Crossing symmetry and Higher spin towers

We consider higher spin operators in weakly coupled gauge conformal field theories. Crossing symmetry of mixed scalar correlators relates different higher spin towers and we study the consequences for the spectrum and structure constants of higher spin operators of different twists. Constraints are obtained to all loops in perturbation theory. The large spin contributions to the structure constants can be resummed into a theory-dependent prefactor times a universal factor, whose structure of poles agrees with the one that would be obtained from a Witten diagram supergravity computation, although only crossing symmetry is assumed. Finally, our results provide an all loop expression for the double null limit of mixed correlators, which is in perfect agreement with the correlator/Wilson loop correspondence.


Introduction
The idea of the conformal bootstrap is that by imposing associativity of the operator product expansion (OPE) for local operators in a unitary conformal field theory (CFT) one can derive constraints for the spectrum and OPE coefficients of the theory [1,2]. For instance, for a four-point function crossing symmetry plus the structure of the OPE expansion schematically implies Fig.1. In a general CFT in higher dimensions, the interplay between direct and Figure 1: Crossing symmetry crossed channels is very complicated: a given operator on one channel will generically map to a very complicated combination of operators on the other channel. The essence of the analytic conformal bootstrap is that for certain operators this map is much simpler. For example, each higher dimensional CFT, strongly [3,4] or weakly [5] coupled possesses double trace higher spin operators which map to the identity operator in the dual channel. In this paper we will concentrate on conformal weakly coupled gauge theories. These theories contain towers of higher spin operators, with small anomalous dimensions, which under crossing, and for high values of the spin, map to each other [5,6]: where τ denotes the twist (dimension minus the spin) of the higher spin tower. In [6] we focused in a four-dimensional CFT, external operators of the form O [2] = T rϕ 2 and singletrace higher spin operators of the schematic form O = T rϕ∂ ϕ, with twist two and spin . In perturbation theory these operators have a small anomalous dimension: It turns out crossing symmetry is powerful enough to fix the behaviour of the anomalous dimension for large values of the spin γ ∼ f (g) log together with the OPE coefficient of O with two external operators Such results [5,6] are valid for high values of the spin but to all orders in perturbation theory! On [6] we have focused on correlators of identical operators. In the present paper we will consider mixed correlators and show that (1) is a particular example of a more general relation, in which two different towers of higher spin operators map to each other Analysing the consequences of crossing in this case is more delicate, since higher spin operators with more than two constituent fields are highly degenerate. For τ > 2 we find that again crossing implies a logarithmic growing for the anomalous dimensions, but this time in the sense of a weighted average (to be defined below). Furthermore, crossing symmetry fixes the large spin behaviour of the OPE coefficient between two scalar operators and a higher spin operator. More precisely, we obtain the following universal behaviour for several families of operators, as will be specified below. This is a natural generalisation of (2). Furthermore, note that the same structure of poles would appear in a Witten diagram supergravity computation of the three-point function for operators of dimensions ∆ p , ∆ q and τ , see for instance [7][8][9]. However, our result arises from crossing symmetry alone, without assuming large N or large R−charges, and is a all-loop result!
Having solved for the structure constants of higher spin operators we can study the mixed correlators under consideration in the double null limit. Our results are in perfect agreement with the picture of [12]. This paper is organised as follows. In the next section we show that crossing symmetry for mixed correlators leads to relations among different higher spin towers. In section three we derive integral relations arising from crossing symmetry for mixed correlators. Although the method is very general, we apply it to a simple model for definiteness. We then show how to solve such integral relations, finding constraints, to all loops in perturbation theory, for the spectrum of higher spin operators and OPE coefficients of the theory. As an interesting application, we compute the double null limit of the correlators under consideration and compare our results to [12]. In section (4) we study in detail the case of weakly coupled N = 4 SYM. In this case the theory possesses a global R−symmetry and crossing symmetry acts on the representations of this symmetry as well. By considering appropriate projections we show that the equations in this case exactly reduce to the equations previously found. We end up with a discussion of our results. Several technical details, needed in the body of the paper, are deferred to the appendices.
2 Crossing symmetry and higher spin towers 2.1 Crossing for mixed scalar operators and higher spin towers Let us start with a discussion of crossing relations for the most general scalar case. This was done for instance in [10]. For four arbitrary scalar operators we can write, see [11] where we have introduced g ∆, (u, v) = g 0,0 ∆, (u, v). In conformal weakly coupled gauge theories there are towers of almost conserved higher spin currents of twist τ = ∆ − = ∆ 0 + γ . In such a situation the crossing relation (10) maps the large spin sector of the higher spin tower HS τ to itself 1 This phenomenon was exploited in [5,6] to find information about the spectrum and OPE coefficients of higher spin operators. Considering instead mixed correlators we see that this is a special case of a more general relation Indeed, consider the contribution to G ijkl (u, v) from operators with twist τ : then such a divergence must come from a tower of higher spin operators, of approximate twist τ . Under crossing this term maps in the dual channel to a term of the form Provided β > 0 this must correspond to an infinite tower of higher spin operators HS τ . Hence crossing leads to the relation (12). Studying different mixed correlators will lead to different constraints involving towers of higher spin operators. Below we will study such constraints in detail, but before let us discuss the properties of different higher spin towers.

Towers of higher spin operators
Although the methods which we will apply in this paper are completely general, we will discuss a specific model for definiteness. Then in section 4 we will focus in a different model and show that the final relation has exactly the same form. Let us consider a conformal weakly coupled gauge theory in four dimensions, with a scalar field ϕ. The simplest gauge invariant operators are traces of such a scalar field and its derivatives. Below we will discuss the towers of higher spin operators that can arise in the light-cone OPE of scalar operators formed only by scalar fields.

Twist 2
These are operators of the form O (2) = T rϕ∂ ϕ + · · · where the derivatives are along a null direction. There is only one primary operator for each even spin and none with spin odd 2 . We denote such a tower by HS 2 . These operators are also called leading-twist operators and their anomalous dimension grows logarithmically with the spin: where we have defined the coupling constant g as the coefficient in front of the logarithmic piece. All other quantities will be expressed in terms of this g.

Twist 3
These are operators of the form O I, = T rϕ 2 ∂ ϕ + · · · . For both, spin odd and even, there is a degeneracy of primary operators, see appendix C. We denote this tower by HS 3 . Again, their anomalous dimension grows logarithmically with the spin, but they grow along a band, as described in [13]. More precisely, for large spin

Twist 4
For twist four and higher we have a new ingredient. On one hand, there are single-trace operators of the form O I, = T rϕ 3 ∂ ϕ + · · · . There are primary operators for both, spin odd and even. Their anomalous dimension grows logarithmically with the spin, again along a band, schematically γ (4) where now 1 ≤ ρ I ≤ 3. In addition, there are double trace operators, of the form [O (2) , 3 , with 1 + 2 + 3 = . For large 1 + 3 (a macroscopic fraction of ) their anomalous dimension again grows logarithmically with the spin. The discussion in [17] and a naive application of the results of [3,4] would imply an anomalous dimension of the form While the regime considered in this paper is not in the range of validity [3,4], the ideas of [17] still apply and we expect this expansion to be true. The main difference with the single trace case is that there are operators with very large spin whose anomalous dimension does not grow logarithmically, the ones with small 1 , 3 . Of course, in a non-planar theory there is really no distinction between single and multi-trace operators, but the behaviour with spin will have the same features: for most operators it will grow logarithmically but there will be some operators for which it wont. We denote the full contribution of all operators HS 4 and the results of this paper will still apply. Higher twists behave in a similar way, except in general we can also have triple trace, etc.

