Renormalization of an Abelian Tensor Group Field Theory: Solution at Leading Order Vincent Lahoche1, Daniele Oriti2, Vincent Rivasseau 3 Abstract Westudyajust-renormalizabletensorialgroupfieldtheoryofranksixwithquarticmelonic interactions and Abelian group U(1). We introduce the formalism of the intermediate field, which allows a precise characterization of the leading order Feynman graphs. We define the renormalization of the model, compute its (perturbative) renormalization group flow and 5 1 write its expansion in terms of effective couplings. We then establish closed equations for the 0 two point and four point functions at leading (melonic) order. Using the effective expansion 2 and its uniform exponential bounds we prove that these equations admit a unique solution b at small renormalized coupling. e F Pacs numbers: 11.10.Gh, 04.60.-m, 02.10.Ox 4 Key words: Quantum Gravity, Group Field Theory, Tensor Models. Report numbers: AEI-2014-058; ICMPA/MPA/2014/23 ] h t - p 1 Introduction e h [ Tensor group field theory (hereafter TGFT) is a background-independent formalism for quantum 3 gravity. Using the powerful quantum field theory language, it offers both a tentative definition of v the fundamental degrees of freedom of quantum spacetime and a precise encoding of their quantum 6 8 dynamics. It combines the results of tensor models [1, 2] about the combinatorics of random 0 discrete spaces and the insights of loop quantum gravity [3] about quantum geometry. More 2 0 in detail, TGFTs are quantum field theories on Lie groups, characterized by a peculiar non-local . 1 pairingoffieldargumentsintheirinteractions,whoseimmediateconsequenceisthattheirFeynman 0 diagrams are dual to cellular complexes rather than simple graphs. The quantum dynamics is thus 5 1 defined, in perturbation theory, by a sum over such cellular complexes (interpreted as discrete : spacetimes) weighted by model-dependent amplitudes, in turn functions of group-theoretic data. v i Historically, group field theories (GFTs) [4, 5] grew out of tensor models for 3d and 4d gravity X [6], themselves a generalization of the matrix model definition of 2d Riemannian quantum gravity r a [7]. In tensor models, the dynamics of a quantum spacetime is given by a sum over equilateral d-dimensional triangulations, generated as the Feynman expansion of the partition function for a finite rank-d tensor, and weighted by (the equilateral restriction of) the Regge action for simplicial gravity. Theyarethusprototypicalmodelsofpurelycombinatorialrandomgeometries. GFTsarise when the domain of the tensors is extended to a group manifold, and the first models [4] make use of these additional data to define amplitudes corresponding to state sum models of topological BF theory (by incorporating appropriate gauge invariance conditions, to which we will return in the following). Soon it was realized [8] that these group-theoretic data gave the boundary states of the [email protected]; LPT-UMR 8627, Universit´e Paris 11, 91405 Orsay Cedex, France, EU. [email protected], Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mu¨hlenberg 1, 14476, Potsdam, Germany [email protected];Laboratoiredephysiqueth´eorique,UMR8627,Universit´eParis11,91405Orsay Cedex,France,andPerimeterInstituteforTheoreticalPhysics,31CarolineSt. N,N2L2Y5,Waterloo,ON,Canada 1 same models the structure of loop quantum gravity states [3]. Later [9], indeed, GFTs were shown to provide a complete definition of the dynamics of the same quantum states as their Feynman amplitudes are given by spin foam models [10], a covariant definition of the dynamics of LQG spin networks, in turn dual to simplicial gravity path integrals [11]. Now, they are understood as a natural second quantized formulation of loop quantum gravity [12], and GFT models incorporating more quantum geometric features of LQG states and simplicial geometry are indeed among