Anomalous dimensions in CFT with weakly broken higher spin symmetry

In a conformal field theory with weakly broken higher spin symmetry, the leading order anomalous dimensions of the broken currents can be efficiently determined from the structure of the classical non-conservation equations. We apply this method to the explicit example of $O(N)$ invariant scalar field theories in various dimensions, including the large $N$ critical $O(N)$ model in general $d$, the Wilson-Fisher fixed point in $d=4-\epsilon$, cubic scalar models in $d=6-\epsilon$ and the nonlinear sigma model in $d=2+\epsilon$. Using information from the $d=4-\epsilon$ and $d=2+\epsilon$ expansions, we obtain some estimates for the dimensions of the higher spin operators in the critical 3d $O(N)$ models for a few low values of $N$ and spin.


Introduction and Summary
The spectrum of a d-dimensional conformal field theory [1] consists of local operators labelled by conformal dimension ∆, a representation R of SO(d), and possibly a representation r G of an internal global symmetry group. The precise determination of the spectrum of an interacting CFT is of fundamental importance. Together with the knowledge of the OPE coefficients, it essentially amounts to a solution of the CFT.
It is well-known that in a unitary CFT d , dimensions of primary operators satisfy certain inequalities known as unitarity bounds [2,3]. For a spin s operator J µ 1 µ 2 ···µs in the symmetric traceless representation of SO(d), the unitarity bound is For a scalar operator O, it reads and one may derive similar bounds for more general representations of SO(d). When these inequalities are saturated, the corresponding operator satisfies some differential equation, and it belongs to a short representation of the conformal algebra. In the case of a scalar operator, the shortening condition is simply the wave equation ∂ 2 O = 0, i.e. O is a free scalar field. For a spin s operator (s ≥ 1), saturation of the bound (1.1) implies that it is a conserved current ∂ µ J µµ 2 ···µs = 0 . (1. 3) The cases s = 1 (an exactly conserved current with ∆ = d − 1) and s = 2 (the conserved stress tensor with ∆ = d) are familiar in any CFT. Conserved currents of higher spins s > 2 are explicitly realized in free field theories. For example, in a free scalar CFT they take the schematic form where brackets denote traceless symmetrization and the coefficients may be determined by the conservation equation, as we review in Section 2.1. It is evident that these operators have exact dimension ∆ s = d − 2 + s in the free theory. As usual, conserved currents correspond to symmetries of the theory. The presence of exactly conserved currents of all spins implies that the CFT has an infinite dimensional higher spin symmetry which includes the conformal symmetry as a subalgebra. Higher spin symmetries turn out to be very constraining. One may prove that if a CFT possesses a spin 4 conserved current, then an infinite tower of conserved higher spin operators is present, and all correlation functions of local operators coincide with those of a free CFT [4].
In an interacting CFT, the higher spin operators are not exactly conserved and acquire an anomalous dimension ∆ s = d − 2 + s + γ s . (1.5) An interesting class of models are those for which the higher spin symmetries are slightly broken. By this we mean that there is an expansion parameter g, playing the role of a coupling constant, such that for small g the anomalous dimensions γ s (g) are small, and in the g → 0 limit one recovers exact conservation of the currents. Explicit examples are weakly coupled fixed points of the Wilson-Fisher type [5], where g corresponds to a power of , or certain large N CFT's, where g is related to a power of 1/N . At the operator level, this implies that the non-conservation equation for the spin s operator takes the form ∂ · J s = gK s−1 (1. 6) where K s−1 is an operator of spin s − 1, and we factored out g to highlight the fact that at g = 0 the current is conserved. The slightly broken higher spin symmetries (1.6) can still be used to put non-trivial constraints on the correlation functions [6]. The equation (1.6) also gives an efficient way to determine the anomalous dimensions γ s to leading order in the small parameter g [7,8]. As we review in Section 2.2, using (1.6) and conformal symmetry one readily finds that γ s (g) ∝ g 2 , where the proportionality constant is simply obtained by computing the two point functions K s−1 K s−1 at g = 0. This method is similar in spirit to the one recently advocated in [9], where the leading anomalous dimension of φ at the Wilson-Fisher fixed point in d = 4 − was reproduced by using conformal symmetry, without explicit input from perturbation theory.
