Spectrum of anomalous dimensions in hypercubic theories

We compute the spectrum of anomalous dimensions of non-derivative composite operators with an arbitrary number of fields $n$ in the $O(N)$ vector model with cubic anisotropy at the one-loop order in the $\epsilon$-expansion. The complete closed-form expression for the anomalous dimensions of the operators which do not undergo mixing effects is derived and the structure of the general solution to the mixing problem is outlined. As examples, the full explicit solution for operators with up to $n=6$ fields is presented and a sample of the OPE coefficients is calculated. The main features of the spectrum are described, including an interesting pattern pointing to the deeper structure.


Introduction
Conformal field theories (CFTs) play a fundamental role in our understanding of the Universe with applications ranging from the discovery of the asymptotic freedom in QCD [1,2] to the potential applications in gravity in the scenario of asymptotic safety [3]. Mathematically, for a given symmetry group, CFT is specified via CFT data, i.e. the scaling dimensions of all the primary operators and the set of OPE coefficients, defined as the constants appearing in the three point functions of the theory.
In this work we study the CFT data of the theory invariant under the hypercubic symmetry group H N realized as a group of symmetries of N -dimensional hypercube. In Nature, this symmetry group appears in the description of critical properties of cubic magnets, like iron [4], in which the magnetic anisotropy induced by the lattice structure is experimentally accessible. It also appears in the description of certain structural phase transitions such as the cubic to tetragonal transition in SrT iO 3 (strontium titanate) [5].
Hypercubic models are usually investigated in the perturbative ǫ-expansion [6] using standard diagrammatic techniques and present-day results involve the computation of anomalous dimensions (γ φ , γ m 2 ) and beta functions to six loops order [7,8]. Recently this theory was also explored non-perturbatively resorting to the conformal bootstrap method [9][10][11][12]. Most of these studies have focused on H 3 [9][10][11] 1 , with the exception of [12], which computed CFT data for composite operator with two fields and arbitrary spin at O(ǫ 3 ) for a generic N . Finally, the model was also used in [13] to investigate the generalized F -theorem conjecture.
The aim of this paper is to compute the spectrum of anomalous dimension for composite operators with arbitrary number of fields n but no derivatives in the H N critical theory to O(ǫ). Our results are complementary to previous works which usually consider small values of n (and/or fix N ) but at the same time consider arbitrary number of derivatives and higher order in the ǫ-expansion. After introducing the model in Sec.2 we lay out the representation theory for H N in Sec.3 with the aim to summarize it in a practical way useful for future investigations of hypercubic models. For the computation of the anomalous dimensions we make use of a recently developed techniques [14,15] based on the Schwinger-Dyson equation and CFT constraints, which we outline in Sec.4. In Sec.5 we present our main results providing complete formula for anomalous dimensions of an infinite number of composite operators with arbitrary n which do not mix with the others and supply an algorithm to obtain the solution to the remaining mixing problem. In Sec. 6 we show explicit results for the anomalous dimension of composite operators up to n = 6 and we give a qualitative description of the spectrum for n > 6. Sec.7 discusses the way to obtain the rest of the CFT data as a further step towards completing the knowledge of the theory and include the computation of some OPE coefficients as example. We give our conclusions in Sec.8.