the most interesting ones. In the meantime, tensor models have witnessed an important resurgence, in the form of colored tensor models [1, 13]. These solved many issues raised by earlier tensor models and allowed a wealth of important mathematical results to be obtained. They triangulate pseudo-manifolds with only local singularities [14], having in particular no tadfaces (i.e. a face which runs several times through a single edge). Most importantly, they admit a large N expansion [15] (where Nd is the size of the tensor), whose leading order is now well understood. The leading graphs in this limit, the melonic graphs, form particularly simple “stacked” triangulations of the sphere in any dimension [16]. Their appearance is a very general phenomenon [17, 18]. Some of these results have immediately been extended to topological GFTs and multiorientable models [19], and beyond the leading order, to define interesting double scaling limits [20]. Incorporating the insights of colored tensor models into GFTs leads to TGFTs. Here, the GFT fields are required to transform as proper tensors under unitary transformations and their interactions are required to have the additional U(N)⊗d invariance, which can be interpreted as a new notion of locality, hence singles out a new theory space [21]. In turn, this invariance requires their arguments to be labeled (ordered). Both facts are crucial for GFT renormalization. GFTrenormalizationisinfactathrivingareaofcurrentresearch. Giventhatthefirstdefinition of the GFT quantum dynamics is in terms of a perturbative expansion around the Fock vacuum, the first aim is to prove renormalizability of specific models, showing therefore their consistency as quantum theories. Second, one is interested in unraveling the phase space of GFT models, looking in particular for a phase in which approximate smooth geometric physics (governed by some possibly modified version of General Relativity) emerges from the collective behavior of their pre-geometric degrees of freedom [22], maybe through a process of condensation. The search for such a geometric phase, and the associated phase transition(s), is common to tensor models [16], loop quantum gravity [23], and spin foam models [24], but also to other related approaches like (causal) dynamical triangulations [25]. Moreover, it has been conjectured [22] to have a direct physical interpretation in a cosmological context [26], and some recent results in GFT support this conjecture [27]. The TGFT framework is well-suited for renormalization, as one can import more or less stan- dard QFT techniques even in such background independent context. One ingredient is the new notion of locality provided by the U(N)⊗d invariance of tensor interactions. The other ingredient, a notion of scale is naturally assumed to be given by the decomposition of GFT fields in group representation. This is fully justified in terms of spectra of the kinetic operator (as in standard QFT) when a Laplacian on the group manifold is used, as suggested by the analysis of radiative corrections to topological GFT models [28] (which correspond to ultra-local truncations of truly propagating models). All these ingredients, it turns out, speak to one another very nicely, as in- deed in TGFT models counter-terms necessary to cure divergences remain of the same form of the initial interactions. More precisely, by precise power counting of divergences, one sees that at large ultraviolet (UV) scales (in the sense of large eigenvalues of the group Laplacian) connected sub- graphs which require renormalization seem local (as defined by tensor invariance) when observed at lower scales. 2 A large amount of results has been already obtained. For models without gauge invariance the proof of renormalizability at all orders, which started with [29], now includes a preliminary classification of renormalizable models [30] and studies of the equations they satisfy [31]. Then Abelian [32, 33] and non-Abelian gauge invariance (whose important role we already emphasized) has been included [34, 35]. The computations of beta functions typically shows UV asymptotic freedom [36, 37] to be a rather generic feature of TGFTs, even if the analysis of more involved models is in fact quite subtle [38]. Renormalizability and UV asymptotic freedom are the two key properties of non-Abelian gauge theories which form the backbone of the quantization of all physical interactions except gravity, hence it is encouraging to find them also in TGFTs, which aim at quantizing gravity. Once renormalizability (and possibly asymptotic freedom) is established, the next stage is to understand the infrared (IR) behavior of the renormalization group flow, in particular phase diagrams and phase transitions. One can prove that the leading “melonic” order of tensor models and of topological GFTs exhibits a phase transition, corresponding to a singularity of the free energy for a certain value of the coupling [16, 39]. The critical susceptibility can be computed at least for simple tensor models to be equal to 1/2. In the same tensor models context, in which the only notion of distance is the graph distance, one sees a phase corresponding to branched polymers, with Hausdorff dimension 2 and spectral dimension 4/3 [40], as in CDT. In GFTs and TGFTs, where the group theoretic data play a prominent role, not only computing observables and critical exponents, but also finding the nature of the transitions and their physical interpretation is much more difficult. Therefore we need more analytic tools. One powerful scheme is provided by functional renor- malization techniques. These have been developed for TGFTs for the first time in [41]. Applied to the (comparatively) easy case of an Abelian rank-3 model, the RG flow equations could be derived and the phase diagram be plotted in the key UV and IR regimes, showing evidence for a phase transition to a condensed phase, at least in some approximation. In this paper we perform a leading order analysis of the correlation functions of a simple TGFT with quartic melonic interactions and U(1) group, in dimension 6, endowed with gauge invariance conditions. This model is just-renormalizable [33], and asymptotically free [37]. Hence it should exist at the level of constructive field theory [42] (see [43] for the construction of a simpler super- renormalizable TGFT). Although we shall not achieve such a complete non-perturbative analysis in this paper, we provide some significant steps in this direction. We define the intermediate field formalism for our model and with a multi-scale analysis we establish its renormalizability, compute the beta function of the model and check its asymptotic freedom. In this way we recover all the results of [33] and [37]. The development of the intermediate field method for our model is in itself, we believe, an interesting result. It is known to be particularly convenient for quartic tensor models [44, 45], and should become a standard tool for TGFT’s as well. One should notice in particular that in our case, due to the gauge conditions, the intermediate fields are of a vector rather than matrix type, a promising new feature. We then define the effective expansion of the model, which sits “in between” the bare and the renormalized expansion. Its main advantage is to be free of renormalons [42]. We check this fact again in our model by establishing uniform exponential upper bounds on effective amplitudes. We also establish closed equations for the leading order (i.e. melonic approximation) to the two- point and four-point functions. Combining all these results proves that these closed equations admit a unique solution for small enough renormalized coupling, and gives full control over the melonic approximation of the theory, bringing it to the level of analysis of the Grosse-Wulkenhaar non-commutative field theory [46]. 3 Similar closed equations have been written for another renormalizable TGFT theory, in dimen- sion 5 and with a simpler propagator without gauge invariance conditions in [47]. The renormal- ization and numerical analysis of these equations have been recently developed in [48]. Our paper is organized as follows. In Section 2 we define the model and its intermediate field representation. In Section 3 we establish and analyse its power-counting with multi-scale analysis. Section 4 describes its renormalization, computes the beta function (in agreement with [37]), introduces the effective expansion and establishes uniform bounds on the corresponding effective amplitudes. Section 5 writes the closed equations for the melonic approximation to the bare and renormalized two point and four-point functions, and completes the proof that these equations have a unique solution at small renormalized coupling, which is in fact the Borel sum of their renormalized expansion. 2 The Model In this section, we shall briefly recall the basics of TGFTs models with closure constraint (gauge invariance) and Laplacian propagator. Then we shall focus on a particular U(1) quartic model at rank six first defined in [32]. Within this section definitions and computations are still formal since we do not introduce cutoffs; this will be done in the next sections. 2.1 General Formalism for TGFTs A generic TGFT is a statistical field theory for a tensorial field, for which the entries are living in a Lie group G, generally compact, such as U(1) or SU2) for the simplest cases. A family of such models was defined and renormalized to all orders in [32, 34, 33, 35]4. The theory is defined by an action and by the following partition function S(φ¯,φ) = S (φ¯,φ)−J¯·φ−φ¯·J, Z(J¯,J) = dµ (φ¯,φ)e−S[φ¯,φ], (2.1) int C (cid:90) where S is the interaction and dµ is a Gaussian measure characterized by its covariance C. The int C fields φ and φ¯ are complex functions φ¯,φ : Gd (cid:55)→ C noted φ(g ,··· ,g ) = φ((cid:126)g), (cid:126)g = (g ,g ,...,g ), 1 d 1 2 d and φ¯((cid:126)g(cid:48)), (cid:126)g(cid:48) = (g(cid:48),g(cid:48),...,g(cid:48)). They should equivalently be also considered as rank-d tensors, that 1 2 d is elements of the tensor space L2(G)⊗d, where L2(G) is the space of functions on G which are square-integrable with respect to the Haar measure. The 2N-point Green functions are obtained ¯ by deriving N times with respect to sources J and N times with respect to anti-sources J ∂2NZ(J¯,J) G ((cid:126)g ,··· ,(cid:126)g ,(cid:126)g(cid:48),···(cid:126)g(cid:48) ) = . (2.2) 2N 1 N 1 N ∂J1((cid:126)g1)∂J¯1((cid:126)g(cid:48)1)···∂JN((cid:126)gN)∂J¯N((cid:126)g(cid:48)N) J=J¯=0 (cid:12) (cid:12) The Gaussian measure is defined by the choice of the action’s kinetic term. TGFTs such as (cid:12) those of [29, 30] use a mass term plus the canonical Laplace-Beltrami operator ∆ on the group Gd, hence correspond to the formal normalized measure 1 dµ (φ¯,φ) = e−Skin[φ¯,φ]Dφ¯Dφ (2.3) C0 Z 0 4Renormalizability has not been yet established for models based on the Lorentz group, which is non-compact. However, at least intuitively, one could expect the additional difficulties present in the non-compact case to be rather of IR nature than of UV nature, from the point of view of TGFT renormalizability; this would imply similar renormalizability results as in the compact group case. 4 with S (φ¯,φ) = [dg]dφ¯((cid:126)g)[(−∆+m2)φ]((cid:126)g), (2.4) kin (cid:90) ¯ where dg is the Haar measure on the group. Although the Lebesgue measure DφDφ in (2.3) is ill-defined, the measure dµ itself is well-defined, and the propagator C in the parametric (or C0 0 Schwinger) representation is ∞ d C ((cid:126)g,g(cid:126)(cid:48)(cid:48)) = dµ (φ¯,φ)φ¯((cid:126)g)φ((cid:126)g(cid:48)) = dαe−αm2 K (g g(cid:48)−1), (2.5) 