Bi-scalar integrable CFT at any dimension

We propose a $D$-dimensional generalization of $4D$ bi-scalar conformal quantum field theory recently introduced by G\"{u}rdogan and one of the authors as a strong-twist double scaling limit of $\gamma$-deformed $\mathcal{N}=4$ SYM theory. Similarly to the $4D$ case, this D-dimensional CFT is also dominated by"fishnet"Feynman graphs and is integrable in the planar limit. The dynamics of these graphs is described by the integrable conformal $SO(D+1,1)$ spin chain. In $2D$ it is the analogue of L. Lipatov's $SL(2,\mathbb{C})$ spin chain for the Regge limit of $QCD$, but with the spins $s=1/4$ instead of $s=0$. Generalizing recent $4D$ results of Grabner, Gromov, Korchemsky and one of the authors to any $D$ we compute exactly, at any coupling, a four point correlation function, dominated by the simplest fishnet graphs of cylindric topology, and extract from it exact dimensions of R-charge 2 operators with any spin and some of their OPE structure constants.


INTRODUCTION
Conformal field theories (CFT) are ubiquitous in two dimensions [1], and quite a few supersymmetric CFTs in D = 3, 4, 6 dimensions are known.But well defined non-supersymmetric CFTs in D > 2, such as 3D Ising or Potts models, or Banks-Zaks model [2], are rare species.The CFTs at D > 2 which in addition are integrable, such as 4D N = 4 SYM and ABJM theories in 't Hooft limit, are true exceptions [3] [4].That's why a new family of integrable CFTs obtained in [5] as a special double scaling limit of γ-deformed N = 4 SYM seems to be an important and instructive example.This theory can be studied via quantum spectral curve (QSC) formalism [6][7][8] or using the integrability of its dominant Feynman graphs via the conformal, SU (2, 2) noncompact spin chain.A nice particular case of this family is the 4D bi-scalar theory, dominated by "fishnet" type Feynman graphs [5,9].
We propose here the following D-dimensional generalization of the 4D bi-scalar theory introduced in [5] where the non-local (for general D) operators in kinetic terms should be understood as an integral kernel The propagator of scalar fields is its functional inverse: The typical structure in the bulk of sufficiently big planar Feynman graphs in this theory is that of the regular square lattice ("fishnet" graphs, proposed in [10] as an integrable lattice model), by the same reasons as in 4D case [5,9], namely, due to the presence of the single chiral interaction vertex in the Lagrangian, and the absence of its hermitian conjugate.For example, the graphs renormalizing local "vacuum" operator tr(φ j ) L are those of the "wheel" type and they can be studied via the integrable conformal SO(2, D) spin chain [11], as was suggested for 4D case in [12].The dimensions of operators of the type tr[φ 3 1 (φ † 2 φ 2 ) n ] have been also studied in 4D [12] by QSC methods.It is not clear whether this method can be generalized to our D dimensional model.But the spin chain methods certainly can.
In general, the propagators of the fishnet graphs of the model (1) are different in two different directions: |x − y| −D/2+ω for φ 1 fields and |x − y| −ω for φ 2 fields.Let us concentrate here on the "isotropic" case ω = D/4.In order to maintain the renormalizability we should add to (1) the following double-trace counterterms [13,14] ) Notice that the first term disappears in the "nonisotropic" case ω = D 4 since the couplings of two terms As it was suggested in [15] and explicitly shown in [16] for the 4D case, the "isotropic" bi-scalar theory with Lagrangian L φ + L dt has two fixed points.We generalize here this result to any dimension, up to two loops, computing the corresponding Feynman graphs (Fig. 1) contributing to the β α1 -function.Its two zeroes are At this critical coupling α 1 (ξ) the bi-scalar theory becomes a genuine non-unitary CFT at any coupling ξ.The operators tr(φ 1 φ 2 ), and tr(φ 1 φ † 2 ) are protected in the planar limit [16].
In this paper, generalizing the 4D results of [16] to any D, we will compute exactly a particular four-point function and read off from it the exact scaling dimensions and certain OPE structure constants of operators of the type tr(φ 1 ∂ S + φ 1 (φ † 2 φ 2 ) n ) + permutations.Their dimensions will be given by a remarkably simple exact relation which reduces of course at 4D to the result of [16].For even D it gives D different solutions ∆(ξ) = ∆ 0 + γ(ξ).
At odd (or non-integer) D there are infinitely many, in general complex, solutions.At weak coupling the two complex conjugate solutions at S = 0 describe anomalous dimensions of the operator tr(φ 1 φ 1 ) at the two fixed points.In a similar way, for any S ∈ 2Z the real weak coupling solution describes the operators of the type tr(φ 1 ∂ S + φ 1 ).For D = 2m, m ∈ N the L.H.S. of ( 6) factorizes into a polynomial of degree 2m and 2m roots of eq.( 6) describe the scaling dimension of the exchanged operators in the OPE channel x 3 → x 4 of (11) together with their shadows ∆ = D − ∆.At ξ = 0 we get the bare dimensions of physical operators (i.e., excluding "shadow" operators) At D = 2 the only solution gives the dimension ∆ = 1 + S 2 − 4ξ 4 of the local twist-2 operators tr(φ 1 ∂ S + φ 1 ) while at 4D the additional ∆ 0 − S = 4 describes twist-4 operators [16].As an example, at D = 6 and S = 0 the possible non-shadow solutions for ( 6) are ∆ 0 = 3, 5, 7.They can be realized as tr(φ 2 1 ) for ∆ 0 = 3, linear combinations of tr(φ 1 ∆φ 1 ), tr(∂ µ φ 1 ∂ µ φ 1 ) for ∆ 0 = 5 and of tr(φ 1 ∆ 2 φ 1 ), tr(∆φ Diagonalizing the mixing matrix of these operators at ξ = 0 we would obtain operators with non-trivial, ξ-dependent anomalous dimensions, as well as the so called log-multiplets, omnipresent in this non-unitary theory [17,18], containing the operators with zero anomalous dimension.Eq.( 6) predicts that all the exchange operators from this set acquire non-trivial anomalous dimensions, whereas the operators belonging to log-multiplets never appear among them.This appears to be true at any even dimension D.
As a general rule, according to the eq.( 6) the operators of the type { tr(φ 2  1 ∂ S + (φ 2 φ † 2 ) n ) + permutations} appear in the multiplets only at D/4 ∈ N, n = 1.We will find below from the exact 4-point function the conformal structure constants of these operators with two scalar fields.

INTEGRABILITY OF D-DIMENSIONAL BI-SCALAR CFT
As it was noticed in [5] and further developed in [9,12,16], the 4D case of the theory (1), with ω = 1, is integrable in the planar limit.On the one hand, this integrability is the direct consequence of integrability of γ-twisted planar N = 4 SYM theory, from which it was obtained in the double scaling limit combining strong imaginary twist and weak coupling.On the other hand, this integrability was explicitly related in [5] to the fact that the bi-scalar theory was dominated by the integrable "fishnet" Feynman graphs [10], [19].
Apart from 4D case, at arbitrary D our bi-scalar model (1) does not have any integrable SYM origin.But the arguments of equivalence to the integrable conformal SO(1, D+1) spin chain do work.Namely, let us introduce the D-dimensional analogue of the 4D "graph-building" 2. Graphical representation of the transfer matrix as a convolution of R-kernels according to formulas ( 8)and ( 9).Black dots are integration points and the weights of propagators are written in the second and third R-kernel.
operator [5]: schematically presented on Fig. 3.It is easy to see that a power of this operator H M L generates a fishnet Feynman graph with topology of a cylinder of length M with the circumference L. Now, in analogy with the 4D observation of [12], we notice that this operator can be related to the transfer-matrix of integrable SO(1, D + 1) conformal Heisenberg spin chain [20] presented on Fig. 2: where we introduced the R-matrix acting as an integral operator with the normalization constant Indeed, in analogy with 4D case [12], at a particular value of spectral parameter this transfer matrix becomes the graph-building operator (7) at any D presented on Fig. 3.

EXACT 4-POINTS CORRELATION FUNCTION
In analogy with 4D results of [16], employing the Ddimensional conformal symmetry of the theory (1),(4) we 3. Graphical representation of the kernel of the graphbuilding operator for generic D and ω.It is otained by setting u = − D 4 in the transfer matrix (8) presented on Fig. 2, so that x jj ′ +1 -type type propagators disappear while x j ′ +1 − yjtype propagators are replaced by δ(x j ′ +1 − yj ) factors.After that, integration over the points yj is equivalent to setting yj = x j ′ +1 .
will compute exactly the four-point correlation function where the notation is introduced for the operators Here G(u, v) is a finite function of 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 ), invariant under the exchange of points x 1 ↔ x 2 and x 3 ↔ x 4 .The OPE expansion leads to the formula where the sums run over operators with scaling dimensions ∆ and even Lorentz spin S.Here C ∆,S is the corresponding OPE coefficient (structure constant) and g ∆,S (u, v) is the known D dimensional conformal block (see (2.9) and sections 4,5 in [21]).If we compute (11), we will identify the conformal data for the operators emerging in the OPE of O(x 1 , x 2 ).
In the planar limit G is given by the set of fishnet Feynman diagrams presented in Fig. 4. Summing up the corresponding perturbation series we encounter a geometric progression involving the combination of operators α 2 V + ξ 4 H 2 , where α 2 = α 2 ± is the double-trace coupling at the fixed point, V is the operator inserting the double-trace vertex which is the D dimensional version of (11) in [16], and the operator H 2 defined by ( 7) adds a scalar loop inside the diagram.Hence we obtain the following representation where x ij ≡ x i − x j .[22] Remarkably, the operators V and H 2 commute with the generators of the conformal group, as in the particu-...
where ∆ = D 2 + 2iν and ∂ 0 ≡ (n∂ x0 ), with n being an auxiliary light-cone vector.The state Φ ∆,S,n belongs to the principal series of the conformal group and can be represented in the form of a conformal three-point correlation function where the operator O ∆,S,n (x 0 ) carries the scaling dimension ∆ and Lorentz spin S, and C ∆,S is the 3-points structure constant.The states (14), satisfy the orthogonality condition [23,24] where Calculating the corresponding eigenvalues of the operators V and H we find where the function h(∆, S) is given by (6).Applying ( 15)-( 17), we can expand the correlation function (13) over the basis of states ( 14).This yields the expansion of G over conformal partial waves defined by the operators where ∆ = D 2 + 2iν, and µ ∆,S = π D /(c 2 (ν, S)h ∆,S ) is related to the norm of the state (15).The fact that the dependence on α 2 disappears from (18), can be understood as follows.Viewed as a function of S, ξ 4 /h ∆,S develops poles at ν = ±iS which pinch the integration contour in (18), for S → 0. The contribution of the operator V is needed to make a perturbative expansion of ( 18) well-defined.For finite ξ 4 , these poles provide a vanishing contribution to (18), but generate a branch-cut −ξ 4 singularity of G(u, v), as in 4D case [16].
At small u, we close the integration contour in (18) to the lower half-plane and pick up residues at the poles located at solutions of ( 6) and satisfying the unitarity bound Re ∆ > S. The resulting expression for G(u, v) takes the expected form (12), with the OPE coefficients given by where the residue is computed w.r.t. the appropriate solution of (6) for each relevant operator.For instance, we can consider tr(φ 2 1 ) † , which is exchanged for any even D; then the perturbative expansion of ( 19) is (20) The relations (6) and (19) define exact conformal data of operators propagating in the OPE channel x 1 → x 2 .

CONCLUSIONS
We showed that the strongly γ deformed N = 4 SYM theory proposed in [5] is just the 4-dimensional representative of a wider, D-dimensional family of theories of two complex scalar fields obtained by modifying the propagators of fields in a D-dependent way.Similarly to the 4D case [16], they turn out to be conformal, if we add to the action certain double-trace terms with specific couplings, and integrable at any D in the planar limit.There are two such complex conjugate values of these couplings and we compute them perturbatively up to two loops.The integrability is explicit due to the domination of sufficiently large orders of perturbation theory by the "fishnet" Feynman diagrams.The cylindric fishnet graphs, related to the renormalization of vacuum tr(φ L 1 ) operators, can be created by multiple application of a "graphbuilding" operator which appears to be an integral of motion of the integrable conformal SO(D + 1, 1) spin chain.We also generalize the bi-scalar model to a CFT with different propagators for the fields φ 1 and φ 2 , leading to "non-isotropic" Feynman graphs.The underlying graph building operator has representations with different conformal spins in two directions on the fishnet graph.In the 2D case the fishnet graphs are described by the same SL(2, C) chain as used for the dynamics of generalizes Lipatov's reggeized gluons [25] but with different value of spin, s = 1/4 in isotropic case, instead of the BFKL reggeized gluon spin s = 0.This spin chain, extensively studied in the literature [26][27][28], is restored in the singular limit ω → 0 of our bi-scalar model (1).In the spirit of [16], we computed here the exact four point correlator at any D as an expansion into conformal blocks with explicit OPE coefficients and dimensions of exchange operators in one of the channels.In 1D case, our results are similar to those of the scalar version of cSYK fermionic theory [29] at q=4.For even D we found a finite, D-dependent number of local exchange operators at a given spin and dimension.The explicit form of this operators can be obtained by the analysis of the mixing matrix for their quantum multiplets [12,30].This becomes more complicated as the dimension grows due to growing rank of the multiplets and the number of transitions, together with log-CFT effects which arise starting from 4D because of chirality.
Although the lagrangian (1) of our theory is nonlocal at general D (apart from the sequence D ∈ 4N in "isotropic" case), it does not prevent the existence of "normal" OPE data in this theory, which is more important for the physical interpretation of this CFT.Moreover it would be interesting to generalize to any D the results for fishnet graphs of the type considered in [31] and to the correlation functions for operators involving more than two scalars.Finally, an important question remains, as in 4D, whether these theories have any string duals at any D, according to the original proposal of G .'t Hooft [32].

FIG. 4 .
FIG.4.General fishnet graphs up to α 2 1 order in the expansion of four point function(13).