O(N) model with cubic anisotropy
The Lagrangian for the O(N ) Ginzburg-Landau model with cubic anisotropy in d = 4 − ǫ is: where the interaction terms can be rewritten in a tensor form as and the tensor δ ijkl defined by: The presence of the coupling g 2 breaks the O(N ) symmetry so that the action is invariant only under H N ⊂ O(N ) where H N is the group of symmetries of an N -dimensional hypercube. At the 1-loop level, the renormalized model predicts four fixed points which read: The first is a trivial gaussian fixed point while the second and the third correspond, respectively, to the φ 4 -theory (N decoupled Ising models) and to the O(N ) model. The fourth, often called "cubic fixed point", corresponds to a theory invariant under H N and the main goal of this paper is to compute the spectrum of anomalous dimension at this last fixed point. For this purpose, it is important to note that for N = 4 the cubic fixed point coincides with the O(N ) symmetric one, for N → ∞ it correspond to the Ising one, while for N = 1 there is only one coupling constant g 1 + g 2 and the cubic fixed point reduces to the free theory. As we will see later, these limits will provide non-trivial cross-checks for our results.
A further consistency check is given by the existence of a special symmetry for N = 2. In fact the interaction term is invariant if we perform a π 4 rotation of the fields [16]: and at the same time transform the coupling constants with This turns the cubic fixed point into the Ising one. Finally, as a consequence of the EOM at the cubic fixed point: the operator g 1 φ i φ 2 + g 2 φ 3 i , which is a primary operator in the free theory, becomes a descendant (a derivative) of the primary operator φ i in the interacting theory, and thus its properties are entirely fixed in terms of those of φ i . This phenomenon is called conformal multiplet recombination [14] and in particular it implies that: where the scaling dimensions of a composite operator with n fields S n is given by: with γ Sn the 1-loop anomalous dimension. We will use Eq.(2.8) as an additional check for our results. Having introduced the model under consideration, the next step is to unveil the operator spectrum at the cubic fixed point, i.e. to find the irreducible representations of the hypercubic group.

Irreducible representations of hypercubic group H N
The aim of this section is to lay out the representation theory for the hypercubic group in a way suited for a streamlined construction of the spectrum of H N composite operators as we are not aware of such construction and few relevant results are somewhat scattered in the literature [17][18][19].
H N is a discrete subgroup of O(N ) and is given by the wreath product S N ⋉ Z N 2 . In order to construct the irreducible representations of S N ⋉ Z N 2 , we start by computing the outer tensor products of the irreducible representations of Z 2 N times. Labelling as [1 2 ] and [2] the two irreducible representations of Z 2 , the irreducible representations of Z N 2 are: In accordance with these representations, the symmetric group S N is divided into direct product S α × S β and then the irreducible representations of S N ⋉ Z N 2 are generated by multiplying those of Z N 2 in Eq.(3.1) with the corresponding direct product 2 S α × S β [20]. For instance, the irreducible representations of Z 3 2 are: and they have to be multiplied by the irreducible representations of S 3 , S 2 × S 1 , S 1 × S 2 and S 3 , respectively. From this construction it is clear that an elegant way to label the irreducible representations of H N is in terms of double-partitions of N, (α, β), which can be represented as ordered pairs of Young diagrams with α and β boxes, respectively [19,21]. For example, the ten irreducible representations of H 3 are: where "∅" stands for the empty partition. The left partition represents α objects, even under Z 2 , which transform under an irreducible representation of S α , while the right partition represents β objects, now odd under Z 2 , which transform under an irreducible representation of S β . The dimension of a given double-partition (α, β) is [21]: where dim(α) (dim(β)) is the dimension of the corresponding representation of the symmetric group S α (S β ) obtained via the standard hook's rule. The defining N −dimensional representation of H N is given by : and the decomposition of the tensor product of the defining representation with an arbitrary representation (α, β) can be obtained through the following formula: where α + are the tableaux obtained by moving one box from β to α, α − are the tableaux obtained moving one box from α to β and the same for β. For example, for the tensor product φ i ⊗ φ j we have: Now, we are in position to construct explicitly composite operator of H N with n fields and no derivatives. The first step is to find the corresponding bi-tableaux by computing the tensor product of the defining representation n times following the rules of (3.5). As a result, we will have some bi-tableaux that never appeared before at smaller n and each such bi-tableau will correspond to unique composite operator. In addition, we will also have bi-tableaux that already appeared at the levels n − 2, n − 4, ..., and "reappeared" at the level n simply as a consequence of moving one box from one side of double-partition to the other and then returning it back to the original place. In this case more than one composite operator of order n will correspond to that tableau and to find the true scaling operators it will be necessary to solve the mixing problem.
The second step is to associate to every bi-tableau the corresponding H N tensor as follows. Let us assume, for the moment, that we are dealing with unique composite operator and consider some generic representation having 0 boxes in the right partition and N boxes in the left partition (α = N, β = 0 ). For example: ... , ∅ .
We start by filling the left tableau with indices that we impose to be all different: Since the left partition represents the objects even under Z 2 , then for every box in the first row we associate an indexed field raised to the zeroth power, for every box in the second row an indexed field raised to the second power and so on, in order of increasing even powers of the fields for subsequent rows. Finally, we have to symmetrize all these indices as usual (symmetrization over all boxes in the same row, and antisymmetrization over all boxes in the same column). This leads to (n = 8 in this case): The same rules also apply to the right partition β, but now we have to associate increasing odd powers of the fields as we increase the number of rows. For instance (with n = 5): Certain bi-tableaux can also appear for the first time at a level n too low to allow the previous constructions and this simply means that the corresponding tensor requires derivatives to be built. 3 Aside from that, the most general H N -tensor will be represented by the bitableau with the left and the right partitions separately comprising the most general Young diagram associated with the corresponding symmetric group. It will have k different types of columns (distinguished by the number of boxes) that we label with the index i. Each type of column can have multiplicity p i . All the indices for the fields have to be different, and the indices of the fields in each column have to be antisymmetrized. The columns associated to the left partition will contain the even powers of the fields while the columns associated to the right partition will contain the odd ones and we label the highest power of the field in the given column by m i . Thus the unique hypercubic composite scaling operators corresponding to this most general bi-tableau can be written compactly as: Now, let us assume that we are dealing with a bi-tableau that "reappeared" at the level n. The corresponding mixing space can be found through the following steps: 1. Write the unique composite operator corresponding to the bi-tableau according to the rules above and then multiply the result with the appropriate power of φ 2 needed to reach the level n. For instance, for n = 6, we have: 2. Then "distribute" φ 2 through the rest of the tensor in all possible ways as follows: 3. Finally, it is also necessary to take into account the mixing between powers of φ 2 and the other H N -scalars. The H N -scalars of order n are formed by products and powers of all the operators of the form: For our example this means that (φ 2 ) 2 will mix with i φ 4 i so that one additional operator has to be added to the mixing space: In the next section we will show how to solve the mixing, find the true scaling operators of H N and compute their anomalous dimensions.