0 C0 α c c (cid:90) (cid:90)0 c=1 (cid:89) where K is the heat kernel associated to the Laplacian operator, and c is our generic notation for α a color index running from 1 to d. In momentum space this propagator becomes diagonal. Let us from now on restrict to the case G = U(1). The Fourier dual of U(1) is Z, hence in momentum space, we note p(cid:126) = (p ,··· ,p ) ∈ Zd, where p ∈ Z is called the strand momentum of color c, and 1 d c we have d 1 C (p(cid:126),p(cid:126)(cid:48)) = δ(p ,p(cid:48)) . (2.6) 0 c c p(cid:126)2 +m2 c=1 (cid:89) In the specific TGFT we study in this paper, we want the field configurations to obey the additional gauge invariance ¯ ¯ φ(g ,g ,...,g ) = φ(hg ,hg ,...,hg ), φ(g ,g ,...,g ) = φ(hg ,hg ,...,hg ) ∀h ∈ G. (2.7) 1 2 d 1 2 d 1 2 d 1 2 d This gauge invariance complicates slightly the writing of the model. In order to implement it, we could introduce the (idempotent) projector P which projects the fields on the subspace of gauge- invariant fields, then equip the interaction vertices and propagators with such projectors. But in this case the tensorial symmetry U(N)⊗D symmetry of the interaction vertex (which provides the analog of a locality principle for renormalization) would be blurred. Hence the best solution, used in [32], consists in implementing the gauge invariance directly on the Gaussian measure by introducing a group-averaged covariance ∞ d C((cid:126)g,(cid:126)g(cid:48)) = dµ (φ¯,φ)φ¯((cid:126)g)φ((cid:126)g(cid:48)) = dαe−αm2 dh K (g hg(cid:48)−1). (2.8) C α c c (cid:90) (cid:90)0 (cid:90) c=1 (cid:89) In other words, we introduce the gauge invariance projector P only in the propagator of the theory5. In momentum space we have. d δ( p ) C(p(cid:126),p(cid:126)(cid:48)) = δ(p ,p(cid:48)) c c . (2.9) c c p(cid:126)2 +m2 c=1 (cid:80) (cid:89) From now on we shall remember that the covariance is diagonal in momentum space, with diagonal values δ( p ) C(p(cid:126)) = c c . (2.10) p(cid:126)2 +m2 (cid:80) hence defining the set P = {p(cid:126) ∈ Z6 | p = 0} of momenta satisfying the gauge constraint, c c all Green functions of our theory can in fact be defined for restricted momenta p(cid:126) ∈ P, or if one (cid:80) prefers, are zero outside P. 5Additional insertions of P on the vertex would result in the same Feynman amplitudes, since P2 =P. 5 TGFT interactions by definition belong to the tensor theory space [17, 18, 21] spanned by U(N)⊗d invariants. Hence the most general polynomial interaction is a sum over a finite set B of such invariants b, also called d-bubbles, associated with different coupling constants t b ¯ ¯ S (φ,φ) = t I (φ,φ), (2.11) int b b b∈B (cid:88) where I is the connected invariant labeled by the bubble b. Graphically, each bubble is associated b with a bipartite d-regular edge-colored graph. Each color c ∈ {1,2,..,d} is associated with a ¯ half-line at each vertex, and each vertex bears respectively a field φ or its complex conjugate φ according to its black or white color. The edge coloring of the bipartite graph allows to visualize the U(N)⊗d invariance by showing the exact pairing of fields and anti-fields argument of the same color. Such graphs also enable to visualize whether the interaction is connected or not. Some examples of connected invariants at ranks d = 3 and d = 6 are shown in Figure 1. Figure 1: Some connected tensor invariants The Feynman amplitudes of the perturbative expansion are associated with Feynman graphs whose vertices belong to the set B of the interaction d-bubbles. A Wick contraction is represented by a dotted line. Figure 2 gives an explicit example for d = 3. Figure 2: A tensorial vacuum (N=0) rank-three Feynman graph For a Feynman graph G, we note V(G), L(G) and E(G) the sets of the vertices (the d-bubbles), internal (dotted) lines and external (dotted) half-lines, and V(G), L(G) and E(G) = 2N(G) the number of elements in these sets. The number of vertices V is also identified with the order of perturbation, also often noted n. The Green functions are given by a sum over Feynman graphs (connected or not) 1 G = (−t )nb(G) A , (2.12) 2N b G s(G) (cid:32) (cid:33) G,E(G)=2N b∈B (cid:88) (cid:89) 6 where n is the number of vertices of type b and s(G) is the graph symmetry factor (dimension b of the automorphism group). Note that expanding each vertex b as a d-regular bipartite edge- colored graph as in Figure 1 and coloring the dotted lines with a new color 0, any such graph G is therefore canonically associated to a unique (d+1)-regular bipartite edge-colored graph, for which the vertices are the black and white nodes, as shown in Figure 2. Hence it defines an associated d-complex, in which in particular faces are easily defined as the bi-colored connected components [17, 18]. These faces are either closed or open if they end up on external half-lines. The connected Green functions or cumulants Gc are obtained by restricting sums such as 2N (2.12) to connected graphs G, and are obtained from the generating functional ¯ ¯ W(J,J) = log[Z(J,J)] (2.13) through ∂2NW(J¯,J) Gc ((cid:126)g ,··· ,(cid:126)g ,(cid:126)g(cid:48),···(cid:126)g(cid:48) ) = . (2.14) 2N 1 N 1 N ∂J1((cid:126)g1)∂J¯1((cid:126)g(cid:48)1)···∂JN((cid:126)gN)∂J¯N((cid:126)g(cid:48)N) J=J¯=0 (cid:12) (cid:12) The vertex functions Γ are obtained by restricting sums such as (2.12) to on(cid:12)e particle irreducible 2N amputatedgraphsG (amputationmeanwereplacealltheexternalpropagatorsfordottedhalf-lines ¯ by 1). They are the coefficients of the Legendre transform of W(J,J). Using the convolution properties of the heat kernel (following from the composition properties of its random path representation), the Feynman amplitude A of G can be expressed in direct G space as [35] ∞ (cid:126) A = dα e−α(cid:96)m2 dh K h(cid:15)(cid:96)f × G  (cid:96) (cid:96) α(f) (cid:96)∈∂f (cid:96)  (cid:96)∈L(G)(cid:90)0 (cid:90) f∈F(G) (cid:18) (cid:19) (cid:89) (cid:89) (cid:89)    (cid:126) K g h(cid:15)(cid:96)fg−1 . (2.15)  α(f) s(f) (cid:96)∈∂f (cid:96) t(f)  f∈F(cid:89)ext(G) (cid:18) (cid:89) (cid:19)   In this expression, F(G) is the set of internal faces of the graph, F (G) the set of external faces, ext and (cid:15) the adjacency matrix which is non zero if and only if the line (cid:96) belongs to the face f and (cid:96)f is ±1 according to their relative orientation. We noted α(f) = α the sum of Schwinger (cid:96)∈∂f (cid:96) parameters along the boundaries-lines of the face f, and g or g the boundary variables in s(f) t(f) (cid:80) the open face f, s for “source” and t for “target” variables. We use also the notation F for the set of faces and F for its cardinal (number of elements). These amplitudes A can be interpreted as lattice gauge theories defined on the cellular com- G plexes dual to the Feynman diagrams G. The group elements h (resp. g , g ) define a discrete (cid:96) s(f) t(f) gauge connection associated to the edges (cid:96) (resp. boundary edges) of the cellular complex, and the ordered products (cid:126) h(cid:15)(cid:96)f (resp. g (cid:126) h(cid:15)efg−1 ) are its holonomies (discrete curvature) (cid:96)∈∂f (cid:96) s(f) e∈∂f e t(f) associated to bulk (resp. boundary) faces of the same complex6. (cid:81) (cid:81) Duetothediagonalcharacterofthepropagatorinmomentumspace,theseFeynmanamplitudes are easier to express in the momentum representation. In particular the momentum conservation 6 In models of 4d quantum gravity that bear a closer relation with loop quantum gravity, and that encode more extensively features of simplicial geometry, additional conditions called simplicity constraints are imposed [10, 11, 12]. Obviously, they complicate the structure of the amplitudes, making them richer. We do not consider these additional constraints here. 7 along faces due to the δ functions in (2.9) ensures that when expressed in momentum space non- zero Green functions of the theory of order 2N must themselves develop into sums over U(N)⊗d tensor-invariants of the momenta of order N; in other words to any entering momentum p must c correspond an exiting momentum with same value p(cid:48) = p . In particular the two point function c c in momentum space is a function G (p(cid:126)) of a single momentum p(cid:126) ∈ Zd, and the connected four 2 point function Gc is a sum over all quartic invariants of the theory. In general the contribution 4 of a given specific tensor invariant is complicated to extract from the Green functions. It requires a somewhat subtle decomposition using Weingarten functions, which we shall not detail here, referring the reader to [44, 45]. 2.2 The Quartic Melonic U(1)-model in dimension 6 After this quick overview of general TGFTs, we come to the particular model studied in this paper, namely the d = 6 Abelian quartic model with melonic interactions. It is the simplest just-renormalizable model (with no simplicity constraints) in the classification of gauge invariant TGFT models [34]. As such, it is also the simplest interesting testing ground for the analytic techniques we develop here. General quartic interactions at rank 6 are of the three types indicated in Figure 3. Melonic interactions correspond to the type 1. They are leading in the 1/N tensorial expansion and are marginal in the renormalization group (RG) sense, the other ones being irrelevant. 1 2 3 Figure 3: The quartic tensor interactions at rank 6 Hence the interaction part of the action considered from now on is the sum of all the bubbles of type 1. There are 6 of them, characterized by a unique index c referring to the special color which colors the two lonely lines of the bubble: 6 ¯ S = λ Tr (φφ). (2.16) int c bc c=1 (cid:88) More explicitly a quartic interaction b with special color 1 writes 1 Tr (φ¯φ) = d(cid:126)gd(cid:126)g(cid:48)φ¯(g ,g ,··· ,g )φ(g(cid:48),g ,··· ,g )φ¯(g(cid:48),g(cid:48),··· ,g(cid:48))φ(g ,g(cid:48),··· ,g(cid:48)) b1 1 2 6 1 2 6 1 2 6 1 2 6 (cid:90) = φ¯(p ,p ,··· ,p )φ(p(cid:48),p ,··· ,p )φ¯(p(cid:48),p(cid:48),··· ,p(cid:48))φ(p ,p(cid:48),··· ,p(cid:48)), (2.17) 1 2 6 1 2 6 1 2 6 1 2 6 p(cid:126),p(cid:126)(cid:48) (cid:88) where the last line is written in Fourier space. Remark that since only fields satisfying the prop- agator constraints p = 0 can contribute, in (2.17) we must have p = p(cid:48). Hence each Tr is a c 1 1 bc (cid:80) 8 function of fields with 9 (rather than 10) independent strand momenta, because p = p(cid:48). We can c c therefore in our model simplify (2.17) into Tr (φ¯φ) = φ¯(p ,p ,··· ,p )φ(p ,p ,··· ,p )φ¯(p ,p(cid:48),··· ,p(cid:48))φ(p ,p(cid:48),··· ,p(cid:48)). (2.18) b1 1 2 6 1 2 6 1 2 6 1 2 6 p(cid:126)∈P,p(cid:126)(cid:88)(cid:48)∈P|p1=p(cid:48)1 From now on we consider only the color-symmetric case λ = λ ∀c = 1,··· ,6. c As remarked, Green functions in momentum space develop into sums of tensor invariants. In particular the connected four point function Gc develops over all quartic invariants (connected 4 or not). Hence it develops over the connected invariants of Figure 3 and over the disconnected invariant which is the square of the quadratic invariant. This may seem dangerous at first sight since to be renormalizable our model should not involve in particular renormalization of invariants of type 2 and 3 which are not part of the initial interaction. As well known, renormalization is best stated in terms of the vertex functions Γ. Hence we shall be particularly interested in computing the two point vertex function or self-energy Γ (p(cid:126)) 2 and the four point vertex function Γ (p(cid:126) ,...,p(cid:126) ). These functions are a priori defined on P or P2. 4 1 4 However we shall see that their divergent part is simpler. More precisely we shall define melonic parts Γmelo(p(cid:126)) and Γmelo(p(cid:126) ,...,p(cid:126) ) for these vertex functions, and even a refined monocolor melonic 2 4 1 4 part Γmelo (p ,p(cid:48)) of Γmelo(p(cid:126) ,...,p(cid:126) ), such that Γ (p(cid:126))−Γmelo(p(cid:126)) and Γ (p ,p(cid:48))−Γmelo (p ,p(cid:48)) 4,mono c c 4 1 4 2 2 4,mono c c 4,mono c c are superficially convergent (hence truly convergent after all divergent strict subgraphs have been renormalized). More precisely we shall prove that Theorem 1 There exist two (ultraviolet-divergent) functions f and g of a single strand momentum p ∈ Z such that 6 Γmelo(p(cid:126)) = −λ f(p ), Γmelo (p ,p(cid:48)) = −λδ(p ,p(cid:48))g(p ). (2.19) 2 c 4,mono c c c c c c=1 (cid:88) and such that Γ (p(cid:126))−Γmelo(p(cid:126)) and Γ (p ,p(cid:48))−Γmelo (p ,p(cid:48)) are superficially convergent (hence 2 2 4,mono c c 4,mono c c truly convergent after all divergent strict subgraphs have been renormalized). All higher order vertex functions are also superficially convergent. In particular Γmelo and Γmelo (p ,p(cid:48)) both depend in fact of a single non-trivial function, respec- 2 4,mono c c tively f and g, of a single strand momentum in Z. We shall prove that the special form (2.19) of the primitive divergencies of the theory is compatible with the renormalization of the couplings in (2.18). Inthenextsectionweintroducetheintermediatefieldrepresentationinwhichthefunctions f and g are particularly simple to represent graphically and to compute. 2.3 The intermediate field formalism The intermediate field formalism is a mathematical trick to decompose a quartic interaction in terms of a three-body interaction, by introducing an additional field (the intermediate field) in the partition function. It is based on the well-known property of Gaussian integrals: +∞ √ dxe−x2/2eiκxy = πe−κ2y2/2. (2.20) (cid:90)−∞ We first apply the general method without exploiting gauge invariance, then stress the simplifica- tionduetogaugeinvariance. Thismeanswestartwith(2.17)whichwewanttoexhibitasasquare. 9 For this we introduce the six auxiliary matrices φ¯(p ,p ,··· ,p )φ(p(cid:48),p ,··· ,p ) = M , p2,···,p6 1 2 6 1 2 6 p1,p(cid:48)1 ¯ which are quadratic in terms of the initial φ and φ and can be thought as partial traces over color (cid:80) indices other than 1. The interaction in (2.17) can be rewritten as Tr (φ¯φ) = tr M2 , (2.21) b1 where tr means a simple trace in (cid:96)2(Z). Using many times (2.20) we can decompose this square interaction tr M2 with a new Hermitian matrix σ corresponds graphically to “pinching” the two 1 special strands of color 1 with this matrix field, as indicated in Figure 4. More precisely √ e−λtr(M2) = dσ1e−tr(σ12)/2ei 2λtr(σ1M). (2.22) dσ e−tr(σ2)/2 (cid:82) 1 1 (cid:82) σ1 2 1 Figure 4: Intermediate field decomposition Thenextstepistomakethisdecompositionsystematicforthesixmelonicinteractions. Writing tr(σ M) = Trφ¯Σ1φ, (2.23) 1 where Σ1 = σ ⊗I⊗I⊗I⊗I⊗I (2.24) 1 acts in the large tensor space (cid:96)2(Z)⊗6 and Tr means a trace in this large tensor space, allows to express the previous intermediate field decomposition as √ eλtrM2 = dσ1e−tr(σ12)/2ei 2λTrφ¯Σ1φ. (2.25) dσ e−tr(σ2)/2 (cid:82) 1 1 Using color permutation, we decompose all(cid:82)six bubbles in this way. An intermediate field σ is c therefore associated to each quartic bubble b with weak color c. The operators c Σc = I···⊗σ ⊗···I (2.26) c commute in the tensor space (cid:96)2(Z)⊗6, as they act on different strands. Introducing Σ = 6 Σc, c=1 we can rewrite the partition function of the original theory as (cid:80) √ Z(J¯,J) = dµ (φ,φ¯)e−Sint(φ¯,φ) = dµ (φ,φ¯)dν(σ)e−φ¯·J−φ·J¯ei 2λTrφ¯Σcφ, (2.27) C C (cid:90) (cid:90) the normalized Gaussian measure dν(σ) being factorized over colors with trivial covariance iden- tity on each independent coefficient (Gaussian unitary ensemble). The tensor integral becomes Gaussian, hence can be computed as a determinant. We find: 10