Large $N$ limit of irreducible tensor models: $O(N)$ rank-$3$ tensors with mixed permutation symmetry

It has recently been proven that in rank three tensor models, the anti-symmetric and symmetric traceless sectors both support a large $N$ expansion dominated by melon diagrams [arXiv:1712.00249 [hep-th]]. We show how to 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 $S_3$. Along the way, we emphasize the role of the irreducibility condition: it prevents the generation of vector modes which are not compatible with the large $N$ scaling of the tensor interaction. This example supports the conjecture 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.


Introduction
The first tensor models governed by a tractable 1/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 ≥ 3 of the tensor degrees of freedom, tensor models turn out 1 scarrozza@perimeterinstitute.ca to be generically dominated by a rather simple but non-trivial class of Feynman graphs, going under the name of melon diagrams [10,11,12].
For some time, this new species of large N expansion has mostly found applications as a tool to generate and understand random geometries in dimension d > 2, either in the context of tensor models themselves [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 geometry are 0-dimensional field theories. It is only recently that the large N melonic limit of tensor degrees of freedom has started to be taken advantage of in the context of quantum mechanics and quantum field theory in more than one dimension. It has first been recognized by Witten [26] that the melonic large 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 models have been investigated in some detail since then [32,33,34,35,36,37,38,39], and have led to a variety of generalizations, including (but not restricted to): higher space-time dimensions [40,41,42,43], higher order 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 several independent copies of a symmetry group G (e.g. O(N ) or U (N )), typically one for each index of the tensor, which greatly simplifies the combinatorial structure of the Feynman diagrams. In particular, it seemed crucial to impose no symmetry at all among the various indices of a given tensor. Because of its implications at the level of the Feynman amplitudes -which can be indexed by colored cell complexes and colored graphs [49] -the tensor models enjoying such a large symmetry have been going under the name of colored or uncolored models 2 .
The colored structure of the Feynman diagrams of colored and uncolored tensor models plays such a heavy role in the original proofs of the existence of their large N expansion [12], that until recently it seemed 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 that both antisymmetric and symmetric traceless O(N ) rank-3 tensors support a melonic large N expansion [1]. The proof relies in part on methods first developed in [51], which successfully tackled a somewhat simpler tensor model involving two symmetric tensors.
As one quickly realizes, a central technical ingredient of the arguments put forward in [50,1] is the irreducibility of the subspace of tensors on which the model is based (indeed, antisymmetric and symmetric traceless tensors both carry an irreducible O(N ) representation). This observation suggests the following question: Is the melonic large N limit generalizable to irreducible tensors of arbitrary rank r ≥ 3?
In this paper, we make a step in the direction of an affirmative answer. We focus on the third available class of rank-3 irreducible tensor spaces, whose elements transform as a two-dimensional representation of the permutation group S 3 . We show that the methods and results of [1] do generalize to this last rank-3 invariant space, providing another instance of melonic large N behaviour in tensor models and tensor field theories.
The paper is organized as follows. In section 2, we first introduce the class of rank-3 tensor models we wish to consider, and identify a potential instability triggered by the trace modes of the tensor. We then define more precisely the mixed symmetry model we will consider in the later sections, in which such trace modes have been correctly removed. In section 3, we introduce a perturbative expansion of the model, indexed by Feynman maps and stranded graphs. Following [1], in section 4 we identify and proceed to a resummation of the infinite family of melon-tadpole maps. This allows to introduce a new perturbative expansion in section 5, which is proven to admit a well-defined 1/N expansion dominated by melon maps. We close with a conclusion and some general comments.

Irreducible O(N) tensor models in rank three
We introduce O(N ) rank three tensor models in general, and define more precisely the theory with mixed symmetry which is the main subject of this note.

