Correlators with $s\ell_2$ Yangian symmetry

Correlators based on $s\ell_2$ Yangian symmetry and its quantum deformation are studied. Symmetric integral operators can be defined with such correlators as kernels. Yang-Baxter operators can be represented in this way. Particular Yangian symmetric correlators are related to the kernels of QCD parton evolution. The solution of the eigenvalue problem of Yangian symmetric operators is described.


Introduction
The symmetry on which quantum integrable systems (QIS) like spin chains are based can be formulated in terms of a Yangian algebra [1], which results from the expansion of the monodromy matrix. In the applications of QIS to gauge field theories, which attracted much attention in the last two decades, the symmetry appears typically in infinite-dimensional representations with action on functions. The first of such applications concerned the high-energy asymptotics of scattering in Quantum Chromodynamics and are based on sℓ 2 [2,3]. In the Regge asymptotics [4] the dynamics reduces to the plane transverse to the scattering axis and in the Bjorken asymptotics [5,6] the dynamics reduces to a light ray. The latter asymptotics is connected with the scale dependence of composite operators. In the case of N = 4 supersymmetric Yang-Mills theory the relation to QIS has been shown [7] to work with the super sℓ(4|4) spin chain for all composite operators in all orders of perturbative expansion and to be connected to strings via the AdS/CFT correspondence, where in the supergravity on the string side classical integrable systems appear. The advances in the methods of scattering amplitude computations have lead in particular for the N = 4 supersymmetric Yang-Mills theory to the discovery of the dual superconformal symmetry [8] and the understanding that the two superconformal symmetries are embedded in the corresponding Yangian algebra [9].
Yangian symmetric correlators (YSC) have been proposed as a convenient formulation of the Yangian symmetry of scattering amplitudes [10]. They are defined by an eigenvalue relation involving the monodromy matrix operator. A number of properties, previously observed in amplitude calculations, have been shown to follow from this monodromy condition. There are convenient methods of construction of YSC, the most important one is based on Yang-Baxter RLL type relations. The construction methods are related to the on-shell graph method developed in the amplitude context. This relation has been discussed in detail in [11].
The tool of YSC is not restricted in its application to super Yang-Mills scattering amplitudes. It may be used wherever Yangian symmetry appears, it is powerful and it will help to exploit the higher symmetry in more problems. The investigation of the particular case of sℓ 2 is important for testing the approach, first of all because this case is relevant in a number of integrable models, in two-dimensional conformal field theories [12] and in particular questions of gauge field theory. The advantage of the method should be tested here in comparison with the standard methods, reproducing known results in alternative ways and going beyond them by relying on the new viewpoint provided. The second reason for devoting attention to the sℓ 2 case is the fact that here Yang-Baxter relations are known in several versions and in most explicit form.
In the present paper we expand the range for Yangian symmetric correlators. We shall show that the approach of YSC is powerful also beyond the investigation of super Yang-Mills scattering amplitudes, where it has been introduced. We shall demonstrate its flexibility by indicating several ways of calculation, several explicit formulations, the opportunity to perform algebraic deformations.
In examples for the sℓ 2 case we shall demonstrate new construction methods beyond the ones known in the amplitude context and the role of the spectral parameters characterizing the representation of the Yangian algebra.
The approach allows to treat the Yangian symmetry in a convenient way. We shall discuss the role of YSC as kernels of symmetric operators. The symmetry of the correlators induces the symmetry of the integral operators.
The approach has a high potential of physical applications. We shall show how the Yangian symmetry of the correlators leads to the exact solution of the spectral problem for the related operators, which is of great importance in view of physics. Among many applications related to the sℓ 2 case we pick up the example of the scale dependence of parton distributions and composite operators. We shall establish a direct relation of the kernels of the leading order scale evolution to YSC.
In Sect. 2 we recall the definition of YSC and formulations of Yang-Baxter relations. We use the explicit expressions of Yang-Baxter (YB) operators together with methods developed for amplitudes [13] to obtain explicit expressions of YSC. We emphasize the specifics of the case of sℓ 2 , which allows several versions in the construction in particular those differing essentially from the ones close to the amplitude methods. A number of explicit examples will be given for illustration in Sect. 3.
We keep arbitrary spectral parameters in the monodromy matrix from which the YSC construction starts, explain their relation to the dilatation weight associated with each point and also how they characterize the representation of the Yangian algebra. Note that in N = 4 super Yang-Mills amplitudes the case of vanishing spectral parameters matters and therefore the case of arbitrary values has been considered in a few papers only, e.g. [14,11,15,16]. We shall see that spectral parameters play an active role in the method.
The symmetry condition defining YSC generalizes to algebraic deformations. We demonstrate this in the trigonometric case in Sect. 4.
In Sect. 5 we study Yangian symmetric operators defined as integral operators with YSC as kernels. We show how particular 4-point YSC result in the Yang-Baxter operators. Generalized YB operators result from particular 2M -point YSC.
It is known how YB relations allow to derive the spectrum of the involved YB R operator. In Sect. 6 we show that this works also for generalized YB operators by demonstrating it in the simplest non-trivial case with M = 3.
The application of QIS and Yangian symmetry to the scale dependence of composite operators and of (generalized) parton distributions [17] is well known in the formulation of quantum spin chain dynamics. In Sect. 5 we present the formulation in terms of YSC. We identify the kernels of the scale evolution of (generalized) parton distributions as 4-point YSC with particular values of the dilatation weights and the spectral parameters. In the presented form of ratio-coordinates interpreted as positions on the light ray these kernels describe the scale dependence of operators composed of two fields with derivatives in the two-point light ray form.
The generalized Yang-Baxter operators constructed from 2M -point YSC are relevant for the scale evolution of operators composed out of M fields. Their spectra, the calculation of which we show in the particular case M = 3, allows to derive exactly the related anomalous dimensions. In the situation of a low number of fields this approach becomes an alternative to the Bethe ansatz calculations used in the spin chain formulation. (2.1) Here R(u − v) stands for Yang's R matrix, which is 4 × 4 in our case and composed of the unit matrix and the matrix representing the permutation of the tensor factors in the tensor product of the two-dimensional fundamental representation spaces, R(u) = uI 4×4 +P . The L matrices are linear in the spectral parameter u, L(u) = uI 2×2 + L. With this ansatz (2.1) implies that the matrix elements of L generate the sℓ 2 algebra. We shall introduce several forms of L below. The Yangian algebra can be introduced by the matrix operator T (u) obeying the above YB relation with L(u) substituted by T (u) and where now the dependence on u is not restricted as above. Then the generators of the Yangian algebra [1] appear in the expansion of T (u) in inverse powers of u and the Yangian algebra relations follow from the fundamental YB relation (2.1). We shall deal with evaluations of the sℓ 2 Yangian of finite order N where T (u) is constructed from the L matrices in the way well know in the context of integrable models. Let the matrix elements of L i (u i ) act on the representation space V i and consider the matrix product defining the monodromy matrix acting on the tensor product V 1 ⊗ · · · ⊗ V N . In the case if the representations V i are of definite weight 2ℓ i the weight parameters are added to the parameter list abbreviated by u. We consider infinite-dimensional representations V i in terms of functions of two or one variables. In the case of V i of definite weight the functions of two variables x i = (x i,1 , x i,2 ) are homogeneous and the weight 2ℓ i is the degree of homogeneity, The one-dimensional variables are related to the two-dimensional as ratios of the latter components, where we have two versions in the following discussion, x i = x i,1 x i,2 and y i = − x i,2 x i,1 . The elements of the tensor product V 1 ⊗ · · · ⊗ V N are then functions of N points, two-or one-dimensional.
N -point functions Φ obeying the condition are called Yangian symmetric correlators (YSC). On l.h.s an operator-valued matrix is acting and on r.h.s. there is the unit matrix I and the eigenvalue depending on the set of parameters u. As solutions we shall accept also expressions involving distributions, extending the representation space V 1 ⊗ · · · ⊗ V N . The δ-distributions are to be understood in Dolbeault sense (cf. [18]), their arguments must not be restricted to real values.
The eigenvalue E(u) is related to the quantum determinant of the monodromy T (u). In the sℓ n case the definition is where the sum is performed over the symmetric group S n of the order n. On r.h.s. T αβ (u− k) are the matrix elements of T (u, δ 1 , ..., δ N ). The quantum determinant qdet(T (u)) is the generating function for the center of Yangian Y (gℓ n ). Because the off-diagonal elements of T (u) annihilate the function Φ it follows immediately from (2.2) that in our case of n = 2

The homogeneous form
We start with two forms of the L operators, L ± i (u) = Iu + L ± i (0), with matrix elements Notice that the elementary canonical transformation Here the transformation acts uniformly for the index values α = 1, 2. With a transformation modified in this respect, i.e. non-trivial for the index value α = 1 only, We have the matrix inversion relations We work with representations on functions of x i = (x i,1 , x i,2 ) or λ,λ so that p i = (∂ i,1 , ∂ i,2 ) or ∂ λ i ,∂ λ i act as derivatives as suggested by notation.
We consider the YB relations of RLL type and find the following expressions for the YB operators: (2.8) Here the integration contour is arbitrary with the restriction that integration by parts is performed with vanishing boundary terms (Appendix A). The expression for R λ 12 is obtained from the one for R ++ 12 (u) by substituting according to the canonical transformation 1 We use here the term helicity form in analogy to the case of sℓ4 Yangian symmetry, where it has been applied to scattering amplitudes.
If the representation spaces V 1 , V 2 are restricted to the subspaces of eigenvalues 2ℓ i of (x i p i ) then the inversion relations (2.6) imply further Yang-Baxter relations. The dependence on the eigenvalue of the restricted L matrices is conveniently formulated by a second argument u + = u + 2ℓ or u − = u − 2ℓ − 2 as The additional YB relations read . The previous relations read in the new form . Each of the relations is characterized by the exchange of a pair of parameters. We shall consider the transformation of monodromy matrices as T (u)R = RT (u ′ ) accompanied by a change of the parameters by permutations σ,σ of the sets u 1 , . . . , u N and u + 1 , . . . , u + N This is conveniently written as the table in two rows .
We shall use also the abbreviation by writing the indices carried by the parameters only.

The ratio-coordinate form
The reduction of the L operators to the subspace of eigenvalue 2ℓ is done for the action on functions of the coordinate components by the restriction to a definite degree of homogeneity.
The conjugated momenta are to be substituted in the case + as and in the case − as Explicitly we have In this way the L operators in the two dual representations of gℓ 2 result in the same form of L operators of sℓ 2 , L(u, u − + 1) = L( u + + 1, u) for u = u − + 1 = u − 2ℓ − 1. Therefore we can omit the superscripts ± in the ratio-coordinate form. The uniformization of both results is achieved also in the notation used in [26]. We substitute for the signature + case v = u + ℓ, but for the signature − case v = u − ℓ − 1. Then the reduced L appears in both cases with the same arguments, The matrix L(v (1) , v (2) ) factorizes as This is one argument in favor of the choice of the parameters v with the particular shift with respect to the original ones u. The other (related) argument is the permutation action discussed below. The L matrix and the sℓ 2 generators are related as (2.14) The inversion relations (2.6) both read in the notation L(v (1) , v (2) ) (2) , −v (1) ). (2.15) This can also be checked directly starting from the factorized form (2.13). The YB relations formulated above (2.7), (2.9), (2.10) can be rewritten in the ratio coordinates with the notation L(v (1) , v (2) ) which does not distinguish signature. The resulting forms of the YB relations are characterized by particular permutations of the parameters v 2 appearing as arguments of the L operators in the product 2 ), [25].
2 ), 1 |v 2 )L 1 (v 2 )L 2 (v 1 |v 2 ). It is appropriate to consider the YB operators related to the elementary permutations [26], 2 ). In the RLL relations the R operators R 1 and R 2 resulting from the above R ++ 12 (u 1 − u 2 ) and R ++ 21 (u + 1 − u + 2 ) interchange the parameters with superscript 1 or 2 respectively. The YB operator S 12 (v 2 results from R +− 12 or R +− 21 . Using (2.13) it is obtained as Notice that here x i are the ratio coordinates and the subscript is the point label. The other elementary permutations of v 2 appear in the RLL relations with 2 ), respectively. They are given by the intertwiner between the representations of weights 2ℓ and −2ℓ − 2. . (2.17) The R operators related to the permutations of parameters of superscript 1 or 2 factorize into the operators S ij related to the elementary permutations. (2) ). This leads to the expressions Whereas the action of the operator S 12 (u) on functions of x 1 , x 2 is multiplicative, the action of the latter operators and of S 11 , S 22 can be defined with the help of the Beta integral representation andN = x∂. We have The action on functions can be defined (up to a constant factor) as N . The resulting parameter set is now conveniently presented as one string of N pairs Restricting the L operator in the helicity form to a subspace of a particular helicity eigenvalue 2h of −∂ λ λ +λ∂ λ related to the coordinate dilatation weight 2ℓ as 2h = 2ℓ + 1, we may introduce another form of the L operator involving only one canonical pair. We change from λ,λ to k, l and calculate the corresponding conjugated operators from We restrict the functions of λ,λ to the form ψ(λ,λ) = l h φ(k).
The result is to be compared with the ratio coordinate reduction L(v (1) , v (2) ) after the canonical transformation x, ∂ → ∂ k , −k. If the ratio coordinate x describes position then k plays the role of momentum.
We add the remark that the factor carrying the weight can be chosen differently, e.g., ψ(λ,λ) = l h k a φ ′ (k). Then the equivalent representation of the algebra on φ ′ (k) is generated by L k′ (0, h) = k −a L k (0, h)k a .

Constructions of symmetric correlators
Any transformation of the monodromy matrix T → T ′ induces a map from a solution Φ of the symmetry condition (2.2) with T to a solution Φ ′ of the symmetry condition with T ′ . Examples of such transformations have been discussed in [10]. They are related to features observed in the construction of scattering amplitudes in gauge field theories [13].
Starting from a solution of the Yangian symmetry condition (2.2) other solutions can be generated by action on Φ with YB operators obeying particular RLL relations with L operators entering the monodromy. This can be implemented both in the formulations with homogeneous coordinates or with coordinate ratios. It is an important tool of constructing YSC and it will be illustrated in the following section.
We act with a sequence of R operators on a trivial correlator composed as a product of elementary 1-point correlators given above. The R operators, in particular the points on which they act non-trivially and their arguments, have to be chosen such that the RLL relations allow to permute them with the product of L operators of the monodromy matrix. In this way we obtain that the particular sequence of R operations applied to the initial trivial correlator produces another Yangian symmetric correlator corresponding to the monodromy resulting by the permutations from the initial monodromy.
In the form (2.14) the generators correspond to infinitesimal Moebius transformations, where the finite transformation can be written as x → x ′ = ax+b cx+d , ad − bc = 1, in the particular case of vanishing weight, 2ℓ = 0. The conformal transformations acting on functions as are generated by S a i . Conformal symmetric correlators are invariant under this transformation and thus obey the condition

Examples of Yangian symmetric correlators
If working in the homogeneous coordinate form each R ++ operator action comes with a contour integral ((2.8), Appendix A). The resulting multiple integral can be rewritten in the standard link integral form with the integration variables being coordinates of the related Grassmannian. This step is not specific to sℓ 2 and the resulting expressions coincide with corresponding ones for amplitude calculations (sℓ 4 ) [13]. The specifics of the sℓ 2 case appears in the next steps to obtain the explicit forms in homogeneous, helicity or ratio coordinates. We have also the alternative to work with the R operators factors (2.16) in the ratio coordinate form directly.

Two-point correlators
We start with two-point correlators obtained from a trivial one by just one R operation. This can be formulated in homogeneous coordinates with R ++ (2.8), in helicity coordinates with R λ , in homogeneous coordinates with R +− or directly in ratio coordinates with S 12 (2.16).

Homogeneous coordinates
The two-point YSC with the monodromy The monodromy results from the initial one T (0) . This is written conveniently by the permutation pattern 1 2 2 1 .
We do the integration with one of the two delta distributions and transform the other one.
We use the notation We express the exponents in terms of the weights . The weights appear as degrees of homogeneity as expected.

Helicity coordinates
With the same monodromy now written in L λ we have the correlator in the helicity form The result appears as a prefactor with exponents 2h = 2ℓ + 1 and a scale independent factor.
Ratio coordinates Another two-point correlator with the monodromy The second factor in the last form is the two-point correlator obtained directly in the framework of the ratio-coordinate form by the action of S 12 . In the notation introduced for the ratio coordinate form the initial monodromy reads here T and the action by S 12 results in the permutation of the second parameter of the first factor with the first of the second, . This is conveniently written by the parameter string displaying just the sequence of arguments. Indeed, the correlator with the monodromy T 2 (characterized by the corresponding parameter string) is

Three-point correlators
We present examples of three-point correlator constructions with representations in homogeneous, helicity and ratio coordinates. The direct construction in ratio coordinates by three steps of action with the elementary permutation operators S ij is described.

Homogeneous coordinates
The three-point YSC with the monodromy characterized by the parameter permutation pattern 1 2 3 2 3 1 obtained by the action with R ++ 21 R ++ 31 is There are two delta distributions with the arguments The equations and the Jacobi determinant is From the permutation pattern we express the exponents in terms of the weights, and obtain the result The symplectic products ij can be substituted by the differences in the ratio coordinates Helicity coordinates The three-point correlator with the same monodromy written in the helicity form in terms

Ratio coordinates
We add examples obtained in the framework of the ratio-coordinate form. We start from the monodromy matrix in ratio coordinates and the trivial constant correlator We abbreviate the monodromy matrix by the string of parameter pairs: As the first step we find the symmetric correlator with the monodromy Because of L 2 acting on a constant results in the unit matrix the RLL relation with S 13 for the product of L 1 L 3 applies here. Next we apply the RLL relation with S 12 and find the correlator with the monodromy We apply now the RLL relation with S 23 to establish the symmetric correlator with the monodromy of the parameter string

4-point correlators
We consider again examples of construction in homogeneous, helicity and ratio coordinate form. The results will be used in the following to define symmetric integral operators and to establish relations to the QCD parton evolution.
We transform to the normal link integral form, where the integration variables coincide with the coefficients in the arguments of the δ-functions. This can be formulated in terms of the link matrices We obtain c We Further, The Jacobi determinant is The result is The exponents can be expressed in terms of the weights and further the weights can be replaced by the helicities 2h = 2ℓ + 1. Because only two of the weights are independent, there appears one difference in the spectral parameters additionally Φ −−++ = 12 1+2ℓ 1 +2ℓ 2 14 1+2ℓ 2 23 1+2ℓ 1 34 12 34 14 23 The appearance of additional parameters was noticed in the amplitude context and proposed as a tool of regularization [14].
The result depends homogeneously on the components of the points i with the weights 2ℓ i . The symplectic products ij can be replaced by the differences in the relative coordinates x 34 x 12 x 34 x 14 x 23 Stepwise construction for kernels We consider a way to an equivalent correlator where the intermediate steps have particular interpretations as kernels (see section 5.2).
The first two R operations result in The monodromy is characterized by the permutation pattern 1 2 3 4 4 3 2 1 .
In the next step we have The resulting permutation pattern allows to calculate the weights, The exponents in the result can be expressed in terms of the weights and a difference of two spectral parameters, x 12 x 34 x 14 x 23 The first factors carrying the weights 2ℓ i are written in the second component of the homogeneous coordinates, but the last factors are written in the differences of the ratio coordinates, i.e. x 12 = Coincidence with the result of the first way (3.5) is obtained by the substitution of the parameters as (3.8) Helicity representation The transformation to the helicity representation is easily done in the integral over the link variables c (3.3) by modifying the delta factors in the integrand We have again four delta distributions with the arguments , but they are not removing the four link integrations. We see the dependence between the related linear equations by multiplying the first by λ 1 , the second by λ 2 , the third byλ 3 , the fourth byλ 4 , and adding them with the result We use the first three δ-functions to do the integrations over c 13 , c 14 , c 23 . The Jacobi factor is Their values are fixed at We transform the fourth delta distribution by taking it together with the third one. The transformation should multiply the matrix of coefficients to bring the second row into the wanted form. Then we get The resulting form of the correlator is where ϕ 4 is given in (3.3). We change to c = c 24 λ 2λ4 and k i = λ iλi in order to separate the one-dimensional momentum dependence from the homogeneity factor.
We substitute the weights owing to the permutation pattern as in (3.4) and use the helicity notation 2h = 2ℓ + 1. Recall that we have here only two independent weights and we choose ε = u 2 − u 3 as the extra parameter. The integration over c is simplest in the case of vanishing momentum transfer in the channel 12 → 34 to which we shall refer later.
Ratio-coordinate construction We add an example of construction in the ratio-coordinate form.
The resulting monodromy is characterized by the string

A 6-point correlator
We construct a 6-point correlator appearing in subsection 5.2 as kernel of a generalised Yang-Baxter operator. We consider the construction in homogeneous coordinates only. The resulting monodromy and the relation between the spectral parameters, the dilatation weights and two extra parameters will be relevant later.
In Appendix B we do six out of the nine integrations by the δ-distributions. The result is

Correlators with deformed symmetry
The symmetry condition for correlators can be extended to the cases with the monodromy matrix operator built from L matrices with algebraic deformations. In applications algebraic deformation may describe particular ways of symmetry breaking.
In the case of trigonometric (q-) deformation and also for the case of elliptic deformation we know the expressions for the L matrices, the YB operators and of its factors in analogy [26]. We discuss the case of quantum deformation only.
The straightforward q-deformations of (2.3) are We have the factorized forms With these forms it is easy to prove that the relation of inversion (2.6) hold with a modification.
Here x 1 , x 2 are the components of a 2-dimensional vector x. In connection with correlators and YB relation we deal with more than one 2-dimensional vectors and we use the index notation x i,1 , x i,2 for the components of x i . As in the undeformed case we introduce the ratio variables for the point i if the corresponding L matrix is chosen with signature + as x i,1 . Then we have for the restriction to the weight 2ℓ in ratio coordinates and similar for L − (u, u − ) in complete analogy to the undeformed case (2.13). We introduce also v (1) , v (2) as above (2.12) and write the inversion relation in analogy, The YB operator factors corresponding to the elementary permutations of the parameters v 2 in the product of L operators are known [26].
2 . (4.5) The latter notation emphasizes that this is the appropriate deformation of (x 1 − x 2 ) w . We use the standard notation for the infinite product (x; q) := ∞ j=1 (1 − xq j−1 ). The other YB operators of elementary permutations are in analogy given in terms of the intertwiner W q , S q11 (v . (4.6) The latter notation emphasizes that the expression can be considered as the deformation of (2.17) with the ratio of Gamma functions. The analogy is useful indeed because there is a q-deformed analogy of the Beta integral [29] allowing to express the action of W q on functions in analogy to (2.18) as The YB relation for S q12 can be checked directly using the factorized form (4.3) [26], see Appendix C.
(4.8) From the deformed S q12 (u) it is straightforward to reconstruct the deformed R +− q12 , The factorized form in homogeneous coordinates allows to repeat the above calculation and to check the YB relation (see Appendix C) The examples of symmetric correlator constructions working in ratio coordinates and the YB operators of elementary permutations work in the deformed case by straightforward analogy. The basic correlator 1 from which we generate the result by R operations has vanishing weights. The resulting correlator has weights calculated from the permutation of parameters to v i . The action of (4.8) on 1 shows that with w = v 1 − v 2 − 1 is the symmetric two-point correlator for the monodromy T 2 . Let us mention an example of a 3-point correlator. We have 3 )S q12 (v 2 )L 1 (v 2 ).
Applying both sides on the constant function 1 we obtain the symmetric 3-point correlator for the monodromy The result of the 3-point correlator (3.1) and the ones of the two steps of its construction lead to the corresponding deformed correlators with the monodromy characterized by the same parameter strings with the substitutions This substitution rule works for all YSC which can be generated by S ij actions (i = j) in this way, in particular for the 4-point YSC (3.10).

Integral operators from symmetric correlators
We consider integral operators with YSC as kernels. The symmetry of the correlators expressed in the monodromy relation (2.2) implies the symmetry of the operators by the relation of inversion and transposition obeyed by the involved L operators.

Generalized Yang-Baxter operators
The YB operators considered above map the tensor product of representation spaces as V ℓ 1 ⊗ V ℓ 2 → V ℓ 1 ⊗ V ℓ 2 and obey a relation with a product of two L matrices of the form We consider generalized R operators mapping tensor products with more factors and obeying the more general relation involving monodromy matrices of M factors.
We have the relation for inversion and operator conjugation of the L matrices We abbreviate .
Then we obtain The result is of the form of (5. Integral operators with YSC kernels act in a highly symmetric way, defining homomorphisms not only of the sℓ 2 Lie algebra action on the functions (global symmetry) but of the related Yangian algebra. The resulting YB relations encode this symmetry property. It is also reflected in the eigenvalue spectrum.

Examples
We specify the construction of Yangian symmetric operators from YSC in the cases M = 2 and M = 3. In the first case we obtain operators with simple commutation relations with a monodromy out of two L operators, in particular the standard RLL Yang-Baxter relation. We use as kernels the YSC from the stepwise construction (3.6, 3.7). In the case M = 3 we use the 6-point YSC (3.12) and the related monodromy (3.11) to obtain the generalized YB relation involving the monodromy out of three L operators.
The cases of monodromies are specified by the permutation pattern meaning in particular the relation between the original parameters u 0 i , u 0+ i and the ones appearing in (5.4), (5.6), u i = u 0 σ(i) , u + i = u 0+ σ(i) . In the following we shall formulate this as substitution rules for the parameters in (5.6) in terms of the original ones and omit the superscript 0 after substitution.

Point permutation operator
In the case of ∆ −−++ (3.6) we have

This means
implies F (u) = 1. We obtain as the specification of (5.6) that the operator∆ defined with this kernel obeyŝ Thus choosing ∆ −−++ as the kernel in (5.5) one represents the operator of permutation P 12 acting as P 12 (x 1 , p 1 )P 12 = (x 2 , p 2 ).
The completely connected 4-point correlator has been constructed by other R sequences, e.g. string with the correlator (3.5). Corresponding to the permutation pattern 1 ′ 2 ′ 1 2 1 2 1 ′ 2 ′ one obtains again the normal YB relation (5.7) with other parameter substitutions and with the weights calculated differently from the original spectral parameters.
Note that the permutation P 12 can be removed by modifying the definition (5.5) of the integral operator by a permutation of 1, 2 or 1 ′ , 2 ′ in the step of relabeling the original 1, 2, 3, 4.

Kernels of QCD parton evolution
The generalized parton distributions [17] contribute to hard-exclusive production processes. The form in (one-dimensional, light-cone component) momenta is normally used. The form in positions (on the light ray) is convenient for representing the conformal symmetry actions and is useful also in the related operator product renormalization. For explaining this schematically (for more details cf. [28]) let us consider composite operators of the form By contracting the indices with parallel light-like vectors x µ 1 , x ν 2 , x 1,µ x µ 1 = x 2,µ x µ 2 = x 1,µ x µ 2 = 0 one projects on the symmetric traceless part. Let us choose the axial gauge x µ 1 A µ = 0, then x µ D µ = x∂. It is natural to consider the generating function of such operator product projections n,m The resulting operator valued function of two points on the light ray can be substituted instead of ψ(x 1 , x 2 ) in the definition of the integral operator (5.5). Operator products involving more than two fields can be treated analogously. Then we deal with functions of M points on the light ray and integral operators with 2M point YSC as kernels. Choosing the kernel appropriately, this results in a convenient formulation of the renormalization of a class of composite operators. In QCD, if choosing the light-cone components of the quark and gluon field, these composite operators are called quasi-partonic [27]. In N = 4 super Yang Mills theory one has analogous composite operators with components of the superfield. With combinations of the scalar components one has in particular the SL(2) sector [20] such that the mixing with the products involving other components is absent. The spin chain treatment of the latter has attracted much attention because of the non-compact representations at the sites.
The ordinary parton distributions [5] contribute to the (inclusive) deep-inelastic scattering structure function. Their evolution kernels correspond to the vanishing momentum transfer limit k 1 + k 2 = 0 of the kernels of generalized parton distributions.
We consider the kernel of the YB operator R 12 (v) in ratio coordinate form (5.9). For real values of the positions and if x 1 ′ , x 2 ′ lie in between x 1 , x 2 we can write this in the simplex integral form. We consider the case ℓ 1 = ℓ 2 = ℓ.
This form is the convenient one for comparison with the evolution kernels of generalized parton distribution as given in [21]. It is also convenient because the Fourier transformation is easily done. The resulting form is the one of [22] and J n 1 ,n 2 ,n 3 is the notation for the terms appearing in the kernels introduced in that papers.
In the limit v → 0 and in the negative integer values for v = −1, −2 we find the particular cases related to the parton kernels [21].
At v → 0 we find in the residues at α 1 = 0 and α 2 = 0 the kernel of the unit operator δ(x 11 ′ )δ(x 22 ′ ) and also At v = −1 we have from (5.9) the kernel At v = −2 we have (5.9) We substitute the weight ℓ or the scaling helicity h for quarks as ℓ = −1, h = − 1 2 and for gluons as ℓ = − 3 2 , h = −1. In both the kernels for gluons and quarks of parallel helicities only one value of the extra Yangian representation parameter v contributes, namely v = 0.
In the kernel of anti-parallel helicity quarks two values of v contribute, v = 0, −1. In the kernel of anti-parallel helicity gluons three values of v contribute, v = 0, −1, −2.
In the case of quarks h = − 1 2 the contribution at v = −2 is singular. Physically such a contribution occurs in the flavor singlet channel only and it is proportional to δ(x 1 ′ − x 2 ′ ). We note that the comparison also works in the case of gluon-quark interaction kernels with h 1 = −1, h 2 = − 1 2 and v → 1 2 , − 1 2 , − 3 2 . The leading order renormalization in the SL(2) sector of N = 4 super Yang Mills corresponds to the kernel (5.9) at h 1 = h 2 = 0 and v → 0.
In the case of ordinary parton distribution kernels where the momentum transfer vanishes we have the result (3.9) of the symmetric correlator. In order to obtain the kernels in one-dimensional momenta we have to remove the factor carrying the scaling weights and to do the same substitution of parameters as leading to R 12 (v) in positions. First we have to do the substitution (3.8), In the last step we have used our convention about the relation between the notations u and v, u i = v i + ℓ i , and the relation of the weights in the considered 4-point correlator, 2ℓ 2 + 2ℓ 4 + 2 = 0. Finally we change to helicities according to 2ℓ i + 1 = 2h i to obtain the substitution rule of ε = u 2 − u 3 by v = v 4 − v 3 after relabeling the points 1234 → 1 ′ 2 ′ 12 as This results in We use the momentum fraction z = k 1 k 1 ′ and consider the same cases as above. The case of parallel helicity appears in the extraordinary situations of quark structure functions of odd chirality and of the photon structure function F γ 3 . The kernels are [23,24] proportional to z (1 − z) , and they result as expected from (5.13) at v = 0 and for h 1 = h 2 = − 1 2 and h 1 = h 2 = −1, respectively.
The ordinary parton distributions have a scale dependence governed by kernels composed as sums of contributions from (5.13) at v = 0, −1, −2 for gluons and v = 0, −1 for quarks. The coefficients in the sums do not follow from the considered symmetry. We find in comparison with the formulation in [27] for gluons (h 1 = h 2 = −1) and for quarks ( w +1 +1 and w The scale evolution of the unpolarized parton distributions are determined by the sum of the splitting kernels with and without spin flip, w +1 +1 + w −1 +1 , w The scale evolution of the polarized (helicity asymmetry) parton distributions is determined by the differences w +1 +1 −w −1 +1 for gluons and w for quarks (flavor non-singlet).
They correspond to our expression (5.13) at v = 0, −1 with the corresponding values of In the expressions written above the regularization at z = 1, z = 0 is not explicit. Recall that our definition of the integral operators assumes integration contours which allow integration by parts (denoted as conjugation T ) without boundary terms. Here the integration in z over the segment (0, 1) is to be understood as the result of the contraction of a closed contour. This results in the conventional formulation for parton splitting kernels in terms of distributions like 1 [1−z] + .

Spectra of symmetric integral operators
We consider operators defined in Sect This implies 2ℓ i = −2 − 2ℓ ′ i = 2 ℓ i . In this way the weights of the resulting function Rψ are the same as the ones of ψ. This ensures the consistency of the eigenvalue problem on functions depending homogeneously on M two-dimensional points with the weights 2 ℓ 1 , . . . , 2 ℓ M , Rψ = λψ. (6.1) The Yangian symmetry of the operator induced by the symmetry of its kernel has strong implications on the spectrum. In the case of the two-point operator obeying the standard RLL relation the calculation of the spectrum is well known (cf. [30]). We recall it for comparison with the generalized case. We describe the solution of the spectral problem in the case of the symmetric 3-point operator. The outlined scheme can be generalized to more points. For applications it is important to have algorithms for calculating the complete spectrum exactly.

The spectrum of the YB operator
The integral operator obtained from the kernel Φ −−++ (3.7) represents a Yang-Baxter operator intertwining the representations 2ℓ 1 and 2ℓ 2 according to (5.7). In this YB relation we change to the ratio coordinate form (5.8) and write the L matrix in terms of sℓ 2 generators S a as (2.14). Let ψ(x 1 , x 2 ) obey the eigenvalue relation (the specification of (6.1)), The YB relation implies We abbreviate the relation as RLψ = λRψ and expand in powers of v. At v 2 we have a trivial relation, at v 1 we find: If ψ is an eigenfunction then it is also (S a 1 + S a 2 )ψ with the same eigenvalue, i.e. we have sℓ 2 irreducible representation subspaces of degeneracy. The conditions at v 0 are related by this global symmetry. Therefore it is sufficient to analyze the condition related to one matrix element, e.g. 12.
We consider the function ψ (0) n = (x 1 − x 2 ) n being a lowest weight vector We calculate where S a = S a 1 − S a 2 . We obtain by (6.2) We have marked the eigenvalue of the representation generated from the lowest weight vector ψ (0) n by the index n. We find that also S − ψ (0) n is an eigenfunction of R 21 with another eigenvalue, R 21 S − ψ (0) n = λ n−1 S − ψ (0) n . By comparison we obtain a recurrence relation for λ n . It is easy to see that the constant function is an eigenfunction and moreover a lowest weight vector. Thus as the result of the analysis we find all eigenfunctions and eigenvalues.
We relabel the points as 1, 2, 3, 4, 5, 6 → 1 ′ , 2 ′ , 3 ′ , 1, 2, 3 and consider the integral operator acting as (5.11). We have shown above that this operator obeys the generalized YB relation (5.12), We transform to the ratio coordinate form, where the L operators are substituted as (2.14), abbreviate the relations asR It implies that the eigenvectors ofR 321 build representations with the lowest weight vectors obeying and appearing explicitly as linear combinations at fixed n = n 1 + n 2 of the basis lowest weight functions The problem reduces to find the particular linear combination obeying the eigenvalue relationR The degeneracy space of lowest weight vectors at given n is n + 1 dimensional.
Because of the global symmetry it is enough to consider one matrix element of this relation; it is convenient to choose the matrix element 12. We shall calculate the expansion coefficients in R −,k (n − 1, n 1 ), at u s , s = 1, 0. We substitute (6.4) into (6.6) and use the result of the action of R 12 and L 12 .
The action ofR 321 has been calculated on r.h.s. by the eigenvalue relation (6.6). For doing this also on l.h.s. we express on both sides ψ (0) n 1 +k,n−1−n 1 −k in the eigenfunction basis by the inversion of the linear transformation (6.4).
We have n(n + 1) conditions at level n. These conditions with both substitutionŝ A ± →Â [s] ± , s = 0, 1, (6.7) are needed to find the (n + 1) × (n + 1) matrix elements of Λ(n). The matricesÂ [s] ± encode the details of the Yangian symmetry. The matrix r(n) transforms the basis ψ The iterative condition (6.11) allows to calculate the matrices Λ(n) with Λ(0) = λ 0 as input. The diagonalization of this matrix then results in the eigenfunctions and eigenvalues in the subspace of the irreducible representations in the threefold tensor product with weight 2ℓ n = 2ℓ 1 + 2ℓ 2 + 2ℓ 3 − n, at the level n = n 1 + n 2 .
By calculating the action of the operators R, L appearing in the relation (5.12) at u 1 and u 0 on the basis (6.5) we shall obtain in Appendix D the explicit forms of the matriceŝ A [1] ± andÂ [0] ± . As the result we find that they have the following non-vanishing elements, at u 1 (Â and at u 0 A convenient algorithm is based on the observation that the matricesÂ ± are near diagonal. A quadratic diagonal matrix Z(n) can be found, Z(n) qq = ℓ 3 − n + q + 1, q = 0, 1, . . . , n − 1, such thatÂ [1] + Z(n) +Â [0] + =Â ′ + is upper triangular and has zero elements on the last row. The two conditions (6.11) withÂ [1] ± andÂ [0] ± can be combined in such a way thatÂ ′ + appears on r.h.s. This linear combination of the conditions allows to calculate easily the matrix elements Λ(n) p p 1 with p = 0, 1, . . . , n and p 1 = 0, 1, . . . , n − 1. After this the matrix Λ(n) with unknown elements on the last column only is substituted in one of the initial conditions, e.g. in the one involvingÂ [1] ± (6.12), to calculate these remaining elements.

Discussion
The approach of Yangian symmetric correlators provides a simple way to treat representations of the Yangian algebra in particular in the case of Jordan-Schwinger type realizations of sℓ n . The considered case of sℓ 2 has the advantage that several explicit forms of the relevant Yang-Baxter operators are known. This provides flexibility in the calculations and allows formulations beyond those parallel to the methods developed for scattering amplitudes. It is convenient to use the forms of the L and R operators both in homogeneous and in ratio coordinates. If the ratio coordinate x describes position by transformation to the helicity form one obtains the corresponding momentum dependence (k = λλ).
The monodromy matrix includes in the expansion in the spectral parameter u (u i = u + δ i ) the generators of the considered Yangian algebra representation. The latter is characterized by the conformal weights 2ℓ i and the shifts δ i , which are the differences of the spectral parameters in L i (u i ), u i+1 − u i . The YSC defined by the monodromy depends on these representation parameters. The described R operator construction results in the weights calculated as spectral parameter differences as well, moreover the sum of all weights is fixed by the elementary correlators from which the construction starts. This means that we deal with particular representations where the representation parameters are not all independent. We have encountered examples with more than one relation constraining the weights. In such cases not all spectral parameter differences or shifts are fixed by the weights, some differences appear as extra parameters. These extra parameters together with the independent weights characterize the representation and the related YSC. In the considered example of a 6-point YSC we have 2 extra parameters. This example can be generalized to 2M -point YSC with M − 1 extra parameters.
The Yangian symmetry condition (2.2) in terms of the monodromy matrix allows generalizations, where the L matrices in the latter are replaced by the ones with trigonometric or elliptic deformations or by other forms related to the standard one by Drinfeld twists. It would be interesting to use such opportunities in applications beyond the well studied integrable models.
The construction of YSC starting from elementary correlators relies on R operators obeying Yang-Baxter relations with the product of two L operators. These relations imply simple rules for the interchange of the R operators with the product of L operators in the monodromy.
Actually the L operators, which have been written in several forms, are the source of the rich symmetry structures. The simplicity of the L operator in the case of sℓ 2 allows to write explicitly several kinds and representations of R operators.
We have considered symmetric operators with YSC as kernels. The Yangian symmetry of the kernel implies the symmetry of the operator. This follows from the L operators by their relations of inversion and transposition. Particular 4-point YSC result in the different kinds of YB operators. The considered 6-point YSC results in a generalized YB operator and the mentioned generalization of the latter to 2M points lead to generalized YB operators where the counterpart of (5.12) involves the products of M factors of L matrices on both sides. The spectral parameter arguments of the generalized YB operators are related to the extra parameters it their kernels.
As a tool of treating Yangian symmetry the YSC approach is expected to be useful in physical applications. In this respect it is important that we have seen explicitly how the symmetry of the correlators implies the symmetry of the operators constructed out of them and how the high symmetry allows to solve exactly the spectral problem. We have formulated this in the non-trivial case of a 3-point generalized YB operators constructed from a particular YSC. The outlined procedure generalizes to more points with increasing complexity.
As an example of applications we have shown the direct relation of particular 4-point YSC to the kernels of the scale evolution of (generalized) parton distributions in leading order QCD. They are the ones appearing as kernels of the YB operator R 12 (v) intertwining the tensor product of the representations with the weights 2ℓ 1 , 2ℓ 2 . We have identified the values of the weights and of the parameter v appropriate for different cases of parton distributions, of gluons or quarks and of different polarization configurations. In QCD the parton kernels are sums of contributions of 4-point YSC with different values of the additional parameter.
These 4-point kernels describe simultaneously the scale dependence of a class of composite operators, actually the ones composed of two quasi-partonic field operators of gluons or quarks. The considered 6-point YSC is to be applied to the scale dependence of composite operators out of 3 quark or gluon field operators.
In this way we have a reformulation in terms of YSC of the known relation of the scale dependence of composite operators to the dynamics of quantum spin chains.
The successful integrable spin chain treatment of all composite operators (with an arbitrary number of fields) of N = 4 super Yang-Mills is related to the fact that here in the renormalization of analogous composite operators only the contribution of one value of the additional parameter, v → 0, appears. Whereas in QCD in general more than one Yangian representation contributes, here one encounters only one contribution.
It would be interesting to consider the YSC based on the sℓ(4|4) superalgebra for a unified treatment of the quantum integrability properties of N = 4 super Yang-Mills, i.e. to have on one hand the known way to the scattering amplitudes and on the other hand an alternative way to the treatment of the operator product renormalization. work of J.F. has been supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS15/215/OHK4/3T /14 and by the Grant of the Plenipotentiary of the Czech republic at JINR, Dubna.

A YB operator R ++ as contour integral
We want to obey the RLL relation with the ansatz We calculate the action of the shift operator on the product of L matrices: The condition that the second term vanishes implies a differential equation on φ(c), because In the last step we have assumed that the integration by parts is done without boundary terms. Thus the condition on φ is It is solved by The condition of vanishing of the third term can be written as We see that it does not imply a further condition on φ(c). Thus we have proved that the form used in calculation obeys the Yang-Baxter relation provided the simple rule of integration by parts. If the latter rule is different then the indicated procedure leads to the appropriate modification.

C q-deformed YB operator S q12
We use the abbreviation Φ(x, λ) = (xq 1−λ ; q 2 ) (xq 1+λ ; q 2 ) defined using the infinite product (x; q) := ∞ j=1 (1 − xq j−1 ). Because S q12 (w) involves the coordinates only, it commutes with the matrix factorsV i in the factorization (4.3). Commuting it through the matrix factorsD i creates extra matrix factors. Commuting S q12 (v 2 ) we see that it commutes with the first factor of L 1 and the third factor of L 2 . We get from the second and third factor of L 1 and the first and second factor of which are the third factor of L 1 and the first factor of L 2 on the right hand side of (4.8).
The proof can be repeated also in terms of the homogeneous coordinates x i,1 , x i,2 . The L + operator (4.1) can be reconstructed from the one in the ratio coordinates (4.3) by inserting x 1 = x 1,1 x 1,2 and 2ℓ 1 → x 1,1 ∂ 1,1 + x 1,2 ∂ 1,2 and in analogous way also the L − operator.

R.
After commuting R(λ) through L + (u)L − (v) we obtain from the second and the third factor of L + (u) and the first and the second factor of L − (v) which proves (4.10).
We have abbreviated using Pauli matrices σS = σ a S a . We consider the matrix element 12, 12 + R

We have suppressed the superscript [s] indicating the power of expansion in u.
We calculate the action of the operators involved on the basis ψ 12 ψ (0) n 1 ,n 2 = −(S 0 1 + S 0 2 ) n 1 ψ The result of the operator action at u 1 can be formulated in terms of the matricesÂ [1] ± (6.10) as written above (6.12).
We shall use again the matrix element 12 of the relation. We calculate the action of these operators on the basis functions of lowest weight (6.5). We start with contributions to R 12 .