Double Copy Relation for AdS

We present a double copy relation in AdS$_5$ which relates tree-level four-point amplitudes of supergravity, super Yang-Mills and bi-adjoint scalars.


INTRODUCTION
Scattering amplitudes in flat space exhibit surprising properties which encode deep lessons for quantum field theories and gravity. While we believe many curvedspacetime generalizations exist, explicit realizations are far from obvious to find. Recently, there has been a lot of activity trying to extend two remarkable flat-space properties, color-kinematic duality [1] and the double copy relation [2], to the simplest curved background -AdS space [3][4][5][6][7] [8]. The flat-space relations relate gauge theory and gravity amplitudes, and have numerous applications in modern amplitude research [9]. Since AdS/CFT maps AdS amplitudes to CFT correlators, generalizations to AdS are especially interesting. While color-kinematic duality has been observed for four points [5][6][7], AdS double copy so far has only worked for three-point functions [3,4]. In fact, it was not clear if the flat-space relation has to be drastically modified at higher points. In this paper, we present an AdS generalization which realizes the double copy construction in four-point amplitudes for the first time. We relate tree-level amplitudes in AdS 5 ×S 5 IIB supergravity, AdS 5 ×S 3 SYM, and nonsupersymmetric AdS 5 ×S 1 bi-adjoint scalars, in a simple way that mirrors the flat space relation. Moreover, our AdS relation works for all amplitudes in these theories, applying to massless and massive particles alike.
We will use the Mellin representation for CFT correlators [10,11]. AdS amplitudes become Mellin amplitudes and enjoy simple analytic structure resembling the flat-space one. Tree-level Mellin amplitudes of AdS supergravity and super gauge theories in various spacetime dimensions were systematically studied in [7,[12][13][14][15][16][17][18][19], and a Mellin color-kinematic relation similar to the flat-space one was pointed out in [7]. Unfortunately, applying the flat-space double copy prescription led to no sensible amplitudes. In this paper, we revisit these results. We will focus on AdS 5 and take advantage of supersymmetry, which allows us to reduce the Mellin amplitudes to simpler reduced Mellin amplitudes. We find that it is in these reduced objects that color-kinematic duality and double copy relation are naturally realized.
Schematically, we will write the reduced amplitude of AdS 5 super gluons with N = 2 superconformal symmetry as a finite sum labelled by integers i, j where the number of terms is determined by the external masses. c s,t,u are standard color factors satisfying c s + c t + c u = 0. The kinematic factors n i,j s,t,u turn out to obey the same relation n i,j s + n i,j t + n i,j u = 0, giving rise to an AdS color-kinematic duality. Replacing c s,t,u with n i,j s,t,u , we recover precisely super graviton reduced amplitudes of AdS 5 ×S 5 IIB supergravity [12,13]. On the other hand, replacing n i,j s,t,u by c s,t,u leads to Mellin amplitudes of conformally coupled bi-adjoint scalars on AdS 5 ×S 1 , which were not studied in the literature. We will prove it by direct calculation. The AdS 5 double copy relation presented here relates theories with varying N = 0, 2, 4 superconformal symmetry. However, it also implies that purely bosonic theories of Einstein gravity, YM, and biadjoint scalars on AdS 5 should be related by double copy, as we will briefly discuss at the end.
Here x ij = x i − x j , and s, t, u are Mandelstam variables satisfying s + t + u = 4 i=1 k i ≡ Σ [21]. We have also extracted a factor of Gamma functions which captures the contribution of double-trace operators universally present in the holographic limit [11] . All dynamic information is contained in M k1k2k3k4 , known as the Mellin amplitude. The four-point function G k1k2k3k4 obeys Bose symmetry which permutes operators. Bose symmetry acts on the Mellin amplitude by interchanging k i , as well as permuting the Mandelstam variables s, t, u -in the same way it acts on a flat-space amplitude. N = 2 superconformal symmetry. We now consider CFTs with N = 2 superconformal symmetry, focusing on the 1 2 -BPS operators. These operators are of the form transforms in the spin j R = k 2 representation of SU (2) R , and has conformal dimensions k = 2, 3, . . .. To conveniently keep track of the SU (2) R indices, we contract them with aux- We then consider their four-point functions (1), and define the Mellin amplitude M N =2 k1k2k3k4 via (2). The N = 2 superconformal symmetry imposes extra constraints on the form of correlators via the superconformal Ward identities [23]. Solving them leads to where G N =2 0,k1k2k3k4 is the protected part of the correlator independent of marginal couplings. The factor R (2) is crossing symmetric, and is fixed by superconformal symmetry to be Here is the SU (2) R cross ratio, and z,z are conformal cross ratios given by All the dynamical information is contained in the simpler reduced correlator H N =2 k1k2k3k4 , which can be viewed as a correlator of operators with shifted conformal dimensions k i + 1 and shifted SU (2) R spins ki 2 − 1. In the regime to be considered, corresponding to AdS tree level, the reduced correlator in fact captures all the information. To make it more precise, let us define a reduced Mellin amplitude via the reduced correlator Note that it is important to shift the u variable toũ = u − 2 so that s + t +ũ = Σ − 2. The shift is to compensate the nonzero weights of the factor R (2) under conformal transformations. As a consequence, Bose symmetry acts differently in the full and reduced Mellin amplitudes, as In the tree-level regime the protected part G N =2 0,k1k2k3k4 does not contribute to the Mellin amplitude [7]. Rather it is generated by a contour pinching mechanism described in [13]. Therefore full amplitudes are completely determined by reduced amplitudes, with the precise relation given by translating both sides of (5) into Mellin space The factor R (2) now becomes a difference operator R (2) [7]. To obtain it, we interpret each monomial U m V n in m,n which acts on functions f (s, t) according to N = 4 superconformal symmetry. The kinematics of N = 4 is similar. The 1 2 -BPS operator, labelled by an integer k = 2, 3, . . ., transforms in the rank-k symmetric traceless representation of the SO(6) R R-symmetry group, and has dimension k. We keep track of the Rsymmetry indices by using null SO(6) vectors t r [24] where t · t = 0. The N = 4 superconformal symmetry dictates that the four-point function is of the form [23, 25] where G N =4 0,k1k2k3k4 is the protected part, and H N =4 k1k2k3k4 is the reduced correlator. Note that the reduced correlator also has shifted quantum numbers, with dimensions k i +2 and SO(6) spin k i − 2 for each operator. The factor R (4) is determined by supersymmetry and doubles the N = 2 factor (6). Here t ij = t i · t j , and The full correlator G N =4 k1k2k3k4 gives rise to the full amplitude M N =4 k1k2k3k4 via (2). The N = 4 reduced amplitude is similarly given by But note here that the shift inũ is by 4, i.e.,ũ = u − 4. The greater shift is due to the higher conformal weights of R (4) . Bose symmetry again permutes s, t, u in M N =4 k1k2k3k4 , and s, t,ũ in M N =4 k1k2k3k4 . At AdS tree level, the protected part again does not contributes to the Mellin amplitude [12,13]. Therefore the full amplitudes are determined by the reduced amplitudes via where we have promoted R (4) into a difference operator R (4) [12,13]. The action of each monomial U m V n in R (4) /((t 12 ) 2 (t 34 ) 2 x 4 13 x 4 24 ) is given by (11) with N = 4.

SUPER GLUON AMPLITUDES
We are now ready to discuss holographic correlators in specific theories. We start with super gluons in AdS 5 preserving N = 2 superconformal symmetry, which can be realized as D3 branes probing F theory singularities [26,27], or as N = 4 SYM with probe flavor D7 branes [28]. In both case, there is an AdS 5 ×S 3 subspace in the holographic description, on which live localized degrees of freedom transforming in the adjoint representation of a color group G F . These degrees of freedom form a vector multiplet, and its Kaluza-Klein reduction gives infinite towers of 1 2 -BPS superconformal multiplets. We refer to the 1 2 -BPS superprimaries as super gluons. At large central charge, gravity decouples and one has only a spin-1 gauge theory. Note S 3 has isometry SO(4) = SU (2) R × SU (2) L . The first factor is identified with the N = 2 R-symmetry group, while the second SU (2) L is a global symmetry suppressed in the above discussion. The operator O k has spin k−2 2 under SU (2) L [27]. We can similarly contract the indices with k − 2 SU (2) L spinorsvā,ā = 1, 2. In reduced correlators, v andv further recombine into null vectors of SO(4) via Pauli matrices, and appear only as polynomials of t ij [7] t r = σ r aā v avā , r = 1, . . . , 4 , t · t = 0 .
To write down the super gluons amplitudes, let us choose, without loss of generality, the ordering k 1 ≤ k 2 ≤ k 3 ≤ k 4 , and distinguish two cases To measure the deviation from the equal weight case k i = Σ 4 , it is useful to introduce the following parameters The reduced Mellin amplitudes are given by [7] [29] which has been rewritten to manifest Bose symmetry. Let us unpack this expression a bit. Here case II) is the extremality, which determines the complexity of the amplitude. After extracting a factor in t ab the reduced Mellin amplitudes are degree-(E − 2) polynomials in σ and τ defined in (15). The color dependence is captured by the color factors where f IJK are the structure constants of the color group G F . Thanks to the Jacobi identity, they satisfy c s + c t + c u = 0. The kinematic factors n i,j s,t,u are given by The non-locality of these expressions is only superficial, and should not raise any alarm. In fact, a similar phenomenon occurs in flat space [30]. Evidently, n i,j s,t,u obey which gives rise to a realization of the color-kinematic duality [1] in AdS. In contrast to the duality pointed out in [7], this new realization has the same form for both massless (k i = 2) and massive (k i > 2) super gluons. Finally, the remaining parameters are given by

SUPER GRAVITON AMPLITUDES
Let us now take a further step with the color-kinematic duality (21), and replace color factors c s,t,u in each monomial σ i τ j by kinematic factors n i,j s,t,u . The result is .
To interpret it as N = 4 reduced amplitudes, we need to replace theũ variable with the N = 4 one, as required by Bose symmetry of M N =4 k1k2k3k4 . Furthermore, we replace the SO(4) vectors t r by SO(6) null vectors [31]. Remarkably, it gives all the super graviton reduced Mellin amplitudes of IIB supergravity on AdS 5 ×S 5 [12,13] up to an overall factor [32]. This generalizes the double copy relation [2] into AdS space for four-point functions [33]. In fact, redefining the super gravitons by O k → O k / √ k gets rid of the normalization factor, and gives the super graviton three-point functions also as the square of the super gluon ones [34].

BI-ADJOINT SCALAR AMPLITUDES
In flat space one can also replace kinematic factors by color factors, and obtains amplitudes of bi-adjoint scalars. We show that the same happens in AdS, and it serves as a nontrivial check. Note that in the above example the superconformal factor R (2) was doubled to R (4) (c.f., (6) and (14)). Going in the opposite direction, we expect R (0) = 1, i.e., the resulting theory has no supersymmetry. Moreover, since the internal spaces changed from S 3 to S 5 , a reasonable guess is that this sequence starts with S 1 , which will soon be confirmed. The symmetry groups are therefore SO(N + 2), and we recall that operators in the reduced amplitudes transform in the rank-(k i − 2) symmetric traceless representation.
Note that for N = 0, the null polarization vectors are two-component. Since we can rescale the null vectors, we are left with two inequivalent choices The dimension k operator O ± k ≡ O k (x, t ± ) has ±(k − 2) charges under U (1) = SO(2), depending on the polarization chosen. Moreover, we assume the scalar interactions are only cubic. Then U (1) charge conservation dictates that at least one of the κ s , κ t , κ u parameters in (18) is zero. For the chosen ordering k 1 ≤ k 2 ≤ k 3 ≤ k 4 , we must impose the condition κ t = 0. This leaves which have identical amplitudes [35]. Noting and replacing n ij s,t,u with the color factors c s,t,u for another color group G F , we find We dropped the tildes because non-supersymmetric theories have only full amplitudes (2), and there is no shift in the u variable. We also included a to-be-determined k i -dependent normalization factor −2N k1k2k3k4 as in the supergravity case. Remarkably, (27) can be rewritten as the sum of three AdS 5 scalar exchange diagrams where S (s) p is the amplitude of exchanging a dimension-p scalar in the s-channel (and similarly for the other two channels) [36] .
Moreover, the weights p s,t,u are precisely those selected by U (1) charge conservation Note that (27) is equivalent to (28) is highly nontrivial, and a priori does not need to happen. We can further fix the normalization N k1k2k3k4 by noting N k1k2k3k4 /(p s − 1) etc, have the interpretation of products of three-point function coefficients C k1k2ps C k3k4ps . The solution, up to a k i -independent overall factor, is Finally, we confirm by direct calculation that the theory is conformally coupled scalars on AdS 5 ×S 1 . The conformal mass on this manifold is M 2 conf = −4 [37]. Decomposing the scalar field φ into S 1 modes φ(z, τ ) = ∞ n=−∞ ϕ n (z)e inτ , we find each mode has mass M 2 n = n 2 −4. This translates into a conformal dimension |n|+2, agreeing with our charge-dimension relation n = ±(k−2). We can further check three-point functions. A cubic vertex φ 3 in AdS 5 ×S 1 gives rise to infinitely many AdS 5 cubic vertices ϕ n1 ϕ n2 ϕ n3 where {n i } conserve the U (1) charge. Using the result of [38], it is straightforward to show that three-point functions are precisely (29). Note that both C k1k2k3 and N k1k2k3k4 can be set to one by redefining O k → √ k − 1O k . Then the double copy relation also holds for three-point functions.

DISCUSSIONS
In this note we found an extension of the double copy relation in curved spacetimes which relates all treelevel four-point functions of AdS 5 ×S 5 IIB supergravity, AdS 5 ×S 3 SYM, and AdS 5 ×S 1 bi-adjoint scalars. Although our result is supersymmetric, it has immediate implications on bosonic Einstein gravity and YM theory in AdS 5 with no internal factor. Thanks to supersymmetry, four-graviton and four-gluon amplitudes can be obtained from the reduced correlators of k i = 2 super gravitons and super gluons by action of differential operators [39]. At tree level these spinning correlators are identical to the ones in bosonic theories because the exchanged fields are the same [40]. Our result then indicates that the bosonic amplitudes should also be related by a double copy construction [41], of which the details we will leave for a future work. Another interesting direction is to extend our results to higher points, although more data of holographic correlators is needed [42]. While the focus here is AdS 5 amplitudes, double copy relations for other backgrounds are also worth exploring. In particular, the AdS 7 case [17] admits similar definitions of reduced amplitudes [43,44]. Finally, it would be interesting to explore extensions at higher genus, where the relevant CFT techniques were developed in [45].