Harald Grosse
email: harald.grosse@univie.ac.at

Fabien Vignes-Tourneret

Quantum field theory on the degenerate Moyal space

Keywords: MSC 2000 81R60, 81T10, 81T15, 81T18 Non-commutative geometry, quantum field theory, renormalization, Moyal space

We prove that the self-interacting scalar field on the four-dimensional degenerate Moyal plane is renormalisable to all orders when adding a suitable counterterm to the Lagrangean. Despite the apparent simplicity of the model, it raises several non trivial questions. Our result is a first step towards the definition of renormalisable quantum field theories on a non-commutative Minkowski space.

Motivations

For the last five years much has been done in order to determine renormalisable noncommutative quantum field theories [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][START_REF] Gurau | Renormalization of non-commutative Φ 4 4 field theory in x space[END_REF][START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF][START_REF] Grosse | Renormalization of the noncommutative φ 3 model through the kontsevich model[END_REF][START_REF] Grosse | A nontrivial solvable noncommutative φ 3 model in 4 dimensions[END_REF][START_REF] Grosse | Exact renormalization of a noncommutative φ 3 model in 6 dimensions[END_REF][START_REF] Langmann | Interacting fermions on noncommutative spaces: Exactly solvable quantum field theories in 2n+1 dimensions[END_REF][START_REF] Langmann | Exact solution of noncommutative field theory in background magnetic fields[END_REF][START_REF] Langmann | Exact solution of quantum field theory on noncommutative phase spaces[END_REF][START_REF] Gurȃu | A translationinvariant renormalizable non-commutative scalar model[END_REF]. Nevertheless all the known models are more or less of the type of a self-interacting scalar field on a Euclidean Moyal space. So it is quite important to extend the list of renormalisable non-commutative models.

The first solution to the uv/ir mixing problem consisted in adding an harmonic potential term to the quadratic part of the Lagrangean [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF]. We propose here to test such a method on a φ 4 -like model on a degenerate four-dimensional Moyal space. By degenerate we mean that the skew-symmetric matrix Θ responsible for the non-commutativity of the space will be degenerate which implies that some of the coordinates will commute.

To understand why we are not addressing a trivial question, let us remind the reader with a precise statement of the problem. We consider scalar quantum field theories on a (degenerate) Moyal space R D Θ . The algebra of functions on such a space is generated by the coordinates {x µ }, µ ∈ 0, D -1 satisfying the following commutation relation

[x µ , x ν ] =ıΘ µν 1 (1.1)
where Θ is a D × D skew-symmetric constant matrix. This algebra is realized as the linear space of Schwartz-class functions S(R D ) equipped with the Moyal-Weyl product: ∀f, g ∈ S(R D ),

(f Θ g)(x) = R D d D k (2π) D d D y f (x + 1 2 Θ • k)g(x + y)e ık•y . (1.2)
In the following we will consider degenerate Θ matrices which means that d out of the D coordinates will be commutative. In [START_REF] Wang | Renormalization of Orientable Non-Commutative Complex Φ 6 3 Model[END_REF] the non-commutative orientable (Φ Φ) 3 model on R 3 Θ has been considered:

S 6 [φ, φ] = d 3 x - 1 2 ∂ µ φ ∂ µ φ + Ω 2 2 (x µ φ) (x µ φ) + 1 2 m 2 φ φ + λ 1 2 (φ φ) 2 + λ 2 3 (φ φ) 3 (x). (1.3)
Being skew-symmetric Θ is necessarily degenerate in odd dimensions such as three. The main result of [START_REF] Wang | Renormalization of Orientable Non-Commutative Complex Φ 6 3 Model[END_REF] is that the complex orientable (Φ Φ) 3 model is renormalisable to all orders. What about its real counterpart? It is indeed a natural question because the graphs responsible for the uv/ir mixing cannot be generated by such a complex interaction.

In an appendix, Z. Wang and S. Wan [START_REF] Wang | Renormalization of Orientable Non-Commutative Complex Φ 6 3 Model[END_REF] exhibited a first problem concerning the real model. The Φ 6 3 model on R 3 Θ leads both to orientable and non-orientable graphs [START_REF] Gurau | Renormalization of non-commutative Φ 4 4 field theory in x space[END_REF][START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF]. The upper bound these two authors were able to prove (the power counting) was not sufficient to discard non-orientable graphs. If that bound is optimal, it would remain logarithmically divergent planar two-point graphs with two broken faces. These graphs are non-local and responsible for the now famous uv/ir mixing.

In section 3 we give a strong argument which tends to prove that the power counting given in [START_REF] Wang | Renormalization of Orientable Non-Commutative Complex Φ 6 3 Model[END_REF] is actually optimal with respect to the behaviour of the planar two-broken face graphs. We also compute the power counting of our model. The section 2 is devoted to definitions and the statement of our main result. In section 4 we perform the renormalisation and identify the missing counterterm. Our conclusions take place in section 5.

Definition of the model

Motivated by the remarks made in section 1, we want to address the question of the renormalisability of the real Φ 4 4 and Φ 6 3 models with degenerate Θ matrix. Whereas we will mainly focus on the Φ 4 model, our results apply to the Φ 6 case as well.

Main result

We consider a real scalar field theory on the Moyal space R 4 Θ with a degenerate matrix:

Θ =     0 θ -θ 0 0 0 0 0     .
(2. [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF] This means that two out of the four coordinates commute with all the other ones. We are going to prove the following Theorem 2.1 The quantum field theory defined by the action

S[φ] = R 4 d 2 xd 2 y 1 2 φ(x, y)(-∆ + Ω 2 θ 2 y 2 + m 2 )φ(x, y) + κ 2 θ 2 R 6 d 2 xd 2 yd 2 z φ(x, y)φ(x, z) + λ 4 R 4 d 2 xd 2 y φ 4 (x, y) (2.2)
where y represents the non-commutative coordinates, is renormalisable to all orders of perturbation.

To this aim, we will treat the new counterterm (the coupling constant of which is κ 2 ) as a perturbation. Such a counterterm linking two propagators will be called κ-insertion or simply insertion. Note that the four-valent vertex in (2.2) has the same form as on the non-degenerate Moyal space except that its oscillation only involves the non-commutative directions.

In the following, we use the momentum space representation. Our new counterterm is then given by:

κ 2 θ 2 R 6 d 2 xd 2 yd 2 z φ(x, y)φ(x, z) = κ 2 (2πθ) 2 R 2 d 2 p φ(p, 0) φ(-p, 0). ( 2.3)
The interaction term reads: .

From (2.4a) one reads that by convention all the momenta are considered incoming. Note once more that the oscillations only involve the non-commutative directions so that the interaction is local in the commutative directions. Let p, q (resp. p, q) denote (two-dimensional) momenta in the commutative (resp. noncommutative) directions. Let Ω := 2θ -1 Ω. The propagator corresponding to the quadratic part of (2.2) is given by:

Ĉ(p, p; q, q) = Ω πθ ∞ 0 dα sinh(2 Ωα) δ(p + q)e -α(p 2 +m 2 ) e -e Ω 4 coth( e Ωα)(p+q) 2 - e Ω 4 tanh( e Ωα)(p-q) 2 .
(2.5)

Feynman graphs

Let G be a Feynman graph of the model (2.2). There are two ways of considering the κ-insertions in it. Either we think of them as vertices. In this case, G is made of fourand two-valent vertices linked to each other by the edges of G. These ones correspond to the propgator (p 2 + Ω 2 y 2 ) -1 . Or we consider that the vertices of G are the only Moyal ones. In this case the insertions belong to generalised lines. These lines are composed of a series of edges related by κ-insertions. When not explicitly stated, we will always consider a generalised line as a single line, whereas it is composed of several edges. In this picture the edges linking two Moyal vertices are called simple lines. Note also that some "external" insertions may well appear, see figure 1 for an example.

κ 2 κ 2 κ 2

Figure 1: Example of graph with insertions

We now fix the notations we use throughout this article:

Definition 2.1 (Graphical notations). Let G be a Feynman graph corresponding to the model (2.2). We define:

• E(G) (resp. Ex(G)) to be the set of internal (resp. external) edges of G. E is the disjoint union of the sets E 0 of internal simple lines and E κ of internal generalised lines. The respective cardinalities of E, E 0 and E κ are denoted by e, e 0 and e κ .

• The number of external lines of G is N (G) =: card Ex.

• The number of external insertions is N κ .

• Let V (G) bet the set of vertices of G. We note v := card V .

• The number of connected components a of G is called k(G).

• Given a spanning tree T (G), let L(G) := E(G) \ T be the set of loop lines in G. Its cardinality is card L = e -v + k =: n(G). By analogy, we write n 0 (resp. n κ ) for the cardinality of L ∩ E 0 (resp. L ∩ E κ ).

a Please note that "connected components" will also be used to denote the quasi-local subgraphs of G in the framework of the multiscale analysis.

Multiscale analysis

We use the multiscale analysis techniques [START_REF] Rivasseau | From Perturbative to Constructive Renormalization[END_REF]. This means that we first slice the propagator in the following way: Ĉ(p, p; q, q) =:

∞ i=0 Ĉi , (2.6a) Ĉ0 (p, p; q, q) := Ω πθ ∞ 1 dα sinh(2 Ωα) δ(p + q)e -α(p 2 +m 2 ) e -e Ω 4 coth( e Ωα)(p+q) 2 - e Ω 4 tanh( e Ωα)(p-q) 2 , (2.6b)
Ĉi (p, q) := Ω πθ

M -2(i-1) M -2i dα sinh(2 Ωα) δ(p + q)e -α(p 2 +m 2 ) e -e Ω 4 coth( e Ωα)(p+q) 2 - e Ω 4 tanh( e Ωα)(p-q) 2 .
(2.6c)

where M > 1. Each propagator Ĉi bears both uv and ir cut-offs. A graph expressed in terms of these sliced propagators is then convergent. The divergences are recovered as one performs the sum over the so-called scale indices (the i in Ĉi ). Now we only study graphs with sliced propagators. Each line of the graph bears an index indicating the slice of the corresponding propagator. A map from the set of lines of a graph G to the natural numbers is called a scale attribution and written µ(G).

Certain subgraphs of G are of particular importance. These are the ones for which the smallest index of the internal lines of G is strictly higher than the biggest index of the external lines. These subgraphs are called connected components or quasi-local subgraphs: let G i be the subgraph of G composed of lines with indices greater or equal to i. G i is generally disconnected. Its connected components (the quasi-local subgraphs) are denoted by G i k . By construction they are necessarily disjoint or included into each other. This means that we can represent them by a tree, the nodes of which are connected components and the lines of which represent inclusion relations. This tree is called Gallavotti-Nicolò tree.

Topology and oscillations

Let G be a graph with v vertices and e internal lines. Interactions of quantum field theories on the Moyal space are only invariant under cyclic permutation of the incoming/outcoming fields. This restricted 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. I[END_REF]. This further justifies the ribbon representation. If g = 0 one has a planar graph, otherwise one has a non-planar graph. For example, the graph of figure 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. A graph with only one broken face is called regular.

Momentum space representation

The expression for the oscillation of a general graph that Filk obtained in [START_REF] Filk | Divergencies in a field theory on quantum space[END_REF] was based on the assumption that the propagator conserves momentum. When one adds an x 2 term to the action, the corresponding new propagator breaks translation invariance and so does not conserve momentum. In [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF], one of us computed the expression for the vertex oscillations of a general graph for any propagator. That was done in x-space. Here we redo it but in momentum space. Whereas the proof follow the same line we give it for completeness.

Definition 2.2 (Line variables).

Let G be a graph and fix a rooted spanning tree T . Let ∈ E(G) be a line which links a momentum p 1 to another one p 2 . When turning around the tree T counterclockwise, one meets p 1 first, say. One defines p := p 1 -p 2 and δp := p 1 + p 2 .