In this paper, we apply this method to the explicit example of interacting scalar field In all these examples, we determine the explicit structure of the non-conservation equation (1.6) and use it to find the leading order anomalous dimensions of higher spin operators in the singlet, symmetric traceless and antisymmetric representations of O(N ). Many of our findings were obtained before by different methods [11,12], but the results in the cubic models in d = 6 − and in the nonlinear sigma model in d = 2 + are new as far as we know. In all examples, we pay particular attention to the large spin behavior of the anomalous dimensions, finding precise agreement with general expectations [13][14][15][16][17]. Combining information from the d = 4 − and d = 2 + expansions, as well as some input from the large spin limit, in Section 7 we also obtain some estimates for the dimension of the singlet higher spin operators in the d = 3 O(N ) models for a few low values of spin s and N .
In addition to their intrinsic interest and their relevance in statistical mechanics, the O(N ) models we study in this note also play an important role in the context of the AdS/CFT correspondence. According to a well understood entry of the AdS/CFT dictionary, exactly conserved currents of spin s in CFT d are dual to massless spin s gauge fields in AdS d+1 .
Interacting higher spin gauge theories in AdS d+1 were explicitly constructed by Vasiliev [18][19][20][21], and a class of them were naturally conjectured [22] to be dual to the singlet sector of the free O(N ) vector model. The exactly conserved currents J s with s = 2, 4, 6, . . . are dual to the corresponding massless gauge fields in the Vasiliev theory, and the scalar operator J 0 = φ i φ i to a bulk scalar field with m 2 = −2(d−2)/ 2 AdS . As further conjectured in [22], one may extend this duality to the interacting case, obtained by adding to the free theory the "double-trace" interaction λ(φ i φ i ) 2 . For d < 4 there is a flow to an interacting IR fixed point 1 which is conjectured to be dual to the same Vasiliev theory but with alternate boundary conditions on the bulk scalar field [23]. A distinguishing feature of large N interacting vector models is that the descendant operator K s−1 appearing in the non-conservation equation is a "double-trace" operator, schematically This implies that the anomalous dimensions are γ s ∼ O(1/N ), which corresponds to a quantum breaking of the higher spin gauge symmetry in the bulk: the higher spin gauge fields acquire masses through loop corrections 2 when the bulk scalar is quantized with the alternate boundary conditions. In representation theory language, the equation (1.6) means that the short representation of the conformal algebra with (∆ = d − 2 + s, s) combines with the representation (∆ = d − 1 + s, s − 1) to form a long multiplet with (∆ > d − 2 + s, s). In the bulk, this phenomenon corresponds to a higher spin version of the Higgs mechanism [24]: the gauge field swallows a spin s − 1 Higgs field to yield a massive spin s field. The fact that the operator on the right-hand side of (1.7) is double-trace implies that the Higgs field is a composite two-particle state, and the breaking is subleading at large N . This is different from theories of Yang-Mills type, where K s−1 in (1.6) is a single trace operator, and the anomalous dimensions are non-zero already at planar level. In the bulk, this would correspond to a tree-level Higgs mechanism.
Let us finally mention that, although in this paper we focus on the scalar O(N ) models, the methods we use can be applied to a variety of interesting theories, such as for instance the critical Gross-Neveu models in 2 < d < 4, conformal QED d with N f fermions, the CP N model, and 3d bosonic and fermionic vector models coupled to Chern-Simons gauge fields [6,25,26]. The latter models have approximate higher spin symmetry at large N for all values of the 't Hooft coupling λ, and one should be able to use (1.6) to find the anomalous dimensions to order 1/N , γ s = f s (λ)/N + . . ., for all values of λ and spin. 3 We leave this to future work.
Note added in proof: After completion of our calculations, while writing up this note, we became aware of [27], which has overlap with some of our results. In this section we will setup the definitions and notations which will then be applied to the particular models. We will also describe the derivation of the master formula which allows to calculate the lowest-order value of the anomalous dimensions without doing any loop calculations.
We first introduce some useful technology for the manipulation of symmetric tensors. For a given a rank s tensor J µ 1 µ 2 ···µs in the symmetric traceless representation, we may introduce an auxiliary "polarization vector" z µ , which can be taken to be null (z 2 = 0), and construct the index-free projected tensor It is evident that the multiplication by z µ selects only the symmetric traceless part of J µ 1 ···µs .
