Exactly solvable magnet of conformal spins in four dimensions

ZMP–HH/19–27 Exactly solv able magnet of conformal spins in four dimensions Sergey Derk acho v a and Enrico Olivucci b a St. Petersbur g Dep artment of the Steklov Mathematic al Institute, 191023 St. Petersbur g, Russia b II. Institute for The or etic al Physics of the University of Hambur g, 22761 Hambur g, Germany W e pro vide the eigenfunctions for a quantum c hain of N conformal spins with nearest-neighbor in teraction and op en b oundary conditions in the irreducible representation of S O (1 , 5) of scaling dimension ∆ = 2 − iλ and spin num bers ` = ˙ ` = 0. The sp ectrum of the model is separated in to N equal con tributions, eac h dep endent on a quantum num ber Y a = [ ν a , n a ] which lab els a repre- sen tation of the principal series. The eigenfunctions are orthogonal and we computed the sp ectral measure b y means of a new star-triangle iden tity . Any p ortion of a conformal F eynmann diagram with square lattice top ology can be represented in terms of separated v ariables, and we reproduce the all-lo op “fishnet” integrals computed by B. Basso and L. Dixon via b ootstrap techniques. W e 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 sup ersymmetric N = 4 Y ang-Mills theory at finite order in the coupling, b y means of a cutting-and-gluing pro cedure on the square lattice. INTR ODUCTION The exactly solv able spin magnets [1, 2] constitute a class of condensed matter mo dels of wide in terest throughout theoretical and mathematical physics. In particular, the integrable chains of nearest-neigh bors in- teracting spins [3, 4] serve as a tool to enco de the sym- metries of lo cal or non-local operators in quantum field theory , providing a rich amoun t of non-p erturbativ e re- sults ranging from the scattering sp ectrum of high-energy gluons in QCD [5–7] to the conformal data of the super- symmetric N = 4 SYM and N = 6 ABJM theories [8]. The archet yp e mo del of this class is the S U (2) Heisenberg magnet of spin 1 2 , which for op en b oundary conditions is describ ed by the Hamiltonian H S U (2) = N − 1 X a =1 ~ σ a · ~ σ a +1 , (1) b eing ~ 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 S O (1 , 5) spin chain [9]. The latter mo del is relev ant for the study of cov ariant quantities in a four-dimensional conformal field theory (CFT) [10]. W e consider the homogeneous mo del in the irreducible unitary representation defined b y the scaling dimension ∆ = 2 − iλ, λ ∈ R , and the S O (4) spins ` = ˙ ` = 0 [11]. The Hamiltonian op erator acts on the Hilb ert spaces V a = L 2 ( x a , d 4 x a ) as H = N − 1 X a =1  2 ln x 2 aa +1 + ( x 2 aa +1 ) − iλ ln( ˆ p 2 a ˆ p 2 a +1 )( x 2 aa +1 ) iλ  + + 2 ln x 2 N 0 + ln( ˆ p 2 1 ) + ( x 2 N 0 ) − iλ ln( p 2 N )( x 2 N 0 ) iλ , (2) where x aa +1 = x a − x a +1 , ˆ p 2 a = − ∂ a · ∂ a and x N +1 = x 0 . The p oin t x 0 is effectively a parameter for the mo del, and we will alwa ys omit it from the set of co ordinates. The spin chain (2) is the four-dimensional v ersion of the op en S L (2 , C ) Heisen b erg magnet which describ es the scattering amplitudes of high energy gluons in the Regge limit of QCD [7, 12]. The integrabilit y of (2) is realized b y the comm utative family of normal op erators Q N ( u ) = Q 12 ( u ) · Q 23 ( u ) · · · Q N 0 ( u ) , (3) lab eled by the sp ectral parameter u ∈ R and where Q ij ( u ) = ( x 2 ij ) − iλ ( ˆ p 2 i ) u ( x 2 ij ) u + iλ . By the in tro duction of the op erator b Q N ( u ) = [ Q N ( u − iλ )] † Q N ( − iλ ) , the Hamiltonian H is recov ered from the expansion Q N ( u ) + b Q N ( u ) = 2 · 1 + u H + o ( u ) . (4) It follows from (4) and from the comm utation relation [ Q N ( u ) , Q N ( v )] = 0 at generic u and v , that the eigen- functions of Q N diagonalize the Hamiltonian (2) as well. The spectra of these ope rators are labeled b y the quan- tum n umbers Y a = 1 + n a 2 + iν a , Y ∗ a = 1 + n a 2 − iν a , ν a ∈ R , n a ∈ N , (5) for a = 1 , . . . , N , and we use to write Y = ( Y 1 , . . . , Y N ). The spectral equation for the op erator (3) reads Q N ( u ) · Ψ αβ ( x | Y ) = τ N ( u, Y ) Ψ αβ ( x | Y ) , where we denote x = ( x 1 , . . . , x N ) and α , β stand for 2 N auxiliary complex spinors | α 1 i , . . . , | α N i and | β 1 i , . . . , | β N i ∈ C 2 . 2 The eigenfunctions form an orthogonal set resp ect to the quan tum num b ers ( Y , α , β ), and the eigenv alue is fac- torized respect to the lab els (5) into equal contributions τ N ( u, Y ) = N Y a =1 τ 1 ( u, Y a ) , (6) τ 1 ( u, Y a ) = 4 u Γ  Y a − i 2 λ  Γ  Y ∗ a + u + i 2 λ  Γ  Y ∗ a + i 2 λ  Γ  Y a − u − i 2 λ  . As a consequence of (4) and (6) we obtained the sp ectrum of the Hamiltonian H as a sum of N independent terms η N ( Y ) = N X a =1  ψ  Y a − i 2 λ  + ψ  Y a + i 2 λ  + ln 4  + c.c. (7) F orm ulas (6),(7) show that the N -b ody system defined in (2) gets separated into N one-particle systems ov er the quan tum num bers (5). In other w ords, the quantities ( Y a , | α a i , | β a i ) are the separated v ariables of the system in the sense of [13–16], and the sp ectrum of (2) and (3) is degenerate in the spinors due to rotation inv ariance. The representation ov er the separated v ariables ( Y , α , β ) is defined for a generic function φ ( x ) = φ ( x 1 , . . . , x N ) by the linear transform e φ ( Y , α , β ) = ˆ d x Ψ αβ ( x | Y ) ∗ φ ( x ) . (8) The in verse transform of (8) provides the expansion of φ ( x ) o ver the basis of eigenfunctions φ ( x ) = X n ˆ d ν µ ( Y ) ˆ D α D β Ψ αβ ( x | Y ) e φ ( Y , α , β ) , (9) where the sum runs ov er the non-negativ e integers n = ( n 1 , . . . , n N ), the integrations d ν = dν 1 · · · dν N are on the real line and the integration in the space of spinors D α = D α 1 · · · D α N is defined as ˆ D α = ˆ C 2 dα e −h α | α i , h α | α i = | α (1) | 2 + | α (2) | 2 . The sp ectral measure in (9) can b e extracted from the scalar product of eigenfunctions and it is giv en by µ ( Y )= 1 N ! N Y a =1 ( n a +1) N Y b 6 = a  ν 2 ab + n 2 ab 4   ν 2 ab + ( n a + n b +2) 2 4  , (10) in the notation ν ab = ν a − ν b and n ab = n a − n b . All considerations done so far can b e extended b y an accurate analytic contin uation of the parameter λ to the imaginary strip ( − 2 i, +2 i ). In particular, at λ = − i eac h site of the c hain carries the represen tation ∆ = 1, ` = ˙ ` = 0 of a bare scalar field in four dimensions. In this x 1 x 3 x 2 x 0 x , 1 x , 2 x , 3 x 1 x 3 x 2 x 0 x , 1 x , 2 x , 3 FIG. 1. On the left the graph-building kernel B 3 ( x | x 0 ), where the lines are propagators 1 /x 2 ij , grey dots are external p oin ts and the blac k ones are in tegrated. On the righ t the p ortion of fishnet ( B 3 ) 4 with tw o fixed p oin ts x 0 (do wn) and ∞ (up). case at the point u = − 1 the op erator Q N ( u ) b ecomes prop ortional to the graph-building in tegral op erator for a F eynmann diagram of square lattice top ology B N φ ( x ) = 1 (2 π ) 4 N ˆ d x 0 φ ( x 0 ) N Y a =1 1 x 2 aa +1 x 2 aa 0 , (11) with x = ( x 1 , . . . , x N ), x 0 = ( x 0 1 , . . . , x 0 N ). Throughout the letter we denote x ab 0 = x a − x 0 b . According to (6) the represen tation of the operator ( B N ) L o ver the separated v ariables factorizes completely a portion of size N × L of the planar fishnet diagram [17] in Fig.1, extending to a 4 D space-time the analogue result in tw o-dimensions of [18]. As a direct application of our results, we computed a sp ecific set of four-point functions of Fishnet CFT [19], pro viding a direct chec k to formula (14) of [20], obtained via argumen ts of AdS/CFT correspondence [21–23]. In the next tw o sections we present the explicit construction of the eigenfunctions of the mo del (2) b y means of newly found integral identities. GENERALIZED ST AR-TRIANGLE IDENTITY Our construction of a basis of eigenfunctions for Q N ( u ) follo ws the logic outlined in [24] for the tw o-dimensional mo del, and requires the formulation of certain conformal in tegral identities in 4 D . First we consider a p ositiv e integer M ≤ N and set x µ 0 = 0 without loss of generalit y . W e will denote x = ( x 1 , . . . , x M ), x 0 = ( x 0 1 , . . . , x 0 M − 1 ). Let us introduce the tensors C αβ µ 1 µ 1 0 µ 2 ...µ M = h α | ¯ σ µ 1 σ µ 0 1 ¯ σ µ 2 · · · ¯ σ µ M | β i , (12) where the symbols σ and ¯ σ are defined in terms of Pauli matrices σ 0 = ¯ σ 0 = 1 , σ k = − ¯ σ k = iσ k , k = 1 , 2 , 3 . 3 The tensors (12) satisfy the light-cone condition t µ 1 ...µ a t ν 1 ...ν a C αβ µ 1 ...µ a ρ...µ M C αβ ν 1 ...ν a ρ ...ν M = 0 , where t µ 1 ...µ a are auxiliary tensors and a = 1 , 1 0 , . . . , M . This property allows to define a family of degree- n ho- mogeneous harmonic polynomials C αβ M ( x | x 0 ) n = h α | ¯ x 11 0 x 1 0 2 ¯ x 22 0 . . . ¯ x M 0 | β i n , (13) where x ij = σ µ x µ ij / | x ij | and ¯ x ij = ¯ σ µ x µ ij / | x ij | . Under a co ordinate inv ersion x µ → x µ /x 2 suc h harmonic p olyno- mials transform co v ariantly and it follo ws that using (13) it is possible to generalize the uniqueness - “star-triangle” - relation for a conformal inv ariant vertex of three scalar propagators [25] (see also [26, 27] and references therein) to an y symmetric traceless represen tation. The core of the generalized iden tity is the mixing op- erator acting on a pair of symmetric spinors | α, α 0 i = | α i ⊗ n ⊗ | α 0 i ⊗ n 0 of degrees n and n 0 as h α, α 0 | R n,n 0 ( z ) | β , β 0 i = Γ( z + n − n 0 2 )Γ( z + n 0 − n 2 ) Γ 2 ( z + n + n 0 2 ) × × ∂ n s ∂ n 0 t (1 + s h α | β i + t h α 0 | β 0 i + st h α | β 0 ih α 0 | β i ) z + n + n 0 2 , (14) where upon differentiation w e set s = t = 0. The op er- ator defined by (14) is a unitary solution of the Y ang- Baxter equation and can b e obtained via the fusion pro- cedure [28] applied to the Y angian R-matrix R 1 , 1 ( z ). Under the uniqueness c onstrain t a + b + c = 4 and for an y n, n 0 ∈ N the follo wing iden tity holds ˆ d 4 x 4 h α | ¯ x 14 x 43 | β i n h α 0 | ¯ x 34 x 42 | β 0 i n 0 ( x 2 14 ) a ( x 2 24 ) b ( x 2 34 ) c = = π 2 ( − 1) n A n,n 0 ( a, b, c ) ( x 2 12 ) (2 − c ) ( x 2 13 ) (2 − b ) ( x 2 23 ) (2 − a ) × × h α ¯ x 12 x 23 , α 0 | R n,n 0 ( c − 2) | β , ¯ x 31 x 12 β 0 i  c − 1 + n + n 0 2   2 − c + n 0 − n 2  . (15) with the coefficient A n,n 0 ( a, b, c ) = = Γ  2 − a + n 2  Γ  2 − b + n 0 2  Γ  3 − c + n 0 − n 2  Γ  a + n 2  Γ  b + n 0 2  Γ  c − 1 + n 0 − n 2  . Setting n 0 = 0, the identit y (15) is equiv alent to (A.11) of [29], and setting further n = 0 it degenerates to the scalar iden tity [25]. W e p oin t out that (15) is the four-dimensional versions of the 2 D star-triangle relation which underlies the solution of the S L (2 , C ) Heisen b erg magnet as in [24, 30]. EIGENFUNCTIONS CONSTRUCTION The eigenfunctions of the op en conformal c hain (2) can b e obtained by a recursive procedure in the num b er of sites of the system. First of all we introduce the integral op erators ˆ Λ αβ M ,Y a = h α | ˆ Λ M ,Y a | β i ˆ Λ αβ M ,Y a · φ ( x ) = ˆ d x 0 Λ αβ M ,Y a ( x | x 0 ) φ ( x 0 ) , (16) through its kernel Λ αβ M ,Y a ( x | x 0 ) = h α | Λ M ,Y a ( x | x 0 ) | β i = = C αβ M ( x | x 0 ) n a ( x 2 M 0 ) 1+ iν a + iλ/ 2 M − 1 Y a =1 ( x 2 a 0 a +1 ) − 1+ iν a + iλ/ 2 ( x 2 aa 0 ) 1+ iν a − iλ/ 2 ( x 2 aa +1 ) iλ , whic h at M = 1 reduces to a conformal propagator of scaling dimension ∆ = 1 + iλ/ 2 + iν a and tensor rank n a Λ αβ 1 ,Y a ( x 1 ) = h α | ¯ x 1 | β i n a ( x 2 1 ) 1+ iν a + iλ/ 2 . Making use of (15) at n = n a , n 0 = 0 w e v erify that Q M ( u ) ˆ Λ αβ M ,Y a = τ 1 ( u, Y a ) ˆ Λ αβ M ,Y a Q M − 1 ( u ) , (17) for an y M > 1, moreov er Q 1 ( u ) Λ αβ 1 ,Y a ( x 1 ) = τ 1 ( u, Y a ) Λ αβ 1 ,Y a ( x 1 ) . (18) The iterativ e application of (17) for the length M going from N to 2, together with the initial condition (18), pro vides a recursive definition of the eigenfunctions of the model with N sites Ψ αβ ( Y | x ) = ˆ Λ α N β N N ,Y N · · · ˆ Λ α 2 β 2 2 ,Y 2 · Λ α 1 β 1 1 ,Y 1 N Y a =1 r ( Y a ) a − 1 √ 2 π 2 N +1 , (19) where the last factor is a suitable normalization and r ( Y ) = Γ  Y − i λ 2  Γ  Y ∗ − i λ 2  Γ  Y + i λ 2  Γ  Y ∗ + i λ 2  . Suc h a function has a simple b eha vior in the p erm utation of t wo separated v ariables ( Y , α, β ), ( Y 0 , α 0 , β 0 ), enco ded b y the exc hange prop ert y ˆ Λ α 0 β 0 M ,Y 0 · ˆ Λ αβ M − 1 ,Y = h α 0 , α | ˆ Λ M ,Y 0 · ˆ Λ M − 1 ,Y | β 0 , β i = = r ( Y ) r ( Y 0 ) h α, α 0 | R ( z ) † ˆ Λ M ,Y · ˆ Λ M − 1 ,Y 0 R ( z ) | β , β 0 i , (20) where z = i ( ν 0 − ν ) and R = R n,n 0 . Any p erm utation of the separated v ariables in (19) can b e decomp osed in to elemen tary steps of type (20), defining a representation of the symmetric group generators s k Y = ( Y 1 , . . . , Y k +1 , Y k , . . . Y N ) , on the space of symmetric spinors s k | α i = R n k ,n k +1 ( i ν k +1 ,k ) | α 1 , . . . , α k +1 , α k , . . . , α N i , 4 x 1 x 2 x 3 α 1 α 2 α 3 x 0 x 1 x 2 x 3 α β x' 1 x' 2 x 0 β 1 x 0 β 2 x 0 β 3 FIG. 2. Graphic represen tation of the integral k er- nel Λ α,β 3 ,Y ( x 1 , x 2 , x 3 | x 0 1 , x 0 2 ) (left) and of the eigenfunction Ψ αβ ( Y | x 1 , x 2 , x 3 ) (right). Solid lines denote ( x 2 ij ) − iλ , while the dashed ones stand for the p olynomials (13) together with the denominators of t yp e ( x 2 i,i 0 ) and ( x 2 i 0 ,i +1 ) carrying the v ariables ν in the p o wer. The external arrows indicate sym- metric spinors and the grey blobs are integrated p oin ts. and allo wing to state the exchange symmetry Ψ αβ ( Y | x ) = Ψ s k ( α,β ) ( s k Y | x ) . (21) The scalar pro duct of tw o eigenfunctions can b e written according to (19) in op eratorial form, so that it can b e reduced to N factorized single-site contributions of the t yp e (Λ α 0 β 0 1 ,Y 0 ) † · Λ αβ 1 ,Y = 2 π 3 n + 1 δ n,n 0 δ ( ν − ν 0 ) h α | α 0 i n h β | β 0 i n , b y the iterativ e application of the property ( ˆ Λ α 0 β 0 M ,Y 0 ) † · ˆ Λ αβ M ,Y = h β 0 , α | ˆ Λ † M ,Y 0 · ˆ Λ M ,Y | α 0 , β i = r ( Y 0 ) r ( Y ) × × π 4 T r n 0 [ h α | R ( z ) | α 0 i ˆ Λ M − 1 ,Y h β 0 | R † ( z ) | β i ˆ Λ † M − 1 ,Y 0 ]  ( ν − ν 0 ) 2 + ( n − n 0 ) 2 4   ( ν − ν 0 ) 2 + ( n + n 0 +2) 2 4  , v alid under the assumption Y 6 = Y 0 and where the trace means the cyclic contraction of indices in the space of primed spinors. As result the scalar pro duct of tw o func- tions (19) tak es the form of an orthogonality relation µ ( Y ) − 1 N ! X π ∈ S N δ ( Y − π ( Y 0 )) h α | π | α 0 ih β 0 | π | β i , (22) where S N are the p ermutations of N ob jects and w e in- tro duced the compact notation δ ( Y − Y 0 ) = N Y a =1 δ n a ,n 0 a δ ( ν a − ν 0 a ) . The relations (21),(22) allow to conjecture the complete- ness of the prop osed eigenfunctions (19) and to define the represen tation of separated v ariables as in (8),(9). CONF ORMAL FISHNET INTEGRALS In analogy with the 2 D results of [18], employing the results of the previous sections w e will compute exactly the four-point correlation function G N ,L = h T r[ φ N 1 ( x 1 ) φ L 2 ( x 2 ) φ † N 1 ( x 3 ) φ † L 1 ( x 4 )] i , (23) for an y N and L , where φ 1 ( x ) , φ 2 ( x ) are the t wo complex scalar N c × N c fields which app ear in the Lagrangian of the conformal fishnet theory [19] in four dimensions L φ = N c tr[ ∂ µ φ † 1 ∂ µ φ 1 + ∂ µ φ † 2 ∂ µ φ 2 + (4 π ) 2 ξ 2 φ † 1 φ † 2 φ 1 φ 2 ] . In the planar limit [31] N c → ∞ the only F eynmann diagram whic h con tributes to the perturbative expansion in the coupling ξ 2 of G N ,L is giv en b y the in tegral ˆ d z (4 π 2 ) N L N Y a =0 1 ( z a,b − z a +1 ,b ) 2 ! L Y b =0 1 ( z a,b − z a,b +1 ) 2 ! , (24) where the integration measure is d z = Q N ,L a,b =1 d 4 z a,b and w e set z 0 b = x 1 , z N +1 b = x 3 , z a 0 = x 4 , z aL +1 = x 2 . Suc h a square-lattice integral can b e expressed via the graph- building op erator (11). Indeed, starting from the fishnet diagram F N ,L = N Y a =1 z 2 aa +1 ! ( B N ) L +1 N Y a =1 δ (4) ( z 0 a − z a ) ! , (25) one can transform it to (24) by the reductions of external p oin ts z a → x 1 , z 0 a → x 3 follo wed b y 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 p oin ts. According to (6) the integral kernel of ( B N ) L in the space of separated v ariables is factorized as f B L N ( Y 1 , . . . , Y N ) = 1 π 2 N L N Y a =1  1 4 ν 2 a + (1 + n a ) 2  L . (26) In order to restore the ( u, v )-dep endence of (24) one has first to expand the r.h.s. of (25) o v er the eigenfunctions via the in verse transform (9). Then, by the appropriate reduction of the external p oin ts and up on in tegration of spinors and normalization b y the bare correlator, we get G N ,L ( u, v ) = X n ∈ Z ˆ d ν µ ( Y ) N Y k =1 | x | − 2 iν k ( ¯ x/x ) ( n k +1) / 2 ( ν 2 k + ( n k + 1) 2 / 4) L + N , where u/v = x ¯ x , v = 1 / p (1 − x )(1 − ¯ x ). After the redefinition n k → a k − 1, ν k → u k , x → z it coincides with the result of [20]. W e shall conjecture further applications of the sep- 5 x 3 x 1 x 2 Y Y Z Z U U FIG. 3. A F eynmann diagram contributing to the planar limit of h T r( φ 2 1 )( x 1 )T r( φ 2 1 )( x 2 )T r( φ † 4 1 )( x 3 ) i at order ξ 28 and its de- comp osition into hexagons. Here M 1 = 1, M 2 = 2, M 3 = 2. Eac h color of a cut corresp onds to the insertion of a different set of separated v ariables, as indicated on the hexagons. arated v ariables transform (9) to the computation of planar fishnet in tegrals. An interesting example in this sense is provided by the three-p oin t function of “v acuum” op erators h T r( φ N 1 )( x 1 )T r( φ L 1 )( x 2 )T r( φ † N + L 1 )( x 3 ) i . (27) In the planar limit the p erturbative expansion of (27) in the coupling constant consist of regular square lattice dia- grams 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” tec hniques [22, 23, 32, 33] w e p erform three cuts on the diagram connecting the punc- tures, and insert along each cut a sum ov er the basis (19), lab eled by the separated v ariables ( Y , α , β ) , ( Z , λ , χ ) , ( U , κ , ω ) , where Y a = [ ν a , n a ] , Z a = [ µ a , m a ] , U a = [ τ a , t a ]. Let M i b e the num ber of φ 2 φ † 2 wrappings around the puncture x i (see Fig.3). The represen tation of the t wo hexagons o ver the separated v ariables reads | H | 2 ∼|A| 2 M 1 + M 3 Y a =1 " 1 ν 2 a + ( n a +1) 2 4 # N M 2 + M 3 Y b =1 " 1 µ 2 b + ( m b +1) 2 4 # L , and the form factor A is giv en b y the o v erlapping of three eigenfunctions of t yp e (19) at different v alues of x 0 A = ˆ d z d z 0 d z 00 Ψ αβ Y ( z , z 0 ) Ψ λχ Z ( z , z 00 ) Ψ κω U ( z 0 , z 00 ) , (28) for z = ( z 1 , . . . , z M 3 ) , z 0 = ( z 0 1 , . . . , z 0 M 1 ) , and z 00 = ( z 00 1 , . . . , z 00 M 2 ). Finally , the F eynmann in tegral is recov- ered b y gluing the t w o hexagons via completeness sums ∼ X n , m , t ˆ d ν d µ d τ µ ( Y ) µ ( Z ) µ ( U ) ˆ D α · · · D ω | H | 2 . An interesting reduction of the correlator (27) is obtained setting L = 0 and degenerating it to the t wo-point func- tion h T r( φ N 1 )( x 