Three-point correlators from string amplitudes: Mixing and Regge spins

This paper has two parts. We first compute the leading contribution to the strong-coupling mixing between the Konishi operator and a double-trace operator composed of chiral primaries by using flat-space vertex operators for the string-duals of the operators. We then compute the three-point functions for protected or unprotected scalar operators with higher spin operators on the leading Regge trajectory. Here we see that the nontrivial spatial structures required by conformal invariance arise naturally from the form of the polarization tensors in the vertex operators. We find agreement with recent results extracted from Mellin amplitudes for four-point functions, as well as with earlier supergravity calculations. We also obtain some new results for other combinations of operators.


Introduction
We can now compute the dimensions of single-trace local operators in planar N = 4 super Yang-Mills (SYM), at least in principle, because of its underlying integrability [1]. However, to fully solve the theory it is also necessary to know the three-point functions between local operators. There are now a number of important results on this subject at both weak [2] and strong coupling [3].
In this paper we continue our study of three-point correlators in N = 4 SYM at strong coupling by using flat-space vertex operators to find the relevant couplings [4,5]. Here we investigate two issues. The first involves the appearance of poles in extremal three-point amplitudes. This leads to a mixing between single-and double-trace operators which we explicitly compute for a Konishi operator mixing with a double-trace operator composed of two chiral primaries.
The second issue we consider is the three-point functions for scalar operators with a higher spin operator. Using results of Schlotterer for flat-space massive string amplitudes [6], we can straightforwardly compute the three-point functions and compare with recent results in [7], where the authors study these correlators by considering Regge amplitudes of four scalar operators. In the limit of large dimensions we find agreement with their results.
In a conformal field theory, the correlator of three local scalar operators must have the form where ∆ i are the operator dimensions. Using Witten diagrams [8], one finds that the structure constant C 123 in a four-dimensional CFT has the form [9] C 123 = (∆ 1 − 1)(∆ 2 − 1)(∆ 3 − 1) The coupling G 123 is given by where ψ J 1 ψ J 2 ψ J 3 is an S 5 overlap integral and V k 1 V k 2 V k 3 is the 3-point string amplitude for the string states dual to the operators, with the momenta k i depending on their dimensions and R-charges. The closed string coupling and tension are translated through the AdS/CFT dictionary to be g c = π 3/2 /N and α = 1/ √ λ, where λ = g 2 Y M N is the 't Hooft coupling. A prominent feature of (1.1) is the presence of poles for extremal correlators, which occur when one of the α i is zero. This corresponds to the dimension of one operator being equal to the sum of the other two. In the case of three chiral primaries it was shown that the pole is canceled by a zero in the coupling G 123 , leaving a finite result [10]. However, if the correlator contains one or more non-primary operators then this might not be the case. In particular, in [5], the coupling for two chiral primaries with opposite R-charges and the Konishi operator was explicitly computed, where it was found that being one of the components that contribute equally to the mixing. In this paper we compute the mixing for large J, and hence the splitting of the dimensions at the extremal point. If two equal dimension operators mix, there will be a log divergence in the two-point function between the operators. Our result at closest approach of the splitting between the two eigenvalues of the mixed operators is where M is the dimension of the J-symmetric, traceless representation of SO (6), In Section 2 we review results from [4,5] which are necessary for the analysis in this paper. In Section 3 we present the details about the three-point correlators between the Konishi operator and two chiral primaries of opposite R-charge. We then show how this leads to the mixing between Konishi and the double-trace operators. In Section 4 we consider the three-point correlators between scalar operators dual to supergravity states and states at the first massive level, and at least one higher spin operator dual to a string state along the leading Regge trajectory. We show here that we can match with the results in [7] as long as the scalars also have large R-charges such that our flat-space approximations are valid. In Section 5 we present our conclusions.

