Large N limit of irreducible tensor models: O(N) rank- 3 tensors with mixed permutation symmetry
aa r X i v : . [ h e p - t h ] J un Large N limit of irreducible tensor models: O ( N ) rank- tensors with mixed permutation symmetry Sylvain Carrozza Perimeter Institute for Theoretical Physics31 Caroline St N, Waterloo, ON N2L 2Y5, Canada
Abstract
It has recently been proven that in rank three tensor models, the anti-symmetric and symmetrictraceless sectors both support a large N expansion dominated by melon diagrams [1]. We show howto extend these results to the last irreducible O ( N ) tensor representation available in this context,which carries a two-dimensional representation of the symmetric group S . Along the way, weemphasize the role of the irreducibility condition: it prevents the generation of vector modes whichare not compatible with the large N scaling of the tensor interaction. This example supports theconjecture that a melonic large N limit should exist more generally for higher rank tensor models,provided that they are appropriately restricted to an irreducible subspace. Contents
The first tensor models governed by a tractable /N expansion were discovered a few years ago [2, 3, 4],and the methods developed in these early works have been generalized in various ways since then[5, 6, 7, 8, 9]. Irrespectively of the rank r ≥ of the tensor degrees of freedom, tensor models turn out [email protected]
1o be generically dominated by a rather simple but non-trivial class of Feynman graphs, going underthe name of melon diagrams [10, 11, 12].For some time, this new species of large N expansion has mostly found applications as a tool togenerate and understand random geometries in dimension d > , either in the context of tensor modelsthemselves [13, 14, 15, 16, 17] or in group field theory [18, 19, 20, 21, 22, 23, 24, 25].In the standard nomenclature of quantum field theories, the tensor models of random geometryare -dimensional field theories. It is only recently that the large N melonic limit of tensor degreesof freedom has started to be taken advantage of in the context of quantum mechanics and quantumfield theory in more than one dimension. It has first been recognized by Witten [26] that the meloniclarge N expansion of tensor models allows to emulate some properties of the celebrated SYK models[27, 28, 29, 30] in the familiar context of large N quantum mechanics (that is without disorder average).Another version of such models was then introduced by Klebanov and Tarnopolsky [31]. Both modelshave been investigated in some detail since then [32, 33, 34, 35, 36, 37, 38, 39], and have led to a varietyof generalizations, including (but not restricted to): higher space-time dimensions [40, 41, 42, 43], higherorder interactions [44], and new asymptotic expansions for matrix-tensor models [45, 46, 47, 48].A key technical ingredient in all these instances of melonic behaviour is the presence of severalindependent copies of a symmetry group G (e.g. O ( N ) or U ( N ) ), typically one for each index of thetensor, which greatly simplifies the combinatorial structure of the Feynman diagrams. In particular, itseemed crucial to impose no symmetry at all among the various indices of a given tensor. Because of itsimplications at the level of the Feynman amplitudes – which can be indexed by colored cell complexesand colored graphs [49] – the tensor models enjoying such a large symmetry have been going underthe name of colored or uncolored models .The colored structure of the Feynman diagrams of colored and uncolored tensor models plays sucha heavy role in the original proofs of the existence of their large N expansion [12], that until recently itseemed unlikely that such results could be generalized to tensor models with less symmetry. However,motivated by the key conjecture and numerical evidence reported in [50], it was recently proven thatboth antisymmetric and symmetric traceless O ( N ) rank- tensors support a melonic large N expansion[1]. The proof relies in part on methods first developed in [51], which successfully tackled a somewhatsimpler tensor model involving two symmetric tensors.As one quickly realizes, a central technical ingredient of the arguments put forward in [50, 1] isthe irreducibility of the subspace of tensors on which the model is based (indeed, antisymmetric andsymmetric traceless tensors both carry an irreducible O ( N ) representation). This observation suggeststhe following question: Is the melonic large N limit generalizable to irreducible tensors of arbitrary rank r ≥ ? In this paper, we make a step in the direction of an affirmative answer. We focus on the third availableclass of rank- irreducible tensor spaces, whose elements transform as a two-dimensional representationof the permutation group S . We show that the methods and results of [1] do generalize to this lastrank- invariant space, providing another instance of melonic large N behaviour in tensor models andtensor field theories.The paper is organized as follows. In section 2, we first introduce the class of rank- tensor modelswe wish to consider, and identify a potential instability triggered by the trace modes of the tensor. Wethen define more precisely the mixed symmetry model we will consider in the later sections, in whichsuch trace modes have been correctly removed. In section 3, we introduce a perturbative expansion ofthe model, indexed by Feynman maps and stranded graphs. Following [1], in section 4 we identify andproceed to a resummation of the infinite family of melon-tadpole maps. This allows to introduce a newperturbative expansion in section 5, which is proven to admit a well-defined /N expansion dominatedby melon maps. We close with a conclusion and some general comments. Colored models, such as [2] and [26], describe the dynamics of several species of fields, each labelled by a color. Incontrast, uncolored models such as [5] and [9, 31] describe the dynamics of a single tensor, hence their name. However,both colored and uncolored models have a symmetry of the type G k ( k ≥ ) resulting from the absence of symmetrizationor antisymmetrization of their indices. Irreducible O(N) tensor models in rank three
We introduce O ( N ) rank three tensor models in general, and define more precisely the theory withmixed symmetry which is the main subject of this note. We consider a bosonic theory in zero dimension with O ( N ) global symmetry. The degrees of free-dom are organized into a real rank- tensor T a a a transforming as a product of three fundamentalrepresentations : ∀ O ∈ O ( N ) , T a a a → [ O · T ] a a a := O a b O a b O a b T b b b . (1)The scalar product h·|·i is the standard one, namely: h T | T ′ i = T a a a T ′ a a a .The partition function is taken to be of the general form : Z N ( λ ) := Z d µ P ( T ) exp (cid:18) λ N / T i i i T i i i T i i i T i i i (cid:19) . (2)where P is an O ( N ) -invariant covariance which will be fixed in subsection 2.4. The N − / scaling ofthe interaction is the standard one, and is necessary to the existence of a melonic large N expansionwith ’t Hooft coupling λ . Remark however that the quartic interaction – which is invariant under O ( N ) , as it should – is not invariant under a larger O ( N ) symmetry, so that our Ansatz can not fallin the class of colored tensor models studied in [9].The quartic interaction defines a map: V ( T ( a ) , T ( b ) , T ( c ) , T ( d ) ) = V a a a ,b b b ,c c c ,d d d T ( a ) a a a T ( b ) b b b T ( c ) c c c T ( d ) d d d , (3)with kernel V a a a ,b b b ,c c c ,d d d := δ a b δ b c δ c d δ d a δ a c δ b d (4)As standard in the literature, we will represent the interaction kernel as an ordinary four-valent vertex,or as a stranded diagram – as shown in Fig. 1. In the stranded representation, each strand representsa contraction by a Kronecker delta, according to (4). As a result of this non-trivial combinatorialstructure, the representation of the vertex kernel as an ordinary vertex is sensitive to its embeddingin the plane. One can also represent such an interaction by the boundary graph shown on the rightpanel of Fig. 1. As this is the complete graph on four vertices, we propose to call such an interactiona complete interaction . By this definition, there exists other complete interactions (for instance the O ( N ) -invariant interaction of [9]) that one might want to include in the action. However, they allbecome equivalent upon reduction to a tensor in a fixed irreducible representation of S , so it issufficient for our purpose to consider only one such interaction. As we now explain, not all choices of propagator P (with reasonable scaling in N ) lead to an interestinglarge N expansion. For instance, let us consider the natural (but eventually unfortunate) choice of freepropagator: h T a a a T b b b i = Z d µ P ( T ) T a a a T b b b = δ a b δ a b δ a b , (5) We assume Einstein’s summation convention throughout the paper. We adopt the standard sign convention of matrix models. Note that the interaction as no reason to be bounded.At this stage, Z N ( λ ) is considered as a formal power series in λ . But as usual, the leading order sector at large N willeventually turn out to be summable. See e.g. [1] for a general definition of boundary graph in tensor models. We will not make further use of this notionin the present paper. It is also sometimes known as a tetrahedral interaction, since the complete graph on four vertices is the one-skeletonof a tetrahedron. ( a ) a a a T ( c ) c c c T ( d ) d d d T ( b ) b b b a a a c c c b b b d d d Figure 1: Graphical representations of the interaction kernel: as an ordinary (embedded) vertex (left);and as a stranded diagram (middle). The associated boundary graph (right) is the complete graph onfour vertices.corresponding to a free action S free = h T | T i . This theory leads to amplitudes which diverge arbitrarilyfast in N and therefore can not support a well-behaved melonic large N expansion. As we will seeshortly, the problem can be boiled down to the presence of vector modes arising from the traces of thetensor T , which couple in an ill-behaved way to the traceless part of T .In more detail, one can decompose the tensor T a a a into: T a a a = T a a a + 1 √ N ( χ a δ a a + ϕ a δ a a + ψ a δ a a ) , (6)where T is a completely traceless tensor ( T aab = T aba = T baa ), while χ , ϕ and ψ are vectors which canbe expressed in terms of the three traces of T . For instance, one finds: ϕ i = √ NN + N − − T aai + ( N + 1) T aia − T iaa ) . (7)With the naive covariance (5), the propagator of ϕ at leading order in /N is appropriately normalized(up to /N corrections) h ϕ i ϕ j i ∼ δ ij . (8)The vector modes interact among themselves, and also with T . We provide two examples, one ofwhich does spoil the large N structure of the model. ϕ i ϕ i ϕ j ϕ j ∼ λN / ( ϕ i ϕ i ) ϕ i ϕ i T abc T abc ∼ λN / ( ϕ i ϕ i ) (cid:0) T abc T abc (cid:1) T T ϕ ϕ ∼ Figure 2: Graphical representations of two of the interactions involving the vector ϕ i : the self-interaction on the left is sufficiently suppressed in /N , while the double-trace interaction on theright leads to an instability.Let us start with the induced ϕ self-interaction: λN / (cid:18) √ N (cid:19) N ( ϕ i ϕ i ) ∼ λN / ( ϕ i ϕ i ) , (9)with one factor of N coming from the contraction of delta functions represented on the left panel ofFig. 2. Since the scaling leading to a sensible large N expansion for such a vector interaction is /N [52], this particular coupling does not preclude the existence of a melonic large N expansion.However, there are also more problematic couplings between T and the vector degrees of freedom,such as: λN / (cid:18) √ N (cid:19) ϕ i ϕ i T abc T abc ∼ λN / ( ϕ i ϕ i ) (cid:0) T abc T abc (cid:1) . (10)4ee again Fig. 2 for a graphical representation. This double-trace interaction does generate arbitrarylarge powers of N , for instance through chains of tadpole graphs, as depicted in Fig. 3. Each tadpoleloop in such a diagram generates a factor of N which is not entirely balanced by the /N / scalingof the interaction. This leads to amplitudes scaling as ∼ N p/ , where p is the number of tadpoles inthe chain. In [1], this type of pathological behaviour is referred to as a trace instability , and has beenexplicitly identified in a symmetric (but not traceless) tensor model. ϕ ϕT T T T Figure 3: Chains of tadpole loops leading to ϕ i two-point functions of arbitrary high order in N .There are two ways of curing such problems. One is to introduce as many independent new ’t Hooftcouplings as needed to tame the problematic large N contributions . For instance, one would need toinclude a new independent coupling to parametrize the "renormalized" interaction ( ϕ i ϕ i ) (cid:0) T abc T abc (cid:1) .This procedure might lead to the definition of interesting vector-tensor models. For the purpose ofthe present paper, we prefer to stick to pure tensor models. We therefore adopt a simpler strategy,consisting in choosing P such that no vector modes at all can propagate in the model. Imposing thetraceless condition by itself is not necessarily sufficient (since mixing between sub-representations maypossibly generate such vector contributions), but working with an irreducible representation is. For the convenience of the reader, we briefly review how the tensor product of three fundamental O ( N ) representations decomposes into irreducible representations (see e.g. [53, 54] for additional details).There are two operations which obviously commute with the O ( N ) action (1): 1) taking the trace overtwo indices; and 2) permuting the indices in an arbitrary way. Moreover, it turns out they are the onlyindependent transformations which needs to be reduced.Let us first discuss the action of the permutation group S , which we take to be: ∀ σ ∈ S , ( σ ⊲ T ) a a a := T a σ (1) a σ (2) a σ (3) . (11)The irreducible representations of S are labelled by Young tableaux with three boxes. One there-fore associates a Young tableau to each tensor subspace with fully reduced permutation symmetry.Graphically, one has:1 ⊗ ⊗ = ⊕ ⊕ ⊕ th index of the tensor. A Young operator is thenuniquely associated to each Young tableau generated on the right-hand side of the equality: it imposesa symmetrization over the indices appearing in a same row, followed by an antisymmetrization of theindices appearing in a same column. In particular, the Young tableau with a single row (resp. column)corresponds to the subspace of completely symmetric (resp. antisymmetric) tensors. The last two sub-representations have mixed symmetry, in the sense that they carry a two-dimensional representation I would like to thank E. Witten for pointing this out. S . They are furthermore equivalent. Finally, the dimensions of these vector spaces are: dim (cid:0) (cid:1) = 16 N (cid:0) N + 3 N + 2 (cid:1) , dim = 16 N ( N −
1) ( N − , (13) dim (cid:18) (cid:19) = dim (cid:18) (cid:19) = 13 N (cid:0) N − (cid:1) . In the rest of the paper, we will construct a tensor model with mixed symmetry. For definiteness, wewill focus on the tableau 1 23 . Its associated Young operator S acts on an arbitrary tensor T as: ( S T ) a a a := 13 ((1 − (1 3)) (1 + (1 2)) ⊲ T ) a a a = 13 ( T a a a + T a a a − T a a a − T a a a ) , (14)yielding a tensor which is in particular anti-symmetric in a ↔ a . The kernel of S is: S a a a ,b b b := 13 ( δ a b δ a b δ a b + δ a b δ a b δ a b − δ a b δ a b δ a b − δ a b δ a b δ a b ) . (15)In order to reduce to an irreducible tensor representation with order N degrees of freedom, onemust further remove the trace modes (which contain only N degrees of freedom each). There are threeof them: one lies in the completely symmetric part, the other two are in the mixed sectors. For thesethree sectors, the reduction is completed by acting with the orthogonal projector on traceless tensors: Q a a a ,b b b := δ a b δ a b δ a b − δ a a N + N − N + 1) δ a b δ b b − δ a b δ b b − δ a b δ b b ) − δ a a N + N − − δ a b δ b b + ( N + 1) δ a b δ b b − δ a b δ b b ) − δ a a N + N − − δ a b δ b b − δ a b δ b b + ( N + 1) δ a b δ b b ) (16)Hence, we have exactly three irreducible tensor representations at our disposal. The symmetrictraceless and antisymmetric sectors have already been treated elsewhere [1]. The irreducible represen-tation associated to the tableau 1 23 (which is equivalent to that of the tableau 1 32 ) is studied inthe remainder of the present paper. In the decomposition (12), the two representations with mixed symmetry are not orthogonal to eachother with respect to h·|·i . This implies that the projector S is not symmetric, and therefore cannotimmediately be used as a propagator. One may instead consider ˜ P := 34 SS ⊤ . (17)The / coefficient has been chosen to ensure that, not only ˜ P ⊤ = ˜ P , but also ˜ P = ˜ P . Hence ˜ P isan orthogonal projector whose image is included in the image of S . One may furthermore check that Tr ˜ P = 13 N (cid:0) N − (cid:1) = dim (cid:18) (cid:19) = rk S . (18) This is in contrast with the symmetric and antisymmetric sectors, which are decomposed in an orthogonal manner. ˜ P is nothing but the orthogonal projector on the subspace of tensors associated to the tableau1 23 . More explicitly, we have: ˜ P a a a ,b b b = 13 ( δ a b δ a b δ a b − δ a b δ a b δ a b )+ 16 ( δ a b δ a b δ a b + δ a b δ a b δ a b ) (19) −
16 ( δ a b δ a b δ a b + δ a b δ a b δ a b ) We finally define the propagator of the model as: P := Q ˜ PQ = ˜ PQ = Q ˜ P , (20)which is the orthogonal projector on traceless tensors with symmetry 1 23 . A direct calculation showsthat: P a a a ,b b b = 13 ( δ a b δ a b δ a b − δ a b δ a b δ a b )+ 16 ( δ a b δ a b δ a b + δ a b δ a b δ a b ) −
16 ( δ a b δ a b δ a b + δ a b δ a b δ a b ) (21) + 12( N −
1) ( δ a b δ a a δ b b + δ a a δ a b δ b b ) − N −
1) ( δ a b δ a a δ b b + δ a a δ a b δ b b ) The Gaussian measure d µ P is degenerate but defines a suitable covariance: Z d µ P ( T ) T a a a T b b b = P a a a ,b b b . (22)As shown in Fig. 4, it is convenient to represent the propagator as a plain line, which decomposesfurther into triplets of strands. Each triplet represents one of the ten patterns of Kronecker deltacontractions appearing in (21), analogously to the representation adopted for the vertex. We will call broken (resp. unbroken ) the last four (resp. the first six) terms of the propagator, and will labelthem accordingly by the letter B (resp. U ). More precisely, we will separate the different broken andunbroken configurations into subclasses B , B , U , . . . , U , as shown in Fig. 4. The Feynman expansion is indexed by combinatorial objects associated to gluings of propagator linesand interaction vertices. We distinguish between the plain line representation, which gives rise to
Feynman maps , and the more fine-grained stranded representation, which yields stranded graphs . The Feynman maps are the combinatorial objects which directly label the various terms obtained byWick contraction in the perturbative Feynman expansion. Since the interaction vertex is naturallyequipped with a cyclic order – or equivalently a local embedding into the plane –, the theory is mostnaturally expanded in terms of combinatorial maps . In the present model the combinatorial maps ofinterest are furthermore -regular, a qualifier we will keep implicit in the rest of the paper. A combinatorial map is a formalization of the notion of graph embedded into an orientable surface – or equivalentlya ribbon graph. The reader is referred to e.g. [55, 56] for further details about these standard definitions. a a a T b b b a a a b b b − ( ) + ( ) + ( ) + ( ) N − + + ( ) N − − + − = U U U U UBB B Figure 4: Graphical representation of the propagator (21). Stranded configurations have been organizedinto several classes of broken ( B ) and unbroken ( U ) configurations.In particular, the full connected two-point function can be expanded as a sum over amplitudes ofconnected two-point maps: C a a a ,b b b := h T a a a T b b b i c = X connected 2 − point maps M λ V ( M ) A ( M ) a a a ,b b b . (23)Its trace C := C a a a ,a a a in turn expands as a sum over vacuum rooted maps : C = N (cid:0) N − (cid:1) X connected rooted maps M λ V ( M ) A ( M ) , (24)where the first term is the trace of the free propagator P .We denote by V ( M ) (resp. E ( M ) ) the number of vertices (resp. edges) in a Feynman map M . If M is a vacuum map, its -regular nature implies that V ( G ) = E ( G ) .Finally, we note that the vertex is invariant under any permutation that preserves the cyclic orderof its external legs up to orientation. For instance: V ( T ( a ) , T ( b ) , T ( c ) , T ( d ) ) = V ( T ( b ) , T ( c ) , T ( d ) , T ( a ) ) = V ( T ( a ) , T ( d ) , T ( c ) , T ( b ) ) , (25)which may be represented pictorially as: ( a ) ( b ) ( c )( d ) ∼ ∼ ( a ) ( c )( d )( b )( a ) ( b )( c )( d ) We also remark that, in contrast to the antisymmetric and symmetric traceless models, the vertex isnot invariant under the full permutation group of its legs. Hence, it is essential to keep track of theembedding information encoded in a Feynman map. This being said, we will often implicitly identifytwo Feynman maps which are in the same equivalent class under reversal of the cyclic orders aroundtheir vertices , since these have the same amplitude. A rooted map is a map with one edge marked by an arrow. In the language of [1], it is the contribution of the ring diagram . Such intermediate combinatorial objects, which are neither abstract graphs nor maps, are sometimes referred to as cyclically ordered graphs [56]. While maps can be thought of as ribbon graphs, which are collections of discs connectedby ribbons, cyclically ordered graphs can be thought of as collections of discs connected by strings. .2 Stranded graphs Each vacuum map M further decomposes into V ( M ) stranded graph configurations G , each corre-sponding to a choice of one among ten terms of (21) for every edge of M . The patterns of identificationsassociated to a choice of stranded configuration results in a collection of closed cycles of strands, whichare called faces . We denote by ˆ G ( M ) the set of stranded configurations of M , and by F ( G ) (resp. B ( G ) , U ( G ) , etc.) the number of faces (resp. broken edges, unbroken edges, etc.) of G ∈ ˆ G ( M ) . Thestranded graphs come with a definite scaling in N , each face bringing one factor of N . More precisely,it is easy to show that: A ( M ) = X G ∈ ˆ G ( M ) ǫ ( G ) R ( G ) N − ω ( G ) , (26)where ω ( G ) is the degree of the stranded configuration Gω ( G ) = 3 + 32 V ( G ) + B ( G ) − F ( G ) , (27)while ǫ ( G ) and R ( G ) are respectively a sign and a normalization factor of order N ǫ = ( − U + U + B , (28) R = 3 U + U U + U (2(1 − /N )) B . The main subtlety in such models – in comparison with their simpler colored and uncolored cousins –is that not all stranded graphs have positive degree. For instance, one may generate arbitrarily negativedegrees by chaining tadpoles as represented in Fig. 5. The specific tadpole configurations giving riseto such pathological amplitudes have earned the name of bad tadpoles in [1]. ∼ N / ∼ N / N / ! p ∼ N p/ N | {z } p Figure 5: Chains of bad tadpole configurations, yielding unbounded degrees.Hence the existence of the /N expansion must rely on delicate cancellations. Given our generaldiscussion of section 2.2, where we emphasized the importance of removing the trace modes, thisfeature is of course to be expected. A quick inspection of Fig. 5 shows that the chains of bad tapolesdo indeed correspond to the propagation of trace modes, and are essentially identical to those of Fig. 3.We therefore expect them to be correctly compensated by terms with opposite signs in the presenttraceless model, which we will confirm explicitly.In more detail, we follow the general method introduced in [1], and introduce a family of mapsrequiring special attention: the melon-tadpole maps. An elementary melon is defined as a connectedtwo-point map with two vertices, while a tadpole is simply a two-point map with a single vertex. Wethen call melon (or melonic ) any connected two-point map obtained from the bare propagator byrecursively replacing edges of the graph by elementary melons. We similarly call melon-tadpole anyconnected two-point map obtained by recursively replacing edges by elementary melons or tadpoles.Finally, we will also call generalized tadpole (resp. generalized melon ) a tadpole (resp. melon) inwhich one of the internal edges has been replaced by a non-trivial two-point graph. These notions areimportant because of the following proposition, which we recall without proof. Note that the stranded graphs we discuss in this paper are always vacuum graphs. roposition 1 ([1], propositions 3 and 4) Let G be a connected stranded graph.i) If G has no melon and no tadpole, then ω ( G ) ≥ .ii) If G has no generalized melon and no generalized tadpole, then ω ( G ) ≥ / . To prove that the model defined in (2) and (21) admits a large N expansion dominated by melonmaps, we can follow [1] and proceed in three steps:1. Prove that the family of melon-tadpoles can be resummed in a controlled manner, and that itssum reduces to the melonic two-point function in the large N limit.2. Define a new perturbative expansion indexed by maps with no melons and no tadpoles, whichby Proposition 1 i) will admit a well-defined large N expansion.3. Prove that generalized tadpoles cannot be leading order; use Proposition 1 ii) and item 1 todeduce that the full two-point function of the model reduces to the melonic two-point functionin the large N limit. In this section we resum the infinite family of melon-tadpole two-point maps, and show that theresulting two-point function is a simple renormalized version of the bare propagator.
To begin with, we remark that because P is an orthogonal projector onto an irreducible O ( N ) ten-sor representation, any two-point map must have an amplitude proportional to the bare propagator.Indeed, by construction any two-point map defines an O ( N ) intertwiner which leaves the space of irre-ducible 1 23 tensors invariant. By Schur’s Lemma it is therefore proportional to the identity on thissubspace. Since it is guaranteed to vanish on the orthogonal complement, it must thus be proportionalto the projector P itself.Graphically, we simply have: ∝ Let us denote by K the melon-tadpole two-point function. Our simple observation guarantees theexistence of a function K ( λ, N ) such that: K = K ( λ, N ) P . (29)In the remainder of this section, we will prove that K ( λ, N ) is a series in √ N and therefore admits awell-defined large N expansion. To get there, we first need to compute the amplitudes of tadpoles andelementary melons. We start by analyzing the tadpole contributions. There are two inequivalent tadpole Feynman maps:10he first (which we call planar) has multiplicity , the second (which we call non-planar) has multi-plicity .The amplitude of a planar tadpole can be shown to evaluate to A ( ) = λ ( N − N − N / ( N − P a a a ,b b b , (30)while that of a non-planar tadpole yields A ( ) = λ N − N / ( N − P a a a ,b b b . (31)We remark that both amplitudes have prefactors scaling as N / , as is to be expected: indeed, giventhat such maps have only one cycle, their stranded configurations can have at most one face, yieldinga maximal scaling NN / . More interestingly, the broken configurations of P are weighted by an overallfactor ∼ N / , which is to be contrasted with the N / we would naively expect from bad tadpoleconfigurations. These problematic configurations simply average to zero, as we were hoping for. Hav-ing recognized that A ( G ) must be proportional to P , we could actually have anticipated it, withoutexplicitly computing the amplitudes. The scaling of broken edges in P being always suppressed by afactor ∼ N with respect to the unbroken ones, it suffices to realize that the unbroken strands are atmost of order N − / to conclude that the broken ones must combine in such a way that they do notcontribute before order N − / .We finally introduce a function f T ( N ) parametrizing the total contribution of tadpole maps: A ( ) + A ( ) = λ ( N − N + 1)2 N / ( N − P a a a ,b b b =: λf T ( N ) P a a a ,b b b . (32)The important fact is that f T ( N ) is a series in √ N with no constant term. We now turn to elementary melon maps. There are six distinct two-point melon maps on two vertices:Due to the invariance of the vertex under cyclic reversal, the melon maps in a same row have the sameamplitude.The amplitude of a melon from the first row is A ( ) = λ N − N − N + 58 N + 120 N − N − N − N P , (33)while a melon from the second row evaluates to A ( ) = λ N − N + 5 N − N + 105 N + 277 N − N ( N − P . (34)In contrast to tadpoles, melons have contributions of order , they will therefore contribute to theleading-order in the large N regime. Their sum is: A ( ) + 4 A ( ) = λ N − N − N − N + 21 N + 4212 N ( N − P := λ f M ( N ) P , (35)where f M ( N ) is a series in /N with non-vanishing constant term.11 .4 Melon-tadpole two-point function As a direct consequence of the recursive definition of melon-tadpoles, the two-point function K verifiesthe Schwinger–Dyson equation of Fig. 6. += K K K K KKK KKKKKK
Figure 6: The Schwinger-Dyson equation of the melon-tadpole two-point function K .Furthermore, we already know that K = K P for some function K ( λ, N ) . As a result, theSchwinger–Dyson equation reduces to the simple algebraic equation: K = 1 + λf T ( N ) K + λ f M ( N ) K . (36)As f T ( N ) → N → + ∞ and f M ( N ) → N → + ∞ / , this equation always has a non-perturbative solution(with K ( λ, N ) → when λ → ) in the large N regime, and for sufficiently small | λ | . K ( λ, N ) canfurthermore be expanded as a series in √ N . Keeping the leading term only, one obtains K ( λ, N ) = K ( λ ) + O (cid:18) √ N (cid:19) , (37)where K ( λ ) is the solution of: K = 1 + λ K . (38)Remarkably, K ( λ ) does not receive any contribution from f T ; it reduces to a sum over melonic two-point amplitudes. Having managed to resum the melon-tadpole family non-perturbatively, we can now rely on an im-proved perturbative expansion, around the free theory associated to the Gaussian measure d µ K .For the connected two-point function and its trace, this gives: C a a a ,b b b = X connected 2 − point maps M without melonwithout tadpole λ V ( M ) K ( λ, N ) E ( M ) A ( M ) a a a ,b b b (39) C − K ( λ, N ) N (cid:0) N − (cid:1) X connected rooted maps M without melonwithout tadpole λ V ( M ) K ( λ, N ) V ( M ) A ( M ) (40)There are only two differences with respect to (23) and (24): 1) the propagator is K = K P ; 2) theFeynman maps being summed over have no melons and no tadpoles. The beauty of this reformulationis that it makes the existence of the large N expansion explicit.12ndeed, since we know from Proposition 1 that ω ( G ) ≥ for any melon-tadpole free stranded graph,we are guaranteed that A ( M ) expands as N times a (formal) series in / √ N , for any melon-tadpole-free vacuum map M . Together with the fact that K ( λ, N ) is itself a series in / √ N , this allows toconclude that both C and C a a a ,b b b have a well-defined large N expansion. The leading order of C is of order N , while that of C a a a ,b b b is of order N , and both expansions are governed by theparameter √ N . We now prove that no Feynman map other than melons contribute to the leading order two-pointfunction. A natural conjecture would be that ω ( G ) ≥ / for any non-melonic stranded diagram G .However, how was realized in [1], this conjecture does not survive closer inspection: there exists maps M which are not melonic, but have vanishing degree stranded configurations. A crucial feature of suchmaps is that they all contain a generalized tadpole. This tells us that the same type of non-trivialcancellations identified in tadpole maps will also be at play here.In more detail, it is convenient to organize the discussion according to the presence or absence ofgeneralized melons and generalized tadpoles (see Fig. 7). Indeed, knowing that the large N expansionof (39) exists, we can be sure that the large N scaling of a generalized melon (resp. a generalizedtadpole) is no higher than that of a melon (resp. a tadpole). Furthermore, the in-depth combinatorialinvestigation of [1] leads to the result of Proposition 1 ii) , which we will rely on.Figure 7: Generalized tadpoles and generalized melons.Consider a two-point map M in the improved perturbative expansion, that is: M is connected,without melon and without tadpole. Let us furthermore suppose that V ( M ) ≥ . We want to provethat M cannot be leading-order.We first argue that if M has a generalized tadpole, then it cannot be leading order. Call T the two-point map attached to the tadpole (which we assume to be planar for definiteness). Then A ( T ) = a ( λ, N ) P with a ( λ, N ) at most of order N (otherwise the /N expansion would not exist:one could construct a graph with arbitrarily high scaling in N by chaining T ). But then A ( T ) ∼ a ( λ, N ) λ √ N P . Calling M ′ the map obtained from M by substituting the generalized tadpole T with a simple propagator, one has A ( M ) ∼ a ( λ, N ) λ √ N A ( M ′ ) . Since A ( M ′ ) scales at most as N ,one concludes that A ( M ) must decay as / √ N or faster.Assume now that M is leading order. By the previous argument together with Proposition 1 ii) , itfollows that M must contain a generalized melon. Let ˜ M be a generalized melon submap of M witha minimal number of vertices. Up to orientation reversal of the vertices, ˜ M has the structure: or G G G G G G with at least one of the subgraphs G i non-empty (otherwise M would have a melon) – say G . In orderto ensure that M is leading order, it is easy to see that G must itself be leading order. But then itmust also contain a generalized melon, which leads to a contradiction given that ˜ M is a generalizedmelon with minimal number of vertices. Hence M cannot be leading order, as we were claiming.13e conclude that all the Feynman maps in the expansion (39) are suppressed by powers of / √ N ,except for the dressed propagator K = K P . Since we have shown that K is itself a series in / √ N whose constant term K is a sum of melonic maps, we may conclude that the full connected two-pointfunction converges to the melonic two-point function in the large N limit.In equation, we have shown that: h T a a a T b b b i c = (cid:16) K ( λ ) + O(1 / √ N ) (cid:17) P a a a ,b b b . (41)where K ( λ ) is the sum of melonic two-point maps. In the -dimensional context of the present paper, K is furthermore a solution of a simple algebraic equation, namely K = 1 + λ K / (it is evenexactly solvable, see for instance [10]). We have proven that irreducible O ( N ) tensor models associated to Young tableaux with the shapeadmit a melonic large N limit. Together with the results of [1], this achieves the proof thatall irreducible rank- O ( N ) tensor models with complete interaction can be organized into a large N expansion dominated by melons. More generally, we conjecture that our construction can be extendedto higher rank irreducible tensors: in rank r , one could consider a complete ( r + 1) -valent interaction.We hope to come back to this question in the near future.It would also be interesting to investigate the vector-tensor models which naturally follow fromthe decomposition of general tensors we relied on in section 2.2. It was invoked to emphasize theinstability introduced by the trace modes of the tensor, which we cured by removing these trace modesaltogether. An alternative strategy would consist in introducing new independent ’t Hooft couplingsparametrizing the various types of interactions generated by the complete tensor interaction. In thisway, one might presumably be able to construct a large N expansion with a leading order includingmixtures of melonic and bubble diagrams.Finally, we would like to emphasize that, even though we restricted our attention to a -dimensionalmodel, the methods of [1] and of the present paper can be applied more generally to bosonic tensor fieldtheories in d ≥ . Provided that the global O ( N ) symmetry is not spontaneously broken, an equationanalogous to (41) will automatically follow. The main difference is that the Schwinger–Dyson equationof the melonic two-point function determining K will not reduce to a simple algebraic equation; it willresult in a more involved melonic integro-differential equation, similar to those found in [31, 40, 42] (in d = 1 and higher). Acknowledgements
I would like to thank Dario Benedetti, Razvan Gurau and Igor Klebanov for useful discussions.This research was supported in part by Perimeter Institute for Theoretical Physics. Research atPerimeter Institute is supported by the Government of Canada through the Department of Innovation,Science and Economic Development Canada and by the Province of Ontario through the Ministry ofResearch, Innovation and Science.
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