}}{\delta\psi_{\rho}\left(-\boldsymbol{q}\right)}+\zeta_{\alpha\beta}$
(45)

Here the hamiltonian $\mathcal{H}$ can be taken to be the same of SSS model,
since all the terms involving derivatives of $\mathcal{H}$ with respect to
${\boldsymbol{\psi}}$ are already projected. Moreover, the two gaussian random
white noises $\boldsymbol{\zeta}$ and $\boldsymbol{\theta}$ have variance

$\displaystyle\langle\theta_{\alpha}\left(\boldsymbol{k},t\right)\theta_{\beta}\left(\boldsymbol{k}^{\prime},t^{\prime}\right)\rangle=2\left(2\pi\right)^{d}\Gamma_{0}\,P_{\alpha\beta}\left(\boldsymbol{k}\right)\,\delta^{(d)}\left(\boldsymbol{k}+\boldsymbol{k}^{\prime}\right)\delta\left(t-t^{\prime}\right)$
(46)
$\displaystyle\langle\zeta_{\alpha\beta}\left(\boldsymbol{k},t\right)\zeta_{\gamma\nu}\left(\boldsymbol{k}^{\prime},t^{\prime}\right)\rangle=4\left(2\pi\right)^{d}\Lambda_{\alpha\beta\gamma\nu}\left(\lambda_{0}^{\perp},\lambda_{0}^{\parallel};\boldsymbol{k}\right)k^{2}\,\delta^{(d)}\left(\boldsymbol{k}+\boldsymbol{k}^{\prime}\right)\delta\left(t-t^{\prime}\right)$
(47)

####  III.2.1 Effects of the constraint

The solenoidal constraint has a double effect on the equations of motion. The
first, and maybe the most trivial, is that the equation of motion for
${\boldsymbol{\psi}}$ is projected orthogonally to $\boldsymbol{k}$, as it
happens in incompressible field theories with no mode-coupling interaction
Forster  _et al._ (1977); Chen  _et al._ (2015). The second effect is less
obvious, and it is represented by the presence of a projection operator
$P_{\nu\rho}\left(\boldsymbol{q}\right)$ in the mode coupling interaction of
the spin dynamics. The existence of this projector is a consequence of the
fact that, in the presence of a solenoidal constraint, the conservative
Hamiltonian force is not simply,

$\mathbb{F}_{\nu}\left(\boldsymbol{q}\right)=-\frac{\delta\mathcal{H}}{\delta\psi_{\nu}\left(-\boldsymbol{q}\right)}$
(48)

but rather,

$\mathbb{F}_{\nu}\left(\boldsymbol{q}\right)=-P_{\nu\rho}\left(\boldsymbol{q}\right)\frac{\delta\mathcal{H}}{\delta\psi_{\rho}\left(-\boldsymbol{q}\right)}$
(49)

The linear part of the force is not affected by this new projector, but the
non-linear terms are. This can be seen by writing explicitly the new force,

$\mathbb{F}_{\nu}\left(\boldsymbol{q}\right)=-\left(r_{0}+q^{2}\right)\psi_{\nu}\left(\boldsymbol{q}\right)+u_{0}P_{\nu\rho}\left(\boldsymbol{q}\right)\int_{\boldsymbol{p},\boldsymbol{h}}\psi_{\rho}\left(\boldsymbol{p}\right)\psi_{\sigma}\left(\boldsymbol{h}\right)\psi_{\sigma}\left(\boldsymbol{q}-\boldsymbol{p}-\boldsymbol{h}\right)$
(50)

where in the first linear term we used the fact that
$P_{\nu\rho}\left(\boldsymbol{q}\right)\psi_{\rho}\left(\boldsymbol{q}\right)=\psi_{\nu}\left(\boldsymbol{q}\right)$.
The linear part of the force contributes to the dynamics of $\boldsymbol{s}$
with the same term as Model G Hohenberg and Halperin (1977)

$\partial_{t}s_{\alpha\beta}\left(\boldsymbol{k}\right)\sim
g_{0}\mathbb{I}_{\alpha\beta\gamma\nu}\int_{\boldsymbol{q}}\left[q^{2}-\left(k-q\right)^{2}\right]\psi_{\gamma}\left(\boldsymbol{k}-\boldsymbol{q}\right)\psi_{\nu}\left(\boldsymbol{q}\right)$
(51)

The factor $q^{2}-\left(k-q\right)^{2}$ arises as a consequence of mode-
coupling and it vanishes as $\boldsymbol{k}\to 0$, thus conserving the total
spin $\boldsymbol{S}(t)=\int
d^{d}x\,\boldsymbol{s}\left(\boldsymbol{x},t\right)=\boldsymbol{s}\left(\boldsymbol{k}=0,t\right)$.
But now, thanks to the presence of the projector, also the non-linear term
contributes to the dynamics of the spin! More precisely, it does so through a
novel dynamical interaction term given by,

$\partial_{t}s_{\alpha\beta}\left(\boldsymbol{k}\right)\sim
2g_{0}u_{0}\mathbb{I}_{\alpha\beta\gamma\nu}\int_{\boldsymbol{q},\boldsymbol{h},\boldsymbol{p}}\psi_{\gamma}\left(\boldsymbol{k}-\boldsymbol{q}\right)P_{\nu\rho}\left(\boldsymbol{q}\right)\psi_{\rho}\left(\boldsymbol{p}\right)\psi_{\sigma}\left(\boldsymbol{h}\right)\psi_{\sigma}\left(\boldsymbol{q}-\boldsymbol{p}-\boldsymbol{h}\right)$
(52)

This is a completely new term, which mixes the static ferromagnetic
interaction (the coupling constant $u_{0}$) with the dynamic mode-coupling
interaction (the coupling constant $g_{0}$); such vertex is absent in the non-
constrained theory, since when $P_{\nu\rho}$ is substituted by
$\delta_{\nu\rho}$, as in the non-solenoidal case, the r.h.s. of Eq. (52)
vanishes. In honour of the historic NYC diner at Houston and Ludlow, we call
this new interaction, the Katz vertex. As we will demonstrate later on, the
Katz vertex is crucial in order to keep closed and self-consistent the RG
calculation and to recover the correct static critical exponents.

The Katz interaction does not vanish when $\boldsymbol{k}\to 0$, meaning that
the equation of motion of $\boldsymbol{s}$ cannot be written as a continuity
equation anymore and thus that the total spin $\boldsymbol{S}(t)$ is no longer
conserved. However, in the following section we will show thats no spin
dissipation is generated by the Katz vertex, suggesting that the violation of
the spin conservation is equivalent to a generalized precession of the total
spin vector.

###  III.3 Field theoretical description

In order to set up the renormalization of the theory we will follow the
procedure proposed by Martin, Siggia, Rose Martin  _et al._ (1973), Janssen
Janssen (1976) and De Dominicis De Dominicis, C. (1976) to write stochastic
differential equations as a field theory formulated using path integrals.
Thanks to this procedure, the behaviour of a field $\boldsymbol{\phi}$
governed by a stochastic differential equation with a deterministic evolution
operator $\boldsymbol{\mathcal{F}}$ and a gaussian noise $\boldsymbol{\theta}$

$\boldsymbol{\mathcal{F}}\left[\boldsymbol{\phi}\right]-\boldsymbol{\theta}=0$
(53)

can be described through a field-theoretical action that correctly reproduces
the statistics, i.e. the correlation and response functions, of Eq. (53),

$\mathcal{S}[\boldsymbol{\hat{\phi}},\boldsymbol{\phi}]=\int_{\tilde{\boldsymbol{k}}}\left[\hat{\phi}_{\alpha}\mathcal{F}_{\alpha}\left[\boldsymbol{\phi}\right]-\hat{\phi}_{\alpha}L_{\alpha\beta}\hat{\phi}_{\beta}\right]$
(54)

where $2L_{\alpha\beta}$ is the variance of the gaussian noise, while
$\boldsymbol{\hat{\phi}}$ is an auxiliary field. From now on, if not
explicitly mentioned, we will work in Fourier space for both the spacial and
temporal dependence, s

$\phi\left(\boldsymbol{x},t\right)=\int_{\tilde{\boldsymbol{k}}}\text{e}^{\text{i}\left(\boldsymbol{k}\cdot\boldsymbol{x}-\omega
t\right)}\phi(\tilde{\boldsymbol{k}})s$ (55)

where $\tilde{\boldsymbol{k}}=\left(\boldsymbol{k},\omega\right)$ and
$\int_{\tilde{\boldsymbol{k}}}=\int_{\boldsymbol{k}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}$.
The presence of the additional field $\boldsymbol{\hat{\phi}}$ is the cost
which has to be paid in order to exploit standard path integral formulation,
which will allow us to use the standard rules of static renormalization and
write the perturbative series in terms of Feynman diagrams. This auxiliary
field takes the name of response field, since the propagator, namely the
response function, can be written as De Dominicis and Giardina (2006):

$G(\tilde{\boldsymbol{k}})=\langle\hat{\phi}(-\tilde{\boldsymbol{k}})\phi(\tilde{\boldsymbol{k}})\rangle$
(56)

The Martin-Siggia-Rose (MSR) action for the stochastic differential equations
(44) and (45) takes the following form