Review of previous results
In this section we collect some relevant results from [4,5].
We assume that the operators are short operators such that the sizes of the string-duals are small compared to the AdS 5 and S 5 radii. We can then approximate the three-point correlator using Witten diagrams [8,9]. The Witten diagram includes an integration over all possible intersection points, but in the case of large dimension operators the integration is dominated by a saddle point. The location of the saddle point can be determined [11,4] as well as the contribution of Gaussian fluctuations. The saddle-point location is itself related to the conservation of constants of the motion, and it is these constants that determine the vertex operators that are inserted in the Witten diagram [4].
The classical constants of the motion are expectation values of the different components of the conformal algebra and are determined by the R-charges, the spins, the dimensions and by the positions of the operators on the boundary. For example, for the scalar operator O i (x i ) in a three-point function the conserved charges include The conformal algebra can be written in a manifestly SO(2, d) covariant way by defining with arbitrary κ, and Casimir − 1 2 M rs M rs = −∆ 2 , r, s = −1 . . . d. Setting d = 4 and combining with the SO(6) R-symmetry Casimir, we have where R IJ are the R-symmetry generators with I, J = 5, . . . , 10.
If ∆ and J are large, then the SO(2, 4) × SO(6) algebra effectively reduces to a 10 dimensional Poincaré algebra. Because of the translation symmetry, we are free to shift the intersection point to x µ = 0, in which case all M µν = 0 and the conserved charges K µ i satisfy If we further choose κ to be then the only non-zero components in (2.2) are M −1m , m = 0, . . . , 4 for all three operators. Assuming ∆ 2 1, and choosing a basis where the only non-zero R-symmetry components are R J,10 , we can then identify the full 10-dimensional flat-space momentum as which satisfies the on-shell condition We can expand the algebra to the full superconformal PSU(2, 2|4), by including the supercharges Q αa and Q ȧ α and the superconformal generators S a α and Sα a , where α anḋ α are space-time spinor indices and raised or lowered a is an SO(6) spinor index. The SU(2, 2) SO(2, 4) covariant supergenerators are where lowered or raisedȧ are SO(2, 4) spinor indices. Defining the supercharges then in the flat-space limit these approach the usual 10-dimensional super-Poincaré generators with where P + is the positive-chirality projector.
where (α,α) are the explicit four-dimensional space-time spinor indices transverse to the trajectory in the AdS 5 part and (ã) are the SO(6) spinor indices for the remaining six dimensions. The operators that satisfy (2.12) are described in detail in [5]. They are linear combinations of operators in the NS-NS and R-R sectors and have a fairly complicated form. However, their flat-space three-point functions were remarkably simple. In particular, the three-point function for two chiral primaries and a Konishi or Konishi-like operator (a primary operator at level one with a nonzero R-charge), is given by (2.14) In the special case where J 1 = −J 2 = J, J 3 = 0, this reduces to (1.5), where ∆ = ∆ 3 .

Extremal correlators and operator mixing
In this section we compute the mixing between the double trace operator O JJ and the Konishi operator O K . We first observe that the coupling in (1.4) near the extremal point J = ∆/2 ≈ λ 1/4 is well approximated by where we have used the AdS/CFT dictionary for g c and α . Using this, we also have that near the extremal point the three-point coupling coefficient in (1.2) is approximately The double trace operator O JJ (x) defined in (1.6) needs to be regulated. Here we use point-splitting and define the regulated operator where I is the identity operator. It is then clear that while the mixing in the two-point function is The point-splitting parameter ε acts as a UV cutoff and leads to an anomalous dimension for the mixed operators. The result in (3.5) is the contribution from one particular R-charge configuration, but there are M equivalent configurations that give the same result up to an unimportant sign that can be absorbed into the definition of the double-traced SO(6) singlet. M is the dimension of the J-symmetric, traceless representation of SO(6) Hence, the anomalous dimension matrix at this order has the form which is an M + 1 dimensional square matrix with elements given by M − 1 of the eigenvalues are 0, corresponding to double-trace operators that are not SO(6) singlets, while the other two eigenvalues are which comes from the mixing of O K with the double-trace singlet. Hence, at the cross-over point J = ∆/2 ≈ λ 1/4 , the mixed operators have dimensions and thus the splitting is We can compare the splitting to the leading correction to the double-trace dimension coming from supergravity, where it was found to be [12] Hence the splitting, which scales as J 7/2 /N , is much larger.

Three-point couplings with spin 4.1 The operators and corresponding string states
In this section we consider three-point functions where at least one of the operators is dual to a string state in the leading Regge trajectory. As in [4,5], the correlators can have one or more chiral primaries, The NS-NS and R-R vertices are given respectively by with the following polarizations We define the untwisting factor as  The primed indices denote a frame in which the directions 0 , . . . , 3 are transverse to the momentum k.
The other operators considered in [4,5] also transform in the [0, J, 0] representation, but have bare dimension ∆ 0 = J + 2 [13]. Operators of this type include where Z = (Φ 5 + iΦ 6 )/ √ 2, and the ellipsis refers to different positions of the Φ I in the trace, such that the corresponding magnon momenta lie at level one [13,14]. For J = 0 we have the Konishi operator which at strong coupling has dimension ∆ K ≈ 2λ 1/4 . These operators correspond at strong coupling to scalar vertex operators at the first massive string level, possibly with some Kaluza-Klein momentum. They can be written as where V 1,k and V 2,k are the NS-NS vertices and V 3,k is in the R-R sector we can write the polarizations in the following way In this work we will also consider relatives of the Lagrangian operator with R-charge J which have protected dimension where F µν is the field strength of N = 4 SYM, and the ellipsis refers to different positions of the F µν in the trace, and also other scalar and spinor terms required by supersymmetry. At strong coupling these operators correspond to the ten-dimensional dilaton with some Kaluza-Klein momentum Finally, an obvious extension to the results in [5] is to consider operators with spin. It is in general nontrivial to find the vertex operator for a generic operator with dimension ∆ and spin S. However, if the operator is a special combination of the three following twist two operators [7] Tr then the dual string state lies on the leading Regge trajectory at level n = 1 2 (S − 2) with scaling dimension ∆ ≈ 2 √ nλ 1/4 . As noted in [6], the vertex operator for a symmetric traceless state in the leading Regge trajectory is given by with ε M 1 ...M S a totally symmetric and traceless tensor. To normalize the vertex operator we need only to compute its two-point function The normalized vertex operator for a string state in the leading Regge trajectory is then

The building blocks
Three-point functions of operators with arbitrary spin can be written as a linear combination of a finite number of conformally invariant building blocks [15], which have been rederived in [16] using the embedding formalism is a polynomial built out of six building blocks which encode the space-time dependence Here we encode symmetric tensors by polynomials obtained by contracting the tensor with a reference vector z µ i . We can recover the tensor by applying the symmetric traceless projector The polynomial Q({x i }) is then written as the following sum where C ijk are the structure constants and the summation range is We can reproduce these structures with our holographic procedure. In [6] Schlotterer derived the coupling of three Regge open string states at levels n 1 , n 2 and n 3 . We can apply this result directly to the left and right movers separately. The three-point function for the left movers is, where s i = n i + 1. We used the following shorthand notations Figure 1: We represent each polarization i by a circle with S i indices. The lines connecting two polarization tensors correspond to contractions of their indices. The leftover indices are contracted with the momenta. The allowed structures correspond to the different ways of connecting the three polarization tensors. and the summation range is the same as in (4.22). We notice then that the superstring amplitude of three closed spin states with polarizations ε 1 , ε 2 and ε 3 will be a sum of terms of the form which we represent diagrammatically in Figure 1. These superstring structures from the coupling of the three string states at the intersection point in the bulk are in fact related to the boundary CFT building blocks.
To understand this relation between the two sets of building blocks, it will be enough to restrict ourselves to the AdS 3 space defined by the bulk direction and the plane on the boundary defined by the three endpoints of the strings. If the polarization is orthogonal to any of those directions then ε i · k j vanishes trivially and ε i · ε j remains proportional to H ij . On the other hand, if the polarizations live in some direction inside the AdS 3 defined above then we must be more careful. At the intersection point the polarization of each string is orthogonal to the momentum which, following [4], is given by with P µ , D , K µ and κ defined in (2.1), (2.4), and (2.6) when one takes the intersection point to the origin. In this case, the momentum at the intersection point can be written in the following way Since the polarization is orthogonal to the momentum at the intersection point, it is useful to change to a basis { O i , P i } in the plane transverse to the momentum. If we let k B i be the projection of k i to the boundary then we can write the basis vectors as We denote this change of basis by M I i . When one looks at the propagation of the string from the intersection point to the boundary, one sees that O i stays the same, but P i becomes proportional to k B i . Finally, we return from the basis { O i , k B i } to the canonical one at the boundary through another change of basis M B i . The relation between the polarization at the intersecton point and at the boundary is then given by where we denoted the polarization at the intersection point by ε I i and the one at the boundary by ε B i . The polarization ε I i is symmetric and traceless, which leads to a polarization at the boundary with the same properties. ε B i will then act as a symmetric and traceless projector ensuring the right tensor structure of Q({x i }). The structure at the intersection point ε I i · k j then corresponds to one of the CFT building blocks at the boundary Analogously, the structure ε I i · ε I j becomes a combination of the CFT building blocks at the boundary Putting all the elements together, terms of the form (4.25) now become proportional to (4.32) Notice that the spacetime dependence from the string theory amplitude combines with the one obtained from the propagation of the strings in AdS, (x 2 12 ) −α 3 (x 2 13 ) −α 2 (x 2 23 ) −α 1 , to produce the spacetime behaviour expected from conformal symmetry in (4.17).
It will also be useful to work out the case where one operator is a scalar, for which we set S 1 = 0. With no loss of generality we choose S 2 ≤ S 3 , so the amplitude will reduce to a sum over (S 2 + 1) structures with coefficients F k Following our analysis on the equivalence of the two sets of building blocks, the three-point function of a scalar with two spin states becomes where D denotes both the contribution from the propagation of the strings in AdS (1.2) and the coupling (1.4) and the structure constants are In the other case of interest, we let two of the operators be scalars, hence S 1 = S 2 = 0. There is now only the structure (α /2) S 3 /2 ( 3 · k S 3 1 )F, with the coefficient F coming from the superstring amplitude. The structure constant is then given by

Structure constants
Now that we have the necessary ingredients, we can compute the three-point coupling for different string states.

Two massless scalars and one Regge spin
We start by computing the three-point functions having two massless scalars, which we can compare with previous results in the literature. When the two massless scalars are dilatons, we can insert the polarizations defined in (4.12) into the results of Appendix A, thus giving the string amplitude between two dilatons and one Regge spin Analogously, if we consider the massless scalars to be two chiral primaries, then we use the polarizations defined in (4.4), obtaining the following amplitude for two chiral operators with R-charges J 1 and J 2 and one Regge spin with some R-charge J 3 When the two chirals have opposite R-charge and the Regge spin has no R-charge, the expression (4.39) becomes (4.38). In this case we can put this result into (1.2) and (1.4), and taking the asymptotics for large 't Hooft coupling, we find the structure constant for two massless scalars with dimension ∆ and a Regge state with spin S and dimension ∆ S ≈ 2(S − 2)λ 1/4 , This reproduces the results obtained in [7], where the authors used Mellin transforms. Note also the presence of poles in the factor csc (π∆ S /2) which are related to mixing with doubletrace operators as discussed in the previous section. Another case we consider is when both scalars have the same dimension ∆, but with the Regge spin also at the massless level, which corresponds to a relative of the graviton with some R-charge J g . When 1 |J g | ∆ then (4.39) becomes simply (4.38) and the structure constant we obtain in that limit is again obtaining agreement with [7] for large scaling dimensions.
If we wish to compute the coupling between three states with R-charges of the same magnitude, then we can compare with [17] where the authors used supergravity to compute the coupling between two chirals and a graviton with some Kaluza-Klein momentum. For chirals with dimension J and a graviton with R-charge J g = J we get where C J 1 C J 2 C J 3 is the unique SO(6) invariant that can be formed from the C J i appearing in the definition of the spherical harmonics on S 5 The structure constant in (4.42) agrees with [17] in the limit of large scaling dimensions. For completeness we have also computed the coupling between one dilaton operator, a chiral and an operator in the leading Regge trajectory and have obtained a vanishing structure constant at leading order.

One massive scalar, one massless scalar and one Regge spin
If we consider the three-point function of a Konishi operator, a dilaton with some R-charge J and a Regge spin with the opposite R-charge, we can use the results of Appendix A and plug in the polarizations from (4.10,4.12), obtaining the amplitude Notice that this amplitude vanishes when we take the Regge spin to S = 2. Going to the limit of large 't Hooft coupling, the structure constant in the case of a Regge spin at the first mass level is If instead of the dilaton we consider a chiral primary, then we use the polarizations of (4.4,4.10) in the results of Appendix A. When the spin operator has S = 2, the structure constant also vanishes at leading order. When considering a Konishi, one chiral with R-charge J and a massive Regge spin the amplitude for J λ 1/4 is When the Regge spin is at the first massive level we obtain Two massive scalars and one Regge spin The two massive scalars we consider here are the Konishi-like operators with the vertex operators (4.8). To compute the correlation function between two of this operators and one Regge spin we use the results derived in Appendix A and plug in the polarization (4.10) thus obtaining, in the case of a massive spin, where all operators can be taken to have no R-charge (4.48) In the limit of large 't Hooft coupling, we get for the structure constant of two Konishi operators with dimension ∆ K ≈ 2λ 1/4 and a massive Regge spin of dimension ∆ S satisfying the triangle inequalities where n is the mass level of the Regge spin. If the massive Regge spin does not obey the triangle inequalities then the structure constant acquires poles, (4.50) When the Regge spin is at the massless level and has an R-charge J λ 1/4 , we get Then the structure constant of a Konishi operator, a Konishi-like operator with some Rcharge J and a relative of the graviton with the opposite R-charge is in the limit of large dimensions.

One massless scalar and two Regge spins
When there are two spins with S 2 ≤ S 3 , there are (S 2 + 1) different structure constants in (4.33). In Appendix A we have computed the amplitude for a dilaton with two spins (A.18), which we can rewrite as with the coefficients given by , (4.54) where Y = k/2 − 2, Y denotes the largest integer not greater than Y and (a) m is the Pochhammer symbol, (a) m = Γ (a+m)/Γ (a) . Analogously, we have computed the amplitude for a chiral operator of dimension ∆ with two spins (A. 19), which we can rewrite as , (4.56) and the values of δ for two massive spins, one massless spin of dimension ∆ and one massive spin, and two spins with S = 2 are respectively. Taking the expression for the structure constant (4.36) for a chiral primary with scaling dimension ∆ and two massive Regge states with the same spin S, in the limit of large coupling constant we have If we take one of the Regge spins to have S = 2 and the opposite R-charge of the scalar, then for ∆ λ 1/4 the structure constant is Finally, if both operators in the Regge trajectory are at the massless level and have dimension ∆ as the massless scalar then (4.60)

One massive scalar and two Regge spins
In Appendix A we have computed the two correlators that make up the amplitude for a Konishi with two spins (A.22,A.25), which we can put together as The coefficient K k is quite lengthy so we refer the reader to (A.28). Taking the expression for the structure constant (4.36) for a Konishi and two massive Regge states with the same spin S, in the limit of large coupling constant where n is the mass level of the Regge spins. If we take one of the Regge spins to be at the massless level with scaling dimension ∆ λ 1/4 and the other at the first massive level then the structure constant becomes Finally, the structure constant for one Konishi and two states with spin S = 2 and dimension ∆ λ 1/4 is (4.64)

