BCJ relations in ${AdS}_5 \times S^3$ and the double-trace spectrum of super gluons

We revisit the four-point function of super gluons in $AdS_5 \times S^3$ in the spirit of the large $p$ formalism and show how the integrand of a generalised Mellin transform satisfies various non-trivial properties such as $U(1)$ decoupling identity, BCJ relations and colour-kinematic duality, in a way that directly mirrors the analogous relations in flat space. We unmix the spectrum of double-trace operators at large $N$ and find all anomalous dimensions at leading order. The anomalous dimensions follow a very simple pattern, resembling those of other theories with hidden conformal symmetries.


Introduction
Understanding properties of (quantum) gravity theories and their relation to gauge theories is a primary goal in modern physics. Most of these relations are obscured in a lagrangian formulation, and seem to manifest all their majesty only through observables such as scattering amplitudes. The study of scattering amplitudes has led to a series of impressive and deep results, for example BCJ dualities [1] and double-copy constructions [2] in various theories, both at tree and loop-level (for a recent review, see [3]). The upshot is that there seems to be an underlying common structure between gravity and gauge theories, yet to be fully understood. While most of these efforts have related to flat space, mainly because of the difficulties in performing such computations in curved backgrounds, a recent series of papers have begun the exploration of these properties in AdS backgrounds, both in Mellin space [4][5][6][7], as well as in position [8,9] and momentum space [10,11]. Many of these developments have use bootstrap methods that have been highly successful in studying supergravity in AdS [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], as well as string corrections [27][28][29][30][31][32][33][34][35][36][37][38][39]. In particular, in [5], AdS versions of colour-kinematic and double copy relations have been found.
In this letter we further explore these relations by focusing on the four-point function of half-BPS operators dual to the scattering of four super gluons in AdS 5 × S 3 , first computed in [4]. In common with [5], we will focus on the 'reduced' Mellin amplitude (which manifests the supersymmetry of the theory) and cast this in a form which makes colour-kinematics and BCJ relations manifest by mirroring directly the form of the flat space amplitude. At leading order, the theory enjoys a hidden 8d conformal symmetry that nicely repackages all Kaluza-Klein modes into a simple reduced Mellin amplitude M p . In notation inspired by the 'large p' limit of [35] it takes the form M p = n s c s s + 1 + n t c t t + 1 + n u c u u + 1 , where the kinematic (n) and colour (c) numerators obey the same Jacobi type relations. We will see that the as-sociated colour-ordered amplitudes also satisfy BCJ relations for all Kaluza-Klein modes. These take the form, where M p (1,2,3,4) are the colour-ordered amplitudes. The fact that these relations directly mirror their flat space counterparts is related to the existence of the 8d conformal symmetry.
In the second part of the letter, we unmix the spectrum of double-trace operators exchanged in the OPE and compute all the anomalous dimensions at leading order. These CFT data are an important part of the bootstrap program for computing one-loop correlators beyond the lowest KK mode correlator. As shown in e.g. [6,14,15,22], the CFT data of the double-trace operators can be used to build the leading discontinuities of the correlator and then from them one can construct the full amplitude with the help of crossing symmetry.
In common with many other cases where the AdS theory exhibits a hidden conformal symmetry, we find that the anomalous dimensions are given by a strikingly simple formula whose form we sketch here, Here the 'effective' spin l 8d and the quantity δ (2) h,j are functions of twist τ , spin l and SU (2) × SU (2) labels [ab] of the double-trace operators and we will give their specific form later on. The results are reminiscent of previous computations in other backgrounds [19,40,42], and suggest that the hidden conformal symmetry, unavoidably, plays a primary role in constraining the data of these SCFTs.
AdS5 × S 3 Mellin transform and the large p formalism The AdS 5 × S 3 background arises in two basic stringy setups. One can either consider a stack of N D3branes probing F-theory 7-brane singularities or a stack of N F D7-branes wrapping an AdS 5 × S 3 subspace in the AdS 5 × S 5 geometry of a stack of N D3-branes. In both cases, the system preserves 8 supercharges, therefore the dual CFT is a 4d N = 2 theory with flavour group G F , which we will keep generic because it is mostly irrelevant for the details considered in this paper. The low-energy degrees of freedom are those of a N = 1 vector multiplet which transforms in the adjoint of G F . Upon reducing on the sphere, it provides an infinite tower of Kaluza-Klein modes organised in different multiplets. In the dual CFT, the super primaries of these multiplets are half-BPS scalar operators of the form O Ia1a2...ap;ā1ā2...āp−2 p .