$\mathcal{S}[\hat{{\boldsymbol{\psi}}},{\boldsymbol{\psi}},\hat{\boldsymbol{s}},\boldsymbol{s}]=\mathcal{S}_{0,\psi}[\hat{{\boldsymbol{\psi}}},{\boldsymbol{\psi}}]+\mathcal{S}_{0,s}[\hat{\boldsymbol{s}},\boldsymbol{s}]+\mathcal{S}_{I}[\hat{{\boldsymbol{\psi}}},{\boldsymbol{\psi}},\hat{\boldsymbol{s}},\boldsymbol{s}]$
(57)

where $\mathcal{S}_{0,\psi}$ and $\mathcal{S}_{0,m}$ are the two Gaussian free
action of the two fields, which reproduce the linear dynamic theory

$\mathcal{S}_{0,\psi}=\int_{\tilde{\boldsymbol{k}}}\left[\hat{\psi}_{\alpha}\left(-\tilde{\boldsymbol{k}}\right)\left(-\text{i}\omega+\Gamma_{0}k^{2}+m_{0}\right)\psi_{\alpha}\left(\tilde{\boldsymbol{k}}\right)-\hat{\psi}_{\alpha}\left(-\tilde{\boldsymbol{k}}\right)\Gamma_{0}P_{\alpha\beta}\left(\boldsymbol{k}\right)\hat{\psi}_{\beta}\left(\tilde{\boldsymbol{k}}\right)\right]$
(58)
$\mathcal{S}_{0,s}=\frac{1}{2}\int_{\tilde{\boldsymbol{k}}}\left[\hat{s}_{\alpha\beta}\left(-\tilde{\boldsymbol{k}}\right)\left(-\text{i}\omega\mathbb{I}_{\alpha\beta\gamma\nu}+k^{2}\Lambda_{\alpha\beta\gamma\nu}\left(\boldsymbol{k}\right)\right)s_{\gamma\nu}\left(\tilde{\boldsymbol{k}}\right)-\hat{s}_{\alpha\beta}\left(-\tilde{\boldsymbol{k}}\right)k^{2}\Lambda_{\alpha\beta\gamma\nu}\left(\boldsymbol{k}\right)\hat{s}_{\gamma\nu}\left(\tilde{\boldsymbol{k}}\right)\right]$
(59)

while $\mathcal{S}_{I}$, which takes contributes from the non-linear dynamic
terms and represents the interaction part of the action, is given by

$\begin{split}\mathcal{S}_{I}=&-g_{0}\,\mathbb{I}_{\rho\beta\gamma\nu}\int_{\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}}}P_{\alpha\rho}(\boldsymbol{k})\hat{\psi}_{\alpha}(-\tilde{\boldsymbol{k}})\psi_{\beta}(\tilde{\boldsymbol{q}})s_{\gamma\nu}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{q}})-g_{0}\,\mathbb{I}_{\alpha\beta\gamma\nu}\int_{\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}}}\boldsymbol{k}\cdot\boldsymbol{q}\,\hat{s}_{\alpha\beta}(-\tilde{\boldsymbol{k}})\psi_{\gamma}(-\tilde{\boldsymbol{q}}+\tilde{\boldsymbol{k}}/2)\psi_{\nu}(\tilde{\boldsymbol{q}}+\tilde{\boldsymbol{k}}/2)+\\\
&-\frac{g_{0}u_{0}}{12}\int_{\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}},\tilde{\boldsymbol{p}},\tilde{\boldsymbol{h}}}K_{\alpha\beta\gamma\nu\sigma\tau}({\boldsymbol{k}},\boldsymbol{q},{\boldsymbol{p}},{\boldsymbol{h}},{\boldsymbol{k}}-\boldsymbol{q}-\boldsymbol{p}-\boldsymbol{h})\hat{s}_{\alpha\beta}(-\tilde{\boldsymbol{k}})\psi_{\gamma}(\tilde{\boldsymbol{q}})\psi_{\nu}(\tilde{\boldsymbol{p}})\psi_{\sigma}(\tilde{\boldsymbol{h}})\psi_{\tau}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{q}}-\tilde{\boldsymbol{h}}-\tilde{\boldsymbol{p}})+\\\
&+\frac{J_{0}}{3}\int_{\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}},\tilde{\boldsymbol{p}}}Q_{\alpha\beta\gamma\nu}(\boldsymbol{k})\hat{\psi}_{\alpha}(-\tilde{\boldsymbol{k}})\psi_{\beta}(\tilde{\boldsymbol{q}})\psi_{\gamma}(\tilde{\boldsymbol{p}})\psi_{\nu}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{q}}-\tilde{\boldsymbol{p}})\end{split}$
(60)

where we have introduced the new parameters,

$\displaystyle m_{0}$ $\displaystyle=\Gamma_{0}r_{0}$ $\displaystyle
J_{0}=\Gamma_{0}u_{0}$ (61)

which are a sort of dynamical generalization of mass and ferromagnetic
coupling constant. In Eq. (60) the following new tensors have been introduced

$Q_{\alpha\beta\gamma\nu}\left(\boldsymbol{k}\right)=P_{\alpha\beta}\left(\boldsymbol{k}\right)\delta_{\gamma\nu}+P_{\alpha\gamma}\left(\boldsymbol{k}\right)\delta_{\beta\nu}+P_{\alpha\nu}\left(\boldsymbol{k}\right)\delta_{\beta\gamma}$
(62)
$\begin{split}K_{\alpha\beta\gamma\nu\sigma\tau}\left({\boldsymbol{k}},{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3},{\boldsymbol{p}}_{4}\right)=&\mathbb{I}_{\alpha\beta\gamma\rho}Q_{\rho\nu\sigma\tau}\left(\boldsymbol{k}-\boldsymbol{p}_{1}\right)+\mathbb{I}_{\alpha\beta\nu\rho}Q_{\rho\gamma\sigma\tau}\left(\boldsymbol{k}-\boldsymbol{p}_{2}\right)+\\\
&+\mathbb{I}_{\alpha\beta\sigma\rho}Q_{\rho\gamma\nu\tau}\left(\boldsymbol{k}-\boldsymbol{p}_{3}\right)+\mathbb{I}_{\alpha\beta\tau\rho}Q_{\rho\gamma\nu\sigma}\left(\boldsymbol{k}-\boldsymbol{p}_{4}\right)\end{split}$
(63)

####  III.3.1 Free theory

The starting point to build the perturbative expansion of the equations of
motion is the free, or Gaussian, dynamic theory, obtained by setting to $0$
all the dynamic non-linear couplings, namely $g_{0}$ and $u_{0}$. From the
gaussian part of the action, given by Eqs. (58) and (59), we can easily derive
the expressions for the bare propagators and correlation functions for the
effective field theory, which are given by:

$\displaystyle\langle\psi_{\alpha}(\tilde{\boldsymbol{k}})\hat{\psi}_{\beta}(\tilde{\boldsymbol{q}})\rangle_{0}$
$\displaystyle=\mathbb{G}_{\alpha\beta}^{0,\psi}(\tilde{\boldsymbol{k}})\hat{\delta}(\tilde{\boldsymbol{k}}+\tilde{\boldsymbol{q}})$
$\displaystyle\langle
s_{\alpha\beta}(\tilde{\boldsymbol{k}})\hat{s}_{\gamma\nu}(\tilde{\boldsymbol{q}})\rangle_{0}$
$\displaystyle=\mathbb{G}_{\alpha\beta\gamma\nu}^{0,s}(\tilde{\boldsymbol{k}})\hat{\delta}(\tilde{\boldsymbol{k}}+\tilde{\boldsymbol{q}})$
(64)
$\displaystyle\langle\psi_{\alpha}(\tilde{\boldsymbol{k}})\psi_{\beta}(\tilde{\boldsymbol{q}})\rangle_{0}$
$\displaystyle=\mathbb{C}_{\alpha\beta}^{0,\psi}(\tilde{\boldsymbol{k}})\hat{\delta}(\tilde{\boldsymbol{k}}+\tilde{\boldsymbol{q}})$
$\displaystyle\langle
s_{\alpha\beta}(\tilde{\boldsymbol{k}})s_{\gamma\nu}(\tilde{\boldsymbol{q}})\rangle_{0}$
$\displaystyle=\mathbb{C}_{\alpha\beta\gamma\nu}^{0,s}(\tilde{\boldsymbol{k}})\hat{\delta}(\tilde{\boldsymbol{k}}+\tilde{\boldsymbol{q}})$
(65)

where
$\hat{\delta}(\tilde{\boldsymbol{h}})=(2\pi)^{d+1}\delta^{(d)}(\boldsymbol{h})\delta(\omega_{h})$.
The subscripted $0$ on thermal averages indicate that they are computed within
the non-interacting theory, namely by setting $u_{0}=g_{0}=0$. The tensors
$\mathbb{G}$ and $\mathbb{C}$ are given by

$\displaystyle\mathbb{G}_{\alpha\beta}^{0,\psi}(\tilde{\boldsymbol{k}})=G_{0,\psi}(\tilde{\boldsymbol{k}})\delta_{\alpha\beta}$
(66)
$\displaystyle\mathbb{C}_{\alpha\beta}^{0,\psi}(\tilde{\boldsymbol{k}})=C_{0,\psi}(\tilde{\boldsymbol{k}})P_{\alpha\beta}(\boldsymbol{k})$
(67)
$\displaystyle\mathbb{G}_{\alpha\beta\gamma\nu}^{0,s}(\tilde{\boldsymbol{k}})=G_{0,s}^{\perp}(\tilde{\boldsymbol{k}})\mathbb{P}_{\alpha\beta\gamma\nu}(\boldsymbol{k})+G_{0,s}^{\parallel}(\tilde{\boldsymbol{k}})\left[\mathbb{I}_{\alpha\beta\gamma\nu}-\mathbb{P}_{\alpha\beta\gamma\nu}(\boldsymbol{k})\right]$
(68)
$\displaystyle\mathbb{C}_{\alpha\beta\gamma\nu}^{0,s}(\tilde{\boldsymbol{k}})=C_{0,s}^{\perp}(\tilde{\boldsymbol{k}})\mathbb{P}_{\alpha\beta\gamma\nu}(\boldsymbol{k})+C_{0,s}^{\parallel}(\tilde{\boldsymbol{k}})\left[\mathbb{I}_{\alpha\beta\gamma\nu}-\mathbb{P}_{\alpha\beta\gamma\nu}(\boldsymbol{k})\right]$
(69)

