Large-N correlation functions in ${\cal N} = 2$ superconformal QCD

We study extremal correlation functions of chiral primary operators in the large-N SU(N) ${\cal N} = 2$ superconformal QCD theory and present new results based on supersymmetric localization. We discuss extensively the basis-independent data that can be extracted from these correlators using the leading order large-N matrix model free energy given by the four-sphere partition function. Special emphasis is given to single-trace 2- and 3-point functions as well as a new class of observables that are scalars on the conformal manifold. These new observables are particular quadratic combinations of the structure constants of the chiral ring. At weak 't Hooft coupling we present perturbative results that, in principle, can be extended to arbitrarily high order. We obtain closed-form expressions up to the first subleading order. At strong coupling we provide analogous results based on an approximate Wiener-Hopf method.


Introduction
References [1,2] computed the exact (extremal) correlation functions of N = 2 chiral primary operators in the 4d N = 2 superconformal gauge theory with SU (2) gauge group coupled to N f = 4 massless hypermultiplets. These correlation functions are highly nontrivial functions of the complexified coupling constant τ = θ 2π + i 4π g 2 and include all-order perturbative and instanton corrections. At the moment, they are the only known example of nontrivial, exactly computed 3-point functions in a 4d QFT. The computation of [1,2] relied on the constraints imposed on the chiral ring correlators by the 4d tt * equations [3], together with input from supersymmetric localization [4], and made use of the relation, proposed in [5], between the sphere partition function Z S 4 and the Zamolodchikov metric on the conformal manifold. The relationship between extremal correlation functions in the N = 2 chiral ring and the sphere partition function was further clarified and extended in [6], which paved the way towards concrete computations in general 4d N = 2 SCFTs with conformal manifolds. More generally, it would be interesting to know if there are also other correlation functions that can be computed in practice by employing similar techniques (see [7] for a recent analogous computation of correlation functions in 3d N = 4 superconformal field theories).
In this paper we consider the family of N = 2 superconformal field theories with gauge group SU (N ) and N f = 2N hypermultiplets. We focus on the large-N 't Hooft-Veneziano limit and explain how correlators of chiral primary operators can be computed as a function of the 't Hooft coupling λ = g 2 N . One reason why these correlators are interesting is that they encode information about a putative string theory dual for this family of large-N theories 1 (see [9] for an earlier discussion of such duality). Moreover, similar techniques could be applied to closely related theories (e.g. N = 2 orbifolds of N = 4 super-Yang-Mills theory (SYM) [10]) with known AdS/CFT duals, and lessons obtained in this paper could be easily extended there as well. A recent discussion of conformal manifolds in the context of the AdS/CFT correspondence from the supergravity point of view can be found in [11].
More generally, having a solid understanding of a large-N correlator as an exact function 1 Since we consider a 't Hooft-Veneziano limit where the ratio N f /N = 2 is fixed and non-vanishing, this duality would have the peculiar feature where mesonic hypermultiplet bilinears would lead to an O(N 2 ) number of gauge invariant operators with low conformal dimension. A related discussion of similar limits in two-dimensional theories can be found in [8]. We thank S. Minwalla for comments related to this feature. of λ in QFT, at leading and subleading orders in the 1/N -expansion, could be a useful guide towards a concrete analysis of various formal aspects of the large-N expansion in a full-fledged 4d gauge theory. N = 2 chiral primary correlators in the 't Hooft limit of the N = 2 SU (N ) , N f = 2N theory were recently considered in [12,13], where 2-point functions of single-trace chiral primaries were computed perturbatively in λ at leading order in 1/N . In this paper we substantially extend these results by computing more general correlation functions at large N . Specifically, we focus on two main classes of observables.
The first are 3-point functions of chiral primary single-trace operators. 3-point functions of chiral primary operators in N = 4 SYM theory have of course been widely studied in the context of holography starting with [14,15]. In the N = 2 theory, all the non-vanishing 3-point functions are extremal, and are especially sensitive to mixing with multi-trace operators [16]. We point out that there is a well-motivated and unambiguous definition of the basis of chiral primary operators near the weak-coupling point based on parallel transport that is formulated in terms of a natural connection on the space of operators in conformal perturbation theory. This definition works particularly well in our class of theories in the large-N limit, where the conformal manifold is essentially one-dimensional. A different basis of chiral primary operators is defined implicitly through the relation with the S 4 partition function Z S 4 [6]. We compute 3-point functions of the form Trϕ k 1 Trϕ k 2 Trϕ k 1 +k 2 in the first few orders in λ around the weak coupling point in both bases. We check that to leading order our methods reproduce the results of [14] for the N = 4 SYM, as expected. Unlike the N = 4 SYM theory, however, in N = 2 theories correlators of chiral primaries receive quantum corrections that we can easily compute up to any desired order in λ.
In order to bypass the subtleties that arise from the mixing between single-trace and multi-trace operators we also consider a new class of observables obtained by certain quadratic combinations of the chiral ring structure constants. These quantities are geometric scalars on the conformal manifold, they are manifestly independent of the choice of basis, and therefore can be meaningfully computed and compared at arbitrary values of the coupling constant.
Furthermore, an infinite subset of them obey a very simple recursion relation, coming from the tt * equations, that can be solved explicitly in terms of 2-point function data immediately available at large N .
We show how both classes of observables can be computed at leading order in the large-N limit from the planar free energy F 0 of the theory on S 4 deformed by higher chiral primary sources. The latter is also the planar free energy of a corresponding matrix model, which arises from localization, and can be determined from the solution of the saddle-point integral t n x n (1.1) where K(x) = 2x ∞ n=1 1 n − n n 2 +x 2 . The sum on the r.h.s. originates from the higher chiral primary source deformations of the theory. It is a polynomial whose degree is suitably adjusted to the correlator we want to compute. The planar free energy F 0 follows directly from the eigenvalue density ρ(x). Eq. (1.1) was first considered in [17] and further used in [12]. We have not been able to solve (1.1) analytically for arbitrary values of λ, so we will limit ourselves to analyzing its solutions in two regimes, at weak and strong coupling λ.
Given a solution of (1.1) (approximate or exact), there is a well-defined procedure [1,6] to recover correlation functions of the physical theory by combining appropriate derivatives of Computations based on the weak coupling expansion of the solutions of (1.1) are pretty straightforward and, technically, they follow closely the logic of [12,17]. At strong coupling the analysis of eq. (1.1) is considerably harder. As was first pointed out in [17] approximate solutions can be obtained with the use of the Wiener-Hopf method. 2 Using this method we estimate the large-λ scaling of 2-point functions of single-trace operators in the chiral ring, extending partial results in [12], and the large-λ scaling of 3-point functions. In section 2 we discuss in detail the correlation functions of interest and we set the conventions that are used in the rest of the paper. In addition, we review the relation between extremal correlation functions in the N = 2 chiral ring, the deformed partition function on S 4 and the matrix model that arises from localization.
In section 3 we discuss general properties of correlation functions in the large-N limit.
We explain what contributions can be extracted from the leading order large-N free energy 2 We should point out that these approximations are not parametrically controlled, so we cannot prove conclusively that the large-λ scalings obtained in this way persist in the exact solution of the saddle-point equations.
of the S 4 partition function and how issues involving the mixing of single-trace and multitrace operators affect our computations. We also define appropriate quadratic combinations of the structure constants and show that they obey a recursion relation, coming from the tt * equations, that can be solved in closed form.
Results specific to the weak coupling expansion of the theory are presented in section 4. We provide closed form expressions for 2-and 3-point functions both at tree level and at the first nontrivial subleading order in perturbation theory. Along the way, we present a method, specific to the large-N limit, that allows us to determine the correlation functions of single-trace operators without going through the full Gram-Schmidt orthogonalization procedure proposed in [6]. In this section we also discuss how the use of parallel transport on the conformal manifold leads to unambiguous perturbative expressions for the single-trace 3-point functions. The basis-independent structure constant squared combinations, defined in section 3, are computed perturbatively in λ at the end of the section.
Finally, partial results in the strong coupling limit of the theory are discussed in section 5. We emphasize the large-λ scaling of 2-and 3-point functions and discuss the technical difficulties associated to the current use of the Wiener-Hopf method.
Four appendices at the end of the paper provide the technical background for the computations presented in the main text. Appendix A summarizes the matrix model that arises from localization and the corresponding saddle-point equations in the large-N limit. In this appendix the reader can also find the derivation of an integral equation obeyed by the density of connected 2-point functions, as well as an explicit solution of this equation at infinite 't Hooft coupling. Appendix C describes the perturbative solution of the saddle-point equations at weak coupling and appendix D the approximate solution based on the Wiener-Hopf method. Appendix B provides the proof of a technically efficient general relation between 3-point functions in the gauge theory and derivatives of the matrix model planar free energy in the large-N limit.