Here I is the colour index, p is the scaling dimension of the operator, a 1 , . . . , a p are symmetrised SU (2) R Rsymmetry indices and similarlyā i are indices of an additional SU (2) L flavour group; these last two groups realise the isometry group of the sphere S 3 . In practice, and as usual in these contexts, is convenient to contract the indices with auxiliary bosonic two-component vectors η andη to keep track of the SU (2) R × SU (2) L indices: In this paper we consider the amplitude of four super gluons, which we denote by A crucial point is that, in these theories, the strength of the self-gluon coupling is larger than the coupling of gluons to gravitons [4]. In light of this, one can perform an expansion in 1/N in which gravity is 1/N suppressed. Schematically, we have The first 'disconnected' term is a sum over products of two-point functions and takes the form of (generalised) free theory. In terms of OPE data it contains the leading order contributions to the three-point functions of the external operators with exchanged two-particle operators. We will refer to the second term as the 'tree-level' amplitude.
The correlator is subject to constraints due to superconformal symmetry. In particular, the superconformal Ward identites [43] allow us to split it into two parts, each separately respecting crossing symmetry, The term G 0, p contains all contributions due to protected multiplets at this order in 1/N . The second term contains all the logarithmic terms which arise due to two-particle operators receiving anomalous dimensions. It contains certain kinematic factors P and I, due to bosonic and femionic symmetries respectively. First, let us define the propagator via where y 2 ij = η i η j η iηj with η i η j = η ia η jb ǫ ab and similarly η iηj =η iāηjb ǫāb. We also introduce cross-ratios via Note that we can write the y,ȳ variables in terms of the η andη variables as The kinematic factors are then given by P ≡ g ks 12 g kt 14 g ku 24 g 13 g 24 where Note that, due to the presence of the factor I = (x − y)(x − y), the remaining function A I1I2I3I4 p has the same degree in y,ȳ. Moreover, since A I1I2I3I4 p is symmetric under y,ȳ exchange, we can write it as a function ofŨ ,Ṽ as well as U and V and the charges p.
The function A I1I2I3I4 p admits a very compact and natural representation, that extends the well known Mellin transform [44,45] to the compact space. The transform makes manifest the so-called large p limit [35] -where p here refers to the charges -and it was found to be very useful in the context of AdS 5 × S 5 [35,37] and AdS 3 × S 3 backgrounds [40]. In our conventions the generalised Mellin transform M is defined via The kernel Γ is factorised into AdS 5 and S 3 contributions and takes the form and Γ t , Γ u defined similarly. Note that the Mellin variables obey the relations, which may be used to eliminate u andũ. Note also that the amplitude A I1I2I3I4 p is polynomial inŨ andṼ . In fact, the integral overs,t can be turned into a discrete sum over a certain domain that in our case is given by The contour integral in s and t requires a little care and we will return to this point in the next section. The double integral (11), when combined with the amplitude M p given in the next section, precisely coincides with the result given in [4]. This generalised AdS 5 × S 3 Mellin transform is quite useful because, as shown in [35], in the large p limit, the integrals localise on a classical saddle point. The authors show that the computation matches with that of four geodesics shooting from the boundary and meeting in a common bulk point at which the particles scatter as if they were in flat space. At the saddle point, the 'boldface' variables s = s +s, t = t +t, u = u +ũ, s + t + u = −3 (15) become proportional to the flat space Mandelstam variables. This explains why, for large p, the Mellin amplitude M I1I2I3I4 p is fixed by the flat space S-matrix with the Mandelstam variables replaced by the bold-face variables s, t, u.
Moreover, as we will see, the integrand M I1I2I3I4 p satisfies BCJ and double-copy relations, directly analogous to the flat space relations, incorporating all Kaluza-Klein modes.
BCJ and colour-kinematics in AdS5 × S 3 Let us consider the field theory amplitude computed in [4] within this formalism. As in [5] we consider the reduced Mellin amplitude M p . In the colour-factor basis, the amplitude M p takes the following very simple form when written in terms of the bold-face variables, Here we have As described above, the large p limit ensures that the amplitude reduces to the flat amplitude with the Mandelstam replaced by bold-face variables where V I1I2I3I4 YM is the field theory gluon amplitude in flat space, see e.g. [46]. Note that this limit somewhat restores the symmetry between AdS and S; in this sense it is a generalisation of the usual flat space limit in which only the AdS (Mellin) variables s, t are taken to be large.
In principle, away from large p, nothing would prevent the amplitude to depend on s,s, · · · separately. However, from (16) we see that in fact the full amplitude M is just a function of the bold face variables. This fact is a consequence of a hidden 8d conformal symmetry of the amplitude. This symmetry allows one to promote the correlator M I1I2I3I4 2222 to a generating function for correlators with arbitrary charges p.
These features are entirely analogous to AdS 3 × S 3 [40,[47][48][49] and AdS 5 × S 5 [20,35] backgrounds where the dynamics is also controlled by hidden conformal symmetries. In other words, A I1I2I3I4 p is generated from A I1I2I3I4 2222 upon acting with a differential operator which takes a very simple form. In fact, we can give a general formula of the operator that interpolates between the three cases. Parametrising the space as AdS θ1+1 ×S θ2+1 , the amplitude A for general p is generated from the one with the lowest charges p = (qqqq) with q = θ1 2 via a differential operator, The operator D θ1,θ2 p takes the following form wherê D θ1,θ2 p,s,t = a={0,ks} and we turned the sphere integral into a sum restricted to the domain T . The operator transforms the gamma functions of A I1I2I3I4 qqqq into those of A I1I2I3I4 p and replaces s, t, u with s, t, u, as it can be easily checked using (a consequence of) Euler's reflection identity.
In fact, as observed above, with the AdS 5 × S 3 background, our variables obey s + t + u = −3. Therefore the the Mellin amplitude M is literally the same function as the flat space amplitude with the Mandelstam variables s, t, u replaced by the shifted bold face variables (s + 1), (t + 1), (u + 1). It follows immediately that all the relations obeyed by the flat space amplitudes also apply to M. Note that it is not trivial that this holds; for example, the analogous relation for AdS 5 ×S 5 is s+t+u = −4 [35]. As an example of the properties obeyed by M we have that which gives an AdS version of the colour-kinematic duality, which was already observed in [5]. Note that (22) captures this duality for all Kaluza-Klein modes. This duality is intimately connected with the so-called BCJ relations between colour-ordered amplitudes. Recall that the full colour-dressed amplitude is: where the partial amplitudes M p (1, 2, 3, 4) are the colour-ordered amplitudes and P(2, 3, 4) are the permutations of points (2,3,4). The translation from one basis to another is: The colour-ordered amplitudes then read as follows, where we used the on-shell relation s + t + u = −3. We stress again that the relations (27)  Having introduced the colour-ordered amplitudes, let us return to the issue of the contour in the Mellin integral (11). It should be noted that the presence of poles at s = −1, t = −1 and u = −1 is potentially a problem for the contour of integration. In fact, since s + t + u = −3, the simultaneous presence of these poles leaves no region in the real s, t plane for the contour to pass through, while separating left moving and right moving sequences of poles in the Mellin integrand. Thus the same property which leads to the direct analogy with the flat space amplitudes also leads to a subtlety in returning to position space from Mellin space. For the colour ordered amplitudes, one does not have all three poles present simultaneously. Thus we propose that the correct definition for the contour is tied to the colour-ordering and we define analogously a colour-ordered correlator, The contour can now be taken to lie slightly below s = −1 and t = −1. Note then that this introduces a subtlety in interpreting the BCJ relations (27) back in position space, since the left and right hand sides of these equations are to be integrated over slightly different contours.
To conclude, let us point out that there is also an AdS version of the double-copy prescription [5]. Replacing colour with kinematic factors we get This is nothing but the SUGRA amplitude in AdS 5 × S 5 [12] rewritten in the large p formalism [35], upon reinterpreting s, t, u as the N = 4 variables, i.e. subject to the constraint u = −s − t − 4. Note also that, similarly to flat space [3], we can use BCJ and colour-kinematic duality to derive an AdS version of the KLT relations: Long disconnected free theory The rest of the letter will be devoted to investigate the structure of the anomalous dimensions of the doubletrace operators exchanged in the OPE at large N . In order to do so, we need two ingredients: the superconformal block decomposition of disconnected generalised free theory and that of the log U discontinuity of the tree-level correlator. The anomalous dimensions are then nothing but the eigenvalues of a certain matrix built out of the block coefficients of these two decompositions. On top of the above mentioned (usual) technology, we also have to deal with the non-trivial flavour structure of the amplitude. However, since all of this just amounts to considering certain symmetric or antisymmetric combinations built out of the correlator, we postpone the discussion on flavour structures to the end of next section. A more detailed discussion can be found in [6].