Consequences of crossing symmetry
Although our methods will be general, we will focus for definiteness on a specific model. We consider a conformal weakly coupled gauge theory in four dimensions and external operators of the form with dimension ∆ = L at tree level. We will assume for simplicity that such operators are protected. Furthermore we denote by G L 1 L 2 L 3 L 4 (u, v) the conformal invariant part of the correlator We will start by considering the simplest case of four identical operators with L = 2. This will serve to introduce some important ingredients. Then we will proceed to discuss a more general case.

Integral relations
Correlator G 2222 (u, v) Let us start by the simplest case G 2222 (u, v). At tree-level we obtain The constants c ij will in general depend on the parameters of the theory, e.g. its central charge, but their explicit form will not be relevant for us. When the coupling constant is turned on these coefficients get dressed by logarithms, and to any order in perturbation theory where the function c ij (log u, log v) is by definition the function in front of u i v j in a small u, v expansion, see [5]. From the structure of divergences, and the powers of u, the contribution c 11 must come from the exchange of an infinite tower of higher spin operators with twist two, or HS 2 . Furthermore c 22 arises solely from the tower HS 4 , while c 21 may receive contributions from both, twist four operators as well as descendants of twist two operators, see appendix A. What do these towers map to under crossing? Crossing symmetry implies We see c 22 gets mapped to the contribution from the identity operator. This is an example of the phenomenon studied in [3,4]. Furthermore, the contribution c 11 maps to itself, so that this corresponds to More precisely, crossing symmetry implies c 11 (log u, log v) = c 11 (log v, log u). In order to study the consequences of this relation let us follow [6] and consider both ways of decomposing c 11 : In order to reproduce the correct divergence at tree level in either sum we require the condition arising from crossing can be written as (28) where inâ( ) and γ (2) ( ) only the contributions that do not vanish at large spin are kept. This integral relation imposes conditions on both, the spectrum and the OPE coefficients. More precisely, at large spin [6]: These results are valid to all loops in perturbation theory. Namely, to all orders in g in the regime g log 1, although they resum all perturbative corrections.
Let us study the correlator G ppqq (u, v) and its permutations, the most relevant case for this paper. In this case there are two distinct crossing relations and they provide different information. Let us start by computing at zero coupling As we turn on the coupling constant c ij → c ij (log u, log v). By crossing symmetry we then obtain 3 where we have introduced the notation c c ij = c ij (log v, log u). Let us focus in the contributions c τ 2 τ 2 , with τ = 2, 4, · · · , 2 min(p, q). From the point of view of the direct channel (30) these can only arise from higher spin towers HS τ . Indeed, descendants of lower twist higher spin towers will not produce a divergent enough term, see appendix A. From the point of view of the dual channel (31) these can only arise from higher spin towers HS p+q−τ , for the same reason. Hence, crossing relates : so that crossing reads For a fixed , the index I labels different operators which are degenerate at tree level. Twist two is the only non-degenerate case. The next step is to convert this relation into an integral equation. This is a bit more subtle than before, as for each spin we have a degeneracy at tree-level. In order to reproduce the correct divergence at tree level we must have The correct divergence in perturbation theory implies a similar behaviour for the quantum OPE coefficients, so that we define Since in general the intermediate operators on both sides of (34) are degenerate at tree level, we introduce the following weighted averages on each side Note that the two averages are with respect to different weights. Using the results of appendix A we arrive to the following integral relation Two comments are in order. First, in the above equation we keep only terms that are not suppressed by powers of the spin in the large spin expansions of γ( ) andâ( ) on both sides. Second, we have chosen a normalization such that in perturbation theorŷ Once a solution is found, we can always multiply both sides by a function of the coupling constant, and the resulting OPE coefficients will still be a solution. Relation (40) imposes non-trivial constraints on the spectrum and OPE coefficients. It turns out these constraints are simpler to analyse in the case in which the tower HS τ is non-degenerate, namely τ = 2.
In this case the equation reduces to Below we will explicitly consider the constraints arising from this relation and comment on the general case. Before that, however, let us comment on the other crossing relation. Compute at tree-level In the quantum theory d ij → d ij (log u, log v). Crossing implies So that we have a relation of the form However, there is a crucial difference with the previous case. According to the results of appendix B: so that the leading divergence as v → 0 does not arise from the leading behaviour of the OPE coefficients but rather from a subleading term, which does not contain (−1) . The consequences of this are that if we were to defineâ( ) as above, not only the leading term would contribute, but also terms which are suppressed in the large spin limit, provided they contain an additional (−1) . The same will happen with the anomalous dimension contributions. For this reason, in the following we will focus on relations (40) and (42).

Solving the integral equation
As we have seen, relation (28) implies a logarithmic behaviour for the anomalous dimension of twist-two operators and fixes completely the large spin behaviour of the OPE coefficients, both results valid to all loops in perturbation theory. In the following we would like to work out the implications of (42).
First, note that at tree-level ... = 1,â = 1, and all anomalous dimensions vanish so that the integral relation is satisfied. As we turn on the coupling it follows that the average anomalous dimensions for twist p+q −2 operators can have at most a logarithmic behaviour, very much as for the twist two case. So that and so on. This is consistent with the analysis of [13]. Note that due to degeneracy in general γ 2 = γ 2 . Each of the quantities of the r.h.s. will have a coupling constant dependence, so that ρ = ρ 1 g + ρ 2 g 2 + · · · (48) This is to be supplemented with the known behaviour for the anomalous dimension of twist two operators. Crossing symmetry implies a similar logarithmic behaviour for the average of the OPE coefficients: a (2) ( ) = 1 + g(a 10 + a 11 log ) + g 2 (a 20 + a 21 log + a 22 log 2 ) + · · · (50) 22 log 2 ) + · · · (51) We could insert all the corresponding expansions into (42), expand order by order in perturbation theory and work out the corresponding constraints. We can also proceed in a more systematic way. First rewrite the integral equation as ∞ 0â (2) x √ v 2 with the following logarithmic behaviour for the anomalous dimensions: (γ (p+q−2) ) n = (ρ log + β) n .
Then introduce the following integral representations: Plugging this into (52) we obtain an equation of the form where P L and P R have a very specific form. It turns out that to any order in perturbation theory the Kernel K 0 (2x)K p−q (2y) is such that the above equation actually implies P L = P R . It turns out this implies the following remarkable property for the average of the spectrum: and furthermore ρ = g So that to any order in perturbation theory the leading logarithmic behaviour of the averaged anomalous dimension of the p + q − 2 higher spin operator behaves as if there were no degeneracy and equals the anomalous dimension of twist two operators! This is not in contradiction with [13], since here we are only talking about a weighted average and in the limit of large spin. Furthermore, crossing also fixes with the understanding that powers of β are to be understood in an averaged sense 4 . α is an arbitrary function of the coupling constant (not fixed by crossing) but independent of the spin. Plugging this back into the integral representations (55) we obtain 4 For instance 2 β → 2 β = 1 + β log 2 + · · · where γ (p+q−2) ( ) = g log + β and again, powers of β (which arise when expanding the expression above) are to be understood in a averaged sense. Two comments are in order. First recall a (2) ( ) arose from a factorised OPE coefficient c ppO (2) c qqO (2) , so that the factor −β should factorize accordingly, namely −β = f (p)f (q). Furthermore note that the rest of the answer factorises as well. Second, note that from our answer we can read off the following universal behaviour at large spin Up to a prefactor which depends on the details of the theory. The result forâ (p+q−2) ( ) has a similar universal behaviour (but in this caseâ (p+q−2) ( ) is itself a sum over many contributions), namely The universal behaviour we have found can be summarised as follows. The OPE coefficient between two scalar operators of weights ∆ p and ∆ q and a higher spin operator of (tree-level) twist τ has the universal behaviour where averages should be understood where it corresponds. This behaviour is also consistent with the most general relation (40), but in this case the prefactor is more complicated. This structure is very reminiscent of the result one would obtain from Witten's diagrams in supergravity. However, in the present paper we have only analysed the consequences of crossing, without any further assumptions.

Comparison to polygonal Wilson loops
The consecutive null limit x 2 i,i+1 → 0 of correlators in conformal gauge theories was studied in [12]. In this limit there are fast particles propagating between consecutive points and the correlator should reduce to the expectation value of a polygonal Wilson loop. For the particular case of a four-point function this limit coincides with the double null limit where u, v → 0 at the same rate. It was argued in [12] (see section 4 of that paper) that in this limit we should obtain lim u,v→0 where G denotes the full correlator (not only its conformal invariant part) and we focus on a given connected contribution, so that the fast particles can frame the Wilson loop, and divide by the corresponding connected piece at tree level. The result (66) can be better understood by choosing coordinates where the insertion points are at the vertices of a large rectangle with sides ∆τ ≈ −1/2 log u and ∆σ ≈ −1/2 log v, see figure 2. As we approach the double null limit we have a fast moving particle going between the different vertices of the polygon. Since this particle is coloured, it sources a colour electric field which is extended on the rectangle (green area in fig. 2). This colour electric flux has constant energy density in the (τ, σ) plane, proportional to Γ cusp , and this produces the leading divergence in the exponential in (66), proportional to the area of the rectangle. In the interacting theory the particles can interact with the flux, and there are further terms. The simplest contribution arises due to corrections to the energies of the propagating particles. These corrections are confined to the edge of the rectangle (red lines in fig. 2) and give rise to the subleading divergences in the exponential in (66), proportional to the perimeter of rectangle. Finally, the factor J(u, v) arises from the fact that the particles are coloured so we can have a back reaction of the colour electric field on the propagation of the particles. Although this is in general a complicated factor some features were studied in [12].
In the present paper we have computed the structure constants of higher spin operators with leading twist. In particular, these operators dominate the correlator in the small u, v limit. Consider the correlator G ppqq above. In the small u, v limit, with their ration fixed, only the functions c ii will survive. Each function corresponds to a different connected contribution. In particular, let us consider c 11 which corresponds to Plugging (61) back into the l.h.s. of (42) we obtain Recall that in our conventions Γ cusp ≡ g. Hence, our result exactly agrees with (66)! Furthermore note that from the point of view of the picture in [12] β in the exponent in (68) arises from corrections to the energy of the particle going from A to B plus corrections to the energy of the particle going from D to C. For the present case the first correction should depend only on p, while the second should depend only on q, leading to a factorised depen-dence, in agreement with the comment below (61-62). Finally, crossing symmetry provides an all loop expression for the factor J(u, v): where powers of β should be understood as averages. This answer satisfies all the general properties for J(u, v) discussed in [12]. Other terms c ii will behave similarly . Hence, our results are in perfect agreement with the correspondence proposed in [12].

A case with global charge: N = SYM
In the simple model studied above we have ignored two important features. On one hand, a generic gauge CFT contains fermions and gauge bosons in addition to scalars. As a result, there will be higher spin operators also formed by non-scalar letters. In particular, some of these will have the same quantum numbers as the higher spin operators studied above, increasing the degeneracy. On the other hand, gauge theories can posses global symmetries, such that scalars and fermions are charged under this global symmetry. Projecting in different representations may split a priori degenerate higher spin operators. In the following we will see how the picture introduced in sections 2 and 3 works for the particular example of N = 4 SYM.

Higher spin towers in N = 4 SYM
Four-dimensional N = 4 SYM posses a global SU (4) R R−symmetry group. Gauge invariant operators are formed by traces of the fundamental fields of the theory: Scalar fields ϕ I in the 6 of SU (4) R ; fermionic fields λ A α andλ Aα , in the 4 and4 of SU (4) R and gauge bosons A µ in the singlet representation; together with their derivatives. We can form the following higher spin operators of spin , classified by their twist and SU (4) representation:
• T rF νµ 1 ∂ µ 2 · · · ∂ µ −1 F ν µ , transforming in the 1 × 1 = 1 As a result, we can consider different towers HS R 2 . Note that HS 20 2 can only be formed by scalars so that it is still non-degenerate.
Again, note that HS 50 3 can only be made by scalars.

Twist p
A similar analysis can be performed for higher and higher twists, with a richer and richer structure. The upshot is that the representation with Dynkin labels [0p0] and hence HS

Crossing symmetry in N = 4 SYM
In four dimensional N = 4 SYM there is a class of half-BPS superconformal primary operators, transforming in the [0, p, 0] of SU (4) R and with protected dimension ∆ = p. They are given by where r i = 1, · · · , 6 and t is a complex six dimensional null vector which encodes the Rsymmetry structure. Superconformal symmetry fixes the structure of the four point function of such operators to be of the form [14] O where we have introduced harmonic cross ratios σ and τ defined as Such correlator can be decomposed into (p 1 +1)(p 1 +2) 2 terms, accordingly to the different SU (4) R representation present in the OPE of [0, , where without loss of generality we assume that p 1 ≤ p 2 ≤ p 3 ≤ p 4 . Each contribution, labelled by (n, m) = [n − m, p 2 − p 1 + 2m, n − m], may be decomposed in conformal blocks as and they can be expanded in collinear conformal blocks as in (73). As previously discussed, the representation (1, 1) is the only one which is non-degenerate and contains only operators built from scalars. 5 As a consequence of crossing symmetry we can write The functionsã where κ is a coefficient which depends on p and q. At this point it is clear that we can apply the same procedure and results of the previous sections provided we project in specific SU (4) R representations, namely So that for each of these representations we obtain a relation exactly as (42).
In [15], the four point function of half-BPS operators of arbitrary dimensions have been computed in planar N = 4 SYM up to three loops. The simplest example with p = q corresponds to the correlator G 2233 (u, v, σ, τ ), so lets list the results for this case. By projecting this four point function in the 20' representation, one can perform the conformal partial wave expansion and extract c 22O (1,1) c 33O (1,1) as well as γ (1,1) ( ), up to two loops in perturbation theory. Using the notation of (50) we obtain a 11 = − log 2 (82) Now using the crossing relations (42) it is possible to compute the coefficients β and β 2 appearing in the expansions (46) of the weighted averages of anomalous dimension of twist-3 operators for any of the three possible representation of SU (4) R : where γ e is Euler Gamma constant. As expected from unitarity, β 2 > β 2 .
We can also obtain results for general p, q. As noticed in [15,16], three point functions of c p 1 p 2 properly normalised are all equal at one loop since there is only one structure at this loop order. This allows computing β for generic p, q giving where ψ 0 denotes the digamma function. Note that for p = q = 2 this agrees with the finite piece of the anomalous dimension of twist two operators in the large spin limit, while for p = 2, q = 3 it agrees with the result given above. Furthermore, it displays the factorised structure discussed section 3.

Conclusions
In the present paper we have studied weakly coupled conformal gauge theories by analytic bootstrap techniques. Weakly coupled gauge theories contain towers of higher spin operators of approximate twist τ . By studying crossing symmetry for mixed correlators we have found that these towers (for large values of the spin) map to each other: This relation takes the form of an integral equation involving the spectrum and structure constants of the higher spin operators. In case of twists higher than two, such operators are degenerate and the integral relation involves weighted averages. Regarding the spectrum, we have found that crossing symmetry is consistent with a logarithmic behaviour, in agreement with [13]. Regarding the structure constants our results take the form Namely, a universal factor times a theory-dependent prefactor f (τ ) pq ( ). The universal factor has a very similar structure to the one that arises when studying Witten diagrams. In particular, it includes a series of poles that start when the full twist of the higher spin operator equals the sum of the dimensions of the other two. In the context of large N MSYM the appearance of analogous poles was analysed in [9,19] and where it was shown to be related to operator mixing. Although our results are in principle only valid in perturbation theory (but to all loops), given the discussions in [9,19] we expect this structure to persist for finite γ , at least in the planar limit. It is very interesting this structure arises naturally by only requiring crossing symmetry. The theory-dependent prefactor, of the schematic form f = −β , depends on the theory under consideration and on averages that are hard to calculate. For the simplest case this prefactor is basically f = −b , where b is the subleading/finite contribution to the anomalous dimension of twist two operators. In this case it does not add any new analytic structure to the answer, and we expect this to be the case in general.
Having solved for the constraints above one can then compute the mixed correlators under consideration in the double null limit. This limit was studied in [12] where it was shown that the expectation value of a polygonal Wilson loop should be recovered. Our results are in perfect agreement with these expectation and furthermore they provide all loop results for certain prefactors that are in general hard to compute.
Some open problems which we consider interesting are the following. The present paper generalises the results of [6] to external operators with arbitrary dimension. This opens up the possibility to compare our results with results at strong coupling, since now we can consider ∆ p and ∆ q large. It would be very interesting to make a detailed comparison to the results of [9] from string vertices. This may also allow to get a handle on the prefactor f (τ ) pq ( ) at strong coupling, ideally to compute it exactly. Regarding this, note that for N = 4SY M the explicit one-loop result (87) grows logarithmically as p or q becomes large. As a result β becomes symmetric under p ↔ . It would be interesting to understand this result.
It would be interesting to explore further the relation to Wilson loops and the picture of [12]. The present paper offers a proof of the correlators/Wilson loop correspondence from crossing symmetry, for the four-dimensional case and gives explicit expressions for all ingredients involved. Can we learn more from this interplay? A related question is to understand our results, and in particular the structure of poles in the universal factor, along the lines of [12,17]. This may provide a finite coupling understanding of the universal factor.
Over the last years there has been progress in the computation of structure constants in planar N = 4SY M by integrability techniques. See for instance [20] for the state of the art. Despite these developments, there are still missing ingredients if one wants to pursue the program to all loops. The present results may be useful in such endeavours. On one hand, the structures found in this paper should be visible in other approaches. Furthermore, the fact that the results of this paper are valid for any length of the external operators means that certain subtleties, such as wrapping, can be pushed away.
It would also be interesting to apply these techniques to other weakly coupled gauge conformal field theories. An interesting example would be β−deformed N = 4 SYM.
Finally, for theories with gravity dual (known or unknown) an interesting question is how much of the structure of the gravity dual can be understood from symmetries of CFT correlators. Or conversely, which CFT theories can admit a gravity dual. There has been a lot of activity in this regard, see for instance [21] for early results in this direction and [22] for a different approach. It is remarkable that our results reproduce the pole structure of Witten diagrams. One may wonder if this would lead to a way to define constructively the would be gravity dual of our CFT's.
In the above expression we have assumed the sum runs over even spins only, which is the case, for instance, if we have identical external operators. In general we can have a sum over all spins. In this case: where the intermediate operator has twist p + q − 2m. For the case p = 3, q = 2, τ = 2 this reduces to the previous case.

C Degeneracy of twist operators
In this appendix we study the degeneracy of primary operators with fixed twist, of the form where the derivative is along a fixed null direction. For such operators the twist coincides with the length L. The degeneracy of such operators can be easily computed by Polya theory. First, let us consider the single letter partition function: Z 1 (q) = q + q 2 + q 3 + · · · = q 1 − q (108) which counts states of the form ∂ n ϕ. The multi-letter partition function, taking into account cyclycity of the trace is given by where (s, L) denotes the largest common divisor of s and L. In order to compute the number of independent primaries, at each level we subtract the number of operators at previous level, so that is the generating function for the number of primaries. For the first few twists we find P 2 (q) = q 2 1 − q 2 (111) P 3 (q) = q 3 ((q − 1)q + 1) (q − 1) 2 (q 2 + q + 1) In particular primary operators with twist two are non-degenerate and have only even spin, while primary operators of twist three and higher are always degenerate. The degeneracy for large values of the spin can be understood from the behaviour near q = 1. We find d L ( ) ∼ L−2 .