The $\Phi^3_4$ and $\Phi^3_6$ matricial QFT models have reflection positive two-point function

We extend our previous work (on $D=2$) to give an exact solution of the $\Phi^3_D$ large-$\mathcal{N}$ matrix model (or renormalised Kontsevich model) in $D=4$ and $D=6$ dimensions. Induction proofs and the difficult combinatorics are unchanged compared with $D=2$, but the renormalisation - performed according to Zimmermann - is much more involved. As main result we prove that the Schwinger 2-point function resulting from the $\Phi^3_D$-QFT model on Moyal space satisfies, for real coupling constant, reflection positivity in $D=4$ and $D=6$ dimensions. The K\"all\'en-Lehmann mass spectrum of the associated Wightman 2-point function describes a scattering part $|p|^2 \geq 2\mu^2$ and an isolated fuzzy mass shell around $|p|^2=\mu^2$.


Introduction
The Kontsevich model [1,2] is of paramount importance because it elegantly proves Witten's conjecture [3] about the equivalence of two approaches to quantum gravity in two dimensions: the Hermitean one-matrix model [4,5,6] versus the intersection theory on the moduli space of Riemann surfaces [7,8,9]. The Kontsevich model is defined by the partition function (we use different notation) where the integral is over the space of self-adjoint (N × N )-matrices with Lebesgue measure dΦ. Perturbative expansion in the 'coupling constant' i 6 yields an asymptotic expansion of log Z[E] in terms of rational functions of the eigenvalues {e i } of the positive self-adjoint matrix E. Kontsevich's main theorem states that log Z[E] is, when viewed as rational function of {e i }, the generating function for the intersection numbers.
In a series of papers [10,11,12], one of us (H.G.) with H. Steinacker turned the Kontsevich model into a quantum field theory on noncommutative Moyal space (inspired by the work of Langmann-Szabo-Zarembo [13]). The action (1.1) was generalised to admit a purely imaginary coupling constant i 6 → i 6 λ, and the interest was turned to correlation functions arising from the formal measure (1.1), With the usual trick of generating Φ kl = ∂ ∂J lk e tr(ΦJ) J=0 under the integral and after absorbing J, λ in a redefinition of E and Φ, the known solution (in the large-N limit) of the Kontsevich action (1.1) allows to formally derive all correlation functions (1.2). The term 'formally' refers to the fact that the large-N limit produces the usual divergences of quantum field theory. Their removal by appropriate choice, depending on dimension {2, 4, 6}, of parameters in E, A, λ was achieved in [10,11,12]. Explicit formulae for the renormalised functions (1.2) at N ∈ {1, 2, 3} and genus g = 0 were given.
In a recent paper [14] we transferred the solution strategy developed by two of us (H.G.+R.W.) for the Φ 4 4 -matricial quantum field theory [15] to the Φ 3 2 -Kontsevich model. This approach uses the Ward-Takahashi identity for the U(N ) adjoint action (first employed in [16]) to derive a system of Schwinger-Dyson equations in which the N-point function depend only on N ′ -point functions with N ′ ≤ N. The initial step for the Kontsevich model is, in a special limit of large matrices coupled with an infinitely strong deformation parameter, an integral equation solved by Makeenko and Semenoff [17]. From this point of origin we explicitly solved all correlation functions.
The present paper extends [14] to D = 4 and D = 6 dimensions. Similar as in [11,12] we can recycle almost everything from two dimensions; only the renormalisation is more involved. We follow the renormalisation philosophy advocated by Wolfhart Zimmermann according to which the theory is defined by normalisation conditions, at the physical energy scale, of a few relevant and marginal couplings. In its culmination due to Zimmermann [18], this BPHZ renormalisation scheme [19,20,18] completely avoids the use of ill-defined (divergent) quantities. That the BPHZ scheme extends smoothly to non-perturbative renormalisation is our first noticeable message.
For reasons explained below we are particularly interested in real Φ 3 coupling constant, i.e. iλ → λ in (1.2). This is a drastic change! The partition function does not make any sense for real coupling constants. Our point of view is to define quantum field theory by quantum equations of motion, i.e. Schwinger-Dyson equations. These equations can formally be derived from the partition function, but then we forget the partition function, declare the equations as exact and construct exact solutions. Whereas the Φ 3 6 -Kontsevich model with imaginary coupling constant is asymptotically free [12], our real Φ 3 6 -model has positive β-function. But this is not a problem; there is no Landau ghost, and the theory remains well-defined at any scale! In other words, the real Φ 3 6 -Kontsevich model could avoid triviality.
It is instructive to compare our exact results with a perturbative BPHZ renormalisation of the model. In D = 6 dimensions the full machinery of Zimmermann's forest formula [18] is required. We provide in sec. 5 the BPHZ-renormalisation of the 1-point function up to two-loop order. One of the contributing graphs has an overlapping divergence with already 6 different forests. Individual graphs show the full number-theoretic richness of quantum field theory: up to two loops we encounter logarithms, polylogarithms Li 2 and ζ(2) = π 2 6 . The amplitudes of the graphs perfectly sum up to the Taylor expansion of the exact result.
The original BPHZ scheme with normalisation conditions at a single scale leads in justrenormalisable models to the renormalon problem which prevents Borel resummation of the perturbation series. We also provide in sec. 5 an example of a graph which shows the renormalon problem. But all these problems cancel as our exact correlation functions are analytic(!) in the coupling constant. Exact BPHZ renormalisation is fully consistent (for the model under consideration)!
The most significant result of this paper is derived in sec. 6. Matrix models such as the Kontsevich model Φ 3 D arise from QFT-models on noncommutative geometry. The prominent Moyal space gives rise to an external matrix E having linearly spaced eigenvalues with multiplicity reflecting the dimension D. In [21] two of us (H.G.+R.W.) have shown that translating the type of scaling limit considered for the matrix model correlation functions back to the position space formulation of the Moyal algebra leads to Schwinger functions of an ordinary quantum field theory on R D . Euclidean symmetry and invariance under permutations are automatic. The most decisive Osterwalder-Schrader axiom [22,23], reflection positivity, amounts for the Schwinger 2-point function to the verification that the diagonal matrix model 2-point function is a Stieltjes function. We proved in [14] that for the D = 2-dimensional Kontsevich model this is not the case. To our big surprise and exaltation, we are able to prove: of a true relativistic quantum field theory [24] (θ, δ are the Heaviside and Dirac distributions). Its Källén-Lehmann mass spectrum ̺( M 2 µ 2 ) [25,26] is explicitly known and has support on a scattering part with M 2 ≥ 2µ 2 and an isolated fuzzy mass shell around M 2 = µ 2 of non-zero width. The original Kontsevich model with purely imaginary coupling constant cannot be positive. This property is shared with the λΦ 4 4 matricial quantum field theory [15] where the stable phase with λ > 0, where the partition functions has a chance to exist, cannot be reflection positive [21]. For λ < 0, where the partition function is meaningless, a lot of evidence has been given [27,28] for reflection positivity of the 2-point function.
The exciting question whether also the higher Schwinger functions of Φ 3 D are reflection positive in D ∈ {4, 6} is left for future investigation.
We dedicate this paper to the memory of Wolfhart Zimmermann. The scientific community owes him, amongst others, the LSZ formalism, momentum space Taylor subtraction under the integral, the forest formula to handle overlapping divergences, operator product expansion and the reduction-of-couplings scenario. Our contribution applies several of these concepts to a non-perturbative setting. We are convinced that Wolfhart Zimmermann would have enjoyed these results.
One of us (H.G.) was several times invited (mostly by Julius Wess) to the MPI in Munich at Föhringer Ring. But during these visits and after seminars we got little by little good contacts to Wolfhart, too. It was an honour to give a summary [29] of our earlier work with R.W. at the Ringberg Meeting celebrating the 80th birthday of Wolfhart Zimmermann. And afterwards I enjoyed phone calls from him showing his interests in our work. We will remember Wolfhart Zimmermann as a deep thinker and very kind person.

The setup
Let the 'field' Φ = Φ * be a self-adjoint operator, of finite rank ≤ N , on some infinitedimensional Hilbert space H. In the end we are interested in a limit N → ∞ to compact operators Φ. Let E be an unbounded self-adjoint positive operator on H with compact resolvent. We consider the following action functional where products and trace are understood in the algebra of finite rank operators and only the projection P EP to the finite-dimensional space (ker Φ) ⊥ = P H contributes to (2.1). The parameter V is a constant discussed later, λ bare is the bare coupling constant (real or complex) and κ, ν, ζ, Z as well as λ bare and the lowest eigenvalue µ 2 bare of 2E are functions of (V, N ) and renormalised parameters (λ r , µ 2 ).
The operator E plays the rôle of a generalised Laplacian for which the Theorem of H. Weyl applies on the recovery of the dimension from the asymptotics of the spectrum: The dimension D need not be an integer. We shall see, however, that renormalisation is only sensitive to the even integer 2[ D 2 ].
• 2[ D 2 ] > 6 breaks the structures and cannot be renormalised. Local details about the eigenvalues of E (e.g. their degeneracy) are not important. In the simplest case all eigenvalues are different; here we would notationally proceed as in [14]. Spaces with larger symmetry, such as spheres and tori, have degenerate eigenvalues. In this paper we notationally assume a D 2 -fold cartesian product of a two-dimensional space with simple eigenvalues {E |n| } ∞ |n|=0 , growing at most linearly in |n|, which on the Ddimensional space arise with multiplicity . This allows us to label the eigenspaces by tuples n := (n 1 , . . . , nD 2 ), n i ∈ N, of natural numbers (which include 0). Indeed, defining |n| := n 1 + · · · + nD 2 , there are precisely different tuples n with the same |n|. But again, these choices 1 only affect the notation in the first part; any E of the same 2[ D 2 ] ≤ 6 leads to a solvable model with the same renormalisation prescription in the large-matrix limit.
From now on we assume that D ∈ {2, 4, 6}, that the projector P commutes with E and that the eigenvalues of E are a discretisation of a monotonously increasing C 1 -function e, E = (E n δ m,n ), 3) The parameter µ > 0 will become the renormalised mass, whereas the bare mass µ bare is a function of (V, N , λ r , µ) identified later. Note that e being increasing amounts to a particular choice of the projection P : Representing Φ = (Φ mn ) m,n∈N 2 in this special eigenbasis, the rank condition becomes Φ mn = 0 if |m| > N or |n| > N . In the very end we are interested in a special limit N → ∞ which is independent of the ordering. In these conventions, and with N : |m| ≤ N }, the action (2.1) takes the following form (simplifications arise for D = 2 and D = 4): The action formally defines a partition function with external field J (which is also a self-adjoint matrix): The integral in the first line of (2.5) does not make any sense for real λ. We rather view the partition function as a tool to derive identities between the (formal) expansion coefficients G |p 1 | are obtained by partial Jderivatives of log Z at J = 0. These external derivatives combine with the internal derivatives in (2.5) to identities between the (N 1 + . . . +N B )-point functions called Schwinger-Dyson equations. Our point of view is to declare these Schwinger-Dyson equations (although their derivation was purely formal) as exact identities between their rigorous solutions G |p 1 | . These Schwinger-Dyson equations alone are not enough. As decisive tool we also need Ward-Takahashi identities which reflect the fact that the partition function is unchanged under renaming the dummy integration variable Φ. This was first pointed out and used in [16]. For actions (2.8) We need a formula for n∈N D/2 ∂J bn ∂Jna , naïvely obtained by multiplication with V Z(Ea−E b ) . But this needs discussion. There is first a kernel W a δ a,b in n∈N D/2 ∂J bn ∂Jna which was identified in [15]. In the present case we can restrict ourselves to a = b where this kernel is absent. Remains the degeneracy problem: There is E a − E b = 0 even for a = b. The solution, already sketched in [15], goes as follows: Because their definition via the action (2.1) and the definition (2.3) of E only involves the spectrum of E, the ( | only depend on |p β i | (and λ), but not on p β i individually. If we can afford to exclude the diagonal a = b, then we can also afford to exclude |a| = |b| and use the following identity, valid for |a| = |b|, (2.10) Recall [14] that for D = 2 we had Z = 1 and ν = ζ = 0, whereas Z = 1 and ζ = 0 for D = 4. These are minor differences: Only the prefactors of terms already present in the equations receive a modification; no new structure is created. This is the essence of multiplicative renormalisation. We will see that the new parameters permit to regularise the integrals up to D = 6. By extrapolation we would need for D ≥ 8 a term ∝ E 3 Φ in the action. But such a term gives rise to 10) which completely destroys the previous structures. Hence, renormalisability is lost for D ≥ 8.
3 Schwinger-Dyson equations and solution for B = 1

Equations
In the same way as in our previous paper [14], namely by inserting (2.5), (2.6) into the corresponding term of (2.7), we derive formulae for the connected N-point functions. We have to discuss separately the cases N = 1, N = 2 and N ≥ 3. The equation for G |a| is straightforward: The equation for G |ab| with |a| = |b| reads: The last line is obtained by inserting the Ward-Takahashi identity (2.10) and comparing the resulting J-derivatives with (2.7). For (N>2) and pairwise different |a i | we have The step to the last line uses the Ward-Takahashi identity (2.10) for pairwise different 1-norms of indices. We start to discuss (3.3) because this unambiguously fixes a particular combination of the renormalisation parameters. We multiply (3.3) by H a 1 a 2 , perform the Jdifferentiations and collect all coefficients of G |a 1 a 2 ...a N | on the lhs: In terms of the functions which necessarily must be finite, we thus obtain . (3.6) The same steps yield for (3.2): Now we describe the renormalisation. We clearly need well-defined functions G |a| , F a , λ r in the limit N → ∞. This in turn forces a particular singular behaviour of κ, ν, ζ, Z, µ bare , λ bare . As familiar from perturbative renormalisation, there is still considerable freedom in choosing finite terms of these singular functions. We follow Zimmermann's prescription [18] and fix the finite terms by normalisation of the first Taylor expansion coefficients of relevant and marginal correlation functions (with i = 1, . . . , D 2 below): Inserting the conditions on G |a| into (3.7) reduces the conditions on G |ab| for D ≥ 4 to We thus conclude in (3.5) The above identities and definitions are now inserted into (3.1). To make sense out of m∈N D/2 N G |am| it is necessary to permit again the case |a| = |b| in (3.7). This is achieved by a continuity argument 2 , G |aa| = 1 2Fa 1+λ r lim |b|→|a| . The limit is well-defined in perturbation theory where G |a| is, before performing the loop sum, a rational function of the {F n } so that a factor F a − F b can be taken out of G |a| − G |b| . The result is (3.12)

Large-(N , V ) limit and integral equations
The action S[Φ], hence log Z and G |n| , W |n| only depend (via E n defined in (2.3)) on the 1-norm |n| = n 1 + · · · + n D/2 and not individually on the components n i of n. Therefore, the sum in (3.12) translates into 2 Alternatively, the limit V → ∞ permits a cheaper solution: We can restrict the sum to For the sake of easier notation we write unrestricted sums. We leave it to the reader's taste to exclude the terms " 0 0 " or to define them by continuity. giving (3.13) We take the limit N , V → ∞ subject to fixed ratio in which the sum converges to a Riemann integral Expressing the 1-norms of discrete matrix elements as |a| =: , and taking the normalisation 2F 0 = µ 2 from (3.9) and the relations (3.5)+(2.3) into account, we arrive at F a |a|=: . The mass µ is the only dimensionful parameter. The previously introduced functions and parameters have in spectral dimension D ∈ {2, 4, 6} the following mass dimensions: 16) where N = N 1 + · · · + N B . In terms of the dimensionless functions the large-(N , V ) limit of µ −4 times (3.13) takes the form We have used here the fact proved later that G |a|a| has a finite large-(N , V ) limit so that 4λ 2 r V 2 G |a|a| from (3.13) does not contribute to the limit. A final transformation and similarly for other capital letters Y (y), T (t) and functions G(X, Y ) =G(x(X), y(Y )) etc., simplifies (3.18) to , Building on [17] we proved in [14, eq. (4.14)] that (3.20) is solved by for some function c(λ, e,ν, Z). The functions c,ν, Z are fixed by the normalisation con- The first condition fixesν(c, Z) to The condition on W ′ (1) in (3.22) then fixes Z(c) to For D = 4 where Z ≡ 1, this equation determines the main function c(λ, e) (in the limit Ξ → ∞ which is safe for ρ(T ) ∝ const) to The formula differs from D = 2 by a power of 2 in the denominator ( For D = 6 we conclude This allows us to compute the β-function of the running coupling constant 3 λ r : We learn that the β-function is -for real coupling constant -strictly positive 4 , with λ b (Ξ) Ξ→∞ −→ +∞, but without developing a Landau pole (a singularity of λ b already at finite Ξ 0 ). We also have Z(Ξ) ∈ [0, 1], as it should. We remark that [12] addresses the case that λ bare , λ r ∈ iR, hence λ 2 r < 0, resulting in negative β-function and asymptotic freedom. We would like to point out, however, that this also implies ρ(T ) < 0 and consequently the senseless result 1 √ Z → −∞. We shall see in the final section that an important positivity property only holds for λ r real.
Inserted into (3.24) we get where the limit Ξ → ∞ is now safe. Eventually, the condition on Obviously,λ = 0 and hence ρ = 0 corresponds to c = 0 for any D. For given e(x), thus ρ(T ), the implicit function theorem provides a unique diffeomorphismλ 2 → c(λ, e) on a neighbourhood of 0 ∈ R or 0 ∈ C. Since we will be able to express all correlation functions in terms of elementary functions of c(λ, e) and ρ(λ, e), this proves analyticity of all correlation functions in these neighbourhoods.
According to [14,Prop 4.2] (which is unchanged due to (3.6)), all N-point functions are explicitly computable from this solution W (X): Proposition 3.1. The connected (N≥1)-point functions in the scaling limit

N -points function with B ≥ 2 boundaries
The derivation of the formula for the (N 1 + . . . +N B )-point function with one N i > 1 proceeds along the same lines as in our previous paper [14]. Key ingredient is the Ward-Takahashi identity (2.10) which compared with [14] contains additional parameters ν, ζ = 0 and Z = 1. However, in identical manner as in the derivation of (3.6), the resulting contributions reconstruct the renormalised functions F a instead of E a in [14]. We thus have for N 1 > 1 (where λ r ≡ λ for D ∈ {2, 4}), its large-(N , V ) limit by Taking the scaling limitG( and transforming variables x i to X i , we essentially obtain [14, eq. (5.7)], for B = 2 and for B ≥ 3 by , (4.8) Explicitly we have for D = 6 (4.10) 5 Linearly spaced eigenvalues and Feynman graphs for D = 6

Expanding the exact result
The noncommutative field theory model discussed in sec. 6 (see also [14, 1) 5 We actually compute 1 √ Z from (3.28) and the integral (3.24). A primitive of the integrand is obtained by computer algebra; the limit Ξ → ∞ is done by hand. Also identities of the type where c(λ) is the inverse solution of .

(5.2)
The first terms of the inverse solution are This gives the following perturbative expansion of the 1-point function: To such a graph Γ we assign an unrenormalised amplitudeG Λ Γ as follows:

Perturbative expansion of the partition function
• Associate a weight (−λ) to each black vertex, weight 1 to each white vertex of Γ.
• Associate weight 1 z 1 +z 2 +1 to an edge of Γ separating faces labelled by z 1 and z 2 . These can be internal or external, also z 1 = z 2 can occur. We list the simplest graphs contributing to the 1-point functionG Λ (x) and their unrenormalised integrals: , , Clearly, all these integrals diverge for Λ → ∞.

Zimmermann's forest formula for ribbon graphs
A key step in proving these claims is the forest formula of Wolfhart Zimmermann [18]. Streamlining previous (and very essential) work of Klaus Hepp [20] on the Bogoliubov-Parasiuk renormalisation description [19], Zimmermann proved that those formal power series µ bare (λ), λ bare (λ), Z(λ), κ(λ), ν(λ), ζ(λ) which enforce the normalisation conditions (3.8) simply amount to a well-defined modification, for every ribbon graph Γ, of the integrand I Γ of Γ (product of vertex weights and edge weights). The modified integrand R(I Γ ) is Lebesgue-integrable over the whole space and by Fubini's theorem can be unambiguously integrated in any order. We describe this modification R for the Φ 3 6 -matrix model under consideration. The situation is far easier than in the papers by Hepp and Zimmermann because in Euclidean space there is no need to discuss the iǫ-limit to tempered distributions. Moreover, the globally assigned face variables give unique edge weights, in contrast to choices of momentum routings in ordinary Feynman graphs that make recursive substitution operations necessary.
Let B Γ , F Γ be the set of external and internal faces of Γ, respectively. A ribbon subgraph γ ⊂ Γ consists of a subset F γ ⊂ F Γ together with all edges and vertices bordering F γ , and not more, such that the (thus defined) set of edges and vertices of γ is connected. The ribbon subgraph defines a unique subset E γ ⊂ B Γ ∪ F Γ \ F γ of neighbouring faces to γ, i.e. any element of E γ has a common edge with γ. We let B γ be the number of connected components of E γ . This number is most conveniently identified when drawing Γ on the 2-sphere where E γ will partition into B γ disjoint regions.
We let f (γ) be the set of those face variables {x, y} which label the faces in E γ . Then for a rational function r γ (f (γ), y 1 , . . . , y l ), where y 1 , . . . , y l label the faces of γ, we let (T ω f (γ) r γ )(f (γ), y 1 , . . . , y l ) be the order-ω Taylor polynomial of r γ with respect to the variables f (γ). We let T ω f (γ) r γ ≡ 0 for ω < 0. A forest U Γ in Γ is a collection of ribbon subgraphs γ 1 , . . . , γ l such that for any pair Here γ i ∩ γ j = ∅ means complete disjointness in the sense that γ i , γ j do not have any common edges or vertices. Similarly, γ j ⊂ γ i means that all faces, edges and vertices of γ j also belong to γ i . We admit U Γ = ∅, but note that Γ itself cannot belong to U Γ because Γ has purely external edges which are not part of a ribbon subgraph. Consequently, a forest is endowed with a partial order which we symbolise by disjoint trees (hence the name). The tree structure identifies for any γ i ∈ U Γ a unique set o(γ i ) = γ i 1 ∪ · · · ∪ γ i k of offsprings, i.e. mutually disjoint ribbon subgraphs γ i h ⊂ γ i such that for any other γ j ∈ U Γ \ (γ i ∪ o(γ i )) either γ j ⊂ o(γ i ), or γ i ⊂ γ j , or γ i ∩ γ j = ∅. A ribbon subgraph γ with o(γ) = ∅ is called a leaf.
Let I Γ be the integrand encoded in a ribbon graph, given as the product of weights of edges of Γ (the constant vertex weights play no rôle). A forest U Γ defines a partition of the integrand I Γ into Then the following holds: Theorem 5.2 (after Zimmermann). The formal power series µ bare (λ), λ bare (λ), Z(λ), κ(λ), ν(λ), ζ(λ) which enforce the normalisation conditions (3.8) amount for any ribbon graph Γ to replace the integrand I Γ as follows: where the sum is over all forests U Γ of Γ including the empty forest ∅. For the planar sector of the Φ 3 D -matrix model, the degree is defined as ω(γ) := D 2 (2 − B γ ) − N γ , where N γ is the number of edges of Γ which connect (within Γ) to vertices of γ but are not in γ itself.

Forest formula applied to 1-point function
We exemplify Zimmermann's rules for the ribbon graphs given in (5.5). The graph Γ 1 has two forests U Γ 1 = ∅ and U Γ 1 = {γ 1 }, where γ 1 is the ribbon subgraph consisting of the face y 1 and its bordering line and vertex. Taking N γ 1 = 1 and B γ 1 = 1 into account, we have to replace we compute the renormalised amplitude of the ribbon graph Γ 1 tõ We confirm the perfect agreement with the λ-expansion (5.4) of the exact formula! The next graph Γ 2 has four forests By γ i we denote the ribbon subgraph with face labelled y i together with its bordering edges and vertices. Note that the faces labelled y 1 , y 2 together do not give rise to a ribbon subgraph because its edge+vertex set would be disconnected. We have N γ 1 = 2, N γ 2 = 1 and B γ 1 = B γ 2 = 1. We have f (γ 1 ) = f (γ 2 ) = {x}. Hence, the Taylor subtraction operator for γ 1 is (−T 1 x )( 1 (x+y 1 +1) 2 ) = − 1 (y 1 +1) 2 + 2x (y 1 +1) 3 , and (−T 2 x )( 1 (x+y 2 +1) ) is analogous to Γ 1 . Then Zimmermann's forest formula factors as follows: In Γ 3 we let the ribbon subgraph γ 2 as before and define γ 12 as consisting of both faces labelled by y 1 and y 2 together with all bordering edges and vertices. There is also the ribbon subgraph γ 1 consisting of y 1 but not γ 2 . Then E γ 2 has B γ 2 = 2 components, namely the disconnected faces labelled by y 2 and by x. In principle we could include γ 1 in the forest formula, but ω(γ 1 ) = −1 forces the Taylor operator T to vanish. Therefore, we can work with the reduced set of forests U Γ = ∅, U Γ = {γ 2 }, U Γ = {γ 12 } and U Γ = {γ 12 , γ 2 }. The external face variables are f (γ 2 ) = {y 1 } and f (γ 12 ) = {x}. The resulting analytic contribution is 6 We see that the whole number-theoretic features of quantum field theory are reproduced! The graph Γ 4 poses no difficulty: (5.13) The graph Γ 5 shows a new quality: the overlapping divergence made of the ribbon subgraphs γ 1 and γ 2 which share a common edge. Overlapping divergences were a problem in the first days of quantum field theory. Now with the forest formula at disposal there is nothing to worry. The definition simply forbids γ 1 and γ 2 in the same forest. The following forests remain: We have N γ 12 = 1, N γ 1 = 3, N γ 2 = 2 and B γ 12 = B γ 1 = B γ 2 = 1, giving ω(γ 12 ) = 2, ω(γ 1 ) = 0, ω(γ 2 ) = 1. The face variables are f (γ 12 ) = {x}, f (γ 1 ) = {x, y 2 }, f (γ 2 ) = {x, y 1 }. It follows: x,y 2 )(Iγ 1 ) x,y 1 )(Iγ 2 ) The sum is indeed the λ 3 -part of (5.4): This coincidence looks remarkable, but it isn't. It is the unavoidable consequence that two correct theorems about the same object must agree.

Renormalons
Renormalised amplitudes typically involve a logarithmic dependence on the external parameters. Our model is no exception, see (5.10). In just renormalisable QFT models one finds that n-fold insertion of these subgraphs into a superficially convergent graph produces an unprotected amplification of the logarithms which let the amplitude of the renormalised graph grow with O(n!). Since there are also O(n!) graphs contributing to order n, there is no hope of a Borel-summable perturbation series. This phenomenon is called the renormalon problem. It is an artefact of the strictly local Taylor subtraction at 0. Constructive renormalisation theory [30] avoids the problem by a reorganisation into an effective series of infinitely many (but related) coupling constants. On the downside we loose the nice forest formula so that it is hard to compute the amplitudes in practice.
In this subsection we convince ourselves that the Φ 3 6 -matrix model under consideration has, in its perturbative expansion, graphs showing the renormalon problem. Consider the finite (N = 4, B = 1)-graph with a single internal face labelled y. We replace the edge between two black vertices by a chain of n vertices and n + 1 edges and attach to every additional black vertex and towards the inner face the simplest 1-point function Γ 1 : Taking the renormalised amplitude (5.10) into account, the amplitude of the total ribbon graph becomes in the simplified case that all external face variables are equal, For large y the integral behaves as This is the renormalon problem for a single graph.
On the other hand we know that the exact formula is analytic inλ. There is a subtle cancellation between different graphs, similar to the log(1 + x) contributions ofG Γ 2...5 which all cancelled in the sum. In a certain sense, this cancellation is another instance of the integrability of the model. 6 From φ ⋆3 D model on Moyal space to Schwinger functions on R D This section parallels the treatment of φ ⋆4 4 in [15,21] and φ ⋆3 2 in [14] where more details are given. The (unrenormalised) φ ⋆3 D -model on Moyal-deformed Euclidean space with harmonic propagation is defined by the action where ⋆ denotes the associative, noncommutative Moyal product parametrised by a skewsymmetric matrix Θ. The Moyal space possesses a convenient matrix basis f mn (x) labelled by pairs of D 2 -tupels m = (m 1 , . . . , m D/2 ) for which we set |m| := m 1 + · · · + m D/2 . The matrix basis satisfies (f kl ⋆ f mn )(ξ) = δ ml f kn (ξ) and R D dξ f mn (ξ) = | det(2πΘ)|δ mn . A convenient regularisation consists in restricting the fields φ to those with finite expansion φ(ξ) = m,n∈N D/2 N Φ mn f mn (ξ), where we recall N D/2 N := {m ∈ N D/2 : |m| ≤ N }. Then (6.1) takes precisely the form of our starting point (2.4) with undone renormalisation (κ = ν = ζ = 0, Z = 1, λ bare = λ, µ bare = µ) and identification Comparing with (2.3), the Moyal space leads to a linear eigenvalue function e(x) = x. According to [21,14] the large-(N , V ) limit of the matrix model induces together with the convention δJmn δJ(ξ) := µ D f mn (ξ) the following relation for connected Schwinger functions on extreme Moyal space (where s β : This shows that out of the Osterwalder-Schrader axioms [22,23], Euclidean invariance and symmetry are automatically fulfilled, whereas clustering does not hold. The remaining section addresses reflection positivity for the Schwinger 2-point function S c (µξ 1 , µξ 2 ).
It should be possible to verify the claim directly. The function tan n φ a+b cos φ has a known primitive so that integration by parts does the job. Below we explain how we obtained the result. Figure 1 shows a plot of the Stieltjes measure ̺(t) for various values of σ.
6.2 Identification of the Stieltjes measure for D = 6 As long as x > 0 the formula (6.5) can be taken literally. But for discussing the Stieltjes property we have to extend it to complex x. Here we already made a choice for the logarithms: When arranging them as in (6.5) we understand that log z has a branch cut along the negative real axis and we choose the standard branch Im(log z) ∈ ]0, π[ for Im(z) > 0 and Im(log z) ∈ ]−π, 0[ for Im(z) < 0. Next, the only reasonable interpretation that applies to c ∈ ]−1, 0] is √ X + c := (2x + 1) + √ −c (2x + 1) − √ −c and √ X := 2x + 1. This shows that x → 2W ′ ((2x + 1)) is holomorphic on C \ ]−∞, 0[ with branch cut along parts of the negative real axis. Such holomorphicity is one of the characterising properties of Stieltjes functions. If we knew thatG(x, x) is Stieltjes, then the measure is recovered from the inversion formulã  We have to distinguish the three cases in (6.5): 1. [0 < t < 1− √ −c]: We have √ X, √ X + c > 0, hence Im(2W ′ (X)) √ Along the branch cut of √ X + c, where this square root has strictly negative imaginary part, the logarithms in (6.5) are well-defined, and we have Positivity will be discussed below.
It is the opposite sign of the last line in (6.19) compared with (6.17) which lets the scattering measure supported at t > 2 arise with the wrong sign! In addition there is an atomic measure from the second term on the rhs of (6.19): Near t = 2 we have Im Adding also the measure on [1 − √ −c, 1 + √ −c] we obtain the representation: Proposition 6.3. The diagonal 2-point function of the renormalised 2-dimensional Kontsevich model Φ 3 2 has, for linearly spaced eigenvalues of E, realλ and in large-(N , V ) limit,