Adrian Tanasa
email: adrian.tanasa@ens-lyon.org

Fabien Vignes-Tourneret

Hopf algebra of non-commutative eld theory

We contruct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum eld theory.

Introduction and motivation

Hopf algebras (see for example [START_REF] Kassel | Quantum Groups[END_REF] or [START_REF] Sorin | Hopf Algebras, An Introduction[END_REF]) are today one of the most studied structures in mathematics. In relation with quantum eld theories (QFT), Hopf algebras were proven to be a natural framework for the description of the forest structure of renormalization -the Connes-Kreimer algebras [START_REF] Connes | Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem[END_REF][START_REF] Connes | Renormalization in Quantum Field Theory and the RiemannHilbert Problem II: The ÿ-Function, Dieomorphisms and the Renormalization Group[END_REF]. Ever since there has been an important amount of work with respect to this new class of Hopf algebras (for a general review see for example [START_REF] Kreimer | Structures in Feynman Graphs -Hopf Algebras and Symmetries[END_REF]). However, this construction was realized so far only at the level of commuta- Recently, NCQFT models were also proven to be renormalizable at any order in perturbation theories, despite the ultraviolet-infrared mixing problem. The non-commutative analogous of the Bogoliubov-Parasiuk-Hepp-Zimmerman (BPHZ) theorem was proven for the Grosse-Wulkenhaar Φ 4 scalar model in [START_REF] Grosse | Power-counting theorem for nonlocal matrix models and renormalisation[END_REF][START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF]. In [START_REF] Gurau | Renormalization of non-commutative φ 4 4 eld theory in x space[END_REF] a general proof in x-space, using multiscale analysis was given. The parametric representation was implemented for this model in [START_REF] Gurau | Parametric representation of noncommutative eld theory[END_REF]. Furthermore, the Mellin representation of the non-commutative Feynman amplitudes was achieved in [START_REF] R Zvan Gur U | Non-Commutative Complete Mellin Representation for Feynman Amplitudes[END_REF]. Finally, the dimensional regularization and renormalization were constructed in [START_REF] Tanas | Dimensional regularization and renormalization of non-commutative QFT[END_REF].

With respect to the form of the associated propagator, a second class of NCQFT models exists. This second class contains the non-commutative Gross-Neveu and the Langmann-Szabo-Zarembo [START_REF] Langmann | Exact solution of quantum eld theory on noncommutative phase spaces[END_REF] models. The associate BPHZ theorem was proven in [START_REF] Vignes-Tourneret | Renormalization of the orientable noncommutative Gross-Neveu model[END_REF] for the non-commutative Gross-Neveu model. Moreover, the parametric representation [START_REF] Rivasseau | Parametric representation of critical noncommutative QFT models[END_REF] and the Mellin representation [START_REF] R Zvan Gur U | Non-Commutative Complete Mellin Representation for Feynman Amplitudes[END_REF] were also implemented for this class of models too. For a recent review on dierent issues of renormalizability of NCQFT the interested reader may report himself to [START_REF] Rivasseau | Non-commutative renormalization. Poincare Seminar, updated and expanded version[END_REF].

Note that even though recent progress has been made in [DGWW07, GW07, BGS07],

physicists do not yet have a renormalizable non-commutative gauge theory.

In this article we construct the Hopf algebra structure associated to the renormalization of these NCQFT models. The paper is organized as follows. In the next section we give some insights on the renormalization of NCQFT with respect to renormalization of commutative QFT. The third section is devoted to the Hopf algebra structure of Feynman diagrams. In the last section we state and prove our main result.

Renormalization of non-commutative quantum eld theory

In this section we briey recall some features of both commutative and non-commutative Euclidean renormalization. We will mainly focus on the Grosse-Wulkenhaar model [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF][START_REF] Gurau | Renormalization of non-commutative φ 4 4 eld theory in x space[END_REF] or non-commutative Φ 4 4 theory. It consists in a scalar quantum eld theory on the four-dimensional Moyal space. Its action is given by

S[φ] = d 4 x - 1 2 φ(-∆)φ + Ω 2 2 x 2 φ 2 + 1 2 m 2 φ 2 + λ 4 φ φ φ φ (x) (2.1)
with x µ = 2(Θ -1 x) µ and Θ a four by four skew-symmetric matrix which encodes the noncommutative character of space time: [x µ , x ν ] = ıΘ µν . It has been shown renormalizable to all orders of perturbation.

Furthermore, as already stated in the previous section, the same renormalization results also hold for the non-commutative Gross-Neveu model [START_REF] Vignes-Tourneret | Renormalization of the orientable noncommutative Gross-Neveu model[END_REF] and the generalized LSZ model [START_REF] Gurau | Renormalization of non-commutative φ 4 4 eld theory in x space[END_REF]:

S GN = d 2 x ψ(-i / ∂ + Ω / x + m + µγ 5 )ψ - 3 A=1 g A 4 (J A J A )(x) ,
(2.2)

J A = ψ Γ A ψ, Γ 1 = 1, Γ 2 = γ µ , Γ 3 = γ 5 , , (2.3)
S gLSZ = d 4 x φ (-ı∂ µ + Ω 1 x µ ) 2 + Ω 2 x 2 + m 2 φ + λ 2 φ φ φ φ (x).
(2.4)

Topology and power counting

Let a graph G with V vertices and I internal lines. Interactions of quantum eld theories on the Moyal space are only invariant under cyclic permutation of the incoming/outcoming elds. This restriced invariance replaces the permutation invariance which was present in the case of local interactions.

A good way to keep track of such a reduced invariance is to draw Feynman graphs as ribbon graphs. Moreover there exists a basis for the Schwartz class functions where the Moyal product becomes an ordinary matrix product [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 2 in the matrix base[END_REF][START_REF] Gracia-Bondía | Algebras of distributions suitable for phase space quantum mechanics[END_REF]. This further justies the ribbon representation. Let us consider the example of gure 2. Propagators in a ribbon graph are made of

- + - + + - + - - + + - (a) x-space repre- sentation o o / / O O / / o o / / o o O O O O / / o o O O O O / / (b) Ribbon representa- tion
Figure 2: A graph with two broken faces double lines. Let us call F the number of faces (loops made of single lines) of a ribbon graph. The graph of gure 2b has V = 3, I = 3, F = 2. Each ribbon graph can be drawn on a manifold of genus g. The genus is computed from the Euler characteristic χ = F -I + V = 2 -2g. If g = 0 one has a planar graph, otherwise one has a non-planar graph. For example, the graph of gure 2b may be drawn on a manifold of genus 0. Note that some of the F faces of a graph may be broken by external legs. In our example, both faces are broken. We denote the number of broken faces by B.

Furthermore let N the number of external legs of the graph. For the commutative φ 4 model one has the following supercial degree of convergence ω = N -4. Thus one has to deal only with the renormalization of the two-and four-point functions. In the case of the Grosse-Wulkenhaar model, the situation is dierent. In [GW05a, GW05b, GR07] it was proven that ω = (N -4) + 8g + 4(B -1).

(2.5) Note that, as proven in [START_REF] Rivasseau | Parametric representation of critical noncommutative QFT models[END_REF] one has the same power counting for the LSZ like model (2.4). The one of the Gross-Neveu model (2.2) is more involved but leads to the same conclusion: one has to deal only with the renormalization of the B = 1, planar two-and four-point graphs hereafter qualied as planar regular.

Locality vs Moyality

A crucial aspect of the uplifting from commutative to non-commutative renormalization is that the principle of locality of renormalized interactions of commutative QFT is replaced with a new principle: renormalized interactions have a non-local Moyal vertex form. This is nothing but the analog of the locality phenomenon which occurs in commutative renormalization. One can thus speak, in the case of non-commutative renormalization, of a new type of renormalization group, where the locality is just replaced by Moyality.

The divergent parts of the planar regular two-and four-point graphs with one broken face (the only divergent graphs) are proportional to the (1PI) tree level terms of the perturbative expansion. Such a new denition of locality was suggested in [START_REF] Kreimer | Structures in Feynman Graphs -Hopf Algebras and Symmetries[END_REF], see equation (62).

Let us also argue here that, despite this uplifting, the combinatorial backbone of renormalization theory is almost the same when dealing with commutative or non-commutative QFT. Thus the combinatorics of non-commutative renormalization will be shown to be encoded by a Hopf algebra.

Renormalization as a factorization issue

The basic operation for renormalization is the disentanglement of a graph Γ into pieces γ and cograph Γ/γ. It is exactly this operation that was present at the level of commutative renormalization and that gave rise to a Hopf algebra structure.

We now argue that this factorization process is also present at the level of noncommutative renormalization. Indeed, consider the dimensional renormalization scheme for the Grosse-Wulkenhaar model. The parametric representation constructed in [START_REF] Gurau | Parametric representation of noncommutative eld theory[END_REF] writes the Feynman amplitude φ(Γ) as

φ(Γ) = K 1 0 L =1 [dt (1 -t 2 ) D 2 -1 ]HU G, V (t) -D 2 e - HV G HU G , (2.6)
where K is some constant,

t = tanh α 2 , = 1, . . . , L, (2.7)
where α are the parameters associated to any of the propagators of the graph. In [START_REF] Gurau | Parametric representation of noncommutative eld theory[END_REF] it was furthermore proved that HU and HV are polynomials in the set of variables t . Considering now a primitive divergent subgraph γ of Γ and rescaling the parameters t of its internal edges, it was proven in [START_REF] Tanas | Dimensional regularization and renormalization of non-commutative QFT[END_REF] that

HU l Γ = HU l γ HU Γ/γ (2.8)
where by the index l we understand the leading terms under the rescaling. A similar factorization theorem was also proven for the exponential part in (2.6) of the Feynman amplitude φ(Γ).

Moreover, in [START_REF] Gurau | Renormalization of non-commutative φ 4 4 eld theory in x space[END_REF] an analogous phenomena of factorization was shown for the Grosse-Wulkenhaar model in position space namely the planar regular graphs contribute to the renormalization of the mass, wave-function, harmonic frequency Ω and coupling constant, see Equation (2.1).

3 Hopf algebra structure of Feynman diagrams

Hopf algebra reminder

In this subsection we recall the general denition of a Hopf algebra (for further details one can refer for example to [START_REF] Kassel | Quantum Groups[END_REF][START_REF] Sorin | Hopf Algebras, An Introduction[END_REF]).

Denition 3.1 (Algebra). A unital associative algebra A over a eld K is a K-linear space endowed with two algebra homomorphisms:

• a product m : A ⊗ A → A satisfying the associativity condition:

∀Γ ∈ A, m • (m ⊗ id)(Γ) =m • (id ⊗ m)(Γ), (3.1)
• a unit u : K → A satisfying:

∀Γ ∈ A, m • (u ⊗ id)(Γ) =Γ = m • (id ⊗ u)(Γ).
(3.2) Denition 3.2 (Coalgebra). A coalgebra C over a eld K is a K-linear space endowed with two algebra homomorphisms:

• a coproduct ∆ : C → C ⊗ C satisfying the coassociativity condition:

∀Γ ∈ C, (∆ ⊗ id) • ∆(Γ) =(id ⊗ ∆) • ∆(Γ), (3.3)
• a counit ε : C → K satisfying:

∀Γ ∈ C, (ε ⊗ id) • ∆(Γ) =Γ = (id ⊗ ε) • ∆(Γ).
(3.4) Denition 3.3 (Bialgebra). A bialgebra B over a eld K is a K-linear space endowed with both an algebra and a coalgebra structure (see Denitions 3.1 and 3.2) such that the coproduct and the counit are unital algebra homomorphisms (or equivalently the product and unit are coalgebra homomorphisms):

∆ • m B =m B⊗B • (∆ ⊗ ∆), ∆(1) = 1 ⊗ 1, (3.5a) ε • m B =m K • (ε ⊗ ε), ε(1) = 1.
(3.5b) Denition 3.4 (Graded Bialgebra). A graded bialgebra is a bialgebra graded as a linear space:

B = ∞ n=0 B (n) (3.6)
such that the grading is compatible with the algebra and coalgebra structures:

B (n) B (m) ⊆ B (n+m) and ∆B (n) ⊆ n k=0 B (k) ⊗ B (n-k) .
(3.7) Denition 3.5 (Connectedness). A connected bialgebra is a graded bialgebra B for which B (0) = u(K).

One can then dene a Hopf algebra:

Denition 3.6 (Hopf algebra). A Hopf algebra H over a eld K is a bialgebra over K equipped with an antipode map S : H → H obeying:

m • (S ⊗ id) • ∆ =u • ε = m • (id ⊗ S) • ∆. (3.8)
Finally we remind a useful lemma:

Lemma 3.1 ([Man03]) Any connected graded bialgebra is a Hopf algebra whose antipode is given by S(1) = 1 and recursively by any of the two following formulas for Γ = 1:

S(Γ) = -Γ - (Γ) S(Γ )Γ , (3.9a) S(Γ) = -Γ - (Γ) Γ S(Γ ) (3.9b)
where we used Sweedler's notation.

Locality and the residue map

In quantum eld theory, Feynman graphs are built from a certain set of edges and vertices

R = R E ∪ R V .
This set is given by the particle content of the model and by the type of interactions one wants to consider. For example, in the commutative φ 4 4 theory (which will be our benchmark until section 4), R E contains only the scalar bosonic line while R V contains the local four-point vertex and the two-point vertices corresponding to the mass and wave-function renormalization:

R E = { }, R V = { , 0 , 1 }.
In the following we will still write R V for the free algebra generated by the elements of R V . Let us now consider the algebra H generated by a certain class of graphs (connected, 1PI etc) made out of the set R.

Denition 3.7 (Subgraph). Let Γ ∈ H, Γ [1] its set of internal lines and Γ [0] its vertices.

A subgraph γ of Γ, written γ ⊂ Γ, consists in a subset γ [1] of Γ [1] and the vertices of Γ [0] hooked to the lines in γ [1] . Note that with such a denition, γ is truncated.

Denition 3.8 (Shrinkable subgraph). Let Γ ∈ H. A subgraph ∅ γ Γ is said shrinkable if res(γ) ∈ R V .
The set of shrinkable subgraphs of Γ will be denoted by Γ.

Note that until now we did not really dene what is the map res. We now do it. First we assume that it is an algebra homomorphism from H to H ∪ R V . Then to compute the graphical residue of a generator of H, we need the following remarks and denitions. The coproduct of H (usually given by (3.15)) drives the combinatorial and algebraic aspects of renormalization if it corresponds to some analytical facts. Before we explain this, let us recall the following denitions.

Denition 3.9. The (unrenormalized) Feynman rules are an homomorphism φ from H to A. The precise denition of A depends on the regularization scheme employed (in dimensional regularization, A is the Laurent series).

Denition 3.10. The projection T is a map from A to A wich has to fulll: ∀Γ ∈ H, Γ primitive

(id A -T ) • φ(Γ) < ∞. (3.10)
This means that if φ(Γ) is supercially divergent (as the cut-o is removed) then its overall divergence is totally included in T • φ(Γ).

External structures The projection T extracts the divergent part of the amplitude φ(Γ). In the case of a two-point graph this divergent part decomposes into two pieces.

The rst one is a mass term whereas the second one contributes to the wave function renormalization (recall that the propagator of the commutative φ 4 theory is (-∆+m 2 ) -1 ).

To distinguish between these two, one introduces external structures [START_REF] Connes | Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem[END_REF][START_REF] Kreimer | Structures in Feynman Graphs -Hopf Algebras and Symmetries[END_REF]. It consists in the following endomorphisms of A (in x-space representation):

σ 0 , φ(Γ) =ρ 0 (Γ)δ y (x), (3.11a) σ 1 , φ(Γ) =ρ 1 (Γ)∆δ y (x), (3.11b) σ 2 , φ(Γ) =ρ 2 (Γ)δ x 2 (x 1 )δ x 3 (x 1 )δ x 4 (x 1 ) (3.11c)
where the ρ i 's are characters on A. If K Γ is the kernel of the amplitude φ(Γ), these characters are given by:

ρ 0 (Γ) = d 4 z K Γ (x, z), (3.12a) ρ 1 (Γ) = 1 8 d 4 z (z -x) 2 K Γ (x, z), (3.12b) ρ 2 (Γ) = d 4 x 2 d 4 x 3 d 4 x 4 K Γ (x, x 2 , x 3 , x 4 ).
(3.12c)

Recall that commutative eld theories are usually translation invariant so that none of the ρ i 's depend on x. With those notations, T = σ 0 + σ 1 on a two-point graph and T = σ 2 on a four-point graph.

There is now a way to relate the analytical operations σ i 's to the graphical map res:

Denition 3.11 (Residue). The residue map res :

H → H ∪ R V is dened by σ i , φ(Γ) =ρ i (Γ) σ i , φ • res(Γ) (3.13)
where i = 0 or 1 for a two-point graph and i = 2 on a four-point graph.

Following equations (3.11) and (3.12) one nds

φ • res(Γ) =δ y (x) + ∆δ y (x) if Γ is a two-point graph, (3.14a) φ • res(Γ) =δ x 2 (x 1 )δ x 3 (x 1 )δ x 4 (x 1 ) if Γ is a four-point graph (3.14b)
which leads to the following graphical denitions:

res( ) = = 0 + 1 -1 , res( ) = .
(3.14c) Equation (3.13) means that the divergent part of a graph Γ looks like another graph called res(Γ). For a renormalizable quantum eld theory the residue of any supercially divergent graph belongs to R V . This is the usual statement according to which all the divergences of a renormalizable eld theory can be absorbed in a redenition of the various coupling constants. If the theory is local then res(Γ) corresponds to the graph obtained from Γ by shrinking all its internal lines to a point. But this is a particular case and we have to dene res as reecting the appropriate projection T . For example, we will see in the next section that the residue of a non-commutative graph is not a local graph anymore.

The T operation is designed to extract the main part of graphs. For the convergent ones there is no good distinction between T • φ(Γ) and (id -T ) • φ(Γ): both are convergent expressions. That's why T is mainly dened on (supercially) divergent graphs. Nevertheless one can dene T to be id A on convergent graphs. Condition (3.10) is then trivially fullled and equation (3.13) is satised with res = id H and ρ the trivial character.

Coassociative coproducts

Using the denitions of section 3.2 we have the following lemma:

Lemma 3.2 (Coassociativity) Let Γ ∈ H. Provided 1. ∀γ ∈ Γ, ∀γ ∈ γ such that res(γ) ∈ R V and res(γ ) ∈ R V , res(γ/γ ) ∈ R V , 2. ∀γ 1 ∈ H, ∀γ 2 ∈ H such that res(γ 1 ) ∈ R V and res(γ 2 ) ∈ R V , there exists gluing data G such that res(γ 1 • G γ 2 ) ∈ R V , the following coproduct is coassociative ∆Γ =Γ ⊗ 1 + 1 ⊗ Γ + ∆ Γ, (3.15a) ∆ Γ = γ∈Γ γ ⊗ Γ/γ. (3.15b)
Remark that Γ/γ is the graph obtained from Γ by replacing γ ⊂ Γ by its residue. Then res(γ) ∈ R V implies Γ/γ ∈ H. We prove this lemma by following closely [START_REF] Connes | Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem[END_REF].

Proof. First note that (∆ ⊗ id)∆ = (id ⊗ ∆)∆ ⇐⇒ (∆ ⊗ id)∆ = (id ⊗ ∆ )∆ which
means that all the following subgraphs can be considered as neither full nor empty. Let Γ a generator of H,

(∆ ⊗ id)∆ Γ =(∆ ⊗ id) γ∈Γ γ ⊗ Γ/γ (3.16) = γ∈Γ γ ∈γ γ ⊗ γ/γ ⊗ Γ/γ (3.17) (id ⊗ ∆ )∆ Γ = γ ∈Γ γ ∈Γ/γ γ ⊗ γ ⊗ (Γ/γ )/γ . (3.18)
By the denitions 3.7, 3.8 and 3.11 it is clear that γ ∈ γ and γ ∈ Γ implies γ ∈ Γ. This implicitly uses the fact that the residue of a graph is independant of the surrounding of this graph and really only depends on the graph itself: res(γ) is the same wether γ is a subgraph or not of another graph. Equation (3.17) can then be rewritten as

(∆ ⊗ id)∆ Γ = γ ∈Γ γ∈Γ|γ γ γ ⊗ γ/γ ⊗ Γ/γ. (3.19)
It is now enough to prove equality between (3.18) and (3.19) at xed γ ∈ Γ. Let us rst x a subgraph γ ∈ Γ such that γ γ and prove that there exists a graph γ ∈ Γ/γ such that γ/γ ⊗ Γ/γ = γ ⊗ (Γ/γ )/γ . Of course the logical choice for γ is γ/γ because then (Γ/γ )/(γ/γ ) = Γ/γ.

We only have to prove that γ = γ/γ ∈ Γ/γ . It is clear that γ/γ is a subset of internal lines of Γ/γ . Then γ/γ ∈ Γ/γ if res(γ) ∈ R V and res(γ ) ∈ R V implies res(γ/γ ) ∈ R V which we assumed.

Conversely let us x γ ∈ Γ/γ and prove that there exists γ ∈ Γ containing γ such that γ/γ ⊗ Γ/γ = γ ⊗ (Γ/γ )/γ . Let us write γ = i∈I γ i for the connected components of γ . Some of these components led to vertices of γ , the others to vertices of (Γ/γ ) \ γ . We can then dene γ as (γ

• G I 1 i∈I 1 γ i ) j∈I 2 γ j with I 1 ∪ I 2 = I. It is clearly a subgraph of Γ and belongs to Γ if ∀γ 1 , γ 2 ∈ H, res(γ 1 ) ∈ R V , res(γ 2 ) ∈ R V there exists gluing data G such that res(γ 1 • G γ 2 ) ∈ R V .
We also assumed it. This ends the proof of Lemma 3.2.

Let us now work out how Lemma 3.2 ts the commutative φ 4 model. In this local eld theory the divergent graphs have two or four external legs. The residue of a given graph is the one obtained by shrinking all its internal lines to a point (see section 3.2) and then only depends on the number of external lines of the graph. Let us check condition 2 of Lemma 3.2 for commutative φ 4 . We consider two graphs γ 1 and γ 2 with two or four external legs. We consider γ 0 = γ 1 • G γ 2 for any gluing data G. Let V i , I i and E i the respective numbers of vertices, internal and external lines of γ i , i ∈ {0, 1, 2}. For all i ∈ {0, 1, 2}, we have

4V i =2I i + E i (3.20a) V 0 = V 1 + V 2 if E 2 = 2 V 1 + V 2 -1 if E 2 = 4 (3.20b) I 0 = I 1 + I 2 + 1 if E 2 = 2 I 1 + I 2 if E 2 = 4 (3.20c)
which proves that E = E 1 . Then as soon as res(γ 1 ) ∈ R V so does res(γ 0 ). Concerning condition 1 note that γ = γ/γ ⇐⇒ ∃G | γ = γ • G γ which allows to prove, in the case of a local theory, that condition 1 also holds and that the coproduct (3.15) is coassociative.

Lemma 3.3 Let H c the linear space of graphs whose residue is R V -valued: In the following all the graphs we are going to insert will be four-point graphs. The case of two-point graphs is easier and left to the reader. Before proving conditions 1 and 2, let us consider the insertion of a regular four-point graph γ 2 into a vertex of another graph γ 1 . Let γ 0 = γ 1 • γ 2 and for all i ∈ {0, 1, 2} let F i , I i , V i , B i the respective numbers of faces, internal lines, vertices and broken faces of γ i . The number of faces of a ribbon graph is the number of closed a single lines. A ribbon vertex is drawn on gure 3a. One sees that the number of faces to which the lines of that vertex belong is at most four. Some of them may indeed belong to the same face. The gluing data necessary to the insertion of γ 2 corresponds to a bijection between the half-lines of the vertex in γ 1 and the external lines of γ 2 . This last one being regular (only one broken face) the typical situation is represented on gure 3b. It should be clear that F = F 2 -1 + F 1 -n for some n 0. F 2 -1 is the number of internal faces of γ 2 i.e. the number of faces of the blob.

H c = {Γ ∈ H : res(Γ) ∈ R V } .
The number n depends on the gluing data. It vanishes if the insertion respects the cyclic ordering of the vertex. For example the following bijection σ does: σ((1 , 2 )) =(2, 3), σ((2 , 3 )) = (3, 4), σ((3 , 4 )) = (4, 1), σ((4 , 1 )) = (1, 2). Moreover by exhausting the 4!/4 possible insertions, one checks that B 0 B 1 . For example, on gure 4a, lines 1 and 3 belong to two dierent broken faces. Figure 4b shows an insertion of a regular four-point graph which increases the number of broken faces by one: now trajectories (1, 4), (1, 2) and 3 are external faces (line (2, 4) is still an internal one).

Let us now turn to proving that the algebra of non-commutative Feynman graphs described above fullls conditions 1 and 2.

1. γ, γ planar implies γ/γ planar thanks to equation (4.6). Furthermore B(γ) = 1 implies B(γ/γ ) = 1 due to the preceding remark. a In the case of external faces, one considers that the corresponding lines are closed. As in the commutative eld theories, the renormalized amplitude φ + of a graph Γ ∈ H is given by: φ + (Γ) =φ - * φ(Γ).

(4.9)

  Figure 1: A Moyal vertex

H

  c is a Hopf subalgebra of H. Proof. Thanks to the denition (3.15), ∆H c ⊂ H c ⊗H c . By induction on the augmentation degree, one also proves that S(H c ) ⊂ H c .

  Figure 3: Insertion procedure

(4. 5 )

 5 As in equations (3.20), I 0 = I 1 + I 2 and V 0 = V 1 + V 2 -1. It follows that the genus of γ 0 satises g(γ 0 ) =g(γ 1 ) + g(γ 2 ) + n.

Figure 4 :

 4 Figure 4: Increasing number of external faces

(4. 7 )

 7 Let φ the unrenormalized Feynman rules and φ -∈ Hom(H, A) the twisted antipode: ∀Γ ∈ H,φ -(Γ) = -T φ(Γ) + γ∈Γ φ -(γ) φ(Γ/γ) .

Acknowledgment We would like to warmly thank Dirk Kreimer for fruitful discussions.

Hopf algebra for non-commutative Feynman graphs

The denition of the Hopf algebra of non-commutative Feynamn graphs which drives the combinatorics of perturbative renormalization is formally the same as in the commutative case [START_REF] Connes | Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem[END_REF]. But before giving the denitions let us dene the residue of a non-commutative graph. As already mentionned it has been proven (rst in [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF]) that the Grosse-Wulkenhaar model (2.1) is renormalizable to all orders of perturbation.

It means that the divergent parts of the divergent graphs are proportionnal to mass, wave-function, x 2 and Moyal vertex terms. Following the procedure exposed in section 3.2, particularly equations (3.13) and (3.14), we nd

if Γ is a four-point planar regular graph (4.1b) which leads to the following graphical denitions:

res( ) = , res( ) = .

(4.1c)

Once more the (graphical) residue of a convergent graph is dened as id H .

Consider now the unital associative algebra H freely generated by 1PI non-commutative Feynman graphs (including the empty set, which we denote by 1). The product m is bilinear, commutative and given by the operation of disjoint union. Let the coproduct We can state the main result of this letter: Theorem 4.1 The quadruple (H, ∆, ε, S) is a Hopf algebra.

Proof. The only thing to prove is the coassociativity of the coproduct (4.2). Once it is done, the denition (4.4) for the antipode follows from the fact that H is graded (by the loop number), connected and from Lemma 3.1. We will use Lemma 3.2 and the fact that for all Γ ∈ H, res(Γ) ∈ R V is equivalent to Γ is planar regular. Then conditions 1 and 2 of Lemma 3.2 are equivalent to:

1. for all γ and γ ⊂ γ both planar regular, γ/γ is planar regular, 2. for all γ and γ ⊂ γ both planar regular, there exits gluing data G such that γ • G γ is planar regular.


Ahmed Lakhoua

Fabien Vignes-Tourneret

Jean-Christophe Wallet
email: jean-christophe.wallet@th.u-psud.fr

One-loop Beta Functions for the Orientable Non-commutative Gross-Neveu Model *

We compute at the one-loop order the β-functions for a renormalisable noncommutative analog of the Gross Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The β-function for the non-commutative counterpart of the Thirring model is found to be non vanishing.

Introduction

In the past few years, some activity has been focused on the study of various classes of field theories defined on Moyal spaces [START_REF] Douglas | Noncommutative field theory[END_REF]. These provide prototypes of non-commutative field theories which are interesting in themselves since they involve some salient features of non-commutative geometry [START_REF] Connes | Noncommutative Geometry[END_REF]. The interest in the study of these field theories was further increased by the claim that somehow similar non-commutative field theories seem to emerge rather naturally from (some limiting regime of) String Theory and matrix theory in magnetic backgrounds [START_REF] Seiberg | String Theory and noncommutative Geometry[END_REF]. Recall that in non-commutative geometry, the commutative algebras of functions defined on differentiable manifolds (roughly the coordinates spaces) are replaced by associative but non-commutative algebras further interpreted as algebras of functions on "non-commutative spaces". Within this later algebraic framework, natural non-commutative analogs of the main geometrical objects usually involved in field theories can be algebraically defined, such as for instance connections, curvatures, vector bundles, so that the construction of various non-commutative analogs of fields theories can be undertaken. The starting relevant configuration spaces for the non-commutative field theories are modules over the associative algebras which are naturally viewed as non commutative analogs of vector bundles. One example of associative algebra among many others is provided by the associative Moyal algebras [START_REF] Gracia-Bondía | Algebras of distributions suitable for phase space quantum mechanics. I[END_REF][START_REF] Varilly | Algebras of distributions suitable for phasespace quantum mechanics. II. topologies on the moyal algebra[END_REF] therefore playing the role of "non-commutative Moyal spaces". At this level, one technical remark is in order. Throughout this paper, we will consider only R D θ algebras, i.e the non-commutative counterpart of the Euclidean R D spaces. For studies related to non-commutative tori, see e.g. [START_REF] Douglas | Noncommutative field theory[END_REF] and references therein. Besides, the ensuing discussion refers to non-commutative field theories defined on free modules.

The simplest non-commutative generalizations on Moyal spaces of the usual scalar theories that have been first investigated were shown to suffer from the so called UV/IR mixing [START_REF] Minwalla | Noncommutative perturbative dynamics[END_REF], a phenomenon that makes the renormalisability very unlikely. Recall that UV/IR mixing results from the existence of potentially dangerous non-planar diagrams which, albeit UV finite, become singular at exceptional (low) external momenta. This triggers the occurrence of UV divergences in higher order diagrams in which they are involved as subdiagrams. This signals that UV and IR scales are related in a non trivial way which should in principle invalidate a Wilson-type renormalisation scheme [START_REF] Wilson | The renormalisation group and the epsilon expansion[END_REF]. An appealing solution to the UV/IR mixing has been recently proposed by Grosse and Wulkenhaar [START_REF] Grosse | Renormalization of Φ 4 -theory on Non Commutative R 4 in the matrix base[END_REF] within the non-commutative ϕ 4 model on the 4-dimensional Moyal space where ϕ is real-valued. They showed that the UV/IR mixing can be suppressed by supplementing the initial action with a harmonic oscillator quadratic term leading to a renormalisable non-commutative quantum field theory. The initial proof [START_REF] Grosse | Renormalization of Φ 4 -theory on Non Commutative R 4 in the matrix base[END_REF] was performed within the matrix-base formalism, roughly a basis for the (Schwarz class) functions for which the associative product of the Moyal algebra is a simple matrix product. This cumbersome proof was simplified through a reformulation into the (position) x-space formalism in [START_REF] Gurau | Renormalization of non-commutative φ 4 4 field theory in x space[END_REF] which exhibits some advantages compared to the matrix-base formulation. For instance, the propagator in x-space can be explicitely computed (as a Mehler kernel [START_REF] Simon | Functional Integration and Quantum Physics[END_REF] ) and actually used in calculations. Besides, it makes easier the comparition of the renormalisation group for non-commutative theories and their commutative counterpart.

At the present time, another renormalisable non-commutative scalar quantum field theory on Moyal spaces has been identified. This is a (complex-valued) scalar theory studied in [START_REF] Gurau | Renormalization of non-commutative φ 4 4 field theory in x space[END_REF] which can be viewed as a modified version of the LSZ model [START_REF] Langmann | Exact solution of quantum field theory on noncommutative phase space[END_REF] (the scalar theory in [START_REF] Grosse | Renormalisation of Φ 4 -theory on noncommutative R 2 in the matrix base[END_REF] is super renormalisable). Note that interesting solvable non-commutative scalar field theories have also been considered in [START_REF] Grosse | Renormalisation of the noncommutative φ 3 model through the Kontsevich model[END_REF]. As far as Fermionic theories are concerned, a non-commutative version on the Moyal plane of the Gross-Neveu model [START_REF] Gross | Asymptotic scale invariance in a massive Thirring model with u(n) symmetry[END_REF], called the orientable non-commutative Gross Neveu model, has been recently considered and shown to be renormalisable to all orders [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF][START_REF] Vignes-Tourneret | Renormalisation des théories de champs non commutatives[END_REF]. It is worth mentioning that this non-commutative field theory still exhibits some UV/IR mixing, even in the presence of the Fermionic version of the harmonic oscillator quadratic term introduced in [START_REF] Grosse | Renormalization of Φ 4 -theory on Non Commutative R 4 in the matrix base[END_REF], which however does not prevent the theory to be renormalisable. Note that in [START_REF] Akhmedov | Non-commutative Gross-Neveu model at large N[END_REF] the large N limit of the non-commutative Gross-Neveu model, with however a restricted interaction, has been studied; renormalisability is shown at this limit together with asymptotic freedom. One should keep in mind that the fact that the orientable Gross Neveu model is renormalisable although it involves some remaining UV/IR mixing [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF] indicates that further investigations are needed to actually clarify the effective role of the various generalizations of the above mentionned harmonic oscillator term and of the related covariance of the considered theory under the Langmann-Szabo duality [START_REF] Langmann | Duality in scalar field theory on non commutative phase spaces[END_REF] and their impact in the control of the UV/IR mixing and renormalisability.

Despite this remaining uncertainty, coupling constant flows and β-functions can be studied in the available renormalisable non-commutative field theories. The β-function for the coupling constant of the non-commutative (real-valued) ϕ 4 model on the 4-dimensional Moyal space has been computed at the one-loop order in [START_REF] Grosse | The beta-function in duality-covariant noncommutative φ 4 -theory[END_REF]. It exhibits a bounded flow, finite fixed point and vanishes when the parameter affecting the harmonic oscillator quadratic term, says Ω, is equal to unity, which corresponds to the (self-dual) point in the parameter space where the field theory is invariant under the Langmann-Szabo duality [START_REF] Langmann | Duality in scalar field theory on non commutative phase spaces[END_REF]. This latter result has been proven very recently to be valid to all orders in [START_REF] Dissertori | Vanishing of Beta function of non commutative Φ 4 Theory to all orders[END_REF]. In the present paper, we compute at the one-loop order the β-functions of the coupling constants involved in the renormalisable non-commutative version on R 2 θ of the Gross-Neveu model considered in [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF]. Recall that the (commutative) Gross-Neveu model exhibits asymptotic freedom together with mass generation phenomenon [START_REF] Gross | Asymptotic scale invariance in a massive Thirring model with u(n) symmetry[END_REF]. In section 2, we recall the principal features of the renormalisable non-commutative version of the Gross-Neveu model and collect the various ingredients relevant for the calculation. The section 3 is devoted to the one-loop computation of the relevant correlation functions. The analysis is carried out within the x-space formalism which appears to be well adapted for the relevant calculations. In section 4, we present the expressions for the β-functions, collect and discuss the main results and finally we conclude.

The Orientable non-commutative Gross-Neveu Model

In this section we recall the main features of the relevant action [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF][START_REF] Vignes-Tourneret | Renormalisation des théories de champs non commutatives[END_REF], fix the conventions and collect the useful material that will be needed throughout this paper. Let R 2 θ denotes the "Moyal plane" [START_REF] Gracia-Bondía | Algebras of distributions suitable for phase space quantum mechanics. I[END_REF][START_REF] Varilly | Algebras of distributions suitable for phasespace quantum mechanics. II. topologies on the moyal algebra[END_REF], which can be viewed in the following as a unital involutive associative algebra over C generated by the coordinate functions on

R 2 such that [x µ , x ν ] ⋆ = ıΘ µν with [a, b] ⋆ = a ⋆ b -b ⋆ a for any (a, b) ∈ R 2 θ .
Here, "⋆" denotes the associative Moyal-Groenwald product on R 2 θ induced by Θ, an invertible constant skewsymmetric matrix, that can be chosen to be Θ = θS, S = 0 -1 1 0 where the parameter θ has mass dimension -2. The Moyal product can be represented as

(a ⋆ b)(x) = 1 (2π) 2 d 2 yd 2 k a(x + 1 2 Θ.k)b(x + y)e ik.y (2.1)
where (Θ.k) µ = Θ µν k ν . We also define X.Θ -1 .Y = X µ Θ -1 µν Y ν . For more mathematical details, see e.g [START_REF] Gracia-Bondía | Algebras of distributions suitable for phase space quantum mechanics. I[END_REF][START_REF] Varilly | Algebras of distributions suitable for phasespace quantum mechanics. II. topologies on the moyal algebra[END_REF].

The action for the orientable non-commutative Gross-Neveu model on R 2 θ [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF] can be written as

S = d 2 x ψ(-i / ∂ + Ω / x + m + κγ 5 )ψ - 3 A=1 g A 4 (J A ⋆ J A )(x) (2.2a) J A = ψ ⋆ Γ A ψ, Γ 1 = 1, Γ 2 = γ µ , Γ 3 = γ 5 (2.2b)
where / a = a µ γ µ , x = 2Θ -1 .x and a summation over the Lorentz indices µ is understood in the interaction term involving J 2 , J 2 ⋆J 2 = µ ( ψ⋆γ µ ψ)⋆( ψ⋆γ µ ψ). The Clifford algebra for the 2D anti-Hermitian gamma matrices satisfy {γ µ , γ ν } = -2δ µν and γ 5 = ıγ 0 γ 1 . The field ψ denotes a 2N-component spinor field where N is the number of colors. The parameters Ω (to be discussed in a while), 0 ≤ Ω < 1, and the g A 's are dimensionless. In (2.2a), the term Ω ψ xψ can be viewed as the Fermionic counterpart of the harmonic oscillator term first introduced in [START_REF] Grosse | Renormalization of Φ 4 -theory on Non Commutative R 4 in the matrix base[END_REF]. Here, two comments are in order. First, notice that the minus sign affecting the four-Fermion interaction term in (2.2a) is a mere generalization of the interaction term in the commutative Gross Neveu model [START_REF] Gross | Asymptotic scale invariance in a massive Thirring model with u(n) symmetry[END_REF] for which asymptotic freedom is obtained. Next, the fact that the model defined in (2.2a) is called "orientable" comes from the present choice for the interactions. Recall that within the present noncommutative framework, six independent four-fermion interactions can in principle be constructed. The three interaction terms in which ψ and ψ alternate, namely a,b ψa ⋆

ψ b ⋆ ψa ⋆ ψ b , a,b ψa ⋆ ψ b ⋆ ψb ⋆ ψ a , a,b ψa ⋆ ψ a ⋆ ψb ⋆ ψ b (
the sum runs over color indices a, b), gives rise after suitable Fierz transformations to the interaction term in (2.2a) from which the diagrams occurring in the loopwise expansion can be given an orientation [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF]. This explains why (2.2a) is called "orientable". Three other terms with adjacent ψ and ψ could in principle be written, namely

a,b ψa ⋆ ψb ⋆ ψ a ⋆ ψ b , a,b ψa ⋆ ψb ⋆ ψ b ⋆ ψ a , a,b ψa ⋆ ψa ⋆ ψ b ⋆ ψ b .
Such interaction terms would result in a field theory with diagrams that cannot be orientated (in addition to the orientable one) [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF]. Although the proper interpretation of the non-orientable interactions is not clear within the present algebraic framework, the corresponding field theory is interesting in itself. Its detailed study has been undertaken in [START_REF] Lakhoua | [END_REF].

