Functional renormalization group for the $U(1)$-$T_{5}^{6}$ tensorial group
field theory with closure constraint

Vincent<EMAIL_ADDRESS>and Dine Ousmane
<EMAIL_ADDRESS>
a) Laboratoire de Physique Théorique, CNRS-UMR 8627, Université Paris-Sud 11,
91405 Orsay Cedex, France

b) Max Planck Institute for Gravitational Physics, Albert Einstein Institute,
Am Mühlenberg 1, 14476, Potsdam, Germany

c) Faculté des Sciences et Techniques/ ICMPA-UNESCO Chair, Université
d’Abomey- Calavi, 072 BP 50, Benin

###### Abstract

This paper is focused on the functional renormalization group applied to the
$T_{5}^{6}$ tensor model on the Abelian group $U(1)$ with closure constraint.
For the first time, we derive the flow equations for the couplings and mass
parameters in a suitable truncation around the marginal interactions with
respect to the perturbative power counting. For the second time, we study the
behavior around the Gaussian fixed point, and show that the theory is
nonasymptotically free. Finally, we discuss the UV completion of the theory.
We show the existence of several nontrivial fixed points, study the behavior
of the renormalization group flow around them, and point out evidence in favor
of an asymptotically safe theory.

###### Contents

  1. 1 Introduction
  2. 2 The $T_{5}^{6}$ TGFT model
    1. 2.1 The model
    2. 2.2 Canonical dimension
  3. 3 Functional renormalization group with closure constraint
    1. 3.1 Truncation and Regularization
    2. 3.2 Flow equations in the UV regime
      1. 3.2.1 Flow equations for $Z$ and $m^{2}$
      2. 3.2.2 Flow equation for $\lambda_{1}$
      3. 3.2.3 Flow equations for $\lambda_{2}$ and $\lambda_{3}$
      4. 3.2.4 Dimensionless renormalized parameters
  4. 4 Fixed points in the UV regime
    1. 4.1 Vicinity of the Gaussian fixed point
    2. 4.2 Non-Gaussian fixed points
  5. 5 Truncation with an interaction of valence 8
  6. 6 Discussion and conclusion

##  1 Introduction

Tensor models (TMs) generalize matrix models and are considered as a
convenient formalism for studying random geometries [1]-[17]. TMs also offer
an alternative to other approaches dealing with quantum gravity (QG) which is
based on new mathematical/statistical tools, for example, the $1/N$ expansion
recently discovered [16]-[17]. On the other hand, group field theory (GFT) is
a quantum field theory over group manifolds and is considered as a second
quantization of loop quantum gravity [18]-[22]. Both TMs and GFT belong to the
so-called background-independent scenario for QG. They aim at describing a
rudimentary phase of the geometry of spacetime, namely, when this geometry is
hypothetically still in a discrete form, or at least not yet continuous. It is
also named ”pre-geometric” phase of our spacetime. Recently TMs and GFT have
been combined to provide a new class of field theories so called tensorial
group field theory (TGFT). TGFTs improve the GFTs in order to allow for
renormalization [23]-[32]. Moreover, it has been shown that several TGFTs
models are asymptotically free in the UV, in other words, near the Gaussian
fixed point [33]-[43].

The renormalization group (RG) method formulated first by Wilson [47]-[48] is
a nonperturbative method which allows to interpolate smoothly between the UV
laws and the IR phenomena in physical systems. The RG has a vast range of
applications. A particularization of the RG, the functional renormalization
group (FRG) is a realization of the RG concept in the framework of quantum
and/or statistical field theory and is one of the best candidates for studying
quantum fluctuations [49]. An important property of this method is that the
FRG could be used in regimes where perturbative calculations are invalid, for
instance, at the vicinity of nontrivial fixed points in the infrared regime.

Recently much interest were focused on the FRG equation of various Matrix and
TGFT models [39]-[46]. The differential equations of the flow were derived
using Wetterich’s equation [49]. The fixed points were given and further
evidence of asymptotically safety and asymptotically freedom were derived
around these fixed points in the UV.

The TGFT of the form $T^{6}_{5}$ on the $U(1)$ group with closure constraint
is proved to be renormalizable [30]. The proof of this claim is performed
using multi-scale analysis. The closure constraint also called gauge
invariance condition can help to define the emergence of the metric on
spacetime after phase transition and therefore makes this type of model
relevant for the understanding the quantum theory of gravitation. This kind of
model with closure constraint namely the six dimensional TGFT with quartic
interactions is studied recently in [39] and [41]. The perturbative
computation of the $\beta$-functions of the $T^{6}_{5}$ model is given in
[38], in which, we have showed that this model is asymptotically free in the
UV. This result seems to be non consistant in the point of view of the FRG
analysis. This paper aims at giving the FRG analysis to a renormalizable
tensor model $T^{6}_{5}$ with closure constraint and for improving the
conclusion given in [38]. In a truncation containing all relevant and marginal
interactions, we find nontrivial fixed points. The FRG flows for coupling
constants and for the mass parameter are solved numerically.

The paper is organized as follows. In section (2) we first present the model
which is analysed in this paper, namely the $T_{5}^{6}$ model with closure
constraints. In section (3) we give the flow equations of the coupling
constants and mass parameter by using the dimensional renormalization
parameters. In section (4) we give the nontrivial fixed points and provide the
numerical solution of the flow equations. In section (5) the validity of the
choice of the truncation of the effective action is discussed. The behavior of
our model in the vicinity of these fixed points is also given. The conclusion
and discussion are made in section (6).

##  2 The $T_{5}^{6}$ TGFT model

This section is devoted to a short review of the particular TGFT that we
present in this paper. First, we define and give some properties of our model.
Next, we discuss the canonical dimension that allows to make sense of the
exponentiation of the functional action in the partition function.

###  2.1 The model

We consider fields $\psi$ and $\bar{\psi}$ acting over the $d$ copies of the
group $U(1)$, i.e. $\psi,\,\bar{\psi}:\,U(1)^{d}\rightarrow\mathbb{C}$. In the
Fourier representation, the fields variables
$T_{\vec{p}},\,\,\bar{T}_{\vec{p}},\,\,\vec{p}\in\mathbb{Z}^{d}$ are maps
$\bar{T},T:\mathbb{Z}^{d}\to\mathbb{C}$, such that,

$\displaystyle\psi(\vec{\mathbf{g}})=\sum_{\vec{p}}T_{\vec{p}}\,e^{i\vec{\theta}\cdot\vec{p}},\quad\vec{\mathbf{g}}=(\mathbf{g}_{1},\cdots,\mathbf{g}_{d}),\,\,\vec{p}=(p_{1},\cdots,p_{d}),\,\vec{\theta}=(\theta_{1},\cdots,\theta_{d}),\,\theta_{j}\in[0,2\pi).$
(1)

The parameter $\theta_{k}$ with $\mathbf{g}_{k}\equiv e^{i\theta_{k}}$ is
related to the parametrization of the $U(1)$ group such that $U(1)\approx
S^{1}$. The theory we consider is described by its generating function or the
vacuum-vacuum transition amplitude:

$Z_{\Lambda}[J,\bar{J}]=e^{W_{\Lambda}[J,\bar{J}]}=\int
d\mu_{C_{\Lambda}}(\bar{T},T)e^{S_{int}[\bar{T},T]+\langle\bar{J},T\rangle+\langle\bar{T},J\rangle}$
(2)

where the notation $\langle.,.\rangle$ means:
$\langle\bar{J},T\rangle=\sum_{\vec{p}\in\mathbb{Z}^{d}}\bar{J}_{\vec{p}}T_{\vec{p}}$,
$d\mu_{C_{\Lambda}}$ is the Gaussian measure with the covariance $C_{\Lambda}$
such that:

$\displaystyle\int\,d\mu_{C_{\Lambda}}T_{\vec{p}}\bar{T}_{\vec{p}\,^{\prime}}=\frac{e^{-(\vec{p}^{2}+m^{2})/\Lambda^{2}}}{\vec{p}^{2}+m^{2}}\delta\bigg{(}\sum_{i=1}^{d}p_{i}\bigg{)}\delta_{\vec{p}\vec{p}^{\prime}}=C_{\Lambda}(\vec{p},\vec{p}\,^{\prime}),$
(3)

and the delta $\delta(\sum_{i=1}^{d}p_{i})$ implements the closure constraint,
see [23, 28]. $\Lambda$ is the UV cutoff which will impose that the modulus of
momentum vectors remains less than $\Lambda$, namely, $|\vec{p}|\leq\Lambda$.
We keep in mind that we will eventually take the limit
$\Lambda\rightarrow\infty$. We define a model by its action at a high (UV)
energy scale. The classical action $S_{int}$ is defined as a sum of tensorial
invariances [1], [3]:

$S_{int}[\bar{T},T]=\sum_{b\in\mathcal{B}}\lambda_{b}\mathrm{Tr}_{b}[\bar{T},T].$
(4)

A tensor invariant is a polynomial in the tensor $T$ and its conjugate
$\bar{T}$ which is invariant under the action of the tensor product of $d$
independent copies of the unitary group $U(N)$. The sum is taken over a finite
set $\mathcal{B}$ of such invariants $d$-bubbles [1] associated with the
couplings $\lambda_{b}$.

The interaction (4) of a tensor field theory in dimension $d=5$ [30] is

$\displaystyle S_{int}[\bar{T},T]=$
$\displaystyle\frac{\lambda_{1}}{2}\sum_{\ell=1}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,4}\mathcal{W}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}$
(5)
$\displaystyle+\frac{\lambda_{2}}{3}\sum_{\ell=1}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{X}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}$
$\displaystyle+\lambda_{3}\sum_{\ell_{i}=1,i=1,2,3}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{Y}^{(\ell_{1},\ell_{2},\ell_{3})}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}},$
(6)

where the symbols $\mathcal{W}^{(\ell)}$, $\mathcal{X}^{(\ell)}$ and
$\mathcal{Y}^{(\ell)}$ are products of delta functions associated to tensor
invariant interactions, and $\lambda_{i}(\Lambda)$ are coupling constants. For
instance:

$\mathcal{W}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4}}=\delta_{p_{1\ell}p_{4\ell}}\delta_{p_{2\ell}p_{3\ell}}\prod_{j\neq\ell}\delta_{p_{1j}p_{2j}}\delta_{p_{3j}p_{4j}}.$
(7)

Such a kernel is called  bubble [1], and can be pictured graphically as a
$6$-colored bipartite regular graph, with black and white vertices
corresponding respectively to the fields $T$ and $\bar{T}$, and each line
corresponding to a Kronecker delta. As an example, the $4$-valent bubble
associated to the kernel $\mathcal{W}^{(1)}$ is depicted on Figure (1) below,
and in the same way, all the interaction bubbles involved in the action
$S_{int}$ are defined as:

$\displaystyle\mathcal{W}^{(\ell)}(\mathbf{g}_{1},\mathbf{g}_{2},\mathbf{g}_{3},\mathbf{g}_{4})$
$\displaystyle=$ (8)
$\displaystyle\mathcal{X}^{(\ell)}(\mathbf{g}_{1},\mathbf{g}_{2},\mathbf{g}_{3},\mathbf{g}_{4},\mathbf{g}_{5},\mathbf{g}_{6})$
$\displaystyle=$ (9)
$\displaystyle\mathcal{Y}^{(\ell_{1},\ell_{2},\ell_{3})}(\mathbf{g}_{1},\mathbf{g}_{2},\mathbf{g}_{3},\mathbf{g}_{4},\mathbf{g}_{5},\mathbf{g}_{6})$
$\displaystyle=$ (10) Figure 1: An example of bubble of order 4

where the index $\ell$ takes values from $1$ to $5$, and refers to the single
color characterizing each bubbles.

###  2.2 Canonical dimension

Another important definition for our purpose concern the notion of  canonical
dimension. We will only give the essential here, and the reader interested in
the details may consult [37]. In our model, the divergence degree for an
arbitrary Feynman graph $\mathcal{G}$ is given by [23, 30]

$\omega(\mathcal{G})=-2L(\mathcal{G})+(F(\mathcal{G})-R(\mathcal{G}))$ (11)

where $L$ is the number of propagators, $F$ the number of faces. Let us pick
an arbitrary orientation for all of the edges $e$ and for all of the faces
$f$. Then $R$ is the rank of the incidence matrix $\epsilon_{fe}$:

$\displaystyle\epsilon_{fe}$ $\displaystyle=\begin{cases}\phantom{-}1&\mbox{
if $e\in f$ and their orientation match}\\\ -1&\mbox{ if $e\in f$ and their
orientation do not match}\\\ \phantom{-}0&\mbox{ otherwise}.\end{cases}$ (12)

Note that the rank $R$ does not depends on the chosen of the orientation.
Denoting by $n_{i}(\mathcal{G})$ the number of bubbles in $\mathcal{G}$ with
$2i$ black and white nodes, the divergences subgraphs are said to be melonic
[16, 23], if and only if they satisfy the following relation:

$F(\mathcal{G})-R(\mathcal{G})=3(L(\mathcal{G})-\sum_{i}n_{i}(\mathcal{G})+1)$
(13)

which, together with the topological relation
$L(\mathcal{G})=\sum_{i}in_{i}(\mathcal{G})-N(\mathcal{G})/2$ leads to:

$\omega(\mathcal{G})=3-\frac{N(\mathcal{G})}{2}-2n_{1}(\mathcal{G})-n_{2}(\mathcal{G})\,,$
(14)

where $N(\mathcal{G})$ denote the number of external lines of $\mathcal{G}$.
For the rest $n_{1}(\mathcal{G})=0$. For $N=4$, $\omega\leq 1$, the value $1$
corresponding to melonic graphs with only 6-point interactions bubble. This
conclusion indicates that perturbatively around the Gaussian Fixed Point
(GFP), the coupling constant $\lambda_{1}$ scales as $\Lambda$ for some cut-
off $\Lambda$, and we associate a canonical dimension $[\lambda_{1}]=1$ to
this constant. In the same way, we deduce that for a generic coupling
$\lambda_{b}$, associated to a melonic bubble with $N_{b}$ external lines:

$[\lambda_{b}]=3-\frac{N_{b}}{2}$ (15)

giving explicitly:

$[m]=1\qquad[\lambda_{1}]=1\qquad[\lambda_{2}]=[\lambda_{3}]=0.$ (16)

##  3 Functional renormalization group with closure constraint

In this section we discuss the physical consequences of the renormalization
group flow by truncating the space of actions. The procedure is standard, and
consists in a systematic projection of the renormalization group flow into a
finite dimensional subspace of generalized couplings. The approximated
trajectory $\\{\Gamma_{k}\\}$ is then described by several functions, which
are solution of a finite coupled system of differential equations, the so-
called  $\beta$-functions. The difficult point of this approach is to justify
the choice of the truncation. For our purpose, we use a standard dimensional
argument, and neglect all the interactions up to the marginal coupling with
respect to the perturbative power counting. As discussed in this section, such
a truncation make sense as long as the anomalous dimension remains small,
giving a consistency condition for the validity of the truncation, which can
be easily checked. However, we will see that, truncation may introduce
singular artefact, as lines of fixed points, which depends on the truncation.

We start this section by evaluating the Wetterich equation, and find the
system of $\beta$-functions studied for our model. The asymptotic behavior in
the UV is also provided.

###  3.1 Truncation and Regularization

The FRG method is based on the following deformation to our original partition
function given in the equation (2) i.e.

$Z_{k}[J,\bar{J}]=\int
d\mu_{C_{\Lambda}}(\bar{T},T)e^{S_{int}[\bar{T},T]-\Delta
S_{k}[\bar{T},T]+\langle\bar{J},T\rangle+\langle\bar{T},J\rangle}$ (17)

where we have added to the original action a IR cut-off $\Delta
S_{k}[\bar{T},T]$, defined as:

$\Delta
S_{k}[\bar{T},T]=\sum_{\vec{p}\in\mathbb{Z}^{5}}R_{k}(|\vec{p}|)\bar{T}_{\vec{p}}T_{\vec{p}}.$
(18)

The cut-off function $R_{k}$, depend on the real parameter $k$ playing the
role of a running cut-off, and is chosen such that:

$\bullet$ $R_{s}(\vec{p})\geq 0$ for all $\vec{p}\in\mathbb{Z}^{d}$ and
$s\in(-\infty,+\infty)$.

$\bullet$ $\lim_{s\to-\infty}R_{s}(\vec{p})=0$, implying:
$\mathcal{Z}_{s=-\infty}[\bar{J},J]=\mathcal{Z}[\bar{J},J].$ This condition
ensures that the original model is in the family (17). Physically, it means
that the original model is recovered when all the fluctuations are integrated
out.

$\bullet$ $\lim_{s\to\ln\Lambda}R_{s}(\vec{p})=+\infty$, ensuring that all the
fluctuations are frozen when $e^{s}=\Lambda$. As a consequence, the bare
action will be represented by the initial condition for the flow at
$s=\ln\Lambda$.

$\bullet$ For $-\infty<s<\ln\Lambda$, the cutoff $R_{s}$ is chosen so that
$R_{s}(|p|>e^{s})\ll 1,$ a condition ensuring that the UV modes $|p|>e^{s}$
are almost unaffected by the additional cutoff term, while
$R_{s}(|p|<e^{s})\sim 1$, or $R_{s}(|p|<e^{s})\gg 1$, will guarantee that the
IR modes $|p|<e^{s}$ are decoupled.

$\bullet$ $\frac{d}{d|\vec{p}|}R_{s}(\vec{p})\leq 0$, for all
$\vec{p}\in\mathbb{Z}^{d}$ and $s\in(-\infty,+\infty)$, which means that high
modes should not be suppressed more than low modes.

The equation describing the flow of the couplings, the so called Wetterich
equation has been established in [49] in the case of a theory with closure
constraint: For a given cut-off $R_{k}$, the effective average action
satisfies the following first order partial differential equation:

$\partial_{k}\Gamma_{k}=\sum_{\vec{p}\in\mathbb{Z}^{5}}\partial_{k}R_{k}(|\vec{p}|)\cdot\big{[}\Gamma^{(2)}_{k}+R_{k}\big{]}^{-1}(\vec{p},\vec{p})\delta\bigg{(}\sum_{i=1}^{5}p_{i}\bigg{)}.$
(19)

where $\Gamma_{k}$, the effective average action and is defined as the
Legendre transform of the free energy $W_{k}:=\ln[Z_{k}]$ as :

$\Gamma_{k}[\bar{T},T]+\sum_{\vec{p}\in\mathbb{Z}^{5}}R_{k}(|\vec{p}|)\bar{T}_{\vec{p}}T_{\vec{p}}:=\langle\bar{J},T\rangle+\langle\bar{T},J\rangle-
W_{k}[J,\bar{J}]$ (20)

and

$\Gamma^{(2)}_{k}(\vec{p},\vec{p}^{\prime}):=\frac{\partial^{2}\Gamma_{k}}{\partial
T_{\vec{p}}\partial\bar{T}_{\vec{p}^{\prime}}}$ (21)

where $T$ denote the mean field $T_{\vec{p}}:=\frac{\partial
W_{k}}{\partial\bar{J}_{\vec{p}}}$, and is a gauge invariant field in the sens
that: $T_{\vec{p}}=T_{\vec{p}}\,\delta\big{(}\sum_{i=1}^{5}p_{i}\big{)}$.

The Wetterich flow equation is an exact differential equation which must be
truncated, i.e. it must be projected to functions of few variables or even
onto some finite-dimensional sub-theory space. However as in nonperturbative
analysis [49], and discussed in introduction of this section the question of
error estimate is very important and nontrivial in functional renormalization.
One way to estimate the error in FRG is to improve the truncation in
successive steps, i.e. to enlarge the sub-theory space by including more and
more running couplings. The difference in the flows for different truncations
gives a good estimate of the error. In addition, one can use different
regulator functions $R_{k}$ in a given (fixed) truncation and determine the
difference of the RG flows in the infrared for the respective regulator
choices. In this section, we adopt the simplest truncation, consisting in a
restriction to the essential and marginal coupling with respect to the
perturbative power counting (i.e. whose canonical dimension is upper or equal
to zero). As mentioned before, such a truncation make sense as long as the
anomalous dimension remains small, and a qualitative argument is the
following. Let us define the anomalous dimension $\eta:=\partial_{s}\ln(Z)$
(see equation (25) below). In the vicinity of a fixed point, $\eta$ can reach
to a non-zero value $\eta_{*}$. As a result, the effective propagator becomes:

$\frac{Z^{-1}}{\vec{p}\,^{2}+(m_{s}^{2}/Z)}\approx\frac{e^{-\eta_{*}s}}{\vec{p}\,^{2}+m_{*}^{2}}\quad,$
(22)

and then modifies the power counting (14), which becomes in the melonic sector
(all the star-quantities refers to the non-Gaussian fixed point that we
consider):

$\omega_{*}(\mathcal{G})=-(2+\eta_{*})L(\mathcal{G})+(F(\mathcal{G})-R(\mathcal{G}))=3-\frac{N}{2}(1-\eta_{*})-3\eta_{*}n_{3}-(1+2\eta_{*})n_{2}-(2+\eta_{*})n_{1}\,.$
(23)

As a result, the canonical dimension (16) turn to be

$[t_{b}]_{*}=3-\frac{N_{b}}{2}(1-\eta_{*})=[t_{b}]+\dfrac{N_{b}}{2}\eta_{*}\quad,$
(24)

from which one can argue that, as long as $\eta_{*}\ll 1$, the classification
in term of essential, inessential and marginal couplings remains unchanged,
and the truncation around marginal couplings with respect to the perturbative
power counting make sense. Note that for more specific explanation the study
of the critical exponent will help to prove whether or not the truncation
given below equation should be improve or not. Unlike to what happens in a
standard local field theory, each line here has several strands (the theory is
non-local). The contractions in the loop of the tadpole concerns only 4
strands out of 5. The last strand circulates freely, and corresponds to an
external momentum. It is by developing on this external variable that we
generate the contribution to the anomalous dimension $\eta$. Thus, the
quantity
$\mathcal{W}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4}}$ does
not explicitly depends on the momentums. The dependence on the momentums is
due to the non-locality of the interactions. Up to these consideration, our
choice of truncation is the following:

$\displaystyle\Gamma_{k}[\bar{T},T]=$
$\displaystyle\sum_{\vec{p}\in\mathbb{Z}^{5}}\bigg{(}Z(k)\vec{p}\,^{2}+m^{2}(k)\bigg{)}T_{\vec{p}}\bar{T}_{\vec{p}}+\frac{\lambda_{1}(k)}{2}\sum_{\ell=1}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,4}\mathcal{W}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}$
(25)
$\displaystyle+\frac{\lambda_{2}(k)}{3}\sum_{\ell=1}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{X}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}$
$\displaystyle+\lambda_{3}(k)\sum_{\ell_{i}=1,i=1,2,3}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{Y}^{(\ell_{1},\ell_{2},\ell_{3})}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}.$

Also note that we have adopted an additional restriction concerning the degree
of the differential operator for the kinetic term, which can be viewed as the
first term in the  derivative expansion. One more time, a consistency check
must be to introduce the next contribution, and evaluate its relative
contribution. We will not considered this question in this paper.

We move on to the extraction of the truncated flow equations for $m^{2}$, $Z$
and $\lambda_{i}$ from the full Wetterich equation (19). We write the second
derivative of $\Gamma_{k}$ as:

$\Gamma_{k}^{(2)}[\bar{T},T](\vec{p},\vec{p}^{\prime})=\left(-Z(k)\vec{p}\,^{2}+m^{2}(k)\right)\delta\bigg{(}\sum_{i=1}^{5}p_{i}\bigg{)}\delta_{\vec{p}\vec{p}^{\prime}}+F_{k,(1)}[\bar{T},T]_{\vec{p},\vec{p}^{\prime}}+F_{k,(2)}[\bar{T},T]_{\vec{p},\vec{p}^{\prime}}$

in such a way that all the field-dependent terms of order $2n$ are in
$F_{k,(n)}$. In particular, $F_{k,(1)}$ depends on $\lambda_{1}(k)$, while
$F_{k,(2)}$ depends on $\lambda_{2}(k)$ and $\lambda_{3}(k)$.

For the regulator $R_{k}$, we adopt the Litim’s cut-off [51], in which we set
$e^{k}\rightarrow k$:

$R_{k}(|\vec{p}|)=Z(k)(k^{2}-\vec{p}\,^{2})\Theta(k^{2}-\vec{p}\,^{2}),$ (26)

and computing the first derivative with respect to $k$, we find:

$k\partial_{k}R_{k}(|\vec{p}|)=\big{\\{}k\partial_{k}Z(k)(k^{2}-\vec{p}\,^{2})+2Z(k)k^{2}\big{\\}}\Theta(k^{2}-\vec{p}\,^{2}).$
(27)

Hence, we are now in position to extract the flow equations for each
couplings, which is the subject of the next section.

###  3.2 Flow equations in the UV regime

We will deduce the flow equation in the UV regime. In this regime, all the
sums can be replaced by integration following the arguments of [41],
essentially because the divergences of the integral approximations are the
same as the exact sums. The method consists to a formal expansion of the r.h.s
of the Wetterich equation (19) in power of couplings, and identification of
the corresponding terms in the l.h.s. The r.h.s involves in general some
contractions between the $F_{k(n)}$ and the effective propagator
$\partial_{k}R_{k}$. And in this UV regime, only the melonic graphs
contribute.

Expanding the r.h.s and the l.h.s of the flow equation, we obtain the
following relations (in matrix notations):

$k\partial_{k}\Gamma_{k,(1)}=-\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(1)}\mathcal{K}_{k}^{-1}\big{]}\,,$
(28)
$k\partial_{k}\Gamma_{k,(2)}=-\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(2)}\mathcal{K}_{k}^{-1}\big{]}+\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}(F_{k,(1)}\mathcal{K}_{k}^{-1})^{2}\big{]}\,,$
(29) $\displaystyle k\partial_{k}\Gamma_{k,(3)}$
$\displaystyle=2\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(1)}\mathcal{K}_{k}^{-1}F_{k,(2)}\mathcal{K}_{k}^{-1}\big{]}-\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}(F_{k,(1)}\mathcal{K}_{k}^{-1})^{3}\big{]}\,.$
(30)

where the subscript $GI$ means ”Gauge Invariant” sums, in the sense that all
the terms summing involve a product with a delta
$\delta\bigg{(}\sum_{i=1}^{5}p_{i}\bigg{)}$, $\Gamma_{k(n)}$ means the term of
order $n$ in the truncation of equation (25), and:

$\mathcal{K}_{k}^{-1}:=\frac{1}{Z(k)\vec{p}\,^{2}+m^{2}(k)+R_{k}(|\vec{p}|)}.$
(31)

####  3.2.1 Flow equations for $Z$ and $m^{2}$

Expanding the trace in the r.h.s of the equation (28), we find, using the
expressions (26) and (27)

$\displaystyle\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(1)}\mathcal{K}_{k}^{-1}\big{]}=\sum_{\vec{p}^{\,2}\leq
k^{2}}\frac{k\partial_{k}Z(k)(k^{2}-\vec{p}^{2})+2Z(k)k^{2}}{[Z(k)k^{2}+m^{2}(k)]^{2}}F_{k(1)}(\vec{p},\vec{p})\delta\bigg{(}\sum_{i=1}^{5}p_{i}\bigg{)}\,.$
(32)

Graphically, this contribution can be pictured as in Figure (2), where the
dashed line represent the contraction with the propagator $\partial_{k}R_{k}$.

Figure 2: Typical contribution to the 2-points observable. The dashed line
represents the contraction with the propagator $\partial_{k}R_{k}$.

The operator $F_{k(1)}$ is a sum over the intermediate colors labeling each
4-valent bubbles: $F_{k(1)}=\sum_{i=1}^{5}F_{k(1)}^{(i)}$, and each term
contribute separately to the wave function and mass flow. As a result, we
focus our attention on the computation of the trace for the case $i=1$. As
explained in [41], we will identify the contribution to the coupling of a
given bubble in the l.h.s by expanding the r.h.s around its local
approximation i.e. around the value $q=0$, where $q$ denote the ”external
momentum” shared by the red lines in the Figure (2). The two-points case is in
a sense the more interesting, because for the wave function contribution, the
local expansion require the first deviation to the exact local approximation,
corresponding to the mass-term. This first deviation is proportional to
$q^{2}$. Then, identifying the terms in front of each powers of $q^{2}$, we
find:

$k\partial_{k}m^{2}(k)=-5\lambda_{1}(k)\frac{k\partial_{k}Z(k)(S_{1}(0)k^{2}-S_{2}(0))+2Z(k)k^{2}S_{1}(0)}{[Z(k)k^{2}+m^{2}(k)]^{2}}$
(33)
$k\partial_{k}Z(k)=\frac{-2\lambda_{1}(k)Z(k)k^{2}S_{1}^{{}^{\prime\prime}}}{[Z(k)k^{2}+m^{2}(k)]^{2}+\lambda_{1}(k)(S_{1}^{{}^{\prime\prime}}k^{2}-S_{2}^{{}^{\prime\prime}})}$
(34)

where the factor $5$ in front of the equation (33) takes into account the
contributions for each $i$, $S_{l}^{{}^{\prime\prime}}$ denote the coefficient
in front of $q^{2}$ in the expansion of $S_{i}(q)$ in power of $q$, and the
sums $S_{1}$ and $S_{2}$ are :

$S_{1}(q)=\sum_{\vec{p}\in\mathbb{Z}^{4}}\delta\bigg{(}\sum_{i=1}^{4}p_{i}+q\bigg{)}\theta(k^{2}-q^{2}-\vec{p}^{\,2}),$
(35)
$S_{2}(q)=\sum_{\vec{p}\in\mathbb{Z}^{4}}\vec{p}^{\,2}\delta\bigg{(}\sum_{i=1}^{4}p_{i}+q\bigg{)}\theta(k^{2}-q^{2}-\vec{p}\,^{2}).$
(36)

Since we will be mostly interested in the large-$k$ limit, we can approximate
the sums by integrals, replacing the Kronecker deltas by Dirac deltas. The
support of the integrals is in the intersection of the hyperplane of equation
$q+\sum_{l=1}^{4}p_{l}=0$ and the $4$-ball of radius $\sqrt{k^{2}-q^{2}}$.
Note that the Kronecker delta of the closure constraint can be rewritten as
$q+\vec{p}\cdot\vec{n}=0$, where $\vec{n}=(1,1,1,1)\in\mathbb{R}^{4}$ is the
vector with all components equals to $1$. Using the rotational invariance of
our integral, we can choose one of our coordinate axis to be in the direction
$\vec{n}$. If we choose the axis $1$ in this direction, our constraint writes
as
$\delta(q+p^{\prime}_{1}|\vec{n}|)=\delta(q+2p^{\prime}_{1})=\delta(q/2+p^{\prime}_{1})/2$,
and we find the following integral approximation:

$\displaystyle S_{1}(q)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\Omega_{3}\bigg{[}k^{2}-\frac{5q^{2}}{4}\bigg{]}^{\frac{3}{2}}$
(37) $\displaystyle S_{2}(q)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\Big{[}\frac{5q^{2}}{4}+\frac{3}{5}\bigg{(}k^{2}-\frac{5q^{2}}{4}\bigg{)}\Big{]}\Omega_{3}\bigg{(}k^{2}-\frac{5q^{2}}{4}\bigg{)}^{\frac{3}{2}},$
(38)

where $\Omega_{d}:=\pi^{d/2}/\Gamma(d/2+1)$ is the volume of the unit
$d$-ball, with the special value $\Omega_{3}=4\pi/3$. Using this integral
approximation, we obtain:

$S_{1}(0)=\frac{2}{3}\pi k^{3},\quad S_{2}(0)=\frac{2}{5}\pi k^{5},\quad
S_{1}^{{}^{\prime\prime}}=-\frac{5}{4}\pi k,\quad
S_{2}^{{}^{\prime\prime}}=-\frac{5}{12}\pi k^{3}$ (39)

giving:

$k\partial_{k}m^{2}(k)=-\frac{4\pi}{3}\lambda_{1}(k)\frac{\eta(k)+5}{[Z(k)k^{2}+m^{2}(k)]^{2}}k^{5}$
(40)
$\eta(k)=\frac{5\pi}{2}\lambda_{1}(k)\frac{k^{3}}{[Z(k)k^{2}+m^{2}(k)]^{2}-\lambda_{1}(k)\frac{5}{6}\pi
k^{3}}$ (41)

with the anomalous dimension $\eta(k)$ defined as:

$\eta(k):=k\partial_{k}\ln(Z(k)).$ (42)

Note that, in this case, and for the other computations, the extraction of the
local approximation in the UV limit brings up a very nice property of the
melonic sector, called  traciality. Traciality is a concept firstly introduced
in a perturbative renormalization framework, ensuring that local approximation
of high subgraphs make sense in the TFGT context [23], [25].

####  3.2.2 Flow equation for $\lambda_{1}$

The flow equation for $\lambda_{1}$ (29) involves two traces that we will
compute separately. The first trace
$\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(2)}\mathcal{K}_{k}^{-1}\big{]}$
involves two typical contributions that we have pictured in Figure (3) below.
Note that the two diagrams pictured on this Figure have the same connectivity
as the 4-valent interaction with intermediate line of color red, and we will
only consider the flow equation for the coupling attached to one of the five
color.

Figure 3: Contributions coming from the 6-points interactions to the 4-point
interaction.

The computation of these two contributions follows exactly the same way as the
extraction of the flow equation for the mass parameter. We expand with respect
to the external momentum $q$ (i.e. the momentum around the external face
sharing the same line as the four internal faces) around $q=0$, the first term
of the expansion giving the relevant contribution. Because the two sums
involve a loop of length one, they can be expressed in terms of the two sums
$S_{1}$ and $S_{2}$, and we find:

$\displaystyle\mathrm{Tr}_{GI}\big{[}k\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(2)\eqref{fig3}a}\mathcal{K}_{k}^{-1}\big{]}=$
$\displaystyle
4\lambda_{3}\frac{k\partial_{k}Z(k)(S_{1}(0)k^{2}-S_{2}(0))+2Z(k)k^{2}S_{1}(0)}{[Z(k)k^{2}+m^{2}(k)]^{2}}$
(43)
$\displaystyle\times\sum_{\\{\vec{p}_{i}\\}\,i=1,...,4}\mathcal{W}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}+\mathcal{O}(q),$
$\displaystyle\mathrm{Tr}_{GI}\big{[}k\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(2)\eqref{fig3}b}\mathcal{K}_{k}^{-1}\big{]}=$
$\displaystyle\lambda_{2}\frac{k\partial_{k}Z(k)(S_{1}(0)k^{2}-S_{2}(0))+2Z(k)k^{2}S_{1}(0)}{[Z(k)k^{2}+m^{2}(k)]^{2}}$
(44)
$\displaystyle\times\sum_{\\{\vec{p}_{i}\\}\,i=1,...,4}\mathcal{W}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}+\mathcal{O}(q).$

The contribution of the last trace
$\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}(F_{k,(1)}\mathcal{K}_{k}^{-1})^{2}\big{]}$
is graphically pictured in Figure (4), where the dotted line means contraction
with Kronecker delta.

Figure 4: Contribution to the 4-point interaction involved two vertices.

Expanding in local approximation, we find, for the extra-local contribution
with $q=0$:

$\displaystyle\mathrm{Tr}_{GI}\big{[}k\partial_{k}R_{k}\mathcal{K}_{k}^{-1}(F_{k,(1),\eqref{fig4}}\mathcal{K}_{k}^{-1})^{2}\big{]}$
$\displaystyle\approx$
$\displaystyle\lambda_{1}^{2}(k)\frac{k\partial_{k}Z(k)(S_{1}(0)k^{2}-S_{2}(0))+2Z(k)k^{2}S_{1}(0)}{[Z(k)k^{2}+m^{2}(k)]^{3}}$
(45) $\displaystyle\times$
$\displaystyle\sum_{\\{\vec{p}_{i}\\}\,i=1,...,4}\mathcal{W}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}.$
(46)

Hence, summing the contributions (43), (44) and (45), and using the integral
approximation for the sums, we find:

$k\partial_{k}\lambda_{1}(k)=-(\lambda_{2}+4\lambda_{3})\frac{4\pi}{15}\frac{\eta(k)+5}{[Z(k)k^{2}+m^{2}(k)]^{2}}k^{5}+\lambda_{1}^{2}(k)\frac{4\pi}{15}\frac{\eta(k)+5}{[Z(k)k^{2}+m^{2}(k)]^{3}}k^{5}$
(47)

####  3.2.3 Flow equations for $\lambda_{2}$ and $\lambda_{3}$

The only contribution to $\lambda_{3}$ is pictured in Figure (5) below,
corresponding to the trace:

$\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(1)}\mathcal{K}_{k}^{-1}F_{k,(2)}\mathcal{K}_{k}^{-1}\big{]}.$

Indeed, it is the only contraction in the melonic sector with the same
connectivity as the 6-point interaction associated with the coupling
$\lambda_{3}$.

Figure 5: Contribution to the flow of $\lambda_{3}$

The local approximation of the diagram can be computed exactly as for the
contribution (3) to $\lambda_{1}$, and we obtain:

$\displaystyle\mathrm{Tr}_{fig\,\eqref{fig5}}(q=0)$ $\displaystyle=$
$\displaystyle\lambda_{1}\lambda_{3}\frac{k\partial_{k}Z(k)(S_{1}(0)k^{2}-S_{2}(0))+2Z(k)k^{2}S_{1}(0)}{[Z(k)k^{2}+m^{2}(k)]^{3}}$
(49)
$\displaystyle\times\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{Y}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}$

and:

$k\partial_{k}\lambda_{3}(k)=\frac{16\pi}{15}\lambda_{1}\lambda_{3}\frac{\eta(k)+5}{[Z(k)k^{2}+m^{2}(k)]^{3}}k^{5}.$
(50)

In the same way, the contribution to the local approximation of the diagram
(6)a to the flow equation for $\lambda_{2}$ writes as:

$\displaystyle\mathrm{Tr}_{fig\,\eqref{fig6}a}(q=0)$ $\displaystyle=$
$\displaystyle\lambda_{1}\lambda_{2}\frac{k\partial_{k}Z(k)(S_{1}(0)k^{2}-S_{2}(0))+2Z(k)k^{2}S_{1}(0)}{[Z(k)k^{2}+m^{2}(k)]^{3}}$
(52)
$\displaystyle\times\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{X}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}$

Figure 6: Contribution to the flow of $\lambda_{2}$

Finally, the contribution coming from the diagram (6)b in Figure (6) involves
a loop with two delta propagators, and:

$\displaystyle\mathrm{Tr}_{fig\,\eqref{fig6}b}(q=0)$ $\displaystyle=$
$\displaystyle\lambda_{1}^{3}(k)\frac{k\partial_{k}Z(k)(S_{1}(0)k^{2}-S_{2}(0))+2Z(k)k^{2}S_{1}(0)}{[Z(k)k^{2}+m^{2}(k)]^{4}}$
(54)
$\displaystyle\times\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{X}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}.$

Grouping together the contributions (52) and (54), we find:

$k\partial\lambda_{2}=\frac{24\pi}{15}\lambda_{1}\lambda_{2}\frac{\eta(k)+5}{[Z(k)k^{2}+m^{2}(k)]^{3}}k^{5}-\frac{12\pi}{15}\lambda_{1}^{3}\frac{\eta(k)+5}{[Z(k)k^{2}+m^{2}(k)]^{4}}k^{5}.$
(55)

####  3.2.4 Dimensionless renormalized parameters

Taking into account the canonical dimension define in Section (2.2), the
renormalized dimensionless couplings are defined as:

$m=\sqrt{Z}k\bar{m}\qquad{\lambda}_{1}=Z^{2}k\bar{\lambda}_{1}\qquad\lambda_{2}=Z^{3}\bar{\lambda}_{2}\qquad\lambda_{3}=Z^{3}\bar{\lambda}_{3}$
(56)

Using the flow equations (41), (40), (47), (55) and (50), we find for the
dimensionless renormalized couplings the following autonomous system:

$\eta(k)=\frac{5\pi}{2}\bar{\lambda}_{1}(k)\frac{1}{[1+\bar{m}^{2}(k)]^{2}-\bar{\lambda}_{1}(k)\frac{5}{6}\pi}$
(57)
$\beta_{m^{2}}=-(2+\eta)\bar{m}^{2}(k)-\frac{4\pi}{3}\bar{\lambda}_{1}(k)\frac{\eta(k)+5}{[1+\bar{m}^{2}(k)]^{2}}$
(58)
$\beta_{\lambda_{1}}=-(1+2\eta)\bar{\lambda}_{1}(k)-(\bar{\lambda}_{2}+4\bar{\lambda}_{3})\frac{4\pi}{15}\frac{\eta(k)+5}{[1+\bar{m}^{2}(k)]^{2}}+\bar{\lambda}_{1}^{2}(k)\frac{4\pi}{15}\frac{\eta(k)+5}{[1+\bar{m}^{2}(k)]^{3}}$
(59)
$\beta_{\lambda_{3}}=-3\eta\bar{\lambda}_{3}(k)+\frac{16\pi}{15}\bar{\lambda}_{1}\bar{\lambda}_{3}\frac{\eta(k)+5}{[1+\bar{m}^{2}(k)]^{3}}.$
(60)
$\beta_{\lambda_{2}}=-3\eta\bar{\lambda}_{2}(k)+\frac{24\pi}{15}\bar{\lambda}_{1}\bar{\lambda}_{2}\frac{\eta(k)+5}{[1+\bar{m}^{2}(k)]^{3}}-\bar{\lambda}_{1}^{3}\frac{12\pi}{15}\frac{\eta(k)+5}{[1+\bar{m}^{2}(k)]^{4}}\,,$
(61)

with the definition : $\beta_{i}:=k\partial_{k}\bar{i}$,
$i\in\\{m^{2},\lambda_{1},\lambda_{2},\lambda_{3}\\}$.

##  4 Fixed points in the UV regime

At vanishing $\beta$-functions we obtain a fixed points. But these fixed
points does not get any quantum correction and is called the Gaussian fixed
point. In the neighborhood of these fixed points, the stability is determined
by the linearized system of $\beta$-functions. All these points are studied in
detail in this section.

###  4.1 Vicinity of the Gaussian fixed point

The autonomous system describing the flow of the dimensionless couplings
admits a trivial fixed point for the values
$\bar{\lambda}_{1}=\bar{\lambda}_{2}=\bar{\lambda}_{3}=\bar{m}=0$ called
Gaussian Fixed Point (GFP). Expanding our equations around this points, we
find the reduced autonomous system:

$\displaystyle\begin{cases}\phantom{-}\beta_{m^{2}}&\approx-2\bar{m}^{2}-\frac{20\pi\bar{\lambda}_{1}}{3}\,,\\\
\phantom{-}\beta_{\lambda_{1}}&\approx-\bar{\lambda}_{1}-\frac{4\pi}{3}(\bar{\lambda}_{2}+4\bar{\lambda}_{3})\big{(}1+\frac{\pi}{2}\bar{\lambda}_{1}+2\bar{m}^{2}\big{)}-\frac{11\pi}{3}\bar{\lambda}_{1}^{2}\,,\\\
\phantom{-}\beta_{\lambda_{2}}&\approx\frac{\pi}{2}\bar{\lambda}_{1}\bar{\lambda}_{2}\,,\\\
\phantom{-}\beta_{\lambda_{3}}&\approx-\frac{13\pi}{6}\bar{\lambda}_{1}\bar{\lambda}_{3}\end{cases}$
(62)

and the anomalous dimension:

$\eta(k)\approx\frac{5\pi\bar{\lambda}_{1}}{2}\,.$ (63)

These equations give the qualitative behavior of the RG trajectories around
the GFP. In order to study its stability, we compute the  stability matrix
$\beta_{ij}:=\partial_{i}\beta_{j}\,i\in\\{m^{2},\lambda_{1},\lambda_{2},\lambda_{3}\\}$,
and evaluate each coefficient at the GFP. We find:

$\beta^{GFP}_{ij}:=\begin{pmatrix}-2&0&0&0\\\ -\frac{20\pi}{3}&-1&0&0\\\
0&-\frac{4\pi}{3}&0&0\\\ 0&-\frac{16\pi}{3}&0&0\end{pmatrix},$ (64)

with eigenvalues $(-2,-1,0,0)$ and eigenvectors
$e_{1}^{GFP}=(\frac{9}{160\pi^{2}},\frac{3}{8\pi},\frac{1}{4},1);\,$
$e_{2}^{GFP}=(0,\frac{3}{16\pi},\frac{1}{4},1);\,$ $e_{3}^{GFP}=(0,0,0,1);\,$
$e_{4}^{GFP}=(0,0,1,0)$. One recall that the  critical exponents are the
opposite values of the eigenvalues of the $\beta_{ij}$, and that the fixed
point can be classified following the sign of their critical exponents. Hence,
we have two relevant directions in the UV, with critical exponents $2$ and
$1$, and two marginal couplings with zero critical exponents. Moreover, note
that the critical exponents are equal to the canonical dimension around the
GFP. Finally, note that the previous system of equations admits other fixed
points, or a  line of fixed points in addition to the Gaussian one, for the
values: $\bar{\lambda}_{1}=\bar{m}=0$;
$\bar{\lambda}_{3}=-\bar{\lambda}_{2}/4$. This fixed point occurs as well as
in the non-perturbation analysis, and we will return on this subject in the
next Section.

For the moment, we are in position to discuss the qualitative flow diagram
around the Gaussian fixed point. First of all, note that all the coefficients
of the beta function of the system (62) are not negative definite. This fact
seems to be a special feature of this model, meaning that the weight of the
anomalous dimension does not dominate the vertex contribution. This fact is a
first difference with respect to the similar non-Abelian $\phi^{6}$ model
studied in [37]. However, the analysis providing in this reference remains
true, and the model is not asymptotically free. We will not repeat the
complete analysis given in [37] , but a qualitative argument is the following.
Exploiting the fact that the hyperplans $\bar{\lambda}_{2}=0$ and
$\bar{\lambda}_{3}=0$ are invariant under the flow, we can look at only a two-
dimensional reduction of the complete system (62). We choose
$\bar{\lambda}_{2}=0$, and plot the numerical integration of the reduced flow
equation in Figure (7) (on the left) below.

Figure 7: Phase portrait in the plans $(\bar{\lambda}_{1},\bar{\lambda}_{3})$
for $\bar{\lambda}_{2}=0$ (one the left) and in the plan
$(\bar{\lambda}_{1},\bar{\lambda}_{2})$, for $\bar{\lambda}_{3}=0$.

In the domain, $\bar{\lambda}_{3}>0$, even if a given trajectory approaches of
the Gaussian fixed point, $\bar{\lambda}_{1}$ reaches a negative value, and it
is ultimately repelled for $k$ sufficiently large. The same phenomenon occurs
for $\bar{\lambda_{2}}$ in the plan $\bar{\lambda}_{2}<0$ (see the Figure(7)
one the right). The issue of the UV completion of a theory which is non-
Asymptotically free is one of the difficult question that we address to the
non-perturbative renormalization group machinery, and the rest of this paper
is essentially devoted to this one.

###  4.2 Non-Gaussian fixed points

Solving numerically the systems (40)-(50), we find some non-Gaussian fixed
points, whose relevant characteristics are summarized in the Table (1) below.
In addition to these non-Gaussian fixed points, the system admits a  line of
fixed points, $LFP$, for the values:

$LFP=\\{\bar{m}^{2}=0,\bar{\lambda}_{1}=0,\bar{\lambda}_{2}=-4\bar{\lambda}_{3}\\}\,,$
(65)

with critical exponents:

$\displaystyle\begin{cases}\phantom{-}\theta^{(1)}&=-2\,,\\\
\phantom{-}\theta^{(2)}&=\,\,\,\,\,0\,,\\\
\phantom{-}\theta^{(3)}&=-\frac{1}{2}\big{(}1+\sqrt{1-\frac{128}{9}\pi^{2}\bar{\lambda}_{2}}\big{)}\,,\\\
\phantom{-}\beta_{\lambda_{3}}&=-\frac{1}{2}\big{(}1-\sqrt{1-\frac{128}{9}\pi^{2}\bar{\lambda}_{2}}\big{)}.\end{cases}$
(66)

FP | $\bar{m}^{2}$ | $\bar{\lambda}_{1}$ | $\bar{\lambda}_{2}$ | $\bar{\lambda}_{3}$ | $\eta$ | $\theta^{(1)}$ | $\theta^{(2)}$ | $\theta^{(3)}$ | $\theta^{(4)}$
---|---|---|---|---|---|---|---|---|---
$FP_{1}$ | -0.3 | 0.005 | 0.0009 | -0.0002 | -6.3 | -299 | 56.1 | -11.7 | 5.8
$FP_{2}$ | -0.7 | 0.008 | 0.0006 | -0.0002 | 0.76 | -7.4-1.9i | -7.4+1.9i | 3.34 | -0.12
$FP_{3}$ | -0.9 | 0.0007 | 3.32.$10^{-6}$ | 0. | 1.3 | -66.7 | -42.63 | -27.7 | 1.80
$FP_{4}$ | -0.8 | 0.04 | -0.02 | 0. | -5.9 | -144.8 | -14.4 | -7.5 | -5.4
$FP_{5}$ | 0.06 | -0.006 | 0.002 | 0. | -0.04 | 1.9 | 1.09 | -0.04 | -0.01
$FP_{6}$ | 1.32 | -0.5 | -0.06 | 0. | -0.6 | 3.0 | -1.23 | -1.13 | -0.39
Table 1: Summary of the properties of the non-Gaussian fixed points. Again,
the critical exponents $\theta^{i}$ are the opposite values of the eigenvalues
of the stability matrix:
$\beta_{*}=:\mathrm{diag}(-\theta^{1}_{*},-\theta^{2}_{*},-\theta^{3}_{*},-\theta^{4}_{*})$.

The denominator of $\eta$,
$D:=[1+\bar{m}^{2}(k)]^{2}-\bar{\lambda}_{1}(k)\frac{5}{6}\pi$ introduce a
singularity in the flow. At the Gaussian fixed point, and in a sufficiently
small domain around, $D>0$. But further away from the GFP, $D$ may cancel,
creating in the $(\bar{\lambda}_{1},\bar{m}^{2})$-plan a singularity line. The
area below this line where $D<0$ is thus disconnected from the region $D>0$
connected to GFP. Then, we ignore for our purpose the fixed points in the
disconnected region, for which $D<0$. A direct computation show that only the
fixed points $FP_{2}$, $FP_{3}$, $FP_{5}$ and $FP_{6}$ are relevant for an
analysis in the domain connected to the Gaussian fixed point.

$\bullet$ The fixed points $FP_{2}$ and $FP_{3}$ are very similar. They have
three irrelevant directions and one relevant direction in the UV. For each of
these fixed points, the three irrelevant directions span a three dimensional
manifold on which the trajectory runs toward the fixed point in the IR, while
the trajectories out side are repelled of this critical surface, as pictured
on Figure (8). This picture, the existence of a separatrix between two
connected regions of the phase space is reminiscent of a critical behavior,
with phase transition between a broken and a symmetric phase, and these
separatrix are  IR-critical surfaces. This interpretation is highlighted for
the two fixed points in the zero momenta limit. Indeed, in both cases, the
contributions in the effective action of the terms proportionals to
$\bar{\lambda}_{2}$ and $\bar{\lambda}_{3}$ can be neglected in comparison
with the contributions of the two first terms, leading in the first
approximation a Ginsburg-Landau equation for $\phi^{4}$ scalar complex theory.
Note that for $FP_{2}$ two critical exponents are complex, providing some
oscillations of the trajectories, and implying that the fixed point is an IR-
attractor in the two-dimensional manifold spanned by the eigenvectors
corresponding to these two critical exponents. Moreover, the fixed point
$FP_{6}$ appears to be an IR fixed point, with coordinates of opposite sign.

$\bullet$ The fixed point $FP_{5}$ has two relevant and two irrelevant
directions in the UV. The relevant directions in the UV span a two-dimensional
manifold corresponding to a  UV-multicritical surface. Such a surface is
interesting for the UV-completion of the theory. Indeed, all the trajectories
in the surface are oriented toward the fixed point in the UV, while the
dimension of the surface give an interesting number of physical parameter,
providing an evidence in favor of the  asymptotic safety of the model in the
UV [52].

Figure 8: Qualitative behavior of the RG trajectories around an IR fixed
point. The critical surface is spanned by the relevant directions in the IR,
and the arrows are oriented toward the IR direction. This illustrates the
senario of asymptotically safety.

$\bullet$ Finally, we have the line of fixed point, for which we will
distinguish four cases:

i In the domain $d_{1}=\\{\bar{\lambda}_{2}<0\\}$ we have two relevant, one
marginal and one irrelevant directions.

ii At the point $d_{2}=\\{\bar{\lambda}_{2}=0\\}$, we recover the GFP, with
two relevant and two marginal directions.

iii In the domain
$d_{3}=\\{\bar{\lambda}_{2}\in]0,\big{(}\frac{3}{8\pi}\big{)}^{2}]\,\\}$ we
have three relevant and one marginal directions. One more time, this section
of the critical line is interesting in view of the UV-completion of the theory
and provide a supplementary evidence if favor of asymptotic safety. Indeed, in
each points, the relevant directions in the UV span a three dimensional UV-
critical surface, in favor of the existence of a non-trivial asymptotically
safe theory with three independent physical parameters. This line of fixed
point has been recently discussed in [43] for a similar model improved by
unconnected interaction bubbles.

iv In the domain
$d_{4}=\\{\bar{\lambda}_{2}>\big{(}\frac{3}{8\pi}\big{)}^{2}\,\\}$ The
situation is very reminiscent of the previous one. We have three eigenvalues
with negative real part and one equal to zero. Hence, we have three relevant
and one marginal directions. The only difference in comparison with the domain
$d_{3}$ is that the eigenvalue have non-zero imaginary parts, giving some
oscillations and attractor phenomena in the trajectories.

Finally, we briefly discuss the values of the anomalous dimensions. With our
conventions, the couplings of the relevant operator are suppressed as a power
of $k$ in the UV limit $k\rightarrow\infty$. The couplings decrease when the
trajectory goes away from the UV regime. However, the power law behavior is
limited to the attractive region of the fixed point, far from its scaling
regime it can deviate from the power law one. And we can evaluate this
deviation. For instance, in the vicinity of $FP_{5}$, one deduce from (24)
that the canonical dimension becomes:

$[t_{b}]_{FP_{5}}\approx 3-1.6\frac{N_{b}}{2}\,,$ (67)

from which we deduce that all the interaction of valence up or equal to four
becomes inessentials. The same phenomenon occurs in the vicinity of $FP_{4}$,
where all the interactions up to these of valence four become inessentials. In
contrary, at the fixed points $FP_{2}$ and $FP_{4}$ the anomalous dimension is
positive, meaning that the power counting is improved with respect to the
Gaussian one, and irrelevent operators are enhanced in the UV.

##  5 Truncation with an interaction of valence 8

This section aims to identify how the adding of interaction of the valence $8$
may modify the flow equation and the fixed point of our model. This means that
the effective action is now truncate to satisfy the following form: let
$e_{ji}\in[1,5]$ is the color of the bubble of valence 8, with $i=1,2,3,4$ and
$j=1,2,\cdots,5,$

$\displaystyle\Gamma_{k}[\bar{T},T]$ $\displaystyle=$
$\displaystyle\sum_{\vec{p}\in\mathbb{Z}^{5}}\bigg{(}Z(k)\vec{p}\,^{2}+m^{2}(k)\bigg{)}\bar{T}_{\vec{p}}T_{\vec{p}}+\frac{\lambda_{1}(k)}{2}\sum_{\ell=1}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,4}\mathcal{W}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}+$
(68) $\displaystyle+$
$\displaystyle\frac{\lambda_{2}(k)}{3}\sum_{\ell=1}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{X}^{(\ell)}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}$
(69) $\displaystyle+$
$\displaystyle\lambda_{3}(k)\sum_{\ell_{i}=1,i=1,2,3}^{5}\sum_{\\{\vec{p}_{i}\\}\,i=1,...,6}\mathcal{Y}^{(\ell_{1},\ell_{2},\ell_{3})}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}$
(70) $\displaystyle+$
$\displaystyle\sum_{i=1}^{4}\lambda_{4,i}(k)\sum_{e_{ji}=1/,e_{ji}\neq
e_{ki}\forall j\neq
k}^{5}\sum_{\\{\vec{p}_{l}\\}\,l=1,...,8}\mathcal{Z}^{i,(e_{ji})}_{\vec{p}_{1},\vec{p}_{2},\vec{p}_{3},\vec{p}_{4},\vec{p}_{5},\vec{p}_{6},\vec{p}_{7},\vec{p}_{8}}T_{\vec{p}_{1}}\bar{T}_{\vec{p}_{2}}T_{\vec{p}_{3}}\bar{T}_{\vec{p}_{4}}T_{\vec{p}_{5}}\bar{T}_{\vec{p}_{6}}T_{\vec{p}_{7}}\bar{T}_{\vec{p}_{8}},$
(71)

where we assume that the last term of the action (68) takes into account all
contributions of melonic interactions of the form $T^{8}$, and the coupling
constants $\lambda_{4,i},\,\,i=1,2,3,4$ are related to the vertex $V_{4,i}$
see figures (9). The set $\\{e_{\cdot\,i}\\}_{j}$ takes into account all the
colors associated to the vertices $V_{4,i}$.

$V_{4,1}$ $V_{4,2}$ $V_{4,3}$ $V_{4,4}$

Figure 9: Graphical representation of the vertices of valence 8

We get for $\Gamma_{k,(3)}$ and $\Gamma_{k,(4)}$ the flow equations:

$\displaystyle k\partial_{k}\Gamma_{k,(3)}$ $\displaystyle=$ $\displaystyle
2\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(1)}\mathcal{K}_{k}^{-1}F_{k,(2)}\mathcal{K}_{k}^{-1}\big{]}-\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}(F_{k,(1)}\mathcal{K}_{k}^{-1})^{3}\big{]}$
(73)
$\displaystyle-\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(3)}\mathcal{K}_{k}^{-1}\big{]}\,,$

and

$\displaystyle k\partial_{k}\Gamma_{k,(4)}$ $\displaystyle=$
$\displaystyle-\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(4)}\mathcal{K}_{k}^{-1}\big{]}+\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}(F_{k,(2)}\mathcal{K}_{k}^{-1})^{2}\big{]}$
(76)
$\displaystyle+\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}(F_{k,(1)}\mathcal{K}_{k}^{-1})^{4}\big{]}$
$\displaystyle-3\mathrm{Tr}_{GI}\big{[}\partial_{k}R_{k}\mathcal{K}_{k}^{-1}F_{k,(2)}\mathcal{K}_{k}^{-1}(F_{k,(1)}\mathcal{K}_{k}^{-1})^{2}\big{]},$

Taking into account the dimensionless renormalized parameter, and grouping all
melonic contributions, the flow equations of the coupling constants
$\bar{\lambda}_{2}$, $\bar{\lambda}_{3}$ and $\bar{\lambda}_{4,i}$, are:

$\displaystyle\begin{cases}\phantom{-}\beta_{m^{2}}&=-(2+\eta)\bar{m}^{2}-\frac{4\pi}{3}\bar{\lambda}_{1}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}\,,\\\
\phantom{-}\beta_{\lambda_{1}}&=-(1+2\eta)\bar{\lambda}_{1}-(\bar{\lambda}_{2}+4\bar{\lambda}_{3})\frac{4\pi}{15}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}+\bar{\lambda}_{1}^{2}\frac{4\pi}{15}\frac{\eta+5}{[1+\bar{m}^{2}]^{3}}\,,\\\
\phantom{-}\beta_{\lambda_{2}}&=-3\eta\bar{\lambda}_{2}+\frac{24\pi}{15}\bar{\lambda}_{1}\bar{\lambda}_{2}\frac{\eta+5}{[1+\bar{m}^{2}]^{3}}-\bar{\lambda}_{1}^{3}\frac{12\pi}{15}\frac{\eta+5}{[1+\bar{m}^{2}]^{4}}-\frac{48\pi}{15}\bar{\lambda}_{4,2}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}-\frac{12\pi}{15}\bar{\lambda}_{4,4}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}\\\
\phantom{-}\beta_{\lambda_{3}}&=-3\eta\bar{\lambda}_{3}+\frac{16\pi}{15}\bar{\lambda}_{1}\bar{\lambda}_{3}\frac{\eta+5}{[1+\bar{m}^{2}]^{3}}-\frac{8\pi}{15}\bar{\lambda}_{4,1}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}-\frac{36\pi}{15}\bar{\lambda}_{4,3}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}-\frac{8\pi}{15}\bar{\lambda}_{4,4}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}\,,\\\
\phantom{-}\beta_{\lambda_{4,1}}&=(1-4\eta)\bar{\lambda}_{4,1}+\frac{16\pi}{15}\bar{\lambda}_{4,1}\bar{\lambda}_{1}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}+\frac{4\pi}{15}\bar{\lambda}_{3}^{2}\frac{\eta+5}{[1+\bar{m}^{2}]^{3}}-\frac{24\pi}{15}\bar{\lambda}_{1}^{2}\bar{\lambda}_{3}\frac{\eta+5}{[1+\bar{m}^{2}]^{4}}+\frac{4\pi}{15}\bar{\lambda}_{1}^{4}\frac{\eta+5}{[1+\bar{m}^{2}]^{5}}\,,\\\
\phantom{-}\beta_{\lambda_{4,2}}&=(1-4\eta)\bar{\lambda}_{4,2}+\frac{32\pi}{15}\bar{\lambda}_{4,2}\bar{\lambda}_{1}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}+\frac{4\pi}{15}\bar{\lambda}_{2}^{2}\frac{\eta+5}{[1+\bar{m}^{2}]^{3}}-\frac{12\pi}{15}\bar{\lambda}_{1}^{2}\bar{\lambda}_{2}\frac{\eta+5}{[1+\bar{m}^{2}]^{4}}\,,\\\
\phantom{-}\beta_{\lambda_{4,3}}&=(1-4\eta)\bar{\lambda}_{4,3}+\frac{24\pi}{15}\bar{\lambda}_{4,3}\bar{\lambda}_{1}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}\,,\\\
\phantom{-}\beta_{\lambda_{4,4}}&=(1-4\eta)\bar{\lambda}_{4,4}+\frac{24\pi}{15}\bar{\lambda}_{4,4}\bar{\lambda}_{1}\frac{\eta+5}{[1+\bar{m}^{2}]^{2}}+\frac{8\pi}{15}\bar{\lambda}_{2}\bar{\lambda}_{3}\frac{\eta+5}{[1+\bar{m}^{2}]^{3}}-\frac{24\pi}{15}\bar{\lambda}_{1}^{2}\bar{\lambda}_{3}\frac{\eta+5}{[1+\bar{m}^{2}]^{4}}\end{cases}$
(77)

with the anomalous dimension given by equation (57). One more time, the system
can be solved numerically, and the fixed points as well as their essential
properties are summarized in the Table (2).

FP | $FP_{1}$ | $FP_{2}$ | $FP_{3}$ | $FP_{4}$ | $FP_{5}$ | $FP_{6}$ | $FP_{7}$ | $FP_{8}$ | $FP_{9}$ | $FP_{10}$
---|---|---|---|---|---|---|---|---|---|---
$\bar{m}^{2}$ | -1.07 | -0.91 | -0.84 | -0.84 | -1.22 | -0.75 | -0.76 | -0.74 | -0.59 | 1.45
$\bar{\lambda}_{1}$ | 0.004 | 0.005 | 0.02 | 0.02 | 0.009 | 0.006 | 0.045 | 0.05 | 0.14 | -0.65
$10^{2}\bar{\lambda}_{2}$ | 0.1 | 0.03 | 0.7 | -0.4 | 0.2 | -0.01 | 2 | 1 | -40 | -16
$10^{3}\bar{\lambda}_{3}$ | -0.3 | 0.04 | -3 | 0. | -0.7 | 0.02 | -10 | -10 | -0.6 | -3
$10^{2}\bar{\lambda}_{4,1}$ | 0.01 | -0.04 | 0.1 | 0.01 | 0.001 | -230 | 0.3 | 0.3 | 0.09 | -0.3
$10^{3}\bar{\lambda}_{4,2}$ | -0.04 | -0.01 | 0. | -1 | -0.06 | -2000. | -0.9 | -1 | 100 | -10
$10^{3}\bar{\lambda}_{4,3}$ | 0. | 0. | -0.3 | -0.9 | 0. | 0. | 0. | 0. | 0. | 0.
$10^{2}\bar{\lambda}_{4,4}$ | 0.003 | 690 | 0.02 | 0.4 | 0.006 | 670 | 0.05 | 0.2 | 0.03 | -0.03
$\eta$ | -6.8 | -6.3 | -6.0 | -6.0 | 2.6 | 0.9 | -5.9 | -5.6 | -0.65 | -0.66
$\theta^{(1)}$ | 307.8 | 289.8 | 179.6 | 180.0 | -110.7 | 24.7 | 142.6 | 137.9 | 121.6 | 3.2
$\theta^{(2)}$ | -245.8 | -112.0 | -12+25i | -45.9 | -50+13i | 14.5 | 19+6i | 20+4i | 30 | 3.1
$\theta^{(3)}$ | 173.0 | -67.8 | -12-25i | 31 | -50-13i | 12.2 | 19-6i | 20-4i | 23.6 | -3.0
$\theta^{(4)}$ | 113.6 | 31+6.7i | 21+10i | 26 | -39 | 7.7 | 11.7 | 13+1.7i | 18.9 | 2.8
$\theta^{(5)}$ | 77+19i | 31-6.6i | 21-10i | -22 | -33 | -6 | 1.2+10i | 13-1.7i | 17.2 | 2.7
$\theta^{(6)}$ | 77-19i | -28.5 | -15 | 10 | 24.8 | 5.7 | 1.2-10i | 11 | 9.8 | 1.0
$\theta^{(7)}$ | -67 | -19 | 6 | 6.1 | -18 | -5 | 9.8 | -5.7 | 6.3 | 0.7
$\theta^{(8)}$ | 7.2 | 5.8 | 2.2 | 4 | -2.3 | -1.3 | 5.5 | 5.4 | 5.2 | 0.3
Table 2: Summary of the properties of the non-Gaussian fixed points in the
$\phi^{8}$ truncation.

Interestingly, note that the line of fixed points has disappeared, that is not
a surprise, because such line of fixed points is generally a pathology of the
crude truncation. Among the fixed points listed in the table, only $FP_{5}$,
$FP_{6}$ and $FP_{10}$ have $D>0$. The over fixed points have a big critical
exponent and become harmful pathology of the model.

$\bullet$ The fixed point $FP_{5}$ has seven irrelevant directions and one
relevant direction in the UV, and seems to be an IR fixed point, whose
irrelevant directions span an IR-critical surface with seven dimensions.

$\bullet$ The fixed point $FP_{6}$ has five relevant directions and three
irrelevant directions in the UV. The relevant directions span an UV-
multicritical surface of dimension five. The existence of a such sub-manifold
is in accordance with the asymptotic safety of the theory.

$\bullet$ The fixed point $FP_{10}$ has seven relevant directions and one
irrelevant direction in the UV. It correspond to an UV fixed point whose
revelant directions span a seven-dimensional UV-critical surface. One more
time, the existence of a such manifold seems to be in accordance with a non-
trivial asymptotically safe theory.

At this stage, it is not obvious to make contact with the fixed points
obtained in the previous truncation. The standard way to highlight these
relation is to consider truncation with higher and higher valence, and seek
convergence of the fixed points. But in our case, the difficulty of a such
computations is very improved by the non-locality of the interactions, and
these conclusions have to be confirmed by more finer analysis. Let us remark
that the study of the critical exponent [49] could upset our analysis for the
choice of the truncations with valances greater than $6$. For instance, the
fixed point $FP_{5}$ leads to a very large critical exponent and this can help
to show that the truncation in this order remains non consistent. For the
fixed point $FP_{10}$ the critical exponent is small, and adding another
interaction of valence more than 8 becomes unnecessary. Unlike, the fixed
point $FP_{10}$ exhibits several relevant directions and these do not appear
in the previous section by using just the truncation with interaction of
valence $6$.

##  6 Discussion and conclusion

In this paper the renomalization group analysis is applied for just
renormalizable $T^{6}_{5}$ TGFT model in the deep UV limit. Using the simplest
approximation consisting in a truncation around the marginal interactions with
respect to the perturbative power counting i.e. around the Gaussian fixed
point, we have derived the flow equations for each couplings. Because we have
focused our attention on the UV sector, the leading contributions to the flow
equations provide to the melonic sector, a consideration which considerably
simplify the computation of the flow equations. In a second time, using of the
appropriate notion of canonical dimension in the UV, we have translated our
flow equation in an autonomous system of differential equations, whose we have
computed numerically the fixed points, as well as the behavior of the flow’s
trajectory around each of them.

We have find two type of fixed points. The IR fixed points, whose relevant
directions in the IR span an IR-critical surface, a picture in favor of phase-
transitions. This is supported, for two of these IR fixed points, by the
negative value of their mass-parameter, and the fact that in their vicinity,
the effective action turn to be a Ginsburg-Landau like equation for a
$\phi^{4}$ scalar complex theory, advocating a condensed phase transition
interpretation. In opposition, the second type of fixed points are UV
critical, and their relevant directions in the UV span critical surface with
dimensions higher or equal to two, a picture in accordance with a well-defined
and non-trivial behavior in the UV for asymptotically safe theories. In all
the case, we observe that anomalous dimensions enhanced or weaken the UV-power
counting for relevant operators with respect to the perturbative power-
counting, a phenomena which seems to indicate a break-down of our crude
truncation in these domains of the phase-space. Moreover, the presence of
pathological effects as a line of fixed point seems to confirm these
suspicions, as well as its disappearance in a higher-truncation, while our
conclusions about asymptotic safety and IR fixed points remain true. The
connection between the new fixed point and these ones obtained in the first
truncation remains however unclear at this stage without more control over the
approximation procedure.

Finally, note that in the complementary IR regime, the flow equations receive
non-melonic contributions. This is due to the fact that, for a very small cut-
off, the sums take values $0$ or $1$. As pointed in [41], the appropriate
rescalling is provided by the standard power counting, and the flow equation
turn as well to be an autonomous system, which can be solved numerically.
However, we have to keep in mind that our model is defined on a compact
manifold ($U(1)^{6}$ in the referenced paper, $U(1)^{5}$ in our case). Then,
no phase transition can be occurs, and all the non-Gaussian fixed point
reached to the Gaussian one when the cut-off tend to $0$, except if the radius
of the circles tend simultaneously to the infinity.

## Acknowledgments

We thank all the referees, who have anonymously contributed to the improvement
of this work. D O Samary research at Max-Planck Institute is supported by the
Alexander von Humboldt foundation.

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