Bi-scalar integrable CFT at any dimension
aa r X i v : . [ h e p - t h ] O c t LPTENS–18/01, ZMP–HH–18/04
Bi-scalar integrable CFT at any dimension
Vladimir Kazakov a,b and Enrico Olivucci c a Laboratoire de Physique Th´eorique de l’ ´Ecole Normale Sup´erieure,24 rue Lhomond, F-75231 Paris Cedex 05, France b PSL University, CNRS, Sorbonne Universit´es,UPMC Universit´e Paris 6 c II. Institut f¨ur Theoretische Physik,Universit¨at Hamburg, Luruper Chaussee 149,22761 Hamburg, Germany
We propose a D -dimensional generalization of 4 D bi-scalar conformal quantum field theory re-cently introduced by G¨urdogan and one of the authors as a particular strong-twist limit of γ -deformed N = 4 SYM theory. Similarly to the 4 D case, the planar correlators of this D -dimensionaltheory are conformal and dominated by “fishnet” Feynman graphs. The dynamics of these graphs isdescribed by the integrable conformal SO (1 , D + 1) spin chain. In 2 D it is the analogue of L. Lipa-tov’s SL (2 , C ) spin chain for the Regge limit of QCD , but with the spins s = 1 / s = 0.Generalizing recent 4 D 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 simplestfishnet graphs of cylindric topology, and extract from it exact dimensions of operators with chiralcharge 2 and any spin, together with some of their Operator Product Expansion structure constants. the paper is dedicated to the memory of L.N. Lipatov INTRODUCTION
Conformal field theories (CFT) are ubiquitous in twodimensions [1], and quite a few supersymmetric CFTsin D = 3 , , D >
2, such as 3D Ising orPotts models, or Banks-Zaks model [2], are rare species,in spite of their rich potential applications ranging fromthe theory of phase transitions to fundamental interac-tions. The CFTs at
D > N = 4 SYM and Aharony-Bergman-Jafferis-Maldacena (ABJM) theories in ’t Hooft limit, aretrue exceptions [3] [4]. That’s why a new family of pla-nar integrable CFTs obtained in [5] as a special dou-ble scaling limit of γ -deformed N = 4 SYM seems tobe an important and instructive example. This theorycan be studied via quantum spectral curve (QSC) for-malism [6–8] or using the integrability of its dominantFeynman graphs via the conformal, SU (2 ,
2) noncom-pact spin chain. A nice particular case of this family isthe 4 D bi-scalar theory, whose planar limit is dominatedby ”fishnet” type Feynman graphs [5, 9].We propose here the following D -dimensional general-ization of the 4 D bi-scalar theory introduced in [5] L φ = N c tr[ φ † ( − ∂ µ ∂ µ ) ω φ + φ † ( − ∂ µ ∂ µ ) D − ω φ + (4 π ) D ξ φ † φ † φ φ ] . (1)where both scalar fields transform under the adjoint rep-resentation of SU ( N c ); ξ is the coupling constant and ω ∈ (cid:0) , D (cid:1) is a deformation parameter. The non-local(for general D, ω ) operators in kinetic terms should be understood as an integral kernel( ∂ µ ∂ µ ) β f ( x ) ≡ ( − β Γ( D + β ) π D Γ( − β ) Z d D y f ( y ) | x − y | D +2 β . (2)The propagator of scalar fields is its functional inverse:( − ∂ µ ∂ µ ) β D ( x ) = δ ( D ) ( x ) , (3) D ( x − y ) = Γ( D − β )4 β π D Γ( β ) | x − y | D − β . The typical structure in the bulk of sufficiently big pla-nar Feynman graphs in this theory is that of the regularsquare lattice (“fishnet” graphs, proposed in [10] as anintegrable lattice spin model), by the same reasons as in4 D case [5, 9], namely, due to the presence of the singlechiral interaction vertex in the Lagrangian, and the ab-sence of its hermitian conjugate. For