Can we make sense out of"Tensor Field Theory"?

We continue the constructive program about tensor field theory through the next natural model, namely the rank five tensor theory with quartic melonic interactions and propagator inverse of the Laplacian on $U(1)^5$. We make a first step towards its construction by establishing its power counting, identifiying the divergent graphs and performing a careful study of (a slight modification of) its RG flow. Thus we give strong evidence that this just renormalizable tensor field theory is non perturbatively asymptotically free.


Introduction
Recently Hairer [Hai14] solved a series of stochastic differential equations such as the KPZ equation or the φ 4 3 equation. An advantage of such equations is that they are better suited to Monte Carlo computations than functional integrals. Since then, in a systematic series of impressive articles, Hairer and his collaborators [BHZ18; CH16; Bru+17] extended their initial programme to cover the BPHZ renormalization [BP57;Hep66;Zim69]. In contrast to dimensional renormalization, BPHZ renormalization is adapted to the program of constructive field theory. It incorporates the multi-scale expansion, a main constructive tool [Fel+85], and a more up-to-date mathematical formulation of renormalization based on Hopf algebras [CK98].
To the attentive observer, constructive field theory, namely the point of view which Hairer called static, is rapidly merging into the regularity structures and corresponding models of Hairer, which he called the dynamic point of view. In the language of quantum field theory, it happens that the equations which Hairer solved were all Bosonic super-renormalizable. Now is time for advancing the next step: the bosonic just-renomalizable quantum field theories. The BPHZ renormalization was initially designed to cover theories such as QED in dimension four, the main theory at the time. But a profound objection were raised, initially by Landau. Now we have a name for that obstacle: QED is not asymptotically free. Fortunately for the future of quantum field theory, the discovery that electroweak and strong interactions are asymptotically free were instrumental in its "rehabilitation" as a fundamental theory.
A famous theorem due to Coleman states that any local Bosonic asymptotically free field theory must include non-Abelian gauge theories. Non-Abelian gauge theories lead to an additional severe problem: the presence of Gribov ambiguities [Gri78] due to gauge fixing. The way out of these difficulties is a main reason for considering the stochastic quantization [PW81], since in this method there are no need to fix the gauge, so no need to solve Gribov ambiguities. But it remains still a tough programme.
On the road to this lofty goal, we propose an intermediate step which might be worth the effort in itself. It escapes Coleman's theorem by being a non-local theory. We have in mind the tensor field theory. Born in the quantum gravity craddle [ADJ91; Sas91; Gro92], random tensor models extend random matrix models and therefore were introduced as promising candidates for an ab initio quantization of gravity in rank/dimension higher than 2. However their study is less advanced since they lacked for a long time an analog of the famous 't Hooft 1/N expansion for random matrix models. Their modern reformulation [Gur17a; Gur16;GR11b] considers unsymmetrized random tensors 1 , a crucial improvement which let the large N limit appear [Gur10b; GR11a;Gur11]. The limit of large matrix models is made of planar graphs. Surprisingly perhaps, the key to the 1/N tensors is made of a new and simpler class of Feynman graphs that we called melonic. They form the dominant graphs in this limit [Bon+11;BGR12b] Random tensor models can be further divided into fully invariant models, in which both propagator and interaction are left invariant by the symmetry (such as U (N ) ⊗d ), and non-local field theories where the propagator is for example the ordinary Laplacien on the torus U (1) ⊗d (which breaks the symmetry) but in which the interaction is left invariant by the symmetry. To our own surprise, such just-renormalizable models turn out to be asymptotically free [BG16;Riv15]. In particular the simplest such model in this category, nicknamed T 4 5 theory is asymptotically free! It made them an ideal playground for advancing the mathematics both in the static sense of constructive theory and in the sense of Hairer's stochastic quantization. This fact now many years old was perhaps overlooked by the theoretical and mathematical physics community.
Also the tensor methods and models in quantum gravity that one of us baptized the tensor track [Riv12a; Riv12b; Riv13; Riv16b; DR18; DR20] was given a big boost from an unexpected corner. Since the advent of the SYK model [Kit15;PR16;MS16] it appears that 1-dimensional quantum random tensor is even richer than the 0-dimensional ordinary random tensor theory [Wit16;Gur17b;CT16;KT17]. It is approximately reparametrization invariant (i.e. conformal), includes holography and it saturates the MSS bound [MSS16].
In fact the real applications, as it often happens, might be elsewhere. Today we probe reality by multiples sensors. That is, we represent that reality by multidimensional big arrays which are, in the mathematical sense, nothing but big tensors. Hence we need to develop better and more versatile algorithms to probe tensors in this limit. Such algorithms could benefit of the modern formulation of random tensors. This is especially true for those separating signal to noise. One example is tensor PCA [RM14; BA+17; BAGJ18; Zar+18], which extends classical matrix PCA to tensors. Such algorithms could be applied in a variety of domains, high energy physics (detection of particule trajectories), spectral imaging or videos, neuroimaging, chemometrics, pharmaceutics, biometrics, social networks and many more. In fact the analysis of big tensors form a bottleneck in such a dazzling kaleidoscope that it is no exaggeration to say that any main progress in this field may create a revolution in artificial intelligence. Now let us come down to earth. The tensor theory new constructive program [Riv16a] is well advanced in the super-renormalizable case [Gur13;DGR14]. In [DR16] the U (1) rank-three model with inverse Laplacian propagator and quartic melonic interactions, which we nickname T 4 3 , was solved. In [RVT18] the U (1) rank-four model T 4 4 was solved. This model looks comparable in renormalization difficulty to the ordinary φ 4 3 theory, but non-locality and the graphs are more complex hence requires several additional non-trivial arguments. The next goal is to treat just-renormalizable asymptotically free Bosonic T 4 5 . In 1979, G. 't Hooft gave a series of lectures entitled Can we make sense out of "Quantum Chromodynamics"? [Hoo79] 3 . He presented there arguments and strategies to control QCD via the study of its singularities in the Borel plane. To this aim, he had to control the flow of the coupling constant in the complex plane. The tensor field theory T 4 5 is a perfect playground for constructive purposes as its flows can be controled precisely thanks to its simple and exponentially bounded divergent sector. In the present paper we make a further step by connecting it, modulo certain hypotheses, to an autonomous non-linear flow of the theory of dynamical systems.
The T 4 5 tensor field theory is precisely defined in Section 1. In particular, we present the cut-offs we use and an alternative representation of the model in terms of an intermediate matrix field. Section 2 is devoted to the three different representations of Feynman graphs we need (tensor graphs, coloured graphs and intermediate field maps) as well as related concepts thereof. In Section 3 we derive the power-counting, identify the families of divergent graphs and give the recursive definitions of the melonic correlation functions. For constructive purposes, we will employ none of the bare, renormalized or even fully effective perturbative expansions. In fact, it will be preferable to fully mass renormalize the correlation functions but use effective wavefunctions and coupling constants. We define all these objects in Section 4. We also prove there that effective wave-functions and coupling constants are analytic functions of the bare coupling. The main result of Section 4 is Theorem 4.3 which consists in a non perturbative definition of the RG flow for the coupling constant. A careful study of an approximation of this flow is carried out in Section 5 using tools and concepts from discrete and continuous holomorphic local dynamical systems. We identify in particular "cardioid-like" domains of the complex plane invariant under this modified RG flow, see Theorems 5.12 and 5.15 and Corollary 5.13. Solving this T 4 5 model means defining its correlation functions non perturbatively in the coupling constant g. More precisely it requires to prove the existence of holomorphic functions of g in a (probably cardioid-like, with a cut on the negative real axis) domain of the complex gplane such that their Taylor expansions coincide with the perturbative expansions of the (formal) correlation functions of the model. Moreover these functions should very probably be proven Borel summable.
To achieve that goal, one expresses the regularized and renormalized correlation functions as series of analytic functions, normally convergent in a domain the size of which is uniformly bounded in the ultraviolet cutoff. The infinite cutoff limit is then well-defined and analytic. These expansions consist in partial resummations of the perturbative series, either expressed in an intermediate field representation ( theory. An update of all currently known approaches to constructive tensor field theory seems necessary [FRVT21].
Throughout this paper, we always use bold characters to denote tuples of at least two variables.
We introduce the normalized Gaussian measure where the covariance C is C n,n = δ n,n C(n), C(n) = 1 n 2 + m 2 , n 2 := n 2 1 + n 2 2 + n 2 3 + n 2 4 + n 2 5 . (1.1) This defines the free tensor fields as random distributions on Z 5 , namely on the dual of rapidly decreasing sequences on Z 5 . But as we are interested in interacting tensor fields, we need to regularise the free measure.

Ultraviolet cutoff. -
In practice we want to restrict the index n to lie in a finite set rather than Z 5 in order to have a well-defined proper (finite dimensional) tensor model. This restriction is an ultraviolet cutoff in quantum field theory language. A colour-factorized ultraviolet cutoff would restrict all previous sums over tensor indices to lie in [−N, N ]. However it is not well adapted to the rotation invariant propagator of (1.1) below, nor very convenient for multi-slice analysis as in [GR14]. Therefore we introduce a rotation invariant cutoff but in contrast with [RVT18] it will be smooth.
Let a, be two positive numbers such that < a. Let χ be a smooth positive function with support [− , ]. We denote by 1 [−a,a] the indicator function of [−a, a]. In order to prepare for multiscale analysis (see Section 4.3), we fix an integer M > 1 (as ratio of a geometric progression M j ) and choose a large integer j max . Our ultraviolet cutoff is defined as It is smooth, positive, compactly supported, and satisfies It is convenient to choose a = 5/2 and = 3/2 so that the UV cutoff κ effectively restricts each colour index to lie in [−N, N ] with The normalized bare Gaussian measure with cutoff j max is where the bare covariance C b is, up to a bare field strength parameter Z b , the inverse of the Laplacian on T 5 with momentum cutoff j max plus a bare mass term n 2 := n 2 1 + n 2 2 + n 2 3 + n 2 4 + n 2 5 .
A random tensor T distributed according to the measure µ C b is almost surely a smooth function on U (1) 5 .