In Eq. (66), (68), (67) and (69) we have,

$\displaystyle G_{0,\psi}\left(\boldsymbol{k},\omega\right)$
$\displaystyle=\frac{1}{-\text{i}\omega+\Gamma_{0}k^{2}+\Gamma_{0}r_{0}}$
$\displaystyle C_{0,\psi}\left(\boldsymbol{k},\omega\right)$
$\displaystyle=\frac{2\Gamma_{0}}{\omega^{2}+\Gamma_{0}^{2}\left(r_{0}+k^{2}\right)^{2}}$
(70) $\displaystyle G_{0,s}^{\perp}\left(\boldsymbol{k},\omega\right)$
$\displaystyle=\frac{2}{-\text{i}\omega+\lambda_{0}^{\perp}k^{2}}$
$\displaystyle C_{0,s}^{\perp}\left(\boldsymbol{k},\omega\right)$
$\displaystyle=\frac{4\lambda_{0}^{\perp}k^{2}}{\omega^{2}+(\lambda_{0}^{\perp}k^{2})^{2}}$
(71) $\displaystyle G_{0,s}^{\parallel}\left(\boldsymbol{k},\omega\right)$
$\displaystyle=\frac{2}{-\text{i}\omega+\lambda_{0}^{\parallel}k^{2}}$
$\displaystyle C_{0,s}^{\parallel}\left(\boldsymbol{k},\omega\right)$
$\displaystyle=\frac{4\lambda_{0}^{\parallel}k^{2}}{\omega^{2}+(\lambda_{0}^{\parallel}k^{2})^{2}}$
(72)

In the diagrammatic framework, bare propagators and correlation functions are
represented in the following way

$\displaystyle\langle\psi_{\alpha}\hat{\psi}_{\beta}\rangle_{0}\quad=$
$\displaystyle\quad\feynmandiagram[small,horizontal=ctob]{c--[fermion]b,};$
$\displaystyle\langle s_{\alpha\beta}\hat{s}_{\gamma\nu}\rangle_{0}\quad=$
$\displaystyle\quad\feynmandiagram[small,horizontal=ctob]{c--[chargedboson]b,};$
(73) $\displaystyle\langle\psi_{\alpha}\psi_{\beta}\rangle_{0}\quad=$
$\displaystyle\quad\feynmandiagram[small,horizontal=ctob]{c--b,};$
$\displaystyle\langle s_{\alpha\beta}s_{\gamma\nu}\rangle_{0}\quad=$
$\displaystyle\quad\feynmandiagram[small,horizontal=ctob]{c--[boson]b,};$ (74)

where the arrows in the propagators aways point in the direction of the
response field.

####  III.3.2 Non-linear terms: the vertices

The four terms that compose $\mathcal{S}_{I}$ represent the non-linear
interactions in the equations of motion. Two of them involve one field
$\hat{{\boldsymbol{\psi}}}$, since they derive from the equation of motion of
${\boldsymbol{\psi}}$, while the other two involve one field
$\hat{\boldsymbol{s}}$, since they derive from the equation of motion of
$\boldsymbol{s}$. In the diagrammatic framework these interactions are
represented by vertices, in which different lines merge together, each
representing one of the fields involved in the interaction. Here we are
representing with a solid line the fields $\psi$ and $\hat{\psi}$, with wavy
lines the fields $s$ and $\hat{s}$. Moreover, an arrow is used to recognize
which legs represent a response field.

The first vertex - namely interaction - involving $\hat{{\boldsymbol{\psi}}}$,
represents the mode coupling non-linearity,

$\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{{\leavevmode\includegraphics{}{}{pgf-
img/Main-article-SMG-
figure0.pdf}}}=g_{0}P_{\alpha\rho}\left(\boldsymbol{k}\right)\mathbb{I}_{\rho\beta\gamma\nu}\hat{\delta}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{q}}-\tilde{\boldsymbol{p}})$
(75)

This interaction represents a purely dynamic interaction, since it is
proportional only to the dynamic coupling $g_{0}$.

The second vertex involving $\hat{{\boldsymbol{\psi}}}$ derives from the
ferromagnetic quartic interaction of the static hamiltonian, ensuring that the
field ${\boldsymbol{\psi}}$ relaxes towards the static equilibrium
distribution, and therefore is proportional to static coupling $u_{0}$. It is
represented by the term

$\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{{\leavevmode\includegraphics{}{}{pgf-
img/Main-article-SMG-
figure1.pdf}}}=-\frac{J_{0}}{3}Q_{\alpha\beta\gamma\nu}\left(\boldsymbol{k}\right)\hat{\delta}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{q}}-\tilde{\boldsymbol{p}}-\tilde{\boldsymbol{h}})$
(76)

The other two vertices involve one field $\hat{\boldsymbol{s}}$, and both
derive from the mode-coupling interaction in the equation for
$\boldsymbol{s}$. The first comes from the linear part of the ”force” defined
in Eq. (49), representing a purely dynamic interaction proportional to
$g_{0}$, and it takes the usual form as in the non-constrained theory

$\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{{\leavevmode\includegraphics{}{}{pgf-
img/Main-article-SMG-
figure2.pdf}}}=\frac{g_{0}}{2}\,\left(p^{2}-q^{2}\right)\,\mathbb{I}_{\alpha\beta\gamma\nu}\hat{\delta}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{q}}-\tilde{\boldsymbol{p}})$
(77)

Here the factor $(p^{2}-q^{2})$, coming from the cross product structure of
the mode-coupling interaction, guarantees that this interaction vanishes when
$\boldsymbol{k}=0$. This is a consequence of the symmetry of the non-
constrained theory, which conserves the total instantaneous spin
$\boldsymbol{S}(t)=\boldsymbol{s}(\boldsymbol{k}=0,t)$.

The last interaction term is the Katz vertex, given by the novel non-linear
contribution peculiar of the solenoidal theory, discussed at the end of Sec.
III.2. This interaction mixes static and dynamic terms, since it represents
the effects of the static quartic interaction on the dynamics of
$\boldsymbol{s}$, mediated by the mode-coupling dynamic interaction.
Therefore, the Katz vertex is proportional to the product of the static
coupling $u_{0}$ and the dynamic coupling $g_{0}$ and it takes the following
form,

$\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{{\leavevmode\includegraphics{}{}{pgf-
img/Main-article-SMG-
figure3.pdf}}}=\frac{g_{0}u_{0}}{12}K_{\alpha\beta\gamma\nu\sigma\tau}\left({\boldsymbol{k}},\boldsymbol{p}_{1},\boldsymbol{p}_{2},\boldsymbol{p}_{3},\boldsymbol{p}_{4}\right)\hat{\delta}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{p}}_{1}-\tilde{\boldsymbol{p}}_{2}-\tilde{\boldsymbol{p}}_{3}-\tilde{\boldsymbol{p}}_{4})$
(78)

The Katz vertex arises as a consequence of the solenoidal constraint, since
the tensor $K_{\alpha\beta\gamma\nu\sigma\tau}$ vanishes in the non-
constrained theory. At variance with the vertex (77), Katz causes a violation
of the spin conservation due to the fact that the order parameter has lost the
$O(d)$ symmetry as a consequence of the solenoidal constraint, meaning that it
does not vanish when $\boldsymbol{k}=0$. However, the fact that the total spin
is not conserved does not mean that it is dissipated. In fact, we will show in
the following sections that no perturbative corrections dissipating the spin
arise after the shell integration as a consequence of the presence of the Katz
vertex, reinforcing the hypothesis according to which $\boldsymbol{s}$ is an
hydrodynamic slow-variable of the system. Note that, were this not the case,
the RG flow would lead to the stable fixed point of solenoidal Model A Cavagna
_et al._ (2019a).

##  IV Renormalization Group calculation

The key idea behind the renormalization group is that, under the assumption of
an infinitely large correlation length, scaling laws and critical exponents
can be obtained by looking at how the parameterss of a theory change by
changing the length-scale at which the system is observed Wilson (1971a). The
RG itself consists in a transformation through which a set of equations
describing the dependence of the couplings from the length-scale, namely the
$\beta$-functions, can be derived.

We will use Wilson’s momentum shell approach Wilson (1971b), which can be
performed by following a two-steps procedure: i) integrating out the short
wavelength details, hence decreasing the cutoff in momentum space; ii)
rescaling space and time, so to formally reinstate the same original cutoff.
The first step is carried out by marginalizing the probability distribution of
the fields over the modes in the momentum shell $\Lambda/b<k<\Lambda$, where
$\Lambda$ is the cutoff. Here $b$ is a parameter larger than $1$, but close to
it, while $\Lambda$ is the cutoff of the theory. The effect of this
integration is to modify the parameters of the theory, and shift the cutoff
from $\Lambda$ to $\Lambda/b$. To compare the coefficients of the new theory
with the bare ones, space, and consequently time, must be rescaled in order to
restore the original cutoff. The iteration of this procedure defines a flow in
the parameter space, from which information on the critical behaviour can be
obtained.