Definition 2.3 (Arches and crossings

). Let = (p 1 , p 2 ), = (p 1 , p 2
) and p k an external momentum. One says that arches over p k if, when turning around the tree counterclockwise, one meets successively p 1 , p k and p 2 . One writes then p ⊃ p k .

Considering the two lines and , if one meets successively p 1 , p 1 , p 2 and p 2 , one says that crosses by the left and writes .

Given a graph G, each of its vertices bears a delta function ensuring momentum conservation. If the propagators were translation-invariant, these delta functions would ensure the overall conservation of the external momenta: for any

v 0 ∈ V (G), v∈V (G) δ v = δ( k∈Ex(G) p k ) v∈V (G)\{v 0 } δ v .
In our case, we can extract an overall delta function which ensures the conservation of external momenta, upto δp variables. This delta function is written δ G . In the following we call rosette factor of a graph G, the complete vertex oscillations of G plus the delta function δ G .

Lemma 2.2 (Tree Reduction) Let a graph G with v(G) = v. The rosette factor after a complete tree reduction is

δ G 2v+2 i=1 p i + l∈T δp l exp(-ıϕ), (2.7) ϕ = 2v+2 i<j=1 p i ∧ p j + l∈T 1 2 p l ∧ δp l + i<T p i ∧ δp l + i>T δp l ∧ p i + T <T δp l ∧ δp l . (2.8)
Proof. We prove it by induction on the number of vertices. Let us assume that we have contracted k -1 tree lines, k < v. These lines form a partial tree T k . We now want to reduce the tree line between our rosette V k and a usual Moyal vertex V 1 :

V k =δ G 2k+2 i=1 p i + l∈T k δp l exp(-ıϕ k ), (2.9
)

ϕ k = 2k+2 i<j=1 p i ∧ p j + l∈T k 1 2 p l ∧ δp l + i<T k p i ∧ δp l + i>T k δp l ∧ p i + T k <T k δp l ∧ δp l , (2.10) V 1 =δ(q 1 + q 2 + q 3 + q 4 ) exp(-ı 4 i<j=1 q i ∧ q j ). (2.11)
Let l 0 the line joining a momentum

p i 0 , 1 i 0 2k + 2 to a momentum of V 1 .
By cyclicity of this one, one can assume that l 0 = (p i 0 , q 1 ). We need to prove:

V k V 1 =δ 4 i=1 q i δ 2k+2 i=1 i =i 0 p i + 4 j=2 q j + l∈T k δp l + δp l 0 exp(-ıϕ k+1 ),
(2.12)

ϕ k+1 = 2k+2 i<j=1 i,j =i 0 p i ∧ p j + i<i 0 p i - i>i 0 p i ∧ 4 j=2 q j + 4 i<j=2 q i ∧ q j + l∈T k 1 2 p l ∧ δp l + 1 2 p l 0 ∧ δp l 0 + i<T k i =i 0 p i ∧ δp l + i>T k i =i 0 δp l ∧ p i + T k+1 <T k+1 δp l ∧ δp l + i<i 0 p i - 4 j=2 q j - i>i 0 p i ∧ δp l 0 + l∈T k , l<i 0 δp l - l∈T k , l>i 0 δp l ∧ 4 j=2 q j (2.13)
with T k+1 = T k ∪ {l 0 }. This would reproduce (2.8).

The statement concerning the delta function in (2.12) is easily obtained. It only consists in the following distributional equality:

δ 2k+2 i=1 p i + l∈T k δp l δ 4 j=1 q j =δ 2k+2 i=1 i =i 0 p i + 4 j=2 q j + l∈T k δp l + δp l 0 δ 4 j=1 q j . (2.14)
Let us now rewrite the oscillations. First of all note that thanks to the delta function in V 1 , the oscillation of the Moyal vertex can be rewritten as exp -ı 4 i<j=2 q i ∧ q j . The complete oscillation in V k V 1 is:

ϕ k+1 = 2k+2 i<j=1 i,j =i 0 p i ∧ p j + l∈T k 1 2 p l ∧ δp l + i<T k p i ∧ δp l + i>T k δp l ∧ p i + T k <T k δp l ∧ δp l + 4 i<j=2 q i ∧ q j + δϕ, (2.15
)

δϕ = i<i 0 p i ∧ p i 0 + p i 0 ∧ j>i 0 p j = i<i 0 p i - j>i 0 p j ∧ (-q 1 + δp l 0 ) (2.16)
where we used p i 0 = -q 1 + δp l 0 . But -q 1 = q 2 + q 3 + q 4 so that

δϕ = i<i 0 p i - i>i 0 p i ∧ 4 j=2 q j + i<i 0 p i - j>i 0 p j ∧ δp l 0 .
(2.17)

We can now write

ϕ k+1 = 2k+2 i<j=1 i,j =i 0 p i ∧ p j + i<i 0 p i - i>i 0 p i ∧ 4 j=2 q j + 4 i<j=2 q i ∧ q j + l∈T k 1 2 p l ∧ δp l + i<T k i =i 0 p i ∧ δp l + i>T k i =i 0 δp l ∧ p i + T k <T k δp l ∧ δp l + i<i 0 p i - j>i 0 p j ∧ δp l 0 + p i 0 ∧ l∈T k , i 0 <l δp l + l∈T k , i 0 >l δp l ∧ p i 0 . (2.18)
Comparing (2.18) and (2.13) we see that it remains to prove

p i 0 ∧ l∈T k , i 0 <l δp l + l∈T k , i 0 >l δp l ∧ p i 0 = l∈T k , l<i 0 δp l - l∈T k , l>i 0 δp l ∧ δp l 0 + 4 j=2 q j + 1 2 p l 0 ∧ δp l 0 - 4 j=2 q j ∧ δp l 0 . (2.19)
We use p i 0 = -q 1 +δp l 0 = q 2 +q 3 +q 4 +δp l 0 and get the equality if 1 2 p l 0 ∧δp l 0 -4 j=2 q j ∧δp l 0 = 0 which is true thanks to q 1 = 1 2 (δp L 0 -p l 0 ). This proves the lemma.

Lemma 2.3 (Rosette Factor)

The rosette factor of a graph G with N (G) = N is given by

δ N k=1 p k + l∈T ∪L δp l exp(-ıϕ) with ϕ = ϕ E + ϕ m + ϕ ∩ + ϕ + ϕ J , (2.20
)

ϕ E = N i<j=1 p i ∧ p j , ϕ m = 1 2 ∈T ∪L p ∧ δp + (T ∪L)⊂L p ∧ δp + 1 2 L L (p ∧ δp + p ∧ δp ), ϕ ∩ = L⊃k p ∧ p k , ϕ = 1 2 L L p ∧ p , ϕ J = (T ∪L)<k δp ∧ p k + (T ∪L)>k p k ∧ δp + (T ∪L)<(T ∪L) δp ∧ δp + 1 2 L L δp ∧ δp .
Proof. Let us first fix an external momentum p k . From lemma 2.2 the linear term in p k is:

k-1 i=1 p i - 2n+2 i=k+1 p j ∧ p k + T <k δp - T >k δp ∧ p k . (2.21) Let a line = (p 1 , p 2 ) ∈ L such that < k. Its contribution to this linear term is (p 1 + p 2 ) ∧ p k = δp ∧ p k . If > k we get p k ∧ δp . Let now ⊃ k, we have (p 1 -p 2 ) ∧ p k = p ∧ p k .
Then the linear term in the external momenta is:

(T ∪L)>k p k ∧ δp + (T ∪L)<k δp ∧ p k + L⊃k p ∧ p k . (2.22)
Let us consider a line = (p 1 , p 2 ) ∈ L. The terms containing p 1 and p 2 are:

i< 1 p i ∧ p 1 + j> 1 p 1 ∧ p j + p 1 ∧ p 2 + i< 2 p i ∧ p 2 + j> 2 p 2 ∧ p j + 1 <T p 1 ∧ δp + 1 >T δp ∧ p 1 + 2 <T p 2 ∧ δp + 2 >T δp ∧ p 2 . (2.23) = i< 1 p i ∧ δp + j> 2 δp ∧ p j + 1 <i< 2 p ∧ p i + T < 1 δp ∧ δp + T > 2 δp ∧ δp + 1 <T < 2 p ∧ δp . (2.24) Let = (p 1 , p 2 ) ∈ L such that < . From (2.23) one reads (p 1 + p 2 ) ∧ δp = δp ∧ δp . If ⊂ , one has p ∧ (p 1 + p 2 ) = p ∧ δp . Finally if , one get p 1 ∧ δp + p ∧ p 2 = 1 2 (p + δp ) ∧ δp + p ∧ 1 2 (δp -p ).
We can now rewrite (2.24) as:

L<L δp ∧ δp + + L⊂L p ∧ δp + 1 2 L L (p ∧ δp + p ∧ δp ) + 1 2 L L p ∧ p + 1 2 L L δp ∧ δp + T <L δp ∧ δp + T >L δp ∧ δp + T ∪⊂L p ∧ δp + 1 2 ∈L p ∧ δp . (2.25)
Using lemma 2.2 together with equations (2.22) and (2.25), one proves the lemma.

3 Power counting

The case κ= 0

As explained in the introduction, we give here a strong argument for the need of a new counterterm. Note also that the bound we obtain seems to be optimal in the sense that exact computations exhibit the same degree of divergence. Remember also that one of our motivations is that it was noticed in [START_REF] Wang | Renormalization of Orientable Non-Commutative Complex Φ 6 3 Model[END_REF] that an harmonic oscillator term is not sufficient to make a scalar theory renormalisable on a degenerate Moyal space. In this article the authors studied a φ 6 model on R 3 Θ (see equation (1.3)) with the x-space representaion. They used the vertex delta functions to improve the usual commutative power counting ω = 1 2 (N -6 + 2v 4 ) where N is the number of external points and v 4 the number of four-valent vertices in the graph under consideration. For non-orientable graphs they got ω = 1 2 (N -2 + 2v 4 ). This upper bound exhibits a logarithmic divergence for the (planar) non-orientable two-point graphs. The authors suggested that a possible solution to this problem may come from the use of the vertex oscillations. We now explain why we think that the solution should be looked for elsewhere.

To take the oscillations into account, a very powerful technique consists in using the matrix basis. On a degenerate Moyal space part of the coordinates commute and we must use a mixed representation. In the commutative directions, we choose the usual x-(or p-)space representation whereas we prefer the matrix basis in the non-commutative directions. On R 3 Θ , let us choose [x 0 , x i ] = 0, i = 1, 2. Each field is expanded as

φ(x) = m,n∈N φ m,n (x 0 )f m,n (x 1 , x 2 ) (3.1)
where the functions f m,n form a basis for the Schwartz-class functions. Then we get a representation of the model which is partly commutative and local (in the x 0 -direction) and partly in the matrix basis. We can now apply the method developped in [START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF] to get an improved power counting namely

ω = 1 2 (N -6 + 8g + 4(b -1) + 2v 4 ) (3.2)
where g is the genus of the graph and b its number of broken faces. The conclusion is that, whereas we took the oscillations into account, there still remains potentially log. divergent two-point graphs with two broken faces. Hence the addition of an harmonic potential in the non-commutative directions does not imply renormalisability on a degenerate Moyal space.

Back to our model (2.2), it is clear that we can easily apply the same kind of mixed representation and get Lemma 3.1 Let G be a Feynman graph corresponding to the model (2.2) at κ = 0. Its degree of convergence obeys the following bound:

ω(G) N -4 + 4g + 2(b -1). (3.3)
Proof. The quadratic parts corresponding to the commutative and non-commutative directions commute with each other. Therefore the Schwinger representation of the corresponding propagators factorize. Then to prove the lemma it is enough to apply the standard bounds in the commutative directions and the method developped in [START_REF] Rivasseau | Renormalization of noncommutative φ 4 -theory by multi-scale analysis[END_REF] in the non-commutative directions.

