Strongly \gamma-deformed N=4 SYM as an integrable CFT

We demonstrate by explicit multi-loop calculation that \gamma-deformed planar N=4 SYM, supplemented with a set of double-trace counter-terms, has two nontrivial fixed points in the recently proposed double scaling limit, combining vanishing 't Hooft coupling and large imaginary deformation parameter. We provide evidence that, at the fixed points, the theory is described by an integrable non-unitary four-dimensional CFT. We find a closed expression for the four-point correlation function of the simplest protected operators and use it to compute the exact conformal data of operators with arbitrary Lorentz spin. We conjecture that both conformal symmetry and integrability should survive in \gamma-deformed planar N=4 SYM for arbitrary values of the deformation parameters.


INTRODUCTION
The most general theory which admits an AdS 5 dual description in terms of a string σ-model [1,2] is believed to be γ-deformed N = 4 SYM [3,4].At the classical level, this σ-model is integrable and conformal.At the quantum level, it admits a solution in terms of the γdeformed quantum spectral curve (QSC γ ) [5][6][7][8][9].It is not obvious, however, that this solution yields the correct description of γ−deformed planar N = 4 SYM at any 't Hooft coupling g 2 = g 2 YM N c , since it automatically implies conformal symmetry and integrability of the theory.Both properties are highly debated, especially due to the loss of supersymmetry for the general deformation parameters γ 1 , γ 2 , γ 3 , breaking the R-symmetry SU (4) → U (1) 3 .
The main danger for both conformality and integrability in this theory comes from the fact that γ−deformed N = 4 SYM is not complete at the quantum level [10][11][12].Namely, in order to preserve renormalizability, it has to be supplemented with new double-trace counter-terms of the kind tr(φ j φ † k ) tr(φ k φ † j ) and tr(φ j φ k ) tr(φ † j φ † k ), with φ j=1,2,3 being a complex scalar field [10,13,14].The corresponding coupling constants run with the scale, thus breaking the conformal symmetry.For example, for the double-trace interaction term α 2 jj tr(φ j φ j ) tr(φ † j φ † j ) the one-loop beta-function is given by [11] where γ ± 1 = ∓ 1 2 (γ 2 ± γ 3 ), γ ± 2 = ∓ 1 2 (γ 3 ± γ 1 ), and γ ± 3 = ∓ 1 2 (γ 1 ± γ 2 ).However, at weak coupling, the betafunction has two fixed points β α 2 jj (α jj ) = 0: At these fixed points, which should persist at arbitrary values of g 2 and arbitrary N c , γ-deformed N = 4 SYM should be a genuine non-supersymmetric CFT.In addition, it is natural to conjecture that the QSC γ formalism gives the integrability description of this theory in the planar limit precisely at the fixed points!To elucidate the role of the double-trace couplings we examine the scaling dimensions of the operators tr(φ J j ).Such operators are protected in the undeformed theory but receive quantum corrections for nonzero deformation parameters γ i .For J ≥ 3 the contribution of the double-trace terms to their anomalous dimensions γ J is suppressed in the planar limit [15].However, this is not the case for J = 2 for which γ J=2 would diverge without the double-trace coupling contribution.At the fixed points (2), we get a finite but complex anomalous dimension This means that we deal with a non-unitary CFT.
In this paper we confirm explicitly, in the double scaling (DS) limit introduced in [15], that γ-deformed planar N = 4 SYM does have a conformal fixed point parameterized by g 2 and the three deformation parameters γ 1 , γ 2 , γ 3 .The existence of this fixed point was first discussed in [12] in the DS limit.
In this paper we compute the beta-functions for the double-trace couplings at 7 loops, confirming that the bi-scalar theory with Lagrangian L φ + L dt given by ( 4) and ( 5) is a genuine non-unitary CFT at any coupling ξ.We examine the two-point correlation functions of the operators tr(φ 1 φ 2 ) and tr(φ 1 φ † 2 ) in this theory and find that they are protected in the planar limit.Moreover, we compute exactly, for any ξ, the four-point function of such protected operators and apply the OPE to show that the scaling dimension of the operator tr(φ 1 φ 1 ) satisfies the remarkably simple exact relation Its solutions define four different functions ∆(ξ).At weak coupling, the solutions ∆ = 2∓2iξ 2 +O(ξ 6 ) describe scaling dimensions of the operator tr(φ i φ i ) (with i = 1, 2) at the two fixed points.The two remaining solutions, ∆ = 4 + ξ 4 + O(ξ 8 ) and ∆ = −ξ 4 + O(ξ 8 ), describe scaling dimensions of a twist-four operator, carrying the same U (1) charge J = 2, and its shadow, respectively.As another manifestation of integrability of the biscalar theory, relation ( 6) can be reproduced [16] by means of the QSC formalism [5][6][7] (see [35]).