In the Gaussian theory, namely when $u_{0}=g_{0}=0$, the shell integration is
harmless since modes at different wavelength are independent Goldenfeld
(1992); hence, only the rescaling step remains, giving to all parameters,
fields, and coupling constants their naive (or engineering) scaling dimension.
However, in the interacting theory, where non-linear interactions are present,
during the shell integration the coupling between long and short wavelength
modes generates perturbative corrections to the parameters of the theory,
which have the effect to correct the scaling dimensions. In order to
explicitly compute these corrections, one performs a perturbative expansion of
the shell integrals in terms of the parameter $\varepsilon=d_{c}-d$, where
$d_{c}$ is the upper critical dimension, namely the dimension above which
mean-field theory is exact. This expansion method, known as
$\varepsilon$-expansion, is nowadays a well established procedure in the
context of perturbative RG techniques Goldenfeld (1992); Parisi (1988); Wilson
and Kogut (1974); Cardy (1996).

###  IV.1 Renormalization group equations

In this section we will explicitly show how the renormalization group changes
the parameters of the model. As stressed out before the RG procedure unfolds
in two steps. In the first step we integrate out the small length scale (large
momenta) fluctuations, namely fluctuations with momenta $\Lambda/b<k<\Lambda$.
The effect of this integration is twofold: i) it changes the cut-off of the
theory, since now only fluctuation $k<\Lambda/b$ are allowed; ii) it changes
the value of the parameters of the model, which acquire corrections due to the
coupling between low and high momenta fluctuations,

$\mathcal{S}_{\Lambda/b}=\int\hat{\psi}\left[-i\omega+\Gamma_{0}(1+\delta\Gamma\ln
b)k^{2}+m_{0}(1+\delta m\ln
b)\right]\psi+\int\hat{s}\left[-i\omega+\lambda^{\perp}_{0}(1+\delta\lambda^{\perp}\ln
b)k^{2}+\lambda^{\parallel}_{0}(1+\delta\lambda^{\parallel}\ln
b)k^{2}\right]s+\dots$ (79)

We omitted the tensorial structure of the action for easier reading. We remark
the fact that all the corrections are proportional to the volume of the
momentum shell, which is always proportional to $\ln b$. The standard way to
carry out this task, and to compute the corrections to the bare parameters of
the model, is using perturbation theory; to be more precise we will compute
the corrections $\delta\mathcal{P}$ using a Feynman diagram expansion.

The second step consist in re-scaling momenta, frequencies and fields555Since
we know, from the static properties of the system, that the anomalous
dimension is zero at one loop order, the scaling dimensions of the fields are
the naive ones Hohenberg and Halperin (1977).,

$\displaystyle k$ $\displaystyle\to bk$ $\displaystyle\omega$
$\displaystyle\to b^{z}\omega$ (80) $\displaystyle\psi$ $\displaystyle\to
b^{d_{\psi}}\psi$ $\displaystyle\hat{\psi}$ $\displaystyle\to
b^{d_{\hat{\psi}}}\hat{\psi}$ (81) $\displaystyle s$ $\displaystyle\to
b^{d_{s}}s$ $\displaystyle\hat{s}$ $\displaystyle\to b^{d_{\hat{s}}}\hat{s}$
(82)

The effect of this second step is to restore the original cut-off of the
theory. Moreover after this step all the parameters of the model acquire a
naive scaling factor, corresponding to their naive/engineering scaling
dimension:

$\displaystyle\Gamma$ $\displaystyle\to b^{z-2}\Gamma$
$\displaystyle\lambda^{\perp}$ $\displaystyle\to b^{z-2}\lambda^{\perp}$
$\displaystyle\lambda^{\parallel}$ $\displaystyle\to
b^{z-2}\lambda^{\parallel}$ (83) $\displaystyle g$ $\displaystyle\to
b^{z-\frac{d}{2}}g$ $\displaystyle J$ $\displaystyle\to b^{2z-d}J$
$\displaystyle m$ $\displaystyle\to b^{z}m$ (84)

where $z$ is the dynamic critical exponents, which determines how the order
parameter relaxes close to the critical point. After these two steps we
completed the RG step, and we ended up with a theory defined by a new set of
parameters:

$\begin{split}\Gamma_{b}&=b^{z-2}\;\Gamma_{0}(1+\delta\Gamma\ln b)\\\
\lambda^{\parallel}&=b^{z-2}\;\lambda^{\parallel}_{0}(1+\delta\lambda^{\parallel}\ln
b)\\\
\lambda^{\perp}_{b}&=b^{z-2}\;\lambda^{\perp}_{0}(1+\delta\lambda^{\perp}\ln
b)\\\ m_{b}&=b^{z}\;m_{0}(1+\delta m\ln b)\\\
g_{b}&=b^{z-\frac{d}{2}}\;g_{0}(1+\delta g\ln b)\\\
J_{b}&=b^{2z-d}\;J_{0}(1+\delta J\ln b)\\\ \end{split}$ (85)

How the parameters change iterating this procedure defines the renormalization
group flow, and the fixed point of this flow rules the critical dynamics of
the system.

Even though we apparently have six equations for six parameters, it is
possible to reduce the complexity of the problem by introducing an appropriate
set of four effective coupling constants and effective parameters, namely,

$f_{0}=\frac{g_{0}^{2}}{\Gamma_{0}\lambda_{0}^{\parallel}}\qquad
u_{0}=\frac{J_{0}}{\Gamma_{0}}\qquad
w_{0}=\frac{\Gamma_{0}}{\lambda^{\parallel}_{0}}\qquad
x_{0}=\frac{\lambda^{\perp}_{0}}{\lambda_{0}^{\parallel}}$ (86)

that, through equations (85), are regulated in a closed manner by the
following four equations,666In fact, $u_{0}$ is not a new effective coupling
constant! We are just going back to the original static ferromagnetic coupling
constant; we are sorry for this back-and-forth, but it was quite inevitable.

$\begin{split}f_{b}&=b^{\epsilon}\,f_{0}\left[1+(2\delta
g-\delta\Gamma-\delta\lambda^{\parallel})\ln b\right]\\\
u_{b}&=b^{\epsilon}\,u_{0}\left[1+(\delta J-\delta\Gamma)\ln b\right]\\\
w_{b}&=w_{0}\left[1+(\delta\Gamma-\delta\lambda^{\parallel})\ln b\right]\\\
x_{b}&=x_{0}\left[1+(\delta\lambda^{\perp}-\delta\lambda_{\parallel})\ln
b\right]\end{split}$ (87)

We point out that if we set
$\lambda_{0}^{\parallel}=\lambda_{0}^{\perp}\equiv\lambda$ the set of
effective parameters (86) coincides with the one of the standard,
unconstrained Model G Hohenberg and Halperin (1977). Once we iterate the RG
transformation, we obtain the final recursive RG flow equations,

$\begin{split}f_{l+1}&=b^{\epsilon}\,f_{l}\left[1+(2\delta
g-\delta\Gamma-\delta\lambda^{\parallel})\ln b\right]\\\
u_{l+1}&=b^{\epsilon}\,u_{l}\left[1+(\delta J-\delta\Gamma)\ln b\right]\\\
w_{l+1}&=w_{l}\left[1+(\delta\Gamma-\delta\lambda^{\parallel})\ln b\right]\\\
x_{l+1}&=x_{l}\left[1+(\delta\lambda^{\perp}-\delta\lambda_{\parallel})\ln
b\right]\end{split}$ (88)

The fixed point value of these parameters will determine the value of the
critical exponents of the model. We notice that the naive scaling dimension of
the effective couplings $f_{0}$ and $u_{0}$ is $\epsilon=4-d$, which suggest
that their fixed point will be of order $\epsilon$. In the next sections we
shall compute all the contributions arising from the shell integral at first
order in $\epsilon$ (one loop), using the Feynman diagrams technique.

Of course, the equations above are rather useless without a determination of
the various perturbative corrections $\delta\Gamma,\delta g,\dots$, which we
now calculate.