The bare model. -
The generating function for the moments of the model is where the scalar product of two tensors A·B means n A n B n , g b is the bare coupling constant (which depends on the cutoff N ), the source tensors J and J are dual respectively to T and T and N is a normalization factor. To compute correlation functions it is common to choose which is the sum of all vacuum bare amplitudes. However following the constructive tradition, we shall limit N to be the exponential of the (infinite) sum of the divergent connected vacuum amplitudes. Remark the Z 2 b scaling factor multiplying g b in (1.2).
To make the interaction c V c (T, T ) in eq. (1.2) explicit, we recall first some notation. Tr, I and , mean respectively the trace, the identity and the scalar product on H ⊗ . I c is the identity on H c , Tr c is the trace on H c and , c the scalar product restricted to H c . The notationĉ means "every color except c". For instance, Hĉ means c =c H c , Iĉ is the identity on the tensor product Hĉ, Trĉ is the partial trace over Hĉ and , ĉ the scalar product restricted to Hĉ. T and T can be considered both as vectors in H ⊗ or as diagonal (in the momentum basis) operators acting on H ⊗ , with eigenvalues T n and T n . An important quantity in melonic tensor models is the partial trace Trĉ [T T ], which we can also identify with the partial product T, T ĉ . It is a (in general non-diagonal) operator in H c with matrix elements in the momentum basis The main new feature of tensor models compared to ordinary field theories is the non-local form of their interaction, which is chosen invariant under independent unitary transformations on each color index. In this paper we consider only the quartic melonic interaction [DGR14], which is a sum over colors 5 This model is globally symmetric under colour permutations. It is just renormalizable like ordinary φ 4 4 but unlike ordinary φ 4 4 it is asymptotically free and using this crucial difference, we aim, in a future work, at making rigorous sense of it.
Mainly in order to prepare for the constructive study of the T 4 5 model, we present here its intermediate field representation [Gur13]. We put g b =: λ 2 b and decompose the five interactions V c in eq. (1.2) by introducing five intermediate Hermitian N × N matrix 4 fields σ t c acting on H c (here the superscript t refers to transposition) and dual to Trĉ [T T ], in the following way where dν is the usual GUE measure, that is dν(σ t c ) = dν(σ c ) is the normalized Gaussian independently identically distributed measure of covariance 1 on each coefficient of the Hermitian matrix σ c . It is convenient to consider C b as a (diagonal) operator acting on H ⊗ , and to define in this space the operator Performing the now Gaussian integration over T and T yields where dν(σ) := c dν(σ c ), and R is the resolvent operator on H ⊗ R(σ) := 1 4 The indices of σ cannot be bigger than the maximal value N of each tensor index.

Feynman graphs
Perturbative expansions in quantum field theory are indexed by graphs called Feynman graphs. Their properties reflect analytical aspects of the action functional. Here we will deal with three different graphical notions.
2.1. Tensor graphs. -The first one corresponds to the Feynman graphs of action (1.2) in which the fields are tensors of rank five. As for random matrix models, Feynman graphs are stranded graphs (so-called ribbon graphs in the matrix case) where each strand represents the conservation of one tensor index. The corresponding Feynman rules are recalled in fig. 1 where an example of such a Feynman graph is also given. Such graphs will be called tensor graphs in the sequel and denoted by emphasized letters such as G. The power-counting, i.e. the behaviour at large N of the amplitude, of a tensor graph G depends on the number F (G) of its cycles, also called faces. Open tensor graphs also have non cyclic strands which we call external paths, see fig. 1b. It will be convenient to express the number of faces in terms of the (reduced) Gurau degree [GS16] of the coloured extension of G. We now explain these notions.