One may always go back to the full tensor by "stripping off" the null vectors and subtracting traces. In practice, this can be done efficiently with with the help of the following differential operator in z-space [8,28,29] Acting once with this operator removes a z µ , thus freeing one index of the tensor, while taking into account the constraint z 2 = 0. The unprojected J µ 1 ···µs can thus be recovered via 3) The symmetrization and tracelessness of the operator obtained this way is ensured by the Similarly, the conservation equation (1.3) of the spin s operator may be written compactly in this notation as Let us now construct the explicit conserved higher spin currents in the free CFT of N real massless scalar fields. They satisfy the free wave equation and there is a O(N ) global symmetry under which φ i transforms in the fundamental representation. This free CFT admits an infinite tower of exactly conserved higher spin operators (1.4), which are bilinears in the scalars with a total of s derivatives acting on the fields.
Projecting indices with the null vector z µ , these operators can be written aŝ where we have introduced the projected derivative∂ = ∂ µ z µ , and c sk are coefficients that will be fixed shortly. Of course, one can separate this operator into irreducible representations of O(N ), as discussed in more detail below. It is convenient to rewrite (2.7) in the following (2.8) where we have encoded the coefficients c sk into the function f s (u, v). Now the conservation equation (2.5) may be turned into a differential equation for the function f s , which, upon using the free equation of motion ∂ 2 φ = 0, reduces to The following ansatz for f s is convenient and results in the ordinary differential equation The solution to this equation is given by the order s Gegenbauer polynomials, φ s (t) = C d/2−3/2 s (t), which are even (odd) for even (odd) s. Hence, up to the overall normalization, one gets the following expressions for the conserved higher spin currentŝ (2.12) One may also write where J s ≡ J ii s denotes the singlet current. It is evident by symmetry that the singlet and symmetric traceless representations only exist for even spin, and the antisymmetric one for odd spins. For s = 1, the antisymmetric operator J The generators can be constructed in a canonical way as follows. First, by contracting a spin s current with a spin s − 1 conformal Killing tensor ζ µ 1 ...µ s−1 , 4 we may obtain an ordinary current J ζ µ,s = J µµ 2 ···µs ζ µ 2 ...µs , which is conserved as it is easily checked. From this, one can get a conserved charge Q ζ s in the usual way. For instance, for s = 2, the singlet current J 2 is proportional to the traceless stress tensor of the CFT, and contracting this with the linearly independent conformal Killing vectors one gets the (d + 2)(d + 1)/2 generators of the conformal algebra. In the interacting theory, all of the currents (2.14), except for J and J 2 , will be broken. In particular, while the free CFT has N (N + 1)/2 conserved "stress tensors", only one of them remains conserved when interactions are switched on.
For what follows, it will be useful to work out the normalization of the two point function of the currents (2.12) in arbitrary dimensions d. Since the currents are bilinear in φ there will be two propagators, which are differentiated by the hatted derivatives at both points. The calculation is drastically simplified by using the Schwinger parametrization of the propagator Owing to the fact that∂x = 0, since z 2 = 0, all hatted derivatives are replaced by −2αx if acting at point x and +2αx at point 0, so that instead of spin sums we have integrals of Gegenbauer polynomials over the parameters α 1 and α 2 for the first and second propagator respectively. Separating the O(N ) indices, we may write The (−1) s comes from the property of C . The spacetime and z µ dependence is of course as required by conformal symmetry for a spin s conserved operator. The normalization factor N s is given by the following expression The norms corresponding to the irreducible representations of O(N ) are then for the singlet, for the symmetric traceless, and for the antisymmetric.