Recall that on the fully non-commutative R 4 Θ , the power counting is ω N -4 + 8g + 4(b -1). The N -4 part is the usual power counting of the commutative φ 4 model whereas the rest has a purely non-commutative origin. On the degenerate fourdimensional Moyal space, only half of the directions are non-commutative so that we gain only half of 8g + 4(b -1) and get (3.3). The consequence is that the planar two-point graphs with two broken faces (N = b = 2, g = 0) diverge logarithmically. They must be renormalised.

Remark. One could ask if the addition of an harmonic potential also in the commutative directions could solve the problem. Unfortunately one can easily convince oneself (for example in p-space) that such an infrared modification of a commutative and local model doesn't change anything to the power counting.

The case κ = 0

In this subsection we are going to compare the power countings of graphs with and without insertions. To this aim, we recall briefly how to get the bound ω(G 0 ) N (G 0 ) -4 on the degree of convergence of a graph G 0 without κ-insertions in momentum space. We first have to perform the so-called momentum routing. This is the optimal way of using the delta functions attached to each Moyal vertex. For this we must choose a spanning rooted tree in G 0 . Then we associate to this tree a set of branch b delta functions which allow to solve v(G 0 ) -1 momenta both in the commutative and non-commutative directions [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF].

In a slice i, a propagator is bounded by Ĉi (p, p; p, q) K e -M -2i p 2 e -M 2i (p+q) 2 -M -2i (p-q) 2 .

(3.4)

For each line l of G 0 , the integration over p l + q l gives a factor M -2i l . We recover the power counting factor of 1 p 2 in four dimensions. Then if l is a loop line, the integrations over p l and p l -q l deliver together M 4i l . If l is a tree line, these integrations are made with a delta function and bring O(1). We get the bound

|A G 0 | K n l∈E(G) M -2i l l∈L(G) M 4i l K n i,k M -ω(G i k ) , ω = N -4. (3.5)
We now turn to the computation of an upper bound on the amplitude of a graph G with κ-insertions. The graph G is equipped with a scale attribution µ(G) which assigns an integer to each edge in E(G). To get the power counting of such a graph we need to perform the momentum routing and pick up a tree T G . Note that each generalised line is considered as one single line so that the tree T G is composed of both simple and generalised lines.

Let us focus on a generalised line between two Moyal vertices. It is made of n( ) insertions and so n + 1 edges k , k ∈ 1, n + 1 . The corresponding analytical expression is:

A :=κ 2n Ĉi 1 (p, p; p, 0) n k=2
Ĉi k (p, 0; p, 0) Ĉi n+1 (p, 0; p, q).

(3.6)

Thanks to the bound (3.4), we have

A κ 2n K n+1 e -( P n+1 k=1 M -2i k )p 2 e -M 2i 1 p 2 -M 2i n+1 q 2 . (3.7)
Let i m := min k∈ 1,n+1 i k , i 1 := max{i 1 , i n+1 } and i 2 := min{i 1 , i n+1 }. Then exchanging p and q if necessary, A is bounded by A (p; p, q) κ 2n K n+1 e -(n+1)M -2im p 2 e -M 2i 1 p 2 -M 2i 2 q 2 . (3.8)

In the following, we consider that a generalised line is a line of scale i := i m ( ). This will be important in the choice of an optimised tree.

Remark (No Moyal vertex). The bound (3.8) does ot depend on the scales i 2 , . . . , i n . This reflects the fact that the corresponding subgraphs are logarithmically divergent. These graphs are made of propagators linked together by κ-insertions but do not contain any Moyal vertex. Let G be such a subgraph with n propagators. The corresponding analytical expression is:

A G = R 2 d 2 p φ(p, 0) φ(-p, 0) n k=2 Ĉi k (p, 0; p, 0). (3.9)
To renormalise such a graph, we expand the propagators around p = 0:

Ĉi (p, 0; p, 0) = Ĉi (0, 0; 0, 0) +

1 0 ds p • ∇ Ĉi (sp, 0; sp, 0) (3.10)
The graph G being only log. divergent, only the zeroth order term is divergent. It contributes to the renormalisation of κ 2 .

In the following of this article, we will not anymore make any reference to these particular subgraphs keeping nevertheless in mind that they are log. divergent and can be renormalised by a change of κ 2 .

Before we state the power counting lemma, we need to give a few definitions: Definition 3.1 (Bridge). Let G be a graph and l ∈ E(G). The line l is a bridge if deleting l increases the number of connected components of G. Definition 3.2 (Admissible generalised line). Let G µ be a graph with scale attribution µ and l ∈ E κ (G). There exists a unique k(l)

∈ N such that l ∈ E κ (G i l k ). The line l is said admissible if it is a bridge in G i l k .
The admissibility of a generalised line depends both on the location of the line in the graph and on the scale attribution as shown in figure 3. Let G be a graph. It is said tree-like if for all l ∈ E κ (G), l is a bridge (independently of the scale attribution). By convention, a graph without insertion is tree-like.

κ 2 i j k k (a) Admissible line κ 2 i k j j (b) Non-admissible line
A tree-like graph is then a tree of generalised lines the nodes of which are graphs with only simple lines and Moyal vertices.

Lemma 3.2 (Power counting)

Let G be a Feynman graph of the model (2.2). Let µ(G) be a scale attribution. Then we have:

• if G is not tree-like, the amplitude A µ G is bounded by |A µ G | K v(G) i,k M -ω(G i k ) , ω N + 4n κ (3.11a)
for the connected components which are not tree-like.

• If G is tree-like, for all connected component G i k , its degree of convergence obeys:

-if G i k is non planar ω N (3.11b) -if g(G i k ) = 0, b(G i k ) 2 ω N -2, (3.11c) -if G i k is planar regular and E κ (G i k ) = ∅ ω N -2 (3.11d) -if G i k is planar regular and E κ (G i k ) = ∅ ω N -4 + 2N κ if N κ < N N -4 + 2(N κ -1) if N κ = N . (3.11e)
This lemma proves that the only graphs (and subgraphs) which need to be renormalised are tree-like. Moreover they are either planar and regular: in that case, if they have four external points, they don't have any insertion. If they are two-point graphs, they have zero, one or two external insertions and possibly internal ones also. Or they are planar with two broken faces.

Proof (1/2): truncated diagrams

In this section we prove all the bounds of lemma 3.2 except the improvement related to the number of external insertions N κ . This is postponed to subsection 3.4.

Let G be a Feynman graph. Its amplitude is given by:

A µ G (℘ 1 , . . . , ℘ N ) = R 8e l∈E(G) d 4 p l 1 d 4 p l 2 Ĉi l (p l 1 , p l 2 ) v∈V (G)
δ v e -ı φ (3.12)

To get the bound (3.11a), we start by bounding the oscillations by one. Then we perform a momentum routing. To this aim we choose an optimised spanning rooted tree c and trade the vertex delta functions for an equivalent set of branch delta functions. This allows to solve one momentum per tree line: at each vertex (except the root) the delta function solves the unique momentum hooked to this vertex which is on the path in the tree between the vertex and the root. For a simple line, let us call the combination p + q a short variable. The momentum routing replaces the solved tree momenta by p L +p Ex +δp where p L (resp. p Ex , δp) is a linear combination of loop momenta of simple line (resp. external momenta, short variables or momenta of generalised lines). As a result we have to integrate

• in the commutative directions, over one momentum per loop line (thanks to the conservation of momentum along the lines)

• and in the non-commutative directions, over one momentum per tree line and two momenta per loop line.

Moreover there remains a global delta function which ensures the exact conservation of the external momenta in the commutative directions and an approximate conservation in the non-commutative directions (see [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF] for details about an equivalent position routing).

c Here "optimised" means that the tree is a subtree in each connected component.

Let l ∈ T ∩ E 0 . In the commutative directions, its corresponding momentum has been solved thanks to the delta function δ b(l) . In the non-commutative directions, this delta function allows to solve p l -q l (see equation (3.4)). We still have to integrate over p l + q l which delivers a factor bounded by M -2i l .

Let ∈ T ∩E κ . We use the delta function corresponding to the branch b( ) to integrate over p and either p or q. The result is bounded by: R 6

d 2 p d 2 p d 2 q A (p; p, q)δ b( ) KM -2(i 2 -im) M -2im . (3.13)
Let l ∈ L ∩ E 0 . We integrate over p, p and q. Thanks to the bound (3.4), the result is bounded by M -2i l M 4i l .

Let ∈ L ∩ E κ . We have to integrate A over p, p and q. The result is bounded by: R 6

d 2 p d 2 p d 2 q A (p; p, q) KM -2(i 1 -i 2 ) M -4(i 2 -im) M -2im . (3.14)
As a consequence, we have

|A µ G | K v(G) l∈E M -2i l l∈L∩E 0 M 4i l l∈Eκ M -2(i 2 -im) l∈L∩Eκ M -2(i 1 -i 2 ) . (3.15)
The last two products are clearly related to the external insertions and contribute to the improvement of the power counting by the factor 2N κ . In subsection 3.4, we will explain how to improve these factors to reproduce completely the bounds of lemma 3.2. Until there we just bound these products by one. Then we have

|A µ G | K v(G) l∈E M -2i l l∈L∩E 0 M 4i l K v(G) i,k M -ω(G i k )
, with ω = N -4 + 4n κ . (3.16) This proves that if a subgraph G i k is divergent, n κ (G i k ) = 0 and all its generalised lines are in the tree. This means that for all l ∈ E κ (G i k ) and for any choice of an optimised tree in G i k , l is in the tree. This implies that l is a bridge in a G i l k . In other words if a connected component is divergent, all its generalised lines are admissible. Nevertheless this doesn't prove yet the bound (3.11a) because an admissible line l is only a bridge in a G i l k but not necessarily in the full graph G, as shown in the figure 3a. We have to improve our bound.

Let us consider a connected component G i k and an admissible generalised line l ∈ E κ (G i k ) which is not a bridge in G i k . This line belongs to the tree T (G i k ) so that we use the delta function δ b(l) to integrate over p l or q l . In the propagator Ĉi l , p l (say) is replaced by p L + p Ex + δp. Usually, we bound Ĉi l by M -2i l but if p L = 0, we can use it to integrate over a loop momentum to get

M -2i l M -2i = M 2i M -4i . The gain is M -4i and makes G i k convergent. Thus for any connected component G i k and any line l ∈ E κ (G i k ) with i l i, G i k divergent implies that l is a bridge in G i k .
All the generalised lines have to be bridges in G = G 0 and so G has to be tree-like to be divergent. If not, the improvement factor M 4 plus the bound (3.16) give the equation (3.11a).

We have proven that if a graph is divergent, all its connected component are tree-like which is equivalent to G itself being tree-like. So let us consider such a graph and prove the bounds (3.11b -3.11d).

We start with the bound (3.16) with n κ = 0 (because G is tree-like). We are going to improve it thanks to the oscillations of A G . In [START_REF] Gurau | Renormalization of non-commutative Φ 4 4 field theory in x space[END_REF], one of us contributed to proving, in x-space, that non-planar graphs are convergent. We just use here the same method but in momentum space: if two line l and l cross each other, there exists an oscillation of the type p l ∧ p l (see lemma 2.3). Note that the presence of generalised lines does not alter that result because they are tree lines and only loop lines may cross each other. We use such an oscillation to integrate over p l say. The gain with respect to the bound (3.16) is M -2(i l +i l ) M -4 min{i l ,i l } . It leads to the bound (3.11b).

Let us now consider a planar connected component but with at least two broken faces. Then there exists an oscillation of the type p l ∧ p e where p e is an external momentum of the subgraph. Once more this oscillation allows to integrate over p e . The gain is M -2i l and gives the bound (3.11c).

Let G be a planar regular tree-like graph. The bound (3.16) for n κ = 0 gives already ω N -4. To improve it and get (3.11d), we must prove the following: given a tree-like graph G and a generalised line l, l is necessarily on the path in the tree between two external points. The graph being tree-like, l is a bridge. G can consequently be depicted as in figure 4 where the two blobs represent any tree-like graphs. To each of these graphs, an odd number of external points are hooked. In particular each of them contains at least one external point. Thanks to the momentum routing, the momentum q (say) equals minus the sum of the momenta entering the right blob. Then we can use the propagator of the line l to integrate over one of these external momenta, in the non-commutative directions. From the bound (3.8), the result is bounded by M -2i l . This makes the improvement from (3.16) to (3.11d).

There now remains to prove how appear the factors N κ in (3.11e). This is the subject of the next section.

Proof (2/2): external insertions

The basic mechanism which implies an improvement of the power counting thanks to the external insertions is the following.

Let us consider a graph G with N κ external insertions and the lowest scale of which is j. It has so N κ external legs which correspond to Ĉi (p, p; p, 0). Thanks to the equation (3.8), the integration over p gives a factor M -2i M -2j . If N κ < N , the graph has N -N κ external momenta and N κ external insertions. We use the global delta function of G to solve one external momentum in function of the others. Then each external insertion delivers a factor M -2i . If N κ = N , the global delta function solves the momentum of one of these insertions. This gives the degree of convergence in (3.11e).

Nevertheless we still have to prove that the procedure which lead to the bound (3.16) reproduces this improvement in all the connected components. Of course this is related to the last two products in equation (3.15). Let us consider a connected component G i k with N κ external insertions. These insertions correspond to generalised lines at lower scales. If these lines are loop lines, the bound (3.14) gives a factor M -2(i 1 -i 2 ) M -4(i 2 -im) which is precisely the gain of two powers per external insertion. But for the generalised lines which are in the tree, the argument is subtler. Each such line bears two momenta, one of them being solved by the momentum routing. This is the "highest" of the two in the tree. The momentum routing solves then at most one such momentum among the momenta corresponding to the external insertions. If N κ = N , this corresponds to the fact that the global delta function δ G i k solves this momentum and that we cannot get a better improvement than 2(N -1). If N κ < N , it could very well happen that an external insertion is on the unique path between G i k and the root of T (G). In this case, we would not get a gain of 2N κ . But we can use the propagator of that external insertion to integrate over an external momentum (which is not another external insertion). The result is M -2i = M -2(i-im) M -2im . This reproduces the bound (3.11e) and is also compatible with (3.11d). This ends the proof of lemma 3.2.

Renormalisation

Thanks to the power counting lemma 3.2 we know which types of graphs are divergent. In this section, we prove that the divergent parts of these graphs reproduce the five terms of the Lagrangean (2.2).

The four-point function

The only divergent four-point graphs are planar regular and contain no κ-insertion (neither internal nor external). The "Moyality" of the corresponding Feynman amplitudes has already been proven in [START_REF] Gurau | Renormalization of non-commutative Φ 4 4 field theory in x space[END_REF] in the case of a non-degenerate Moyal space using the xspace representation. The only differences here are that we use the momentum space representation and that our non-commutative space is half commutative. Nevertheless, our proof would be so close to the one in [START_REF] Gurau | Renormalization of non-commutative Φ 4 4 field theory in x space[END_REF] that we do not feel the need to reproduce it here.

The two-point function 4.2.1 The planar regular case

Let G be a planar regular two-point graph. We distinguish mainly three different cases:

1. E κ (G) = ∅ and N κ (G) = 0, 2. E κ (G) = ∅ and N κ (G) = 0 and 3. N κ (G) = 0.
No insertion at all As in the case of the four-point function, there is no major difference between our degenerate model and the case treated in [START_REF] Gurau | Renormalization of non-commutative Φ 4 4 field theory in x space[END_REF]. The two-point graphs contribute to the flow of the mass, wave-functions d and oscillator frequency Ω.

With internal insertions Let G be a connected planar regular tree-like graph with E κ = ∅ and N κ = 0. Let i be the lowest of its scales. We now prove that its divergent part renormalises κ 2 .

For all line l ∈ E(G), let p l L be a linear combination of loop momenta in the commutative directions. Let p l L (resp. δp l ) be a linear combination of loop momenta (resp. short variables and momenta of generalised lines) in the non-commutative directions. Finally let δp be the sum of all the short variables of G plus the sum of all the momenta of the generalised lines. The amplitude of G integrated over external fields and after a momentum routing is given by:

A µ G =: R 6 d 2 p d 2 p d 2 q φ(p, p) φ(-p, q)A(p, p, q) (4.1) = R 4+2(v-1)+6n d 2 p d 2 p φ(p, p) φ(-p, -p -δp) e ı φ l∈L(G) d 2 p l d 2 p l d 2 q l Ĉi l (p l , p l ; p l , q l ) l∈T ∩E 0 d 2 q l Ĉi l (p + p l L , q l + p + p l L + δp l ; p + p l L , q l ) l∈T ∩Eκ d 2 q l Ĉi l (p, p + δp l ; p, q l ).
The oscillation is of the type φ = p ∧ δp. Note that after the momentum routing, there is no loop momenta in the generalised lines. This is because all such lines are bridges. This allows to bound the external momentum p by |p| M -i . We then perform a Taylor expansion of the external fields around p = 0. In the commutative directions, as usual, we expand the tree propagators around p = 0. Thanks to the power counting lemma, we know that such an amplitude is only log. divergent. As a consequence, only the zeroth order term of these expansions is divergent.

A µ G = R 2 d 2 p φ(p, 0) φ(-p, 0) R 2+2(v-1)+6n d 2 p e ı φ l∈L(G) d 2 p l d 2 p l d 2 q l Ĉi l (p l , p l ; p l , q l ) l∈T ∩E 0 d 2 q l Ĉi l (p l L , q l + p + p l L + δp l ; +p l L , q l ) l∈T ∩Eκ d 2 q l Ĉi l (0, p + δp l ; 0, q l ) + convergent contributions. (4.2)
The planar regular tree-like graphs with internal insertions contribute to the renormalisation of κ 2 .

With external insertions Let G be a connected planar regular tree-like graph with N κ = 0. Let i be the lowest of its scales. We now prove that its divergent part renormalises κ 2 . Let us first consider that G has only one external insertion. In this case, its amplitude is

A µ G = R 6 d 2 p d 2 p d 2 q φ(p, 0) Ĉ(p, 0; p, p)A(p, p, q) φ(-p, q) (4.3)
where A is defined by the equation (4.1). In contrast with the previous case, even if E κ = ∅, the external momentum p (and consequently q) is bounded by M -i . This is due to the external insertion. Then we can safely expand the external field around q = 0, and A and the external propagator around p = 0. This leads to

A µ G = R 2 d 2 p φ(p, 0) φ(-p, 0) R 4
d 2 p d 2 q Ĉ(0, 0; 0, p)A(0, p, q) + convergent terms. (4.4)

If N κ (G) = 2, the amplitude is

A µ G = R 6
d 2 p d 2 p d 2 q φ(p, 0) Ĉ(p, 0; p, p)A(p, p, q) Ĉ(-p, q; -p, 0) φ(-p, 0). (4.5)

We expand A and the external propagators around p = 0. The divergent part of this expansion renormalises κ 2 too. Ĉi l (p l , p l ; p l , q l ) l∈T d 2 q l Ĉi l (p + p l L , q l + p + p l L + δp l ; p + p l L , q l ) where the oscillation is of the type φ = p ∧ (δp + p L ) (see lemma 2.3). The oscillation between the external momentum p and some loop momenta p L allows to prove that p is actually bounded by M -i (see [START_REF] Vignes-Tourneret | Renormalization of the orientable non-commutative Gross-Neveu model[END_REF] for the details). We can then perform the same types of expansions as before to get a renormalisation of κ 2 . If G contains internal or external insertions, it should now be clear that its amplitude contributes also to the flow of κ 2 . Indeed, whatever the reason, if one can control the size of the (non-commutative) external momentum, one can expand the external fields around 0 as was done in the preceding cases.

Conclusion

Motivated by the work of Wan and Wang [START_REF] Wang | Renormalization of Orientable Non-Commutative Complex Φ 6 3 Model[END_REF] and by the possibility of defining a renormalisable model on non-commutative Minkowski space, we addressed here the problem of the renormalisability of a self-interacting quantum field on a degenerate Moyal space. On such a space, part of the coordinates are commutative. Contrary to the non-commutative Φ 4 4 model [START_REF] Grosse | Renormalisation of φ 4 -theory on noncommutative R 4 in the matrix base[END_REF], the harmonic oscillator term is not sufficient to make the model renormalisable. We proved that the model contains indeed additionnal divergencies of the type (Tr φ) 2 . By adding such a counterterm, we defined a renormalisable model (see (2.2)).

The interest for such a study is twofold. On one side, the appearence of such counterterms (of the type "product of traces") is quite natural on non-commutative spaces and has already been noticed in different works, see [START_REF] Gayral | Quantum field theory on projective modules[END_REF] for an example. It is often mentioned that the studied models are renormalisable at one-loop order provided one adds such a term. Our work is the first study to all orders of such a quantum field theory.

One the other side, a (non-commutative) model on a degenerate space could open a way towards non-commutative Minkowski space. There already exists lots of works about quantum field theory on non-commutative Minkowski space concerning mainly causality, unitarity, definition of the appropriate Feynman rules etc. But no renormalisable model is known. On commutative spaces, using a regularization à la Feynman, one can prove the perturbative renormalisability of a Minkowskian model from the corresponding Euclidean version. This results in the perturbative definition of the time-ordered Green functions. Could we do an equivalent on a non-commutative space? It turns out that with a proper (i.e. preserving unitarity) definition of a time-ordered product on noncommutative Minkowski space [START_REF] Denk | Time ordered perturbation theory for non-local interactions; applications to NCQFT[END_REF][START_REF] Bahns | On the unitarity problem in space/time noncommutative theories[END_REF], one is lead to the conclusion that such a product is not equivalent to the use of a Feynman propagator. Therefore the usual techniques employed on commutative spaces may not apply. However one could address a simpler question namely the perturbative renormalisability of a field theory on a non-commutative Minkowski space with a commuting time e . This is where our present proposition could enter into the game.

R 4 d 2 4 j=1d 4 4 i<j=1p

 42444 xd 2 y φ 4 (x, y) = 1 π 4 | det Θ| p j (2π) 4 φ(p j ) δ i ∧ p j and p i ∧ p j :

  Let us consider the example of figure 2. Propagators in a ribbon graph are made of double lines. Let us call f the number of faces (loops made of single lines) of a ribbon graph. The graph of figure 2b has v = 3, e = 3, f = 2. Each ribbon graph can be drawn on a manifold of genus g. The genus is computed from the Euler characteristic

Figure 2 :

 2 Figure 2: A graph with two broken faces

Figure 3 :

 3 Figure 3: Scale attribution and admissibility, i > j > k

Figure 4 :

 4 Figure 4: A tree-like graph

4. 2 . 2

 22 The planar irregular caseLet G be a planar irregular (b(G) = 2) two-point graph. Let us first treat the case of a graph without any insertion. Its amplitude would beA µ G = R 4+2(v-1)+6nd 2 p d 2 p φ(p, p) φ(-p, -p -δp) e ı φ l∈L(G) d 2 p l d 2 p l d 2 q l

b Given a spanning rooted tree T (G) and a line l ∈ T , the branch b(l) is the set of vertices v such that l is on the unique path in the tree between v and the root of T .

d The coefficients in front of the Laplacean in the commutative and non-commutative directions renormalise separately.


Luc Sibille

Didier Marot

Philippe Poullain

Franck Lominé

Phenomenological interpretation of internal erosion in granular soils from a discrete fluid-solid numerical model

INTRODUCTION

Four main types of internal erosion can be identified in water-retaining structures made of soil (Fell & Fry 2007[START_REF] Bonelli | Erosion in Geomechanics Applied to Dams and Levees[END_REF]): the concentrated leak erosion (erosion of a crack or a hole), the backward erosion (progressing from the surface of the downstream part of the structure), the suffusion (taking place in the soil bulk), and contact erosion occurring at an interface between fine and coarse soil layers respectively. In all these processes of internal erosion, soil particles are firstly detached by the action of the water flow and secondly transported in the existing hole (or crack), or the pore network of the soil. A third step of self filtration may occur in the case of suffusion (or even for contact erosion [START_REF] Béguin | Etude multi-échelle de l'érosion de contact au sein des ouvrages hydrauliques en terre[END_REF])) when transported particles are redeposited within the interstitial space of the soil (possibly leading to a partial clogging of the latter [START_REF] Reddi | Comparison of internal and surface erosion using flow pump test on a sand-kaolinite mixture[END_REF][START_REF] Sail | Suffusion tests on cohesionless granular matter[END_REF].

Hydraulic shear stress is classically considered in the cases of contact erosion and concentrated leak erosion (as hole erosion) as the hydraulic loading parameter fixing the erosion regime [START_REF] Béguin | Pore-scale flow measurements at the interface between a sandy layer and a model porous medium: Application to statistical modeling of contact erosion[END_REF][START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF][START_REF] Bonelli | The scaling law in the hole erosion test with a constant pressure drop[END_REF][START_REF] Haghighi | Improvement of hole erosion test and results on ref-erence soils[END_REF]. Concerning the erosion by suffusion the development of suffusion has been described in the literature from different hydraulic parameters such as the global [START_REF] Skempton | Experiments on piping in sandy gravels[END_REF] or local [START_REF] Moffat | A large permeameter for study of internal stability in cohesionless soils[END_REF]) hydraulic gradient, the hydraulic shear stress [START_REF] Reddi | Comparison of internal and surface erosion using flow pump test on a sand-kaolinite mixture[END_REF], or the pore fluid velocity [START_REF] Perzlmaier | Hydraulic criteria for internal erosion in cohesionless soil[END_REF].

Nevertheless, a look at the work in the field of sediment transport in river beds and overland flows shows that several authors use the stream power (representing an estimation of the power dissipated by viscous shear in the fluid near the fluid/soil interface) to characterize solid particle detachment and/or transport [START_REF] Bagnold | The flow of cohesionless grains in fluids[END_REF][START_REF] Bagnold | An empirical correlation of bedload transport rates in flumes and natural rivers[END_REF][START_REF] Low | Effect of sediment density on bed-load transport[END_REF][START_REF] Govers | Evaluation of transporting capacity formulae for overland flow[END_REF][START_REF] Ferro | Evaluating overland flow sediment transport capacity[END_REF]). More recently, attempts have also been made in the field of soil internal erosion to characterize the hydraulic loading driving the erosion development by the flow power (the power consumed by the water to seep through the soil) for suffusion [START_REF] Marot | Study of scale effect in an internal erosion mechanism: centrifuge model and energy analysis[END_REF][START_REF] Rogoz | Seepage power as a factor determining the piping and silting up the loose water-bearing soils[END_REF] and hole erosion (Regazzoni andMarot 2013, Marot et al. 2011). A possible advantage of such an approach could be constituted by the rather simple estimation of the flow power through a soil sample or a soil structure (a dam or a dike for instance) from the water head drop and the seepage flow rate [START_REF] Rochim | Characterization of suffusion susceptibility of granular soils[END_REF].

In this paper, we take advantage of recent developments in coupling numerical discrete element model (DEM) with computational fluid dynamics to perform direct numerical simulations with a hole erosion model in order to investigate the possibility to characterize internal soil erosion from the flow power. In the numerical model, the DEM, used to describe the motion of each particle of a granular soil, is coupled with the lattice Boltzmann method to solve the dynamics of the interstitial fluid flow [START_REF] Lominé | Modelling of fluid-solid interaction in granular media with coupled lb/de methods: application to piping erosion[END_REF]). With such a numerical model, assumptions about fluidsolid interactions are considerably reduced and very few mechanical parameters are introduced. That is why we are speaking about direct simulations (or numerical experiments). The numerical model is presented in a first section of the paper. Then interpretations of the particle detachment step is discussed in a second section, from the numerical data, in terms of both hydraulic shear stress and flow power. Then we attempt in a last section to apply the expression identified from the numerical experiments to a case of erosion by suffusion characterized in laboratory on glass bead mixtures. As in suffusion transported particles may be filtrated, contrary to the hole erosion, the description is completed to take into account this latter step characterizing the suffusion.

DETACHMENT OF SOLID PARTICLES / INITIATION OF EROSION

A micro hydro-mechanical model of hole erosion

Numerical experiments of hole erosion have been performed with a two-dimensional coupled DEM-LBM numerical model (cf. Figure 1). The solid granular phase is described with the discrete element method in which each solid particle is represented with a circular disk. Contact forces between two particles are computed according to simple contact laws and can be decomposed into a normal contact force, F n , normal to the contact plane, and a shear contact force, F s , tangent to the contact plane. The relation between F n and the normal component of the relative displacement of particles at contact point is purely elastic. The relation between F s and the relative shear (or tangential) displacement is also elastic up to a sliding limit represented by a Coulomb criterion defined by a contact friction angle φ c . Beyond this limit the contact is sliding in the tangential direction and energy is dissipated by dry friction. Contact strength can be increased by considering additional normal and shear cohesion terms, C n and C s , as represented in Figure 2. Fluid dynamics is solved between solid particles with the lattice Boltzmann method (LBM) with a relatively fine discretization of the fluid domain (the ratio of the mean solid particle diameter to the fluid lattice space is about 17). The coupling between the LBM and the DEM is based on the balance of momentum between the fluid and the solid phase computed around each boundary of solid particles [START_REF] Lominé | Modelling of fluid-solid interaction in granular media with coupled lb/de methods: application to piping erosion[END_REF]. Not any assumption is introduced in the definition of coupling (except the no slip condition at the boundary between the two phases). Consequently, the mechanical parameters introduced in the coupled method are very limited: the fluid viscosity, and the parameters defining the contact interaction law between solid particles (stiffnesses, cohesions and friction angle). Hence, for instance, hydraulic conductivity of drag forces constitute output, and not input, of the numerical method.

The bi-dimensional model is displayed in Figure 1 [START_REF] Lominé | Modelling of fluid-solid interaction in granular media with coupled lb/de methods: application to piping erosion[END_REF]. It consists of a tube filled, in its central part, of solid grains, circular and cohesive, characterized by a mean diameter d mean . The grains along the tube axis have been manually removed in order to create an initial hole with an aperture d h ≈ 4 d mean . The contact generated in the initial configuration between two grains, or between a grain and a wall of the tube, are cohesive. Normal and shear cohesions, C n and C s (defined in Figure 2 are chosen such that C n = C s = C. The cohesion is brittle, hence if cohesion breaks, contacts become purely frictional (see Figure 2). Finally, any new contact created during the simulation is itself purely frictional.

The tube walls are impermeable with respect to the fluid and constitute a rigid boundary for the solid particles. A pressure drop ∆P is imposed between the inlet and the outlet of the tube to generate the fluid flow through the hole, with a Reynold number from 0.50 to 40.0. The effect of gravity is discarded. These simulations reproducing about 200 s of physical time, and including 800 solid grains and more that 300,000 fluid nodes, were performed for 10 different values of pressure drop, and 7 values of contact cohesion.

Under the action of the fluid flow, solid grains are detached from the boundary of the hole and carried with the fluid until the outlet section of the tube (Figure 3). The grains crossing the latter section are con-

Particle detachment and hydraulic shear stress

The classical description of the hole erosion (or concentrated leak erosion) consists in representing the hydraulic loading by the hydraulic shear stress τ s developing on the solid/fluid interface of the hole. Then the erosion rate per unit surface ˙ depends on the excess shear stress with respect to the critical stress below which particle detachment does not occur [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF]:

˙ = k d (τ s -τ c ) if τ s > τ c (1)
where k d is an erosion coefficient fixing the kinetics of particle detachment for shear stresses exceeding the threshold. It is worth noting that such a description of particle detachment driven by the hydraulic shear stress has been also considered in the case of the erosion by suffusion by [START_REF] Bonelli | Micromechanical modeling of internal erosion[END_REF], who represented the pore of soils as small tubes. Then, they assumed solid particles are detached from the peripheral surface of the tubes as in a hole erosion problem. Results obtained from the numerical model and presented in Figure 5 are in agreement with this interpretation. A parametric study with respect to the contact cohesion C showed the critical stress for particle detachment increases with C, apparently linearly; whereas the erosion coefficient k d seems independent of C (Sibille et al. 2015a). However, these conclusions should be confirmed by further studies involving a wider range of variation of C values. The use of flow power (or stream power) to characterize the transport of sediments in river beds has been suggested since the 50s by [START_REF] Bagnold | The flow of cohesionless grains in fluids[END_REF][START_REF] Bagnold | An empirical correlation of bedload transport rates in flumes and natural rivers[END_REF], but also more recently for overland flow sediment transport [START_REF] Low | Effect of sediment density on bed-load transport[END_REF][START_REF] Govers | Evaluation of transporting capacity formulae for overland flow[END_REF][START_REF] Ferro | Evaluating overland flow sediment transport capacity[END_REF]. In both cases (river and overland flow), the stream power is defined as the product between the hydraulic shear stress on the bed soil and the mean flow velocity, giving an estimation of the power dissipated by viscous shear in the fluid near the soil bed interface. [START_REF] Bagnold | An empirical correlation of bedload transport rates in flumes and natural rivers[END_REF] and [START_REF] Govers | Evaluation of transporting capacity formulae for overland flow[END_REF] showed that the sediment transport rate can be expressed as a power function of the stream power. Recently, [START_REF] Marot | Study of scale effect in an internal erosion mechanism: centrifuge model and energy analysis[END_REF] and [START_REF] Regazzoni | A comparative analysis of interface erosion tests[END_REF] suggested that internal erosion in soils could be driven by the power dissipated by the fluid seepage throughout the porous solid phase. Therefore, we investigate in this section the possibility to interpret the internal erosion of soils from energetic terms representative of the water flow through the soil.

Nevertheless, it is generally accepted that internal erosion in soils comprises three steps: detachment of solid particles from the initial granular skeleton, their transport within the interstitial space and eventually a possible deposition (or filtration) of the transported particles in soil pores. In the framework of sediment transport in rivers and overland flow, there is no filtration step and the interpretation in terms of stream power introduced above aims to characterize only the detachment and transport of particles. Consequently, the analysis carried out here after in the framework of internal erosion, is voluntarily limited in a first time to the step of detachment for the sake of separation of problems. The discrete numerical model has indeed been designed to avoid deposition or filtration, because even if some particles may temporarily settle, detachment and transport are far more predominant with an irreversible increase of the hole width. -

S i +So p v. n dS = S G [p v. n G -( σv . n G ) . v] dS + + V σ v ij ∂v i ∂x j dV (2)
with: n the outer unit normal vector to inlet and outlet surfaces, and n G the outer unit normal vector to the boundary surface of solid grains.

The left-hand side of equation ( 2) represents the power supplied to the fluid to flow within the granular assembly. This term is called the flow power P F . The integral over surface S G , that will be denoted I G , represents the power transferred from the fluid to the solid particles, and the remaining volume integral is the power dissipated by viscous stresses in the fluid bulk P V . Then:

P F = I G + P V .
One could be interested in describing the action of the interstitial fluid flow on the solid phase of the soil from the term I G . From a study on laboratory tests of suffusion and hole erosion it seems difficult to estimate the proportion of the flow power transferred to the solid phase to deform it and detach some particles, because this transfer represents a quasi negligible fraction in suffusion case and is highly variable in hole erosion cases (Sibille et al. 2015a).

Consequently we paid attention to the flow power P F which can be easily estimated, in an engineering context, from a flux and a pressure drop. In addition, the flow power is mainly dissipated by viscosity (term P V ) in the neighborhood of the solid interfaces (constituted by the solid grains included in the intact granular skeleton and the detached grains currently transported by the fluid), at least for the seepage configurations we studied (see Figure 7). Hence the flow power P F can be seen as an indicator of the fluid-solid interactions.

Description of particle detachment from the flow power

In the framework of the lattice Boltzmann method, we defined the fluid boundary nodes (FB nodes) as the computational nodes of the fluid domain constituting its boundary on the fluid/solid interface [START_REF] Lominé | Modelling of fluid-solid interaction in granular media with coupled lb/de methods: application to piping erosion[END_REF]. Hence, the power dissipated by viscosity at FB nodes, P F B V , represents the power dissipation occurring the most closely with the solid particles. The plot of the erosion rate in terms of P F B V in Figure 8a shows the erosion rate varies as a power function of the excess power dissipated by viscosity at the direct vicinity of solid particles:

ṁ ∝ (P F B V -P F B * V ) β V
(3)

where P F B * V is a threshold power below which particle detachment does not occur.

P F B V cannot be directly quantified from a practical point of view. Then an identical interpretation can be carried out by considering the total flow power P F as displayed in Figure 8b. Although the scaling of the erosion rate with P F is slightly less satisfying than with P F B V , the erosion rate can also be expressed satisfyingly by:

ṁ ∝ (P F -P * F ) β F (4)
where P * F constitutes a threshold value for particle detachment. In this last case the hydraulic loading represented by the flow power P F , computed as the product between the fluid flux and the pressure drop, is quite easy to determine.

It is worth noting that imposed pressure drops, and thus fluid velocities, have been kept voluntarily small in these numerical simulations leading to a flow regime characterized by a Reynolds number between 0.50 and 40.0. Such a flow regime may be quite far from what is expected in hole erosion tests, usually characterized by a Reynolds number from 2,000 up to 20,000 [START_REF] Bonelli | Erosion in Geomechanics Applied to Dams and Levees[END_REF]. This point constitutes a restriction of the numerical model. Nevertheless such an interpretation, of particle detachment for a hole erosion, based on flow power, has been satisfyingly applied on experimental data from a laboratory hole erosion test (Sibille et al. 2015a).

DESCRIPTION OF INTERNAL EROSION INCLUDING A FILTRATION STEP

We assume here, that the description of particle detachment based on the flow power, presented in the previous section holds in the case of erosion by suffusion of granular materials [START_REF] Bonelli | Micromechanical modeling of internal erosion[END_REF]. In other words it is assumed the mechanisms of detachment of particles involved in the suffusion are similar to the one represented with the numerical model of hole erosion. Nevertheless, suffusion may be accompanied by an additional step of filtration, possibly dominating the step of particle detachment, and limiting the erosion of particles after a given time.

Suffusion tests were carried out on a glass bead mixture composed of small beads (diameter ranging from 0.1 to 0.2 mm) and bigger beads (diameter ranging from 1.2 to 3.4 mm). This is therefore a gapgraded material with fines (small beads) constituting 40 % of the total mass. Tests were performed with a oedo-permeameter sketched in Figure 9 and described by [START_REF] Sail | Suffusion tests on cohesionless granular matter[END_REF]. The sample made of beads is built inside the rigid cell of the oedo-permeameter, above a wire mesh with 1.25 mm pore opening size avoiding the migration of the biggest particles. The sample is then compressed under a vertical stress of 25 kPa and slowly saturated with an upward flow.

(-) (hours) N1 250 1-2-3-3.2 1-1-1-1 N2 450 1-2-3-4-4.8 1-1-1-1-1 N3 450 1-2-3-3.7-4.7 1-1-1-1-1 N4 600 1-2-3-4.9-5.5 1-1-1-1-1 N5 250 0.1-0.2-0.4-0.8-1-2.2-3 1-1-1-1-1-1-1 N6 250 0.1-0.2-0.4-0.8-1 1-1-5-4-1
Figure 9: Sketch of the oedo-permeameter used to perform suffusion tests on glass bead assemblies

Once the saturation phase is finished the bead samples are subjected to a downward water flow under a controlled hydraulic gradient i increasing by steps. An erosion test consists in successive steps of about 60 min (except for test N6) with a constant hydraulic gradient i. Tests are presented in the Table 1 and discussed in detail in (Sibille et al. 2015b). They differ from the size of the increments of hydraulic gradient between each step, and the length of the tested samples.

Beads carried with the effluent at the outlet of the cell are collected using a sampling device [START_REF] Sail | Suffusion tests on cohesionless granular matter[END_REF]. Time series of the cumulative mass of collected beads are shown in Figure 10. Steps of the hydraulic gradient are visible, since at the beginning of each step, mass increases rapidly and tends to stabilize at the end of the step. The decreasing of the erosion rate during a hydraulic step is related to the limitation of detachable fine beads (this limitation being itself dependent on the hydraulic gradient [START_REF] Bonelli | Micromechanical modeling of internal erosion[END_REF], and the possible development of filtration (re-deposition of transported particles within the sample itself). Consequently, we assume that only the erosion rate at the initiation of each step of hydraulic gradient is characteristic of particle detachment. Erosion rate per unit volume, ṁ, is plotted according to the flow power per unit volume, PF , for the six tests (N1 to N6) in Figure 11. The highest erosion rates displayed in the latter plot correspond to the initiation of increments of hydraulic gradient and are assumed to be representative of the detachment of solid particles, independently of the quantity of potentially erodible 4), with the power law :

ṁupper limit = α ref PF β F (5)
with α ref = 0.003 and β F = 0.8, which can be seen as intrinsic parameters to the tested material and representing its erodability. The threshold flow power P * F is very low for this kind of material and as been neglected here for the sake of simplicity.

The decrease in erosion rate during each step of hydraulic gradient emphasizes the necessity to take into account the history of hydraulic loading (the amplitude but also the duration of each increment). Then

ṁ = α ref 1 ∆ Ē PF t * + 1 PF β F (6)
where t * is a characteristic time relative to the material (taken here equal to 130 s by calibration from test N2). For the tests discussed here, the hydraulic conductivity of the material was only slightly affected by the erosion. Consequently, by assuming the hydraulic conductivity constant and equal to k 0 , the one initially measured on the glass bead assembly, the erosion rate per unit volume, ṁ, can be computed at any time. The cumulative eroded mass is directly deduced from the latter, integrated over the time, and compared with the experimental data in Figure 10. The description proposed is able to capture the main features of the erosion process. However, the prediction of eroded mass is not totally in agreement with the experimental data.

Although tests N2 and N3 have been performed with the same parameters (see Table 1) and thus stand for the repeatability of tests, the cumulative mass of particles collected is about 25 % larger for test N2 than N3. Obviously, the model is not able to describe such a difference since the input parameters are identical (or at least almost identical with an initial hydraulic conductivity k 0 = 1.24 • 10 -4 m/s for N2, whereas k 0 = 1.50 • 10 -4 m/s for test N3). Consequently, due to the discrepancies between the experimental data, it is difficult to conclude here about the ability of prediction of the model. Nevertheless it is worth noting this model stays rather simple, involving four parameters (α ref , β F , t * , k 0 ) identifiable from a single suffusion test performed by increasing the hydraulic gradient by steps.

CONCLUSION

An alternative description of hole erosion was suggested on the basis of direct numerical simulations performed with a coupled DEM-LBM model. This interpretation states the detachment rate of solid particles can be represented from the seepage power according to a power law. This is not in opposition with a more common interpretation based on the hydraulic shear stress, since the seepage power mainly depends on the work of the hydraulic shear stress at the vicinity of fluid-solid interfaces. However, a rough determination of the seepage power can be easily assessed from a drop of water head and an estimation of the seepage flow rate. This can present an advantage for the application of this interpretation in engineering problems and in the field. This description has been then extended to erosion by suffusion from the interpretation of suffusion tests performed on glass bead mixtures. At the initiation of suffusion, detachment and transport of solid particles can also be described from the seepage power according to a power law. Additional terms are needed for larger time periods to take into account the finite quantity of potentially erodible particles (with respect to the hydraulic loading applied) and the filtration step. This results in a simple and rough phenomenological model involving four parameters. This model is able to capture the main features observed experimentally.

Figure 1 :

 1 Figure 1: Sketch of the numerical model of hole erosion; hatched regions correspond to the locations of the cohesive granular assembly such that an initial horizontal hole is formed

Figure 3 :

 3 Figure 3: Snapshots of a simulation of erosion (at t = 0.7, 13.3 and 55.0 s from top to bottom), the colour scale is related to the fluid velocity whereas the translation velocity of solid particles is represented with arrows

  Simulated erosion rate for a cohesion C/d = 0.506 N/m in terms of the hydraulic shear stress estimated on hole boundaries, (a) linear scale and (b) logarithmic scale (in the latter the excess shear stress τ s -τ c is used instead of the shear stress and points with shear stress values lower than the threshold τ c have been removed)

  2.3.1 Fluid power of a seepage through a granular medium Let us consider an incompressible viscous fluid. We define at a position x of its volume V , the fluid density ρ, its static pressure p and velocity v, and the tensor of viscous stresses σv . The energy conservation equation of a volume V of fluid seeping through solid grains in a tube (Figure6) delimited by inlet and outlet surfaces, S i and S o respectively, writes (without taking into account the gravity)(Sibille et al. 2015a):

Figure 6 :

 6 Figure 6: Water seepage through a granular assembly enclosed in a tube: the boundaries of the fluid domain comprise outer boundaries (inlet and outlet sections S i and S o , and lateral tube surface S l ) and inner boundaries S G formed by the solid grains

  (a) Erosion rate as a function of the power dissipated by viscosity at fluid boundary nodes P F B V (for a cohesion C/d = 1.27 N/m), the dashed line represents an approximation with a power law; (b) similar interpretation but in terms of the flow power P F .

Figure 10 :

 10 Figure 10: Cumulative eroded mass produced from suffusion tests on glass bead mixtures, symbols represent experimental data whereas continuous lines represent a model prediction

Figure 11 :

 11 Figure 11: Identification of the maximum erosion rate per unit volume as a function of the flow power per unit volume

Table 1 :

 1 Properties of suffusion tests on glass bead assemblies.

	Test	Initial sample length	Successive increments of global hydraulic gradient	Duration of hydraulic loading increments
		(mm)


Anis Tlili

Arnaud Voituriez

Angela Marinetti

Pierre Thuéry

Thibault Cantat

Synergistic effects in ambiphilic phosphino-borane catalysts for the hydroboration of CO_2

Featuring a carbon atom at the +IV oxidation state, CO2 is a stable molecule and its conversion to either fuels or chemicals necessitates the use of a mild reductant and an efficient catalyst, to promote the cleavage of C-O bonds and the formation of novel C-H and/or C-C bonds. 1 In this respect, hydroboranes (R2BH) have a mild reduction potential (ca. -0.5 V vs NHE) well-suited for the reduction of CO2 to the methanol level. [START_REF]CRC Handbook of Chemistry and Physics[END_REF] In contrast to borohydrides, the reduction of CO2 with hydroboranes requires a catalyst. [START_REF] Chong | [END_REF] In fact, the catalytic hydroboration of CO2 has first been reported in 2010 by Guan and coworkers who showed that nickel(II) pincer complexes were able to catalyze the reduction of CO2 to methoxyboranes in the presence of 9-BBN, catBH and pinBH. 4 Shortly after, Sabo-Etienne and Bontemps described efficient ruthenium catalysts for this transformation and a variety of inorganic and organometallic catalysts have been reported since then. 5 Notably, the polarized B-H bond in hydroboranes can also be activated by simple organic compounds: the organocatalyzed hydroboration of CO2 has been unveiled in 2013 by Fontaine and coworkers with P/B Frustrated Lewis Pairs (FLPs), and by our group with guanidines and amidines bases as well as using an isolated FLP-CO2 adduct. 6 This collection of catalysts has been completed with several bases among which phosphines proved to be potent catalysts for this reduction process. 7 In this context, we have also shown recently that proazaphosphatranes enable the first catalytic methylation of amines involving CO2 reduction with hydroboranes. 7a Thus, since the hydroboration of CO2 is a reaction of fundamental importance, it becomes crucial now to better apprehend the role of the organocatalysts in these reduction processes. In particular, the possible positive influence of combining a Lewis base with a Lewis acid in the same catalytic system is still to be established. To address this question, we report herein the synthesis of simple phosphine, borane and phosphino-borane compounds based on the same ferrocene backbone (Chart 1). Comparing the catalytic performances of the separated Lewis acids and bases and their combination as intermolecular or intramolecular systems demonstrates the benefits of the ambiphilic systems, for the first time.

Chart 1. Candidates for the catalytic hydroboration of CO2

The ferrocene backbone is a well-suited platform to assemble Lewis acids and bases by substituting the cyclopentadienyl ligands on iron. This approach has been exemplified over the last years by the groups of Bourissou, Aldridge and Erker to access novel FLPs and ambiphilic ligands. 8 Additionally, rotation around the ferrocene axis provides some flexibility that can adapt the catalyst to the geometric constraints associated with the catalytic pathway. In this respect, ferrocene based FLPs differ from the system used by Fontaine et al. for the hydroboration of CO2, where a benzene ring maintains a rigid structure between the P and B centres. 6a In this study, diphenylphosphinoferrocene, 1, 9a was isolated by quenching the lithiated ferrocene 9b with chlorodiphenylphosphine (Scheme 1). Similarly, the corresponding ferrocenyl-pinacolborane 2-Bpin was obtained in a moderate 25% yield by reacting bis(pinacolato)diboron with lithiated ferrocene. Crystals of 2-Bpin were obtained by recrystallization from dichloromethane (see ESI). When dimesitylboron fluoride was used as an electrophile, 2-BMes2 was obtained in a significantly better yield of 71% (Scheme 1).

Scheme 1. Synthesis of 1, 2-Bpin and 2-BMes2

The phosphino-borane derivatives 3-Bpin and 3-BMes2 were prepared using an established procedure based on a sequential halogen/lithium exchange starting from 1,1'-dibromoferrocene (Scheme 2). 8a Thus, 1-bromo-1'-(diphenylphosphino)ferrocene 4 was obtained in 87% isolated yield. Subsequently, a second bromine/lithium exchange was conducted and the lithiated ferrocene was quenched with bis(pinacolato)diboron, yielding the desired product 3-Bpin in a low 18 % yield. Other pinacol-boron electrophiles including 2-isopropoxy-4,4,5,5-tetramethyl-1,3,2dioxaborolane, were also tested, without success. Alternative routes were explored to improve the yields of 3-Bpin. Unfortunately, inverting the reaction sequence by quenching 1-bromo-1'-lithioferrocene with bis(pinacolato)diboron first proved unproductive. Also, quenching 4 with trimethyl borate and subsequent reaction with pinacol failed.

As reported by et al., bis(pinacolato)diboron with fluoride yielded 3-BMes2 in 82% yield. 8a It is noteworthy that the P/B derivatives 3-BMes2 and 3-Bpin display 31 P NMR chemical shifts similar to those of (diphenylphosphino)ferrocene (1: δ = -16.2; 3-Bpin: δ = -17.4; 3-BMes2: δ = -19.3 ppm). This suggests the absence of any interaction between the two heteroatoms, as noted previously by Bourissou et al. Moreover, no interaction with CO2 has been observed either by 31 P or 1 H NMR, when a solution of 3-BMes2 is exposed to an atmosphere of CO2. [START_REF] Similarly | no adduct forms when 3-BMes2 is reacted with a stoichiometric amount of 9-BBN[END_REF] Having in hand a homogeneous series of P-, B-and P/B-based catalysts, their reactivity has been investigated in the catalytic hydroboration of CO2. At room temperature, phosphine 1 (1 mol%) exhibits a negligible catalytic activity in the conversion of CO2 and 9borabicyclo[3.3.1]nonane (9-BBN) to methoxyborane MeO-BBN (Table 1, Entry 1). Indeed a Turn Over Number (TON) lower than 5 was observed after 28 h at 25 °C. Increasing the reaction temperature to 70 °C did not improve the catalytic performances of 1 (Entry 11 in Table 1). These results are in line with the results of Stephan et al. on the phosphine catalysed hydroboration of CO2. 7b Similarly, the borane 2-BMes2 is inefficient both at 25 and 70 °C (Entries 2 and 12 in Table 1). In stark contrast, a mixture of 1 and 2-BMes2 provides a competent catalytic system, enabling the reduction of CO2 to MeOBBN with a TON of 41 after 28 h at 25 °C (TOF = 1.5 h -1 ). This result demonstrates for the first time the synergistic and positive influence of combining a Lewis base and a Lewis acid in the reduction of CO2 with hydroboranes. Replacing 2-BMes2 with the less acidic 2-Bpin borane somewhat deactivates the catalytic system and a TON of 36 (TOF = 1.3 h -1 ) is measured under the same conditions. To further establish the benefit of an ambiphilic structure, the catalytic potential of the intramolecular Lewis pairs 3-BMes2 and 3-Bpin has been determined. 3-BMes2 presents a better activity than the mixture of 1 and 2-BMes2 and a TON of 100 was reached after 28 hours at 25 °C (TOF = 3.6 h -1 ).

It is worth noting here, that the reaction course follows a profile similar to the one observed previously for guanidines and N/B FLPs. 6f Indeed, monitoring the product distribution over time, using 1 H and 13 C NMR spectroscopy, reveals that CO2 is first reduced to the boryl formate HCOO-BBN which is present in only a very low concentration and does not accumulate (Fig. S1). This intermediate is subsequently reduced to the acetal form H2C(OBBN)2, with a conversion rate of 90% after just 1 hour. Finally, reduction of the acetal provides MeO-BBN. This step is slow and requires 27 hours at 25 °C to yield the methoxyborane species quantitatively. The kinetics of the reaction improved at 70 °C. These conditions revealed a remarkable catalytic activity of 3-BMes2, allowing the full conversion of CO2 to the corresponding methoxyborane with a TON of 100, in only 2 hours (Table 1, entry 7). Reducing the loading of catalyst 3-BMes2 to 0.1 and 0.05 mol%, afforded an excellent TON of 1980, after 20 h at 70 °C, and a TOF of 250 h -1 , while the isolated components of the catalytic system, namely 1 and 2-BMes2, are unreactive (Table 1, Entries 9-12).

Dialkylboranes, other than 9-BBN, were also tested as potential reductants. As expected, the hydroboration of CO2 with catecholborane is slower than with 9-BBN because of the lower Lewis acidity of the borane. In fact, under the optimized reaction conditions (48 h at 70 °C), 3-BMes2 promotes the reduction of CO2 to MeOBcat with a modest TOF of 23. Replacing catecholborane with BH3 . SMe2 did not lead to any significant product formation.

Again, replacing the BMes2 unit with a Bpin fragment somewhat slows down the catalytic reaction and a TOF of 3.0 h -1 is measured with 3-Bpin, at 25 °C (vs 3.6 h -1 for BMes2). The difference of reactivity between 3-BMes2 and 3-Bpin is more pronounced at 70 °C, with TOFs values of 50 and 29 h -1 , respectively (Entries 7-8 in Table 1). This behaviour directly follows from the Arrhenius rate law.

Previous studies by the groups of Stephan and Cantat have established that basic guanidines, N-heterocyclic carbenes and phosphines can promote the hydroboration of CO2. 6f, 11 Mechanistic and computational investigations have shown that basic catalysts proceed through the activation of the B-H bond, by coordination of the Lewis base to the hydroborane. 6f Therefore we propose here that the benefit of combining a trivalent phosphorus function with a remote Lewis acid, as in 3-BMes2, results from a concomitant H-transfer / boraneformate adduct formation, in the first step of the CO2 reduction process (Scheme 3). In this approach, linking together the Lewis acid and the Lewis base functions should reduce the volume of the nonproductive region of the conformational space between the catalysts [a] reactions performed in J-Young tubes; monitored by NMR spectroscopy with 0.0048 mmol of catalyst, 0.48 mmol of 9-BBN in d8-THF (0.40 mL) and exposed to an atmosphere of CO2 (1 bar), [b] The TON is obtained by 1 H NMR monitoring (CH3O-BBN formation), with mesitylene as internal standard.

Table 1 .

 1 Catalytic reduction of CO2 to methoxyborane CH3OBBN

	Entry a Catalyst	T°C	Time	TON b	TOF
		(mol%)		(h)		(h -1 ) b
	1	1 (1 mol%)	25 °C	28h	<5	<0.2
	2	2-BMes2 (1 mol%)	25 °C	28h	0	0
	3	3-Bpin (1 mol%)	25 °C	28h	84	3.0
	4	3-BMes2 (1 mol%)	25 °C	28h	100	3.6
	5	1 (1 mol%)	25 °C	28h	36	1.3
		+ 2-Bpin (1 mol%)
	6	1 (1 mol%)	25 °C	28h	41	1.5
		+ 2-BMes2 (1 mol%)
	7	3-BMes2 (1 mol%)	70 °C	2h	100	50
	8	3-Bpin (1 mol%)	70 °C	2h	57	29
	9	3-BMes2 (0.1 mol%)	70 °C	4h	1000	250
	10	3-BMes2 (0.05 mol%)	70 °C	20h	1980	99
	11	1 (1 mol%)	70 °C	2h	<2	<1
	12	2-BMes2 (1 mol%)	70 °C	2h	0	0

| Scheme 2. Synthesis of

3-Bpin and 3-BMes2

Acknowledgements

We thank the CHARMMMAT Laboratory of Excellence, CEA, CNRS, Institut de Chimie des Substances Naturelles, and the ANR for financial support. The European Research Council is acknowledged for an ERC Starting Grant to T.C (Grant Agreement no. 336467). † Footnotes should appear here. These might include comments relevant to but not central to the matter under discussion, limited experimental and spectral data, and crystallographic data. Electronic Supplementary Information (ESI) available: Fig. S1 and experimental procedures. See DOI: 10.1039/c000000x/

| 3

and the substrates and thus increase further the reaction rates. 12 In fact, experimentally, catalysts 3-BMes2 and 3-Bpin are about twice as reactive as the corresponding bimolecular systems at 25 °C. This difference of reactivity corresponds roughly to a decrease of ca. 1 kcal/mol in activation energy, consistent with an entropic effect.

Scheme 3. Proposed synergistic effect of phosphino-borane catalysts in the hydroboration of CO2

Overall, we have demonstrated for the first time that the combination of phosphines and boranes offers a synergistic effect in the catalytic reduction of CO2 with hydroboranes. While the phosphine and the borane fragments, taken separately, do not catalyse the reduction of CO2, the P/B derivatives 3, combining the two functions, display high catalytic activity.

Notes and references

a NIMBE, CEA, CNRS, Université Gif-sur-Yvette, France E-mail: thibault.cantat@cea.fr. Fax: (+): (+33) 1 6908 6640 Homepage: http://iramis.cea.fr/Pisp/thibault.cantat/index.html. b Institut de Chimie des Substances Naturelles, CNRS UPR 2301, Université Paris-Sud, Université Paris-Saclay, 1 av. de la Terrasse, 91198 Gif-sur-Yvette (France) E-mail: arnaud.voituriez@cnrs.fr.

angela.marinetti@cnrs.fr.