Coloured graphs. -
Strands of a tensor graph correspond to indices of the original tensor fields T and T . Each such index is labelled by an integer from 1 to 5 recalling that T is an element of H 1 ⊗ H 2 ⊗ · · · ⊗ H 5 . We can then associate bijectively to any tensor graph G a bipartite 6-regular properly edge-coloured graph G called its coloured extension. See fig. 2 for a pictorial explanation of the bijection as well as an example. Such edge-coloured graphs, with or without the constraint of being 6-regular, will be called coloured graphs for simplicity and their symbols will be written in normal font. A (D + 1)-regular coloured graph will simply be called (D + 1)-coloured graph.
We will need several different notions associated to coloured graphs. The coloured extension of a closed (resp. open) tensor graph will also be considered closed (resp. open). In this work, edges of coloured graphs will bear a "colour" in [5] := {0, 1, . . . , 5}. We will also write [5] * for the set {1, 2, . . . , 5}. Let G be a coloured graph. We let col(G) be the set of colours labelling at least one edge of G. Let c be an element of [5]. We will often writeĉ for [5] \ {c}. We denote by E c (G) the set of edges of G of colour c. The elements of E c (G) are the c-edges of G. If C is a subset of [5], we denote c∈C E c (G) by E C (G). Let E be any subset of the edges of G, we let 5 What we call external edges are actually half-edges and open graphs are in fact pre-graphs. But we do not insist on being so precise with our terminology.  An n-bubble B is a bubble such that col(B) has cardinality n. A 2bubble of a closed coloured graph is therefore a cycle whose edges bear two alternating colours. Cyclic 2-bubbles of G whose colour set belong to {0, i} , i ∈ [5] * correspond to the faces of the corresponding tensor graph G. By extension, cyclic 2-bubbles are often also called faces and their number denoted F (G). The number of {0, c}-bubbles will be written F 0c and we define . Similarly, we denote by F ∅ (G) the total number of faces of G, both colours of which are different from 0. Non cyclic 2-bubbles of G represent the external paths of G. The interaction vertices of a tensor graph G are in bijection with thê 0-bubbles of its coloured extension G.
The (reduced) Gurau degree δ(G) of a closed (D + 1)-coloured graph 6 G is defined as follows [GS16]: is the number of vertices of G and C(G) its number of connected components. It is a non negative integer. One can indeed show that it is the sum of the genera of some maps associated to G [Gur11]. Let S (c) [D] be the set of cyclic permutations of [D] and τ be such a permutation. Let J τ (G) (J τ if the context is clear) be the map whose underlying graph is G and whose cyclic ordering of the edges around vertices is given by τ . Such maps are called jackets in the tensor field literature [BG+10], see fig. 3 for an example. Then, we have g Jτ .  In order to classify the divergent graphs of tensor models, one needs an extension of the Gurau degree to open coloured graphs and the notion of boundary graph of a coloured graph. To start with, we need a slightly generalised version of jacket 7 .
where E(G) is the number of external edges of G. ♦ Proof. -Let τ be a cyclic permutation of [D] and J τ (G) be the corresponding jacket of G. By Euler relation, where e(G) denotes the number of internal edges of G. Faces of J τ can be divided into two parts: the ones which are faces of G, the other ones which are not. The latter will be called external and their number will be denoted by F ext (J τ ). What are these external faces exactly? Let us look at the upper left graph of fig. 4, considered as a map. Recall that in the definition of a jacket, we removed external edges. If one does so for this map, it will contain an external face which goes all around it. This face is not a face of G because it is bordered by edges of three different colours, 0, i and j with i, j = 0. It will be convenient to define external faces of G. Non-cyclic 2-bubbles b of G are bordered by two external vertices, namely its two vertices incident with external edges. If the edges of b bear colours 0 and i, we call b an external path of G of colour i. An external face of G of colour ij is then defined as a cyclic and alternating sequence of adjacent external paths of colour i and j respectively. The important point to notice is that the external faces of G are in bijection with the faces of ∂G.
External faces of G of colour ij are faces of J τ if and only if τ contains the sequence i0j or j0i. Then, a given external face of G belongs to exactly 2(D − 2)! jackets. Each face of G belongs to exactly 2(D − 1)! jackets so that Similarly, using that the total number of jackets of ∂G is (D − 1)!, that 2e(∂G) = DV (∂G) and . This concludes the proof.
Coloured graphs of vanishing degree are said to be melonic. They form the dominant family of the 1/N -expansion of coloured tensor models [Gur11].

Intermediate field maps. -
The third graphical notion we will deal with corresponds to the Feynman graphs of action (1.3) viz., Feynman graphs of the intermediate field representation of our model. As the intermediate field representation is a multi-matrix model, its Feynman graphs are ribbon graphs or maps. As each field σ c bears a colour index c (and the covariance is diagonal in this colour space), the edges of these maps bear a colour too. The Tr log interaction term implies that there is no constraint on the degrees of the vertices of these maps nor on the properness of their edge-colouring. Such maps will be called coloured maps. As for coloured graphs, we let col(G) be the set of colours labelling at least one edge of G.
There is a bijection between the Feynman maps of the intermediate field representation and the Feynman graphs of the original tensorial action (1.2). A precise description of this bijection can be found in [BLR17]. Let us remind the reader of its most salient features. Firstly, note that we will in fact explain a bijection between coloured graphs and coloured maps. Let G be a 6-coloured graph of the T 4 5 model and let G be the corresponding coloured map. In eacĥ 0-bubble of G, there are two sets of four parallel edges. Each set will be called a partner link.
Edges of G are in bijection with the0-bubbles (or interaction vertices) of G. Each such bubble has a distinguished colour, namely the colour common to the two edges which do not belong to a partner link. We label the corresponding edge of G with it. Partner links of G are in bijection with half-edges of G. Let us now describe the vertices of G. They form cycles of half-edges. But there is a subtlety due to external edges of G. Each maximal alternating sequence of adjacent 0-edges and partner links in G form either a cycle or a (non cyclic) path in case of external (0-)edges. In any case, we represent such a sequence as a vertex in G. If a sequence is not cyclic, we add a cilium, i.e. a mark, to the corresponding vertex of G. See fig. 5 for an illustration of this bijection. In the sequel, we will use (at least) two features of this bijection between coloured graphs of our model and coloured maps: we also used V (G) = 4V (G) and set D = 5. The original proof of eq. (3.1) can be found in [OSVT13]. After substituting the combinatorial relation 2L + E = 4V , the divergence degree of G can be written as • divergent four-point graphs are trees such that the unique path between their two cilia is monochrome, •  Proof. -Gurău and Schaeffer [GS16] defined two very convenient coloured graphs we will need. The first one is a chain. Chains can be broken or unbroken. In our case, chains of an intermediate field map G are paths of the form (e 1 , v 1 , e 2 , v 2 , . . . , e n−1 , v n−1 , e n ) where the e i 's are edges of G, the v i 's are vertices of G such that for all i between 1 and n − 1, the degree of v i in G is two. Such a chain is unbroken if all its edges bear the same colour. It is broken otherwise. The second simple but very useful object is that of trivial coloured graphs or ring graphs. They consist in a single loop and no vertex. This loop bears a colour. In our case, this will always be the colour 0. Ring graphs are melonic by convention and are represented by an isolated vertex in the intermediate field representation. According to eq. (3.2), the divergence degree of a 4-point graph G is bounded above by zero. It vanishes if and only if C(∂G) = 1 and δ(G) = 0. Divergent four-point graphs are thus trees with two cilia in the intermediate field representation. Now, recursively remove all degree one vertices of this tree which do not bear a cilium. One gets a non trivial path P with a cilium at each end. This path has the same power counting as the initial tree G. It is melonic and its boundary graph is connected if P is monochrome, disconnected otherwise. Thus, according to eq. (3.2), G is superficially divergent if and only if the unique path between its two cilia is monochrome.
Let us now consider a Feynman graph G such that E(G) = 2. The divergence degree of such a graph is bounded above by two. The coloured extension of its boundary graph is the unique 6-coloured graph with two vertices. It is thus connected i.e. C(∂G) = 1. Then ω(G) = 2 if and only if δ(G) = 0. Moreover, as proven in [BGR13], if δ(G) > 0 then δ(G) D−2 where D + 1 is the number of colours of G. In our case, D equals five and the smallest possible positive degree is three. Consequently the only superficially divergent 2-point graphs have vanishing degree.
Let us finally treat the case of a closed (E = 0) superficially divergent Feynman graph and work in the intermediate field representation. Note that the divergence degree of such a graph is bounded above by five. As a consequence, it has excess at most one. Indeed, adding an edge to a connected graph G increases the number of its corners by two (hence the number of edges of G increases by two) while the number of faces of G can at most increase by one. Thus the divergence degree decreases by at least three. A connected closed graph G with maximal divergence degree (five) is melonic and corresponds, in the intermediate field representation, to a tree. According to the argument above, a superficially divergent closed graph has an excess smaller or equal to one.
Let us focus on divergent graphs G of excess one. They are maps with exactly two faces i.e. maps with a unique cycle and trees attached to the vertices of this cycle. In order to further classify such divergent graphs, as in [GS16], we first remove recursively all vertices of degree one. This does not change the degree of the graph. The result is a cycle C i.e. a ring graph into which a maximal proper chain Ch has been inserted. According to [GS16,p. 288] the Gurau degree of the coloured graph C is 3 if C is monochrome (the chain Ch is then non-separating and unbroken with a single resulting face). It is 5 otherwise (Ch is then a non-separating broken chain).
The T 4 5 model (1.2) has the power counting of a just renormalizable theory (and can be proven indeed perturbatively renormalizable by standard methods). However the structure of divergent subgraphs is simpler both than in ordinary φ 4 4 or in the Grosse-Wulkenhaar model [GW04] and its translation-invariant renormalizable version.
Melonic graphs with zero, two and four external legs are divergent, respectively as N 5 , N 2 and log N . In the sequel we will only consider 1PI (i.e. one particle-irreducible or 2-edge connected) graphs as they represent the only necessary renormalizations. Melonic graphs are trees in the intermediate field representation. The condition that they are 1PI exactly corresponds to the ciliated vertices being of degree one in the tree (cilia do not count). Melonic vacuum graphs are always 1PI.
The divergent melonic graphs of the theory are obtained respectively from the fundamental melonic graphs of fig. 6, by recursively inserting the fundamental 2-point melon on any bold line, or, in the case of the four-point function, also replacing any interaction vertex by the fundamental 4-point melon so as to create a "melonic chain" of arbitrary length (see fig. 7 for a chain of length two), in which all vertices must be of the same colour (otherwise the graph won't be divergent).

Melonic correlation functions. -
Let us call G mel E,b and Γ mel E,b respectively the connected and one-particle irreducible melonic functions (i.e. sum over the melonic Feynman amplitudes) of the theory with E external fields. With a slight abuse of notation, the bare melonic twopoint function G mel 2,b (n, n) = δ n,n G mel 2,b (n) is related to the bare melonic self-energy Σ mel b (n, n) = δ n,n Σ mel b (n) by the usual equation of the single integer n c : is uniquely defined by the last two equations and the following one (see fig. 9) .
. At fixed cutoff N = 2M jmax , these equations define Σ mel b , G mel 2,b and Γ mel 4,b (hence also G mel 4,b ) at least as analytic functions for g b Z 2 b sufficiently small, because the species of melonic graphs is exponentially bounded as the number of vertices increases, see Section 4.4. However this does not allow to take the limit N → ∞ since the radius of convergence shrinks to zero in this limit. In short we need to now renormalize.

Renormalized 1PI functions. -
The renormalization consists in a melonic BPHZ scheme which is given by BPHZ-like normalization conditions at zero external momenta, but restricted to the divergent sector, namely melonic graphs 8 .
The full melonic two-point function is therefore

Mass renormalization. -Let us start by performing the mass renormalization, and postpone the wave-function and four-point coupling constant renormalization to the next section.
Indeed mass renormalization is simpler as it does not involve renormalons [Riv91]. So throughout this section we keep the bare coupling constant g b , and the bare wave-function normalization Z b . The mass renormalization subtracts recursively the value of all subinsertions at 0 external momentum. Hence, recalling eq. (3.3), the monochrome melonic mass-renormalized self-energy Σ mel mr obeys the closed equation .
The sum over p in the equation above diverges only logarithmically as j max → ∞. The total mass counterterm is where we used that Σ mel mr (0) = 0. Remark that δ c m is independent of c, so that in fact

Effective renormalization. -
We want to perform only the effective (or "useful") part of the coupling constant and wave-function renormalizations, that is when the inner loop slice is higher than the external one.
Provided M 2 > 2, η j (j > 0) is positive, smooth, and satisfies As a consequence, we define

The decomposition of each propagator in the amplitude A mr (G) of any Feynman graph G allows to write
A mr (G) =

Definition 4.1 (Effective wave-function). -The effective wave-function constant Z j is
where Σ mel mr; j (n) = c Σ mel mr; j (n c ) is the sum of mass-renormalized amplitudes of all 1PI melonic 2-point graphs, all internal scales of which are greater than or equal to j, namely Σ mel mr; j (n c ) : . ♠ Note that with these notations, Z jmax = Z b and Z −1 = Z r = 1.

Analyticity. -
This section is devoted to proving that the effective wave-functions and coupling constants are analytic functions of the bare coupling g b (in a disk of radius going to 0 as j max → ∞). According to fig. 9, the generating function for the number of 1PI divergent 2-point graphs is This can be proven either by solving the associated equation for Σ mel , of divergent 4-point graphs is given by which, from eq. (4.1), gives are asymptotically equal to 20 n 64 √ πn 3 . Remembering the definitions of Z j and g j Z 2 j (see Section 4.3.2), we have

.3)
A n is the sum of the derivatives of the mass-renormalized amplitudes of the 1PI divergent melonic 2-point graphs of order n. B n is the sum of the mass-renormalized amplitudes of the 1PI divergent melonic 4-point graphs of order n. According to their generating functions, the number of such graphs is bounded by a constant to the power n. Moreover there certainly exist p, q ∈ N such that the amplitudes of these graphs are bounded by (j max ) pn M 2qnjmax . Recall that Then, by the implicit function theorem, Z b is an analytic function of g b in a neighbourhood of 0 (which shrinks to {0} as j max → ∞). Let us now define F and G on Ω 1 × Ω 2 where Ω 1 (resp. Ω 2 ) is a complex neigbourhood of 0 (resp. of 1) such that The amplitude of any divergent graph is a finite sum (because our UV cutoff is compactly supported) of analytic functions of Z b in Ω 2 . A n and B n are thus analytic functions of Z b . Series in eqs. (4.2) and (4.3) converge normally so that F and G are analytic on Ω 1 × Ω 2 . Finally Z j and g j Z 2 j are holomorphic functions of g b around 0, by composition of F and G respectively with Z b (g b ). This proves that, at fixed UV cut-off j max , both g j Z 2 j and Z j are analytic functions of g b in a neighbourhood of 0.
Note also that g j is an invertible function of g b in a neighbourhood of 0.

Asymptotic freedom. -Our aim is to prove
where β j = β 2 + O(M −j ), β 2 is a negative real number and O(g 3 j ) = g 3 j f (g j ) where f is analytic around the origin (in a domain which shrinks to {0} as j max → ∞). ♦ Proof. -Let us define α (j) 1 , α (j) 2 and γ (j) 1 as coefficients of the Taylor expansions of g j Z 2 j and Z j : We thus have Inserting the previous equation into the Taylor expansion of g j+1 at order 2, we get Let us now compute the coefficients α (j) 1 , α (j) 2 and γ (j) 1 : where C j+1 (p) := η j+1 (p 2 )/(Z b p 2 + m 2 r ) and A (j) 4,2 (n c , n c ) is the mass-renormalized amplitude, "down to scale j", of the rightmost graph of fig. 6. To get α (j) 1 and α (j) 2 , we need the Taylor expansion of Z b at first order: 4,2 equals A (j) 4,2 evaluated at Z b = 1. We have thus 4,2 (0, 0).
Before computing the flow equation for g j , we need the first order Taylor coefficient γ (j) 1 of Z j : where, once again,Ã (j) 4,2 (resp. K j ) equalsÃ (j) 4,2 (resp. K j ) evaluated at Z b = 1. Finally, we get .
We now prove that K j (like K j+1 ) is of order M −2j and that the sum of the other terms in β j equals a positive constant plus O(M −j ). First, we note that η j (p 2 ) = h(M −2j p 2 ) where h(p 2 ) = κ(p 2 ) − κ(M 2 p 2 ). Remark also that the support of h is [M −2 , 2].
The above sum is a Riemann sum of the compactly supported C 1 function h(p 2 )/p 2 . Its difference with the corresponding integral (which vanishes) is of order of the mesh, that is M −j . Thus where we used η j+1 = η j+1 + η j+2 and η i η j = 0 if |i − j| > 1. We get The analyticity of g j+1 − g j − β j g 2 j as a function of g j follows from the analyticity of g j and Z j as functions of g b (see Section 4.4).
We have proven that for all j, g j is a holomorphic function of g b in a neighbourhood of 0 which goes to {0} as j max → ∞. This defines g j+1 as a holomorphic function of g j , in a neighbourhood of 0 which goes to {0} as j max → ∞. Moreover the first two coefficients of the expansion g j+1 in powers of g j have a finite limite as j max → ∞.

Holomorphic RG flow
In Section 4.5 we proved that where f is holomorphic on a neighbourhood Ω jmax of the origin and β j = β 2 + O(M −j ), β 2 < 0. Note that a priori Ω jmax → {0} as j max → ∞. But the first two Taylor coefficients of h jmax,j have in fact finite limits as the ultraviolet cutoff is removed. In order to know if such a result holds true at all orders, which would prove that h jmax,j is holomorphic in a domain uniform in j max , we need a better understanding of the series g j+1 (g j ). In the sequel, we assume it.
Assumption 1. -The series g j+1 (g j ) is holomorphic in a domain uniform in j max .
The dynamics defined by h jmax,j is not autonomous, its Taylor coefficients depend on j. Nevertheless, far from the infrared cutoff (here m 2 r ), the behaviour of β j suggests that the dynamics becomes autonomous. In the sequel, we assume it.
Assumption 2. -The discrete RG flow g j+1 = h(g j ) is defined by the iteration of a (unique) holomorphic map h, tangent to the identity, and such that h(z) = z + β 2 z 2 + O(z 3 ), β 2 < 0. (5.1) Throughout this section, we will be interested in Cauchy problems with complex initial data. In particular, we will prove appropriate uniform boundedness of their solutions with respect to their initial data. In other words, we would like to approach results such as "for all > 0, there exists a complex neighbourhood Ω of 0 such that g r = g −1 ∈ Ω implies for all j 0, |g j | < ".

Parabolic holomorphic local dynamics. -
Our first objective is to understand the qualitative behaviour of the approximate RG flow (5.1) by invoking the theory of holomorphic dynamical systems. To this aim, we need to recall some classical definitions and theorems, see [Aba10] for example.
In other words, the stable set of f is the set of all points z ∈ U such that the orbit {f •k (z) : k ∈ N} is well-defined. If z ∈ U \ {K f } , we shall say that z (or its orbit) escapes from U . Clearly, p ∈ K f and so the stable set is never empty (but it can happen that K f = {p}). The RG flow we consider here is thus a parabolic dynamical system (λ = 1).

Definition 5.5 (Multiplicity)
. -Let f ∈ End(C, 0) be a holomorphic local dynamical system with a parabolic fixed point at the origin. Then we can write: with a r+1 = 0. r + 1 is called the multiplicity of f . ♠ Definition 5.6 (Directions). -Let f ∈ End(C, 0) be tangent to the identity of multiplicity r + 1 2. Then a unit vector v ∈ S 1 is an attracting (resp. repelling) direction for f at the origin if a r+1 v r is real negative (resp. real positive). ♠ Clearly, there are r equally spaced attracting directions, separated by r equally spaced repelling directions: if a r+1 = |a r+1 |e iα , then v = e iθ is attracting (resp. repelling) if and only if It turns out that to every attracting direction is associated a connected component of K f \ {0}.  Leau-Fatou flower). -Let f ∈ End(C, 0) be a holomorphic local dynamical system tangent to the identity with multiplicity r + 1 2 at the fixed point. Let v ± 1 , . . . , v ± r ∈ S 1 be the attracting (resp. repelling) directions of f at the origin. Then, 1. for each attracting (resp. repelling) direction v ± j there exists an attracting (resp. repelling petal) P ± j , so that the union of these 2r petals together with the origin forms a neighbourhood of the origin. Furthemore, the 2r petals are arranged cyclically so that two petals intersects if and only if the angle between their central directions is π/r.
2. K f \ {0} is the (disjoint) union of the basins centered at the r attracting directions.
3. If B is a basin centered at one of the attracting directions, then there is a function ϕ : Furthermore if P is the corresponding petal, then ϕ| P is a biholomorphism with an open subset of the complex plane containing a right-half plane -and so f | P is holomorphically conjugated to the translation z → z + 1. ♦ As a consequence of Theorem 5.9, if z belongs to an attracting petal P of a holomorphic local dynamical system tangent to the identity, then its entire orbit is contained in P and moreover f •n (z) goes to 0 (as n → ∞), tangentially to the corresponding attracting direction. A typical trajectory can be seen on fig. 11. Note that Theorem 5.9 asserts the existence of attracting Figure 11. Attracting (green) and repelling (red) petals of a dynamics of multiplicity 4, and a typical trajectory. and repelling petals whose union with the origin forms a neighbourhood of the origin. The intersection properties of these petals implies that their asymptotic opening angle (i.e. their opening angle close to 0) is strictly bigger than π/r for a system of multiplicity r. But in fact, with a bit more work, one can construct petals whose asymptotic opening angle is 2π/r, see [CG93]. Such attracting petals are tangent at 0 to their two neighbouring repelling directions.
In case of the system (5.1), we have a parabolic dynamical system of multiplicity 2 so that there is only one attracting (resp. repelling) petal corresponding to the attractive (resp. repelling) direction (1, 0) (resp. (−1, 0)). The asymptotic opening angle of the attracting petal is 2π which makes it very similar to cardioid-like domains obtained by Loop Vertex Expansion In the next sections, we get more quantitative results on the RG trajectories in case g r is real, on the shapes of attracting petals, and on the size of the Nevanlinna-Sokal disk they contain. To this aim, we will study continuous dynamical systems, more precisely linear ODEs, rather than iterations of holomorphic maps. This is justified by the following argument. As we saw in Section 4.5, there is evidence that the discrete RG flow of the T 4 5 tensor field is well approximated by the dynamical system (5.1), at least in the deep ultraviolet. From Theorem 5.9 a trajectory starting in the unique attracting petal P + remains forever in this petal. Moreover, in P + , the dynamics is conjugated to the translation z → z + 1. But this translation is the time 1 flow of the constant vector field Y = ∂ z . Thus there exists a holomorphic vector field X such that h in eq. (5.1) is the time 1 flow of X. The Taylor coefficients of X can be computed recursively via the equation e X (z) = h(z). One finds X = β 2 z 2 + (β 3 . As a consequence we will consider, in the next sections, ODEs of the form g = β 2 g 2 + β 3 g 3 + O(g 4 ), β 2 ∈ R − keeping in mind that the above β 3 corresponds in fact to β 3 − 2β 2 2 in the notation of eq. (5.1).
We now prove a uniform bound on |g| for g r in the following compact domain Ω of the complex plane.
Definition 5.10 (Domain of uniform boundedness of a quadratic flow). -Let be a positive real number. In polar coordinates (z = ρe iθ , ρ ∈ R + , θ ∈ [−π, π]), The set Ω is made of three parts. On {Re z 0}, Ω is a closed half-disk of radius , centered at 0. On {Re z 0} ∩ {Im z 0}, Ω is a closed half-disk of radius 2 centered at i 2 . On {Re z 0} ∩ {Im z 0}, Ω is a closed half-disk of radius 2 centered at −i 2 . See fig. 13 for a picture of Ω .
-Ω contains the cardioid domain C := ρ cos 2 ( θ 2 ) which is the typical domain of analyticity of correlation functions predicted by Loop Vertex Expansion.
Remark. -In fact, by the holomorphic (on C * ) change of coordinate z → 1/z, one can even prove that g r ∈ Ω implies g(t) ∈ Ω for all t > 0.
-The partial derivatives of the right-hand side of eq. (5.5a) with respect to x and y are continuous so that the Cauchy-Lipschitz theorem applies. As a consequence, for any complex initial data g r , there exists a unique maximal continuously differentiable solution g defined on [0, T ) for some T > 0. Moreover if Im g r is positive (resp. negative), then for all t ∈ [0, T ), Im g(t) is positive (resp. negative). In particular g(t) = 0.
It is easy to check that the partial derivatives of h are continuous on D so that h is continuously differentiable on D. Then, as (0, 0) ∈ D, by the Cauchy-Lipschitz theorem, there exists a unique (continuously differentiable) solution φ to φ = h(t, φ) such that φ(0) = 0. In particular, φ is defined on [0, T ) for some T > 0.

Differential flow of higher degree. -
Let us now consider more general complex differential equations and prove that for sufficiently small initial conditions, their solutions are uniformly bounded as well. Let U be a complex neighbourhood of 0. Let f be the following function: f : R + × U → C (t, z) → β 2 z 2 + β 3 z 3 + z 4 h(z) (5.14) where h is holomorphic on U . We are interested in the following Cauchy problem: g = f (t, g) (5.15a) g(0) = g r ∈ C. (5.15b) Definition 5.14 (Disks). -Let r be real and positive. We will denote by D r the open disk of radius r centered at 0. An open disk S r of radius r centered at r will be called a Nevanlinna-Sokal disk. S r is the set of complex numbers z such that Re 1 z > 1 2r or equivalently |z| < 2r cos(arg z). ♠ Proof. -By the Cauchy-Lipschitz theorem, for all g r ∈ U ∩ R, there exists a unique solution of the Cauchy problem (5.15) defined on [0, T ) for some T > 0. Let a : U ∩ R → R be defined as f (t, x) := β 2 x 2 a(x), so that a(x) = 1 + β 3,2 x + 1 β 2 x 2 h(x). Let g c be the smallest positive zero of a in U ∩ R if it exists and sup U ∩ R otherwise. As f (t, x) is negative if x ∈ (0, g c ), by unicity of the solutions of the Cauchy problem (5.15), for all t ∈ [0, T ), f (t, g(t)) is negative. As a consequence, 0 < g(t) < g r and g is in fact defined on R + (and decreasing).