PERTURBATIVE CONFORMALITY OF BI-SCALAR THEORY
In order to compute the beta-functions for the doubletrace couplings, we consider the following two-point correlation functions of dimension 2 operators The reason for this choice is that, in the planar limit, each G i receives contributions from Feynman diagrams involving double-trace interaction vertices of one kind only.As a consequence, G i depends only on two coupling constants, ξ and α i .For arbitrary values of the couplings α i , the renormalized correlation function G i (x) satisfies the Callan-Symanzik evolution equation depending on the beta-function for the coupling α i .
To compute the correlation functions (7) we employ dimensional regularization with d = 4 − 2ǫ.We start with G 2 and G 3 .In the planar limit, they receive contributions from Feynman diagrams shown in Fig. 1(left).They consist of a chain of scalar loops joined together through single-or double-trace vertices.In momentum space, their contribution to G i forms a geometric progression.In configuration space, the bare correlation function is given by where i = 2, 3 and G (0) i denotes the Born level contribution.To obtain a finite result for the correlation function, we have to replace bare couplings with their renormalized values and perform renormalization of the operators in ( 7) by multiplying G i by the corresponding Z Gi −factor.Requiring Z Gi G i to be finite for ǫ → 0 leads to the following expression for Z i in the minimal subtraction scheme: In the standard manner, we use this relation to find the exact beta-function for the coupling α i (for i = 2, 3) We deduce from this relation that the beta-functions vanish for α 2 i = ξ 2 , which also implies Z i = 1.As follows from (8), the correlation function at the fixed point is i , so that the operators tr(φ 1 φ 2 ) and tr(φ 1 φ † 2 ) are protected.
The calculation of the correlation function G 1 is more involved.In the planar limit it receives contributions from Feynman diagrams shown in Fig. 1(right) and those known as "wheel" diagrams [17].In momentum space, their contribution to G 1 factorizes into a product of Feynman integrals I L that form a geometric progression, Here the sum over ℓ runs over double-trace vertices and I L (p) denote (2L + 1)−loop scalar "wheel" integrals with 2L internal vertices shown in Fig. 1(right).
In dimensional regularization, I L (p) takes the form I L (p 2 ) = (p 2 ) −(2L+1)ǫ /ǫ 2L+1 (c 0 + c 1 ǫ + . . .), with L−dependent coefficients c i .Expressions for I L at L = 0, 1 are known in the literature [18], we computed I L for L = 2, 3. Using the obtained expressions we determined the expression for the bare correlation function G 1 (x) up to 7th order in perturbation theory.Going through the renormalization procedure, we use (9) to express G 1 (x) in terms of renormalized coupling constants ξ 2 , α 2  1 and require Z G1 G 1 (x) to be finite for ǫ → 0. This fixes the coupling renormalization factor Z 1 and allows us to compute the corresponding beta-function For β 1 we obtained in the minimal subtraction scheme where the functions a, b, and c are given by At weak coupling their expansion runs in powers of ξ 4 .
Similarly to (11), β 1 is a quadratic polynomial in the double-trace coupling α 2 1 .This property follows from the structure of irreducible divergent subgraphs of Feynman diagrams shown in Fig. 1.As a consequence, β 1 has two fixed points For these values of the double-trace coupling, the correlation function scales as G 1 (x) ∼ 1/(x 2 ) ∆ , where the scaling dimension ∆ ± = ∆(α 2 1,± ) is given by [36] ∆ Notice that, in distinction from (15), the expansion of ∆ ± runs in powers of ξ 4 .It is straightforward to verify that ∆ ± satisfy the exact relation (6).Curiously, there exists the following relation between the functions ( 14) and the scaling dimensions at the fixed point: It can be understood as follows.For generic complex ξ, we find using (13) that for µ → 0 and µ → ∞ the coupling α 1 (µ) flows into one of the fixed points α ± .Then, in the vicinity of a fixed point, for µ → ∞, the Callan-Symanzik equation fixes the form of renormalized G 1 (p) where α 2 1 (µ), α 2 1 (p) are in the vicinity α 2 ± , and γ(α) = ∆ − 2 is the anomalous dimension.We recall that the bare correlation function G 1 (p) is given by the geometric progression (12), so that as a function of α 2  1 it has a simple pole at some α 2 1 .After the renormalization procedure, α 2  1 is effectively replaced by a renormalized coupling constant defined at the scale µ 2 = p 2 .The requirement for G 1 (p) to have a simple pole in α 2 1 (p) fixes the exponent on the left-hand side of ( 18) to be 1, leading to (17).

EXACT CORRELATION FUNCTION
can exploit the conformal symmetry to compute exactly the four-point correlation function which is obtained from the two-point function G 1 (x) defined in ( 7) by point splitting the scalar fields inside the traces.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 .It admits the conformal partial wave expansion where the sum runs over operators with scaling dimensions ∆ and even Lorentz spin S.Here C ∆,S is the corresponding OPE coefficient and g ∆,S (u, v) is the conformal block [19].Having computed (19), we can identify the conformal data of the operator tr[φ 2 1 (x)] by examining the leading asymptotic behaviour of G for x 2 12 → 0. In the planar limit G is given by the same set of Feynman diagrams as G 1 (see Fig. 1), with the only difference that two pairs of scalar lines joined at the left-and rightmost vertices are now attached to the points x 1 , x 2 and x 3 , x 4 respectively.The fact that the contributing Feynman diagrams have a simple iterative form allows us to obtain the following compact representation for G: Here ± is the double-trace coupling at the fixed point, and V, H are integral operators 22) where Φ(x 1 , x 2 ) is a test function.Expanding (21) in powers of α 2 and ξ 4 we find that the operator H adds a scalar loop inside the diagram whereas V inserts a doubletrace vertex.The operators V and H are not well-defined separately, e.g. for an arbitrary Φ(x i ) the expressions for α 4 V 2 Φ(x i ) and ξ 4 HΦ(x i ) are given by divergent integrals.However, at the fixed point, their sum is finite by virtue of conformal symmetry.
A remarkable property of the operators V and H is that they commute with the generators of the conformal group.This property fixes the form of their eigenstates where ∆ = 2 + 2iν and ∂ 0 ≡ (n∂ x0 ), with n being an auxiliary light-cone vector.state Φ ∆,S,n belongs to the principal series of the conformal group and admits a representation in the form of the conformal three-point correlation function where the operator O ∆,S,n (x 0 ) carries the scaling dimension ∆ and Lorentz spin S. The states (23) satisfy the orthogonality condition [20,21] where ∆ ′ = 2 + 2iν ′ , Y (x 00 ′ ) = (n∂ x0 )(n ′ ∂ x 0 ′ ) ln x 2 00 ′ , and .
Calculating the corresponding eigenvalues of the operators (22) we find where the function h(∆, S) is given by Applying ( 25)-( 27) we can expand the correlation function (21) over the basis of states (23).This yields the expansion of G over conformal partial waves defined by the operators O ∆,S (x 0 ) in the OPE channel O(x 1 )O(x 2 ) where ∆ = 2 + 2iν, and µ ∆,S = 1/c 2 (ν, S) is related to the norm of the state (25).The fact that the dependence on the double-trace coupling α 2 disappears from ( 29) can be understood as follows.At weak coupling, expansion of G(u, v) runs in powers of ξ 4 /h ∆,S .Viewed as a function of S, ξ 4 /h ∆,S develops poles at ν = ±iS which pinch the integration contour in (29) for S → 0. The contribution of the operator V is needed to make a perturbative expansion of (29) well-defined.For finite ξ 4 , these poles provide a vanishing contribution to (29) but generate a branch-cut −ξ 4 singularity of G(u, v).
At small u, we close the integration contour in (29) to the lower half-plane and pick up residues at the poles located at and satisfying Re ∆ > S. The resulting expression for G(u, v) takes the expected form (20) with the OPE coefficients given by .

CONCLUSIONS
We demonstrated by explicit multi-loop calculation that the strongly γ−deformed planar N = 4 SYM has two nontrivial fixed points whose position depends on the properly rescaled 't Hooft coupling.We also provided evidence that, at the fixed points, it is described by an integrable non-unitary four-dimensional conformal field theory.Namely, we found a closed expression for the four-point correlation function of the simplest protected operators and used it to compute the exact conformal data (scaling dimensions and OPE coefficients) of twist−2 and twist−4 operators with arbitrary Lorentz spin.In general, correlation functions in this theory are dominated by fishnet graphs [15,22] which admit a description in terms of integrable noncompact Heisenberg spin chains [23][24][25].Following [15,24,28], the integrability can be used to compute these correlation functions and also the amplitudes [26,27].
We conjecture that both conformal symmetry and integrability should survive in γ−deformed planar N = 4 SYM for arbitrary values of the deformation parameters γ i .The underlying integrable non-unitary CFT 4 can be studied using the QSC γ formalism [5][6][7][8].
The integrable non-unitary CFTs of the kind considered here also exist in lower/higher dimensions.The known examples include a two-dimensional effective theory describing the high-energy limit of QCD [29], where the two-dimensional fishnet graphs can also be studied [30], the three-dimensional strongly γ-deformed ABJM model [28], and a six-dimensional three-scalar model [31] for which the "mother" gauge theory is not known (see also [37]).According to [31], the latter two theories are self-consistent CFTs and do not require adding double-trace counter-terms.
It would be interesting to find the dual string description of the bi-scalar theory.It might be nontrivial due to the tachyon in γ−deformed AdS 5 × S 5 [32].

FIG. 1 .
FIG.1.Feynman diagrams contributing to two-point correlation functions G2, G3 (left) and G1 (right) in the planar limit.Interaction vertices in the left diagram describe either the single-trace coupling ξ 2 or the double-trace coupling α 2 i (with i = 2, 3) depending on the choice of Gi.The right diagram consists of the chain of scalar loops joined together through the double-trace coupling α 2 1 .Each internal scalar loop is built using the single-trace coupling ξ 2 .