Exact 2-and 3-point functions in the N = 2 chiral ring
In this paper we focus on extremal correlation functions of N = 2 chiral primary operators in a specific class of N = 2 superconformal field theories defined as N = 2 SYM theory with gauge group SU (N ) coupled to 2N hypermultiplets (in short, SU (N ) N = 2 superconformal-QCD, or N = 2 SCQCD). We will mostly follow the conventions of [1,18], where one can also find a detailed description of generic properties of the N = 2 chiral primary operators and further useful references to the literature.
We begin with a quick summary of the operators of interest tailored to the specific features of the N = 2 SCQCD theories and the goals of this paper. Then, we proceed to define the correlation functions that will play a central role in our discussion and to summarize recent developments that allow their exact non-perturbative computation. Along the way, we emphasize the implications of the recent developments on 2-and 3-point functions.
Operator notation. In the course of the paper we will consider the N = 2 theory either on R 4 or S 4 . To keep the distinction between these cases explicit at all times, we will refer to the chiral primary operators on R 4 as O K and the corresponding operators on S 4 as O S 4 K . K is an appropriate multi-index that labels the operator. Moreover, for notational economy we will frequently refer to single-trace generators Tr[ϕ k ] on R 4 as k inside correlation functions, double-trace operators Tr[ϕ k 1 ]Tr[ϕ k 2 ] as k 1 k 2 , etc. The S 4 operators may acquire a further label, O R 4 , that refers to specific linear combinations of single/multi-trace operators to be defined.

Correlation function notation.
Correlation functions on R 4 will be denoted as · · · R 4 (or simply as · · · without index), correlation functions on S 4 as · · · S 4 and correlation functions on the associated matrix model as · · · .

SU (N ) N = 2 chiral ring
We begin by considering the SU (N ) N = 2 SCQCD theory on flat space, R 4 . The N = 2 chiral primary operators are, by definition, local superconformal primary operators annihilated by all four left-chiral Poincaré supercharges Q iα , where i = 1, 2 is an SU (2) R index andα = ± a spinor index. In the N = 2 SCQCD theory these operators have a simple description as generic multi-trace operators of the adjoint complex scalar field ϕ in the N = 2 vector multiplet. Using a multi-index K = {n } we denote them as where n are arbitrary non-negative integers. The proportionality symbol refers to an overall normalization factor that will be fixed later. Corresponding multi-trace operators built out of the complex conjugate field ϕ will be denoted as O K ; those are N = 2 anti-chiral primary operators annihilated by all four right-chiral Poincaré supercharges Q i α .
The scaling dimension ∆ K of each of the operators O K is half their This relation holds non-perturbatively for generic values of the exactly marginal coupling constant of the theory τ = θ 2π + i 4π g 2 , where as usual θ is the theta-angle of the theory and g the gauge coupling.
It is clear from the definition (2.1) that the full class of chiral primary multi-trace operators O K can be generated by Operator Product Expansion (OPE) multiplication from a finite set of N − 1 single-trace operators Tr ϕ +1 , = 1, . . . , N − 1. In what follows we will adopt a normalization convention, consistent with the so-called holomorphic gauge [1], where the leading term in the OPE between two chiral primary operators, The

Extremal correlation functions in the N = 2 chiral ring
The main interest of the paper lies in the so-called extremal correlation functions, defined as correlation functions of chiral and anti-chiral primary operators with a single anti-chiral The U (1) R charge conservation requires the R-charge relation

7)
3 In this expression δ J I is the obvious multi-index Kronecker delta.
otherwise the correlator vanishes.
In [1,2] it was argued that all extremal correlation functions can be reduced to the computation of the 2-and 3-point functions, respectively ∆ is the common scaling dimension of the two insertions in the 2-point function (2.8), and ∆ K etc. the scaling dimensions of each operator in the 3-point function (2.9). The interesting datum in each of these correlation functions is the position independent, but generally coupling constant dependent, numerator g KL in the 2-point functions and C KLM in the 3-point functions. In the rest of the text it will be convenient to refer to these coefficients using the notation There is a simple well-known relation between the 2-and 3-point function coefficients and the OPE coefficients C M KL in the N = 2 chiral ring Notice that by using the convention (2.5) equation (2.11) reduces to As an explicit illustration of this relation, consider the computation of the 3-point function of single-trace operators where following the aforementioned convention we denote the single trace operator Tr[ϕ k ] simply as k in a correlation function. Equation (2.12) implies that this is equal to which is a 2-point function between a double-trace operator and a single-trace operator.

2-and 3-point functions from S 4 partition functions and matrix models
So far we exclusively discussed correlation functions of the N = 2 SCQCD theory on R 4 . In recent developments, however, a concrete relation has been put forward between the 2-point function coefficients O K , O M of the theory on R 4 and the derivatives of a suitably deformed partition function of the theory on the four-sphere S 4 [5,6,19,20]. The latter is further related by supersymmetric localization [4] to the partition function of a corresponding matrix model.
Let us briefly review the main elements of this relation and set up the appropriate notation. For additional explanations and details we refer the reader to the original work in [5,6,19,20].

Deformed partition functions on S 4 and their localization
The first step of the procedure starts, quite generally, by placing the N = 2 superconformal field theory on S 4 in a manner that preserves the supergroup of a general massive theory, osp(2|4). In addition, we deform the theory by F-term interactions that are upper components of short multiplets containing the N = 2 chiral primary fields O K . It is enough for our purposes to consider deformations restricted to the single-trace chiral primary fields Tr ϕ k . In superspace form the deformations of interest are (2.15) where E is the N = 2 chiral density. τ 2 ≡ τ = θ 2π + i 4π g 2 is the exactly marginal deformation of the N = 2 SCQCD theory. Now consider the partition function of this theory (2.16) The finite part of this quantity is physical [5,20] and depends non-trivially on the complex couplings (τ n ,τ n ). Interestingly, although this quantity is given by a complicated path integral, it can be reduced by supersymmetric localization to a corresponding matrix integral that can be analyzed with standard methods [4]. The precise form of the matrix integral in the case of the N = 2 SCQCD theories is presented in appendix A.

Relation between the S 4 partition function and 2-point functions on R 4
Recently, [6] put forward a concrete general prescription that relates the τ n -derivatives of Z S 4 to the flat-space 2-point function coefficients O K , O M . One way to summarize the prescription is the following.
The sum I runs over the set S ∆ , which is defined to include all the chiral primaries of scaling dimension ∆ I < ∆, ∆ I = ∆ mod 2. The coefficients a I , b I are clearly dimensionful and therefore proportional to an appropriate power of the sphere radius. They are fully fixed by implementing the Gram-Schmidt orthogonalization procedure, The key statement of [6] is the relation 5 where the matrix A has, by definition, the elements In eq. (2.20) we assumed that the matrix A is invertible, which is a prerequisite for the prescription of [6] to work properly. 4 At this point we will include an index R 4 or S 4 in the notation of the correlation functions to denote explicitly whether we refer to correlation function on R 4 or S 4 . 5 Notice that the conformal mapping between the sphere and the plane introduces an additional factor of 4 ∆ in the relation between the sphere and the plane 2-point functions. To avoid clutter in the equations, we absorb this factor in the normalization of the operators O K . Of course this has no effect on the normalized correlators that we discuss in the rest of the paper.
Tr ϕ +1 n Tr ϕ +1 n . (2.22) · · · denotes a correlation function in the matrix model of appendix A.  of the corresponding matrix model.

Formulae for 3-point functions
As an explicit example consider again the 3-point function of three single-trace operators.
The above prescription gives Obviously, the structure of the sum on the r.h.s becomes increasingly complicated with increasing scaling dimension.
Explicit evaluation gives the following simple 2-and 3-point function formulae where the final result is expressed directly in terms of derivatives of the matrix model free energy F .
The correlation functions of operators with higher scaling dimensions can be expressed similarly solely in terms of F , but the final expression is considerably more complicated.
Further simplifications occur, however, in the large-N limit, which is the main topic of the following sections.

Correlation functions at large N
In this section we study extremal correlation functions in the large-N limit and their relation to the matrix model. Due to large-N factorization, the behavior of correlators involving multi-trace operators is dominated at large N by the factorized answer. Therefore, we introduce a notion of "connected" 2-point functions, which involves, as usual, the full 2-point functions minus the factorized pieces. We argue that these correlators can be determined by the leading contribution to the free energy in the large-N limit, which in turn can be computed by the saddle-point method.
At a later part of this section we specialize to the two main classes of observables that we are interested in. First, we study in detail the relation between single-trace 3-point functions and the free energy, and discuss some useful simplifications that occur at large N . We also discuss in detail issues related to mixing between single-and multi-trace operators, which in principle can affect the results of our computation, and propose one way to get around these difficulties by using the natural connection provided by conformal perturbation theory.
Lastly, we define a new interesting class of observables, which are quadratic combinations of the structure constants that enjoy many useful properties. Most notably, these observables are manifestly free from ambiguities related to the choice of basis of chiral operators, so they are not affected by the subtleties associated to large-N mixing between single-and multitrace operators. In addition, the tt * equations provide a very simple recursion relation for these observables, which can be solved in closed form in terms of simple geometric data on the conformal manifold.
The explicit analysis of these quantities at weak and strong coupling is the subject of subsequent sections.

Correlation functions and the matrix model free energy at large N
We consider the large-N limit at fixed 't Hooft coupling constant λ = g 2 N . Similarly, we rescale the sources of the higher Casimir operators so that the parameters 6 are kept fixed in the limit, as was done in [12]. The free energy (2.24) has the following large-N expansion F 0 is the leading large-N contribution. It can be evaluated using the saddle-point approximation, details of which we review in appendix A.
In the previous section, we reviewed how generic 2-point functions (of single-trace or multi-trace operators) in the chiral ring on R 4 can be expressed in terms of an algebraic functional of derivatives of the free energy F . In the large-N limit, and after the Gram-Schmidt procedure has been properly applied, the result contains a finite number of derivatives of F with respect to the parameters g n . The leading contribution to this result comes from F 0 , and may scale with N in different ways depending on the specifics of the operator insertions.
For instance, the 2-point function of two single-trace operators In [12], a different convention for the couplings was used, namely g there n = π n 2 g here n . We find that our choice is more convenient for the purpose of this paper, as it avoids various explicit factors of π that would otherwise appear in intermediate formulae. This effectively corresponds to a different overall normalization of the chiral operators compared to [12], which of course does not have any effect on the normalized correlators.
scales at large N as a constant. Similarly, the 2-point function of a multi-trace operator with a single-trace operator scales like The leading order behavior is dominated by factorization, unless the single-trace 2-point functions above vanish. Clearly, at this order the 2-point function (3.5) does not contain any new information beyond (3.3). However, the connected version of k 1 k 2 , k 3 k 4 , is far more interesting and scales with a subleading power of N , as O(N −2 ). The leading contribution to the connected correlator is also determined by suitable combinations of derivatives of the free energy term F 0 . Hence, quantities like (3.6) are also accessible within the saddle-point approximation of the matrix model and contain useful information about the large-N gauge theory. We will consider observables related to (3.6) in subsection 3.3.
More generally, we can consider the 2-point function of multi-trace operators

Single-trace 2-and 3-point functions
Next let us take a closer look at the Gram-Schmidt procedure, [6], at large N . It was argued in [12] that mixing between single-and multi-trace operators can be ignored for the purpose of computing single-trace 2-point functions in flat space from the sphere correlators.
As a consequence, the 2-point functions on the plane can be easily calculated from F 0 . Using standard formulae for the Gram-Schmidt diagonalization in terms of matrix determinants, we thus obtain 7 where M k is the k × k matrix given by As a trivial check, eq. The single-trace 3-point functions can also be determined in terms of F 0 . More concretely, we are interested in computing (3.10) The following (streamlined) procedure leads to the desired result. First, we perform the Gram-Schmidt orthogonalization procedure by diagonalizing the matrix of sphere 2-point functions of single-trace operators only. This leads to the following formal identification where c k k = 1 and the remaining c k 's are determined from the condition Our claim is that The proof of this statement is presented in appendix B. This formula is non-trivial because in principle it differs from the prescription presented in subsection 2.3.3. According to that prescription , to compute (3.10) we would have to perform the Gram-Schmidt diagonalization for the operator O R 4 k 1 O R 4 k 2 directly, whose expression in terms of sphere operators differs from the square of (3.11). The reason why we can work directly with the single-trace operators at large N is due to large-N factorization of correlators, as explained in appendix B.

Mixing with multi-trace operators
The single-trace 3-point functions are affected by the following subtlety: R-charge conservation implies that the only non-vanishing 3-point functions are "extremal", or more specifically k 3 = k 1 + k 2 in (3.10). This means that the single-trace 3-point functions, unlike the 2-point functions, are sensitive to mixing with multi-trace operators [16]. As a consequence, different choices for the basis of operators away from the weakly coupled point will lead to different answers for these 3-point functions. This is best illustrated in an example. Let us consider the operator  (4.20), so α(0) = 0.
As we move away from the weakly coupled point, however, it is not obvious a priori that the scheme we are employing matches the one of ordinary perturbation theory in flat space.
There are two possibilities to get around this issue. One is to work with quantities that are manifestly free from ambiguities related to the choice of basis. This is the approach that is described in the following subsection. Alternatively, one can fix the basis of operators away from the weakly-coupled point using the following well-motivated procedure. Conformal perturbation theory provides us with a preferred connection on the space of operators [3].
This connection can be used to parallel transport the operators away from the weaklycoupled point. While this procedure depends on the path chosen to connect the points on the conformal manifold, at large N a preferred path emerges, since the conformal manifold effectively becomes one-dimensional. We explain how to implement this procedure explicitly, and provide various examples, in section 4.3.

Basis-independent 3-point functions
So far we have discussed correlation functions in a specific basis of chiral operators, where both 2-and 3-point functions are non-trivial. In general, however, it is customary to work in a different basis, where the 2-point functions are unit normalized, and all the non-trivial information is encoded in the coupling-constant dependence of higher-point correlators. Alternatively, we can work directly with quantities that are manifestly free from ambiguities arising from the choice of basis. In geometric language, we can look at scalar quantities on the conformal manifold. The simplest such quantity that can be constructed solely from the chiral ring data is This object is closely related to the "properly normalized" 3-point functions defined in [1].
For example, in the case of gauge group SU (2), the chiral ring is generated by and the 2-point functions in the chiral ring are given by whereĈ 2m+2n 2m 2n are the 3-point functions written in a basis where the 2-point functions are unity. These quantities were computed exactly in [1].
In order to compute (3.15) at large N , we would need to compute the 2-point functions of all the chiral operators (both single-and multi-trace) to the appropriate order in N and then take the large-N limit. The leading order O(N 0 ) term in |C (∆ 1 ,∆ 2 ) | 2 is a combinatoric constant determined by large-N factorization, so to get non-trivial results we have to consider the terms of order 1/N 2 in (3.15).
It is easy to see that these 1/N 2 corrections are captured by the leading order free energy F 0 . Indeed, we can work in a basis where C K IJ = δ K I+J , so that (schematically) where the (anti-)chiral operators are sent to the same point. In the large-N limit, the first two terms in the expression above give the O(N 0 ) factorized contribution to |C (∆ 1 ,∆ 2 ) | 2 , while the connected 4-point function behaves as O(1/N 2 ). The free energy is the generating function for the connected correlators, hence we conclude that the 1/N 2 correction to |C (∆ 1 ,∆ 2 ) | 2 can indeed be computed (at least in principle) from the leading term in the free energy F 0 .
In fact, in the case where ∆ 1 = 2, ∆ 2 = ∆, we can derive an explicit relation for the 1/N 2 correction to |C (2,∆) | 2 in terms of the single-trace 2-point functions (3.8). This is possible because this quantity obeys a very simple recursive relation coming from the tt * equations that can be solved explicitly.

tt * equations and |C
We recall the general tt * equations for a (complex) 1-dimensional moduli space If we contract the indices appropriately and define the quantity where n (∆) is the number of chiral primary operators of dimension ∆. This recursion equation can be solved explicitly, and it gives where the sum runs over even conformal dimensions only.
Let us now examine how the recursion equation (3.21) and its solution (3.22) behave in the large-N limit. We begin by analyzing the "curvature" term (3.20). Recall that the large-N limit is taken by keeping the 't Hooft coupling constant fixed Since instanton corrections are suppressed in this limit, we can assume that all the quantities (in particular, the 2-point functions) depend on λ only. Therefore The matrix of 2-point functions can be written as (3.25) and its inverse has a similar expansion, where the leading term is just (g 0,K ∆ M ∆ ) −1 . Therefore we have (3.26) where the ellipses indicate higher order terms in 1/N and R (∆) 0 is given by We can now consider the behavior of |C (2,∆) | at large-N . Using (3.22) we see that It is clear that the 1/N 2 correction to |C (2,∆) | 2 can be determined directly from F 0 using equation (3.8). We will give explicit formulae for |C (2,∆) | 2 as a perturbative series in λ around λ = 0 in section 4.4.

Weak coupling results
In this section we analyze in detail the correlators described in the previous section around the weak coupling point λ = 0. We begin by implementing explicitly the Gram-Schmidt diagonalization procedure at tree-level. This allows us to compute the flat-space tree-level 2-point functions and 3-point functions of single-trace operators. At this order, the computation is identical to N = 4 SYM and indeed we reproduce the results of [14].
We then show that the first non-trivial subleading corrections to these correlators can also be computed explicitly in flat space, thanks to a simplifying property of the 1-loop determinant of the matrix model noticed in [6]. Using the techniques of appendix C, we also present several examples of 3-point functions computed to much higher order in λ. We also deal with the subtleties related to mixing with multi-trace operators, anticipated in section 3.2.1, by introducing a notion of parallel transport of operators, which is natural in conformal perturbation theory.
Finally, we analyze the squared structure constants |C (2,∆) | 2 defined in (3.15); we explicitly compute the leading and subleading (order λ 2 ) results for general ∆ and we present results to higher orders in λ in examples.

Single trace 2-and 3-point functions at tree-level
Here we consider the Gram-Schmidt procedure at large N and at tree-level in λ. For simplicity we present the procedure for the sector of even chiral primaries. We start with the matrix of 2-point functions of single-trace operators on the sphere, which is given by [12] We will also need the 3-point functions on the sphere [12] A more general formula for even and odd chiral primary three-point functions on the sphere is given in appendix C. Performing the Gram-Schmidt procedure we find These coefficients agree with the Chebyshev prescription of [13]. We can then use our previous formulae to compute 2-and 3-point functions on R 4 . We find If we normalize canonically the 2-point functions, we find which agrees with the results of [14].

Single-trace 2-and 3-point functions at higher orders in λ
Using ( which means that the ζ(3) correction to a correlation function can be obtained from the tree-level result by applying an appropriate derivative operator. We now find this operator explicitly and study its behavior at large N .
We can rewrite (4.8) in the form where Z 0 [A] is the matrix model partition function at tree-level with the insertion of the operator A. Notice that in converting the expansion of the 1-loop determinant into a derivative, we have assumed that A itself does not depend on τ in its representation inside the integral. In particular, this means that this equation cannot be applied directly to connected correlators, since the mixing of an operator with the identity on the sphere will exhibit a non-trivial dependence on τ . It is easy to show that the equation above is equivalent to where A 0 is the expectation value of A at tree-level and we have defined to avoid clutter in the equations. A connected correlator will then satisfy (4.12) In the large-N limit, only the first and third terms in the brackets will contribute, since they scale like N 0 while the second term scales like N −2 .
Translating these formulae to the language of connected correlation functions on the sphere we have (4.13) Analogously, one can derive the ζ(3) correction to the connected 3-point function as (4.14) Remarkably, these formulae are valid even for flat-space correlators. To prove this, consider for definiteness the case of 2-point functions (4.13) applied to the case k 1 = k 2 = k.
Employing the Gram-Schmidt procedure as in section 2.3.2 we deduce that the leading perturbative correction to the flat-space 2-point function can be written as where in the last equality we used that the operators O R 4 k are orthogonal to all the operators of lower conformal dimension by definition. Applying (4.13) to the previous expression, and using the same argument to move the derivative in front of the sum, we find (4.17) The same argument applies to the 3-point functions as well. Consequently, if we know the tree-level 2-and 3-point functions on the plane, we can determine their ζ(3) correction using (4.13) and (4.14). The final result is We first notice that the tree-level results match the ones computed in [14], as they should.  (7)) 32768π 8 λ 4 + . . . , (4.24) . . .
In particular, notice that while the λ 2 corrections are absent in the normalized 3-point functions that do not involve the operator O 2 , the λ 3 and higher corrections are present.

3-point functions in the parallel transported basis
As we anticipated in section 3.2.1, the 3-point functions defined above are sensitive to mixing with multi-trace operators. At the weakly coupled point, the canonical trace basis is particularly convenient, and we can ask if this basis can be extended (or transported) in a canonical way across the conformal manifold. Since conformal perturbation theory defines a natural connection on the space of operators, it is sensible to use this connection to parallel transport the canonical trace basis defined at the weakly coupled point to other points in the conformal manifold. In general, this is an ambiguous operation, because the parallel transported basis will depend on the particular path chosen due to curvature. Fortunately, at large N , the conformal manifold becomes effectively 1-dimensional, and a preferred path emerges, the one "along λ".
Let us define the vielbein-like objects e I k (λ) such that Here the index I is allowed to run over all the chiral primaries (both single-and multi-trace) of dimension ∆ = k and the arbitrary proportionality constant can be chosen so that the diagonal part of the 2-point functions is unity. A choice of e I k (λ) corresponds to a choice of basis of chiral primaries that agree (up to an overall normalization) with the tree-level trace basis when λ = 0. The parallel transported basis will be determined by demanding that where ∇ λ is the covariant derivative along λ. The parallel transported 3-point functions will then be given by It is clear that these correlators will agree with (4.20) at leading order in λ.
In order to implement this procedure explicitly in the present case, we need to determine the connection from the S 4 partition function. To do so, we will assume that the basis of operators on the plane implicitly defined by the Gram-Schmidt procedure is a holomorphic basis. 8 This assumption passes an important consistency check, namely that the resulting 2-point functions do obey the tt * equations written in a holomorphic basis [6]. However, as we noted previously at the end of section 2.3.2, we are not aware of a complete proof of this statement. If the assumption is correct, then it is easy to show [1] that the connection along λ is 8 Strictly speaking, the basis that we are considering is holomorphic only when the operators are multiplied by the factor e R c K , where K is the Kähler potential of the theory. For more details and a discussion on the effect of Kähler ambiguities, we refer the reader to [1].
where g is the matrix of 2-point functions at the appropriate level. Using this and the explicit results up to order λ 2 of the previous subsection, it would be easy to compute (4.27) up to this order. However, due to the property in equation (4.8), this correction will vanish.
To exemplify how the procedure works in practice, we explicitly work out the case k 1 = k 2 = 2, k 3 = 4 up to order λ 3 . The relevant matrices of 2-point functions are then (4.30) We now have everything we need to solve (4.26). The result is The tree-level piece is of course the same as before. We also notice that the λ 2 correction is absent, as expected from (4.8), but the correlator does receive quantum corrections from order λ 3 and higher.
We start by giving an analytic expression of the curvatures R The piece of the curvature coming from the 2-point functions of the operators k k n k will then be given by where we used large-N factorization. Therefore, the leading large-N contribution to the curvature on the space of chiral primaries of conformal dimension ∆ is given by where n (∆) is the dimension of the space of chiral primaries at level ∆ (https://oeis.org/ A182746) and p (∆) is the number of 2's in all partitions of ∆ that do not contain 1 as part (https://oeis.org/A182716). Combining this result with (3.28) gives us the 1/N 2 piece to order λ 2 exactly. As before, it is very easy to compute higher-order corrections by following the algorithm of section 3.2, but we do not have a closed form expression for them.
The case where ∆ = 2 is particularly interesting, as it gives the leading contribution to the conformal block expansion of the 4-point function in the chiral channel, namely x 1 → x 2 , x 3 → x 4 . In the case of N = 4 SYM, the analogous object was studied both from the field theory side [21] and the gravity side [22], and the results turned out to be independent of the coupling constant consistent with the N = 4 nonrenormalization theorem [14][15][16][23][24][25][26][27][28][29]. In our case, we can explicitly see that this quantity does depend non trivially on the coupling constant, its first perturbative corrections around in [30] that follows from conformal bootstrap techniques.

Partial strong coupling results
The t n x n . in [17] it was argued that µ = µ + = −µ − → ∞ in the strong coupling limit, λ → ∞, at t n = 0. It was further argued that the leading order relation between λ and µ in this limit is The presence of small higher single-trace couplings t n (n ≥ 2) will not affect this qualitative behavior of µ ± , but the specifics of the dependence of µ ± on the general t n , that generalizes Tr ϕ 2 , Tr ϕ 2 R 4 ∼ (log λ) 2 , Tr ϕ 2 , Tr ϕ 2 , Tr ϕ 4 R 4 ∼ λ (log λ) 3 .

(5.5)
Many more explicit results like this can be obtained from the computations of appendix D.
Determining the precise numerical prefactor in these expressions is hard. As we detail in appendix D the employed Wiener-Hopf approximations are not based on a well-controlled 9 The behavior of Tr ϕ 2 , Tr ϕ 2 R 4 at strong coupling, determined by the same approximation method, was reported also in [12] and agrees with our results below. expansion in terms of a parametrically small number. In fact, performing this computation at the next iteration we found corrections to the leading order coefficients that are numerically comparable to the leading contribution. In any case, focusing on the leading scaling of the term on the r.h.s. of (5.6) we find

A. Deformed matrix integrals from supersymmetric localization
The function H(x) is defined as

A.1. Large-N limit and the saddle-point equations
In the large-N limit we can further set the instanton contributions Z inst → 1, and express the single-trace couplings τ n in terms of their 't Hooft combinations (3.1). Since the end result depends only on the imaginary part of the couplings τ n we set Then, the large-N (U (N )) partition function takes the form In the large-N limit it is also convenient to introduce the density of eigenvalues which is normalized so that where − denotes a principal value integral. The function K controls the measure of the matrix integral, and the function f controls its potential.
The undeformed version of these equations, with g n = 0 for all n > 2, was analyzed previously in [17]. An analysis of the deformed equations, with g n = 0 for all odd n, was initiated more recently in [12]. We revisit this analysis below and extend it in several directions.
Following [17], it is possible, for a general single-cut configuration, to recast the saddlepoint equations (A.7) into the form To obtain this result we used the identity [12] − Integrating this equation further over the domain of eigenvalues gives, in conjunction with the normalization condition (A.6), We used the identities [12] µ + where The simultaneous solution of the above equations determines µ ± and ρ(x) parametrically in terms of the single-trace coupling constants g n . Technical aspects of the solution are discussed in detail in the two subsequent appendices in different regimes. Once a solution is known the free energy is determined by the expression which can be manipulated further, using the saddle-point equations, to the more convenient form that does not contain directly the special function H(x).
It is useful to highlight here the following properties of the saddle-point equations and their solutions.
A. Self-contained system of equations. There are obviously enough equations ((A.10), (A.13)) to obtain the eigenvalue density ρ(x) and the bounds of the support µ ± when µ − = −µ + . This occurs when the deformation is even, i.e. when only g n with n even are present.
In that case the density is symmetric around the origin, ρ(x) = ρ(−x). More generally, e.g.
when odd interactions are allowed in the potential V , the eigenvalue density is not symmetric and µ + = −µ − . In that case, there is an additional non-trivial condition [12,31] that can be used to fix the relation between µ + and µ − . It follows from the requirement that the resolvent at large x. To ensure this drop-off the resolvent cannot contain any terms proportional to x n with n > 0. This is equivalent to the requirement This equation is automatic when µ + = −µ − and the density is even. ng n π n/2 b r b n−k−2 µ r

B. Special cases vs generic
In the presence of a non-trivial function K, however, an exact solution is not known in closed form. In appendices C and D we analyze the density ρ and free energy F perturbatively in the strong and weak g 2 = 8π 2 λ regime, respectively. Although it is possible to perform a comprehensive analysis of the weak-λ regime at any desired order of perturbation theory, current techniques do not provide an equally satisfactory treatment of the strong coupling side.

C. Universality.
Once a solution of the saddle-point equations is obtained, the result can be inserted into the free energy F to obtain a generating function for the connected correlation functions of the model. Remarkably, the final expression of the correlation functions can be written solely in terms of µ ± (the limits of the eigenvalue support), and is independent of the details of the potential function f . We will give an example of this property of universality in a moment for connected 2-point functions. The explicit dependence on the single-trace couplings that appear in the function f arises from the corresponding dependence of the quantities µ ± .

A.2. Aside: density for connected 2-point functions of single-trace operators
In the previous subsection, and in appendices C, D, we evaluate the free energy of the deformed matrix model at general values of the single-trace couplings g n and take derivatives at g n = 0 for n > 2, and g 2 = 8π λ free. This allows us to compute connected correlation functions of the matrix model at any value of g 2 and g n = 0 for n > 2.
In this appendix we consider a different evaluation of the same connected correlation functions that does not go through the computation of the free energy in the deformed matrix model with g n = 0 (n > 2). We focus on the 2-point functions of single-trace operators.
The result is expressed as a double integral over the eigenvalues with an appropriate 2-point function densityρ 2 (x, y) There are well-known results in the literature regardingρ 2 (and its higher n-point generalizations) -for a recent comprehensive review we refer the reader to [32]. For example, in the case of the standard Hermitian matrix model where V is a real polynomial of some degree, the leading large-N contribution to the connected 2-point function densityρ 2 takes the form We have assumed a single-cut solution of the saddle-point equations in the symmetric interval [−µ, µ]. In accordance to point C in the previous subsection, this expression is universal, namely independent of the details of the potential V .

Integral equation for ρ 2
At this point we take the opportunity to derive a general integral equation obeyed by ρ 2 (we are not aware of a similar derivation in the literature). In an attempt to be fairly general, let us consider the large-N limit of the matrix integral For starters, the effective action for the eigenvalues a i is The leading-order saddle-point equations are We dropped the term and performing the Gaussian integrations over δ i we obtain where P and Q O are respectively the vector and matrix After a few straightforward algebraic manipulations one can show that up to order O(N −2 ) dP is the matrix with elements Then, in the continuum limit explicit evaluation gives where m n are the moments Combining this equation with the definition of Q (A.32) (in its continuous form) we deduce that the leading O(N −1 ) part of Q −1 obeys the more explicit integral equation Requiring the boundary conditions we can perform two integrations by part to recast (A.42) as on both sides of (A.46) we obtain which is an integral equation for the derivative of F 2 , ∂ x F 2 (x, y). Solving this equation, or its progenitor (A.46), and applying a second derivative on the second argument of F 2 we obtain the density ρ 2 that allows us to determine the connected two-point function of single-trace operators at any value of the 't Hooft coupling at leading order in the 1/N expansion.
As an additional comment on the boundary conditions (A.43), notice that the first one is a natural consequence of the assumption that the inverse Q −1 (x, y) is regular at x = ±µ (or not singular enough -an example is provided by the standard Hermitian matrix model, in exact agreement with the known result (A.23). is currently not known. It is straightforward, however, to obtain the analytic solution at infinite coupling. Since, the strong coupling regime is hard to analyze with existing methods, it is of some value to report here the analytic profile ofρ 2 at infinite coupling.
For this purpose it is convenient to return to eq. (A.46). Since ∂ x F 2 (x, y) is a function of the difference x − y, we can perform a simple Fourier transformation which gives in real space As usual, ψ(z) = Γ (z) Γ(z) is the logarithmic derivative of the Γ-function. Unfortunately, this datum is not enough to determine the large-λ behavior of the connected single-trace two-point functions. When we apply the formula (A.21) at large but finite λ we need to know also the first subleading (in 1 λ ) correction toρ 2 . The mere knowledge of (A.52) appears to provide the correct leading large-µ scaling of the connected two-point functions but fails to capture the exact numerical coefficient.

Towards higher connected n-point functions
Working directly with the densities of connecting n-point functions (as we did above for n = 2) is a direction with a potential for promising results. In the past considerable results have been obtained in standard matrix models through the analysis of the matrix model loop equations which provide recursive equations between the generating functions of connected n-point functions (see [32] for a modern review and references to the original literature).
Such results can be extended to more general cases (like (A.24)) -see, for instance, the recent work [33]. It would be interesting to pursue this approach for the matrix models of interest in this paper. We hope to return to this aspect in a future publication. (3.12) In this appendix we prove formula (3.12). Let us first recall that the 3-point function we are interested in is given by the formula

B. Proof of
where, as always, the R 4 indices on the operators indicate that we have applied the full Gram-Schmidt procedure, i.e. the primed operators are orthogonal to all the operators of lower conformal dimension, both single-and multi-trace. We want to prove that in the large-N limit the 2-point function above is given by the simpler expression where the operators O R 4 k are orthogonal to all the single-trace operators of lower conformal dimension only, and O R 4 is simply the product of the operators O R 4 k 1 and O R 4 k 2 .
We first examine the difference between the operators where the last three terms on the r.h.s. involve operators of conformal dimension less than Let us analyze the double-trace coefficients c 1 , 2 k 1 ,k 2 . They are given by Using large N -factorization, we obtain where c k is given by From this we conclude that Also recall that up to O(1/N ) corrections the linear combinations O R 4 k involve only singletrace operators.
If we now examine again equation (B.1), we see that the single-trace operators on the r.h.s. of (B.7) do not contribute since O R 4 k 3 is orthogonal to all of them, while the higher-trace operators are suppressed in the large-N limit. This proves (3.12). Notice that it is important that the operator O R 4 k 1 O R 4 k 2 does not appear in the r.h.s. of the definition of O R 4 k 3 , otherwise it could give a contribution of the same order as (B.2) by large-N factorization. For correlators that respect R-charge conservation, namely k 3 = k 1 +k 2 , the higher-trace operators in the full appear. In the case k 3 > k 1 + k 2 , applying (3.12) would give a non-zero result, inconsistent with R-charge conservation. However, this is precisely the case where we cannot ignore the in the expansion of O R 4 k 3 ; its effect is to precisely cancel the non-zero contribution coming from (3.12).
In summary, we have proven that the formula (3.12) gives the correct 3-point function in the large-N limit for operators that respect R-charge conservation, which are the only non-zero 3-point functions on the plane.

C. Large-N matrix model at weak coupling
In this short appendix we collect some explicit expressions for the three-point functions at tree-level. Similar results were presented in [12]. Here we generalize the results of appendix C in [12] to obtain three-point functions for single-trace operators of both even and odd powers at tree-level. This serves as input for the closed form diagonalization procedure discussed in section 4.
As reviewed in appendix A (see equations (A.7),(A.8),(A.9)) in the continuum limit we obtain the saddle-point equation This equation can be inverted to get an expression for ρ(x) by applying the integral operator . Further use of the integral identities (A.11), (A.14) provides implicit relations for the moments up to an arbitrary but finite number of loops The moments appear linearly in this expression, hence by truncating the perturbation theory to a finite order and solving the corresponding system of linear equations we obtain expressions for the moments accurate up to that order.
In what follows it will be useful to define a separate expression for the first line of the previous expression The function φ q (g n ) is equal to the tree-level contribution to the moments. For quick reference, we note here its second derivative with all couplings set to zero, except g 2 (and hence appropriately µ − set to −µ + ) (C.4)

C.1. Results at tree-level
At tree-level the moments simply reduce to m q = φ q (g n ). In addition, when all the couplings except g 2 have been set to zero the endpoints of the eigenvalue distribution are given by the expressions µ + (g 2 ) = −µ − (g 2 ) = 2 πg 2 . To specify the derivatives of the endpoints with respect to the coupling g n we require two conditions. The first condition comes from the normalization of the eigenvalue density The second condition, (A. 19), turns into the constraint and 14) The combination of these results with the formulae developed in section 4.2 yields results on the tree-level flat-space three-point functions and their first subleading correction.

D. Large-N matrix model at strong coupling
The analysis of [17] at infinite coupling shows that the size of the cut of the eigenvalue distribution grows with the value of the 't Hooft coupling. In that case, a more sophisticated treatment of the saddle-point equations is needed. Ref. [17] proposed an approximate Wiener-Hopf approach. Similar to the weak coupling case we can extend this method by adding appropriate polynomial sources to the free energy. In this appendix we consider only the case of even source terms. This extension was also suggested and partially implemented in [12].
For the benefit of the reader, and in order to set up the appropriate notation, we begin with a quick review of the Wiener-Hopf method used in [17].

D.1. Wiener-Hopf method
Employing the following integral identity for the function K(x) we can rewrite our saddle-point equation as a slightly generalized version of equation (4.15) of [17] ρ(x) − dy ρ(y)(x − y) coth π(x − y) + x coth(πx) = 1 2π where the coefficients b k are given in (A.12). As an integral equation this expression strictly holds true only within the interval [−µ, µ]. This means that as an operator the integration kernel is singular. In the Wiener-Hopf method we can exploit the knowledge that ρ(x) is zero outside of the interval. Accordingly, the goal is to find an expression whose inverse Fourier transfrom vanishes outside of the interval. A general method for this was given in [34], but it requires solving a very non-trivial factorization problem. We will not attempt to solve this factorization problem here. Instead, following [17], we will aim to find a function that is guaranteed to vanish only for x > µ. We will then argue that as long as µ 1 the other endpoint will not be of major concern. As a result, we need to find an expression of the form where f (k) is a function that is analytic in the lower half-plane. Computing the inverse Fourier transform by means of contour integration shows us that this satisfies the boundary condition.
To find an expression of the form (D.3) we proceed by first computing the Fourier trans- where F (k) is the Fourier transform of the right-hand side of (D.2), and where everything that is not strictly determined by the integral equation is encapsulated by the two free functions X − (k) and X + (k). Note that if X ± (k) is analytic in, respectively, the upper or lower half-plane then X ± (x) = 0 for either x > µ or x < −µ.
The key step in what follows is that the integration kernel in Fourier space has a known analytic decomposition cosh(k) Most notably, these functions are respectively analytic in either the upper or lower half-plane and they go to zero sufficiently fast as k → ∞. Also note that their poles are located at respectively ∓iν l where ν l = π(l + 1 2 ). At these poles the residues are given by r l = (−2) l+1 Γ 2 l 2 + 5 4 √ π l + 1 2 Γ(l + 1) . (D.7) Next we multiply the integrand of (D.4) with G + (k) and subtract the poles in the lower half-plane by means of the integral transform This operation provides the expression In the regime of very large, but finite µ, the last term can be argued to be small. Treating this term perturbatively leads to an approximation scheme where we have an accurate description only at the right endpoint of the eigenvalue density. However, since we add only even sources to the action we can find a valid description for ρ(x) in the interval [0, µ], assume that ρ(x) is an even function, and reflect it around x = 0. In this way, one finds [17]

D.2. Computing the moments
The condition that arises from the normalization of the eigenvalue distribution was found in [17] by exploiting the fact that the eigenvalue density is an even function By closing the contour in the lower half-plane, [17] finds ∞ l=0 r l e −µν l F (−iν l ) ν l + ν 0 = r 0 2ν 0 e −µν 0 . (D.14) Moreover, we have the eigenvalue density in Fourier space we can find an expression for the moments as follows In the last step we have eliminated all terms that oscillate rapidly in the µ 1 regime.
From the form of ρ(k) it is clear that we can close the contour in the positive half-plane and pick up all the poles, which are the ones originating from G − (k) and the higher order poles at zero. Let us consider the manipulations of each of the terms appearing in (D. 15) individually.
First, the term (−1) n (2n)! iπ ∞ −∞ dk 2 sinh 2 ( k 2 ) k 2n+1 cosh(k) F (k) (D. 16) is simply given by evaluating the residue at zero The last term is only slightly more complicated and is therefore killed due to the normalization condition (D.14).
Putting everything together we find the following expression for the moments r l e −µν l ν l (1−F (−iν l )),

(D.21)
where h n is an unimportant set of constants. We want to calculate correlators which are derivatives of the moments with respect to coupling constants, so the explicit value of the constants h n is not needed.
Next let us focus on the infinite sum in the rightmost term of the last equation, (D.21).
We rewrite this term by means of the normalization condition (D.14) and by removing terms that are exponentially suppressed in the regime µ 1, Continuing to ignore the exponentially suppressed terms the hypergeometric function contained within F (−iν l ) has the following asymptotic expansion

D.3. Computing correlators
Unlike the computation of correlators in the weak coupling regime, in this subsection we will bypass the evaluation of the planar free energy by means of the following relation ∂ ∂g 2n F = π n µ −µ dz z 2n ρ(z) = π n m 2n . (D.28) Since the correlators are, by default, given by higher derivatives of the free energy we can obtain them by taking derivatives of the moments [12] Tr  This relation also provides a non-trivial check for the moments as they naturally have to satisfy the following commutativity property π m ∂ ∂g 2n m 2m = π n ∂ ∂g 2m m 2n . (1)) suggests that µ ∼ 2 π log(λ) in the strong coupling regime where λ 1. This scaling was already conjectured in [17] by means of an interpolation between the weak coupling and infinite 't Hooft coupling regime.
Using this proposed behavior for µ as a function of λ and the orthogonalization procedure of [6] we find the leading order behavior for the correlators quoted in section 5.