Basso-Dixon Correlators in Two-Dimensional Fishnet CFT

We compute explicitly the two-dimensional version of Basso-Dixon type integrals for the planar four-point correlation functions given by conformal fishnet Feynman graphs. These diagrams are represented by a fragment of a regular square lattice of power-like propagators, arising in the recently proposed integrable bi-scalar fishnet CFT. The formula is derived from first principles, using the formalism of separated variables in integrable SL(2,C) spin chain. It is generalized to anisotropic fishnet, with different powers for propagators in two directions of the lattice.


Introduction
Recently, B. Basso and L. Dixon obtained an elegant explicit expression for a specific, conformal planar Feynman graph of fishnet type [1], having N rows and L columns, and thus (N + 1)(L + 1) − 4 loops. This graph is presented on Fig.1. It has four external fixed coordinates and, similarly to the conformal 4-point functions, has a non-trivial dependence on two cross-ratios u, v. This Basso-Dixon (BD) formula takes the form of an N × N determinant of explicitly known "ladder" integrals [2,3]. It is one of very few examples of explicit results for Feynman graphs with arbitrary many loops.
The BD formula appeared in the context of its application to the four dimensional conformal theory which emerged as a specific double scaling limit of γ-deformed N = 4 SYM theory combining weak coupling and strong imaginary γ-twists [4,5]. In particular, in one-coupling reduction of this theory -the so called bi-scalar, or "fishnet" CFT -the BD integral represents indeed a particular single-trace correlation function (described below). In general, the bulk structure of planar graphs in fishnet CFT is that of the regular square lattice of massless propagators. Such a graph represents an integrable two-dimensional statistical mechanical system [6] which can be studied via integrable quantum spin chain with the symmetry of 4D conformal group SU (2, 2) [4,5,7,8].
Two of the current authors recently proposed the D-dimensional generalization of bi-scalar fishnet theory [9]. Its action is given in terms of two interacting complex N c × N c matrix scalar fields X(x), Z(x): where ω is an arbitrary "anisotropy" parameter producing different powers of propagators along two axis of the fishnet square lattice and ξ is the coupling constant. 1 At D = 4, ω = 0 it reduces to the local bi-scalar action following from the double scaling limit of N = 4 SYM [4]. The BD-type integral corresponds to the following single-trace correlation function: I BD L,N (z 0 , z 1 , w 0 , w 1 ) = tr X L (z 0 )Z N (z 1 )X †L (w 0 )Z †N (w 1 ) . (

1.2)
It is easy to see that, due to the chiral nature of interaction of two scalars, this correlation function is given in the planar limit by a single, fishnet-type planar graph of BD-type drawn in Fig.1. Explicitly, this Feynman graph is given by expression where we have N · L integration variables belonging to the L × N lattice of positive integers L L,N = {1 ≤ L ≤ L, 1 ≤ n ≤ N }, and we take equal coordinates at each of the four boundaries of this rectangular lattice: {z j,0 = z 0 , z j,N +1 = w 0 , z 0,k = z 1 , z L+1,k = w 1 } are imposed for j = 1, ..., L and k = 1, ..., N .
This integral was computed explicitly in D = 4, for "isotropic" case ω = 0, in [1]. The derivation is based on certain assumptions, typical for the S-matrix bootstrap methods inherited from the  Basso-Dixon type diagram I BD L,N (z0, z1, w0, w1) (on the left), its reduction GL,N (z|w) (in the middle) and generalization DL,N (z|w) (on the right) described in sec.2. We integrate only the coordinates in the vertices marked by black blobs. Sending w0 → ∞ in the original Basso-Dixon type diagram, we remove the upper row of propagators and obtain the reduced diagram (in the middle). Using conformal invariance of the original graph (on the left), we can always restore it from the graph on the right, by inversion and shift of coordinates w1, z1, z0. Further on, we generalize the middle diagram by splitting the end point coordinates of left and right columns of external propagators, to separate coordinates z1 → (z1, z2, . . . , zN ) and w1 → (w1, w2, . . . , wN ), and then add at the left a column of vertical propagators [zi − zi+1] −γ , thus getting the generalized configuration (on the right).
integrability of planar N = 4 SYM [15]. It would be important to derive this formula from the first principles, based on the conformal spin chain interpretation of fishnet graphs, but in four dimensions such a derivation is so far missing.
Our derivation is based on integrable SL(2, C) spin chain methods worked out in [16][17][18], using the Sklyanin separation of variables (SoV) method [19][20][21]. The result can be presented in explicit form, in terms of N × N determinant of a matrix with the elements which are explicitly computed in terms of hypergeometric functions of cross-ratios 3 . Our main formula looks as follows: and Formula (1.6) is also generalized in sections 4, 5 to the principal series representations of SL(2, C), see (3.1).
In the next section, we will define the basic building blocks for construction of the Basso-Dixon configuration in operatorial way. In section 3, we will introduce the generalized "graph-building" operator related to the transfer-matrix of the integrable open SL(2, C) quantum spin chain. We will diagonalize there this operator by means of the SoV method and describe the full system of its eigenfunctions. In section 4 the result for 2d Basso-Dixon-like N × L graph will be presented in terms of an N × N determinant of the matrix constructed from 1 × M such graph called the ladder graph. In section 5, the ladder graph will be computed explicitly, in terms of the hypergeometric functions and their derivatives, thus completing the explicit result for the full two-dimensional Basso-Dixon-like N × L graph presented above. The ladder graph is employed to compute the so-called simple wheel graph in two dimensions. A particular case of N = L = 1 (the two-dimensional "cross" graph) will be explicitly given in terms of the elliptic functions of the cross ratio.
2 Transformations of Basso-Dixon type graph and L ↔ N duality In order to apply powerful methods of SL(2, C) spin chain integrability, such as the separation of variables (SoV), we will use the conformal symmetry to reduce the BD graph on Fig.1 to a more convenient quantity for our purposes. First of all, we send w 0 → ∞ and drop the corresponding propagators containing this variable: where we take {z j,0 = z 0 , z 0,k = z 1 , z L+1,k = w 1 } for j = 1, ..., L and k = 1, ..., N . We can always restore the original quantity I BD L,N (z 0 , z 1 , w 0 , w 1 ) from G L,N (z 1 , w 1 |z 0 ), presented on Fig.2 where all the external legs on the left and on the right of Fig.2(left) have different coordinates: {z j,0 = z 0 , z 0,k = z k , z L+1,k = w k } for j = 0, 1, ..., L and k = 1, ..., N . We introduced in the r.h.s. of (2.1) the vector notations: z = {z 1 , z 2 , . . . , z N }, w = {w 1 , w 2 , . . . , w N }. Notice that, after point-splitting, we multiplied, for the future convenience, the middle diagram of Fig.2 by the vertical propagators on the left, without altering the essential part of the quantity, since the coordinates in the left column are exterior and they are not integrated.
The last expression (2.1), representing the diagram on the right of Fig.2, is the most appropriate for the application of integrability methods. Namely, we can represent it as a consecutive action of a "comb" transfer matrix "building" the graph, as shown on the Fig.3. In the next section, we will define yet a more general transfer matrix Λ N (x)(z|w) depending on a spectral parameter x and diagonalize it by means of eigenfunctions using separation of variable (SoV) method of Sklyanin. The lattice of propagators can be inhomogeneous in L-direction, since each transfer matrix, corresponding to an open spin chain of length N "building" the BD configuration by L consecutive applications, as on Fig.3, can have its own spectral parameter. Its particular, homogeneous case will give the explicit formula for 2D BD graph. 4 Now we will comment on the obvious L ↔ N duality of the original BD diagram: where we explicitly introduced among the arguments the anisotropy parameter γ. It is useful to represent the same quantity in a more explicitly conformally symmetric way: Then the L ↔ N duality reads as follows: 3 "Graph building" operator Λ N (x|z 0 ) and its diagonalization Our main goal in the rest of this paper is the computation of the quantity B (γ) L,N (η) directly related to the BD integral by (2.3). To that end, we define a more general transfer matrix of an open SL(2, C) spin chain, building the generalized BD graph. The explicit computations will be carried out for values of γ corresponding to the principal series of representations of SL(2, C). Then the original quantity (1.4) is obtained by analytic continuation to real γ = 1 2 + ω in the final result. First of all, we fix our parameters: • Definition of the conformal spin: where n s ∈ Z is the SO(2) spin and ν s ∈ R, so that 1 + 2iν s is the scaling dimension in the principal series of representations [22].
• Definition of the x k -parameters which will play the role of spin chain inhomogenieties in spectral parameter, and then also of Sklyanin separated variables: where n k ∈ Z and ν k ∈ R.
• The spin s and the parameter x (or y) will enter almost everywhere in special combinations 5 , so that for simplicity we shall use the shorthand notations and define the α , β , γ-parameters Now let us define the integral operator Λ N (y|z 0 ) by its explicit action on a function Φ(z 1 , . . . , z N ) by the formula where by definition z N +1 = z 0 , and we introduced the symbol [z] α ≡ z α (z * )ᾱ (see the details for this notation in App. A). Note that the operator Λ N (y|z 0 ) maps the function of N variables to the function of N + 1 variables, but the last variable z 0 plays some special role of an external variable. The diagrammatic representation for the kernel of the integral operator Λ N (y|z 0 ) is shown schematically on the Fig.4. The operators Λ N (y|z 0 ) form a commutative family and the proof of the commutation relation is equivalent to the proof of the corresponding relation for the kernels which is demonstrated on the Fig.5. The proof is presented there diagrammatically, with the help of cross relation (A.7). In this way, we proved the integrability of our open spin chain since both operators on each side of the last relation contain different spectral parameter, y 1 or y 2 .
We shall use the similar notations Λ k (y) for k = 2 , . . . , N − 1 for operators whose action on a function Φ(z 1 , . . . , z k ) is defined in a similar way The variable z k+1 plays here a special role and the diagrammatic representation for the kernel of Λ k (y) is the same as for Λ N (y|z 0 ) with the evident substitutions N → k and z 0 → z k+1 .

Eigenfunctions of the operator
The eigenfunctions of the operator Λ N (y|z 0 ) are constructed explicitly and they admit the following representation where the operatorsΛ N −k (x k ) differ from the operators Λ N −k (x k ) by a simple factor with r N −k defined according to and where we introduce a shorthand vector notation for the whole set of variables The presence of the pre-factor (3.10) in the definition ofΛ N −k (x) operators (3.9) is crucial to prove the exchange relationΛ from which follows that Ψ(x|z) are symmetric functions of the x-variables The vector of variables x is used as quantum numbers (separated variables) to label the eigenfunction and z is the set of complex coordinates in our initial representation. We will prove that where and the function a(α, β, γ) is defined in App. A. We should note that functions Ψ(x|z) are generalized eigenfunctions of the operator A + z 0 B where A, B are standard matrix elements of the monodromy matrix [21,23]. Note that the detailed notation for the eigenfunction should be Ψ N (x|z) but we shall skip N almost everywhere for sake of brevity.
In the simplest case N = 1 we have The relation (3.16) can be derived by using the chain integration rule (A.4). The general proof of the relations (3.14)-(3.15) is based on the exchange relation The proof of the relation (3.17) for N = 3 is shown in Fig.6 and the generalization is obvious. Notice that after exchange, the operator defining the eigenfunction enters with the reduced length N of the effective spin chain. Using the exchange relation step by step it is easy to derive the formula Then the proof that Ψ(x|z) from (3.8) is eigenfunction of the operator Λ N (x|z 0 ) with the eigenvalues given by (3.14) is reduced to the relation (3.16) in the form 6 We will see that these eigenfunctions form the complete orthonormal basis. Using them, as well as the explicit eigenvalues of Λ N (y|z 0 ) give above, we will compute the Basso-Dixon type two-dimensional integral.

Orthogonality and completeness
The functions Ψ(x|z) form a complete orthonormal basis in the Hilbert space H N . Any function Φ ∈ H N can be expanded w.r.t. this basis as follows (2) Star-triangle transformations insideΛ2(x1) and two lines β and 1 − β1 ending at z0 joint to the one line (3) Movement of the line with index β1 − β upstairs using cross relations leads toΛ2(x1) Λ2(y|z0), (4).
The symbol D N x stands for the measure in the principal series representation of SL(2, C) group Depending on the value of spin in the quantum space, n s = s −s, the sum over n k goes over all integers (integer n s ) or half-integers (half-integer n s ). The coefficient function C(x) is given by the scalar product The weight function µ(x) is the so-called Sklyanin measure [19,20]. It is related to the scalar product of the eigenfunctions Here the delta function δ N (x − x ) is defined as follows: where summation goes over all permutations of N elements and we define These formulae were obtained in [17,18] and the corresponding diagrammatic calculations are discussed at length in these papers. The completeness condition for the functions Ψ(x|z) has the following form A similar formula was proven in the case of SL(2, R) Toda spin chain by [25], in the case of modular XXZ magnet in [26] and for b-Whittaker functions in [27]. It is commonly believed to work for our SL(2, C) spin chain as well, though the proof is still missing.

SoV representation of generalized Basso-Dixon diagrams and reductions
We have now the necessary instrumentary to reduce the Basso-Dixon type Feynman integrals to the SoV form. First we present the most general, inhomogeneous generalization of our construction and then reduce it to homogeneous anisotropic, or even isotropic case. The last one will be the 2D analogue of the standard fishnet graph considered in D = 4 dimensions in [1]. We will suggest for it an explicit determinant representation.

SoV representation for general inhomogeneous lattice
Using the completeness (3.26) and the relation (3.14) we can represent the most general "graphgenerating" kernel, operator which "builds" a lattice formed by a repeated action of the operator (3.5). The integral kernel of the operator (4.1) in coordinate representation looks as followŝ The graphical representation for the left hand side (4.2) for this general case is given in the left picture on Fig.7. This operator is represented there in the form of a lattice with inhomogeneities defined by spectral parameters y 1 , y 2 , . . . , y L+1 . Later in this section we will perform the reduction of this formula to the homogeneous lattice of propagators as in the Basso-Dixon integral (1.3) by taking equal spectral parameters in each column: y 1 = y 2 = · · · = y L+1 = y, or even a more particular case of homogeneous but anisotropic lattice of propagators (different powers in two directions), putting y = s − 1. But so far we consider the most general configuration. The diagram in Fig.7 (right) can be reduced to a generalized Basso-Dixon diagram. First, we have to perform amputation of the most left vertical lines, then the reduction of all z k → z 1 in the function Ψ(x|z) and finally the reduction of all w k → w 1 in the function Ψ(x|w) in the right hand side of (4.2). We will see that such a reduction leads to a significant simplification of the eq. (4.2), allowing to perform at the end all the integrations and summations over separated variables explicitly.
Let us start from the function Ψ N (x 1 , x 2 . . . x N |z). All the needed steps are illustrated in the Fig. 8 for N = 3. Before the reduction z k → z 1 we have to perform the amputation of the factors After amputation and reduction z k → z 1 we obtain the diagram for the action of the operator Λ N (x) for x = s − 1 on the function Ψ (N −1) (x 2 , x 3 . . . x N |z). It is an eigenfunction for this operator, with the eigenvalue λ(y 1 , x 2 ) λ(y 1 , x 3 ) · · · λ(y 1 , x N ) . The next step is similar but for a reduced chain N → N −1 and we obtain the next eigenvalue which is λ(y 2 , x 3 ) λ(y 2 , x 4 ) · · · λ(y 2 , x N ) , etc.
After all these manipulations we obtain the following formula for the reduction of the amputated eigenfunction where we introduced and used the factor r n (x k ,x k ) defined in (3.10). The reduction z k → z 1 for the eigenfunction Ψ(x|z) without amputations of the lines is shown step by step in the Fig.9. First of all we use the star-triangle relation and reduce the triangle to the corresponding delta-function. This elementary reduction is shown on the right in Fig.2. Using this elementary reduction it is possible to reduce the first layer of the diagram for the general eigenfunction Ψ(x|z) to the product of the corresponding delta-functions After integrations in the corresponding vertices in the second layer all delta-functions disappear so that it is possible to repeat the same procedure. After all iterations one obtains the following expression for the reduced eigenfunction  Ψ(x1, x2, x3|z1, z2, z3) and then reduction in the limit z k → z1 to the simple power [z0 − z1] −α 1 −α 2 −α 3 . We perform amputation of [z1 − z2] and [z2 − z3] lines in (1), then (2) we reduce the first row z2, z3 → z1 leading to (3). We can open the triangle in (3) to a star, so that integrations in upper-left, and then lower-left vertex are performed using chain relation and star-triangle relation. At the next step (4) we join propagators with coinciding coordinates on the left, and performing the last integration (5) via chain relation, the eigenfunction is reduced to a simple line (6).
Note that we have to perform such reduction also in the function Ψ(x|w) so that it remains to perform the complex conjugation and evident substitution z → w. Using the rules of the complex conjugation and substituting z → w we obtain Finally, as a result of amputation-reduction on Ψ(x|z) and reduction of Ψ(x|w), by the use of (4.

3) and (4.7) the projector Ψ(x|z)Ψ(x|w) is transformed into
We point out that the way we reduce the N coordinates z = {z k } to a single point in the functions Ψ(x|z) and Ψ(x|z) can be alternatively obtained by inserting the complete basis (3.26) between two Λkernels in (4.1), and repeating their diagonalization after the reduction of the last kernel Λ N (y L+1 |z 0 ) and the amputation and reduction of the first Λ N (y 1 |z 0 ). From formula (4.8) we obtain the following representation for the two-dimensional analogue of generalized Basso-Dixon diagram: We recall that α k = 1 − s − x k , β k = 1 − s + x k and x k = n k 2 + iν k ,x k = − n k 2 + iν k . Introducing the amputated cross ratio we rewrite the last expression for inhomogeneous and anisotropic 2D Basso-Dixon type integral in a concise form and by superscript y we mean the vector of inhomogeneity parameters y = (y 1 , y 2 , . . . , y N ).

Determinant representation
We notice that in (4.12) we deal with the multiple integral of a special type which can be transformed, similarly to the eigenvalue reduction of the hermitian one-matrix integral [28,29], to the determinant form where we introduced the momenta with the weight function given in our case by the expression where λ(x) and λ(y, x) are defined in eqs.(4.4), (3.15). So for any pair of integers L, N the problem is reduced to the computation of momenta (4.14), which we will do explicitly in the section 5 after the reduction to Basso-Dixon configuration of the general formula (4.11).

Reductions
In particular case, leading to the homogenous Basso-Dixon lattice configuration, we put y 1 = y 2 = · · · = y L = y and obtain for the reduced quantity B y (z 0 )(z|w) | y1=y2=···=y L =y ≡ B(y; z 0 )(z|w) = Λ L (y|z 0 )(z|w) (4.16) the following SoV representation: The further reduction of this expression, β k → 0, or y k = y → s − 1, will lead to anisotropic Basso-Dixon type D = 2 integral (1.4) with parameters γ = 2s−1,γ = 2s−1. After this reduction we obtain the second diagram in Fig.7, with the different propagators [z − z ] 1−2s and [z − z ] 2s−2 in vertical and horizontal directions of the lattice. In this case, we have to substitute into the formula (4.2) representing this diagram the reduced eigenvalues This leads, after the identification of external coordinates: z k → z 1 , w k → w 1 , described above, to the following representation for the two-dimensional analog of (anisotropic) Basso-Dixon diagram B L ,N (η) in terms of the multiple integral over N separated variables Notice that the parameters of the representation (s, s) can be chosen in the principal series (3.1), or even in the imaginary strip ν (s) ∈ (−i/2 , 0) by analytic continuation. With the choice of parameters n s = 0 and ν (s) = −i/4 ± i ω/2 in (3.1) we describe the 2D Basso-Dixon type integral with real propagators |z − z | −1∓ω , where ± signs corresponds to two different axis of the square lattice shaped Feynman graph, according to the bi-scalar Lagrangian (1.1). The isotropy of the lattice is restored at s =s = 3/4, that is ω = 0.
The determinant formula (4.13) reads for this reduction as follows where and where λ(x) is defined in eqs.(4.4).

Explicit computation of ladder integral
In this section, we will explicitly compute the momenta m ik given by (4.21) in terms of hypergeometric functions, which leads to explicit expressions of Basso-Dixon type integrals via the determinant representation (4.20). Some details of the derivation can be found in Appendix C.
Noticing that we are led to compute the following sum and integral 7 : where in the last line we substituted explicit parameters. We will compute the integral over ν by residues. The structure of the poles and zeroes is shown in the Fig. 10. We can close the integration contour on the upper/lower half-plane under the condition |η|< 1, respectively |η|> 1, ensuring the exponential suppression of the integrand at ±i∞. Consider first the case |η|< 1. In the upper halfplane there is one infinite sequence of poles of the order M. After the change of variables n → −n+n s +1 in the sum over n and ν → ν + ν s in the integral over ν, the integral (5.2) reads We close the contour in the upper half-plane and calculate the ν-integral as the sum of residues. Due to the mechanism illustrated in fig.10, this is equivalent to take residues at the points ν = in 2 + ik , k = 0 , 1 , 2 , . . ., i.e. the series of the poles created by the function Γ M ( n 2 + iν). The residue at the point ν = in 2 + ik can be represented in the following form Using this formula one obtains the following relation Remarkably enough, since we take derivative at ε = 0 the last double sum can be equivalently rewritten in a factorized form, setting p = n + k − 1 and we obtain the following expression for the ladder integral where γ = 2s − 1 and the function F M (λ , ε|η) is given by the hypergeometric series Therefore we can write in a more compact notation, for |η|< 1: In the opposite case of |η|> 1 the same kind of computation can be repeated picking residues in the lower half plane. After redefinition n → −n + 2n s + 2, this is equivalent to pick the series of poles ν = 2is + in 2 − ik , k = 0 , 1 , 2 , . . ., and the residues are It follows from (5.5) that the final expression of the ladder for |η|> 1 is the same as (5.4) after replacing η with 1/η. For a generic cross-ratio |η|≶ 1 the M -ladder is, respectively and it shows explicitly the invariance under exchange z 1 ↔ w 1 ; in fact The result (5.6), obtained under the assumption of (s,s) in the principal series of SL(2, C), can be remarkably extended by analytic continuation to s =s ∈ (1/2 , 1), that is setting γ =γ ∈ (0, 1) in (5.6). The direct computation of ladder integrals is more involved in this last case, since analytic continuation leads to the failure of the cancelation of poles by zeros presented on Fig.10, and integration in (5.2) must be carried out under an appropriate contour deformation prescription. The explicit result for the particular choice of weights γ =γ = 1/2, corresponding to the isotropic fishnet theory (the case considered by Basso and Dixon in [1] for D = 4) reads: Moreover in the isotropic case γ = 1 − γ, and for the simple "cross" N = 1, L = 1 diagram (computed below in terms of elliptic functions), the duality (2.4) is a mere consequence of (5.7) For the sake of duality in the more involved anisotropic case we will need also the relation between ladders with exchange of γ ↔ 1 − γ. This relation can be easily checked and looks as follows and due to B the duality (2.4) is also proved. In the simplest particular case M = 1 we can simply put ε = 0 everywhere and then reduce to the simple power which is precisely the single propagator in the trivial case of the Basso-Dixon type formula, with no integrations.
In order to get a better feeling of the structure of our result (5.4) at generic N + L, it is instructive to compute the first non-trivial graph G L=1,N =1 (z 1 , w 1 , z 0 ) -the two-dimensional "cross" integral. In four dimensions, the cross integral can be computed in terms of the Bloch-Wigner function (di-logarithm function) [2]. We will see that in our two-dimensional case the answer for cross can be expressed through elliptic functions. Since it involves only N = 1 separated variable, it is simply related to the ladder I 2 : For M = 2 the ladder integral (5.4) reads: Choosing the conformal weights for isotropic fishnets γ =γ = 1/2, the ladder simplifies to We can recall the expression of the 2D conformal cross integral [30] (e.g. see the formula (1.7) of [31]); after amputation of one line by sending w 0 to infinity, we get In order to compare with (5.10) we should set h = 1/2, that is σ = 0. Due to the vanishing of B(1/2), this expression is an ill-defined sum of two divergent terms. The issue is solved by taking the limit σ → 0 in (5.12), which gives the well defined function and reproduces the result of plugging (5.11) into (5.10). The problem reduces to computing F (σ|η) and ∂ σ | σ=0 F (σ|η) which reduce to elliptic integrals. Then the cross integral can be presented in explicit form: where here: and K(x) is the elliptic K integral: This result for the cross integral suggests that even for any L, N the formula for two-dimensional Basso-Dixon integral can be presented in terms elliptic poly-logarithms encountered [32] in various Feynman graph calculations.  operators tr(X l )(z), tr(Z l )(z). As explained in [4,9], the perturbative expansions of their correlators consist, for l > 2, of only of the "globe"-shaped fishnet Feynman integrals: where we set z j,0 ≡ z, z j,N +1 ≡ w, and the expansion itself reads: For any value of the coupling ξ 2 the correlators (6.1) are conformal, thus it is possible to define the scaling dimension of the fields X and Z as: where the anomalous dimension γ is an expansion in the log-divergence of F l,N graphs, i.e. its coefficient of 1 ε in dimensional regularization. Since this divergence is the same for the corresponding wheel graph, obtained after amputation of |z j,N − z j,N +1 | propagators: where W (1) l,N stands for the 1/ε-divergence coefficient in the expansion of the (l, N ) wheel in dimensional regularization. 8 The simple case N = 1 can be worked out explicitly, since the integral (6.4) can be regarded as a ladder with periodic boundary conditions and L = l − 1, see Fig.11. In the formalism of integral operators (3.5) we can write: We can insert inside the trace in (6.5) a complete basis (3.23) in order to get an integral over one separated variable: The integration over z is the scalar product of two eigenfunctions with the same weights x, thus carrying the log-divergence of (6.5), or the 1 divergence which is the leading one at N = 1 in the -regularization. We can easily extract it: and the resulting W l,1 reads: The L-ladder at η = 1 is a finite quantity only for L = l − 1 > 1, and it isn't otherwise possible to close the integration contour in (5.2). Indeed the asymptotic expansion of λ 1 in ν is The divergence of the wheel diagram at l = L + 1 = 2 is in agreement with our expectations: in order to renormalize correlators (6.1) at l = 2 the specific double-trace counterterms are needed [9, 12-14, 33, 34]. More explicitly, fixing the propagators along the frames and spokes to be the same (ω = 0), we get: The quantity (6.7) can be computed numerically and, hopefully, expressed in terms of Elliptic Multiple Zeta Values.

Conclusions and prospects
In this paper, we derived an explicit formula for the two-dimensional analogue of Basso-Dixon integral given by conformal fishnet Feynman graph represented by regular square lattice of rectangular L × N shape, presented on Fig.1 and Fig.2(left). The definition of this integral and the result are presented at the end of Introduction (sec.1). Our result represents a slightly more general quantity then Basso-Dixon graph: it concerns the anisotropic fishnet, i.e. with different powers for vertical and horizontal propagators, corresponding to arbitrary spins s,s of principal series representation of SL(2, C) group, or for the analytic continuation to s =s belonging to the real interval 1 2 , 1 . The particular case of isotropic fishnet, a-la Basso-Dixon, corresponds to the case s =s = 3/4. In two-dimensional case the fishnet graph is built from propagators 1 |z1−z2| . Such graph is a particular case of single-trace correlators introduced in [35,36] for the study of planar scalar scattering amplitudes in the bi-scalar fishnet CFT [4,9]. In the simplest case N = L = 1 (cross integral) we managed to present the result in terms of elliptic functions. It seems plausible that even for general L, N the result can be expressed in terms of elliptic functions. A probable full basis of such functions, in terms of which our quantity could be presented, are the so-called multiple elliptic poly-logarithmic functions (see [37] and references therein). It would be interesting to obtain it for a few smallest N, L.
Interestingly, in the case s → 1/2 (or, alternatively, s → 1, which is an equivalent SL(2, C) representation for the graph's propagators) this fishnet corresponds to one of the conservation laws of Lipatov integrable (open) spin chain hamiltonian [38,39] describing the system of reggeized gluons for the Regge (BFKL) limit of QCD [18,[40][41][42]. It would be interesting to understand what kind of BFKL physics it can describe.
The Basso-Dixon type configuration represents only one set of possible physical quantities which can be, in principle, analyzed and computed in the planar bi-scalar fishnet CFT due to integrability. To fix the OPE rules in such a theory, we have to compute the spectrum of anomalous dimensions and the structure constants of all local operators. Some of them have been analyzed and even computed in the literature. In particular, the so-called wheel graphs, corresponding to operators tr X L , have been computed in D = 4 dimensions in [4,43] up to two wrappings at any L. In [7] they have been computed in particular cases of L = 2, 3 (L = 4 case is to appear [44]) to any reasonable loop order (for any wrapping there exists an iterative analytic procedure) or numerically with a great precision, by means of the Quantum Spectral Curve method [45][46][47][48]. We think that, to give a more general result for any L in rather explicit form, we have to employ a powerful technique of separated variables, similarly to the one we employed here in two dimensions for Basso-Dixon type graphs. The first step would be to compute the wheel graphs in two dimensions using the techniques of this paper. To advance to D > 2 dimensions by integrable spin chain methods, we have to understand the construction of separated variables for higher rank symmetries, such as SU (2, 2). Some recent results in this direction might provide the necessary computational tools [49][50][51][52][53][54]. It would be also good to generalize our techniques, at least in two dimensions, to the computation of multi-magnon operators related to "multi-spiral" Feynman graphs [5].
The computation of structure constants is an even more complicated task. Certain explicit results for OPE of short protected operators have been obtained for fishnet CFT in [9,33,34] (see also [55,56] in BFKL limit) using solely the conformal symmetry. The calculation of more complicated structure constant is a difficult task demanding the most sophisticated techniques, such as SoV method. Since for the 2D case the SoV formalism is well developed it would be interesting to apply the methods of the current paper to computations of more complicated structure constants at least in two dimensions.
Finally, it would be good to understand the role of separated variables in the non-perturbative structure of the bi-scalar fishnet CFT. A good beginning would be to understand in terms of SoV the strong coupling limit for long operators of the theory and to relate it to the classical limit of the dual non-compact sigma model which will probably arise in two-dimensional case similarly to the one which was observed in four-dimensional bi-scalar fishnet CFT in [57].