###  IV.2 Self-energies

We start computing the perturbative corrections to the Gaussian parameters of
the action, namely $m_{0},\Gamma_{0}$,
$\lambda^{\perp}_{0},\lambda^{\parallel}_{0}$. Diagrammatically these
perturbative corrections are given by the two-field vertex function, also
referred as self-energies:

$\displaystyle\Sigma_{\alpha\beta}(\tilde{\boldsymbol{k}})=$ (89)
$\displaystyle\Pi_{\alpha\beta\gamma\nu}(\tilde{\boldsymbol{k}})=$ (90)

The self-energies $\Sigma$ and $\Pi$ are given by the following Feynman
diagrams expansion:

$\Sigma_{\alpha\beta}=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{{\leavevmode\includegraphics{}{}{pgf-
img/Main-article-SMG-
figure6.pdf}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{{\leavevmode\includegraphics{}{}{pgf-
img/Main-article-SMG-
figure7.pdf}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
41.9605pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure8.pdf}}}}$ (91)
$\Pi_{\alpha\beta\gamma\nu}=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
42.86923pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure9.pdf}}}}$ (92)

which corrects the action as follows:

$\delta\mathcal{S}=\int\hat{v}_{\alpha}(-k,-\omega)\Sigma_{\alpha\beta}v_{\beta}(k,\omega)+\hat{s}_{\alpha\beta}(-k,-\omega)\Pi_{\alpha\beta\gamma\nu}(k,\omega)s_{\gamma\nu}(k,\omega)$
(93)

Therefore $\Sigma_{\alpha\beta}$ corrects $\Gamma_{0}$ and $r_{0}$, while
$\Pi_{\alpha\beta\gamma\nu}$ corrects $\lambda_{0}^{\perp}$ and
$\lambda_{0}^{\parallel}$. These diagrams can be written as integrals, using
the standard Feynman diagrammatic rules, and read

$\begin{split}\Sigma_{\alpha\beta}(\tilde{\boldsymbol{k}})=&+g_{0}^{2}P_{\alpha\rho}(\boldsymbol{k})\int_{\frac{\Lambda}{b}}^{\Lambda}\frac{d^{d}p}{(2\pi)^{d}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}P_{\tau\mu}(\boldsymbol{p}_{+})\mathbb{G}_{\sigma\tau}^{0,\psi}(\tilde{\boldsymbol{p}}_{+})\mathbb{C}_{\rho\sigma\mu\beta}^{0,s}(\tilde{\boldsymbol{p}}_{-})\\\
&+g_{0}^{2}P_{\alpha\rho}(\boldsymbol{k})\int_{\frac{\Lambda}{b}}^{\Lambda}\frac{d^{d}p}{(2\pi)^{d}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}(k^{2}-p_{-}^{2})\mathbb{G}_{\rho\sigma\tau\beta}^{0,s}(\tilde{\boldsymbol{p}}_{+})\mathbb{C}_{\sigma\tau}^{0,\psi}(\tilde{\boldsymbol{p}}_{-})-\\\
&-J_{0}Q_{\alpha\beta\sigma\tau}(\boldsymbol{k})\int_{\frac{\Lambda}{b}}^{\Lambda}\frac{d^{d}p}{(2\pi)^{d}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\mathbb{C}_{\sigma\tau}^{0,\psi}(\tilde{\boldsymbol{p}})\end{split}$
(94)
$\Pi_{\alpha\beta\gamma\nu}(\tilde{\boldsymbol{k}})=g_{0}^{2}\mathbb{I}_{\alpha\beta\sigma\mu}\mathbb{I}_{\rho\tau\gamma\nu}\int_{\frac{\Lambda}{b}}^{\Lambda}\frac{d^{d}p}{(2\pi)^{d}}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}(p_{+}^{2}-p_{-}^{2})P_{\iota\rho}(\boldsymbol{p}_{+})\mathbb{G}_{\mu\iota}^{0,\psi}(\tilde{\boldsymbol{p}}_{+})\mathbb{C}_{\sigma\tau}^{0,\psi}(\tilde{\boldsymbol{p}}_{-})$
(95)

where
$\tilde{\boldsymbol{p}}_{+}=\tilde{\boldsymbol{p}}+\frac{\tilde{\boldsymbol{k}}}{2}$,
while
$\tilde{\boldsymbol{p}}_{-}=\tilde{\boldsymbol{p}}-\frac{\tilde{\boldsymbol{k}}}{2}$.
The integration over the frequency is performed explicitly, as $\omega$ has no
cutoff, while the integration in the wave vector is performed by using a thin-
shell approximation, which is valid for $b\sim 1$; in this way we get:

$\begin{split}\Sigma_{\alpha\beta}(\tilde{\boldsymbol{k}})=&-(m_{0}+\Gamma_{0}k^{2})\delta_{\alpha\beta}\frac{3(1+3w_{0}+2x_{0})f_{0}}{4(1+w_{0})(x_{0}+w_{0})}\ln{b}+\\\
&+\delta_{\alpha\beta}\frac{9}{2}u_{0}(m_{0}-\Gamma_{0}\Lambda^{2})\ln{b}\end{split}$
(96)
$\begin{split}\Pi_{\alpha\beta\gamma\nu}(\tilde{\boldsymbol{k}})=&-\lambda_{0}^{\perp}k^{2}\mathbb{P}_{\alpha\beta\gamma\nu}(\boldsymbol{k})\frac{f_{0}}{6x_{0}}\ln{b}-\\\
&-\lambda_{0}^{\parallel}k^{2}\left[\mathbb{I}_{\alpha\beta\gamma\nu}-\mathbb{P}_{\alpha\beta\gamma\nu}(\boldsymbol{k})\right]\frac{f_{0}}{3}\ln{b}\end{split}$
(97)

Here we evaluate $\Sigma_{\alpha\beta}$ and $\Pi_{\alpha\beta\gamma\nu}$ a the
relevant order in the momenta, namely up to order $k^{2}$. From (96), (97) it
is possible to read the perturbative correction to the parameters
$\Gamma_{0},m_{0},\lambda_{0}^{\perp},\lambda_{0}^{\parallel}$:

$\displaystyle\delta\Gamma$
$\displaystyle=\frac{3(1+3w_{0}+2x_{0})}{4(1+w_{0})(x_{0}+w_{0})}f_{0}$ (98)
$\displaystyle\delta m\ $
$\displaystyle=-\frac{9}{2}u_{0}\frac{m_{0}-\Gamma_{0}\Lambda^{2}}{m_{0}}+\frac{3(1+3w_{0}+2x_{0})}{4(1+w_{0})(x_{0}+w_{0})}f_{0}$
(99) $\displaystyle\delta\lambda^{\perp}$ $\displaystyle=\frac{f_{0}}{6x_{0}}$
(100) $\displaystyle\delta\lambda^{\parallel}$ $\displaystyle=\frac{f_{0}}{3}$
(101)

####  IV.2.1 Absence of spin dissipation

It is important at this point to emphasize a key result: the self-energy
$\Pi_{\alpha\beta\gamma\nu}(k,\omega)$ vanishes when $k\to 0$. This fact
implies that the renormalization group is not generating a dissipative term
for the spin, namely it is not generating a linear term in the spin equation
of motion which is finite at $k=0$. This result is strictly related to the
particular structure of the mode coupling vertex of the spin equation of
motion (77), and in particular to the fact that this vertex vanishes at zero
external momentum $k$. We believe that this result, which we proved here only
at one loop level, could be valid at all orders in perturbation theory.
Indeed, the most general diagram that can generate a dissipation is given by,

$\Pi_{\alpha\beta\gamma\nu}=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
42.09343pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure10.pdf}}}}$ (102)

where the blob represents the sum of all 1-particle irriducible diagrams
compatible with the given external legs, namely the renormalized mode coupling
spin vertex, in which all the possible diagrammatic corrections (at all
orders) are taken into account. If the renormalized spin mode coupling vertex
vanishes at zero external momentum $k$, then this diagram is zero too at
$k=0$, implying that no dissipation is generated. Therefore, as far as the
structure of the spin mode coupling vertex is preserved under RG, no spin
dissipation can be generated. Even though in this work we explicitly showed
that the structure of this vertex is preserved under RG only at one loop
level, there are some hints suggesting that this result could remain valid at
all order in perturbation theory.

It can be shown, by computing the Fokker-Planck equation Hohenberg and
Halperin (1977) from the Langevin equations (44) and (45), that the
probability density of the system approaches the equilibrium Gibbs-Boltzmann
stationary state, $P\sim\exp\left(-\mathcal{H}\right)$. This non-perturbative
result relies on the specific structure of the mode coupling vertices, which
means that if the structure of the mode coupling interactions (77) and (75)
were different, the system would not have a stationary equilibrium
distribution. As this result is not perturbative, it is reasonable to think
that the RG does not violate it; indeed, it would be very strange if under
coarse-graining an equilibrium model flowed to an out-of-equilibrium one.
Therefore, we believe that the structure of the mode coupling vertices is
preserved by the RG at all orders in perturbation theory. Finally, since an
essential requirement to generate a spin dissipation is that the RG changes
the structure of the mode coupling vertices, we believe that no dissipation is
generated at all orders in perturbation theory. Note also that this result
ensures that in the presence of a small bare dissipation, the same crossover
observed in Cavagna  _et al._ (2019a, b) between a conservative and a
dissipative dynamics is recovered.

Finally, we remark once again that the absence of spin dissipation does not
imply that the total spin is instantaneously conserved: due to the presence of
the novel Katz vertex (52), which is not zero at $k=0$, conservation is
broken, even though it is still true that the mean value of the total spin
vector is conserved, suggesting that a generalized spin precession occurs:
similarly to a standard angular momentum without dissipation but in presence
of an external force, which performs a periodic precession with constant time
average, so the total spin in presence of the solenoidal constraint has a non-
dissipative time dynamics that conserves its mean value.

###  IV.3 Mode coupling vertex corrections

The corrections to the mode coupling constant $g_{0}$ are given by the
following vertex functions,

$\displaystyle V_{\alpha\beta\gamma\nu}^{\hat{\psi}\psi
s}(\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}})$
$\displaystyle=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
45.65002pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure11.pdf}}}}$ $\displaystyle
V_{\alpha\beta\gamma\nu}^{\hat{s}\psi\psi}(\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}})$
$\displaystyle=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
5.88058pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure12.pdf}}}}$ (103)

These two vertex functions correct the action as follows:

$\delta\mathcal{S}=\int_{\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}}}\hat{\psi}_{\alpha}(-\tilde{\boldsymbol{k}})\psi_{\beta}(\tilde{\boldsymbol{q}})s_{\gamma\nu}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{q}})V^{\hat{\psi}\psi
s}_{\alpha\beta\gamma\nu}(\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}})+\int_{\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}}}\hat{s}_{\alpha\beta}(-\tilde{\boldsymbol{k}})\psi_{\gamma}(\tilde{\boldsymbol{k}}/2-\tilde{\boldsymbol{q}})\psi_{\nu}\
(\tilde{\boldsymbol{k}}/2+\tilde{\boldsymbol{q}})V^{\hat{s}\psi\psi}_{\alpha\beta\gamma\nu}(\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}})$
(104)

We are interested in how these vertex function change the value of the mode
coupling constant, hence we can compute $V^{\hat{\psi}\psi s}$ at the zeroth
order in the momenta, and $V^{\hat{s}\psi\psi}$ at the second order in the
momenta (because the $g_{0}\hat{s}\psi\psi$ term in the action is of second
order in the wave vector).

#### A first key consistency check: perturbative renormalization vs symmetry
generator

The coupling $g_{0}$ is the parameter conjugated to the generator of the
rotational symmetry (the spin), hence it plays a central role in the
definition of the Poisson structure and for this reason it cannot take
perturbative contributions from the RG calculation. Therefore, we expect both
$V^{\hat{\psi}\psi s}$ and $V^{\hat{s}\psi\psi}$ to be zero, and fortunately
this is indeed the case in our calculation. From the technical point of view,
however, the fact that these vertex functions are zero is extremely nontrivial
and it is worth showing, as it is a vital consistency check of the calculation
and in particular of the necessity of the new Katz vertex. The vertex function
$V^{\hat{\psi}\psi s}$ is given by the following Feynman diagrams:

$V^{\hat{\psi}\psi
s}_{\alpha\beta\gamma\nu}=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
6.19406pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure13.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
5.88058pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure14.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
6.19406pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure15.pdf}}}}+\
\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
18.2661pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure16.pdf}}}}$ (105)

To compute the corrections to $g_{0}$ we must compute $V^{\hat{\psi}\psi s}$
at the zeroth order it the momenta. As it happens in the non-constrained case
Hohenberg and Halperin (1977), these four diagrams, at zero external momenta
$\tilde{\boldsymbol{k}}$, $\tilde{\boldsymbol{q}}$, sum up to zero.

$V^{\hat{\psi}\psi s}_{\alpha\beta\gamma\nu}(0,0)=0$ (106)

The vertex function
$V^{\hat{s}\psi\psi}_{\alpha\beta\gamma\nu}(\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}})$,
on the other hand, is given by the following nonzero Feynman diagrams:

$V^{\hat{s}\psi\psi}_{\alpha\beta\gamma\nu}=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
18.2661pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure17.pdf}}}}+\
\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
2.19403pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure18.pdf}}}}+\
\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
18.2661pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure19.pdf}}}}+\
\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
24.19997pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure20.pdf}}}}$ (107)

The first two diagrams (the triangles) cancel each other, exactly as in the
non-solenoidal case. The other two diagrams are, on the other hand, specific
to the solenoidal case. The first diagram, which vanishes in the non-
constrained theory Cavagna  _et al._ (2019b), is nonzero when the solenoidal
constraint is present, due the suppression of the longitudinal $\psi$ mode,

$\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
42.86923pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure21.pdf}}}}=-\frac{g_{0}u_{0}}{8}\mathbb{I}_{\alpha\beta\sigma\tau}\left(k_{\sigma}k_{\gamma}\delta_{\tau\nu}+k_{\sigma}k_{\nu}\delta_{\tau\gamma}\right)+\dots$
(108)

where the ellipses stand for higher order in the momentum expansion,
representing corrections to RG-irrelevant interactions. This vertex
corrections not only would give to $g_{0}$ a perturbative correction due to
the shell integration, but it would generate a novel interaction term too,
since it has not the same tensorial structure as the original interaction
(77). Therefore, if no other diagram canceling it were present, the RG would
not have a closed structure, meaning that the equations of motion would not be
eigenstates of the RG transformation, since the shell integration generates
new interaction terms that were not present in the bare theory. This
unpleasant scenario, in which new relevant interactions arise during the RG
flow, is avoided by the key presence of a new Feynman diagram formed by a
bubble connection of two lines of the Katz vertex, namely,

$\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
44.06407pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure22.pdf}}}}=\frac{g_{0}u_{0}}{8}\mathbb{I}_{\alpha\beta\sigma\tau}\left(k_{\sigma}k_{\gamma}\delta_{\tau\nu}+k_{\sigma}k_{\nu}\delta_{\tau\gamma}\right)+\dots$
(109)

where, as before, the ellipses stand for higher order in the momentum
expansion. The presence of this diagram is fundamental, since it exactly
cancels the contributions of diagram (108), therefore curing the anomalies
that the latter carries and making the solenoidal RG calculation self-
consistent. We therefore consider the following diagrammatic equation a key
result of our calculation:

$\boxed{\begin{split}\\\
\qquad\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
42.86923pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure23.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
44.06407pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure24.pdf}}}}=0\qquad\\\ \\\ \end{split}}$ (110)

Moreover, although the Katz vertex contributes to the dynamic behaviour of
$\boldsymbol{s}$ also at $\boldsymbol{k}=0$, namely violating the conservation
of the total spin, the last diagrams vanishes at $\boldsymbol{k}=0$. The fact
that the simplest one-loop diagram that can be constructed starting from the
Katz vertex does not give any perturbative contribution at vanishing momenta
suggests that no spin dissipation should arise, not even at higher orders in
the $\varepsilon$-expansion. This is related to the fact that the Katz vertex
violates the conservation of the spin leading to a precession of it, but
keeping fixed its average value.

###  IV.4 Ferromagnetic vertex corrections

In contrast to the mode coupling vertex, the ferromagnetic coupling does have
perturbative corrections due to the shell integration. The coupling $J_{0}$ is
corrected by the four-field vertex function $V^{\hat{\psi}\psi\psi\psi}$,

$V_{\alpha\beta\gamma\nu}^{(\hat{\psi}\psi\psi\psi)}(\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}},\tilde{\boldsymbol{p}})=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
32.20932pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure25.pdf}}}}$ (111)

that corrects the action as follows:

$\delta\mathcal{S}=\int_{\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}},\tilde{\boldsymbol{p}}}\hat{\psi}_{\alpha}(-\tilde{\boldsymbol{k}})\psi_{\beta}(\tilde{\boldsymbol{p}})\psi_{\gamma}(\tilde{\boldsymbol{q}})\psi_{\nu}(\tilde{\boldsymbol{k}}-\tilde{\boldsymbol{q}}-\tilde{\boldsymbol{p}})\
V^{\hat{\psi}\psi\psi\psi}_{\alpha\beta\gamma\nu}(\tilde{\boldsymbol{k}},\tilde{\boldsymbol{q}},\tilde{\boldsymbol{p}})$
(112)

At one loop, only the following Feynman diagrams contribute to the vertex
function $V^{\hat{\psi}\psi\psi\psi}$ in a non-trivial way:

$V_{\alpha\beta\gamma\nu}^{\hat{\psi}\psi\psi\psi}=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
32.20932pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure26.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
32.20932pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure27.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
20.11682pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure28.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
32.20932pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure29.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
32.20932pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure30.pdf}}}}$ (113)

The first term is the classic fish diagram of the standard ferromagnetic
theory; the second diagram is generated by joining a mode-coupling vertex with
the Katz vertex; the last three diagrams are of purely mode-coupling origin.
Computing $V^{\hat{\psi}\psi\psi\psi}$ at the zeroth order in the external
momenta, gives,

$V^{\hat{\psi}\psi\psi\psi}_{\alpha\beta\gamma\nu}=-\frac{J_{0}}{3}Q_{\alpha\beta\gamma\nu}\left[-\frac{17}{2}u_{0}+\frac{3(1+3w_{0}+2x_{0})}{4(1+w_{0})(x_{0}+w_{0})}f_{0}\right]\ln
b$ (114)

so that the perturbative corrections to the coupling $J_{0}$ are,

$\delta
J=-\frac{17}{2}u_{0}+\frac{3(1+3w_{0}+2x_{0})}{4(1+w_{0})(x_{0}+w_{0})}f_{0}$
(115)

####  IV.4.1 A second key consistency check: statics vs dynamics

The dynamical RG calculation (at equilibrium) must of course contain in itself
the static RG calculation; more specifically, if a coupling constant is
present also in the static case, its dynamical renormalization must be exactly
the same as its static renormalization. This is clearly the case for the
ferromagnetic coupling, which is perfectly well-defined also within a purely
static framework. Therefore, in this section we show that this consistency
between statics and dynamics is achieved by our calculation.

First of all we recall that the actual ferromagnetic coupling, namely the
coupling constant that appears in the static Hamiltonian, is
$u_{0}=J_{0}/\Gamma_{0}$ (see (61)). Hence, the static ferromagnetic coupling
$u_{0}$ gets perturbative corrections both from $J_{0}$ and $\Gamma_{0}$,

$\delta u=\delta J-\delta\Gamma$ (116)

and from equations (115) and (98) we have,

$\delta u=-\frac{17}{2}u_{0}$ (117)

Hence, the static coupling $u_{0}$ does not receive any perturbative
corrections from the dynamic coupling $g_{0}$, which is healthy. But the
crucial check is whether the recursive relation we get for $u$ from the
dynamic RG is the same as the static one, equation (27). Fortunately, it is.
From equation (88) and (117) we obtain,

$u_{l+1}=b^{\varepsilon}\,u_{l}\left[1-\frac{17}{2}u_{l}\ln b\right]$ (118)

which is exactly the same as the static RG recursive equation (27). We stress
that this key consistency is recovered in an extremely nontrivial way; in
particular, the Katz vertex plays a crucial role. The cancellation of the
dynamical coupling $g_{0}$ in the perturbative correction of the static
coupling is achieved through the following diagrammatic identity:

$\boxed{\begin{split}&\\\
\quad&\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
22.96227pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure31.pdf}}}}\
+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
22.96227pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure32.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
22.96227pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure33.pdf}}}}+\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
22.96227pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure34.pdf}}}}=\frac{\partial}{\partial
k^{2}}\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
22.96227pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure35.pdf}}}}\\\ &\\\ &\qquad\qquad D_{1}=-\frac{f_{0}}{4(w_{0}+x_{0})}\\\
&\qquad\qquad
D_{2}+D_{3}+D_{4}=\frac{(2+5w_{0}+3x_{0})}{2(1+w_{0})(x_{0}+w_{0})}f_{0}\\\
&\qquad\qquad D_{5}=\frac{3(1+3w_{0}+2x_{0})}{4(1+w_{0})(x_{0}+w_{0})}f_{0}\\\
&\end{split}}$ (119)

where the l.h.s of the equation is computed at zero external momenta. The
diagram $D_{1}$ is the product of the interplay between the mode-coupling
vertex and the Katz vertex, which is therefore crucial in recovering the
correct static behaviour.

There is a second, and subtler, consistency check related to the
renormalization of the ferromagnetic coupling constant. The coupling $u_{0}$
not only appears in front of the ferromagnetic vertex, but - due to the
static-dynamic coupling induced by the solenoidal constraint - it also appears
in front of the Katz vertex, $V^{\hat{s}\psi\psi\psi\psi}$, which is indeed
proportional to $g_{0}u_{0}$ (see equation (78)); we know that $g_{0}$ does
not acquire perturbative corrections, hence any diagrammatic correction to the
Katz vertex must be billed to $u_{0}$; but $u_{0}$ has been already corrected
by its natural ferromagnetic vertex $V^{\hat{\psi}\psi\psi\psi}$, in the
static-compliant way that we have just seen, equation (117). Hence, it seems
we have two potentially independent corrections to $u_{0}$, one coming from
the bona fide ferromagnetic vertex, and a second one from the Katz vertex! If
these diagrammatic corrections were different from each other, we would have a
serious problem, as there would be a bifurcation of the ferromagnetic
interaction, with highly dubious physical interpretation, not to mention the
impossible recovery of the equilibrium static results. Once again,
fortunately, the calculation does not disappoint us, even though in a very
nontrivial way. The Katz vertex function $V^{\hat{s}\psi\psi\psi\psi}$ has (at
one loop) the following non-vanishing diagrammatic contribution,

$V_{\alpha\beta\gamma\nu\sigma\tau}^{\hat{s}\psi\psi\psi\psi}=\leavevmode\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\immediate\hbox{\lower
22.96227pt\hbox{{\leavevmode\includegraphics{}{}{pgf-img/Main-article-SMG-
figure36.pdf}}}}$ (120)

Computing this Feynman diagram we get,

$V_{\alpha\beta\gamma\nu\sigma\tau}^{\hat{s}\psi\psi\psi\psi}=-\frac{g_{0}u_{0}}{12}\left[1-\frac{17}{2}u_{0}\right]K_{\alpha\beta\gamma\nu\sigma}$
(121)

and therefore the correction to $u_{0}$ that we obtain from the Katz vertex is
exactly the same as from the ferromagnetic vertex, namely $\delta
u=-17/2u_{0}$, which saves the day.

##  V The critical dynamics of solenoidal Model G

We now have all that we need to finally calculate the dynamical critical
exponent $z$ at one loop in a mode-coupling theory subject to a solenoidal
constraint, that is in solenoidal Model G.

###  V.1 The recursive RG equations and the $\beta$-functions

First we collect all the perturbative contributions, we plug them into
equations (88) and write the recursive relations for the effective parameters
and coupling constants of the theory,

$\displaystyle u_{l+1}$
$\displaystyle=b^{\varepsilon}u_{l}\left[1-\frac{17}{2}u_{l}\ln{b}\right]$
(122) $\displaystyle f_{l+1}$
$\displaystyle=b^{\varepsilon}f_{l}\left[1-f_{l}\left(\frac{1}{3}+\frac{3\left(1+3w_{l}+2x_{l}\right)}{4\left(1+w_{l}\right)\left(x_{l}+w_{l}\right)}\right)\ln{b}\right]$
(123) $\displaystyle w_{l+1}$
$\displaystyle=w_{l}\left[1-f_{l}\left(\frac{1}{3}-\frac{3\left(1+3w_{l}+2x_{l}\right)}{4\left(1+w_{l}\right)\left(x_{l}+w_{l}\right)}\right)\ln{b}\right]$
(124) $\displaystyle x_{l+1}$
$\displaystyle=x_{l}\left[1-\frac{f_{l}}{3}\left(1-\frac{1}{2x_{l}}\right)\ln{b}\right]$
(125)

where we are now working at $T=T_{c}$, namely at $r=m=0$. Since the
calculation here is performed at one loop, the static coupling $u$ does not
contribute to the renormalization of the dynamic parameters, therefore
completely decoupling the dynamic behaviour from the static one. Because we
have already abundantly checked that the renormalization of $u$ is compatible
with the statics, we simply drop this equation from now on.

The derivatives of a parameter $\mathcal{P}$ with respect to $\ln{b}$ is known
as the $\beta$-function
$\beta_{\mathcal{P}}=\frac{\partial\mathcal{P}}{\partial\ln{b}}$, and measure
how the parameter change when an infinitesimal RG transformation is performed.
The $\beta$-functions of the effective parameters $f$, $w$ and $x$, obtained
from Eqs. (123), (124) and (125), are given by:

$\displaystyle\beta_{f}$
$\displaystyle=f\left(\varepsilon-\frac{1}{3}-\frac{3\left(1+3w+2x\right)}{4\left(1+w\right)\left(x+w\right)}\right)$
(126) $\displaystyle\beta_{w}$
$\displaystyle=wf\left(\frac{3\left(1+3w+2x\right)}{4\left(1+w\right)\left(x+w\right)}-\frac{1}{3}\right)$
(127) $\displaystyle\beta_{x}$
$\displaystyle=\frac{f}{3}\left(\frac{1}{2}-x\right)$ (128)

The zeros of the $\beta$-functions give the fixed points of the RG flow, which
have a crucial role in ruling the critical behaviour of the theory. Since we
are interested in a genuine mode-coupling dynamical regime, we will not
consider the trivial fixed points with $f^{*}=0$, since they lead to the
overdamped dynamics with $z=2$ (at one loop) typical of Model A Hohenberg and
Halperin (1977). We find a set of non-trivial fixed points, one stable and the
other unstable. The stable fixed point is given by,

$f^{*}=\frac{3\varepsilon}{2}\quad,\quad
w^{*}=\frac{21+\sqrt{697}}{8}\quad,\quad x^{*}=\frac{1}{2}$ (129)

As expected the effective coupling constant is of order $\varepsilon=4-d$ at
this fixed point.

###  V.2 Anisotropy in the spin dynamics

In Model G Hohenberg and Halperin (1977), the absence of anisotropic
interactions leads to the implicit assumption that all the different
directions of the fields, both in real and Fourier space, meaning that
transverse and parallel modes must be equal and therefore
$\lambda=\lambda^{\perp}=\lambda^{\parallel}$, namely $x=1$. However, at the
new stable fixed point of the solenoidal Model G described by Eq. (129)
$x^{*}=1/2$, meaning that the anisotropy due to the suppression of the
longitudinal $\psi$-mode leads to a different dynamic behaviour of the two
$s$-modes $\boldsymbol{s}^{\perp}$ and $\boldsymbol{s}^{\parallel}$, in such a
way that ${\lambda^{\parallel}}^{*}=2{\lambda^{\perp}}^{*}$. This result
directly follows from the fact that the perturbative corrections
$\delta\lambda^{\perp}$ and $\delta\lambda^{\parallel}$, given in Eqs. (100)
and (101) respectively, obey the relation

$\delta\lambda^{\parallel}=2x\,\delta\lambda^{\perp}$ (130)

It is not yet clear to us whether this factor $2$ between $\lambda^{\perp}$
and $\lambda^{\parallel}$ can be guessed through a direct analysis of the
equations of motion, or if it valid only in the long wavelength and long time
dynamic behaviour.

In any case, since the diagram contributing to $\delta\lambda^{\perp}$ and
$\delta\lambda^{\parallel}$ does not involve neither propagators nor
correlators of the field $s$, we may observe that this result is a pure
consequence of suppression of the $\psi^{\parallel}$ modes. In fact, even if
we had naively left $\lambda_{0}=\lambda_{0}^{\perp}=\lambda_{0}^{\parallel}$
at a bare level, the RG transformation would have led to two different
perturbative corrections $\delta\lambda^{\perp}$ and
$\delta\lambda^{\parallel}$, meaning that the infrared behaviour of this
theory has two different diffusive coefficients for the
$\boldsymbol{s}^{\perp}$ and $\boldsymbol{s}^{\parallel}$ modes.