Computation
To compute the spectrum of anomalous dimensions γ we resort to a method first proposed in [14] (and recently generalized in [15]) which makes use of constraints from conformal symmetry combined with the Schwinger-Dyson equation (SDE) in order to obtain nontrivial consistency conditions which allow to extract the value of γ. The key idea is to consider three-point functions of the form: and use the EOM (2.7) to rewrite them as 4 : Here S n is a composite operator of order n, i.e. a product of n fields transforming under an irreducible representation of H N . Matching the results for the computation of the three-point functions Eq.(4.1) and Eq.(4.2), gives a recursion relation for the anomalous dimension of the scaling operators γ Sn , whose general solution at the cubic fixed point is the following eigenvalue equation: Here V j 1 ,j 2 ,i 1 ,i 2 is the tensor defined in Eq.(2.2) and the tensors S are given by: Eq.(4.3) can be recast in a more practical form: where 5 : The scaling operators are found with the techniques described in the previous section. For the operators which mix Eq.(4.5) determines uniquely the mixing matrix.

Towers
In order to cross-check our results for anomalous dimensions of H N operators, we need to recall the corresponding ones for O(N ) and Ising models. with the corresponding dimension given by: 5 Eq.(4.5) with D given by Eq.(4.6) was already found in [22].
During our analysis, we found it useful to check the decomposition of O(N ) tensors to those of H N by monitoring the dimension of the representations using Eq.(3.3) and Eq. (5.4). Explicit examples of this decomposition will be presented in Sec.6.

Decoupled Ising model
For N = 2, ∞ the cubic fixed point coincides with N decoupled copies of the Ising model. In this case only the representations coming from the smallest Young tableaux in Eq.(5.1) survive which is a Tr if n is even and if n is odd. Since in the decoupled Ising model the field has only one component, these representations altogether simply collapse to the "tower" of composite scaling operators φ n and their leading order anomalous dimensions in d = 4 − ǫ, are given by [14]:

The hypercubic tower
Finally, we turn our attention to the anomalous dimensions at the cubic fixed point. As discussed above, at every n there will be a set of bi-tableaux which appear for the first time and each such bi-tableau corresponds to only one scaling operator. This class of operators is given by Eq.(3.7) and they will compose our "hypercubic tower". The corresponding anomalous dimensions can be found by applying the differential operator D defined in Eq.(4.6): and for the most general bi-tableau having k different types of columns (labeled by the index i) with multiplicity p i and the highest power of the field in the column given by m i we obtain:  1 and p 3 ) boxes. This defines the following tower of operators: The dimensions and the γ's of the operators in this tower at the level n are: Aside from Eq.(5.7), it is clear that the content of the previous sections provides an algorithmic way to compute γ for every non-derivative composite operator of arbitrary order n. However, as we will see, at large n the mixing of operators corresponding to the same double-partition becomes complex making the analytic progress difficult. In the next section we will present explicit results up to n = 6.

Spectrum of anomalous dimensions
The indices of the operators in this section, with exception of those in Sec.6.1, have to be understood to be all different, e.g. φ i φ j means φ i φ j together with the condition i = j.

n=1
The anomalous dimension of φ i appears only at O(ǫ 2 ). In the spirit of Sec.4 it can be computed at the leading order by considering the following two point function: and using the EOM (2.7) to rewrite it as: Matching the evaluation of these two point functions gives:

3N
As expected, the dimensions of these representations add up to N (N +1) 2 which is Eq.(5.2) for n = 2, while the 2-index traceless symmetric O(N ) tensor decomposes under H N as i.e. into a traceless diagonal symmetric tensor φ i 2 −φ j 2 and a off-diagonal symmetric tensor The results in this subsection were already obtained in [12].
The Finally, for this operator, Eq.(2.8), which comes from the multiplet recombination phenomenon, is also satisfied.
The sum of the dimensions of these representations is N (N +1)(N +2)(N +3) 24 while the decomposition of the 4-index traceless symmetric O(N ) tensor under H N is These results were already obtained in [22] and here we already solved the mixings explicitly. The reader may easily check the consistency of the results for N = 1, 2, 4, ∞.
6.5 n=5 They pass all the consistency checks in a way similar to Sec.6.3 for N = 1, 2, 4, ∞. Namely, for N = 1 it is easy to check that 4 by 4 mixing matrix for the vectors has one zero eigenvalue while for N = 2, ∞ it has one eigenvalue equal to 10/3 as required by Eq.(5.5).
We are not aware of any explicit results for n ≥ 5 in the literature.

