Four point function of $\mathcal{N}=4$ stress-tensor multiplet at strong coupling

In this short note we use the flat space limit and the relation between the 4-pt correlation function of the bottom and top components of the stress tensor multiplet to constraint its stringy corrections at strong coupling in the planar limit. Then we use this four point function to compute corrections to the anomalous dimension of double trace operators of the Lagrangian density and to compute energy-energy correlators at strong coupling.


Introduction
Correlation functions of local operators in a CFT are the closest analogue of scattering amplitudes and the presence of a gravity dual turns this analogy more concrete [1,2,3]. In this case a correlation function in the CFT corresponds to sending excitations into the bulk of AdS and measure the out states. Another important property of a correlation function is that it organizes the CFT data, i.e. dimension of operators and OPE coefficients into non-trivial functions of the cross ratios, for the specific case of four operators. The structure of the four point function is so restrictive that the CFT data contained in it cannot take arbitrary values. Moreover, different configurations of the points will probe physics which is associated with certain singularities of the correlation function. For example the leading contribution, in a correlation function, when two points come close together is dominated by operators with low dimension. Other limits of the four point functions include the Lorentzian OPE limit, the Regge limit and the large spin limit [4,5,6,7,8,9,10,11,12]. In this note we will study a different kinematical regime, the flat space limit, that explores the connection between correlation function and scattering amplitudes [13,14,15,16,17]. Let us emphasize that the physical interpretation of this limit just makes sense when there is a gravity dual.
The main result of this paper is the computation of stringy corrections to the four point of a primary operator in N = 4 SYM defined by O(x, y) = y I y J tr(φ I φ J )(x) (2) where the auxiliary variables y I satisfy y 2 = 0 and the fields φ I , with I = 1 . . .
where G (0) is the tree level contribution and the other part contains the quantum corrections. The factor R encodes the polarizations vectors y i and its specific form is given in appendix A. It is convenient to perform the analysis in terms of Mellin amplitudes 1 where b 1 and b 2 are undetermined coefficients that cannot be fixed from the flat space limit alone, is the supergravity result [18] and λ is the 't Hooft coupling 2 . In fact, we show that the λ − 3+n 2 correction to M F (s, t) is a polynomial of maximum degree n and we are able to determine all coefficients of degree n in s and t. To obtain this result we use three properties of N = 4 SYM: • Analytic structure implied by OPE and dimension of unprotected operators in the planar limit; • Flat space limit of AdS and the relation between the 4 point function of the Lagrangian density and the Virasoro Shapiro scattering amplitude [19]; • Relation between the four point function of the Lagrangian density and the one of the operator O(x, y) In section 2 we derive the result (6), in section 3 we use the data of the four point function to derive stringy corrections to the anomalous dimension of double trace operators and in section 4 we use the four point function to the computation of energy-energy correlators.

Lagrangian four-point function at strong coupling
In this section we shall study the four point function of the Lagrangian density of N = 4 SYM which is on the same supermultiplet of O(x, y). The advantage of analyzing this object is that the flat space limit has already been computed for this correlation function [19,9] and the four point function of the Lagrangian is related by supersymmetry [20,21] to the correlation function of O(x, y). The Mellin amplitude, M L (s, t), is defined by where the integration runs parallel to the imaginary axis. Notice that the Mellin amplitude M L (s, t) should satisfy, since the correlation functon is invariant under permutation of the external points. In [20] the correlator of Lagrangian density was related to the correlation function of the primary operator O(x, y). More concretely we have, 2 More specifically λ can be written in terms of the AdS radius and string length l s as λ = R 4 where the function H(u, v) is related to F (u, v) in (3) by a eight-order differential operator 3 The action of the differential operator (11) on the function F (u, v) is mapped to a difference equation in the Mellin representation, and the functions q a,b (s, t) are given in appendix B. For example we can pick the supergravity result for F (u, v) that was first computed in [18,22,23] and check that (12) implies the supergravity result which agrees with the previous computation [24,19] and where u = 16 − s − t. In the strong coupling limit it is natural to divide the 1/λ corrections to the Mellin amplitude from the supergravity approximation, The OPE limit determines the singular behavior of a correlation function as two points come close together. This is associated with the exchange of an operator, which in this integral representation (8) is controlled through the poles of the integrand. Let us just point out that the explicit poles coming from the Γ functions correspond to the contribution of double trace operators of the external fields that appear in the OPE [19,17]. For example in the supergravity approximation the Mellin amplitude M SUGRA L (s, t) has poles corresponding to the exchange of a graviton [19]. In general we expect that the exchange of an operator with dimension ∆ and J will give rise to a pole in the Mellin amplitude of the form 4 , 3 Notice that [20] uses different notation, our F (u, v) is their F(u, v). See Appendix A for more details. Moreover, we redefined our differential operator by a constant factor such that it maps the SUGRA result M F (s, t) to M L (s, t). 4 See for instance formula (119) of [9].
where the residue C 12∆ C 34∆ Q(s) contains information about the OPE coefficient, we have omitted poles coming from other channels and m is an integer that is associated with contribution of descendants operators with twist ∆ − J + 2m. The poles of corrections M λ L (s, t) are associated with unprotected operators. However, at strong coupling these operators gain a large anomalous dimension ∆ i ≈ λ 1 4 . Thus the Mellin amplitude M λ L (s, t) is a meromorphic function of s and t at each order in the 1/λ expansion. In appendix A.2, we show that the 1/λ corrections to M F (s, t) have to be regular functions in s and t. The simplest regular function is a constant, thus using (12) we get The correlation function of four Lagrangians in N = 4 SYM is related to the scattering amplitude of four dilatons in superstring theory through the flat space limit. This is the limit of the four point function that focus the interaction region in AdS to be small, thus it only probes flat space physics. In [19,25] it was shown that this leads to the relation, We should emphasize that this is a prediction/constraint for M L (s, t) and gives information which is not easily accessible by other methods. From (15) we know that the corrections to the Mellin amplitude do not have poles. The flat space limit relation (19) constraints the 1/λ corrections to be polynomial functions in s and t at each fixed order in 1/λ. The flat space limit relation allows us to write the Mellin amplitude M L (s, t) as with l n (s, t) and f n (s, t) polynomials of degree n. 5 Let us emphasize that l 4 (s, t) was completely determined, up to a constant, just from the relation between the four point function of O and L, the regular behavior of the corrections to M F (s, t) and the existence of a flat space limit. In particular, the large s and t behavior of (12) tells us that l n (s, t) satisfies 5 It will be clear in the following why M F (s, t) has this form.
where f n (s, t) is defined by f n (s, t) = lim b→∞ b −n f n (bs, bt). Notice that since the flat space limit is sensitive just to the highest power of s and t it is possible to extract all the coefficients of f n (s, t). In fact, as we show in Appendix B.2, we can rewrite the flat space limit in terms for M F (s, t) Again, this relation gives non-trivial information about M F (s, t) and in particular was used to derive (6). Notice that this data was obtained using just the flat space limit and analytic properties of Mellin amplitudes. A natural extension is the study of Regge limit of this four point function and try to access more structure of this correlation function. Recently there has been advances in the strong coupling computation of the relevant CFT data of this regime [9,26,27,28], namely the BFKL spin j(ν).

Extracting CFT data
As emphasized in the introduction a four point function contains information about the CFT data, i.e. OPE coefficients and dimension of operators. These can be extracted using the OPE limit once the explicit expression for the correlation function is known. The goal of this section is to extract the CFT data from the four point function computed in the previous section. Recall that a four point function can be expanded in conformal blocks as, where G ∆,J (u, v) is a conformal block. The Lorentzian OPE corresponds to the limit of u → 0 while v is kept fixed, in this regime the conformal block can be written as [29], where the functions g m (v) can be determined recursively from the Casimir equation and encode the contribution of descendant operators with twist ∆ − J + 2m.
One of the operators that will flow in the OPE are the double trace operators of the externals fields, represented schematically as The dimension of this operator has the following form The anomalous dimension γ can be thought as the gravitational binding energy of a two particle state with angular momentum J [30]. The anomalous dimension γ(J, m, λ) can be read by studying the ln u piece of the correlation function. In the Mellin representation of the four point function the ln u, associated with the anomalous dimension of the double trace operators terms, is generated by the explicit double poles of Γ functions in (8).
Unprotected operators gain a large anomalous dimension as was remarked before, for example the dimension of the Konishi operator is given by So these operators give subleading contribution, in the Lorentzian OPE limit, compared to the double trace of the previous paragraph. In terms of Mellin amplitudes we can say that the poles associated with these operators run off to infinity. Notice that several operators will give sensibly the same contribution to the four point function, for example operators with higher twist. Thus it is hard to disentangle the CFT data from each of these unprotected operators. However we can speculate about the coupling dependence of the OPE coefficients. According to (17) an unprotected operator, with dimension ∆ ≈ cλ 1/4 + O(λ 1/4 ) and spin J, will have a Mellin amplitude of the form 6 , where Q J,m (s) are polynomials of maximum degree J in s for any m, they are usually called Mack polynomials. Thus we conclude that the OPE coefficient of the Konishi operator for example has to decrease at least as fast as C 12K C 34K ≈ λ −5/4 , otherwise this would not be consistent 7 with the fact that corrections to the Mellin amplitude start with λ −3/2 .

Anomalous dimension of double trace operators
The goal of this subsection is to determine stringy corrections to the anomalous dimension γ(J, m, λ) of the double trace operators. We will employ the same methods used in the computation of the leading term of γ, see section 6.1.1 of [35]. To check the normalization of our four point function we will compute the OPE coefficient of two Lagrangians and the stress energy tensor and compare with result predicted from Ward identity [36]. To compute this we just need to know the expression for Q 2,m (s) since the stress energy tensor has spin two with P ν,J (s, t) defined in (166) of [9] 8 . Recall the conformal Ward identity imposes that the OPE coefficient in this case should be given by From the conformal block decomposition (24) it can be seen that we need to determine the leading order OPE coefficient of the double trace operators before we are able to compute the anomalous dimension γ(J, m, λ). For simplicity we will focus on the case with m = 0. The OPE coefficients can be determined from the disconnected diagrams of the four point function of L(x). In [34] these OPE coefficients were determined for any value dimension of the external operators and space time dimension, in this particular case the leading order OPE coefficients are for J even integer, notice that odd spins do not contribute since the external operators are all the same. Following our analysis of the conformal block decomposition (24) the anomalous dimension with m = 0 will give a contribution to the four point function of the form The function g 0 (v) is related to the Mack polynomial Q J,0 (s), see for instance (117) and (119) of [9]. Using the explicit expression for Q J,0 (s) and its orthogonality condition 9 we obtain a formula relating the Mellin amplitude to the anomalous dimension of double trace operators, In fact, this is just the limit of ∆ 3 → ∆ 1 of (172) of [35] with the appropriate shifts in the Mellin integration variables. Let us compute the first correction in λ −3/2 to the anomalous dimension. The first stringy correction to M L (8, t) is completely determined from the flat space limit, For integer J, the hypergeometric function is a polynomial in t, so the integral (33) is of Mellin Barnes type, Notice that we can absorb the first correction in 1/λ of the Mellin amplitude in the Γ functions, 9 Check formula (123) of [9] with c 0 = 1370, c 1 = −504, c 2 = 21. The hypergeometric, for positive integer spin J, is expressed in terms of a finite sum of Γ functions that are easily absorbed in the integrand of (33) Recall that the α 3 correction to interaction of the Lagrangian involves a term of the form α 3 (∂ 8 L 4 ), this is the reason why the correction to the anomalous dimension vanishes for J > 4. Let us emphasize that the next order correction to γ can be determined up to a constant. Recently the anomalous dimension for double trace operators of this same four point function were studied [37] but in the limit of 1 m.

Event shapes in N = SYM
The supermultiplet that contains O(x) has in its components the R-current and the stress energy tensor. The special feature about this super multiplet is that the four point function of any of its components can be obtained, in principle, from the correlation function of its lowest component, the scalar primary operator O(x). This feature can be explored to compute physical observables such as energy-energy or charge-charge correlators [38,39,40] defined as where D i (n i ) is interpreted as detector that measures the flow of either charge or energy and σ tot is just normalization that is obtained for the case where there are no detectors. The subscript W in the correlation function is to remark that the operators are not time-ordered. The physical picture is the following, we excite the vacuum with a particular operator O and measure the flow of energy/charge at infinity with the our detectors D i (n i ). The detectors can be defined in terms of stress energy tensor T µν and R-current where A and B are R-charge indices. Since the energy density E is on the same supermultiplet as O(x, y) this observable can be rewritten in terms of F (u, v) for the special case of two calorimeters 10 [38], where n and n are polarization null vectors encoding the direction of the calorimeter and q is a momentum vector. The integration runs parallel to the imaginary with Re(s + t) > 2 and Re(s), Re(t) < 2 .Here θ measures the angle between the two calorimeters at infinity in the center of mass frame. The goal of this section is to explore the consequences of the strong coupling expansion of the four point function (6) in the computation of energy-energy and charge charge correlation.
The leading order result for F(z) using M SUGRA F (s, t) is given by, which agrees with [41,38,39,40] 11 . Let us proceed and write the 1 λ corrections to the function F(z) in terms of f n (s, t), where we have introduced the superscript to denote that it just contains the 1/λ corrections.
Since the correction of order λ − 3 2 is just a constant the integral in s is divergent. This is not unexpected 12 since we have obtained f n (s, t) using the flat space limit which assumes that s is of order √ λ. However in the computation of energy-energy correlators we are integrating over s up to infinity. So to obtain a sensible result we should re-sum all orders in s and then do the integral. This comes naturally by changing variables to x = s/ √ λ, In appendix C, we re-sum the first order, which will contribute to the next-to-leading to the event shape function F(z). From the results (78) we conclude that, Let us be more concrete about the corrections to the event shape function. The polynomial f n (s, t) has the generic form, f n (s, t) = k n,0,0 s n + s n−1 k n,1,0 t + k n,1,1 + s n−2 (k n,2,0 t 2 + k n,2,1 t + k n,2,2 ) + . . .
with k n,i,j constants and where the . . . represent subleading terms in s. Let us point out that k n,i,0 can be determined from the flat space limit. Thus we conclude that the next orders of the event shape are given by, where the coefficients u i,0 can be determined, in principle, using the flat space limit result 13 . The polynomials Q n (z) satisfy the recursion relation Notice that the Q n (z) when integrated from 0 to 1 give zero, so the event shape function satisfies automatically the conservation of energy, valid for all λ. 12 The computation of this observable involve an integration over the detectors working time. It had already been noticed [41,38] that there could be an order of limits issue in doing the perturbative expansion before or after the integration over the detectors working time. 13 We tried to compute the coefficient u 1,0 and we have obtained infinity. This may happen because the integral over the detector working time does not commute with the perturbative expansion around strong coupling.