The action (2.2a) has been shown to be renormalisable to all orders in [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF]. Notice that the proof relies rather heavily on the orientability of the diagrams. In the massive case (m =0), the term κ ψγ 5 ψ in (2.2a), even not present at the classical level would be generated by higher order corrections (at the two-loop order) [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF]. The Feynman graphs can be computed from the propagator and interaction vertex derived from (2.2a). In the following, we will work within the x-space formalism [START_REF] Gurau | Renormalization of non-commutative φ 4 4 field theory in x space[END_REF] which proves convenient in the present analysis. The propagator in the x-space C(x, y) = (-i / ∂ + Ω / x + m) -1 (x, y) can be written as [19,[START_REF] Vignes-Tourneret | Renormalisation des théories de champs non commutatives[END_REF]

C(x, y) = ∞ 0 dtC(t; x, y); C(t; x, y) = C(t; y -x) exp(ıΩx ∧ y), (2.3a)
where x ∧ y = 2x.Θ -1 .y and

C(t; u) = Ω πθ e -m 2 t sinh(2 Ωt) e -e Ω 2 coth(2 e Ωt)u 2 ı Ω coth(2 Ωt)/ u + Ω / u + m e -2γ 5 e Ωt , (2.3b)
with Ω = 2Ω θ . Note that the following formula e -αγ 5 = cosh(α)1 -sinh(α)γ 5 (2.4) holds for any real parameter α. The propagator C is diagonal in its color indices. The interaction vertices can be read off from the RHS of

d 2 x ( ψ ⋆ Γ A ψ ⋆ ψ ⋆ Γ A ψ) = 1 π 2 θ 2 4 i=1 d 2 x i ψ(x 1 )Γ A ψ(x 2 ) ψ(x 3 )Γ A ψ(x 4 ) (2.5a) × δ(x 1 -x 2 + x 3 -x 4 )e -ı P i<j (-1) i+j+1 x i ∧x j .
We will denote the vertex kernel as

V (x 1 , x 2 , x 3 , x 4 ) =δ(x 1 -x 2 + x 3 -x 4 )e -ı P i<j (-1) i+j+1 x i ∧x j . (2.5b)
The graphical representation of the vertex is depicted on the figure 1. The non-locality of the interaction is conveniently represented by the rhombus appearing on fig. 1 whose vertices correspond to the x i 's occurring in (2.5). It is usefull to represent the alternate signs in the delta function of (2.5) by plus-and minus-signs, as depicted on the figure. By convention, a plus-sign (resp. minus-sign) corresponds to an incoming field ψ (resp. outgoing field ψ). This permits one to define an orientation on the diagrams obtained from the loop expansion. Notice that external lines are not drawn explicitly as we will deal essentially with amputated Green functions. In the computation of the relevant diagrams, recall that a factor

3 A=1 g A 4π 2 θ 2 D(x); D(x) = 4 i=1 d 2 x i V (x 1 , x 2 , x 3 , x 4 ) (2.6)
must appear in the amplitude for each involved (square) vertex. Besides, the contraction between ψ and ψ used in the computation of the amplitudes is defined by C(x, y) = D ψDψe -S free ψ(x) ψ(y). Furthermore, the following formulas among the γ matrices will be useful

γ µ γ ν = -δ µν 1 -ıǫ µν γ 5 γ µ γ ν γ ρ =(δ µν δ ρ σ -δ µρ δ ν σ + δ νρ δ µ σ )γ σ (2.7a) γ 5 γ µ = -ıǫ µ ν γ ν γ 5 γ µ γ ν = -ıǫ µν 1 -δ µν γ 5 (2.7b) Tr(γ µ γ ν ) = -2δ µν Tr(γ µ γ ν γ ρ γ σ ) =2(δ µν δ ρσ -δ µρ δ νσ + δ µσ δ νρ ) (2.7c) Tr(γ 5 γ µ ) =0 Tr(γ 5 γ µ γ ν ) = -2ıǫ µν (2.7d) Tr(γ 5 γ µ γ ν γ ρ γ σ ) =2ı(δ µν ǫ ρσ -δ µρ ǫ νσ + δ µσ ǫ νρ + δ νρ ǫ µσ -δ νσ ǫ µρ + δ ρσ ǫ µν ) (2.7e)
where 1 is the identity in the Clifford algebra while the trace of an odd number of γ's vanishes. In (2.7), the Levi-Civita symbol ǫ µν satisfies ǫ 01 = +1. We are now in position to evaluate the one-loop contributions to the two-and four-point functions.

3 Calculation of the Correlation Functions

The Two-point Function

Within the orientable non-commutative Gross Neveu model, the two-point Function receives contributions at the one-loop level from tadpole diagrams. There are obviously two types of tadpole diagrams depending whether the contraction between ψ and ψ is performed within one among the two operators J defined in (2.2b) that forms the whole vertex or the contraction takes place between the two operators J . Typical tadpoles are depicted on the figures 2 and 3. The diagram on fig. 2 corresponds to a tadpole of the first type mentionned above. The corresponding amputated amplitude is given by

A 1 = - 3 A=1 g A 4π 2 θ 2 D(y) ψ(y 1 )Γ A ψ(y 2 ) Tr(C(y 4 , y 3 )Γ A ) (3.1)
where the trace runs over spinor and color indices. It can be easily realized that the contribution from the other tadpole diagram, obtained through the substitution y 1 ↔ y 3 , y 2 ↔ y 4 is equal to the one given in (3.1) which reflects the invariance of the phase factor in the vertex kernel (2.5b) under cyclic permutation. Therefore, both diagrams are taken into account simply by multiplying the RHS of (3.1) by 2. In the same way, the figure 3 represents a typical tadpole of the second type mentioned above. The corresponding amplitude can be written as

A 2 = 3 A=1 g A 4π 2 θ 2 D(y) ψ(y 1 )Γ A C(y 2 , y 3 )Γ A ψ(y 4 ) (3.2)
while the amplitude stemming from the other tadpole diagram obtained from y 1 ↔ y 3 , y 2 ↔ y 4 is equal to the RHS of (3.2) so that again taking into account both diagrams amounts to multiply the RHS of (3.2) by a factor 2. By further making use of (2.3), (2.5) and the useful identity

d 2 x (2π) 2 e ıαx∧z = θ 2 4α 2 δ(z), (3.3)
it can be easily seen that two among the four integrals over the space variables y i 's can be explicitely performed thanks to delta functions so that (3.1) and (3.2) can be reexpressed as

A 1 = - 3 A=1 g A 4(1 + Ω) 2 d 2 y 1 d 2 y 2 δ(y 1 -y 2 ) ∞ 0 dt ψ(y 1 )Γ A ψ(y 2 ) Tr( C(t; y 1 -y 2 )Γ A ), (3.4) A 2 = 3 A=1 g A 4(1 -Ω) 2 d 2 y 1 d 2 y 4 δ(y 1 -y 4 ) ∞ 0 dt ψ(y 1 )Γ A C(t; y 1 -y 4 )Γ A ψ(y 4 ), (3.5)
which, upon integrating over the remaining delta functions, reduce respectively to

A 1 = - 3 A=1 g A 4(1 + Ω) 2 d 2 ydt ψ(y)Γ A ψ(y) Tr( C(t; 0)Γ A ), (3.6)
A 2 = 3 A=1 g A 4(1 -Ω) 2 d 2 ydt ψ(y)Γ A C(t; 0)Γ A ψ(y).
(3.7) Relation (3.6) combined with (2.3b) yields

A 1 = - 3 A=1 mΩg A 4πθ(1 + Ω) 2 d 2 ydt ψ(y)Γ A ψ(y) e -tm 2 coth(2 Ωt) Tr(Γ A ) -Tr(γ 5 Γ A ) (3.8)
Then, by further inspecting the remaining integrals over the Schwinger parameter t, it is easy to see that the second term in (3.8) gives rise to a finite contribution and can therefore be ignored in the present analysis while the first term is UV logarithmically divergent as it can be realized from

I 1 def = ∞ 0 dt e -m 2 t coth(2 Ωt) = lim ǫ→0 (- θ 4Ω log ǫ) + . . . (3.9)
where the ellipses denote finite contributions. Finally, since Tr(Γ A ) is non vanishing only when Γ A = 1, (3.8) yields

A 1 = - mΩg 1 2πθ(1 + Ω) 2 I 1 d 2 y ψ(y)ψ(y) + . . . (3.10)
where again the ellipses denote finite contributions. In a similar way, we find that the logarithmically diverging part of (3.7) can be written as

A 2 = mΩ 4πθ(1 -Ω) 2 I 1 (g 1 -2g 2 + g 3 ) d 2 y ψ(y)ψ(y) + . . . (3.11)
Notice that the finite parts of (3.10) and (3.11) both involve a term equal to δµ i dy ψ(y)γ 5 ψ(y), i = 1, 2 with δµ 1 =

Ωg 3 2πmθ(1+Ω) 2 and δµ 2 = -Ω(g 1 +2g 2 +g 3 ) 4πmθ(1-Ω) 2 .