Rank-3 tensor models with complete interaction
We consider a bosonic theory in zero dimension with O(N ) global symmetry. The degrees of freedom are organized into a real rank-3 tensor T a 1 a 2 a 3 transforming as a product of three fundamental representations 3 : The scalar product ·|· is the standard one, namely: T |T ′ = T a 1 a 2 a 3 T ′ a 1 a 2 a 3 . The partition function is taken to be of the general form 4 : where P is an O(N )-invariant covariance which will be fixed in subsection 2.4. The N −3/2 scaling of the interaction is the standard one, and is necessary to the existence of a melonic large N expansion with '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 ) 3 symmetry, so that our Ansatz can not fall in the class of colored tensor models studied in [9]. The quartic interaction defines a map: 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 represents a contraction by a Kronecker delta, according to (4). As a result of this non-trivial combinatorial structure, the representation of the vertex kernel as an ordinary vertex is sensitive to its embedding in the plane. One can also represent such an interaction by the boundary graph 5 shown on the right panel of Fig. 1. As this is the complete graph on four vertices, we propose to call such an interaction a complete interaction 6 . By this definition, there exists other complete interactions (for instance the O(N ) 3 -invariant interaction of [9]) that one might want to include in the action. However, they all become equivalent upon reduction to a tensor in a fixed irreducible representation of S 3 , so it is sufficient for our purpose to consider only one such interaction.

Vector modes and trace instability
As we now explain, not all choices of propagator P (with reasonable scaling in N ) lead to an interesting large N expansion. For instance, let us consider the natural (but eventually unfortunate) choice of free propagator: a2 a1 a3 c2 c3 c1 b1b2b3 d3d2d1 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 on four vertices.
corresponding to a free action S free = 1 2 T |T . This theory leads to amplitudes which diverge arbitrarily fast in N and therefore can not support a well-behaved melonic large N expansion. As we will see shortly, the problem can be boiled down to the presence of vector modes arising from the traces of the tensor T , which couple in an ill-behaved way to the traceless part of T .
In more detail, one can decompose the tensor T a 1 a 2 a 3 into: where T 0 is a completely traceless tensor (T 0 aab = T 0 aba = T 0 baa ), while χ, ϕ and ψ are vectors which can be expressed in terms of the three traces of T . For instance, one finds: With the naive covariance (5), the propagator of ϕ at leading order in 1/N is appropriately normalized The vector modes interact among themselves, and also with T 0 . We provide two examples, one of which does spoil the large N structure of the model. Figure 2: Graphical representations of two of the interactions involving the vector ϕ i : the selfinteraction on the left is sufficiently suppressed in 1/N , while the double-trace interaction on the right leads to an instability.
Let us start with the induced ϕ 4 self-interaction: with one factor of N coming from the contraction of delta functions represented on the left panel of Fig. 2. Since the scaling leading to a sensible large N expansion for such a vector interaction is 1/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 0 and the vector degrees of freedom, such as: See again Fig. 2 for a graphical representation. This double-trace interaction does generate arbitrary large powers of N , for instance through chains of tadpole graphs, as depicted in Fig. 3. Each tadpole loop in such a diagram generates a factor of N 3 which is not entirely balanced by the 1/N 5/2 scaling of the interaction. This leads to amplitudes scaling as ∼ N p/2 , where p is the number of tadpoles in the chain. In [1], this type of pathological behaviour is referred to as a trace instability, and has been explicitly identified in a symmetric (but not traceless) tensor model. There are two ways of curing such problems. One is to introduce as many independent new 't Hooft couplings as needed to tame the problematic large N contributions 7 . For instance, one would need to include a new independent coupling to parametrize the "renormalized" interaction (ϕ i ϕ i ) T 0 abc T 0 abc . This procedure might lead to the definition of interesting vector-tensor models. For the purpose of the 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 the traceless condition by itself is not necessarily sufficient (since mixing between sub-representations may possibly generate such vector contributions), but working with an irreducible representation is.

Irreducible tensors
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 over two indices; and 2) permuting the indices in an arbitrary way. Moreover, it turns out they are the only independent transformations which needs to be reduced.
Let us first discuss the action of the permutation group S 3 , which we take to be: The irreducible representations of S 3 are labelled by Young tableaux with three boxes. One therefore associates a Young tableau to each tensor subspace with fully reduced permutation symmetry. Graphically, one has: In this representation, a box i stands for the i th index of the tensor. A Young operator is then uniquely associated to each Young tableau generated on the right-hand side of the equality: it imposes a symmetrization over the indices appearing in a same row, followed by an antisymmetrization of the indices 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 subrepresentations have mixed symmetry, in the sense that they carry a two-dimensional representation 7 I would like to thank E. Witten for pointing this out.
of S 3 . They are furthermore equivalent. Finally, the dimensions of these vector spaces are: In the rest of the paper, we will construct a tensor model with mixed symmetry. For definiteness, we will focus on the tableau 1 2 3 . Its associated Young operator S acts on an arbitrary tensor T as: yielding a tensor which is in particular anti-symmetric in a 1 ↔ a 3 . The kernel of S is: In order to reduce to an irreducible tensor representation with order N 3 degrees of freedom, one must further remove the trace modes (which contain only N degrees of freedom each). There are three of them: one lies in the completely symmetric part, the other two are in the mixed sectors. For these three sectors, the reduction is completed by acting with the orthogonal projector on traceless tensors: Hence, we have exactly three irreducible tensor representations at our disposal. The symmetric traceless and antisymmetric sectors have already been treated elsewhere [1]. The irreducible representation associated to the tableau 1 2 3 (which is equivalent to that of the tableau 1 3 2 ) is studied in the remainder of the present paper.

Tensor model with mixed permutation symmetry
In the decomposition (12), the two representations with mixed symmetry are not orthogonal to each other with respect to ·|· 8 . This implies that the projector S is not symmetric, and therefore cannot immediately be used as a propagator. One may instead consider The 3/4 coefficient has been chosen to ensure that, not onlyP ⊤ =P, but alsoP 2 =P. HenceP is an orthogonal projector whose image is included in the image of S. One may furthermore check that HenceP is nothing but the orthogonal projector on the subspace of tensors associated to the tableau 1 2 3 . More explicitly, we have: We finally define the propagator of the model as: which is the orthogonal projector on traceless tensors with symmetry 1 2 3 . A direct calculation shows that: The Gaussian measure dµ P is degenerate but defines a suitable covariance: As shown in Fig. 4, it is convenient to represent the propagator as a plain line, which decomposes further into triplets of strands. Each triplet represents one of the ten patterns of Kronecker delta contractions 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 label them accordingly by the letter B (resp. U ). More precisely, we will separate the different broken and unbroken configurations into subclasses B 1 , B 2 , U 1 , . . . , U 4 , as shown in Fig. 4.

Feynman expansion and amplitudes
The Feynman expansion is indexed by combinatorial objects associated to gluings of propagator lines and 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.

Feynman maps
The Feynman maps are the combinatorial objects which directly label the various terms obtained by Wick contraction in the perturbative Feynman expansion. Since the interaction vertex is naturally equipped with a cyclic order -or equivalently a local embedding into the plane -, the theory is most naturally expanded in terms of combinatorial maps 9 . In the present model the combinatorial maps of interest are furthermore 4-regular, a qualifier we will keep implicit in the rest of the paper. 9 A combinatorial map is a formalization of the notion of graph embedded into an orientable surface -or equivalently a ribbon graph. The reader is referred to e.g. [55,56] for further details about these standard definitions. In particular, the full connected two-point function can be expanded as a sum over amplitudes of connected two-point maps: Its trace C := C a 1 a 2 a 3 ,a 1 a 2 a 3 in turn expands as a sum over vacuum rooted maps 10 : where the first term is the trace of the free propagator P 11 . 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 4-regular nature implies that 2V (G) = E(G).
Finally, we note that the vertex is invariant under any permutation that preserves the cyclic order of its external legs up to orientation. For instance: which may be represented pictorially as: We also remark that, in contrast to the antisymmetric and symmetric traceless models, the vertex is not invariant under the full permutation group of its legs. Hence, it is essential to keep track of the embedding information encoded in a Feynman map. This being said, we will often implicitly identify two Feynman maps which are in the same equivalent class under reversal of the cyclic orders around their vertices 12 , since these have the same amplitude.

Stranded graphs
Each vacuum map M further decomposes into 10 2V (M) stranded graph 13 configurations G, each corresponding to a choice of one among ten terms of (21) for every edge of M. The patterns of identifications associated to a choice of stranded configuration results in a collection of closed cycles of strands, which are called faces. We denote byĜ(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 ∈Ĝ(M). The stranded graphs come with a definite scaling in N , each face bringing one factor of N . More precisely, it is easy to show that: where ω(G) is the degree of the stranded configuration G while ǫ(G) and R(G) are respectively a sign and a normalization factor of order N 0 The main subtlety in such models -in comparison with their simpler colored and uncolored cousinsis that not all stranded graphs have positive degree. For instance, one may generate arbitrarily negative degrees by chaining tadpoles as represented in Fig. 5. The specific tadpole configurations giving rise to such pathological amplitudes have earned the name of bad tadpoles in [1]. Figure 5: Chains of bad tadpole configurations, yielding unbounded degrees.
Hence the existence of the 1/N expansion must rely on delicate cancellations. Given our general discussion of section 2.2, where we emphasized the importance of removing the trace modes, this feature is of course to be expected. A quick inspection of Fig. 5 shows that the chains of bad tapoles do 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 present traceless model, which we will confirm explicitly.
In more detail, we follow the general method introduced in [1], and introduce a family of maps requiring special attention: the melon-tadpole maps. An elementary melon is defined as a connected two-point map with two vertices, while a tadpole is simply a two-point map with a single vertex. We then call melon (or melonic) any connected two-point map obtained from the bare propagator by recursively replacing edges of the graph by elementary melons. We similarly call melon-tadpole any connected 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) in which one of the internal edges has been replaced by a non-trivial two-point graph. These notions are important because of the following proposition, which we recall without proof.
To prove that the model defined in (2) and (21) admits a large N expansion dominated by melon maps, 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 its sum 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, which by 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 to deduce that the full two-point function of the model reduces to the melonic two-point function in the large N limit.

Resummation of the melon-tadpole family
In this section we resum the infinite family of melon-tadpole two-point maps, and show that the resulting two-point function is a simple renormalized version of the bare propagator.

General form of a two-point function
To begin with, we remark that because P is an orthogonal projector onto an irreducible O(N ) tensor 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 irreducible 1 2 3 tensors invariant. By Schur's Lemma it is therefore proportional to the identity on this subspace. Since it is guaranteed to vanish on the orthogonal complement, it must thus be proportional to the projector P itself. Graphically, we simply have: ∝ Let us denote by K the melon-tadpole two-point function. Our simple observation guarantees the existence of a function K(λ, N ) such that: In the remainder of this section, we will prove that K(λ, N ) is a series in 1 √ N and therefore admits a well-defined large N expansion. To get there, we first need to compute the amplitudes of tadpoles and elementary melons.

Tadpole maps
We start by analyzing the tadpole contributions. There are two inequivalent tadpole Feynman maps: The first (which we call planar) has multiplicity 2, the second (which we call non-planar) has multiplicity 1.
The amplitude of a planar tadpole can be shown to evaluate to while that of a non-planar tadpole yields We remark that both amplitudes have prefactors scaling as 1 N 1/2 , as is to be expected: indeed, given that such maps have only one cycle, their stranded configurations can have at most one face, yielding a maximal scaling N N 3/2 . More interestingly, the broken configurations of P are weighted by an overall factor ∼ 1 N 3/2 , which is to be contrasted with the 1 N 1/2 we would naively expect from bad tadpole configurations. These problematic configurations simply average to zero, as we were hoping for. Having recognized that A(G) must be proportional to P, we could actually have anticipated it, without explicitly computing the amplitudes. The scaling of broken edges in P being always suppressed by a factor ∼ 1 N with respect to the unbroken ones, it suffices to realize that the unbroken strands are at most of order N −1/2 to conclude that the broken ones must combine in such a way that they do not contribute before order N −3/2 .
We finally introduce a function f T (N ) parametrizing the total contribution of tadpole maps: The important fact is that f T (N ) is a series in 1 √ N with no constant term.

Elementary melon maps
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 same amplitude.
The amplitude of a melon from the first row is while a melon from the second row evaluates to In contrast to tadpoles, melons have contributions of order 1, they will therefore contribute to the leading-order in the large N regime. Their sum is: where f M (N ) is a series in 1/N with non-vanishing constant term.

Melon-tadpole two-point function
As a direct consequence of the recursive definition of melon-tadpoles, the two-point function K verifies the Schwinger-Dyson equation of Fig. 6. Figure 6: The Schwinger-Dyson equation of the melon-tadpole two-point function K.
Furthermore, we already know that K = KP for some function K(λ, N ). As a result, the Schwinger-Dyson equation reduces to the simple algebraic equation: where K 0 (λ) is the solution of: Remarkably, K 0 (λ) does not receive any contribution from f T ; it reduces to a sum over melonic twopoint amplitudes.

Existence of the large N expansion
Having managed to resum the melon-tadpole family non-perturbatively, we can now rely on an improved perturbative expansion, around the free theory associated to the Gaussian measure dµ K . For the connected two-point function and its trace, this gives: There are only two differences with respect to (23) and (24): 1) the propagator is K = KP; 2) the Feynman maps being summed over have no melons and no tadpoles. The beauty of this reformulation is that it makes the existence of the large N expansion explicit.
Indeed, since we know from Proposition 1 that ω(G) ≥ 0 for any melon-tadpole free stranded graph, we are guaranteed that A(M) expands as N 3 times a (formal) series in 1/ √ N , for any melon-tadpolefree vacuum map M. Together with the fact that K(λ, N ) is itself a series in 1/ √ N , this allows to conclude that both C and C a 1 a 2 a 3 ,b 1 b 2 b 3 have a well-defined large N expansion. The leading order of C is of order N 3 , while that of C a 1 a 2 a 3 ,b 1 b 2 b 3 is of order N 0 , and both expansions are governed by the parameter 1 √ N .

Dominance of melon diagrams
We now prove that no Feynman map other than melons contribute to the leading order two-point function. A natural conjecture would be that ω(G) ≥ 1/2 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 such maps is that they all contain a generalized tadpole. This tells us that the same type of non-trivial cancellations 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 of generalized melons and generalized tadpoles (see Fig. 7). Indeed, knowing that the large N expansion of (39) exists, we can be sure that the large N scaling of a generalized melon (resp. a generalized tadpole) is no higher than that of a melon (resp. a tadpole). Furthermore, the in-depth combinatorial investigation of [1] leads to the result of Proposition 1 ii), which we will rely on. 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) ≥ 1. We want to prove that 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 0 (otherwise the 1/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 with at least one of the subgraphs G i non-empty (otherwise M would have a melon) -say G 1 . In order to ensure that M is leading order, it is easy to see that G 1 must itself be leading order. But then it must also contain a generalized melon, which leads to a contradiction given thatM is a generalized melon with minimal number of vertices. Hence M cannot be leading order, as we were claiming.
We conclude that all the Feynman maps in the expansion (39) are suppressed by powers of 1/ √ N , except for the dressed propagator K = KP. Since we have shown that K is itself a series in 1/ √ N whose constant term K 0 is a sum of melonic maps, we may conclude that the full connected two-point function converges to the melonic two-point function in the large N limit.
In equation, we have shown that: where K 0 (λ) is the sum of melonic two-point maps. In the 0-dimensional context of the present paper, K 0 is furthermore a solution of a simple algebraic equation, namely K 0 = 1 + λ 2 K 0 4 /12 (it is even exactly solvable, see for instance [10]).

Conclusion
We have proven that irreducible O(N ) tensor models associated to Young tableaux with the shape admit a melonic large N limit. Together with the results of [1], this achieves the proof that all irreducible rank-3 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 extended to 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 from the decomposition of general tensors we relied on in section 2.2. It was invoked to emphasize the instability introduced by the trace modes of the tensor, which we cured by removing these trace modes altogether. An alternative strategy would consist in introducing new independent 't Hooft couplings parametrizing the various types of interactions generated by the complete tensor interaction. In this way, one might presumably be able to construct a large N expansion with a leading order including mixtures of melonic and bubble diagrams.
Finally, we would like to emphasize that, even though we restricted our attention to a 0-dimensional model, the methods of [1] and of the present paper can be applied more generally to bosonic tensor field theories in d ≥ 1. Provided that the global O(N ) symmetry is not spontaneously broken, an equation analogous to (41) will automatically follow. The main difference is that the Schwinger-Dyson equation of the melonic two-point function determining K 0 will not reduce to a simple algebraic equation; it will result in a more involved melonic integro-differential equation, similar to those found in [31,40,42] (in d = 1 and higher).