Acknowledgements
The author wish to thank João Penedones for suggesting this problem, numerous discussions and comments on the draft. We wish to thank Gregory Korchemsky, Emeri Sokatchev and S. Caron-Huot for useful discussions. We also wish to thank João Caetano, Pedro Vieira and Miguel Costa for useful comments on this manuscript. The research leading to these results has received funding from the

A Four point function of O(x, y)
The goal of this appendix is to analyze the four point function of O(x, y) using just the structure of (3) and the strong coupling behavior of the dimension of operators that can flow in the OPE. The four point function (3) can be decomposed in six different R-charge channels, where r labels the differents representations r = 20 ⊗ 20 = 1 ⊕ 15 ⊕ 20 ⊕ 84 ⊕ 105 ⊕ 175. This particular four point function has received considerably amount of attention in the last years. At weak coupling it is known up to six loops in terms of integrals [42] and in terms of polylogarithms is known up to three loops [43,44,45]. At strong coupling it has been computed to leading order [18,22].

A.1 SU (4) channels
The function R that contains the polarizations in the four point function of the scalar primary operator O(x, y) is given by, so the amplitudes A r (u, v) in (51) are written as A.2 Symmetry and analyticity properties of the Mellin amplitude M F (s, t) The factor R is permutation symmetric and has weight one at each point, so it must satisfy, This symmetry is translated in terms of Mellin amplitudes as (5). Each amplitude in (51) can be written in terms of the Mellin amplitude M F (s, t) given in (4). Imposing that absence of poles in the Mellin amplitudes corresponding to the amplitudes A r would not lead to constraints on M F (s, t). Let us study how these constraints come about by analyzing the channel 105 of the four point function. Following the notation of [9] we write the Mellin amplitude of the channel 105 as with M 105 (s, t) given by The absence of poles in M 105 (s, t) allows M F (s, t) to have double poles at t = 0 and t = 2, i.e.
with h(s, t) a regular function in s and t. However we now that M F (s, t) satisfies, Notice that M F (s, t) cannot have poles, otherwise it is not possible to satisfy Thus we conclude that the absence of poles in the channel M 105 implies that M F (s, t) is a meromorphic function of s and t. In particular this is useful to study the 1/λ corrections to the four point function.

B Dilaton Four point function from F (u, v)
The goal of this section is derive the relation between F (u, v) and the four point function of Lagrangians. We will use eqs (1.3), (2.23), (3.1) of [20]. The dilaton can be written in terms of fields L + and L − , Then we just have to use, together with the fact that a four point function with unequal number of fields L + and L − gives zero. Using this we get (10).

C Borel Ressumation
In section 4 we postponed the summation of the leading contribution to the energy-energy correlator. The goal of this section is to explain how to perform the Borel ressumation that is necessary to extract the 1/λ to the event shape (43). The starting point is the function f ∞ (x, t, λ) defined by, The 1 λ correction to the event shape F(z) comes from the integral over x of f ∞,0 (x). This function is written in terms of the coefficients of the highest powers in s of f n (s, t) as The coefficient of s n in f n (s, t) can be read from (23), c n = Γ(6 + n)ζ 3+n 2 n+11 N 2 even n , c n = 0 odd n where ζ n is the Riemann Zeta function. Notice that odd terms in c n vanish, so f ∞,0 (x) is an even function. A direct substitution of the coefficients in the sum does not work since the sum diverges. Let us define the Borel transform of both functions as, The original series can be obtained by integrating against e −z , Bf ∞,0 (zx)e −z dz.
The Borel transform of f ∞,0 is given by, The integral in (44) can be rewritten as where we have used that f ∞,0 (x) is an even function to change the integration limits. To change the order of integration we need to keep track where the pole in the z plane is In particular for large x the poles will accumulate near the origin and pinch the contour and for this reason we introduce an ±i . The integral in (76) can then be written as, Using the delta function representation we can perform the integral in z to obtain