The Four-point Function

The whole set of diagrams contributing to the 4-point Function can be conveniently determined by finding all the possible ways to draw two contractions among the spinor fields involved in the correlation function

A,B g A g B 16π 4 θ 4 D(y)D(z) 0| ψ(y 1 )Γ A ψ(y 2 ) ψ(y 3 )Γ A ψ(y 4 ) ψ(z 1 )Γ B ψ(z 2 ) ψ(z 3 )Γ B ψ(z 4 )|0 (3.12)
while forbidding the graphs involving vacuum-vacuum subdiagrams. By fixing the first contraction to occur between ψ(y 1 ) and ψ(z 2 ), one easily finds that the remaining contraction can be built from 5 different ways in (3.12) generating 5 different diagrams. These diagrams can be classified into two different types [START_REF] Grosse | Renormalization of Φ 4 -theory on Non Commutative R 4 in the matrix base[END_REF] among which only the planar regular diagrams [START_REF] Gurau | Renormalization of non-commutative φ 4 4 field theory in x space[END_REF], plagued with UV logarithmic divergences, are relevant for the computation of the β-functions [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF]. These diagrams are depicted on the figures 4 and 5. At this level, one comment is in order. Recall that the power-counting of a non-commutative field theory depends on the topology of its Feynman diagrams. These ones may be equivalently represented by ribbon diagrams. For example, the graph of the figure 6 corresponds also to the figure 7. From the ribbon representation, one can easily compute the genus g (through the Euler characteristic) and the number of broken faces B (defined as the number of faces to which external legs belong). For the non-commutative Φ 4 theory [START_REF] Grosse | Renormalization of Φ 4 -theory on Non Commutative R 4 in the matrix base[END_REF], the superficial degree of convergence is

ω =N -4 + 8g + 4(B -1) (3.13)
with N the number of external legs. From (3.13), one infers that the only divergent diagrams have g = 0 and B = 1 (the planar regular ones) which are therefore the diagrams relevant for the calculation of the β functions. In the Gross-Neveu case [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF], the powercounting is slightly more complicated but the same conclusion holds. The amputated amplitudes corresponding to the figures 4 and 5 are given respectively by The other type of one-loop diagrams, namely the broken-face diagrams, are UV finite [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF][START_REF] Grosse | Renormalization of Φ 4 -theory on Non Commutative R 4 in the matrix base[END_REF] and can therefore be ignored in the present analysis. For instance, one obtains the broken-face diagram depicted on the figure 6. Its corresponding amputated amplitude is given by

B 1 = - A,B g A g B 16π 4 θ 4 D(y)D(z) Tr C(z 2 , y 1 )Γ A C(y 2 , z 1 )Γ B ψ(y 3 )Γ A ψ(y 4 ) ψ(z 3 )Γ B ψ(z 4 ), (3.14) B 2 = A,B g A g B 16π 4 θ 4 D(y)D(z) ψ(y 3 )Γ A C(y 4 , z 3 )Γ B ψ(z 4 ) ψ(z 1 )Γ B C(z 2 , y 1 )Γ A ψ(y 2 ). (3.15)
B BF = A,B g A g B 16π 4 θ 4 D(y)D(z) ψ(z 1 )Γ B C(z 2 , y 1 )Γ A ψ(y 2 ) ψ(z 3 )Γ B C(z 4 , y 3 )Γ A ψ(y 4 ) (3.16)
and can be explicitely verified to be UV finite.

We now extract the potentially diverging part of (3.14) relevant for the computation of the β-functions. First, integrating over y 2 and z 2 and making use of (2.3a) and the relation (3.3), we find that (3.14) can be cast into the form with

B 1 = d 2 x 1 d 2 x 2 d 2 x 3 d 2 x 4 F 1 (x 1 , x 2 ; x 3 , x 4 ) (3.17a)
F 1 = - A,B g A g B 16π 2 θ 2 (1 Ω) 2 δ(Y 1 + Z 1 )e -ı(x 1 ∧x 2 +x 3 ∧x 4 ) ψ(x 1 )Γ A ψ(x 2 ) ψ(x 3 )Γ B ψ(x 4 ) × ∞ 0 dt 1 dt 2 d 2 u Tr C(t 1 ; u + Z 1 )Γ A C(t 2 ; Y 1 -u)Γ B e ı 2 (1-Ω)u∧(Y 1 -Z 1 ) (3.17b)
where we have defined Y 1 = x 1 -x 2 and Z 1 = x 3 -x 4 . Then, by further making use of the properties for the traces of products of γ matrices given in (2.7) combined with (2.3b) and (2.4), we find that (3.17b) can be rewritten as

F 1 = - A,B g A g B Ω 2 Ω 2 16π 4 θ 4 (1 + Ω) 2 δ(Y 1 + Z 1 )e -ı(x 1 ∧x 2 +x 3 ∧x 4 ) ψ(x 1 )Γ A ψ(x 2 ) ψ(x 3 )Γ B ψ(x 4 ) (3.18) × dt 1 dt 2 d 2 X e -m 2 (t 1 +t 2 ) (c 1 c 2 ) 2 e -e Ω 2 (c 1 +c 2 )X 2 -ı 2 (1-Ω)X∧(Y 1 -Z 1 ) Tr( / XΓ A / XΓ B ) + . . .
in which c i = coth(2 Ωt i ), i = 1, 2 and the ellipses denote finite contributions. Finally, from the explicit computation of the trace in (3.18), one easily infers that the only non vanishing contributions are those where Γ A and Γ B are equal to each other, leading to (X 2 = X µ X µ )

B 1 = - NΩ 2 Ω 2 8π 2 θ 2 (1 + Ω) 2 dxdX V(X) g 2 1 X 2 ψ ⋆ ψ ⋆ ψ ⋆ ψ -g 2 3 X 2 ψ ⋆ γ 5 ψ ⋆ ψ ⋆ γ 5 ψ -g 2 2 (2X µ X µ -X 2 ) ψ ⋆ γ µ ψ ⋆ ψ ⋆ γ µ ψ (x) + . . . , (3.19a)
V(X) = ∞ 0 dt 1 dt 2 e -m 2 (t 1 +t 2 ) (c 1 c 2 ) 2 e -e Ω 2 (c 1 +c 2 )X 2 . ( 3.19b)
A somehow similar analysis applied to (3.15) permits one to extract the potentially diverging part relevant for the calculation of the β-functions. It takes the form

B 2 = d 2 x 1 d 2 x 2 d 2 x 3 d 2 x 4 F 2 (x 1 , x 2 ; x 3 , x 4 ) (3.20a)
with

F 2 = A,B g A g B Ω 2 Ω 2 16π 4 θ 4 -Ω) 2 e -ı(x 4 ∧x 1 +x 2 ∧x 3 ) δ(Y 2 + Z 2 ) d 2 X V(X) (3.20b) × ψ(x 1 )Γ A / XΓ B ψ(x 2 )( ψ(x 3 )Γ B / XΓ A ψ(x 4 ) + . . . in which Y 2 = x 4 -x 1 , Z 2 = x 2 -x 3 , V(X)
is given by (3.19b) and the ellipses denote finite contributions. By repeated use of (2.7a) and (2.7b), the amplitude (3.20b) can be cast into the form

B 2 = Ω 2 Ω 2 16π 2 θ 2 (1 -Ω) 2 dxdX V(X) (3.21) × (g 2 1 + g 2 3 )(X µ ) 2 ψ ⋆ γ µ ψ ⋆ ψ ⋆ γ µ ψ + 2g 2 2 (X µ ) 2 ψ ⋆ γ ν ψ ⋆ ψ ⋆ γ ν ψ + 2(g 1 g 2 + g 2 g 3 )(X µ ) 2 ψ ⋆ ψ ⋆ ψ ⋆ ψ + ψ ⋆ γ 5 ψ ⋆ ψ ⋆ γ 5 ψ + 2g 1 g 3 (X µ ) 2 ψ ⋆ γ µ+1 ψ ⋆ ψ ⋆ γ µ+1 ψ (x) + . . .
where in the last line, µ ∈ Z 2 .

It appears that B 1 and B 2 are plagued with UV logarithmic divergences as proven in [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF]. This can be easily verified by performing the integral over X in (3.19a) and (3.21) and then studying the behaviour of the resulting expressions when the Schwinger parameters become close to zero. Then using

∞ 0 dt 1 dt 2 e -(t 1 +t 2 )m 2 coth(αt 1 ) coth(αt 2 ) coth(αt 1 ) + coth(αt 2 ) 2 = lim ǫ→0 + -log ǫ α 2 + O(1), (3.22)
the diverging parts of (3.19a) and (3.21) are thus respectively given by

B div 1 = lim ǫ→0 - N log ǫ 32π(1 + Ω) 2 dx g 2 1 ψ ⋆ ψ ⋆ ψ ⋆ ψ -g 2 3 ψ ⋆ γ 5 ψ ⋆ ψ ⋆ γ 5 ψ (x) (3.23) B div 2 = lim ǫ→0 - log ǫ 64π(1 -Ω) 2 dx (g 1 + g 3 ) 2 + 4g 2 2 ψ ⋆ γ µ ψ ⋆ ψ ⋆ γ µ ψ (3.24) + 4(g 1 g 2 + g 2 g 3 ) ψ ⋆ ψ ⋆ ψ ⋆ ψ + ψ ⋆ γ 5 ψ ⋆ ψ ⋆ γ 5 ψ (x).

The β Functions

After having obtained the divergent parts of the relevant graphs at one loop, we are now in position to write down the beta functions. First of all note that the two-point function only enter the beta functions through the wave-function renormalisation. At one-loop order, only the tadpoles (3.10) and (3.11) contribute to the two-point function. It turns out that their divergent parts are exactly local and then only renormalise the mass. Then the beta functions at one-loop order are only computed from the four-point graphs. Notice that, as a byproduct, Ω is not renormalised to the one-loop order. The interaction part of the effective action is

Γ int eff = g 1 4 ψ ⋆ ψ ⋆ ψ ⋆ ψ + g 2 4 ψ ⋆ γ µ ψ ⋆ ψ γ µ ψ + g 3 4 ψ ⋆ γ 5 ψ ⋆ ψ ⋆ γ 5 ψ + 4 2! - N log ǫ 32π(1 + Ω) 2 g 2 1 ψ ⋆ ψ ⋆ ψ ⋆ ψ -g 2 3 ψ ⋆ γ 5 ψ ⋆ ψ ⋆ γ 5 ψ - log ǫ 64π(1 -Ω) 2 (g 1 + g 3 ) 2 + 4g 2 2 ψ ⋆ γ µ ψ ⋆ ψ ⋆ γ µ ψ + 4(g 1 g 2 + g 2 g 3 ) ψ ⋆ ψ ⋆ ψ ⋆ ψ + ψ ⋆ γ 5 ψ ⋆ ψ ⋆ γ 5 ψ . (4.1)
On the second line of (4.1), the factor 1/2! comes from the expansion of the exponential of S int and the factor 4 is just the number of Wick contractions leading to the considered graphs. By definition,

β i = d d(-log ǫ) g i ({g jR }) (4.2a)
where g jR stands for the renormalized constants. This gives

β 1 = - g 2 1R 4π(1 + Ω) 2 - 1 2π(1 -Ω) 2 (g 1R g 2R + g 2R g 3R ) (4.2b) β 2 = - g 2 2R 2π(1 -Ω) 2 - 1 8π(1 -Ω) 2 (g 1R + g 3R ) 2
(4.2c)

β 3 = g 2 3R 4π(1 + Ω) 2 - 1 2π(1 -Ω) 2 (g 1R g 2R + g 2R g 3R ).
(4.2d)

In [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF], it has been proven that the action (2.2a) is renormalisable to all orders. From our one-loop computation, it seems that the stable manifold of parameters may well be reduced. Indeed from the expressions for the beta functions, it is clear that at one-loop, if g 1 = g 3 = 0 then the interaction g 2 ψ ⋆ γ µ ψ ⋆ ψ ⋆ γ µ ψ is stable. An other possibility is g 2 = 0 = g 1 + g 3 which is also stable. These two points are also true in the commutative case [START_REF] Gross | Asymptotic scale invariance in a massive Thirring model with u(n) symmetry[END_REF].

The table 1 summaries similarities and differences between the commutative and noncommutative Gross-Neveu and Thirring models. Remind that the commutative Gross-Neveu and Thirring models are given respectively by the following Lagrangians:

L GN = ψ( / p + m)ψ -λ ψψ ψψ (4.3a) L Th = ψ( / p + m)ψ -λ ψγ µ ψ ψγ µ ψ (4.3b)
In table 1, the trajectories in the graphs correspond to contractions of spinor and color indices. For example the first and third graphs, in the non-commutative case, correspond respectively to figures 4 and 5. On stable (which means that no new interaction vertex is created by radiative corrections). On R 2 θ the Gross-Neveu is not stable anymore. The third graph generates indeed a vertex of the form ψ ⋆ γ µ ψ ⋆ ψ ⋆ γ µ ψ. This was not the case on R 2 thanks to a compensation between two graphs. On R 2 θ one of them is now finite (it has two broken faces). Note that the same phenomenon occurs for the second graph which becomes finite on R 2 θ . A wellknown fact about the commutative Gross-Neveu and Thirring model is their asymptotic freedom. This is also a feature of the non-commutative model (2.2a). For g 1 = g 2 = g 3 (the bare values) or for g 1 = g 3 = 0 or for g 2 = 0 = g 1 + g 3 , we find that the model is asymptotically free in the UV region. A representative example is shown on the figure 8 when g 1 = g 2 = g 3 .

In [START_REF] Grosse | The beta-function in duality-covariant noncommutative φ 4 -theory[END_REF], the one-loop beta function for the non-commutative Φ 4 model has been com-puted. It was shown that for Ω < 1, the flow is bounded contrary to the commutative case where the theory is asymptotically free in the IR. Note that this is not the case for the non-commutative Gross-Neveu model. Moreover, at Ω = 1, the beta function of Φ 4 vanishes at any order (asymptotically in the UV region) [START_REF] Dissertori | Vanishing of Beta function of non commutative Φ 4 Theory to all orders[END_REF]. In the present case, the limit Ω → 1 -is singular as it can be seen from equation e.g. (3.24). This singularity hides the actual behaviour of the β-functions at Ω = 1 so that further investigations are needed to determine whether or not vanishing β can also be observed whenever Ω = 1, as it is the case for the Φ 4 model. Finally recall that for N = 1, the beta function of the equivalent Thirring and Gross-Neveu models on R 2 vanishes. This feature is also lost on R 2 θ .

Figure 1 :

 1 Figure 1: Graphical representation for the vertex in the x-space, obtained from (2.5). The plussign (resp.minus-sign) appearing in the rhombus corresponds to incoming (resp. outgoing) external line associated with ψ (resp. ψ).

Figure 2 :

 2 Figure 2: A typical tadpole obtained from the contraction between Fermion fields occurring within one operator J . The other tadpole of the same type is obtained through the substitution y 1 ↔ y 3 , y 2 ↔ y 4 . Both diagrams give rise to equal contributions to the amputated two-point Green function.

Figure 3 :

 3 Figure 3: A tadpole obtained from the contraction between Fermion fields taking place between two operators J , the other one being obtained through y 1 ↔ y 3 , y 2 ↔ y 4 .

Figure 4 :

 4 Figure 4: A UV logarithmically divergent planar diagram contributing to the β-functions, related to the amplitude (3.14).

Figure 5 :

 5 Figure 5: The other one-loop planar diagram contributing to the β-functions, related to the amplitude (3.15).

Figure 6 :

 6 Figure 6: The broken-face diagram corresponding to the amputated amplitude (3.16).

Figure 7 :

 7 Figure 7: Ribbon diagram corresponding to the amputated amplitude (3.16), which is equivalent to the broken-face diagram depicted on the figure 6

Table 1 :

 1 R 2 both the Gross-Neveu and the Thirring models are Commutative Comparition between the Gross Neveu and Thirring models and their non-commutative counterparts.

Figure 8 :

 8 Figure 8: UV Asymptotic Freedom for the model (2.2a) when g 1 = g 2 = g 3 . The vertical (resp. horizontal) axis represents g 1 (resp.log ǫ). Notice that the UV region corresponds to the leftmost values. The units are arbitrary.

Acknowledgement Discussions with J. Magnen, V. Rivasseau and R. Wulkenhaar at various stages of this work are gratefully acknowledged.

* Work supported by ANR grant NT05-3-43374 "GenoPhy".