Exactly solvable magnet of conformal spins in four dimensions

We provide the eigenfunctions for a quantum chain of $N$ conformal spins with nearest-neighbor interaction and open boundary conditions in the irreducible representation of $SO(1,5)$ of scaling dimension $\Delta = 2 + i \lambda$ and spin numbers $\ell=\dot{\ell}=0$. The spectrum of the model is separated into $N$ equal contributions, each dependent on a quantum number $Y_a=[\nu_a,n_a]$ which labels a representation of the principal series. The eigenfunctions are orthogonal and we computed the spectral measure by means of a new star-triangle identity. Any portion of a conformal Feynmann diagram with square lattice topology can be represented in terms of separated variables, and we reproduce the all-loop"fishnet"integrals computed by B. Basso and L. Dixon via bootstrap techniques. We conjecture that the proposed eigenfunctions form a complete set and provide a tool for the direct computation of conformal data in the fishnet limit of the supersymmetric $\mathcal{N}=4\,$ Yang-Mills theory at finite order in the coupling, by means of a cutting-and-gluing procedure on the square lattice.


INTRODUCTION
The exactly solvable spin magnets [1,2] constitute a class of condensed matter models of wide interest throughout theoretical and mathematical physics. In particular, the integrable chains of nearest-neighbors interacting spins [3,4] serve as a tool to encode the symmetries of local or non-local operators in quantum field theory, providing a rich amount of non-perturbative results ranging from the scattering spectrum of high-energy gluons in QCD [5][6][7] to the conformal data of the supersymmetric N = 4 SYM and N = 6 ABJM theories [8]. The archetype model of this class is the SU (2) Heisenberg magnet of spin 1 2 , which for open boundary conditions is described by the Hamiltonian being σ a the vector of Pauli matrices acting on the space V a = C 2 . Generalizations of (1) to other symmetry groups are known, including the non-compact SO(1, 5) spin chain [9]. The latter model is relevant for the study of covariant quantities in a four-dimensional conformal field theory (CFT) [10]. We consider the homogeneous model in the irreducible unitary representation defined by the scaling dimension ∆ = 2 + iλ, λ ∈ R, and the SO(4) spins =˙ = 0 [11]. The Hamiltonian operator acts on the Hilbert spaces V a = L 2 (x a , d 4 x a ) as 2 ln x 2 aa+1 + 1 (x 2 aa+1 ) iλ ln(p 2 ap 2 a+1 )(x 2 aa+1 ) iλ + + 2 ln x 2 N 0 + ln(p 2 1 ) , where x aa+1 = x a − x a+1 ,p 2 a = −∂ a · ∂ a and x N +1 = x 0 . The point x 0 is effectively a parameter for the model, and we will always omit it from the set of coordinates. The spin chain (2) is the four-dimensional version of the open SL(2, C) Heisenberg magnet which describes the scattering amplitudes of high energy gluons in the Regge limit of QCD [7,12]. The integrability of (2) is realized by the commutative family of normal operators labeled by the spectral parameter u ∈ C and where By the introduction of the operator the Hamiltonian H is recovered from the expansion It follows from (4) and from the commutation relation [Q N (u), Q N (v)] = 0 at generic u and v, that the eigenfunctions of Q N diagonalize the Hamiltonian (2) as well. The spectra of these operators are labeled by the quantum numbers for a = 1, . . . , N , and we use to write Y = (Y 1 , . . . , Y N ). The spectral equation for the operator (3) reads where we denote x = (x 1 , . . . , x N ) and α,β stand for 2N auxiliary complex spinors |α 1 , . . . , |α N and |β 1 , . . . , |β N ∈ C 2 . arXiv:1912.07588v2 [hep-th] 10 Jan 2020 The eigenfunctions form an orthogonal set respect to the quantum numbers (Y, α, β), and the eigenvalue is factorized respect to the labels (5) into equal contributions As a consequence of (4) and (6) we obtained the spectrum of the Hamiltonian H as a sum of N independent terms Formulas (6), (7) show that the N -body system defined in (2) gets separated into N one-particle systems over the quantum numbers (5). In other words, the quantities (Y a , |α a , |β a ) are the separated variables of the system in the sense of [13][14][15][16], and the spectrum of (2) and (3) is degenerate in the spinors due to rotation invariance.
The representation over the separated variables (Y, α, β) is defined for a generic function φ(x) = φ(x 1 , . . . , x N ) by the linear transform The inverse transform of (8) provides the expansion of φ(x) over the basis of eigenfunctions where the sum runs over the non-negative integers n = (n 1 , . . . , n N ), the integrations dν = dν 1 · · · dν N are on the real line and the integration in the space of spinors Dα = Dα 1 · · · Dα N is defined aŝ The spectral measure in (9) can be extracted from the scalar product of eigenfunctions and it is given by in the notation ν ab = ν a − ν b and n ab = n a − n b .
All considerations done so far can be extended by an accurate analytic continuation of the parameter λ to the imaginary strip (−2i, +2i). In particular, at λ = i each site of the chain carries the representation ∆ = 1, =˙ = 0 of a bare scalar field in four dimensions. In this case at the point u = −1 the operator Q N (u) becomes proportional to the graph-building integral operator for a Feynmann diagram of square lattice topology According to (6) the representation of the operator (B N ) L over the separated variables factorizes completely a portion of size N × L of the planar fishnet diagram [17] in Fig.1, extending to a 4D space-time the analogue result in two-dimensions of [18]. As a direct application of our results, we computed a specific set of four-point functions of Fishnet CFT [19], providing a direct check to formula (14) of [20], obtained via arguments of AdS/CFT correspondence [21][22][23].
In the next two sections we present the explicit construction of the eigenfunctions of the model (2) by means of newly found integral identities.

GENERALIZED STAR-TRIANGLE IDENTITY
Our construction of a basis of eigenfunctions for Q N (u) follows the logic outlined in [24] for the two-dimensional model, and requires the formulation of certain conformal integral identities in 4D. First we consider a positive integer M ≤ N and set x µ 0 = 0 without loss of generality. We will denote where the symbols σ andσ are defined in terms of Pauli matrices The tensors (12) satisfy the light-cone condition where t µ1...µa are auxiliary tensors and a = 1, 1 , . . . , M . This property allows to define a family of degree-n homogeneous harmonic polynomials where Under a coordinate inversion x µ → x µ /x 2 such harmonic polynomials transform covariantly and it follows that using (13) it is possible to generalize the uniqueness -"star-triangle" -relation for a conformal invariant vertex of three scalar propagators [25] (see also [26,27] and references therein) to any symmetric traceless representation. The core of the generalized identity is the mixing operator acting on a pair of symmetric spinors |α, α = |α ⊗n ⊗ |α ⊗n of degrees n and n as where upon differentiation we set s = t = 0. The operator defined by (14) is a unitary solution of the Yang-Baxter equation and can be obtained via the fusion procedure [28] applied to the Yangian R-matrix R 1,1 (z). It follows that for any n, n ∈ N and under the constraint a + b + c = 4 the following identity holdŝ Setting n = 0, the identity (15) is equivalent to (A.11) of [29], and setting further n = 0 it degenerates to the scalar identity [25]. We point out that (15) is the four-dimensional versions of the 2D star-triangle relation which underlies the solution of the SL(2, C) Heisenberg magnet as in [24,30].

EIGENFUNCTIONS CONSTRUCTION
The eigenfunctions of the open conformal chain (2) can be obtained by a recursive procedure in the number of sites of the system. First of all we introduce the integral operatorsΛ αβ M,Ya = α|Λ M,Ya |β which at M = 1 reduces to a propagator in the irreducible represenation of scaling dimension ∆ = 2 + iλ + 2iν a and tensors rank n a of the principal series Making use of (15) at n = n a , n = 0 we verify that for any M > 1, moreover The iterative application of (17) for the length M going from N to 2, together with the initial condition (18), provides a recursive definition of the eigenfunctions of the model with N sites where the last factor is a suitable normalization and Such a function has a simple behavior in the permutation of two separated variables (Y, α, β), (Y , α , β ), encoded by the exchange propertŷ where z = i(ν − ν) and R = R n,n . Any permutation of the separated variables in (19) can be decomposed into elementary steps of type (20), defining a representation of the symmetric group generators Graphic representation of the integral kernel Λ α,β 3,Y (x1, x2, x3|x 1 , x 2 ) (left) and of the eigenfunction Ψ αβ (Y|x1, x2, x3) (right). Solid lines denote (x 2 ij ) −iλ , while the dashed ones indicate the presence of polynomials (13). The external arrows indicate symmetric spinors and the grey blobs are integrated points.
on the space of symmetric spinors s k |α = R n k ,n k+1 (i ν k+1,k ) |α 1 , . . . , α k+1 , α k , . . . , α N , and allowing to state the exchange symmetry The scalar product of two eigenfunctions can be written according to (19) in operatorial form, so that it can be reduced to N factorized single-site contributions of the type by the iterative application of the property , valid under the assumption Y = Y and where the trace means the cyclic contraction of indices in the space of symmetric spinors. As result the scalar product of two functions (19) takes the form of an orthogonality relation (22) where S N are the permutations of N objects and we introduced the compact notation The relations (21), (22) allow to conjecture the completeness of the proposed eigenfunctions (19) and to define the representation of separated variables as in (8), (9).

CONFORMAL FISHNET INTEGRALS
In analogy with the 2D results of [18], employing the results of the previous sections we will compute exactly the four-point correlation function for any N and L, where φ 1 (x) , φ 2 (x) are the two complex scalar N c × N c fields which appear in the Lagrangian of the conformal fishnet theory [19] in four dimensions In the planar limit [31] N c → ∞ the only Feynmann diagram which contributes to the perturbative expansion in the coupling ξ 2 of G N,L is given by the integral where the integration measure is dz = N,L a,b=1 d 4 z a,b and we set z 0b = x 1 , z N +1b = x 3 , z a0 = x 4 , z aL+1 = x 2 . Such a square-lattice integral can be expressed via the graphbuilding operator (11). Indeed, starting from the fishnet diagram one can transform it to (24) by the reductions of external points z a → x 1 , z a → x 3 followed by a conformal transformation. Therefore, as a functions G N,L (u, v) of the cross-ratios u = x 2 12 x 2 34 /(x 2 13 x 2 24 ) and v = x 2 14 x 2 23 /(x 2 13 x 2 24 ), the planar limit of (23) is equal to F N,L with reduced external points. According to (6) the integral kernel of (B N ) L in the space of separated variables is factorized as In order to restore the (u, v)-dependence of (24) one has first to expand the r.h.s. of (25) over the eigenfunctions via the inverse transform (9). Then, by the appropriate reduction of the external points and upon integration of spinors and normalization by the bare correlator, we get . After the redefinition n k → a k − 1, ν k → u k , x → z it coincides with the result of [20].
We shall conjecture further applications of the sep- 3. A Feynmann diagram contributing to the planar limit of Tr(φ 2 1 )(x1)Tr(φ 2 1 )(x2)Tr(φ † 4 1 )(x3) at order ξ 28 and its decomposition into hexagons. Here M1 = 1, M2 = 2, M3 = 2. Each color of a cut corresponds to the insertion of a different set of separated variables, as indicated on the hexagons. arated variables transform (9) to the computation of planar fishnet integrals. An interesting example in this sense is provided by the three-point function of "vacuum" operators In the planar limit the perturbative expansion of (27) in the coupling constant consist of regular square lattice diagrams drawn on a three-punctured sphere S 2 \{x 1 , x 2 , x 3 } as explained in [32] and exemplified in Fig.3. In the same spirit of "hexagonalisation" techniques [22,23,32,33] we perform three cuts on the diagram connecting the punctures, and insert along each cut a sum over the basis (19), labeled by the separated variables where Y a = [ν a , n a ], Z a = [µ a , m a ], U a = [τ a , t a ]. Let M i be the number of φ 2 φ † 2 wrappings around the puncture x i (see Fig.3). The representation of the two hexagons over the separated variables reads , and the form factor A is given by the overlapping of three eigenfunctions of type (19) at different values of x 0 for z = (z 1 , . . . , z M3 ), z = (z 1 , . . . , z M1 ), and z = (z 1 , . . . , z M2 ). Finally, the Feynmann integral is recovered by gluing the two hexagons via completeness sums ∼ n,m,tˆd ν dµ dτ µ(Y) µ(Z) µ(U)ˆDα · · · Dω |H| 2 .
An interesting reduction of the correlator (27) is obtained setting L = 0 and degenerating it to the two-point function Tr(φ N 1 )(x 1 )Tr(φ † N 1 )(x 3 ) , for which the planar fishnet lies on a cylinder and it is conformally equivalent to a "wheel" diagram [19,[34][35][36]. As a general fact the diagrams describing the planar limit of (27) develop UV divergences, which in our representation should be contained in the form factor (28). The elaboration of a regularization technique at this level is an intriguing task as it would enable the direct computation of several conformal data in the Fishnet CFT at finite order in the coupling.

CONCLUSIONS
We formulated and solved the spin chain of SO(1, 5) conformal spins for any number of sites N and for open boundary conditions, in the principal series representation of zero spin [11]. Its integrability is realized by a commuting family of spectral parameter-dependent operators Q N (u) which generate the conserved charges of the model. The spectrum of the model is separated into N symmetric contributions, each depending on quantum numbers which for this reason we call separated variables. We explained how to construct the eigenfunctions and prove their orthogonality, extending the logic of [24] to a four dimensional space-time by means of new integral indentities which generalize the star-triangle relation [25] to symmetric traceless tensors. Our results can be analytically continued from the representation of the principal series to real scaling dimensions, recovering the graph-building operator -introduced in 2D by the authors and V. Kazakov [18] -for the Feynmann diagrams of Fishnet CFT [19,37]. The variant of this graph-builder with periodic boundary was first introduced in [19] and coincides with theB-operator of the Fishchain holographic model [38][39][40]. Following the same steps as [18], we computed the planar limit of the fishnet correlator studied by B. Basso and L. Dixon providing a direct check of the formula (14) of [20]. The separation of variables (SoV) for non-compact spin magnets is a topic which recently attracted great attention [41][42][43][44][45][46], and SoV features appear in remarkable results of AdS/CFT integrability, for instance [47,48]. It has not escaped our notice that the properties of the proposed eigenfunctions immediately suggest their role in the SoV of the periodic SO(1, 5) spin chain [29], in full analogy with [30]. Moreover it would be interesting to apply our methods to the computation of other classes of Feynmann integrals, for example introducing fermions as in [49,50], or considering any space-time dimension and extending our results to the theory proposed in [51]. In the latter context, the functions (19) for N = 2 sites have been derived in a somewhat different form and applied to the formulation of the Thermodynamic Bethe Ansatz equations [52]. Finally we have conjectured how, by means of a cuttingand-gluing procedure inspired by [32], certain planar twoand three-point functions of the Fishnet CFT at finite coupling get factorized into simple contributions over the separated variables. This observation puts as a compelling future task the regularization of such formulas, in order to compare the results based on the AdS/CFT correspondence to a direct computation.