A new representation for two- and three-point correlators of operators from sl(2) sector

We construct a new representation for two- and three-point correlators of operators from sl(2) sector of planar N = 4 SYM. The spin and twist of operators are arbitrary. We start with the correlation function of light-ray operators and carry out a projection to particular local operators using the method of Separated Variables. With the same calculation we obtain polynomials which are dual to wave functions of sl(2,R) spin-chain.


Introduction
The operator product expansion in N = 4 SYM theory, as in any CFT , is completely characterized by its 2-point and 3-point correlators, or, in other words, by the spectrum ∆ j (λ) of anomalous dimensions of local operators O j and by the structure constants 1 C ijk (λ). Both the dimensions and the structure constants are in general complicated functions of coupling λ = g 2 N c and of quantum numbers of the operators. For the N =4 SYM spectral problem, there has been a lot of progress in the last years [1] allowing to study it numerically, at any coupling. Recently, these developments have culminated in the formulation of a well defined system of Riemann-Hilbert equations [2]. However, for the correlation functions the situation is far more complicated, and one is here in the early stage of a case-by-case study in a weak or strong coupling regime.
A significant progress was achieved for su (2) sector in the weak-coupling regime. The method of "tailoring" of Bethe states which was proposed in [3] has been greatly evolved [5][6][7][8][9][10][11][12]. It was applied to su(3) sector [13] as well as to higher loops in su(2) case [7,11]. On the other hand, the case of noncomtact sl(2) sector has been much less investigated. Some interesting results concerning sl(2) sector one can find in [14][15][16][17][18][19]. One can find in [14] an interesting all-loop prediction for the case of two protected operators and one twist-2 operator with large spin. One loop prediction for two BPS and one sl(2) operator of arbitrary spin and twist is presented in [12].
In this note we propose a new approach to the calculation of correlation functions of sl (2) operators O s,L = trD s + Z L with arbitrary spin s and twist L in the leading order in the coupling. We are starting with the calculation of correlation function of nonlocal light-ray operators which serve as generating functions for local operators O s,L . In order to make projection on particular local operators we use Sklyanin's method of Separated Variables (SoV) [20][21][22][23][24][25]. To generate wave-functions in SoV representation, we act a few times on the correlator of light-ray operators by Q-operator which was constructed in [26]. Further, we are using the scalar product [23] on the space of wave-functions in SoV representation in order to make projection on particular states. As a check for our method, we compare our formula in the case of twist-2 operators with a direct calculation involving explicit expression for the wave-function through Gegenbauer polynomials.
2 Light-ray operator as a generating function In this section we collect some facts about light-ray operators. For more details, see review [27].
Let us introduce the light-ray operator O(z − ): where n + is a light-ray vector 2 n 2 gauge link between x and y (see in Fig.1). We will further omit these gauge links, because we take into consideration only Born level approximation.
where D + = n µ + D µ is covariant derivative in n + direction. This decomposition is an expansion over local operators. There is a distinguished basis of local operators. They diagonalize dilatation operator which has the form of sl(2, R) spin chain Hamiltonian in the one-loop approximation. All primary operators can be constructed using the Bethe 2 We use the basis {n+, n−, e 1⊥ , e 2⊥ }, where n 2 + = n 2 − = 0, (n+n− = 1) and any vector can be decomposed in the following way x = x−n+ + x+n− + x ⊥ . ansatz technique. The full basis contains primaries as well as their descendants. Formally, the decomposition of O(z − ) reads as follows: Explicit form of polynomials Ψ l,s,{α} (p − ) can be found by applying the Bethe ansatz technic. All descendants have the following form: S + i are defined as a sum of sl(2, R) generators in the spin-1 2 representation: The coefficient r l,s,{α} is a normalizing constant defined by the following condition: As it was established in the paper [28], polynomials Ψ s,{α} (p − ) and Φ s,{α} (z − ) are related by the dual symmetry 3 : where {u k } is the set of Bethe roots, and Q s,

Q-operator
One can obtain the explicit form of the conformal operators O s,{α} and their anomalous dimensions γ s,{α} by diagonalizing the dilatation operator in the sl(2) sector of N =4 SYM.
Having used (2.3) and (2.4), this spectral problem can be reformulated as an eigenproblem for the Hamiltonian H which acts [29] on the space of polynomials φ(z − ): The Baxter approach to this eigenproblem is based on the existence of operatorQ(u) which depends on complex variable u, acts on the space of polynomials, and satisfies a set of conditions: . The operatorQ(u), satisfying all conditions mentioned above, was constructed in [26](see also [28]) and it reads as follows: 14) The operatorQ(u) is SL(2, R) invariant, and thus, we get the following action ofQ(u) on descendants:Q

Two-point correlation function
Let us consider the correlator of two light-ray operators O(x 0 , x − ),Ō(y 0 , y − ) stretched along n + direction. In the tree-level approximation both of them should have the same number L of fields Z andZ. Extra labels x 0 and y 0 indicate the starting points for these operators: We fix the propagator for Z-field in the planar limit in the following way: The correlator in the tree-level approximation simply reads as follows: where σ is a cyclic permutation of (1, ..., L), and the sum goes over L different cyclic permutations.
On the other hand, one can expand O(x 0 , x − ),Ō(y 0 , y − ) over local operators using (2.3), and rewrite the correlator of two nonlocal operators as a sum of 2-point correalators of local operators: Now let us act on x − -coordinates byQ x − (u) 4 on both sides of (3.4): to both sides of (3.4), we get: Now let us introduce Ω s,{α} (u) = Ω s,{α} (u 1 , ..., u L−1 ) -the wave function in the Sklyanin's Separated Variables(SoV): The SoV representation for the sl(2, R) spin chain was constructed in [23]. The authors have explicitly established unitary transformation to Separated Variables along with the Sklyanin's measure defining the scalar product in the SoV representation. They have also proved equivalence of SoV and ABA methods. The orthogonality condition for the wave functions in the SoV representation reads as follows: where |(s, {α}) u = Ω L,(s,{α}) (u), N s,{α} is a coefficient, and the labelμ means that the scalar product is defined by the measureμ(u), which has the following form: Now let us obtain 2-point correlator of particular operators from the correlator of two light-ray operators (3.4).
We are interested in the particular primary operator O s,{α} with spin s. Thus, we can expand ω(x 0 , x − , y 0 , y − ) in the series and collect the terms, such as P s (x − )Q s (y − ) , where P s (x − ) and Q s (y − ) are homogenous polynomials of order s. It can be easily done by one extra integration. Namely, one can replace , the projection of the function ω L (x 0 , x − , y 0 , y − ) on the states with spin s: (3.10) This projection corresponds to the contribution of all operators with spin s. In the general case of arbitrary twist L we have several primary operators of the spin s. Moreover, the descendants Ψ k,s−k,{α} also have spin s and contribute to (3.10). To separate one particular primary operator with quantum numbers (s, {α}) we use orthogonality of the wave-functions in the SoV representation. As a first step, we generate wave functions and then we use orthogonality: 3.1 Discussion of (3.12) At first, we should stress that the representation (3.12) gives us in one calculation both 2-point correlator and the polynomial Φ s,{α} which is dual to the wave function Ψ s,{α} .
The second comment concerns normalization. One can multiply functions Φ s,{α} by any constant c and, at the same time, multiply two-point correlator by 1 c 2 . Thus, the left-hand side of (3.12) will not be changed. This freedom in the normalization is not surprising, because we have fixed only the action of Ψ s, In order to obtain the two-point correlator of particular operators, it is sufficient to act in (3.12) just by one operatorQ x − , and take the scalar product with (s, {α}) u | = Ω s,{α} (u). Indeed, all other terms disappear due to orthogonality of local operators. Nevertheless, we choose the form as in (3.12) because it is symmetric and well adopted for the normalization of three-point correlation functions. Now let us notice, that the product of Γ-functions in the measure (3.9) is exactly canceled by Γ-functions which comes from operatorQ x − (u). For this reason we introduce a new operator Q Nevertheless, this action can be explicitly formulated: wherex l− is a linear combination of all coordinates x k− . Operation "hat" inx l− has an elegant graphical representation.   The formula (3.12) can be rewritten in the following way: where operators Q x − (u) and Q y − (v) act as in the (3.13), and ...|... µ is defined as follows:  As we noticed before, only one term from the operator O gives nonzero contrubution to the correlator in the lowest order in the coupling. This term has the form c trZ L 2 −lZL 1 −l , where c is a coefficient. In this paper we are concentrated on the sl (2) operators. Due to this reason, we introduce extra normalisation 1 N which cancels this coefficient, irrelevant to our discussion. We will specify N a bit later.
The 3-point correlator can be calculated in the same way as in the 2-point case. Its expression reads as follows 6 : where and the function ω s 1 ,s 2 ,0 L 1 ,L 2 ,M (x 0 , x − , y 0 , y − ; z 0 ) has the following form: where σ x and σ y are cyclic permutations of {1, ..., L 1 } and {1, ..., L 2 } correspondingly. Labels M and 0 mean the twist M = L 1 +L 2 −2l and spin of operator O correspondingly. The three-point correlator of operators with spin is a sum of different tensor structures [30,31]. It means that the correlator is characterised by the set of structure constants. As was demonstrated in the [15](see also Appendix A),by choosing a special kinematics, one can collapse all those tensor structures into one. To achieve this goal we restrict positions of all local operators to the two-dimension subspace = {n + , n − } spanned by two light-ray vectors n + , n − . In this case, 3-point correlator has the following form: where symbol " " means that we restrict positions of all operators to two-dimension space .For two-point correlators we get: The normalized 3-point structure constant is defined as: As it was mentioned above, one can introduce a normalization factor 1 N , and cancel the contribution of operator O . Namely, we specify this factor as a normalized structure constant N = C (L 1 ,(0,{∅})),(L 2 ,(0,{∅})), of three operators trZ L 1 ,trZ L 2 and O . It can be easily calculated: The ratio of two normalized 3-point correlators can be expressed in the following way: (4.14) The last multiplier of all expressions (4.12)-(4.14) was introduced to cancel coordinate dependence.

Case of three sl(2) operator
In order to construct three-point correlator of three sl(2) operators, let us introduce three 6-dimension vectors p 1 , p 2 , p 3 with zero norm |p i | 2 = 0 and nonzero pairwise scalar products (p m , p k ) = 0. Then we can introduce three sl(2) sectors 7 which consists of operators with the form O Lm,(sm,αm) = trD sm p J m φ J and {φ J } are 6 scalar fields in N =4 SYM. The propagator between two fields U k (x) and U m (y) has the following form: Thus, we construct nonzero three-point correlator: We can obtain the representation similar to (4.1) where 18) and the function ω s 1 ,s 2 ,s 3 L 1 ,L 2 ,L 3 (x − , y − , z − ) has the following form: where . σ x , σ y and σ z are cyclic permutations of {1, ..., L 1 }, {1, ..., L 2 } and {1, ..., L 3 } correspondingly. For discussion on how this method works for numerical calculations see Appendix B.

Case of twist-2 operators. Comparing with direct calculation
In the special case of twist-2 operators one can calculate left-hand side of (4.11), using explicit form of operators O 2,(s 1 ,{α 1 }) (x),O 2,(s 2 ,{α 2 }) (y) through the Gegenbauer polinomials: The direct calculation gives us: On the other hand the measure in this case is µ(u) = 1, and Q-function is Hahn polynomial Q s (u) ∼ 3 F 2 (−s, s + 1, 1 2 − iu; 1, 1; 1). Using these explicit formulas and Appendix B, we have checked that the right hand side of (4.11) exactly coincides with (4.23).

Conclusions
In this paper we have proposed a new approach to the leading order calculation of two-(3.14) and three-point (4.1), (4.11),(4.17) correlation functions of sl(2) operators. It is important to stress that our construction gives us in one calculation both 2-,3-point correlator and polynomials Φ s,{α} which are dual to wave functions Ψ s,{α} . As the initial data we use only Baxter Q-function. This approach is based on the decomposition (2.3) and Sklyanin's method of separated variables. SoV representation is one of the most general methods in Integrability. We suppose that this approach can be efficiently applied to the study of large spin s case [32], because number of integrals doesn't depend on s. It would be interesting to generalize our construction to su(2) case, and clarify a connection with the method proposed in [3].
The wave function in SoV representation is constructed from Q-functions. On the other hand, recently proposed P − µ system [2] gives us information on a variety of Q-functions at any coupling. It would be very tempting to construct SoV representation for N =4 SYM [33], and then, use it for generalization of our method. Another important point to stress in this context concerns nonlocal light-ray operators (2.1). They are our starting objects, and they preserve their form at any coupling constant.
Our construction can also be useful for the calculation of correlators of generalized operators, such as O ω = trZD −1+ω + Z with ω → 0. Operators O ω play an important role in the BFKL physics, and they were recently understood [34] as nonlocal light-ray operators realizing principal series representation of sl(2, R). Corresponding noncompact spin-chain can not be solved by ABA technic due to the absence of the extremal-weight vector. However, it was solved by authors of [35], who have applied both methods of Baxter Q-operator and Separation of Variables. This appendix is a reminder of the formulas obtained in the paper [31], with some precisions for our particular cases. According to its methods, a formula for correlation function of any three primary operators with dimensions ∆ i and spins l i was obtained, using the embedding formalism. Below we give their expression in original notations and apply it to the particular case, when all operators are restricted to two-dimensional plane = {n + , n − }. Embedding formalism implies the embedding of physical space ) is realized linearly. The vector x from V lifts up to M by the formula x ↔ P x = (1, x 2 , x) , which sets the one-to-one correspondence of vectors from V and light-rays in M. Scalar product of two vectors P 1 = (P 1+ , P 1− , p 1 ) and P 2 from M sets as (P 1 · P 2 ) = − P 1+ P 2− +P 1− P 2+ 2 + p 1 p 2 , where p 1 p 2 means the scalar product in V. In the paper [31], three vectors of polarization Z i ↔ z i were introduced which contract tensor indices of each operator: φ(x, z) = φ a 1 ,...,a l z a 1 ...z a l . In our case this corresponds to the projection of all indexes on n + direction. Thus in our case all indices have the same polarization z 1 = z 2 = z 3 = n + . The formula for three-point correlation function reads in these notations as follows: Φ(P 1 , Z n + )Φ(P 2 , Z n + )Φ(P 3 , Z n + ) = Step 1. As a first step, we should calculate functions ω s L (x 0 , x − , y 0 , y − ), ω s 1 ,s 2 ,0 L 1 ,L 2 ,M (x 0 , x − , y 0 , y − ; z 0 ), ω s 1 ,s 2 ,s 3 L 1 ,L 2 ,L 3 (x 0 , x − ; y 0 , y − ; z 0 , z − ). This calculation corresponds to the calculation of residue of an rational function. It can be easily carried out by Mathematica.
Step 2. The next step is to act by operators Q x − (u) which are defined in (3.13). After proceeding through Step 1 we get polynomial expression w.r.t. x − . It is easy to see that all integrals which can appear, have very simple form: where k and m are integer numbers. One can rewrite Beta-function in the following way: .
Step 3. Finally, we should carry out an integration with measure µ (3.16). It is easy to see, that all integrals have the following form: