Razvan Gurau
email: razvan.gurau@th.u-psud.fr

Jacques Magnen
email: magnen@cpht.polytechnique.fr

Vincent Rivasseau
email: vincent.rivasseau@th.u-psud.fr

Fabien Vignes-Tourneret

Renormalization of Non-Commutative Φ 4 4 Field Theory in x Space

In this paper we provide a new proof that the Grosse-Wulkenhaar noncommutative scalar Φ 4 4 theory is renormalizable to all orders in perturbation theory, and extend it to more general models with covariant derivatives. Our proof relies solely on a multiscale analysis in x space. We think this proof is simpler. It also allows direct interpretation in terms of the physical positions of the particles and should be more adapted to the future study of these theories (in particular at the non-perturbative or constructive level).

Power Counting in x-Space

Model, Notations

Beware that throughout this paper we will use many different notations for position variables. To avoid any confusion for the reader we summarize these notations at the end of the paper.

Introduction

In this paper we recover the proof of perturbative renormalizability of non-commutative Φ 4 4 field theory [START_REF] Grosse | Power-counting theorem for non-local matrix models and renormalisation[END_REF][START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF][START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF] by a method solely based on x space. In this way we avoid completely the sometimes tedious use of the matrix basis and of the associated special functions of [START_REF] Grosse | Power-counting theorem for non-local matrix models and renormalisation[END_REF][START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF][START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF] and we recover the more physical direct space representation of fields and particles. Moreover our proof works for the optimal range ]0, 1] of the parameter Ω which was restricted to a much smaller interval in a previous proof. We also extend the corresponding BPHZ theorem to the more general complex Langmann-Szabo-Zarembo φ ϕ φ ϕ model with covariant derivatives, hereafter called the LSZ model. This model has a slightly more complicated propagator, and is exactly solvable in a certain limit [START_REF] Langmann | Exact solution of quantum field theory on noncommutative phase spaces[END_REF].

Our method builds upon previous work of Filk and Chepelev-Roiban [START_REF] Filk | Divergencies in a field theory on quantum space[END_REF][START_REF] Chepelev | Convergence theorem for non-commutative Feynman graphs and renormalization[END_REF]. These works however remained inconclusive [START_REF] Chepelev | Renormalization of quantum field theories on noncommutative R d . i: Scalars[END_REF], since these authors used the right interaction but not the right propagator, hence the problem of ultraviolet/infrared mixing prevented them from obtaining a finite renormalized perturbation series.

The Grosse Wulkenhaar breakthrough was to realize that the right propagator in non-commutative field theory is not the ordinary commutative propagator, but has to be modified to obey Langmann-Szabo duality [START_REF] Langmann | Duality in scalar field theory on noncommutative phase spaces[END_REF][START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF].

Non-commutative field theories (for a general review see [START_REF] Douglas | Noncommutative field theory[END_REF]) deserve a thorough and systematic investigation. Indeed they may be relevant for physics beyond the standard model. They are certainly effective models for certain limits of string theory [START_REF] Connes | Noncommutative geometry and matrix theory: Compactification on tori[END_REF][START_REF] Seiberg | String theory and noncommutative geometry[END_REF]. Also they form almost certainly the correct framework for a microscopic ab initio understanding of the quantum Hall effect which is currently lacking. We think that x space-methods are probably more powerful for the future systematic study of the noncommutative Langmann-Szabo covariant field theories.

Fermionic theories such as the two dimensional Gross-Neveu model can be shown to be renormalizable to all orders in their Langmann-Szabo covariant versions, using either the matrix basis or the direct space version developed here [START_REF] Vignes-Tourneret | Perturbative renormalizability of the noncommutative Gross-Neveu model[END_REF]. However the x-space version seems the most promising for a complete non perturbative construction, using Pauli's principle to controll the apparent (fake) divergences of perturbation theory. In the case of φ 4 4 , recall that although the commutative version is until now fatally flawed due to the famous Landau ghost, there is some hope that the non-commutative field theory treated at the perturbative level in this paper may also exist at the constructive level [START_REF] Grosse | The beta-function in duality-covariant noncommutative φ 4 -theory[END_REF][START_REF] Rivasseau | Non-commutative renormalization[END_REF]. Again the x-space formalism is probably better than the matrix basis for a rigorous investigation of this question.

In the first section of this paper we establish the x-space power counting of the theory using the Mehler kernel form of the propagator in direct space given in [START_REF] Gurau | Propagators for noncommutative field theories[END_REF]. In the second section we prove that the divergent subgraphs can be renormalized by counterterms of the form of the initial Lagrangian. The LSZ models are treated in the Appendix. Note that we do not prove here the exact topological power counting for irrelevant graphs. This should be doable with our methods but is not necessary for our theorem.

The simplest noncommutative ϕ 4 4 theory is defined on R 4 equipped with the associative and noncommutative Moyal product (2.1)

The renormalizable action functional introduced in [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF] is

S[ϕ] = d 4 x 1 2 ∂ µ ϕ ∂ µ ϕ+ Ω 2 2 (x µ ϕ) (x µ ϕ)+ 1 2 µ 2 0 ϕ ϕ+ λ 4! ϕ ϕ ϕ ϕ (x) , (2.2)
where xµ = 2(θ -1 ) µν x ν and the Euclidean metric is used.

In four dimensional x-space the propagator is [START_REF] Gurau | Propagators for noncommutative field theories[END_REF] C(x, x ) = Ω 2

[2π sinh Ωt] where Ω = 2θ -1 Ω and the (cyclically invariant) vertex is [START_REF] Filk | Divergencies in a field theory on quantum space[END_REF] V (x 1 , x 2 , x 3 , x 4 ) = δ(x 1 -x 2 + x 3 -x 4 )e ı P 1 i<j 4 (-1) i+j+1 x i θ -1 x j (2.4) where we note a xθ -1 y ≡ 2 θ (x 1 y 2 -x 2 y 1 + x 3 y 4 -x 4 y 3 ). The main result of this paper is a new proof in configuration space of Theorem 2.1 (BPHZ Theorem for Noncommutative Φ 4 4 [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF][START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF]) The theory defined by the action (2.2) is renormalizable to all orders of perturbation theory.

Let G be an arbitrary connected graph. The amplitude associated with this graph is (with selfexplaining notations):

A G = v,i=1,...4 dx v,i l dt l v δ(x v,1 -x v,2 + x v,3 -x v,4 )e ı P i<j (-1) i+j+1 x v,i θ -1 x v,j l Ω 2 [2π sinh( Ωt l )] 2 e - e Ω 2 coth( e Ωt l )(x 2 v,i(l) +x 2 v ,i (l) )+ e Ω sinh( e Ωt l )
x v,i(l) .x v ,i (l) -µ 2 0 t l .

(2.5)

For each line l of the graph joining positions x v,i(l) and x v ,i (l) , we choose an orientation and we define the "short" variable u l = x v,i(l) -x v ,i (l) and the "long" variable v l = x v,i(l) + x v ,i (l) . With these notations, defining Ωt l = α l , the propagators in our graph can be written as:

l Ωdα l [2π sinh(α l )] 2 e -e Ω 4 coth( α l 2 )u 2 l - e Ω 4 tanh( α l 2 )v 2 l - µ 2 0 e Ω α l . (2.6)
a Of course two different θ parameters could be used for the two symplectic pairs of variables of R 4 .

Orientation and Position Routing

A rule to solve the δ functions at every vertex is a "position routing" exactly analog to a momentum routing in the ordinary commutative case, except for the additional difficulty of the cyclic signs which impose to orient the lines. It is well known that there is no canonical such routing but there is a routing associated to any choice of a spanning tree in G. Such a tree choice is also useful to orient the lines of the graph, hence to fix the exact sign definition of the "short" line variables u l , and to optimize the multiscale power counting bounds below.