1 )T r( φ † N 1 )( x 3 ) i , for which the planar fish- net lies on a cylinder and it is conformally equiv alent to a “wheel” diagram [19, 34–36]. As a general fact the diagrams describing the planar limit of (27) dev elop UV div ergences, whic h in our represen- tation should be con tained in the form factor (28). The elab oration of a regularization tec hnique at this level is an in triguing task as it would enable the direct compu- tation of several conformal data in the Fishnet CFT at finite order in the coupling. CONCLUSIONS W e formulated and solv ed the spin chain of S O (1 , 5) conformal spins for any n umber of sites N and for op en b oundary conditions, in the principal series represen ta- tion of zero spin [11]. Its integrabilit y is realized by a comm uting family of spectral parameter-dependent op- erators Q N ( u ) whic h generate the conserved charges of the mo del. The sp ectrum of the mo del is separated in to N symmetric contributions, each dep ending on quantum n umbers whic h for this reason we call separated v ari- ables. W e explained ho w to construct the eigenfunctions and prov e their orthogonality , extending the logic of [24] to a four dimensional space-time by means of new inte- gral inden tities whic h generalize the star-triangle relation [25] to symmetric traceless tensors. Our results can b e analytically con tinued from the repre- sen tation of the principal series to real scaling dimen- sions, recov ering the graph-building op erator - intro- duced in 2 D by the authors and V. Kazak ov [18] - for the F eynmann diagrams of Fishnet CFT [19, 37]. The v ariant of this graph-builder with p erio dic boundary was first in troduced in [19] and coincides with the ˆ B -op erator of the Fishc hain holographic mo del [38–40]. F ollo wing the same steps as [18], w e computed the planar limit of the fishnet correlator studied b y B. Basso and L. Dixon pro viding a direct c heck of the form ula (14) of [20]. The separation of v ariables (SoV) for non-compact spin magnets is a topic which recently attracted great atten- tion [41–46], and SoV features app ear in remark able re- sults of AdS/CFT integrabilit y , for instance [47, 48]. It has not escaped our notice that the properties of the pro- p osed eigenfunctions immediately suggest their role in the SoV of the p eriodic S O (1 , 5) spin chain [29], in full analogy with [30]. Moreo ver it would be in teresting to apply our metho ds to the computation of other classes of F eynmann in tegrals, for example introducing fermions as in [49, 50], or considering an y space-time dimension and extending our results to the theory proposed in [51]. In the latter context, the functions (19) for N = 2 sites ha v e b een derived in a somewhat different form and applied to the formulation of the Thermo dynamic Bethe Ansatz equations [52]. Finally we hav e conjectured how, by means of a cutting- and-gluing procedure inspired by [32], certain planar tw o- and three-p oint functions of the Fishnet CFT at finite 6 coupling get factorized in to simple con tributions o v er the separated v ariables. 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