^{1}

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^{1,2}

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^{3}.

Tensor models admit the large

The consistent formulation of the quantum theory of gravity (QG) is one of the fundamental and tedious problems of modern physics, which remains unsolved, and most probably intertwines between quantum mechanics and general relativity (GR). It has evolved a lot since the past two decades due to the appearance of new background independent approaches such as loop quantum gravity, dynamical triangulations, and noncommutative geometry (see

The renormalization of TGFT models started with the work given in

The Feynman graphs of TM can be organized as a series in

Recently one new breakthrough is done in the context of FRG of TGFT models, to improve the truncation and to choose the regulator in the appropriate way

The paper is organized as follows: In Sec.

A tensorial group field theory (TGFT) is a field theory defined on a direct product of group manifolds. In this paper we focus on an Abelian group field theory, defined on

Among their properties, the tensorial interactions have revealed a new and nontrivial notion of locality, said

A connected tensorial invariant bubble interaction is said to be local. In the same footing, any interacting action expanded as a sum of such diagrams is said to be local.

The quantum theory is then defined from the partition function

Strictly speaking the term “quantum” is abusive, we should talk about statistical model, or quantum field theory in the Euclidean time.

:A face is defined as a maximal and bicolored connected subset of lines, necessarily including the color 0. We distinguish two cases:

The closed or internal faces, when the bicolored connected set correspond to a cycle.

The open or external faces when the bicolored connected set does not close as a cycle.

The boundary of a given face is then the subset of its dotted edges, and its length is defined as the number of internal dotted edges on its boundary.

A typical Feynman graph contributing to the perturbative expansion of the connected two-point functions. The dotted edges correspond to free propagator contractions between pair of fields.

To complete this definition, we provide what we call internal/external edges and interior/boundary vertices.

On a given Feynman graph, the set of edges split into internal and external edges. External edges come from the Wick contraction with external fields and internal edges come from the Wick contractions between vertex fields. Moreover, a vertex is said to be a boundary vertex if at least one of the external edges is hooked to him. It is an interior vertex otherwise. Finally, we define the interior of a Feynman diagram as the set of internal vertices with dotted edges.

The quantum fluctuation can then be integrated out from higher to lower scales, generating a sequence of effective theories which describes a curve into the theory space. Along the trajectories, all the coupling constants move from their initial definition, their running describing the

To make this more concrete, the regulator is chosen of the form

Now let us give an important aspect about the construction of the flow and the difference between symmetric and nonsymmetric phases. Usually, these terms refer to the value of the mean field

As long as the effective two-points function

This definition comes from the fact that in a perturbative regime, all the 1PI correlations functions of the form

Note that the last condition holds for theory with

Finally let us end this section by introducing the notion of

Consider

An opening Feynman graph with four external edge and its boundary graph. The strand in the interior of

This section addresses the renormalization of a TGFT model, mixing standard melons and a new interacting sector called pseudo-melonic. First, we recall some properties of renormalizable theories and consider the purely melonic sector, which has been shown to be just renormalizable for

Basically, a renormalizable sector is a proliferating family of divergent graphs having the same combinatorial structure and the same power counting, such that their divergences can be canceled from a finite set of counterterms. For our model with a kinetic Laplacian term as a boundary condition in the UV, the degree of divergences

If the degree of divergence depends only on the number of external edges, and decrease with him, the theory is said to be

If the degree of divergence depends on the number of external edges, and decreases both with the number of vertices and external edges, the theory is said to be

If the degree of divergence depends on the number of external edges, and increases with the number of vertices and/or the number of external edges, the theory is said to be

The adjective “superficially” refers to the fact that such a classification remains heuristic without a rigorous proof for finiteness of renormalized amplitudes. The renormalization of TGFT models by considering only the melonic sector is given in

We now give a precise definition of a sector.

Families and sectors.

A family

A leading sector

A subleading sector is a sector whose families have the same boundary graphs as the families of the leading sector, but smallest degree of divergence.

Then all the diagrams in a given family behave like

Let

Drawing an edge between

Contracting it, deleting the end nodes

Figure

The sum of two connected melonic bubbles.

Among the motivations for these definitions, we recall that the sum of connected tensorial invariant

which is nothing but a contraction of a 0-dipole see definition (10).

does not change theFrom the definition of families and sectors, we define the notion of divergent families and sectors as follows.

A family is said to be divergent if

Among the families and sectors, we must make the difference between the case for which the divergent degree increases, decreases or is constant with respect to the number of vertices. Note that when the divergent degree depends on the number of vertices, the families have a short length. In contrast, when a family is made of an infinite number of graphs, the divergent degrees, for fixed

Note that the subleading order sectors must have a divergent sector, requiring to be separately renormalized. The power-counting renormalizability is a first requirement to prove that a given theory is renormalizable. In particular, a renormalizable theory requires the definition of a finite set of

Then let us recall some useful definitions that we will use in the rest of this work. We start by giving the definition of

A 3-dipole (on the left) and its contraction (on the right).

Another important aspect, especially concerning the power counting theorem is the notion of

In general, there are more than one spanning tree in a given graph.

are then related byConsider the family

This definition makes sense due to the fact that for a family whose rosettes are

First let us consider an example of the melonic sector and let us adopt the following definition useful particularly in the next section and also compatible with

The melonic sector

The melonic sector has then the set

Vacuum melonic diagrams.

For a purely quartic model containing only the set

For models involving higher melonic bubbles, obtaining as sums of elementary quartic melon, the vacuum diagrams are obtained from quartic vacuum diagrams by contraction of some 0-dipole. As illustration, Fig.

A quartic vacuum melonic diagrams with three vertices (on left), and a vacuum melonic diagram with one 3-valent melonic interaction obtained from the first diagram by the contraction of the edge

Note that the elementary procedure allows to replace one edge with a two-point tadpole does not change the degree of divergence: To be more precise we add two dotted edges and therefore decrease the degree by 4, but this variation is exactly compensated from the creation of four internal faces. It is easy to check that

A four-point melonic diagram with two vertices. We have

For 1PI melonic diagrams with more external edges, we get

Let

Among the interesting properties of melonic diagrams, their rosettes are

Counting the number of edges in a Feynman graph with

In standard field theory, interactions can be classified following their dimensions. However, in TGFT, there is no background space-time, and roughly speaking the action

Let

In words, the renormalizable interaction scales logarithmically with the UV cutoff, which is another way to say that their weight on the power counting is zero—as in Eq.

Let

Note that

Deleting a

The canonical dimension of the different boundaries bubbles for a given sectors have to be closed. From the bubbles with valence

In this subsection we consider a new leading order sector that we call

The pseudo-melonic sector

An elementary investigation suggests that

The set

We prove this lemma recursively. Let

A connected Feynman graph

For any pseudo-melonic graph

For any pseudo-melonic graph,

Let us consider an edge

Illustration of the third configuration: The two pseudo-melonic connected tensorial invariants

Let us consider a (

Let

We proceed by induction on the number of elementary pseudo-melonic bubbles. For

Only one external edge is hooked to the vertex

Two external edges are hooked to the vertex

Three external edges are hooked to

Note that the last case with four external edges hooked to

(a) An illustration for the first item: One external edge is hooked to

From Lemma 2, if

Contracting the edges of the trees

For any nonvacuum pseudo-melonic diagram

Any pseudo-melonic diagram can be obtained from a purely quartic pseudo-melonic diagram from the contraction of 0-dipole edges. However, 0-dipole contractions do not change the

As a direct consequence, we deduce that a necklace bubble cannot have more than three colored edges between a pair of black and white nodes. Otherwise, it will be possible to obtain a four-dipole, which violates the bound

Let us consider a nonvacuum pseudo-melonic graph

For a fixed configuration

From

The pseudo-melonic sector is just renormalizable up to 3-valent pseudo-melonic interactions. Moreover, power counting graphs with

For a purely 3-valent model, the power counting reduces to

The complete set of just-renormalizable interactions, having zero canonical dimension.

A direct inspection shows that all the divergences have not taken into account in the leading divergent set of pseudo-melons. Indeed, for

In other words, adding a new vertex has no additional cost, which is explicit in Eq.

One-vertex two-point function examples for leading and subleading orders.

This section aims at building the effective vertices for a renormalizable sector beyond the standard melonic Feynman graphs. We deduce the effective vertices for a model including melonic and pseudo-melonic renormalizable interactions as initial conditions in the UV. For the rest of this paper, we consider 5-dimensions TGFT with these two leading contributions, and we restrict our attention on the UV sector

In this section we investigate the leading sector for a theory mixing melonic and pseudo-melonic renormalizable sectors. The theory from which we start in the deep UV is the following:

Structure of general pseudo-melonic interactions.

As a first step we will investigate the general structure for the leading order (LO) graphs. Let

We denote by

Obviously, for any LO graph

The quartic sector is most conveniently studied in the Hubbard-Stratonovich (HS) representation. For tensorial theories, HS representation has been largely discussed in the literature

It is also called

HS correspondence between some quartic Feynman graphs.

We will investigate the structure of the LO quartic graphs in the HS representation. We have the first result.

Leading order quartic vacuum Feynman graphs are trees in the HS representation, with power counting

We proceed recursively on the number of colored and bicolored edges. Let

Elementary moves on the tree

Investigating separately each case, we get:

We add a colored edge between two vertices, creating a loop (a) or we add a tadpole colored edge on a single vertex (b). From these moves, we create two internal propagator edges of color 0, and at most one internal face. The variation of the power counting is then

We add a bicolored edge between two vertices, creating a loop (a’) or we add a tadpole bicolored edge on a single vertex (b’). From these moves, we create two internal propagator edges of color 0, and at most two internal faces. Taking into account the canonical dimension of the quartic pseudo-melonic vertices, the variation of power counting is then

We add a new monocolored leaf (c) or a monocolored bridge (d). From these moves, we create two internal propagator edges of color 0 and 4 internal faces. The variation of power counting is then

We add a new bicolored leaf (c’) or a biocolored bridge (d’). From these moves, we create two internal propagator edges of color 0 and 3 internal faces. The variation of power counting is then

As a result, only the moves

Nonvacuum 1PI two-point graphs are then obtained from vacuum graphs from cutting an internal dotted edge. Obviously, cutting a dotted edge on the boundary of an internal face with length upper than 1 creates a 1PR graph. Then we have to cut a dotted edge with ends points hooked on the same vertex, corresponding to the leafs on the HS representation. Opening an internal dotted edge deletes five internal faces, the variation of power counting is then

1PI four-point graphs are then obtained from deleting another tadpole. However, we have to distinguish between four-point diagrams with melonic and pseudo-melonic boundaries.

For the melonic boundary, we have to distinguish two cases: The first one when at least one of the two boundary vertices is a melonic vertex, the second one when the two boundary vertices are pseudo-melonic. Let us start with the first case, and assume that the first deleted dotted edge is a tadpole over a melonic vertex. The second move can delete another tadpole, and will be optimal if the deleted dotted edge is on the boundary of one of the five opened faces from the first move. The second move then discards only four faces, implying

For pseudo-melonic boundaries, the two opening tadpoles have to be pseudo-melonic, meaning that the four external dotted edges are hooked on two pseudo-melonic vertices. The optimal cutting deletes a single dotted tadpole edge in the boundary of two external faces. Then, this move delete only three internal faces, such that the total variation for power counting is

The four possible boundaries. The three first ones (a) and (b) and (c) have a quartic melonic boundary while the last one (d) has a quartic pseudo-melonic boundary. The target of the external faces running through the interiors of the diagrams are pictured as internal colored edges between boundary vertices.

We now move on the full nonbranching sector, including 3-valent nonbranching interaction bubbles. As recalled in the previous section, the LO graphs may be obtained from the contraction of some 0-dipoles, corresponding to the connected sum of two quartic pseudo-melons. In the HS representation, the 0-dipoles are the arcs on vertices having more than one colored or bicolored edges. Moreover, in the nonbranching sector, the contracted 0-dipoles have to be the arcs between two bicolored edges with the same couple of colors. See Fig.

Contraction of a 0-dipole forming an arc (

As a result, from Proposition 4, we deduce the following.

Leading order vacuum Feynman graphs are trees in the HS representation, whose edges may be simple colored or bicolored edges as well as breaking bicolored edges. The power counting remains the same as for purely quartic sector:

The last condition on the invariance of the power counting may be easily checked.

In this section we investigate the structure of the LO effective vertices with 2, 4, 6 and 8 points. We will use the method discussed on

Let us start with the two-point function. We denote as

The leading order self-energy

From the previous section, we know that LO 1PI two-point graphs may be obtained from the cutting of an internal tadpole edge. They correspond to leafs on the HS representation, and the final vertex on the leaf can be hooked to a single colored edge, to a bicolored edge or to a breaking bicolored edges. As a result, the opening tadpole may be localized on a quartic melonic vertex, either on a quartic pseudo-melonic or on a 3-valent pseudo-melonic vertex. Opening a melonic tadpole on a LO graph means that there are two half dotted edges hooked to this vertex, and connecting it to the rest of the diagram. But it is easy to check that the remaining part of the diagram hooked to this vertex is nothing but the LO two-point function

The structure of the 1PI four-point function may be obtained from the same strategy in terms of elementary essential or marginal couplings, as well as effective two-point functions. All the configurations for boundary vertices are pictured in Fig.

The structure of the graph is a tree in the HS representation.

The connectivity of the external faces between the boundary vertices has to be ensured following their respective nature.

First, let us consider a LO four-point graph having quartic melonic boundary, such that the external edges are fully connected to melonic vertices. Such a configuration corresponds to Fig.

A tree with two cilia contributing to the LO four-point function with melonic boundary. The red arrows follow the path of the skeleton corresponding to the color 1, joining together the boundary vertices.

The decomposition

The zero-momenta effective skeleton function for purely melonic graphs,

Let us consider a LO tree having purely melonic red skeleton. To each vertex in the way of this red path are hooked some connected components

The translation of the final diagram

The two configurations for a melonic insertion. The black node labeled by 1 can be contracted on the right or on the left.

In addition to purely melonic configurations, we have to take into account bicolored edge insertions along the skeleton. Moreover from Fig.

A skeleton with two pseudo-melonic insertion. The red arrows follow the path of the skeleton corresponding to the color 1, joining together the boundary melonic vertices.

Taking into account bicolored insertions introduces another difficulty. Quartic pseudo-melons are not the only source of bicolored edges. We have to take into account a new type of edge, coming from breaking edges. To understand how they occur in the way of the skeleton, let us consider the example pictured in Fig.

As a result, summing over all trees having the same skeleton, we get, as for Eq.

A typical contribution for

Therefore we have an analogous lemma to Lemma 3.

The zero momenta purely pseudo-melonic function

The graphs structure is very reminiscent of the pure melonic case. The essential difference comes from the fact that we have to distinguish two elementary building block configurations, respectively made of chains of type-1 and type-2 pseudo-melons. For the first case, when elementary quartic type-1 pseudo-melons form a chain with zero momenta running throughout the boundaries of the external faces, we have the formal sum:

To obtain the complete four-point kernel

The boundary vertices are both of type-1.

The boundary vertices are both of type-2.

One boundary vertex is of type-1, the second of type-2.

As we will see, a large part of contributions to the effective function

From these elementary “pure” building block functions, we can easily deduce all the allowed configurations for each configuration in Figs.

The complete zero momenta skeleton functions

The components

Computation of

The skeletons of trees contributing to

The value of each diagram being independent to the choice of the selected values for these indices, the complete elementary pattern including pseudo-melonic contributions may be written as

Computation of

The next contributions coming from Figs.

Typical contributions to

Finally, for the computation of

This elementary pattern has to be completed with melonic insertions between the two effective boundaries, leading to the contribution:

Computation of

To complete the proof we have to compute the kernels

In the same way, we defined the function

Finally, the last function that we have to compute is

The

The effective essential and marginal couplings at scale

The equations for the LO four-point function are obtained above. We have to compute the same equations for six- and eight-point effective vertices in this section. We will investigate successively the effective six- and eight-point vertices having melonic, pseudo-melonic or intertwining boundary graphs. Indeed, we will see that, in addition to the melonic and pseudo-melonic boundaries, we get mixing boundaries, having intermediate canonical dimensions between melons and pseudo-melons as mentioned in our Introduction. These mixing boundaries correspond to connected sums of elementary quartic melons and/or pseudo-melons, with colors respecting the LO tadpole deletions leading to six- and eight-point functions from LO four-point functions. We will detail all of them for each case. However before starting this investigation, let us make a remark about the existence of mixing configurations. An elementary example, corresponding to the connected sum of a quartic melon and a quartic pseudo-melon is given in Fig.

Contraction of a zero dipole

The four possible LO boundary graphs for melonic, pseudo-melonic and interwining graphs. A nonbranching 3-valent melon (a), a 3-valent branching melon (b), a nonbranching pseudo-melon (d) and an intertwiner diagram (c). The canonical dimensions are also indicated.

The first graph of Fig.

As for pseudo-melons, nonbranching melons are obtained as connected sums of the same elementary quartic melon. These graphs are described in

The intertwining configurations are relevant for our analysis. Indeed, we will use two kinds of equations in the next section: The flow equation

Let us start with nonbranching melonic six-point effective vertices. As for four-point functions, the six-point effective vertex has to be a sum indexed with the color of its external face, being as well the color of the skeleton in the corresponding tree in HS representation,

Typical tree configurations contributing to the LO six-point functions with melonic boundaries. In (a), the three arms of the red skeleton are hooked on the same vertex

Similarly, effective vertices having boundaries of type (b) and (c) in Fig.

The effective zero-momenta tripod functions

Note that, as for four-point functions, the upper index 3 refers to the number of cilia.

The bare coupling

As for the trees having melonic boundary, the LO trees contributing to the effective tripod function

The effective marginal coupling

However, using Lemma 5, we are able to provide the proof of Proposition 8.

We will proceed as for the proof of the previous lemma, listing all the allowed configurations compatible with the corresponding boundary diagram, pictured in Fig.

In the same way, the configurations contributing to effective six-point vertex functions having branching melonic and intertwining boundaries can be easily obtained from a list of all allowed configurations. As for the previous case, some of these contributions can be resumed as effective pseudo-melonic effective vertices, such that the relevant LO contributions for each case are pictured in Figs.

Expressing this relationship in detail, we get for zero external momenta effective functions:

All the possible configurations having a nonbranching melon as boundary graph. The grey bubbles denote effective vertices whose boundary graphs are explained. The trajectories of the external faces running in the interior of the diagrams are pictured with colored arrows.

Two configurations contributing to the six-point effective vertices having branching melonic boundaries.

Three configurations contributing to the intertwining six-point effective vertices.

Let us now focus on the leading order eight-point functions. They can be obtained as for four- and six-point functions from deleting a new tadpole in the boundary of an opened face running through the interior of the diagrams. Once again all the eight-point function can be classified from their power counting and their boundary graphs. Moreover, in the flow equations, the melonic interactions become closed from six-point functions. Then, the eight-point functions are relevant only to close the pseudo-melonic interactions. As a result, only the leading order functions having a boundary graph whose one-loop contraction leads to a pseudo-melonic boundary will contribute in our flow equations. Figure

From these considerations, it is easy to check that there are only two relevant boundaries for leading order eight-point graphs, pictured in Fig.

One-loop contraction on the melonic “pole” in a eight-point boundary graph. The corresponding eight-point function has divergent degrees bounded as

Allowed boundaries for relevant eight-point functions with their respective canonical dimensions. As for six-point boundaries, type (a) intertwines between melons and pseudo-melons and has a single melonic “pole,” whereas type (b) is a purely nonbranching eight-point graph.

In both cases, the corresponding eight-point functions are labeled with a pair of indices:

The four arms hooked to external deleted pseudo-melonic tadpole can be hooked to the same vertex.

The four arms can be hooked to form effective tripods, hooked together with a common bicolored path.

Figure

The zero-momenta skeleton functions

Like for four- and six-point functions, we have to list all leading order configurations and, for each of them, compute their respective boundary graphs and their divergent degrees. From this analysis, and taking into account effective summations like for six-point graphs, we easily check that there are only five configurations whose complete set is given in Fig.

Two allowed topologies for configurations of the four arms of the leading order eight-point functions. In case (1), the four arms are hooked to the same vertex. In case (2), the arms form effective tripods, hooked together with a bicolored path.

Three configurations for the arms of the four skeletons. In (a), the four arms are hooked to the same vertex. In (b) two arms are hooked to the same breaking edge. In (c) the two remaining edges are hooked to another common breaking edge.

Moving on to the configurations having intertwining boundaries, it is easy to check that the only ones allowed such boundaries are pictured in Fig.

The three leading order configurations for eight-point functions having nonbranching pseudo-melonic boundaries.

The four leading order configurations for eight-point functions having intertwining boundaries.

The leading effective vertex being obtained, let us now focus on the building of the leading order renormalization group flow corresponding to the initial conditions

From Definition 3, local interactions correspond to pure connected tensorial invariants. Therefore, we have defined the effective couplings at scale

The effective couplings at scale

The effective mass at scale

We have the following proposition.

In the deep UV sector (

Deriving the exact flow equation

However, among these contractions, some of them contribute to the melonic sector, and some others contribute to the pseudo-melonic one; and we have to classify them with respect to their boundary graphs. For instance, in the first line, the two first diagrams have melonic boundaries, while the two first diagrams in the second line have pseudo-melonic boundaries. The diagrams in the third and fourth lines require to be careful analyzed because their boundary graphs depend on the selected indices

Disconnected contributions can be taken into account; and mixing sectors involving their contributions have been investigated in

Finally, using renormalization conditions

Finally, following the same strategy, we get for the six-point function

Until now we have no explicitly introduced counterterms. However, fields and couplings in the original action

The flow equations deduced in the previous section are “exact” i.e., up to the limit of our approximation scheme. However, the anomalous dimension, formally given from Eq.

In the symmetric phase, and in the range of momenta contributing significantly in the domain defined by the distribution

Note that this definition is compatible with usual truncations in the symmetric phase, that is with derivative and mean field expansion considered in the literature

In addition to this definition, we recall the usual definition for

In the deep UV, the dimensionless and renormalized couplings,

Now, let us give the set of WT identities, which will help to extract the anomalous dimension

The Ward-Takahashi identity was extensively discussed in the literature

Let us consider the unitary transformations

We can translate this conclusion as follows: In the vicinity of the unity we can write

In practice, in the symmetric phase, the Ward identity allows to link up

Assuming we are in the symmetric phase and taking the derivative of Eq.

To prove formulas

To complete this proposition, we may compute the derivative in both sizes, for

In dimensional regularization, the continuum limit corresponds to

The Ward identity allows to compute the derivative of the four-point functions at zero momenta, or in other words, to keep additional information coming from the momentum dependence of the effective vertex. Note that such a dependence does not appear in the standard crude truncations, as it can be easily checked from Appendix

In the melonic sector, and using

We discard the boundary terms for this discussion.

In the deep UV and for purely melonic sector, and with approximation 25, the Ward identity rely the beta functions and anomalous dimension as

This equation is an additional constraint on the flow, and among their consequences, it adds a strong constraint on the fixed points. Indeed, let us define a fixed point

Even to close this discussion, let us return on the expression

We lack the final subtraction in the Zimmerman forest formula.

Then, we expect thatWe recall that

In the symmetric phase and in the continuum limit, the anomalous dimension

All the dimensionless sums involved in these propositions may be easily computed using integral approximation. The details are given in Appendix

Finally, from the approximation

In the symmetric phase, and taking the continuum limit, the beta functions for

Once again, these relations which depend only on the choice of the regulator and on the continuum limit can be used to analyze the robustness of the fixed points obtained from flow equation

To conclude this section, we will investigate the structure of the phase space, and the existence of nontrivial fixed points. We divide these investigations into two parts. In a first time we will study the vicinity of the Gaussian fixed point, and argue in favor of an asymptotic safety scenario. In a second time, we will move on to the research of nontrivial fixed points in accordance to this scenario. Due to the complicated structure of the flow equations, we use numerical methods for this purpose. Another important simplification comes from the choice of the regulator

Let us remark that Definition 25 is not supported with a rigorous proof. However, despite the fact that we expected the reliability on the results deduced from our method, the approximation scheme that we will use to extract information, especially for non-Gaussian fixed points, is suspected to increase the dependence on the regulator. Due to this difficulty, a rigorous discussion on the reliability of our results, especially on the choice of the regulator function will be considered for a future work, and in this section we only investigate the plot of using the Litim regulator

The Gaussian point

Phase portrait in the plan

It is easy to show that any trajectory starting in this region (which excludes the vertical axis) has to cross the

In polar coordinates, denoting by

To investigate the non-Gaussian fixed points, we have to solve the complete autonomous system given from Proposition 12. However, the complicated structure of these equations requires approximations to solve them. Due to this difficulty, we limit our investigations on the fixed points which can be reached from a perturbative analysis. Our strategy is the following. Increasing the terms keeping in the perturbative expansion of the exact flow equations, we get a list of fixed points which progressively converge to a single one:

In this paper we have built a version of the nonperturbative renormalization group flow including nontrivial dependence of the effective vertices on the relevant and marginal operators in a sector mixing melonics and pseudo-melonics interactions. This allows to use them to solve the renormalization group flow of the full operators in the UV, and close the infinite hierarchy equation coming from the flow equations. The resulting flow equations seem to indicate the existence of a nontrivial UV attractive fixed point including only purely melonic interactions. However, we showed that the Ward identity is strongly violated at this fixed point. As a result, our unique fixed point seems to be unphysical, and the possible existence of other nontrivial fixed points far away of our investigation procedure seems to be necessary.

The importance to include the constraint coming from Ward identity in the resolution of the flow equation is not a novelty, and is well known in gauge theory, especially in the QCD nonperturbative approach. Note that it is not the only limitation of our results. Despite the fact that we do not crudely truncate the flow, we have made some approximations whose consistency have to be supported in forthcoming works. In particular our investigations have been limited on the symmetric phase, ensuring convergence of any expansion around vanishing means field. Moreover, we have retained only the first terms in the derivative expansion of the two-point function, and only considered the local potential approximation, i.e., potentials which can be expanded as an infinite sum of melonic and pseudo-melonic local interactions. Some deviations from ultralocality could be introducing nontrivial effects having the same power counting with ultralocal interactions with higher valences, and then to contribute with them on the same footing. All these difficulties are not taken into account in our conclusions, and will be discussed in some works in progress.

In this section we provide the proof of the useful sums involved in the flow equations. First of all, let us write

As in the previous paragraph the sum

In this section the flow equation for the

The truncation is a projection of the RG flow into a finite dimensional subspace of the infinite dimensional full theory space. In the case where

Now using the results of Sec.

Solving numerically the system

Summary of the non-Gaussian fixed points.

Let us remember that the fixed points are chosen in the domain

For the fixed point FP1, two pairs of complex conjugate critical exponents, with real parts having opposite signs. The fixed point is then focused attractive in the plan defined from two eigenvectors, and focused repulsive in the complementary plan, spanned from the two remaining eigenvectors.

For the fixed point FP2, we have one relevant direction, one irrelevant direction, and a focusing attractive behavior in the complementary plan.

For the fixed point FP3, we get one relevant and four irrelevant directions.