Let n be the number of vertices of G, N the number of its external fields, and L the number of internal lines of G. We have L = 2n -N/2. Let T be a rooted tree in the graph (when the graph is not a vacuum graph it is convenient to choose for the root a vertex with external fields but this is not essential). We orient first all the lines of the tree and all the remaining half-loop lines or "loop fields", following the cyclicity of the vertices. This means that starting from an arbitrary orientation of a first field at the root and inductively climbing into the tree, at each vertex we follow the cyclic order to alternate entering and exiting lines as in Figure 1.

Figure 1: Orientation of a tree

Every line of the tree by definition of this orientation has one end exiting a vertex and an other entering another one. This may not be true for the loop lines, which join two "loop fields". Among these, some exit one vertex and enter another; they are called well-oriented. But others may enter or exit at both ends. These loop lines are subsequently referred to as "clashing lines". If there are no clashing lines, the graph is called orientable. If not, it is called non-orientable.

We will see below that non-orientable graphs are irrelevant in the renormalization group sense. In fact they do not occur at all in some particular models such as the LSZ model treated in the Appendix, or in the most natural noncommutative Gross-Neveu models [START_REF] Vignes-Tourneret | Perturbative renormalizability of the noncommutative Gross-Neveu model[END_REF].

For all the well-oriented lines (hence all tree propagators plus some of the loop propagators) we define in the natural way u l = x v,i(l) -x v ,i (l) if the line enters at x v,i(l) and exits from x v ,i (l) . Finally we fix an additional (completely arbitrary) auxiliary orientation for all the clashing loop lines, and fix in the same way u l = x v -x v with respect to this auxiliary orientation.

It is also convenient to define the set of "branches" associated to the rooted tree T . There are n -1 such branches b(l), one for each of the n -1 lines l of the tree, plus the full tree itself, called the root branch, and noted b 0 . Each such branch is made of the subgraph G b containing all the vertices "above l" in T , plus the tree lines and loop lines joining these vertices. It has also "external fields" which are the true external fields hooked to G b , plus the loop fields in G b for the loops with one end (or "field") inside and one end outside G b , plus the upper end of the tree line l itself to which b is associated. In the particular case of the root branch, G b 0 = G and the external fields for that branch are simply all true external fields. We call X b the set of all external fields f of b.

We can now describe the position routing associated to T . There are n δ functions in (2.5), hence n linear equations for the 4n positions, one for each vertex. The momentum routing associated to the tree T solves this system by passing to another equivalent system of n linear equations, one for each branch of the tree. This equivalent system is obtained by summing the arguments of the δ functions of the vertices in each branch. Obviously the Jacobian of this transformation is 1, so we simply get another equivalent set of n δ functions, one for each branch.

Let us describe more precisely the positions summed in these branch equations, using the orientation. Fix a particular branch G b , with its subtree T b . In the branch sum we find a sum over all the u l short parameters of the lines l in T b and no v l long parameters since l both enters and exits the branch. This is also true for the set L b of well-oriented loops lines with both fields in the branch. For the set L b,+ of clashing loops lines with both fields entering the branch, the short variable disappears and the long variable remains; the same is true but with a minus sign for the set L b,-of clashing loops lines with both fields exiting the branch. Finally we find the sum of positions of all external fields for the branch (with the signs according to entrance or exit). For instance in the particular case of Figure 2, the delta function is

δ (u l 1 + u l 2 + u l 3 + u L 1 + u L 3 -v L 2 + X 1 -X 2 + X 3 + X 4 ) .
(2.7)

- + - + + - + - + - + - + - + - X 2 X 1 X 3 X 4 L 1 L 2- L 3 l 1 l 2 l 3 Figure 2: A branch
The position routing is summarized by: Lemma 2.1 (Position Routing) We have, calling I G the remaining integrand in (2.5):

A G = v δ(x v,1 -x v,2 + x v,3 -x v,4 ) I G ({x v,i }) (2.8) = b δ   l∈T b ∪L b u l + l∈L b,+ v l - l∈L b,- v l + f ∈X b (f )x f   I G ({x v,i }),
where (f ) is ±1 depending on whether the field f enters or exits the branch.

Using the above equations one can at least solve all the long tree variables v l in terms of external variables, short variables and long loop variables, using the n -1 non-root branches. To this end, recall that the unique X i which is at the upper end of each tree line should be written in (2.7) as 1/2(v l ± u l ). There remains then the root branch δ function. If G b is orientable, this δ function of branch b 0 contains only short and external variables, since L b,+ and L b,-are empty. If G b is non-orientable one can solve for an additional "clashing" long loop variable. We can summarize these observations in the following lemma: Lemma 2.2 The position routing solves any long tree variable v l as a function of:

• the short tree variable u l of the line l itself,

• the short tree and loop variables with both ends in G b(l) ,

• the long loop variables of the clashing loops with both ends in G b(l) (if any),

• the short and long variables of the loop lines with one end inside G b(l) and the other outside.

• the true external variables x hooked to G b(l) .

The last equation corresponding to the root branch is particular. In the orientable case it does not contain any long variable, but gives a linear relation among the short variables and the external positions. In the non-orientable case it gives a linear relation between the long variables w of all the clashing loops in the graph some short variables u's and all the external positions.

From now on, each time we use this lemma to solve the long tree variables v l in terms of the other variables, we shall call w l rather than v l the remaining n + 1 -N/2 independent long loop variables. Hence looking at the long variables names the reader can check whether Lemma 2.2 has been used or not.

Multiscale Analysis and Crude Power Counting

In this section we follow the standard procedure of multiscale analysis [START_REF] Rivasseau | From Perturbative to Constructive Renormalization[END_REF]. First the parametric integral for the propagator is sliced in the usual way :

C(u, v) = C 0 (u, v) + ∞ i=1 C i (u, v), (2.9)
with

C i (u, v) = M -2(i-1) M -2i Ωdα [2π sinh α] 2 e -e Ω 4 coth α 2 u 2 - e Ω 4 tanh α 2 v 2 - µ 2 0 e Ω α l
(2.10)

Lemma 2.3 For some constants K (large) and c (small):

C i (u, v) KM 2i e -c[M i u +M -i v ] (2.11)
(which a posteriori justifies the terminology of "long" and "'short" variables).

The proof is elementary. For i 1, it relies only on second order approximation of the hyperbolic functions near the origin. This bound is also true for the first slice i = 0 with K depending on µ 0 . Taking absolute values, hence neglecting all oscillations, leads to the following crude bound:

|A G | µ du l dv l l C i l (u l , v l ) v δ v , (2.12)
where µ is the standard assignment of an integer index i l to each propagator of each internal line l of the graph G, which represents its "scale". We will consider only amputated graphs. Therefore we have no external propagators, but only external vertices of the graph; in the renormalization group spirit, the convenient convention is to assign all external indices of these external fields to a fictitious -1 "background" scale.

To any assignment µ and scale i are associated the standard connected components G i k , k = 1, ..., k(i) of the subgraph G i made of all lines with scales j i. These tree components are partially ordered according to their inclusion relations and the (abstract) tree describing these inclusion relations is called the Gallavotti-Nicolò tree [START_REF] Gallavotti | Renormalization theory in four-dimensional scalar fields. i[END_REF]; its nodes are the G i k 's and its root is the complete graph G (see Figure 3).