Let us begin with disconnected free theory. The only correlators with non-zero disconnected contributions are with pairwise equal charges and their spacetime dependence can be computed by performing simple Wick contractions. We have However, due to the non-trivial colour structure of the amplitude, only representations with a definite parity under t ↔ u exchange enter the OPE. In practice, we need to decompose the following combinations of diagrams G ± disc,pqpq = δ pq g p 14 g p 23 η 1η4 2 η 2η3 2 ± g p 13 g p 24 η 1η3 2 η 2η4 2 (31) Now, following [43], we first extract the unprotected contribution and then decompose it in long superblocks [52], whose form is given in the appendix. The block decomposition reads where L τ are the long superblocks. We find that the coefficients take a particularly simple form, Here the A and B factors are given by , while δ is given by Here, h,h and j,j label, respectively, the conformal and internal representations. We can also express them in terms of the more common quantum labels τ = (τ, b, l, a) where τ, l are twist and spin, and b, a can be seen as the analogues of twist and spin on the sphere. Note the different ways the two internal SU (2) factors enter the coefficients. On the one hand, SU (2) L only comes in through the function Bj. On the other hand, the decomposition under the R-symmetry group SU (2) R produces also the function δ and, in particular the combination δ (2) h,j δ (2) h,j . This object is the eigenvalue of a Casimir operator operator acting on the blocks, Here the differential operator D 4 is given by where D ± x is [51], (38) and the functions G τ,l , H b,a , F ± h , that appear in the long superblocks, are defined in the appendix. Note that F − j (ȳ) is a spectator in (37). The presence of δ suggests that the hidden symmetry in free theory is realised not on the correlator of the O p but on a correlator of superconformal descendants of O p , obtained by action of the Casimir. A more detailed discussion can be found in [20] for AdS 5 × S 5 and in [42] for AdS 2 × S 2 background, where the logic is exactly the same. In these last two cases, D 4 is replaced by D 8 and D 2 , respectively.
In the rest of the section we would like to highlight some features common to various AdS θ1+1 × S θ2+1 backgrounds. To start with, the coefficients of long disconnected free theory are very similar in all these theories, (c.f. formulas in [40]). In fact, upon shifting (p, q) → (p + 1, q + 1) in (33), they are the same when written in h−type variables, except for the function δ which depends on the theory, where δ (4) h,h,j,j ≡ δ (2) h,j δ (2) h,j . Note also that the dictionary between h labels and τ labels depends on the theory. By borrowing the results from [40], we can write down the general dictionary interpolating between the three backgrounds The existence of such formulas for disconnected graphs interpolating between different theories turns out to be a particular case of a more general formula for all freetheory diagrams which can be proved through a Cauchy identity [41].

Anomalous dimensions and residual degeneracy
We will not give too many details of the computation, which can be found in [16,19] for the similar AdS 5 × S 5 case; analogous computations in AdS 3 × S 3 can be found in [40]. The main difference with the N = 4 case is that here double-trace operators have a flavour structure. Because of this, there will be two types of anomalous dimensions, those of operators exchanged in symmetric or antisymmetric channels.
At large N , the operators acquiring anomalous dimensions are of the schematic form, where P ij is an appropriate projector that projects onto symmetric or antisymmetric representations of the gauge group exchanged in the OPE. For any given quantum numbers τ = (τ, b, l, a), the number of operators exchanged in the OPE can be represented with the number of pairs (pq) filling a rectangle [19], The rectangle R τ consists of d = µ(t − 1) allowed lattice points where The picture below shows an example with µ = 4, t = 9.
This representation turns out to be particularly useful when we take into account 1/N corrections. In fact, operators on the same vertical line will continue to be degenerate at this order. To see this, let us consider the OPE at genus zero. This is best cast in a matrix form [16]. First, arrange a d × d matrix of correlators with the pairs (p 1 , p 2 ) and (p 3 , p 4 ) running over the same R τ . Here, we denote by A ± p the inverse Mellin transform of the following Mellin amplitudes, The OPE equations then read Here L ± τ is a (diagonal) matrix of CPW coefficients of disconnected free theory defined by (32), while M ± τ is a matrix of CPW coefficients of the log U discontinuity of Finally, η ± τ is a diagonal matrix of anomalous dimensions and C ± τ = O p O q K ± rs is a matrix of three-point functions with two half-BPS and one double-trace operator. Here, we denote with K ± rs the true two-particle operator in interacting theory, that differs by O ± pq , precisely because there is mixing. Note that, since A ± p can be written as a function ofŨ andṼ , the SU (2) L × SU (2) R representations contributing to M ± τ can be reorganised into SO (4) representations, while this is not so for the disconnected contribution L ± τ . It is simple to show, with some linear algebra, that the anomalous dimensions are the eigenvalues of the matrix M τ L ± τ −1 . By computing them for various quantum numbers, we find that the anomalous dimensions follow a very simple pattern, where l 8d is and can be interpreted as a sort of effective 8d spin, the definition being dictated by the partial wave decomposition of the flat amplitude in 8d [20]. Note that (47) only depends on p, not q, or in other words, operators on the same vertical line in the rectangle will acquire the same anomalous dimensions. We stress again that these are the anomalous dimensions associated to the double-trace operators exchanged in the amplitudes M ± p : the gauge group enters the anomalous dimensions only through an overall constant which does not play any significant role in the computation.