example, the graphsrenormalizing local “vacuum” operator tr( φ j ) L are thoseof the “wheel” type and they can be studied via the in-tegrable conformal SO (2 , D ) spin chain [11], as was sug-gested for 4 D case in [12]. The dimensions of operators ofthe type tr[ φ ( φ † φ ) k ] have been also studied in 4 D [12]by QSC methods. It is not clear whether this methodcan be generalized to our D dimensional model. But thespin chain methods certainly can.In general, the propagators of the fishnet graphs ofthe model (1) are different in two different directions: | x − y | − D +2 ω for φ fields and | x − y | − ω for φ fields.Let us concentrate here on the “isotropic” case ω = D/ FIG. 1. Loop expansion of h tr( φ )( x )tr( φ ) † (0) i planargraphs up to 2-loops. to (1) the following double-trace counterterms [13, 14] L dt / (4 π ) D = α
21 2 X i =1 tr( φ i φ i ) tr( φ † i φ † i ) −− α tr( φ φ )tr( φ † φ † ) − α tr( φ φ † )tr( φ φ † ) , (4)Notice that the first term disappears in the “non-isotropic” case ω = D since the couplings of two termsin the first line of (4) would become dimensionful.As it was suggested in [15] and explicitly shown in [16]for the 4D case, the “isotropic” bi-scalar theory with La-grangian L φ + L dt has two fixed points. We generalizehere this result to any dimension, up to two loops, com-puting the corresponding Feynman graphs (Fig.1) con-tributing to the β α -function. Its two zeroes are α ( ξ ) = ∓ i ξ − J ( D ) ξ + O ( ξ ) (5)where the real coefficient J ( D ) depends on the ǫ − co-efficient of the down-left graph of Fig.1 in dimensionalregularization. For example: J (4) = 1 / J (2) = 2 ln 2, J (1) = π +4 ln 22 √ π [17]. At this critical coupling α ( ξ ) thebi-scalar theory becomes a genuine non-unitary CFT atany coupling ξ . The operators tr( φ φ ), and tr( φ φ † ) areprotected in the planar limit as in [16].In this paper, generalizing the 4 D results of [16] toany D , we will compute exactly a particular four-pointfunction and read off from it the exact scaling dimensionsand certain OPE structure constants of operators of thetype tr( φ ∂ S + φ ( φ † φ ) k ) + permutations . Their dimen-sions will be given by a remarkably simple exact relation h ∆ ,S ≡ Γ (cid:0) D − ∆ − S (cid:1) Γ (cid:0) D − ∆ − S (cid:1) Γ (cid:0) D + ∆+ S (cid:1) Γ (cid:0) − D + ∆+ S (cid:1) = ξ , (6)which reduces of course at 4 D to the result of [16]. Foreven D it gives D different solutions ∆( ξ ) = ∆ + γ ( ξ ).At odd (or non-integer) D there are infinitely many, ingeneral complex, solutions. At weak coupling the twocomplex conjugate solutions at S = 0 [18] γ = ± i ξ Γ (cid:0) D (cid:1) ± i ξ Γ (cid:0) D (cid:1) (cid:18) π − ψ (1) (cid:18) D (cid:19)(cid:19) + O ( ξ ) describe anomalous dimensions of the operator tr( φ φ )at the two fixed points. In a similar way, for any S ∈ Z the real weak coupling solution γ = − ξ Γ( S )Γ (cid:0) D (cid:1) Γ (cid:0) D + S (cid:1) + 2 ξ Γ( S ) Γ (cid:0) D (cid:1) Γ (cid:0) D + S (cid:1) ×× (cid:18) ψ (0) (cid:18) D (cid:19) − ψ (0) (cid:18) D S (cid:19) + ψ (0) ( S ) + γ E (cid:19) + O ( ξ ) . describes the operators of the type tr( φ ∂ S + φ ), where ∂ S + = (ˆ n · ∂ ) S with ˆ n being an auxiliary light-like vector.For D = 2 m, m ∈ N the L.H.S. of (6) factorizes intoa polynomial of degree 2 m and 2 m roots of eq.(6) de-scribe the scaling dimension of the exchanged operatorsin the OPE channel x → x of (11) together with theirshadows ˜∆ = D − ∆. At ξ = 0 we get for the bare di-mensions of physical operators (i.e., excluding “shadow”operators) ∆ − S = { m, m + 2 , · · · , m − } , At D = 2 there is a single solution with the dimension∆ = 1 + p S − ξ of the local twist-2 operators of thetype tr( φ ∂ S + φ ), while at 4 D the additional ∆ − S = 4describes twist-4 operators [16].As an example, at D = 6 and S = 0 the pos-sible non-shadow solutions for (6) are ∆ = 3 , , φ ) for ∆ = 3, linearcombinations of tr( φ ∆ φ ), tr( ∂ µ φ ∂ µ φ ) for ∆ =5 and of tr( φ ∆ φ ), tr(∆ φ ∆ φ ), tr( ∂ µ φ ∂ µ ∆ φ ),tr( ∂ µ ∂ ν φ ∂ µ ∂ ν φ ) for ∆ = 7.Diagonalizing the mixing matrix of these operatorsat ξ = 0 we would obtain operators with non-trivial, ξ -dependent anomalous dimensions, as well as theso called log-multiplets [12, 19], omnipresent in thisnon-unitary theory [20, 21], containing the operatorswith zero anomalous dimension. Eq.(6) predicts that allthe exchange operators from this set acquire non-trivialanomalous dimensions, whereas the operators belongingto log-multiplets never appear among them. Thisappears to be true at any even dimension D .As a general rule, according to the eq.(6) the operatorsof the type { tr( φ ∂ S + φ ( φ φ † ) k ) + permutations } appearin the multiplets only at D/ ∈ N , k = 1. We willfind below from the exact 4-point function the conformalstructure constants of these operators with two scalarfields. INTEGRABILITY OF D -DIMENSIONALBI-SCALAR CFT As it was noticed in [5] and further developed in[9, 12, 16], the 4 D case of the theory (1), with ω = 1,is integrable in the planar limit. On the one hand, this x' x y . . . u - (cid:1) + D / - u - D / - u + D / u + (cid:0) + D / x' x' x' x' L x x x L y y y L FIG. 2. Graphical representation of the transfer matrix asa convolution of R-kernels according to formulas (8)and (9).Black dots are integration points and the weights of propaga-tors are written in the second and third R-kernel. integrability is the direct consequence of integrability of γ -twisted planar N = 4 SYM theory, from which it wasobtained in the double scaling limit combining strongimaginary twist and weak coupling. On the other hand,this integrability was explicitly related in [5, 12] to thefact that the bi-scalar theory was dominated by the in-tegrable “fishnet” Feynman graphs [10],[22].Apart from 4 D case, at arbitrary D our bi-scalar model(1) does not have any integrable SYM origin. But thearguments of equivalence to the integrable conformal SO (1 , D +1) spin chain do work. Namely, let us introducethe D -dimensional analogue of the 4 D ”graph-building”operator [5] at general ω -deformation H L Φ( x , . . . , x L ) =1 π DL Z d D x ′ . . . d D x L ′ Φ( x ′ , . . . , x L ′ ) | x ′ | D − ω . . . | x LL ′ | D − ω × | x ′ ′ | ω . . . | x L ′ ′ | ω (7)schematically presented on Fig.3. It is easy to see that apower of this operator H ML generates a fishnet Feynmangraph with topology of a cylinder of length M with thecircumference L . Now, in analogy with the 4 D observa-tion of [12], we notice that this operator can be related tothe transfer-matrix of integrable SO (1 , D + 1) conformalHeisenberg spin chain [23] presented on Fig.2: T ( u ) = Tr ( R ( u ) R ( u ) . . . R L ( u )) (8)where u is the spectral parameter and the R -matrix actsas an integral operator[ R Φ]( x , x )( u ) = c ( u, D, ω ) ×× Z d D x ′ d D x ′ Φ( x ′ , x ′ )( x ) − u − D ( x ′ ) D + u + ω ( x ′ ) D + u − ω ( x ′ ′ ) − u + D , (9)with the normalization constant c ( u, D, ω ) = 4 u π D Γ (cid:0) u + D + ω (cid:1) Γ (cid:0) u + D − ω (cid:1) Γ (cid:0) − u − D + ω (cid:1) Γ (cid:0) − u + D − ω (cid:1) . Indeed, in analogy with 4 D case [12], at a particular valueof spectral parameter this transfer matrix becomes the x' x' x' x . . . x' x' x' L x x x x L x D / - ω D / - ω ω ωωω D / - ω D / - ω D / - ω FIG. 3. Graphical representation of the kernel of the graph-building operator for generic D and ω . It is otained by setting u = − D in the transfer matrix (8) presented on Fig. 2, so that x jj ′ +1 –type type propagators disappear while x j ′ +1 − y j -typepropagators are replaced by δ ( D ) ( x j ′ +1 − y j ) factors. Afterthat, integration over the points y j is equivalent to setting y j = x j ′ +1 . graph-building operator (7) at any D H L = π − DL (cid:20) (4 π ) D Γ (cid:18) D (cid:19)(cid:21) L lim ǫ → ǫ L T (cid:18) − D ǫ (cid:19) . (10)presented on Fig. 3. Thus this operator is one ofthe conserved charges of the equivalent spin chain:[ T ( u ) , T ( u ′ )] = [ T ( u ) , H L ] = 0. EXACT 4-POINTS CORRELATION FUNCTION
In analogy with 4 D results of [16], employing the D -dimensional conformal symmetry of the theory (1),(4) wewill compute exactly the four-point correlation function G = h O ( x , x ) ¯ O ( x , x ) i = G ( u, v )(2 π ) D ( x x ) D , (11)where the notation is introduced for the operators O ( x, y ) = tr[ φ ( x ) φ ( y )] and ¯ O ( x, y ) = tr[ φ † ( x ) φ † ( y )].Here G ( u, v ) is a finite function of cross-ratios u = x x / ( x x ) and v = x x / ( x x ), invariant un-der the exchange of points x ↔ x and x ↔ x . TheOPE expansion leads to the formula G ( u, v ) = X ∆ X S/ ∈ Z + C ,S u (∆ − S ) / g ∆ ,S ( u, v ) , (12)where the sums run over operators with scaling dimen-sions ∆ and even Lorentz spin S . Here C ∆ ,S is thecorresponding OPE coefficient (structure constant) and g ∆ ,S ( u, v ) is the known D dimensional conformal block(see (2.9) and sections 4,5 in [24]). If we compute (11), wewill identify the conformal data for the operators emerg-ing in the OPE of O ( x , x ).In the planar limit G is given by the set of fishnetFeynman diagrams presented in Fig. 4. Summing up thecorresponding perturbation series we encounter a geo-metric progression involving the combination of opera-tors α V + ξ H , where α = α ± is the double-tracecoupling at the fixed point, V is the operator insertingthe double-trace vertex V Φ( x , x ) = 2 π D Z d D x ′ d D x ′ δ ( D ) ( x ′ ′ ) Φ( x ′ , x ′ ) | x ′ | D/ | x ′ | D/ , which is the D dimensional version of (11) in [16], and theoperator H defined by (7) adds a scalar loop inside thediagram. Hence we obtain the following representation G = 1(2 π ) D Z d x ′ d x ′ ( x ′ x ′ ) D h x , x | − α V − ξ H | x ′ , x ′ i ++( x ↔ x ) . (13)where x ij ≡ x i − x j . [25]Remarkably, the operators V and H commute withthe generators of the conformal group, as in the particu-lar 4 D case [16]. This fixes the form of their eigenstatesΦ ∆ ,S,n ( x , x ) = 1( x ) D (cid:18) x x x (cid:19) (∆ − S ) / (cid:18) ∂ ln x x (cid:19) S , (14)where ∆ = D + 2 iν and ∂ ≡ (ˆ n · ∂ x ). The state Φ ∆ ,S,n belongs to the principal series of the conformal group andcan be represented in the form of a conformal three-pointcorrelation function C ∆ ,S Φ ∆ ,S,n ( x , x ) = h tr[ φ ( x ) φ ( x )] O ∆ ,S,n ( x ) i , where the operator O ∆ ,S,n ( x ) carries the scaling dimen-sion ∆ and Lorentz spin S , and C ∆ ,S is the 3-pointsstructure constant. The states (14), satisfy the orthogo-nality condition [26, 27] Z d D x d D x ( x ) D Φ ∆ ′ ,S ′ ,n ′ ( x ′ , x ′ ) Φ ∆ ,S,n ( x , x )= c ( ν, S ) δ ( ν − ν ′ ) δ S,S ′ δ ( D ) ( x ′ )( nn ′ ) S + c ( ν, S ) δ ( ν + ν ′ ) δ S,S ′ Y S ( x ′ ) / ( x ′ ) D − S − iν , (15)where ∆ ′ = D + 2 iν ′ , Y ( x ′ ) = ( n∂ x )( n ′ ∂ x ′ ) ln x ′ ,and c ( ν, S ) = 2 S +1 S ! | Γ(2 iν ) | (cid:0) ν + ( D + S − (cid:1) − π − (3 D/ (cid:12)(cid:12) Γ (cid:0) D − iν (cid:1)(cid:12)(cid:12) Γ( D + S ) , (16) c ( ν, S ) = 2( − S Γ (cid:0) D + S − iν (cid:1) π − ( D +1) Γ (cid:0) D + S + iν (cid:1) Γ(2 iν )Γ( D + 2 iν − × Γ( D + S + 2 iν − S !Γ( D + S − iν )Γ( D + S )Calculating the corresponding eigenvalues of the opera- ... x x x x x x x x x x x x FIG. 4. General fishnet graphs up to α order in the expansionof four point function (13). tors V and H we find V Φ ∆ ,S,n ( x , x ) = δ ( ν ) δ S, Φ ∆ ,S,n ( x , x ) , H Φ ∆ ,S,n ( x , x ) = h − ,S Φ ∆ ,S,n ( x , x ) , (17)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 O ∆ ,S ( x ) in the OPE channel O ( x , x ) G ( u, v ) = X S/ ∈ Z + Z ∞−∞ dνµ ∆ ,S u (∆ − S ) / g ∆ ,S ( u, v ) h ∆ ,S − ξ , (18)where ∆ = D + 2 iν , and µ ∆ ,S = 2 π D /c ( ν, S ) is relatedto the norm of the state (15). The fact that the depen-dence on α disappears from (18), can be understoodas follows. Viewed as a function of S , ξ /h ∆ ,S developspoles at ν = ± iS which pinch the integration contour in(18), for S →
0. The contribution of the operator V isneeded to make a perturbative expansion of (18) well-defined. For finite ξ , these poles provide a vanishingcontribution to (18), but generate a branch-cut p − ξ singularity of G ( u, v ), as in 4 D case [16].At small u , we close the integration contour in (18)to the lower half-plane and pick up residues at the poleslocated at solutions of (6) and satisfying the unitaritybound Re ∆ > S . The resulting expression for G ( u, v )takes the expected form (12), with the OPE coefficientsgiven by C ,S = Γ( D + S ) S ! Γ(∆ − − D ) Res (cid:18) d ∆ h ∆ ,S − ξ (cid:19) × Γ( S − ∆ + D ) Γ (cid:0) ( S + ∆) (cid:1) Γ (cid:0) ( S − ∆ + D ) (cid:1) Γ( S + ∆ − . (19)where the residue is computed w.r.t. the appropriatesolution of (6) for each relevant operator. For instance,we can consider tr( φ ) † , which is exchanged for any even D ; then the perturbative expansion of (19) is C φ = 2 + 4 iξ Γ (cid:0) D (cid:1) (cid:18) ψ (0) (cid:18) D (cid:19) − ψ (0) (cid:18) D (cid:19) + γ E (cid:19) + O ( ξ )(20)The relations (6) and (19) define exact conformal data ofoperators propagating in the OPE channel x → x .Finally, we discuss an interesting D → ∞ limit ofthe theory. We should then rescale the coupling ξ = ξ ∞ p Γ( D/ ξ ∞ is fixed. Anomalous dimension γ ∞ of tr( φ ) has finite limit since for S = 0 in eq.(6) itis given by − γ ∞ sin (cid:16) πγ ∞ (cid:17) = 2 πξ ∞ while γ ∞ vanishes for operators with higher spin S = 0.As concerns the expansion (12), the number of exchangedoperators becomes a countable infinity, diverging linearlyin D . Finally, the OPE structure constant (20) for tr( φ )trivially reduces to its bare value in this limit. CONCLUSIONS
We showed that the strongly γ deformed N = 4 SYMtheory proposed in [5] is just the 4-dimensional repre-sentative of a wider, D -dimensional family of theoriesof two complex scalar fields obtained by modifying thepropagators of fields in a D -dependent way. Similarlyto the 4 D case [16], they turn out to be conformal andintegrable at any D , at least in the planar limit, if weadd to the action certain double-trace terms with spe-cific couplings. The conformality of our theory at finite N c remains an open question, though it is quite plau-sible that the planar conformal point simply shifts tosome other complex values of couplings. There are twosuch complex conjugate values of these couplings and wecompute them perturbatively up to two loops. The inte-grability is explicit due to the domination of sufficientlylarge orders of perturbation theory by the “fishnet” Feyn-man diagrams. The cylindric fishnet graphs, related tothe renormalization of “vacuum” tr( φ L ) operators, canbe created by multiple application of a “graph-building”operator which appears to be an integral of motion ofthe integrable conformal SO (1 , D + 1) spin chain. Wealso generalize the bi-scalar model to a CFT with dif-ferent propagators for the fields φ and φ , leading to“non-isotropic” fishnet Feynman graphs. The underly-ing graph building operator has representations with dif-ferent conformal spins in two directions on the fishnetgraph. In the 2D case the fishnet graphs are describedby the same SL (2 , C ) chain as used for the dynamics ofgeneralized Lipatov’s reggeized gluons [28] but with dif-ferent value of spin, s = 1 / s = 0. This spin chain, extensively studied in the liter-ature [29–34], is restored in the singular limit ω → D as anexpansion into conformal blocks with explicit OPE coef-ficients and dimensions of exchange operators in one of the channels. In 1 D case, our results are similar to thescalar version of conformal Sachdev-Ye-Kitaev fermionictheory [35] at q = 4. For even D we found a finite, D -dependent number of local exchange operators at a givenspin and dimension. It would be very interesting to com-pute some of the discussed quantities (dimensions, struc-ture constants) in the next 1 /N c approximation, simi-larly to [36–38], if the conformality of the theory holdsat any N c . The explicit form of this operators can beobtained by the analysis of the mixing matrix for theirquantum multiplets [12, 19]. This becomes more compli-cated as the dimension grows due to growing rank of themultiplets and the number of transitions, together withlog-CFT effects which arise starting from 4 D , due to thechirality.Although the lagrangian (1) of our theory is nonlocal atgeneral D (apart from the sequence D ∈ N in ”isotropic”case), it does not prevent the existence of “normal” OPEdata in this theory, which is more important for the phys-ical interpretation of this CFT. Moreover it would be in-teresting to generalize to any D the results for fishnetgraphs of the type considered in [39] and to the corre-lation functions for operators involving more than twoscalars. Finally, an important question remains, as in4 D , whether these theories have any string duals at any D , according to the original proposal of G .’t Hooft [40]. ACKNOWLEDGMENTS
We thank B. Basso, J. Caetano, S. Derkachov, N. Gro-mov, G. Korchemsky, F. Levkovich-Maslyuk for numer-ous useful discussions. The work of V.K. was supportedby the European Research Council (Programme “Ideas”ERC-2012-AdG 320769 AdS-CFT-solvable). The workof E.O. is supported by the German Science Foundation(DFG) under the Collaborative Research Center (SFB)676 Particles, Strings and the Early Universe and theResearch Training Group 1670. [1] P. Di Francesco, P. Mathieu and D. Senechal,
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