Discussion
In the first part of this article we studied how the poles appearing in extremal three-point functions relate to the mixing of single-and double-trace operators of N = 4 SYM. We have made this relation quantitative for the case of the Konishi operator and the double-trace operator made of chiral primaries with large R-charge.
In the paper's second part we computed three-point functions involving higher spin states. For correlators with symmetric traceless operators, conformal symmetry allows for a number of different spacetime dependent structures. Here we have successfully matched the building blocks appearing in the superstring amplitudes with the structures allowed by conformal symmetry. The set of operators we considered are the chiral primaries, the scalar primaries dual to string states at the first massive level, Lagrangian-like operators dual to the dilaton with some Kaluza-Klein momentum and twist two operators dual to strings in the leading Regge trajectory. We have also successfully matched our results for the three-point functions of two massless scalars and a spin state in the leading Regge trajectory with recent calculations using the Mellin amplitude formalism, and older supergravity calculations. Having checked with these known cases we went on to compute new correlation functions involving two higher spin states and Konishi operators.
For the future, there are many interesting directions to explore. For example, it would be interesting to generalize our procedure to include operators with small scaling dimensions. When the operators are dual to strings at massive levels, then our methods are valid, since their dimensions scale as 2 √ nλ 1/4 . However, we are forced to give large R-charges to protected operators, which makes it impossible to study three-point functions involving, say, the chiral primary in the 20. Another direction of study would be to compute sub-leading corrections to our results in the α expansion. Besides the corrections coming from the expansions of the dimensions, we would also need to calculate loop corrections to the worldsheet amplitude.
Another possibility is to study four-point correlators of short operators. In principle the main ideas behind our methods should still hold. When doing the saddle point approximation one would obtain two intersection points in the bulk, corresponding to the product of two three-point functions and exchange of some operator. It is possible that under some approximations this computation might become feasible.
Perhaps yet another future application of our results involves the comparison to upper bounds of leading twist operators using bootstrap methods. In [18] the authors showed that the upper bounds are consistent with the supergravity correction in (3.12) for the J = 2 double-trace state at large values of the central charge, c = 1 4 (N 2 − 1). There is evidence that the upper bounds occur at special values of the coupling [19], which corresponds to an 't Hooft coupling of order λ ∼ N . For this value the splitting is large compared to the supergravity correction but also only affects double-traces with J ∼ N 1/4 and so a present comparison cannot be made. If instead one could also expand the bootstrap analysis to include the 't Hooft parameter as an input, then one could in principle make such a comparison if at the same time one can find the splitting for low J values.
Finally, a bigger goal is to understand the structure of the full AdS 5 × S 5 vertex operators and to use the underlying integrability of the string world-sheet to compute holographic three-point functions.

Acknowledgments
We thank T. Bargheer and V. Gonçalves for discussions. This research is supported in part by Vetenskapsrådet under grant #2012-3269. JAM thanks the CTP at MIT for hospitality during the course of this work.

A Vertex function contractions Two massless scalars and one Regge spin
When computing the three-point function of one Regge spin and two massless scalars, we need to compute only two amplitudes. The first, with three NS-NS vertices, is W 1 W 1 O n which can be computed by using the result for the left movers (4.23) and taking n 1 = n 2 = 0. Combining with the right mover part we have The other correlator is W 2 W 2 O n which gives One massless scalar, one massive scalar and one Regge spin To compute the three-point function of one massless scalar, one scalar at the first massive level and one Regge spin, we will need three amplitudes, two with three NS-NS vertices and one with two vertices in the R-R sector. Once again, for V 1 W 1 O n , we can use the left mover result (4.23) and take n 1 = 1 and n 2 = 0. Including also the right movers we obtain We also need to compute V 2 W 1 O n which is The amplitude with R-R vertices is

Two massive scalars and one Regge spin
In this case we will need three amplitudes with three NS-NS vertices and one with two R-R vertices. For V 1 V 1 O n we just take (4.23) with n 1 = n 2 = 1, thus obtaining The other correlators with NS-NS vertices are and (A.14) The correlator with two R-R vertices is

One massless scalar and two Regge spins
When considering the correlator of two Regge spins with a massless scalar it is enough to use (4.23) with n 1 = 0 which, for n 2 ≤ n 3 , gives for the left mover contractions ) .

(A.17)
For the right movers we will have the same formula, with the sum running over k R instead. If we contract with the polarization of the dilaton and denote k = k L + k R then we have If instead we contract with the polarization of the chiral operator then we obtain One massive scalar and two Regge spins For this correlator there are two amplitudes we must consider. The first one can be taken by using (4.23) with n 1 = 1 which, for n 2 ≤ n 3 , gives for the left mover contractions For the right movers we will have an analogous formula, so contracting with the polarization of the Konishi opeartor and using k = k L + k R leads to (n 2 !n 3 !) 2 (α /2) Combining with the contractions from the right movers and contracting indices from the scalar polarization, we get for k = k L + k R (n 2 !n 3 !) 2 (α /2) )η k 23 l!(n 2 − l)!(n 3 − l)!(k − l)!(n 2 − k + l)!(n 3 − k + l)! Putting the two correlators together we get with the coefficients K k given by n 2 !n 3 ! 16l!(s 2 − l)!(s 3 − l)!(k − l)!(s 2 − k + l)!(s 3 − k + l)! × × K 1 (k, l, S i , ∆ i ) + (k − l) (s 2 − k + l + 1)(s 3 − k + l + 1)