n ≥ 6
Even though the techniques described in this paper allow to compute the whole spectrum of γ's up to arbitrary n, we stop giving exhaustive results and simply provide a qualitative description of the spectrum for n ≥ 6. At every n, all non-mixing scaling operators together with their γ's, are given by Eq.(3.7) and Eq.(5.7). For instance, at n = 6 these are 7 : The first six operators listed above correspond to bi-tableaux with maximum two rows and thus their γ's can be computed using Eq.(5.14). In fact, there they correspond to Moreover, at n = 6 we find, for the first time, a unique operator whose γ cannot be computed by using Eq.(5.14) but only resorting to the full Eq.(5.7). This is the last operator in Eq.(6.8), which corresponds to the following bi-tableau with three rows in the left partition (k = 1, m i = 4, p i = 1): ... , ∅ . (6.9) Returning to the general n, apart from non-mixing scaling operators all the other irreducible representations will be given by those at the level n−2 and all the corresponding eigenvectors can be generated as explained in Sec.3. At every even n, the lowest dimensional irreducible representations are given by the scalar sector. This is given by Eq. needed in order to obtain the free Gaussian theory. Of course, this happen also in the case of the vectorial (dim = N ) irreducible representations, which appear at every odd n. The scalar sector for n = 6 is: and it is easy to check that for N = 1 one eigenvalue is 0. Further insights on the spectrum of anomalous dimensions can be gained by looking at: where d Sn and γ Sn are the dimensions and the anomalous dimensions of the composite operators S n , respectively, and the sum runs over all the irreducible representations at the level n. In fact from Eq.(4.6) it follows that, for any n, W n has to be proportional to N − 1. This can be easily checked for n ≤ 5 using our previous results while for n = 6 it can be checked by noting that the only irreducible representations which contribute to W 6 (N = 1) are Eq.(6.10) and: which has dimension 2 × 1 2 N (N − 3). Moreover, the values of W n exhibit an interesting pattern 8 : which is indicative of a general formula for W n . We conclude this section by giving the decomposition of the 6-index traceless symmetric tensor of O(N ) under H N : (6.14)

Towards complete CFT data
The description of the spectrum of anomalous dimensions is an important step towards the computation of complete CFT data for the hypercubic model. The other important set of data is provided by the structure constants C ijk defined by: The simplest class of C ijk corresponds to C 112 = φ i φ j S 2 , where S 2 is a scaling operator of order n = 2. These coefficients were already computed up to O(ǫ 2 ) in [12] using the conformal bootstrap, while all the C 113 are trivially 0. We therefore present a novel computation for a sample of C 114 = φ i φ j S 4 coefficients. For this purpose, techniques developed in [15] and described in Sec.4 are again useful and allow the computation of several classes of leading order structure constants. Moreover, this task is noticeably simplified by the group-theoretic analysis of Sec.3. Coefficients C 114 can be computed via the following formula [15]: Here, V ijkl and S mjkl are again given, respectively, by Eq.(2.2) and Eq.(4.4). We obtain: Up to now, we have considered only scaling operators with no derivatives. Aside diagrammatic techniques, the spectrum of γ's for operators with derivatives can be computed using conformal bootstrap as in [12] or generalizing the techniques of Sec.4. Work along this direction was initiated in [26] and we plan to investigate this generalization in future.
Another future direction is to extend this analysis to the next-to-leading order in the ǫ-expansion. At the moment, this requires approaches quite different from those of Sec.4, such as exploiting the conformal bootstrap machinery. It is an open problem to see if, by extending the EOM technique of Sec.4 to higher order correlation functions, one can access next-to-leading order in the ǫ-expansion in a relatively simple way.

Conclusion
In this paper we have computed the spectrum of anomalous dimensions at the leading order in ǫ-expansion in the O(N ) model with cubic anisotropy in d = 4 − ǫ. We derived the complete closed-form expression for the anomalous dimensions of the operators which do not undergo mixing effects and delineated the structure of the general solution to the mixing problem for arbitrary n. We have examined the general features of the spectrum uncovering the interesting pattern of the values for the sum of the anomalous dimensions weighted by the size of the corresponding representations pointing to the deeper structure. A plethora of examples were provided including the explicit solution for operators with n ≤ 6.
For this purpose, in Sec.3, we have systematized, for the the first time in the literature, the representation theory for H N with the aim of providing a solid ground for future analyses of models with hypercubic symmetry. For instance, it can be useful for extending the conformal bootstrap analysis of model with H 3 , recently performed in [9,10], to the general H N case. It would be also instrumental in extensions of our results to operators with derivatives and/or higher orders in ǫ-expansion.
As another future direction, it would be interesting to compute the spectrum of anomalous dimensions in models with different symmetries such as the models based on the hypertetrahedral symmetry group H tetrahedral ≃ S N +1 ⊗ Z 2 .