Anomalous dimensions of the weakly broken currents
Let us consider a CFT with a parameter g playing the role of a coupling constant, such that in the g = 0 limit there are exactly conserved currents J s . When a non-zero coupling g is turned on, the currents will be no longer conserved for general s and acquire anomalous dimensions The non-conservation of the currents means that a non-zero operator of spin s − 1 must appear on the right hand side of (1.3), or equivalently (2.5) where we have pulled out an explicit factor of g in front of the descendant to stress that the right hand side vanishes when g = 0. Here g is assumed to be a small expansion parameter, and may be either a power of in the Wilson-Fisher type models, or a power of 1/N in the large N approach. We now proceed by noting that in a CFT the form of the two-point function of the spin s operators is fixed by conformal symmetry to be (2.25) Acting on this two-point function with ∂ µ D µ z on both operators (with different projection vectors z 1 and z 2 ), one gets, using the form of the non-conservation equation (2.23) On the other hand, differentiating the right hand side of (2.24), setting z 1 = z 2 at the end, and dividing by the two-point function of J s, one finds the relation [7,8] The right-hand side being proportional to γ s is not a coincidence and follows from the conservation of the higher-spin current at zero coupling (2.5). From a CFT standpoint, (2.27) is an exact relation. In practice, when doing perturbation theory in g, one computes the correlators on the left hand side in powers of the coupling. It is then evident that (2.27) allows to gain an order in perturbation theory. To obtain the anomalous dimensions of the broken currents to leading order in g, one has simply to evaluate ratio of correlators in the free theory, g = 0. In particular, this only involves finite tree-level correlators, avoiding the issues of regularization and renormalization.
The considerations above are general and apply to any CFT with weakly broken higher spin operators (2.23). For the explicit examples discussed in the rest of the paper, it will be useful to determine the general form of the descendants K s−1 in the scalar theories. Applying the divergence operator to the higher spin currents (2.12), we find in terms of the function Of course, this is zero in the free theory where ∂ 2 1 = ∂ 2 2 = 0. In the interacting theory, (2.28) allows to determine the form of the descendent once the equation of motion for φ is known in the specific model of interest. The function h s (u, v) can be evaluated more explicitly using the recurrence relations between the Gegenbauer polynomials, and one finds As discussed above, the vanishing at d = 3 is superficial and is a consequence of the normalization of the currents (2.12).
Note that the methods described in this section can also be used to fix the leading order anomalous dimension of a nearly free field. For instance, in the case of a scalar field, the equation of motion takes the form where V is some operator of spin zero and bare dimension d/2 + 1. By an analogous calculation as the one described above for the higher spin operators, one can show that to leading order γ φ ∝ g 2 , where the proportionality constant is related to the two point function V V at g = 0. We will use this method in the next Section to reproduce the well-known anomalous dimension of φ at the Wilson-Fisher fixed point, see also [9]. The analogous calculations in the large N approach and in the nonlinear sigma model in d = 2 + are given in Section 4 and 6 respectively.
We now apply the general formulae obtained in the previous section to the case of the critical The one-loop beta function is well known and reads and thus there is a IR critical point at λ * = 8π 2 N +8 + O( 2 ). Before moving on to the higher spin operators, let us show how to reproduce the leading order anomalous dimension of φ using the classical equations of motion, following the methods reviewed above, see also [9].
In the free theory, when λ = 0, the elementary field φ i has canonical dimension ∆ 0 = d/2−1, thus saturating the unitarity bound and obeying ∂ 2 φ i = 0. The tree-level two-point function of φ i is given by When we turn on the interaction, the equation of motion is modified to 4) and the two-point function receives corrections. At the conformal point, the exact two-point function is constrained by conformal symmetry to be Applying the equation of motion twice, i.e. taking the ∂ 2 1 ∂ 2 2 on both sides and taking the ratio one gets: The fact that the right-hand side is proportional to γ φ is expected and is due to the shortening condition at zero coupling, ∂ 2 φ i = 0. To get the leading order in for γ φ we notice that in the left hand side λ 2 * ∼ 2 , so in the two-point function ratio we can just plug d = 4 propagators In the right-hand side we get since it is evident that γ φ ∼ 2 . Now, the two-point function in the numerator of (3.6), evaluated at tree level, yields Finally, taking the ratio by the free propagator and equating the right-hand side (3.8), we recover the well-known result One can contrast this to the usual calculation, which is technically quite different. There, the leading order correction to the two-point function of the φ i field is given by extracting the logarithmic divergence of the standard two-loop diagram in Fig. 1.
We may now proceed to studying the higher spin currents using the same method. We use the definition of the currents (2.12) and the descendant (2.28). To lowest order the dependence is fixed by the the critical coupling λ * , so we can use d = 4 everywhere. The  11) and the descendant: Note that this form is redundant in the sense that we could combine∂ 3 +∂ 4 into∂ 3 acting on , but it makes all the symmetries of the diagrams we will need to calculate explicit.
This vanishes for the singlet as it is the conserved stress-energy tensor. It does not vanish for the symmetric traceless representation, as the corresponding operator acquires an anomalous dimension in the interacting theory. As another example, the spin 3 descendant of the spin 4 singlet current iŝ The master formula (2.27) at the leading order yields the following for γ s : For the two-point function of the currents one has in d = 4, according to (2.17): for the singlet and similarly where we used the fact that the singlet and symmetric traceless representations exist for even spins only, and the antisymmetric one for odd spins.
To obtain the anomalous dimensions via eq. (3.15), we have to compute the two-point function of the descendant at tree level. Each descendant (3.12) consists of a differential operator acting on four φ fields. We simply have to compute the free field Wick contractions between the fields (contractions of fields on the same descendant are of course excluded) and then act with the differential operator in    + 2))) of the N = 1 supersymmetric minimal models for p = 2s. One may wonder if there is a deeper significance to these apparent coincidences. We thank Igor Klebanov for bringing this to our attention.
The latter vanishes for s = 1, corresponding to the exact conservation of the current in the adjoint of O(N ). All these results are in agreement with [11].
It is worth mentioning the analysis of these results done in [16,17], where the asymptotic s → ∞ behavior for the twist, τ s = ∆ s − s and consequently the anomalous dimension of higher-spin currents is established. Namely one should get for the twist of the higher spin operator of the form φ∂ s φ where τ min is the minimal twist among the operators present in the φφ OPE. First of all, we see that the limiting value is equal to 2 (N +2) 2(N +8) 2 = 2γ φ as follows from expanding (3.27) to order . Second, the leading correction behaves as 1 s 2 , which is a manifestation of having a tower of operators with twist 2, which at this order in are the higher-spin currents and the φ i φ i operator (see [16] for more details). In principle, the coefficient c τ min is determined by the certain three-point functions of these operators with φ. We will go into more detail about this in the next two sections.

The large N critical O(N ) model
To develop the 1/N expansion of the φ 4 theory, one may introduce a Hubbard-Stratonovich auxiliary field σ, so that the action (3.1) may be rewritten as (4.1) In the IR limit for d < 4, the last term becomes unimportant and can be dropped. 6 To develop perturbation theory, it is convenient to rescale σ so that the action becomes The σ field then acquires an effective non-local propagator upon integrating out φ i7 The equation of motion for φ is and the equation of motion for σ is formally φ i φ i = 0 after we drop the last term of (4.2) in the IR limit. The role of this equation is to "subtract" from the theory the operator φ i φ i , which is replaced by σ. This fact will play an important role in our calculation below.
Before turning to the higher spin currents let us calculate as a warmup the anomalous dimension γ φ of the φ field, without computing Feynman diagrams. Using the equations of motion (4.4) and acting on the φ two-point function with ∂ 2 1 ∂ 2 2 , we get From this, the leading order value of γ φ immediately follows which is a well-known result. It is quite remarkable how simple the calculation is, provided we know C σσ . It is also helpful that one completely avoids (to the lowest order) the is-sues of regularization and renormalization, which are actually somewhat thorny in the 1/N expansion.
Let us now turn to the higher spin currents. These have the same form (2.12), and using (2.28) and the equations of motion (4.4), one ends up with where the coefficients C s k are given explicitly by not the equation for σ whose role is to formally project out J 0 = φ i φ i from the theory. This implies that in fact the form of the descendant (4.9) only applies as written to the non-singlet currents. For the singlets, one obtains the correct descendant by the prescription that the term s = 0 should be dropped from the sum. As an example, for s = 2 we get from (4.9) For the symmetric traceless K At d = 3, we see thatK 3 ∝Ĵ 2∂ σ − 2/5(∂Ĵ 2 )σ, in agreement with [6].
We can now compute the anomalous dimensions using (2.27). Let us first discuss the case of the non-singlet currents, where we can use directly the form (4.8), equivalent to (4.9).
The descendant two-point function can be computed similarly to the previous section and one ends up with , where γ φ is the anomalous dimension of φ field. This is the correct result, in agreement [12].
Let us now turn to the case of the singlet currents, which is slightly more involved due to the J 0 projection discussed above. The correct singlet descendant is given by (here as usual we denote by J s = J ii s the singlet operators) , (4.16) in agreement with [12]. One may also check that setting d It is again interesting to mention the s → ∞ behavior [16,17]. The expansion at large s We see a tower of higher spin currents (1/s d−2 ) in the singlet channel, as well as the σ (1/s 2 ) in both the singlet and the traceless parts. The higher-spin contribution vanishes for the symmetric traceless. The 1/s 2 coming from σ is universal and can be calculated. The coefficient c τ min in (3.27) is given by the formula (3.18) of [16] which reads in terms of twoand three-point function coefficients  In this model, a novel feature in the calculation of the anomalous dimension of the higher spin operators is that the free theory contains two independent towers of conserved, O(N ) singlet, higher spin currents: where we have normalized J s,φ so that both currents have J s J s ∼ O(1). Once interactions are turned on, we expect these operators to mix non-trivially, and one should determine the appropriate eigenstates of the dilatation operator.
The equations of motion are It is evident that the equations of motion will induce the mixing between the currents, since ∂ · J s,φ ∂ · J s,σ = 0 due to the g 1 -dependent interactions in (5.3). The descendant operators (in this case we find it more convenient to include the coupling constants into the definition of the K s−1 's) can be computed in a straightforward way by following similar steps as in the previous sections. Explicitly, they are given bŷ We can then use the general relation (2.27), suitably generalized to the present case with non-trivial mixing, to obtain the following anomalous dimension mixing matrix where the non-diagonal terms comes from the non-zero two-point function K s−1,φKs−1,σ .
From this mixing matrix, one can compute the two eigenvalues to leading order in and finite N , using the expression for the fixed point couplings given in [10]. The resulting finite N expressions are easy to get, but rather lengthy. At large N , using the expressions for the fixed point couplings [10] one finds that the eigenvalues are given by The higher order corrections can be obtained to any desired order, but for simplicity we have listed here only the leading order in 1/N . We see that the γ 1 eigenvalue vanishes at s = 2, and one can check that this is true for any N . This eigenvalue then corresponds to the tower of "single-trace" higher spin currents which include the stress-energy tensor. Indeed, one can explicitly verify that γ 1 matches the 1/N expansion result (4.16) expanded in d = 6 − .
The dimension corresponding to the second eigenvalue is which suggests that this should match the "double-trace" operator σ∂ s σ ∼ φ 2 ∂ s φ 2 in the large N approach. Indeed, one can match (5.10) with the result given in [12] for The order 1/N is seen to exactly match the large N result (4.13). Furthermore, we checked that the 1/N 2 term also matches with the result obtained in [34] using large N methods for arbitrary d. 10 Let us now study the large spin limit of these results. For the eigenvalues of the singlet mixing matrix, the large spin expansion can be written in a simple form, valid for finite N , in terms of the fixed point couplings where we have used the known expression for the one-loop anomalous dimensions of φ and σ in the cubic model [10]. The leading terms are precisely consistent with the expected large spin limit. The subleading 1/s 2 contributions are clearly coming from the exchange of the σ field, which has ∆ σ = τ = 2. We can check explicitly the prediction of the formula (4.18) for the coefficients of the 1/s 2 terms. For the γ 1 eigenvalue, we should take that The two-point function coefficients are C φφ =C σσ = 1 4π 3 . The three point function coefficient is given at the lowest order by a diagram with one g 1 φ i φ i vertex in the middle. The diagram is given by the integral: Combining the factors we get c τ min = 2 Overall, we then get 384π 3 in the leading order. The same applies for the eigenvalue γ 2 , corresponding to τ s,σ , where we get: is the leading order dimension (and thus twist) of σ.
The coefficient of 1/s 2 is reproduced the three-point function σσσ .
The cubic model in d = 6 − also admits non-unitary fixed points which are of interest in statistical mechanics. The simplest case is the N = 0 model, which just consists of a single scalar field σ with cubic interaction g 2 /6σ 3 . This model has a non-unitary fixed point at As pointed out by Fisher [35], this theory is related to the Lee-Yang edge singularity of the with Sp(2) global symmetry. The IR stable fixed point occurs at [36] g * 2 = 2g * 1 , where the first equality holds to all orders in perturbation theory. For such a relation between couplings, one can verify that the model has an enhanced "supersymmetry" OSp(1|2) which implies that the dimension of θ and σ are equal. It turns out that this OSp(1|2) invariant fixed point is equivalent to the q → 0 limit of the q-state Potts model [37]. The dimension of the Sp(2) invariant higher spin currents at the fixed point can be obtained from (5.6) setting N = −2 and g 2 = 2g 1 . This yields the two eigenvalues = − (s − 2)(s + 5) 15(s + 1)(s + 2) We see that the first eigenvalue corresponds to the tower which includes the stress tensor of the theory, since it vanishes at s = 2 (this eigenvalue corresponds to an OSp(1|2) singlet).
In the large spin limit one gets The equality of the leading terms is a consequence of ∆ θ = ∆ φ , as follows from the OSp(1|2) symmetry. One may also obtain the dimension of the non-singlet currents, which are the same as in (5.11), with g 1 given in (5.19). We start with the action with an auxiliary field inserted to resolve the sphere constraint on the φ i field, φ i φ i = 1/g 2

Nonlinear sigma model
To develop perturbation theory, one may resolve the constraint by introducing a set of N − 1 independent fields. A convenient parametrization is φ a = ϕ a , a = 1, . . . , N − 1; In terms of the ϕ a fields, the action is To leading order in perturbation theory and in d = 2 + , the coupling constant has the beta and there is a UV fixed point at [39,40] The factor of N − 2 is due to the fact that the O(2) model is conformal and has a trivial beta function in d = 2. Consequently, the perturbative UV fixed point in d = 2 + only exists for N > 2.
Before moving onward to the higher spin operators, we will calculate the anomalous dimension of the ϕ field (or equivalently φ i in the action (6.1)) using the classical equations of motion. To leading order in g, they are given by 11 In full analogy with the discussion in d = 4 − , we can apply the equations of motion (6.6) to the two-point function of ϕ field, obtaining To specialize to the expansion in d − 2 = we need to mention several important points: γ will be of the order and not 2 , unlike in d = 4 − . This means that the third term in the right-hand side is of the order as well. Second, the bare propagator is , (6.8) and applying derivatives to it produces powers of . They will combine with Γ( /2) = 2 + O(1). Since g 4 * ∼ 2 at the critical point, only terms of order 2 are needed from the two-point function in the numerator, since we have three Γ s on top and one on bottom, which amounts to 1/ 2 . Having said that, the relevant term of the two-point function is easy to calculate. The first and the second derivatives of the propagator are: It is evident then that the only way to get O( 2 ) is to contract ϕ a and ϕ b which would be undifferentiated, and the other ϕ's accordingly so that the from (6.10) is picked up two times. Overall one gets for the left hand side of (6.7) (6.11) The right-hand side of (6.7) yields 8γ φ ( + 2γ φ ) to leading order in . Solving the resulting quadratic equation for γ φ gives which is the well-known result [39,40].
Let us now move to the higher spin operators, restricting to the case of the O(N ) singlets.
The form of the higher spin currents is most easily written in terms of the constrained fields φ i , i = 1, . . . , N appearing in the action (6.7). In terms of these fields, they take the same form (2.12)Ĵ where we have set d = 2 since we will only perform a leading order calculation.
It turns out that due to the properties of C −1/2 s (x), in the currents (6.13) all terms have both φ i (x 1 ) and φ i (x 2 ) differentiated at least once, so that after resolving the constraint (6.2), we have in terms of ϕ â One may check, for instance, that for s = 2 this yields the correct stress tensor coming from (6.3). The reason that we have to keep the term of order g 2 is that, when we compute the descendant by ∂ µ D µ zĴ s , both terms in (6.14) yield a contribution of order g 2 (because the first term is a conserved current at g = 0, but the second is not). Using the general equation (2.28), we have (recall that s is even) When acting with ∂ 2 on the first term, we use the equation of motion (6.6). When acting on the second term, on the other hand, we can actually use the free equation of motion ∂ 2 ϕ a = 0 to this order, so that ∂ 2 1 ϕ a ϕ a (x 1 ) = 2∂ µ ϕ a ∂ µ ϕ a . The final result for the descendant to order g 2 can then be written in the form where H k = k n=1 1/n is the harmonic number. The 1/N expansion of this result precisely matches the expansion of (4.16) in d = 2 + . In the large spin limit, we see the logarithmic behavior (since H k ∼ log(k) at large k) Also, we note that the leading order in γ s is 2 , although the leading order anomalous dimension of the φ field is (6.12). This may seem to contradict the expected s → ∞ behavior. The simple resolution of this "paradox" is suggested by looking at the large N result (4.17) for the singlet currents, expanded near d = 2 + : We see that for the singlets the 2γ φ term is canceled by the expansion of the second term, coming from the higher-spin current tower, and the log(s) is exactly what one gets from expanding the harmonic number H s−2 . As for the non-singlet operators, from (4.17) one gets γ s(ij) = N + O( 1 s 2 ), and it is evident that the leading order is indeed 2γ φ as expected [13][14][15][16][17]. Thus, a finite N calculation of the anomalous dimensions of the non-singlet operators should yield a result starting at order , unlike (6.17). We leave the more detailed discussion of the non-singlet currents for future work.
where the coefficients are fixed by matching the known perturbative expansions in d = 4 − and d = 2 + . Rather than performing this procedure on γ s (d) itself, guided by the expected large spin behavior [13][14][15][16][17], we find it more convenient to consider the quantity From the results (3.24), (6.17), we can obtain the expansion of this quantity to order 2 .
Further information in d = 4 − can be obtained using the result of [32], who derived the anomalous dimensions of the higher spin operators in the O(N ) theory to order 3 . 12 For the singlet currents, it reads where H s is the harmonic number. This vanishes at s = 2, as expected. It is also interesting to check the large spin behavior, which yields (using the known result for γ φ to order λ 3 , see 12 For N = 1, the result is known to order 4 [41]. e.g. [42]) We see that a logarithmic term arises at subleading order in the coupling constant, consistently with general expectations [16,43]. Using the value of the critical coupling [42] λ * = 8π 2 N + 8 we can obtain the expansion of γ s around d = 4 to order 3 . Further using the expansions of γ φ near d = 2 and d = 4, we can get the function f (d) defined in (7.2) to the same order where for simplicity we did not write explicitly the O( 3 ) in d = 4 − , it can be read off from (7.4). This allows to construct Padé approximants (7.1) with a maximum value n + m = 6.
The results for s = 4 given in Table 1 appear to be consistent with the ones given in [47].  While we do not expect these to be high precision results, we observe that they appear to be quite close to the estimates derived in [43]. For the spin 4 operator, [52] obtained the slightly lower value γ 4 = 0.0208 (12). For N = 2, following a similar procedure and using 2γ N =2 φ = 0.0381 [53], we obtain γ N =2 s=4 = 0.0252 , γ N =2 s=6 = 0.0315 γ N =2 s=8 = 0.0340 , γ N =2 s=10 = 0.0353 .

(7.9)
In all cases, we observe that the anomalous dimensions of the higher spin operators is rather small (similarly to what happens for the anomalous dimension of φ). From the results in Table 1, and (7.8),(7.9), we also notice some non-monotonic behavior as a function of N , with a maximum between N = 3 and N = 4. A qualitatively similar non-monotonic behavior can be observed in the sphere free energy [54,55] and C T [44]. It would be interesting to understand better the origin of this behavior and the relation between these quantities.

1318681.
A Technical details on the computation of the descendant 2-point function The calculation of the descendant two-point functions is carried out using the relation of Gegenbeauer polynomials to hypergeometric functions (see appendix of [56]). We will illustrate the technique to calculate the function A s (see eq. 3.20) as a function of s in terms of Gegenbauer integrals. The calculation of the other structures is similar. After introducing the Schwinger parametrization, the h s function can be written as: α n = α n α 1 + α 2 + α 3 + α 4 ,h s (x) = C The goal now is to calculate the integral of functionsh. First we study the integral of twõ h with the same argument: The idea now is to employ the Rodrigues formula for the Gegenbauer polynomial: C ν s (1 − 2α) = 4 s s! Γ(s + ν)Γ(s + 2ν) Γ(ν)Γ(2s + 2ν) (α(1 −α)) −ν+1/2 d s dα s (α(1 −α)) s+ν−1/2 (A.5) We again split the integral into two parts: This is the main idea of the calculation, the rest is basically collecting all the coefficients and applying the same method to the other integrals which will appear (all of them will be of the same type though). For the sake of reference the coefficients ofh s (1 − 2α) are obtained most easily by using the relation of Gegenbauer polynomials to the hypergeometric function: (s − 1)s(s + 1)(s + 2) 8(2s + 1) (A.11)