###  V.3 The dynamical critical exponent $z$

To find the dynamic critical exponent, following Hohenberg and Halperin (1977)
and Cardy (1996), we require that the kinetic coefficient of the primary
field, $\Gamma$, is non-singular at the RG fixed point, thus ensuring that the
effective RG theory has a non-singular characteristic time scale. This amounts
to imposing the condition,

$\Gamma^{*}=\lim_{l\to\infty}\Gamma_{l}=O(1)$ (131)

and thus we need to write explicitly the recursive RG equation for the kinetic
coefficient; that equation can be found in (85), complemented by its
perturbative corrections, equation (98), thus giving,

$\Gamma_{l+1}=b^{z-2}\,\Gamma_{l}\left[1+\frac{3(1+3w_{l}+2x_{l})f_{l}}{4(1+w_{l})(x_{l}+w_{l})}\ln
b\right]$ (132)

By using the obvious expansion, $b^{x}(1+y\ln b)=b^{x+y}$, we finally obtain
from (131) and (132) that the general expression for the dynamic critical
exponent is,

$z=2-\frac{3\left(1+3w^{*}+2x^{*}\right)f^{*}}{4\left(1+w^{*}\right)\left(x^{*}+w^{*}\right)}$
(133)

and once we use the one-loop values of the parameters at the stable fixed
point (129), we obtain

$z=\frac{d}{2}$ (134)

Despite the difference of all effective parameters and coupling constants at
the stable fixed point, this is exactly the same dynamic critical exponent as
the standard unconstrained Model G Hohenberg and Halperin (1977). This result
is somewhat surprising. The solenoidal constraint does change the static
universality class: the static critical exponents are different from the
Landau-Ginzburg class, and define the novel dipolar ferromagnet class (see
Table I and Aharony and Fisher (1973)). According to the common wisdom in the
theory of critical phenomena, we would expect a change also in the dynamic
universality class, as universality is normally broader at the static level
than at the dynamic level. For example, Ising-like ferromagnets, have the same
static critical exponents, while the dynamical critical exponents varies
depending on whether the order parameter is conserved (Model B) or not (Model
A). Here, we see something different: the dynamic universality class does not
change due to the solenoidal constraint. Even though we have derived this
conclusion perturbatively, this is probably a non-perturbative result due to
the great power of the symmetry; the lack of diagrammatic renormalization of
the coupling constant conjugated to the generator of the rotations, $g_{0}$,
leads to the recursive relation,

$g_{l+1}=b^{z-\frac{d}{2}}g_{l}$ (135)

If we now ask that the central charge of the symmetry does not change at all,
we get $z=d/2$, at the non-perturbative level.

The fact that the solenoidal constraint does not alter the dynamical critical
exponent is an encouraging result, since it suggests that using the
incompressibility condition to simplify the dynamical equations of active
matter with mode coupling interactions is a reasonable approximation, as it
does not change dramatically the dynamical behaviour of the system.

##  VI Conclusions

We have studied the effects of a solenoidal constraint on the critical
dynamics of a field ${\boldsymbol{\psi}}$ with $O(d)$-symmetry in the presence
of mode-coupling interaction with the generator of the rotational symmetry
$\boldsymbol{s}$, which we called spin; more succinctly, we have studied
solenoidal Model G. The presence of the constraint leads to the suppression of
the $\psi$-mode parallel to the wave-vector $\boldsymbol{k}$, namely
$\psi^{\parallel}(\boldsymbol{k})=\boldsymbol{k}\cdot{\boldsymbol{\psi}}\left(\boldsymbol{k}\right)/\mathinner{\\!\left\lvert
k\right\rvert}$, violating the $O(d)$-symmetry and modifying the static
behaviour. The equations of motion of the constrained theory have been derived
starting from the symmetries and Poisson-bracket relations between the
hydrodynamic variables, namely the order parameter ${\boldsymbol{\psi}}$ and
the spin $\boldsymbol{s}$. We performed a one-loop renormalization group
calculation to investigate the long wave-length and long time behaviour in the
critical region. The closed structure of the RG transformation and the
consistency of the RG flow with the static behaviour of dipolar ferromagnets
provide a self-consistent proof that no RG-relevant interaction has been
omitted and that the equations of motion we derived are correct.

Two main dynamic effects arise as a consequence of the solenoidal constraint.
The first, and most predictable one, is the projection on the plane orthogonal
to $\boldsymbol{k}$ of the equation of motion for the order parameter
${\boldsymbol{\psi}}$. On the contrary, no similar projection of the equation
for $\boldsymbol{s}$ can be performed in order to obtain the spin dynamics;
instead, the suppression of $\psi^{\parallel}(\boldsymbol{k})$ leads to a
novel non-linear interaction - the Katz vertex - combining the effect of the
static ferromagnetic coupling of the field ${\boldsymbol{\psi}}$ and the mode-
coupling dynamic interaction. The presence of this new mixed interaction is
the second, less intuitive, effect of the constraint, which adapts the spin
dynamics to the presence of the constraint by making the static quartic
coupling contributing to the torque-like interaction
$\partial_{t}\boldsymbol{s}\sim
g_{0}{\boldsymbol{\psi}}\times\delta_{\psi}\mathcal{H}$. Moreover, the Katz
vertex contributes to the time-derivative of the spin also at zero wave
vector, therefore violating the conservation of the total spin.

The lack of conservation of the spin is not something strange: the order
parameter is not rotational-invariant, due to the solenoidal constraint, and
therefore the generator of its rotations is not a conserved quantity. It is
however crucial to understand whether the spin is dissipated or not, since the
presence of a dissipation generated by the RG would make the spin stop being
an hydrodynamic variable, therefore suppressing any inertial behaviour in the
critical region. The torque-like nature of the Katz vertex, which is the only
dynamic term violating the spin conservation, indicates that this violation
gives rise to a generalized precession of the total spin, rather than a
dissipation. At one loop, the perturbative expansion confirms this
interpretation, since the self-energy of the spin $\Pi$ does not contain any
perturbative corrections at $\boldsymbol{k}=0$. Moreover, we show that the
presence of any dissipative term in the linear dynamics of the spin can arise
only if the dynamic mode-coupling vertex in the equation of $\boldsymbol{s}$
did not vanish when $\boldsymbol{k}=0$, which seems not to be the case for
this theory.

Our RG calculation passed several nontrivial consistency checks. First of all,
the fact that the equations of motion appear to be eigenstates of the RG, in
the sense that the shell integration step does not generate new interaction
terms, ensures that we did not miss any relevant coupling in the problem
description. Secondly, the RG recursive relations we found for the dynamic
theory reproduce the behaviour of dipolar ferromagnets, ensuring that the
static behaviour is correctly reproduced by the dynamics. We must remark that
both these results directly follow from the presence of the new Katz vertex,
in absence of which the theory would not describe correctly the dynamics of
the system; therefore meaning that the this new non-trivial interaction plays
a crucial role in making the dynamic behaviour of the spin compatible with the
constraint.

From the study of the RG recursive relations, the dynamic behaviour has been
shown to be characterized by a critical exponent $z=\frac{d}{2}$, which is the
same as the non-constrained theory. This result was somewhat unexpected. In
general, static properties are more robust compared to dynamical properties;
models with different dynamical critical behaviours are often characterized by
the same static behaviour, such as Model A and Model G of Hohenberg and
Halperin (1977). On the contrary, in our model the solenoidal constraint
changes the static universality class, leaving unchanged the critical dynamics
of the system. This suggests that the dynamic critical behaviour of
homogeneous systems does not change when a solenoidal constraint, i.e.
incompressibility, is enforced; this indicates that we can try and understand
their dynamic critical behaviour by studying their incompressible version.
Homogeneous systems are governed by equations of motion in which
incompressibility is, in general, not required, but where density
fluctuations, and therefore density-velocity couplings, are negligible.
Incompressibility, though, completely suppresses density fluctuations and
therefore represents a stronger requirement than homogeneity. Moreover,
requiring incompressibility, hence imposing a solenoidal constraint on the
velocity field, generates long-range interactions that could change the
properties of a system; while this is indeed the case for the static behaviour
of our theory, the long-range interactions are not sufficient to modify its
dynamic universality class. Theoretical evidences of this fact have already
been discussed in homogeneous active systems Cavagna  _et al._ (2021),
therefore suggesting that the solenoidal constraint does not significantly
affect the critical dynamic behaviour in the presence of neither activity nor
mode-coupling terms. This result is very encouraging, as it allows to study
the homogeneous phase of the off-equilibrium Inertial Spin Model under a
incompressible hypothesis, where the absence of the density field leads to a
great simplification of the calculation.

Our result is an important stepping stones towards developing an RG theory for
natural swarms. The Katz interaction vertex in the spin dynamics that we
derived in this work, which emerges as an effect of the solenoidal constraint,
must certainly characterize also the equations of motion of an incompressible
out-of-equilibrium field theory, in which terms coupling the order parameter
to its generator of rotations are present. It would have been extremely
difficult to derive the Katz vertex had one tackled the problem directly in
the presence of activity. Despite this step forward, though, the complexity of
the present calculation suggests that significant more theoretical efforts
will be needed to carry out the full-fledged out-of-equilibrium mode-coupling
RG study of natural swarms.

We warmly thank M. Testa, C. Castellani, and U. Tauber for fruitful
discussions. L. Di Carlo also thanks I. Basile for interesting exchanges about
the work. This work was supported by ERC Advanced Grant RG.BIO (contract
n.785932) to AC, and ERANET-CRIB Grant to AC and TSG. TSG was also supported
by grants from CONICET, ANPCyT and UNLP (Argentina).

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