Note that the numerator is doubled with respect to the AdS 5 × S 3 case, as a consequence of the fact that supersymmetry is also doubled. Finally, let us also point out that the object δ (2) h,j appearing ubiquitously is, perhaps with no much surprise, nothing but the anomalous dimension of the two-derivative sector in AdS 2 × S 2 [42].
We conclude the section by commenting on the flavour structure of the correlator. One way to deal with it is to decompose t, u channel flavour structures (of both disconnected and tree-level correlators) in a basis of representations appearing in the tensor product of two adjoint representations in the s channel. We then read off the coefficients associated to each flavour structure which are of the form where a runs over all symmetric (antisymmetric) representations in adj ⊗ adj with the proportionality coefficient depending on the specific group as well as the exchanged representation. Examples of such coefficients are given in [6]. The unmixing procedure can then be consistently carried for each a separately. For the symmetric (antisymmetric) representations the relevant doubletrace operators exchanged are of the type O + pq (O − pq ) with the respective anomalous dimensions proportional to η + τ (η − τ ). Actually, the only antisymmetric representation exchanged is the adjoint itself.

Outlook and conclusions
In the first part of this letter we have discussed colourkinematics and BCJ relations between colour-ordered amplitudes of super gluons in AdS 5 × S 3 , by making use of the large p formalism [35]. We believe this formalism makes clearer the direct parallel with the flat space versions of these relations and that they hold for all Kaluza-Klein modes. This, in turn, shows that, like in flat space, there is a precise relation between colour-kinematic duality and BCJ relations.
In the second part of the paper we have computed the anomalous dimensions in the large N limit. As a consequence of the 8d hidden conformal symmetry, and in common with the analogous problems in AdS 5 × S 5 and AdS 3 × S 3 , the anomalous dimensions turn out to have a residual degeneracy which is nicely captured by the vertical columns of the rectangular lattice R τ described in (42) and below. We were also able to give a number of formulae which interpolate results on the spectrum between different backgrounds of different dimensions.
These results open a lot of exciting possibilities. Firstly, as we mentioned already in the introduction, the knowledge of the anomalous dimensions can be of use in bootstrapping loop corrections, beyond the lowest charge correlator studied in [6], and help in further exploring whether some features of the double-copy relations persist beyond tree level. Moreover, following the procedure described in [50], one can imagine treating the theory of gluons as an effective model and introduce higher order D n F 4 interactions, analogous to the higher curvature corrections present for gravitons in e.g. AdS 5 ×S 5 . Much like the curvature corrections responsible for completing the Virasoro-Shapiro amplitude in AdS 5 × S 5 [34,37], such terms will induce a splitting of the residual degeneracy in the anomalous dimensions. Finally, the computation of open and closed string amplitudes in AdS might give a clue on how KLT and world-sheet monodromy relations work in a curved spacetime. by the ERC Consolidator grant 648630 IQFT. RG is supported by an STFC studentship. MS is supported by a Mayflower studentship from the University of Southampton.

Appendix: superconformal blocks
We quickly review here the superconformal block technology needed in this letter. The long superconformal blocks [52], which capture the the non-protected multiplets exchanged in the OPE, are the simplest. They are the product of ordinary conformal and internal blocks for both SU (2) factors. In our notation they take the form where G τ,l (x,x) = with Here, G τ,l (x,x) are the standard 4d conformal blocks (up to a shift by 2 in the twist τ ) and H b,a (y,ȳ) are the internal blocks. The latter are the product of two SU (2) spherical harmonics, one corresponding to the Rsymmetry group SU (2) R and the other corresponding to the flavour group SU (2) L . Finally, τ, l are, respectively, twist and spin, and b, a label the different representation of SO(4) and can be viewed as the analogues of twist and spin on the sphere. Notice that the internal blocks are not invariant under y ↔ȳ exchange. In fact, the two SU (2) have a different nature; in paricular, notice that free theory is also not invariant under y ↔ȳ. This means that, unlike other theories like AdS 5 ×S 5 and AdS 5 ×S 3 , the decomposition is extended to spherical harmonics with label a < 0. In particular, for given charges p i , we decompose a function in spherical harmonics labelled by two quantum numbers that we denote by [ab]. The values of a run over the following set: