A tree-level 3-point function in the su(3)-sector of planar N=4 SYM

We classify the 3-point functions of local gauge-invariant single-trace operators in the scalar sector of planar N=4 supersymmetric Yang-Mills involving at least one su(3) operator. In the case of two su(3) and one su(2) operators, the tree-level 3-point function can be expressed in terms of scalar products of su(3) Bethe vectors. Moreover, if the second level Bethe roots of one of the su(3) operators is trivial (set to infinity), this 3-point function can be written in a determinant form. Using the determinant representation, we evaluate the structure constant in the semi-classical limit, when the number of roots goes to infinity.


Introduction
Computing n-point functions of local composite gauge-invariant operators in N = 4 supersymmetric Yang-Mills theory, SYM 4 , is an important problem because these n-point functions are among the basic objects on which the AdS/CFT correspondence can be tested 1 . It is also a hard problem, even at tree-level, if only because of the combinatorial complexity of the operators involved. However, developments over the past few years, starting with [3] and subsequent works, raise the hope that the methods of classical and quantum integrability can be used to solve this problem, at least in the planar limit. For a comprehensive review of integrability in SYM 4 and AdS/CFT, see [4] and references therein. For shorter review, see [5].
In this work, we focus on operators {O} that are composed of fundamental fields in the scalar sector of SYM 4 . Representing these fundamental fields as matrices in the adjoint representation of su(N c ), {O} are traces of products of N c × N c matrices. Further, in the planar limit that we are interested in, N c → ∞, multi-trace operators are suppressed by factors of 1/N c , and one can take {O} to be single-trace operators.

The weak-coupling limit
In weakly-coupled, perturbative Yang-Mills theory, the computation of the n-point functions is a welldefined problem. Following Okuyama and Tseng [6], it is sufficient at tree-level to count all possible planar sets of Wick contractions between the operators involved. Apart from normalization factors, the essential object in a 3-point function is the structure constant which, up to a normalisation is a tri-linear form in the Hilbert space of states, which we call the cubic vertex, in analogy with string field theory. In [7], Roiban and Volovich showed that these n-point functions reduce to scalar products of spin-chain states constructed using the algebraic Bethe Ansatz [8]. A systematic study in the case of three operators that belong to (different) su(2) sectors was presented by Escobedo, Gromov, Sever and Vieira [9]. The tree-level correlation function of three su (2) operators was expressed in [9] in terms of scalar products of off-shell Bethe states 2 of XXX spin- 1 2 chains. This method is known as "tailoring". Furthermore, it was shown in [10] that the 3-point function can be recast in terms of scalar products of an off-shell state and an on-shell state and thereby can be evaluated in determinant form. We refer to this method as "freezing".

The semi-classical limit
We are interested in computing the correlation functions of long operators that are dual to semiclassical string states in AdS 5 × S 5 . The semi-classical (heavy) operators {O sc } are associated with classical solutions of the string σ-model [11][12][13]. For a review see [14]. In spin-chain terms, the operators {O i } are eigenstates of the spin-chain Hamiltonian. In other words, they are functions of rapidity variables that satisfy Bethe equations. For such operators, the Bethe roots condense into several cuts (macroscopic Bethe strings) in the complex rapidity plane [11]. The n-point functions of semiclassical operators {O sc } are particularly interesting, as they can be compared with the corresponding correlation functions computed on the string theory side. Computing n-point functions of {O sc } in the string theory was addressed in [15][16][17][18][19][20][21][22]. However, the only case when the complete answer is known is that of two heavy and one light operators [16,23,24]. The same configuration (heavy-heavylight) was considered on the gauge theory side by Escobedo et al. [25,26], and in the sl(2) sector by Georgiou [27]. They used a coherent state approximation for the two heavy operators in the su (3) sector. Comparison with the Frolov-Tseytlin limit [28] of the string theory result [16,23,24] showed a perfect match.
The general case, when all three asymptotically-long operators are non-BPS, the complete answer for the three-point function is known only for weak coupling, and for special choice of the operators. In [29], Gromov, Sever and Vieira presented a thorough analysis of the case of one BPS and two non-BPS heavy fields from the su(2) sector. In spin-chain terms, BPS operators are characterized by trivial Bethe roots that are set to infinity. The main result of [29] is an analytic contour integral derived from Korepin's sum expression for the scalar product of two off-shell states [30]. In [31,32], the determinant expression obtained in [10] was used to solve the problem in the general case of three heavy non-BPS operators. In [33][34][35], it was argued that this solution gives the 3-point function at one and two loops. At one loop this conjecture was verified in [33,35].

Outline of contents
In Section 2, we classify the 3-point functions such that at least one operator is from the su(3) sector. In 3, we recall the formulation of the su(2) 3-point function, including the cubic vertex, in determinant form. In 4, we generalize the freezing method of ref. [10] to the case where two of the operators belong to su(3) sectors while the third belongs to an su(2) sector. Then we take one set of Bethe roots of one of the su(3) operators to be trivial (sent to infinity) and use the result of [36] to write the 3point function in a determinant form. 3 In 5, we recall the how the su(2) 3-point function was written in [31,32,37] in terms of certain functionals in order to be able to compute its semi-classical limit. In 6, we write the su(3) 3-point function in terms of the quantities defined in the previous section. In 7, we compute the semi-classical limit of the su(3) 3-point function of three non-BPS operators, under an assumption that allows us to compute the semi-classical limit of the norm of an su(3) Bethe eigenstate. Appendix A contains a brief introduction to the nested coordinate Bethe Ansatz which is needed for the 'tailoring' approach to the su(3) 3-point function. Appendix B includes details of the 'tailoring' approach to the su(3) 3-point functions. Appendix C discusses the properties of the functional forms that are needed to obtain the semi-classical limits.  2) where N c is the number of colors, λ is the 't Hooft coupling and the three factors N i depend on the normalization of the operators {O i (x i )}. The structure constant C 123 (λ) does not depend on the normalization. The factors L i , equal to the number of fundamental operators in O i , account for the cyclic rotations of the trace operator O i . The structure constant C 123 (λ) has the perturbative expansion To compute the tree-level structure constant C 123 using the method of [9], one needs the 1-loop wave functions. At 1-loop level, the operator O i is represented by a Bethe eigenstate with energy ∆ i . In a su(2) sector, there is only one non-trivial configuration of 3-point functions. In the presence of one or more operators from an su(3) sector, the structure of the 3-point functions becomes richer and we need to classify the set of possible non-trivial configurations of structure constants. An example of a set of planar contractions is given in Fig. 1. The contractions XX , YȲ and ZZ are represented respectively by solid lines, (red) wavy lines and dashed lines.
Let us introduce some conventions. There are several possible choices of an su(3) sector, which correspond to a choice of three distinct complex scalar fields X, Y , Z,X,Ȳ ,Z, with pairs of mutually conjugated fields, like Z andZ, excluded. When only two types of non-conjugate scalar fields are chosen, the composite operator belongs to an su(2) sector. If O 1 , O 2 and O 3 belong to su(α), su(β) and su(γ) sectors respectively, then the corresponding 3-point function of type {α, β, γ}. By permutation invariance, the order of α, β, γ is irrelevant.
We represent the different classes of correlation functions schematically by specifying the different types of Wick contractions between pairs of operators. For example, the correlation functions corresponding to Fig. 1      that it contains only the scalar fields X, Y and Z. Since we are interested in the large N c limit, only planar contractions are retained.
We start by classifying the type-{3, 3, 3} structure constants. In this case, there are two nontrivial inequivalent configurations, as is shown in Fig. 3. Deleting one line, that is, one type of Wick contractions, from each of these two configurations, one obtains type-{2, 3, 3} structure constants. There are three such configurations, as shown in Fig. 2 and Fig. 4.

Tailoring the tree-level structure constants
Following [9], we construct the structure constant in three steps. 1. We split the algebraic Bethe Ansatz representation of each spin chain into two: a left sub-chain and a right sub-chain. 2. Considering each spin chain to be an in-state, we "flip" each left sub-chain from an in-state to an out-state. 3. We take the scalar products of the left sub-chain state of O i with the right sub-chain state of O i+1 mod 3 (i = 1, 2, 3). Finally, we normalize the three external states. Further details on the tailoring procedure are in appendix B. We give our setup data in the table below.

Operators Length Rapidities
No. of Rapidities Partitions of Rapidities The lengths of the left subchains are The structure constant reads 5) where N i are the norms of the Bethe states 4 : The H F factors are given by In the previous formula we used the following notations: we denote the scattering factors as and, given a function F (x, y) and two sets of variables u, v, we define (2.9) Proportionality factor between ABA and CBA Bethe state: |u alg = c u |u cor is given by While the formula (2.5) can be explicitly used for a small numbers of magnons, it is not adapted for taking the classical limit where the number of magnons is large. The main obstruction for taking the classical limit of (2.5) is that the scalar products involved are between off-shell states, and there is no closed form expression such as a determinant for this scalar product. In the following sections, we restrict our attention to a particular situation where the 3-point function can be written in terms of a scalar products of an off-shell state and an on-shell state.

The su(2) cubic vertex in terms of scalar products
In preparation for the computation of the type-{2, 3, 3} su(3) 3-point function that we are interested in, we review an analogous computation of a type-{2, 2, 2} su(2) 3-point function in [10]. Consider It is advantageous to generalize the problem slightly by introducing inhomogeneities associated with the sites of the three spin chains. Thus the i-th chain is characterized by inhomogeneities The three sets of inhomogeneities are not independent, because the inhomogeneities associated with two sub-chains whose fundamental fields are contracted should match. The independent inhomogeneities associated with the contractions between the i-th left subchain and the j-th right sub-chain are denoted by θ (ij) . The cardinality of the set θ (ij) is L ij . In this notation (23) . (3.1) The planarity of the ZZ contractions between the operators O 2 and O 3 and the XX contractions between the operators O 1 and O 3 selects the component of O 3 with N 3 = L 13 successiveX's and L 3 − N 3 successive Z's, as in Fig. 1. Consequently, the correlation function is given by the product of two factors: • The probability to find the component Tr[Z L 23X L 13 ] in the state |w .
• The contribution of the remaining contractions can be recast as the scalar product of an on-shell state of rapidities u and an off-shell state of rapidities v, in a spin chain of length L 1 .
We present below the derivation of the two factors using the language of the six-vertex model.

The su(2) Bethe states as six-vertex-model partition functions
The three type of vertex configurations, a, b, c represented in Fig. 6 have weights The rapidities u and z = θ + i/2 are associated respectively with the horizontal and with the vertical lines. These weights are given by the three types of non-zero elements of the L-matrix, which in our case coincides with the R-matrix, Here I is the identity matrix and P is the permutation matrix.

The su(2) scalar/ inner product in terms of the 6-vertex model
With the normalisation of the basis is obtained simply by gluing two such partition functions, as shown in Fig. 8, and summing over the free indices. In addition to the sesquilinear form (3.6), which is 'the scalar product' , we define a bilinear form, which we call 'the inner product' The vertex representation of the inner product (3.7) is obtained by gluing two lattices as the one shown in Fig  It follows from the hermitian conjugation properties of the creation (B) and the annihilation (C) operators (see the historical note [38]) that for N -magnon states where the set of rapidities v * is obtained from v by complex conjugation. Since the Hamiltonian of the XXX chain is hermitian, the sets of rapidities of the Bethe eigenstates are symmetric under complex conjugation. Therefore the normalisation factor N 1 = | u, u | in (2.1) is equal to the (squared) norm u|u of the Bethe eigenstate.
The structure constant C 123 is equal, up to the normalization factor, to the cubic vertex made of the wave functions of the Bethe states in the representation (3.4), where the form of the cubic vertex depends on the choice of the three su(2) sectors. In our particular case  factorizes out and can be cut out. Furthermore, the right piece is connected to the rest only with type-2 contractions and factorizes out. The lattice splits into two pieces which do not talk to each other and can be evaluated separately.
The cubic vertex u, v, w can be evaluated using the fact that it gives the partition function of the six-vertex model on a lattice obtained by gluing three rectangular lattices with dimensions L 1 × N 1 , L 2 × N 2 and L 3 × N 3 as shown in Fig. 9. The indices 1 and 2 are identified with Z and X or their complex conjugates, depending on the operator under consideration. First we notice that in the part of the lattice that has vertical lines labeled by θ (23) , represented by the shaded area in Fig. 9, there is only one six-vertex configuration, and therefore its contribution to the cubic vertex factorizes out. The factor is a pure phase if the sets v and θ (23) are symmetric under complex conjugation. We will assume that this is the case and will ignore this phase factor. Therefore we can delete this part of the lattice. Next, we observe that the sub-lattice associated with the operator O 3 factorizes because all lines that connect it with the rest of the lattice are of type 2. (This factorisation is obvious in the expression (3.10) for the cubic vertex.) These operations are schematically represented in Fig. 10.
The problem boils down to the calculation of two independent six-vertex partition functions, which give the two non-trivial factors in the structure constant. These two factors will be computed using the freezing procedure. The freezing procedure for the first factor works as follows. One starts from a rectangular lattice corresponding to the scalar product ṽ|u . Both sets of rapidities have cardinality N 1 . The first N 2 rapiditiesṽ coincide with the rapidities v characterizing the operator O 3 , the rest N 3 = N 1 −N 2 = L 1 −L 12 of the rapiditiesṽ will be denoted byṽ N 2 +1 =z L 12 +1 , . . . ,ṽ N 1 =z L 1 , or symbolically,ṽ = v ∪z.

The su(2) freezing procedure
If we adjust the rapidity of the last magnon to the value of the last inhomogeneity,z L 1 = θ L 1 − i/2, then the vertex at the low right corner is necessarily of type c. Then the only possibility for the rest of the vertices on the last row and the last column is that they are type b. This is what we call "freezing". Hence last row and the last column form a hooked index line carrying the index 2, as shown in Fig. 11, left. This procedure is repeated N 3 times, the rapidities of the lowest N 3 rows fixed toz = θ (13) The result is that the rightmost N 3 indices below the lowest horizontal u-line are fixed to the value 2. After removing the frozen part of the lattice, shaded in blue in the figure, we obtain that the first factor in the cubic vertex equals the scalar product v ∪z, u . The contribution of the frozen part of the lattice is the product of all c-vertices on the diagonal, which equals (−1) L 13 , a factor we will ignore.
In a similar way we compute the second factor . The freezing procedure is shown in Fig. 11, right. We start with a scalar product w,z for a chain of length L 3 = N 3 + L 23 . We freeze the rapidities of Figure 11: The freezing procedure for the two factors in C 0 123 the bra state toz = θ (13) − i/2. The frozen area (shaded in blue) gives a contribution, which is a pure phase if both sets w and θ (23) are symmetric under complex conjugation. We will assume that this is the case and will ignore this factor. The rest of the lattice gives the second factor in the expression for the cubic vertex. We find up to a factor which takes into account the contribution of the deleted and added pieces of the lattice. This factor is a pure phase, since the set w is symmetric under complex conjugation, and can be ignored.

The su(3) cubic vertex in terms of scalar products
As before, we consider that the three operators, In the configuration we are considering, w 2 = ∅, since O 3 is an su(2) operator. We refer to Section 6 for the equations obeyed by these rapidities.
Again, we have two types of contributions to the correlation function: • the contribution of the ZZ contractions between the operators O 2 and O 3 , through the factor z 1 , w 1 , withz 1 = θ (13) − i/2 and #z 1 = N 3 , • the remaining contractions, which can be recast as the inner product v ∪z, u between an on-shell vector of a spin chain with length L 1 and rapidities u = {u 1 , u 2 } and an off-shell state with the same length and rapidities Below we evaluate, using the freezing argument, the {2, 3, 3} type structure constant (Fig. 1), We will show that the corresponding cubic vertex is given by Here , denotes, as before, the su(2) inner product, and , su(3) denotes the su(3) inner product.

The su(3) Bethe states in terms of the 15-vertex-model
In order to generalize the freezing procedure to su (3) An example for such a vertex configuration is given in Fig. 13. The first L 1 vertical lines carry rapidities θ The structure constant factorizes as in the su(2) case. The two factors can be cast in the form of scalar products of an on-shell and an off-shell Bethe states by applying the freezing procedure.

The su(3) freezing procedure
Consider the scalar product of two su(3) states of the first chain, ṽ, u = {ṽ 1 ,ṽ 2 }, {u 1 , u 2 } , as represented in figure 14. Our purpose is to freeze the rightmost N 3 indices to the value 3, as imposed by the planarity of the contractions in the three point function. This can be done by setting the last N 3 rapidities of first level magnons at their freezing values, This will insure that the corresponding frozen region contains only red and black lines propagating from the top to the bottom of the diagram. The number of black or red lines is not fixed by the freezing, only their sum is fixed. In order to force all the lines in the frozen region to be black, we have to apply once again the freezing procedure to the magnons of second level, by fixing

The su(2) structure constant in terms of the A -functional
In the su(2) case, the rapidities of the on-shell Bethe states satisfy the Bethe equations The Bethe equations (5.1) follow from the requirement that the eigenvalue T (u) of the T -matrix, has vanishing residues at the Bethe roots. 6 Here Q u and Q θ are the Baxter polynomials where the pseudomomentum p u , known also as the counting function, is defined modulo π by The T -matrix (5.2) is normalized according to the vertex representation with weights (3.2). The eigenvalues of the diagonal elements of the monodromy matrix on the vacuum, A(u) and D(u), are given by .

(5.7)
The inner product of an on-shell Bethe state u and an off-shell state v is given in a determinant form by [40] is the Slavnov determinant. The Slavnov kernel Ω(u, v) is defined by 7 The Slavnov determinant cannot be directly evaluated in the classical limit. This can be done using the representation in terms of the A -functional introduced in [29]. The Slavnov determinant S u,v was expressed in terms of the A -functional first for the limit u → ∞ [29] and then in the general case [32] [31]. Later a more compact expression was found in [37].
The A -functional, whose properties are listed in Appendix C, is defined for any function f (u) and any set of points in the complex plane u = {u 1 , . . . , u N } as follows, The Slavnov determinant is expressed in terms of this functional as [37] Let us express the scalar products in the expression for the cubic vertex in terms of the Afunctional. We find for the two inner products in (4 ]. (5.14) Proof: Using the properties of the A -functional (Appendix C), we transform ].

(5.16)
Ignoring the factors that are pure phases, we find for the cubic vertex ] . (5.17) The second factor has been evaluated in a different ways in [10] and [31], where it was used that it equal to a partial domain wall partition function of the six-vertex model. The norm u, u is most easily computed by taking the expression for the inner product v, u , Eq. (5.12), in the limit v → u.
The Bethe equations (6.2) follow from the requirement that that the su(3) T -matrix in the fundamental representation, has vanishing residues at the Bethe roots. For given distribution of the roots u 1 and u 2 , the pseudomomenta p i u (z) are determined modulo 2π by 9 see e.g. [42]. In terms of the three pseudomomenta, the Bethe equations (5.5) read It is convenient to introduce the functions P 1 u (z) and P 1 u (z), associated with the two nodes of the Dynkin graph of su (3), and related to the quasimomenta p i u (z), i = 1, 2, 3, by In terms of these functions, which we will also call pseudomomenta, the Bethe equations take the more standard form The functions P 1 and P 2 can be expressed in terms of the su(3) Cartan matrix {M ab } = 2 −1 −1 2 as , a = 1, 2. (6.8) Let us stress that the values of the local conserved charges are determined only by the level-1 roots u 1 . The duality transformations change the level-2 roots u 2 , but leave invariant the level-1 roots u 1 , which carry the physical information [42].
The norm of an on-shell Bethe state The squared norm of an on-shell Bethe state has been computed for the case of su(3) by Reshetikhin 10 [39] and is expressed as the determinant of the matrix of the derivatives of the two quasi-momenta: 11 where the determinant is with respect to the double indices A = {a, j} and B = {b, k}. The normalizationn factor c u is given by (2.2). The matrix of the derivatives of the two quasimomenta is explicitly , (6.10) 9 Here we used the conventions of Eqs. (5.3) and (5.4). 10 A conjecture for su(n) is proposed by EGSV in [9]. 11 Here it is assumed that the set of the Bethe roots is symmetric under complex conjugation. where Instead of taking the derivatives, we will compute the norm as the limit of the determinant depending on two sets of rapidities, u and v, which has the limit (6.10) when v → u. We define the The expression for the norm, which we are going to evaluate in the classical limit, is The inner product u, v in the limit u 2 → ∞ Unlike the su(2) case, the inner product of an on-shell Bethe state with an of-shell Bethe state is not generically a determinant. Determinant representations exist in some particular cases [36,43,44]. We will use the determinant expression obtained by Wheeler [36], when the rapidities of the second type of magnons of the Bethe eigenstate are sent to infinity. We assume that M is odd; then one can send to infinity the u 2 roots one by one. As a result the second level Bethe equations become trivial and the first level Bethe equations take the same form as for su (2). The inner product u, v su (3) θ factorizes into two su(2) inner products [36] lim (6.14) Using (C.13), we write (6.14) in the form The su(3) structure constant in terms of the A -functional. Combining (5.13), (5.14), (6.14) in Eq. (4.2), we get ] . (6.16) 7 The semi-classical limit of the su(3) 3-point function 7

.1 The Sutherland limit
The classical , or thermodynamical limit is attained for long spin chains (L 1) with macroscopically many excitations N, M ∼ L, and in the low energy regime (δE ∼ 1/L) [11,12,45]. Such spin chains correspond to "heavy" operators, which are traces of products of many SYM fields. In this limit the roots scale as u a,j ∼ L. In the condensed matter literature the classical limit has been studied by Sutherland [46] and by Dhar and Shastry [47], and is known as Sutherland scaling limit. In the classical limit the roots are organized in several macroscopic strings, which condense into cuts in the complex rapidity plane. The three quasimomenta p 1 , p 2 , p 3 become the three branches of the same meromorphic function. The three sheets of the corresponding Riemann surface are joined among themselves along the cuts defined by the long Bethe strings. In the classical limit, the Bethe state is characterised by the resolvents as well as the resolvent for the inhomogeneities The two resolvents, G u 1 and G u 2 , can be expressed in terms of the three quasimomenta p 1 u , p 2 u and p 3 u , which become the three branches of a single meromorphic function on the tri-foliated Riemann surface, Let C α ij be the cuts joining the i-th and the j-th sheets. Then the Bethe equations (6.5) become boundary conditions on these cuts, depending on the mode numbers n α ij : where / p denotes the half-sum of the values of the function p on both sides of the cut.

Stacks
In addition, there is the possibility of configurations called stacks (bound states of rapidities associated with different nodes [48]), which represent pairs of roots belonging to the nodes 1 and 2 and at distance O(1) from each other [42,45,49]. We can have macroscopic strings of stacks, which in the classical limit become two cuts that merge into one cut. Since the roots that form the string of stacks belong to two different nodes, they correspond to a cut type 1-2 and a cut type 2-3, where we understand that the cut of type i-j joins the i-th and the j-th sheets of the Riemann surface. The result of merging of the two cuts is a cut of the type 1-3. Therefore, in order to have a description of the generic Bethe state in the classical limit, we must assume also the existence of cuts of type 1-3. The boundary condition on these cuts is obtained by taking the limit of (7.5) and has the form The bosonic duality transformations [42] in the classical limit corresponds simply to the exchange of the Riemann sheets 2 and 3.

The semi-classical norm
The determinant (6.13) can be computed in the classical limit under the assumption that there are only 1-2 and 2-3 type cuts, which are separated at macroscopic distance ∼ L. With this assumption, the off diagonal elements of Ω(u, v) ∼ (u − v) −2 ∼ 1/L 2 , and the only matrix elements of order one are those in a strip of width ∼ 1/ √ L along the diagonal. As a consequence, the non-diagonal blocks do not contribute in the classical limit and the determinant is simply the product of the determinants of the diagonal blocks, Let us evaluate the norm assuming that there there are no cuts relating the first and the third sheet of the Riemann surface. We can use the expression for the classical limit of the norm in the su(2) sector (Appendix (C)): The norm of the classical Bethe state is then 12 Li 2 e 2iGu 2 (z)−iGu 1 (z) . (7.11) 12 We conjecture that in the most general case, when some of the roots can form bound states ("stacks"), this logarithm of the norm is given by Li2 where Cij (i, j = 1, 2, 3) denote the contour (or contours) surrounding the cuts between the i-th and the j-th sheets.
7.4 Semi-classical limit of the structure constant Substituting (7.8) in (6.16) we find for the structure constant in the classical limit The last line is evaluated in the classical limit according to (7.11).

Conclusions and outlook
We have analyzed the tree-level 3-point functions of single-trace operators of the planar N = 4 SYM theory in the su (3) sector. Each of the three operators is an eigenstate of the dilatation operator, and it is characterized by a set of charges (angular momenta) and a set of rapidities. We have classified the possible configurations of Wick contractions and given the general expression of the 3-point functions in terms of rapidities associated to each operator. This expression, obtained using the tailoring technique of EGSV [9], is not adapted for taking the classical limit. In some particular situation, when one of the operators belongs to an su(2) sector, we are able to express the 3-point function using the alternative method of freezing proposed in [10]. By further specializing the second group of rapidities corresponding to one of the operators, we can use a result of [36] to express the scalar products as determinants. Finally, the semi-classical limit of the determinants can be taken using the results from [31,32]. The simplest classical operator from the su(3) sector are the three-spin solution obtained by C. Kristjansen [50]. There are two obvious directions to explore. First, one can try to evaluate the general sum over partitions in (2.5) quasi-classically. One can either try to perform a quasiclssical evaluation of the Korepin sum over partitions for the scalar product of two off-shell Bethe states, in the spirit of [29], or refine the coherent state approximation [25,51,52]. Another direction is to explore the non-compact sector of the theory. There are several recent papers which are relevant for that, [53][54][55][56][57][58][59][59][60][61][62]. At each site of the spin chain, there is a spin with K different polarizations. The Hilbert space of the spin chain is H = (C K ) ⊗L . In the Hamiltonian (A.1) I n,n+1 is the identity operator in the space C K n ⊗ C K n+1 and P n,n+1 is the permutation operator in C K n ⊗ C K n+1 . The Dynkin diagrams of the Lie algebra give a convenient way to label the excitations. For the su(K) algebra, the Dynkin diagram is given by Fig.(15).
where u a is the set of rapidities of node a and n a labels the positions of the excitation of node a (a = 1, · · · , K − 1). The summation is taken over all possible positions of excitations. The nested ket state |n 1 , · · · , n K−1 is constructed in the following steps: 1. Start with an initial value of length L; 4. Repeat the above steps for all the K − 1 nodes.
The procedure for the su(3) case is explained in Fig.(16). We would like to stress the important feature that the excitations of node a should only be created from the excitations of node a − 1 (the reduced spin chain). Therefore we have N a ≤ N a−1 . The positions should obey 1 ≤ n 1,1 < · · · < n 1,N 1 ≤ L, 1 ≤ n a,1 < · · · < n a,Na ≤ N a−1 , for 2 ≤ a ≤ r .
The beauty of this method is that at each step, the operation is simple and the same: to create one kind of excitation in a reduced spin chain of length N a . Here we see the nested feature of this method.
The wave function ψ(u 1 , · · · , u K−1 ) is constructed by nested Bethe ansatz. It is given in terms of a series of wave functions ψ a , a = 2, · · · , K − 1.The Bethe wave function reads where the wave functions ψ a , a = 2, · · · , K − 1 are given by with P a the permutation (P a,1 , . . . , P a,Na ) of (1, · · · , N a ). We define ψ K (P K−1 ) = 1. M a,b is the Cartan matrix of the Lie algebra, for su(K) being Our choice of normalization is A a (1, 2 · · · , N a ) = 1 (A. 6) and the coefficients A a obey the relation A a (· · · , i, j, · · · ) A a (· · · , j, i, · · · ) = S 2 (u a,i , u a,j ) . (A.7) Here we used the following definition, In order that (A.2) is an eigenstate of the spin chain Hamiltonian, the rapidities should satisfy the Bethe ansatz equations : where V a are the Dynkin labels. We consider the fundamental representation in this paper, where V a = δ a,1 .

B The su(3) tailoring prescription
We consider the operators in su(3) sector with definite one-loop anomalous dimensions. In the spin chain language, these operators are represented by the Bethe eigenstates of the su(3) spin chain.
One first write the Bethe state as the entangled state of two subchain states. This operation is called "cutting". After cutting operation, each subchain state also takes the form of Bethe states. In order to perform Wick contraction, one needs to "flip" one of the subchains. Flipping is an operation that takes a ket state into the corresponding bra state with the same wave function. One can flip either the left or the right subchain. In this paper, we always flip the right subchain. The last step is to calculate the scalar product of Bethe states, this is called the "gluing" operation. Consider a generic su(2) Bethe state |u of a spin chain with length L. We define the first l sites from the left to be the left subchain and the rest L − l sites to be the right subchain. |u can be written as an entangled state of the subchains |u = min{N,l} k=0 1≤n 1 <···<n k ≤l l≤n k+1 <···≤L ψ(u)|n 1 , · · · , n k l ⊗ |n k+1 − l, · · · , n N − l r (B.1) where k is the number of magnons in the left subchain. Note that one needs to re-label the positions of the magnons in the right subchain, see either be in the left subchain or in the right subchain. After cutting a Bethe eigenstate, the resulting two subchains states still take the form of Bethe state. Hence the two subchain states have their own Bethe wave functions ψ l (u ) and ψ r (u ), where u and u is a partition of u, In general, ψ(u) = H(u , u ) ψ l (u ) ψ r (u ) where H(u , u ) is a partition-dependent factor which shall be called H-factor from now on. Formally, the cutting of a Bethe state can be written as Usually the expression for the H-factor is long. In order to make the expression more compact, we introduce some short-hand notations. Given a function F (x, y) and two sets of variables u, v, we define For a constant c, we define With the notations from (A.8) the H-factor for the su(2) spin chain is given by The cutting operation can be generalized to su(K) Bethe state. Let us denote the nested Bethe state by |u where u = (u 1 , · · · , u K−1 ). We have with the H-factor where u 0 is defined as u 0 = {0 l } with l the length of left subchain. In order to perform Wick contraction, we need to "flip" the right subchain from a ket state into a bra state. The flipping operation is different from Hermitian conjugate. Given a state |ψ = e iθ |XZXZZ (B.9) the Hermitian conjugate and flipping (denoted by superscript F) lead to (|ψ ) † = e −iθ XZXZZ| (|ψ ) F = e +iθ ZZXZX | .
For an su(2) Bethe state |u , the flipped state is proportional to the hermitian conjugate of the Bethe state |u * (|u ) F = u * |F (u) (B.10) where u * is the complex conjugate of u and we call the proportionality F (u) the F -factor. For an su(2) Bethe state, the F -factor reads F (u) = S 1 (u, 0) L+1 S > 2 (u, u) (B.11) where L is the length of the spin chain. For an su(K) Bethe state | u , the F -factor is given by S 1 (u n , u n−1 )S > 2 (u n , u n ) (B.12) where u 0 is defined by u 0 = {0 L+1 }. From now on, by "tailor" (denoted by T ) a (nested) Bethe state, we mean first cut the state and then flip the right subchain state. We define the product of the corresponding H-factor and the F -factor to be the H F factor. For an su(K) Bethe state |u 1 , · · · , u K−1 where u ∪ u = u, u ∩ u = ∅ (B.14) and the H F -factor reads H u F = K−1 n=1 S 1 (u n , u n−1 )S > 2 (u n , u n ) . (B.15) C The functionals A ±

C.1 Definition
For any set of points u = {u j } N j=1 in the complex plane and for any complex function f (z), we define the pair of functionals A ± u [f ], which are completely symmetric polynomials of degree N of the variables f (u 1 ), . . . , f (u N ). The functional A ± u is defined as a sum of monomials labeled by all possible partitions of the set u into two disjoint subsets u and u , with u ∪ u = u, The expansion (C.1) was thoroughly studied by Gromov, Sever and Vieira [29]. The expansion (C.1) is summed up by the following operator expression, where the different operatorÂ ± [u] is defined aŝ The operator functionalÂ ± [u] is formally obtained from the c-functional A ± [u] aŝ (C.4)

C.2 Properties
It was found in [29] that for constant function f (u) = κ, the expansion (C.1) does not depend on the positions of the rapidities u and the functional A ± u [f ] is given in this case by The functional A ± [f ] have other remarkable properties [31,32]: 1) Determinant formula, Eq. (5.11), which is obtained by substituting ∆ u = det jk (u k−1 j ) in the definition (C.3).
2) Functional relations between A + and A − : 3) Reduction formula: Aplying N times the factorisation property, one obtains the representation Relation to the Slavnov determinant. The determinant of the kernel Ω(u, v) defined as where the set u satisfies the "on-shell" condition is evaluated as [37] det jk Ω f (u j , v k ) det jk (C.13)

C.3 Classical limit.
In the classical limit, the Bethe roots condensate in one or several disjoint cuts. Let C k be a contour encircling the k-th cut anticloskwise and leaving outside all other singularities of f and G u . The filling fraction of the k-th cut is (C.14) We consider the limit L → ∞ with all α k finite. Then the leading, linear in L, term of log A ± is given by the contour integral While there is not yet a rigorous proof of this formula, it has passed a number of analytical and numerical checks. A heuristic derivation of (C.15) for f (u) = u−i/2 u+i/2 L , was presented in [29]. When f (u) = κ, it was shown in [29] that the quasiclassical formula (C. 15) gives the exact answer (C.5). Moreover, the integral (C.15) satisfies the functional equation (C.6) thanks to the functional equation for the dilogarithm, Li 2 ( 1 ω ) = −Li 2 (ω) − π 2 6 − 1 2 log 2 (−ω). (C. 16)