Xavier Frogneux

Niklas Von Wolff

Pierre Thuéry

Guillaume Lefèvre

Thibault Cantat

CO_2 Conversion to Esters by Fluoride-mediated Carboxylation of Organosilanes and Halide Derivatives

the carboxylation of benzylsilane derivatives featuring a directing group (eq. 3). [10] The group of Kondo developed the carboxylation of alkynylsilanes (eq. 4). [11] Early on, Effenberger and Spiegler reported the carboxylation of electron defficient arylsilanes using KF or CsF under high pressures of CO2 (50 atm), presumably to avoid the formation of the free aryl anion (eq. 5). [12] These few examples highlight the small scope of reactive organosilanes, which is limited to the generation of stabilized carbanions having a corresponding pKa lower than 35. Additionally, the esterification step must be carried out in a second step to prevent a direct quenching of the carbon nucleophile by the electrophile (RCHN2 or RX) in the formation of the ester product. Aromatic and heteroaromatic esters being key intermediates in the synthesis of polymers, pharmaceutical intermediates or final products, [13] we sought to enlarge the scope of ester derivatives available from CO2 and organosilanes, under metal-free conditions. We show herein that a wide range of heteroaromatic silanes can be used in the direct synthesis of esters from CO2, organohalides and a fluoride source; and the methodology has been transposed to the formation of a polyester material from CO2. Mechanistic studies show that this reactivity can be enabled by a unique catalytic influence of CO2 in the activation of the C-Si bond.

In the absence of any transition metal, the controlled activation of commercially available phenyltrimethylsilane (1a) and 2-(trimethylsilyl)pyridine (2a) with fluoride ions is challenging because the corresponding C6H5 -and C5H5N -anions are unstabilized carbanions with a pKa of 45 and 44, respectively. Indeed, phenyltrimethylsilane (1a) exhibits no reactivity when subjected to an atmosphere of CO2, in the presence of CH3I 1, entries 6-7). It is worth noting that a procedure to recycle the Ph3SiF by-product is possible by the addition of TBAF to re-form TBAT. [14] Carrying the reaction in THF or CH2Cl2 allowed for a nearly quantitative conversion to the ester. [a] reaction conditions: NMR tube, 2a (0.1 mmol), additive (0.1 mmol), 3a (0.1 mmol), solvent (0.3 mL), CO2 (1 bar). NMR yield determined with mesitylene as internal standard.. [b]

TBAF=[n-Bu4N][F]; TMAF=[Me4N][F]. [c] Isolated yield.
The formation of 4aa represents the first example of the direct carboxylation of a pyridylsilane reagent. Importantly, the organosilane is compatible with both CO2 and the halogenoalkane and the reaction proceeds in one pot. In fact, changing the nature of the electrophile enables the formation of a large diversity of picolinic esters (Scheme 2). Primary and secondary alkyl and allyl iodides provided very good yields to the desired esters (4ab 90%, 4ac 85%, 4ae 93%, 4af 99%) after 40 h at RT or 18 h at 40 °C (Scheme 2). Iodobenzene proved to be unreactive in this reaction indicating that the formation of the O-C bond occurs via a nucleophilic substitution. Importantly, the alkylation of the pyridine ring with the organic halide was not observed and the carbon nucleophile, released by C-Si bond activation, selectively reacts with CO2. Bromoalkanes and chloroalkanes being less electrophilic, their utilization requires longer reaction times at RT (72 h) or mild heating (70 °C), as exemplified in the formation of 4ae, 4af and 4ag in >93 %yield from various organic halides (Scheme 2). Interestingly, using a difunctional electrophilic species such as 1,4-diiodobutane, diester 4ah is formed in 85% yield and crystals suitable for X-ray diffraction analysis could be isolated (see SI and Figure S6). The substituents borne by the silicon center have a notable influence on the reactivity of the pyridylsilane reagent and different kinetic rates were measured when one methyl group in 2a is replaced with a vinyl (2a2), an allyl (2a3) or a phenyl (2a4) group. While the presence of an allyl group slows down the activation of the C-Si bond, introducing a phenyl or vinyl group at the silicon center enhances the kinetics of the carboxylation reaction (see SI).

Taken together, these data suggest that the formation of picolinic esters proceeds via a fluoride mediated activation of the C-Si bond, which releases a carbanion able to undergo carboxylation in the presence of CO2 prior to the esterification step with the electrophile. Noticeably, no reaction occurred for arylsilanes that do not feature a nitrogen atom on the ortho position of the silyl group and both phenyldimethylvinylsilane (1a2) and 3-(trimethylsilyl)pyridine (2l) are unreactive under the applied conditions (eq. 6 and 7). These results are puzzling given that the three anions C6H5 -, o-C5H4N -and m-C5H4N -have similar pKas and hence similar stability. The mechanism of this novel transformation was thus investigated from stepwise reactions and DFT calculations so as to rationalize these findings.

Experimentally, under an argon atmosphere, silane 2a2 is stable in the presence of TBAT and only a small amount of pyridine (6%) is detected by 1 H NMR after 22 h at 70 °C. However, in the presence of CO2, the formation of a picolinate species 2-PyCO2 -is observed (68% yield), at room temperature (see SI). After addition of CH3I, the ester 4aa is obtained as the sole pyridine based product, along with NBu4I. These results suggest that the pyridine anion C5H4N -does not form under the applied conditions and that CO2 is required to facilitate the C-Si bond activation.

The different reactivities of 1a2, 2a2 and 2l were further investigated using DFT calculations (Scheme 3). For both the phenyl (1a2) and pyridine (2a2) derivatives, the formation of the carboxylate anion Ar-CO2 -is favored thermodynamically (ΔG < -19.7 kcal.mol -1 ) and the esterification step is also mildly exergonic. Using Me3SiF2 -as a fluoride source model, the formation of the pentavalent anionic species PhSiMe2(CHCH2)F -(APh) from 1a2 is endothermic yet accessible at RT (ΔG = 21.8 kcal.mol -1 ). Nevertheless, the direct carboxylation of APh to Ph-CO2 -(TS2) or the generation of the C6H5 -anion (TS3) is incompatible with the reaction conditions, with high-energy transition states (TSs) lying at 38.1 and 38.9 kcal.mol -1 , respectively. Replacing the phenyl group with pyridine core only stabilizes these two TSs by 2.9 kcal.mol -1 , that the C-Si bond activation follows a different pathway for 2a2. Formation of anion A2-Py is accompanied with an increase of negative charge on the nitrogen atom of the pyridine core, from 0.48 in 2a2 to 0.52 in A2-Py, thereby enabling fixation of CO2 to the N atom. The formation of the resulting CO2adduct B involves an energy barrier of 7.5 kcal.mol -1 (TS4). We found that introducing an electron withdrawing group (EWG), namely CO2, significantly facilitates the C-Si bond cleavage and the release of a stabilized anion (C), via an accessible TS5 at 28.6 kcal.mol -1 . C readily undergoes carboxylation at the carbanionic position to yield D, prior to a decarboxylation of the N-CO2 -moiety to 2-PyCO2 -. Overall, this mechanism requires a low energy demand of 28.6 kcal.mol -1 and it highlights the unique role of CO2 in this transformation, which acts both as a reactant and a catalyst able to facilitate the C-Si bond cleavage. In fact, whereas carboxylation of electron defficient arylsilanes is possible, [12] the method reported herein uses reversible CO2coordination as a mean of traceless heteroarene activation. The driving force for the fluoride-mediated carboxylation of 2a2 derives from the stabilization of anion C, which is reflected in the decreased pKa of the pyridine-CO2 adduct at the 2 position (30.2 vs 43.4 in pyridine, Scheme 3). In contrast, a carbanion at the 3 position of the pyridine-CO2 adduct is characterized by a pKa of 32.0 and its formation from 2l is 3.9 kcal.mol -1 higher in energy than for 2a2, in line with their different reactivities.

Based on these findings, the esterification of a variety of heteroaromatic silanes was carried out, with CO2, CH3I and TBAT (Scheme 4). Introducing a trifluoromethyl EWG on the 4 or 5 positions of the pyridine ring somewhat deactivates the reaction at RT and heating at 70 °C is desirable to reach good conversions into 4fa and 4ga (>99% and 71%, respectively). This behavior results from a destabilization of the pyridine-CO2 adduct (B). In contrast, the presence of a methyl group on positions 4, 5 and 6 is beneficial and excellent conversions (72-99%) to the desired methyl picolinates were obtained. Diester 4ha could be obtained from 2,6-(di-trimethylsilyl)pyridine with an excellent 97% yield. The carboxylation of other heterocycles is also feasible: 2-(thienyl)trimethylsilane affords 4ia in a good 75% yield while the furyl derivative 4ja is obtained under the same conditions in a low 27% yield. For these substrates, it is likely that activation of the heteroaryl-silane with CO2 is unnecessary as the aromatic anions are stable enough (pKa < 35). [16] Nonetheless, the pyrrole derivative N-methyl-(2-triethylsilyl)pyrrole remained unreacted, even after 40 h at 70 °C. The corresponding N-methyl-pyrrole Scheme 3 Fluoride-mediated Computed pathway for the carboxylation of 1a2 and 2a2 (PBE0-GD3, PCM THF) anion presents a high pKa of 39.6. [16] This lack of reactivity can thus be explained by the steric hindrance on the nitrogen center and its absence of nucleophilic character for the coordination of CO2. From a practical perspective, heteroaromatic esters 4 are usually prepared by carboxylation of the corresponding ortho-lithiated aromatics (e.g. 2-lithiopyridine) with the subsequent addition of an electrophile or via oxidation of methylated heteroaromatics to the carboxylate derivative, followed by condensation of an alcohol. [15] Given that novel catalytic methods recently appeared to form heteroaromatic silanes by C-H bond functionalization, without the need for a lithiation step, [17] the carboxylation strategy presented herein offers an attractive pathway to the synthesis of esters 4. Furthermore, the carboxylation of pyridylsilane derivatives proceeds using CO2 as a traceless activator (via the formation of stabilized carbanion C). This enables the carboxylation of pyridylsilanes at RT but also the use of an electrophile in an onestep procedure, for the first time. To further highlight these findings, the polymerization version of the present transformation was attempted (eq. 8).

Under an atmosphere of CO2, disilane 2h reacts at RT with 3h to afford a polyester material (4hh). GPC and NMR data reveal that 4hh is characterized by a succession of picolinate and butyl moieties. The polymerization exceeds 4 resulting from the incorporation of ca. 8 CO2 molecules (see SI). While many efforts are currently devoted to promoting the incorporation of CO2 in polycarbonate materials, the formation of oligomer 4hh represents the first example of CO2 conversion to a polyester chain and it offers a novel venue for the utilization of CO2 in polymer chemistry. [18] Current efforts are devoted to expand this proof-of-concept.

Scheme 2 .

 2 Scheme 2. Synthesis of picolinic esters by fluoride mediated carboxylation of 2-(trimethylsilyl)pyridine (2a) with various electrophiles. NMR yield. [a] Average isolated yield over two runs on 1 mmol scale.[b] 3h (0.5 eq.).

Table 1 .

 1 Screening of the reaction conditions for the direct esterification of 2a

	using CO2 and CH3I
	Entry [a]	Additive b]	Solvent	Yield (%)
	1	CsF	THF	traces
	2	TBAF.3H2O	THF	7
	3	TMAF	THF	traces
	4	Cs2CO3	THF	0
	5	tBuOK	THF	0
	6	TASF	THF	46
	7	TBAT	THF	93 (82)[c]
	8	TBAT	CH3CN	12
	9	TBAT	CH2Cl2	95
	10	TBAT	toluene	2
	11	-	THF	0

Acknowledgements

For financial support of this work, we acknowledge CEA, CNRS, the University Paris-Saclay (Fellowship to X.F.), the CHARMMMAT Laboratory of Excellence and the European Research Council (ERC Starting Grant Agreement no. 336467). T.C. thanks the Fondation Louis -Institut de France for its formidable support. We thank Dr. G. Carrot (CEA) for collecting GPC data.

Keywords: CO2 • organosilicon compounds • mechanism • DFT calculations

A novel methodology is described to convert CO2 to hetero-aromatic esters in the presence of organosilanes and organic halides, using fluoride anions as promoters for C-Si bond activation. CO2 exhibits a unique catalytic influence of CO2 in the C-Si bond cleavage of pyridylsilanes, serving as a traceless activator.

Xavier Frogneux, Niklas von Wolff, Pierre Thuéry, Guillaume Lefèvre* and Thibault Cantat*