More precisely for an arbitrary subgraph g one defines:

i g (µ) = inf l∈g i l (µ) , e g (µ) = sup l external line of g i l (µ) . (2.13)
The subgraph g is a G i k for a given µ if and only if i g (µ)

i > e g (µ). As is well known in the commutative field theory case, the key to optimize the bound over spatial integrations is to choose the real tree T compatible with the abstract Gallavotti-Nicolò tree, which means that the restriction T i k of T to any G i k must still span G i k . This is always possible (by a simple induction from leaves to root). In figure 3a, an example of such a compatible tree is given with bold lines. We pick such a compatible tree T and use it both to orient the graph as in the previous section and to solve the associated branch system of δ functions according to Lemma 2.2 We obtain:

|A G,µ | K n l M 2i l du l dv l l e -c[M i l u l +M -i l v l ] b δ b . K n l M 2i l du l dw l l e -c[M i l u l +M -i l v l (u,w,x) ] δ b 0 . (2.14)
The key observation is to remark that any long variable integrated at scale i costs KM 4i whereas any short variable integrated at scale i brings KM -4i , and

G 0 1 = G G 1 1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 11} G 2 1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 11} G 3 1 = {1, 2, 3, 4, 11} G 4 1 = {1, 2} G 4 2 = {3, 4} G 3 2 = {5, 6, 7} G 4 3 = {5, 6, 7}
(c) The "Gallavotti-Nicolò" tree

the variables "solved" by the δ functions bring or cost nothing. For an orientable graph the optimal solution is easy: we should solve the n -1 long variables v l 's of the tree propagators in terms of the other variables, because this is the maximal number of long variables that we can solve, and they have highest possible indices because T has been chosen compatible with the Gallavotti-Nicolò tree structure. Finally we still have the last δ b 0 function (equivalent to the overall momentum conservation in the commutative case). It is optimal to use it to solve one external variable (if any ) in terms of all the short variables and the external ones. Since external variables are typically smeared against unit scale test functions, this leaves power counting invariant b . The non-orientable case is slightly more subtle. We remarked that in this case the system of branch equations allows to solve n long variables as a functions of all the others. Should we always choose these n long variables as the n -1 long tree variables plus one long loop variable? This is not always the optimal choice. Indeed when several disjoint G i k subgraphs are non-orientable it is better to solve more long clashing loop variables, essentially one per disjoint non-orientable G i k , because they spare higher costs than if tree lines were chosen instead. We now describe the optimal procedure, using words rather than equations to facilitate the reader's understanding.

Let C be the set of all the clashing loop lines. Each clashing loop line has a certain scale i, therefore belongs to one and only one G i k and consequently to all G j k ⊃ G i k . We now define the set S of n long variables to be solved via the δ functions. First we put in S all the n -1 long tree variables v l . Then we scan all the connected components G i k starting from the leaves towards the root, and we add a clashing line to S each time some new non-orientable component G i k appears. We also remove p -1 tree lines from S each time p 2 non-orientable components merge into a single one. In the end we obtain a new set S of exactly n long variables.

More precisely suppose some G i k at scale i is a "non-orientable leaf", which means that is contains some clashing lines at scale i but none at scales j > i. We then choose one (arbitrary) such clashing line and put it in the set S. Once a clashing line is added to S in this way it is never removed and no other clashing line is chosen in any of the G j k at lower scales j < i to which the chosen line belongs. (The reader should be aware that this process allows nevertheless several clashing lines of S to belong to a single G i k , provided they were added to different connected components at upper scales.) When p 2 non-orientable components merge at scale i into a single non-orientable G i k , we can find p -1 lines in the part of the tree T i k joining them together, (e.g. taking them among the first lines on the unique paths in T from these p components towards the root) and remove them from S.

We see that if we have added in all q clashing lines to the set S, we have eliminated q -1 tree lines. The final set S thus obtained in the end has exactly n elements. The non trivial statement is that thanks to inductive use of Lemma 2.2 in each G i k , we can solve all the long variables in the set S with the branch system of δ functions associated to T .

We perform now all remaining integrations. This spares the corresponding M 4i integration cost for each long variable in S. For any line not in S we see that the net power counting is 1, since the cost of the long variable integration exactly compensates the gain of the short variable integration. But for any line in S we earn the M -4i power counting of the corresponding short variable u without paying the M 4i cost of the long variable.

Gathering all the corresponding factors together with the propagators prefactors M 2i leads to the following bound:

|A G,µ | K n l M 2i l l∈S M -4i l . (2.15)
Remark that if the graph is well-oriented this formula remains true but the set S consists of only the n -1 tree lines.

In the usual way of [START_REF] Rivasseau | From Perturbative to Constructive Renormalization[END_REF] we write

l M 2i l = l i l i=1 M 2 = i,k l∈G i k M 2 = i,k M 2l(G i k ) (2.16)
and

l∈S i l i=1 M -4i l = i,k l∈G i k ∩S M -4 , (2.17)
and we must now only count the number of elements in

G i k ∩ S. If G i k is orientable, it contains no clashing lines, hence G i k ∩ S = T i k , and the cardinal of T i k is n(G i k ) -1. If G i
k contains one or more clashing lines and p clashing lines l 1 , ... , l p in G i k have been chosen to belong to S, then p -1 tree variables in T i k have also been

removed from S and G i k ∩ S = T i k ∪ {l 1 , ... , l p } -{p -1 tree variables}, hence the cardinal of G i k ∩ S is n(G i k ). Using the fact that 2l(G i k )-4n(G i k ) = -N (G i k )
we can summarize these results in the following lemma: Lemma 2. [START_REF] Langmann | Exact solution of quantum field theory on noncommutative phase spaces[END_REF] The following bound holds for a connected graph (with external arguments integrated against fixed smooth test functions):

|A G,µ | K n i,k M -ω(G i k ) (2.18) for some (large) constant K, with ω(G i k ) = N (G i k ) -4 if G i k is orientable and ω(G i k ) = N (G i k ) if G i k is non-orientable.
This lemma is optimal if vertices oscillations are not taken into account, and proves that non-orientable subgraphs are irrelevant. But it is not yet sufficient for a renormalization theorem to all orders of perturbation.

Improved Power Counting

Recall that for any non-commutative Feynman graph G we can define the genus of the graph, called g and the number of faces "broken by external legs", called B [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF][START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF]. We have g 0 and B 1. The power counting established with the matrix basis in [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF][START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF], rewritten in the language of this paper c is:

ω(G) = N -4 + 8g + 4(B -1) , (2.19)
hence we must (and can) renormalize only 2 and 4 point subgraphs with g = 0 and B = 1, which we call planar regular. They are the only non-vacuum graphs with ω 0.

In the previous section we established that

ω(G) N -4 , if G orientable , ω(G) N , if G non orientable . (2.20)
It is easy to check that planar regular subgraphs are orientable, but the converse is not true. Hence to prove that orientable non-planar subgraphs or orientable planar subgraphs with B 2 are irrelevant requires to use a bit of the vertices oscillations to improve Lemma 2.4 and get: c Beware that the factor i in [START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF] is now 2i, and that the ω used here is the convergence rather than divergence degree. Hence there is both a sign change and a factor 2 of difference between the ω's of this paper and the ones of [START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF].

Lemma 2.5 For orientable subgraphs with g 1 we have ω(G) N + 4 .

(2.21)

For orientable subgraphs with g = 0 and B 2 we have

ω(G) N . (2.22)
This lemma although still not giving (2.19) is sufficient for the purpose of this paper. For instance it implies directly that graphs which contain only irrelevant subgraphs in the sense of (2.19) have finite amplitudes uniformly bounded by K n , using the standard method of [START_REF] Rivasseau | From Perturbative to Constructive Renormalization[END_REF] to bound the assignment sum over µ in (2.12). The rest of this subsection is essentially devoted to the proof of this Lemma 2.5.

We return before solving δ functions, hence to the v variables. We will need only to compute in a precise way the oscillations which are quadratic in the long variables v's to prove (2.21) and the linear oscillations in vθ -1 x to prove (2.22). Fortunately an analog problem was solved in momentum space by Filk and Chepelev-Roiban [START_REF] Filk | Divergencies in a field theory on quantum space[END_REF][START_REF] Chepelev | Convergence theorem for non-commutative Feynman graphs and renormalization[END_REF], and we need only a slight adaptation of their work to position space. In fact in this subsection short variables are quite inessential but it is convenient to treat on the same footing the long 1/2 v and the external x variables, so we introduce a new global notation y for all these variables. The vertices rewrite as v δ(y 1 -y 2 + y 3 -y 4 + 1 2 i u i )e ı P i<j (-1) i+j+1 y i θ -1 y j +yQu+uRu .

(2.23)

for some inessential signs i and some symplectic matrices Q and R. Since we are not interested in the precise oscillations in the short u variables we will note in the sequel quite sloppily E u any linear combination of the u variables. Let's consider the first Filk reduction [START_REF] Filk | Divergencies in a field theory on quantum space[END_REF], which contracts tree lines of the graph. It creates progressively generalized vertices with even number of fields. At a given induction step and for a tree line joining two such generalized vertices with respectively p and q -p + 1 fields (p is even and q is odd), we assume by induction that the two vertices are

δ(y 1 -y 2 + y 3 ... -y p + E u )δ(y p -y p+1 + ... -y q + E u ) (2.24)
e ı P 1 i<j p (-1) i+j+1 y i θ -1 y j + P p i<j q (-1) i+j+1 y i θ -1 y j +yQu+uRu .

Using the second δ function we see that:

y p = y p+1 -y p+2 + .... + y q -E u . (2.25)
Substituting this expression in the first δ function we get:

δ(y 1 -y 2 + ... -y p+1 + .. -y q + E u )δ(y p -y p+1 + ... -y q + E u ) (2.26)
e ı P 1 i<j p (-1) i+j+1 y i θ -1 y j + P p i<j q (-1) i+j+1 y i θ -1 y j +yQu+uRu .

The quadratic terms which include y p in the exponential are (taking into account that p is an even number):

p-1 i=1 (-1) i+1 y i θ -1 y p + q j=p+1
(-1) j+1 y p θ -1 y j .

(2.27)

Using the expression (2.25) for y p we see that the second term gives only terms in yLu. The first term yields:

p-1 i=1 q j=p+1 (-1) i+1+j+1 y i θ -1 y j = p-1 i=1 q-1 k=p (-1) i+k+1 y i θ -1 y k , (2.28)
which reconstitutes the crossed terms, and we have recovered the inductive form of the larger generalized vertex.

One should be aware that y p has disappeared from the final result, but that all the subsequent y s>p have changed sign. This complication arises because of the cyclicity of the vertex. As p was chosen to be even (which implies q odd) we see that q -1 is even as it should. Consequently by this procedure we will always treat only even vertices. We finally rewrite the product of the two vertices as:

δ(y 1 -y 2 + ... + y p-1 -y p+1 + .. -y q + E u )δ(y p -y p-1 + ... -y q + E u ) e ı P 1 i<j q (-1) i+j+1 y i θ -1 y j +yQu+uRu (2.29)
where the exponential is written in terms of the reindexed vertex variables. In this way we can contract all lines of a spanning tree T and reduce G to a single vertex with "tadpole loops" called a "rosette graph" [START_REF] Chepelev | Convergence theorem for non-commutative Feynman graphs and renormalization[END_REF]. In this rosette to keep track of cyclicity is essential so rather than the "point-like" vertex of [START_REF] Chepelev | Convergence theorem for non-commutative Feynman graphs and renormalization[END_REF] we prefer to draw the rosette as a cycle (which is the border of the former tree) bearing loops lines on it (see Figure 4). Remark that the rosette can also be considered as a big vertex, with r = 2n + 2 fields, on which N are external fields with external variables x and 2n + 2 -N are loop fields for the corresponding n + 1 -N/2 loops. When the graph is orientable (which is the case to consider in Lemma 2.5, the fields alternatively enter and exit, and correspond to the fields on the border of the tree T , which we meet turning around counterclockwise in Figure 1. In the

δ(z 1 -z 2 + ... -z r + E u )e ı P 1 i<j r (-1) i+j+1 z i θ -1 z j +zQu+uRu .
(2.30)

The initial product of δ functions has not disappeared so we can still write it as a product over branches like in the previous section and use it to solve the y l variables in terms of the z variables and the short u variables. The net effect of the Filk first reduction was simply to rewrite the root branch δ function and the combination of all vertices oscillations (using the other δ functions) as the new big vertex or rosette factor (2.30).

The second Filk reduction [START_REF] Filk | Divergencies in a field theory on quantum space[END_REF] further simplifies the rosette factor by erasing the loops of the rosette which do not cross any other loops or arch over external fields. Here again the same operation is possible. Consider indeed such a rosette loop l (for instance loop 2 in Figure 4). This means that on the rosette cycle there is an even number of vertices in betwen the two ends of that loop and moreover that the sum of z's in betwen these two ends must be zero, since they are loop variables which both enter and exit between these ends. Putting together all the terms in the exponential which contain z l we conclude exactly as in [START_REF] Filk | Divergencies in a field theory on quantum space[END_REF] that these long z variables completely disappears from the rosette oscillation factor, which simplifies as in [START_REF] Chepelev | Convergence theorem for non-commutative Feynman graphs and renormalization[END_REF] to

δ(z 1 -z 2 + ... -z r + E u )e ı(zIz+zQu+uRu) , (2.31)
where I ij is the antisymmetric "intersection matrix" of [START_REF] Chepelev | Convergence theorem for non-commutative Feynman graphs and renormalization[END_REF] (up to a different sign convention). Here I ij = +1 if oriented loop line i crosses oriented loop line j coming from its right, I ij = -1 if i crosses j coming from its left, and I ij = 0 if i and j do not cross. These formulas are also true for i external line and j loop line or the converse, provided one extends the external lines from the rosette circle radially to infinity to see their crossing with the loops. Finally when i and j are external lines one should define I ij = (-1) p+q+1 if p and q are the numbering of the lines on the rosette cycle (starting from an arbitrary origin). If a node G i k of the Gallavotti-Nicolò tree is orientable but non-planar (g 1), there must therefore exist two intersecting loop lines in the rosette corresponding to this G i k , with long variables w 1 and w 2 . Moreover since G i k is orientable, none of the long loop variables associated with these two lines belongs to the set S of long variables eliminated by the δ constraints. Therefore, after integrating the variables in S the basic mechanism to improve the power counting of a single non planar subgraph is the following:

dw 1 dw 2 e -cM -2i 1 w 2 1 -cM -2i 2 w 2 2 -iw 1 θ -1 w 2 +w 1 .E 1 (x,u)+w 2 E 2 (x,u) = dw 1 dw 2 e -cM -2i 1 (w 1 ) 2 -cM -2i 2 (w 2 ) 2 +iw 1 θ -1 w 2 +(u,x)Q(u,x) = KM 4i 1 dw 2 e -(M 2i 1 +M -2i 2 )(w 2 ) 2 = K. (2.32)
In these equations we used for simplicity M -2i instead of the correct but more complicated factor ( Ω/4) tanh(α/2) (see 2.6) (of course this does not change the argument) and we performed a unitary linear change of variables w 1 = w 1 + 1 (x, u), w 2 = w 2 + 2 (x, u) to compute the oscillating w 1 integral. The gain in (2.32) is M -4(i 1 +i 2 ) , which is the difference between K and the normal factor M 4(i 1 +i 2 ) that the w 1 an w 2 integrals would have cost if we had done them with the regular e -cM -2i 1 w 2 1 -cM -2i 2 w 2 2 factor for long variables. Beware that in (2.32) our constant c depends on θ and that our bounds are singular in the limit θ → 0.

This basic argument must then be generalized to each non-planar leaf in the Gallavotti-Nicolò tree. This is done exactly in the same way as the inductive definition of the set A of clashing lines in the non-orientable case. In any orientable non-planar 'primitive" G i k node (i.e. not containing sub non-planar nodes) we can choose an arbitrary pair of crossing loop lines which will be integrated as in (2.32) using this oscillation. The corresponding improvements are independent. This leads to an improved amplitude bound:

|A G,µ | K n i,k M -ω(G i k ) , (2.33
)

where now ω(G i k ) = N (G i k ) + 4 if G i
k is orientable and non planar (i.e. g 1). This bound proves (2.21).

Finally it remains to consider the case of nodes G i k which are planar orientable but with B 2. In that case there are no crossing loops in the rosette but there must be at least one loop line arching over a non trivial subset of external legs in the G i k rosette (see line 6 in Figure 4). We have then a non trivial integration over at least one external variable, called x, of at least one long loop variable called w. This "external" x variable without the oscillation improvement would be integrated with a test function of scale 1 (if it is a true external line of scale 1) or better (if it is a higher long loop variable) d . But we get now

dxdwe -M -2i w 2 -iwθ -1 x+w.E 1 (x ,u) = KM 4i dxe -M +2i x 2 = K , (2.34)
so that a factor M 4i in the former bound becomes O(1) hence is improved by M -4i . This proves (2.22) hence completes the proof of Lemma 2.5.

This method could be generalized to get the true power counting (2.19). One simply needs a better description of the rosette oscillating factors when g or B increase. We conjecture that it is in fact possible to "disentangle" the rosette by some kind of "third Filk move". Indeed the rank of the long variables quadratic oscillations is exactly the genus [START_REF] Chepelev | Renormalization of quantum field theories on noncommutative R d . i: Scalars[END_REF], and the rank of the linear term coupling these long variables to the external ones is exactly B -1. So one can through a unitary change of variables on the long variables inductively disentangle adjacent crossing pairs of loops in the rosette. This means that it is possible to diagonalize the rosette symplectic form through explicit moves of the loops along the rosette. Once oscillations are factorized in this way, the single improvements shown in this section generalize to one improvement of M -8i per genus and one improvement of M -4i per broken face. In this way the exact power counting (2.19) should be recovered by pure x-space techniques which never require the use of the matrix basis. This study is more technical and not really necessary for the BPHZ theorem proved in this paper.

Renormalization

In this section we need to consider only divergent subgraphs, namely the planar two and four point subgraphs with a single external face (g = 0, B = 1, N = 2 or 4). We shall prove that they can be renormalized by appropriate counterterms of the form of the initial Lagrangian. We compute first the oscillating factors Q and R of the short variables in (2.31) for these graphs. This is not truly necessary for what follows, but is a good exercise.

The Oscillating Rosette Factor

In this subsection we define another more precise representation for the rosette factor obtained after applying the first Filk moves to a graph of order n. We rewrite in terms of u l and v l the coordinates of the ends of the tree lines l, l = 1, . . . , n -1 (those contracted in the first Filk moves), but keep as variables called s 1 , . . . , s 2n+2 the positions of all external fields and all ends of loop lines (those not contracted in the first Filk moves).

We start from the root and turn around the tree in the trigonometrical sense. We number separately all the fields as 1, . . . , 2n + 2 and all the tree lines as 1, . . . , n -1 in the order they are met, but we also define a global ordering ≺ on the set of all the fields and tree lines according to the order in which they are met (see Figure 5). In this way we know whether field number p is met before or after tree line number q. For example, in Figure 5, field number 8 ≺ tree line number 6.

Lemma 3.1 The rosette contribution after a complete first Filk reduction is exactly:

δ(s 1 -s 2 + • • • -s 2n+2 + l∈T u l )e ı P 0 i<j 2n+2 (-1) i+j+1 s i θ -1 s j e -ı P l≺l u l θ -1 u l e -ı P l (l) u l θ -1 v l 2 e ı P l,i≺l (-1) i s i θ -1 u l +ı P l,i l u l θ -1 (-1) i s i , (3.1)
where (l) is -1 if the tree line l is oriented towards the root and +1 if it is not.

Proof: We proceed by induction. We contract the tree lines according to their ordering. In this way, at any step k we contract a generalized vertex with 2k + 2 external fields corresponding to the contraction of the k -1 first lines with a usual four-vertex with r = 4, and obtain a new generalized vertex with 2k + 4 fields. We suppose inductively that the generalized vertex has the above form and prove that it keeps this form after the contraction. We denote the external coordinates of this vertex as s 1 , . . . , s 2k+2 and the coordinates of the four-vertex as t 1 , . . . , t 4 . We contract the propagator (s p , t q ) with associated variables v = s p + t q and u = (-1) p+1 s p + (-1) q+1 t q . We also note that, since the tree is orientable, p + q is odd.

Adding the arguments of the two δ functions gives the global δ function. We have the two equations: Using the invariance of the t vertex we can always eliminate the contribution of t q in the phase factor. We therefore have:

s 1 -s 2 + • • • -s 2k+2 + u s = 0 , t 1 -t 2 + t 3 -t 4 = 0 . ( 3
ϕ = [s 1 -s 2 + • • • + (-1) p s p-1 ]θ -1 (-1) p s p +(-1) p s p θ -1 [(-1) p+2 s p+1 + • • • -s 2k+2 ] = [s 1 -s 2 + • • • + (-1) p s p-1 ]θ -1 [-u + (-1) q+1 t q ] +[-u + (-1) q+1 t q ]θ -1 [(-1) p+2 s p+1 + . . . . -s 2k+2 ]. (3.3)
As (-1) q+1 t q = 4 i=1,i =q (-1) i t i we see that the sθ -1 t q terms in the above expression reproduce exactly the crossed terms needed to complete the first exponential. We rewrite the other terms as:

[s 1 -s 2 + • • • + (-1) p s p-1 ]θ -1 (-u) + (-u)θ -1 [(-1) p+2 s p+1 + • • • -s 2k+2 ] = [s 1 -s 2 + • • • + (-1) p s p-1 ]θ -1 (-u) +(-u)θ -1 [-s 1 + s 2 • • • + (-1) p s p - s u s ] = 2[s 1 -s 2 + • • • + (-1) p s p-1 ]θ -1 (-u) + (-u)θ -1 (-1) p s p + uθ -1 s u s = 2 i≺l (-1) i s i θ -1 u + (-1) p+1 uθ -1 v 2 + s uθ -1 u s . (3.4)
where we have used (-1) p s p = (-1) p (v -u)/2. Note that further contractions will not involve s 1 . . . s p-1 . After collecting all the contractions and using the global delta function we write:

2 l,i≺l (-1) i s i θ -1 u l = l,i≺l (-1) i s i θ -1 u l + l,i l u l θ -1 (-1) i s i + l,l u l θ -1 u l , (3.5)
and the last term is zero by the antisymmetry of θ -1 . We note L the set of loop lines, and analyze now further the rosette contribution for planar graphs. We call now x i , i = 1, . . . , N the N external positions. We choose as first external field 1 an arbitrary entering external line. We define an ordering among the set of all lines, writing l ≺ l if both ends of l are before the first end of l when turning around the tree as in Figure 5 where l 1 ≺ l 2 . Analogously we define l ≺ j when j is an external vertex (l 1 ≺ x 4 in Figure 5). We define l ⊂ l if both ends of l lie in between the ends of l on the rosette (l 2 ⊂ l 4 in Figure 5). We count a loop line as positive if it turns in the trigonometric sense like the rosette and negative if it turns clockwise. Each loop line l ∈ L has now a sign (l) associated with this convention, and we now explicit its end variables in terms of u l and w l .

With these conventions we prove the following lemma:

Lemma 3.2
The vertex contribution for a planar regular graph is exactly:

δ( i (-1) i+1 x i + l∈T ∪L u l )e ı P i,j (-1) i+j+1 x i θ -1 x j e ı P l∈T ∪L, l≺j u l θ -1 (-1) j x j +ı P l∈T ∪L, l j (-1) j x j θ -1 u l e -ı P l,l ∈T ∪L, l≺l u l θ -1 u l -ı P l∈T u l θ -1 v l 2 (l)-ı P l∈L u l θ -1 w l 2 (l) e -ı P l∈L, l ∈L∪T ; l ⊂l u l θ -1 w l (l) . (3.6)
Proof We see that the global root δ function has the argument:

i (-1) i+1 x i + l∈L∪T u l . (3.7)
Since the graph has one broken face we always have an even number of vertices on the external face between two external fields. We express all the internal loop variables as functions of u's and w's. Using lemma 3.1, we regroup the terms which still contain the external points which we relabel x in one quadratic and one linear form in the external positions. The quadratic term can be written as:

i<j (-1) i+j+1 x i θ -1 x j . (3.8)
The linear term in the external vertices is:

i<j (-1) i+1 s i θ -1 (-1) j x j + i>j (-1) j x j θ -1 (-1) i+1 s i + l∈T,l j (-1) j x j θ -1 u l + l∈T,l≺j u l θ -1 (-1) j x j = l ∈L,l j u l θ -1 (-1) j x j + l ∈L,l j (-1) j x j θ -1 u l + l∈T,l j (-1) j x j θ -1 u l + l∈T,l≺j u l θ -1 (-1) j x j . (3.9)
Consider a loop line from s p to s q with p < q. Its contribution to the vertex amplitude decomposes in a "loop-loop" term and a "loop-tree" term. The first

Renormalization of the Four-point Function

Consider a 4 point subgraph which needs to be renormalized, hence is a node of the Gallavotti-Nicolò tree. This means that there is (i, k) such that N (G i k ) = 4. The four external positions of the amputated graph are labeled x 1 , x 2 , x 3 and x 4 . We also define Q, R and S as three skew-symmetric matrices of respective sizes

4 × l(G i k ), l(G i k ) × l(G i k ) and [n(G i k ) -1] × l(G i k )
, where we recall that n(G) -1 is the number of loops of a 4 point graph with n vertices. The amplitude associated to the connected component

G i k is then A(G i k )(x 1 , x 2 , x 3 , x 4 ) = ∈T i k du C (x, u, w) l∈G i k , l ∈T du l dw l C l (u l , w l ) δ x 1 -x 2 + x 3 -x 4 + l∈G i k u l e ı( P p<q (-1) p+q+1 xpθ -1 xq+XQU +U RU +U SW ) . (3.14)
The exact form of the factor p<q (-1) p+q+1 x p θ -1 x q follows from Lemma 3.2. From this Lemma and (3.15) below would also follow exact expressions for Q, R and S, but we wont need them. The important fact is that there are no quadratic oscillations in X times W (because B = 1) nor in W times W (because g = 0). C l is the propagator of the line l. For loop lines C l is expressed in terms of u l and w l by formula (2.6), (with v replaced by our notation w for long variables of loop lines). But for tree lines ∈ T i k recall that the solution of the system of branch δ functions for T has reexpressed the corresponding long variables v in terms of the short variables u, and the external and long loop variables of the branch graph G which lies "over" in the rooted tree T . This is the essential content of the subsection 2.2. More precisely consider a line ∈ T i k with scale i( ) i; we can write

v = X + W + U (3.15)
where

X = e∈E( ) ,e x e (3.16)
is a linear combination on the set of external variables of the branch graph G with the correct alternating signs ,e ,

W = l∈L( ) ,l w l (3.17)
is a linear combination over the set L( ) of long loop variables for the external lines of G (and ,l are other signs), and

U = l ∈S( ) ,l u l (3.18)
is a linear combination over a set S of short variables that we do not need to know explicitly. The tree propagator for line then is

C (u , X , U , W ) = M -2(i( )-1) M -2i( ) Ωdα e -e Ω 4 {coth( α 2 )u 2 l +tanh( α 2 )[X +W +U ] 2 } [2π sinh(α )] 2 . (3.19)
To renormalize, let us call e = max e p , p = 1, ..., 4 the highest external index of the subgraph G i k . We have e < i since G i k is a node of the Gallavotti-Nicolò tree. We evaluate A(G i k ) on external fields e ϕ e (x p ) as:

A(G i k ) = 4 p=1 dx p ϕ e (x p ) A(G i k )(x 1 , x 2 , x 3 , x 4 ) = 4 p=1 dx p ϕ e (x p ) e ıExt ∈T i k du C (u , tX , U , W ) (3.20) l∈G i k l ∈T du l dw l C l (u l , w l ) δ ∆ + t l∈G i k u l e ıtXQU +ıU RU +ıU SW t=1
.

with ∆ = x 1 -x 2 + x 3 -x 4 and Ext = 4 p<q=1 (-1) p+q+1 x p θ -1 x q . This formula is designed so that at t = 0 all dependence on the external variables x factorizes out of the u, w integral in the desired vertex form for renormalization of the ϕ ϕ ϕ ϕ interaction in the action (2.2). We now perform a Taylor expansion to first order with respect to the t variable and prove that the remainder term is irrelevant. Let U = l∈G i k u l , and e For the external index to be exactly e the external smearing factor should be in fact p ϕ e (x p )p ϕ e-1 (x p ) but this subtlety is inessential.

R(t) = - ∈T i k Ω 4 tanh( α 2 ) t 2 X 2 + 2tX W + U ≡ -t 2 AX.X -2tAX.(W + U ) . ( 3
where C (u , U , W ) is given by (3.19) but taken at X = 0. The first term, denoted by τ A, is of the desired form (2.4) times a number independent of the external variables x. It is asymptotically constant in the slice index i, hence the sum over i at fixed e is logarithmically divergent: this is the divergence expected for the four-point function. It remains only to check that (1-τ )A converges as i-e → ∞. But we have three types of terms in (1-τ )A, each providing a specific improvement over the regular, log-divergent power counting of A:

• The term U.∇δ(∆ + tU). For this term, integrating by parts over external variables, the ∇ acts on external fields ϕ e , hence brings at most M e to the bound, whether the U term brings at least M -i .

• The term XQU . Here X brings at most M e and U brings at least M -i .

• The term R (t). It decomposes into terms in AX.X, AX.U and AX.W .

Here the A brings at least M -2i( ) , X brings at worst M e , U brings at least M -i and X W brings at worst M e+i( ) . This last point is the only subtle one: if ∈ T i k , remark that because T i k is a sub-tree within each Gallavotti-Nicolò subnode of G i k , in particular all parameters w l for l ∈ L( ) which appear in W must have indices lower or equal to i( ) (otherwise they would have been chosen instead of in T i k ). In conclusion, since i( ) i, the Taylor remainder term (1 -τ )A improves the power-counting of the connected component G i k by a factor at least M -(i-e) . This additional M -(i-e) factor makes (1 -τ )A(G i k ) convergent and irrelevant as desired.

Renormalization of the Two-point Function

We consider now the nodes such that N (G i k ) = 2. We use the same notations than in the previous subsection. The two external points are labeled x and y. Using the global δ function, which is now δ x -y + U , we remark that the external oscillation e ıxθ -1 y can be absorbed in a redefinition of the term e ıtXQU , which we do from now on. Also we want to use expressions symmetrized over x and y. The full amplitude is

A(G i k ) = dxdyϕ e (x)ϕ e (y)δ x -y + U (3.23) l∈G i k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u ,
+ 1 0 ds(1 -s)(y -x) µ (y -x) ν ∇ µ ∇ ν ϕ e (x + s(y -x)) 2 ,
and substitute into (3.23). The first term A 0 is a symmetric combination with external fields at the same argument. Consider the case with the two external legs at x, namely the term in [ϕ e (x)] 2 . For this term we integrate over y. This uses the δ function. We perform then a Taylor expansion in t at order 3 of the remaining function

f (t) = e ıtXQU +R(t) , (3.26)
where we recall that R(t) = -[t 2 AX.X + 2tAX.(W + U )]. We get

A 0 = 1 2 dx[ϕ e (x)] 2 e ı(U RU +U SW ) l∈G j k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , U , W ) f (0) + f (0) + 1 2 f (0) + 1 2 1 0 dt (1 -t) 2 f (3) (t) . ( 3

.27)

In order to evaluate that expression, let A 0,0 , A 0,1 , A 0,2 be the zeroth, first and second order terms in this Taylor expansion, and A 0,R be the remainder term. First,

A 0,0 = dx [ϕ e (x)] 2 e ı(U RU +U SW ) l∈G i k ,l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , U , W ) (3.28)
is quadratically divergent and exactly of the expected form for the mass counterterm. Then

A 0,1 = 1 2 dx[ϕ e (x)] 2 e ı(U RU +U SW ) l∈G i k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , U , W ) ıXQU + R (0) (3.29)
vanishes identically. Indeed all the terms are odd integrals over the u, w-variables. A 0,2 is more complicated:

A 0,2 = 1 2 dx[ϕ e (x)] 2 e ı(U RU +U SW ) l∈G i k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , U , W ) -(XQU ) 2 -4ıXQU AX.(W + U ) -2AX.X + 4[AX.(W + U )] 2 . (3.30)
The four terms in (XQU ) 2 , XQU AX.W , AX.X and [AX.W ] 2 are logarithmically divergent and contribute to the renormalization of the harmonic frequency term Ω in (2.2). The terms in x µ x ν with µ = ν do not survive by parity and the terms in (x µ ) 2 have obviously the same coefficient. The other terms in XQU AX.U , (AX.U )(AX.W ) and [AX.U ] 2 are irrelevant. Similarly the terms in A 0,R are all irrelevant.

For the term in A 0 (y) in which we have dx[ϕ e (y)] 2 we have to perform a similar computation, but beware that it is now x which is integrated with the δ function so that Q, S, R and R change, but not the conclusion.

Next we have to consider the term in (y -x) µ .∇ µ ϕ e (x) 2 in (3.25), for which we need to develop the f function only to first order. Integrating over y replaces each y -x by a U factor so that we get a term

A 1 = 1 2 dx U µ .∇ µ ϕ e (x) 2 e ı(U RU +U SW ) l∈G i k ,l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , U , W ) f (0) + 1 0 dtf (t)dt (3.31)
The first term is

A 1,0 = 1 2 dx U µ .∇ µ ϕ e (x) 2 e ı(U RU +U SW ) l∈G i k ,l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , U , W ) (3.32)
The terms with µ = ν do not survive by parity. The other ones reconstruct a counterterm proportional to the Laplacian. The power-counting of this factor A 1,0 is improved, with respect to A, by a factor M -2(i-e) which makes it only logarithmically divergent, as should be for a wave-function counterterm. The remainder term in A x 1,R has an additional factor at worst M -(i-e) coming from the 1 0 dtf (t)dt term, hence is irrelevant and convergent. Finally the remainder terms A R with three or four gradients in (3.25) are also irrelevant and convergent. Indeed we have terms of various types:

• There are terms in U 3 with ∇ 3 . The ∇ act on the variables x, hence on external fields, hence bring at most M 3e to the bound, whether the U 3 brings at least M -3i .

• Finally there are terms with 4 gradients which are still smaller.

Therefore for the renormalized amplitude A R the power-counting is improved, with respect to A 0 , by a factor M -3(i-e) , and becomes convergent.

Putting together the results of the two previous section, we have proved that the usual effective series which expresses any connected function of the theory in terms of an infinite set of effective couplings, related one to each other by a discretized flow [START_REF] Rivasseau | From Perturbative to Constructive Renormalization[END_REF], have finite coefficients to all orders. Reexpressing these effective series in terms of the renormalized couplings would reintroduce in the usual way the Zimmermann's forests of "useless" counterterms and build the standard "old-fashioned" renormalized series. The most explicit way to check finiteness of these renormalized series in order to complete the "BPHZ theorem" is to use the standard "classification of forests" which distributes Zimmermann's forests into packets such that the sum over assignments in each packet is finite [START_REF] Rivasseau | From Perturbative to Constructive Renormalization[END_REF] f . This part is completely standard and identical to the commutative case. Hence the proof of Theorem 2.1 is completed.

A The LSZ Model

In this section we prove the perturbative renormalizability of a generalized Langmann-Szabo-Zarembo model [START_REF] Langmann | Interacting fermions on noncommutative spaces: Exactly solvable quantum field theories in 2n+1 dimensions[END_REF]. It consists in a bosonic complex scalar field theory in a fixed magnetic background plus an harmonic oscillator. The quartic interaction is of the Moyal type. The action functional is given by

S = 1 2 φ -D µ D µ + Ω 2 x 2 + µ 2 0 ϕ + λ φ ϕ φ ϕ (A.1)
where D µ = ∂ µ -ıB µν x ν is the covariant derivative. The 1/2 factor is somewhat unusual in a complex theory but it allows us to recover exactly the results given After resolving the v , ∈ T variables in terms of X , W and U , they can be included in the vertices oscillations by a redefinition of the Q, S and R matrices (see (3.14)). For the four-point function, we can then perform the same Taylor subtraction as in the Φ 4 case. The two-point function case is more subtle. Let us consider the generic amplitude

A(G i k ) = dxdy φ e (x)ϕ e (y)δ x -y + U (A.6) l∈G i k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , X , U , W ) e ıXQU +ıU RU +ıU SW .
The symmetrization procedure (3.24) over the external fields is not possible anymore, the theory being complex. Nevertheless we can decompose φ(x)ϕ(y) in a symmetric and an anti-symmetric part: which is the complex equivalent of (3.24).

In the anti-symmetric part of A, called A a , the linear terms φ∇ϕ do not compensate:

A φ(x)ϕ(y) = -((y -x).∇) 3 φ(x + s(y -x))ϕ(x) . (A.9)

We decompose A a into five parts following the Taylor expansion (A.9):

A 1+ a = dxdy φ(x)(y -x).∇ϕ(x)δ x -y + U (A.10) where we performed the integration over y thanks to the delta function. The changes have been absorbed in a redefinition of X , U and Q. From now on X (and X) contain only x (if x is hooked to the branch b(l)) and we forget the primes for Q and U . We expand the function f defined in (3.26) up to order 2: The zeroth order term vanishes thanks to the parity of the integrals with respect to the u and w variables. The first order term contains φ(x) U µ ∇ µ ϕ(x) (ıXQU + R (0)) .

l∈G i k , l ∈T
A 1+ a = φ(x) U.∇ϕ(x)
(A.12)

The first term leads to (U 1 ∇ 1 + U 2 ∇ 2 )ϕ(x 1 U 2 -x 2 U 1 ) with the same kind of expressions for the two other dimensions. Due to the odd integrals, only the terms of the form (U 1 ) 2 x 2 ∇ 1 -(U 2 ) 2 x 1 ∇ 2 survive. We are left with integrals like To prove that these terms give the same coefficient (in order to reconstruct a x ∧ ∇ term), note that, apart from the (u 1 ) 2 , the involved integrals are actually invariant under an overall rotation of the u and w variables. Then by performing rotations of π/2, we prove that the counterterm is of the form of the Lagrangian. The R (0) and the remainder term in A 1+ a are irrelevant.

Let us now study the other terms in A a .

A 1- a = -dx U.∇ φ(x) ϕ(x) The f (0) term renormalizes the wave-function. The remainder term in (A.15) is irrelevant. A 2- a doubles the A 2+ a contribution. Finally the last remainder terms (the last two lines in (A.9)) are irrelevant too. This completes the proof of the perturbative renormalizability of the LSZ models.

Remark that if we had considered a real theory with a covariant derivative which corresponds to a neutral scalar field in a magnetic background, the angular momentum term wouldn't renormalize. Only the harmonic potential term would. It seems that the renormalization "distinguishes" the true theory in which a charged field should couple to a magnetic field. It would be interesting to study the renormalization group flow of these kind of models along the lines of [START_REF] Grosse | The beta-function in duality-covariant noncommutative φ 4 -theory[END_REF].

B Notations of positions

• The letter x is used for the four initial positions of a vertex

• the letter X is used solely for external positions of the considered graph or subgraphs

• the letters v and u are used for the sum and difference of two positions joined by an internal line

• the letter w is used solely as an other name for a v variable which corresponds to a loop line (not a tree line) once a tree has been chosen

• the letter y is used for the collective of long and external variables.

• z is to y what w is to v namely a name for the external variables or long loop variables

• s and t are names for external variables and ends of loop lines variables in rosette vertices.

Hence the same complete set of 4n variables for a graph with n vertices depending on context can be noted x ; X, u and v ; y and u ; X, u, w and the v of the tree lines ; z, u and the v of the tree lines. The s and t are only used in subsection 3.1.

(

  a b)(x) = d 4 k (2π) 4 d 4 y a(x+ 1 2 θ•k) b(x+y) e ık•y .

Figure 4 :

 4 Figure 4: A typical rosette

Figure 5 :

 5 Figure 5: Total ordering of the tree lines and fields

4 p=1 1 0

 41 dx p ϕ e (x p ) e ıExt ∈T i k du C (u , U , W ) l∈G i k l ∈T du l dw l C l (u l , w l ) e ıU RU +ıU SW (3.22) δ(∆) + dt U.∇δ(∆ + tU) + δ(∆ + tU)[ıXQU + R (t)] e ıtXQU +R(t) .

  X , U , W ) e ıXQU +ıU RU +ıU SW . First we write the identity ϕ e (x)ϕ e (y) = 1 2 [ϕ e (x)] 2 + [ϕ e (y)] 2 -[ϕ e (y) -ϕ e (x)] 2 , (3.24) we develop it as ϕ e (x)ϕ e (y) = 1 2 [ϕ e (x)] 2 + [ϕ e (y)] 2 -(y -x) µ .∇ µ ϕ e (x) (3.25)

  x)ϕ(y) + φ(y)ϕ(x) + φ(x)ϕ(y) -φ(y)ϕ(x)) def = (S + A) φ(x)ϕ(y). (A.7)The symmetric part of A, called A s , will lead to the same renormalization procedure as the Φ 4 case. Indeed, x)ϕ(x) + φ(y)ϕ(y) -( φ(x) -φ(y)) (ϕ(x) -ϕ(y))} (A.8)

ds( 1 -

 1 x)ϕ(y) -φ(y)ϕ(x)) )(y -x).∇ϕ(x) -(y -x).∇ φ(x)ϕ(x) s) 2 φ(x)((y -x).∇) 3 ϕ(x + s(y -x))

  du l dw l C l (u l , w l ) ∈T i k du C (u , X , U , W ) e ıXQU +ıU RU +ıU SW = dx φ(x) U.∇ϕ(x) l∈G i k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , X , U , W ) e ıXQ U +ıU RU +ıU SW

1 0dt ( 1 -

 11 l∈G i k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , U , W ) e ıU RU +ıU SW f (0) + f (0) + t)f (t) . (A.11)

(u 1 ) 2

 2 l∈G i k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , U , W ) e ıU RU +ıU SW . (A.13)

1 0

 1 l∈G i k , l ∈T du l dw l C l (u l , w l ) ∈T i k du C (u , X , U , W ) e ıXQU +ıU RU +ıU SW . (A.14)Once more we decouple the external variables form the internal ones by Taylor expanding the function f . Up to irrelevant terms, this only doubles the x ∧ ∇ term in A 1+ a . ) (U.∇) 2 ϕ(x)l∈G i k , l ∈T du l dw l C l (u l , w l ) (A.15) ∈T i k du C (u , U , W ) e ıU RU +ıU SW f (0) + dt f (t) .

b In the case of a vacuum graph, there are no external variables and we must therefore use the last δ b0 function to solve the lowest possible short variable in terms of all others. In this way, we loose the M -4i factor for this short integration. This is why the power counting of a vacuum graph at scale i is not given by the usual formula M (4-N )i = M 4i below at N = 0, but is in M 8i , hence worse by M 4i . This is of course still much better than the commutative case, because in that case and in the analog conditions, that is without a fixed internal point, vacuum graphs would be worse than the others by an ... infinite factor, due to translation invariance! In any case vacuum graphs are absorbed in the normalization of the theory, hence play no role in the renormalization.

d Since the loop line arches over a non trivial (i.e. neither full nor empty) subset of external legs of the rosette, the variable x cannot be the full combination of external variables in the "root" δ function.

f One could also use the popular inductive scheme of Polchinski, which however does not extend yet to non-perturbative "constructive" renormalization

Acknowledgment We thank V. Gayral and R. Wulkenhaar for useful discussions on this work.

one is: i<p (-1) i+1 s i θ -1 (-1) p s p + p<i i =q (-1) p s p θ -1 (-1) i+1 s i + s p θ -1 s q + i<q i =p (-1) i+1 s i θ -1 (-1) q s q + q<i (-1) p s q θ -1 (-1) i+1 s i = i<p (-1) i+1 s i θ -1 [(-1) p s p + (-1) q s q ] + q<i [(-1) p s p + (-1) q s q ]θ -1 (-1)

(3.10)

Taking into account that (-1) i+1 s i + (-1) j+1 s j = u l if s i and s j are the two ends of the loop line l , we can rewrite the above expression as:

where l is fixed in all the above expressions. Summing the contributions of all the lines (being careful not to count the same term twice) we get the final result:

-

(3.12)

We still have to add the "loop-tree" contribution. It reads:

Collecting all the factors proves the lemma.

in [START_REF] Gurau | Propagators for noncommutative field theories[END_REF] with Ω 2 → ω 2 = Ω 2 + B 2 . By expanding the quadratic part of the action, we get a Φ 4 -like kinetic part plus an angular momentum term:

Here the skew-symmetric matrix B has been put in its canonical form

In x space, the interaction term is exactly the same as (2.4). The complex conjugation of the fields only selects the orientable graphs.

At Ω = 0, the model is similar to the Gross-Neveu theory. Its renormalization is therefore harder [START_REF] Vignes-Tourneret | Perturbative renormalizability of the noncommutative Gross-Neveu model[END_REF] and is not treated in this paper. If we additionally set B = θ -1 we recover the integrable LSZ model [START_REF] Langmann | Interacting fermions on noncommutative spaces: Exactly solvable quantum field theories in 2n+1 dimensions[END_REF].

A.1 Power Counting

The propagator corresponding to the action (A.1) has been calculated in [START_REF] Gurau | Propagators for noncommutative field theories[END_REF] in the two-dimensional case. The generalization to higher dimensions e.g. four, is straightforward:

Note that the sliced version of (A.4) obeys the same bound (2.11) as the ϕ 4 propagator. Moreover the additional oscillating phases exp ıxθ -1 y are of the form exp ı u l θ -1 v l . Such terms played no role in the power counting of the Φ 4 theory. They were bounded by one. This allows to conclude that Lemmas 2.4 and 2.5 hold for the generalized LSZ model. Note also that in this case, the theory contains only orientable graphs due to the use of complex fields.

A.2 Renormalization

As for the noncommutative Φ 4 theory, we only need to renormalize the planar (g = 0) two and four-point functions with only one external face.

Recall that the oscillating factors of the propagators are exp ı sinh Bt 2 sinh ωt u l θ -1 v l . (A.5)